Hybrid models of cell populations. Application to cancer modelling and treatment – Bimod

This project is devoted to the development of a new class of mathematical models in biology called hybrid models. They describe the evolution of cell populations (tissue, organ, organism) on the basis of coupled discrete-continuous approaches. Biological cells are considered as individual (discrete) objects. They can divide, die by apoptosis or necrosis, change their type. Cells can also interact mechanically with each other and with the surrounding medium. Cell behaviour (proliferation, apoptosis, differentiation) is determined by intra-cellular regulatory networks described by systems of ordinary differential equations for concentrations of proteins and other bio-chemical substances. On the other hand, they can be influenced by the extra-cellular medium that provides nutrients, growth factors, hormones and proteins, submitted or not to the influence of circadian clocks, etc. that can either come from outside and be consumed by cells or be produced by the cells. Concentrations of substances in the extra-cellular matrix are described by partial differential equations of reaction-diffusion or reaction-diffusion-convection types. Thus, discrete representation of cell population is coupled with continuous models for intra-cellular and extra-cellular regulation.

This description of cell populations is the most adequate and close to biological reality for the description of cancer dynamics. It will allow us to fill the growing gap between cell biology and mathematical modelling. We will apply it to model various biological phenomena with the emphasis on cancer modelling and treatment of two types of cancers: colorectal cancer and acute myeloblastic leukaemia. It should be noted that hybrid models belong to so-called multi-scale models in biology where different scales, intra-cellular, tissue, organ, organism are taken into account in their interactions. It is also important to emphasize that discrete description of cell populations should be completed by continuous approaches where they are represented as a continuous medium and described by partial differential equations structured either with respect to age, maturity or various intra-cellular variables: physiologically structured equations, or with respect to space: reaction-diffusion equations.. Continuous models are more amenable to analytical investigations than hybrid models.

Intra-cellular and extra-cellular regulation will be described for the cell types susceptible for malignant mutations. In particular, for various immature blood cells in the bone marrow, which can be at the origin of acute myeloblastic leukemia. The pharmacokinetics-pharmacodynamics of various cancer treatments will be described and parameter identification, with and without treatment will be carried out, as far as possible from experimental data. Intra-cellular and extra-cellular regulations, and their external control by differentiable functions representing continuous drug infusion schemes, will be included into the hybrid models in order to study various protocols of cancer treatments. Optimisation of these treatments will be considered under both constraints of limiting unwanted toxicity to healthy tissues and of avoiding emergence of drug resistant tumoral cell subpopulations. The developed methods will also be used to study other problems related to cell population dynamics including morphogenesis and blood flows