Cancer immunotherapy, mathematical modeling and optimal control.

Clinical immunologists, among other problems, routinely face a question: what is the best time and dose for a certain therapeutic agent to be administered to the patient in order to decrease/eradicate the pathological condition? In cancer immunotherapies the therapeutic agent is something able to elicit an immune response against cancer. The immune response has its own dynamics that depends on the immunogenicity of the therapeutic agent and on the duration of the immune response. The question then is "how can we decide when and how much of the drug to inject so to have a prolonged and effective immune response to the cancer?". This question can be addressed in mathematical terms in two stages: first one construct a mathematical model describing the cancer-immune interaction and secondly one applies the theory of optimal control to determine when and to which extent to stimulate the immune system by means of an immunotherapeutic agent administered in discrete variable doses within the therapeutic period. The solution of this mathematical problem is described and discussed in this article. We show that the method employed can be applied to find the optimal protocol in a variety of clinical problems where the kinetics of the drug or treatment and its influence on the physiologic/pathologic functions have been described by a system of ordinary differential equations.

[1]  R. B. Martin,et al.  Optimal control drug scheduling of cancer chemotherapy , 1992, Autom..

[2]  R S Acharya,et al.  Development of optimal drug administration strategies for cancer-chemotherapy in the framework of systems theory. , 1984, International journal of bio-medical computing.

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

[4]  Benedetto Piccoli,et al.  Optimal Control in a Model of Dendritic Cell Transfection Cancer Immunotherapy , 2006 .

[5]  Benedetto Piccoli,et al.  Optimal vaccine scheduling in cancer immunotherapy , 2006 .

[6]  E. Gilboa,et al.  Immunotherapy of cancer with dendritic-cell-based vaccines , 1998, Cancer Immunology, Immunotherapy.

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

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

[9]  Stephen Barnett,et al.  Introduction to Mathematical Control Theory , 1975 .

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

[11]  Urszula Ledzewicz,et al.  OPTIMAL CONTROL FOR A CLASS OF COMPARTMENTAL MODELS IN CANCER CHEMOTHERAPY , 2003 .

[12]  Mauro Garavello,et al.  Hybrid Necessary Principle , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[13]  Roland Bulirsch,et al.  Variational Calculus, Optimal Control and Applications , 1998 .

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