Reconstruction of Mutation Laws in Heterogeneous Tumours with Local and Nonlocal Dynamics

TL;DR

This paper develops methods to reconstruct mutation laws in heterogeneous tumours using local and nonlocal models, based on macroscopic data, and employs regularisation techniques to address ill-posed inverse problems.

Contribution

It introduces novel reconstruction techniques for mutation laws in tumour models considering local and nonlocal dynamics, with regularisation for stability.

Findings

Successful reconstruction of mutation laws under various assumptions.

Demonstrated stability of solutions with added measurement noise.

Applicable to both local and nonlocal tumour invasion models.

Abstract

Cancer cell mutations occur when cells undergo multiple cell divisions, and these mutations can be spontaneous or environmentally-induced. The mechanisms that promote and sustain these mutations are still not fully understood. This study deals with the identification (or reconstruction) of the usually unknown cancer cell mutation law, which lead to the transformation of a primary tumour cell population into a secondary, more aggressive cell population. We focus on local and nonlocal mathematical models for cell dynamics and movement, and identify these mutation laws from macroscopic tumour snapshot data collected at some later stage in the tumour evolution. In a local cancer invasion model, we first reconstruct the mutation law when we assume that the mutations depend only on the surrounding cancer cells (i.e., the ECM plays no role in mutations). Second, we assume that the mutations depend on the ECM only, and we reconstruct the mutation law in this case. Third, we reconstruct the mutation when we assume that there is no prior knowledge about the mutations. Finally, for the nonlocal cancer invasion model, we reconstruct the mutation law that depends on the cancer cells and on the ECM. For these numerical reconstructions, our approximations are based on the finite difference method combined with the finite elements method. As the inverse problem is ill-posed, we use the Tikhonov regularisation technique in order to regularise the solution. Stability of the solution is examined by adding additive noise into the measurements.