An Irreversible Two-Compartment Model with Age-Dependent Turnover Rates

Abstract

In the literature of ruminant nutrition, the sequential irreversible two-compartment model is generally accepted for describing the passage of indigestible matter through the gastrointestinal tract of ruminants (Blaxter, McGraham and Wainman, 1956; Grovum and Williams, 1973). The model in common use is deterministic with linear kinetics and constant transfer rates. The stochastic analog of this model assumes exponential lifetimes of particles in the compartments, which implies constant hazard rates. One generalization of the above model is that suggested by Matis (1972), in which a gamma lifetime distribution and hence age-dependent hazard rates are introduced into the first compartment. This model has been applied to fecal output data in systems with various nonsteady-state phenomena, for instance, digestive breakdown of particle sizes and initial incomplete mixing (see, for example, Ellis, Matis and Lascano, 1979). The present research generalizes the previous models by incorporating such age-dependent phenomena into both of the conceptual compartments. A broad overview of compartmental analysis has led Wise (1974) to question the assumption of the so-called 'homogeneous' compartments. As an alternative he has postulated the power-law models which have been successfully fitted to many kinetic data sets. A recent, more general approach involves the incorporation of nonexponential lifetimes through the semi-Markov framework (see, for example, Weiner and Purdue, 1977; Marcus and Becker, 1977). Although this approach is elegant, the nonparametric estimation of the lifetime distributions in the compartments is extremely difficult in practice. The only applications of the semi-Markov model to experimental data known to the authors are those made by Marcus (1979). In the present research a specific parametric form, namely sequential gammas, is used for lifetime distributions, and an analytical form of the model is obtained. This resulting model is tractable for fitting experimental data, and provides a rich family of models.