Abstract
An optimal control problem for cancer chemotherapy is considered that includes immunological activity. In the objective a weighted average of several quantities that describe the effectiveness of treatment is minimized. These terms include (i) the number of cancer cells at the terminal time, (ii) a measure for the immunocompetent cell densities at the terminal point (included as a negative term), (iii) the overall amount of cytotoxic agents given as a measure for the side effects of treatment and (iv) a small penalty on the terminal time that limits the overall therapy horizon which is assumed to be free. This last term is essential in obtaining a well-posed problem formulation. Employing a Gompertzian growth model for the cancer cells, for various scenarios optimal controls and corresponding responses of the system are calculated. Solutions initially follow a full dose treatment, but then at one point switch to a singular regimen that only applies partial dosages. This structure is consistent with protocols that apply an initial burst to reduce the tumor volume and then maintain a small volume through lower dosages. Optimal controls end with either a prolonged period of no dose treatment or, in a small number of scenarios, this no dose interval is still followed by one more short burst of full dose treatment.
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References
Bell DJ, Jacobson DH (1975) Singular optimal control problems. Academic Press, London
Benson DA (2004) A Gauss pseudospectral transcription for optimal control, Ph.D. dissertation, Department of Aeronautics and Astronautics, MIT
Benson DA, Huntington GT, Thorvaldsen TP, Rao AV (2006) Direct trajectory optimization and costate estimation via an orthogonal collocation method. J Guid Control Dyn 29(6): 1435–1440
Bonnard B, Chyba M (2003) Singular trajectories and their role in control theory. In: Mathématiques & applications, vol 40. Springer, Paris
Bressan A, Piccoli B (2007) Introduction to the mathematical theory of control. American Institute of Mathematical Sciences, Springfield
Burden T, Ernstberger J, Fister KR (2004) Optimal control applied to immunotherapy. Discrete Contin Dyn Syst Ser B 4: 135–146
Castiglione F, Piccoli B (2006) Optimal control in a model of dendritic cell transfection cancer immunotherapy. Bull Math Biol 68: 255–274
de Pillis LG, Radunskaya A (2001) A mathematical tumor model with immune resistance and drug therapy: an optimal control approach. J Theor Med 3: 79–100
de Vladar HP, González JA (2004) Dynamic response of cancer under the influence of immunological activity and therapy. J Theor Biol 227: 335–348
d’Onofrio A (2005) A general framework for modeling tumor–immune system competition and immunotherapy: mathematical analysis and biomedical inferences. Phys D 208: 220–235
d’Onofrio A (2006) Tumor–immune system interaction: modeling the tumor-stimulated proliferation of effectors and immunotherapy. Math Models Methods Appl Sci 16: 1375–1401
d’Onofrio A, Gandolfi A, Rocca A (2009) The dynamics of tumour–vasculature interaction suggests low-dose, time-dense antiangiogenic schedulings. Cell Prolif 42: 317–329
d’Onofrio A, Ledzewicz U, Maurer H, Schättler H (2009) On optimal delivery of combination therapy for tumors. Math Biosci 222: 13–26. doi:10.1016/j.mbs.2009.08.004
Ergun A, Camphausen K, Wein LM (2003) Optimal scheduling of radiotherapy and angiogenic inhibitors. Bull Math Biol 65: 407–424
Fister KR, Hughes Donnelly J (2005) Immunotherapy: an optimal control approach. Math Biosci Eng (MBE) 2(3): 499–510
Golubitsky M, Guillemin V (1973) Stable mappings and their singularities. Springer, New York
Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer, New York
Huntington GT (May 2007) Advancement and analysis of a Gauss pseudospectral transcription for optimal control. Ph.D. dissertation, Department of Aeronautics and Astronautics, MIT
Kirschner D, Panetta JC (1998) Modeling immunotherapy of the tumor–immune interaction. J. Math Biol 37: 235–252
Kuznetsov VA, Makalkin IA, Taylor MA, Perelson AS (1994) Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis. Bull Math Biol 56: 295–321
Ledzewicz U, Marriott J, Maurer H, Schättler H (2010) Realizable protocols for optimal administration of drugs in mathematical models for anti-angiogenic treatment. Math Med Biol 27: 157–179
Ledzewicz U, Maurer H, Schättler H (2009) Bang-bang and singular controls in a mathematical model for combined anti-angiogenic and chemotherapy treatments. In: Proceedings of the 48th IEEE conference on decision and control, Shanghai, China, pp 2280–2285
Ledzewicz U, Naghnaeian M, Schättler H (2010) Bifurcation of singular arcs in an optimal control problem for cancer immune system interactions under treatment. In: Proceedings of the 49th IEEE conference on decision and control. Atlanta, USA, pp 7039–7044
Ledzewicz U, Naghnaeian M, Schättler H (2011a) An optimal control approach to cancer treatment under immunological activity. Appl Math 38(1): 17–31
Ledzewicz U, Naghnaeian M, Schättler H (2011b) Dynamics of tumor–immune interactions under treatment as an optimal control problem. AIMS Proc (in press)
Ledzewicz U, Schättler H (2002) Analysis of a cell-cycle specific model for cancer chemotherapy. J Biol Syst 10: 183–206
Ledzewicz U, Schättler H (2007) Anti-angiogenic therapy in cancer treatment as an optimal control problem. SIAM J Control Optim 46: 1052–1079
Ledzewicz U, Schättler H (2008) Optimal and suboptimal protocols for a class of mathematical models of tumor anti-angiogenesis. J Theor Biol 252: 295–312
Norton L, Simon R (1977) Growth curve of an experimental solid tumor following radiotherapy. J Natl Cancer Inst 58: 1735–1741
Norton L (1988) A Gompertzian model of human breast cancer growth. Cancer Res 48: 7067–7071
Rao AV, Benson DA, Huntington GT, Francolin C, Darby CL, Patterson MA (2008) User’s Manual for GPOPS: a MATLAB package for dynamic optimization using the Gauss pseudospectral method, University of Florida report
Stepanova NV (1980) Course of the immune reaction during the development of a malignant tumour. Biophysics 24: 917–923
Swierniak A (2008) Direct and indirect control of cancer populations. Bull Pol Acad Sci Tech Sci 56(4): 367–378
Swierniak A, Ledzewicz U, Schättler H (2003) Optimal control for a class of compartmental models in cancer chemotherapy. Int J Appl Math Comput Sci 13: 357–368
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Ledzewicz, U., Naghnaeian, M. & Schättler, H. Optimal response to chemotherapy for a mathematical model of tumor–immune dynamics. J. Math. Biol. 64, 557–577 (2012). https://doi.org/10.1007/s00285-011-0424-6
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DOI: https://doi.org/10.1007/s00285-011-0424-6