Optimal treatment strategy of cancers with intratumor heterogeneity

Haifeng Zhang; Jinzhi Lei; Haifeng Zhang; Jinzhi Lei

doi:10.3934/mbe.2022625

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 12: 13337-13373. doi: 10.3934/mbe.2022625

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Optimal treatment strategy of cancers with intratumor heterogeneity

Haifeng Zhang ¹,
Jinzhi Lei ^{2
,
,}

1.
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
2.
School of Mathematical Sciences, Center for Applied Mathematics, Tiangong University, Tianjin 300387, China

Academic Editor: Kevin Flores

Received: 23 June 2022 Revised: 29 July 2022 Accepted: 03 September 2022 Published: 14 September 2022

Intratumor heterogeneity hinders the success of anti-cancer treatment due to the interaction between different types of cells. To recapitulate the communication of different types of cells, we developed a mathematical model to study the dynamic interaction between normal, drug-sensitive and drug-resistant cells in response to cancer treatment. Based on the proposed model, we first study the analytical conclusions, namely the nonnegativity and boundedness of solutions, and the existence and stability of steady states. Furthermore, to investigate the optimal treatment that minimizes both the cancer cells count and the total dose of drugs, we apply the Pontryagin's maximum(or minimum) principle (PMP) to explore the combination therapy strategy with either quadratic control or linear control functionals. We establish the existence and uniqueness of the quadratic control problem, and apply the forward-backward sweep method (FBSM) to solve the optimal control problems and obtain the optimal therapy scheme.
- mathematical oncology,
- drug resistance,
- heterogeneity,
- optimal control
Citation: Haifeng Zhang, Jinzhi Lei. Optimal treatment strategy of cancers with intratumor heterogeneity[J]. Mathematical Biosciences and Engineering, 2022, 19(12): 13337-13373. doi: 10.3934/mbe.2022625

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Abstract

Intratumor heterogeneity hinders the success of anti-cancer treatment due to the interaction between different types of cells. To recapitulate the communication of different types of cells, we developed a mathematical model to study the dynamic interaction between normal, drug-sensitive and drug-resistant cells in response to cancer treatment. Based on the proposed model, we first study the analytical conclusions, namely the nonnegativity and boundedness of solutions, and the existence and stability of steady states. Furthermore, to investigate the optimal treatment that minimizes both the cancer cells count and the total dose of drugs, we apply the Pontryagin's maximum(or minimum) principle (PMP) to explore the combination therapy strategy with either quadratic control or linear control functionals. We establish the existence and uniqueness of the quadratic control problem, and apply the forward-backward sweep method (FBSM) to solve the optimal control problems and obtain the optimal therapy scheme.

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