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

¹ Sorbonne Universités, CNRS, Universités Paris-Diderot, INRIA, Laboratoire Jacques-Louis Lions, 75005 Paris, France
² Università degli Studi di Genova, DIME, Genova 16129, Italy
³ University of Konstanz, Department of Mathematics and Statistics, Universitätsstraße 10, 78457, Konstanz, Germany
⁴ University of Melbourne, School of Mathematics and Statistics, Melbourne, VIC 3010, Australia
⁵ University of St Andrews, School of Mathematics and Statistics, St Andrews, KY16 9SS, UK

^* Corresponding author: tl47@st-andrews.ac.uk

Received: 3 August 2018
Accepted: 30 January 2019

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 i.e. the optimal dosing regimen to minimise the average size of the cancer cell population during the course of treatment.