Optimisation of Cancer Drug Treatments Using Cell Population Dynamics - Mathematical Methods and Models in Biomedicine - page 269

Delay differential models can also be viewed as deriving from age- structured
PDE models since they can be obtained by integrating PDEs along characteristics.
Thus Bernard et al. [ 23 ] proposed a model composed of delayed differential
equations to model tumour and normal cell population dynamics in the phases of
the cell cycle under circadian control and chemotherapy. They compared the efficacy
and toxicity of constant and chronomodulated schedules of 5-FU, a phase-specific
drug used in the treatment of colorectal cancer.

3.5 Mixed Models, Both Spatially and Physiologically

Structured

We call “mixed models”, models that include both spatial and physiological
dynamics. Such models are useful to investigate spatial changes induced by a phase-
specific chemotherapy combined or not with an antiangiogenic agent. This kind of
models has not been highly developed. Bresch and co-workers [ 25 , 30 , 31 ] developed
a multiscale model of tumour growth that includes cell age in the proliferative phases
of the cell cycle and tissue motion of tumour cells. On the basis of the model
developed by Bresch et al., coupled with an angiogenesis model, one of us and
her co-workers [ 25 ] investigated the effects of an innovative antiangiogenic drug
on tumour vasculature and hence on tumour growth. This multiscale model takes
into account some molecular events such as cell cycle dynamics and cell receptor
binding. This model could be coupled to a model of phase-specific drug, such as 5-
FU, to analyse tumour and endothelial cell dynamics under drug infusion. It could
also be interesting to determine optimal drug schedules that would maximise tumour
cell death under constraints of minimising endothelial cell death to ensure drug
delivery to tumour cells (remember that endothelial cells are cells that constitute
the vessel wall, see Sect. 3.4 for details).
Alarc on et al. [ 2 ] proposed a more complex multiscale model of vascular tumour
growth that integrates tissue, cell and intracellular scales. For instance, this model
accounts for vascular network, blood flow, cell-cell interaction, cell-cycle, VEGF
production and integrates several kinds of models (ODE models, cellular automata,
etc). The authors investigated the effects of low and high concentrations of protein
p27 on the dynamics of tumour and normal cell populations.

4 Control and Its Missions: Representing the Action of Drugs

In the previous section, we have described some dynamic models for cell pop-
ulations frequently used in the study of cancer growth and treatments. These
models can thus be seen as controlled dynamic systems with drug effects as