Comparison of Approaches to Modeling of Cell Population Dynamics

Abstract

This paper reviews structured cell population models. A typical formulation is the partial differential equation (PDE) \[ \frac{{\partial p}}{{\partial t}} + \frac{{\partial p}}{{\partial a}} + \frac{{\partial ( {gp} )}}{{\partial a}} = B - D, \] the Lotka–von Förster equation, generalized by Webb, where t is the chronological time, a is cell age, $p = p( {a,t} )$ is the population density, and g is the cell growth rate, while B and D are birth and death terms, respectively. Essentially, it is a transport equation with additional nonlocal boundary conditions. Another approach, apparently not involving a transport equation, and related to the theory of branching processes has been originally derived by Kimmel and analyzed by the authors. It is shown that this latter approach fits in the framework of PDE models and has comparable generality. The relationships between both types of models are generally nontrivial and seem important for their applicability.