Numerical Solution of Tumor-Immune Model with Targeted Chemotherapy by Multi Step Differential Transformation Method

Dhar, Biplab; Gupta, Praveen Kumar

doi:10.1007/978-3-030-42363-6_47

Cited by 4 publications

(5 citation statements)

References 12 publications

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“…We seek help of multi‐step differential transformation method (DTM) (Ms‐DTM) 2,26‐28 to carry out the numerical calculations of the system ()–() under the initial conditions and parameter values given in Table 1. The Ms‐DTM is an algorithm for deducing precisely approximate solutions in a sequence of intervals for system of both weak and strong non‐linear differential equations.…”

Section: Numerical Simulationmentioning

confidence: 99%

“…However, the disease is a multi‐factorial sickness; the hazard elements of which incorporate utilization of tobacco items, unfortunate eating routine, air contamination, and much more. There are several kinds of treatment amongst which targeted chemotherapy is efficient 1‐5 . Targeted chemotherapeutic drugs are one of the drugs that block the growth and spread of malignant tumors by interfering with specific molecules that are involved in growth of malignant tumors.…”

Section: Introductionmentioning

confidence: 99%

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A numerical approach of tumor‐immune model with B cells and monoclonal antibody drug by multi‐step differential transformation method

Dhar

Gupta

2020

Math Methods in App Sciences

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Monoclonal antibody (mAb) drugs are used to kill malignant tumors by making them able to see into the immune system. In this article, we developed a mathematical model with the help of tumor‐immune system with antibodies effect and efficacy of mAb drug. We defined the existence of solutions of the model, and by the Lyapunov method, we showed that the solutions are bounded; thereafter, we illustrated the local and global stability phenomenon, which is described by the Lyapunov method and localization of compact invariant sets (LCIS) method for disease‐free equilibrium points including the biological significance. In actual fact, this model suggested some ideas about the malignant tumors reciprocate to mAb drug dosage and its efficacy, eventually to overcome malignancy. It is likely to control the steadiness of the tumor, which is very important for any kind of treatment. The numerical calculations for the proposed model have been carried out by multi‐step differential transformation method (Ms‐DTM). The simulation of the model is depicted through Figures 1–5, which shows the effect of interaction rate of tumor cells and antibodies and dose of the drug for various values of efficacy.

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Section: Numerical Simulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A numerical approach of tumor‐immune model with B cells and monoclonal antibody drug by multi‐step differential transformation method

Dhar

Gupta

2020

Math Methods in App Sciences

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“…This method allows us to write a semi-analytical solution of the model which depends on the Taylor series [17]- [21]. The DTM has been successfully applied to solve various mathematical models including disease transmission models [18], [22]. Arenas et al [17] used the differential transformation method for solving non autonomous system of differential equations.…”

Section: Introductionmentioning

confidence: 99%

Application of Differential Transformation Method for Solving HIV Model with Anti-Viral Treatment

Bunga

Ndii

2020

BAREKENG

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Mathematical models have been widely used to understand complex phenomena. Generally, the model is in the form of system of differential equations. However, when the model becomes complex, analytical solutions are not easily found and hence a numerical approach has been used. A number of numerical schemes such as Euler, Runge-Kutta, and Finite Difference Scheme are generally used. There are also alternative numerical methods that can be used to solve system of differential equations such as the nonstandard finite difference scheme (NSFDS), the Adomian decomposition method (ADM), Variation iteration method (VIM), and the differential transformation method (DTM). In this paper, we apply the differential transformation method (DTM) to solve system of differential equations. The DTM is semi-analytical numerical technique to solve the system of differential equations and provides an iterative procedure to obtain the power series of the solution in terms of initial value parameters.. In this paper, we present a mathematical model of HIV with antiviral treatment and construct a numerical scheme based on the differential transformation method (DTM) for solving the model. The results are compared to that of Runge-Kutta method. We find a good agreement of the DTM and the Runge-Kutta method for smaller time step but it fails in the large time step.

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“…The mixed chemo–immunotherapy model considering the innate and the adaptive immune response is presented in [3 ]. The tumour–immune interaction model with the impact of small molecules drug is presented in [6 ]. The multi‐step differential transformation method is used to solve the tumour–immune interaction model in [5 ].…”

Section: Introductionmentioning

confidence: 99%

“…In this study, we will analyse and expand a model of cancer during the treatment by adding immunotherapy and chemotherapy from a new perspective. Many models have been present to show the tumour-immune interaction considering chemoimmunotherapy [3][4][5][6]. The mixed chemo-immunotherapy model considering the innate and the adaptive immune response is presented in [3].…”

Section: Introductionmentioning

confidence: 99%

Pseudo‐spectral method for controlling the drug dosage in cancer

Nazari

Skandari

2020

IET Systems Biology

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A mixed chemotherapy-immunotherapy treatment protocol is developed for cancer treatment. Chemotherapy pushes the trajectory of the system towards the desired equilibrium point, and then immunotherapy alters the dynamics of the system by affecting the parameters of the system. A co-existing cancerous equilibrium point is considered as the desired equilibrium point instead of the tumour-free equilibrium. Chemotherapy protocol is derived using the pseudo-spectral (PS) controller due to its high convergence rate and simple implementation structure. Thus, one of the contributions of this study is simplifying the design procedure and reducing the controller computational load in comparison with Lyapunov-based controllers. In this method, an infinite-horizon optimal control problem is proposed for a non-linear cancer model. Then, the infinite-horizon optimal control of cancer is transformed into a non-linear programming problem. The efficient Legendre PS scheme is suggested to solve the proposed problem. Then, the dynamics of the system is modified by immunotherapy is another contribution. To restrict the upper limit of the chemo-drug dose based on the age of the patients, a Mamdani fuzzy system is designed, which is not present yet. Simulation results on four different dynamics cases how the efficiency of the proposed treatment strategy.

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Numerical Solution of Tumor-Immune Model with Targeted Chemotherapy by Multi Step Differential Transformation Method

Cited by 4 publications

References 12 publications

A numerical approach of tumor‐immune model with B cells and monoclonal antibody drug by multi‐step differential transformation method

A numerical approach of tumor‐immune model with B cells and monoclonal antibody drug by multi‐step differential transformation method

Application of Differential Transformation Method for Solving HIV Model with Anti-Viral Treatment

Pseudo‐spectral method for controlling the drug dosage in cancer

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