Geodesic
Travelling waves for low-grade glioma growth and response to a chemotherapy model
International Journal of Applied Mathematics and Computer Science, Tome 33 (2023) no. 4, pp. 569-581.

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Low-grade gliomas (LGGs) are primary brain tumours which evolve very slowly in time, but inevitably cause patient death. In this paper, we consider a PDE version of the previously proposed ODE model that describes the changes in the densities of functionally alive LGGs cells and cells that are irreversibly damaged by chemotherapy treatment. Besides the basic mathematical properties of the model, we study the possibility of the existence of travelling wave solutions in the framework of Fenichel’s invariant manifold theory. The estimates of the minimum speeds of the travelling wave solutions are provided. The obtained analytical results are illustrated by numerical simulations.
Keywords: low grade glioma, generalized model, partial differential equation, wave solution, chemotherapy model
Mots-clés : glejak, model uogólniony, równanie różniczkowe cząstkowe, rozwiązanie falowe, model chemioterapii 
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Bartłomiejczyk, Agnieszka; Bodnar, Marek; Bogdańska, Magdalena U.; Piotrowska, Monika J. Travelling waves for low-grade glioma growth and response to a chemotherapy model. International Journal of Applied Mathematics and Computer Science, Tome 33 (2023) no. 4, pp. 569-581. http://geodesic.mathdoc.fr/item/IJAMCS_2023_33_4_a4/

[1] [1] Adenis, L., Plaszczynski, S., Grammaticos, B., Pallud, J. and Badoual, M. (2021). The effect of radiotherapy on diffuse low-grade gliomas evolution: Confronting theory with clinical data, Journal of Personalized Medicine 11(8): 818.

[2] [2] Belmonte-Beitia, J. (2016). Existence of travelling wave solutions for a Fisher-Kolmogorov system with biomedical applications, Communications in Nonlinear Science and Numerical Simulation 36: 14-20, DOI: 10.1016/j.cnsns.2015.11.016.

[3] [3] Bodnar, M., Bogdańska, M.U. and Piotrowska, M. (2019). Mathematical analysis of a generalised model of chemotherapy for low grade gliomas, Discrete and Continuous Dynamical Systems B 24(5): 2149-2167, DOI: 10.3934/dcdsb.2019088.

[4] [4] Bodnar, M. and Vela Pérez, M. (2019). Mathematical and numerical analysis of low-grade gliomas model and the effects of chemotherapy, Communications in Nonlinear Science and Numerical Simulation 72: 552-564, DOI: DOI: 10.1016/j.cnsns.2019.01.015.

[5] [5] Bogdańska, M.U., Bodnar, M., Belmonte-Beitia, J., Murek, M., Schucht, P., Beck, J. and Pérez-García, V.M. (2017). A mathematical model of low grade gliomas treated with temozolomide and its therapeutical implications, Mathematical Biosciences 288: 1-13, DOI: 10.1016/j.mbs.2017.02.003.

[6] [6] Bogdańska, M.U., Bodnar, M., Piotrowska, M.J., Murek, M., Schucht, P., Beck, J., Martínez-González, A. and Pérez-García, V.M. (2017). A mathematical model describes the malignant transformation of low grade gliomas: Prognostic implications, PLOS ONE 12(8): 1-24, DOI: 10.1371/journal.pone.0179999.

[7] [7] Bressloff, P.C. (2013). Waves in Neural Media: From Single Neurons to Neural Fields, Springer, New York.

[8] [8] Fenichel, N. (1979). Geometric singular perturbations theory for ordinary differential equations, Journal of Differential Equations 31: 53-98, DOI: 10.1016/0022-0396(79)90152-9.

[9] [9] Fife, P.C. (1979). Mathematical Aspects of Reacting and Diffusing Systems, Springer, Berlin.

[10] [10] Gourley, S.A. and Bartuccelli, M.V. (2000). Existence and construction of travelling wavefront solutions of Fisher equations with fourth-order perturbations, Dynamics and Stability of Systems 15(3): 253-262, DOI: 10.1080/026811100418710.

[11] [11] Gugat, M. and Wintergerst, D. (2018). Transient flow in gas networks: Traveling waves, International Journal of Applied Mathematics and Computer Science 28(2): 341-348, DOI: 10.2478/amcs-2018-0025.

[12] [12] Henry, D. (1981). Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin/Heidelberg.

[13] [13] Joiner, M.C. and van der Kogel, A. (2019). Basic Clinical Radiobiology, CRC Press, Boca Raton.

[14] [14] Jones, C.K.R.T. (1995). Geometric singular perturbation theory, in R. Jonson (Ed), Dynamical Systems, Lecture Notes in Mathematics, Vol. 1609, Springer, Berlin/Heidelberg, pp. 44-118, DOI: 10.1007/BFb0095239.

[15] [15] Keles, G.E., Lamborn, K.R. and S. Berger, M. (2011). Low-grade hemispheric gliomas in adults: A critical review of extent of resection as a factor influencing outcome, Journal of Neurosurgery 95(5): 735-45, DOI:10.3171/jns.2001.95.5.0735.

[16] [16] Kowal, M., Skobel, M., Gramacki, A. and Korbicz, J. (2021). Breast cancer nuclei segmentation and classification based on a deep learning approach, International Journal of Applied Mathematics and Computer Science 31(1): 85-106, DOI: 10.34768/amcs-2021-0007.

[17] [17] Murray, J.D. (1989). Mathematical Biology, Springer, Berlin.

[18] [18] Pallud, J., Blonski, M., Mandonnet, E., Audureau, E., Fontaine, D., Sanai, N., Bauchet, L., Peruzzi, P., Frénay, M., Colin, P., Guillevin, R., Bernier, V., Baron, M.-H., Guyotat, J., Duffau, H., Taillandier, L. and Capelle, L. (2013). Velocity of tumor spontaneous expansion predicts long-term outcomes for diffuse low-grade gliomas, Neuro-Oncology 15(5): 595-606.

[19] [19] Pallud, J., Capelle, L., Taillandier, L., Badoual, M., Duffau, H. and Mandonnet, E. (2013). The silent phase of diffuse low-grade gliomas. Is it when we missed the action?, Acta Neurochirurgica 155(12): 2237-2242.

[20] [20] Pallud, J., Taillandier, L., Capelle, L., Fontaine, D., Peyre, M., Ducray, F., Duffau, H. and Mandonnet, E. (2012). Quantitative morphological magnetic resonance imaging follow-up of low-grade glioma, Neurosurgery 71(3): 729-740.

[21] [21] Pouratian, N. and Schiff, D. (2010). Management of low-grade glioma, Current Neurology and Neuroscience Reports 10(3): 224-31.

[22] [22] Pérez-García, V.M., Bogdanska, M., Martínez-González, A., Belmonte-Beitia, J., Schucht, P. and Pérez-Romasanta, L.A. (2014). Delay effects in the response of low-grade gliomas to radiotherapy: A mathematical model and its therapeutical implications, Mathematical Medicine and Biology: A Journal of the IMA 32(3): 307-329.

[23] [23] Sakarunchai, I., Sangthong, R., Phuenpathom, N. and Phukaoloun, M. (2013). Free survival time of recurrence and malignant transformation and associated factors in patients with supratentorial low-grade gliomas, Journal of the Medical Association of Thailand 96(12): 1542-9.

[24] [24] Swanson, K., Rostomily, R. and Alvord, E. (2008). A mathematical modelling tool for predicting survival of individual patients following resection of glioblastoma: A proof of principle, British Journal of Cancer 98(1): 113-119, DOI: 10.1038/sj.bjc.6604125.

[25] [25] Wang, C.H., Rockhill, J.K., Mrugala, M., Peacock, D.L., Lai, A., Jusenius, K., Wardlaw, J.M., Cloughesy, T., Spence, A.M., Rockne, R., Alvord Jr, E.C. and Swanson, K.R. (2009). Prognostic significance of growth kinetics in newly diagnosed glioblastomas revealed by combining serial imaging with a novel biomathematical model, Cancer Research 69(23): 9133-9140, DOI: 10.1158/0008-5472.CAN-08-3863.