Research article

Numerical solution for a problem arising in angiogenic signalling

  • Received: 10 November 2018 Accepted: 24 December 2018 Published: 07 January 2019
  • Since the process of angiogenesis is controlled by chemical signals, which stimulate both repair of damaged blood vessels and formation of new blood vessels, then other chemical signals known as angiogenesis inhibitors interfere with blood vessels formation. This implies that the stimulating and inhibiting effects of these chemical signals are balanced as blood vessels form only when and where they are needed. Based on this information, an optimal control problem is formulated and the arising model is a system of coupled non-linear equations with adjoint and transversality conditions. Since many of the numerical methods often fail to capture these type of models, therefore, in this paper, we carry out steady state analysis of these models before implementing the numerical computations. In this paper we analyze and present the numerical estimates as a way of providing more insight into the postvascular dormant state where stimulator and inhibitor come into balance in an optimal manner.

    Citation: Kolade M. Owolabi, Kailash C. Patidar, Albert Shikongo. Numerical solution for a problem arising in angiogenic signalling[J]. AIMS Mathematics, 2019, 4(1): 43-63. doi: 10.3934/Math.2019.1.43

    Related Papers:

  • Since the process of angiogenesis is controlled by chemical signals, which stimulate both repair of damaged blood vessels and formation of new blood vessels, then other chemical signals known as angiogenesis inhibitors interfere with blood vessels formation. This implies that the stimulating and inhibiting effects of these chemical signals are balanced as blood vessels form only when and where they are needed. Based on this information, an optimal control problem is formulated and the arising model is a system of coupled non-linear equations with adjoint and transversality conditions. Since many of the numerical methods often fail to capture these type of models, therefore, in this paper, we carry out steady state analysis of these models before implementing the numerical computations. In this paper we analyze and present the numerical estimates as a way of providing more insight into the postvascular dormant state where stimulator and inhibitor come into balance in an optimal manner.


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    [1] J. T. Betts, Practical method for optimal control using nonlinear programing, SIAM, Philadelphia, 2001.
    [2] T. Boehm, J. Folkman, T. Browder, et al. Antiangiogenic therapy of experimental cancer does not induce acquired drug resistance, Nature, 390 (1997), 404-407. doi: 10.1038/37126
    [3] B. Bonnard and M. Chyba, Singular trajectories and their role in control theory, Springer Verlag, 2003.
    [4] A. Bressan and B. Piccoli, Introduction to the mathematical theory of control, Vol. 2, American Institute of Mathematical Sciences, 2007.
    [5] R. L. Burden and J. D. Faires, Numerical analysis, PWS-KENT Publishing Company, Boston, 2004.
    [6] J. H. E. Cartwright and O. Piro, The Dynamics of Runge-Kutta Methods, Int. J. Bifurcat. Chaos, 2 (1192), 427-449.
    [7] B. Czako, J. Sápi, L. Kovács, Model-based optimal control method for cancer treatment using model predictive control and robust fixed point method, 2017 IEEE 21st International Conference on Intelligent Engineering Systems (INES), (2017), 271-276.
    [8] W. Cheney and D. Kincaid, Numerical mathematics and computing, Thomson, Belmont, California, 2004.
    [9] S. Davis and G. D. Yancopoulos, The angiopoietins: Yin and Yang in angiogenesis, Curr. Top. Microbiol., 237 (1999), 173-185.
    [10] A. L. Dontchev and W. W. Hager, The Euler approximation in state constrained optimal control, Math. Comput., 70 (2001), 173-203.
    [11] A. Ergun, K. Camphausen and L. M. Wein, Optimal scheduling of radiotherapy and angiogenic inhibitors, B. Math. Biol., 65 (2003), 407-424. doi: 10.1016/S0092-8240(03)00006-5
    [12] J. Folkman, Endogenous angiogenesis inhibitors, APMIS, 112 (2004), 496-507. doi: 10.1111/j.1600-0463.2004.apm11207-0809.x
    [13] J. Folkman, Antiangiogenesis: new concept for therapy of solid tumors, Ann. Surg., 175 (1972), 409-416. doi: 10.1097/00000658-197203000-00014
    [14] P. Hahnfeldt, D. Panigrahy, J. Folkman, et al. Tumor development under angiogenic signaling: a dynamical theory of tumor growth, treatment response, and postvascular dormancy, Cancer Res., 59 (1999), 4770-4775.
    [15] R. K. Jain, Normalizing tumor vasculature with anti-angiogenic therapy: a new paradigm for combination therapy, Nat. Med., 7 (2001), 987-989. doi: 10.1038/nm0901-987
    [16] R. K. Jain and L. L. Munn, Vascular normalization as a rationale for combining chemotherapy with antiangiogenic agents, Principles of Practical Oncology, 21 (2007), 1-7.
    [17] R. E. Kalman, Contribution to the theory of optimal control, Buletin Sociedad Matematica Mexicana, 5 (1960), 102-119.
    [18] R. S. Kerbel, Inhibition of tumor angiogenesis as a strategy to circumvent acquired resistance to anti-cancer therapeutic agents, BioEssays 13 (1991), 31-36.
    [19] H. Khan, A. Szeghegyi and J. K. Tar, Fixed point transformation-based adaptive optimal control using NLP. In: Proc. of the 2017 IEEE 30th Jubilee Neumann Colloquium, Budapest, Hungary, (2017), 35-40.
    [20] M. Klagsburn and S. Soker, VEGF/VPF: the angiogenesis factor found?, Curr. Biol., 3 (1993), 699-702. doi: 10.1016/0960-9822(93)90073-W
    [21] R. S. Kerbel, A cancer therapy resistant to resistance, Nature, 390 (1997), 335-336. doi: 10.1038/36978
    [22] U. Ledzewicz and H. Schättler, A synthesis of optimal controls for a model of tumor growth a under angiogenic inhibitors, Proc. 44th IEEE Conference on Decision and Control, (2005), 934-939.
    [23] U. Ledzewicz and H. Schättler, Anti-angiogenic therapy in cancer treatment as an optimal control problem, Summer Research Fellowship, 2006.
    [24] U. Ledzewicz and H. Schättler, Anti-angiogenic therapy in cancer treatment as an optimal a control problem, SIAM J. Control Optim., 46 (2007), 1052-1079. doi: 10.1137/060665294
    [25] U. Ledzewicz and H. Schättler, Analysis of a mathematical model for tumor anti-angiogenesis, Optimal Control Applications and Methods, 29 (2008), 41-57. doi: 10.1002/oca.814
    [26] U. Ledzewicz and H. Schättler, Optimal and suboptimal protocols for a class of mathematical a models of tumor anti-angiogenesis, J. Theor. Biol., 252 (2008), 295-312. doi: 10.1016/j.jtbi.2008.02.014
    [27] U. Ledzewicz, J. Munden and H. Schȧttler, BA and-bang Controls for anti-angiogenesis under logistics growth of the tumor, International Journal of Pure and Applied Mathematics, 9 (2008), 511-516.
    [28] U. Ledzewicz, J. Munden, and H. Schättler, Scheduling of angiogenic inhibitors for Gomapertzian and logistic tumor growth models, Discrete Cont. Dyn-B, 12 (2009), 415-438. doi: 10.3934/dcdsb.2009.12.415
    [29] U. Ledzewicz, J. Marriott, H. Maurer, et al. Realizable protocols for optimal administration of drugs in mathematical models for anti-angiogenic treatment, Math. Med. Biol., 27 (2009), 157-179.
    [30] U. Ledzewicz and B. Cardwell, Robustness of optimal controls for a class of mathematical models for tumor anti-angiogenesis, Math. Biosci. Eng., 8 (2011), 355-369. doi: 10.3934/mbe.2011.8.355
    [31] U. Ledzewicz, H. Maurer and H. Schättler, Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy, Math. Biosci. Eng., 8 (2011), 307-323. doi: 10.3934/mbe.2011.8.307
    [32] S. Lenhart and J. T. Workman, Optimal control applied to biological models, Chapman Hall/CRC, 2007.
    [33] E. Naevdal, Solving Continuous-Time Optimal-Control Problems with a Spreadsheet, Journal of Economic Education, 34 (2003), 99-122. doi: 10.1080/00220480309595206
    [34] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, et al. The mathematical theory of optimal processes, MacMillan, New York, 1964.
    [35] S. S. Samaee, O. Yazdanpanah and D. D. Ganji, New approaches to identification of the Lagrange multiplier in the variational iteration method, J. Braz. Soc. Mech. Sci., 37 (2015), 937-944. doi: 10.1007/s40430-014-0214-3
    [36] B. Sebastien, Mathematical and numerical analysis of a model for anti-angiogenic therapy in metastatic cancers, ESAIM, 46 (2011), 207-237.
    [37] A. Swierniak, Comparison of six models of antiangiogenic therapy, Applicationes Mathematicae, 36 (2009), 333-348. doi: 10.4064/am36-3-6
    [38] A. Swierniak, Direct and indirect control of cancer populations, B. Pol. Acad. Sci-Tech, 56 (2008), 367-378.
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