Subject Areas : Electrical Engineering

Amine Moustafid ¹ ( Department of Mathematics and Informatics, Ain Chock Faculty, Hassan II University, Casablanca, Morocco )

Received: 2021-10-05 Accepted : 2021-12-25 Published : 2021-12-01

Keywords:

References:

[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.