Mass concentration in a nonlocal model of clonal selection

TL;DR

This paper introduces a mathematical model for clonal selection in cancer, demonstrating how self-renewal potential influences tumor growth and showing solutions tend to concentrate at optimal self-renewal points.

Contribution

It extends existing leukemia models by incorporating nonlocal feedback, proving mass concentration at maximum self-renewal, and establishing global stability of the model solutions.

Findings

Solutions tend to Dirac measures at maximum self-renewal points

Total mass converges to a stable equilibrium

Model stability proven in the space of Radon measures

Abstract

Self-renewal is a constitutive property of stem cells. Testing the cancer stem cell hypothesis requires investigation of the impact of self-renewal on cancer expansion. To understand better this impact, we propose a mathematical model describing dynamics of a continuum of cell clones structured by the self-renewal potential. The model is an extension of the finite multi-compartment models of interactions between normal and cancer cells in acute leukemias. It takes a form of a system of integro-differential equations with a nonlinear and nonlocal coupling, which describes regulatory feedback loops in cell proliferation and differentiation process. We show that such coupling leads to mass concentration in points corresponding to maximum of the self-renewal potential and the model solutions tend asymptotically to a linear combination of Dirac measures. Furthermore, using a Lyapunov function constructed for a finite dimensional counterpart of the model, we prove that the total mass of the solution converges to a globally stable equilibrium. Additionally, we show stability of the model in space of positive Radon measures equipped with flat metric. The analytical results are illustrated by numerical simulations.