In classic kinetics, the rate constants are defined by the data and these models are often 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 allometric scaling, and (4) complex dosing regimens and saturable processes such as metabolism and binding are easily accommodated.
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 believed to be important in describing a chemical's disposition.
Physiologic modeling has enormous potential predictive power compared with classic compartmental modeling. 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 and blood flow). 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, respiration rate, etc., for rodents and humans are different. Other parameters, such as solubility in tissues, are similar in the rodent 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. Because physiologic models represent real, measurable values, such as blood flows and ventilation rates, the same model structure can resolve such disparate kinetic behaviors among species.
The basic unit of the physiologic model is the lumped compartment (Figure 7–3), which is a single region of the body with a uniform xenobiotic concentration. A compartment may be a particular functional or anatomical portion of an organ, a single blood vessel with 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.
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 are often 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 Figure 7–3, the toxicant enters the vascular subcompartment at a certain rate in mass per unit of time (e.g., mg/h). The rate of entry is a product of the blood flow rate to the tissue (Qt, L/h) and the concentration of the toxicant in the blood entering the tissue (Cin, mg/L). Within the compartment, the toxicant 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 a different net rate (Flux2). Some toxicants can bind to cell components; thus, within a compartment there may be both free and bound toxicants. The toxicant leaves the vascular space at a certain venous concentration (Cout). Cout is equal to the concentration of the toxicant in the vascular space.
The most common types of parameters, or information required, in physiologic models are anatomical, physiologic, thermodynamic, and transport.
Anatomical parameters are used to physically describe the various compartments. The size of each of the compartments in the physiologic model must be known. The size is generally specified as a volume (milliliters or liters) because a unit density is assumed even though weights are most frequently obtained experimentally. If a compartment contains subcompartments such as those in Figure 7–3, those volumes also must be known. Volumes of compartments often can be obtained from the literature or from specific toxicokinetic experiments.
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 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 parameters relate the total concentration of a xenobiotic in a tissue (Ct) 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 × partition coefficient, or Ct = CfPt. Knowledge of the value of Pt, a partition or distribution coefficient, permits an indirect calculation of the free concentration of xenobiotic or Cf = Ct/Pt.
Table 7–1 compares the partition coefficients for a number of toxic volatile organic chemicals. The larger values for the fat/blood partition coefficients compared with those for other tissues suggest that these chemicals distribute into fat to a greater extent than they distribute into other tissues.
Table 7–1 Partition coefficients for four volatile organic chemicals in several tissues. ||Download (.pdf)
Table 7–1 Partition coefficients for four volatile organic chemicals in several tissues.
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.
The passage of a chemical 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 (Flux1 in Figure 7–3) or across the cell membrane (Flux2 in Figure 7–3). For simple diffusion, the net flux (mg/h) from one side of a membrane to the other is described as Flux = permeability coefficient × driving force, or:
The permeability coefficient [PA] is often called the permeability–area cross-product for the membrane (L/h) and is a product of the cell membrane permeability constant (P, μm/h) for the xenobiotic and the total membrane area (A, μm2). The permeability constant takes into account the rate of diffusion of the specific xenobiotic and the thickness of the cell membrane. C1 and C2 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.
A perfusion-limited compartment is also 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, uptake of xenobiotic by tissue subcompartments is limited by the rate at which the blood containing a xenobiotic arrives at the tissue and 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 in Figure 7–3, 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 are usually lumped together as a single compartment that is often called the extracellular space.
As indicated in Figure 7–3, 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), and the intracellular compartment is in equilibrium with the extracellular compartment, and these tissue subcompartments are usually lumped as a single compartment. This flow-limited tissue compartment is shown in Figure 7–4. 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, Ct the concentration of free xenobiotic in the compartment (VtCt equals the amount of xenobiotic in the compartment), Vt(dCt/dt) the change in the amount of xenobiotic in the compartment with time expressed as mass per unit of time, Qt the blood flow to the tissue, Cin the xenobiotic concentration entering the compartment, and Cout the xenobiotic concentration leaving the compartment. These mass balance differential equations require that input into one equation must be balanced by outflow from another equation in the physiologic model.
Schematic representation of a compartment that is blood flow-limited. Rapid exchange between the extracellular space (salmon) and intracellular space (bisque) 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 = Ct/Pt. In this case, the differential equation describing the rate of change in the amount of a xenobiotic in a tissue becomes
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.
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 of a xenobiotic across cell membranes is slow compared with blood flow to the tissue. In this case, the permeability–area cross-product is small compared with blood flow, or PA << Qt. Figure 7–5 shows the structure of such a compartment. 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 is thus diffusion-limited. Two mass balance differential equations are necessary to describe this compartment:
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.
Here, Qt is blood flow and C the 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](C1 – C2).
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: (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 Figure 7–6, 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 via 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 via 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):
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 (salmon) 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; 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, for example, Calv = Cart/Pb.
Because of these assumptions, at steady state 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.
The liver is often 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 in Figure 7–4, 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 (mg/h), Cf the free concentration of xenobiotic in the liver (mg/L), Vl the liver volume (L), and Kf the first-order rate constant for metabolism (h–1).
In physiologic models, the Michaelis–Menten expression for saturable metabolism, which employs two parameters, Vmax and KM, is written as:
where Vmax is the maximum rate of metabolism (mg/h) and KM the Michaelis constant, or xenobiotic concentration at one-half the maximum rate of metabolism (mg/L). Because many xenobiotics are metabolized by enzymes that display saturable metabolism, the above equation is a key factor in the success of physiologic models for 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 using physiologic models. Metabolism can be also included in other compartments in much the same way as described for the liver.