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 …
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.
We present a delay differential model with optimal control that describes the interactions of the tumour cells and immune response cells with external therapy. The intracellular delay is incorporated into …
We present a delay differential model with optimal control that describes the interactions of the tumour cells and immune response cells with external therapy. The intracellular delay is incorporated into the model to justify the time required to stimulate the effector cells. The optimal control variables are incorporated to identify the best treatment strategy with minimum side effects by blocking the production of new tumour cells and keeping the number of normal cells above 75% of its carrying capacity. Existence of the optimal control pair and optimality system are established. Pontryagin’s maximum principle is applicable to characterize the optimal controls. The model displays a tumour-free steady state and up to three coexisting steady states. The numerical results show that the optimal treatment strategies reduce the tumour cells load and increase the effector cells after a few days of therapy. The performance of combination therapy protocol of immunochemotherapy is better than the standard protocol of chemotherapy alone.
Herein, 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. A …
Herein, 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. A discrete time-delay is considered to justify the time-needed for the effector cells to develop a suitable response to the tumour cells. The control variables are included to justify the best treatment strategy 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. The numerical simulations show that the optimal treatment strategy reduces the load of tumour cells increases the effector cells after few days of therapy.
<abstract><p>Herein, we discuss an optimal control problem (OC-P) of a stochastic delay differential model to describe the dynamics of tumor-immune interactions under stochastic white noises and external treatments. The required …
<abstract><p>Herein, we discuss an optimal control problem (OC-P) of a stochastic delay differential model to describe the dynamics of tumor-immune interactions under stochastic white noises and external treatments. The required criteria for the existence of an ergodic stationary distribution and possible extinction of tumors are obtained through Lyapunov functional theory. A stochastic optimality system is developed to reduce tumor cells using some control variables. The study found that combining white noises and time delays greatly affected the dynamics of the tumor-immune interaction model. Based on numerical results, it can be shown which variables are optimal for controlling tumor growth and which controls are effective for reducing tumor growth. With some conditions, white noise reduces tumor cell growth in the optimality problem. Some numerical simulations are conducted to validate the main results.</p></abstract>
We have considered a tumor growth model with the effect of tumor-immune interaction and chemotherapeutic drug. We have considered two immune components—helper (resting) T-cells which stimulate CTLs and convert them …
We have considered a tumor growth model with the effect of tumor-immune interaction and chemotherapeutic drug. We have considered two immune components—helper (resting) T-cells which stimulate CTLs and convert them into active (hunting) CTL cells and active (hunting) CTL cells which attack, destroy, or ingest the tumor cells. In our model there are four compartments, namely, tumor cells, active CTL cells, helper T-cells, and chemotherapeutic drug. We have discussed the behaviour of the solutions of our system. The dynamical behaviour of our system by analyzing the existence and stability of the system at various equilibrium points is discussed elaborately. We have set up an optimal control problem relative to the model so as to minimize the number of tumor cells and the chemotherapeutic drug administration. Here we used a quadratic control to quantify this goal and have considered the administration of chemotherapy drug as control to reduce the spread of the disease. The important mathematical findings for the dynamical behaviour of the tumor-immune model with control are also numerically verified using MATLAB. Finally, epidemiological implications of our analytical findings are addressed critically.
An optimal control problem for combination of cancer chemotherapy withimmunotherapy in form of a boost to the immune system is considered as amulti-input optimal control problem. The objective to be …
An optimal control problem for combination of cancer chemotherapy withimmunotherapy in form of a boost to the immune system is considered as amulti-input optimal control problem. The objective to be minimized is chosenas a weighted average of (i) the number of cancer cells at the terminal time,(ii) a measure for the immunocompetent cell densities at the terminal point(included as a negative term), the overall amounts of (iii) cytotoxic agentsand (iv) immune boost given as a measure for the side effects oftreatment and (v) a small penalty on the free terminal time that limits theoverall therapy horizon. This last term is essential in obtaining a mathematically well-posedproblem formulation. Both analytical and numerical results about thestructures of optimal controls will be presented that give some insightsinto the structure of optimal protocols, i.e., the dose rates and sequencing ofdrugs in these combination treatments.
In this paper, we investigate strategies for administering chemo- and immunotherapy to force a tumor-immune system to its healthy equilibrium. To solve this problem, we use Pontryagin’s Maximum Principle applied …
In this paper, we investigate strategies for administering chemo- and immunotherapy to force a tumor-immune system to its healthy equilibrium. To solve this problem, we use Pontryagin’s Maximum Principle applied to a modified Stepanova model. This model directly accounts for the detrimental effects of chemotherapy on immune cell density. Because the parameter for this interaction is unknown, we run simulations while varying the parameter to observe the effect on the system. Our results show that combined dosages of chemo- and immunotherapy over the first days of the treatment period are sufficient to force the system to its healthy equilibrium.
Abstract In this study, we discuss a cancer model considering discrete time-delay in tumor-immune interaction and stimulation processes. This study aims to analyze and observe the dynamics of the model …
Abstract In this study, we discuss a cancer model considering discrete time-delay in tumor-immune interaction and stimulation processes. This study aims to analyze and observe the dynamics of the model along with variation of vital parameters and the delay effect on anti-tumor immune responses. We obtain sufficient conditions for the existence of equilibrium points and their stability. Existence of Hopf bifurcation at co-axial equilibrium is investigated. The stability of bifurcating periodic solutions is discussed, and the time length for which the solutions preserve the stability is estimated. Furthermore, we have derived the conditions for the direction of bifurcating periodic solutions. Theoretically, it was observed that the system undergoes different states if we vary the system’s parameters. Some numerical simulations are presented to verify the obtained mathematical results.
In this article, we propose the interaction of tumor cells with the immune system in the presence of chemotherapy. The existence, uniqueness, non‐negativity, and boundedness of the solutions have been …
In this article, we propose the interaction of tumor cells with the immune system in the presence of chemotherapy. The existence, uniqueness, non‐negativity, and boundedness of the solutions have been established. The conditions for the existence and stability of equilibrium points have been presented in both drug‐free and treated systems. The local stability of the co‐existing equilibrium point is proved using the Routh–Hurwitz rule, and the global stability is proved using the Lyapunov function. We have used quadratic optimal control to minimize the number of tumor cells and the side effects of chemotherapy on the immune system and healthy cells. We have demonstrated the existence of optimal control and derived the corresponding optimality system using Pontryagin’s maximum principle. The optimal system is solved using the forward‐backward sweep method with fourth‐order Runge–Kutta approximation. Reduction in tumor cell growth has been observed due to the increase in recruitment of immune cells activated by tumor cell antigenicity and the rate of conversion of resting immune cells into active immune cells. Additionally, the impact of administering varying chemotherapy doses on reducing tumor cell growth has been noted. Finally, a comparison between controlled and uncontrolled dynamics has been conducted to comprehend the effect of optimal control.
We present a competition model of cancer tumor growth that includes both the immune system response and drug therapy. This is a four‐population model that includes tumor cells, host cells, …
We present a competition model of cancer tumor growth that includes both the immune system response and drug therapy. This is a four‐population model that includes tumor cells, host cells, immune cells, and drug interaction. We analyze the stability of the drug‐free equilibria with respect to the immune response in order to look for target basins of attraction. One of our goals was to simulate qualitatively the asynchronous tumor‐drug interaction known as “Jeffs phenomenon.” The model we develop is successful in generating this asynchronous response behavior. Our other goal was to identify treatment protocols that could improve standard pulsed chemotherapy regimens. Using optimal control theory with constraints and numerical simulations, we obtain new therapy protocols that we then compare with traditional pulsed periodic treatment. The optimal control generated therapies produce larger oscillations in the tumor population over time. However, by the end of the treatment period, total tumor size is smaller than that achieved through traditional pulsed therapy, and the normal cell population suffers nearly no oscillations.
Abstract Tumors are the most threatening issue everywhere throughout the world. The development of tumor cells is dubious in the human body because of its unusual phenomena. The Rough set …
Abstract Tumors are the most threatening issue everywhere throughout the world. The development of tumor cells is dubious in the human body because of its unusual phenomena. The Rough set is a rising and the most special mathematical device to manage uncertain circumstances. A scientific model is given for tumor cells population development with carrying capacity and by the Rough set in uncertain circumstances. In this methodology, the mathematical analysis of the nonlinear behavior of tumor cells population is set up via carrying capacity and simulation by using Euler’s method. The accuracy of the carrying capacity of the number of tumors cells 99.53% correct according to our model. The paper is an interface between mathematical modeling, numerical computation, simulation, and implementation of application on biomedical systems, which is an oriented idea to biology.
In this article, we analyze the dynamics of the non-linear tumor-immune delayed (TID) model illustrating the interaction among tumor cells and the immune system (cytotoxic T lymphocytes, T helper cells), …
In this article, we analyze the dynamics of the non-linear tumor-immune delayed (TID) model illustrating the interaction among tumor cells and the immune system (cytotoxic T lymphocytes, T helper cells), where the delays portray the times required for molecule formation, cell growth, segregation, and transportation, among other factors by exploiting the knacks of soft computing paradigm utilizing neural networks with back propagation Levenberg Marquardt approach (NNLMA). The governing differential delayed system of non-linear TID, which comprised the densities of the tumor population, cytotoxic T lymphocytes and T helper cells, is represented by non-linear delay ordinary differential equations with three classes. The baseline data is formulated by exploiting the explicit Runge-Kutta method (RKM) by diverting the transmutation rate of Tc to Th of the Tc population, transmutation rate of Tc to Th of the Th population, eradication of tumor cells through Tc cells, eradication of tumor cells through Th cells, Tc cells' natural mortality rate, Th cells' natural mortality rate as well as time delay. The approximated solution of the non-linear TID model is determined by randomly subdividing the formulated data samples for training, testing, as well as validation sets in the network formulation and learning procedures. The strength, reliability, and efficacy of the designed NNLMA for solving non-linear TID model are endorsed by small/negligible absolute errors, error histogram studies, mean squared errors based convergence and close to optimal modeling index for regression measurements.
Cancer chemotherapy has been the most common cancer treatment. However, it has side effects that kill both tumor cells and immune cells, which can ravage the patient’s immune system. Chemotherapy …
Cancer chemotherapy has been the most common cancer treatment. However, it has side effects that kill both tumor cells and immune cells, which can ravage the patient’s immune system. Chemotherapy should be administered depending on the patient’s immunity as well as the level of cancer cells. Thus, we need to design an efficient treatment protocol. In this work, we study a feedback control problem of tumor‐immune system to design an optimal chemotherapy strategy. For this, we first propose a mathematical model of tumor‐immune interactions and conduct stability analysis of two equilibria. Next, the feedback control is found by solving the Hamilton–Jacobi–Bellman (HJB) equation. Here, we use an upwind finite‐difference method for a numerical approximate solution of the HJB equation. Numerical simulations show that the feedback control can help determine the treatment protocol of chemotherapy for tumor and immune cells depending on the side effects.
To find optimal methods to inhibit tumors, we propose a tumor-lymphocyte immune optimal model with immuno-chemotherapy. Firstly, we investigate the therapeutic effects of high-dose single immunotherapy and high-dose single chemotherapy …
To find optimal methods to inhibit tumors, we propose a tumor-lymphocyte immune optimal model with immuno-chemotherapy. Firstly, we investigate the therapeutic effects of high-dose single immunotherapy and high-dose single chemotherapy for tumor logistic growth, respectively. Furthermore, we apply the optimal control theory to investigate the optimal control problem of immuno-chemotherapy to eliminate tumors, maximize the remaining number of lymphocytes and minimize the cost caused by drugs over a finite time interval. The necessary and sufficient conditions for the existence of optimal control are also discussed. Finally, the numerical results indicate that the effect of immuno-chemotherapy with strong killing rate to tumors and weak killing rate to immune cells is the most effective strategy in inhibiting tumor growth.
Failure in cancer treatment often stems from drug resistance, which can manifest as either intrinsic (pre-existing) or acquired (induced by drugs). Despite extensive efforts, overcoming this resistance remains a challenging …
Failure in cancer treatment often stems from drug resistance, which can manifest as either intrinsic (pre-existing) or acquired (induced by drugs). Despite extensive efforts, overcoming this resistance remains a challenging task due to the intricate and highly individualized biological mechanisms involved. This paper introduces an innovative extension of an already well-established mathematical model to account for tumour resistance development against chemotherapy. This study examines the existence and local stability of model solutions, as well as exploring the model asymptotic dynamics. Additionally, a numerical analysis of the optimal control problem is conducted using an objective functional. The numerical simulations demonstrate that a constant anti-angiogenic treatment leads to a concatenation of bang-bang and singular intervals in chemotherapy control, resembling a combined protocol comprising maximal tolerated dose and metronomic protocols. This observation lends support to the hypothesis that mean-dose chemotherapy protocols may help circumvent acquired drug resistance. Lastly, a sensitivity analysis is undertaken to scrutinize the dependence of model parameters on the outcomes of the previously examined therapeutic protocols.
This paper examines the dynamics of a time-delay differential model of the tumour immune system with random noise. The model describes the interactions between healthy tissue cells, tumour cells, and …
This paper examines the dynamics of a time-delay differential model of the tumour immune system with random noise. The model describes the interactions between healthy tissue cells, tumour cells, and activated immune system cells. We discuss stability and Hopf bifurcation of the deterministic system. We then explore stochastic stability, and the dynamics of the system in view of environmental fluctuations. Criteria for persistence and sustainability are discussed. Using multiple Lyapunov functions, some sufficient criteria for tumour cell persistence and extinction are obtained. Under certain circumstances, stochastic noise can suppress tumour cell growth completely. In contrast to the deterministic model which shows no stable tumour-free state, the white noise can either lead to tumour dormancy or tumour elimination. Some numerical simulations, by using Milstein's scheme, are carried out to show the effectiveness of the obtained results.
In this article, a mathematical model of the COVID-19 pandemic with control parameters is introduced. The main objective of this study is to determine the most effective model for predicting …
In this article, a mathematical model of the COVID-19 pandemic with control parameters is introduced. The main objective of this study is to determine the most effective model for predicting the transmission dynamic of COVID-19 using a deterministic model with control variables. For this purpose, we introduce three control variables to reduce the number of infected and asymptomatic or undiagnosed populations in the considered model. Existence and necessary optimal conditions are also established. The Grünwald-Letnikov non-standard weighted average finite difference method (GL-NWAFDM) is developed for solving the proposed optimal control system. Further, we prove the stability of the considered numerical method. Graphical representations and analysis are presented to verify the theoretical results.
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 …
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.
Mathematical models have long been considered as important tools in cancer biology and therapy. Herein, we present an advanced non-linear mathematical model that can predict accurately the effect of an …
Mathematical models have long been considered as important tools in cancer biology and therapy. Herein, we present an advanced non-linear mathematical model that can predict accurately the effect of an anticancer agent on the growth of a solid tumor.Advanced non-linear mathematical optimization techniques and human-to-mouse experimental data were used to develop a tumor growth inhibition (TGI) estimation model.Using this mathematical model, we could accurately predict the tumor mass in a human-to-mouse pancreatic ductal adenocarcinoma (PDAC) xenograft under gemcitabine treatment up to five time periods (points) ahead of the last treatment.The ability of the identified TGI dynamic model to perform satisfactory short-term predictions of the tumor growth for up to five time periods ahead was investigated, evaluated and validated for the first time. Such a prediction model could not only assist the pre-clinical testing of putative anticancer agents, but also the early modification of a chemotherapy schedule towards increased efficacy.
In the present study, we developed a modified immune-tumor-normal cell model, considering Lotka-Volterra-type competitions between the cell populations and the chemotherapy drugs. The local stability of the model has been …
In the present study, we developed a modified immune-tumor-normal cell model, considering Lotka-Volterra-type competitions between the cell populations and the chemotherapy drugs. The local stability of the model has been examined at each equilibrium point. Also, the global stability of the model at tumor-free equilibrium has been looked at, and a range of drug administration rates has been found for which the tumor-free state is asymptotically stable globally. Also, the growth of tumor cells was kept to a minimum by setting up an optimal control policy for how drugs are given. We found that the optimal control strategy helped eliminate tumor cells with fewer adverse side effects because it kept the number of normal and immune cells high. The optimal control strategy also reduces the time needed for the treatment strategy. Finally, numerical simulations are performed to verify some of our theoretical results.
<abstract><p>This study addresses a cancer eradication model involving effector cells in the presence of gene therapy, immunotherapy, and chemotherapy. The main objective of this study is to understand the optimal …
<abstract><p>This study addresses a cancer eradication model involving effector cells in the presence of gene therapy, immunotherapy, and chemotherapy. The main objective of this study is to understand the optimal effect of immuno-chemotherpay in the presence of gene therapy. The boundedness and positiveness of the solutions in the respective feasible domains of the proposed model are verified. Conditions for which the equilibrium points of the system exist and are stable have been derived. An optimal control problem for the system has been constructed and solved to minimize the immuno-chemotherapy drug-induced toxicity to the patient. Amounts of immunotherapy to be injected into a patient for eradication of cancerous tumor cells have been found. Numerical and graphical results have been presented. From the results, it is seen that tumor cells can be eliminated in a specific time interval with the control of immuno-chemotherapeutic drug concentration.</p></abstract>
Cancer is one of the deadliest diseases in the world. From a medical point of view, cancer is commonly known as a malignant disease characterized by the abnormal growth of …
Cancer is one of the deadliest diseases in the world. From a medical point of view, cancer is commonly known as a malignant disease characterized by the abnormal growth of cells. On the other hand, there are several methods for cancer treatment, such as chemotherapy, radiation therapy, immunotherapy, surgery, etc. In this study, a nonlinear mathematical model of tumor growth has been investigated to examine the interactions between the immune system and the tumor. Furthermore, basic mathematical analysis, including the local stability of the tumor-free and co-existing equilibrium points, is addressed. In addition, pharmaceutical measures, including immunotherapy and chemotherapy, are considered to suppress the development of the tumor. The adequate prescription of drug measures during the treatment period is presented to the patient in the framework of an optimal control approach. The side effects of the mentioned treatment approaches are reduced via the optimal control strategy and the numerical simulation results deal with the proposed goals.
The long-term efficacy of targeted therapeutics for cancer treatment can be significantly limited by the type of therapy and development of drug resistance, inter alia. Experimental studies indicate that the …
The long-term efficacy of targeted therapeutics for cancer treatment can be significantly limited by the type of therapy and development of drug resistance, inter alia. Experimental studies indicate that the factors enhancing acquisition of drug resistance in cancer cells include cell heterogeneity, drug target alteration, drug inactivation, DNA damage repair, drug efflux, cell death inhibition, as well as microenvironmental adaptations to targeted therapy, among others. Combination cancer therapies (CCTs) are employed to overcome these molecular and pathophysiological bottlenecks and improve the overall survival of cancer patients. CCTs often utilize multiple combinatorial modes of action and thus potentially constitute a promising approach to overcome drug resistance. Considering the colossal cost, human effort, time and ethical issues involved in clinical drug trials and basic medical research, mathematical modeling and analysis can potentially contribute immensely to the discovery of better cancer treatment regimens. In this article, we review mathematical models on CCTs developed thus far for cancer management. Open questions are highlighted and plausible combinations are discussed based on the level of toxicity, drug resistance, survival benefits, preclinical trials and other side effects.
The long-term efficacy of targeted therapeutics for cancer treatment can be significantly limited by the type of therapy and development of drug resistance, inter alia. Experimental studies indicate that the …
The long-term efficacy of targeted therapeutics for cancer treatment can be significantly limited by the type of therapy and development of drug resistance, inter alia. Experimental studies indicate that the factors enhancing acquisition of drug resistance in cancer cells include cell heterogeneity, drug target alteration, drug inactivation, DNA damage repair, drug efflux, cell death inhibition, as well as microenvironmental adaptations to targeted therapy, among others. Combination cancer therapies (CCTs) are employed to overcome these molecular and pathophysiological bottlenecks and improve the overall survival of cancer patients. CCTs often utilize multiple combinatorial modes of action and thus potentially constitute a promising approach to overcome drug resistance. Considering the colossal cost, human effort, time and ethical issues involved in clinical drug trials and basic medical research, mathematical modeling and analysis can potentially contribute immensely to the discovery of better cancer treatment regimens. In this article, we review mathematical models on CCTs developed thus far for cancer management. Open questions are highlighted and plausible combinations are discussed based on the level of toxicity, drug resistance, survival benefits, preclinical trials and other side effects.
Adoptive T cell based immunotherapy is gaining significant traction in cancer treatment. Despite its limited success, so far, in treating solid cancers, it is increasingly successful, demonstrating to have a …
Adoptive T cell based immunotherapy is gaining significant traction in cancer treatment. Despite its limited success, so far, in treating solid cancers, it is increasingly successful, demonstrating to have a broader therapeutic potential. In this paper we develop a mathematical model to study the efficacy of engineered T cell receptor (TCR) T cell therapy targeting the E7 antigen in cervical cancer cell lines. We consider a dynamical system that follows the population of cancer cells, TCR T cells, and IL-2. We demonstrate that there exists a TCR T cell dosage window for a successful cancer elimination that can be expressed in terms of the initial tumor size. We obtain the TCR T cell dose for two cervical cancer cell lines: 4050 and CaSki. Finally, a combination therapy of TCR T cell and IL-2 treatment is studied. We show that certain treatment protocols can improve therapy responses in the 4050 cell line, but not in the CaSki cell line.
<abstract><p>Herein, we discuss an optimal control problem (OC-P) of a stochastic delay differential model to describe the dynamics of tumor-immune interactions under stochastic white noises and external treatments. The required …
<abstract><p>Herein, we discuss an optimal control problem (OC-P) of a stochastic delay differential model to describe the dynamics of tumor-immune interactions under stochastic white noises and external treatments. The required criteria for the existence of an ergodic stationary distribution and possible extinction of tumors are obtained through Lyapunov functional theory. A stochastic optimality system is developed to reduce tumor cells using some control variables. The study found that combining white noises and time delays greatly affected the dynamics of the tumor-immune interaction model. Based on numerical results, it can be shown which variables are optimal for controlling tumor growth and which controls are effective for reducing tumor growth. With some conditions, white noise reduces tumor cell growth in the optimality problem. Some numerical simulations are conducted to validate the main results.</p></abstract>
In this article, we propose the interaction of tumor cells with the immune system in the presence of chemotherapy. The existence, uniqueness, non‐negativity, and boundedness of the solutions have been …
In this article, we propose the interaction of tumor cells with the immune system in the presence of chemotherapy. The existence, uniqueness, non‐negativity, and boundedness of the solutions have been established. The conditions for the existence and stability of equilibrium points have been presented in both drug‐free and treated systems. The local stability of the co‐existing equilibrium point is proved using the Routh–Hurwitz rule, and the global stability is proved using the Lyapunov function. We have used quadratic optimal control to minimize the number of tumor cells and the side effects of chemotherapy on the immune system and healthy cells. We have demonstrated the existence of optimal control and derived the corresponding optimality system using Pontryagin’s maximum principle. The optimal system is solved using the forward‐backward sweep method with fourth‐order Runge–Kutta approximation. Reduction in tumor cell growth has been observed due to the increase in recruitment of immune cells activated by tumor cell antigenicity and the rate of conversion of resting immune cells into active immune cells. Additionally, the impact of administering varying chemotherapy doses on reducing tumor cell growth has been noted. Finally, a comparison between controlled and uncontrolled dynamics has been conducted to comprehend the effect of optimal control.
The mathematical study of cancer received great attention in recent years. The theoretical and numerical investigations of models employed for studying the growth of cancer cells and treatment of cancer …
The mathematical study of cancer received great attention in recent years. The theoretical and numerical investigations of models employed for studying the growth of cancer cells and treatment of cancer can help the clinical practitioners in adapting new strategies to face the challenges posed by the deadly disease. Chemotherapy is the most common method of treatment for cancer. The resistance of tumor cells toward the administered drug and the toxic effect of anti-cancer agents on healthy cells are major hurdles to the success of therapy. In this paper, we study this as an optimal control problem in which the amount of drug injected is taken as control. The evolution of different types of cells — sensitive tumor cells, resistant tumor cells and normal cells — under treatment are analyzed. The pharmacokinetics of the drug is also incorporated into the mathematical model. We propose a treatment protocol that assures the death of a maximum number of tumor cells but also manages to keep the normal cells at a sufficient level. We also validated our theoretical results numerically.
In this paper, the suggested transform namely (Sadiq- Emad- Jinan) integral transform and denoted by "SEJI" integral transform is applied to solve a linear second-order delay differential equation with the …
In this paper, the suggested transform namely (Sadiq- Emad- Jinan) integral transform and denoted by "SEJI" integral transform is applied to solve a linear second-order delay differential equation with the initial-value problem for the equation. Firstly, we introduce the form of this equation and provide examples of its use in a wide variety of scientific disciplines. Then the proposed transform is defined and describe its essential features as well as the value of the SEJI integral transform for derivatives. After that, proved a theorem for getting the precise answer by using "SEJI integral transform". The theoretical results were illustrating by a concrete example as presented above. As a result, an appropriate conclusion was introduced for this work.
Abstract In this study, a cancer eradication model involving effector cells in the presence of gene therapy, immunotherapy, and chemotherapy has been proposed. Boundedness and positiveness of the solutions in …
Abstract In this study, a cancer eradication model involving effector cells in the presence of gene therapy, immunotherapy, and chemotherapy has been proposed. Boundedness and positiveness of the solutions in the respective feasible domains of the proposed model are verified. Conditions for which the equilibrium points of the system are stable have been derived. Amounts of immunotherapy to be injected to a patient for eradication of cancerous tumor cells have been found. To minimize the chemotherapy drug-induced toxicity to the patient, an optimal control problem for the system has been constructed and solved. Numerical and graphical results have been presented. Through the investigation, it was seen that tumor cells can be eliminated in a specific time interval with the control of chemotherapeutic drug concentration.
The long-term efficacy of targeted therapeutics for cancer treatment can be significantly limited by the type of therapy and development of drug resistance, inter alia. Experimental studies indicate that the …
The long-term efficacy of targeted therapeutics for cancer treatment can be significantly limited by the type of therapy and development of drug resistance, inter alia. Experimental studies indicate that the factors enhancing acquisition of drug resistance in cancer cells include cell heterogeneity, drug target alteration, drug inactivation, DNA damage repair, drug efflux, cell death inhibition, as well as microenvironmental adaptations to targeted therapy, among others. Combination cancer therapies (CCTs) are employed to overcome these molecular and pathophysiological bottlenecks and improve the overall survival of cancer patients. CCTs often utilize multiple combinatorial modes of action and thus potentially constitute a promising approach to overcome drug resistance. Considering the colossal cost, human effort, time and ethical issues involved in clinical drug trials and basic medical research, mathematical modeling and analysis can potentially contribute immensely to the discovery of better cancer treatment regimens. In this article, we review mathematical models on CCTs developed thus far for cancer management. Open questions are highlighted, and plausible combinations are discussed based on the level of toxicity, drug resistance, survival benefits, preclinical trials and other side effects.
Time delays and fractional order play a vital role in biological systems with memory. In this paper, we propose an epidemic model for Zika virus infection using delay differential equations …
Time delays and fractional order play a vital role in biological systems with memory. In this paper, we propose an epidemic model for Zika virus infection using delay differential equations with fractional order. Multiple time delays are incorporated in the model to consider the latency of the infection in a vector and the latency of the infection in the infected host. We investigate the necessary and sufficient conditions for stability of the steady states and Hopf bifurcation with respect to three time delays τ 1 , τ 2 , and τ 3 . The model undergoes a Hopf bifurcation at the threshold parameters , , and . Some numerical simulations are given to show the effectiveness of obtained results. The numerical simulations confirm that combination of fractional order and time delays in the epidemic model effectively enriches the dynamics and strengthens the stability condition of the model.
Abstract In this paper, a novel coronavirus (2019-nCov) mathematical model with modified parameters is presented. This model consists of six nonlinear fractional order differential equations. Optimal control of the suggested …
Abstract In this paper, a novel coronavirus (2019-nCov) mathematical model with modified parameters is presented. This model consists of six nonlinear fractional order differential equations. Optimal control of the suggested model is the main objective of this work. Two control variables are presented in this model to minimize the population number of infected and asymptotically infected people. Necessary optimality conditions are derived. The Grünwald–Letnikov nonstandard weighted average finite difference method is constructed for simulating the proposed optimal control system. The stability of the proposed method is proved. In order to validate the theoretical results, numerical simulations and comparative studies are given.
In this article, optimal control for variable order fractional multi-delay mathematical model for the co-infection of HIV/AIDS and malaria is presented. This model consists of twelve differential equations, where the …
In this article, optimal control for variable order fractional multi-delay mathematical model for the co-infection of HIV/AIDS and malaria is presented. This model consists of twelve differential equations, where the variable order derivative are in the sense of Caputo. Three control variables are presented in this model to minimize the number of the co-infected individuals showing no symptoms of AIDS, the infected individuals with malaria, and the individuals asymptomatically infected with HIV/AIDS. Necessary conditions for the control problem are derived. The Grünwald-Letnikov nonstandard finite difference scheme is constructed to simulating the proposed optimal control system. The stability of the proposed scheme is proved. In order to validate the theoretical results numerical simulations and comparative studies are given.
Abstract In this paper we present a model of the macrophage T lymphocyte interactions that generate an anti-tumor immune response. The model specifies i) induction of cytotoxic T lymphocytes, ii) …
Abstract In this paper we present a model of the macrophage T lymphocyte interactions that generate an anti-tumor immune response. The model specifies i) induction of cytotoxic T lymphocytes, ii) antigen presentation by macrophages, which leads to iii) activation of helper T cells, and iv) production of lymphoid factors, which induce a) cytotoxic macrophages, b) T lymphocyte proliferation, and c) an inflammation reaction. Tumor escape mechanisms (suppression, antigenic heterogeneity) have been deliberately omitted from the model. This research combines hitherto unrelated or even contradictory data within the range of behavior of one model. In the model behavior, helper T cells play a crucial role: Tumors that differ minimally in antigenicity (i.e., helper reactivity) can differ markedly in rejectability. Immunization yields protection against tumor doses that would otherwise be lethal, because it increases the number of helper T cells. The magnitude of the cytotoxic effector cell response depends on the time at which helper T cells become activated: early helper activity steeply increases the magnitude of the immune response. The type of cytotoxic effector cells that eradicates the tumor depends on tumor antigenicity: lowly antigenic tumors are attacked mainly by macrophages, whereas large highly antigenic tumors can be eradicated by cytotoxic T lymphocytes only.
We investigate a mathematical population model of tumor-immune interactions. The populations involved are tumor cells, specific and non-specific immune cells, and concentrations of therapeutic treatments. We establish the existence of …
We investigate a mathematical population model of tumor-immune interactions. The populations involved are tumor cells, specific and non-specific immune cells, and concentrations of therapeutic treatments. We establish the existence of an optimal control for this model and provide necessary conditions for the optimal control triple for simultaneous application of chemotherapy, tumor infiltrating lymphocyte (TIL) therapy, and interleukin-2 (IL-2) treatment. We discuss numerical results for the combination of the chemo-immunotherapy regimens. We find that the qualitative nature of our results indicates that chemotherapy is the dominant intervention with TIL interacting in a complementary fashion with the chemotherapy. However, within the optimal control context, the interleukin-2 treatment does not become activated for the estimated parameter ranges.
We present a competition model of cancer tumor growth that includes both the immune system response and drug therapy. This is a four‐population model that includes tumor cells, host cells, …
We present a competition model of cancer tumor growth that includes both the immune system response and drug therapy. This is a four‐population model that includes tumor cells, host cells, immune cells, and drug interaction. We analyze the stability of the drug‐free equilibria with respect to the immune response in order to look for target basins of attraction. One of our goals was to simulate qualitatively the asynchronous tumor‐drug interaction known as “Jeffs phenomenon.” The model we develop is successful in generating this asynchronous response behavior. Our other goal was to identify treatment protocols that could improve standard pulsed chemotherapy regimens. Using optimal control theory with constraints and numerical simulations, we obtain new therapy protocols that we then compare with traditional pulsed periodic treatment. The optimal control generated therapies produce larger oscillations in the tumor population over time. However, by the end of the treatment period, total tumor size is smaller than that achieved through traditional pulsed therapy, and the normal cell population suffers nearly no oscillations.
Abstract This paper presents a new technique for numerical treatments of Volterra delay integro-differential equations that have many applications in biological and physical sciences. The technique is based on the …
Abstract This paper presents a new technique for numerical treatments of Volterra delay integro-differential equations that have many applications in biological and physical sciences. The technique is based on the mono-implicit Runge — Kutta method for treating the differential part and the collocation method (using Boole’s quadrature rule) for treating the integral part. The efficiency and stability properties of this technique have been studied. Numerical results are presented to demonstrate the effectiveness of the methodology.
There is an increasing interest in the use of therapeutic devices which deliver chemotherapeutic agents in a continuous manner. In this paper the Gompertz model of cancer growth with a …
There is an increasing interest in the use of therapeutic devices which deliver chemotherapeutic agents in a continuous manner. In this paper the Gompertz model of cancer growth with a loss term depending on a cancer chemotherapeutic agent is applied to human multiple myeloma. Three different performance criteria are introduced which measure the influence of the anti-cancer drug in driving the tumor population level to a desired target level. Engineering optimal control theory is used to produce expressions for the continuous-time optimal control. A comparison is made between the natures of the controller for the three problems considered. Parameter values used in the models are based on patient data. Results of the present study may be useful in the construction of algorithms for use with drug delivery devices that incorporate a microprocessor. Use of such devices may be useful in improving the treatment schedules and treatment outcome of cancer patients.
This paper presents a critical review of selected topics related to the modelling of cancer onset, evolution and growth, with the aim of illustrating, to a wide applied mathematical readership, …
This paper presents a critical review of selected topics related to the modelling of cancer onset, evolution and growth, with the aim of illustrating, to a wide applied mathematical readership, some of the novel mathematical problems in the field. This review attempts to capture, from the appropriate literature, the main issues involved in the modelling of phenomena related to cancer dynamics at all scales which characterise this highly complex system: from the molecular scale up to that of tissue. The last part of the paper discusses the challenge of developing a mathematical biological theory of tumour onset and evolution.
In this paper, we look at a model depicting the relationship of cancer cells in different development stages with immune cells and a cell cycle specific chemotherapy drug. The model …
In this paper, we look at a model depicting the relationship of cancer cells in different development stages with immune cells and a cell cycle specific chemotherapy drug. The model includes a constant delay in the mitotic phase. By applying optimal control theory, we seek to minimize the cost associated with the chemotherapy drug and to minimize the number of tumor cells. Global existence of a solution has been shown for this model and existence of an optimal control has also been proven. Optimality conditions and characterization of the control are discussed.
We present a delay differential model with optimal control that describes the interactions of the tumour cells and immune response cells with external therapy. The intracellular delay is incorporated into …
We present a delay differential model with optimal control that describes the interactions of the tumour cells and immune response cells with external therapy. The intracellular delay is incorporated into the model to justify the time required to stimulate the effector cells. The optimal control variables are incorporated to identify the best treatment strategy with minimum side effects by blocking the production of new tumour cells and keeping the number of normal cells above 75% of its carrying capacity. Existence of the optimal control pair and optimality system are established. Pontryagin’s maximum principle is applicable to characterize the optimal controls. The model displays a tumour-free steady state and up to three coexisting steady states. The numerical results show that the optimal treatment strategies reduce the tumour cells load and increase the effector cells after a few days of therapy. The performance of combination therapy protocol of immunochemotherapy is better than the standard protocol of chemotherapy alone.
This review will outline a number of illustrative mathematical models describing the growth of avascular tumors. The aim of the review is to provide a relatively comprehensive list of existing …
This review will outline a number of illustrative mathematical models describing the growth of avascular tumors. The aim of the review is to provide a relatively comprehensive list of existing models in this area and discuss several representative models in greater detail. In the latter part of the review, some possible future avenues of mathematical modeling of avascular tumor development are outlined together with a list of key questions.
Immunotherapy with Interleukin-2: A Study Based on Mathematical Modeling The role of interleukin-2 (IL-2) in tumor dynamics is illustrated through mathematical modeling, using delay differential equations with a discrete time …
Immunotherapy with Interleukin-2: A Study Based on Mathematical Modeling The role of interleukin-2 (IL-2) in tumor dynamics is illustrated through mathematical modeling, using delay differential equations with a discrete time delay (a modified version of the Kirshner-Panetta model). Theoretical analysis gives an expression for the discrete time delay and the length of the time delay to preserve stability. Numerical analysis shows that interleukin-2 alone can cause the tumor cell population to regress.
Cancer is a complex disease in which a variety of factors interact over a wide range of spatial and temporal scales with huge datasets relating to the different scales available. …
Cancer is a complex disease in which a variety of factors interact over a wide range of spatial and temporal scales with huge datasets relating to the different scales available. However, these data do not always reveal the mechanisms underpinning the observed phenomena. In this paper, we explain why mathematics is a powerful tool for interpreting such data by presenting case studies that illustrate the types of insight that realistic theoretical models of solid tumour growth may yield. These range from discriminating between competing hypotheses for the formation of collagenous capsules associated with benign tumours to predicting the most likely stimulus for protease production in early breast cancer. We will also illustrate the benefits that may result when experimentalists and theoreticians collaborate by considering a novel anti-cancer therapy.
We consider the problem of minimizing the tumor volume with a priori given amounts of anti-angiogenic and cytotoxic agents. For one underlying mathematical model, optimal and suboptimal solutions are given …
We consider the problem of minimizing the tumor volume with a priori given amounts of anti-angiogenic and cytotoxic agents. For one underlying mathematical model, optimal and suboptimal solutions are given for four versions of this problem: the case when only anti-angiogenic agents are administered, combination treatment with a cytotoxic agent, and when a standard linear pharmacokinetic equation for the anti-angiogenic agent is added to each of these models. It is shown that the solutions to the more complex models naturally build upon the simplified versions. This gives credence to a modeling approach that starts with the analysis of simplified models and then adds increasingly more complex and medically relevant features. Furthermore, for each of the problem formulations considered here, there exist excellent simple piecewise constant controls with a small number of switchings that virtually replicate the optimal values for the objective.
Herein, 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. A …
Herein, 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. A discrete time-delay is considered to justify the time-needed for the effector cells to develop a suitable response to the tumour cells. The control variables are included to justify the best treatment strategy 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. The numerical simulations show that the optimal treatment strategy reduces the load of tumour cells increases the effector cells after few days of therapy.
There exists an extensive literature on delay differential models in biology and biomedicine, but only a few papers study such models in the framework of optimal control theory. In this …
There exists an extensive literature on delay differential models in biology and biomedicine, but only a few papers study such models in the framework of optimal control theory. In this paper, we consider optimal control problems with multiple time delays in state and control variables and present two applications in biomedicine. After discussing the necessary optimality conditions for delayed optimal control problems with control-state constraints, we propose discretization methods by which the delayed optimal control problem is transformed into a large-scale nonlinear programming problem. The first case study is concerned with the delay differential model in [21] describing the tumour-immune response to a chemo-immuno-therapy. Assuming L¹-type objectives, which are linear in control, we obtain optimal controls of bang-bang type. In the second case study, we introduce a control variable in the delay differential model of Hepatitis B virus infection developed in [7]. For L¹-type objectives we obtain extremal controls of bang-bang type.