Viability Control of Chemo-Immunotherapy and Radiotherapy by Set-Valued Analysis

Abstract

In this paper we set-valued analyze the problem of asymptotic stabilizing the tumor size. A mathematical model of exponential tumor growing caused by carcinogenic substance is considered, with chemotherapy, immunotherapy, and radiotherapy effects. We control the model to be viable in therapeutic domains, and reverse the exponential growing of the tumor size. The obtained controls derive from the derivative cone of therapeutic domains as solution of minimizing problem.

Keywords

• chemotherapy
• immunotherapy
• radiotherapy
• anti-angiogenic therapy
• virotherapy
• viability theory
• set-valued analysis
• tube
• exponential stability
• contingent derivative
• contingent cone
• differential equations

References

1. El Haout, S., Fatani, M., Farha, N., AlSawaftah, N., Mortula, M., Husseini, G.: Modeling the Effects of Chemotherapy and Immunotherapy on Tumor Growth. Journal Of Biomedical Nanotechnology. 17, 2505-2518, 2021
2. Pratap, J.: An optimal control strategy for mathematically modeling cancer combination therapy. ArXiv Preprint ArXiv:2101.12120, 2021
3. Nave, O.: A mathematical model for treatment using chemo-immunotherapy. Heliyon. 8, e09288, 2022
4. Unni, P., Seshaiyer, P.: Mathematical modeling, analysis, and simulation of tumor dynamics with drug interventions. Computational And Mathematical Methods In Medicine, 2019
5. Sharma, S., Samanta, G.: Analysis of the dynamics of a tumor{immune system with chemotherapy and immunotherapy and quadratic optimal control. Differential Equations And Dynamical Systems. 24, 149-171, 2016
6. Rodrigues, D., Mancera, P., Carvalho, T., Gonccalves, L.: A mathematical model for chemoimmunotherapy of chronic lymphocytic leukemia. Applied Mathematics And Computation. 349, 118-133, 2019
7. Hellal, H., Elabsy, H., Elkaranshawy, H.: Mathematical Model for Combined Radiotherapy and Chemotherapy that Fits with Experimental Data. Journal Of Physics: Conference Series. 2287, 012013, 2022
8. Bekker, R., Kim, S., Pilon-Thomas, S., Enderling, H.: Mathematical modeling of radiotherapy and its impact on tumor interactions with the immune system. Neoplasia. 28, 100796, 2022

9. Butuc, I., Mirestean, C., Iancu, D.: A nonlinear model in the dynamics of tumourimmune system combined with radiotherapy. UPB Sci. Bull., Ser. A. 81, 311-322, 2019
10. Pinho, S., Bacelar, F., Andrade, R., Freedman, H.: A mathematical model for the effect of anti-angiogenic therapy in the treatment of cancer tumours by chemotherapy. Nonlinear Analysis: Real World Applications. 14, 815-828, 2013
11. Esen,  O., Cetin, E., Esen, S.: A mathematical immunochemoradiotherapy model: A multiobjective approach. Nonlinear Analysis: Real World Applications. 9, 511-517, 2008
12. Kuznetsov, M., Kolobov, A.: Algorithm of optimization of fractionated radiotherapy within its combination with antiangiogenic therapy by means of mathematical modeling. ITM Web Of Conferences. 31, 02001 ,2020
13. Kuznetsov, M., Gubernov, V., Kolobov, A.: Analysis of anticancer efficiency of combined fractionated radiotherapy and antiangiogenic therapy via mathematical modelling. Russian Journal Of Numerical Analysis And Mathematical Modelling. 33, 225-242, 2018
14. Hutchinson, L., Mueller, H., Gaffney, E., Maini, P., Wagg, J., Phipps, A., Boetsch, C., Byrne, H., Ribba, B.: Modeling longitudinal preclinical tumor size data to identify transient dynamics in tumor response to antiangiogenic drugs. CPT: Pharmacometrics & Systems Pharmacology. 5, 636-645, 2016
15. Rosch, K., Scholz, M., Hasenclever, D.: Modeling combined chemo-and immunotherapy of high-grade non-Hodgkin lymphoma. Leukemia & Lymphoma. 57, 1697-1708, 2016
16. Yang, J., Tang, S., Cheke, R.: Modelling pulsed immunotherapy of tumour{immune interaction. Mathematics And Computers In Simulation. 109, 92-112, 2015
17. Saleem, M., Farman, M., Ahmad, A., Meraj, M. 31.: Mathematical model based assessment of the cancer control by chemo-immunotherapy. Pure And Applied Biology (PAB). 7, 678-683, 2018
18. Moustad, A.: Set-valued analysis of anti-angiogenic therapy and radiotherapy. Mathematical Modelling And Numerical Simulation With Applications. 2, 187-196, 2022
19. Kassara, K., Moustad, A.: Angiogenesis inhibition and tumor-immune interactions with chemotherapy by a control set-valued method. Mathematical Biosciences. 231, 135-143, 2011
20. Moustad, A.: General anti-angiogenic therapy protocols with chemotherapy. International Journal Of Mathematical Modelling & Computations. 11, 2021
21. Moustad, A.: Feedback protocols for anti-angiogenic therapy in the treatment of cancer tumors by chemotherapy. International Journal on Optimization and Applications. 2, 17-24, 2022
22. Moustad, A.: General Chemotherapy Protocols. Journal Of Applied Dynamic Systems And Control. 4, 18-25, 2021
23. Dehingia, K., Sarmah, H., Hosseini, K., Sadri, K., Salahshour, S., Park, C.: An optimal control problem of immuno-chemotherapy in presence of gene therapy. AIMS Mathematics. 6, 11530-11549, 2021
24. Piretto, E., Delitala, M., Ferraro, M.: Combination therapies and intra-tumoral competition: Insights from mathematical modeling. Journal Of Theoretical Biology. 446, 149-159, 2018
25. Abdeljalil, S., Essid, A., Aouadi, S.: Analysis of a tumor growth model with a nonlocal boundary condition. ArXiv Preprint ArXiv:2205.12167, 2022
26. Roy, S., Pal, S.: Optimal personalized therapies in colon-cancer induced immune response using a Fokker-Planck framework. ArXiv Preprint ArXiv:2209.03812, 2022
27. Malinzi, J., Eladdadi, A., Sibanda, P.: Modelling the spatiotemporal dynamics of chemovirotherapy cancer treatment. Journal Of Biological Dynamics. 11, 244-274, 2017
28. Nono, M., Ngouonkadi, E., Bowong, S., Fotsin, H.: Spatiotemporal dynamics and optimal control of glioma virotherapy enhanced by MEK Inhibitors. Results In Control And Optimization. 7, 100101, 2022
29. Pomeroy, A., Schmidt, E., Sorger, P., Palmer, A.: Drug independence and the curability of cancer by combination chemotherapy. Trends In Cancer, 2022
30. Abaid Ur Rehman, M., Ahmad, J., Hassan, A., Awrejcewicz, J., Pawlowski, W., Karamti, H., Alharbi, F.: The Dynamics of a Fractional-Order Mathematical Model of Cancer Tumor Disease. Symmetry. 14, 1694, 2022
31. Panwar, V., Uduman, P.: Existence a nd Uniqueness of Solutions for Mixed Immunotherapy and Chemotherapy Cancer Treatment Fractional Model with Caputo- Fabrizio Derivative. Progress In Fractional Differentiation And Applications An International Journal. 8, 243-251, 2022
32. Farayolaa, M., Shaea, S., Siama, F., Mahmudb, R., Ajadic, S.: Mathematical modeling of cancer treatments with fractional derivatives: An Overview. Malaysian Journal Of Fundamental And Applied Sciences. 17, 389-401, 2021
33. Baleanu, D., Jajarmi, A., Sajjadi, S., Mozyrska, D.: A new fractional model and optimal control of a tumor-immune surveillance with non-singular derivative operator. Chaos: An Interdisciplinary Journal Of Nonlinear Science. 29, 083127 ,2019
34. Yildiz, T., Arshad, S., Baleanu, D.: Optimal chemotherapy and immunotherapy schedules for a cancer-obesity model with Caputo time fractional derivative. ArXiv Preprint ArXiv:1804.06259, 2018
35. Sweilam, N., Rihan, F., Seham, A.: A fractional-order delay differential model with optimal control for cancer treatment based on synergy between anti-angiogenic and immune cell therapies. Discrete & Continuous Dynamical Systems-S. 13, 2403, 2020
36. Erjaee, G., Ostadzad, M., Amanpour, S., Lankarani, K.: Dynamical analysis of the interaction between effector immune and cancer cells and optimal control of chemotherapy. Nonlinear Dynamics, Psychology, And Life Sciences. 17, 449-463, 2013
37. Shamsaraa, E., Esmailyb, H., Bahrampourc, A.: Mathematical Model of Optimal Chemotherapy and Oncolytic Virotherapy. Filomat. 34, 5195-5206, 2020
38. Rihan, F., Lakshmanan, S., Maurer, H.: Optimal control of tumour-immune model with time-delay and immuno-chemotherapy. Applied Mathematics And Computation. 353, 147-165, 2019
39. Das, P., Das, S., Upadhyay, R., Das, P.: Optimal treatment strategies for delayed cancer-immune system with multiple therapeutic approach. Chaos, Solitons & Fractals. 136, 109806, 2020
40. Moussa, K., Fiacchini, M., Alamir, M.: Probabilistically certied region of attraction of a tumor growth model with combined chemo-and immunotherapy. International Journal Of Robust And Nonlinear Control. 2022
41. Deepak, K., Sanjeev, K.: A mathematical model of radioimmunotherapy for tumor treatment. African Journal Of Mathematics And Computer Science Research. 3, 101-106, 2010
42. Kok, H., Rhoon, G., Herrera, T., Overgaard, J., Crezee, J.: Biological modeling in thermoradiotherapy: present status and ongoing developments toward routine clinical use. International Journal Of Hyperthermia. 39, 1126-1140, 2022
43. Ahmadi, E., Zarei, J., Razavi-Far, R., Saif, M.: A dual approach for positive T-S fuzzy controller design and its application to cancer treatment under immunotherapy and chemotherapy. Biomedical Signal Processing And Control. 58, 101822, 2020
44. Roguia, S., Mohamed, N.: An optimized rbf-neural network for breast cancer classication. International Journal Of Informatics And Applied Mathematics. 1, 24-34, 2019
45. Engeland, C., Heidbuechel, J., Araujo, R., Jenner, A.: Improving immunovirotherapies: the intersection of mathematical modelling and experiments. ImmunoInformatics, 100011, 2022
46. Bunonyo, K., Ebiwareme, L.: Tumor growth mathematical modeling and application of chemo-immunotherapy and radiotherapy treatments. Int J Stat Appl Math. 7(2), 123-132, 2022
47. Aubin, J.P.: Viability Theory. Modern Birkhauser Classics, Birkhauser Boston. 2009
48. Grimes, W., Achache, M.: An Infeasible Interior-point Algorithm for Monotone Linear Complementarity Problems. International Journal Of Informatics And Applied Mathematics. 4, 53-59, 2021