The combined effects of optimal control in cancer remission

We investigate a mathematical model depicting the nonlinear dynamics of immunogenic tumors as envisioned by Kuznetsov et?al. 1. To understand the dynamics under what circumstances the cancer cells can be eliminated, we implement the theory of optimal control. We design two types of external treatment strategies, one is Adoptive Cellular Immunotherapy and another is interleukin-2. Our aim is to establish the treatment regimens that maximize the effector cell count and minimize the tumor cell burden and the deleterious effects of the total amount of drugs. We derive the existence of an optimal control by using the boundedness of solutions. We characterize the optimality system, in which the state system is coupled with co-states. The uniqueness of an optimal control of our problem is also analyzed. Finally, we demonstrate the numerical illustrations that the optimal regimens reduce the tumor burden under different scenarios.

[1]  D. Kirschner,et al.  Modeling immunotherapy of the tumor – immune interaction , 1998, Journal of mathematical biology.

[2]  Wolfgang Hackbusch,et al.  A numerical method for solving parabolic equations with opposite orientations , 1978, Computing.

[3]  D. Schwartzentruber.  In vitro predictors of clinical response in patients receiving interleukin‐2-based immunotherapy , 1993, Current opinion in oncology.

[4]  S. Kamidono,et al.  Rejection of Mouse Renal Cell Carcinoma Elicited by Local Secretion of Interleukin‐2 , 1996, Japanese journal of cancer research : Gann.

[5]  S. Rosenberg,et al.  Cancer Immunotherapy Using Interleukin-2 and Interleukin-2-Activated Lymphocytes , 1986 .

[6]  F. Hanson,et al.  Optimal control of drug delivery to brain tumors for a distributed parameters model , 2005, Proceedings of the 2005, American Control Conference, 2005..

[7]  J. M. Murray,et al.  Optimal control for a cancer chemotherapy problem with general growth and loss functions. , 1990, Mathematical biosciences.

[8]  G. W. Swan,et al.  General applications of optimal control theory in cancer chemotherapy. , 1988, IMA journal of mathematics applied in medicine and biology.

[9]  A. Radunskaya,et al.  Mixed Immunotherapy and Chemotherapy of Tumors: Modeling, Applications and Biological Interpretations , 2022 .

[10]  Matjaz Perc,et al.  Detecting and controlling unstable periodic orbits that are not part of a chaotic attractor. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  G. W. Swan.  Role of optimal control theory in cancer chemotherapy. , 1990, Mathematical biosciences.

[12]  Subhas Khajanchi,et al.  Bifurcation analysis of a delayed mathematical model for tumor growth , 2015 .

[13]  Sandip Banerjee,et al.  Stability and bifurcation analysis of delay induced tumor immune interaction model , 2014, Appl. Math. Comput..

[14]  F Castiglione,et al.  Cancer immunotherapy, mathematical modeling and optimal control. , 2007, Journal of theoretical biology.

[15]  R. Kaempfer,et al.  Prediction of response to treatment in superficial bladder carcinoma through pattern of interleukin-2 gene expression. , 1996, Journal of clinical oncology : official journal of the American Society of Clinical Oncology.

[16]  D L S McElwain,et al.  A history of the study of solid tumour growth: The contribution of mathematical modelling , 2004, Bulletin of mathematical biology.

[17]  S. P. Chakrabarty,et al.  A CONTROL THEORY APPROACH TO CANCER REMISSION AIDED BY AN OPTIMAL THERAPY , 2010 .

[18]  A. Perelson,et al.  Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis. , 1994, Bulletin of mathematical biology.

[19]  Ping Bi,et al.  Periodic and chaotic oscillations in a tumor and immune system interaction model with three delays. , 2014, Chaos.

[20]  D. Longo,et al.  Phase I study of subcutaneously administered interleukin-2 in combination with interferon alfa-2a in patients with advanced cancer. , 1996, Journal of clinical oncology : official journal of the American Society of Clinical Oncology.

[21]  Jon Ernstberger,et al.  OPTIMAL CONTROL APPLIED TO IMMUNOTHERAPY , 2003 .

[22]  K Renee Fister,et al.  Immunotherapy: an optimal control theory approach. , 2005, Mathematical biosciences and engineering : MBE.

[23]  Matjaž Perc,et al.  Determining the flexibility of regular and chaotic attractors , 2006 .

[24]  Ami Radunskaya,et al.  A mathematical tumor model with immune resistance and drug therapy: an optimal control approach , 2001 .

[25]  Subhas Khajanchi,et al.  A Mathematical Model to Elucidate Brain Tumor Abrogation by Immunotherapy with T11 Target Structure , 2014, PloS one.

[26]  Mark A J Chaplain,et al.  Mathematical modelling of the spatio-temporal response of cytotoxic T-lymphocytes to a solid tumour. , 2004, Mathematical medicine and biology : a journal of the IMA.

[27]  Subhas Khajanchi,et al.  Dynamic behavior of a Beddington-DeAngelis type stage structured predator-prey model , 2014, Appl. Math. Comput..

[28]  John Carl Panetta,et al.  Optimal Control Applied to Competing Chemotherapeutic Cell-Kill Strategies , 2003, SIAM J. Appl. Math..

[29]  W. Fleming,et al.  Deterministic and Stochastic Optimal Control , 1975 .

[30]  Guanrong Chen,et al.  Synchronization transitions on small-world neuronal networks: Effects of information transmission delay and rewiring probability , 2008 .

[31]  L. Preziosi,et al.  Modelling and mathematical problems related to tumor evolution and its interaction with the immune system , 2000 .

[32]  A. Murugan,et al.  Chemotherapy for tumors: an analysis of the dynamics and a study of quadratic and linear optimal controls. , 2007, Mathematical biosciences.