Bilinear controlled model in adaptive cancer therapy

Ellina Grigorieva; Evgenii Khailov

doi:10.3934/dcdsb.2025015

Article Contents

2025, Volume 30, Issue 11: 4324-4339. Doi: 10.3934/dcdsb.2025015

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Bilinear controlled model in adaptive cancer therapy

Ellina Grigorieva^1, , and
Evgenii Khailov^2,

1.
School of Sciences, Texas Woman's University, Denton, TX 76204, USA
2.
Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Moscow, 119991, Russia

^*Corresponding author: Ellina Grigorieva
^*Corresponding author: Ellina Grigorieva

Received: October 2024

Early access: February 2025

Published: November 2025

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Abstract

Most patients diagnosed with prostate cancer (PC) are cured by surgery or radiation therapy. However, those who have already started the process of metastasis or who have relapsed after local therapy, require additional hormonal (androgen) therapy aimed at systemic treatment, carried out either in a continuous (CAS) or intermittent (IAS) scenario. The interactions of a population of androgen-dependent cancer cells and two populations of androgen-independent cancer cells in prostate cancer with and without hormonal therapy are modeled using appropriate systems of linear differential equations. Considering IAS as an effective treatment, prolonging the therapeutic advantage in the sensitivity of prostate cancer cells to pharmacological intervention and thereby improving the patient's quality of life, such linear systems are combined into a bilinear control system by introducing a control function. The control takes two values $ 1 $ or $ 0 $, which model the interaction of cancer cells when the hormonal treatment is "on" or "off", respectively. Our objective is to minimize the prostate-specific antigen (PSA) level. By solving the optimal control problem, we find how to optimally switch the periods "on" and "off" in order to minimize the cancer load at the end of a given treatment period. We solve this minimization problem analytically using two approaches: the attainability set and the Pontryagin maximum principle. Both approaches allow us to find the qualitative behavior of the optimal control, estimate its maximum number of switchings, and replace the original minimization problem with corresponding finite-dimensional nonlinear optimization problems, which are then solved numerically using MAPLE. In addition, the original optimal control problem is solved only numerically using BOCOP. Numerical results obtained for various values of parameters and initial conditions of the bilinear control system are discussed.

Keywords:

Adaptive cancer treatment,
bilinear control system,
quasi-commutativity condition of matrices,
attainability set,
optimal control,
Pontryagin maximum principle,
switching function.

Mathematics Subject Classification: Primary: 34H05, 93C10; Secondary: 93C95.

Citation:

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Figure 1. Graphs of optimal solutions $ x_{1}^{*}(t) $, $ x_{2}^{*}(t) $, $ x_{3}^{*}(t) $ and control $ u_{*}(t) $: upper row: $ x_{1}^{*}(t) $, $ x_{2}^{*}(t) $; lower row: $ x_{3}^{*}(t) $, $ u_{*}(t) $

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Figure 2. Graphs of optimal solutions $ x_{1}^{*}(t) $, $ x_{2}^{*}(t) $, $ x_{3}^{*}(t) $ and control $ u_{*}(t) $: upper row: $ x_{1}^{*}(t) $, $ x_{2}^{*}(t) $; lower row: $ x_{3}^{*}(t) $, $ u_{*}(t) $

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Figure 3. Graphs of optimal solutions $ x_{1}^{*}(t) $, $ x_{2}^{*}(t) $, $ x_{3}^{*}(t) $ and control $ u_{*}(t) $: upper row: $ x_{1}^{*}(t) $, $ x_{2}^{*}(t) $; lower row: $ x_{3}^{*}(t) $, $ u_{*}(t) $

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Figure 4. Example 1. Graphs of optimal solutions $ x_{1}^{*}(t) $, $ x_{2}^{*}(t) $, $ x_{3}^{*}(t) $ and control $ u_{*}(t) $: upper row: $ x_{1}^{*}(t) $, $ x_{2}^{*}(t) $, $ x_{3}^{*}(t) $; lower row: $ {\rm psa}_{*}(t) $, $ u_{*}(t) $

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Figure 5. Example 2. Graphs of optimal solutions $ x_{1}^{*}(t) $, $ x_{2}^{*}(t) $, $ x_{3}^{*}(t) $ and control $ u_{*}(t) $: upper row: $ x_{1}^{*}(t) $, $ x_{2}^{*}(t) $, $ x_{3}^{*}(t) $; lower row: $ {\rm psa}_{*}(t) $, $ u_{*}(t) $

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Figure 6. Example 3. Graphs of optimal solutions $ x_{1}^{*}(t) $, $ x_{2}^{*}(t) $, $ x_{3}^{*}(t) $ and control $ u_{*}(t) $: upper row: $ x_{1}^{*}(t) $, $ x_{2}^{*}(t) $, $ x_{3}^{*}(t) $; lower row: $ {\rm psa}_{*}(t) $, $ u_{*}(t) $

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