Model compartmental

These models have relatively few parameters, and the parameters have a limited physiological or anatomical meaning. For example, a compartmental volume relates the quantity of the drug to its concentration in a compartment, and does not refer to an anatomically- or physiologically-defined area of the body. 

The rate of transport from a certain compartment is governed by the concentration in that compartment and a proportionality constant, denoted (elimination or distribution) clearance (dimension volmne time ) as formulated below. 

Usually, it is assumed that there is no net transport between two compartments if the concentrations in both compartments are equal in this specific case CL21 = CL12. 

Similar equations can be written for compartment 2. The same principle can be applied to any compartmental model, irrespective of its complexity. 

The peripheral compartment is characterized by a volume of distribution V2 (usually it is assumed that there is no net transport between the two compartments if the concentrations in both compartments are equal in this case k i V2 = ki2 Fj). Elimination may take place from both compartments and is characterized by rate constants kio and IC20, respectively. 

The choice of model should be based on biological, physiological, and pharmacokinetic plausibility. For example, compartmental models may be used because of their basis in theory and plausibility. It is easy to conceptualize that a drug that distributes more slowly into poorly perfused tissues than rapidly perfused tissues will show multi-phasic concentration-time profiles. Alternatively, the Emax equation, one of the most commonly used equations in pharmacodynamic modeling, can be developed based on mass balance principles and receptor binding kinetics. 

Unfortunately, not enough time is spent teaching students to think critically about the models they are using. Most mathematics and statistics classes focus on the mechanics of a calculation or the derivation of a statistical test. When a model is used to illustrate the calculation or derivation, little to no time is spent on why that particular model is used. We should not delude ourselves, however, into believing that once we have understood how a model was developed and that this model is the true model. It may be in physics or chemistry that elementary equations may be true, such as Boyle s law, but in biology, the mathematics of the system are so complex and probably nonlinear in nature with multiple feedback loops, that the true model may 

Pharmacokinetic models are typically modeled by examining the number of phases in the concentrationtime profile after single dose administration with the number of compartments equaling the number of phases. If after bolus administration or after extravas-cular administration and absorption is essentially completed, the concentration-time profile when plotted on a log-scale shows only one phase, a 1-compartment open model is chosen. If two phases are shown, a 2-compart-ment open model is chosen. 

Suppose one does a study wherein a single dose of the drug is given both orally and intravenously on two separate occasions and finds that the oral concentrationtime data were best fit using a four-term polyexponential equation, whereas after intravenous administration the concentration-time profile was best fit with a two-term polyexponential equation. In this case there are 27 possible compartmental models to choose from. Whereas most books on pharmacokinetics present the 1- and 2-compartment model, the situation is clearly not that simple (see Wagner s (1993) text for examples). 

So how then does a pharmacokineticist choose an appropriate compartmental model Wagner suggests collecting the following data to aid in choosing a class of model  

Mixing tanks in series with linear transfer kinetics from one to the next with the same transit rate constant kt have been utilized to obtain the characteristics of flow in the human small intestine [173,174]. The differential equations of mass transfer in a series of m compartments constituting the small intestine for a nonabsorbable and nondegradable compound are 

Solving the system of (6.10) and (6.11) in terms of the fraction of dose absorbed, we obtain 

Analysis of experimental human small-intestine transit time data collected from 400 studies revealed a mean small-intestinal transit time (TSi) = 199 min [173]. Since the transit rate constant kt is inversely proportional to (TSj), namely, kt = to/ (TSi), (6.12) was further fitted to the cumulative curve derived from the distribution frequency of the entire set of small-intestinal transit time data in order to estimate the optimal number of mixing tanks. The fitting results were in favor of seven compartments in series and this specific model, (6.10) and (6.11) with to = 7, was termed the compartmental transit model. 

The incorporation of a passive absorption process in the compartmental transit model led to the development of the compartmental absorption transit model (CAT) [175]. The rate of drug absorption in terms of mass absorbed qa (t) from the small intestine of the compartmental transit model is 

Recall that kt is equal to 7/ (7 sl), while ka can be expressed in terms of the effective permeability and the radius R of the small intestine [55]  

The concentration of drug within the compartment, c, is related to the total mass within the compartment, M = cV, so that  

The half-life for drug residence within the compartment is related to the first-order rate constant  

Simple pharmacokinetic models can be used to tailor therapies to the patient. After measuring the half-life of clearance in a patient, subsequent doses can be provided at intervals calculated to maintain the plasma concentration above a 

Murine monoclonal antibody Dose (mg/kg body weight) Serum retention half-life (h) 

The one-compartment model can be extended to include slow absorption of drug the slow absorption step may represent entry of drug through the gastrointestinal tract or leakage into the circulation after subcutaneous injection. Absorption is added by modifying the mass balance equation—Equation 7-1—to yield  

---

The measurement of carotenoid absorption is fraught with difficulties and riddled with assumptions, and it is therefore a complex matter. Methods may rely on plasma concentration changes provoked by acute or chronic doses, oral-faecal mass balance method variants and compartmental modelling. 

Pharmacokinetics is closely related to pharmacodynamics, which is a recent development of great importance to the design of medicines. The former attempts to model and predict the amount of substance that can be expected at the target site at a certain time after administration. The latter studies the relationship between the amount delivered and the observable effect that follows. In some cases the observable effect can be related directly to the amount of drug delivered at the target site [2]. In many cases, however, this relationship is highly complex and requires extensive modeling and calculation. In this text we will mainly focus on the subject of pharmacokinetics which can be approached from two sides. The first approach is the classical one and is based on so-called compartmental models. It requires certain assumptions which will be explained later on. The second one is non-compartmental and avoids the assumptions of compartmental analysis. 

The synthetic data have been obtained by adding random noise with standard deviation of about 0.4 )0.g 1 to the theoretical plasma concentrations. As can be seen, the agreement between the estimated and the computed values is fair. Estimates tend to deteriorate rapidly, however, with increasing experimental error. This phenomenon is intrinsic to compartmental models, the solution of which always involves exponential functions. 

We now turn our attention to the graphical determination of the various parameters of our two-compartmental model, i.e. the plasma volume of distribution Vp,... 

Fig. 39.14. (a) Catenary compartmental model representing a reservoir (r), absorption (a) and plasma (p) compartments and the elimination (e) pool. The contents X, Xa, Xp and X,. are functions of time t. (b) The same catenary model is represented in the form of a flow diagram using the Laplace transforms Xr, Xa and Xp in the j-domain. The nodes of the flow diagram represent the compartments, the boxes contain the transfer functions between compartments [1 ]. (c) Flow diagram of the lumped system consisting of the reservoir (r), and the absorption (a) and plasma (p) compartments. The lumped transfer function is the product of all the transfer functions in the individual links. 

Fig. 39.16. Paradigm for the fitting of sums of exponentials from a compartmental model (c) to observed concentration data (o) as contrasted by the results of statistical moment analysis (s). (After Thom [13].)...

The ICRP (1994b, 1995) developed a Human Respiratory Tract Model for Radiological Protection, which contains respiratory tract deposition and clearance compartmental models for inhalation exposure that may be applied to particulate aerosols of americium compounds. The ICRP (1986, 1989) has a biokinetic model for human oral exposure that applies to americium. The National Council on Radiation Protection and Measurement (NCRP) has also developed a respiratory tract model for inhaled radionuclides (NCRP 1997). At this time, the NCRP recommends the use of the ICRP model for calculating exposures for radiation workers and the general public. Readers interested in this topic are referred to NCRP Report No. 125 Deposition, Retention and Dosimetry of Inhaled Radioactive Substances (NCRP 1997). In the appendix to the report, NCRP provides the animal testing clearance data and equations fitting the data that supported the development of the human mode for americium. 

Respiratory Tract Clearance. This portion of the model identifies the principal clearance pathways within the respiratory tract. The model was developed to predict the retention of various radioactive materials. Figure 3-4 presents the compartmental model and is linked to the deposition model (see Figure 3-2) and to reference values presented in Table 3-5. This table provides clearance rates, expressed as a fraction per day and also as half-time (Part A), and deposition fractions (Part B) for each compartment for insoluble... 

Stirred tank models have been widely used in pharmaceutical research. They form the basis of the compartmental models of traditional and physiological pharmacokinetics and have also been used to describe drug bioconversion in the liver [1,2], drug absorption from the gastrointestinal tract [3], and the production of recombinant proteins in continuous flow fermenters [4], In this book, a more detailed development of stirred tank models can be found in Chapter 3, in which pharmacokinetic models are discussed by Dr. James Gallo. The conceptual and mathematical simplicity of stirred tank models ensures their continued use in pharmacokinetics and in other systems of pharmaceutical interest in which spatially uniform concentrations exist or can be assumed. 

Pharmacokinetics emerged as a discipline in the 1960s with its foundation in compartmental modeling, although earlier origins of pharmacokinetics can be traced [1], Mammillary compartmental models provided the framework for phar-... 

Extensive literature is available on general mathematical treatments of compartmental models [2], The compartmental system based on a set of differential equations may be solved by Laplace transform or integral calculus techniques. By far... 

Model development is intimately linked to correctly assigning model parameters to avoid problems of identifiability and model misspecification [27-29], A full understanding of the objectives of the modeling exercise, combined with carefully planned study protocols, will limit errors in model identification. Compartmental models, as much as any other modeling technique, have been associated with overzealous interpretation of the model and parameters. 

K Godfrey. Compartmental Models and Their Application. London Academic Press, 1983. 

The cumulative curve obtained from the transit time distribution in Figure 9 was fitted by Eq. (48) to determine the number of compartments. An additional compartment was added until the reduction in residual (error) sum of squares (SSE) with an additional compartment becomes small. An F test was not used, because the compartmental model with a fixed number of compartments contains no parameters. SSE then became the only criterion to select the best compartmental model. The number of compartments generating the smallest SSE was seven. The seven-compartment model was thereafter referred to as the compartmental transit model. 

Soil modeling follows three different mathematical formulation patterns (1) Traditional Differential Equation (TDE) modeling (2) Compartmental modeling and... 

In a series of recent papers Q - 4), we have advocated the use of the fugacity concept as an aid to compartmental modeling of chemicals which may be deliberately or inadvertantly discharged into the environment. The use of fugacity instead of concentration may facilitate the formulation and interpretation of environmental models it can simplify the mathematics and permit processes which are quite different in character to be compared... 

A Compartmental Model for Lead Biokinetics with Multiple Pool for Blood Lead... 

Absorbed lead is distributed in various tissue compartments. Several models of lead pharmacokinetics have been proposed to characterize such parameters as intercompartmental lead exchange rates, retention of lead in various pools, and relative rates of distribution among the tissue groups. See Section 2.3.5 for a discussion of the classical compartmental models and physiologically based pharmacokinetic models (PBPK) developed for lead risk assessments. 

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... 

Based on the descriptions of spatial variation in each environmental compartment, multimedia models can be categorized into multimedia compartmental models (MCMs) [3-20], spatial multimedia models (SMs) [21-24] and spatial multimedia compartmental models (SMCMs) [25-27]. MCMs assume homogeneous landscape properties in each medium and assume all environmental compartments are well mixed. SMs are collections of single-media models in which the output of one model serves as the input to the others. Each individual model in the SMs is a spatial model describing the variation of environmental properties in one or more directions. SMCMs are similar to MCMs, but consider one or more environmental compartments as nonuniform regions. 

Ten Cate, A., Bermingham, S. K., Derksen, J. J., and Kramer, H. M. J., Compartmental Modeling of a 1,100 L Crystallizer Based on Large Eddy Flow Simulation . Proceedings of the 10th European Conference on Mixing, Delft, Netherlands, 255-264 (2000). 

---