This article is part of the monograph on Mathematical Modeling in Environmental Health Studies.
Address correspondence to A.J. Schwab, McGill University Medical Clinic, Montreal General Hospital, 1650 Cedar Ave., Montreal, Quebec, Canada H3G 1A4. Telephone: (514) 937-6011, ext. 2918. Fax: (514) 937-6961. E-mail: aschwa@po-box.mcgill.ca
This work was supported by the Medical Research Council of Canada and the Fast Foundation.
Received 20 January 2000; accepted 16 May 2000.
Pharmacokinetic assessments are essential elements of drug development and toxicological research. After exposure of an organism to a particular environment--drugs, xenobiotics, toxicants--the resulting concentrations may be measured conveniently in only a few locations such as blood, plasma, urine, or exhaled air. However, the biological effects of environmental agents depend on concentrations at the site(s) of action. These can be receptors or other proteins located within a cell or on its surface. Tissue concentrations of drug or toxicants, however, are not easily assessed experimentally, especially when metabolism occurs. The partition of the chemical between extra- and intracellular spaces and within cells may be critical in the determination of the biological activity or toxicity of an agent. Biopsy material can sometimes be acquired for measuring concentrations within selected tissues, but it is not easily obtained. In some cases, data may be obtained by destructive sampling at the end of an experiment, but evaluation is often hampered by the sampling procedure and factors such as interindividual variability.
A complementary approach to appraise time-dependent distribution and concentration profiles of substances is through mathematical modeling based on scientific principles and mathematical procedures aimed at quantitatively explaining and predicting the pharmacological and toxic activities of chemicals (1-4). Existing models can be classified as simple compartmental models (with one or more compartments), physiologically based pharmacokinetic (PBPK) models, and noncompartmental approaches (black-box models with model-free parameter assessment). Use of PBPK models is popular with risk assessment (5-10). The classical PBPK approach portrays individual organs as well-mixed compartments that are interconnected by the circulation, with instantaneous, flow-limited access of substrates to the organ and constant tissue partition coefficients. For simplification, some of the noneliminatory compartments of similar flow and drug partition characteristics are often grouped. This kind of modeling approach offers an overly simplistic view of the organ. Important time-dependent phenomena such as the kinematics of blood perfusion, carrier-mediated transport across cellular membranes (which can be active or passive), binding to cellular proteins, and uptake into organelles have been neglected. To implement more realistic versions of PBPK models, the peculiarities of these time-dependent phenomena must be described in fuller detail.
In this article, we describe experimental and theoretical methods useful in identifying the mechanisms underlying the determinants of drug and metabolite kinetics in intact organs. We present the multiple indicator dilution (MID) method, a dynamic tracer method based on comparing the temporal data obtained from a studied substance prone to degradation and excretion with that of an inert reference indicator substance. As simplifications of this approach, the dispersion model and steady-state organ models are also introduced. The utility of these methods for risk assessment efforts is discussed.
Distribution and disposal of a xenobiotic within the body can be viewed as an input-output system with dose as the input and the measured concentrations as the output; in other instances, pharmacodynamic measurements constitute the output. In many cases, metabolic reactions and transport processes proceed according to first-order or linear kinetics such that their rates are proportional to the concentration of the transported or reacting species. This occurs when concentrations are low and fall below the Michaelis constants or the dissociation constants of the processes involved. Systems in which all processes obey linear kinetics with first-order rate constants that do not change with time or concentration are known as time-invariant linear systems. A special case arises when a steady state is maintained in all respects except for the movement and transformation of the tracer (nonisotopic steady state). In this case, observable rates of tracer movement and transformation are proportional to the amount of tracer (or its concentration) in the source compartment (
11,12), resulting in linear systems.
Linear systems play an important role in a variety of scientific and engineering contexts such as electrical circuits or economics. Theoretical exploration has revealed some general properties independent of the detailed structure of these systems (11). The output from a time-invariant linear system, e.g., the measured concentrations or effects at different times, can be represented by a time-dependent output function g(t). This can be derived from the equivalently defined input function f(t) by the following equation:
[1]
In the special case where the input is the unit impulse function (approximated experimentally by a rapid injection), the output is h(t), or the impulse response of the system. The impulse response is a property of the system that does not depend on the input. The integral in Equation 1 is the convolution integral, and the function g is the convolution of the functions f and h. The impulse response can then be determined from the input and output functions by an operation known as deconvolution. Several numerical algorithms for deconvolution can be found in the literature (13-15).
The mathematical treatment of linear systems can be simplified by using Laplace transformations. The Laplace transform f~(s) of a function f(t) is defined as the integral
[2]
The Laplace transform of the convolution of two functions f(t) and h(t) is then found as the product of their Laplace transforms, f~(s) and h~(s). Thus, Equation 1 implies
[3]
For two systems connected in series, the output from the first system constitutes the input into the second system. In this case, the impulse response of the combined system is the convolution of the impulse responses of the individual subsystems. Because the convolution operation is commutative, the order of connection is unimportant, i.e., two systems connected in series can be interchanged. The following explorations will be restricted to linear systems.
The MID technique (
16-18) [also referred to as the indicator diffusion method (
19)] is a method often utilized to explore the kinetics and disposition of substances within intact organs. The classical MID approach does not make specific assumptions on the details of heterogeneous perfusion of capillaries but utilizes indicators for the assessment of transit time distributions. Within the MID experiment, a mixture of labeled substances (noneliminated indicators--
51Cr-labeled erythrocytes,
125I-labeled albumin, [
14C] or [
3H]sucrose, [
2H]
2O--and labeled tracer substrate) are injected rapidly into the inlet blood vessel of the test organ. Timed samples are then collected at the outflow and analyzed for the noneliminated indicators, the tracer, and its potential metabolites. The method was introduced by Goresky and colleagues for dog liver
in vivo (
20,21). The basis for this work was a flow-permeability model within the liver (
18), yielding the starting point for a mathematical analysis that includes convection and membrane permeation of solutes. It was successively adapted to perfused rat livers (
22-36) in which recirculation is avoided and sampling can be extended to longer collection periods. Analogous work has been performed for the heart (
37-40), the lung (
17,41), the kidney (
42-44), and the brain (
45,46).
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Figure 1. Schematic representation of a distributed-in-space system for defining the influx (k1), efflux (k-1), and cellular sequestration (k2) coefficients described in Table 1. The capillary unit functions in analogy to a chromatographic column in which the vascular space corresponds to the mobile phase and the cellular space to the stationary phase.
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Within the organ, blood flow generates convection of plasma and movement of formed blood elements, resulting in the convective movement of indicators. Metabolism or excretion occurs upon transport of solutes into the organ. Diffusion, carrier-mediated transport for cellular influx and efflux, binding to plasma proteins or tissue constituents, and structural modifications via biochemical means (mostly enzymatic) further shape the distribution and tissue transit of the solutes. A comprehensive analysis of the complex interplay between these factors needs an elaborate mathematical description. The general methods have been derived from concepts developed in chemical engineering (47). Usually this involves partial differential equations in time and space. The complexity is simplified by subdividing the space into a finite number of discrete compartments with uniform concentrations in each of the compartments, calling for ordinary instead of partial differential equations (compartmental system analysis). This concept has been widely used in biological and other contexts (48). It is adequate whenever the events under study are considerably slower than convection or diffusion within compartments. In many cases, however, the influence of membrane permeation, metabolism, and blood flow on the overall disposition of solute cannot be readily separated. More specifically, removal of substances because of metabolism or excretion pursuant to entry gives rise to concentration gradients along the capillary flow path. Under such circumstances, the classic form of compartmental system analysis would fail. Instead, a distributed-in-space concept (Figure 1) involving systems of partial differential equations (49,50) must be used. This concept will be covered in greater detail in the ensuing sections.
In the ensuing section a general overview of the mathematics of space-distributed systems as applicable to MID experiments is given. The focus of the exploration rests on the liver as the main biotransformation organ.
The microcirculation of an organ such as the liver consists of a collection of capillaries or sinusoids that may or may not be interconnected, forming a capillary bed (Figure 2). Blood enters the capillary bed via a set of feeding arteries and arterioles and leaves via a set of drainage venules and veins. The strategy for modeling the movement of tracers through the hepatic sinusoidal bed is to initially explore the movement along a single path of blood flow through the organ with a single transit time. The movement within the whole capillary bed is then modeled as a combination of many flow paths acting in parallel, although the structure of real capillary beds is generally more like a network than like an array of independent parallel paths. This is a legitimate strategy if the transport and metabolism of the indicator follow linear kinetics (51,52). The tracer concentration at the outflow of the organ becomes the flow-weighted average of the outflow concentrations from the individual flow paths. Because different flow paths through an organ show different transit times, this average is formulated such that it takes into account the distribution of transit times among the paths. Each flow path consists of two portions: a capillary-bed portion, where exchange with extravascular space occurs, and a noncapillary-bed portion consisting of arteries and arterioles as well as venules and veins, where all indicators are confined to the vasculature. Note that because of the commutative property of convolution, the order of serial connection is irrelevant in linear systems, such that arteries, arterioles, venules, and veins (the large vessels) can be lumped together and described by a single distribution of transit times. Thus, flow paths of given total transit times, 
, vary in capillary transit time, c, according to the conditional probability distribution, P(c|) (53). One of the following special cases usually applies simplifying the analysis (54): uniform capillary transit time independent of with variable large-vessel transit time, or variable capillary transit time c = - t0 with uniform large-vessel transit time, t0. As a less constrictive alternative, capillary and large-vessel transit times are both assumed variable, and the special cases to be considered are random coupling, in which capillary and noncapillary transit times are stochastically independent from each other, or flow coupling, in which for each total flow-path transit time , there is a single capillary transit time c (Figure 2) (53). A special case of flow coupling is a linear relation between capillary and large-vessel transit time (55).
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Figure 2. Modes of coupling between large vessels and capillaries.
In MID experiments, the influence of the injection and collection devices (the injector and the catheters connecting the injector with the organ and the organ with the fraction collector) must be taken into account. The impulse response of the latter is assumed to be randomly coupled with that of the organ itself. With linear kinetics, the impulse response of the organ can be obtained from the measured outflow profile by deconvolution of the impulse response of the injection and collection devices alone, the latter being determined experimentally in control experiments in absence of the organ (34,56). This procedure is convenient in the case of the liver or the lung where the organ noncapillary (large-vessel) transit time is considered uniform (18,57).
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A simple scenario exists for the rapid exchange of indicator between the vascular space and the extravascular space. If the process is fast compared to vascular convection (flow limitation), the extravascular concentration is equal to the vascular concentration at any position along the flow path. The resulting differential equation is
[4]
where vF is velocity of blood flow in the capillary, t is time, x is distance from the origin along the flow path, c(x,t) is concentration within a capillary at position x and time t, 
is the ratio of the extravascular to the vascular distribution space, q0 is the amount of tracer injected at t = 0, A is the cross-sectional area of the sinusoid, and 
is the impulse function. The term q0(x)(t)/A indicates rapid injection of material at t = 0 and x = 0. The values of vF, , and A are assumed to be constant throughout the flow path. Equation 4 has the solution
[5]
where Qs = vFA is volume flow through the capillary. According to this equation, a bolus introduced into the origin of the capillary flow path travels along the path at a velocity vF/(1 + ), which is less than the average velocity of perfusate, vF. Because the output from a linear system is the convolution of the input and the impulse response, this result can be generalized to the propagation of any input function. A flow-limited indicator thus behaves according to the delayed-wave principle (18), as depicted schematically in Figure 3.
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Figure 3. Schematic representation of the delayed-wave principle in a single capillary. The extravascular volume accessible to a flow-limited indicator is assumed twice as large as the intravascular volume at any point in the exchange area. (A) Vascular indicator at uniform intervals of time propagated uniformly along sinusoid with the velocity of flow. (B) Delayed propagation of the flow-limited indicator travelling at one third the velocity of the vascular indicator. (C) Propagation of flow-limited substance spreading into the same extravascular volume and undergoing simultaneous removal. The concentration wave emerges damped as well as delayed.
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The impulse response of the sinusoidal portion of a flow path with sinusoidal transit time c is given by the dose-normalized concentration at the outflow after an impulse input, obtained from Equation 5 by setting x = vFc. Arteries and arterioles will cause a delay of the input into the sinusoids, and venules and veins will cause a delay of the output. In linear systems, these delays are combined into a single large-vessel delay. With the assumption that all flow paths share a common large-vessel transit time, t0, as originally proposed by Goresky for the liver (18), the impulse response of the total flow path is delayed by t0 relative to that of the sinusoidal portion alone. Let f()d be the fraction of total organ blood flow emerging from paths with sinusoidal transit times between c and c + dc. The mixed outflow concentration from the whole organ, C(t), normalized to the amount of the injected dose will then be the flow-weighted average of the dose-normalized outflow concentrations, according to the integral
[6]
where Q is the total flow into the organ. Assuming that the value of is the same for all capillary paths, Equation 5, when substituted into Equation 6, yields the following:
[7]
For a vascular reference indicator such as labeled erythrocytes, = 0, thus
[8]
from which, after replacing t with t + t0,
f(t) = QCvas(t + t0). [9]
Substitution of Equation 9 into Equation 6 yields
[10]
According to Equation 10, the outflow profile of the test indicator superimposes onto that of the vascular reference after a linear scaling operation. This superposition principle was introduced by Goresky in his classic publication (18). Four indicators were injected simultaneously into the portal vein of an anesthetized dog: 51Cr-labeled erythrocytes, an indicator confined to the vascular space; Evans blue (a label for albumin); [14C]sucrose and [22Na]Cl, indicators for the extracellular water space; and [3H]2O, an indicator that enters erythrocyte water, plasma water, and cellular water spaces. Although outflow profiles of these indicators differed in duration and intensity because of differences in their distribution spaces (Figure 4), the venous outflow profiles were similar after superposition according to Equation 10 (Figure 5), confirming the validity of the model and the assumption of a uniform value for .
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Figure 4. Outflow profiles for 51Cr-labeled erythrocytes ( ), Evans blue as an indicator for albumin ( ), [14C]sucrose ( ), [22Na]+ ( ), and 3HOH ( ) in the liver of an anesthetized dog. Broken lines represent monoexponential extrapolation of data to correct for recirculation. Adapted from Goresky (18), with permission.
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Figure 5. Superposition of the adjusted diffusible indicator curves upon the erythrocyte curve from the experiment illustrated in Figure 4. Different concentration and time scales were used for each indicator separately, as indicated by the symbols shown at the left lower corner. Adapted from Goresky (18), with permission.
Interpretation of the result is strongly related to the structure of the hepatic sinusoids. The endothelial cells lining the sinusoids are equipped with fenestrae that connect the sinusoidal space with the space of Disse, an interstitial space situated between the endothelial and the parenchymal cells (Figure 6). The size of the fenestrae is small such that erythrocytes are confined to the vascular space, whereas albumin, sodium, and sucrose penetrate the interstitial space freely. The Disse space contains a fibrous meshwork consisting mainly of fibronectin (58) that partly excludes albumin such that the distribution volume of albumin is less than that of sodium or sucrose. These tracers are considered extracellular, as they do not enter parenchymal cells or erythrocytes within the time frame of a MID experiment. However, water is transported rapidly across the cellular membranes through the action of aquaporins (59) and penetrates the whole liver. This includes the parenchymal and endothelial cells that are stationary and also the cellular blood components. For evaluation of outflow profiles for D2O, the distribution space ratio (Equations 5 and 10) must be interpreted differently, namely, as the ratio of the stationary (parenchymal + Disse) space to the moving (total blood) space.
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Figure 6. Interior of a sinusoid of a mouse liver with fenestrated hepatic endothelial cells. Scanning electron micrograph courtesy of Anne-Marie Steffan, Laboratoire de Virologie INSERM U74, 67 000 Strasbourg, France.
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In the case of linear kinetics with irreversible cellular uptake of an indicator along a single capillary path, mass transfer is described by the differential equation:
[11]
where P is permeability of the cellular membrane, S is its surface area, and Vp is the volume of the plasma space in the capillary lumen. The solution to Equation 11 is
[12]
According to this equation, a bolus introduced into the origin of the capillary flow path travels along the path at a velocity vF while its amplitude diminishes exponentially in intensity. A reference indicator not taken up exhibits zero permeability (P = 0) and its amplitude remains unchanged.
The relation between dose-normalized outflow concentrations of the test and vascular reference indicators from the whole organ, C(t), is evaluated using Equation 6 in a way analogous to that demonstrated for the flow-limited case (Equation 10). This yields
[13]
According to Equation 13, the delayed-wave transport equation is modified by the exponential damping of the wave along the flow path (Figure 3, lower panel). This can be reformulated as
[14]
or
[15]
where Cref(t) is the outflow profile of an appropriate reference tracer defined as
[16]
The simplest case is the irreversible sequestration of a substrate either by metabolism or by excretion. An example is the sequestration of the dye, sulfobromophthalein (BSP), in the dog liver in vivo (60) (Figure 7). Because this dye, like Evans blue, is tightly bound to albumin, it occupies the same fraction of the Disse space, such that the values of for BSP and for albumin are almost identical. Therefore, Evans blue (as a label for albumin) was used as the appropriate reference. Plotting the natural logarithm of the ratio of the dose-normalized concentrations of albumin and BSP against time yields a straight line (Figure 7, inset). This is expected from Equation 15 and confirms the linear uptake kinetics of BSP.
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Figure 7. Outflow profiles for 51Cr-labeled erythrocytes (), Evans blue as an indicator for albumin (), and [32S]sulfobromophthalein ( ) in the liver of an anesthetized dog. Adapted from Goresky (60), with permission. Inset: Natural logarithm of the ratio of the albumin-to-sulfobromophthalein data.
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It is generally assumed that only the part of a solute like BSP that is not bound to plasma proteins can permeate cellular membranes. Equations 11, 15, and 16 therefore contain an apparent permeability P = fuPu that depends on the fraction of the solute that is not bound to proteins (unbound fraction, fu), and its permeability Pu. This formulation assumes rapid binding and release of the solute from plasma proteins. If binding is very tight, dissociation of the solute-protein complex may become rate limiting. In this case, the apparent permeability depends on the dissociation rate constant and may also be limited by diffusion of the solute within the Disse space (61-63).
A similar situation occurs with indicators that are distributed rapidly throughout the cells and are irreversibly sequestered. Examples are metabolism of monohydric alcohols in the dog liver (64) and salicylamide or octanoate in perfused rat liver (24,29). These are lipophilic substances that accumulate in the cellular lipids. It is probable that many of the environmental agents of interest show similar modes of hepatic disposition.
The binding of substrates to erythrocytes and albumin or other plasma proteins often occurs. In these instances, interpretation of the outflow profile is facilitated when the outflow profile is related to that of a reference indicator. The reference indicator is one that distributes between the vascular and the extravascular space in a way identical to the substance studied but one that does not enter the hepatocyte. For solutes present as unbound species in plasma, an appropriate reference indicator would be sucrose, cobalt-EDTA, or sodium (
65,66), small molecular weight compounds that occupy the plasma space and most of the Disse space. For solutes bound very tightly to plasma proteins with a negligible unbound fraction, albumin, the high molecular weight indicator that is partially excluded from the interstitial space (
18) is the appropriate reference indicator (
67). In the case of partial binding to plasma proteins, a hypothetical reference indicator outflow profile may be constructed using the superposition principle (Equation 10), with an interpolated interstitial distribution space intermediate between those of sucrose and albumin (
33,34). The extravascular-to-vascular space ratio for the reference indicator,
ref, is then obtained from that of albumin,
Alb, and that of an indicator that is not bound to plasma proteins such as sodium or sucrose,
Suc, according to the following equation:
ref = fuSuc + (1 - fu)Alb, [17]
where fu is the fraction of the substrate not bound to plasma proteins. In the case where the substance under study undergoes rapid exchange between erythrocytes and plasma, the outflow profile of an indicator for erythrocytes is included in constructing the appropriate reference curve (33).
A more general scenario describes exchange between the interstitial space and the cellular space where sequestration occurs. Transport across cellular membranes may proceed either by simple diffusion in case of a lipid-soluble indicator or by carrier-mediated transport. In the liver, equilibration between plasma and interstitial space in the lateral direction is quasi-instantaneous, such that interstitial concentrations and vascular concentrations are assumed to be equal at any location along the flow path. Partial differential equations for the vascular and cellular concentrations,
c(
x,t) and
cc(
x,t), at time
t and position
x, are then (
25)
[18]
and
[19]
where k1, k-1, and k2 are rate constants as defined in Table 1, 
is the ratio of the cellular to the vascular space, and fu and ft are the fractions of tracer not bound to proteins in plasma and tissue, respectively.
The system of partial differential equations (Equations 18 and 19) has the solution (65)
[20]
where S(·) is a unit step function and Qs is perfusate flow rate. The whole organ C(t) is evaluated using Equation 6 in a way analogous to that demonstrated above (32).
[21]
This equation was used for evaluating the hepatic extraction of galactose in dog liver (65). In the case of a metabolically inert tracer insignificantly metabolized in the MID time frame, such as rubidium (66) or inorganic sulfate (22), k2 was set to zero.
Equation 21 has two terms: the first term, which is analogous to the expression for irreversible removal (Equation 14), represents the throughput material that has never left the plasma space; the second term represents material that has entered hepatocytes and returned to the plasma space, later in time. The throughput component dominates the outflow of substances that are poorly permeable substrates such as acetaminophen sulfate, whereas for substances such as salicylamide sulfate that penetrate the cell more readily, the throughputs are much smaller (Figure 8). Proper analysis of MID experiments thus demonstrates that substances of similar chemical structures or charges can have quite different permeation properties, presumably because of differences in carrier-mediated transport.
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Figure 8. Outflow profiles for (A) [14C]sucrose () and [3H]acetaminophen sulfate () or (B) [58Co]EDTA () and [14C]salicylamide sulfate () in the perfused rat liver. Adapted from Pang et al. (29) and Xu et al. (33), with permission. The dotted lines represent spline approximations to the sucrose or CoEDTA data. Theoretical curves on how acetaminophen sulfate or salicylamide sulfate behaved (continuous lines) in the perfused rat liver preparation are resolved into throughput and returning components.
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According to the above principles, throughput and returning components were obtained for two organic anions that undergo carrier-mediated entry, hippuric acid (34) and the glutathione conjugate of BSP, sulfobromophthalein gluatathione conjugate (BSPGSH), that is removed by biliary excretion (25,26). The poor removal of hippuric acid, a nonmetabolizeable substrate, is not due to poor sinusoidal entry but to lack of biliary excretion because of impaired transport across the canalicular membrane of hepatocytes. Excretion of BSPGSH appeared to be rate-limited by sinusoidal entry. Optimal parameters of barrier-limited transport and cellular sequestration of various substrates and xenobiotics are compiled in Table 2.
Similar analytical solutions have been obtained for situations involving two barriers between blood and cells (37), the occurrence of a barrier between plasma and erythrocytes (68), and formation of a metabolic product (27,69). Analytical solutions are not available for more complex cases such as multistep metabolic systems. Although algorithms based on analytical solutions have been used in the past for calculating outflow profiles, approximations based on finite-difference methods can be more efficient and more versatile alternatives (70,71).
When reversible reactions occur within the cell or when metabolic products appear in the outflow, the above equations must be extended to include terms related to metabolic intermediates and/or products. This may be achieved in a concise and general fashion by applying a notation including vectors and matrices (
30,72,73), as illustrated by the following equation:
[22]
where c is a vector of concentrations, A is a compartmental matrix composed of transfer coefficients, and q0 is a vector of injected amounts. W is a diagonal matrix whose elements are velocities: they are zero for intracellular moieties, are equal to vF for vascular moieties, and equal to vF/(1 + ref) for moieties shared between the vascular and the interstitial or the cellular space. The previously discussed models for describing single-capillary events are special cases of this general scheme.
An example for a more complex situation evaluated in this fashion is shown with the metabolism of acetaminophen in the perfused rat liver (30). MID experiments showed an early peak of labeled acetaminophen in the presence of red blood cells that disappeared when the erythrocytes were omitted from the perfusion medium (Figure 9). The model used for interpretation of the data (Figure 10) necessitates consideration of the binding of acetaminophen to erythrocytes as well as intracellular metabolism to acetaminophen sulfate (the only metabolite detected at the low acetaminophen concentrations used). Because labeled acetaminophen entrapped by erythrocytes escapes into plasma with a very slow transfer coefficient of 0.05 s-1, that part of the acetaminophen (10% at hematocrit of 0.15, and 15% of at hematocrit of 0.3) was carried through the sinusoidal bed without leaving the erythrocytes (Figure 11). Data for the metabolite, acetaminophen sulfate, were also described well by the model (Figure 12). Similar results were obtained for hepatic metabolism of l-lactate in the anesthetized dog liver (74).
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Figure 9. Outflow fractions of [3H]acetaminophen () and noneliminated reference indicators in perfused rat liver, MID experiments conducted at hematocrit = 0.3 (A), 0.15 (B), and hematocrit = 0 (absence of erythrocytes) (C). From Pang et al. (30), with permission.
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Figure 10. Schematic presentation of red cell carriage of acetaminophen (with the erythrocyte and plasma as separate vascular components), its transfer into liver cell, distribution into an intracellular pool, and metabolism to the sulfate conjugate, with return to plasma and biliary excretion. Adapted from Pang et al. (30), with permission.
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Figure 11. Fitted theoretical outflow profiles for [3H]acetaminophen (same data as Figure 9) calculated according to the scheme presented in Figure 10 for different hematocrits. The acetaminophen in the erythrocytes, which never left them during their passage along the vasculature (an erythrocyte throughput), is shaded. From Pang et al. (30), with permission.
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Figure 12. Fitted theoretical outflow profiles of [3H]acetaminophen and [3H]acetaminophen sulfate (same data as Figure 9) calculated according to the scheme presented in Figure 8 for a hematocrit of 0.3 (A), 0.15 (B), and 0 (C). From Pang et al. (30), with permission.
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An alternative to providing transit time distributions by using reference indicators is to apply a physical model for tracer distribution within the hepatic microvasculature. A popular model using this idea is the dispersion model (
75-77), a stochastic model based on assumptions of concurrent convective and random-walk (pseudodiffusive) movements in the direction of flow. It is characterized by the following differential equation for a noneliminated indicator:
[23]
where C is the outflow concentration normalized to the amount of dose; DN is a dimensionless number known as the dispersion number (its inverse is called the Peclet number); T is dimensionless time (time after injection divided by the mean transit time); and X is the dimensionless distance or position along the flow path (where X = 0 represents the point of inflow and X = 1 the point of outflow of the sinusoidal bed).
There are two variants of the model, depending on the boundary conditions at the inflow and the outflow: these are either closed (reflective) (Danckwerts conditions) or closed at the inflow and open (transparent) at the outflow (mixed conditions). The solution to the closed-boundary variant must be obtained using a recursive algorithm (78). The mixed conditions give rise to a simple expression for the form of the outflow profile of a vascular indicator after rapid injection (impulse input):
[24]
where V is the volume of distribution. Equation 24 represents the inverse Gaussian or random-walk function (79). Curves for both types of boundary conditions are shown in Figure 13. At moderate values of DN at between 0.1 and 0.3, as observed for the perfused liver preparation (75), the solutions for both kinds of boundary conditions are quite similar, whereas they diverge from each other at higher DN numbers (Figure 13). The closed boundary conditions, however, give a more realistic picture of the events within the organ (80).
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Figure 13. Outflow profiles predicted by the dispersion model with closed (A) or mixed (B) boundary conditions at varying dispersion numbers.
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The dispersion model provides reasonable approximations to experimental outflow profiles of noneliminated indicators when the appropriate correction for the effect of the nonexchanging part of the flow paths is applied (Figure 14). However, it underestimates the experimental data at longer collection times (75). In the case of linear kinetics, the dispersion model is mathematically equivalent to a model representing the sinusoidal (or capillary) bed by a set of interconnected tubes (81).
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Figure 14. Fits of the dispersion model with correction for the effects of injection and collection devices to experimental data from a perfused rat liver. From Schwab et al. (75), with permission.
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Theoretically, the dispersion model is pertinent to the description of transport and metabolic kinetics in the sense that it provides an expression for the distribution of transit times. With linear kinetics, outflow profiles are independent of the location where mixing occurs (51,52). Models with interconnected flow paths are therefore equivalent to models with independent flow paths. The inverse Gaussian function in Equation 24 can thus be used to represent the outflow profile of a reference indicator for obtaining approximations to the metabolized outflow profiles (82). For example, in the expression for irreversible cellular uptake (Equation 14), the reference profile Cref(t) is obtained from Equation 24 with the substitutions V = Vc(1 + ) (for the space of distribution of the reference indicator) and T = tQ/V. The resulting outflow profile is the same as that formulated for the dispersion model of hepatic elimination (76,77). The concept of a barrier between the vascular and the (stationary) extravascular space extends into the two-compartment dispersion model (Turner capacitance model) (77,83). The outflow profile predicted by this model can be obtained by substituting the right-hand side of Equation 24 for Cref(t) in Equation 21, with appropriate substitutions as above. This equivalence can be verified by comparing the Laplace transforms (77). The dispersion model therefore can be regarded as a special case of the Goresky approach.
A direct comparison between the Goresky model (GM) and the two-compartment dispersion model (DM) has been made (82). The two models were applied for the interpretation of the same data set for hippurate, a noneliminated compound that undergoes bidirectional transport at the sinusoid (Figure 15), and for the BSPGSH that enters hepatocytes via the organic anion transporter polypeptide 1, oatp1 (84) and is excreted unchanged by the multidrug resistance-associated protein 2, Mrp2 (26,85,86) (Figure 16). Both models yielded similar values for the influx coefficients using experimental reference curves. The dispersion model thus provides a convenient but less reliable method of analyzing outflow profiles. However, often a reference indicator was not used. Moreover, reliable values for efflux and cellular sequestration coefficients could not be obtained using the dispersion model mostly because of the inability of the model in describing the dispersion of the late-in-time data (see fits to tail of the reference curves in Figures 15A, 16A). In principle, any function of a shape similar to the reference curve can be used as an approximation for the distribution of capillary transit times (87). With a sum of (potentially complex) exponential terms, an analytical solution of the integrals according to Equation 6 can be obtained (72,88). King et al. (89) used a lagged normal density distribution (90) for capillary flow, a quantity that is proportional to the inverse of capillary transit time at constant capillary volume.
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Figure 15. Fit of the barrier-limited model according to Goresky (GM) and the dispersion (DM) model of hepatic elimination to outflow profiles of hippurate in perfused rat liver. From Tirona et al. (82), with permission.
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Figure 16. Fit of the barrier-limited model according to Goresky (GM) and the dispersion (DM) model of hepatic elimination to outflow profiles of the glutathione conjugate of sulfobromophthalein (BSPGSH) in perfused rat liver. From Tirona et al. (82), with permission.
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Elimination of xenobiotics generally occurs in highly perfused organs such as the liver and the kidney. On the time scale of whole-body kinetics, elimination by these organs can be regarded as being in the steady state. This allows for simplifications in the mathematical treatment of whole-organ kinetics. The concept of steady-state approximation is a fundamental principle that applies to a variety of situations involving physical, chemical, and biological processes. Its mathematical basis is Tihonov's theorem (
91), which governs the rules for separating processes according to a hierarchy of time scales. Processes following the faster time scale are considered to be in a quasi-steady state after a short initial period. Approximate equations for these processes are obtained by setting the time derivatives of the state variables to zero. In the case of linear systems, the steady-state solution can also be obtained by integrating the impulse response from zero to infinite time (
16). A useful parameter is the steady-state extraction ratio,
E, defined as the fraction of a substance delivered to an organ that is extracted:
[25]
where cin is the concentration at the inflow and cout is the concentration at the outflow of the organ.
In the case of the liver, three steady-state models corresponding to three different models of organ perfusion or elimination have been proposed: the one-compartment (well-stirred or venous equilibration) model (92), the parallel-tube (sinusoidal perfusion) model (93), and the dispersion model (94).
In the well-stirred model, all tracer within the organ is assumed to be in rapid equilibrium. Thus, all concentrations are proportional to the concentration in the venous outflow. The steady-state extraction ratio is then given by
[26]
where RN = PS/Q for a substance that is irreversibly removed from the vasculature, and RN = (fuPinS/Q)k-1/(k-1 + k2) for a substance that enters parenchymal cells.
The parallel-tube model assumes a constant capillary transit time for all capillary flow paths. The concentration of a substance that is irreversibly eliminated from the vascular space follows the following differential equation within a single flow path, derived from Equation 11 by setting the time derivative to zero:
[27]
Integration over an entire flow path yields, for the steady-state extraction ratio:
E = 1 - e-PS/Q, [28]
where the plasma flow rate is given by Q = VpvF/x.
If the vascular and the extravascular spaces are separated by a barrier, the steady-state concentrations in these spaces can be obtained for the parallel-tube model by setting time derivatives in Equations 18 and 19 to zero. This results in the ordinary differential equation
[29]
and the algebraic equation
[30]
yielding the solution
E = 1 - e-RN. [31]
Solving Equation 31 for cc yields
[32]
For a single flow path, the concentrations in both spaces decrease in a monoexponential fashion in the direction of flow. Depending on the values of the transfer coefficients, the concentration in the extravascular space will be higher or lower than that in the vascular space. In particular, high intracellular concentrations due to high k1/k-1 ratios are indicative of active transport. It is thus possible to predict intracellular concentrations after modeling the fitted parameters k1 and k-1 from MID experiments.
The dispersion model of hepatic elimination with closed boundary conditions yields
[33]
where
The values for the steady-state extraction ratio computed according to Equation 33 are intermediate between those for the one-compartment and the parallel-tube models (94).
Similar expressions can easily be derived for situations with multiple barriers or multiple metabolic reactions arranged in parallel (competing pathways) or in sequence. Expressions for intracellular concentrations and metabolic flux rates using compartmental, parallel-tube, and dispersion models have been derived by St-Pierre et al. (95). Generalization of this principle for metabolic networks with arbitrary structure using compartmental system analysis is straightforward if the organ is assumed to behave as a well-stirred phase. For a single flow path, and, by extension, for an organ with a uniform capillary transit time (parallel-tube model), systems of linear differential equations are obtained that can easily be solved by eigenvalue analysis (30,96).
Application of these mathematical procedures allows prediction of vascular and cellular concentrations and their variations along the flow path. As an example, predicted acinar profiles of vascular and cellular concentrations of acetaminophen and its metabolite, acetaminophen sulfate, are shown in Figure 17, using the parameters obtained from MID experiments with labeled acetaminophen in the perfused rat liver. These simulations predict considerable accumulation of the metabolite, acetaminophen sulfate, within hepatocytes (Figure 17B). Although the vascular metabolite concentration was highest in the perivenous region of the acinus, the cellular concentration was highest in the periportal region. Acetaminophen sulfate is not believed to show toxic effects. However, toxicity of acetaminophen has been observed at higher concentrations with increased production of the toxic metabolite, N-acetyl-p-benzoquinone imine by cytochrome P450 (97). It can be hypothesized that similar mechanisms might lead to intracellular accumulation of this metabolite or of toxic metabolites in general.
Figure 17. Predicted acinar steady-state concentration profiles of acetaminophen (A) and its metabolite, acetaminophen sulfate (B), calculated for a flow path with an erythrocyte transit time of 10 seconds. Optimized parameters derived from fitting the MID profiles (from data shown in Figure 9A) were used for these simulations. Adapted from from Pang et al. (30), with permission.
To construct a whole-body PBPK model, the individual organs must be combined in a suitable fashion. PBPK models are recirculatory models in which the output of the systemic (peripheral) circulation is used as input for the pulmonary (central) circulation and the output of the pulmonary circulation is used as input for the systemic circulation (Figure 18). The output of the systemic circulation is obtained as the sum of the outputs of the individual organs expressed as mass fluxes, with splanchic and hepatic circulation in series. In the case of linear kinetics, the organ outputs (venous concentrations) are obtained as convolutions of the inputs (arterial concentrations) with organ impulse responses. The latter correspond to outflow profiles obtained from MID experiments. The results of MID experiments should therefore provide useful information for the building blocks of PBPK models. However, these two concepts have not been used in conjunction in the past.
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Figure 18. Recirculatory model using separate representations of transit time distributions for central and peripheral circulation. The output of the peripheral circulation is used as input for the central circulation and vice versa. From Weiss (101), with permission.
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In implementing this principle, Ebling et al. (98) used compartmental representations of individual organs. The parameters of the organ models were obtained from tissue contents measured at different times after venous bolus injection. With this concept, the whole body is represented by a multicompartment model. The use of Laplace transformations, however, allows a more general (nonparametric) implementation of recirculatory models (99-101), as outlined in the following.
The output from the systemic circulation is obtained by convolution of the impulse response of systemic circulation fs(t) with the concentration at the input to the systemic circulation; the latter is equal to the measured concentration at the arterial sampling site, C(t). Given that with Laplace transformation convolutions become products, the Laplace transform of the output from the systemic circulation is C~(s)f~s(s). Input to the pulmonary circulation is the sum of the output of the systemic circulation and the bolus input. Given that the Laplace transform of a unit impulse function is 1, the Laplace transform of the input to the pulmonary circulation is D/CO + C~(s)f~s(s), where D is the dose amount and CO is cardiac output. The measured concentration at the arterial sampling site, C(t), is the convolution of the concentration of the input to the pulmonary circulation with the impulse response of the pulmonary circulation fp(t); its Laplace transform is
[34]
Solving for C~(s) yields:
[35]
Weiss et al. (100) and Weiss (101) describe such a recirculatory model using inverse Gaussian distributions as empirical representations of the impulse responses of the pulmonary and the systemic circulation (Figure 18). The time dependence of the arterial concentrations C(t) were obtained by numerical inversion of the Laplace transform C~(s) (102). In this implementation, the different organs involved in the systemic circulation thus were not treated separately. The model was used to describe the disposition kinetics of sorbitol, a hydrophilic compound that does not partition significantly into cells, such that organs are not explicitly included in the model.
The principle outlined above should allow integration of results of MID experiments in PBPK models, as it can easily accommodate organ impulse responses not derived from compartmental models.
There is increasing emphasis that transport--influx into and efflux from the organ--as well as the drug-metabolizing activities are important in the determination of drug transit time and exposure. It becomes mandatory to understand the role of transport--either from
in vivo or
in vitro experiments such as isolated enzymes, single cells, or reconstituted transport systems--in xenobiotic metabolism within the whole organ. Although molecular cloning experiments have inevitably identified the role of transporters for the influx, efflux, and canalicular excretion of substrates (
103,104), the relative importance of these transporters for uptake, efflux, or excretion in the whole organ is uncertain because the quantitative expression of the transporters is unknown. The MID technique preserves the vascular architecture, the endothelial barrier, and the polarity of parenchymal cells, discriminating between the basolateral and apical parts of the plasma membrane for net influx and excretion. MID studies therefore allow an overall appraisal of membrane transport, tissue partition, and the excretion and metabolizing activities of the whole organ, with separate estimation of Michaelis constants for these processes (
23). As illustrated in the examples, the MID method has great utility in describing carrier-mediated transport (
25,26,65,105,106), metabolic conversion (
107), red cell carriage (
23,68,108), and tissue binding (
25,26). By performing several MID experiments on the same specimen at differing steady-state concentrations of the unlabeled substrates, dependence of permeability, vascular binding, metabolic conversion, and excretion on concentration may be assessed. As shown in Table 2, comparison of the derived parameters from MID studies readily reveals the slowest of the steps that would qualify as rate-limiting steps in the overall removal of the compound. In summary, the MID method is a dynamic method that can provide information that is not obtainable from steady-state studies. For example, it can be used to reveal the existence of a transport barrier, to determine concentrations and metabolic rates in inaccessible compartments such as the cytosol or mitochondria, and to clarify some aspects of metabolic control.
The compartmental structure of the PBPK models currently used in risk assessment studies presumes a uniform concentration within the organ and that the tissue concentration equals that in venous blood. In organs in which substantial elimination or metabolism of a substance occurs, MID experiments provide a more physiological picture that considers concentration gradients along the flow paths of the liver. Concentrations near the arterial inflow site of the capillary bed are similar to the arterial concentration and can be considerably higher than the venous concentration, whereas the concentrations of metabolites can be lower in the capillaries than within the cell of the formation organ (69). Concomitantly, intracellular concentrations of substrates and metabolites can develop similar acinar gradients. An understanding of the kinetics could explain the cause for localized toxic effects, as observed in the case of hepatic toxicity of carbon tetrachloride (109).
Integration of results of MID experiments into PBPK models could be achieved in future studies using the concept of recirculatory models according to Equation 35. This integration can be facilitated by using parametric approximations of the outflow profiles such as inverse Gaussian functions, as suggested by the dispersion model. Whenever the quasi-steady-state approximation is applicable to whole-body models, steady-state models of organ elimination can be used. Since xenobiotic metabolism occurs mainly in liver or kidney, which are well-perfused organs with rapid turnover of metabolites, such an approximation will very often be feasible. Nevertheless, the quasi-steady-state approach has never been systematically implemented in PBPK models.
The utility of the MID method lies in its role in determining the basic mechanisms of the interaction of organs with vascular components. MID experiments are also useful tools for providing some of the kinetic parameters needed for PBPK models. The usual strategy in PBPK model discrete ue compartments relies on tissue partition coefficients obtained from vial equilibrium assays, whereby samples of homogenized tissues are analyzed under equilibrium conditions [either from the head space of gas chromatography sample vials (110,111) or by equilibration with an organic phase (112)]. The MID technique, because it is a dynamic method, provides values of kinetic parameters such as permeability surface area products for cellular influx and efflux. Parameters obtained experimentally from intact organs are more reliable because they conform more to the in vivo situation, especially for compounds taken up into cells by transport systems or involved in energy-dependent processes. Barrier-limited transport between a blood and a tissue compartment within an organ have previously been incorporated into PBPK models (98,113-115) using permeability parameters (diffusional resistances) that were optimized using the available whole-body pharmacokinetic data. Permeability parameters obtained from MID studies may further be used for validation of flow limitation or barrier limitation at different organs. Barrier limitation is especially pertinent when lipophilic compounds are metabolized within cells to form hydrophilic products, which are mostly retained within the cells. Accumulation of metabolites will occur under such circumstances (100); this may have implications for their biological action if the intracellular space is the target site. Similarly, MID experiments are suitable for determining rates of intracellular metabolic processes. Such measurements are complementary to measurements with cell extracts of subcellular fractions, as the whole organ provides a more realistic microenvironment for the participating enzymes. Finally, intact organs allow the quantification of excretory processes such as biliary excretion.
Although MID and PBPK methods have been used for some time, there is almost no overlap in the substances investigated. PBPK methods have been used mainly for lipophilic xenobiotics, whereas MID data are available for use with both lipophilic (29,30) and hydrophilic substances (27,32,33) undergoing carrier-mediated transport. Accordingly, MID experiments may be applied in a fruitful fashion to enhance PBPK models. Future efforts to use MID for characterizing the organ disposition of environmentally relevant substances will certainly improve the relevance of biological risk assessment.
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Last Updated: October 4, 2000
