Stability Analysis of Delayed Immune Response BCG Infection in Bladder Cancer Treatment Model by Stochastic Perturbations

Computational and Mathematical Methods in Medicine, Jul 2018

We present a revised mathematical model of the immune response to Bacillus Calmette-Guérin (BCG) treatment of bladder cancer, optimized according to biological and clinical data accumulated during the last decade. The improved model accounts for cytotoxic T lymphocyte differentiation as an integral element of the delayed immune response, as well as the logistic growth terms for cancer cell proliferation. Three equilibria are demonstrated for the proposed model, which is assumed to be influenced by white noise stochastic perturbations that are directly proportional to the system state deviation from an equilibrium. Stability conditions for all equilibria are analyzed using the Kolmanovskii-Shaikhet general method of Lyapunov functionals construction.

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Stability Analysis of Delayed Immune Response BCG Infection in Bladder Cancer Treatment Model by Stochastic Perturbations

Stability Analysis of Delayed Immune Response BCG Infection in Bladder Cancer Treatment Model by Stochastic Perturbations Leonid Shaikhet and Svetlana Bunimovich-Mendrazitsky Department of Mathematics, Ariel University, Ariel 40700, Israel Correspondence should be addressed to Svetlana Bunimovich-Mendrazitsky; li.ca.leira@ubanaltevs Received 28 January 2018; Accepted 11 June 2018; Published 9 July 2018 Academic Editor: Michele Nichelatti Copyright © 2018 Leonid Shaikhet and Svetlana Bunimovich-Mendrazitsky. 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. Abstract We present a revised mathematical model of the immune response to Bacillus Calmette-Guérin (BCG) treatment of bladder cancer, optimized according to biological and clinical data accumulated during the last decade. The improved model accounts for cytotoxic T lymphocyte differentiation as an integral element of the delayed immune response, as well as the logistic growth terms for cancer cell proliferation. Three equilibria are demonstrated for the proposed model, which is assumed to be influenced by white noise stochastic perturbations that are directly proportional to the system state deviation from an equilibrium. Stability conditions for all equilibria are analyzed using the Kolmanovskii-Shaikhet general method of Lyapunov functionals construction. 1. Introduction Bladder cancer (BC) is 7th most common cancer (the 4th most common for men) with approximately 356,000 new cases each year and more than 145,000 deaths per year. The highest incidence occurs in industrialized and developed areas such as Europe, North America, and Australia (Jemal et al., [1]). Tobacco smoking is the main BC risk factor, accounting for at least 50% of BC cases. Roughly 10% of all BC cases have been related to occupational exposure to chemicals and dye, mostly in industrial areas processing paint, metal, and petroleum products (Bunimovich-Mendrazitsky et al. [2]). The treatment of the BC has improved during last 40 years due to development of high definition of cystoscopy, newly technology in the bladder drugs instillation. However, the prognosis of advanced bladder cancer has not improved during the last years (Alexandroff et al. [3]). The high rates of recurrence, invasive surveillance strategies, and high treatment costs combine to make bladder cancer the single most expensive cancer in both England and the United States (Eylert et al. [4]). BC is most frequently treated with intravesical instillations of an adjuvant immunotherapy with the Bacillus Calmette-Guérin (BCG) bacteria. BCG immunotherapy, originally established by Morales et al. [5], is administered after surgical removal of the tumor at a point where no visual lesions or morphologically evident malignant cells present in random biopsies (Brandau and Suttman, [6]). BCG immunotherapy has proven its superiority over chemotherapy in reducing tumor recurrence rates for patients with high grade or high risk nonmuscle invasive BC. While Lamm et al. [7] found BCG to even reduce disease progression, there is a need to understand why the standard BCG treatment protocol is not effective for nonresponding or relapsing patients. The BCG treatment protocol remains to be optimized specifically for those patients who do not achieve remission from treatment with the standard regimen. In last three decades, it has been commonly accepted that a qualitative understanding of dynamic cancer treatment requires a mathematical framework in which the essential features of this complex process are represented (Byrne, [8]). The model proposed in [9] was the first mathematical model to describe tumor-immune system interactions in the bladder as a result of continuous BCG therapy. Bunimovich-Mendrazitsky et al. [9–12] have modeled the use of BCG in noninvasive bladder cancer, identifying stability points of the mathematical system to ensure durability of the simulated results, and found a tolerable bacteria threshold and effective treatment regimens to minimize undesirable side effects. A system of ordinary differential equations (ODE) was used for effective description of BCG treatment dynamics. In this manuscript, we present an improved BCG model based upon that of B-M et al. [9] which describes the tumor-immune system interactions in the bladder in response to BCG therapy, updated according to newly published biological and clinical data. Three equilibria for the optimized model are demonstrated. One of the main problems encountered in mathematical models described by differential equations is that of their stability. In this work, equilibria stability is analyzed using the Kolmanovskii-Shaikhet general method of Lyapunov functionals construction [13–17] and the method of linear matrix inequalities (LMIs). Equilibria stability in probability of a system of non (...truncated)


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Leonid Shaikhet, Svetlana Bunimovich-Mendrazitsky. Stability Analysis of Delayed Immune Response BCG Infection in Bladder Cancer Treatment Model by Stochastic Perturbations, Computational and Mathematical Methods in Medicine, 2018, 2018, DOI: 10.1155/2018/9653873