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Physiologically structured cell population dynamic models with applications to combined drug delivery optimisation in oncology

Jean Clairambault 1, 2 Olivier Fercoq 3, 4
2 BANG - Nonlinear Analysis for Biology and Geophysical flows
LJLL - Laboratoire Jacques-Louis Lions, Inria Paris-Rocquencourt
3 MAXPLUS - Max-plus algebras and mathematics of decision
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
4 Operational Research and Optimization Group, The School of Mathematics, The King's Buildings
School of Mathematics - Operational Research and Optimization
Abstract : Optimising drug delivery in the general circulation targeted towards cancer cell pop- ulations, but inevitably reaching also proliferating healthy cell populations imposes to design optimised drug infusion algorithms in a dynamic way, i.e., controlling the growth of both populations simultaneously by the action of the drugs in use, wanted for cancer cells, and unwanted for toxic side effects on healthy cells. Towards this goal, we design models and methods, with optional representation of circadian clock control on proliferation in both populations, according to three axes [15, 16]: a) representing the oncologist's main weapons, drugs, and their fates in the organism by molecular-based pharmacokinetic-pharmacodynamic equations; b) representing the cell populations under attack by drugs, and their proliferation dynamics, including in the models molecular and functional targets for the drugs at stake, by physiologically structured equations; c) using numerical algorithms, optimising drug delivery under different constraints at the whole organism level, representing impacts of multiple drugs with different targets on cell populations. In the present study, two molecular pharmacological ODE models, one for oxali- platin, and one for 5-Fluorouracil, have been designed, using law of mass action and enzyme kinetics, to describe the fate of these two cytotoxic drugs in the organism. An age-structured PDE cell population model has been designed with drug control. targets to represent the effects of oxaliplatin and 5-Fluorouracil on the cell cycle in proliferating cell populations. The models for proliferating cell population dynam- ics involve possible physiological fixed (i.e., out of reach of therapeutic influence) circadian clock control, and varying drug control to be optimised, connected with pharmacological models. Concentrations of drugs, represented by outputs of ODEs, are assumed to be homogeneous in the cell populations under attack by cytotoxic drugs. The possi- bility to describe the effects of other drugs, cytostatic (including in this category anti-angiogenic drugs, considered as acting on the G1 phase, choking its entries and slowing it down), is also presented, but not put in pharmacokinetic equations and actual simulations in this study, that is focused on the combination of 5-FU and oxaliplatin, a classic therapeutic association in the treatment of colorectal cancer. We then set conditions to numerically solve drug delivery optimisation problems (maximisation of cancer cell kill under the constraint of preserving healthy cells over a tolerability threshold) by considering a trade-off between therapeutic and toxic effects. The observed effects on proliferation are growth exponents, i.e., first eigenvalues of the linear PDE systems, in the two populations, healthy and cancer. The solutions to an optimisation problem taking into account circadian clock control are presented as best delivery time schedules for the two drugs used in combined treatments, to be implemented in programmable delivery pumps in the clinic.
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Submitted on : Sunday, November 11, 2012 - 11:38:46 PM
Last modification on : Wednesday, December 9, 2020 - 3:06:01 PM
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Jean Clairambault, Olivier Fercoq. Physiologically structured cell population dynamic models with applications to combined drug delivery optimisation in oncology. Bachar, Mostafa and Batzel, Jerry and Chaplain, Mark. Mathematical modelling of cancer growth and treatment, Springer, 2012, Springer Lecture Notes in Mathematics Biosciences (LNMBIOS subseries). ⟨hal-00750633⟩



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