Set-valued analysis of anti-angiogenic therapy and radiotherapy




Anti-angiogenic therapy, radiotherapy, viability theory, set-valued analysis


The aim of the paper is to study a cancer model based on anti-angiogenic therapy and radiotherapy. A set-valued analysis is carried out to control the tumor and carrying capacity of the vasculature, so in order to reverse tumor growth and augment tumor repair. The viability technique is used on an augmented model to solve the control problem. Obtained control is a selection of set-valued map of regulation and reduces tumor volume to around zero. A numerical simulation scheme with graphical representations and biological interpretations are given.


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Özköse, F., Şenel, M.T., & Habbireeh, R. Fractional-order mathematical modelling of cancer cells-cancer stem cells-immune system interaction with chemotherapy. Mathematical Modelling and Numerical Simulation with Applications, 1(2), 67-83, (2021).

Ghori, M.B., Naik, P.A., Zu, J., Eskandari, Z., & Naik, M.U.D. Global dynamics and bifurcation analysis of a fractional-order SEIR epidemic model with saturation incidence rate. Mathematical Methods in the Applied Sciences, 45(7), 3665-3688, (2022).

Sinan, M., Leng, J., Anjum, M., & Fiaz, M. Asymptotic behavior and semi-analytic solution of a novel compartmental biological model. Mathematical Modelling and Numerical Simulation with Applications, 2(2), 88-107, (2022).

Naik, P.A., Yavuz, M. & Zu, J. The role of prostitution on HIV transmission with memory: A modeling approach. Alexandria Engineering Journal, 59(4), 2513-2531, (2020).

Gholami, M., Ghaziani, R.K., & Eskandari, Z. Three-dimensional fractional system with the stability condition and chaos control. Mathematical Modelling and Numerical Simulation with Applications, 2(1), 41-47, (2022).

Naik, P.A., Owolabi, K.M., Yavuz, M., & Zu, J. Chaotic dynamics of a fractional order HIV-1 model involving AIDS-related cancer cells. Chaos, Solitons & Fractals, 140, 110272, (2020).

Naik, P.A., Eskandari, Z., & Shahraki, H.E. Flip and generalized flip bifurcations of a two-dimensional discrete-time chemical model. Mathematical Modelling and Numerical Simulation with Applications, 1(2), 95-101, (2021).

Naik, P.A., Zu, J., & Naik, M.U.D. Stability analysis of a fractional-order cancer model with chaotic dynamics. International Journal of Biomathematics, 14(06), 2150046, (2021).

Naik, P.A., Yavuz, M., Qureshi, S., Zu, J., & Townley, S. Modeling and analysis of COVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan. The European Physical Journal Plus, 135(10), 1-42, (2020).

Joshi, H., & Jha, B.K. Chaos of calcium diffusion in Parkinson’s infectious disease model and treatment mechanism via Hilfer fractional derivative. Mathematical Modelling and Numerical Simulation with Applications, 1(2), 84-94, (2021).

Joshi, H., & Jha, B.K. On a reaction–diffusion model for calcium dynamics in neurons with Mittag–Leffler memory. The European Physical Journal Plus, 136(6), 1-15, (2021).

Joshi, H., & Jha, B.K. Generalized Diffusion Characteristics of Calcium Model with Concentration and Memory of Cells: A Spatiotemporal Approach. Iranian Journal of Science And Technology, Transactions A: Science, 46(1), 309-322, (2022).

Ledzewicz, U., & Schättler, H. The structure of optimal protocols for a mathematical model of chemotherapy with antiangiogenic effects. SIAM Journal on Control and Optimization, 60(2), 1092-1116, (2022).

Ledzewicz, U., & Schättler, H. Combination of antiangiogenic treatment with chemotherapy as a multi-input optimal control problem. Mathematical Methods in the Applied Sciences, 45(5), 3058-3082, (2022).

Ghita, M., Ghita, M., Copot, D., Birs, I.R., Muresan, C., & Ionescu, C.M. Optimizing radiotherapy with chemotherapy using PKPD modeling for lung cancer. 2022 IEEE 20th Jubilee World Symposium on Applied Machine Intelligence and Informatics (SAMI), 299-304, (2022).

O’Reilly, M.S. The combination of antiangiogenic therapy with other modalities. Cancer Journal (Sudbury, Mass.), 8, S89-99, (2002).

Gasparini, G., Longo, R., Fanelli, M., & Teicher, B.A. Combination of antiangiogenic therapy with other anticancer therapies: results, challenges, and open questions. Journal Of Clinical Oncology, 23(6), 1295-1311, (2005).

Shannon, A.M., & Williams, K.J. Antiangiogenics and radiotherapy. Journal of Pharmacy and Pharmacology, 60(8), 1029-1036, (2008).

Senan, S., & Smit, E.F. Design of clinical trials of radiation combined with antiangiogenic therapy. The Oncologist, 12(4), 465-477, (2007).

O’Reilly, M.S. The interaction of radiation therapy and antiangiogenic therapy. The Cancer Journal, 14(4), 207-213, (2008).

O’Reilly, M.S. Radiation combined with antiangiogenic and antivascular agents. Seminars in Radiation Oncology, 16(1), 45-50, (2006).

Mazeron, R., Azria, D., & Deutsch, E. Angiogenesis inhibitors and radiation therapy: from biology to clinical practice. Cancer Radiotherapie: Journal de la Societe Francaise de Radiotherapie Oncologique, 13(6-7), 568-573, (2009).

Mortezaee, K., Parwaie, W., Motevaseli, E., Mirtavoos-Mahyari, H., Musa, A., Shabeeb, D., Esmaely, F., Najafi, M., & Farhood, B. Targets for improving tumor response to radiotherapy. International Immunopharmacology, 76, 105847, (2019).

Moustafid, A. General chemotherapy protocols. Journal of Applied Dynamic Systems and Control, 4(2), 18-25, (2021).

Kassara, K., & Moustafid, A. Feedback protocol laws for immunotherapy. PAMM: Proceedings in Applied Mathematics and Mechanics, 7(1), 2120033-2120034, (2007).

Moustafid, A. General anti-angiogenic therapy protocols with chemotherapy. International Journal of Mathematical Modelling & Computations, 11(3), (2021).

Kassara, K., & Moustafid, A. Angiogenesis inhibition and tumor-immune interactions with chemotherapy by a control set-valued method. Mathematical Biosciences, 231(2), 135-143, (2011).

Kienle-Garrido, M.L., Breitenbach, T., Chudej, K., & Borzì, A. Modeling and numerical solution of a cancer therapy optimal control problem. Applied Mathematics, 9(8), (2018).

Wein, L.M., Cohen, J.E., & Wu, J.T. Dynamic optimization of a linear–quadratic model with incomplete repair and volumedependent sensitivity and repopulation. International Journal of Radiation Oncology, Biology, Physics, 47(4), 1073-1083, (2000).

Ledzewicz, U., D’Onofrio, A., & Schättler, H. Tumor development under combination treatments with anti-angiogenic therapies. In Mathematical Methods and Models in Biomedicine, Springer, 311-337, (2013).

Chudej, K., Huebner, D., & Pesch, H.J. Numerische Lösung eines mathematischen Modells für eine optimale Krebskombinationstherapie aus Anti-Angiogenese und Strahlentherapie. Tagungsband ASIM 2016-23 Symposium Simulationstechnik, Dresden, 52, 169-176, (2016).

Mellal, L., Folio, D., Belharet, K., & Ferreira, A. Modeling of optimal targeted therapies using drug-loaded magnetic nanoparticles for liver cancer. IEEE Transactions On Nanobioscience, 15(3), 265-274, (2016).

Ergun, A., Camphausen, K., & Wein, L.M. Optimal scheduling of radiotherapy and angiogenic inhibitors. Bulletin of Mathematical Biology, 65(3), 407-424, (2003).

Ledzewicz, U., & Schättler, H. Multi-input optimal control problems for combined tumor anti-angiogenic and radiotherapy treatments. Journal of Optimization Theory And Applications, 153, 195-224, (2012).

Jarrett, A.M., Faghihi, D., Hormuth, D.A., Lima, E.A., Virostko, J., Biros, G., Patt, D., & Yankeelov, T. Optimal control theory for personalized therapeutic regimens in oncology: Background, history, challenges, and opportunities. Journal of Clinical Medicine, 9(5), 1314, (2020).

Ledzewicz, U., Maurer, H., & Schättler, H. Optimal combined radio-and anti-angiogenic cancer therapy. Journal of Optimization Theory and Applications, 180(1), 321-340, (2019).

Chudej, K., Wagner, L., & Pesch, H. Numerical solution of an optimal control problem in cancer treatment: Combined radio and anti-angiogenic therapy. IFAC-PapersOnLine, 48(1), 665-666, (2015).

Schättler, H., & Ledzewicz, U. Optimal control for mathematical models of cancer therapies: An application of geometric methods. New York: Springer, Vol. 42, (2015).

Nastitie, N., & Arif, D.K. Analysis and optimal control in the cancer treatment model with combining radio and antiangiogenic therapy. IJCSAM (International Journal of Computing Science And Applied Mathematics), 3(2), 55-60, (2017).

Ledzewicz, U., & Schättler, H. On the role of the objective in the optimization of compartmental models for biomedical therapies. Journal Of Optimization Theory And Applications, 187(2), 305-335, (2020).



DOI: 10.53391/mmnsa.2022.015

How to Cite

Moustafid, A. (2022). Set-valued analysis of anti-angiogenic therapy and radiotherapy. Mathematical Modelling and Numerical Simulation With Applications, 2(3), 187–196.



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