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PART 111 Generalizations and Extrapolations

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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

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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

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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,

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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.

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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-

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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.

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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.)

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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.

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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).

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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

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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-

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]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.

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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

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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.

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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

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]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

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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

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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.

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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.

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|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.