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Journal of Applied Mathematics
Volume 2014 (2014), Article ID 363109, 9 pages
Research Article

Optimal Finite Cancer Treatment Duration by Using Mixed Vaccine Therapy and Chemotherapy: State Dependent Riccati Equation Control

Mechanical Engineering, Center of Excellence in Robotics and Control, KNTU, Pardis Street, Vanak Square, Tehran 16569 83911, Iran

Received 21 November 2013; Revised 28 January 2014; Accepted 2 February 2014; Published 13 March 2014

Academic Editor: Zhijun Liu

Copyright © 2014 Ali Ghaffari et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The main objective of this paper is to propose an optimal finite duration treatment method for cancer. A mathematical model is proposed to show the interactions between healthy and cancerous cells in the human body. To extend the existing models, the effect of vaccine therapy and chemotherapy are also added to the model. The equilibrium points and the related local stability are derived and discussed. It is shown that the dynamics of the cancer model must be changed and modified for finite treatment duration. Therefore, the vaccine therapy is used to change the parameters of the system and the chemotherapy is applied for pushing the system to the domain of attraction of the healthy state. For optimal chemotherapy, an optimal control is used based on state dependent Riccati equation (SDRE). It is shown that, in spite of eliminating the treatment, the system approaches the healthy state conditions. The results show that the development of optimal vaccine-chemotherapy protocols for removing tumor cells would be an appropriate strategy in cancer treatment. Also, the present study states that a proper treatment method not only reduces the population of the cancer cells but also changes the dynamics of the cancer.