Synthesis of optimal control in a mathematical model of tumour–immune dynamics

Yegorov, Ivan; Todorov, Yancho

doi:10.1002/oca.2103

Cited by 9 publications

(7 citation statements)

References 21 publications

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“…At each iteration, the Riccati differential equations are solved via backward integration with the final values of S = diag (10,10,10,10). The solution of the Riccati differential equations for the 11th iteration is illustrated in Figure 12, which is the solution of the following equation: P [11] = −P [11] A m (…”

Section: Drug Delivery Scenario For the Reference Patient Via Saamentioning

confidence: 99%

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Controller design for personalized drug administration in cancer therapy: Successive approximation approach

Babaei

Salamcı

2017

Optim Control Appl Methods

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Summary A novel controller design approach is developed to determine personalized chemotherapy drug delivery protocols for cancer treatment. The methodology combines successive approximation approach for optimal control and model reference adaptive control to realize the proposed drug administration scenario for patients without prior knowledge of model parameters in the nonlinear cancer dynamics. Although many approaches have been proposed to determine the optimal drug delivery protocol for eradicating tumor in the nonlinear cancer model, the main shortcoming of these approaches is the requirement of nonlinear model dynamics, which is unknown to physicians in reality. To overcome this deficiency, we first determine the optimal drug delivery protocol for a reference patient with known mathematical model and parameters via successive approximation approach technique. Then, using the proposed approach, we adapt the reference patient's drug administration scenario to unknown cancer patients by suggesting a new adaptation mechanism for the unknown nonlinear plant dynamics. An efficient and robust approach is proposed here for the physicians to prescribe a personalized chemotherapy protocol for a cancer patient by regulating the drug delivery protocol of a reference patient. The efficacy of the proposed algorithm in eradicating the tumor lumps with different sizes in several patients is verified using numerical simulations in which the unknown parameters are randomly selected in the Monte Carlo approach.

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Section: Drug Delivery Scenario For the Reference Patient Via Saamentioning

confidence: 99%

“…1,7,8 To avoid toxicity and weakening of the patient's immune system in some optimal control studies, the constraints are considered for drug maximum dose and lower bond for a minimum population of immune cells. 6,[9][10][11][12] The survival chance of the cancer patient has a direct dependence on the appropriate drug delivery protocol.…”

Section: Introductionmentioning

confidence: 99%

Controller design for personalized drug administration in cancer therapy: Successive approximation approach

Babaei

Salamcı

2017

Optim Control Appl Methods

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“…(), where the optimal chemotherapy is determined by applying the state‐dependent Riccati equation based OC technique. An OC problem for a model of tumor–immune dynamics under the influence of chemotherapy is addressed in Yegorov and Todorov (). The authors use the Pontryagin maximum principle to analyze the proposed system of ordinary differential equations.…”

Section: Oc Formulationmentioning

confidence: 99%

On a multiobjective optimal control of a tumor growth model with immune response and drug therapies

Rocha

Costa

Fernandes

2016

Int Trans Operational Res

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In this paper, we consider a tumor growth mathematical model that includes an immune system and drug therapies. Immuno-and chemodrug administration as well as periodic administration of radiation are integrated in the model. We have set an optimal control (OC) problem relative to the model so that the average number of tumor cells and immuno-and chemotherapeutic drug administration are simultaneously minimized in a multiobjective context. The external injection of two immunostimulators and a chemotherapeutic drug has been considered as control, and the objective functionals aim to reflect the severity of both types of drug administration. The multiobjective approach to the OC of the tumor growth model provides good-quality approximate solutions and has shown to be a valuable procedure to identify good trade-offs between conflicting objectives.

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“…Many optimal control strategies have been proposed for highlighting the advantage of optimization of some parameters in some biomathematical models . In this context, some applied mathematicians advise to rely dosage problems; also, on their optimal control approaches, they see beneficial for developing therapeutic strategies that aim to reduce tumor volume with lesser side‐effects, see, for instance, in the case of chemotherapy, the work in Yegorov and Todorov, and in the case of immunotherapy, papers in Hamdache et al…”

Section: Introductionmentioning

confidence: 99%

“…Many optimal control strategies have been proposed for highlighting the advantage of optimization of some parameters in some biomathematical models. [9][10][11] In this context, some applied mathematicians advise to rely dosage problems; also, on their optimal control approaches, they see beneficial for developing therapeutic strategies that aim to reduce tumor volume with lesser side-effects, see, for instance, in the case of chemotherapy, the work in Yegorov and Todorov, 12 and in the case of immunotherapy, papers in Hamdache et al 13,14 Some research papers on mathematical medicine have developed mathematical models which describe the evolution of tumors 15,16 and analyze the different interactions between the tumor, immune-system, and treatment. 17,18 In this paper, we formulate a minimization problem from which we aim to find the optimal dose and duration of the BCG immunotherapy procedure in the case of superficial bladder cancer, based on a prevalidated mathematical model 19 in the form of 4 ordinary differential equations which describe the superficial bladder tumor-immune interactions after the injection of BCG in the bladder of a hypothetical patient.…”

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confidence: 99%

Optimal duration and dosage of BCG intravesical immunotherapy: A free final time optimal control approach

Alkama

Larrache

Rachik

et al. 2018

Math Methods in App Sciences

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In this paper, we propose a free final time optimal control approach applied to 4 ordinary differential equations which describe the tumor‐immune interactions after the injection of the bacillus Calmette‐Guérin (BCG) in the bladder of a hypothetical patient. The main goal of this optimal control strategy is to find the optimal dosage amount needed in each instillation of BCG for stimulating the immune‐system and then killing superficial bladder tumors and to determine the optimal duration of treatment, adequate for stopping the intravesical therapy with lesser side‐effects. For this, we introduce into the model of interest, a control function which represents the dose of BCG immunotherapy procedure and we formulate a minimization problem where the final time is considered free (nonfixed). The characterization of the sought optimal control noted u∗ is derived based on Pontryagin's maximum principle, while the formulation of the sought optimal final time noted tfinal∗ is based on formulae of sensitivity which are obtained conditions from the derivative of the objective function with respect to tfinal∗. We investigate the resolution of the free final time optimal control problem in 3 possible cases: (a) when tfinal∗ is quadratic in the final gain function, (b) when the final gain function does not depend on tfinal∗, and (c) when tfinal∗ is linear in the final gain function. Finally, we obtain the sought optimal dose of BCG, and we conclude that in case (a), we obtain an optimal duration which is more beneficial regarding the activation of immune cells while cases (b) and (c) both provide an optimal duration which is more adequate for the minimization of the tumoral population.

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Synthesis of optimal control in a mathematical model of tumour–immune dynamics

Cited by 9 publications

References 21 publications

Controller design for personalized drug administration in cancer therapy: Successive approximation approach

Controller design for personalized drug administration in cancer therapy: Successive approximation approach

On a multiobjective optimal control of a tumor growth model with immune response and drug therapies

Optimal duration and dosage of BCG intravesical immunotherapy: A free final time optimal control approach

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