On the stochastic modeling of tracer kinetics

Virtually all of the mathematical models in current use to describe tracer kinetics are deterministic (non-stochastic). However, in this paper we suggest that the "real world" of tracer kinetics is stochastic, and we formulate a unified one-compartment model structure that incorporates multiple sources of stochasticity. The stochastic models are presented with various combinations of a probabilistic transfer mechanism, a random rate coefficient, and a random initial condition, and the mean value functions and covariances are derived for these models with both time-independent and time-varying rate coefficients. The covariances have unique forms that are helpful in model identification and many of the means are non-exponential functions. Outstanding among the conclusions are 1) a proof that a deterministic model is not always equivalent in an average sense to its stochastic counterpart; 2) the existence of many rich, practical, and realistic alternatives to the exponential decay function; and 3) the notion that models are identifiable from their covariance structure.