A Gradient Discretisation Method for Anisotropic Reaction–Diffusion Models with Applications to the Dynamics of Brain Tumors

Yahya Alnashri; Hasan Alzubaidi

doi:10.1515/cmam-2020-0081

Article

A Gradient Discretisation Method for Anisotropic Reaction–Diffusion Models with Applications to the Dynamics of Brain Tumors

Yahya Alnashri and Hasan Alzubaidi

Published/Copyright: April 2, 2021

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Computational Methods in Applied Mathematics Volume 21 Issue 4

Abstract

A gradient discretisation method (GDM) is an abstract setting that designs the unified convergence analysis of several numerical methods for partial differential equations and their corresponding models. In this paper, we study the GDM for anisotropic reaction–diffusion problems, based on a general reaction term, with Neumann boundary condition. With natural regularity assumptions on the exact solution, the framework enables us to provide proof of the existence of weak solutions for the problem, and to obtain a uniform-in-time convergence for the discrete solution and a strong convergence for its discrete gradient. It also allows us to apply non-conforming numerical schemes to the model on a generic grid (the non-conforming $ℙ ⁢ 1$ finite element scheme and the hybrid mixed mimetic (HMM) methods). Numerical experiments using the HMM method are performed to assess the accuracy of the proposed scheme and to study the growth of glioma tumors in heterogeneous brain environment. The dynamics of their highly diffusive nature is also measured using the fraction anisotropic measure. The validity of the HMM is examined further using four different mesh types. The results indicate that the dynamics of the brain tumor is still captured by the HMM scheme, even in the event of a highly heterogeneous anisotropic case performed on the mesh with extreme distortions.

Keywords: A Gradient Discretisation Method (GDM); Gradient Schemes; Convergence Analysis; Existence of Weak Solutions; Anisotropic Reaction–Diffusion Models; Dirichlet and Neumann Boundary Conditions; Non-Conforming Finite Element Methods; Finite Volume Schemes; Hybrid Mixed Mimetic (HMM) Method; Brain Tumor Dynamics; Fractional Anisotropy

MSC 2010: 35K57; 65N12; 65M08

A Appendix

We show here that any solution to the truncated model given in Section 6.1.2 with initial condition in $[ 0 , 1 ]$ remains in $[ 0 , 1 ]$ at any time.

Proposition A.1.

Let $c ¯$ be a solution to the model (1.1) with the function F defined by (6.5). If $c ¯ ⁢ ( 𝐱 , 0 ) ∈ [ 0 , 1 ]$ , for $𝐱$ in Ω, then $c ¯ ⁢ ( 𝐱 , t ) ∈ [ 0 , 1 ]$ , for $( 𝐱 , t ) ∈ Ω × ( 0 , T )$ .

Proof.

Since $F ⁢ ( 0 ) = F ⁢ ( 1 ) = 0$ , it is known that $u ¯ ≡ 1$ is a super-solution to the model (1.1) with the function F. We have

$- ∇ ⁡ u ¯ ⋅ 𝐧 ≥ ∇ ⁡ c ¯ ⋅ 𝐧$

and

$u ¯ ⁢ ( 𝒙 , 0 ) ≥ c ¯ ⁢ ( 𝒙 , 0 ) for 𝒙 in Ω .$

From the comparison principle, it follows that $c ¯ ⁢ ( 𝒙 , t ) ≤ 1$ . Similarly, with taking the sub-solution $u ¯ ≡ 0$ , we can conclude $c ¯ ⁢ ( 𝒙 , t ) ≥ 0$ , which completes the proof. ∎

References

[1] R. Achouri, Travelling wave solutions, Master’s thesis, School of Manchester, 2016. Search in Google Scholar

[2] Y. Alnashri and J. Droniou, Gradient schemes for an obstacle problem, Finite Volumes for Complex Applications VII. Methods and Theoretical Aspects, Springer Proc. Math. Stat. 77, Springer, Cham (2014), 67–75. 10.1007/978-3-319-05684-5_5Search in Google Scholar

[3] G. Arioli and H. Koch, Existence and stability of traveling pulse solutions of the FitzHugh–Nagumo equation, Nonlinear Anal. 113 (2015), 51–70. 10.1016/j.na.2014.09.023Search in Google Scholar

[4] R. Bammer, A. Burak and M. E. Moseley, In vivo MR tractography using diffusion imaging, Eur. J. Radiology 45 (2002), 223–234. 10.1016/S0720-048X(02)00311-XSearch in Google Scholar

[5] P. J. Basser, J. Mattiello and D. LeBihan, MR diffusion tensor spectroscopy and imaging, Biophys. J. 66 (1994), 259–267. 10.1016/S0006-3495(94)80775-1Search in Google Scholar

[6] P. K. Burgess, P. M. Kulesa, J. D. Murray and E. C. Alvord, Jr., The interaction of growth rates and diffusion coefficients in a three-dimensional mathematical model of gliomas, J. Neuropathy Exp. Neurology 56 (1997), 704–713. 10.1097/00005072-199706000-00008Search in Google Scholar

[7] C. Cherubini, A. Gizzi1, M. Bertolaso, V. Tambone and S. Filippi, A bistable field model of cancer dynamics, Commun. Comput. Phys. 11 (2012), 1–18. 10.4208/cicp.270710.220211aSearch in Google Scholar

[8] O. Clatz, P.-Y. Bondiau, H. Delingette, M. Sermesant, S. K. Warfield, G. Malandain and N. Ayacher, Brain tumor growth simulation, Technical report, INRIA, 2004. Search in Google Scholar

[9] M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7 (1973), 33–75. 10.1051/m2an/197307R300331Search in Google Scholar

[10] G. C. Cruywagen, D. E. Woodward, P. Tracqui, G. T. Bartoo, J. D. Murray and E. C. Alvord, Jr., The modeling of diffusive tumours, J. Biol. Syst. 3 (1995), 937–945. 10.1142/S0218339095000836Search in Google Scholar

[11] K. Das, R. Singh and S. C. Mishra, Numerical analysis for determination of the presence of a tumor and estimation of its size and location in a tissue, J. Thermal Biol. 38 (2013), no. 1, 32–40. 10.1016/j.jtherbio.2012.10.003Search in Google Scholar PubMed

[12] J. Droniou and R. Eymard, Uniform-in-time convergence of numerical methods for non-linear degenerate parabolic equations, Numer. Math. 132 (2016), no. 4, 721–766. 10.1007/s00211-015-0733-6Search in Google Scholar

[13] J. Droniou, R. Eymard and P. Feron, Gradient schemes for Stokes problem, IMA J. Numer. Anal. 36 (2016), no. 4, 1636–1669. 10.1093/imanum/drv061Search in Google Scholar

[14] J. Droniou, R. Eymard, T. Gallouët, C. Guichard and R. Herbin, The Gradient Discretisation Method, Math. Appl. (Berlin) 82, Springer, Cham, 2018. 10.1007/978-3-319-79042-8Search in Google Scholar

[15] J. Droniou, R. Eymard, T. Gallouët and R. Herbin, A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods, Math. Models Methods Appl. Sci. 20 (2010), no. 2, 265–295. 10.1142/S0218202510004222Search in Google Scholar

[16] J. Droniou, R. Eymard, T. Gallouet and R. Herbin, Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations, Math. Models Methods Appl. Sci. 23 (2013), no. 13, 2395–2432. 10.1142/S0218202513500358Search in Google Scholar

[17] J. Droniou, R. Eymard and R. Herbin, Gradient schemes: generic tools for the numerical analysis of diffusion equations, ESAIM Math. Model. Numer. Anal. 50 (2016), no. 3, 749–781. 10.1051/m2an/2015079Search in Google Scholar

[18] R. Eymard, P. Féron, T. Gallouët, R. Herbin and C. Guichard, Gradient schemes for the Stefan problem, Int. J. Finite Vol. 10 (2013), 1–37. Search in Google Scholar

[19] R. Eymard, C. Guichard and R. Herbin, Small-stencil 3D schemes for diffusive flows in porous media, ESAIM Math. Model. Numer. Anal. 46 (2012), no. 2, 265–290. 10.1051/m2an/2011040Search in Google Scholar

[20] R. Eymard, C. Guichard, R. Herbin and R. Masson, Gradient schemes for two-phase flow in heterogeneous porous media and Richards equation, ZAMM Z. Angew. Math. Mech. 94 (2014), no. 7–8, 560–585. 10.1002/zamm.201200206Search in Google Scholar

[21] A. Giese, R. Bjerkvig, M. E. Berens and M. Westphal, Cost of migration: Invasion of malignant gliomas and implications for treatment, J. Clinical Oncology 21 (2003), 1624–1636. 10.1200/JCO.2003.05.063Search in Google Scholar PubMed

[22] A. Giese, L. Kluwe, B. Laube, H. Meissner, M. E. Berens and M. Westphal, Migration of human glioma cells on myelin, Neurosurgery 38 (1996), 755–764. 10.1227/00006123-199604000-00026Search in Google Scholar

[23] R. Herbin and F. Hubert, Benchmark on discretization schemes for anisotropic diffusion problems on general grids, Finite Volumes for Complex Applications V, ISTE, London (2008), 659–692. Search in Google Scholar

[24] J. G. Heywood and R. Rannacher, Finite-element approximation of the nonstationary Navier–Stokes problem. IV. Error analysis for second-order time discretization, SIAM J. Numer. Anal. 27 (1990), no. 2, 353–384. 10.1137/0727022Search in Google Scholar

[25] T. Hillen and K. J. Painter, Transport and anisotropic diffusion models for movement in oriented habitats, Dispersal, Individual Movement and Spatial Ecology, Lecture Notes in Math. 2071, Springer, Heidelberg (2013), 177–222. 10.1007/978-3-642-35497-7_7Search in Google Scholar

[26] T. Hines, Mathematically modeling the mass-effect of invasive brain tumor, SIAM Undergrad. Res. Online 74 (2010), 684–700. 10.1137/09S010526Search in Google Scholar

[27] M. Ibrahim and M. Saad, On the efficacy of a control volume finite element method for the capture of patterns for a volume-filling chemotaxis model, Comput. Math. Appl. 68 (2014), no. 9, 1032–1051. 10.1016/j.camwa.2014.03.010Search in Google Scholar

[28] R. Jaroudi, F. Åström, B. T. Johansson and G. Baravdish, Numerical simulations in 3-dimensions of reaction–diffusion models for brain tumour growth, Int. J. Comput. Math. 97 (2020), no. 6, 1151–1169. 10.1080/00207160.2019.1613526Search in Google Scholar

[29] R. Jaroudi, G. Baravdish, B. T. Johansson and F. Åström, Numerical reconstruction of brain tumours, Inverse Probl. Sci. Eng. 27 (2019), no. 3, 278–298. 10.1080/17415977.2018.1456537Search in Google Scholar

[30] A. Jbabdi, E. Mandonnet, H. Duffau, L. Capelle, K. R. Swanson, M. Pelegrini Issac, R. Guillevin and H. Benali, Simulation of anisotropic growth of low-grade gliomas using diffusion tensor imaging, Magn. Resonance Medicine 54 (2005), 616–624. 10.1002/mrm.20625Search in Google Scholar PubMed

[31] C. Kuttler, Reaction–diffusion equations with applications, Lecture notes (2011). Search in Google Scholar

[32] X. Li and W. Huang, A study on nonnegativity preservation in finite element approximation of Nagumo-type nonlinear differential equations, Appl. Math. Comput. 309 (2017), 49–67. 10.1016/j.amc.2017.03.038Search in Google Scholar

[33] S. Mori, Introduction to Diffusion Tensor Imaging, Elsevier, Amsterdam, 2007. 10.1016/B978-044452828-5/50019-3Search in Google Scholar

[34] P. Mosayebi, D. Cobzas, A. Murtha and M. Jagersand, Tumor invasion margin on the riemannian space of brain fibers, Medical Imag. Anal. 16 (2011), 361–373. 10.1016/j.media.2011.10.001Search in Google Scholar PubMed

[35] J. D. Murray, Mathematical Biology. II: Spatial Models and Biomedical Application, Springer, New York, 2004. Search in Google Scholar

[36] H. Ninomiya, Entire solutions and traveling wave solutions of the Allen–Cahn–Nagumo equation, Discrete Contin. Dyn. Syst. 39 (2019), no. 4, 2001–2019. 10.3934/dcds.2019084Search in Google Scholar

[37] A. D. Norden and P. Y. Wen, Glioma therapy in adults, Neurologist. 12 (2006), 279–292. 10.1097/01.nrl.0000250928.26044.47Search in Google Scholar PubMed

[38] K. J. Painter and T. Hillen, Mathematical modelling of glioma growth: The use of diffusion tensor imaging (DTI) data to predict the anisotropic pathways of cancer invasion, J. Theoret. Biol. 323 (2013), 25–39. 10.1016/j.jtbi.2013.01.014Search in Google Scholar PubMed

[39] E. M. Rutter, T. L. Stepien, B. J. Anderies, J. D. Plasencia, E. C. Woolf, A. C. Scheck, G. H. Turner, Q. Liu, D. Frakes, V. Kodibagkar, Y. Kuang, M. C. Preul and E. J. Kostelich, Mathematical analysis of glioma growth in a murine model, Sci. Rep. 7 (2017), 1–16. 10.1038/s41598-017-02462-0Search in Google Scholar PubMed PubMed Central

[40] A. H. V. Schapira, Neurology and Clinical Neuroscience, Elsevier, Philadelphia, 2007. Search in Google Scholar

[41] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University, Oxford, 1997. 10.1093/oso/9780198548522.001.0001Search in Google Scholar

[42] J. Smoller, Shock Waves and Reaction–Diffusion Equations, Springer, New York, 1983. 10.1007/978-1-4684-0152-3Search in Google Scholar

[43] S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Westview Press, Boulder, 2001. Search in Google Scholar

[44] P. C. Sundgren, Q. Dong, D. Gomez-Hassan, S. K. Mukherji, P. Maly and R. Welsh, Diffusion tensor imaging of the brain: Review of clinical applications, Neuroradiology 46 (2004), 339–350. 10.1007/s00234-003-1114-xSearch in Google Scholar PubMed

[45] K. R. Swanson, E. C. Alvord, Jr. and J. D. Murray, A quantitative model for differential motility of gliomas in grey and white matter, Cell Proliferation 33 (2000), 317–329. 10.1046/j.1365-2184.2000.00177.xSearch in Google Scholar PubMed PubMed Central

[46] K. R. Swanson, E. C. Alvord, Jr. and J. D. Murray, Virtual brain tumours (gliomas) enhance the reality of medical imaging and highlight inadequacies of current therapy, British J. Cancer 86 (2002), 14–18. 10.1038/sj.bjc.6600021Search in Google Scholar PubMed PubMed Central

[47] P. Tracqui, G. C. Cruywagen, D. E. Woodward, G. T. Bartooll, J. D. Murray and E. C. Alvord, Jr., A mathematical model of glioma growth: The effect of chemotherapy on spatio-temporal growth, Cell Proliferation 28 (1995), 17–31. 10.1111/j.1365-2184.1995.tb00036.xSearch in Google Scholar PubMed

[48] X. Zeng, M. A. Saleh and J. P. Tian, On finite volume discretization of infiltration dynamics in tumor growth models, Adv. Comput. Math. 45 (2019), no. 5–6, 3057–3094. 10.1007/s10444-019-09727-4Search in Google Scholar

[49] H. Zhou, Z. Sheng and G. Yuan, Positivity preserving finite volume scheme for the Nagumo-type equations on distorted meshes, Appl. Math. Comput. 336 (2018), 182–192. 10.1016/j.amc.2018.04.058Search in Google Scholar

[50] B. Zinner, Existence of traveling wavefront solutions for the discrete Nagumo equation, J. Differential Equations 96 (1992), no. 1, 1–27. 10.1016/0022-0396(92)90142-ASearch in Google Scholar

Received: 2020-05-17

Revised: 2021-02-28

Accepted: 2021-03-03

Published Online: 2021-04-02

Published in Print: 2021-10-01

You are currently not able to access this content.