Modified Swanson’s Equation to Detect the Growth of Glioblastomas Multiforme (GBM) Tumour
Abstract
The concentration of glioblastomas multiforme (GBM) tumour equation is numerically solved in terms of net rates of proliferation and invasion. The GBM tumour cells concentration evolution is known as a reaction-diffusion process and the diffusion coefficients differ according to the brain’s white and grey matter composition. An Integer order non-linear equation can produce significant errors as most of the method used to solve the equation require linearization, discretization and perturbation which involved more computational work. Hence, a non-linear diffusion logistic density model proposed by Özuğurlu (2015) is modified to non-linear fractional-order partial differential equation (FPDE) that described in the Caputo sense. This study presents another alternative to investigate the GBM tumour growth using the theory of fractional calculus (FC). Q-homotopy analysis transform method (q-HATM) and Laplace Adomian decomposition method (LADM) were applied to solve the proposed model. From the results obtained, the fractional order (α) has greater flexibility which allows more degree of freedom in design and analysis. Results derived are in more accurate manner. Q-HATM is more efficient as it proposes a modest way to adjust the stability and the convergence region of the solution using an auxiliary parameter (h) and asymptotic parameter (n≥1).
References
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Veeresha, P., Prakasha, D. G., & Baskonus, H. M. (2019). New numerical surfaces to the mathematical model of cancer chemotherapy effect in Caputo fractional derivatives. Chaos: An Interdisciplinary Journal of Non-linear Science, 29(1), 013119. doi:10.1063/1.5074099
Veeresha, P., Prakasha, D. G., & Baskonus, H. M. (2019). Solving smoking epidemic model of fractional-order using a modified homotopy analysis transform method. Mathematical Sciences. doi:10.1007/s40096-019-0284-6
Zhang, J., Zhang, X., & Yang, B. (2018). An approximation scheme for the time fractional convection–diffusion equation. Applied Mathematics and Computation, 335, 305–312. doi:10.1016/j.amc.2018.04.019
A. Prakash, M. Goyal and S. Gupta, “Q-homotopy analysis method for fractional Bloch model arising in nuclear magnetic resonance via the Laplace transform”, Indian Journal of Physics. 2019.
Abdelmajid, B., Omar, B., Mohamed, Y. & Abdelhadi, R. (2018). Brain tumor temperature extraction from MRI imaging using bioheat equation. Procedia Computer Science, 127, 336-343.
Atangana, A., & Bildik, N. (2013). Approximate Solution of Tuberculosis Disease Population Dynamics Model. Abstract and Applied Analysis, 2013, 1–8.
Bozkurt, F. (2014). Mathematical modeling and stability analysis of the brain tumor glioblastoma multiforme (GBM). International Journal of Modeling and Optimization, 4(4). doi:10.7763/IJMO.2014.V4.383
D. Kumar, J. Singh, A. Prakash and R. Swroop, “Numerical simulation for system of time-fractional linear and non-linear differential equations”, Progress in Fractional Differentiation and Application. vol. 5, no 1, pp. 65–77, 2019.
Fu, H., & Wang, H. (2017). A preconditioned fast finite difference method for space-time fractional partial differential equations. Fractional Calculus and Applied Analysis, 20(1). doi:10.1515/fca-2017-0005
Gaxiola, O. G. & Jaquez, R. B. (2016). Applying Adomian decomposition method to solve Burgess equation with a non-linear source. International Journal of Applied and Computational Mathematics, 3(1), 213-224. doi: 10.13140/RG.2.1.3758.9203
J. Manimaran, L. Shangerganesh, A. Debbouche and V. Antonov, “Numerical solutions for time-fractional cancer invasion system with nonlocal diffusion”, The European Physical Journal Plus. vol. 7, 2019.
Jaroudi, R. (2017). Inverse mathematical models for brain tumour growth. Journal Department of Science and Technology, 1(1787), 19-28. Retrieved February 27, 2019 from https://liu.diva-portal.org/smash/get/diva2:1149624/FULLTEXT01.pdf
Kumar, D., Chaudhary, S., & Srinivas Kumar, V. V. K. (2019). Fractional Crank–Nicolson–Galerkin finite element scheme for the time‐fractional non-linear diffusion equation. Numerical Methods for Partial Differential Equations. doi:10.1002/num.22399
Landrove, M. M. (2017). Reaction-diffusion models for glioma tumor growth. Medical Physics. Retrieved February 19, 2019 from https://arxiv.org/abs/1707.09409v1
Murray, J. D. (2003). Basic mathematical model of glioma growth and invasion. In S.S, Antman, J.E, Marsden, L, Sirovich, & S, Wiggins (Ed.) Mathematical biology ii: spatial models and biomedical applications (interdisciplinary applied mathematics (18)) (3rd ed., pp. 542-545). Berlin, Heidelberg: Springer-Verlag.
Murray, J. D. (2012). Glioblastoma brain tumours: estimating the time from brain tumour initiation and resolution of a patient survival anomaly after similar treatment protocol. Journal of Biology Dynamics, 6(sup 2), 118-127. doi:10.1080/17513758.2012.678392
Mustaf, M., Sali, A. F., Illzam, E. M., Sharifa, A. M., & Nang, M. K. (2018). Brain cancer: Current concepts, diagnosis and prognosis. Journal of Dental and Medical Sciences, 17(3), 41-46. doi:10.9790/0853-1703084146
Özuğurlu, E. (2015). A note on the numerical approach for the reaction–diffusion problem to model the density of the tumor growth dynamics. Computers & Mathematics with Applications, 69(12), 1504–1517. doi:10.1016/j.camwa.2015.04.018
Rothman, J. (2018, March 05). Brain Tumors: Who Gets Them and What is the Survival Rate? Retrieved December 1, 2018, from https://www.everydayhealth.com/brain-tumor/brain-tumor-statistics.aspx
Saad, K. M., & Al-Sharif, E. H. M. (2017). Analytical study for time and time-space fractional burgers’ equation. Advances in difference equations. SpringerOpen Journal, 300. doi:10.1186/s13662-017-1358-0
Swanson, K. R., Alvord, E. C., & Murray, J. D. (2000). A quantitative model for differential motility of gliomas in grey and white matter. Cell Proliferation, 33(5), 317–329. doi:10.1046/j.1365-2184.2000.00177.x
Szeto, M. D., Chakraborty, G., Hadley, J., Rockne, R., Muzi, M., Alvord, E. C., … Swanson, K. R. (2009). Quantitative Metrics of Net Proliferation and Invasion Link Biological Aggressiveness Assessed by MRI with Hypoxia Assessed by FMISO-PET in Newly Diagnosed Glioblastomas. Cancer Research, 69(10), 4502–4509. doi:10.1158/0008-5472.can-08-3884
Takeuchi, Y., Yoshimoto, Y., & Suda, R. (2017). Second order accuracy finite difference methods for space-fractional partial differential equations. Journal of Computational and Applied Mathematics, 320, 101–119. doi:10.1016/j.cam.2017.01.013
Troparevsky, M. I., Seminara, S. A., & Fabio, M. A. (2019). A Review on Fractional Differential Equations and a Numerical Method to Solve Some Boundary Value Problems. Non-linear Systems-Theoretical Aspects and Recent Applications. doi: 10.5772/intechopen.86273
Veeresha, P., Prakasha, D. G., & Baskonus, H. M. (2019). New numerical surfaces to the mathematical model of cancer chemotherapy effect in Caputo fractional derivatives. Chaos: An Interdisciplinary Journal of Non-linear Science, 29(1), 013119. doi:10.1063/1.5074099
Veeresha, P., Prakasha, D. G., & Baskonus, H. M. (2019). Solving smoking epidemic model of fractional-order using a modified homotopy analysis transform method. Mathematical Sciences. doi:10.1007/s40096-019-0284-6
Zhang, J., Zhang, X., & Yang, B. (2018). An approximation scheme for the time fractional convection–diffusion equation. Applied Mathematics and Computation, 335, 305–312. doi:10.1016/j.amc.2018.04.019
Published
2021-03-01
How to Cite
MOHD, Fashareena; SULAIMAN, Hanifah; ALIAS, NorAzlina.
Modified Swanson’s Equation to Detect the Growth of Glioblastomas Multiforme (GBM) Tumour.
International Journal of Advanced Research in Engineering Innovation, [S.l.], v. 3, n. 1, p. 1-18, mar. 2021.
Available at: <https://myjms.mohe.gov.my/index.php/ijarei/article/view/12441>. Date accessed: 14 sep. 2024.
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