AM  Vol.10 No.4 , April 2019
Optimal Control of Cancer Growth
Abstract: The purpose of the present paper is to apply the Pontryagin Minimum Principle to mathematical models of cancer growth. In [1], I presented a discrete affine model T of cancer growth in the variables C for cancer, GF for growth factors and GI for growth inhibitors. One can sometimes find an affine vector field X on whose time one map is T. It is to this vector field we apply the Pontryagin Minimum Principle. We also apply the Discrete Pontryagin Minimum Principle to the model T. So we prove that maximal chemo therapy can be optimal and also that it might not depending on the spectral properties of the matrix A, (see below). In section five we determine an optimal strategy for chemo or immune therapy.
Cite this paper: Larsen, J. (2019) Optimal Control of Cancer Growth. Applied Mathematics, 10, 173-195. doi: 10.4236/am.2019.104014.

[1]   Chr, J. (2016) Larsen Models of Cancer Growth. Journal of Applied Mathematics and Computing, 53, 615-643.

[2]   Seierstad, A. and Sydsaeter, K. (1988) Optimal Control Theory with Economic Applications North Holland.

[3]   Goodwin, G. (2005) Constrained Control and Estimation. An Optimization Approach. Springer Verlag, Berlin.

[4]   Swierniak, A., Kimmel, M. and Smieja, J. (2009) Mathematical Modelling as a Tool for Planning Anti Cancer Therapy. European Journal of Pharmacology, 625, 108-121.

[5]   Laird, A.K. (1964) Dynamics of Cancer Growth. British Journal of Cancer, 18, 490-502.

[6]   Adam, J.A. and Bellomo, N. (1997) A Survey of Models for Tumor-Induced Immune System Dynamics. Birkhauser, Boston.

[7]   Geha, R. and Notarangelo, L. (2012) Case Studies in Immunology: A Clinical Companion. Garland Science, Hamden.

[8]   Marks, F., Klingmüller, U. and Müller-Decker, K. (2009) Cellular Signal Processing. Garland Science, Hamden.

[9]   Molina-Paris, C. and Lythe, G. (2011) Mathematical Models and Immune Cell Biology. Springer Verlag, Berlin.

[10]   Murphy, K. (2012) Immunobiology. 8th Edition, Garland Science, Hamden.

[11]   Rees, R.C. (2014) Tumor Immunology and Immunotherapy. Oxford University Press, Oxford.

[12]   Larsen, J.C. (2016) Hopf Bifurcations in Cancer Models. JP Journal of Applied Mathematics, 14, 1-31.

[13]   Larsen, J.C. (2017) A Mathematical Model of Adoptive T Cell Therapy. JP Journal of Applied Mathematics, 15, 1-33.

[14]   Larsen, J.C. (2016) Fundamental Concepts in Dynamics.

[15]   Larsen, J.C. (2017) The Bistability Theorem in a Cancer Model. International Journal of Biomathematics, 11, Article ID: 1850004.

[16]   Larsen, J.C. (2016) The Bistability Theorem in a Model of Metastatic Cancer. Applied Mathematics, 7, 1183-1206.

[17]   Larsen, J.C. (2017) A Study on Multipeutics. Applied Mathematics, 8, 746-773.

[18]   Larsen, J.C. (2017) A Mathematical Model of Immunity. JP Journal of Applied Mathematics.

[19]   Larsen, J.C. (2018) Models of Cancer Growth Revisited. Applied Mathematics, 9, Article ID: 84308.

[20]   Uspensky (1948) Theory of Equations. McGraw-Hill, New York.

[21]   Alexendarian, A. (2013) On Continuous Dependence of Roots of Polynomials on Coefficients.