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.