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Fractional model of brain tumor with chemo-radiotherapy treatment

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Abstract

Fractional calculus is recognized as a technique with many uses, along with studying biological systems. This article frames the mathematical model for the nonlinear fractional differential equations system involving caputo fractional derivative for Chemo-Radiation therapy of a brain tumor. The system is investigated for the model’s stability analysis, existence, and uniqueness. The impact of a fractional differential equation on the analysis of the described model is examined by utilizing Caputo Fractional operator. Stability analysis is discussed under three categories: without any therapy, with chemotherapy, and with chemo-radiotherapy treatment. However, numerical simulations have been utilized to investigate the model on fractional order derivative. The graphs have been displayed for the three treatments using various values for the fractional order. This analysis suggests that combination therapy could lead to tremendous success in treating gliomas.

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References

  1. Cooper, G.M. (ed.): The Development and Causes of Cancer. The Cell: A Molecular Approach. Sinauer Associates, Sunderland (2000)

    Google Scholar 

  2. Peiffer, J., Kleihues, P., Scherer, H.J.: Pioneer in glioma research. Brain Pathol. 9, 241–245 (1999). https://doi.org/10.1111/j.1750-3639.1999.tb00222.x

    Article  Google Scholar 

  3. Forst, D.A., et al.: Low-grade gliomas. Oncologist 19, 403–413 (2014). https://doi.org/10.1634/theoncologist.2013-0345

    Article  Google Scholar 

  4. Bondiau, P.Y., et al.: Biocomputing: numerical simulation of glioblastoma growth using diffusion tensor imaging. Phys. Med. Biol. 53, 879–893 (2008). https://doi.org/10.1088/0031-9155/53/4/004

    Article  Google Scholar 

  5. Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J. Neuropathol. Exp. Neurol. 66, 1–9 (2007). https://doi.org/10.1097/nen.0b013e31802d9000

    Article  Google Scholar 

  6. Badoua, M., et al.: Oedema-based model for diffuse low-grade gliomas: application to clinical cases under radiotherapy. Cell Prolif. 47, 369–380 (2014). https://doi.org/10.1111/cpr.12114

    Article  Google Scholar 

  7. Shua, Y., Huanga, J., Donga, Y., Takeuchib, Y.: Mathematical modeling and bifurcation analysis of pro and anti-tumor macrophages. Appl. Math. Model. 88, 758–773 (2020). https://doi.org/10.1016/j.apm.2020.06.042

    Article  MathSciNet  Google Scholar 

  8. Panga, L., Liub, S., Zhangc, X., Tian, T.: Mathematical modeling and analysis of tumor-volume variation during radiotherapy. Appl. Math. Model. 89, 1074–1089 (2021). https://doi.org/10.1016/j.apm.2020.07.028

    Article  MathSciNet  Google Scholar 

  9. Duan, W.L., Fang, H., Zeng, C.: The stability analysis of tumor-immune responses to chemotherapy system with Gaussian white noises. Chaos Solitons Fract. 127, 96–102 (2019). https://doi.org/10.1016/j.chaos.2019.06.030

    Article  MathSciNet  Google Scholar 

  10. Rockne, R., Alvord, E.C., Rockhill, J.K., Swanson, K.R.: A mathematical model for brain tumor response to radiation therapy. J. Math. Biol. 58, 561–578 (2009). https://doi.org/10.1007/s00285-008-0219-6

    Article  MathSciNet  MATH  Google Scholar 

  11. Pinho, S., Freedman, S.H., Nani, F.: Chemotherapy model for the treatment of cancer with metastasis. Math. Comput. Model. 36, 77–803 (2002). https://doi.org/10.1016/S0895-7177(02)00227-3

    Article  MathSciNet  MATH  Google Scholar 

  12. Ledzewicz, U., Naghnaeian, M., Schättleri, H.: Optimal response to chemotherapy for a mathematical model of tumor-immune dynamics. Math. Comput. Model. 64, 557–77 (2012). https://doi.org/10.1007/s00285-011-0424-6

    Article  MathSciNet  MATH  Google Scholar 

  13. Ghaffari, A., Bahmaie, B., Nazari, M.: A mixed radiotherapy and chemotherapy model for treatment of cancer with metastasis. Math. Comput. Model. 39, 4603–17 (2016). https://doi.org/10.1002/mma.3887

    Article  MathSciNet  Google Scholar 

  14. Liuand, Z., Yang, C., Nazari, M.: A mathematical model of cancer treatment by radiotherapy followed by chemotherapy. Math. Comput. Simul. 124, 1–15 (2016). https://doi.org/10.1016/j.matcom.2015.12.007

    Article  MathSciNet  Google Scholar 

  15. Barazzuol, L., Burnet, N.G., Jones, B., Jefferies, S.J., Kirby, N.F.: A mathematical model of brain tumour response to radiotherapy and chemotherapy considering radiobiological aspects. J. Theor. Boil. 262, 553–65 (2010). https://doi.org/10.1016/j.jtbi.2009.10.021

    Article  MATH  Google Scholar 

  16. Pinho, S.T.R., Barcelar, F.S., Andrade, R.F.S., Freedman, H.I.: A mathematical model for the effect of anti-angiogenic therapy in the treatment of cancer tumors by chemotherapy. Nonlinear Anal. Real World Appl. 14, 815–828 (2013). https://doi.org/10.1016/j.nonrwa.2012.07.034

    Article  MathSciNet  MATH  Google Scholar 

  17. Spratt, J.S., Spratt, T.L.: Rates of growth of pulmonary metastases and host survival. Ann. Surg. 159, 161–171 (1964). https://doi.org/10.1097/00000658-196402000-00001

    Article  Google Scholar 

  18. Borges, F.S., et al.: Model for tumor growth with treatment by continuous and pulsed chemotherapy. Biosystems 116, 43–48 (2014). https://doi.org/10.1016/j.biosystems.2013.12.001

    Article  Google Scholar 

  19. Iarosz, K.C., et al.: Mathematical model of brain tumor with glia-neuron interactions and chemotherapy treatment. J. Theor. Biol. 368, 113–121 (2015). https://doi.org/10.1016/j.jtbi.2015.01.006

    Article  MATH  Google Scholar 

  20. Nass, T., Efferth, T.: Drug targets and resistance mechanisms in myeloma. Cancer Drug Resist. 1, 87–117 (2018). https://doi.org/10.20517/cdr.2018.04

    Article  Google Scholar 

  21. Sun, X., Bao, J., Shoa, Y.: Mathematical modeling of therapy-induced cancer drug resistance: connecting cancer mechanisms to population survival rates. Sci. Rep. 6, 22498 (2016). https://doi.org/10.1038/srep22498

    Article  Google Scholar 

  22. Ionescu, C., Lopes, A., Copot, D., Machado, J.H.T., Bates, J.H.T.: The role of fractional calculus in modelling biological phenomena: a review. Commun. Nonlinear Sci. Numer. Simul. 51, 141–159 (2017). https://doi.org/10.1016/j.cnsns.2017.04.001

    Article  MathSciNet  MATH  Google Scholar 

  23. Tarasov, V.E.: Review of some promising fractional physical models. Int. J. Mod. Phys. B 27(09), 1330005 (2013). https://doi.org/10.1142/S0217979213300053

    Article  MathSciNet  MATH  Google Scholar 

  24. Hassani, H., Avazzadeh, Z., Tenreiro Machado, J.A., Agarwal, P., Bakhtiar, M.: Optimal solution of a fractional HIV/AIDS epidemic mathematical model. J. Comput. Biol. 29(3), 276–291 (2022). https://doi.org/10.1089/cmb.2021.0253

    Article  MathSciNet  Google Scholar 

  25. Singh, R., Rehman, A.U., Masud, M., Alhumyani, H.A., Mahajan, S., Pandit, A.K., Agarwal, P.: Fractional order modeling and analysis of dynamics of stem cell differentiation in complex network. AIMS Math. 7(4), 5175–5198 (2022). https://doi.org/10.3934/math.2022289

    Article  MathSciNet  Google Scholar 

  26. Wang, F.Z., Khan, M.N., Ahmad, I., Ahmad, H., Abu-Zinadah, H., Chu, Y.M.: Numerical solution of traveling waves in chemical kinetics: time-fractional fishers equations. Fractals 30(2), 22400051 (2022). https://doi.org/10.1142/S0218348X22400515

    Article  MATH  Google Scholar 

  27. Rashid, S., Sultana, S., Karaca, Y., Khalid, A., Chu, Y.M.: Some further extensions considering discrete proportional fractional operators. Fractals 30(1), 2240026 (2022). https://doi.org/10.1142/S0218348X22400266

    Article  MATH  Google Scholar 

  28. Ganji, R.M., Jafari, H., Moshokoa, S.P., Nkomo, N.S.: A mathematical model and numerical solution for brain tumor derived using fractional operator. Results Phys. 28, 104671 (2021). https://doi.org/10.1016/j.rinp.2021.104671

    Article  Google Scholar 

  29. Ghaziani, R.K., Alidousti, J., Eshkaftaki, A.B.: Stability and dynamics of a fractional order Leslie–Gower Prey–Predator model. Appl Math Modell. 40, 2075–86 (2019). https://doi.org/10.1016/j.apm.2015.09.014

    Article  MathSciNet  MATH  Google Scholar 

  30. Majee, S., Adak, S., Jana, S., et al.: Complex dynamics of a fractional-order sir system in the context of COVID-19. J. Appl. Math. Comput. 68, 4051–4074 (2022). https://doi.org/10.1007/s12190-021-01681-z

    Article  MathSciNet  MATH  Google Scholar 

  31. Avazzadeh, Z., Hassani, H., Ebadi, M.J., et al.: Optimal approximation of fractional order brain tumor model using generalized Laguerre polynomials. Iran J. Sci. 47, 501–513 (2023). https://doi.org/10.1007/s40995-022-01388-1

    Article  MathSciNet  Google Scholar 

  32. Maayah, B., Arqub, O.A., Alnabulsi, S., Alsulami, H.: Numerical solutions and geometric attractors of a fractional model of the cancer-immune based on the Atangana-Baleanu-Caputo derivative and the reproducing kernel scheme. Chin. J. Phys. 80, 463–483 (2022). https://doi.org/10.1016/j.cjph.2022.10.002

    Article  MathSciNet  Google Scholar 

  33. Abu Arqub, O., Alsulami, H., Alhodaly, M.: Numerical Hilbert space solution of fractional Sobolev equation in (1+1)-dimensional space. Math. Sci. 16, 1–12 (2022). https://doi.org/10.1007/s40096-022-00495-9

    Article  MathSciNet  Google Scholar 

  34. Arqub, O.A., Maayah, B.: Adaptive the Dirichlet model of mobile/immobile advection/dispersion in a time-fractional sense with the reproducing kernel computational approach: formulations and approximations. Int. J. Mod. Phys. B 37(18), 2350179 (2023). https://doi.org/10.1142/S0217979223501795

    Article  Google Scholar 

  35. Maayah, B., Moussaoui, A., Bushnaq, S., Arqub, A.: The multistep Laplace optimized decomposition method for solving fractional-order coronavirus disease model (COVID-19) via the Caputo fractional approach. Demonstr. Math. 55(1), 963–977 (2022). https://doi.org/10.1515/dema-2022-0183

    Article  MathSciNet  MATH  Google Scholar 

  36. Maayah, B., Moussaoui, A., Bushnaq, S., Arqub, A.: The extended Laplace transform method for mathematical analysis of the Thomas–Fermi equation. Chin. J. Phys. 55(6), 2548–2558 (2017). https://doi.org/10.1016/j.cjph.2017.10.001

    Article  MathSciNet  Google Scholar 

  37. Ali, R., Ghosh, U.N., Mandi, L., et al.: Application of Adomian decomposition method to study collision effect in dusty plasma in the presence of polarization force. Indian J. Phys. 97, 2209–2216 (2023). https://doi.org/10.1007/s12648-023-02588-0

    Article  Google Scholar 

  38. Fatoorehchi, H., Djilali, S.: Stability analysis of linear time-invariant dynamic systems using the matrix sign function and the Adomian decomposition method. Int. J. Dyn. Control. 11, 593–604 (2023). https://doi.org/10.1007/s40435-022-00989-3

    Article  Google Scholar 

  39. Fatoorehchi, H., Rach, R., Sakhaeinia, H.: Explicit Frost–Kalkwarf type equations for calculation of vapour pressure of liquids from triple to critical point by the Adomian decomposition method. Can. J. Chem. Eng. 95, 2199–2208 (2017). https://doi.org/10.1002/cjce.22853

    Article  Google Scholar 

  40. Fatoorehchi, H., Alidadi, M., Rach, R., Shojaeian, A.: Theoretical and experimental investigation of thermal dynamics of Steinhart–Hart negative temperature coefficient thermistors. ASME J. Heat Transf. 141(7), 072003 (2019). https://doi.org/10.1115/1.4043676

    Article  Google Scholar 

  41. Duan, N., Sun, K.: Power system simulation using the multistage Adomian decomposition method. IEEE Trans. Power Syst. 32(1), 430–441 (2017). https://doi.org/10.1109/TPWRS.2016.2551688

    Article  Google Scholar 

  42. Hasan, B., Mahmut, E., Vedat, A., Roza, H.B.: Numerical solution of a viscous incompressible flow problem through an orifice by Adomian decomposition method, applied mathematics and computation. Appl. Math. Comput. 153(3), 733–741 (2004). https://doi.org/10.1016/S0096-3003(03)00667-2

    Article  MathSciNet  MATH  Google Scholar 

  43. Jose, T., Kun, T., Antonio, M.B., Celso, G.: Mathematical model of brain tumor growth with drug resistance. Commun. Nonlinear Sci. Numer. Simul. 103, 106013 (2021). https://doi.org/10.1016/j.cnsns.2021.106013

    Article  MATH  Google Scholar 

  44. Simbawa, E., Al-Johani, N., Al-Tuwairqi, S.: Modeling the spatiotemporal dynamics of oncolytic viruses and radiotherapy as a treatment for cancer. Comput. Math. Methods Med. 358, 3642654 (2020). https://doi.org/10.1155/2020/3642654

    Article  MATH  Google Scholar 

  45. El-Sayed, A.M.A., El-Mesiry, A.E.M., El-Saka, H.A.A.: On the fractional- order logistic equation. Appl. Math. Lett. 20, 817–823 (2007). https://doi.org/10.1016/j.aml.2006.08.013

    Article  MathSciNet  MATH  Google Scholar 

  46. Diethelm, K., Freed, A.D.: The FracPECE subroutine for the numerical solution of differential equations of fractional order. Forsch. Wiss. Rechn. 1999, 57–71 (1998)

    Google Scholar 

  47. Diethelm, K.: An algorithm for the numerical solution of differential equations of fractional order. Electron. Trans. Numer. Anal. 5(1), 1–6 (1997)

    MathSciNet  MATH  Google Scholar 

  48. Garrappa, R.: Numerical solution of fractional differential equations: a survey and a software tutorial. Mathematics 6(2), 16 (2018). https://doi.org/10.3390/math6020016

    Article  MATH  Google Scholar 

  49. Li, C., Tao, C.: On the fractional Adams method. Comput. Math. Appl. 58(8), 1573–1588 (2009). https://doi.org/10.1016/j.camwa.2009.07.050

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Coimbatore, Tamil Nadu, 641020, India

    S. Sujitha & T. Jayakumar

  2. Department of Science and Humanities, Sri Krishna College of Technology, Coimbatore, Tamil Nadu, 641042, India

    D. Maheskumar

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  1. S. Sujitha
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Sujitha, S., Jayakumar, T. & Maheskumar, D. Fractional model of brain tumor with chemo-radiotherapy treatment. J. Appl. Math. Comput. 69, 3793–3818 (2023). https://doi.org/10.1007/s12190-023-01901-8

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