Optimisation of Cancer Drug Treatments Using Cell Population Dynamics
Cancer is primarily a disease of the physiological control on cell population proliferation. Tissue proliferation relies on the cell division cycle: one cell becomes two after a sequence of molecular events that are physiologically controlled at each step
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1 Introduction Cancer is primarily a disease of the physiological control on cell population proliferation. Tissue proliferation relies on the cell division cycle: one cell becomes two after a sequence of molecular events that are physiologically controlled at each step of the cycle at so-called checkpoints, in particular at transitions between phases of the cycle [105]. Tissue proliferation is the main physiological process occurring in development and later in maintaining the permanence of the organism in adults, at that late stage mainly in fast renewing tissues such as bone marrow, gut and skin. Proliferation is normally controlled in such a way that tissue homeostasis is preserved. By tissue homeostasis we mean permanence in the mean of tissue in volume, mass and function to ensure satisfaction of the needs of the whole organism. In cancer tissues, this physiological control, which also relies on the so-called checkpoints in the division cycle of individual replicating cells, is disrupted, leading to an overproduction of cells that eventually results in the development of tumours. Anticancer drugs all attack the cell division cycle, either by slowing it down (possibly until quiescence, i.e., non-proliferation, cells remaining alive), or by blocking it at checkpoints, which in the absence of cell material repair eventually leads to cell death. Various mathematical models have been proposed to describe the action of anticancer drugs in order to optimise it, that is to minimise the number of cancer cells or a related quantity, as the growth rate of the cancer cell population. F. Billy () • J. Clairambault INRIA BANG team, BP 105, F78153 Rocquencourt, and LJLL, UPMC, 4 Place Jussieu, F75005 Paris, France e-mail: [email protected]; [email protected] O. Fercoq INRIA MAXPLUS team, CMAP, Ecole Polytechnique, F91128 Palaiseau, France e-mail: [email protected] U. Ledzewicz et al., Mathematical Methods and Models in Biomedicine, Lecture Notes on Mathematical Modelling in the Life Sciences, DOI 10.1007/978-1-4614-4178-6 10, © Springer Science+Business Media New York 2013
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The constraints at stake, met everyday in the clinic of cancers, are related mainly to resistance to treatment in cancer cell populations and to unwanted toxicity in healthy tissues. We briefly review some of these models, namely ordinary differential equation (ODE) models, partial differential equation (PDE) models with spatial structure, phase structured cellular automata and physiologically structured PDE models. We do not claim to be exhaustive in a field where so much has been published in the last 50 years. However, we present the main models used in cancer treatment in the last decades, together with the biological phenomena that can be described by each of them. We then present some techniques used for the identification of the parameters of population dynamic models used in chemotherapy. We also briefly review theoretical therapeutic optimisation methods that can be used in the context of differe
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