A phenotype-structured model for the tumour-immune response

¹ Laboratory of Biomathematics, University Djillali Liabes, BP 89, 22000 Sidi Bel Abbes, Algeria, and Dynamical Systems and Applications Laboratory, Department of Mathematics, Faculty of Sciences, University Abou Bekr Belkaid, BP 119, 13000 Tlemcen, Algeria
² Université Paris Cité, MAP5 (UMR 8145), FP2M, CNRS FR 2036, 75006 Paris, France
³ Laboratoire Jacques-Louis Lions, Sorbonne Université, UPMC Univ. Paris 06, CNRS UMR 7598, 4, place Jussieu, BC 187, 75252 Paris Cedex 05, France, and Team Mamba, INRIA Paris, 2 rue Simone Iff, CS 42112, 75589 Paris cedex 12, France

^* Corresponding author: pouchol.camille@gmail.com

Received: 11 February 2023
Accepted: 27 July 2023

Abstract

This paper presents a mathematical model for tumour-immune response interactions in the perspective of immunotherapy by immune checkpoint inhibitors (ICIs). The model is of the nonlocal integro-differential Lotka-Volterra type, in which heterogeneity of the cell populations is taken into account by structuring variables that are continuous internal traits (aka phenotypes) present in each individual cell. These represent a lumped “aggressiveness”, i.e., for tumour cells, malignancy understood as the ability to thrive in a viable state under attack by immune cells or drugs – which we propose to identify as a potential of de-differentiation–, and for immune cells, ability to kill tumour cells, in other words anti-tumour efficacy. We analyse the asymptotic behaviour of the model in the absence of treatment. By means of two theorems, we characterise the limits of the integro-differential system under an a priori convergence hypothesis. We illustrate our results with a few numerical simulations, which show that our model reproduces the three Es of immunoediting: elimination, equilibrium, and escape. Finally, we exemplify the possible impact of ICIs on these three Es.