An age-and-cyclin-structured cell population model with proliferation and quiescence for healthy and tumoral tissues

We present a nonlinear model of the dynamics of a cell population divided in a proliferative and a quiescent compartments. The proliferative phase represents the complete cell division cycle ($G_{1}-S-G_{2}-M$) of a population committed to divide at its end. The model is structured by the time spent by a cell in the proliferative phase, and by the amount of \emph{cyclin~D/(CDK4 or 6)} complexes. Cells can transit from one compartment to the other, following transition rules which differ according to the tissue state: healthy or tumoral. The asymptotic behaviour of solutions of the nonlinear model is analysed in both cases, exhibiting tissue homeostasis or tumour exponential growth. The model is simulated by numerical solutions which confirm its theoretical predictions.