Linear Compartmental Models

some fundamental hypotheses, or as commonly called laws, were employed to expand the transfer rates appearing in (8.1). Fick s law is largely used in current modeling (cf. Section 2.3 and equation 2.14). It assumes that the transfer rate of material by diffusion between regions l (left) and r (right) with concentrations c and cr, respectively, is 

This law may be applied to the transfer rates Rji (t) of the previous equation for all pairs j and i of compartments corresponding to l and r and for the elimination rate R (t), where the concentration is assumed nearly zero in the region outside the compartmental system. One has for the compartment i, 

The constants k are called the fractional flow rates. They have the dimension of time-1 and they are defined as follows  

In contrast to the clearance, the fractional flow rates indicate the direction of the flow, i.e., kji kij, the first subscript denoting the start compartment, and the second one, the ending compartment. The fractional flow rates and the volumes of distribution are usually called microconstants. 

When the volume of the compartment being cleared is constant, the assumption that the fractional flow rate is constant is equivalent to assuming that the clearance is constant. But in the general case, in which the volume of distribution cannot be assumed constant, the use of the fractional flow rates k is unsuitable, because the magnitude of k depends as much upon the volume of the compartment as it does upon the effectiveness of the process of removal. In contrast, the clearance depends only upon the overall effectiveness of removal, and can be used to characterize any process of removal whether it be constant or changing, capacity-limited or supply-limited [308]. 

---

Consider a tablet that is taken regularly once a day. We want to find the optimal quantity of the drug (i.e., the only active ingredient) in the tablet in order to keep the drug concentration in the blood within a given therapeutic range [Cj, Cy] as strictly as possible. To predict the drug concentration we use the linear compartmental model shown in Fig. 2.9, one of the most popular models in pharmacokinetics. 

X2° = X30 = 0 assumed to be known exactly. The only observed variable is = x. Jennrich and Bright (ref. 31) used the indirect approach to parameter estimation and solved the equations (5.72) numerically in each iteration of a Gauss-Newton type procedure exploiting the linearity of (5.72) only in the sensitivity calculation. They used relative weighting. Although a similar procedure is too time consuming on most personal computers, this does not mean that we are not able to solve the problem. In fact, linear differential equations can be solved by analytical methods, and solutions of most important linear compartmental models are listed in pharmacokinetics textbooks (see e.g., ref. 33). For the three compartment model of Fig. 5.7 the solution is of the form... 

Let Xj(f) be the mass of a drug in the ith compartment. The notation for input, loss, and transfers is summarized in Figure 8.3. Because this notation describes the compartment in full generality, it is a little different from that used in earlier chapters. This difference is necessary to understand how one passes to the linear compartmental model. In Figure 8.3, the rate constants describe mathematically the mass transfer of material among compartments interacting with the ith compartment (Fji is the transfer of material from compartment i to compartment j, F j is the transfer of material from compartment j to compartment i), the new input F q (this corresponds to Xq in Chapter 4), and loss to the environment Fqi from compartment i. The mathematical expression describing the rate of... 

Figure 1.18 Indistinguishable parent-metabolite models for isoniazid. Parent drug is administered into Compartment 1 with parent and metabolite concentrations measured in Compartment 1 and 2, respectively. The models in the first column are pairwise symmetric to those in the second column by interchanging Compartments 2 and 4. The same is true for the third and fourth columns. All 18 of these models are kinetically indistinguishability and identifiable. Reprinted from Mathematical Biosciences, vol. 103, Zhang L-Q, Collins KC, and King PH Indistinguishability and identifiability analysis of linear compartmental models, pp. 77-95. Copyright (1971) with permission of Elsevier.

Audoly, S., D Angio, L., Saccomani, M.P., and Cobelli, C. Global identifiability of linear compartmental models— A computer algebra algorithm. IEEE Transactions on Biomedical Engineering 1998 45 36-47. 

Merino, J.A., de Biasi, J., Plusquellec, Y., and Houin, G. Minimal experimental requirements for structural iden-tifiability in linear compartmental models The software IDEXMIN . Arzneimittel Forschung/Drug Research 1996 46 324-328. 

Zhang, L.-Q., Collins, J.C., and King, P.H. Indistinguishabil-ity and identifiability analysis of linear compartmental models. Mathematical Biosciences 1991 103 77-95. 

The LSA linear drug disposition definition does not impose any restrictions on the mathematical form of the drug input response. This is in contrast to linear compartmental models. For example, a two-compartment model implies a biexponential response to an IV bolus injection ... 

Vajda, S. (1982). On parameter and structural identifiability—Nonunique observability, reconstructibility and equivalence of linear compartmental models. IEEE Trans. Autom. Control, 27, 1136. 

FIGURE9.3 The linear compartmental model of glucose kinetics by Cobelli et al. [1984b]. 

An interesting application of linear compartmental models at the organ level is in describing the exchange of materials between blood, interstitial fluid, and cell of a tissue from multiple tracer indicator dilution data. Compartmental models provide a finite difference approximation in the space dimension of a system described by partial differential equations, which may be easier resolvable from the data. These models are discussed in facquez [1996] and an example of a model describing glucose transport and metabolism in the human skeletal muscle can be found in Saccomani et al. [1996]. 

To examine the quality of model predictions to observed data, in addition to visual inspection, various statistical tests on residuals are available to check for presence of systematic misfitting, nonrandomness of the errors, and accordance with assumed experimental noise. Model order estimation, that is, number of compartments in the model, is also relevant here, and for linear compartmental models, criteria such as F-test, and those based on the parsimony principle such as the Akaike and Schwarz criteria, can be used if measurement errors are Gaussian. 

The linear model. Equation 9.6, has become very useful in applications due to an important result the kinetics of a tracer in a constant steady-state system, linear or nonlinear, are linear with constant coefficients. An example is shown in Figure 9.3 where the three-compartment model by Cobelli et al. [1984b] for studying tracer glucose kinetics in steady state at the whole-body level is depicted. Linear compartmental models in conjunction with tracer experiments have been extensively used in studying distribution of materials in living systems both at whole-body, organ and cellular level. Examples and references can be found in Carson et al. [1983], Jacquez [1996], and Cobelli et al. [2000], Carson and Cobelli [2001]. 

---