Optimal control of tumour-immune model with time-delay and immuno-chemotherapy

Rihan, Fathalla A.; Lakshmanan, S.; Maurer, Helmut

doi:10.1016/j.amc.2019.02.002

Cited by 45 publications

(37 citation statements)

References 33 publications

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“…To include external chemotherapy in model (3), we should consider extra two variables namely amount of chemotherapy u(t) and normal cells N(t) with control variable v(t). The formulation as an optimal control problem allows us to: (i) investigate the dynamical system of interacting cell populations being affected by the treatments; (ii) Optimize the application of the control such that the quantity of the treatments is optimized; and (iii) Minimize the tumour size at some of end-time; See [15]. We aim to design an efficient treatment protocol, where we employ the tools of optimal control theory.…”

Section: Delay Differential Model Of Tumour-immune System With Chemotmentioning

confidence: 99%

“…Recently, Rihan et. al [15] proposed a delay differential model with optimal control for tumour-immune system with external treatments.…”

Section: Introductionmentioning

confidence: 99%

“…The objective was to find a control strategy that minimizes the total drug administered subject to the constraint that patient recovers. Recently, the authors in [15], studied an optimal control problem of delay differential model to describe the dynamics of tumour-immune interactions in presence of immuno-chemotherapy. The model includes constant delays in the mitotic phase to justify the time required to stimulate the effector cells and for the effector cells to develop a suitable response to the tumour cells.…”

Section: Introductionmentioning

confidence: 99%

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Delay Differential Model for Tumour-Immune System With Chemo-therapy and Optimal Control

Rihan

2020

ITM Web Conf.

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In this paper, we present a mathematical model of tumour-immune interactions in presence of chemotherapy treatment. The model is governed by a system of delay differential equations with optimal control variables. The control variables are included to justify the best strategy of treatments with minimum side effects, by reducing the production of new tumour cells and keeping the number of normal cells above the average of its carrying capacity. Existence of optimality and optimality conditions are also proved. The numerical simulations show that the optimal treatment strategy reduces the load of tumour cells and increases the effector cells after few days of therapy.

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Section: Delay Differential Model Of Tumour-immune System With Chemotmentioning

confidence: 99%

“…Recently, Rihan et. al [15] proposed a delay differential model with optimal control for tumour-immune system with external treatments.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Delay Differential Model for Tumour-Immune System With Chemo-therapy and Optimal Control

Rihan

2020

ITM Web Conf.

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“…The adjoint equations (20) are modified by replacing the standard Hamiltonian H by the augmented Hamiltonian . Then only the adjoint equation for z changes to .…”

Section: Time-delayed Optimal Control Problem With State Constraintsmentioning

confidence: 99%

“…To our knowledge, this is the first time that an optimal control problem with delays in state and controls, subject to state constraints, is applied to an HIV model with HIV treatment drugs and immunotherapy. Let us mention that optimal chemo-immuno therapies of cancer involving time delays in the control variables have recently been developed by Rihan et al 20 and Göllmann and Maurer. 21 A maximum principle for time-delayed optimal control problems with state constraints has recently been given in the work of Vinter.…”

Section: Introductionmentioning

confidence: 99%

Optimal control of HIV treatment and immunotherapy combination with state and control delays

Silva

Maurer

2019

Optim Control Appl Methods

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Summary In this paper, we propose and analyze an optimal control problem where human immunodeficiency virus treatment and immunotherapy are described by two control functions that are subject to time delays representing pharmacological and absorption delays, respectively. The goal is to propose effective optimal control solutions for the combination of human immunodeficiency virus treatment and immunotherapy, ensuring a functional behavior of the immune system. The incubation period is mathematically represented by a time delay in the virus load, and the local asymptotic and Hopf bifurcation analysis of the CTL‐equilibrium point of the uncontrolled delayed system is studied. We obtain optimal controls of bang‐singular type both for the nondelayed and delayed optimal control problem with and without state constraints. We study boundary arcs of state constraints and junction properties of the control and adjoint variables at entry and exit points of boundary arcs. Moreover, we derive an explicit formula of the multiplier associated with the state constraint.

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Optimal control and the Pontryagin's principle in chemical engineering: History, theory, and challenges

2022

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In the mid‐1950s, Pontryagin et al. published a principle that became a fundamental concept in optimal control (OC) theory. The principle provides theoretical and practical methods to find the solution of OC problems, in particular, open‐loop control problems. In chemical engineering, the principle has played an important role as a decision making framework for more than 60 years. This study gathers the main contributions on the application of the Pontryagin's principle to the dynamic optimization of chemical processes. A concise overview of the optimality conditions for a wide class of constrained OC problems is provided. Numerical methods to solve the necessary conditions and strategies to address inequality constraints are summarized. The information and illustrative case study presented in this work can be used as a guide to implement the principle in different settings. Opportunities for further application of the principle in relevant chemical engineering problems are also discussed.

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Optimal control of tumour-immune model with time-delay and immuno-chemotherapy

Cited by 45 publications

References 33 publications

Delay Differential Model for Tumour-Immune System With Chemo-therapy and Optimal Control

Delay Differential Model for Tumour-Immune System With Chemo-therapy and Optimal Control

Optimal control of HIV treatment and immunotherapy combination with state and control delays

Optimal control and the Pontryagin's principle in chemical engineering: History, theory, and challenges

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