Analysis and Implementation of an Age Structured Model of the Cell Cycle

Sammanfattning: In an age-structured model originating from cancer research, the cell cycle is divided into two phases: Phase 1 of variable length, consisting of the biologically so called G1 phase, and Phase 2 of fixed length, consisting of the so called S, G2 and M phases. A system of nonlinear PDEs along with initial and boundary data describes the number densities of cells in the two phases, depending on time and age (where age is the time spent in a phase). It has been shown that the initial and boundary value problem can be rewritten as a coupled system of integral equations, which in this M.Sc. thesis is implemented in Matlab using the trapezoidal and Simpson rule. In the special case where the cells are allowed to grow without restrictions, the system is uncoupled and possible to study analytically, whereas otherwise, a nonlinearity has to be solved in every step of the iterative equation solving. The qualitative behaviour is investigated numerically and analytically for a wide range of model components. This includes investigations of the notions of crowding, i.e. that cell division is restricted for large population sizes, and quorum sensing, i.e. that a small enough tumour can eliminate itself through cell signalling. In simulations, we also study under what conditions an almost eliminated tumour relapses after completed therapy. Moreover, upper bounds for the number of dividing cells at the end of Phase 2 at time t are determined for specific cases, where the bounds are found to depend on the existence of so called cancer stem cells. Lastly, a careful error analysis of the Matlab implementation is performed both in a linear and in a nonlinear case.