Abstract

Review Article

Different optimization strategies for the optimal control of tumor growth

Abd El Moniem NK*, Sweilam NH and Tharwat AA

Published: 10 December, 2019 | Volume 3 - Issue 1 | Pages: 052-062

In this article different numerical techniques for solving optimal control problems is introduced, the aim of this paper is to achieve the best accuracy for the Optimal Control Problem (OCP) which has the objective of minimizing the size of tumor cells by the end of the treatment. An important aspect is considered, which is, the optimal concentrations of drugs that not affect the patient’s health significantly. To study the behavior of tumor growth, a mathematical model is used to simulate the dynamic behavior of tumors since it is difficult to prototype dynamic behavior of the tumor. A tumor-immune model with four components, namely, tumor cells, active cytotoxic T-cells (CTLs), helper T-cells, and a chemotherapeutic drug is used. Two general categories of optimal control methods which are indirect methods and direct ones based on nonlinear programming solvers and interior point algorithms are compared. Within the direct optimal control techniques, we review three different solutions techniques namely (i) multiple shooting methods, (ii) trapezoidal direct collocation method, (iii) Hermit- Simpson’s collocation method and within the indirect methods we review the Pontryagin’s Maximum principle with both collocation method and the backward forward sweep method. Results show that the direct methods achieved better control than indirect methods.

Read Full Article HTML DOI: 10.29328/journal.acst.1001010 Cite this Article Read Full Article PDF

Keywords:

Optimal control for differential equations; Tumor-immune model; Optimal control direct methods; Opti-mal control indirect methods; Nonlinear programming

References

  1. Gibbs WW. Untangling the roots of cancer. Scientific America. 2003.
  2. Abbas AK, Lichtman AH, Pillai S. Cellular and molecular immunology. Saunders Elsevier. 2014.
  3. Curiel T. Tregs and rethinking cancer immunotherapy. Journal of Clinical Investigation. 2007; 117: 1167-1174. PubMed: https://www.ncbi.nlm.nih.gov/pubmed/17476346
  4. Kirschner D, P Panetta. Modeling immuno therapy of the tumor–immune interaction. Journal of Mathematical Biology. 1998; 37: 235-252. PubMed: https://www.ncbi.nlm.nih.gov/pubmed/9785481
  5. Kirschner DE, TL Jackson, JC Arciero. A mathematical model of tumorimmune evasion and siRNA treatment. Discrete and continous dynamical systems series- B. 2003; 37: 39-58.
  6. K Leon, K Garcia, J Carneiro. A Lage. How regulatory CD25(+)CD4(+) T cells impinge on tumor immunobiology? On the existence of two alternative dynamical classes of tumors. Journal of Theoretical Biology. 2007; 247: 122-137.
  7. De Pillis LG, Radunskaya AE. Mixed immunotherapy and chemotherapy of tumors: modeling, applications and biological interpretations. Journal of Theoretical Biology. 2006; 238.
  8. Schattlera H, Urszula L. Optimal Control for Mathematical Models of Cancer Therapies. Springer Publishing Co., USA. 2015.
  9. Sharma S, Samanta GP. Dynamical Behaviour of a Tumor-Immune System with Chemotherapy and Optimal Control. Journal of Nonlinear Dynamics. 2013: 1-13, 2013.
  10. Sweilam NH, Al-Ajami TM. Legendre spectral-collocation method for solving some types of fractional optimal control problems. Journal of Advanced Research, 2015; 393-403. PubMed: https://www.ncbi.nlm.nih.gov/pubmed/26257937
  11. García-Heras J, Soler M, Sáez FJ. A Comparison of Optimal Control Methods for Minimum Fuel Cruise at Constant Altitude and Course with Fixed Arrival Time. Procedia Engineering. 2014; 80:231-244.
  12. Rao AV, Benson DA, Darby C, Patterson MA, Francolin C, et al. Algorithm 902: GPOPS, A MATLAB software for solving multiple-phase optimal control problems using the gauss pseudospectral method. ACM Transactions on Mathematical Software (TOMS), 2010; 37: 1-39.
  13. Biral F, Bertolazzi E, Bosetti P. Notes on Numerical Methods for Solving Optimal Control Problems. IEEJ Journal of Industry Applications. 2015; 5:154-166
  14. Betts JT. A Survey of Numerical Methods for Trajectory Optimization. Control and Dynamics. 1998; 21:193-207.
  15. Rao AV. A survey of numerical methods for optimal control. Advances in the Astronautical Sciences. 2009; 135: 497-528.
  16. Joshi HR. Optimal control of an HIV immunology model. Optimal Control Applications and Methods. 2002; 23: 199-213.
  17. Zaman G, Han Kang Y, Jung IH. Stability analysis and optimal vaccination of an SIR epidemic model. BioSystems. 93: 240-249. 2008. PubMed: https://www.ncbi.nlm.nih.gov/pubmed/18584947
  18. Pillis LG, Radunskaya AE. A mathematical model of immune response to tumor invasion. MIT. 2003; 1661-16668.
  19. De Pillis LG, W Gu, Fister KR, Head T, Maples K, et al. Chemotherapy for tumors: An analysis of the dynamics and a study of quadratic and linear optimal controls. Mathematical Biosciences. 2007; 209: 292-315.
  20. Bellman RE. Dynamic Programming. Courier Corporation. 2003.
  21. Pontryagin LS. Mathematical Theory of Optimal Processes. CRC Press. March 1987.
  22. Anita S, Arnautu V, Capasso V. An introduction to optimal control problems in life sciences and economics: from mathematical models to numerical simulation with MATLAB®. Modeling and simulation in science, engineering and technology. Birkhäuser. New York. 2011.
  23. Sweilam NH, AL-Mekhla M. On the Optimal Control for Fractional Multi-Strain TB Model. Optimal Control, Applications and Methods. 2016.
  24. Karush W. Minima of Functions of Several Variables with Inequalities as Side Constraints. Ph.D. Department of Mathematics. University of Chicago. Chicago. 1939
  25. H Kuhn, A Tucker. Nonlinear Programming. 1951; 481-492, California. University of California Press. Berkeley.
  26. Bryson AE, Ho YC. Applied optimal control. Hemisphere Publication Corporation. 1975.
  27. Aktas Z, Stetter HJ. A classification and survey of numerical methods for boundary value problems in ordinary differential equations. International journal for numerical methods in engineering. 1977; 11: 771-796.
  28. Shampine LF, Gladwell I, Thompson S. Solving ODEs with MATLAB. Cambridge University Press. 2003.20.
  29. Lenhart S and Workman JT. Optimal Control Applied to Biological Models. Chapman & Hall/CRC Mathematical and Computational Biology. CRC Press. Taylor & Francis Group. 2007.
  30. Mitter SK. The successive approximation method for the solution of optimal control problems. Automotica. 1996; 3:135-149.
  31. Hackbusch W. A numerical method for solving parabolic equations with opposite orientations. Computing. 1978; 20: 229-240.
  32. Victor VM. Practical Direct Collocation Methods for Computational Optimal Control. In Modeling and Optimization in Space Engineering. Volume 73 of Springer Optimization and Its Applications. Springer New York. 2013; 33-60.
  33. Chachuat B. Nonlinear and Dynamic Optimization: From Theory to Practice - IC-32: Spring Term. EPFL. 2009.
  34. Binder T, Blank L, Bock HG, Bulirsch R, Dahmen W, et al. Introduction to Model Based Optimization of Chemical Processes on Moving Horizons. In Introduction to Model Based Optimization of Chemical Processes on Moving Horizons. Springer Berlin Heidelberg. 2001; 295-339.
  35. Bock H, Plitt K. A multiple shooting algorithm for direct solution of optimal control problems. In 9th IFAC. Pergamon Press. 1984; 242-247.
  36. Diehl M, Findeisen R, Schwarzkopf S, Uslu I, Allgöwer F, et al. An Efficient Algorithm for Nonlinear Model Predictive Control of Large-Scale Systems Part I: Description of the Method. At-Automatisierungstechnik Methoden und Anwendungen der Steuerungs-, Regelungs-und Informationstechnik, 2002; 50: 557.
  37. Dickmanns ED, Well KH. Approximate solution of optimal control problems using third order hermite polynomial functions. In Marchuk GI, editor. Optimization Techniques IFIP Technical Conference Novosibirsk, number 27 in Lecture Notes in Computer Science, pages. Springer Berlin Heidelberg. 1974; 158-166.
  38. Törn A, Žilinskas A, Goos G, Hartmanis J, Barstow D, et al. Global Optimization, volume 350 of Lecture Notes in Computer Science. Springer Berlin Heidelberg. Berlin. Heidelberg. 1989.
  39. Biegler LT. Nonlinear programming: concepts, algorithms, and applications to chemical processes. Number 10 in MOS-SIAM series on optimization. SIAM. Philadelphia. 2010.
  40. Betts JT. Practical methods for optimal control and estimation using nonlinear programming. Advances in design and control. Society for Industrial and Applied Mathematics. Philadelphia. 2nd edition. 2010.
  41. Matthew PK. Transcription Methods for Trajectory Optimization A beginners tutorial. Technical report. Cornell University. 2015.
  42. E Hairer, Norsett SP, Wanner G. Solving Ordinary Differential Equations I Nonstiff Problems. Springer-Verlag Berlin Heidelberg, 2008.

Figures:

Figure 1

Figure 1

Figure 1

Figure 2

Figure 1

Figure 3

Figure 1

Figure 4

Figure 1

Figure 5

Figure 1

Figure 6

Figure 1

Figure 7

Figure 1

Figure 8

Figure 1

Figure 9

Figure 1

Figure 10

Figure 1

Figure 11

Figure 1

Figure 12

Figure 1

Figure 13

Similar Articles

Recently Viewed

Read More

Most Viewed

Read More

Help ?