Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment (1987)

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PART 111 Generalizations and Extrapolations

Allometry: Body Size Constraints in Animal Design Stan [. Lindstedt INTRODUCTION In 1928 the English philosopher and physiologist J. B. S. Haldane wrote a captivating essay entitled, "On Being the Right Size." He began by discussing the dimensions of the giant Pope and Pagan from Pilgrim's Progress. If the giants were 10 times Christian's height, they would have been 1,000 (10 x 10 x 10) times his mass. Being of similar shape, Haldane concludes that the cross-sectional area of their bones would be but 100 times those of Christian. "As the human thigh bone breaks under about 10 times the human weight, Pope and Pagan would have broken their thighs every time they took a step. This is doubtless why they were sitting down in the picture I remember. But it lessens one's respect for Christian and Jack the Giant Killer" (Haldane, 19281. In the case of giants, as is a moose from the perspective of a mouse, bones are not built with geometric similarity, i.e., dimensional proportionality, rather bones of all species must be built with similarity of structural strength. This is why bones of larger animals are relatively more robust than those of smaller animals; the skeleton accounts for nearly 20% of an elephant's mass but less than 5% of a shrew' s. Perhaps no single factor is more dominant in constraining animal design than body size. Size-induced patterns have been identified for all aspects of animal design and function from structural dimensions, to life history characteristics, to pharmacokinetics. An animal's body size is certainly among it most prominent of all distinguishing features. Among the mam- 65

66 STAN L. LINDSTEDT mars, the 136,000-kg (150-ton) blue whale is 75 million times the mass of the 2-g Etruscan shrew, yet both share the same skeletal architecture, suite of organ systems, biochemical pathways, and even temperature of operation. What engineers have known for a long time, however, is only now gaining widespread consideration among biologists: there are trade- offs that must accompany size changes. The scaling up of a bridge, or a mammal, to a size 100,000 times larger than the original requires more than just the creation of bigger parts. Rather, those parts must be rede- signed if they are to perform the same functions throughout a size range spanning several orders of magnitude in body mass. Further, changes in size result in shifts in optimal or preferred frequencies of use. It is be- coming increasingly apparent that there are a suite of body size-dependent physical laws that dictate many features of animal design. Size-dependent constraints of design are expressed in the form of al- lometric equations. Allometry (literally, "of another measure") describes the disproportionate changes in size (or function) that occur when separate isolated features in animals are compared across a range of body sizes. If all characteristics varied in direct proportion, that is, if the large animal were merely a scaled up exact replica of the small one, they would be built with isometry. Quantitatively, allometry takes the form of power law equations relating some variable of structure of function (Y) as a dependent function of body mass (M) in the form Y = aMb, where a and b are derived empirically. On logarithmic coordinates this equation describes a straight line with a slope of b. Therefore, the value of b describes the nature of the rela- tionship. When b is near 1.0, Y scales as a fixed percentage of body mass (i.e., isometrically). Virtually all volumes or capacities, for instance, lung, gut, and heart volume, scale isometrically. Blood volume in all mammals is about 7% of body mass. If b is greater than 1, Y increases more rapidly than does mass in going from small to large animals. To maintain a constant safety factor of bone strength (discussed above), the mass of the skeleton varies with an exponent near 1.1 in both birds and mammals (Anderson et al., 19794. If b is between 0 and 1, a unit increase in mass is accompanied by a fractional increase in Y. The well-known Kleiber equation describing resting metabolism in homeotherms scales with an exponent of 3/4 (Klei- ber, 1932~; thus, weight-specific metabolism (rate of metabolism per unit mass) is much higher in small than in large animals. Finally, if b is negative, its absolute value is highest in the smallest animals. The exponent b is close to—1/4 for biological rates such as heart and respiratory rates. Allometric equations are often limited in their use as descriptive tools to identify patterns of foe and function. Recently, many of these equa- tions have been compiled and presented in encyclopedic form (see, for example, Peters, 19831. Although these equations are most valuable for

Body Size and Pharmacokinetics 67 identifying interspecific patterns, there may be equally important infor- mation hidden within the variance around the equations. In looking for those species that do not conform to body size predictions, allometry can identify those animals that Calder (1984) refers to as "adaptive deviants," suggesting that the process of deviation from body size-expected patterns must be the result of selection rather than random variance. Finally, in addition to identification of patterns and those animals that deviate from the patterns, allometry may be most useful as a tool to identify interspecific constraints of design and function. Many of these are discussed in two recent books (Carder, 1984; Schmidt-Nielsen, 1984~. - Allometric equations can furnish much more than empirical patterns that lack a conceptual foundation. Many structural relationships can best be explained allometrically. For instance, as the surface area of an animal, or any object, varies as the 2/3 power of its volume fi.e., suface area (Y) = aM2'31 any increase in mass results in a decrease in relative surface area. Here we encounter an allometric "law" of sorts, namely, that size changes often require a concomitant mechanism to permit surface area to increase linearly with mass (i.e., as a fixed percentage of mass). A suf- ficient number of allometric equations have been generated to confirm quantitatively what Haldane (1928) proposed nearly 60 years ago: "The higher animals are not larger than the lower because they are more com- plicated. They are more complicated because they are larger.... Com- parative anatomy is largely the story of the struggle to increase surface in proportion to volume. " The design of the mammalian respiratory system provides a quantitative example of Haldane's declaration. Lungs with alveoli and pulmonary capillaries, and circulatory systems with capillaries and red cells, are prominent examples of structures that are necessary only as a price for large body size. Interestingly, in all cases the surface areas of the above-named structures vary with body mass exponents (b) near 1.0. Across 6 orders of magnitude in body size, metabolically important surfaces generally increase in direct proportion to volume. How can these apparent design patterns be of use in risk assessment? SIZE, DESIGN, AND PHARMACOKINETICS A hopeful result of the linking of pharmacokinetics and risk assessment will be to identify interspecific principles of design. Once identified, these may be extremely valuable in making species extrapolations with the eventual goal of risk assessment in humans. In suggesting a potential role for allometric analysis in that process, I will focus on three diverse ex- amples from my own research which illustrate the value of allometry as a unique mechanism for identification of design constraints. In all cases,

68 STAN L. Ll NDSTEDT apparent principles of design surface only when viewed across a range of species. Aerobic Energetics of Muscle In viva The structural, contractile, and metabolic properties of skeletal muscle have been well characterized. The maximal cross-sectional force of all muscles is roughly constant, and mechanisms of cross-bridge cycling, adenosine triphosphate (ATP) synthesis, and use are well characterized (for reviews, see Peachey et al., 1983), yet in vivo muscle possesses some properties that are unpredictable from in vitro experimentation. In mam- mals, the external (weight-specific) work performed by locomotory mus- cles does not vary systematically with body size; however, the energy required to perform that work does. Small animals expend more energy for a given force production than do large animals. As a result, the en- ergetic cost of locomotion (energy spent to move a unit mass a unit distance) and, therefore, the efficiency of locomotion are strongly body size dependent. There is an energetic cost associated with small body size, the source of which could be the obligate scaling of muscle contraction times. While all skeletal muscle can produce roughly the same maximal cross-sectional force, the power required to do so increases with increasing contraction velocity. Because differences in energetics span more than an order of magnitude, by examining this question allometrically, we have the luxury of a fa- vorable signal-to-noise ratio that is not present within any single species. It is possible to estimate the in vivo rate of muscle shortening by calculating a series of equations and combining these algebraically (after Stahl, 19621. Thus, ratios or products of allometric equations can be formed to predict functions that are either unmeasurable or impractical to measure. By doing so we found that variance in muscle biochemistry, structure, contractile properties, and whole animal maximum oxygen uptake all followed an identical pattern. Hence, there is a parallel (causal?) relationship among the rate of muscle shortening and the energy supplying oxygen consump- tion, volume density of ATP-synthesizing mitochondria, and the activity of myosin ATPase (Lindstedt et al., 1985) (Figure 11. When examined over a broad range of body sizes, constraints of design surface suggest a common scheme of muscle structure and function. In this example, allometry is useful for making estimates for problems too demanding to be currently solved with direct measurements; it can thus be more than a descriptive tool.

Body Size and Pharmacokinetics ^ 100— In . 1~,, 10— . cat 0 1 - - E 0.1— cat o .> - 1 —100 ~ n` .—1 V 0 —10 at: ~ . _ . —0.1 0 ~ ~7J~°sln am_ ~ a: ~ - ~ ~mt^~ - ~ 100— _, 1 0— - ~~aCV,,It,,, o 0 0.1— 1— 0.001 0.01 0.1 1 10 100 1000 BODY MASS (kg) 69 - 1e - 1~ i_ In - 100 ~ -10 ~, — 1 _ ¢> -0.1 ~ In FIGURE l An allometic analysis of the design of the knee extensor muscles in mammals demonstrates remarkable structure-function consistency. Among terrestrial mammals, there is a constant relationship among oxygen consumption (VO2m`2X), volume density (Vv) of mitochon- odria, myosin ATPase activity, and the rate of muscle shortening. Although these parameters vary greatly as a function of body mass, within any given mammal their ratios remain nearly constant. (Used with permission of the American Journal of Physiology.) Conflict of Physiological and Chronological Time Four decades ago, Brody (1945) introduced the concept of "physio- logical time," in acknowledgment of a variable biological time scale that exists among organisms. Hill (1950) refined the concept by suggesting that within an organism physiological time may be as constant as is chro- nological time. Hill's speculation was that all physiological events are likely entrained to the same body size-dependent clock. Thus, critical biological times, such as gestation period or time for growth to maturity, as well as other temporally linked biological events, may all be constant if compared per unit of physiological time. Hill ended this most interesting paper by suggesting that to s to a large animal may be physiologically equivalent to l s in a small one, implying that they differ only in pace of life, not in absolute number of life's events. A sufficient number of physiological rates and times have been measured to permit a quantitative examination of Hill's hypothesis. Among physi-

70 STAN L. LINDSTEDT ological events, all apparently occur with a body mass-dependent metric (Figure 21. In fact, we (Lindstedt and Calder, 1981) found that virtually all biological times do indeed vary with nearly the same body mass ex- ponent (mean = 0.24) in both birds and mammals (Figure 31. It is assumed that this parallel scaling is the result of a common body size-dependent clock to which these events are all entrained (a periodengeber; see Lind- stedt, 19851. Thus, whether the result of function is in a single tissue (e.g., muscle contraction times), single organ (e.g., cardiac cycle), organ systems (e.g., inulin or PAH clearance), the entire organism (e.g., growth times), or even populations of organisms (population doubling time), all biological times seem to vary as a consistent and predictable function of body mass. There are at least two consequences of the regularity of physiological time that have direct bearing on pharmacokinetics and risk assessment. 1o2 1o1 _ 10° In ~ ° 10 ' [L' 10 2 10-3 10 Brood Cat ,~O myth\ _ _ 10-5 .001 .01 1 00 1 000 1 00000 - .1 1 10 BODY MASS (kg) FIGURE 2 The duration of physiologically critical time periods are examined over a range of body masses. Equations are from many sources and have been uniformly extrapolated (when necessary) to encompass the range of body mass in terrestrial mammals.

Body Size and Pharmacokinetics 71 1o8 107 1o6 104 103 In 3 LL o1 10° 10-1 10-2 10-3 = 10-4 ~' Prod ~ i_ __--~~~~~~ - ~^tract\°~_ - F~ ~ I 1 1 1 19 1O9 10O9 1 kg 10kg 100kg 1000 kg BODY MASS FIGURE 3 Biological times are shown for birds (dashed lines) as well as mammals (solid lines). These include strictly physiological times (muscle contraction) to strictly life history or ecological times (time to achieve reproductive maturity or population doubling). The slopes of these lines are nearly identical (mean = 0.24) in spite of the apparent unrelated nature of the times and methods of measurement. (Used with permission of the Quarterly Review of Biology.)

72 STAN L. LINDSTEDT The first is that if volumes and capacities vary as linear functions of mass and times as mass raised to the 1/4 power, then volume rates (i.e., volume divided by time, such as clearance rates) must vary as M~IM~'4 = M3'4. Application of this physiological volume rate scaling will be discussed below. The second is that ratios of venous temporal events are essentially constant, independent of body size, and independent of their absolute rate of occurrence (Figure 41. Mammals may indeed burn roughly the same 109 1o8 107 In In o I_ A a) E ~ 104 F 1o6 105 103 1o2 1o1 10° Tlme to 50% Growth: Muscle Contraction Tlme 5.9 x 10 M Population Cycle Tlme: ATP Cleavage Tlme 3.3 x 106 M Tlme to Reproductive Maturity: Blood Clrculatlon Tlme _ 8.4 x 105 M-0.03 Post Embryonic Mass Doubilng: Renal Blood Clearance (GFR) _ 46x 104M -0.02 Tlme to Burn Blood Vol. of O2: Respiratory Cycle Lifespan: Gestation P-rl^~ 233 M~00 65 M-O 05 Respiratory Cycle: Heart Cycle 4.5 M 1 1 1 1 1 1 1 0.001 0.01 0.1 1 10 100 1000 10000 BODY MASS (k9) FIGURE 4 Ratios of any two physiological times are nearly constant, varying little from shrew to elephant. Thus, all mammals experience about four to five heart beats per breath, and all burn roughly the same number of calories per gram of tissue per lifetime. This constancy of times suggests that there is a body size-dependent clock to which all of life's events are entrained. Thus, physiological time is just as critical as chronological time and may be the best measure of time for cross-species comparisons.

Body Size and Pharmacokinetics 73 number of calories per gram of tissue per lifetime, which itself is composed of a constant maximum number of cardiac, respiratory, and other cycles (Lindstedt, 1985; Lindstedt and Calder, 1981~. Thus, a second allometric law is identified; even though chronological time is body mass independent (b = 0), physiological time is not (b ~ 1/4~. All animals must contend with identical lengths of days and seasons, but these span disproportionately longer physiological periods in small than in large animals (i.e., chronological/physiological time = M0IM~'4 = M- ~'4~. This simple principle may explain several interactions of time and size, from fasting survival times to the structure of the mammalian lung (Lindstedt, 1984; Lindstedt and Boyce, 19851. The ubiquitous pres- ence and regularity of physiological time among animals argues strongly for its consideration as a constraint of design. That physiological time is just as critical as chronological time should have a profound effect on interpretation and extrapolation of experimental data. In summary, through the use of allometric equations, a pattern surfaces, namely, that among the noise of various physiological rate functions there is a strong signal linking events as diverse as muscle contraction times occurring over periods of milliseconds with population doubling times requiring years. So salient is this pattern that physiological time must be recognized an identifiable characteristic of organisms. Species Extrapolations, Physiological Time, and Pharmacokinetics How does a consideration of allometry in general, and physiological time specifically, have an impact on pharmacokinetics? Like other rate functions, those involving the biochemistry of drug metabolism are in- separably bound to physiological but not chronological time. Uptake, processing, and excretion of drugs all transpire at rates that are generally directly proportional to one another and with body mass exponents char- acteristicofphysiologicaltime,b ~ 1/4 (Boxenbaum, 1982;Calder, 1984; Dedrick et al., 1970; Weiss et al., 19771. Virtually all pharmacokinetic variables from tissue dose (M. E. Anderson, this volume) to first-order kinetics (E. J. O'Flaherty, this volume) can best be interpreted through the perspective of physiological time. I naively present one caution and one possible application. One danger exists in the way in which we perform and interpret our experiments. Doses (per kilogram) and incubation times are virtually al- ways fixed, independent of both body mass and physiological time. Per- haps we are losing some information by comparing processes that transpire at mass-specific rates (b = - 1/41i using concentrations and times that Because rate is the reciprocal of time, the exponents describing biological rates are likewise the reciprocal of those describing times (1IM"4 = M - "4).

74 STAN L. LINDSTEDT are invariant (b = 0~. This distinction becomes especially important when results are extrapolated over several orders of magnitude in body mass, for example, from rodents to humans. A simple example might best il- lustrate this point. Miners used canaries in caves as biomonitors because of their apparent increased sensitivity to toxicants; when the bird dropped dead the miners would make a hasty retreat from the mine. In fact, are canaries more sensitive or are they affected by the same absolute levels of toxicants over shorter (physiological) time periods? Pharmacokinetic effects are not simple dose effects, but are dose effects integrated over physiological time periods. Thus, when examined over physiological rather than chronological time, species variability in pharmacokinetic rates may disappear (see, e.g., Mordenti, 19851. In a now classic paper, Dedrick et al. ( 1970) first made use of biological time scaling in their interspecific analysis of methotrexate pharmacoki- netics. They found that plasma concentrations could be normalized if dose were divided by body mass raised to the 1/4 power (b = 1/41. Amid the noise of interspecific variation there emerged a strong signal; there is a hidden Mi'4 scaling linking diverse animals to a single pharmacokinetic pattern. In general, dose varies as a function of rate of clearance; hence, it would be expected that dose = quantity/time. If we correct for physio- logical rather than chronological time, then we must divide by Mi'4. A simple example best illustrates the point. Dedrick et al. (1973) carefully examined the clearance of 1-~-D-arabinofuranosylcytosine (Ara-C) in four mammalian species. They found that clearance rates varied nearly as mass raised to the 3/4 power, as would be predicted based on the volume and rate allometry discussed above. Thus, while absolute rates of clearance varied by nearly 1,000-fold, they did so in an identifiable pattern. If those data are normalized for size and physiological time scaling, clearance can be expressed per unit of body mass per unit of physiological time (Figure 51. By this manipulation, interspecific differences for this particular com- pound can be adequately explained solely as a function of size-dependent . . . time sea, sing. Several compounds have been examined through the perspective of what has since been called the Dedrick plot (Boxenbaum and Ronfeld, 1983~. A key to understanding this type of manipulation is not that it makes all animals equal, rather that their differences actually become more pro- nounced when removed from inevitable, size-dependent variation resulting from physiological time scaling. For instance, if the resultant slope is not close to O (as it is in Figure 5), then we know immediately that there is a difference in the way small and large animals process a given drug. In the case of clearance data, a positive slope would imply that large animals are clearing the compound in faster physiological times; a slope below would imply that smaller animals are doing so. In any case, the appropriate

100 i_ _; O z In CI: I ~ y O — 1 c5 a: 10 0.1 ' _ o * - / 0.01 0.1 10 100 BODY MASS (kg) FIGURE 5 Clearance of 1-~-D-arabinofuranosylcytosine (Ara-C) in four species of mammals: mouse, monkey, dog, and man. Data are expressed in conventional units of milliliters per minute (asterisks and dashed line), as well as physiological time units of milliliters per kilogram phys- iological minute (open circles and solid line). Interspecific variation in the clearance rate of Ara- C virtually disappears when size-specific time scaling is considered (data are from Dedr~ck et al., 1973). means of comparison is against a physiological rather than a chronological metric. Perhaps physiological time can form the bases of comparison for risk assessment across species. The exponentially expanding pharmacokinetic data base may be outgrowing useful paradigms for interpretation. If com- parisons across widely different taxa are possible, perhaps an equivalent means of comparison might be physiological time. In collaboration with several individuals at the Environmental Protection Agency (EPA) in Du- luth, Minn., we have made a first attempt at such a comparison for the insecticide fenvalerate. Our preliminary results are somewhat promising because we have been able to directly equate results from different tem- peratures and modes of delivery in invertebrates (lobsters and flies) and vertebrates (rats and three species of fishes). We plotted the 50% lethal

76 STAN L. LINDSTEDT doses (LDsoS) as a function of physiological time, calculated as the time to metabolize a given quantity of oxygen proportional to body mass (roughly the reciprocal of weight-specific metabolism). The results suggest that the huge variability among animals may be largely accounted for (r2 = 0.82) when physiological time is considered (Figure 61. The exception was the fly, which should have a much higher relative sensitivity (i.e., by insec- ticide design) and is about four orders of magnitude more sensitive than predicted by this analysis. While this approach is only a first rough attempt with a limited sample size, it may provide one potential means of cross- species pharmacokinetic comparison. 1000 100 - Lu! 1 0 to J 1 He LU LL to u, 9 0.1 0.01 — · Rat Bluegill O Fly r2 = 0.82 · Trout · Fathead Minnow · Lobster 0.01 0.1 1 10 100 1000 BIOLOGICALTIME SCALE (min) FIGURE 6 One method of comparing animals from diverse taxonomic groups may be on a single continuum of physiological time. Here LD50 values for fenvalerate are plotted as a function of one measure of time, namely, the time required to metabolize one body mass equivalent of O2. These preliminary data suggest that sensitivity to fenvalerate decreases with decreasing body mass. They also suggest that the fly may be several orders of magnitude more sensitive to the drug than are the other species represented on this graph. For comparison purposes, man would be predicted to fall very near the point for trout (data are from the EPA laboratory in Duluth, Minn.).

Body Size and Pharmacokinetics 77 CONCLUSIONS In an essay entitled, "Comparative Physiology, Compared to What?," F. E. Yates (1979) summarized what should be the value of "comparative physiology." As a consequence of making comparisons (in design or function), physical laws and constraints often surface suggesting design principles that operate across species. Allometry is one of the most directly comparative of all tools available to biologists. MacArthur (1972) began his now classic book Geographical Ecology with the affirmation, "To do science is to search for repeated patterns in nature, not simply to accu- mulate facts." While the accumulation of facts will always be a necessary goal of science, the process of pattern identification continues to occur only infrequently. Physiological time has emerged as one of those patterns in nature. It may prove to be of significant value in understanding interspecific phar- macokinetics. It is not proposed as a substitute for direct measurements; however, cross-species comparisons should be made by normalizing for physiological time first, assuming it is the "default value." If differences persist, those can be fairly attributed to species differences. In planning and executing experiments, researchers must be aware of the two important time scales that have an impact on animals and ask how each may affect interpretation and extrapolation of results. Lack of consideration of this pervasive principle is no less of an assumption. For instance, in designing experiments to assess lifetime exposure, perhaps the best technique for the extrapolation from risk to an experimental rat to a potential risk to humans is some measure of physiological time equivalence such as number of heartbeats (suggested by Blancato and S. L. Friess, personal com- munication) or a measure of drug or metabolite turnover time. If so, 2 years may be much shorter than researchers assume. Further, perhaps shrews or small birds, which have the most rapid physiological time scales, may be the ideal experimental animals for certain studies. SU M MARY One of the products of science is an exponentially increasing pool of knowledge. Hopefully one goal of science is to combine emergent facts in such a way to form patterns from which new conceptual hypotheses and predictions can be derived. One of the potentially valuable techniques for the identification of patterns is allometry. Allometry literally means "of a different measure," in distinction to isometry. Functionally, allometry is an attempt to quantify the extreme importance body size has in dictating virtually all aspects of an animal's morphology, physiology, and ecology. Allometric equations can identify

78 STAN L. LINDSTEDT patterns of design and function that would be otherwise obscure. Further, they may be useful in identifying species that deviate from the pattern. Finally, allometric equations may be extremely useful in suggesting in- terspecific constraints of design and function. In this paper I present three examples. 1. Using a very simple model of a single mass and spring system, it is possible to couple the geometry of the mammalian knee with a descrip- tion of muscle energetics to make an allometr~c estimation of locomotory energetics. A running mammal behaves like a tuned spring system with an optimal (and therefore most efficient) stride frequency, somewhat anal- ogous to a resonating frequency (see Taylor, 19851. Rather than using allometry merely as a descriptive tool, we are able to make mechanical and bioenergetic predictions starting with a series of allometr~c equations. 2. If we examine a variety of rate functions across a wide range of species, another design constraint seems to surface, that of physiological time. While all animals experience the same chronological time of days or seasons, each animal apparently also possesses an internal body size- dependent clock to which physiological rates are entrained. From the rate of muscle contraction to life-span itself, physiological processes seem to be set to the same "periodengeber." 3. As an extension of this concept, we propose that species extrapo- lations may best be made as a function of physiological time (rather than dose or some other aspect of chronological time). Preliminary results for Fenvalerate suggest potential utility of this approach. Physiological time is not a substitute for measurements, rather it is a strong enough design pattern that it should be considered in designing and interpreting any experiment. REFERENCES Anderson, J. F., H. Rahn, and H. D. Prange. 1979. Scaling of supportive tissue mass. Q. Rev. Biol. 54:139-148. Boxenbaum, H. 1982. Interspecies scaling, allometry, physiological time, and the ground plan of pharmacokinetics. J. Pharmacokinet. Biopharm. 10:201 -227. Boxenbaum, H., and R. Ronfeld. 1983. Interspecies pharmacokinetic scaling and the Dedrick plots. Am. J. Physiol. 245:R768-R774. Brody, S. 1945. Bioenergetics and Growth. New York: Reinhold. Calder, W. A., III. 1984. Size, Function, and Life History. Cambridge, Mass.: Harvard University Press. Dedrick, R., K. B. Bischoff, and D. S. Zaharko. 1970. Interspecies correlation of plasma concentration history of methotrexate. Cancer Chemother. Rep. 54:95-101. Dedrick, R. L., D. D. Forrester, J. N. Cannon, S. M. Eldareer and B. Mellett. 1973. Pharmacokinetics of 1-~-D-arabinofuranosylcytosine (Arc-C) deamination in several spe- cies. Biochem. Pharmacol. 22:2405-2417.

Body Size and Pharmacokinetics 79 Haldane, J. B. S. 1928. On being the right size. Pp. 952-957 in The World of Mathematics II, J. R. Newman, ed. (Reprinted 1956 by Simon and Schuster, New York.) Hill, A. V. 1950. The dimensions of animals and their muscular dynamics. Sci. Prog. 38:209-230. Kleiber, M. 1932. Body size and metabolism. Hilgardia 6:315-353. Lindstedt, S. L. 1984. Pulmonary transit time and diffusing capacity in mammals. Am. J. Physiol. 246:R384-R388. Lindstedt, S. L. 1985. Birds. Pp. 1-21 in Non-Mammalian Models for Research in Aging, F. A. Lints, ed. Basel: S Karger. Lindstedt, S. L., and M. S. Boyce. 1985. Seasonality, body size and survival time in mammals. Am. Nat. 125:873-878. Lindstedt, S. L., and W. A. Calder III. 1981. Body size, physiological time, and longevity of homeotherrnic animals. Q. Rev. Biol. 56:1-15. Lindstedt, S. L., H. Hoppeler, K. M. Bard, and H. A. Thronson, Jr. 1985. Estimate of muscle shortening rate during locomotion. Am. J. Physiol. 249:R699-R703. MacArthur, R. H. 1972. Geographical Ecology. New York: Harper and Row. Mordenti, J. 1985. Forecasting cephalosporin and monobactam antibiotic half-lives from data collected in laboratory animals. Antimicrob. Agents Chemother. 27:887-891. Peachy, L. D., R. H. Adrian, and S. R. Geiger. 1983. Handbook of Physiology, Skeletal Muscle. American Physiological Society. Baltimore: Willaims & Wilkins. Peters, R. H. 1983. The ecological implications of body size. Cambridge: Cambridge University Press. Schmidt-Nielsen, K. 1984. Scaling: Why is animal size so important? Cambridge: Cam- bridge University Press. Stahl, W. R. 1962. Similarity and dimensional methods in biology. Science 137:205-212. Taylor, C. R. 1985. Force development during sustained locomotion: A determinant of gait, speed, and metabolic power. J. Exp. Biol. 115:253-262. Weiss, M. W., W. Sziegoleitt, and W. Forster. 1977. Dependence of pharmacokinetic parameters on the body weight. Int. J. Clin. Pharmacol. 15:572-575. Yates, F. E. 1979. Comparative physiology: Compared to what? Am. J. Physiol. 237:R1- R2.

Prediction of in viva Parameters of Drug Metabolism and Distribution from in vitro Studies Grant R. Wilkinson Physiologically based pharmacokinetic modeling is critically dependent on accurate estimates of the determining anatomical, physiological, and drug-specific parameters. Anatomical parameters, such as organ size and blood flow rates, are usually constant and well-established; if not, they can often be estimated directly or by allometry (see S. L. Linstedt, this volume). On the other hand, factors like drug binding to plasma and tissue constituents and elimination by various organs are not so readily available. In many instances these characteristics must be determined experimentally. In the past, this has largely involved in vivo measurements such as the determination of tissue/plasma partition coefficients or the apparent kinetic parameters of metabolism. Potentially, however, data obtained in vitro may, if appropriately interpreted, be applicable to the in vivo situation and can be rigorously incorporated into a physiologically based pharma- cokinetic model to allow quantitative prediction. This approach was an implicit hope of early investigators (Bischoff and Dedrick, 1968; Dedrick et al., 1973), but unfortunately, it has not been explored as extensively as might be expected. The ever increasing number of drugs and other chemicals to which animals and humans are exposed and the need to assess their disposition, however, along with any toxicity, will necessitate that such in vitro studies become more widely applied. This is particularly the case in humans, in which direct exposure to a putative toxic agent may be limited, for example, to accidents or occupational activities for which the dose and other details of the exposure are not well known. Over the past several years, a limited number of in vitrolin vivo extrapolation studies 80

Extrapolation from In Vitro Systems 81 have been undertaken that demonstrate the utility of the involved ap- proaches, as well as their limitations. In this chapter I will review some of the findings and their potential for further application. IN VITRO PREDICTION OF IN VIVO DRUG METABOLISM Numerous in vitro systems are available to determine the metabolism of a drug, including simple aqueous solutions for measurement of nonen- zymatically mediated reactions, purified enzyme preparations, crude tissue homogenates, microsomes, isolated intact and cultured cells, and tissue slices. Such variety indicates that no single system provides all the nec- essary inflation required for a complete understanding of the deter- mining factors of drug biotransformation. Moreover, the purpose and experimental design of the large majority of studies with such systems is frequently not suitable for extrapolation of the findings to the in viva situation. A critical requirement is that the kinetic parameters of metab- olism, expressed in terms of a specific pathwayLs) or the overall process, must be available. Generally, these are interpreted according to the well- known Michaelis-Menten mechanism so that the rate of metabolism t~dES]~°'I dt)V] can be described by Equation d[S]tot Vmax [S]t°t v = - - , dt K /fU + [S]tOt (1) where Van, is the maximal rate of metabolism, Km is the concentration of unbound substrate at half the maximal velocity, tS]tOt is the total substrate concentration, and fU is the fraction of the substrate that is unbound in the reaction medium. By normalizing the rate to the substrate concentra- tion, metabolism can be expressed as a clearance value (Equation 2) which is frequently termed total intrinsic clearance (Gillette, 1971, 1984; Wilk- inson and Shand, 19751: d [S]'°' V dt [S]t°' KmlfU + [S]t°' Vmax f u V = Or tat Km +fu[s]tot — flint (2) Alternatively, the metabolic clearance rate can be related to the unbound drug concentration bv the free intrinsic clearance rate (CL'i°' — f u CLiUnt) Account can also be taken that metabolism often involves more than one pathway, so that the overall intrinsic clearance in any system represents the sum of the individual processes (Equation 31: ion V i- ~ Kmi + f ES]

82 GRANT R. WILKINSON In many instances the driving concentration for metabolism, the un- bound concentration, is far less than the Km of the involved enzymeks), and the kinetics become independent of substrate concentration. Under such linear or first-order conditions, the free intrinsic clearance reaches a constant maximum value equal to the ratio of the overall Vmax to Km (Equation 4~: i=n V CL urn = A, i . (4) i=l Kmi In principle, therefore, the means exists to relate readily determinable Michaelis-Menten enzyme kinetic constants to a well-understood concept and parameter of pharmacokinetics. Such Vma,~ and Km estimates can be directly incorporated into the differential mass-balance equation for total drug in a particular metabolizing organts) by using a perfusion-limited physiological model (Equation 51: ACT L Vm=(CTl Kp ~ ] (5) where C, KP, and V represent the total substrate concentration, tissue/ plasma partition coefficient, and volume of the eliminating tissue (T), respectively; and Q is the blood flow to the organ. This approach has been applied to studies with cytosine arabinoside, when the in vitro kinetics of deamination in a number of organs and in different species allowed good predictions of the in viva situation (Dedrick et al., 1972, 19731. Similar findings were made with three barbiturates (Igari et al., 19821. One possible reason for the paucity of this type of data is that Equation 5 provides no intuitive insights into drug elimination relative to the con- ventional quantitative estimates of this process, such as (organ) clearance (CL) and extraction ratio (E). Unless the equation is solved along with others in the model and the resulting blood concentration/time data are analyzed by conventional pharmacokinetic techniques, there is no easy way to assess the validity of the in vitro values relative to those determined from experimental data. Under steady-state conditions, Equation 5 can, however, be rearranged to provide an estimate of the extraction ratio (Equation 6) or clearance (QE) of an eliminating organ: Vmar KmlfU + CTlKp Q Vmax KmlfU + CTlKp which under first-order conditions simplifies to: (6)

Extrapolation from In Vitro Systems 83 Cr tot Guru E = Hint J B Quint , (7) Q + CL,t°t Q + fg CLUnt where fB refers to the unbound fraction of drug in blood which is related to the more frequently measured value in the plasma (fp) by the blood/ plasma concentration ratio tfB = fpl(BlP)~. Equations 6 and 7 therefore provide the means to incorporate in vitro estimates of enzyme kinetics into a perfusion-limited model of an eliminating organ and to evaluate their appropriateness by comparing the predicted extraction ratio or clear- ance to that measured in vivo. This model has been termed the venous- equilibration or well-stirred model of organ elimination (Pang and Row- land, 1977; Rowland et al., 1973; Wilkinson and Shand, 19751. The feasibility of such an in vitrolin vivo predictive approach was first inves- tigated with several drugs that had widely divergent extraction ratios (0.2 to >0.9) undergoing drug oxidation (Rane et al., 19771. The Michaelis- Menten kinetics of overall substrate disappearance were determined in a 9,000 x g supernate of rat liver; and after correction for dilution and protein recovery, the first-order, intrinsic clearance value (Vma,~lKm), along with the unbound fraction, was incorporated into Equation 7, resulting in an in vitro predicted hepatic extraction ratio. Comparison of this with an experimentally determined steady-state extraction ratio in the isolated per- fused rat liver showed excellent agreement (r = 0.988, p ~ 0.0011. Subsequently, the same approach was applied to other drugs and metabolic pathways, as well as other hepatic preparations (Table 11. The prediction of metabolic elimination by other organs such as the lung and intestinal mucosa was also studied (Table 21. Despite the relatively small number of these studies, some tentative conclusions can be drawn from their findings. First, and most important, the in vitro predictive approach using Equation 7 appears to provide a reasonable estimate of the in vivo situation for a number of drugs across a broad range of metabolic activity. This is particularly the case when the compounds are metabolized by the liver and involve the cytochrome P-450 system. Examples of other hepatic routes of metabolism are not sufficiently large to draw any conclusions, although the in vitro under- estimation of glucuronidation (Rane et al., 1984) is consistent with the known lability of glucuronyltransferase activity. The situation in the lung is less clear, which may reflect the anatomical and biochemical hetero- geneity of this organ. The type of in vitro system on which the prediction is based may also be an important factor. Successful predictions require that the conditions of metabolism in vitro be identical to those in vivo. Unfortunately, these are never known, and this is the crux of the problem. Microsomal and purified enzyme preparations are usually investigated with a maximal

84 ._ au Cal at: En 5 Ct Cal ._ Cal ~ _ O ~ c: Ct `_— ~ I ,= O ~ Cal S:: ~ ·_ Ct O ~ ·— ._ Cal Cal ~ LO Cal O Cal X — o ._ Cal ~ m o · _ _ ~ _ ;2 ~ — O m ~ o . - o ._ S. i:. Cal o - m ~ En ~ So ~ 00 00 0~ _ - ^ 00 ~ ~ 0Oo ~ ~ ~ <~N ~ ~ Pt ~ ~ Em ~ 'A ~ (~\ ~ {: ~ ~D _ Ce ~ C~ Ce ~ _ ~ _ ~ _ _ _ _ ~ _ ~ _ C,, ~ _ ~ . _ ~ ~ ~ ~ ~ ~ ~ 04 ~ ~ C) ~ ~ .,_ ~ ~ ~ ~ ~ ~ ~ . _ ~ Ct ° Ct C';: ~ Ce :Ct Ct ~ Ct C~ O Ct ~ ~ v~ pt pt := e~ ~ ~ ::; ~ ~ ~ ~ C: ~ . O Ct ._ ~ 2 c oCt o.° .°.°.°.°.° .° .°,- ~ 3 — ~ ,~ c~s c~s ,,: ce ,~ ~ ,~ ~ t .s ~ ~ ~ ~ ~ ~ ~ ~ V ~ C.) X X X X X X X ~ X _ ~ ~ ~ O O O O O O O ~ O ~ s: ~ ~ o o o ._ ._ ._ Ct Ct c<5 ._ ._ ._ X X X 00 0 U~ 00 ~ ON ~ o o o 0 _t r~ 0 oo oo oo — ~ r~ u~ 0 ~ ~ ~ ~ 0 — ~ ~ u~ O .~ c~ ~ ~ ~ ~ cr~ 0 0 ~ ~D o0 oo oo ~ ~ ~ ~ oo ~ O 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 := ~ A A A o ° ~ · C} 1 ~1 0 u~ ~ u~ ~ - 1 _ o 0 ~ oo 0 C) ~ ~ ~ ~ ~ o o oo ~ C~ oo oo ~ ~ oo ~ ~ . . _= ooooooooooooooo ._ Ct Ct C) oc o ~ ~ U~ ~ o oo . . . . o o o o C~ ~ ~ ~ ~ C) ~ c~ (e (e (t ~V :~ :e {e ce C) ~ C) C) ~ C) ~ ~ C) ~ C~ C~ ~ =, .=, `, ~ C~ ~ ~ ~ C) o~ oc ~ C) oc oc ~ X X o ~ X X ~ o o o o X ~ X o C) X o X O ~ O ~ ca ~ v, O4 ~ O cn Ce o o ),o., Ce o o X ',o., -° i.°, s.°., ° s° ° -° ~ ° ~ ° =~= .= ~ ~ -s ~ ~ ·— c~ _ ?~ D~ ~ ~ ~ ~ _ o o — u — 2 — c 2 C · X ~ j C ' U 2 ._ D D ct O t: ~ 'e ~ ~ o r~ ~

85 a: o ._ so ._ ._ Cal no ~9 C) ._ 4 - EM - o o ._ X ~ o ~ ., ~ ~ ~ o 3 ce m - Cal O ~ · = - Sol` o Cot m m C > O Ct ~ 3 — Ct Ct ._ o o ._ .~ ._ _ ~ . ° Ct en To Go o o o ~ Do Do Do — ~ at ~ - ~ Do _ _ ~ ~ O ~ ~ p~ o ^ mm == ~ Ct Ct ~ ~ ~ .— . ~ ~L) .— ._ _ C) C~ Ct S~ C~ cl o l ~ o ~ ~ . - o o x ~ o ~ ~ =' ~ ~ ·x x mv~o o ~s o ~ ~ ~ c~ ~ ~ ~ - ~ o o o o o u) o o o o ~ tl tl Ol ~ ~ ~ ~ _ C~ ~ O — ~ . . . . . O O O O O C~ C) C~ C~ C~ o ~ X X ~ — ·~ ~ 8 o ~ ~ .= ~ o X ~ Ce U~ ~ Ct C~ CL ~ C) D ~ U 3: ~ e ~ m m Ct ~ a' - Ct C) ._ P" au o S~ ._ ;^ e~ ._ 0D ._ r: C) _ Ct D O s~ O C: C) 5 5

86 GRANT R. W'EK! NSON concentration of substrate and under optimal biochemical conditions, in which the pH of the medium and the concentrations of cofactors may be much different from those in the in viva situation. Purification steps may also affect the biochemical parameters. The latter may account, for ex- ample, for the almost 50% underestimate of phenytoin's hepatic extraction using microsomes (Collins et al., 1978), but not when a 9,000 x g supernate was used (Rane et al., 1977~; preparation of microsomes reduces the Vmax of phenytoin by about one-half without affecting the Km (Kutt and Fouts, 19711. The use of intact cells such as hepatocytes may not necessarily overcome such problems because at least one additional factor is involved, namely, the ability of the drug and metabolites to diffuse into and out of the cells (deLannoy and Pang, 1986~. If such diffusion is slow, then the apparent Km value, but presumably not Vma,~, may differ from that observed in cell-free preparations. Other complicating factors may also occur so that the Vma,` or Km values alone or together may be different from those in other in vitro systems (Gillette, 1984~. Unfortunately, no general rules appear to exist that would indicate when hepatocytes, for example, reflect the in vivo situation more closely than do other prepa- rations (Pang et al., 19851. In addition to these factors, it must be recognized that the Michaelis- Menten mechanism is probably an oversimplification of actual intracellular events. The relationship between Vma,` and Km may be quite complex and modified by changing the concentrations of, for example, cosubstrate, activators, and even products of the substrate and cosubstrate. Moreover, a multienzyme system is more frequently operative in a drug's overall metabolism than is a single enzyme. These enzymes may have widely different characteristics, including reaction mechanisms, so that any over- all kinetic value like intrinsic clearance (Equation 4) may not accurately reflect in in vivo determinants (Gillette, 19841. A further consideration in extrapolating in vitro biochemical data to the in vivo pharmacokinetic situation is the validity of the venous equilibration model (Equation 7) to describe organ elimination. In this model the organ is considered to be a single well-mixed compartment, and diffusion from the blood into the tissue is not considered to be rate-limiting. Accordingly, the concentration of unbound drug in the emergent blood is assumed to be in equilibrium with that in the tissue, and the concentration profile across the organ is constant. Two alternative perfusion-limited models have been developed, however, that have the merit that an arterio-venous concentration gradient exists across the organ. The sinusoidal perfusion or parallel tube model conceives of the organ as a large number of identical cylindrical tubes that are arranged in parallel with the cells (hepatocytes), each of which has the same eliminating activity, surrounding the cylinder. The relationship between intrinsic clearance and extraction is then given

Extrapolation from In Vitro Systems 87 by Equation 8 (Bass et al., 1976; Pang and Rowland, 1977; Winkler et al., 1979~: ru cru _J B inr E= 1 —e Q (8) By contrast, the dispersion model of elimination conceives of the organ as a packed-bed chemical reaction with nonideal flow (Roberts and Row- land, 19861. Two parameters characterize this model; the efficiency num- ber (RN) which describes the efficiency of drug removal and is equivalent under first-order conditions tOfgCL'n~/Q, and an axial dispersion number DN. The latter is a measure of dispersion or spread of residence times of drug molecules moving through the organ; the higher the value of DN, the greater the degree of axial dispersion, reflective of functional heter- ogeneity. The mathematical solution for the extraction ratio is quite com- plex (Equation 9~: 4a ( 1 + a)2 exp [ (a—1 )12L)N] - ( 1—a)2 exp—[ (a + 1 )12DN] (9) where a = (1 + 4RNDN)~/2. Interestingly, when DIN X, i.e., when there is extensive axial dispersion, the model devolves to the venous- equilibration situation (Equation 7), whereas when DN ~ O. i.e., axial dispersion is negligible, Equation 9 reduces to a form similar to that of the sinusoidal perfusion model (Equation 8). In global terms, all three models predict similar relationships between the biological detet~inants of drug elimination, but they differ at the finer level. For example, an in vitro estimate of intrinsic clearance will predict a larger extraction ratio when incorporated into a sinusoidal perfusion model than if the same value is used in the venous equilibration model; the dispersion model provides an intermediate value. Moreover, the model discrepancies become larger as the intrinsic clearance increases, i.e., the greater the drug's extraction ratio. Discrimination studies to determine which model, if any, provides the best in vitrolin vivo predictions are, however, very limited. Recently, Roberts and Rowland (1985) concluded that the dispersion model was more consistent with published data for 10 drugs than either of the other two models, especially when the extraction ratio was greater than about 80%. Further studies of this aspect of pre- diction are clearly required. Another modeling consideration requiring further investigation relates to the known functional heterogeneity of organs such as the liver. It is well established, for example, that hepatic perfusion is not uniform and intraorgan shunting is present; also, enzyme activity is not uniformly

88 GRANT R. WILKINSON distributed in hepatocytes. The venous-equilibration model cannot take such factors into account, whereas they are implicit in the dispersion model; and the sinusoidal perfusion model can be modified to reflect such heterogeneity (Bass, 1980, 1983; Bass et al., 1978; Sawada et al., 19851. Studies have begun to address this problem (Pang, 1983; Pang et al., 1986), particularly with regard to the formation and elimination of different metabolites; however, it is not a factor that can be currently taken into account in in vitrolin viva predictions. Other unresolved problems of in vitro extrapolation of drug metabolism include consideration of metabolites. In the past, the primary focus has been on unchanged drug and its disposition, but the formation and sub- sequent fate of a metabolite, stable or reactive, may in certain cases be a more important consideration. Complications due to such factors as sui- cide-substrate inactivation of metabolizing enzymes and cofactor depletion need to be considered when certain types of nonlinear pharmacokinetics are likely. Finally, it is important to note that there appears to be only a single reported study (Baarnhielm et al., 1986) where the metabolism of a drug in humans has been predicted from in vitro data. Human liver preparations, for example, are becoming increasingly available, and it would be a reasonable extension to apply modeling approaches to this critical area. This is particularly important with respect to interspecies extrapolation by using physiologically based pharmacokinetic models. The well-established variability in metabolism between species logically pre- cludes the use of any allometric scaling factor. Accordingly, studies in human tissuefs) will be required. IN VITRO PREDICTION OF IN VIVO DRUG BINDING AND DISTRIBUTION Drugs and other xenobiotics invariably bind in a reversible fashion, described by the law of mass action, to a variety of blood and tissue constituents. Such binding determines the unbound fraction in the blood/ plasma, and also distribution from the intravascular space into tissues. Accordingly, for physiologically based pharmacokinetic modeling to be successful, estimates of this binding and distribution are required. The various in vitro techniques and interpretative approaches for char- acterizing the kinetics of drug binding to plasma proteins are well-estab- lished, as are various factors that modulate such binding (Reidenberg and Erill, 1986; Tillement and Lindenlaub, 19861. A limited number of ex- amples exist in which in vitro parameters of linear or nonlinear plasma binding has been explicitly incorporated into physiologically based phar- macokinetic models of specific compounds (Bischoff and Dedrick, 1968; Engasser et al., 1981; Igari et al., 1983; Tsuji et al., 19831. But this factor

Extrapolation from In Vitro Systems 89 is generally ignored, and only total drug is considered. Given the potential importance of plasma binding on the pharmacokinetics of drug elimination and distribution (Pang and Rowland, 1977; Wilkinson, 1983; Wilkinson and Shand, 1975), and the fact that only the unbound drug is considered to be biologically active, the omission of this factor in modeling is re- grettable. But, more importantly from the extrapolation standpoint is the validity of the assumption that the binding parameters established in vitro reflect those in viva. The conventional concept of plasma binding and drug transport out of the vascular space is that only unbound drug is able to penetrate mem- br-anes, and therefore, the unbound fraction and concentration in the cap- illary blood are the important determinants. Moreover, binding equilibrium is maintained within the capillary so that the kinetics of binding is the same as that in systemic blood, and this can be estimated by an appropriate in vitro technique such as equilibrium dialysis. An increasing number of experimental studies with xenobiotics that are normally very extensively bound to plasma proteins (>99 percent), however, appear to be incon- sistent with this dogma. Instead, it appears that bound drug is in some fashion intimately involved in uptake by such organs as the liver, heart, and brain, beyond its role as a passive store for replenishing the unbound moiety subsequent to its extraction. The mechanism involved in such transport is not well understood and may well differ, depending on the drug and the experimental situation (Jones et al., 1986; Pardridge, 1986; Weisiger, 19861. Nevertheless, it is clear that in this type of situation conventional modeling approaches will be of little value in predicting drub disposition. For a noneliminating organ, the mass balance equation for drug transport Is: d—VT = Q(Cin COW{) (10) Generally, it as assumed that distribution is perfusion limited, so that the emergent venous blood is in equilibrium with the average total concen- tration in the tissue (CT), SO that Equation 10 can be modified to the familiar form: dCT ~ ~ VT = Q niacin _ CT'\ KpJ (1 1) where Kp is the tissue/blood (plasma) drug concentration ratio, also called the partition coefficient or solubility of drug in the tissue. Accordingly, knowledge of this parameter is critical in the physiological modeling procedure.

90 GRANT R. WILKINSON In practice, Kp is generally determined by direct measurement after drug administration and analysis of the arterial blood (plasma) and the tissues of interest. Sampling is usually perfo~ed after intravenous drug infusion to steady-state, although it is also possible to determine Kp after an in- travenous bolus dose subsequent to the attainment of pseudo-equilibrium distribution. Theoretically, the partition coefficient during the terminal phase of elimination (Kp app) is greater than the value obtained at steady state, the difference being smaller the more rapidly drug distributes into the tissue and the faster the rate of elimination (Equation 121: Kp Kp,app = 1 Az kT (12) where Az and kT are the first-order rate constant during the terminal elim- ination phase and for tissue uptake, respectively (Rowland, 1986~. In most instances, it is likely that kT > Az, so that the difference between Kp and Kp~app is small, and experimental comparisons support this assumption (tin et al., 1982; Schuhmann et al., 1987~. It has also been demonstrated that determination of a partition coefficient is further complicated if elim- ination occurs in the tissue (Chen and Gross, 1979; Lam et al., 19824. In this case, the appropriate blood concentration used to estimate Kp should be that in the emergent venous blood of the organ rather than the arterial level (Rowland, 1986~. Under steady-state conditions, and recognizing that total drug concen- tration can be defined in terms of the unbound fraction and concentration, Equation 11 can be rearranged to show that Kp is equal to the ratio of the unbound fractions in the blood and tissue ~B/fT). Thus, in theory the in vitro estimation of the extent of a drug's plasma and tissue binding could be used to predict the in vivo partition coefficient. If valid, this approach would have considerable value in physiologically based pharmacokinetics. The measurement of tissue binding and determining factors has not been as extensively explored as that involving plasma proteins. This may partly be explained by methodological problems, but mainly through the studies of Kurz and Fichtl ~ 1983) and colleagues (Fichtl and Schuhmann, 1986; Schuhmann et al., 1987) these now appear to be mostly resolved. Tissue binding can be readily determined by ultrafiltration (Kurz and Fichtl, 1983) or equilibrium dialysis (Igari et al, 1982) of tissue homogenates, with appropriate correction for the effects of dilution (Kurz and Fichtl, 19831. Such binding demonstrates many of the characteristics of binding to plasma proteins, including saturability. Importantly, there does not, in general, appear to be any useful correlation between the two phenomena. Thus, plasma binding cannot be used to predict tissue binding. Also, binding to

Extrapolation from In Vitro Systems 91 tissue constitutents appears to involve mechanisms besides simple hydro- phobic interactions as related to lipid solubility measured by partitioning into organic solvents. Good correlations (Figure 1) have been found between predicted Kp values based on in vitro measurement offp end fT and those determined in vivo following intravenous infusion to steady-state of a number of drugs with widely different physicochemical and partitioning characteristics (Schuhmann et al., 19871. The best prediction was for muscle fir = 0.93), in which all of the data centered around a line with a slope of unity. A similar finding was obtained for the liver, except for drugs with a high extraction ratio in which the in vivo value was lower than predicted; this probably was due to the use of the arterial rather than the more appropriate (~/P)m loo - 10 01 1 l (T/P)m | 100 - 10 - 1 0.1 MUSCLE (T/P)m ~- 100 - 10 - oul:' / B.~4P T e 1 /PHe PRO i/ BUP/ SALO/ ~S/BZ ( TtP)C 0.1 - 01 1 10 100 Ll VER (T/P)m I , 100- au'~^ON 10- TP: ·~54' /— O PRO 1 - /7/ (T/P)C 0.1 . . I 0.1 1 10 100 LUNGS SAL O/ 0W O PRO ·1~1 Outgo PTS 0/ · D'Z O8UP ( T/P)c / ~- 0.1 1 10 100 KIDNEYS PHB5 o PBZ / l IMP— opp0 QUI`o BuPo TPL`g ~ PTB O/ AMP / ( T./ P)' 0.1 1 10 100 FIGURE 1 Correlation between experimentally measured (TlP)m and in vitro-predicted (TIP)c tissue partition coefficients for several drugs and tissues in the rabbit. Reproduced with permission from Schuhmann et al. (1987).

92 GRANT R. WILKINSON hepatic venous plasma concentration to estimate the in vivo partition coef- ficient. In the lung and kidney, the predicted Kp values of highly lipophilic, cationic drugs such as buphenine, quinidine, imipramine, and propranolol were lower than those measured in vivo. This probably reflects the loss of saturable uptake process in these homogenized tissues and the slow equilibration time for distribution in vivo. These and other data (Harashima et al., 1983; Lin et al., 1982), although limited, indicate that it is possible to predict tissue partitioning in vivo for several tissues from their in vitro binding data, particularly if the drug is anionic or neutral. With cationic drugs, distribution into muscle, and possibly liver, is also well predicted; but discrepancies are possible with organs such as lung and kidney, in which active uptake processes may be present in vivo. A similar situation would be expected if tissue distribution is not perfusion limited; i.e., diffusion across the capillary epithelium is slow. Interestingly, preliminary findings indicate that binding of drugs to various different tissues are well-correlated and similar (Fichtl and Schuhmann, 19861. This raises the possibility that drug distribution to a variety of organs could be predicted by using a single tissue such as muscle. As an extension of this hypothesis, Schuhmann et al. ( 1987) have recently found that the unbound volume of distribution of drugs in vivo could be predicted by their binding to rabbit muscle tissue in vitro. Moreover, interspecies differences in tissue binding do not appear to be as pronounced as those in plasma binding. Partitioning of a large number of drugs into muscle tissue of rats, rabbits, and humans is very similar (Fitchtl and Schuhmann, 19861. If further substantiated, this would simplify consid- erably the interspecies extrapolation problem. CONCLUSION In summary, the concepts and experimental techniques for predicting the in vivo distribution of a drug into tissues and its elimination by me- tabolism by using in vitro data have been established. Experimental data to substantiate and validate these approaches are still relatively limited, but they are sufficiently encouraging to suggest practical feasibility. Clearly, much more investigation is required, particularly with respect to delin- eation of the approaches' limitations. In this regard it is important to define the goals and recognize the purposes of any in vitrolin vivo prediction. As the precision and accuracy requirements become more rigorous, the more difficult it will be to make a prediction. On the other hand, much useful information and insight can be gained from appropriately interpreted in vitro data, even though the prediction is not necessarily exact. Finally, it must be appreciated that interindividual variability, particularly in drug

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Extrapolation from In Vitro Systems 95 Rane, A., J. Sawe, B. Lindberg, J.-O. Svensson, M. Garle, R. Erwald, and H. Jorulf. 1984. Morphine glucuronidation in the rhesus monkey: A comparative in vivo and in vitro study. J. Pharmacol. Exp. Ther. 229:571-576. Reidenberg, M. M., and S. Erill. 1986. Drug-Protein Binding. New York: Praeger. Rowland, M. 1986. Physiologic pharmacokinetic models and interanimal species scaling. Pharmacol. Ther. 29:49-68. Roberts, M. S., and M. Rowland. 1985. Correlation between in vitro microsomal enzyme activity and whole organ hepatic elimination kinetics: Analysis with a dispersion model. J. Pharm. Pharmacol. 38:177-181. Roberts, M. S., and M. Rowland. 1986. A dispersion model of hepatic elimination. 1-3. J. Pharmacokinet. Biopharm. 14:227-308. Rowland, M., L. Z. Benet, and G. G. Graham. 1973. Clearance concepts in pharmaco- kinetics. J. Pharmacokinet. Biopharm. 1: 123- 136. Sawada, Y., Y. Sugiyama, Y. Miyamoto, T. Iga, and M. Hanano. 1985. Hepatic drug clearance model: Comparison among the distributed, parallel-tube and well-stirred mod- els. Chem. Pharm. Bull. 33:319-326. Schuhmann, G., B. Fichtl, and H. Kurz. 1987. Prediction of drug distribution in vivo on the basis of in vitro binding data. Biopharm. Drug. Dispos. 8:73-76. Skanberg, I. 1980. Metabolism of two beta-adrenoceptor antagonists, alprenolol and me- toprolol, in different species. In vitro and in vivo correlations. Acta Universitatis Up- saliensis (Abstracts of Upsalla Dissertations from the Faculty of Pharmacy), Vol. 50. Smith, B. R., and J. R. Bend. 1980. Prediction of pulmonary benzo(a)pyrene 4,5-oxide clearance: A pharmacokinetic analysis of epoxide-metabolizing enzymes in rabbit lung. J. Pharmacol. Exp. Ther. 214:478-482. Tillement, J.-P., and E. Lindenlaub, ed. 1986. Symposia Medica Hoeschst, Vol. 20, Protein Binding and Drug Transport. Stuttgart: F. K. Schattauer Verlag. Tsuji, A., T. Yoshikawa, K. Nishide, H. Minami, M. Kimura, E. Nakashima, T. Terasaki, E. Miyamoto, C. H. Nightingale, and T. Yamana. 1983. Physiologically based phar- macokinetic model for p-lactam antibiotics. I. Tissue distribution and elimination in rats. J. Pharm. Sci. 72:1239-1252. Weisiger, R. A. 1986. Non-equilibrium drug binding and hepatic drug removal. Pp. 293- 310 in Symposia Medica Hoeschst, Vol. 20, Protein Binding and Drug Transport, J.- P. Tillement and E. Lindenlaub, eds. Stuttgart: F. K. Schauttauer Verlag. Wiersma, D. A., and R. A. Roth. 1980. Clearance of 5-hydroxytryptamine by rat lung and liver: The importance of relative perfusion and intrinsic clearance. J. Pharmacol. Exp. Ther. 212:97-102. Wilkinson, G. R. 1983. Plasma and tissue binding considerations in drug disposition. Drug Metab. Rev. 14:427-465. Wilkinson, G. R., and D. G. Shand. 1975. A physiological approach to hepatic drug clearance. Clin. Pharmacol. Ther. 18:377-390. Winkler, K., L. Bass, S. Keiding, and N. Tygstrup. 1979. The physiologic basis for clearance measurements in hepatology. Scand. J. Gastroentrol. 14:439-448.

Dose, Species, and Route Extrapolation: General Aspects James R. Gillette The objective of any pharmacokinetic study is to describe as simply as possible the factors that govern the time course of the concentrations of biologically active Formosa of a substance at putative action sites and to relate these concentrations to the incidence and magnitude of the biological responses. The biologically active Formosa may be the parent substance and/or one or more of its metabolites. In developing pharmacokinetic models to achieve this objective, investigators must consider a host of interrelated complex factors and events that could conceivably affect the time course of the parent substance and its biologically active metabolites at the action sites and to decide whether such factors are sufficiently important to include in the model. DIFFERENT PROBLEMS AND OBJECTIVES, DIFFERENT MODELS Clearly, the complexity of suitable models differs with the substance. A model that is suitable for describing the pharmacokinetics of substances, like ozone, which might be expected to act almost solely at the site of administration (such as lung) would clearly differ from a model designed to describe the pharmacokinetics of a substance (like halogenated hydro- carbons and heavy metals) that persists for several years in the body or in the food chain and causes toxicity in remote organs of the body. A model designed to describe the pharmacokinetics of a parent substance would be inadequate to describe the pharmacokinetics of a toxic metab- 96

General Aspects of Extrapolation 97 elite. A model designed to describe the pharmacokinetics of a stable metabolite would differ from one designed to describe the pharmacoki- netics of a suicide inhibitor. A model designed to describe the kinetics of a single dose of a substance may or may not be appropriate for the description of the kinetics of a substance entering the body by a series of repeated doses or at a continuous rate. Different Mechanisms When a substance enters the body by a series of repeated doses until a steady state is achieved, the investigator must also consider whether the incidence or magnitude of the biological response is most closely related to the maximum concentration, the average concentration, the minimum concentration, or the total dose of the biologically active form of the toxicant. The response will be most closely related to the maximum con- centration of the unbound biologically active forms (1) when the response is manifested virtually instantaneously after the active forms interact re- versibly with receptor sites, (2) when the response is due to irreversible interaction of a substance (or enzyme) that turns over rapidly in the body, or (3) when the biologically active Formosa causes physiological or bio- chemical changes that exceed the capacity of homeostatic mechanisms to adjust to them and result in irreversible damage. The response will be most closely related to the average concentration of the unbound biolog- ically active Formosa when the response is due to certain irreversible mech- anisms when the rates of replacement or repair of the action sites are slow. The response will be most closely related to the minimum concentration of the unbound active Formosa (1) when the response is caused by non- competitive, but reversible, inhibition of an enzyme that alters the con- centration of a vitally important endogenous substance, or (2) with certain kinds of irreversible mechanisms. The response will be more closely related to the total dose of the biologically active forms when the response is due to an irreversible accumulation of toxic products in the body. In this context, the accumulation of transformed cells that serve as clones for tumors can be viewed as the accumulation of toxic products. GENERAL PHYSIOLOGICALLY BASED PHARMACOKINETIC MODELS Unquestionably, the approach used in developing physiologically based pharmacokinetic models is the most versatile of all the approaches used in solving pharmacokinetic problems. As pointed out by K. B. Bischoff (this volume), one simply writes an appropriate set of differential equa- tions, based on the conservation of mass law, which describe the rates of

98 JAMES R. GILLEUE changes in the amounts of the substance and its metabolites in various compartments in blood, extracellular fluid, and cells within an organ and allows the computer programs to integrate the equations by iterative pro- cedures. The investigator thus is not limited to first-order reactions that have simple analytic solutions. A complete description of all of the events that can occur within an organ, however, can be very complex. Sets of equations could be written to describe the rates of change in the amounts of various metabolites within cells and other compartments of the organ. Other sets of equations could be written to include changes in concentration of the substance and metabolites as they pass from the proximal to the distal portion of capillaries in a given organ. Thus, the number of differential equations used by the investigator to describe a physiologically based pharmacokinetic model can be virtually infinite. The solution of such a model, however, would also require knowl- edge of the values of all of the rate constants that relate the amount of the substance and its metabolites in the various compartments to the rates of transfer into and out of each of the various compartments. Most of these rate constants, however, are not only unknown but also virtually unattainable. Simplification of Models Clearly, the investigator wishes to simplify the system to a minimum number of differential equations that still would provide all of the relevant information. How the model can be simplified and still be valid, however, will differ with the substance and the organ and the time frame within which the investigator wishes to focus attention. Most simplifications are based on the realization that when a substance is constantly infused through an organ most reactions that occur within the intraorgan compartment will approach "virtual equilibria" and "vir- tual steady states" within the time required for the blood to pass through the capillary bed of the organ. For example, theoretical calculations of the rates of formation and dissociation of complexes of various small molecular weight substances with various proteins, including enzymes, will usually be considerably faster than the transit time of blood through organs. Thus, these complexes are usually represented by steady-state . . cone ltlons. Rates of Formation of Complexes The validity of this conclusion can be illustrated by the events that occur immediately after a substrate is added to a solution containing an enzyme. Under these conditions the following differential equation can be written:

General Aspects of Extrapolation 99 d[ES]ldt= {[Et] [ES]} [S]k1 - [ES](k2 + k3), (1) d[ES]ldt = [Et][S]k1 - [ES](kl[S] + k2 + k3), or (la) d[ES]Idt = [Et][S]k1 - [ES]k1 {[S] + [(k2 + k3 )/kl]} (lb) But (k2 + k3)lkl may be set to Km and thus, d [ES]ldt = [Et] [S] k1 - [ES] k1 ( [S] + Km) (1c) Integration of Equation to gives: [ES] = ~ t~ ~ (S] [1 — e-ki(tS]+Km)~] (ld) Inspection of the exponent reveals that the half-time required to approach the maximum concentration of [ES] is dependent on the value of kit, as well as on [S] and Km. Although the value of kl may vary markedly, most values of kl between proteins and small molecular weight substances are between 1Os M- ~ s- ~ to 108 M- 1 s- ~Taylor, 1972). Thus, for an enzyme in which the Km is greater than 10-5, the half-time for the approach to a steady state usually should be less than a second; thus, a virtual steady state usuallY would be achieved within 5 s. After a virtual steady state is achieved, the rate of metabolism of the substrate can be frequently ex- pressed by the usual form of the Michaelis-Menten equation. Equations analogous to Equation la can be written to describe the rate at which a substance may become reversibly bound to each of the com- ponents in blood, extracellular fluid, and cells in an organ. The time required to approach a virtual steady state depends largely on the second- order rate constant for the formation of the complex and the equilibrium constant of the complex. Since the average residence time of blood within capillaries is probably in the range of 1 to 10 s in most organs, one can be reasonably assured that virtual steady-state values can be assumed when either the unbound concentration or the 1 IKa values for reversible binding are greater than 1o-s M. But one would worry when they are less than 10-8 M. Equilibrium constants rather than rate constants of association and dissociation are thus generally used in the models, but the rate constant of dissociation for ultrahigh-affinity binding sites may still be important during the terminal phases of elimination, when ES] decreases below the 1/Ka values or during constant exposure to very small doses. Diffusional Barriers and Modified Fick's Law Whether an investigator wishes to incorporate diffusional barriers in the model depends on the time required to achieve the virtual steady state in "filling" the extracellular and intracellular spaces. According to Fick's

|00 JAMES R. GILLEUE Law of Diffusion, the rate of passive transfer of a substance across a thin membrane is usually written as, dS = D (Ci —CU2), (2) where D is the diffusivity index, A is the area of the membrane, and X is the thickness of the membrane. Since a membrane is usually considered to be lipoidal in character, the value of D is usually considered to be proportional to the partition of the substance between the cell membrane and water, which is frequently estimated from the partitioning of the substance between oil and water. When the substance is a weak acid or a weak base, it is usually assumed that only the neutral form of the substance is able to pass through semipermeable membranes and that the concentration of the neutral form can be estimated from the total concen- tration of the unbound substance by means of the Henderson-Hasselbalch equation. The ionized forms of many substances, however, can pass through semipermeable membranes, even though the diffusivity index for the ion- ized form may be orders of magnitude smaller than that for the neutral form. Thus, for weak acids with pKa values that are several units smaller than 7, and weak bases with pKa values that are several units larger than 7, the rate of diffusion can be governed predominantly by the concentration of the ionized form, despite its low diffusivity index. Moreover, mem- branes can differ markedly in their structure and function. For example, capillaries usually have intercellular spaces, which allow passage of both polar and nonpolar substances. Moreover, the plasma membranes of cells frequently permit the passage of very small substances through the mem- brane. Thus, the value of the effective D depends on a number of the diffusion indices representing different parts of the membrane (Pang and Gillette, 19791. For these reasons a modified version of Fick's Law of Diffusion is more appropriate (Figure 11. Thus, the passage of the sub- stance across a membrane can be described by: dSldt = (DA/X) Cy— (D2AIX)C2, (2a) in which both Do and D2 incorporate not only the heterogenous nature of biological membranes but also the effects of different pH values and other factors that affect electrochemical gradients at the two interfaces of the membrane. In Equation 2a, both Cat and C2 are the concentrations of the unbound substance at the two interfaces of the membrane; they do not include the amount reversibly bound to proteins and other components in the compartments. The rate at which the concentration of unbound substance changes in compartment 2 can be expressed by:

General Aspects of Extrapolation 101 MODIFIED FICKS LAW OF DIFFUSION [1]o [Reversibly Bound] O At Equilibrium ~11 [NJo Dl ~ [1]j D'' ON DIN ~ [N]; dt = [I]o ( x ) + [N]o ( x ) [I]' ( x ) - [N]j ( x ) [Reversibly B ound ] j D'A DNA D''A DNA [I]o ( x ) + [N]o ( x ) = [1]; ( x ) + [N]' ( x ) [I]o ( D ) + [N]o = N= Neutral: 1= ionized [It (D—) + [N]l FIGURE 1 Modified version of Fick's Law of Diffusion. V2 dC2ldt = (D1A/X) C1 - (D2AIX)CU2, (2b) where V2 is the apparent volume of distribution of the unbound substance in compartment 2. Since C2 is expressed as the concentration of unbound substance within the compartment, V2 would be defined as the total amount of substance in compartment 2 at any given time divided by C2 at that time. Under conditions in which Cat is constant, integration of Equation 2b would give the equation: C2U = 1 1 ~ 1 — e (D2A/XV2U)~- (2c)

i02 JAMES R. GILLEUE Thus, the approach to a virtual equilibrium in the system depends on the apparent volume of compartment 2, but the ratio of the concentrations of unbound substance in the two compartments at equilibrium does not. When a substance leaves compartment 2 by first-order processes, such as metabolism, Equation 2c would be modified to: Cu (DiA/X)C~ t1 (D2AIX) + CL20 —{[(D2 A/X) + CL2o]/v2}t i, (2d) where CL20 represents the metabolic clearance. When a substance leaves the compartment by passage into another fluid, such as lymph or the cerebral spinal fluid, the CL20 is replaced by CL2olfu. Thus, inclusion of the clearance term actually decreases the time required to come to a virtual steady state. But because of the ratio of the concentrations of unbound substance in the two compartments, a virtual steady state would be de- pendent on the relative values of (Di A/X) and CL20. Whether the approach to a virtual steady state for the diffusion of substance is less than a few seconds, and therefore can be excluded from consideration by the investigator, depends not only on the substance but also on the organ and even perhaps on the types of cells within an organ. Obviously, a lipid-soluble substance in an organ in which the apparent volume of distribution is small is unlikely to be diffusion limited. Indeed, the attempts to demonstrate diffusion-limited metabolism with such sub- stances have invariably failed. With polar substances, however, the con- clusion is less certain. Obviously, diffusion limitations do take place with substances that are apparently excluded from the brain; in such cases, the low brain concentrations are maintained at low levels predominantly by the cerebral spinal flow. Moreover, the rate of metabolism of polar sub- stances in liver can be diffusion limited (deLannoy and Pang, 19861. Simple PB-PK Models Frequently, pharmacokineticists that develop physiologically based pharmacokinetic models (PB-PK models) simply assume that the substance in the capillaries of an organ distribute instantaneously to all compartments within the organ, that metabolism within the cell is diffusion independent, and that the approach to virtual steady states in all organs is limited solely by blood flow rates. They describe the approach to a virtual steady state of the entire organ by a very simple differential equation. For a well- stirred model, in which the total concentration of a substance in capillary blood is assumed to equal its total concentration in venous blood leaving the organ,

General Aspects of Extrapolation 103 Vj a'Ci Q C Ri RiKm(i) + Ci QiCi Vma'~(i) Ci _ CLinr(~) Ci (3) where Vi is the actual volume of the organ, Ci is the amount of a substance in the organ at any given time divided by the volume of the organ, Qi is the blood flow rate through the organ, VmaX(i' and Km(i, are Michaelis- Menten parameters, and CLint(i) is the intrinsic clearance of other processes of elimination from the organ. Ri is the ratio of Ci to the total concentration in the blood within the organ, Cv(i' Basic Parameters fu and R Although within the limitations of the assumptions Equation 3 is correct as written, it nevertheless is deceptive in that certain relationships may not necessarily be obvious. Equation 3 could just as easily be written as: RiVi dCV(i' = Qi tCa — Cv(i)] Vma,`(i, Cv(i) Km(i' ~ Cv(i) CLint~i' Cv(i' (3a) In this form of the equation it becomes obvious that the Km value and the CLint~i~ value are expressed in terms of the total concentration of the sub- stance. But it is the unbound concentration in capillary blood, which in turn is assumed to be its unbound concentration within the cells of the organ, that governs the activity of the enzyme within the cells. Thus, there would be no way of relating a Km value or the CLint value obtained from in vitro experiments with the Km and the CLint values used in Equa- tions 3 and 3a. To emphasize this point, Equation 3 more properly should be written as: Vi dC = QiC _ QiCi CLint(i~fU Ci Ri RiVidC ti' = Qi [Ca — Cv(i)] Vma,~(i, f Cv(i) K +fUC(i) , or Vmat(i, Ci [Ri Km(i)lfU] + Ci (3b) CLint(i, fu Cv(i) ~ (3c) where fU is the unbound fraction of the substance in blood. The term Ci represents the total concentration of the substance in the organ. It thus does not necessarily represent the total concentration of the substance in any compartment within the organ. Moreover, Ri is assumed

104 JAMES R. GILLETTE to be a constant, but because reversible binding of substances to proteins and other components in blood plasma, erythrocytes, and tissue cells may be nonlinear, it should be regarded as a mathematical function, which under certain conditions may be virtually a constant. Furthermore, the rate of metabolism is assumed to be independent of the rates at which the substance diffuses through capillary and cellular membranes. When the rate of metabolism of polar substances in liver is thought to be diffusion limited (deLannoy and Pang, 1986), the equations would have to be modified. Non l i near Ki netics a nd Lost Concepts When the concentration of substances approach or exceed the Km values of enzymes or transport systems or the 1/Ka values of binding sites in blood or tissues, the meanings of such terms as organ clearances, organ availabilities, organ extraction ratios, volumes of distribution, total body clearances, and biological half-life become obscure. Moreover, the use- fulness of measurements of area under the curves of the substance in blood and tissues in evaluating many of these parameters, as well as bioavail- ability, is largely lost. When the concentrations of substances and their metabolites are small, however, physiologically based, linear compartment models can be developed, which provide information that cannot be ob- tained from either general pharmacokinetic approach alone. I NTERFACE BETWEEN PB-PK MODELS AN D CLEARANCES When Cacti' never exceeds a value equal to about 10% of the smallest Km of any drug-metabolizing enzyme in an organ, the rate of metabolism by the enzyme can be written as a linear equation: fit Car ~ is V,?~arlKm = fU Car ~ j' CLint ~ (4) and we can rewrite Equation 3c as: At = QitCa — Cv~i'] —fuCv~i'fsum CLin,). (4a) Organ Availabilities (F), Extraction Ratios (E), and Clearances (CL) After a virtual steady state is achieved in the organ, RiVi dCV`j' can be set equal to 0, and the resulting equation can be written as: Qjl1 — tCv(;~/Cal) = EfUcv~i~lca] Sum CLjn, = CLj. (4b) The term Cv~i'/Ca is frequently called the organ availability, Fj, and can be obtained by another rearrangement of the equation:

General Aspects of Extrapolation 105 Fi = Cv(i)/Ca = Qil(Qi + fU Sum Clingy. (4c) Substitution of Equation 4c into Equation 4b provides the following equa- tion: CLi = QjEl — F] = Q Ej = QQ~fu S CnE ' (4d) where Ei is referred to as the extraction ratio of the substance. Equation 4d is the same equation for a well-stirred model discussed by G. R. Wilkinson (this volume) and is the link between the differential equations used in general physiologically based pharrnacokinetic models and the equations used in the physiologically based linear compartment pharrna- cokinetic models. Physiologically Based Linear Compartmental Pharmacokinetic Models In these models investigators simply assume that all elimination pro- cesses that occur in organs approach virtual steady states rapidly. They visualize how the individual organ clearances can be incorporated into a total body clearance term and obtain a first-order elimination constant by dividing the total body clearance by an apparent volume of an empirical central compartment that includes not only the blood and the organs of elimination but also the nonelimination organs that would be included in this apparent volume of distribution. Validity of the Assumption of Virtual Steady State In order to gain an insight into the time required to attain a virtual steady state in an organ, let us assume that QCa remains constant during the approach to virtual steady state and integrate Equation 4a. We would obtain: where, cv = QiCa (1 — e-k') (5) Qi + fU Sum CLint k = Qi + f Sum CLin,~i~ (Sa) Thus, the value of k, which governs the time required to approach a virtual steady state, is dependent on Ri, even though the steady-state value of Cv is not. The longest time required to approach a virtual steady state can be estimated by setting k = QilRiVi. The effect of Ri on the maximal to s

}06 JAMES R. GILLEUE for the approach to virtual steady state in the organ can be estimated from the relationship, tots = Ri in 21(QilVi). For example, most estimates of the blood rate through the liver of different animals, including humans, are usually between 1 to 3 ml min- ~ g- i. On setting Qi/Vi = 1.0 min- i, we can calculate that in 21(QilVi) = 0.693 min- i. Thus, we can calculate that the longest tots for a substance with a Ri = 10 in liver would be about 7 min. Similar estimates can be made for other organs in the body. The assumption that the major organs of elimination (namely liver, lung, and kidney) attain virtual steady states by the time that the first blood sample is taken thus will be reasonably valid for most substances. Calculation of Other Compartmental Mode' Parameters The rate constants for the other compartments in the model can also be derived with the same parameters that are required for the solution of the simple physiologically based pharmacokinetic models. Each of the organs (or group of organs) would represent a separate compartment. The re- spective rate constants for the substance entering the organ from the central compartment would be (QilRCVc), and the rate constants for the substance leaving the organ and entering the central compartment would be QiFI RiVi. After substitution of these rate constants into the appropriate La Place transform equation, the hybrid rate constants used in compartmental models can be obtained by finding the roots of the resulting equation. These in turn can be used to calculate the coefficients of the various phases. When there are many compartments in the model, however, the process can be quite laborious. Approximations of Terminal Half-Lives As long as the rate constant of absorption is not the smallest rate constant, an approximation of the half-life of the terminal phase can be obtained rather easily from the relationship: A in 2 < to s <~errnina~' < (A + B) in 2, where (4e) A = (Rc VC + Sum Ri Vi~lCL, and B = Sum (Ri Ail Qi) (4f) (4g) In Equation 4f the value of A is the transit time of a linear model. During the workshop on which the volume is based, Leslie Benet suggested this value as an estimate of the minimum half-life of a substance in the body.

General Aspects of Extrapolation 107 Approximate Time Required to Approach Steady State During repeated administration of a substance, the amount of substance eliminated from the body during the dosage interval approaches the amount of substance administered during the dosage interval. The time required to approach within 95% of the steady-state value can be estimated by multiplying the sum of A and B in Equation 4e by 3. LINEAR PHARMACOKINETIC SYSTEMS Even though all processes that govern the pharmacokinetics of sub- stances in organs do not always follow first-order kinetics, physiologically based pharmacokinetic models of first-order systems provide insights of relationships that are not always obvious. From these relationships, the investigator can identify parameters that are useful in comparing data for consistency and in making inferences and extrapolations that would be difficult, if not impossible, to obtain from direct experimentation. Among the most useful are the relationships: _ F Dosel(AUC)a _ {F(Dosel~)lt(AUC)al~)ss C (single dose) (repeated doses) (constant infusion) ' (6) where F is the fraction of the dose that reaches arterial blood, (AUC)a is the area under the concentration-time curve in arterial blood, ~ is the dosage interval between doses when the substance is administered repet- itively, ko is the constant rate of infusion, and Ca(ss' is the steady-state concentration ultimately achieved during constant infusion; Ca(ssy implies that all organs and compartments within the organs are in steady state. These relationships should be valid as long as all processes that govern the parameters follow first-order kinetics and remain constant during the time frame under consideration. The term steady state, as used in ~F(Do.~el~)lt(AUC)al~)ss, simply means that the administration of a substance is repeated until the amount of substance eliminated from the body during the dosage interval equals the amount of substance administered at regular intervals. The relationship may express a variety of different dosage schedules. For example, an individual may be exposed to a substance for 8 in/day for 5 days a week. In this situation the dosage interval would be 1 week, and the dose would be the total dose received during the week.

Age JAMES R. GILLETTE Total Body Clearance In these equations, the total body clearance that governs the value of Caress' includes all of the processes by which the substance is eliminated from the body and not just the metabolic clearances by various organs in the body. Almost invariably investigators would consider the contributions of excretion into urine of the unchanged substance and perhaps the con- tribution of biliary excretion. But they frequently ignore elimination of volatile substances into exhaled air and through the skin, and of lipid- soluble substances directly into the gastrointestinal tract across the intes- tinal wall. Indeed, Fick's Law of Diffusion suggests that any site of administration that requires the passage of a substance across a biological membrane should also be considered as an organ of elimination. In fact, even saliva can be visualized as a fluid of elimination; indeed, assays of substances in saliva have been used as indirect measures of the concen- tration of unbound drug in blood (although with mixed success) (Horning et al., 19771. One conclusion that results from these considerations is that the unbound concentration of an absorbed substance in arterial blood can never be greater than the maximum concentration of the substance at the site of administration, unless there are active transport systems or ion trapping effects due to pH differences. The maximum concentrations of metabo- lites, however, may exceed the concentration of the parent substance. importance of the Unbound Concentration of Substances Because substances are usually bound to many components within tis- sues of an organ, in addition to possible reversible binding to action sites, measurements of the total concentration of the substance within an organ is proportional to the magnitude of the response only to the extent that the total concentration of the substance is proportional to the concentration of the substance in water in the immediate environment of the action site. Investigators interested in biologically relevant concentrations thus should focus attention on the factors that govern the concentration of the unbound substances at action sites rather than on factors that govern the total concentration of the substance in organs containing the action sites. Classification of Organs; Routes of Administration Although the concept of AUC values have usually been restricted to the concentration of substances in arterial blood, the concept can be used to estimate AUC values of substances in blood within capillaries of various organs and within the total organ. In this approach it is useful to classify

General Aspects of extrapolation 1 09 the organs according to different categories (Gillette, 1985) (Figure 2~. After a substance is absorbed from the site of administration, it enters a series of organs before it reaches the arsenal circulation. All of the organs through which the substance has passed have been called Arst-pass organs. In turn, the first-pass organs can be classified into two groups: elimination organs and nonelimination organs. The elimination organs contain either mechanisms that remove the substance from the organ, such as metabolism or excretion into bile or expired air, or nonelimination organs, which do not contain any of these processes. After the substance reaches the arterial circulation, it is distributed to other organs in the body, which have been called non-first-pass organs. In turn, the non-first-pass organs can be divided into rapidly equilibrated and slowly equilibrated organs according to their QIVR values. Rapidly and slowly equilibrated organs are further subdivided as nonelimination and elimination organs. Some organs, such as the brain, are visualized as elimination organs, because substances can be removed from the organ in fluids other than blood, even though the clearances by these processes do not contribute to the total body clearance of the substance. According to this classification, some organs such as the kidney are always non-first-pass organs. The lungs are always first- pass organs, except in certain experimental procedures. Some organs, such as the intestinal mucosa, the liver, and the skin, can be either first- ,\\\\\\\\\\\\ _ ~ First Pass Organs .__ ~ 1 ~ 1 1 Nonfirst Pass Organs Site of Administration Venous Blood and Nonelimination Organs Lung and Other Elimination Organs _ . , ~ , ~ I Arterial Blood Kidney and Other ation organs Slowly Equilibrated Organs I Rapidly Equilibrated I Organs FIGURE 2 Relationships among sites of administration, first-pass organs, and non-f~rst-pass organs in physiologically based models.

1 TO JAMES R. GILLE0E pass organs or non-first-pass organs, depending on the route of admin- istration. Non-First-Pass, Nonelimination Organs Under steady-state conditions, there is no net flux of a substance be- tween the blood and the tissues within a nonelimination organ. It follows, therefore, that the steady-state concentration of the substance within the capillaries of nonelimination organs that are also non-first-pass organs is identical to that in arterial blood. It also follows from the identities shown in Equation 6 that this is also true for the AUC values after a single dose and the steady-state AUC values during repetitive administration. This principle is the basis for the rationale that measurements of the AUC in venous blood passing through nonelimination, non-first-pass organs pro- vide valid estimates of the AUC value in arterial blood, even though the shape of the time course of the concentrations in the arterial and the venous blood may differ. It also follows from this principle that the value of R for a given organ in linear reversible binding models can be estimated from the ratio of the area under the concentration curve of the substance in the organ to the area under the concentration curve in the blood. The value of R obtained from this relationship should be constant, provided that the concentration of reversibly bound substance to components in both blood and the tissues in the organ are directly proportional to the concentration of unbound substance (Gillette, 1984, 1985. Nonlinear, dose-dependent changes in the elimination of the substance in other organs, however, would not change the value of R in a nonelimination organ. As long as changes in reversible binding to various components in different organs and blood do not result in changes in the steady-state concentration of unbound substance in arterial blood, they do not affect the steady-state concentration of unbound substance at action sites in non- first-pass organs that are not elimination organs. The steady-state con- centration of unbound substance in arterial blood, however, is governed only by the available dose and the clearances of the elimination organs. Inspection of Equation 4d reveals that under steady-state conditions, the equation for the concentration of unbound substance in blood or in the organ does not contain any term for reversible binding of the substance in the organ cells. Thus, alterations of reversible binding to tissues other than the blood usually do not affect the average concentration of unbound substance in blood. Indeed, they do so only when the organ clearance is nonlinear and the changes in tissue binding result in oscillations in the concentration that affect any of the organ clearances. The principle also predicts that the steady-state concentration of sub- stance in both extracellular and cellular water (i.e., unbound concentration)

General Aspects of Extrapolation 111 in the organ is identical to that in blood, provided that the classical equation of Fick's Law of Diffusion is valid. It would not be valid, however, if there were ion trapping of weak acids and weak bases due to differences in pH between cellular and extracellular water of tissues in the organ or if there were active transport systems in cellular membranes (Figure 11. Range of Maximum and Minimum Unbound Concentrations in Nonelimination Organs and Repetitive Administration Linear compartmental models, based on the assumption that the classical forte of Fick's Law of Diffusion is valid (i.e., Cat = C2 at equilibrium), reveal that after a substance is administered repetitively until a steady state is achieved, the maximum concentration of unbound substance in any nonelimination compartment cannot exceed the maximum concentration of unbound substance in the central compartment during the dosage in- terval. In fact, when the maximum concentration of unbound substance is reached in any nonelimination organ, the concentrations of unbound substance in blood and the organ should be equal, a principle that provides another way of measuring R (i.e., R = Ci~m~zx~lCb~a~ ~ '. Moreover, the minimum concentration of unbound substance in any nonelimination com- partment cannot be less than the minimum concentration of unbound substance in the central compartment. Thus, under steady-state conditions the maximum and minimum concentrations of the unbound substance in the central compartment serve as the boundaries between which the con- centration of unbound substance must be in any nonelimination compart- ment in the body (Gillette, 1984, 1985) (Figure 31. When these principles are applied to physiologically based pharmaco- kinetic models, it follows that during a dosage interval under steady-state conditions, the maximum and minimum concentrations of unbound sub- stance in arterial blood will be between the maximum and minimum concentrations of unbound substance in any non-first-pass, nonelimination organ. The maximum concentration of unbound substance in such organs will be between the maximum concentration and the AUC/; value of the unbound substance in arterial blood, and the minimum concentration of unbound substance in such organs will be between the minimum concen- tration and the AUC/~ value of unbound substance in arterial blood. In organs with small R in 21(QIV) values, the changes will tend to mimick the arterial concentrations rather closely. In organs with large R in 21(QI V) values, the concentration of unbound substance will remain rather constant during the dosage interval and will mimick the AUC/~ value. From these considerations, it becomes evident that much information can be gained concerning the concentration of unbound substance in non-first- pass, nonelimination organs from studies of the concentration of unbound

2 JAMES R. GILLETTE 20 / 10 G 5 LOWER BOUNDARY 2 1 UPPER BOUNDARY 1 1 1 1 1 1 1 i , \ _ _ _ _ _ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 6 9 12 TIME FIGURE 3 Steady-state maximum and minimum concentrations of the diffusible form of substances administered intravenously. The concentrations represent the boundaries between which the concentrations must lie in non-first-pass, nonelimination organs when the substance enters and leaves cells solely by passive diffusion. Blood (------); shallow compartment (0----O); deep compartment (~-------~). substance in arterial blood, even when the R values for various organs are not known. Non-First-Pass, Elimination Organs According to the well-stirred model, the concentration of a substance In blood within the capillaries of an elimination organ is assumed to be the same as the concentration of the substance in venous blood leaving the organ. Thus, under steady-state conditions, the concentration of the substance in blood within the capillaries can be estimated from the arterial

General Aspects of Extrapolation 113 concentration and the ratio C0u~lCin = F. It follows, therefore, that the AUCorg within the capillaries can be estimated from the AUC of the sub- stance in arterial blood by the relationship: AUCorg = (Accra Forg (7) As pointed out by G. R. Wilkinson (this volume), many investigators have questioned the validity of the well-stirred model as a universally valid concept. Some have suggested the parallel tube model, in which the concentration of the substance in blood decreases gradually as the sub- stance passes from the proximal to the distal ends of the capillaries. Others have suggested models that provide estimates between the well-stirred model and the parallel tube model. In the parallel tube model the average concentration of the substance in blood within the capillaries EC`av'] can be estimated from the relationship: C<av' = (sin — Cou~)lln (CinlCOU~), or (7a) C~av'/Cin = (1 — F)/ln (1/F). (7b) As F approaches 1.0, however, both C(av' and COu~ approach Cin, and thus, the distinctions among the models become less important when the value of 1 —F is small. Portions of the blood pass through several organs from the time it leaves the arterial circulation until it reenters the arterial circulation. For example, blood passes through the intestinal mucosa into the portal vein and then through the liver and lungs before it reenters the arterial circulation. When several of these organs serve as elimination organs, adjustments must be made to relate the arterial concentration to the concentration entering the various sequential organs (Gillette, 1982; Rowland and Tozer, 19801. Fl PST-PASS, NON ELI M I NATION ORGANS A substance can pass through several organs from the site of admin- istration to the arterial circulation. If a substance is not eliminated in an organ in this pathway, that organ would be considered a first-pass, non- . . . elimination organ. The steady-state rate at which a substance enters a first-pass organ depends on two rates: the rate at which the substance enters the organ directly from the site of administration and the rate at which the substance enters the organ from the arterial circulation. If none of the first-pass organs were elimination organs, to determine the concentration of the substance entering a first-pass, nonelimination organ, under steady-state . . . cone strops, we can write:

114 JAMES R. GILLETTE Gin = fRatetabs' + Rate~arty~ssl Qorg = fRate~abs~ssl Qorg; + Caress' (8) . The blood passing through the site of administration, however, may have passed through an elimination organ before it reached the first-pass, non- elimination organ, and some of the blood coming from the arterial cir- culation may have passed through organs of elimination before it reached the first-pass, nonelimination organ. Thus, adjustments may have to be made for both elements governing the concentration of substance entering the organ: Cin = [F~ Rate~absy~ssy~lQorg + F2Carssy' (8a) where Fit is the fraction of the absorbed dose that enters the organ, and F2 reflects a rather complex function that represents an effective availa- bility, which has been called a poly-input availability (Gillette, 19821. As with non-first-pass, nonelimination organs, the steady-state concen- tration of the substance in blood within the capillaries of the organ and in venous blood leaving the organ is the same as that entering the organ. Thus, all of the conclusions that were valid for non-first-pass, noneli- mination organs are also valid for first-pass, nonelimination organs, after the input concentration has been modified. In addition, however, the term Fit Rate~absy~ssylQorg represents the minimum AUC value that the substance can attain even when the clearance by organs not contributing to Fit are so large that Ca is virtually negligible. Indeed, the relative contributions of the two terms to the total steady-state concentration represents the difference by which different routes of administration may affect the steady- state concentration of the substance in first-pass organs. First-Pass, Elimination Organs According to the well-stirred model, the steady-state concentration of the substance in blood within the capillaries of a first-pass, elimination organ can be described by: Coat = Forg Cin ~ (9) where Cin is described by Equation 8a. According to the parallel tube model, the average concentration of the substance in blood within the capillaries of the organ, however, should be described by: Crave = Cin t(1 — F)/ln (1/F)~. ROUTE-TO-ROUTE EXTRAPOLATION (9a) When the various pathways along which blood must pass from the various sites of administration to the arterial circulation are considered,

General Aspects of Extrapolation 115 it becomes evident that the lungs are a first-pass organ by almost any route of administration. Thus, the general equation for the steady-state concentration of a substance in arterial blood takes the form of: koFpreFL Ca(ss) — CL (10) CL is the total body clearance, which includes the clearance by the lung which in turn depends on the intrinsic clearances of exhalation and me- tabolism, Fpre is the fraction of the dose entering the body that enters the lung, and Fit is the availability of the substance in lung. For a well-stirred model: ~ Qco + CL<exhaiationy + CL<mety (lOa) where Qco is the cardiac output, and the other clearances are intrinsic clearances of exhalation and metabolism. The effects of various routes of administration on the steady-state con- centration of the substance in arterial blood thus depends on whether the route of administration changes Fpre Lungs and Skin Administration When a volatile substance is inhaled and absorbed into blood in the lungs, the value of Fpre would be 1 .O. When the same substance is absorbed through the skin, the value of Fpre depends on the fraction of the absorbed dose that escapes metabolism in skin and enters the lungs. Usually the fraction of the dose that is eliminated from the body by the skin by diffusion and metabolism is assumed to be negligible, and thus, the value of Fpre in this situation is assumed to be 1.0. Thus, the steady-state concentration obtained by two routes of administration (skin and lung) should be identical as long as all processes follow first-order kinetics and equal amounts of the substance are absorbed by the two routes of administration. Whether an inhalation experiment provides sufficient information to predict the steady-state concentrations in blood depends on how the ex- periment is performed. Placing animals in a closed container and mea- suring the area under the concentration curve of the substance in the container provides an estimate only of the total metabolic clearance of the substance. It does not estimate the total body clearance, because the unchanged substance passes from the animals back into the container and thus does not measure the clearance of exhalation. Estimation of the total body clearance requires either measurements of the AUC in the blood of the animals or a combination of estimations of the amount of unchanged substance that would be exhaled as well as the amount metabolized.

1~6 JAMES R. GILLETTE Oral Administration When a substance is administered orally, the steady-state concentration in arterial blood depends on where the substance is absorbed. If it is absorbed only in the mouth, the effect on Fpre would depend only on the fraction of the dose that escaped metabolism in the membrane lining of the mouth, but if the substance is absorbed in the stomach or the gas- trointestinal tract (excluding certain parts of the rectum), the value of Fpre depends not only on the fraction of the absorbed dose that escaped me- tabolism by these tissues but also by the liver, because the venous blood draining these organs enters the portal circulation. Biliary Excretion Whether biliary excretion contributes to Fpre depends on whether the substance excreted into bile is reabsorbed. If it were completely reab- sorbed, biliary excretion would not contribute to Fpre, although biliary excretion would affect the shape of the concentration-time curve during single or repeated administration of the substance. Location of Organs of Elimination The effects of different routes of administration on the steady-state concentration of the substance in first-pass organs depends on which organs contribute to the total body clearance of the substance. If the substance were eliminated from the body by only one organ in the body, the steady- state concentration of the substance in that organ would be independent of the route of administration, even though the organ may be a first-pass organ after one route of administration but a non-first-pass organ by an- other. The concentration of the substance in first-pass, nonelimination organs, however, would still depend on the route of administration. The ratio of the steady-state concentrations of the substance in blood within f~rst-pass elimination organs becomes much more complicated when the substance is eliminated by several organs of the body. For example, the well-stirred model predicts that when the same amount of a substance is absorbed and the substance is eliminated by the lungs, kidney, and liver, the ratio of the steady-state concentrations of the substance in blood within the sinusoids of the liver would be: CH SS(P.O ~ 1 CH.ss(inh) _ ~1 + Q (ELFR + ER) + (Q(O QH QR)EL] (11) If the substance were eliminated from the body solely by lung and liver the equation would become:

General Aspects of Extrapolation 1 1 7 CH SS(P.O.) CH ss(inh) 1 | (0co QH)EL1 F I 1 + Q I, (lla) = where Qco is the cardiac output, the E terms are extraction ratios, and the F terms are organ availabilities. Notice that Equation 10 does not contain any terms representing the elimination of the substance by non-first-pass organs and Equation 11 does not contain any terms representing the extraction of the substance by the liver. Thus, the effects of different routes of administration on the steady- state concentrations of a substance in an organ depend solely on the elimination of the substance in organs other than the one under consid- eration. Inspection of Equations 10 and 11 also reveals that the effects of dif- ferent routes of administration depend on the organ availabilities (i.e., the F values). It follows, therefore, that when the apparent total body clear- ance, as measured by the dose/A UC value, is considerably smaller than the slowest blood flow rates through any organ of elimination, the value of all elimination organs is virtually 1.0 and the elimination of the sub- stance is flow independent. Under these conditions any difference between the AUC values observed after different routes of administration can be attributed to differences in the fraction of the dose that is absorbed from the site of administration or to the effects of different rates of absorption on saturable processes of elimination. Linear Steady-State Models for Metabolite Formation and Elimination The rate of formation of a metabolite in cells within an organ obviously depends on the concentration of its precursor in the cells, and thus, the discussion of the factors that govern the concentration of the parent sub- stance in the previous sections are relevant to the fot~ation of its metab- olites. The pharmacokinetics of metabolites, however, are far more subtle and frequently not intuitively obvious. If a metabolite were formed directly from the parent substance in a homogenous group of cells within a single non-first-pass organ, a linear, well-stirred, steady-state concentration of the metabolite in arterial blood can be described by Equation 12 (Gillette, 1985; Pang and Gillette, 19791: CM _ koF`l Fort CL ins F akss)— CLSCLM (12) where Fit represents the fraction of the dose of the substance that enters the arterial circulation and reaches the organ in which the metabolite is

~ l~ JAMES R. GILLEUE formed. FSrg is the organ availability for the substance, and ~ is the unbound fraction of the substance. CLiSn'>M is the intrinsic clearance of the enzymes that catalyze the formation of the metabolite; FM is an availability term that describes the fraction of the metabolite formed in the cells that leaves the cells and eventually enters the arterial circulation; and CLS and CLM are the total body clearances of the substance and its metabolite, respectively. In Equation 12, the terms ho FS FSrg fUICLS are those that govern the steady-state concentration of unbound parent substance within the organ. The steady-state rate of formation of the metabolite can be expressed by the steady-state concentration of the unbound substance and the intrinsic clearances of the enzymes that catalyze the formation of the metabolite. These terms also can be expressed as the rate of infusion times the fraction of the dose that is converted to the metabolite in the organ, FS~M. Thus, Equation 12 can be written as: C (ss) = Lo FSrgM FM (12a) This form of the equation points out the invalidity of the assumption of many investigators that only the activity of the enzymefs) that catalyzes the formation of a toxic metabolite governs the steady-state concentration of the toxic metabolite. Instead, Equation 12a demonstrates that as long as FM and CLM remain constant, changes in the value of CLiSn~ M affect the steady-state concentration of the toxic metabolite in arterial blood only to the extent that such changes result in changes in FSrg>M Indeed, if the formation of the metabolite were the sole mechanism of elimination of the parent substance, FSrgM would be 1.0, and the steady-state concen- tration of the metabolite in arterial blood would be completely independent of the activity of the enzyme that catalyzed the formation of the metabolite, as long as the rate of infusion did not exceed the Vma,: of the enzyme (see Figure 71. TABLE 1 Functional Classification of Metabolites 1. Ultrashort-lived metabolites Metabolites never leave the enzyme (suicide enzyme inhibitors). 2. Short-lived metabolites Metabolites never leave the cell. 3. Intermediately lived metabolites Metabolites never enter aortic blood. 4. Long-lived metabolites Negligible amounts of metabolites excreted into air, bile, and urine. Ultralong-lived metabolites Metabolites extensively excreted into air, bile, or urine.

General Aspects of Extrapolation 1 19 Stable Metabolites When the value of FM approaches 1.0, the general conclusions of the relationships between the steady-state concentration of unbound substance in blood to the concentration of unbound substance in nonelimination organs are also applicable to the metabolite. For example, under steady- state conditions the maximum concentration of unbound metabolite in an organ that does not participate in either the formation or the elimination of the metabolite will never exceed the highest concentration of the un- bound metabolite in blood, and the minimum concentration of unbound metabolite under these conditions will always exceed the minimum con- centration of metabolite in blood during a dosage interval, unless there are ion-trapping effects or active transport systems in the organ. Categories of Unstable Metabolites With many toxic metabolites, however, the value of FM may approach 0, in which case the metabolite would be undetectable in blood. It is, therefore, useful to devise functional categories of metabolites according to the extent to which the metabolites can leave the enzyme that catalyzes their formation, leave the cells in which they are generated, and leave the organ in which they are generated (Table 11. The approaches used to study the kinetics of metabolites that may be appropriate to one of these cate- gories may not be appropriate for studying another category. For example, when it can be demonstrated that a metabolite never leaves the cells in which it was generated, the total body clearance of the metabolite would be restricted to the cells in which the metabolite was generated, and the investigator would focus attention on such intracellular mechanisms. Sev- eral equations describing the kinetics of metabolites in different categories have been devised (Gillette, 19851. Such considerations are especially relevant in focusing attention on potential target organs of toxic, chemi- cally reactive metabolites. If toxic, reactive metabolites never leave the cells in which they were formed, then the direct toxic effects would be restricted to those cells containing the enzymes that catalyzed their for- mation. Alterations in the activities of the enzymes that catalyze the for- mation of the metabolite in cells of different organs thus may alter the relative toxicities in different target organs and would be unpredictable from pharmacokinetic studies solely of the parent substance. By contrast, the interorgan differences in the toxicity of long-lived metabolites depend largely on the biochemistry and physiology of the target organ.

120 JAMES R. GILLETTE DOSE EXTRAPOLATIONS Increasing the dose of a substance can have profound effects on virtually every aspect of pharmacokinetics, including the rates and extents of ab- sorption from various sites of administration, the extent of reversible binding of the substances to binding components in blood and other organs, the rates of formation of different metabolites, and the rates of excretion of the substance and its metabolites into bile and urine. Unknown Biologically Active Forms It is important to realize that when the biologically active Formosa is not known, only the dose-dependent effects on the rate and extent of absorption can be unequivocably related to the incidence and intensity of toxicities. It is frequently impossible to decide whether the response will be directly proportional to dose, greater than proportional to dose, or less than proportional to dose when increasing doses result in the approach to saturation of various enzymes and active transport systems. With nonlinear models the concentration of unbound parent substance becomes propor- tionately greater as the dose is increased. But the increase in the concen- tration of various metabolites may be either greater or less than proportional to the dose. It is also possible that under certain conditions the concen- tration of a metabolite may remain directly proportional to the dose, even when the concentration of its precursor is not. Under other conditions, the concentration of a metabolite may be larger or smaller than proportional to the dose, even though the concentration of its precursor is virtually proportional to the dose. When pharmacokinetic studies are made a part of a toxicity study, such discrepancies frequently aid the investigator in elucidating the biologically active forms that cause toxicity. But without such prior knowledge, the regulator is faced with a host of possible interpretations that are frequently contradictory. When considering the effects of increasing doses and concentrations of compounds, such as enzymes, on saturable processes, many investigators focus attention on a single enzyme, apparently without realizing that knowledge of the Michaelis-Menten parameters of a single enzyme alone seldom, if ever, provides sufficient information to predict either the con- centration of the parent substance or any of its metabolites. Dose-Dependent Absorption of Insoluble Substances According to Fick's Law of Diffusion, the rate of absorption by passive diffusion should always be directly proportional to the concentration of

General Aspects of Extrapolation 121 diffusible forms of a substance, provided that the substance does not alter the characteristics of the membrane. Most dose-dependent kinetics of absorption occur with insoluble compounds, in which a steady state is established between the rate of dissolution of the crystals of the substance and the rate of diffusion across the pulmonary membranes, gastrointestinal membranes, or skin (Pang and Gillette, 1979; Rowland and Tozer, 19801. The question then arises as to which of these processes becomes the predominant rate-limiting step in governing the rate of absorption. Two extreme cases are represented in Figure 4. In the first case, the substance has a high oil/water partition ratio and therefore would be rapidly absorbed, but the rate of absorption may be limited by the rate of diffusion from the bulk liquid phase to the membrane interface. Thus, the rate of ab- sorption of a substance in the gastrointestinal tract frequently may be limited by the motility of the gastrointestinal tract. In a well-mixed ab- sorption compartment, however, the rate of absorption may be limited by the rate of dissolution of the substance from its crystalline form. In this case, however, the rate of absorption would be dependent more on the size of crystals and other physicochemical characteristics of the dosage preparation than on the factors that govern Fick's Law of Diffusion. RATE OF ABSORPTION Which is the Rate Governing Step? ~ '_ Solvent I Particle ~ I Phase Slowly Absorbed Material ! // c a Rapidly Absorbed Material Rate of Dissolution ~ Important ho \\\\\\\\ ~ Capillary | Membrane Tissues (1 ) \\\\\\ _~ I ss*---t- ss -t- ss -t ~,,,,,,,,,,,,,,,,,,,~ , Capillary Sues_ ~ (2) 1 it Ss. - - it- Ss. -t - Ss. t Ss. ,,,,,,,,,,,,,,,,, SS., Steady state level of diffusible form when CL by other mechanisms is zero. FIGURE 4 Rates of dissolution and rates of diffusion as rate-limiting steps in absorption.

i22 JAMES R. GILLEUE Fraction of the Dose Absorbed and Bioavailability The fraction of the dose that is absorbed, however, depends not only on the rate of absorption diffusion and facilitative and active transport systems, but also the rate at which the substance is removed from the site of administration removed by other processes. Bioavailability and Gastrointestinal Absorption In the gastrointestinal tract, the substance may be eliminated with the feces. In addition, however, the substance may be unstable at pH values occurring in the stomach, or it may be metabolized by intestinal flora and/ or intestinal mucosa. These latter processes confuse the interpretation of pharmacokinetic studies of metabolites, because it is frequently difficult to determine whether a metabolite is formed in the gastrointestinal tract and then absorbed or is absorbed and then converted to a metabolite within other organs of the body (Rowland and Tozer, 19801. Biliary excretion and passive diffusion from the body through the gas- trointestinal wall further complicate the interpretation of bioavailability studies, particularly when these processes represent major contributions to the total body clearance. Bioavailabi l ity and Absorption Through Skin Studies on the absorption of insoluble substances through the skin are especially difficult to interpret and apply to risk assessment studies. Al- though it might be assumed that a substance must dissolve into a liquid vehicle before it can be absorbed, the solubility in that vehicle seldom is known. Because it is the concentration of the substance in the vehicle as well as the partition ratio of the substance between the vehicle and the lipoidal membrane that presumably governs the rate of diffusion, extrap- olations to different vehicles in which the substance may dissolve to different extents would be difficult. Moreover, the fraction of the dose that is absorbed would depend on the time at which the substance is washed off or otherwise removed from the skin. Whatever the actual kinetics may be, the assumption that the rate of absorption would be directly propor- tional to the total amount of material suspended in the dosage form will not always be valid. Because the rate of absorption of substances across thin membranes is usually directly proportional to the oil/water partition ratios, it might be assumed that this would also be true for thick membranes, such as the skin. But when a membrane contains several layers of cells, the substance must pass through not only lipoidal membranes but also aqueous phases

General Aspects of Extrapolation 123 before it reaches the capillary beds within the membrane. When the mem- brane consists of several layers, the rate of diffusion depends not only on the diffusivity through the lipoidal layer but also the diffusivity through aqueous layers. The rate of diffusion of substances with very high oil/ water partition ratios, therefore, can be limited by its concentration and diffusivity through the aqueous phases (Figure 51. Thus, there frequently is an optimum oil/water partition ratio for absorption through thick mem- branes (Houston et al., 19741. The rate of absorption through thick membranes, however, is seldom limited by the blood flow rate through the membrane. Under conditions ire which this would likely occur, the substance simply diffuses more deeply into the membrane and enters capillaries in the deeper regions of the membrane. 50 o ~ _ ~ O 10 of a) LL at _ o 1 50 ~ — G O 10 Z ~ UJ cr of _ o Cat 1 VERY LIPID SOLUB LE SUB STANCES - S A \ S A T Q 1 U R E R O E U D S A Q \EU O~ U S A Q U E O YOU Say A Q U E o U S S T 1 R R E D VERY POLAR SUB STANCES U L U L U D S D \_ 1 FIGURE 5 Diffusion through thick membranes. 5x 104 o 104 ~ Z 5X103 of o 5X10-3 o 10-3 ~ Z 5x10 - Cal at o - ._ Q ._ -

i24 JAMES R. GILLE0E Dose Dependencies in Reversible Binding of Substances to Proteins and Other Macromolecules in Blood and Organs Increases in the dose of a substance may result in concentrations of unbound substances in blood and organs that approach or exceed the 1/Ka values of the various binding components in blood and the organs. Because the effects are dependent on concentration, however, the effect is dependent on the rate and the extent of absorption of the substance from the site of administration, as well as the rates of distribution and elimination of the substance. The effects of increases in concentration on the value of the organ partition ratio R depend on the Ka values of the various binding compo- nents, the number of binding sites with a given Ka value, and the location of the binding sites. Suppose, for example, that the sites with the highest Ka value were located in the blood, the sites with the second highest Ka value were located in the tissues of the organ, the sites with the third highest Ka value were in the blood, and the sites with the fourth highest Ka value were in the tissues of the organ. As the concentration of unbound substance is increased from a very low level, the value of R would first increase, then decrease, then increase, and then decrease. Because the relative distribution of binding sites within an organ may differ with the organ, the pattern of changes may differ with the organ. Thus, as long as the relative as well as the total number of the binding sites and their respective Ka values remain unknown, the effects of an increase in the concentration of the substance on the value of R in various organs are unpredictable. The situation becomes even more complex if we include competitive binding with the metabolites of the substance. It follows, therefore, that the effects of increasing doses on the shape of the time course of the total concentration of the substance in various organs and blood are unpredictable. Moreover, the relationship between the total and the unbound concentrations in various organs varies not only as the con- centration approaches the maxima in various organs but also as the con- centrations in the organs decline after the administration of bolus doses of the substance. Thus, the time course of the unbound concentration should be simulated separately in order to relate them to the response. The effects of concentration-dependent changes in the reversible binding to blood components on the maximum, minimum, and average concen- trations of unbound substance after a steady state is achieved during re- peated administration of the substance largely depend on the influence that reversible binding of the substance to blood components has on the rate of absorption of the substance and the extraction ratios and location of the elimination organs. If saturation of the binding sites decreases the rate of absorption without having much effect on the extraction ratio of

General Aspects of Extrapolation 125 a first-pass organ, the maximum concentration may be less than propor- tional to the dose. But if saturation of the binding sites has little influence on the rate of absorption of the substance and markedly decreases the extraction ratio of a first-pass organ, the maximum concentration may be greater than proportional to the dose. Thus, the effects of saturation of binding sites to blood components is unpredictable. Saturation of binding sites in tissues within the organs, however, causes greater increases than expected in the magnitude of the oscillations between the maximum and minimum concentrations of unbound substance in the blood and the organs during the dosage interval, which may be relevant to the magnitude of rapid responses initiated by reversible binding to receptor sites. DOSE-DEPENDENT CHANGES IN METABOLISM OF SUBSTANCES AND THEIR METABOLITES Michaelis-Menten Kinetics When pharmacokineticists discuss the implications of dose-dependent metabolism of substances, they usually focus attention on the classical Michaelis-Menten equation and note that when the concentration of the substance approaches infinity, the rate of metabolism approaches a con- stant, namely, Max. But they frequently fail to emphasize the importance of other mechanisms of elimination of the parent compound in governing the concentration of the substance under steady-state conditions. More- over, most substances can be metabolized by several enzymes in the body, each one of which has its own substrate specificity and its own set of Michaelis-Menten parameters for any given substrate. It may be useful for the purpose of illustration to visualize a model in which a substance is constantly infused intravenously until a steady state is achieved and the substance is eliminated from the body by being excreted unchanged into urine and into exhaled air and by metabolism in the liver. In this model, the value offs is assumed to be 1.0 in order to preclude any complicating effects on the clearances caused by dose-dependent, reversible binding. The steady-state equation for such a system would be: Ca~ss' = CLH + CLR + (Qco — CLH — CL8) Er (13) where Qco is the cardiac output. If the substance were metabolized either by a single enzyme or by a group of enzymes with the same Km value, we can write the equation for hepatic clearances as:

] 26 JAM ES R. GI LLEUE CL QH Sum Vm</ (14) CL = CL`in!(H) (14a) where Clint = Sum(Vmax/Km) The term COu~`,ss' requires the solution of the quadratic equation: COU[(SS) = 0 5 [C'n(ss) — Km [1 + (CLin,/Q)] + ({Cin(ss) — Km [1 + (CLintlQ)]}2 - 4 KmCin(ss)) ] (14b) Simulations based on these equations would indicate that when CLint ~ O., the hepatic clearance approaches hepatic blood flow at very low values of COu~¢ss'. But when COu~`ss' approaches virtual infinity, the hepatic clear- ance approaches 0. Substitution of these values into Equation 13 would thus provide an estimate of the maximum range of effects that an increase in the rate of infusion would have on the steady-state concentration of the substance. Studies designed to estimate only the Michaelis-Menten parameters Vmax and Km or the clearance of a single organ, however, do not provide sufficient information to predict steady-state concentrations of a given substance in arterial blood and other organs at different doses. The values of the clearances of other organs that contribute to the total body clearance and dose-dependent effects on these clearances must also be known and reported to accomplish this objective. Intraorgan Localization of Enzymes Although the clearance of a substance by an organ in most cases can be adequately described by well-stirred models, the kinetics describing the formation and elimination of metabolites requires more sophisticated models when the organ availabilities of either the parent substance or the metabolite are small. Under such conditions the intraorgan localization of the enzymes (Baron and Kawabata, 1983) may be important. As pointed out by G. R. Wilkinson (this volume), the changes in liver metabolites of highly extracted substances caused by changes in either the blood flow rate or the fraction of unbound drug are not accurately predicted by either the well-stirred model or the parallel tube model. Instead, the experimental availabilities are usually between the values predicted by the two models. The parallel tube model, however, provides an easy way of illustrating the effects of an increase in the dose (and thus the steady-state concen-

General Aspects of Extrapolation 127 tration of the parent substance) on the exit concentrations of metabolites formed by different enzymes preferentially distributed in different zones of an organ (Figure 61. For example, imagine that an enzyme (Figure 6, enzyme 1) that formed one metabolite (Mi) was localized preferentially in the periportal zone of the liver (the proximal end of the tube) and that another enzyme (Figure 6, enzyme 3) that formed another metabolite (M3) was localized in the pericentral zone (the distal end of the tube). Also imagine that the intrinsic clearance of enzyme 1 was considerably greater than the hepatic blood flow rate. When the rate of infusion is slow, virtually all of the substance entering the liver would be converted to Me, because very little of the substance would reach enzyme 3. When the infusion rate is increased, however, the concentration may approach or exceed the Km value of enzyme 1, and the available fraction reaching enzyme 3 would increase. Thus, the pattern of metabolites exiting the liver would change as the infusion rate increased, even when the Km values of the two enzymes are virtually identical. If the Km values of the two enzymes differed, the effects of an increase in the infusion rate would depend on which enzyme had the lower Km value. The increase in the hepatic availability of the substance caused by increasing the infusion rate would become evident at lower infusion rates when enzyme 1 has the lower Km value than when enzyme 3 has the lower Km value. Moreover, the shape of the changes in the relative proportions IDEALIZED DISTRIBUTION OF ENZYMES · · ~ 1 1 1 1 1 1 1 1 ~\1 ~ ~ ~ 1 1 ~ ~ ~ I ~ ~1\ \ \ aa all 1 1 1 1 11N~ _= ENZYME 1 = · CONDITION 1 =— CONDITION 2 = ENZYME 2 = 0 ENZYME 3 = ~1 FIGURE 6 Idealized intraorgan distribution of enzymes.

128 JAMES R. GILLETTE of the metabolite depends on which of the enzymes has the lower Km value as both enzymes approach saturation. The parallel tube model also predicts that the location of enzymes that catalyze the metabolism of a metabolite affects the apparent organ avail- ability of the metabolite, when the intrinsic clearance of the enzyme that catalyzes the metabolism of the metabolite greatly exceeds the hepatic blood flow. If enzyme 1 generates the metabolite and enzyme 3 metab- olizes it (model 1), very little of the metabolite would escape the liver. But if enzyme 3 generates the metabolite and enzyme 1 metabolizes it (model 2), virtually all of the metabolite would escape the liver. If the Km value of the enzyme that generates the formation of the metabolite were always larger than the Km value of the enzyme that catalyzes the metabolism of the metabolite, then an increase in the rate of infusion would result in increases in the availability of the metabolite in model 1, but not in model 2. An increase in the rate of infusion would decrease the hepatic clearance of the metabolite in both models. It is unlikely that enzymes would be segregated as markedly as has been assumed in models 1 and 2, and thus, any equation that described these events would not be universally valid. Nevertheless, the distribution of enzymes within organs is known to influence the availability of me- tabolites, although the extent of the influence is unpredictable (Pang, 1983~. Fraction of the Dose Principle Although many studies of toxic metabolites focus attention on the en- zymes that catalyze the formation of the toxic metabolizes, it is important to realize that any alteration in the activity of the various enzymes that catalyze the metabolism of any of the precursors of the toxic metabolize will affect the steady-state concentration of the toxic metabolize only to the extent that the alterations cause a change in the fraction of the dose that is converted to the metabolize. According to this general principle, approaches to the saturation of enzymes that catalyze the metabolism of a precursor to innocuous metabolites are just as important as approaches to the saturation of enzymes that catalyze the reactions that lead to the toxic metabolites. This principle can be illustrated by simulations of mod- els based on Equations 12 and 13, in which a substance is constantly infused intravenously into a rat until a steady state is achieved and the substance is eliminated from the body by a combination of excretion unchanged into urine and metabolism by two enzymes in the liver (Figures 7 and 81. Enzyme l is assumed to form a toxic metabolite, whereas enzyme 2 is assumed to form a nontoxic metabolite. The renal clearance is assumed to be independent of the concentration of the parent substance, and the

General Aspects of Extrapolation 129 C E in co F 2.0 lo LL cr i_ 1.0 0.0 0.05 0.04 - u' u' 0.03 - lll 0 0.02 m - -? 0.01 CL`p'= 0.01 ml/mint 3.0 ~ CL`M' ~ = 1 .00 ml/mint Enz (2) Vm = 0.3 ,umol/min Km=0.1 mM l Enzyme ( 1 ) Vm Km rLmol/min mM A 0.003 0.01 B 0.03 0.1 C 0.3 1.0 B - A 0.00 0.0 0.1 0.2 0.3 0.4 0.5 INFUSION RATE, ,umol./min. FIGURE 7 Importance of alternative mechanisms of elimination: low renal clearance. Simu- lations of the steady-state concentrations of parent compound and toxic metabolites when the toxic metabolites are formed by different enzymes (1A, 1B, and 1C) with different V,n`,` and K,'' values. CLp is the renal clearance of the parent compound, and enzyme 2 catalyzes the formation of a nontoxic metabolite. CLM is the total body clearance of the toxic metabolite (M,).

i30 JAMES R. GILLETTE 0.S~ CL<p'= 10.0 ml/mint CL`M''=1.00 ml/mint 0.6 :E 0.0 0.05 0.04 0.03 LL t 0 0.02 m 111 0.01 0.00 / // A,B /C Enz (2) Vm = 0.3 ,umol/min Km=0.1 mM C / - Enzyme (1 ) Vm Km 1lmol/min mM A 0.003 0.01 B 0.03 0.1 C 0.3 1.0 B - - A ~ 1 ~ INFUSION RATE, ,umol./min. 63 10 FIGURE 8 Importance of alternative mechanisms of elimination: high renal clearance. Sim- ulations of the steady-state concentrations of parent compound and toxic metabolites when the toxic metabolites are formed by different enzymes (1A, 1B, and 1C) with different Vm,~,` and Km values. CLp is the renal clearance of the parent compound, and Enzyme 2 catalyzes the formation of a nontoxic metabolite. CLMi is the total body clearance of the toxic metabolite (Ma).

General Aspects of Extrapolation 131 Vma,` and Km values of enzyme 2 are the same in all simulations; the intrinsic clearance (Vma,rlKm) value of enzyme 2 was set at 3.0 ml/min, which is well below the hepatic blood flow in rats (about 25 ml per 300- g rat). Enzyme 1 is assumed to have any one of three different Km values, but the intrinsic clearances were set at 0.3 ml/min in the three situations. The total body clearance of the toxic metabolite is assumed to be the same in all simulations and independent of the concentration of the metabolite. Thus, in these simulations low infusion rates would result in the same steady-state concentrations of the toxic metabolite, regardless of the Km value of enzyme 1. First Set of Simulations In this set of simulations, the renal clearance was set at 0.01 ml/min, which is about the rate of urine formation in rats (Figure 71. Thus, the value is about the expected renal clearance of a lipid-soluble substance that entered the glomerular filtrate, but was reabsorbed with water during the concentration of the filtrate. Because the sum of the intrinsic clearances of enzymes 1 and 2 was set at 3.3 ml/min, at low infusion rates the substance is eliminated from the body predominantly by metabolism in the liver. Enzyme ~ Km Equals Enzyme 2 Km In simulation B (Figure 7), the Km of enzyme 1 is set at the same Km as enzyme 2. In this situation the effects of an increase in the infusion rates on the rates of formation of the metabolites formed by the two enzymes would be identical. The increase in the steady-state concentration of the metabolite remains virtually directly proportional to the rate of infusion until the rate of infusion approaches the sum of the VmaX values of the two enzymes (see Gillette, 1986; O'Flaherty, 1986~. This follows from the definition of a steady state; i.e., the rate of elimination of the substance must equal the rate of infusion. Because in this particular model the rate of elimination is predominantly due to metabolism of the substance up to an infusion rate that equals the sum of the Vmax values, the substance simply accumulates in the animal until it reaches a concentration at which the rate of elimination by the combination of metabolism and excretion equals the rate of infusion. The fraction of the dose that is converted to the toxic metabolite over this range of infusion rates, however, would remain virtually constant; and therefore, the steady-state concentrations of metabolites 1 and 2 would be directly proportional to the rate of infusion. Most of the increase in the steady-state concentration of the precursor occurs within a relatively small range of infusion rates. The equation Vm<,`/

|32 JAMES R. GILLEUE (Km + fu S), which defines the effective Clint, predicts that as the con- centration of unbound substance increases from very low values up to a value equal to the Km, the value of the effective Clint decreases by 50%. If the total body clearance were governed solely by the enzyme, we can calculate that an infusion rate equal to 50% of the Vmax value of the enzyme would result in a steady-state concentration of the substance that would be only double that predicted by the relationship kol(Vma,:/Km). When the clearances by other routes of elimination are very small compared with Vma,~lKm values of the enzymes, however, the steady-state concentration of the substance would approach values predicted by kolCL when the infusion rates exceed the Vma,: value of the enzyme. Thus, only a doubling of the infusion rate would be required to go from O.SVma,; to Vma,~. Such large increases in the concentration of the parent substance probably would result in toxic effects of the parent substance rather than a delayed toxicity of the metabolite. The narrowness of the dosage range, however, implies the existence of a virtual dose threshold. This discussion should not be construed as indicating that the concen- tration of a substance administered repetitively may never be severalfold larger than the Km value of the dominant enzyme during the dosage interval under steady-state conditions. For example, if a substance were injected into a one-compartment model repeatedly until a steady state were achieved, the following equation based on the integrated form of the Michaelis- Menten equation can be written, (Dose/) ss fU Css ~ 15) in which: Css = t(Co - C~/ln (colc~]ss, (lSa) where CO is the concentration of the substance immediately after the injection of the substance at the beginning of the dosage interval, and Car is the concentration of the substance at the end of the dosage interval under steady-state conditions. Thus, as long as (Dosel~) < Vma,` A, the system theoretically can attain a steady state regardless of the dosage interval and the volumes of dis- tribution that relate the amount of the substance in the compartment to the concentrations of the substance. The equation is not valid for multi- compartment systems, and thus, iterative approaches would be needed to simulate not only the approach to steady states but also the concentration changes within a dosage interval. Nevertheless, Equation 15 may be useful for estimating a range of permissible dosage schedules in toxicity studies from in vitro estimates of Vial,,; and Km values. Caution should be used, however, because the estimates of V, and Km obtained in vitro do not

General Aspects of Extrapolation 133 always reflect the Vmax and Km values in vivo and because the in viva Vma, and Km values may change during the course of the study. When the pharmacokinetic study is not an integral part of the toxicity study, even greater caution should be used in interpreting the results. Differences in the Vma,; values between the groups of animals in either enzymes 1 or 2 that changed the Sum VmaX value would affect the dose-response curve whether the toxicity was caused by the parent substance or a metabolite of enzyme 1, because such differences could result in a change in the proportion of the dose that was metabolized by the enzymes and would affect the dose at which the steady-state concentration of the substance would no longer be linearly related to the dose. Differences in the Vma,; value in enzyme 2 without proportional increases in enzyme 2 would also affect the dose-response curve of a toxicity caused by a metabolite of enzyme 1. In simulation A (Figure 7) the Km value of enzyme 1 was set at 0.01 mM, an order of magnitude below that of enzyme 2. Thus, the steady- state rate of formation would approach Vm`,` (enzyme 1) at rather low rates of infusion. With further increases in the rate of infusion, the fraction of the dose converted to the toxic metabolite would be decreased, but the value of FIRM would tend to decline linearly until Vma,` (enzyme 2) was reached and then would decline rapidly with further increases in the in- fusion rate. In such situations, it would obviously be better to focus attention on the value of Vmax (enzyme 1) and to ignore the variations in the relationships between the rate of synthesis of the toxic metabolite and the dose. When the pharrnacokinetic studies are an integral part of the toxicity, it may be possible to distinguish between toxicities caused by the metab- olites of enzyme 1, enzyme 2, and the parent substance. The maximum concentration and thus, possibly, the maximum intensity of the toxicity caused by a metabolite of enzyme 1 would occur well below the Km value of enzyme 2. By contrast, the intensity of a toxicity caused by a metabolite of enzyme 2 may be proportional to the dose, and the dose-intensity relationship of a toxicity caused by the parent substance may be very sharp and appear to have a dose threshold. When the pharmacokinetic studies are not a part of the toxicity studies and the toxic form is unknown, such issues become clouded and risk estimators would not know which phar- macokinetic parameters should be used in their calculations. Moreover, they could be led to the wrong conclusion if the pharmacokinetic studies were focused only on the parent substance. In simulation C, the Vmax of enzyme 1 equals that of enzyme 2, but the Km of enzyme 1 is an order of magnitude larger than that of enzyme 2. With increasing rates of infusion, the steady-state concentration of the metabolite of enzyme 1 would deviate from a linear relationship, but would

|34 JAMES R. GILLEUE approach an asymptote up to a rate of infusion that equalled the sum of the Vma,: values of the two enzymes. By contrast, the steady-state con- centration of the parent substance would approach concentrations governed by ko/CL more gradually than in simulations A and B. Thus, as the perfusion rate is increased, the fraction of the dose converted to a metab- olite by enzyme 1 increases until the Vmax of enzyme 1 is reached and then decreases. Thus, attempts to relate the toxicity to any of the forms of the toxicant would be difficult. Second Set of Simulations For the simulations shown in Figure 8, the clearance for the excretion of unchanged substance into urine (or air) was increased to 10 ml/mint This is about what would be expected for the clearance of a polar substance that is excreted by an active transport system in kidneys, but for the purpose of illustration it has been assumed that an increase in the infusion rate would not affect the clearance of the unchanged substance. The Vma,` and Km values for enzyme 1 and enzyme 2 were identical to those used in the simulations shown in Figure 7. In this set of simulations, the maximal fraction of the dose that undergoes metabolism by the two enzymes at low infusion rates would be about 25% and would decrease as the infusion rates were increased in all three simulations. As a result the steady-state concentrations of the parent substance are virtually proportional to the infusion rates regardless of the degree of saturation of the enzymes. A1- though an increase in the renal clearance does not significantly change the shape of curve A, the steady-state rate of infusion required to achieve the Vma,: of enzyme 1 is significantly larger in Figure 8. Moreover, the shapes of curves B and C as well as the rates of infusions required to achieve Vma,~ values of enzymes 1 and 2 are changed. From comparisons of the simulations in Figures 7 and 8, it should be obvious that studies of the kinetic parameters of all processes of elimination of the substance and its biologically active metabolites are needed to predict their steady-state concentrations. Knowledge of the kinetic pa- rameters of enzymes without knowledge of the clearance by other pathways of elimination of the parent substance is also necessary. Pharmacokinetic studies that do not provide such information in a readily understandable form are thus virtually uninterpretable and largely useless to the risk assessor. Concentration-Dependent Metabolite Elimination An increase in the rate of infusion of a substance may also result in dose-dependent elimination of a metabolite and thereby affect the steady-

General Aspects of Extrapolation 135 state concentration of the metabolite. For example, if the metabolite were eliminated from the body predominately by an enzyme with a smaller Km value than that of the enzyme that catalyzed its formation, the steady- state concentration of the metabolite would increase. When the metabolite is formed and eliminated by different enzymes, the maximum extent to which the steady-state concentration of the metabolite can increase depends on the clearance values of other pathways of elimination of the metabolite. Indeed, relationships can be written as: Climax = Vs ~M/ t(VM~O/ EM]max) + CL ss max mar ss other (16) which can be solved by a quadratic equation. Without an alternative pathway of elimination, a steady state would not be achievable when the rate of formation of the metabolite exceeds VMaX I, and the concentration of the metabolite would increase to infinity. If the metabolite were formed and eliminated by the same enzyme, however, a steady state could be achievable, because the parent substance and the metabolite would serve as mutual competitive inhibitors as the steady-state concentration of the substance is increased. If the metabolite were eliminated solely by the enzyme, the steady-state concentration of the metabolite can be predicted by the equation: EM]ss/ ES]ss = (vmajelKm~s~Ml (vma'clKm)M~o (17) Thus, the steady-state concentration of the metabolite would be directly proportional to the steady-state concentration of the substance regardless of the infusion rate of the substance. But because the metabolite serves as a competitive inhibitor of the metabolism of the parent substance, it affects the relationship between the steady-state concentration of the parent substance and the rate of infusion. Dose-Dependent Cofactor Depletion Although most simulations of dose-dependent kinetics utilize a sim- plistic form of the Michaelis-Menten equation, in which both Vma,` and Km are viewed as constants, it is important to realize that there are many different enzyme mechanisms. Equations that relate the rate of metabolism to the substrate concentration for some of these mechanisms may contain terms in which the substrate concentration is raised to the second or even greater powers, but fortunately, these mechanisms are rarely encountered in the metabolism of foreign substances. It is important, however, to realize that the concentrations of cofactors within cells may limit the rates of metabolism of foreign compounds. In such instances, the kinetics of the processes that govern the concentration of the cofactors within cells can play a dominant role.

136 JAMES R. GILLETTE In many enzyme mechanisms, the relationship between the rate of metabolism of a substrate and the concentration of the cofactor in the immediate environment of the enzyme can be expressed by the following Michaelis-Menten equation: E~K~k[C] ~ K4+KstC~J RateS~M = = (K2 + K3E ] ) + tSi Vmax PS] Km + [S] (18) where E, is the total amount of enzyme; K1, K2, K3, K4, and Ks are ratios of sets of rate constants, the meanings of which vary with the mechanism of the enzyme; and k is a first-order rate constant. Inspection of Equation 18 reveals that the simple Michaelis-Menten equation would be valid only when either of two situations occur: (1) The rate of metabolism of substrate may change fC] but the value of [C]K3 is always much greater than that of K2, and the value of PC]Ks is always much greater than that of K4. In this situation the rate of metabolism would be virtually independent of EC]. (2) The rate of metabolism of the substrate does not perceptibly change EC]. It is evident, therefore, that the validity of the simple form of the Michaelis-Menten equation depends on the kinetics of the mechanisms that govern the concentration of the cofactor within cells. In the absence of the foreign substance, the amount of a cofactor is presumably governed by its rate of synthesis from endogenous precursors (Pre kPre~c), the rate at which the cofactor is converted back to its pre- cursors (VctC]ss kC spree, and the rate at which it is consumed by other endogenous reactions in the cells (VcEC]ss kC~°~. If the amount of the cofactor can be assumed to remain in a virtual steady state, the equation can be written as: Vc[C]ss = kc~pre + I' (19) where Vc is the volume of distribution of the cofactor. Immediately after the injection of the foreign compound, however, the rate of change in the concentration of the cofactor can be written as: dt = Pre kPr~c _ vctC] (kC~pre + kC~°) —RateS~M (20) where Rates~M is defined by equation 18. Integration of Equation 20 can be performed only by an iterative pro- cedure, but it describes the approach to a virtual steady state, after which time:

General Aspects of Extrapolation 137 Pre kPr~c Ire + kC >a + (Rates~MlVctclss) V |clss 7 f~nr`, I 7 C. an I ~n_~_.~MIT' ran ~ (~20a`) Inspection of this equation reveals that when kC~Pre + keen ~ Rates ~M/vctciss' simulations based on the simple Michaelis-Menten equa- tion are valid. But when kC~Pre + kC~° <( Rates ~MIVctC]ss' the rate of metabolism is governed by the rate of synthesis of the precursor. Studies of acetaminophen metabolism performed in various laboratories have demonstrated that the rates of formation of all three major metabolites of the drug may be governed by dose-dependent cofactor depletion. The formation of acetaminophen sulfate may be limited by the cofactor phos- pho-adenosyl-phospho sulfate (PAPS), and ultimately on the total body pool of sulfate (Galinsky and Levy, 1981; Hjelle et al., 1985; Levy et al., 19821. The formation of acetaminophen glucuronide may be limited by uridine diphosphate glucuronate (UDPGA) (Hjelle et al., 1985; Jollow et al., 1982; Price and Jollow, 1982, 1984), and ultimately on the processes that govern intracellular uridine triphosphate (UTP) and glucose concen- trations. Acetaminophen is also converted to N-acetyl imidoquinone, which reacts both enzymatically and nonenzymatically with glutathione (Mitchell et al., 19731. The rate of formation of the glutathione conjugate thus may be limited by glutathione. Since the hepatotoxicity caused by acetami- nophen is thought to be mediated by N-acetyl imidoquinone (Mitchell et al., 1973), it is noteworthy that dose-dependent depletion of cofactors for all three reactions would tend to increase the virtual steady-state concen- tration of the toxic form of the drug. Suicide Inhibitors Occasionally a substance is converted to a metabolite that never leaves the enzyme that catalyzes its formation and thereby causes virtual irre- versible inhibition of the enzyme. Among the substrates that inactivate enzyme by this mechanism are many cholinesterase and monoamine ox- idase inhibitors. A general equation that describes this kind of inhibitor is: —= kE—E k c f I l dt ° ( Km + f t51) (21) where ko is the steady-state rate of synthesis of the enzyme, E is amount of active enzyme at any given time, kE is the first-order rate constant that governs the normal catabolism of the enzyme, kc is the catalytic constant of the enzyme, and F is the fraction of the metabolite that results in the inactivation of the enzyme. The kinetics of inactivation depend on the processes that govern the concentration of the precursor substance. Thus,

138 JAMES R. GILLETTE the factors that govern the amount of active enzyme at any given time frequently are complex and are best estimated by iterative procedures rather than integral calculus. In certain situations, however, solutions may be obtained by integral calculus. For example, iffU ES] were held constant, the following equation can be written: Et = EkoEIk] + [Eo—(kglk)] e-k', where ED is the initial amount of active enzyme, and kCFJUtS]ss E Km + fUtS]ss (21a) (2 lb) Inspection of Equation 21a reveals that the approach to the new steady state should be first order and that at the new steady state the amount of active enzyme should equal koEIk. Moreover, provided that the rate of synthesis of the enzyme does not change, the effectiveness of the suicide inhibitor can be estimated from the relationship EsslEo = kElk. In cell-free systems, there presumably would be neither synthesis nor catabolism of the enzymes. Moreover, the value of ES] can be held con- stant. Thus, under these conditions Equation 21 can be modified to: dE EkCFES]ss dt Km + tS]ss' which may be integrated to: where. (2 lo) E = EOe-k(inact)t k — kcr[s~SS (inact) Km + fUt5]55 (21d) Thus, the amount of active enzyme decreases exponentially, but the rate constant depends on tS]ss. Plots of 11k(inaCt) versus 1/tS]55, however, provide estimates of the relationships between kc' F. and Km' i.e.: 1lk(inact) = k F ( tS]ss) (21e) Under another set of conditions in which (1) most of the precursor of the suicide inhibitor is eliminated from the body predominantly by a mechanism other than metabolism by the inhibitable enzyme, (2) the elim- ination of the precursor can be described by a linear one-compartment system, (3) both koE and kE are negligible during the elimination of the precursor from the body, the approximate amount of active enzyme may be estimated by:

General Aspects of Extrapolation 139 In (<EIE ~ (k F) 1n(fu LS]O e ke/' + Km) (~21f) where key is the first-order elimination constant and LSO] is the concentration of the precursor. initial If the value of SO ~ Km' however, the approximate amount of active enzyme may be estimated by: 1~ tr /~i _ kCF~ I AUC lo Ill \=O/=J v Hem (21g) In- Equation 21g, the precursor need not be restricted to a single com- partment. Equation 21g permits direct comparisons between In vivo and in vitro values, because kCFlKm can be estimated in vitro from Equation 21e. SPECI ES-TO-SPECI ES EXTRAPOLATIONS It may be useful to describe categories of the various factors that govern the pharmacokinetics of substances and their biologically active metab- olites according to the extent to which we believe they would contribute to differences between individuals in the human population and to dif- ferences between experimental animals and subpopulations of human beings. Extrapolations in the Absorption of Substances Some of the factors that govern the pharmacokinetics of a substance and its metabolites are dependent predominantly on the physical chemical characteristics of the substances and the physiological characteristics of animals. For example, the rates of absorption of a substance may be dependent on its solubility, the rates of dissolution of the dosage form of the substance, and the lipid/aqueous buffer partition ratio. In the gastroin- testinal tract, the extent of absorption may be limited by the residence time of the substance in the tract. Although the residence time may be affected by species differences in the length of the intestines, it is fre- quently governed to a greater extent by either diarrhea or constipation and to differences in binding to constituents in food and the gastrointestinal motility. Indeed, the extent of absorption of a substance in a given in- dividual may vary markedly from day to day, depending on the times at which the substance is swallowed and the meal is eaten, as well as on the composition of the meal. Some substances are unstable under the acidic conditions that exist in the stomach or, as in the case of nitrite, react with other substances to

140 JAMES R. GILLETTE form precursors of mutagens. Other substances are metabolized by bac- terial flora that either inactivate the parent substance or convert the sub- stance to biologically active metabolites. Daily changes in the relative residence times of such substances in the stomach, the small intestine, and the large intestine can result in marked daily differences in the extent to which these reactions can occur in an individual. Moreover, there are species and perhaps individual differences in the intestinal flora (Drasar et al., 19701. The extent to which various reactions occur within the intestines thus can vary with the species. But the extent to which metab- olism by intestinal flora would contribute to differences in formation of toxic metabolites also depends on the consumption of antibiotics. Extrapolations of ~nterorgan Distribution of Substances As pointed out above, most phamacokineticists who use physiologically based pharmacokinetic models assume that a substance entering an organ is distributed virtually instantaneously and that equilibration of the sub- stance between that in blood and that within nonelimination organs is governed solely by the blood flow rate and the partition ratio between the total concentration of the substance and the total concentration in blood, i.e., R. Although models in which this assumption has been used have been surprisingly successful in predicting the time course of substances in various organs, various tests of the assumption suggest that the pro- portionality with blood flow rates through slowly equilibrating organs is really an indirect measure of other phenomena that govern the distribution of the substance. Indeed, if the assumption were entirely valid, then venous blood taken from the arms of human subjects should reflect the concen- tration of the substance in the muscles of the arms and hands rather than the concentration in arterial blood, assuming, of course, that the arterio- venous shunts in muscles are virtually nonfunctional. The findings that the concentrations of substances in venous blood are reasonably good estimators of the concentrations of substances in central compartments thus raise doubts concerning the validity of the assumption. Moreover, the assumption predicts that vigorous exercise, which results not only in an increase in cardiac output but also in a shift in the relative rates of blood flow to muscles and vital organs, should markedly hasten the en- trance of substances into muscles. But whether exercise does change the pharmacokinetics of drugs to the extent that would be expected is ques- tionable. By contrast, it has been suggested that changes in the hepatic blood flow caused by drugs or meals can affect the bioavailability of highly extracted drugs. Regardless of what governs the distribution of substances into slowly equilibrating nonelimination organs, the fact that the kinetics appear to be related to baseline blood flow rates through such

General Aspects of Extrapolation 141 organs suggests that some allometr~c extrapolations between animal spe- cies may be valid. Valid and Invalid Extrapolations of Allometric Methods In most instances, there are little interspecies differences in the gross composition of individual organs. Therefore, as long as the tissue to blood partition ratios in any given organ depend predominantly on the partition of substances between cell water and neutral fat in adipocytes and on reversible binding to various components in an organ and blood, it seems likely that little difference in the partition ratios of different organs will be found between different animal species. When the partition ratio is markedly affected by the presence of active transport systems and/or the presence of metabolic enzymes, however, the validity of the assumption that allometric methods provide accurate extrapolations is less certain. In most instances, the organs in which such active transport systems and metabolic enzymes exist are usually small and therefore usually make only minor contributions to the total body volume of distribution of the sub- stance. But clearly, the presence of active transport systems in certain cells within an organ can govern which cells are at risk. Indeed, much of the tissue specificity in the toxicity of paraquat is thought to be due in part to the transport systems in certain lung cells (Rose et al., 19761. Relevance and irrelevance of Distributional Rate Constants Because tissue to blood partition ratios are components of the rate constants that describe the entrance and exit of substances in nonelimi- nation organs, they can have a profound influence on the time course of the substance in different organs after a single administration of the sub- stance. Indeed, some substances may have remarkably long half-lives in the body, even when the hepatic clearance approaches hepatic blood flow, because the partition ratios in many of the organs are very high. Moreover, the combination of blood flow and partition ratios can have a profound influence on the approach to steady state during repeated administration of the substance. After a steady state is achieved and the substance is withdrawn, the time required for the substance to be virtually eliminated from the body also depends on the half-life of the terminal phase of elimination. When the investigator is primarily concerned with these phases of the time course of the concentration of a substance and its metabolites, then knowledge of the values of the organ to blood partition ratios are important. In these instances it is important to establish the validity of allometric approaches of extrapolating the value of the blood flow rates and the values of Rorgan/b~ooc~ ratios from species to species.

142 JAMES R. GILLETTE When investigators are primarily concerned with studies in which the substance and its biologically active metabolites are in steady states during most of the time of the study, however, knowledge of the blood flow rates and the Rorgan/b~ooy values may diminish in importance. Under steady- state conditions, species differences in the Rorgan/b~ooa' ratios govern the magnitude of the oscillations between the maximum and minimum con- centration of unbound substance in organs; the larger the values of Rorgan/ bloody the smaller the magnitude of the oscillation. Moreover, species dif- ferences in the VRorgan/b~ooy ratios, particularly in rapidly equilibrating organs, affect the magnitude of the oscillations in the maximum and minimum concentrations of unbound substance in blood. But the mag- nitude of the oscillations affects the average concentration of unbound substance in nonelimination organs only to the extent that they affect the clearance (CL) and the availability values of the organs of elimination. SPECIES DIFFERENCES IN THE ELIMINATION OF FOREIGN COMPOUNDS To the extent that species differences in the magnitude of toxicities are due to pharmacokinetic factors, it is likely that most differences are due to factors that govern the total body clearances of substances and the formation and elimination of biologically active metabolites. Elimination by Excretion into Urine, Air, and Bile To the extent that substances are excreted unchanged from the body by processes that depend predominantly on physiological processes of animals and the physical chemical characteristics of the substance, it seems likely that allometric methods for extrapolating from one species to another provide reasonably valid results. For example, substances eliminated into exhaled air are governed predominantly by the air/water partition ratio of the substance, the total volume of distribution, the rate of respiration, and the cardiac output (McDougal et al., 19861. Excretion of unbound sub- stances into urine is frequently governed predominantly by the glomerular filtration rate and the lipid solubility of the substance. Substances that are actively transported into bile (Smith, 1973) and urine, however, are subject to marked species differences; but the rates of excretion of many of these substances may be limited by the renal and hepatic blood flow rates. When this occurs species differences may appear to be related when allometric methods are used to estimate blood flow rates. When the rate of elimination of substances by the liver is limited by the blood flow rate, however, care should be used in the interpretation of extrapolations based on allometric methods. In such situations, the

General Aspects of Extrapolation 143 terminal half-life of the substance may appear to be predicted by the allometric methods, but there still may be marked species differences in the bioavailability of substances administered orally that may not be pre- dicted by the allometric methods. Elimination by Metabolism It is doubtful that allometric methods are universally valid for predicting species differences in the metabolism of foreign compounds, because there can be marked differences in the metabolism of foreign compounds even between individuals and strains of the same species. Before discussing species differences in metabolism, it is useful to discuss factors that con- tribute to these intraspecies differences. Despite the emphasis that phar- macokineticists place on flow-limited metabolism, it is important to realize that the metabolism of most foreign compounds is slow compared with the blood flow rates. Thus, Rorgan/b~ooa'' fat in blood, or blood flow rates may be largely irrelevant to discussions of the species differences in pharmacokinetic factors that contribute to species differences in the tox- icities of substances that are administered repetitively to test animals during toxicity studies lasting several weeks or months. INDIVIDUALS AND STRAIN DIFFERENCES IN METABOLISM OF FOREIGN COMPOUNDS During the past three decades, it has become obvious that foreign com- pounds are seldom metabolized in the body by a single enzyme. Many compounds are metabolized by combinations of reactions, such as oxi- dation, dehydrogenation, hydrolysis, sulfation, glucuronidation, and glu- tathionyl conjugation reactions. Each of these general reactions can be catalyzed by several different enzymes. Indeed, several isozymes catalyze the transfer of sulfate from PAPS to phenols and alcohols (Jakoby et al., 19801; several catalyze the transfer of glucuronic acid from UDPGA to phenols and alcohols (Burchell et al., 19851; several catalyze the transfer of glutathione to various kinds of foreign compounds (Jakoby et al., 19801; and several catalyze the hydrolysis of esters and amides (Heymann, 19801. Even the term cytochrome P-450 is now known to represent at least nine different isozymes in rat liver (Levin et al., 1985~. Each of the isozymes of the different categories of enzymes has its own substrate specificity. Some of the isozymes are highly specific and catalyze the metabolism of only a few substances, but others catalyze the metabolism of many dif- ferent substrates. For each substrate, each isozyme has its own set of rate constants that govern the apparent Vm<ZX and Km values. A substrate can be metabolized by several different enzymes and isozymes that may or

144 JAMES R. GILLETTE may not have similar apparent Km values. Each isozyme can form several metabolites from a given substrate, but the relative rates of formation of the metabolites may differ with the isozyme. In some instances, the pattern of metabolites formed by different isozymes is virtually identical; in other instances, it is markedly different. ISOZYMES WITH DIFFERENT Km VALUES The total rate of metabolism at any given concentration of a substrate within cells is governed not only by the amounts of the various isozymes that are able to metabolize the substrate but also by their apparent Km values. For example, the 7-hydroxylation of 2-acetylaminofluorene in rabbit liver microsomes is catalyzed by at least two isozymes with Km values of 0.5 and 147 AM (Thorgeirsson, 19851. Thus, the relative con- tributions of the various isozymes to the total metabolism of the substrate may vary with the substrate concentration. At high substrate concentra- tions, the rate may be governed predominantly by isozymes with large Vmax and Km values, whereas at low substrate concentrations the rate may be governed predominantly by isozymes with low VmaX and Km values. The presence of such enzymes is frequently difficult to detect from in vivo pharmacokinetic studies, especially when the studies are interrupted before the concentrations of unbound substance have not declined to levels below the Km values of the high-affinity enzymes. SEX DIFFERENCES The amounts of many of the isozymes in various organs are under hormonal control. For example, the amounts of some isozymes of cyto- chrome P-450 may be altered by the administration of growth hormone (Gustafsson et al., 1985), testosterone, or estradiol. Thus, the relative distribution of the cytochrome P-450 isozymes (at least in rat liver) changes as the animals mature and results in marked sex differences in the me- tabolism of some compounds but not in others. INDUCERS The amounts of some isozymes may also be changed by foreign com- pounds entering the body from the environment. Many foreign compounds increase the amounts of some isozymes (Conney, 1982) but decrease the amounts of others, presumably by interacting either directly or indirectly with regulatory genes. Different inducers affect different regulatory genes and thus may cause increases and decreases in several isozymes, but the changes in the amounts of different isozymes vary with the inducer (Thomas

General Aspects of Extrapolation 145 et al., 1986~. There are strain differences not only in structural genes that govern the amino acid sequence of the various isozymes but also in reg- ulatory genes. Thus, at a given dose, an inducer may cause marked al- terations in the pattern of isozymes in one strain of animals but not in another. INTERORGAN DIFFERENCES IN METABOLISM Although liver is rightly considered the major organ involved in the metabolism of most foreign compounds, drug-metabolizing enzymes, in- cluding isozymes of cytochrome P-450, are also present in other organs. However, the relative distribution of the isozymes varies with the organ. For example, 4-ipomeanol, a substance produced in moldy sweet potatoes that causes lethal pulmonary injury in cattle, rats, mice, and hamsters, is converted to a toxic metabolite in the Clara cells in rat lungs (Boyd, 1980) by an isozymeLs) that has a Km value about one-tenth that of the predom- inant isozymets) in rat liver that catalyzes the formation of the metabolite in rat liver (Boyd et al., 19781. Moreover, inducers generally alter the isozymes in different organs to different extents. For example, pretreat- ment of rats with 3-methylcholanthrene increases the rate of formation of the toxic metabolite of ipomeanol in liver microsomes, but not that in lung microsomes. As a result, liver becomes a target organ, but the toxicity in lung is diminished (Boyd, 19801. Nevertheless, 3-methylcholanthrene does increase the amount of some of the isozymes in extrahepatic organs. For example, pretreatment of rats with 3-methylcholanthrene increases the rate of metabolism of 3,4-benzoLa~pyrene by microsomes of many organs, including intestinal mucosa and lung. By contrast, pretreatment of rats with phenobarbital also increases the activity of the isozymes that catalyze the formation of the toxic metabolite of ipomeanol in liver mi- crosomes, but not the isozymes that catalyze its formation in lung. But, because pretreatment with phenobarbital causes an even greater increase in the glucuronidation of ipomeanol in viva, it decreases the amount of metabolite formed in both liver and lung and thus decreases the toxicity in lung, while the liver remains unaffected. INTERSTRAIN DIFFERENCES Interstrain differences in the relative amounts of various isozymes may result in either small or large differences in rates of metabolism, depending on the substrate and the animal species. In rats and mice interstrain dif- ferences in the metabolism of several substrates, such as hexobarbital, aminopyrine, and acetanilide, by liver microsomes appear to be between two- and threefold (Testa and Jenner, 19761. Whether this would be true

|46 JAMES R. GILLEUE for all substrates, however, is open to debate. Within a group of eight strains of rabbits, the metabolism of hexobarbital, aminopyrine, and an- iline by liver microsomes also varied between two- and threefold, but there was a 20-fold interstrain difference in the metabolism of amphet- amine. Polymorphisms in Animals A given kind of metabolite of some substances may be formed pre- dominantly by a single enzyme in the body. When this occurs a change in a structural gene would cause the synthesis of a different allozyme that may or may not have marked differences in the metabolism of the substrate. On one hand, purification of certain allozymes of cytochrome P-450 have virtually identical kinetic characteristics. But on the other hand, allozymes of N-acetyltransferase, which catalyzes the acetylation of many drugs and other foreign compounds, have been isolated from phenotypically different rabbits and have been found to have markedly different kinetic charac- teristics (Andrea' and Weber, 19861. The Vma,; values of the allozyme isolated from slow acetylators (rr) for both p-aminobenzoic acid and pro- cainamide were an order of magnitude below the Vmax values of the al- lozyme isolated from fast acetylators (RR). But the Km value of the rr allozyme for p-aminobenzoic acid was also about an order of magnitude smaller than that of the RR allozyme, whereas the Km value of the rr allozyme for procaine amide was double that of the RR allozyme. Thus, the intrinsic clearances (VmaXlKm) of the two allozymes were only slightly different forp-aminobenzoic acid but markedly different for procainamide. There is therefore little difference between the phenotypes in the acety- lation of p-aminobenzoic acid in viva, but more than an order of magnitude difference in the acetylation of procainamide. Polymorphisms and Environmental Differences in Humans A combination of environmental and genetic polymorphisms are known to contribute to the individual variability in the metabolism of foreign compounds in humans. For example, smoking and consumption of certain vegetables and open-fire-broiled meat are known to alter the metabolism of many foreign compounds (Conney et al., 1980; Kappas et al., 19771. Moreover, polymorphisms are also known to exist (Kalow, 19621; those associated with the acetylation of arylamines, the hydrolysis of succinyl choline and paroxon, and the hydroxylation of debrisoquine (Smith, 1985) have been the most extensively studied. There is also suggestive evidence of other polymorphisms in the human population in the metabolism of

General Aspects of Extrapolation 147 foreign compounds, but these have been less well documented. The com- bination of both environmental and genetic differences undoubtedly ac- counts for the observation by clinicians that individual differences in drug metabolism can range from 4- to 40-fold, depending on the drug (Sjoqvist et al., 1976; Vesell and Penno, 19831. In fact, the debrisoquine poly- morphism has been reported to cause as much as a 300-fold difference in the hydroxylation of debrisoquine and quanoxan between two scientists in the same laboratory (Idle, 19801. The variance in metabolism can be even greater in extrahepatic tissues; placentas from a group of women, composed of both smokers and nonsmokers, had as much as a 400-fold difference in the rates of metabolism of benzoka~pyrene (Conney, 19821. Interspecies Differences in the Metabolism of Foreign Compounds Over the past several decades there has been a plethora of studies of the metabolism of foreign compounds in various animal species. But there does not appear to be any consistent relationship that would justify the assumption that species differences can always be related by allometric methods. The half-lives of some drugs are longer in humans than in rats, which would tend to support the concept that allometric methods might be useful in making such extrapolations. But the half-lives of other drugs are virtually identical in rats and humans in other studies, and the half- lives of still other drugs are longer in rats than in humans (Smith and Caldwell, 19771. Species differences in the pattern of metabolites excreted into urine are even more extensive. For example, in a literature survey in which the patterns of the urinary metabolites of 23 compounds excreted by rats and humans were compared, only four patterns were considered sufficiently similar to rate a good correlation, whereas eight patterns were considered completely different (Smith and Caldwell, 19771. When the toxicity of a substance is caused by a minor pathway of metabolism, studies of pharmacokinetic parameters and the pattern of urinary metabolism may not always provide sufficient information to pre- dict toxicities. For example, at doses of acetaminophen that cause hepa- totoxicity in mice, but not in Sprague-Dawley rats, there are such small differences in both the half-life of the drug and the pattern of its urinary metabolites that it would have been difficult to conclude from those data alone that the species difference in toxicity was due to differences in the metabolism (Gillette, 19771. Other studies, including the measurements of declines in glutathione concentrations and covalent binding to hepatic proteins, were needed to reveal the relevant species differences in metab- olism that caused the species differences in toxicity (Mitchell et al., 1973; Potter et al., 19741.

|48 JAM ES R. GI LLEUE There also may be species differences in the effects of inducers. For example, pretreatment of mice with phenobarbital increases the hepato- toxicity of acetaminophen by causing a preferential increase in the for- mation of the chemically reactive metabolite, but pretreatment of hamsters with phenobarbital decreases the hepatotoxicity of acetaminophen by pref- erentially accelerating the glucuronidation of the drug (Gillette, 19771. Thus, it is not always possible to extrapolate the effects of inducers from one species to another. Molecular biologists have begun to classify the isozymes of cytochrome P-450 according to the degree of similarity in their amino acid sequences. Isozymes having less than 36% homology are considered to represent different families. According to this classification there are at least eight families of cytochrome P-450 isozymes in mammals (Nebert et al., in press). Within each family there may be several subfamilies" Whether a given inducer increases the synthesis of the same family of isozymes, or whether it even increases all of the members of the same family, has not yet been established. But it is clear that allozymes in different animal species and the isozymes induced in the different species do not necessarily have the same substrate specificity. Moreover, when the isozymes do metabolize the same substrate, they do not always form the same relative amounts of metabolites or possess the same Vmax and Km values. For example, isozymes analogous to cytochrome P-450c and cytochrome P- 450d induced by 3-methylcholanthrene in rats convert propranolol almost exclusively to desisopropyl propranolol in rabbit liver microsomes, almost exclusively to 4-hydroxypropranolol and 5-hydroxylpropranolol in guinea pig microsomes, and to all three metabolites in liver microsomes of rats and mice (H. A. Sasame, unpublished results). GENERAL COMMENTS The utopian objective of a quantitative risk assessment is to be able to predict the incidence rates of toxicities in a human population that is exposed to low doses of toxicants solely from the results of a single toxicity study in a single strain of test animal subjected to relatively large doses of toxicants. Various mathematical models have been developed by stat- isticians as aids in making extrapolations from high doses to low doses and from populations of small laboratory animals to human populations. That there are uncertainties in the validity of the values obtained with such mathematical models is recognized by nearly everyone. But there is considerable disagreement among scientists concerning the magnitude of the uncertainty. At one extreme, some scientists apparently believe that the values should be accurate within a very narrow range (say, about twofold), and thus focus attention on factors that would affect the cal-

General Aspects of Extrapolation 149 culated value very little. At the other extreme, some scientists apparently believe that the uncertainties are so great that any estimate obtained with the mathematical models is virtually useless. To many scientists, it seems evident that the degree of inaccuracy of the values calculated by quantitative risk assessment methods differs with the toxicant, the mechanism of the toxicity evoked by the toxicant, and the situation. It therefore is important to delineate those kinds of toxicants and mechanisms of toxicity in which we can be reasonably confident that the degree of inaccuracy is small from those in which the degree of uncertainty is likely to be large. To this end, however, it is necessary to understand precisely what the problems are and whether the various math- ematical models adequately address the problems. One of the problems is that the shape of a dose-response curve may be a composite of at least three different kinds of dose-response relationships. Indeed, the meaning of a dose-response curve is largely dependent on how the response is recorded. The meaning of each type can be illustrated by an idealized experiment in which a drug acts by binding reversibly to a set of receptor sites in a group of identical animals. 1. In this idealized situation, the magnitude of the response is presum- ably directly proportional to the number of receptor sites occupied by the drug. Thus, if the action sites act independently of one another, the fraction of the total number of receptor sites that are occupied by the substance at any given concentration of unbound substance will usually follow the Law of Mass Action, i.e.: (Effect/Maximum Effect) = [DlI(Kd + fD]), (22) where ED] is the concentration of the unbound drug, and Ky is the dis- sociation constant of the drug-receptor site complex. Notice that at low concentrations of the drug the magnitude of the effect should be directly proportional to the concentration of the unbound drug. Thus, the claim that there is an inherent dose threshold for the action of a reversibly acting drug is fallacious. What is meant by a no-observed-effect level (NOEL) is either that the magnitude of the effect at very low concentrations can become so small that it cannot be detectable or that the response is not always directly proportional to the number of receptor sites occupied. Consider, however, that in a group of completely identical animals, the magnitude of the response should be the same in all animals. 2. A dose-duration of response curve depends on the rates at which the drug enters the blood and rises to its maximum concentration in the im- mediate environment of the action sites and then declines to a concentration that exerts a magnitude of response below which the effect is no longer detectable. Consider, however, that a given dose administered identically

150 JAMES R. GILLETTE to a group of identical animals should result in identical durations of response in all of the animals in the group. 3. In a dose-incidence of response curve, the proportion of the group of animals receiving a given dose of the drug is evaluated. Inherent in this type of dose-response curve is the assumption that there is some dose that results in a magnitude of response below which the response either is not detectable or is not biologically significant. In the idealized exper- iment, however, there would be no observable effect in any of the animals until a certain critical dose is administered, but at slightly larger doses the response would be observable in all of the animals. Hence the slope of the dose-incidence of response curve in the idealized experiment would be infinite. In the real world, the slope of the dose-incidence of response curve thus is a measure of the precision with which the substance is administered and the response recorded, and the homogeneity of the pop- ulation of animals in response to the substance. Although the mathematics required to describe other mechanisms of response would be different, the basic meanings of the dose-magnitude of response and the dose-incidence of response curves are applicable to all mechanisms of response. Many risk assessments for carcinogens are based on the multistage model, which is described by the equation: P= 1 —e-X where (23) X = qO + qld + q2d2 (23a) In Equation 23 P is defined as the probability of occurrence of a tumor in an individual animal exposed to a daily dose (d) of a chemical for a lifetime, and the q values are parameters estimated from the experiment. At first glance Equation 23 may appear to be reasonable, but in light of the above discussion, Equation 23 is based on inconsistent logic, because a dose-incidence of response curve implies the existence of a NOEL below which tumors would be undetectable. Indeed, it seems more logical to treat data obtained in a study of carcinogenesis as a dose-magnitude of response curve in which the number of tumors observed during the course of the study is related to the numbers of cells at risk (cells). Accordingly, we could write an equation: Tumors/Cells = X = qO + qid + q2d2 + .... (24) According to this view, however, the development of tumors would be considered a rare event, and thus a group of 100 virtually identical animals receiving the same dose in a bioassay for carcinogenesis would be visu-

General Aspects of Extrapolation 151 alized as a single animal, which is 100 times as large and the number of animals actually employed. The two approaches thus are conceptually very different and lead to different interpretations of the slope of a dose-response curve; in the dose- incidence of response curve, the slope is a measure of the homogeneity of the animals used in the bioassay, whereas in the dose-magnitude of response curve the slope assumes no heterogeneity within the population. Nevertheless, the two approaches can be interrelated by normalizing the number of cells to the number of cells at risk per animal, that is: (Tumors/Cells)/(Cells/Animal) = (Tumors/Animal) = X. (25) Whereas, at low values of X: P = (1 —e-X)~X. (26) Thus, in practice the two concepts usually provide virtually the same estimate. But the practice of counting an animal with multiple tumors only once in a bioassay, though defensible as a practical matter, is not defensible from a theoretical point of view. Another problem that must be considered is that Equations 23a and 24 probably are not completely valid in describing all of the possible ways in which the value of X is curvilinear with increasing doses. An increase in dose can alter not only pharmacokinetic parameters but also pharma- codynamic factors. For example, perhaps high doses of a substance may cause so much damage to DNA that the rates of repair approach maximal values. Some promoters can act by releasing growth factors, but the rate or extent of release of these factors may not be directly proportional to the dose. On the other hand, the bioassay coupled with pharmacokinetic studies may not always reveal the presence of high-affinity, low-capacity enzymes that can be of predominant importance at low doses but only of trivial significance at high doses. Nor will the effects of dose-dependent phar- macokinetic parameters be accurately predicted by a function of X rep- resented by a polynomial. In calculating the P (probablity) values, statisticians do incorporate interanimal variability within error functions. Moreover, they can differ- entiate between models through the use of maximum likelihood methods, but marked interanimal variability decreases their ability to distinguish between plausible models in the bioassay. Whatever the problems that occur in the interpretation of experiments may be, however, it is important to remember that the calculated values for P represent the values for the animals used in the bioassay under the conditions of the experiment. In bioassays lasting several days, weeks, months, or years, either the processes that govern the pharmacodynamic factors or the processes that

|52 JAMES R. GILLEUE govern the pharmacokinetic factors may be altered during repeated ad- ministration of high doses of the substance under investigation. Indeed, studies of alterations in the concentrations of several hormones that are thought to influence the manifestation of tumors are sometimes incorpo- rated into the protocol of the bioassay. But it is also true that repeated administration of substances can alter processes that govern relevant phar- macokinetic parameters, such as kidney and pulmonary function or the activity of enzymes that govern the total body clearances of a parent compound and its biologically active metabolites and the pattern of me- tabolism of various tissues. Moreover, if short-lived metabolites are thought to cause the toxicity, it may be necessary to assess the activity of enzymes that catalyze the formation of the metabolite, inactivation of the enzymes in potential target cells and organs, as well as the activity of enzymes that account for most of the total body clearances of the precursors of the ultimate carcinogen. Thus, pharmacok~netic studies performed solely with untreated animals may be largely irrelevant to the quantitative risk as- sessment process. Instead, the pharmacokinetic studies should be repeated at intervals throughout the course of the bioassay. If the pharmacokinetic factors do change during repeated administration, then the investigator is faced with the problem of deciding whether the changes would markedly influence the magnitude of toxic response, and if they do how they would affect the shape of the dose-magnitude of response curve. In the extrapolation of the calculated values of tumors/cells in Equation 24 from high doses in animals to low doses in humans, it is important to remember that the sequence is an extrapolation from high doses to low doses for the animals in the bioassay and then an extrapolation from low doses in the experimental animals to low doses in the human population. Thus, to the extent that pharmacokinetic parameters contribute to inter- species differences in the magnitude of the response, the differences usu- ally can be expressed by differences in the parameters of linear models. Hence, most of the complexities in pharmacokinetics due to dose-depen- dent nonlinearities that are important in understanding the shape of the dose-magnitude of response curve in the animal assay are usually not relevant in extrapolating from low doses in animals to low doses in the human population. Nevertheless, in making an extrapolation from the animals used in the bioassay to the human population, investigators are faced with two prob- lems: ( 1 ) Are the pharmacodynamic and pharmacokinetic factors that gov- ern the magnitude of response at low doses of the toxicant in the test animals markedly different from those in the mean of the human popu- lation? (2) What is the range of the differences in these factors in the human population relative to the range of the differences in the animal population used in the bioassay?

General Aspects of Extrapolation 153 Within this context, it is important to stress that any given pharrnaco- kinetic and toxicity study in research animals is performed in animals that are phenotypically virtually identical and maintained under environmental conditions that are kept as homogeneous as possible. By contrast, the human population is composed of individuals that are genetically heter- ogeneous, are of different ages and sizes, are either female or male, may suffer from various diseases, have different personal habits including smoking or exercise, eat different diets, and live in different environments that are heterogeneously distributed throughout the world. All of these factors have been shown or suspected to contribute to differences in the pharmaco- dynamics or the pharmacokinetics of foreign compounds. It is also im- portant to realize that the way that pharmacokinetic data are reported in the literature stresses the mean value within given populations. Outliers are usually ignored, on the assumption that they result from analytical or sampling errors. The possibility that they represent polymorphisms in either the pharmacodynamic or pharmacokinetic factors of toxicants is frequently ignored, even though such subpopulations may include virtually all of the individuals suffering from the toxicity of a substance in the human population. Elucidation of such polymorphisms, however, can only occur by studying the human population. Unfortunately, science has not yet been able to develop a universally valid approach for predicting the range of variability within the human population in the disposition of all foreign compounds. Indeed, the range of variability within the human population is known to vary with the foreign compound. In attempting to extrapolate estimates from animals to the mean of the human population, various investigators have pointed out that interspecies differences in many physiological processes, including cardiac output, organ sizes, blood flow rates, and basal metabolism rates, may be related to the surface area of the animals and physiological time. In recent years, a few pharmacokinetic studies have provided data that would tend to support the idea that such extrapolations may be valid for those com- pounds. But the finding that interspecies differences in the pharmacoki- netics of some compounds can be related by allometric methods should not be construed as meaning that allometric methods are valid for pre- dicting interspecies differences in the pharmacokinetics of all foreign compounds. Indeed, the predominance of evidence in the field of drug metabolism that has accumulated in the past indicates that allometric methods would be virtually useless for the prediction of metabolism of many substances as a mean of the human population, much less the range of values within the human population. It therefore seems important to delineate situations in which allometric methods are likely to provide reasonably valid extrapolated values from those in which the validity of extrapolations obtained by such methods would be highly questionable.

54 JAMES R. GILLEUE It seems likely that interspecies differences in those processes that de- pend predominantly on the physical chemical properties (such as solubility in water and lipid/aqueous partition ratios) and basic physiological pro- cesses (such as rates of respiration, blood flow rates through various organs, and renal clearances, particularly those dependent predominantly on glomerular filtration rates) may be predictable by allometric methods with a reasonable degree of precision (e.g., within a fivefold range of confidence). Thus, allometric methods should provide reasonably valid predictions when interspecies differences in the toxicity are predominantly due to pharmacokinetic factors (as opposed to pharmacodynamic factors) and are caused solely by a parent compound that is eliminated predomi- nantly unchanged into exhaled air, urine, or possibly, feces. Moreover, allometric methods can also provide reasonably valid predictions when the toxicity is caused solely by a parent compound that is eliminated by flow-limited metabolism in non-first-pass organs; under these conditions marked interspecies differences in the intrinsic clearances of the enzymes that metabolize the compound can occur without having marked inter- species differences in the total body clearance or the biological half-life of the compound. I am less hopeful that allometric methods will always provide valid interspecies extrapolations when toxic parent substances are eliminated from the body predominantly by enzymes having intrinsic clearances much less than the blood flow rates through the organs of elimination or when the toxicity is caused by metabolites of the toxicant. In such cases allo- metric methods may provide reasonably valid interspecies extrapolations for some toxicants, but not for others. To differentiate between these broad categories, however, it would be necessary to perform well-integrated pharmacokinetic and toxicity studies that would elucidate whether the toxicity is caused by the parent substance, one or more of its metabolizes, or a combination of the parent substance and its metabolize. When the mechanism of toxicity is not known and the sources of in- terspecies and intraspecies differences in the response to the toxicant are not clearly understood, extrapolations based on allometric methods rep- resent only first guesses of the incidence rates of the toxicity in the human population that we hope will be reasonably valid for most compounds. Perhaps some of the extrapolations may be validated by studies of the toxicants in humans that are accidentally exposed. In such cases Bayesian approaches to evaluate the mean values and ranges of pharmacodynamic and pharmacokinetic parameters in humans may be useful. At the present time, however, the hope of the risk assessor is not whether the extrapolated incidence rate is completely accurate. Instead, the hope is that the assessment is not so far wrong that the incidence rate exceeds

General Aspects of Extrapolation 155 a de minimus value. Although it would be nice to think that the de minimus value could be set at an arbitrary level (such as 1:1 ,000,000), as a practical matter it is really limited to the ability of epidemiological methods to detect toxicities over background values. Even though allometnc methods will not always provide accurate assessments, there does not appear to be any reasonable alternative to using them. Thus, in the absence of complete information concerning the mechanisms of toxicity and the pharmacoki- netic and pharrnacodynamic factors that govern the manifestations of the toxicities, it is perhaps advisable to try to establish a consensus among scientists of arbitrary broad ranges of uncertainty, even though such ranges cannot be rigorously defended by science. When the mechanism of toxicity becomes known and if it can be established that intraspecies and inter- species differences in the pharrnacodynamic and pharmacokinetic factors can be predictable within narrower ranges of uncertainty for given tox- icities caused by given toxicants, the arbitrary range of uncertainty can be narrowed for that toxicity and toxicant. Unless such a system is es- tablished, I believe that the field of quantitative risk assessment will remain highly controversial. Whatever system is ultimately established, I believe it inevitable that some mistakes will be made and therefore that quantitative risk assessments will never replace the need for epidemiological studies or toxicity reporting systems. REFERENCES Andres, H. H., and W. W. Weber. 1986. N-Acetylation pharmacogenetics: Michaelis- Menten constants for arylamine drugs as predictors of their N-acetylation rates in vivo. Drug Metab. Dispos. 14:382-385. Baron, J., and T. T. Kawabata. 1983. Intratissue distribution of activating and detoxicating enzymes. Pp. 105-135 in Biological Basis of Detoxication, J. Caldwell and W. B. Jakoby, eds. New York: Academic Press. Boyd, M. R. 1980. Biochemical mechanisms in chemical-induced lung injury: Roles of metabolic activation. CRC Crit. Rev. Toxicol. 103- 176. Boyd, M. R., L. T. Burka, B. J. Wilson, and H. A. Sasame. 1978. In vitro studies on the metabolic activation of the pulmonary toxin, 4-isomeanol by rat lung and liver microsomes. J. Pharmacol. Exp. Ther. 207:677-686. Burchell, B., M. R. Jackson, S. M. E. Kennedy, L. McCarthy, and G. C. Barr. 1985. Characterization and regulation of hepatic UDP-glucuronyltransferase. Pp. 212-220 in Microsomes and Drug Oxidations, A. R. Boobis, J. Caldwell, F. DeMatteis, and C. R. Elcombe, eds. Conney, A. H. 1982. Induction of microsomal enzymes by foreign chemicals and carci- nogenesis by polycyclic aromatic chemicals. G. H. Clowes Memorial Lecture. Cancer Res. 82:4875-4917. Conney, A. H., M. K. Buening, E. J. Pantuck, C. B. Pantuck, J. A. Fortner, K. E. An- derson, and A. Kappas. 1980. Regulation of human drug metabolism by dietary factors. Pp. 147-167 in Environmental Chemicals, Enzyme Function and Human Disease, Ciba Foundation Symposium, Vol. 76.

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158 JAMES R. GILLETTE Taylor, P. W. 1972. Fast reactions flow and relaxation methods. Pp. 351-380 in Methods in Pharmacology, Vol. 2, Physical Methods, A. Schwartz, series ea., C. F. Chignell, ed. New York: Appleton-Century-Crofts. Testa, B., and P. Jenner. 1976. Pp. 361-384 in Drug Metabolism: Chemical and Biological Aspects. New York: Marcel Dekker. Thomas, P. E., L. M. Reik, S. L. Maines, S. Bandiera, D. E. Ryan, and W. Levin. 1986. Antibodies as probes of cytochrome P-450 isozymes. Pp. 95-106 in Biological Reactive Intermediates III, J. J. Kocsis, D. J. Jollow, C. M. Witmer, J. O. Nelson, and R. Snyder, eds. New York: Plenum. Thorgeirsson, S. S. 1985. Kinetics of acetylaminofluorene hydroxylation reactions in mi- crosomes and drug oxidations. Pp. 320-329 in A. R. Boobis, J. Caldwell, F. DeMatteis, and C. R. Elcombe, eds. Microsomes and Drug Oxidations. London: Taylor & Francis. Vesell, E. S., and M. B. Penno. 1983. Intraindividual and interindividual variations. Pp. 369- 410 in Biological Basis of Detoxication, J. Caldwell and W. B. Jakoby, eds. New York: Academic Press.

Dose, Species, and Route Extrapolation Using Physiologically Based Pharmacokinetic Models Harvey ]. Clewe1/t III and Melvin E. Andersen I NTRODUCTION We can distinguish four types of extrapolations involved in assessing the expected human risk associated with exposure to environmental chem- icals. They consist of ( 1 ) predicting the low-dose response in experimental animals based on observed responses at very much higher doses, (2) predicting the response in the human population based on the results in the test species, (3) predicting the risks associated with the anticipated human exposure route based on toxicity observed when a different route of exposure is used in the animal toxicity studies, and (4) predicting human response at realistic, discontinuous environmental exposures based on animal results in well-controlled, much more easily characterized expo- sures. In this paper, these are called, respectively, the dose, species, route, and exposure scenario extrapolations. Each of these extrapolative steps is important and must be conducted on the basis of the best available, sci- entifically justifiable procedures if the final exposure limits are to have validity and enjoy consensus support from government, industry, and the concerned public. Before considering the application of physiologically based pharmacokinetic modeling for performing these extrapolations, we will briefly review current practices in each of the four areas. Much of the introductory material for this paper is from H. J. Clewell and M. E. An~lersen. 1985. Risk assessment extrapolations and physiological modeling. Toxicol. Ind. Health. 1:1 1 1- 131. 159

]60 HARVEY 3. CLEWELL Ill AND MELVIN E. ANDERSEN Dose Many risk assessments are based on the results of animal experimen- tation conducted at very high daily doses compared with those likely to be encountered in human exposures. Typically, extrapolations are based on administered dose, and a linear extrapolation through zero is utilized to predict the incidence of a particular effect in the low-dose region. A variety of statistical models is used for this extrapolation, and the estimated risk can differ by several orders of magnitude, depending on the model. Often the response data are extrapolated by using a parametric fit (e.g., the multistage model), and the linear, nonthreshold model is assumed to apply in the low-dose region. It is hoped that if nonlinearities exist, the linear model will tend to err conservatively. The potential problems are twofold. On the one hand, saturation of metabolism could lead to situations in which a linear extrapolation was not conservative, particularly if a reactive metabolic intermediate was the active moiety (Ramsey and Reitz, 19811. On the other hand, in cases in which an effective threshold for toxicity appears to exist, the use of an overly conservative estimate could result in an unnecessarily restrictive regulatory decision. Species Lack of controlled human exposure data on most toxic chemicals makes it necessary to infer human susceptibility from animal results. The dose administered to the test species is generally converted to an equivalent human dose on the basis of either body weight (in milligrams per kilogram) or surface area, where surface area is taken to be proportional to body weight raised to the 2/3 power. The latter scaling factor is generally justified on the basis of the studies by Freireich et al. ( 1966), who examined the interspecies differences in toxicity of a variety of antineoplastic drugs. This "surface area adjustment" is often appropriate for a toxic chemical detoxified by metabolism. It has come to be applied routinely, however, even in the case of chemicals with toxic metabolites for which such a surface area dependence would not be expected (M. E. Andersen, this volume). For a chemical that demonstrates significant interspecies vari- ation in toxicity in animal experiments, the most susceptible species is generally used as the reference for this extrapolation. Because of the recognized uncertainty involved, a safety factor of 10 to 1,000 or more may often be applied. Route In some cases, there are no animal data corresponding to the expected human exposure route. For example, in developing a surface water stan-

Physiologically Based Models for Extrapolation 161 card for a compound that has only been studied via inhalation, the rela- tionship between the inhalation and oral routes of administration must be estimated. The surface water concentration is then calculated from the acceptable oral dose by using some assumed level of human consumption. Methods for relating inhalation and oral doses for systemic toxicants usu- ally assume some sort of direct correspondence based on total administered dose, calculated uptake, or achieved blood levels (EPA, 19841. Again, a safety factor is often applied, reflecting the increased uncertainty. Exposure Scenario There are many other ways in which animal studies may differ from expected human exposure scenarios, chiefly relating to the frequency and duration of exposure. Examples include estimating lifetime carcinogenic risk from studies of less than lifetime duration, correlating 50% fetal doses (LDsoS) for acute toxicity to no-observed-adverse-effect levels for chronic exposure, and adjusting workplace exposure limits to reflect changes in work shifts. The chosen relationships may be statistical, semiempirical, or just plain intuitive. Pharmacokinetically Based Extrapolations In contrast to these rule-of-thumb approaches, the fundamental as- sumption in conducting pharmacokinetic extrapolations is that a particular effective tissue dose in one species or by one route of administration is equally effective in another species or if obtained by a different route of administration. For all of these questions of how to extrapolate from animal experiments to estimate human risk, we must know how to calculate the effective dose of a chemical reaching appropriate target tissues under any exposure condition. The basic issue, then, is the relationship between the administered dose and some delivered or effective dose. In the past it has been tacitly assumed that the effective dose was the same as the admin- istered dose. It is now clear that the relationship between these two expres- sions of dose is complex (Andersen, 19811. There are many factors involved in determining this relationship, and they are functions of both the chemical and the organism, leading to a complexity which defies any general de- scription. For example, saturation of metabolism can lead to apparently nonlinear dose-response behavior. For some chemicals, this relationship can be further complicated by induction or inhibition of the relevant en- zyme systems as well as by depletion of necessary cofactors. Metabolic first-pass effects can lead to variations in bioavailability both within and between routes. The direction of the effect of these factors can be to either increase or decrease relative toxicity, depending on whether the toxicity results from the parent chemical, a stable metabolite, or a transient in-

162 HARVEY J. CLEWELL Ill AND MELVIN E. ANDERSEN termediate. What is needed is a framework with which to describe the actions of all these important factors in a quantitative fashion. The de- velopment of such a framework is one of the purposes of physiologically based pharmacokinetic modeling. PHYSIOLOGICALLY BASED PHARMACOKINETIC MODELING Pharmacokinetics is the study of the time course for the absorption, distribution, metabolism, and elimination of a chemical substance in a biological system. Implicit in any pharmacokinetic description is the as- sumption that the response of some target tissue can be related to the concentration profile of the active form of the substance in that tissue. Pharmacokinetic models generally are divided into two categories: com- partmental and physiological. A typical compartmental model attempts to relate the blood or tissue concentration profile of the parent or the me- tabolite to the administered dose of the parent chemical by using a set of mathematical equations. The parameters for these equations are deter- mined from experiments following the time course of the chemical in body fluids and occasionally in specific tissues. A simple model might consist of just two compartments: a central compartment in equilibrium with the blood, and a peripheral compartment whose concentration can be related to the central compartment by rate constants in each direction. The volumes of the compartments and the values of the rate constants are adjusted to fit the experimental data, after which the model can be used for inter- polation and limited extrapolation. Physiologically based pharmacokinetic models differ from the conven- tional compartmental models in that they are based to a large extent on the actual physiology of the organism (Figure 11. Instead of compartments defined by the experimental data itself, actual organ and tissue groups are used with weights and blood flows from the literature (Bischoff and Brown, 1966; Himmelstein and Lutz, 1979~. Instead of composite rate constants determined by fitting the data, actual physical-chemical and biochemical constants of the compound are used. The result is a model that predicts the qualitative behavior of the experimental time course without being based on it. Refinement of the model to incorporate additional insights gained from comparison with experimental data yields a model that can be used for quantitative extrapolation well beyond the range of experi- mental conditions. The chief advantage of a physiologically based model is its greater predictive power. Because fundamental metabolic parameters are used, dose extrapolation over ranges in which saturation of metabolism occurs is possible. Because known physiological parameters are used, a different species can be modeled by simply replacing the appropriate constants

Physiologically Based Models for Extrapolation 163 QALV _ C INH QT CVF I Alveolar Space Lung C VEN Blood Fat Tissue Group Muscle Tissue Group CVM Richly Perfused CVR Tissue Group QALV CALV QT C ART QF J C ART ~ QM J CART QR J C ART Metabolizing- ~ Q L Tissue ~ C ART Grolip _J CVL - ~ Mmabolitm KM FIGURE 1 Diagram of a generic physiologically based pharmacokinetic model for volatile organic chemicals. All of the models described in this paper are adaptations of this simple model, which was used by Ramsey and Andersen (1984) to investigate the pharmacokinetics of styrene. In this description, groups of tissues are defined with respect to their volumes, blood flows (Q), and partition coefficients for the chemical. The uptake of vapor is determined by the alveolar ventilation (Qatar), cardiac output (Q.), blood: air partition coefficient, and the concentration gradient between arterial and venous pulmonary blood (Car, and Oven) Metabolism is described in the liver, with a saturable pathway defined by a maximum velocity (Vmax) and affinity (Km) and, when necessary, with a first-order pathway (data not shown). The mathematical description assumes equilibration between arterial blood and alveolar air, as well as between each of the tissues and the venous blood exiting from that tissue. Reproduced with permission from Ramsey and Andersen (1984).

164 HARVEY J. CLEWELL Ill AND MELVIN E. ANDERSEN (Dedrick, 19731. Similarly, the behavior for a different route of admin- istration can be determined by adding equations which describe the nature of the new input function. The extrapolation from one exposure scenario (say, a single 6-h exposure) to another (e.g., a repetitive 6-h exposure, 5 days a week for the life of the animal) is relatively easy and only requires a little ingenuity in writing the equations for the dosing regimen in the kinetic model (D. J. Paustenbach, H. J. Clewell, M. L. Gargas, and M. E. Andersen, this volume). Because measured physical-chemical and biochemical parameters are used, the behavior for a different chemical can quickly be estimated by determining the appropriate constants. An important result is the ability to reduce the need for extensive range-finding experiments with new chemicals. The process of selecting the most informative experimental design is also facilitated by the availability of a predictive pharmacokinetic model. Perhaps the most desirable feature of a physiologically based model is that it provides a conceptual framework for employing the scientific method in which hypotheses can be described in terms of biological pro- cesses, predictions can be made on the basis of the description, and the hypotheses can be revised on the basis of comparison with experimental data. The trade-off against the greater predictive capability of physiologically based models is an increased number of parameters and equations. Values for many of the parameters, however, particularly the physiological ones, are already available in the literature; and several techniques have been developed for rapidly determining the compound-specific parameters. There is even a prospect that predictive models can be developed based entirely on data obtained from in vitro studies. For volatile liquids, the type of chemicals with which we are most familiar, tissue partition coefficients can be determined by a simple in vitro technique called vial equilibration (Sato and Nakajima, 1979b), and tissue metabolic constants can be de- termined by a modification of the same technique (Sato and Nakajima, 1979a). Alternatively, other rapid in vivo approaches for determining metabolic constants can be used either based on steady-state (Andersen et al., 1984b) or gas uptake (Andersen et al., 1980; Filser and Bolt, 1979; Gargas et al., 1 986a,b) experiments. The resulting set of constants together with the general physiological parameters provide a model of parent chem- ical behavior and rate of metabolism which can be predictive of kinetic behavior at various concentrations, for various dose routes, in a variety of species, with any number of exposure scenarios. In essence, the same approach can be used with nonvolatile xenobiotics; but at present exper- iments to determine solubility, tissue binding, and metabolic constants are not as easily conducted with these chemicals as they are with the gases and volatile liquids. Nonetheless, there are now several very good ex-

Physiologically Based Models for Extrapolation 165 amples of physiologically based models which describe the kinetics of important nonvolatile environmental contaminants, including kepone, po- lybrominated biphenyls, and polychlor~nated dibenzofurans (Bungay et al., 1981; King et al., 1983; Tuey and Matthews, 1980~. DOSE-ROUTE EXTRAPOLATION We have described a generic physiologically based pharmacokinetic model for volatile organic chemicals (Clewell and Andersen, 1986~. This generic model predicts blood and tissue concentrations of parent chemical and the rate of metabolism in the liver. Nonlinear kinetic behavior, the high-dose to low-dose extrapolation problem, is accounted for by including terms for two metabolic pathways in the liver tissue, one of which is saturable and the other of which is strictly first order. Dose-route extrapolations can be conducted by using the inhalation description and adding appropriate equations representative of other routes of administration. Intravenous (i.v.) injection is easily described by a constant rate of infusion into mixed venous blood. Oral administration in a water vehicle can be modeled by first-order uptake from a bolus gas- trointestinal dose, with the incoming chemical presumed to appear in the liver. This is done because portal blood flow goes to the liver before it is recirculated in the systemic circulation. Predictions of the inhalation, i.v., and oral kinetics of dibromomethane (Figure 2) from the model agree very well with data collected in our laboratory (Clewell and Andersen, 1986~. In this case, oral kinetics were not independently predicted. The inhalation model with first-order input from the gut was manipulated by adjusting the uptake rate constant until the best description of the oral uptake data was obtained. It is worth noting that the similar data for oral administration in an oil vehicle could not be successfully simulated by assuming either first-order or zero-order input (see also Ramsey and An- dersen, 19841. Additional work is needed to understand the effect of administering a chemical dissolved in oil on its uptake kinetics. This simple, generic description provides a prototypical predictive model for a wide variety of very important gases and vapors materials such as trichloroethylene, perchloroethylene, benzene, chloroform, methylene chloride all of which have been identified as water-borne environmental contaminants in various groundwater samples. We have now collected the partition coefficients and metabolic constants for all these chemicals and used these constants to construct simple, four-compartment, physiologi- cally based models. In our laboratory, it now takes about 2 weeks to develop the metabolic and solubility constants needed to develop a phys- iological model for these volatile chemicals. Of course, for some chemicals metabolism may not be adequately represented by the two pathways de-

|66 HARVEY J. CLEWELL Ill AND MELVIN E. ANDERSEN A DIHALOMETHANE MODEL ~ INHALATION MODEL ~ ~ = O2 |' GSH , ~ ~ _, ~co) r . 1 l [l BbC OD l l l HBCO l l EXHALATION CO INPUT. 80.0- 60.0- c' 40.0 (BROMIDE) ., EXTRACELWLAR WATER 20.0~ | URINARY l | EXCRETION I 80.0- C 24.0_ 60.0- 1'` 18.0- `, 40.0- \ 50mg/kg-oral ot2.0- 20.0- \~` 6.00- ~ 1 Omg/kg . 0.00- . 0. 0 0.75 1.50 2.25 3.00 T B hi 65.5mg/kg-iv x\ a\ so 13.1 x: x \ x ~ rev o.oo O.oo 0.75 1.50 T . , . 2.25 3.00 D /~0 pmm \ ~_~ . A / xt 200 ppm 0.00 it_ 0.00 2.00 4.00 6.00 8.00 FIGURE 2 Dose-route extrapolation. (A) For the dihalomethanes, metabolism precedes by two pathways. A saturable oxidative pathway yields carbon monoxide (CO) and halide ion (bromide in the case of dibromomethane) and a first-order glutathione (GSH) conjugation pathway produces halide but not CO. By varying the input function in a physiologically based pharmacokinetic model that includes both of these metabolic pathways, the time course of Dibromomethane can be predicted for a variety of exposure routes. (B) Dibromomethane concentration (in milligrams per liter) in mixed venous blood of Fischer 344 rats given intravenous injections of 65.5 and 13.1 mg/kg. Solid lines are the predictions of the model, and points represent the average of two (top curve) or three (bottom curve) animals. (C) Dibromomethane concentration (in milli- grams per liter) in mixed venous blood of Fischer 344 rats given oral doses of 50 and 10 ma/ kg administered as a saline solution. Solid lines are the predictions of the model assuming a first-order uptake from the stomach (10/h), and points represent the average of four (top curve) or two (bottom curve) animals. (D) Dibromomethane concentration (in milligrams per liter) in mixed venous blood of Fischer 344 rats during and after 4-h inhalation exposures at 200 ppm and 100 ppm. Solid lines are the predictions of the model, and points represent the average of three animals. Reproduced with permission from Clewell and Andersen (1986). scribed above. In this case additional experiments must be performed to characterize the metabolic conversion of the particular chemical under study. This basic model has several tissue groups which are lumped together according to their perfusion and solubility characteristics. Each of these

Physiologically Based Models for Extrapolation ~ 67 several compartments is described by a single mass-balance differential equation. It would be possible to describe individual tissues in each of the lumped compartments. For instance, fat could be broken down to perirenal, epidydimal, brown fat, etc. This detail is usually unnecessary unless some particular tissue in a lumped compartment is the target tissue. One might want to separate brain from other well-perfused tissues if the model were for a chemical that had a toxic effect on the central nervous system. More biochemical and physiological detail could then be incor- porated for the target tissue. With brain there might be reason to explicitly define a blood:brain barrier or to include terms which describe saturable binding of the chemical to specific receptors in the tissue. Increasing the number of subcompartments does increase the number of differential equa- tions required to define the model. However, within reason, the number of equations does not pose any problem. It is relatively straightforward to solve the small groups of equations that describe most physiological models of xenobiotic disposition, requiring only a personal microcomputer and readily accessible software. By adding another compartment, the basic inhalation model can easily be extended to predict the kinetics for uptake of the chemical through the skin. In this case a diffusion-limited compartment is used to represent the skin as a portal of entry. With this minimal change the model has been used to describe the absorption of chemical vapors through the skin of rats fitted with masks to prevent inhalation exposure (McDougal et al., 19861. The dermal absorption of dibromomethane vapors (Figure 3) was very accurately predicted by this model with a skin permeation coefficient of 1.32 cm/in. EXPOSURE SCENARIO EXTRAPOLATION The basic inhalation model can also be enlarged to focus on kinetics of metabolites or on the amount of chemical metabolized in a given exposure scenario. For example, our more complete model of the brom- inated dihalomethanes (Andersen et al., 1984a; Gargas et al., 1986b) not only tracks the parent chemical (e.g., dibromomethane or bromochloro- methane) but also two metabolites: carbon monoxide (CO) and bromide ion (Br). Two metabolic pathways are still described: a saturable oxidative pathway that produces carbon dioxide (CO2), CO, and Br and a first-order conjugative pathway which produces only CO2 and Br (Figure 21. The metabolite model includes a fairly complete description of the fate of the CO produced and is able to predict the fraction of hemoglobin tied up as carboxyhemoglobin at any time, as determined by the current rate of CO production (and inhalation of CO, if appropriate), the competition of oxygen and CO for hemoglobin, and the rate of exhalation of unbound

|68 HARVEY ]. CLEWELL Ill AND MELVIN E. ANDERSEN lo3 lo2 - 101 - 10-' . 0.0 1.0 l Wl 1 O,OOO ppm 59OOO "m 1,000 ppm : ~ ~ ]500 ppm l , . 2.0 3.0 4.0 TIME (HOURS) FIGURE 3 Dose-route extrapolation (continued). Dibromomethane concentration (in milligrams per liter) in mixed venous blood of Fischer 344 rats whose skin only was exposed to dibro- momethane vapor at 10,000, 5,000, 1,000, and 500 ppm. Rats were protected against inhalation of the vapors by masks. Solid lines show the predictions of the model using a skin permeation coefficient of 1.32 cm h, and symbols represent the mean and standard deviation for five or six animals. To describe uptake of vapors through the skin, a diffusion-limited skin compartment was added to the basic dihalomethane model. Reproduced with permission from McDougal et al. (1986). CO. This model has also been successfully applied to a variety of exposure routes: inhalation, intravenous, oral, and dermal. As a challenge of the ability of this model to accurately predict kinetics for very different ex- posure scenarios, we performed short-duration, high-concentration ex- posures of rats to methylene chloride and bromochloromethane (Andersen et al., 1984a). The model correctly predicted the appearance of an in- creased fraction of carboxyhemoglobin in the blood which was maintained for several hours after the exposure was terminated (Figure 41. SPECIES EXTRAPOLATION To demonstrate interspecies extrapolation with the dihalomethane model, predictions of the model were compared with data available from studies

Physiologically Based Models for Extrapolation 169 20 z m o ~J 1 o S 12 X o m ce at 8 o o ~` O ,,6 To "` n . o CH2 CL2 —O—CH2BRCL \ o ~ O \O to 4 O O o ~ ~ ~ _ TIME (HRS) FIGURE 4 Exposure scenario extrapolation. Percent carboxyhemoglobin in the blood of Fischer 344 rats following a very short (0.5 h), high-concentration (5,000 ppm) exposure to bromoch- loromethane (CH2BrCl) or methylene chloride (CH2C12). The shaded area indicates the exposure period, lines are the predictions of the model, and points represent the average of two (CH2BrCl) or three (CHC12) animals. To predict the time course of blood carboxyhemoglobin, the basic dihalomethane model was expanded to include a description of the fate of the CO produced by the oxidative pathway, including competition with oxygen for binding to hemoglobin and ex- halation of unbound CO. The longer postexposure metabolism of bromochloromethane as com- pared with methylene chloride reflects its greater tissue solubility. Adapted from Andersen et al. (1984a). in which human volunteers were exposed to methylene chloride (Andersen et al., 1986b). For the purpose of this extrapolation, tissue solubilities in humans were made the same as in the rat, tissue volumes were considered proportional to body weight, and flows were considered proportional to body weight raised to the 0.7 power (see Adolph, 19491. The human blood:air partition coefficient of 9.7 was determined in our laboratory (Andersen et al., 1986b). Metabolism of methylene chloride to carbon monoxide was scaled by assuming that affinity (Km) did not change from species to species, while Vmax in humans could be estimated from human exposure data and was found to be about 119 math (Andersen et al., 1986b). Usually, it is more difficult to determine how to scale Vma,; and Km because there is less reason to believe that biochemical constants for xenobiotic metabolism vary coherently from species to species (Dedrick and Bischoff, 19801. This is especially true for molecules with structures

]70 HARVEY J. CLEWELL Ill AND MELVIN E. ANDERSEN that are much more complex than that of methylene chloride. It may be possible to estimate metabolic parameters on small biopsied human tissue samples or to evaluate them in many different mammalian species to determine if there is extensive interspecies variation. The approach utilized with methylene chloride provided a very good prediction of human phar- macokinetics (Figure 5), both for the parent chemical and for exhaled carbon monoxide. DOSE EXTRAPOLATION The versatility of the physiological approach for examining the kinetics of volatile compounds can also be appreciated by considering another experimental design for conducting an inhalation exposure. In the usual exposure design, animals are exposed to a constant concentration of chem- ical for a specified period of time. A less frequently used exposure design is to expose animals in a small chamber into which is injected a known amount of test chemical. The chamber concentration in this experiment declines from the initial value as the chemical is absorbed into the tissues of the animals, and there is further loss from the chamber if the chemical is metabolized. These types of experiments are concerned with the uptake of chemical into the rats from the chamber atmosphere. In view of this, the technique has been referred to as gas uptake. From a mathematical point of view, this differs from the constant concentration exposure model by the addition of one more mass-balance equation describing the chamber atmosphere itself. While this exercise looks like a form of dose-route extrapolation in which the input function is slightly altered, this exposure system actually provides a very convenient method to estimate the kinetic constants for metabolism of a variety of volatile organic chemicals (Gargas et al., 1 986a). In addition, because metabolism of many of these chemicals is readily saturated, this procedure also enables us to illustrate the ex- trapolation from high-concentration behavior in which metabolism is sat- urated to lower concentrations and in which metabolism is first order with respect to the chemical substrate. Gargas et al. (1986b) described the closed-chamber kinetics of meth- ylene chloride (Figure 61. There is a marked concentration dependence on the observed rate of loss of this chemical from the chamber. At high concentrations, oxidative metabolism is saturated, and uptake is primarily determined by the solubilities of the test chemical in the blood and tissues of the rats and by the rate of the first-order metabolic pathway. At the intermediate concentrations, at which there is extensive curvature, we observe the transition of the oxidative pathway from saturation to first- order behavior; and the shape of the curve is critically dependent on the capacity of the oxidative pathway. Finally, at the lowest initial chamber

Physiologically Based Models for Extrapolation 17 1 10 - At c\! O ='~E O Z 0.1 O ~ O O m ~ 1 0.01 40 C] o Z ^ O ~ -: Q c Q Be _ O A 24 m O D: a: z ~ lo A: I Z X O 32 16 8 o - - - - - 1 1 1 1 1 10 15 20 25 30 0 5 1 TIME (hours) \ \ - - 1 1 1 1 1 1 0 5 10 15 20 25 30 TIME (hours) FIGURE 5 Species extrapolation. Methylene chloride concentration (in milligrams per liter) in mixed venous blood (A) and carbon monoxide concentration (in parts per million) in exhaled air (B) from human volunteers during and after a 6-h inhalation exposure to 350 ppm methylene chloride. Solid lines are the predictions of the dihalomethane model, and points represent the average of four individuals (Andersen et al., 1986b). The model was scaled to humans on the basis of both direct human data and established allometric relationships, as described in the text.

i72 HARVEY J. CLEWELL Il! AND MELVIN E. ANDERSEN 1 0000 1 000 100 10 _~ ~ . ~ ma*_ O ...... ah..... - _.... _. ~ . _- 0 2 4 TIME (HRS) FIGURE 6 Dose extrapolation. Concentration (in parts per million) of methylene chloride in a closed, recirculated chamber containing three Fischer 344 rats. Initial chamber concentrations were (from top to bottom) 3,000, 1,000, 500, and 100 ppm. Solid lines show the predictions of the model for a Vmar of 4.0 mg/kglh, a Km of 0.3 mg/liter, and a first-order rate constant of 2.0/kg/h, while symbols represent the measured chamber atmosphere concentrations. The closed- chamber experiment was described mathematically by adding a compartment to the basic model that followed the chamber atmosphere. The initial decrease in chamber concentration results from the uptake of chemical into the animal tissues. Subsequent uptake is a function of the metabolic clearance in the animals, and the complex behavior reflects the transition from partially saturated metabolism at higher concentrations to linearity in the low-concentration regime. Re- produced with permission from Gargas et al. (1986b). concentration, the entire system is linear and the uptake curves are pre- dominantly determined by the animal's ventilation rate and the binding affinity of the metabolizing enzyme for the substrate. These curves can be quantitatively analyzed with a physiological model to estimate the values of the kinetic constants for metabolism. The tissue partition coef- ficients are determined experimentally, and physiological parameters are estimated based on a combination of literature values and laboratory ex- perience derived from the modeling process. The physiological model is then exercised for various values of the kinetic constants until a single choice of constants provides good agreement with the entire set of uptake data. With methylene chloride the uptake curves were best described by contributions from both a saturable and a strictly first-order component

Physiologically Based Models for Extrapolation 173 for metabolism. The kinetic constants for the capacity-limited oxidative pathway were 4.0 mg/kg/h and 0.3 mg/liter for V,~,` and Km' respectively. The first-order constant had a value of 2/kg/in (Gargas et al., 1986b). In our laboratory in Dayton, Ohio, we have now estimated kinetic constants for about 30 chemicals by means of these gas-uptake techniques (Gargas et al., 1986a). The technique is very rapid and straightforward for chem- icals with appropriate tissue partition coefficients and vapor pressure char- acteristics. SUICIDE ENZYME INHIBITION Saturable metabolism is by no means the only mechanism by which nonlinearities arise in the kinetics of disposition of various volatile chem- icals. In fact, we have studied two other capacity-limited metabolic sys- tems with these same gas-uptake methods. The first is suicide enzyme inhibition caused by metabolism of cis- and trans-1,2-dichloroethylene. Both of these materials are initially metabolized by microsomal oxidation, but reactive metabolites produced during their metabolism react with and destroy the active site of the metabolizing enzyme (Andersen et al., 1986c). This behavior was only uncovered because the uptake curves were ana- lyzed with a physiological pharmacokinetic model instead of a compart- mental model. These chloroethylenes are metabolized by a single, high- affinity, saturable pathway. With the appropriate values of tissue partition coefficients and physiological parameters and assuming a single saturable pathway, it was impossible to generate a good fit to the experimental data. When an attempt was made to fit the high-concentration data preferentially (Figure 7), the actual data points fell off more rapidly at the beginning of the experiment, and then the decline of chemical in the chamber slowed down more than could be accounted for by the standard physiological model. This indicated that the rate constant of metabolism was decreasing with time. At the same time, the model's consistent underprediction of metabolic clearance for the two lower concentration experiments indicated that this time-dependent decrease was less severe for lower concentrations of chemical. Together these observations suggested that enzyme destruc- tion was occurring. Other experiments with mixed atmospheres in the chamber confirmed the loss of chloroethylene-metabolizing capacity (An- dersen et al., 1986c). The use of the physiological model allowed us to investigate the nature of the inhibition and the relative rates at which enzyme inactivation pro- ceeded. We tested four possibile mechanisms for the suicide inactivation, and only one was able to reproduce the time course behavior observed by gas-uptake analysis. The successful description assumed that reactive me-

]74 HARVEY J. CLEWELL Ill AND MELVIN E. ANDERSEN 1o1 10° PPM - -1 10 — - —2 0.00 1.20 Volt* ~ _ o ~ °~ ~ o ~ X _ _ o o o o ~ o o 2.40 3.60 TIME (HRS) - - - - 4.80 6.00 FIGURE 7 Enzyme inhibition. Concentration (in parts per million) of trans-1,2-dichloroethy- lene (trans-DCE) in a closed, recirculated chamber containing three Fischer 344 rats (Andersen et al., 1986c). Initial chamber concentrations were (top to bottom) 12, 8, and 5 ppm. Symbols represent the measured chamber atmosphere concentrations, and the solid lines are the best result that could be obtained from an attempt to fit all of the data with a single set of metabolic constants by using the closed-chamber model described in the legend to Figure 6. As described in the text, the systematic discrepancy between the model and the data provided an indication that the simple description of metabolism in the model was inadequate for this chemical. tabol~tes produced by 1,2-dichloroethylene metabolism reacted with the enzyme-substrate complex to inactivate metabolizing enzyme. In addition, it was also necessary to include enzyme resynthesis in the model to obtain an accurate representation of the experimental data (Figure 8~. These interactions were included simply by enlarging the mass-balance equation for the liver to include the biochemical events involved with enzyme inactivation. For these chemicals that are very efficient suicide inhibitors, one of the important kinetic constants for risk assessment is the resynthesis rate for new enzyme since resynthesis becomes the limiting step for me- tabolism at high concentrations of inhibitor. In the case of these studies with trans-1,2-dichloroethylene, the estimated zero-order resynthesis rate in male Fischer 344 rats was 2.5% of the steady-state activity per hour. These experiments also showed the bans isomer to be a much more active suicide inhibitor than the cis isomer of dichloroethylene.

Physiologically Based Models for Extrapolation 175 1ol 10° - PPM -1 10 - Ah_ J -—~- _. Am_ - - - 1o-2 l l l o.oo 1.50 3~00 4.50 6.00 TIME (HRS) FIGURE 8 Enzyme inhibition (continued). Symbols represent the same experimental data as described in the legend to Figure 7. In this case, however, the lines show the predictions of a closed-chamber model in which the mathematical description included inactivation of the me- tabolizing enzyme by reactive metabolites assumed to be produced during the metabolism of trans-DCE. Four alternative mechanisms for the suicide inactivation were actually tested, and only one was able to coherently describe the entire data set. In the successful description, the rate of enzyme inactivation was proportional to a second-order rate constant (k8) times the square of the instantaneous rate of metabolism, representing the reaction of free metabolite with the enzyme-substrate complex. The model also included a zero-order rate of enzyme resynthesis (k5) during the exposure. The curves shown were obtained with a Vm`~` of 3.0 mg/kglh, a Km of 0.1 mg/liter, a ka of 400, and a k5 of 0.025/h. The suggestion of enzyme inactivation was also demonstrated experimentally by the inhibition of trichloroethylene metabolism after preexposure to 10-20 ppm trans-DCE (Andersen et al., 1986c). GLUTATH ION E DEPLETION Another example of capacity-limited metabolism was observed during studies of the gas-uptake behavior of several chemicals that are known to produce depletion of hepatic glutathione. The conjugation pathway for the reaction of methylene chloride and glutathione regenerates glutathione, but with other volatile chemicals, such as 1,2-dichloroethane and allyl chloride, the conjugation reaction consumes glutathione. In gas-uptake

176 HARVEY3. CLEWELL Ill AND MELVIN E. ANDERSEN experiments with allyl chloride, both the oxidative and the conjugative (glutathione) pathways appear to be dose dependent (Figure 91. The data points were obtained in the uptake studies, and the smooth curves in Figure 9 are the best-fit curves, assuming a saturable pathway and a first-order pathway with a rate constant that is independent of concentration. With this model there were systematic errors about the predicted curves. The prediction at high concentrations was lower than the data points, and at intermediate concentrations the prediction was uniformly higher than the 1 0000 - 1 000 _ _ _ AL 100= ~ _ _ _ 0.00 1.00 2.00 art. ~ a*** ****C_ Be,,< _ ~ Be* -*~ 3.00 4.00 5.00 6.00 TIME ( HRS) FIGURE 9 Cofactor depletion. Concentration (in parts per million) of allyl chloride in a closed, recirculated chamber containing three Fischer 344 rats (Andersen et al., 1986a). Initial chamber concentrations were (top to bottom) 5,000, 2,000, l,O00, and 500 ppm. Symbols represent the measured chamber atmosphere concentrations. The curves represent the best result that could be obtained from an attempt to fit all of the data with the same set of metabolic constants by using the closed-chamber model described in the legend to Figure 6. In this case, the apparent dose- dependent nature of the discrepancy between the model and data suggested the presence of a second capacity limitation on metabolism not included in the original description. Because this ;~;~ti~n `~c rmncictent with other experimental evidence that the metabolism of allot chloride ilt(.ll~tl~n was ~ivn~l5~nt wlin {fitly ~_4A&~ TV-- I_ _________ consumes glutathione, the mathematical model of the closed-chamber experiment was expanded to include a more complete description of the glutathione-dependent pathway.

Physiologically Based Models for Extrapolation 177 data. A much better fit could be obtained by setting the first-order rate constant to a lower value at the higher concentration. This approach pro- vided a better correspondence between data and the smooth curves from the model, but did not provide biological information about the mecha- nism~s) by which the rate of this second pathway diminishes with con- centration. To generate a model for examining the biological basis of the kinetic behavior, it was necessary to model the time dependence of hepatic glu- tathione. The basic model used for this description (Andersen et al., 1 986a) had a zero-order production of glutathione and a first-order consumption rate that was increased by reaction of the glutathione with allyl chloride. In the final model, glutathione resynthesis was regulated by controlling the concentration of the rate-limiting enzyme for glutathione biosynthesis. The production of this enzyme was inversely related to the instantaneous glutathione concentration. This description, coupling the loss of allyl chlo- ride from the chamber and depletion of the glutathione concentration in the liver, provided a much improved correspondence between the data and the predicted behavior (Figure 101. Of course, the improvement in fit was obtained at the expense of adding several new glutathione-related constants. While this does add more freedom to the model for fitting the uptake data, it also suggests that we can generalize the behavior and predict both allyl chloride and hepatic glutathione concentrations during constant concentration inhalation exposures. Model output for expected end-ex- posure hepatic glutathione concentrations compared very favorably with actual data that was obtained by J. Waechter at Dow Chemical Co., Midland, Mich. (Table 1~. Once again, the experiments with allyl chloride can be considered as they relate to estimations of risk. At high concen- trations, the ability of a tissue to produce the glutathione conjugate be- comes a function of the maximum resynthesis rate of the glutathione cofactor in that particular tissue. If the glutathione conjugate is the toxic moiety, tissue toxicity may well be dependent on the tissue capacity for glutathione resynthesis at high substrate concentrations. CONCLUSION The mathematical structure of physiological pharmacokinetic models is somewhat more complex than that of simpler one-, two-, or three-com- partment models that have closed-form solutions. The solution of these physiological models requires numerical integration of a series of nonlin- ear, simultaneous differential equations. One advantage of these more computationally demanding models is that they can be explicitly designed to allow for the processes of extrapolation that are so necessary for rational risk assessements. These extrapolations are high dose to low dose, dose

]78 HARVEY J. CLEWELL Ill AND MELVIN E. ANDERSEN 10000 = 1 000 - 100 = 10 1 -**? At\ o.oo 1.20 2.40 3.60 4.80 6.00 TIME (HRS) FIGURE 10 Cofactor depletion (continued). Symbols represent the same experimental data as described in the legend to Figure 9. The curves show the predictions of the expanded model (Andersen et al., 1986a), which not only included depletion of glutathione by reaction with allyl chloride but also provided for the regulation of glutathione biosynthesis on the basis of the instantaneous glutathione concentration, as described in the text. route, interspecies, and dose rate. A second advantage is that these models can easily be expanded to include more detailed information on the chem- istry and biochemistry of the test chemical and the test animal. This progressive expansion of a simple model to include more detail was seen with both the suicide inhibitors and glutathione depletion. In both cases, the crucial role of the model in the conduct of the scientific method was apparent. The mathematical model gives quantitative form to the inves- tigator's conception of the biological system, allowing him or her to develop testable, quantitative hypotheses, to design informative experi- ments, and to recognize inconsistencies between theory (model) and data. As the models become more complex, they necessarily contain larger

Physiologically Based Models for Extrapolation 179 TABLE 1 Predicted Glutathione Depletion Caused by Inhalation Exposures to Allyl Chloride Depletion (~M) Concentration (ppm) Observed Predicted 0 7,080 + 120 7'088a 10 7,290 + 130 6,998 0 7,230 + 80 7,238a 100 5,660 + 90 5,939 0 7,340 + 180 7,341a 1,000 970 + 10 839 0 6,890 + 170 6,890a 2 000 464 + 60 399 Note: Glutathione depletion data were graciously supplied by John Waechter, Dow Chemical Co., Midland, Mich. aFor the purpose of this comparison, the basal glutathione consumption rate in the model was adjusted to obtain rough agreement with the controls in each experiment. This basal consumption rate was then used to simulate the associated exposure. numbers of physiological, biochemical, and biological constants. The task with model development is to keep the description as simple as possible and to keep an eye out to ensure identifiability of new parameters that are added to the models. Every attempt should be made to obtain or verify model constants from experimental studies separate from the modeling exercises themselves. In the final analysis, complex models may be the price we have to bear for living in a complex world. This complexity can be handled by good experimental design; competent, critical accumulation of necessary model input; and honest attempts to develop generalizable descriptions that support extrapolation to exposure conditions relevant to exposed human populations. Toxicological studies are the cornerstone of any risk analysis and pro- vide dose-response curves on which risk analyses must be based. In con- trast, pharmacokinetic models are interpretive tools to be used in conjunction with toxicity and mechanistic studies. The use of predictive physiological kinetic models in risk assessment is predicated on a very simple premise: An effective dose in one species or under a particular condition is expected to be equally effective in another species or under some altered exposure condition. While this is obviously not true for some chemicals, like dioxin with its great species specificity (Kociba and Schwetz, 1982), the premise does appear valid for many chemicals and especially for those solvent chemicals whose toxicity is due to formation of reactive metabolites. Physiological models can be used to redefine the dose-effect relationship

18O HARVEY ]. CLEWELL Ill AND MELVIN E. ANDERSEN based on effective dose. It is important to remember that kinetic studies alone cannot determine which parameters one should regard as the ap- propriate measure of effective dose. Mechanistic studies and biological plausibility are required to develop the argument that one measure of dose is more correlated with toxicity than another. The kinetic analysis can then be used to clarify the relationship between effect and the more mean- ingful measure of effective dose. Clearly, physiologically based pharrnacokinetic models will not remove all of the uncertainty from the risk assessment process. In fact, in a way they introduce new uncertainties: the adequacy of the model, the accuracy of the parameters in the model, and the appropriateness of the chosen measure of effective dose. The rationale for using physiologically based pharmacokinetic models in risk assessment is that they provide a docu- mentable, scientifically defensible means of bridging part of the gap be- tween animal bioassays and human risk estimates. In particular, they move the risk assessment from the administered dose to a dose more closely associated with the toxic effect by explicitly describing their relationships as a function of dose, species, route, and exposure scenario. The next step, from the dose at the target tissue to the actual toxic event, is the subject of pharmacodynamic modeling; and the nature of this relationship is an area of considerable uncertainty. Nevertheless, risk estimates must continue to be made as the need arises, on the basis of what is known at that time, and in the most scientifically defensible manner available. Every effort must be made to apply scientific principles throughout the risk assessment process; to document the assumptions, the decisions, and the uncertainties at each step; and to provide this information to the risk manager in a form which allows him or her to weigh the predicted risks along with the uncertainties in the assessment to arrive at a final decision concerning acceptable exposure levels. Substituting conservatism for sci- ence throughout the risk assessment process severely restricts the utility of the results. Many important risk management decisions such as pr~or- itizing hazardous waste sites or deciding which solvent to use for an industrial process require accurate comparative risk estimates of the relevant chemicals, not just individually conservative estimates. REFER ENCES Adolph, E. F. 1949. Quantitative relations in the physiological constitutions of mammals. Science 109:579-585. Andersen, M. E. 1981. Saturable metabolism and its relationship to toxicity. CRC Crit. Rev. Toxicol. 9: 105-150. Andersen, M. E., M. L. Gargas, R. A. Jones, and L. J. Jenkins, Jr. 1980. Determination of the kinetic constants of metabolism of inhaled toxicant in vivo based on gas uptake measurements. Toxicol. Appl. Pharmacol. 54:100-116.

Physiologically Based Models for Extrapolation 181 Andersen, M. E., R. L. Archer, H. J. Clewell, and M. G. MacNaughton. 1984a. A phys- iological model of the intravenous and inhalation pharmacokinetics of three dihalome- thanes CH2Cl2, CH2rCl, and CH2Br2 in the rat. Toxicologist 4(1):111. (Abstract 443.) Andersen, M. E., M. L. Gargas, and J. C. Ramsey. 1984b. Inhalation pharmacokinetics: Evaluating systemic extraction, total in vivo metabolism and the time course of enzyme induction for inhaled styrene in rats based on arterial blood:inhaled air concentration ratios. Toxicol. Appl. Pharmacol. 73:176-187. Andersen, M. E., H. J. Clewell, M. L. Gargas, and R. B. Conolly. 1986a. A physiological pharmacokinetic model for hepatic glutathione (GSH) depletion by inhaled halogenated hydrocarbons. Toxicologist 6(1): 148. (Abstract 598.) Andersen, M. E., H. J. Clewell, M. L. Gargas, F. A. Smith, and R. H. Reitz. 1986b. Physiologically-based pharmacokinetics and the risk assessment for methylene chloride. Toxicol. Appl. Pharmacol. 87:185-205. Andersen, M. E., M. L. Gargas, and H. J. Clewell. 1986c. Suicide inactivation of mi- crosomal oxidation by cis- and trans-dichloroethylene (C-DCE and T-DCE) in male Fischer rats in vivo. Toxicologist 6(1): 12. (Abstract 47.) Bischoff, K. B., and R. G. Brown. 1966. Drug distribution in mammals. Chem. Eng. Prog. Symp. Ser. 62(66):33-45. Bungay, P. M., R. L. Dedrick, and H. B . Matthews. 1981. Enteric transport of chlordecone (Kepone) in the rat. J. Pharmacokinet. Biopharm. 9:309-341. Clewell, H. J., and M. E. Andersen. 1986. A multiple dose-route physiological pharma- cokinetic model for volatile chemicals using ACSL/PC. Pp. 95-101 in Languages for Continuous System Simulation, F. D. Cellier, ed. San Diego: Society for Computer Simulation. Dedrick, R. L. 1973. Animal scale-up. J. Pharmacokinet. Biopharm. 1:435-461. Dedrick, R. L., and K. B. Bischoff. 1980. Species similarities in pharmacokinetics. Fed. Proc. 39:54-59. EPA (Environmental Protection Agency). 1984. National primary drinking water regula- tions; volatile synthetic organic chemicals. Fed. Regist. 49:24330-24355. (40 CFR Part 141.) Filser, J. G., and H. M. Bolt. 1979. Pharmacokinetics of halogenated ethylenes in rats. Arch. Toxicol. 42:123-136. Freireich, E. J., E. A. Gehan, D. P. Rall, L. H. Schmidt, and H. E. Skipper. 1966. Quantitative comparison of toxicity of anticancer agents in mouse, rat, hamster, dog, monkey and man. Cancer Chemother. Rep. 50:219-244. Gargas, M. L., M. E. Andersen, and H. J. Clewell. 1986a. A physiologically based sim- ulation approach for determining metabolic constants from gas uptake data. Toxicol. Appl. Pharmacol. 86:341-352. Gargas, M. L., H. J. Clewell, and M. E. Andersen. 1986b. Metabolism of inhaled di- halomethanes in vivo: Differentiation of kinetic constants for two independent pathways. Toxicol. Appl. Pharmacol. 82:211-223. Himmelstein, K. J., and R. J. Lutz. 1979. A review of the application of physiologically based pharmacokinetic modeling. J. Pharmacokinet. Biopharm. 7:127-145. King, F. G., R. L. Dedrick, J. M. Collins, H. B. Matthews, and L. S. Birnbaum. 1983. Physiological model for the pharmacokinetics of 2,3,7,8-tetrachlorodibenzofuran in sev- eral species. Toxicol. Appl. Pharmacol. 67:390-400. Kociba, R. J., and B. S. Schwetz. 1982. Toxicity of 2,3,7,8-tetrachlorodibenzo-p-dioxin (TCDD). Drug Metab. Rev. 13:387-406.

|82 HARVEY J. CLEWELL Ill AND MELVIN E. ANDERSEN McDougal, J. N., G. W. Jepson, H. J. Clewell, M. G. MacNaughton, and M. E. An- dersen. 1986. A physiological pharmacokinetic model for dermal absorption of vapors in the rat. Toxicol. Appl. Pharmacol. 85:286-294. Ramsey, J. C., and M. E. Andersen. 1984. A physiological model for the inhalation pharmacokinetics of inhaled styrene monomer in rats and humans. Toxicol. Appl. Phar- macol. 73:159-175. Ramsey, J. C., and R. H. Reitz. 1981. Pharmacokinetics and threshold concepts. Pp. 239- 256 in American Chemical Society Symposium Series 160, Pesticide Chemist and Mod- ern Toxicology, S. K. Bandel, G. J. Marco, L. Goldberg, and M. L. Leng, eds. Wash- ington, D.C.: American Chemical Society. Sato, A., and T. Nakajima. 1979a. A vial equilibration method to evaluate the drug metabolizing enzyme activity for volatile hydrocarbons. Toxicol. Appl. Pharmacol. 47:41- 46. Sato, A., and T. Nakajima. 1979b. Partition coefficients of some aromatic hydrocarbons and ketones in water, blood and oil. Br. J. Ind. Med. 36:231-234. Tuey, D. B., and H. B. Matthews. 1980. Distribution and excretion of 2,2',4,4',5,5'- hexabromobiphenyl in rats and man: Pharmacokinetic model predictions. Toxicol. Appl. Pharmacol. 53:420-431.