  We consider an age-size structured cell population model based on the cell
cycle length. The model is described by a first order partial differential
equation with initial-boundary conditions. Using the theory of semigroups of
positive operators we establish new criteria for an asynchronous exponential
growth of solutions to such equations. We discuss the question of exponential
size growth of cells. We show how to incorporate into our description models
with constant increase of size and with target size division. We also present
versions of the model when the population is heterogeneous.
