Modeling nonlinear compartments

As might be apparent, this method is tedious and quickly becomes cumbersome. Other methods, such as determining the rank of the Jacobian matrix (Jacquez, 1996) or Taylor series expansion for nonlinear compartment models (Chappell, Godfrey, and Vajda, 1990), are even more difficult and cumbersome to perform. There... 

On some occasions, the body does not behave as a single homogeneous compartment, and multicompartment pharmacokinetics are required to describe the time course of drug concentrations. In other instances certain pharmacokinetic processes may not obey first-order kinetics and saturable or nonlinear models may be required. Additionally, advanced pharmacokinetic analyses require the use of various computer programs, such as those listed on the website http //www.boomer.org/pkin/soft.html. 

The mass transfer coefficients may also be expressed in units of time-1 by multiplying by the appropriate compartmental volume term. Irreversible drug elimination from the tissue requires the addition of an expression to the differential equation that represents the subcompartment in which elimination occurs. For instance, hepatic drug elimination would be described by a linear or nonlinear expression added to the intracellular liver compartment mass balance equation since this compartment represents the hepatocytes. Formal elimination terms are given below for the simplified tissue models. 

PBPK and classical pharmacokinetic models both have valid applications in lead risk assessment. Both approaches can incorporate capacity-limited or nonlinear kinetic behavior in parameter estimates. An advantage of classical pharmacokinetic models is that, because the kinetic characteristics of the compartments of which they are composed are not constrained, a best possible fit to empirical data can be arrived at by varying the values of the parameters (O Flaherty 1987). However, such models are not readily extrapolated to other species because the parameters do not have precise physiological correlates. Compartmental models developed to date also do not simulate changes in bone metabolism, tissue volumes, blood flow rates, and enzyme activities associated with pregnancy, adverse nutritional states, aging, or osteoporotic diseases. Therefore, extrapolation of classical compartmental model simulations... 

In addition to the mechanistic simulation of absorptive and secretive saturable carrier-mediated transport, we have developed a model of saturable metabolism for the gut and liver that simulates nonlinear responses in drug bioavailability and pharmacokinetics [19]. Hepatic extraction is modeled using a modified venous equilibrium model that is applicable under transient and nonlinear conditions. For drugs undergoing gut metabolism by the same enzymes responsible for liver metabolism (e.g., CYPs 3A4 and 2D6), gut metabolism kinetic parameters are scaled from liver metabolism parameters by scaling Vmax by the ratios of the amounts of metabolizing enzymes in each of the intestinal enterocyte compart-... 

Pharmacokinetic/pharmacodynamic model using nonlinear, mixed-effects model in two compartment, best described time course of concentration strong correlation with creatinine clearance predicted concentration at the efi ect site and in reduction of heart rate during atrial fibrillation using population kinetic approach... 

Extrapolation between species should ideally take into account metabolic routes, i.e., the absence or presence of metabolites, as well as the relative rate of formation of the individual metabolites. In PBPK models (Section 4.3.6), both aspects (nonlinearity, formation of active metabolites) are incorporated. This modeling technique uses compartments that correspond to actual tissues or tissue groups of the body. Size, blood flow, air flow, etc. are taken into account, in addition to specific compound-related parameters such as partition coefficients and metabolic rate data. Based on such studies, target-organ concentrations of active metabolites can be predicted in experimental animals and humans, thus providing the best possible basis for extrapolation (Feron et al. 1990). 

For a better idea of the toxicity of VOCs, we can look more closely at some studies of TCE (Bogen et al., 1998). In vitro uptake of C-14-labeled trichloroethylene (TCE) from dilute (similar to 5-ppb) aqueous solutions into human surgical skirt was measured using accelerator mass spectrometry (AMS). The AMS data obtained positively correlate with (p approximate to 0) and vary significantly nonlinearly with (p = 0.0094) exposure duration. These data are inconsistent (p approximate to 0) with predictions made for TCE by a proposed EPA dermal exposure model, even when uncertainties in its recommended parameter values for TCE are considered but are consistent (p = 0.17) with a 1-compartment model for exposed skin-surface. This study illustrates the power of AMS to facilitate analyses of contaminant biodistribution and uptake kinetics at very low environmental concentrations. Eurther studies could correlate this with toxicity. 

The relatively simple two-enzymes/two-compartments model is thus represented in (4.101) via the above set of eight coupled ordinary nonlinear differential equations (4.103) to (4.106). This system of IVPs has the eight state variables hj(t), sy(t), S2j(t), ssj(t) for j = 1,2 that depend on the time t. The normalized reaction rates rj t) are given in equations (4.107) and (4.108). The system has 26 parameters that describe the dynamics for all compounds considered in the two compartments. A specific list of validated experimental parameter values follows in Section 4.4.5. 

For the sake of simplicity, simple monophasic pharmacokinetics (one compartment and one half-life) was assumed in the above example and in many other examples in this report. In real life, most chemicals express biphasic or polyphasic pharmacokinetics (several compartments and several half-lives). Squeezing a polyphasic pharmacokinetic behavior into a one-compartment model by assuming a single half-life may lead to negligible errors for some chemicals and serious misinterpretation of biomarker concentrations for others. The same can be said about nonlinear processes, such as metabolic induction, inhibition, and saturation. A good way to check the accuracy of a simple pharmacokinetic model is to verify its performance by comparing with a physiologically based pharmacokinetic (PBPK) model that may encompass the mentioned factors. 

Pharmacokinetic models. An important advance in risk assessment for hazardous chemicals has been the application of pharmacokinetic models to interpret dose-response data in rodents and humans (EPA, 1996a Leung and Paustenbach, 1995 NAS/NRC, 1989 Ramsey and Andersen, 1984). Pharmacokinetic models can be divided into two categories compartmental or physiological. A compartmental model attempts to fit data on the concentration of a parent chemical or its metabolite in blood over time to a nonlinear exponential model that is a function of the administered dose of the parent. The model can be rationalized to correspond to different compartments within the body (Gibaldi and Perrier, 1982). 

The structural submodel describes the central tendency of the time course of the antibody concentrations as a function of the estimated typical pharmacokinetic parameters and independent variables such as the dosing regimen and time. As described in Section 3.9.3, mAbs exhibit several parallel elimination pathways. A population structural submodel to mechanistically cover these aspects is depicted schematically in Fig. 3.14. The principal element in this more sophisticated model is the incorporation of a second elimination pathway as a nonlinear process (Michaelis-Menten kinetics) into the structural model with the additional parameters Vmax, the maximum elimination rate, and km, the concentration at which the elimination rate is 50% of the maximum value. The addition of this second nonlinear elimination process from the peripheral compartment to the linear clearance process usually significantly improves the fit of the model to the data. Total clearance is the sum of both clearance parts. The dependence of total clearance on mAb concentrations is illustrated in Fig. 3.15, using population estimates of the linear (CLl) and nonlinear clearance (CLnl) components. At low concentra-... 

From an examination of Eq. (6) for a two-compartment model it is evident that Vss is dependent on the quantification of K12 and K21. For this model K12 and K21 can be determined by nonlinear regression analysis of plasma concentration-time data, either by deriving them from the fitted values of the coefficients and exponentials of the bi-exponential expression describing the concentration-time data, or by coding them directly into the modeling program. For the case where tissue elimination exists, it is possible to code into the model the existence of a K20, but the convergence process will not be able to resolve the appropriate micro rate constant. 

The biexponential rate equation associated with this model was fitted to the experimental data using a nonlinear least squares procedure. Pharmacokinetic constants for the two-compartment model were calculated by standard methods. The fraction amount absorbed as a function of time was estimated by the Loo-Riegelman method using the macroscopic rate constants calculated from the intravenous data. The slope of the linear part of the Loo-Riegelman plot combined with the total amount absorbed (quantitated by depletion analysis of the saturated donor solution) was used to calculate the zero-order rate constant for buccal permeability. 

A mechanism-based PK/PD model for rHu-EPO was used to capture the physiological knowledge of the biological system. An open, two-compartment disposition model with parallel linear and nonlinear clearance, and endogenous EPO at baseline, was used to describe recombinant human erythropoietin (rHu-EPO) disposition after intravenous administration [35]. The pharmacodynamic effect of rHu-... 

Other processes that lead to nonlinear compartmental models are processes dealing with transport of materials across cell membranes that represent the transfers between compartments. The amounts of various metabolites in the extracellular and intracellular spaces separated by membranes may be sufficiently distinct kinetically to act like compartments. It should be mentioned here that Michaelis-Menten kinetics also apply to the transfer of many solutes across cell membranes. This transfer is called facilitated diffusion or in some cases active transport (cf. Chapter 2). In facilitated diffusion, the substrate combines with a membrane component called a carrier to form a carrier-substrate complex. The carrier-substrate complex undergoes a change in conformation that allows dissociation and release of the unchanged substrate on the opposite side of the membrane. In active transport processes not only is there a carrier to facilitate crossing of the membrane, but the carrier mechanism is somehow coupled to energy dissipation so as to move the transported material up its concentration gradient. 

If the hazard rate of any single particle out of a compartment depends on the state of the system, the equations of the probabilistic transfer model are still linear, but we have nonlinear rate laws for the transfer processes involved and such systems are the stochastic analogues of nonlinear compartmental systems. For such systems, the solutions for the deterministic model are not the same as the solutions for the mean values of the stochastic model. 

The two-compartment model and the model of the enzymatic reaction (cf. Sections 9.1.2 and 8.5.1, respectively) will be presented as typical cases for linear and nonlinear models, respectively. For these simulations, the model parameters were set as follows ... 

To illustrate how to proceed using the cumulant generating functions, the well-known two-compartment model and the enzymatic reaction will be presented as examples of linear and nonlinear systems, respectively. In these examples, there are two interacting populations (m = 2) and the cumulant generating function is... 

A very general scheme for relating effects to concentration, of which both the effect-compartment and the indirect-effect models are special cases, was outlined by Sheiner and Verotta [452], The models presented in the study can be considered to be a special case of that unified scheme. As judiciously presented by these authors, both direct-response and indirect-response models are composed of one nonlinear static submodel and one dynamic submodel, but the placement of the submodels in the global model differs ... 

The classical three-compartment model describes pharmacokinetics of 5-HT1A receptor agonists. By means of a sigmoidal function E (c), the 5-HT1A agonist concentration c (t) influences the set-point signal that dynamically interacts with the body temperature. By using x (t) and y (t) as dimensionless state variables for the set-point and temperature, respectively, the model is expressed by the set of two nonlinear differential equations ... 

Another cutoff model, which deals with nonlinearity in biological systems, is one defined by McFarland (191). It attempts to elucidate the dependency of drug transport on hydrophobicity in multicompartment models. McFarland addressed the probability of drug molecules traversing several aqueous lipid barriers from the first aqueous compartment to a distant, final aqueous compartment. The probability of a drug molecule to access the final compartment n of a biological system was used to define the drug concentration in this compartment. 

Some fractional transfer functions of compartmental models may actually be functions, (i.e., the model may actually be nonlinear). The most common example is when a transfer or loss is saturable. Here a Michaelis-Menten type of transfer function can be defined, as was shown in Chapter 2 for the elimination of phenytoin. In this case, loss from compartment 1 is concentration dependent and saturable, and one can write... 

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