Inverse problem approaches for mutation laws in heterogeneous tumours with local and nonlocal dynamics

Maher Alwuthaynani; Raluca Eftimie; Dumitru Trucu; Maher Alwuthaynani; Raluca Eftimie; Dumitru Trucu

doi:10.3934/mbe.2022171

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 4: 3720-3747. doi: 10.3934/mbe.2022171

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Inverse problem approaches for mutation laws in heterogeneous tumours with local and nonlocal dynamics

1.
Division of Mathematics, University of Dundee, Dundee DD1 4HN, Scotland, UK
2.
Laboratoire Mathématiques de Besançcon, UMR-CNRS 6623, Université de Bourgogne Franche-Comté, 16 Route de Gray, Besançcon 25000, France

Academic Editor: Kevin Flores

Received: 20 October 2021 Revised: 05 December 2021 Accepted: 22 December 2021 Published: 10 February 2022

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.
- inverse problems,
- mutation identification,
- Tikhonov regularisation,
- tumour growth
Citation: Maher Alwuthaynani, Raluca Eftimie, Dumitru Trucu. Inverse problem approaches for mutation laws in heterogeneous tumours with local and nonlocal dynamics[J]. Mathematical Biosciences and Engineering, 2022, 19(4): 3720-3747. doi: 10.3934/mbe.2022171

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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.

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