Drinking Water and Health, Volume 8 Pharmacokinetics in Risk Assessment (1987) / Chapter Skim
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II. Mathematical Modeling
II. Mathematical Modeling
Pages 25-62
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PART 11 Mathematical Modeling
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From page 27... ...
where M is mass, C is concentration, D is the diffusion constant with dimensions distance2/time, A is the cross-sectional area of the diffusion volume, and dx is the distance over which the infinitesimally small concentration difference dC is measured. When Fick's First Law is restated for diffusion across a membrane barrier of thickness dx, the concentration gradient dC is approximated by the concentration difference across the membrane, Car —C2, and DAIRY is the first-order transfer constant kit for diffusion across the membrane, with dimensions distance3/time, or volume/time: dt MIX ~ ~ 2)
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From page 28... ...
Thus, diffusion out of a single compartment is a first-order process whose rate constant, k', depends both on the chemical and on the tissue and has the dimensions volume/time. If both sides of Equation 2 are divided by V so that it expresses the rate of change of concentration, not of mass since concentration is what is measured in viva then the rate constant becomes the elimination rate constant ke' with dimensions time(Figure 11.
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7 h, and 96 h. Porte and Havel compared these half-lives with turnover times reported for different pools of cholesterol, and concluded that the slowest, 96-h component represented metabolic turnover plus equilibration with very slowly exchanging compartments; the 7-h component represented formation of plasma ester cholesterol; the 65-min component represented exchange of free cholesterol between plasma and red blood cells; and the 30-min component represented exchange of free cholesterol between plasma and liver.
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With general dissemination of explicit mathematical solutions of multicompartment models and recognition of the implications of these solutions, in the mid- 1960s a reaction set in. If there is only one compartment, the half-life is the half-life of elimination, and dose/CO is the physiological volume of distribution.
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(5) Simultaneous integration of these two equations gives the explicit solution of the two-compartment model, in which the intercepts AD and Be and the kinetic rate constants cx and ~ are expressed in terms of the rate constants kit and kin for transfer between compartment 1 and compartment 2, the elimination rate constant ke, and the volumes of the compartments: Car = AOe-at + BOe-~t D(or—ken)
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From page 32... ...
Because physiologically based pharmacokinetic modeling has received so much recent attention, it is important to make the point that physiological pharmacokinetics and classical pharmacokinetics are not fundamentally incompatible. Although the philosophy behind the two approaches is different and dictates their application for different purposes, there is a direct link between the two approaches.
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. The form of this equation suggests that it should be possible to substitute physiological values of flow rates and volumes, and values of partition coefficients, in order to obtain predictive, physiologically based pharmacokinetic models.
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The need for interspecies conversion of laboratory animal data, in particular, has led to the development of physiological pharmacokinetic models, in which the unspecified compartments of the classical pharmacokinetic models are replaced by actual organs and tissues with their known blood flows. Because tissue volumes, blood flow rates, and enzyme activities can be varied only within physiological limits in these models, the models are not fit to experimental data in the classical sense.
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But classical pharmacokinetic models still have their place. Specifically, they are amenable to statistical treatment and, thus, to hypothesis testing, whereas the purely physiological pharmacokinetic models are not as readily treated statistically.
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One is the scientific intellectual satisfaction of having quantitative predictive models based on underlying knowledge, rather than the more empirical, curvefitting approaches often used. The latter are always needed to some extent, of course, but should be minimized if possible.
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Useful insights into the quantitative operation of the body were obtained, although specific organ levels were usually not considered. Physiological pharmacokinetics, however, also attempts to predict the various organ and tissue levels, even extra- versus intracellular concentrations.
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Mammalian species share a remarkable geometric similarity. The same blood flow diagram could be used for all mammals, and most organs and tissues are similar fractions of the body weight.
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Despite these observations, it does not appear safe to make generalizations concerning rates of unknown metabolic reactions. Many exceptions could be found to the apparent tendency of the intrinsic rate to decrease with increasing body size; quite different qualitative pathways can dominate in different species, and toxicity sometimes can correlate with the concentration of an active intermediate that represents only a minor elimination pathway.
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40 KENNETH B BISCHOFF Right heart L Lung f Upper body } Left heart 1 ''in 1 WK< ,, ,~ \\,// Small Intestine | l l Trunk t r Large Intestine Lower extremity FIGURE 1 Flow diagram for mammals.
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It is most important, however, to state that this approximation is not fundamental to the physiological pharmacokinetic approach, and is commonly used merely for the lack of sufficient information. Another basis for the initial determination of the number of required body regions is the speed or the time scale of events.
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~ ~ . ~ _m 1100 Kidney 22 1 14 84 160 Lower 130 80 8710 16700 FIGURE 2 Flow diagram for disposition of rapidly acting substances anesthetic agents or tracers (numbers are volume estimates, in cubic centimeters)
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For most of the discussion here, however, only relatively long time scales will be considered. If one carries this lumping to the extreme, the entire body can be assumed to have uniform (water)
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Only the fully lumped models for body regions will be considered, but the physiochemical complications will be retained. A complete set of anatomically and physiochemically complicated mass balances could be formulated, but these are probably much too involved for any practical use at this time.
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Mass Balance: Blood Pool dCB WB VB d t = ~ 01 V dCB* (unbound drug)
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SIMPLIFICATIONS OF MASS BALANCES Since the membrane permeabilities are not commonly available, Equations 3 to 5 given above are usually simplified into blood and tissue regions,
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A corollary is that the free concentration is essentially identical throughout the region, since this is the driving force for mass transfer across the membrane, for simple passive diffusion. Mathematically, a very large permeability in Equations 3 and 6 is PA x, and CBi— CTi—Ci (equal free concentrations)
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Given the parameter values, sets of mass balances like the various combinations given above can be readily solved for the important interconnected body regions. Naturally, this question of parameter values is the crucial one, but it is also a key advantage of the physiological pharmacokinetics approach.
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5.615 QB Blood Pool not In Equlilbrlum with Tissues B QV 4.080 QL 1.275 Q .A 0.260 Viscera V Lean Tissue L Adipose Tissue A
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Figure 5 is the lumped, longer time scale model used, with an important compartment being the adipose tissue because the drug is highly lipid soluble. This drug is also strongly and nonlinearly bound, and so the detailed mass balances (Equations 1 and 7)
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52 KENNETH B BISCHOFF BLOOD 100 y E - o he Cal He o c,, 10 On 2 Dog Human _ ~ .g ·: , 0 1/2 1 2 TIME, hours FIGURE 6 Comparison of data with model predictions for thiopental.
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(llf) \ For gut lumen: dCGL 1 dCi = -Y dt 4 t-~i dt (1 la)
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Figure 11 shows one prediction of the concentrations of cytosine arabinoside and its metabolite, uracil arabinoside. A final reduction in the complexity of the models is possible when the excretion/metabolism processes are relatively slow compared with the intercompartment blood flows.
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In terms of Equations 9 and 10 this implies that: r1 < |
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It appears that this quite often gives a good estimate of the overall drug concentrations throughout the body, even though the drug may be membrane limited in certain specific organs. Thus, a combination of the analyses illustrated in the examples, plus the use of Equation 6 for the specific region, might be a reasonable scheme for both the overall drug distribution and the details of, for example, tumor uptake.
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For example, the major barrier to the transport of many drugs is not the capillary membrane but the cellular membrane; in this case, tissue can be defined as intracellular space, and blood can be defined as vascular plus interstitial space. The important point is that the physi
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The reviews quoted above provide many examples of this; all of these are based on flow diagrams similar to those shown in Figure 7 or 10, with appropriate modifications for the specific drug.
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The above types of reaction-diffusion models lead to partial differential equations, which are more difficult mathematically, but with modern computing technology they should not cause severe problems in calculations. A related issue is the use of more realistic descriptions of the local blood flows in the microcirculation.
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1983. Physiologically based pharmacokinetic modeling: Principles and applications.
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From page 61... ...
1980. Physiological pharmacokinetics: An in vivo approach to membrane transport.
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