Analysis of a molecular structured population model with possible polynomial growth for the cell division cycle

Modelling Physiological and Pharmacological Control on Cell Proliferation to Optimise Cancer Treatments

“…stand here for source and loss terms, that can be related to exchanges with previous and subsequent phases. In this transport equation [22,23], the parameter Γ 0 denotes the evolution speed of physiological age a with respect to time t, which is assumed to be constant in this model; if for example Γ 0 = 0.5, it means that physiological age a evolves twice as slowly as real time t. Similarly, the function Γ 1 represents the evolution speed of Cyclin D/(Cdk4 or 6) with respect to time in G 1 phase, i.e., Γ 0 times the speed dx/da of x with respect to physiological age a (the 32 J. Clairambault…”

“…Proliferation-quiescence models, classical [119,120,182,183], or more recent [22,23,73], all rely on the idea that in proliferating tissues, not all cells proliferate, so that one can divide the population into two subpopulations, without any unrelevant space allocation (cells are all mixed up, as can be seen for example under the microscope in a bone marrow cell population sample), but according to their actual (or not) proliferating status. The main underlying biological hypothesis is that when environmental conditions are not adequate, proliferating cells become quiescent; and conversely, when proliferation is wanted, quiescent cells may be reintroduced in the proliferative compartment.…”

“…They should represent known exchanges, from a biological point of view, between G 0 (quiescence) and G 1 phases, and only the parameters of the exchange functions should be different from tumour to healthy tissue, as sketched in [22,23]. Making use of exactly the same models (same targets or drugs), but with different parameters, it is possible to obtain for the cell populations asymptotic behaviours of qualitatively very different types, with convergence to a stable equilibrium in healthy tissues, and in tumours growth of the population according to exponential, polynomial or linear laws [22,23,73]. …”

“…stand here for source and loss terms, that can be related to exchanges with previous and subsequent phases. In this transport equation [22,23], the parameter Γ 0 denotes the evolution speed of physiological age a with respect to time t, which is assumed to be constant in this model; if for example Γ 0 = 0.5, it means that physiological age a evolves twice as slowly as real time t. Similarly, the function Γ 1 represents the evolution speed of Cyclin D/(Cdk4 or 6) with respect to time in G 1 phase, i.e., Γ 0 times the speed dx/da of x with respect to physiological age a (the 32 J. ClairambaultCell proliferation control and cancer treatment coupling between the two structure variables being modelled by another equation, that describes Cyclin D /p27 kinetics). The function Γ 1 in the advection term may be chosen to represent, in a more or less stiff way, the transition to the next phase: the stiffer is its passage from low to high values (or the opposite), the more synchronised is the population of cells represented here by its density p(t, a, x).…”

Oscillations in a molecular structured cell population model

“…In [6] and [7] a nonlinear cell population model for both, tumoral and healthy tissue is introduced in which cells are structured with respect to age and with respect to the content of a group of proteins called cyclin and CDK (cyclin dependent kinases) which plays an important role in the regulation of the cell cycle (see [15]). Cells are divided into two categories: proliferating cells which grow and divide, giving birth at the end of the cycle to new cells, or else transit to the quiescent compartment, and quiescent cells which do not age, nor divide nor change their cyclin content.…”