Research Article
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Fractional-Order Mathematical Modeling and Stability Analysis of Tumor-Immune System Relation Including Chemotherapy Drug Effect

Year 2024, Volume: 8 Issue: 2, 85 - 101

Esen Kaya , Fatma Özköse , M. Tamer Senel

https://doi.org/10.29002/asujse.1384833

Abstract

Worldwide, cancer is the second most common cause of death. Chemotherapy is a widely used strategy to tumor treatment that is particularly effective in controlling the growth of cancerous tumors and their size. We created a fractional-order mathematical model that illustrates tumor growth in the presence of chemotherapy to obtain a more profound comprehension of the complexities of chemotherapy mechanisms. This all-inclusive paradigm addresses both the impacts of medication therapy and the immune system's reaction. We demonstrated the positivity and boundedness of the solutions by looking at their existence and uniqueness in order to demonstrate the biological significance of the system. Our approach involves identifying equilibrium points and investigating stability requirements within a range of model parameters in order to characterize the dynamic features of this differential equation model. Additionally, we ran numerical simulations with various parameter values.
To illustrate the memory effect of the fractional derivative, we also simulated the system's dynamic behavior for various orders of fractional derivatives. To put it another way, we came to the conclusion that the chemotherapeutic treatment is quite effective on populations and that the memory effect happens when ϑ, decreases from 1. The purpose of this research is to assist physicians in adopting the appropriate safety measures when diagnosing and treating cancer.

Keywords

Existence-uniqueness Numerical solutions Stability Numerical simulation Fractional order derivative

Supporting Institution

Erciyes Üniversitesi

References

  • [1] Galluzzi, L., Morselli, E., Kepp, O., Vitale, I., Rigoni, A., Vacchelli, E., Kroemer, G. (2010). Mitochondrial gateways to cancer, Molecular aspects of medicine, 31(1), 1-20. DOI:10.1016/j.mam.2009.08.002.
  • [2] Hanahan, D., Weinberg, R. A. (2000). The hallmarks of cancer, cell, 100(1), 57-70. DOI: 10.1016/S0092-8674(00)81683-9.
  • [3] Koudelakova, V., Kneblova, M., Trojanec, R., Drabek, J., Hajduch, M. (2013). Non-small cell lung cancer-genetic predictors, Biomedical Papers of the Medical Faculty of Palacky University in Olomouc, 157(2). DOI: 10.5507/bp.2013.034.
  • [4] Salgia, R., Pharaon, R., Mambetsariev, I., Nam, A., Sattler, M. (2021). The improbable targeted therapy: KRAS as an emerging target in non-small cell lung cancer (NSCLC), Cell Reports Medicine, 2(1). DOI: 10.1016/j.xcrm.2020.100186.
  • [5] Siegel, R. L., Miller, K. D., Jemal, A. (2018). Cancer statistics, 2018, CA: a cancer journal for clinicians, 68(1), 7-30. DOI: 10.3322/caac.21442.
  • [6] El-Gohary, A. (2008). Chaos and optimal control of cancer self-remission and tumor system steady states, Chaos, Solitons & Fractals, 37(5), 1305-1316. DOI: 10.1016/j.chaos.2006.10.060.
  • [7] El-Gohary, A., Alwasel, I. A. (2009). The chaos and optimal control of cancer model with complete unknown parameters, Chaos, Solitons & Fractals, 42(5), 2865-2874. DOI: 10.1016/j.chaos.2009.04.028.
  • [8] Kirschner, D., Panetta, J. C. (1998). Modeling immunotherapy of the tumor–immune interaction, Journal of mathematical biology, 37, 235-252. DOI: 10.1007/s002850050127.
  • [9] Özköse, F., Yılmaz, S., Yavuz, M., Öztürk, İ., Şenel, M. T., Bağcı, B. Ş., Önal, Ö. (2022). A fractional modeling of tumor–immune system interaction related to lung cancer with real data, The European Physical Journal Plus, 137, 1-28. DOI: 10.1140/epjp/s13360-021-02254-6.
  • [10] Özköse, F., Şenel, M. T., Habbireeh, R. (2021). 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. DOI: 10.53391/mmnsa.2021.01.007.
  • [11] López-Alvarenga, J. C., Minzoni-Alessio, A., Olvera-Chávez, A., Cruz-Pacheco, G., Chimal-Eguia, J. C., Hernández-Ruíz, J., Quispe-Siccha, R. M. (2023). A Mathematical Model to Optimize the Neoadjuvant Chemotherapy Treatment Sequence for Triple-Negative Locally Advanced Breast Cancer, Mathematics, 11(11), 2410. DOI: 10.3390/math11112410.
  • [12] Song, G., Liang, G., Tian, T., Zhang, X. (2022). Mathematical modeling and analysis of tumor chemotherapy, Symmetry, 14(4), 704. DOI: 10.3390/sym14040704.
  • [13] Dalir, M., Bashour, M. (2010). Applications of fractional calculus, Applied Mathematical Sciences, 4(21), 1021-1032.
  • [14] Loverro, A. (2004). Fractional calculus: history, definitions and applications for the engineer, Rapport technique, Univeristy of Notre Dame: Department of Aerospace and Mechanical Engineering, 1-28.
  • [15] Podlubny I. (1999). Fractional differential equations, Academic Pres, New York.
  • [16] Oldham, K. B., Spanier, J. (1974). The fractional calculus, Academic Press, New York.
  • [17] Miller, K. S. (1993). An introduction to the fractional calculus and fractional differential equations, John Willey & Sons.
  • [18] Hilfer, R. (2000). Applications of fractional calculus in physics. World scientific. DOI: 10.1142/3779.
  • [19] Öztürk, I., Özköse, F. (2020). Stability analysis of fractional order mathematical model of tumor-immune system interaction, Chaos, Solitons & Fractals, 133, 109614. DOI: 10.1016/j.chaos.2020.109614.
  • [20] Özköse, F., Habbireeh, R., Şenel, M. T. (2023). A novel fractional order model of SARS-CoV-2 and Cholera disease with real data, Journal of Computational and Applied Mathematics, 423, 114969. DOI: 10.1016/j.cam.2022.114969.
  • [21] Yavuz, M., Özköse, F., Susam, M., Kalidass, M. (2023). A new modeling of fractional-order and sensitivity analysis for hepatitis-b disease with real data, Fractal and Fractional, 7(2), 165. DOI: 10.3390/fractalfract7020165.
  • [22] Sabbar, Y., Yavuz, M., Özköse, F. (2022). Infection eradication criterion in a general epidemic model with logistic growth, quarantine strategy, media intrusion, and quadratic perturbation. Mathematics, 10(22), 4213. DOI: 10.3390/math10224213.
  • [23] Özköse, F., Yavuz, M., Şenel, M. T., Habbireeh, R. (2022). Fractional order modelling of omicron SARS-CoV-2 variant containing heart attack effect using real data from the United Kingdom, Chaos, Solitons & Fractals, 157, 111954. DOI: 10.1016/j.chaos.2022.111954.
  • [24] Özköse, F., Yavuz, M. (2022). Investigation of interactions between COVID-19 and diabetes with hereditary traits using real data: A case study in Turkey, Computers in biology and medicine, 141, 105044. DOI: 10.1016/j.compbiomed.2021.105044.
  • [25] Diethelm, K., Freed, A. D. (1998). The FracPECE subroutine for the numerical solution of differential equations of fractional order, Forschung und wissenschaftliches Rechnen, 1999, 57-71.
  • [26] Diethelm, K. (1997). An algorithm for the numerical solution of differential equations of fractional order, Electron. Trans. Numer. Anal, 5(1), 1-6.
  • [27] Garrappa, R. (2010). On linear stability of predictor–corrector algorithms for fractional differential equations, International Journal of Computer Mathematics, 87(10), 2281-2290. DOI: 10.1080/00207160802624331.
  • [28] Garrappa, R. (2018). Numerical solution of fractional differential equations: A survey and a software tutorial, Mathematics, 6(2), 16. DOI: 10.3390/math6020016.
  • [29] Li, C., Tao, C. (2009). On the fractional Adams method, Computers & Mathematics with Applications, 58(8), 1573-1588. DOI: 10.1016/j.camwa.2009.07.050.

Kemoterapi İlaç Etkisi Dahil Tümör-Bağışıklık Sistemi İlişkisinin Kesirli Mertebeden Matematiksel Modellenmesi ve Kararlılık Analizi

Year 2024, Volume: 8 Issue: 2, 85 - 101

Esen Kaya , Fatma Özköse , M. Tamer Senel

https://doi.org/10.29002/asujse.1384833

Abstract

Kanser dünya çapında ikinci en sık ölüm nedenidir. Kemoterapi, özellikle kanserli tümörlerin büyümesinin ve boyutlarının kontrol edilmesinde etkili olan, tümör tedavisinde yaygın olarak kullanılan bir stratejidir. Kemoterapi mekanizmalarının karmaşıklığının daha derinlemesine anlaşılmasını sağlamak için kemoterapi varlığında tümör büyümesini gösteren kesirli dereceli bir matematiksel model oluşturduk. Bu her şeyi kapsayan paradigma, hem ilaç tedavisinin etkilerini hem de bağışıklık sisteminin tepkisini ele alır. Sistemin biyolojik önemini ortaya koymak için çözümlerin varlığına ve tekliğine bakarak çözümlerin pozitifliğini ve sınırlılığını ortaya koyduk. Yaklaşımımız, bu diferansiyel denklem modelinin dinamik özelliklerini karakterize etmek için denge noktalarının belirlenmesini ve bir dizi model parametresi dahilinde stabilite gereksinimlerinin araştırılmasını içerir. Ek olarak çeşitli parametre değerleriyle sayısal simülasyonlar yürüttük.
Kesirli türevin hafıza etkisini göstermek için, ayrıca kesirli türevlerin çeşitli dereceleri için sistemin dinamik davranışını da simüle ettik. Başka bir deyişle, kemoterapötik tedavinin popülasyonlar üzerinde oldukça etkili olduğu ve hafıza etkisinin ϑ, 1'den düştüğünde ortaya çıktığı sonucuna vardık. Bu araştırmanın amacı, kanseri tedavi etmek ve teşhis koyarken uygun güvenlik önlemlerini alma konusunda hekimlere yardımcı olmaktır.

Keywords

Varlık-teklik Sayısal çözümler Kararlılık Sayısal simülasyon Kesirli mertebeden türev

References

  • [1] Galluzzi, L., Morselli, E., Kepp, O., Vitale, I., Rigoni, A., Vacchelli, E., Kroemer, G. (2010). Mitochondrial gateways to cancer, Molecular aspects of medicine, 31(1), 1-20. DOI:10.1016/j.mam.2009.08.002.
  • [2] Hanahan, D., Weinberg, R. A. (2000). The hallmarks of cancer, cell, 100(1), 57-70. DOI: 10.1016/S0092-8674(00)81683-9.
  • [3] Koudelakova, V., Kneblova, M., Trojanec, R., Drabek, J., Hajduch, M. (2013). Non-small cell lung cancer-genetic predictors, Biomedical Papers of the Medical Faculty of Palacky University in Olomouc, 157(2). DOI: 10.5507/bp.2013.034.
  • [4] Salgia, R., Pharaon, R., Mambetsariev, I., Nam, A., Sattler, M. (2021). The improbable targeted therapy: KRAS as an emerging target in non-small cell lung cancer (NSCLC), Cell Reports Medicine, 2(1). DOI: 10.1016/j.xcrm.2020.100186.
  • [5] Siegel, R. L., Miller, K. D., Jemal, A. (2018). Cancer statistics, 2018, CA: a cancer journal for clinicians, 68(1), 7-30. DOI: 10.3322/caac.21442.
  • [6] El-Gohary, A. (2008). Chaos and optimal control of cancer self-remission and tumor system steady states, Chaos, Solitons & Fractals, 37(5), 1305-1316. DOI: 10.1016/j.chaos.2006.10.060.
  • [7] El-Gohary, A., Alwasel, I. A. (2009). The chaos and optimal control of cancer model with complete unknown parameters, Chaos, Solitons & Fractals, 42(5), 2865-2874. DOI: 10.1016/j.chaos.2009.04.028.
  • [8] Kirschner, D., Panetta, J. C. (1998). Modeling immunotherapy of the tumor–immune interaction, Journal of mathematical biology, 37, 235-252. DOI: 10.1007/s002850050127.
  • [9] Özköse, F., Yılmaz, S., Yavuz, M., Öztürk, İ., Şenel, M. T., Bağcı, B. Ş., Önal, Ö. (2022). A fractional modeling of tumor–immune system interaction related to lung cancer with real data, The European Physical Journal Plus, 137, 1-28. DOI: 10.1140/epjp/s13360-021-02254-6.
  • [10] Özköse, F., Şenel, M. T., Habbireeh, R. (2021). 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. DOI: 10.53391/mmnsa.2021.01.007.
  • [11] López-Alvarenga, J. C., Minzoni-Alessio, A., Olvera-Chávez, A., Cruz-Pacheco, G., Chimal-Eguia, J. C., Hernández-Ruíz, J., Quispe-Siccha, R. M. (2023). A Mathematical Model to Optimize the Neoadjuvant Chemotherapy Treatment Sequence for Triple-Negative Locally Advanced Breast Cancer, Mathematics, 11(11), 2410. DOI: 10.3390/math11112410.
  • [12] Song, G., Liang, G., Tian, T., Zhang, X. (2022). Mathematical modeling and analysis of tumor chemotherapy, Symmetry, 14(4), 704. DOI: 10.3390/sym14040704.
  • [13] Dalir, M., Bashour, M. (2010). Applications of fractional calculus, Applied Mathematical Sciences, 4(21), 1021-1032.
  • [14] Loverro, A. (2004). Fractional calculus: history, definitions and applications for the engineer, Rapport technique, Univeristy of Notre Dame: Department of Aerospace and Mechanical Engineering, 1-28.
  • [15] Podlubny I. (1999). Fractional differential equations, Academic Pres, New York.
  • [16] Oldham, K. B., Spanier, J. (1974). The fractional calculus, Academic Press, New York.
  • [17] Miller, K. S. (1993). An introduction to the fractional calculus and fractional differential equations, John Willey & Sons.
  • [18] Hilfer, R. (2000). Applications of fractional calculus in physics. World scientific. DOI: 10.1142/3779.
  • [19] Öztürk, I., Özköse, F. (2020). Stability analysis of fractional order mathematical model of tumor-immune system interaction, Chaos, Solitons & Fractals, 133, 109614. DOI: 10.1016/j.chaos.2020.109614.
  • [20] Özköse, F., Habbireeh, R., Şenel, M. T. (2023). A novel fractional order model of SARS-CoV-2 and Cholera disease with real data, Journal of Computational and Applied Mathematics, 423, 114969. DOI: 10.1016/j.cam.2022.114969.
  • [21] Yavuz, M., Özköse, F., Susam, M., Kalidass, M. (2023). A new modeling of fractional-order and sensitivity analysis for hepatitis-b disease with real data, Fractal and Fractional, 7(2), 165. DOI: 10.3390/fractalfract7020165.
  • [22] Sabbar, Y., Yavuz, M., Özköse, F. (2022). Infection eradication criterion in a general epidemic model with logistic growth, quarantine strategy, media intrusion, and quadratic perturbation. Mathematics, 10(22), 4213. DOI: 10.3390/math10224213.
  • [23] Özköse, F., Yavuz, M., Şenel, M. T., Habbireeh, R. (2022). Fractional order modelling of omicron SARS-CoV-2 variant containing heart attack effect using real data from the United Kingdom, Chaos, Solitons & Fractals, 157, 111954. DOI: 10.1016/j.chaos.2022.111954.
  • [24] Özköse, F., Yavuz, M. (2022). Investigation of interactions between COVID-19 and diabetes with hereditary traits using real data: A case study in Turkey, Computers in biology and medicine, 141, 105044. DOI: 10.1016/j.compbiomed.2021.105044.
  • [25] Diethelm, K., Freed, A. D. (1998). The FracPECE subroutine for the numerical solution of differential equations of fractional order, Forschung und wissenschaftliches Rechnen, 1999, 57-71.
  • [26] Diethelm, K. (1997). An algorithm for the numerical solution of differential equations of fractional order, Electron. Trans. Numer. Anal, 5(1), 1-6.
  • [27] Garrappa, R. (2010). On linear stability of predictor–corrector algorithms for fractional differential equations, International Journal of Computer Mathematics, 87(10), 2281-2290. DOI: 10.1080/00207160802624331.
  • [28] Garrappa, R. (2018). Numerical solution of fractional differential equations: A survey and a software tutorial, Mathematics, 6(2), 16. DOI: 10.3390/math6020016.
  • [29] Li, C., Tao, C. (2009). On the fractional Adams method, Computers & Mathematics with Applications, 58(8), 1573-1588. DOI: 10.1016/j.camwa.2009.07.050.
There are 29 citations in total.

Details

Primary Language English
Subjects Numerical Analysis
Journal Section Research Article
Authors

Esen Kaya 0000-0002-4613-1722

Fatma Özköse 0000-0002-7021-8342

M. Tamer Senel 0000-0003-1915-5697

Publication Date
Submission Date November 1, 2023
Acceptance Date November 1, 2024
Published in Issue Year 2024Volume: 8 Issue: 2

Cite

APA Kaya, E., Özköse, F., & Senel, M. T. (n.d.). Fractional-Order Mathematical Modeling and Stability Analysis of Tumor-Immune System Relation Including Chemotherapy Drug Effect. Aksaray University Journal of Science and Engineering, 8(2), 85-101. https://doi.org/10.29002/asujse.1384833

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