Evolutionary Dynamics in Vascularised Tumours under Chemotherapy: Mathematical Modelling, Asymptotic Analysis and Numeri
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Evolutionary Dynamics in Vascularised Tumours under Chemotherapy: Mathematical Modelling, Asymptotic Analysis and Numerical Simulations Chiara Villa1 · Mark A. J. Chaplain1
· Tommaso Lorenzi1,2
Received: 12 December 2019 / Accepted: 16 June 2020 / © The Author(s) 2020
Abstract We consider a mathematical model for the evolutionary dynamics of tumour cells in vascularised tumours under chemotherapy. The model comprises a system of coupled partial integro-differential equations for the phenotypic distribution of tumour cells, the concentration of oxygen and the concentration of a chemotherapeutic agent. In order to disentangle the impact of different evolutionary parameters on the emergence of intra-tumour phenotypic heterogeneity and the development of resistance to chemotherapy, we construct explicit solutions to the equation for the phenotypic distribution of tumour cells and provide a detailed quantitative characterisation of the long-time asymptotic behaviour of such solutions. Analytical results are integrated with numerical simulations of a calibrated version of the model based on biologically consistent parameter values. The results obtained provide a theoretical explanation for the observation that the phenotypic properties of tumour cells in vascularised tumours vary with the distance from the blood vessels. Moreover, we demonstrate that lower oxygen levels may correlate with higher levels of phenotypic variability, which suggests that the presence of hypoxic regions supports intra-tumour phenotypic heterogeneity. Finally, the results of our analysis put on a rigorous mathematical basis the idea, previously suggested by formal asymptotic results and numerical simulations, that hypoxia favours the selection for chemoresistant phenotypic variants prior to treatment. Consequently, this facilitates the development of resistance following chemotherapy.
Mark A. J. Chaplain
[email protected] Chiara Villa [email protected] Tommaso Lorenzi [email protected] 1
School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, United Kingdom
2
Department of Mathematical Sciences “G. L. Lagrange”, Dipartimento di Eccellenza 2018-2022, Politecnico di Torino, 10129 Torino, Italy
C. Villa et al.
Keywords Vascularised tumours · Evolutionary dynamics · Intra-tumour heterogeneity · Resistance to chemotherapy · Mathematical oncology · Non-local partial differential equations Mathematics Subject Classification (2010) 35Q92 · 92C50 · 92D25 · 35K55
1 Introduction Previous empirical and theoretical work has suggested that spatial variation in oxygen levels can foster the emergence of intra-tumour phenotypic heterogeneity [4, 26, 29, 32, 50, 56, 58, 73]. In particular, it has been hypothesised that the nonlinear interplay between impaired oxygen delivery caused by structural abnormalities present in the tumour vasculature [21, 25, 42, 43, 61, 75, 76], limited oxygen diffusion and oxygen consumption by tumour cells may lead to the creation of distinct ecological niches in vascularised t
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