Nonlinear Compartmental Models

However, in practice, one can find the moments directly using conditional expectations (cf. Appendix D)  

Besides the deterministic context, the predicted amount of material is subjected now to a variability expressed by the second equation. This expresses the random character of the fractional flow rate, and it is known as process uncertainty. Extensive discussion of these aspects will be given in Chapter 9. 

Therefore, the transfer rates and the fractional flow rates are functions of the vector q (/,) and t. The dependence on t may be considered as the exogenous environmental influence of some fluctuating processes. If no environmental dependence exists, it is more likely that the transfer rates and the fractional flow rates depend only on q(t). Nevertheless, since q(t) is a function of time, the 

Until now, the compartmental model was considered as consisting of compartments associated with several anatomical locations in the living system. The general definition of the compartment allows us to associate in the same location a different chemical form of the original molecule administered in the process. In other words, the compartmental analysis can include not only diffusion phenomena but also chemical reaction kinetics. 

One source of nonlinear compartmental models is processes of enzyme-catalyzed reactions that occur in living cells. In such reactions, the reactant combines with an enzyme to form an enzyme-substrate complex, which can then break down to release the product of the reaction and free enzyme or can release the substrate unchanged as well as free enzyme. Traditional compartmental analysis cannot be applied to model enzymatic reactions, but the law of mass-balance allows us to obtain a set of differential equations describing mechanisms implied in such reactions. An important feature of such reactions is that the enzyme 

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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. 

Vajda, S., Godfrey, K. R., and Rabitz, H. (1989). Similarity transformation approach to identifiability analysis of nonlinear compartmental models. Math. Biosci 93,217-248. Walter, E. (1982). Identifiability of State Space Models, Lect. Notes Biomath. No. 46. Springer-Verlag, New York... 

Chapter 9 shows how compartmental models may be used to describe physiological systems, for example, pharmacokinetics. The production, distribution, transport, and interaction of exogenous materials, such as drugs or tracers, and endogenous materials, such as hormones, are described. Examples of both linear and nonlinear compartmental models are presented, as well as parameter estimation, optimal experiment design, and model validation. 

The fcy s are called fractional transfer coefficients. Equation 9.3 describes the generic nonlinear compartmental model. If the fcy s do not depend on the compartmental masses qfs, the model becomes... 

K is thus a (column) diagonally dominant matrix. This is a very important property, and in fact the stability properties of compartmental models are closely related to the diagonal dominance of the compartmental matrix. For instance for the linear model. Equation 9.6, one can show that all eigenvalues have nonpositive real parts and that there are no purely imaginary eigenvalues this means that all solutions are bounded and if there are oscillations they are damped. The qualitative theory of linear and nonlinear compartmental models have been reviewed in Jacquez and Simon [1993], where some stability results on nonlinear compartmental models are also presented. 

Almost aU the biological models are nonhnear dynamic systems, including for example saturation or threshold processes. In particular, nonlinear compartmental models. Equation 9.5, are frequently found in biomedical applications. For such models the entries of K are functions of q, most commonly fcy is a function of only few components of q, often q, or qj. Examples of fcy function of q,- or qj are the Hill and... 

FIGURE 9.4 The nonlinear compartmental model of insulin control on glucose distribution and metabolism by Caumo and Cobelli [1993]. The dashed line denotes a control signal. 

Additional examples and references on nonlinear compartmental models can be found in Carson et al. [1983], Godfrey [1983], Jacquez [1996], and Carson and Cobelli [2001]. 

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