A fractional mathematical model approach on glioblastoma growth: tumor visibility timing and patient survival

Year 2024, Volume: 4 Issue: 1, 66 - 85, 31.03.2024

Nurdan Kar , Nuri Özalp

https://doi.org/10.53391/mmnsa.1438916

Cited By: 1

Abstract

In this paper, we introduce a mathematical model given by
\begin{equation}
{ }^c \mathfrak{D}_t^\alpha u = \nabla \cdot \mathrm{D} \nabla u + \rho f(u) \quad \text{in } \Omega,
\end{equation}
where $f(u)=\frac{1}{1-u/\mathrm{K}}, \, u/\mathrm{K} \neq 1, \, \mathrm{K} > 0$, to enhance established mathematical methodologies for better understanding glioblastoma dynamics at the macroscopic scale. The tumor growth model exhibits an innovative structure even within the conventional framework, including a proliferation term, $f(u)$, presented in a different form compared to existing macroscopic glioblastoma models. Moreover, it represents a further refined model by incorporating a calibration criterion based on the integration of a fractional derivative, $\alpha$, which differs from the existing models for glioblastoma. Throughout this study, we initially discuss the modeling dynamics of the tumor growth model. Given the frequent recurrence observed in glioblastoma cases, we then track tumor mass formation and provide predictions for tumor visibility timing on medical imaging to elucidate the recurrence periods. Furthermore, we investigate the correlation between tumor growth speed and survival duration to uncover the relationship between these two variables through an experimental approach. To conduct these patient-specific analyses, we employ glioblastoma patient data and present the results via numerical simulations. In conclusion, the findings on tumor visibility timing align with empirical observations, and the investigations into patient survival further corroborate the well-established inter-patient variability for glioblastoma cases.

Keywords

Glioblastoma, Tumor visibility, Survival, Fractional mathematical model, recurrence

Supporting Institution

Scientific and Technological Research Council of Turkiye (TUBITAK)

Project Number

E-21514107-115.99-301087

References

• [1] Yabo, Y.A., Niclou, S.P. and Golebiewska, A. Cancer cell heterogeneity and plasticity: a paradigm shift in glioblastoma. Neuro-Oncology, 24(5), 669-682, (2022).
• [2] Louis, D.N., Perry, A., Wesseling, P., Brat, D.J., Cree, I.A., Figarella-Branger, D. et al. The 2021 WHO classification of tumors of the central nervous system: a summary. Neuro-Oncology, 23(8), 1231-1251, (2021).
• [3] Carlsson, S.K., Brothers, S.P. and Wahlestedt, C. Emerging treatment strategies for glioblastoma multiforme. EMBO Molecular Medicine, 6(11), 1359-1370, (2014).
• [4] Kelley, P.J. and Hunt, C. The limited value of cytoreductive surgery in elderly patients with malignant gliomas. Neurosurgery, 34(1), 62-67, (1994).
• [5] Piroth, M.D., Pinkawa, M., Holy, R., Klotz, J., Schaar, S., Stoffels, G. et al. Integrated boost IMRT with FET-PET-adapted local dose escalation in glioblastomas. Strahlentherapie und Onkologie, 188(4), 334-339, (2012).
• [6] Silbergeld, D.L. and Chicoine, M.R. Isolation and characterization of human malignant glioma cells from histologically normal brain. Journal of Neurosurgery, 86(3), 525-531, (1997).
• [7] Harpold, H.L.P., Alvord Jr, E.C. and Swanson, K.R. The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology & Experimental Neurology, 66(1), 1-9, (2007).
• [8] Pandya, N.M., Dhalla, N.S. and Santani, D.D. Angiogenesis-a new target for future therapy. Vascular Pharmacology, 44(5), 265-274, (2006).
• [9] Das, S. Functional Fractional Calculus for System Identification and Controls. Springer: New York, (2008).
• [10] Curtin, L., Whitmire, P., White, H., Bond, K.M., Mrugala, M.M., Hu, L.S. et al. Shape matters: morphological metrics of glioblastoma imaging abnormalities as biomarkers of prognosis. Scientific Reports, 11(1), 23202, (2021).
• [11] Konukoglu, E., Clatz, O., Bondiau, P.Y., Delingette, H. and Ayache, N. Extrapolating glioma invasion margin in brain magnetic resonance images: suggesting new irradiation margins. Medical Image Analysis, 14(2), 111-125, (2010).
• [12] Burgess, P.K., Kulesa, P.M., Murray, J.D. and Alvord Jr., E.C. The interaction of growth rates and diffusion coefficients in a three-dimensional mathematical model of gliomas. Journal of Neuropathology & Experimental Neurology, 56(6), 704-713, (1997).
• [13] Jacobs, J., Rockne, R.C., Hawkins-Daarud, A.J., Jackson, P.R., Johnston, S.K., Kinahan, P. et al. Improved model prediction of glioma growth utilizing tissue-specific boundary effects. Mathematical Biosciences, 312, 59-66, (2019).
• [14] Jbabdi, S., Mandonnet, E., Duffau, H., Capelle, L., Swanson, K.R., Pelegrini-Issac, M. et al. Simulation of anisotropic growth of low-grade gliomas using diffusion tensor imaging. Magnetic Resonance in Medicine, 54(3), 616-624, (2005).
• [15] Konukoglu, E., Clatz, O., Menze, B.H., Stieltjes, B., Weber, M.A., Mandonnet, E. et al. Image guided personalization of reaction-diffusion type tumor growth models using modified anisotropic eikonal equations. IEEE Transactions on Medical Imaging, 29(1), 77-95, (2010).
• [16] Le, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J. et al. MRI based Bayesian personalization of a tumor growth model. IEEE Transactions on Medical Imaging, 35(10), 2329-2339, (2016).
• [17] Lipkova, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C. et al. Personalized radiotherapy design for glioblastoma: integrating mathematical tumor models, multimodal scans, and Bayesian inference. IEEE Transactions on Medical Imaging, 38(8), 1875-1884, (2019).
• [18] Rockne, R., Alvord Jr, E.C., Reed, P.J. and Swanson, K.R. Modeling the growth and invasion of gliomas, from simple to complex: the goldie locks paradigm. Biophysical Reviews and Letters, 03(01n02), 111-123, (2008).
• [19] Rockne, R., Alvord Jr, E.C., Rockhill, J.K. and Swanson, K.R. A mathematical model for brain tumor response to radiation therapy. Journal of Mathematical Biology, 58, 561-578, (2009).
• [20] Rockne, R., Rockhill, J.K., Mrugala, M., Spence, A.M., Kalet, I., Hendrickson, K. et al. Predicting the efficacy of radiotherapy in individual glioblastoma patients in vivo: a mathematical modeling approach. Physics in Medicine and Biology, 55(12), 3271-3285, (2010).
• [21] Swanson, K.R, Alvord Jr, E.C. and Murray, J.D. A quantitative model for differential motility of gliomas in grey and white matter. Cell Proliferation, 33(5), 317-329, (2000).
• [22] Swanson, K.R., Bridge, C., Murray, J.D. and Alvord Jr, E.C. Virtual and real brain tumors: using mathematical modeling to quantify glioma growth and invasion. Journal of the Neurological Sciences, 216(1), 1-10, (2003).
• [23] Tracqui, P., Cruywagen, G.C., Woodward, D.E., Bartoo, G.T., Murray, J.D. and Alvord Jr, E.C. A mathematical model of glioma growth: the effect of chemotherapy on spatio-temporal growth. Cell Proliferation, 28(1), 17-31, (1995).
• [24] Woodward, D.E., Cook, J., Tracqui, P., Cruywagen, G.C., Murray, J.D. and Alvord Jr, E.C. A mathematical model of glioma growth: the effect of extent of surgical resection. Cell Proliferation, 29(6), 269-288, (1996).
• [25] Ucar, E., Ozdemir, N. and Altun, E. Fractional order model of immune cells influenced by cancer cells. Mathematical Modelling of Natural Phenomena, 14(3), 308, (2019).
• [26] Ozdemir, N., Ucar, S. and Eroglu, B.B.I. Dynamical analysis of fractional order model for computer virus propagation with kill signals. International Journal of Nonlinear Sciences and Numerical Simulation, 21(3-4), 239-247, (2020).
• [27] Ozkose, F., Yilmaz, S., Yavuz, M., Ozturk, I., Senel, M.T., Bagci, B.S. et al. A fractional modeling of tumor–immune system interaction related to Lung cancer with real data. The European Physical Journal Plus, 137, 1-28, (2022).
• [28] Rahman, M., Arfan, M. and Baleanu, D. Piecewise fractional analysis of the migration effect in plant-pathogen-herbivore interactions. Bulletin of Biomathematics, 1(1), 1-23, (2023).
• [29] Mustapha, U.T., Ado, A., Yusuf, A., Qureshi, S. and Musa, S.S. Mathematical dynamics for HIV infections with public awareness and viral load detectability. Mathematical Modelling and Numerical Simulation with Applications, 3(3), 256-280, (2023).
• [30] Kawarada, H. On solutions of Initial-Boundary Problem for ut = uxx + 1/(1 − u). Publications of the Research Institute for Mathematical Sciences, 10(3), 729-736, (1975).
• [31] Acker, A. and Walter, W. On the global existence of solutions of parabolic differential equations with a singular nonlinear term. Nonlinear Analysis, Theory, Methods and Applications, 2(4), 499- 504, (1978).
• [32] Ke, L. and Ning, S. Quenching for degenerate parabolic equations. Nonlinear Analysis, 34, 1123-1135, (1998).
• [33] Levine, H.A. and Montgomery, J.T. The quenching of solutions of some nonlinear parabolic equations. SIAM Journal on Mathematical Analysis, 11(5), 842-847, (1980).
• [34] Levine, H.A. Levine, The phenomenon of quenching: a survey Trends in the theory and practice of nonlinear analysis. In Proceedings, Sixth International Conference on Document Analysis and Recognition (Vol. 110) pp. 275-286, North-Holland, Amsterdam, (1985).
• [35] Levine, H.A. Quenching, nonquenching, and beyond quenching for solution of some parabolic equations. Annali di Matematica Pura ed Applicata, 155, 243–260, (1989).
• [36] Deng, K. and Levine, H.A. On the blow up of ut at quenching. In Proceedings American Mathematical Society, pp. 1049-1056, 106(4), (1989, August).
• [37] Guo, J.S. On the quenching behavior of the solution of a semilinear parabolic equation. Journal of Mathematical Analysis and Applications, 151, 58-79, (1990).
• [38] de Pablo, A., Quiros, F. and Rossi, J.D. Nonsimultaneous quenching. Applied Mathematics Letters, 15(3), 265-269, (2002).
• [39] Ji, R., Zhou, S. and Zheng, S. Quenching behavior of solutions in coupled heat equations with singular multi-nonlinearities. Applied Mathematics and Computation, 223, 401-410, (2013).
• [40] Jia, Z., Yang, Z. and Wang, C. Non-simultaneous quenching in a semilinear parabolic system with multi-singular reaction terms. Electronic Journal of Differential Equations, 2019(100), 1-13, (2019).
• [41] Mu, C., Zhou, S. and Liu, D. Quenching for a reaction-diffusion system with logarithmic singularity. Nonlinear Analysis: Theory, Methods & Applications, 71(11), 5599-5605, (2009).
• [42] Zheng, S. and Wang, W. Non-simultaneous versus simultaneous quenching in a coupled nonlinear parabolic system. Nonlinear Analysis: Theory, Methods & Applications, 69(7), 2274- 2285, (2008).
• [43] Xu, Y. and Zheng, Z. Quenching phenomenon of a time-fractional diffusion equation with singular source term. Mathematical Methods in the Applied Sciences, 40(16), 5750-5759, (2017). [CrossRef]
• [44] Xu, Y. and Wang, Z. Quenching phenomenon of a time-fractional Kawarada equation. Journal of Computational and Nonlinear Dynamics, 13(10), 101010, (2018).
• [45] Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. Theory and Applications of Fractional Differential Equations (Vol. 204). Elsevier: Amsterdam, (2006).
• [46] Oldham, K.B. and Spariier, J. The Fractional Calculus. Academic Press, New York-London, (1974).
• [47] Podlubny, I. Fractional Differential Equations. Academic Press: San Diego, (1999).
• [48] Samko, S.G., Kilbas, A.A. and Marichev, O.I. Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, Switzerland, (1993).
• [49] Du, M., Wang, Z. and Hu, H. Measuring memory with the order of fractional derivative. Scientific Reports, 3, 3431, (2013).
• [50] Caputo, M. Elasticità e Dissipazione. Bologna: Zanichelli, (1969).
• [51] Caputo, M. and Mainardi, F. A new dissipation model based on memory mechanism. Pure and Applied Geophysics, 91, 134-147, (1971).
• [52] Yang, W., Warrington, N.M., Taylor, S.J., Whitmire, P., Carrasco, E., Singleton, K.W. et al. Sex differences in GBM revealed by analysis of patient imaging, transcriptome, and survival data. Science Translational Medicine, 11(473), eaao5253, (2019).
• [53] Bi, J., Khan, A., Tang, J., Armando, A.M., Wu, S., Zhang, W. et al. Targeting glioblastoma signaling and metabolism with a re-purposed brain-penetrant drug. Cell Report, 37, 109957, (2021).
• [54] Skaga, E., Kulesskiy, E., Fayzullin, A., Sandberg, C.J., Potdar, S., Kyttälä, A. et al. Intertumoral heterogeneity in patient-specific drug sensitivities in treatment-naïve glioblastoma. BMC Cancer, 19, 628, (2019).