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Casarett & Doull's Essentials of Toxicology > Unit 2. Disposition of Toxicants > Chapter 7: Toxicokinetics >

Key Points

Toxicokinetics is the study of the modeling and mathematical description of the time course of disposition (absorption, distribution, biotransformation, and excretion) of xenobiotics in the whole organism.

The apparent volume of distribution (Vd ) is the apparent space into which an amount of chemical is distributed in the body to result in a given plasma concentration.

Clearance describes the rate of chemical elimination from the body in terms of the volume of fluid containing chemical that is cleared per unit of time.

The half-life of elimination (T½) is the time required for the blood or plasma chemical concentration to decrease by one-half.

Introduction

Toxicokinetics is the study of the modeling and mathematical description of the time course of disposition (absorption, distribution, biotransformation, and excretion) of xenobiotics in the whole organism. In the classic model, chemicals are said to move throughout the body as if there were one or two compartments that might have no apparent physiologic or anatomic reality. In physiologically based toxicokinetic models, the body is represented as a series of mass balance equations that describe each organ or tissue on the basis of physiologic considerations. There is no inherent contradiction between the classic and physiologically based approaches, yet certain assumptions differ between the two models. Ideally, physiologic models can predict tissue concentrations, whereas classic models cannot.

Classic Toxicokinetics

The least invasive and simplest method for gathering information on the absorption, distribution, metabolism, and elimination of a compound is sampling blood or plasma over time. If one assumes that the concentration of a compound in blood or plasma is in equilibrium with concentrations in tissues, changes in plasma chemical concentrations reflect changes in tissue chemical concentrations. Compartmental pharmacokinetic models consist of

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a central compartment representing plasma and tissues that equilibrate rapidly with chemical, connected to one or more peripheral compartments that represent tissues that equilibrate more slowly with chemical (). Chemical is administered into the central compartment and is distributed between central and peripheral compartments. Chemical elimination occurs from the central compartment, which is assumed to contain rapidly perfused tissues that are capable of eliminating chemical (e.g., kidneys and liver). Compartmental pharmacokinetic models require no information on tissue physiology or anatomic structure, and they are valuable in predicting plasma chemical concentrations at different doses, establishing the time course of chemical in plasma and tissues and the extent of chemical accumulation with multiple doses, and determining effective doses and dose regimens in toxicity studies.

Figure 7-1. Compartmental pharmacokinetic models where ka is the first-order extravascular absorption rate constant into the central compartment (1), kel is the first-order elimination rate constant from the central compartment (1), and k12 and k21 are the first-order rate constants for distribution of chemical into and out of the peripheral compartment (2) in a two-compartment model.

One-Compartment Model

The simplest toxicokinetic analysis entails measurement of the plasma concentrations of a xenobiotic at several time points after the administration of a bolus intravenous injection. If the data obtained yield a straight line when they are plotted as the logarithms of plasma concentrations versus time, the kinetics of the xenobiotic can be described with a one-compartment model (). Compounds whose toxicokinetics can be described with a one-compartment model equilibrate rapidly, or mix uniformly, between blood and the various tissues relative to the rate of elimination. The one-compartment model depicts the body as a homogeneous unit. This does not mean that the concentration of a compound is the same throughout the body, but it does assume that the changes that occur in the plasma concentration reflect proportional changes in tissue chemical concentrations.

Figure 7-2. Concentration versus time curves of chemicals exhibiting behavior of a one-compartment pharmacokinetic model (top) and a two-compartment pharmacokinetic model (bottom) on a linear scale (left) and a semilogarithmic scale (right).

Elimination rate constants, kel and are determined from the slope of the log-linear concentration versus time curve. Half-life (T½) is the time required for blood or plasma chemical concentration to decrease by one-half. C 0 is the concentration of the chemical at t = 0 determined by extrapolating the log-linear concentration time curve to the Y-axis (t = 0).In the simplest case, a curve of this type can be described by the expression

where C is the blood or plasma chemical concentration over time t, C 0 is the initial blood concentration at time t = 0, and kel is the first-order elimination rate constant with dimensions of reciprocal time (e.g., t – 1).

Two-Compartment Model

After the rapid intravenous administration of some chemicals, the semilogarithmic plot of plasma concentration versus time yields a curve rather than a straight line, and this implies that there is more than one dispositional phase. In these instances, the chemical requires a longer time for tissue concentrations to reach equilibrium with the concentration in plasma, and a multicompartmental analysis of the results is necessary (). A multiexponential mathematical equation then best characterizes the elimination of the xenobiotic from the plasma.

Generally, a curve of this type can be resolved into two monoexponential terms (a two-compartment model) and is described by

where A and B are proportionality constants and and are the first-order

distribution and elimination rate constants, respectively (). During the distribution ( ) phase, concentrations of the chemical in the plasma decrease more rapidly than they do in the

postdistributional elimination ( ) phase. The distribution phase may last for only a few

minutes or for hours or days. The equivalent of kel in a one-compartment model is in a two-compartment model.

Elimination

Elimination includes biotransformation, exhalation, and excretion. The elimination of a chemical from the body whose disposition is described by a one-compartment model usually occurs through a first-order process; that is, the rate of elimination at any time is proportional to the amount of the chemical in the body at that time. First-order reactions occur at chemical concentrations that are not sufficiently high to saturate elimination processes.

The equation for a monoexponential model, C = C 0 x e –kel

 x t, can be transformed to a logarithmic equation that has the general form of a straight line, y = mx + b:

where log C 0 represents the y-intercept or initial concentration and (kel /2.303) represents the slope of the line. The first-order elimination rate constant (kel ) can be determined from the slope of the log C versus time plot (i.e., kel  = 2.303 x slope). The first-order elimination rate

constants, kel and , have units of reciprocal time (e.g., min– 1 and h– 1) and are independent of dose.

Mathematically, the fraction of dose remaining in the body over time (C/C 0) is calculated by using the elimination rate constant by rearranging the equation for the monoexponential function and taking the antilog to yield

Apparent Volume of Distribution

In a one-compartment model, all chemical is assumed to distribute into plasma and tissues instantaneously. The apparent volume of distribution (Vd ) is a proportionality constant that relates the total amount of chemical in the body to the concentration of a xenobiotic in plasma and typically is described in units of liters or liters per kilogram of body weight. Vd is the apparent space into which an amount of chemical is distributed in the body to result in a given plasma concentration. For example, the apparent volume of distribution of a chemical in the body is determined after intravenous bolus administration and is defined mathematically as the quotient of the amount of chemical in the body and its plasma concentration. Vd is calculated as

where Doseiv is the intravenous dose or known amount of chemical in body at time zero,

is the elimination rate constant, and AUC 0 ∞ is the area under the chemical

concentration versus time curve from time zero to infinity. The product, x AUC 0 ∞, is the concentration of xenobiotic in plasma.

For a one-compartment model, Vd can be simplified by the equation Vd  = Doseiv /C 0, where C 0 is the concentration of chemical in plasma at time zero. C 0 is determined by extrapolating the plasma disappearance curve after intravenous injection to the zero time point (). Vd is called the apparent volume of distribution. The magnitude of the Vd term is chemical-specific and represents the extent of distribution of chemical out of plasma and into other body tissues. Thus, a chemical with high affinity for tissues also will have a large volume of distribution. Conversely, a chemical that predominantly remains in the plasma will have a low Vd that approximates the volume of plasma. Once the Vd for a chemical is known, it can be used to estimate the amount of chemical remaining in the body at any time if the plasma concentration at that time also is known by the relationship Xc  = Vd  x Cp , where Xc is the amount of chemical in the body and Cp is the plasma chemical concentration.

Clearance

Clearance describes the rate of chemical elimination from the body in terms of the volume of fluid containing chemical that is cleared per unit of time. Thus, clearance has the units of flow (milliliters per minute). A clearance of 100 mL/min means that 100 mL of blood or plasma containing xenobiotic is cleared completely each minute.

The overall efficiency of the removal of a chemical from the body can be characterized by clearance. High values of clearance indicate efficient and generally rapid removal, whereas low clearance values indicate slow and less efficient removal of a xenobiotic from the body. Total body clearance is defined as the sum of clearances by individual eliminating organs:

where Clr depicts renal, Clh hepatic, and Cli intestinal clearance. After bolus intravenous administration, total body clearance is defined as

Clearance also can be calculated if the volume of distribution and elimination rate constants are known and can be defined as Cl = Vd  x kel for a one-compartment model and Cl = Vd

x  for a two-compartment model.

Half-Life

The half-life of elimination (T½) is the time required for the blood or plasma chemical concentration to decrease by one-half and is dependent on both volume of distribution and clearance. T½ can be calculated if Vd and Cl are known:

Because of the relationship T½ = 0.693 x kel , the half-life of a compound can be calculated

after kel (or ) has been determined from the slope of the line that designates the elimination phase on the log C versus time plot. The T½ also can be determined by means of visual inspection of the log C versus time plot, as is shown in . For compounds eliminated by first-order kinetics, the time required for the plasma concentration to decrease by one-half is constant. After seven half-lives, 99.2 percent of a chemical is eliminated, and this can be practically viewed as complete elimination. The half-life of a chemical obeying first-order elimination kinetics is independent of the dose and does not change with increasing dose.

Saturation Toxicokinetics

As the dose of a compound increases, its volume of distribution or its rate of elimination may change owing to saturation kinetics. Biotransformation, active transport processes, and protein binding have finite capacities and can be saturated. When the concentration of a chemical in the body is higher than the KM (chemical concentration at one-half Vmax , the maximum metabolic capacity), the rate of elimination is no longer proportional to the dose. The transition from first-order kinetics to saturation kinetics is important in toxicology because it can lead to prolonged residency time of a compound in the body or increased concentration at the target site of action, which may result in increased toxicity.

Nonlinear toxicokinetics are indicated by the following: (1) The decline in the levels of the chemical in the body is not exponential, (2) AUC 0 ∞ is not proportional to the dose, (3) Vd ,

Cl, kel (or ), or T½ changes with increasing dose, (4) the composition of excretory products changes quantitatively or qualitatively with the dose, (5) competitive inhibition by other chemicals that are biotransformed or actively transported by the same enzyme system occurs, and (6) dose–response curves show a nonproportional change in response with an increasing dose, starting at the dose level at which saturation effects become evident.

The elimination of some chemicals from the body is readily saturated. Important

characteristics of zero-order processes are as follows: (1) An arithmetic plot of plasma concentration versus time yields a straight line, (2) the rate or amount of chemical eliminated at any time is constant and is independent of the amount of chemical in the body, and (3) a true T½ or kel does not exist but differs depending on the dose.

By comparison, the important characteristics of first-order elimination are as follows: (1) The rate at which a chemical is eliminated at any time is directly proportional to the amount of that chemical in the body at that time. (2) A semilogarithmic plot of plasma concentration

versus time yields a single straight line. (3) The elimination rate constant (kel or ), apparent volume of distribution (Vd ), clearance (Cl), and half-life (T½) are independent of dose. (4) The concentration of the chemical in plasma and other tissues decreases similarly by

some constant fraction per unit of time, the elimination rate constant (kel or ).

Bioavailability

The extent of absorption of a xenobiotic can be determined experimentally by comparing the plasma AUC 0 ∞ after intravenous and extravascular dosing. The resulting index quantifies the fraction of the dose absorbed systemically and is called bioavailability (F). Bioavailability can be determined by using different doses provided that the compound does not display dose-dependent or saturable kinetics. Pharmacokinetic data after intravenous administration are used as the reference from which to compare extravascular absorption because all chemical is delivered (or is 100 percent bioavailable) to the systemic circulation. For example, bioavailability after an oral exposure can be determined as follows:

where AUCpo , AUCiv , Dosepo , and Doseiv are the respective areas under the plasma concentration versus time curves and doses for oral and intravenous administration. Bioavailabilities for various chemicals range in value between 0 and 1. Complete absorption of chemical is demonstrated when F = 1. When F < 1, incomplete absorption of chemical is indicated. The fraction of a chemical that reaches the systemic circulation is of critical importance in determining toxicity. Several factors can alter this systemic availability greatly, including (1) limited absorption after oral dosing, (2) intestinal first-pass effect, (3) hepatic first-pass effect, and (4) mode of formulation, which affects, for example, dissolution rate or incorporation into micelles (for lipid-soluble compounds).

Physiologic Toxicokinetics

In classic kinetics, the rate constants are defined by the data, and these models often are referred to as data-based. In physiologically based models, the rate constants represent known

or hypothesized biological processes. The advantages of physiologically based models are that (1) these models can provide the time course of distribution of xenobiotics to any organ or tissue, (2) they allow estimation of the effects of changing physiologic parameters on tissue concentrations, (3) the same model can predict the toxicokinetics of chemicals across species by means of allometric scaling, and (4) complex dosing regimes and saturable processes such as metabolism and binding are accommodated easily.

Basic Model Structure

Physiologic models often look like a number of classic one-compartment models that are linked together. The actual model structure, or how the compartments are linked together, depends on both the chemical and the organism being studied. It is important to realize that there is no generic physiologic model. Models are simplifications of reality and ideally should contain elements that are believed to be important in describing a chemical's disposition.

Physiologic modeling has enormous potential predictive power. Because the kinetic constants in physiologic models represent measurable biological or chemical processes, the resultant physiologic models have the potential for extrapolation from observed data to predicted situations.

One of the best illustrations of the predictive power of physiologic models is their ability to extrapolate kinetic behavior from laboratory animals to humans. Simulations are the outcomes or results (such as a chemical's concentration in blood or tissue) of numerically integrating model equations over a simulated time period, using a set of initial conditions (such as intravenous dose) and parameter values (such as organ weights). Whereas the model structures for the kinetics of chemicals in rodents and humans may be identical, the parameter values, such as organ weight, heart beat rate, and respiration rate, for rodents and humans are different. Other parameters, such as solubility in tissues, are similar in rodents and human models because the composition of tissues in different species is similar. Because the parameters underlying the model structure represent measurable biological and chemical determinants, the appropriate values for those parameters can be chosen for each species, forming the basis for successful interspecies extrapolation.

Compartments

The basic unit of the physiologic model is the lumped compartment (), which is a single region of the body with a uniform xenobiotic concentration. A compartment may be a particular functional or anatomic portion of an organ, a single blood vessel with the surrounding tissue, an entire discrete organ such as the liver or kidney, or a widely distributed tissue type such as fat or skin. Compartments consist of three individual well-mixed phases, or subcompartments. These subcompartments are (1) the vascular space through which the compartment is perfused with blood, (2) the interstitial space that forms the matrix for the cells, and (3) the intracellular space consisting of the cells in the tissue.

Figure 7-3. Schematic representation of a lumped compartment in a physiologic model.

The blood capillary and cell membranes separating the vascular, interstitial, and intracellular subcompartments are depicted in black. The vascular and interstitial subcompartments often are combined into a single extracellular subcompartment. Qt is blood flow, Cin is chemical concentration into the compartment, and Cout is chemical concentration out of the compartment.As shown in , the xenobiotic enters the vascular subcompartment at a certain rate in mass per unit of time (e.g., milligrams per hour). The rate of entry is a product of the blood flow rate to the tissue (Qt , in liters per hour) and the concentration of the xenobiotic in the blood entering the tissue (Cin , in milligrams per liter). Within the compartment, the xenobiotic moves from the vascular space to the interstitial space at a certain net rate (Flux1 ) and from the interstitial space to the intracellular space at different net rate (Flux2 ). Some xenobiotics can bind to cell components; thus, within a compartment there may be both free and bound xenobiotics. The xenobiotic leaves the vascular space at a certain venous concentration (Cout ). Cout is equal to the concentration of the xenobiotic in the vascular space.

Parameters

The most common types of parameters, or information required, in physiologic models are anatomic, physiologic, thermodynamic, and transport.

Anatomic

Anatomic parameters are used to describe the various compartments physically. The size of each of the compartments in the physiologic model must be known. The size generally is specified as a volume (milliliters or liters) because a unit density is assumed even though weights most frequently are obtained experimentally. If a compartment contains subcompartments such as those shown in , those volumes also must be known. Volumes of compartments often can be obtained from the literature or from specific toxicokinetic experiments.

Physiologic

Physiologic parameters encompass various processes, including blood flow, ventilation, and elimination. The blood flow rate (Qt , in volume per unit time, such as mL/min or L/h) to individual compartments must be known. Additionally, information on the total blood flow rate or cardiac output (Qc ) is necessary. If inhalation is the route for exposure to the xenobiotic or is a route of elimination, the alveolar ventilation rate (Qp ) also must be known. Blood flow rates and ventilation rates can be taken from the literature or can be obtained experimentally. Renal clearance rates and parameters to describe rates of biotransformation are required if these processes are important in describing the elimination of a xenobiotic.

Thermodynamic

Thermodynamic parameters relate the total concentration of a xenobiotic in a tissue (C) to the concentration of free xenobiotic in that tissue (Cf ). Two important assumptions are that (1) total and free concentrations are in equilibrium with each other and (2) only free xenobiotic can enter and leave the tissue. Whereas total concentration is measured experimentally, it is the free concentration that is available for binding, metabolism, or removal from the tissue by blood. The extent to which a xenobiotic partitions into a tissue is directly dependent on the composition of the tissue and independent of the concentration of the xenobiotic. Thus, the relationship between free and total concentration becomes one of proportionality: total = free x partition coefficient, or C = Cf x P. Knowledge of the value of P, a partition or distribution coefficient, permits an indirect calculation of the free concentration of xenobiotic or Cf · Cf

= C/P.

A more complex relationship between the free concentration and the total concentration of a chemical in tissues occurs when the chemical may bind to saturable binding sites on tissue components. In these cases, nonlinear functions relating the free concentration in the tissue to the total concentration are necessary.

Transport

The passage of a xenobiotic across a biological membrane is complex and may occur by passive diffusion, carrier-mediated transport, facilitated transport, or a combination of processes. The simplest of these processes—passive diffusion—is a first-order process. Diffusion of xenobiotics can occur across the blood capillary membrane (Flux 1 in ) or across the cell membrane (Flux 2 in ). Flux refers to the rate of transfer of a xenobiotic across a boundary. For simple diffusion, the net flux (milligrams per hour) from one side of a membrane to the other is described as Flux = permeability coefficient x driving force, or

The permeability coefficient [PA] often is called the permeability-area cross-product for the membrane (in units of liters per hour) and is a product of the cell membrane permeability constant (P, in micrometers per hour) for the xenobiotic and the total membrane area (A, in square micrometers). The cell membrane permeability constant takes into account the rate of diffusion of the specific xenobiotic and the thickness of the cell membrane. C 1 and C 2 are the free concentrations of xenobiotic on each side of the membrane. For any given xenobiotic, thin membranes, large surface areas, and large concentration differences enhance diffusion.

There are two limiting conditions for the transport of a xenobiotic across membranes: perfusion-limited and diffusion-limited.

Perfusion-Limited Compartments

A perfusion-limited compartment also is referred to as blood flow–limited or simply flow-limited. A flow-limited compartment can be developed if the cell membrane permeability coefficient [PA] for a particular xenobiotic is much greater than the blood flow rate to the tissue (Qt ). In this case, the rate of xenobiotic uptake by tissue subcompartments is limited by the rate at which the blood containing a xenobiotic arrives at the tissue, not by the rate at which the xenobiotic crosses the cell membranes. In most tissues, transport across vascular cell membranes is perfusion-limited. In the generalized tissue compartment shown in , this means that transport of the xenobiotic through the loosely knit blood capillary walls of most tissues is rapid compared with delivery of the xenobiotic to the tissue by the blood. As a result, the vascular blood is in equilibrium with the interstitial subcompartment and the two subcompartments usually are lumped together as a single compartment that often is called the extracellular space.

As is indicated in , the cell membrane separates the extracellular compartment from the intracellular compartment. The cell membrane is the most important diffusional barrier in a tissue. Nonetheless, for molecules that are very small (molecular weight <100) or lipophilic, cellular permeability generally does not limit the rate at which a molecule moves across cell membranes. For these molecules, flux across the cell membrane is fast compared with the tissue perfusion rate ([PA] Qt ), the intracellular compartment is in equilibrium with the extracellular compartment, and these tissue subcompartments usually are lumped as a single compartment. This flow-limited tissue compartment is shown in . Movement into and out of the entire tissue compartment can be described by a single equation:

where Vt is the volume of the tissue compartment, C is the concentration of free xenobiotic in the compartment (Vt x C equals the amount of xenobiotic in the compartment), Vt x dC/dt is the change in the amount of xenobiotic in the compartment with time expressed as mass per unit of time, Qt is blood flow to the tissue, Cin is xenobiotic concentration entering the compartment, and Cout is xenobiotic concentration leaving the compartment. These mass balance differential equations require that input into one equation be balanced by outflow from another equation in the physiologic model.

Figure 7-4. Schematic representation of a compartment that is blood flow–limited.

Rapid exchange between the extracellular space (blue) and intracellular space (light blue) maintains the equilibrium between them as symbolized by the dashed line. Qt is blood flow, Cin is chemical concentration into the compartment, and Cout is chemical concentration out of the compartment.

In the perfusion-limited case, Cout , or the venous concentration of xenobiotic leaving the tissue, is equal to the free concentration of xenobiotic in the tissue, Cf . As was noted above, Cf (or Cout ) can be related to the total concentration of xenobiotic in the tissue through a simple linear partition coefficient, Cout = Cf = C/P. In this case, the differential equation describing the rate of change in the amount of a xenobiotic in a tissue becomes Vt x dC/dt = Qt

x (Cin – C/P)

In a flow-limited compartment, the assumption is that the concentrations of a xenobiotic in all parts of the tissue are in equilibrium. Additionally, estimates of flux are not required to develop the mass balance differential equation for the compartment. Given the information required to estimate flux, this simplifying assumption significantly reduces the number of parameters required in the physiologic model.

Diffusion-Limited Compartments

When uptake into a compartment is governed by cell membrane permeability and total membrane area, the model is said to be diffusion-limited, or membrane-limited. Diffusion-limited transport occurs when the flux, or the transport of a xenobiotic across cell membranes, is slow compared with blood flow to the tissue. In this case, the permeability-area cross-product [PA] is small compared with blood flow, Qt , or PA << Qt . shows the structure of a compartment of this type. The xenobiotic concentrations in the interstitial and vascular spaces are in equilibrium and make up the extracellular subcompartment where uptake from the incoming blood is flow-limited. The rate of xenobiotic uptake across the cell membrane (into the intracellular space from the extracellular space) is limited by cell membrane permeability and thus is diffusion-limited. Two mass balance differential equations are necessary to describe this compartment:

Qt is blood flow, and C is free xenobiotic concentration in entering blood (in), exiting blood (out), extracellular space (1), or intracellular space (2). Both equations contain terms for flux, or transfer across the cell membrane, [PA] x (C 1 – C 2).

Figure 7-5. Schematic representation of a compartment that is membrane-limited.

Perfusion of blood into and out of the extracellular compartment is depicted by thick arrows. Transmembrane transport (flux) from the extracellular to the intracellular subcompartment is depicted by thin double arrows. Qt is blood flow, Cin is chemical concentration into the compartment, and Cout is chemical concentration out of the compartment.

Specialized Compartments

Lung

The inclusion of a lung compartment in a physiologic model is an important consideration because inhalation is a common route of exposure to many toxic chemicals. The assumptions inherent in lung compartment description are as follows: (1) Ventilation is continuous, not cyclic, (2) conducting airways function as inert tubes, carrying the vapor to the gas exchange region, (3) diffusion of vapor across the lung cell and capillary walls is perfusion-limited, (4) all xenobiotic disappearing from inspired air appears in arterial blood (i.e., there is no storage of xenobiotic in the lung tissue and insignificant lung mass), and (5) vapor in the alveolar air and arterial blood within the lung compartment are in rapid equilibrium.

In the lung compartment depicted in , the rate of inhalation of xenobiotic is controlled by the ventilation rate (Qp ) and the inhaled concentration (Cinh ). The rate of exhalation of a xenobiotic is a product of the ventilation rate and the xenobiotic concentration in the alveoli (Calv ). Xenobiotic also can enter the lung compartment in venous blood returning from the heart, represented by the product of cardiac output (Qc ) and the concentration of xenobiotic in venous blood (Cven ). Xenobiotic leaving the lungs in the blood is a function of both cardiac output and the concentration of xenobiotic in arterial blood (Cart ). Putting these four processes together, a mass balance differential equation can be written for the rate of change in the amount of xenobiotic in the lung compartment (L):

Figure 7-6. Simple model of gas exchange in the alveolar region of the respiratory tract.

Rapid exchange in the lumped lung compartment between the alveolar gas (blue) and the pulmonary blood (light blue) maintains the equilibrium between them as symbolized by the dashed line. Qp is alveolar ventilation (L/h), Qc is cardiac output (L/h), Cinh is inhaled vapor concentration (mg/L), Cart is concentration of vapor in the arterial blood, and Cven is concentration of vapor in the mixed venous blood. The equilibrium relationship between the chemical in the alveolar air (Calv ) and the chemical in the arterial blood (Cart ) is determined by the blood/air partition coefficient Pb , e.g., Calv = Cart /Pb .Because of some of these assumptions, the rate of change in the amount of xenobiotic in the lung compartment becomes equal to zero (dl/dt = 0). Calv can be replaced by Cart /Pb , and the differential equation can be solved for the arterial blood concentration:

The lung is viewed here as a portal of entry and not as a target organ, and the concentration of a xenobiotic delivered to other organs by the blood, or the arterial concentration of that xenobiotic, is of primary interest.

Liver

The liver often is represented as a compartment in physiologic models because hepatic biotransformation is an important aspect of the toxicokinetics of many xenobiotics. A simple compartmental structure for the liver is assumed to be flow-limited, and this compartment is similar to the general tissue compartment shown in , except that the liver compartment contains an additional process for metabolic elimination. One of the simplest expressions for this process is first-order elimination:

where R is the rate of metabolism (milligrams per hour), Cf is the free concentration of xenobiotic in the liver (milligrams per liter), Vl is the liver volume (liters), and Kf is the first-order rate constant for metabolism in units of h–1.

In physiologic models, the Michaelis-Menten expression for saturable metabolism, which employs two parameters, Vmax and KM , is written as follows:

where Vmax is the maximum rate of metabolism (in milligrams per hour) and KM is the Michaelis constant, or xenobiotic concentration at one-half the maximum rate of metabolism (in milligrams per liter). Because many xenobiotics are metabolized by enzymes that display saturable metabolism, this equation is a key factor in the success of physiologic models for the simulation of chemical disposition across a range of doses.

Other, more complex expressions for metabolism also can be incorporated into physiologic models. Bisubstrate second-order reactions—reactions involving the destruction of enzymes, the inhibition of enzymes, or the depletion of cofactors—have been simulated by using physiologic models. Metabolism also can be included in other compartments in much the same way as was described for the liver.

Conclusion

This chapter has overviewed the simpler elements of physiologic models and the important

and often neglected assumptions that underlie model structures. The field of physiologic modeling is expanding rapidly. Physiologic models of a parent chemical linked in series with one or more active metabolites, models describing biochemical interactions among xenobiotics, and more biologically realistic descriptions of tissues previously viewed as simple lumped compartments are just a few of the most recent applications. Finally, physiologically based toxicokinetic models are beginning to be linked to biologically based

toxicodynamic models to simulate the entire exposure dose response paradigm that is basic to the science of toxicology.

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Shargel L, Yu ABC: Applied Biopharmaceutics and Pharmacokinetics, 3d ed. Norwalk, CT: Appleton & Lange, 1993.

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