General Chemotherapy Protocols
الموضوعات :
1 - Department of Mathematics and Informatics, Ain Chock Faculty, Hassan II University, Casablanca, Morocco
الکلمات المفتاحية: Chemotherapy, Viability theory, Set-valued analysis,
ملخص المقالة :
In this paper we treat the problem of cancer control by chemotherapy, through general model in ordinary differential equation form of tumor dynamics. The model is augmented by an ordinary linear differential equation of chemotherapy drugs, and the control problem is reset in the framework of the viability theory. Set-valued analysis method is applied to design procedures leading to the formulation of treatment protocols, which are single-valued selections of set-valued maps, and divide in two categories according to the advancement of initial state cancer, which is characterized by specific set-valued map, upon the strict negativity of the dynamic tumor function at the initial state. Protocols corresponding to non-advanced stage cancer, ensures the decreasing of tumor cells, unlike the ones of advanced stage. Logistic model is considered from the literature to illustrate effects of feedback protocols, by which tumor cells is controlled to be on exponentially decreasing all over chemotherapy horizon, under normalized carrying capacity to reach infinitesimal values.
[1] W. Krabs, S. Pickl, "An optimal control problem in cancer chemotherapy", Applied Mathematics and Computation, Vol. 217: 1117-1124, 2010.
[2] N. N. Subbotina, N. G. Novoselova, "chemotherapy of a malignant tumor growing according to the Gompertz law", International Federation of Automatic Control, Vol. 51, No. 32: 855-860, 2018.
[3] M. Leszczyński, U. Ledzewicz, H. Schättler, "Optimal control for a mathematical model for chemotherapy with pharmacometrics", Mathematical Modelling of Natural Phenomena, Vol. 15, 2020.
[4] G. W. Swan, "General applications of optimal control theory in cancer chemotherapy", Institute of Mathematics and its Applications Mathematics Applied in Medicine & Biology, Vol. 5: 303-316, 1988.
[5] L. A. Fernández, C. Pola, "Catalog of the optimal controls in cancer chemotherapy for the Gompertz model depending on PK/PD and integral constraint", Discrete and continuous dynamical systems series B, Vol. 19, No. 6: 1563-1588, 2014.
[6] S. P. Zanjani, M. J. Mahjoob, S. Amanpour, R. Kheirbakhsh, M. H. Akhoundzadeh, "A supplemental treatment for chemotherapy : Control simulation using a mathematical model with estimated parameters based on in vivo experiment", International Federation of Automatic Control, Vol. 49, No. 26: 277-282, 2016.
[7] E. Sharifnia, Y. B. Lari, M. Eghtesad, "Nonlinear tracking control for reducing cancerous tumor volume using three different cell kill models", International Conference on Mechanical Engineering, 2017.
[8] J. M. Greene, C. S.Tapia, E. D. Sontag, "Control structures of drug resistance in cancer chemotherapy", Institute of Electrical and Electronics Engineers Conference on Decision and Control, 5195-5200, 2018.
[9] K. R. Fister, J. C. Panetta, "Optimal control applied to competing chemotherapeutic cell-kill strategies", Society for Industrial and Applied Mathematics Journal on Applied Mathematics, Vol. 63, No. 3: 1954-1971, 2003.
[10] J. L. Boldrini, M. I. S. Costa, "The influence of fixed and free final time of treatment on optimal hemotherapeutic protocols", Mathematical and Computer Modelling, Vol. 27, No. 6: 59-72, 1998.
[11] R. Eidukeriius, "Deterministic and stochastic modelling of tumor growth and optimal chemotherapy", Applied Mathematics and Computer Science, Vol. 4, No. 2: 233-237, 1994.
[12] G. W. Swan, "Optimal Control Applications in the Chemotherapy of Multiple Myeloma", Institute of Mathematics and its Applications Journal of Mathematics Applied in Medicine & Biology, Vol. 2: 139-160, 1985.
[13] P. A. Orlando, R. A. Gatenby, J. S. Brown, "Cancer treatment as a game : integrating evolutionary game theory into the optimal control of chemotherapy", Physical Biology Vol. 9, No. 6, 2012.
[14] A. Remesh, "Tachling the problems of tumor chemotherapy by optimal drug scheduling", Journal of Clinical and Diagnostic Research, Vol. 7, No. 7: 1404-1407, 2013.
[15] M. H. N. Skandari, M. H. Farahi, "Optimal control for general n-compartmental models in cancer chemotherapy using measure theoretical approach", International journal of sensing, computing & control, Vol. 2, No. 1: 27-37, 2012 .
[16] M. Leszczyński, E. Ratajczyk, U. I. Ledzewicz, H. Schättler, "Sufficient conditions for optimality for a mathematical model of drug treatment with pharmacodynamics", Opuscula Mathematica, Vol. 37, No. 3: 403-419, 2017.
[17] F. M. F. Loho Pereira, M. R. M. Pinho, C. E. Pedreira, M. H. R. Fernandes, "Design of multidrug cancer chemotherapy via optimal control", International Federation of Automatic Control Proceedings Volumes, Vol. 23: 231-236, 1990.
[18] A. Świerniak, U. Ledzewicz, H. Schättler, "Optimal control for a class of compartmental models in cancer chemotherapy", International Journal of Applied Mathematics and Computer Science, Vol. 13, No. 3: 357-368, 2003.
[19] A. Świerniak, Z. Duda, "Singularity of optimal control in some problems related to optimal chemotherapy", Mathematical and Computer Modelling, Vol. 19, No. 6-8: 255-262, 1994.
[20] O. El Kahlaoui, M. Rachik, N. Yousfi, M. Laklalech, O. I. Kacemi, "On the analysis of a biological system : Compartment model approach", Applied Mathematical Sciences, Vol. 1, No. 41: 2001-2022, 2007.
[21] O. Balatif, M. Rachik, I. Abdelbaki, Z. Rachik, "Optimal control for multi-input bilinear systems with an application in cancer chemotherapy", International Journal of Scientific and Innovative Mathematical Research, Vol. 3, No. 2: 22-31, 2015 .
[22] M. Siket, G. Eigner, D. A. Drexler, I. Rudas, L. Kovács, "State and parameter estimation of a mathematical carcinoma model under chemotherapeutic treatment", Applied Sciences, Vol. 10, No. 24, 2020.
[23] M. I. S. Costa, J. L. Boldrini, R. C. Bassanezi, "Optimal chemotherapy : A case study with drug resistance, saturation effect, and toxicity", Institute of Mathematics and its Applications Journal of Mathematics Applied in Medicine & Biology, Vol. 11, No. 1: 45-59, 1994.
[24] J. C. Panetta, "A logistic model of periodic chemotherapy", Applied Mathematics Letters, Vol. 8: 83-86, 1995.
[25] A. D’Onofrio, A. Gandolfi, S. Gattoni, "Chemotherapy of vascularised tumours : Role of vessel density and the effect of vascular ”pruning” ", Journal of Theoretical Biology, Vol. 2: 253-264, 2010.
[26] Á. G. Lópeza, K. C. Iaroszb, A. M. Batistad, J. M. Seoanea, R. L. Vianae, M. A. F. Sanjuána, "Nonlinear cancer chemotherapy : Modelling the Norton-Simon hypothesis", Communications in Nonlinear Science and Numerical Simulation, Vol. 70: 307-317, 2019.
[27] P. Castorina, D. Carcò, C. Guiot, T. S. Deisboeck, "Tumor growth instability and its implications for chemotherapy", American Association for Cancer Research, Vol. 69, No. 21: 8507-8515, 2009.
[28] S. Xu, "Qualitative analysis of a general periodic system", Communications of the Korean Mathematical Society, Vol. 33 No. 3: 1039-1048, 2018.
[29] J. C. Panetta, "A logistic model of periodic chemotherapy with drug resistance", Applied Mathematics Letters, Vol. 10, No. 1: 123-127, 1997.
[30] H. V. Jain, H. M. Byrne, "Qualitative analysis of an integro-differential equation model of periodic chemotherapy", Applied Mathematics Letters, Vol. 25: 2132-2136, 2012.
[31] T. Henson, J. Surendrakumari, "Application of integro-differential equation in periodic chemotherapy", International Journal of Computing Algorithm, Vol. 3: 812-814, 2014.
[32] A. D’Onofrio, A. Gandolfi, S. Gattoni, "The Norton–Simon hypothesis and the onset of non-genetic resistance to chemotherapy induced by stochastic fluctuations", Physica A, Vol. 391: 6484-6496, 2012.
[33] B. A. Costa, J. M. Lemos, "Drug administration design for cancer Gompertz model based on the Lyapunov method", Controlo, 131-141, 2016.
[34] M. Sharifi, H. Moradi, "Nonlinear composite adaptive control of cancer chemotherapy with online identification of uncertain parameters", Biomedical Signal Processing and Control, Vol. 49: 360-374, 2019.
[35] A. Al Khoei, M. R. Zakerzadeh, M. Ayati, "Chemotherapy treatment scheduling via fuzzy system approach", 4th International Conference on Electrical, Computer, Mechanical and Mechatronics Engineering, 2016.
[36] H. Moradi, G. Vossoughi, H. Salarieh, "Optimal robust control of drug delivery in cancer chemotherapy : A comparison between three control approaches", computer method and program in biomedicine, Vol. 112: 69-83, 2013.
[37] A. Petrovski, S. Shakya, J. McCall, "Optimising cancer chemotherapy using an estimation of distribution algorithm and genetic algorithms", GECCO ’06 : Proceedings of the 8th annual conference on Genetic and evolutionary computation, 413-418, 2006.
[38] A. Petrovski, J. McCall, "Multi-objective optimisation of cancer chemotherapy Using Evolutionary Algorithms", The Open Access Institutional Repository at The Robert Gordon University, 531-545, 2001.
[39] Y. Liang, K. S. Leung, T. S. K. Mok, "A novel evolutionary drug scheduling model in cancer chemotherapy", Institute of Electrical and Electronics Engineers Transactions on Information Technology in Biomedicine, Vol. 10, No. 2: 237-245, 2006.
[40] R. Luus, F. Hartig, F. J. Keil, "Optimal drug scheduling of cancer chemotherapy". Periodica Polytechnica Chemical Engineering, Vol. 38, No. 1-2: 105-109, 1994.
[41] R. Luus, F. Hartig, F. J. Keil, "Optimal drug scheduling of cancer chemotherapy by direct search optimization", Hungarian Journal of Industrial Chemistry Veszprém, Vol. 23: 55-58, 1995.
[42] J. T. Tsai, W. H. Ho, "Optimized drug schedulling for cancer chemotherapy using improved immune algorithm", International Journal of Innovative, Computing, Information and Control, Vol. 9, No. 7: 2821-2838, 2013.
[43] E. Shochat, D. Hart, Z. Agur, "Using computer simulations for evaluating the efficacy of breast cancer chemotherapy protocols", Mathematical Models and Methods in Applied Sciences, Vol. 9, No. 4: 599-615, 1999.
[44] H. Agahi, M. J. Yazdanpanah, "Constrained control approach for monotone systems : Application to tumour chemotherapy", IET Control Theory & Applications, Vol. 13: 996-1005, 2019.
[45] L.G. De Pillis, A. Radunskaya, "A mathematical tumor model with immune resistance and drug therapy : an optimal control approach", Computational and Mathematical Methods in Medicine, Vol. 3, No. 2: 79-100, 2001.
[46] K. Kassara, A. Moustafid, "Angiogenesis inhibition and tumor-immune interactions with chemotherapy by a control set-valued method", Mathematical Biosciences, Vol. 231: 135-143, 2011.
[47] L. G. De Pillis, W. Gu, K. R. Fister, T. Head, K. Maples, A. Murugan, T. Neal, K. Yoshida, "Chemotherapy for tumors : An analysis of the dynamics and a study of quadratic and linear optimal controls", Mathematical Biosciences, Vol. 209, No. 1: 292-315, 2007.
[48] J. P. Aubin, "Dynamic economic theory : A viability approach", Springer, Berlin, 1997.
[49] S. Saravi, M. Saravi, "A short survey in application of ordinary differential equations on cancer research", American Journal of Computational and Applied Mathematics, Vol. 10, No. 1: 1-5, 2020.
[50] A. Moustafid, "General anti-angiogenic therapy protocols with chemotherapy", International Journal of Mathematical Modelling Computations, 2021.
