The Poisson process as a model for compartment digesta flow in ruminants

The Poisson process, the simplest stochastic flow process, was used to develop a multicompartment model of ruminant digesta flow with Gamma distributed retention times. Although mathematically the model is a generalization of many previously published models, the physiological model differs substantially in asserting that the distributed delay time and the exponential rate (scale) parameters, including the scale parameter of the Gamma distribution, are determined by total digesta flow, and thus invariant with respect to the fraction marked. The shape factor of the Gamma distribution is shown to be sufficient to explain the difference between markers in rate of marker excretion. Consequently, the parameters of multiple markers can be simultaneously estimated with the constraint that the exponential scale parameters and the delay time are invariant with respect to marker. This constraint leads to a measure of pure error to strengthen statistical tests for model rejection. Steady-state digesta retention time is estimated from the transient marker retention parameters, eliminating the necessity of speculating on what fraction of digesta the marked fraction represents. Tests of various models, using simulations and animal experiments indicate that, even if a model is correct, it is not possible to obtain reliable parameter estimates by fitting to a single marker. Even with multiple markers some caution must be used in interpreting parameter estimates derived from least squares fitting.