Numerical study of the mathematical model of the evolution of drug resistance during cancer chemotherapy with the least squares support vector machine approach

Document Type : Original Paper


Department of Mathematics, Shiraz University of Technology, Shiraz, Iran


Tumor resistance to chemotherapy and targeted drugs is one of the main factors of treatment failure. Experimental evidence in recent years shows that the progression of cancer cells to drug resistance does not have to happen by chance, but may be caused by the treatment itself. In this regard, understanding the clinical consequences of resistance caused by treatment in the process of chemotherapy helps to develop appropriate solutions. In this paper, to investigate this issue, we first introduce the general mathematical model of drug resistance in the chemotherapy process in the form of a device of nonlinear ordinary differential equations. Then we perform the numerical simulation of the dynamic behavior of the model in three different cases using the least squares support vector machine approach. In this study, the effects of three drugs with different drug resistance induction coefficients are considered. Finally, we will examine the results of these simulations according to the type of drug prescribed in tumor growth control.


Main Subjects

[1] Aggarwal, C.C., 2018. Neural Networks and Deep Learning. Springer. doi: 10.1007/978-3-319-
[2] Banerjee, S. and Sarkar, R.R., 2008. Delay-induced model for tumor-immune interaction and control of malignant tumor growth. Biosystems, 91(1), pp.268-288. doi:10.1016/j.biosystems.2007.10.002
[3] Bekisz, S. and Geris, L., 2020. Cancer modelling: from mechanistic to data-driven approaches, and from fundamental insights to clinical applications. Journal of Computational Science, 46, pp.101198.doi: 10.1016/j.jocs.2020.101198
[4] Borges, F., Iarosz, K., Ren, H., Batista, A., Baptista, M., Viana, R., Lopes, S. and Grebogi, C., 2014.
Model for tumour growth with treatment by continuous and pulsed chemotherapy. Biosystems, 116,
pp.43-48. doi: 10.1016/j.biosystems.2013.12.001
[5] Franssen, L.C., Lorenzi, T., Burgess, A.E. and Chaplain, M.A., 2019. A mathematical framework for modelling the metastatic spread of cancer. Bulletin of Mathematical Biology, 81(6), pp.1965-2010. doi:10.1007/s11538-019-00597-x
[6] Géron, A., 2022. Hands-on Machine Learning with Scikit-Learn, Keras, and TensorFlow, O’Reilly
Media. Inc..
[7] Gestel, T.V., Suykens, J.A.K., Baesens, B., Viaene, S., Vanthienen, J., Dedene, G., Moor, B.D. and
Vandewalle, J., 2004. Benchmarking least squares support vector machine classifiers. Machine Learning, 54, pp.5-32. doi: 10.1023/B:MACH.0000008082.80494.e0
[8] Greene, J.M., Gevertz, J.N. and Sontag, E.D., 2019. Mathematical approach to differetiate spontaneous and induced evolution to drug resistance durig cancer treatment. JCO Clinical Cancer Informatics, 3,pp.1-20. doi: 10.1200/CCI.18.00087
[9] Mehrkanoon, S., Falck, T. and Suykens, A.K., 2012. Approximate solutions to ordinary differential equations using least squares support vector machiens. IEEE Transactions on Neural Networks and Learning Systems, 23(9), pp.1356-1367. doi: 10.1109/TNNLS.2012.2202126
[10] Mehrkanoon, S. and Suykens, J.A.K., 2015. Learning solution to partial differential equations using LS-SVM. Neurocomputing, 159, pp.105-116. dio: 10.1016/j.neucom.2015.02.013
[11] Mohri, M., Rostamizadeh, A. and Talwalkar, A., 2018. Foundations of Machine Learning. MIT Press.
[12] Padmanabhan, R., Meskin, N. and Ala-Eddin, A.M., 2021. Mathematical Models of Cancer and
Different Therapies. Springer: Singapore. dio: 10.1007/978-981-15-8640-8
[13] Pakniyat, A., Parand, K. and Jani, M., 2021. Least squares support vector regression for differential equations on unbounded domains. Chaos Solitions and Fractals, 151, pp.111232. doi:
[14] Parand, K., Aghaei, A.A., Jani, M. and Ghodsi, A., 2021. A new approach to the numerical solution of Fredholm intergal equations using least squares-support vector regression. Mathematics and Computers in Simulation, 180, pp.114-128. doi: 10.1016/j.matcom.2020.08.010
[15] Pinho, S., Freedman, H. and Nani, F., 2002. A chemotherapy model for the treatment of cancer with metastasis. Mathematical and Computer Modelling, 36(7-8), pp.773-803. doi: 10.1016/S0895-7177(02)00227-3
[16] Pillis, L.D. and Radunskaya, A., 2001. A mathematical tumor model with immune resistance and drug therapy: an optimal control approach. Computational and Mathematical Methods in Medicine,3(2), pp.79-100. doi: 10.1080/10273660108833067
[17] Pillis, L.D. and Radunskaya, A., 2003. The dynamics of an optimally controlled tumor model:
a case study. Mathematical and Computer Modelling, 37(11), pp.1221-1244. doi: 10.1016/S0895-
[18] Sun, X. and Hu, B., 2018. Mathematical modeling and computational prediction of cancer drug resistance. Briefings in Bioinformatics, 19(6), pp.1382-1399. doi: 10.1093/bib/bbx065
[19] Suykens, A.K., Gestel, T.V., Brabanter, J.D., Moor, B.D. and Vandewalle, J., 2002. Least-Squares
Support Vector Machines. World Scientific.
[20] Schölkopf, B., Luo, Zh. and Vovk, V., 2013. Empirical Inference: Festchrifl in Honor of Vladimir N. Vapnik. Springer. doi: 10.1007/978-3-642-41136-6
[21] Tse, S.M., Liang, Y., Leung, K.S., Lee, K.H. and Mok, T.S.K., 2007. A memetic algorithm for
multiple-drug cancer chemotherapy schedule optimization. IEEE Transactions on Systems, Man, and Cybernetics. Part B , 37(1), pp.84-91. doi: 10.1109/TSMCB.2006.883265
[22] Wu, H.K., Chen, P.J. and Hsieh, J.G., 2006. Simple algorithms for least square support vector
machines. IEEE International Conference on Systems; Man and Cybernetics, 6, pp.5106-5111. doi: