Evolution of cancer cell populations under cytotoxic therapy and treatment optimisation: insight from a phenotype-structured model

Abstract

We consider a phenotype-structured model of evolutionary dynamics in a population of cancer cells exposed to the action of a cytotoxic drug. The model consists of a nonlocal parabolic equation governing the evolution of the cell population density function. We develop a novel method for constructing exact solutions to the model equation, which allows for a systematic investigation of the way in which the size and the phenotypic composition of the cell population change in response to variations of the drug dose and other evolutionary parameters. Moreover, we address numerical optimal control for a calibrated version of the model based on biological data from the existing literature, in order to identify the drug delivery schedule that makes it possible to minimise either the population size at the end of the treatment or the average population size during the course of treatment. The results obtained challenge the notion that traditional high-dose therapy represents a 'one-fits-all solution' in anticancer therapy by showing that the continuous administration of a relatively low dose of the cytotoxic drug performs more closely to the optimal dosing regimen to minimise the average size of the cancer cell population during the course of treatment. * BDH acknowledges support from the Australian Research Council (DP140100339). LA and TL gratefully acknowledge support of the project PICS-CNRS no. 07688 and the French "ANR blanche" project Kibord: ANR-13-BS01-0004.

Discrete and continuum phenotype-structured models for the evolution of cancer cell populations under chemotherapy

“…However, the stochastic individual-based model presented here, as well as the related formal method to derive the corresponding deterministic continuum model, can be easily adapted to drug doses that vary over time. In this regard, it would be interesting to investigate whether the delivery schedules for the chemotherapeutic agent obtained through numerical optimal control of the nonlocal PDE for the population density function [2,62] would remain optimal also for the individual-based model. Another track to follow might be to investigate the effect of stress-induced epimutations triggered by the selective pressure that chemotherapeutic agents exert on cancer cells [25].…”

Evolutionary Dynamics in Vascularised Tumours under Chemotherapy: Mathematical Modelling, Asymptotic Analysis and Numerical Simulations

“…Substituting (2) and (10) into (1) yields ∂n ∂t = β ∂ 2 n ∂y 2 + a − b(y − h) 2 − ζρ(t, x) n, n ≡ n(t, x, y), (t, x, y) ∈ (0, ∞)× ×R. (31) Building upon the results presented in [5,19,51], we make the ansatz (22). Substituting this ansatz into (31) and introducing the notation v(t, x) := 1/σ 2 (t, x) we find…”

“…Finally, as similarly done by [5] and [64], it would be relevant to address numerical optimal control of the model equations in order to identify possible delivery schedules of the chemotherapeutic agent that make it possible to minimise the number of tumour cells at the end of the treatment or the average number of tumour cells during the course of treatment. In particular, it would be relevant to verify whether the results presented in [5] for a spatially homogeneous model-which indicate that continuous administration of a relatively low dose of the chemotherapy performs more closely to the optimal dosing regimen to minimise the average number of tumour cells during the course of treatment-carry through when spatial reaction-diffusion dynamics of the chemotherapeutic agent are incorporated into the model. In this regard, it would be interesting to assess the impact of molecular properties of the chemotherapeutic agent (e.g.…”

“…The main novelty of our work is that we allow tumour cells to undergo spontaneous epimutations (i.e., heritable phenotypic changes that occur randomly due to non-genetic instability and are not induced by any selective pressure [39]) and we do not impose any smallness assumptions on the rate at which such phenotypic changes occur. In this more general scenario, building upon the method of proof presented in [5,8,19,51], we carry out an analytical study of evolutionary dynamics. In particular, we construct explicit solutions to the equation for the phenotypic distribution of tumour cells and, considering the case where the concentrations of oxygen and chemotherapeutic agent are stationary, we provide a detailed quantitative characterisation of the long-time asymptotic behaviour of such solutions.…”

Stepping From Modeling Cancer Plasticity to the Philosophy of Cancer

“…In these equations, the structuring variable, i.e., the parameter-like one that codes for the biological variability of interest, is assumed to store the heterogeneity of the cell population with respect to the expression of drug resistance. It was chosen to be a positive real variable representing the expression of a resistance phenotype continuously from 0 (totally sensitive) to 1 (totally resistant) (Perthame, 2007 , 2015 ; Lavi et al, 2013 ; Lorz et al, 2013 , 2015 ; Chisholm et al, 2015 , 2016a , b ; Lorenzi et al, 2016 ; Almeida et al, 2018 , 2019 ; Cho and Levy, 2018a , b ; Clairambault, 2019 ; Clairambault and Pouchol, 2019 ; Nguyen et al, 2019 ).…”