Bezier Control Points Method to Solve Scheduling of Injections of Immunotherapeutic Agents

Fateme Ghomanjani

Department of Applied Mathematics, Ferdowsi University of Mashhad, Iran.

DOI: 10.4236/ica.2012.31003                

Abstract

Cancer immunotherapy aims at enhancing immune system to defend against the tumor. However, it is associated with injecting small doses of tumor-bearing molecules or even using drugs. The problem is that how to schedule these injections effectively and/or how to apply drugs in a way to decrease toxic side effects of drugs such that the tumor growth to be stopped or at least to be limited. Here, the theory of optimal control has been applied to find the optimal schedule of injections of an immunotherapeutic agent against cancer. The numerical method employed works for any dynamic linear system and has almost precise solution. In this work, it was tested for a well known model of the tumor immune system interaction.

Keywords

Immunotherapy; Bezier Control Points; Constrained Optimal Control

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	A. Cappuccio, F. Castiglione and B. Piccoli, “Determination of the Optimal Therapeutic Protocols in Cancer Immunotherapy,” Mathematical Biosciences, Vol. 209, No. 1, 2007, pp. 1-13. doi:10.1016/j.mbs.2007.02.009
[2]	B. Piccoli and F. Castiglione, “Optimal Vaccine Scheduling in Cancer Immunotherapy,” Physica A, Vol. 370, No. 2, 2006, pp. 672-680. doi:10.1016/j.physa.2006.03.011
[3]	C. P. Neuman and A. Sen, “A Suboptimal Control Algoithm for Constrained Problems Using Cubic Splines,” Automatica, Vol. 9, No. 5, 1973, pp. 601-613. doi:10.1016/0005-1098(73)90045-9
[4]	G. Elnagar, M. Kazemi and M. Razzaghi, “The Pseudospectral Legendre Method for Discretizing Optimal Control Problem,” IEEE Transactions on Automatic Control, Vol. 40, No. 10, 1965, pp. 1793-1796. doi:10.1109/9.467672
[5]	H. Jaddu, “Spectral Method for Constrained Linearquadratic Optimal Control,” Mathematicas and Computers in Simulation, Vol. 58, No. 2, 2002, pp. 159-169. doi:10.1016/S0378-4754(01)00359-7
[6]	H. R. Sirisena, “Computation of Optimal Controls Using a Piecewise Polynomial Parameterization,” IEEE Transactions on Automatic Control, Vol. 18, No. 4, 1973, pp. 409-411. doi:10.1109/TAC.1973.1100329
[7]	H. R. Sirisena and F. S. Chou, “State Parameterization Approach to the Solution of Optimal Control Problems,” Optimal Control Applications and Methods, Vol. 2, No. 3, 1981, pp. 289-298. doi:10.1002/oca.4660020307
[8]	I. Troch, F. Breitenecker and M. Graeff, “Computing Optimal Controls for Systems with State and Control Constraints,” Proceedings of the IFAC Control Applications of Nonlinear Programming and Optimization, Paris, 7-9 June 1989, pp. 39-44.
[9]	J. Vlassenbroeck, “A Chebyshev Polynomial Method for Optimal Control with State Constraints,” Automatica, Vol. 24, No. 4, 1988, pp. 499-504. doi:10.1016/0005-1098(88)90094-5
[10]	K. Teo, C. Goh and K. Wong, “A Unified Computational Approach to Optimal Control Problem,” Longman, Harlow, 1981.
[11]	M. Evrenosoglu and S. Somali, “Least Squares Methods for Solving Singularity Perturbed Two-Points Boundary Value Problems Using Bezier Control Point,” Applied Mathematics Letters, Vol. 21, No. 10, 2008, pp. 10291032. doi:10.1016/j.aml.2007.10.021
[12]	P. A. Frick and D. J. Stech, “Solution of Optimal Control Problems on a Parallel Machine Using the Epsilon Method,” Optimal Control Applications and Methods, Vol. 16, 1995, pp. 1-17.
[13]	R. Pytlak, “Numerical Methods for Optimal Control Problems with State Constraints,” Springer-Veriag, Berlin, 1999.
[14]	V. Yen and M. Nagurka, “Optimal Control of Linearly Constrained Linear Systems via State Parameterization,” Optimal Control Applications and Methods, Vol. 13, No. 2, 1992, pp. 155-167. doi:002/oca.4660130206
[15]	V. Kuznetsov, I. Makalkin, M. Taylor and A. Perelson, “Nonlinear Dynamics of Immunogenic Tumors Parameter Estimation and Global Bifurcation Analysis,” Bulletin of Mathematical Biology, Vol. 56, No. 2, 1994, pp. 295-321.
[16]	D. Kirschner and J. C. Panetta, “Modeling Immunotherapy of Tumor-Immune Interaction,” Journal of Mathematical Biology, Vol. 37, No. 3, 1998, pp. 235-252. doi:10.1007/s002850050127
[17]	T. Burden, J. Ernstberger and K. R. Fister, “Optimal Control Applied to Immunotherapy,” Discrete and Continuous Dynamical Systems Series B, Vol. 4, No. 1, 2004, pp. 135-146.
[18]	A. Ghaffari and N. Naserufar, “Optimal Therapeutic Protocols in Cancer Immunotherapy,” Computers in Biology and Medicine, Vol. 40, No. 3, 2010, pp. 261-270. doi:10.1016/j.compbiomed.2009.12.001