2016, 13(2): 443-460. doi: 10.3934/mbe.2015011

A multiscale model for glioma spread including cell-tissue interactions and proliferation

1. 

WWU Münster, Institute for Computational und Applied Mathematics and Cluster of Excellence EXC 1003, Cells in Motion, Orleans-Ring 10, 48149 Münster, Germany, Germany

2. 

Technische Universität Kaiserslautern, Felix-Klein-Zentrum für Mathematik, Paul-Ehrlich-Str. 31, 67663 Kaiserslautern

Received  July 2015 Revised  November 2015 Published  December 2015

Glioma is a broad class of brain and spinal cord tumors arising from glia cells, which are the main brain cells that can develop into neoplasms. They are highly invasive and lead to irregular tumor margins which are not precisely identifiable by medical imaging, thus rendering a precise enough resection very difficult. The understanding of glioma spread patterns is hence essential for both radiological therapy as well as surgical treatment. In this paper we propose a multiscale model for glioma growth including interactions of the cells with the underlying tissue network, along with proliferative effects. Our current accounting for two subpopulations of cells to accomodate proliferation according to the go-or-grow dichtomoty is an extension of the setting in [16]. As in that paper, we assume that cancer cells use neuronal fiber tracts as invasive pathways. Hence, the individual structure of brain tissue seems to be decisive for the tumor spread. Diffusion tensor imaging (DTI) is able to provide such information, thus opening the way for patient specific modeling of glioma invasion. Starting from a multiscale model involving subcellular (microscopic) and individual (mesoscale) cell dynamics, we perform a parabolic scaling to obtain an approximating reaction-diffusion-transport equation on the macroscale of the tumor cell population. Numerical simulations based on DTI data are carried out in order to assess the performance of our modeling approach.
Citation: Christian Engwer, Markus Knappitsch, Christina Surulescu. A multiscale model for glioma spread including cell-tissue interactions and proliferation. Mathematical Biosciences & Engineering, 2016, 13 (2) : 443-460. doi: 10.3934/mbe.2015011
References:
[1]

D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini, Unified analysis of discontinuous galerkin methods for elliptic problems, SIAM Journal on Numerical Analysis, 39 (2002), 1749-1779. doi: 10.1137/S0036142901384162.

[2]

P. Bastian, M. Blatt, A. Dedner, C. Engwer, R. Klöfkorn, M. Ohlberger and O. Sander, A generic grid interface for parallel and adaptive scientific computing. part I: Abstract framework, Computing, 82 (2008), 103-119. doi: 10.1007/s00607-008-0003-x.

[3]

P. Bastian, M. Blatt, A. Dedner, C. Engwer, R. Klöfkorn, R. Kornhuber, M. Ohlberger and O. Sander, A generic grid interface for parallel and adaptive scientific computing. part II: Implementation and tests in DUNE, Computing, 82 (2008), 121-138. doi: 10.1007/s00607-008-0004-9.

[4]

A. M. Belkin, G. Tsurupa, E. Zemskov, Y. Veklich, J. W. Weisel and L. Medved, Transglutaminase-mediated oligomerization of the fibrin(ogen) $\alpha$C domains promotes integrin-dependent cell adhesion and signaling, Blood, 105 (2005), 3561-3568.

[5]

N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, Complexity and mathematical tools toward the modeling of multicellular growing systems, Mathematical and Computer Modelling, 51 (2010), 441-451. doi: 10.1016/j.mcm.2009.12.002.

[6]

N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, On the asymptotic theory from microscopic to macroscopic tissue models: An overview with perspectives, Math. Models Methods Appl. Sci., 22 (2012), 1130001, 37 pp. doi: 10.1142/S0218202512005885.

[7]

N. Bellomo, D. Knopoff and J. Soler, On the difficult interplay between life, "complexity", and mathematical sciences, Math. Models Methods Appl. Sci., 23 (2013), 1861-1913. doi: 10.1142/S021820251350053X.

[8]

T. Beppu, T.Inoue, Y. Shibata, A. Kurose, H. Arai, K. Ogasawara, A. Ogawa, S. Nakamura and H. Kabasawa, Measurement of fractional anisotropy using diffuson tensor MRI in supratentorial astrocytic tumors, Journal of Neuro-Oncology, 63 (2003), 109-116.

[9]

K. Böttger, H. Hatzikirou, A. Chauviere and A. Deutsch, Investigation of the migration/proliferation dichotomy and its impact on avascular glioma invasion, Mathematical Modelling of Natural Phenomena, 7 (2012), 105-135. doi: 10.1051/mmnp/20127106.

[10]

S. Coons, Anatomy and growth patterns of diffuse gliomas, in The gliomas (eds. M. Berger and C. Wilson), W.B. Saunders Company, Philadelphia, (1999), 210-225.

[11]

G. D'Abaco and A. Kaye, Integrins: Molecular determinants of glioma invasion, J. of Clinical Neurosci., 14 (2007), 1041-1048.

[12]

C. Daumas-Duport, P. Varlet, M. L. Tucker, F. Beuvon, P. Cervera and J. P. Chodkiewicz, Oligodendrogliomas. Part I: Patterns of growth, histological diagnosis, clinical and imaging correlations: A study of 153 cases, Journal of Neuro-Oncology, 34 (1997), 37-59.

[13]

T. Demuth and M. E. Berens, Molecular mechanisms of glioma cell invasion and migration, Journal of Neuro-Oncology, 70 (2004), 217-228.

[14]

M. Descoteaux, R. Deriche, T. R. Knische and A. Anwander, Deterministic and Probabilistic Tractography Based on Complex Fibre Orientation Distributions, IEEE Transactions on Medical Imaging, 28 (2009), 269-286.

[15]

M. Descoteaux, High Angular Resolution Diffusion MRI: From Local Estimation to Segmentation and Tractography, Ph.D. thesis, Université Nice-Sophia Antipolis, 2008.

[16]

C. Engwer, T. Hillen, M. Knappitsch and C. Surulescu, Glioma follow white matter tracts: A multiscale DTI-based model, Journal of Math. Biol., 71 (2015), 551-582. doi: 10.1007/s00285-014-0822-7.

[17]

C. Engwer, A. Hunt and C. Surulescu, Effective equations for anisotropic glioma spread with proliferation: A multiscale approach and comparisons with previous settings, IMA J. Math. Medicine and Biol., 32 (2015), doi:10.1093/imammb/dqv030, 2015. doi: 10.1093/imammb/dqv030.

[18]

R. Erban and H. Othmer, From signal transduction to spatial pattern formation in E. coli: A paradigm for multiscale modeling in biology, Multiscale Modeling and Simulation, 3 (2005), 362-394. doi: 10.1137/040603565.

[19]

A. Ern, A. F. Stephansen and P. Zunino, A discontinuous Galerkin method with weighted averages for advection-diffusion equations with locally small and anisotropic diffusivity, IMA Journal of Numerical Analysis, 29 (2009), 235-256. doi: 10.1093/imanum/drm050.

[20]

E. R. Gerstner, P.-J. Chen, P. Y. Wen, R. K. Jain, T. T. Batchelor and G. Sorensen, Infiltrative patterns of glioblastoma spread detected via diffusion MRI after treatment with cediranib, Neuro-Oncology, 12 (2010), 466-472.

[21]

A. Giese, M. A. Loo, N. Tran, D. Haskett, S. Coons and M. Berens, Dichotomy of astrocytoma migration and proliferation, International Journal of Cancer, 67 (1996), 275-282.

[22]

A. Giese and M. Westphal, Glioma invasion in the central nervous system, Neurosurgery, 39 (1996), 235-252. doi: 10.1097/00006123-199608000-00001.

[23]

D. Grünbaum, Advection-diffusion equations for internal state-mediated random walks, SIAM Journal for Applied Mathematics, 61 (2000), 43-73. doi: 10.1137/S0036139997332075.

[24]

D. Hanahan and R. A. Weinberg, Hallmarks of cancer: The next generation, Cell, 144 (2011), 646-674. doi: 10.1016/j.cell.2011.02.013.

[25]

H. Hatzikirou, D. Basanta, M. Simon, K. Schaller and A. Deutsch, 'Go or Grow': the key to the emergence of invasion in tumour progression?, Math Med Biol, 29 (2012), 49-65. doi: 10.1093/imammb/dqq011.

[26]

T. Hillen, Transport equations with resting phases, Europ. J. Appl. Math., 14 (2003), 613-636. doi: 10.1017/S0956792503005291.

[27]

T. Hillen, $M^5$ Mesoscopic and macroscopic models for mesenchymal motion, Journal of Mathematical Biology, 53 (2006), 585-616. doi: 10.1007/s00285-006-0017-y.

[28]

T. Hillen and H. Othmer, The diffusion limit of transport equations derived from velocity jump processes, SIAM Journal on Applied Mathematics, 61 (2000), 751-775. doi: 10.1137/S0036139999358167.

[29]

K. S. Hoek, O. M. Eichhoff, N. C. Schlegel, U. Döbbeling, S. Hemmi and R. Dummer, In vivo switching of human melanoma cells between proliferative and invasive states, Cancer Res., 68 (2008), 650-656. doi: 10.1158/0008-5472.CAN-07-2491.

[30]

J. Kelkel and C. Surulescu, On some models for cancer cell migration through tissue networks, Mathematical Biosciences and Engineering, 8 (2011), 575-589. doi: 10.3934/mbe.2011.8.575.

[31]

J. Kelkel and C. Surulescu, A multiscale approach to cell migration in tissue networks, Mathematical Models and Methods in Applied Sciences, 23 (2012), 1150017, 25 pp. doi: 10.1142/S0218202511500175.

[32]

D. A. Lauffenburger and J. L. Lindermann, Receptors. Models for Binding, Trafficking and Signaling, Oxford University Press, 1993.

[33]

K.R. Legate, S.A. Wickström, and R. Fässler, Genetic and cell biological analysis of integrin outside-in signaling, Genes Dev., 23 (2009), 397-418. doi: 10.1101/gad.1758709.

[34]

T. Lorenz and C. Surulescu, On a class of multiscale cancer cell migration models: Well-posedness in less regular function spaces, Mathematical Models and Methods in Applied Sciences, 24 (2014), 2383-2436. doi: 10.1142/S0218202514500249.

[35]

Y. Matsukado, C. MacCarty, J. Kernohan, et al., The growth of glioblastoma multiforme (astrocytomas, grades 3 and 4) in neurosurgical practice, Journal of Neurosurgery, 18 (1961), 636-644. doi: 10.3171/jns.1961.18.5.0636.

[36]

G. Meral, C. Stinner and C. Surulescu, On a multiscale model involving cell contractivity and its effects on tumor invasion, Discr. Cont. Dyn. Syst. B, 20 (2015), 189-213. doi: 10.3934/dcdsb.2015.20.189.

[37]

G. Meral and C. Surulescu, Mathematical Modelling, Analysis and numerical simulations for the influence of the heat shock proteins on tumour invasion, Journal of Math. Anal. Appl., 408 (2013), 597-614. doi: 10.1016/j.jmaa.2013.06.017.

[38]

N. Oppenheimer-Marks and P. E. Lipsky, Adhesion molecules and the regulation of the migration of lymphocytes, in Adhesion Molecules and Chemokynes in Lymphocyte Trafficking (ed. A. Hamann), Harwood Acad. Publ. (1997), 55-88.

[39]

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. Theoretical Biol., 323 (2013), 25-39. doi: 10.1016/j.jtbi.2013.01.014.

[40]

M. Sidani, D. Wessels, G. Mouneimne, M. Ghosh, S. Goswami, C. Sarmiento, W. Wang, S. Kuhl, M. El-Sibai, J. M. Backer and R. Eddy, D. Soll and J. Condeelis, Cofilin determines the migration behavior and turning frequency of metastatic cancer cells, The Journal of Cell Biology, 179 (2007), 777-791. doi: 10.1083/jcb.200707009.

[41]

C. Stinner, C. Surulescu and G. Meral, A multiscale model for pH-tactic tumor invasion with time-varying carrying capacities, IMA Journal of Applied Mathematics, 80 (2015), 1300-1321. doi: 10.1093/imamat/hxu055.

[42]

C. Stinner, C. Surulescu and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Analysis, 46 (2014), 1969-2007. doi: 10.1137/13094058X.

[43]

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, Neurocarciology, 46 (2004), 339-350. doi: 10.1007/s00234-003-1114-x.

[44]

Z. Szymanska, J. Urbanski and A. Marciniak-Czochra, Mathematical modelling of the influence of heat shock proteins on cancer invasion of tissue, Journal of Mathematical Biology, 58 (2009), 819-844. doi: 10.1007/s00285-008-0220-0.

[45]

D. S. Tuch, Q-ball imaging, Magnetic resonance in Medicine, 52 (2004), 1358-1372. doi: 10.1002/mrm.20279.

[46]

D. S. Tuch, T. G. Reese, M. R. Wiegell and V. J. Wedeen, Diffusion MRI of complex neural architecture, Neuron, 40 (2003), 885-895. doi: 10.1016/S0896-6273(03)00758-X.

[47]

J. H. Uhm, C. L. Gladson and J. S. Rao, The role of integrins in the malignant phenotype of gliomas, Frontiers in Bioscience, 4 (1999), 188-199.

[48]

S. Wagner, S. M. Rampersad, Ü Aydin, J. Vorwerk, T. F. Oostendorp, T. Neuling, C. S. Herrmann, D. F. Stegeman and C. H. Wolters, Investigation of tDCS volume conduction effects in a highly realistic head model, Journal of neural engineering, 11 (2014), 016002, 14pp. doi: 10.1088/1741-2560/11/1/016002.

[49]

M. F. Wheeler, An elliptic collocation-finite element method with interior penalties, SIAM Journal on Numerical Analysis, 15 (1978), 152-161. doi: 10.1137/0715010.

[50]

M. Wrensch, Y. Minn, T. Chew, M. Bondy and M. S. Berger, Epidemiology of primary brain tumors: Current concepts and review of the literature, Neuro-Oncology, 4 (2002), 278-299.

show all references

References:
[1]

D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini, Unified analysis of discontinuous galerkin methods for elliptic problems, SIAM Journal on Numerical Analysis, 39 (2002), 1749-1779. doi: 10.1137/S0036142901384162.

[2]

P. Bastian, M. Blatt, A. Dedner, C. Engwer, R. Klöfkorn, M. Ohlberger and O. Sander, A generic grid interface for parallel and adaptive scientific computing. part I: Abstract framework, Computing, 82 (2008), 103-119. doi: 10.1007/s00607-008-0003-x.

[3]

P. Bastian, M. Blatt, A. Dedner, C. Engwer, R. Klöfkorn, R. Kornhuber, M. Ohlberger and O. Sander, A generic grid interface for parallel and adaptive scientific computing. part II: Implementation and tests in DUNE, Computing, 82 (2008), 121-138. doi: 10.1007/s00607-008-0004-9.

[4]

A. M. Belkin, G. Tsurupa, E. Zemskov, Y. Veklich, J. W. Weisel and L. Medved, Transglutaminase-mediated oligomerization of the fibrin(ogen) $\alpha$C domains promotes integrin-dependent cell adhesion and signaling, Blood, 105 (2005), 3561-3568.

[5]

N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, Complexity and mathematical tools toward the modeling of multicellular growing systems, Mathematical and Computer Modelling, 51 (2010), 441-451. doi: 10.1016/j.mcm.2009.12.002.

[6]

N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, On the asymptotic theory from microscopic to macroscopic tissue models: An overview with perspectives, Math. Models Methods Appl. Sci., 22 (2012), 1130001, 37 pp. doi: 10.1142/S0218202512005885.

[7]

N. Bellomo, D. Knopoff and J. Soler, On the difficult interplay between life, "complexity", and mathematical sciences, Math. Models Methods Appl. Sci., 23 (2013), 1861-1913. doi: 10.1142/S021820251350053X.

[8]

T. Beppu, T.Inoue, Y. Shibata, A. Kurose, H. Arai, K. Ogasawara, A. Ogawa, S. Nakamura and H. Kabasawa, Measurement of fractional anisotropy using diffuson tensor MRI in supratentorial astrocytic tumors, Journal of Neuro-Oncology, 63 (2003), 109-116.

[9]

K. Böttger, H. Hatzikirou, A. Chauviere and A. Deutsch, Investigation of the migration/proliferation dichotomy and its impact on avascular glioma invasion, Mathematical Modelling of Natural Phenomena, 7 (2012), 105-135. doi: 10.1051/mmnp/20127106.

[10]

S. Coons, Anatomy and growth patterns of diffuse gliomas, in The gliomas (eds. M. Berger and C. Wilson), W.B. Saunders Company, Philadelphia, (1999), 210-225.

[11]

G. D'Abaco and A. Kaye, Integrins: Molecular determinants of glioma invasion, J. of Clinical Neurosci., 14 (2007), 1041-1048.

[12]

C. Daumas-Duport, P. Varlet, M. L. Tucker, F. Beuvon, P. Cervera and J. P. Chodkiewicz, Oligodendrogliomas. Part I: Patterns of growth, histological diagnosis, clinical and imaging correlations: A study of 153 cases, Journal of Neuro-Oncology, 34 (1997), 37-59.

[13]

T. Demuth and M. E. Berens, Molecular mechanisms of glioma cell invasion and migration, Journal of Neuro-Oncology, 70 (2004), 217-228.

[14]

M. Descoteaux, R. Deriche, T. R. Knische and A. Anwander, Deterministic and Probabilistic Tractography Based on Complex Fibre Orientation Distributions, IEEE Transactions on Medical Imaging, 28 (2009), 269-286.

[15]

M. Descoteaux, High Angular Resolution Diffusion MRI: From Local Estimation to Segmentation and Tractography, Ph.D. thesis, Université Nice-Sophia Antipolis, 2008.

[16]

C. Engwer, T. Hillen, M. Knappitsch and C. Surulescu, Glioma follow white matter tracts: A multiscale DTI-based model, Journal of Math. Biol., 71 (2015), 551-582. doi: 10.1007/s00285-014-0822-7.

[17]

C. Engwer, A. Hunt and C. Surulescu, Effective equations for anisotropic glioma spread with proliferation: A multiscale approach and comparisons with previous settings, IMA J. Math. Medicine and Biol., 32 (2015), doi:10.1093/imammb/dqv030, 2015. doi: 10.1093/imammb/dqv030.

[18]

R. Erban and H. Othmer, From signal transduction to spatial pattern formation in E. coli: A paradigm for multiscale modeling in biology, Multiscale Modeling and Simulation, 3 (2005), 362-394. doi: 10.1137/040603565.

[19]

A. Ern, A. F. Stephansen and P. Zunino, A discontinuous Galerkin method with weighted averages for advection-diffusion equations with locally small and anisotropic diffusivity, IMA Journal of Numerical Analysis, 29 (2009), 235-256. doi: 10.1093/imanum/drm050.

[20]

E. R. Gerstner, P.-J. Chen, P. Y. Wen, R. K. Jain, T. T. Batchelor and G. Sorensen, Infiltrative patterns of glioblastoma spread detected via diffusion MRI after treatment with cediranib, Neuro-Oncology, 12 (2010), 466-472.

[21]

A. Giese, M. A. Loo, N. Tran, D. Haskett, S. Coons and M. Berens, Dichotomy of astrocytoma migration and proliferation, International Journal of Cancer, 67 (1996), 275-282.

[22]

A. Giese and M. Westphal, Glioma invasion in the central nervous system, Neurosurgery, 39 (1996), 235-252. doi: 10.1097/00006123-199608000-00001.

[23]

D. Grünbaum, Advection-diffusion equations for internal state-mediated random walks, SIAM Journal for Applied Mathematics, 61 (2000), 43-73. doi: 10.1137/S0036139997332075.

[24]

D. Hanahan and R. A. Weinberg, Hallmarks of cancer: The next generation, Cell, 144 (2011), 646-674. doi: 10.1016/j.cell.2011.02.013.

[25]

H. Hatzikirou, D. Basanta, M. Simon, K. Schaller and A. Deutsch, 'Go or Grow': the key to the emergence of invasion in tumour progression?, Math Med Biol, 29 (2012), 49-65. doi: 10.1093/imammb/dqq011.

[26]

T. Hillen, Transport equations with resting phases, Europ. J. Appl. Math., 14 (2003), 613-636. doi: 10.1017/S0956792503005291.

[27]

T. Hillen, $M^5$ Mesoscopic and macroscopic models for mesenchymal motion, Journal of Mathematical Biology, 53 (2006), 585-616. doi: 10.1007/s00285-006-0017-y.

[28]

T. Hillen and H. Othmer, The diffusion limit of transport equations derived from velocity jump processes, SIAM Journal on Applied Mathematics, 61 (2000), 751-775. doi: 10.1137/S0036139999358167.

[29]

K. S. Hoek, O. M. Eichhoff, N. C. Schlegel, U. Döbbeling, S. Hemmi and R. Dummer, In vivo switching of human melanoma cells between proliferative and invasive states, Cancer Res., 68 (2008), 650-656. doi: 10.1158/0008-5472.CAN-07-2491.

[30]

J. Kelkel and C. Surulescu, On some models for cancer cell migration through tissue networks, Mathematical Biosciences and Engineering, 8 (2011), 575-589. doi: 10.3934/mbe.2011.8.575.

[31]

J. Kelkel and C. Surulescu, A multiscale approach to cell migration in tissue networks, Mathematical Models and Methods in Applied Sciences, 23 (2012), 1150017, 25 pp. doi: 10.1142/S0218202511500175.

[32]

D. A. Lauffenburger and J. L. Lindermann, Receptors. Models for Binding, Trafficking and Signaling, Oxford University Press, 1993.

[33]

K.R. Legate, S.A. Wickström, and R. Fässler, Genetic and cell biological analysis of integrin outside-in signaling, Genes Dev., 23 (2009), 397-418. doi: 10.1101/gad.1758709.

[34]

T. Lorenz and C. Surulescu, On a class of multiscale cancer cell migration models: Well-posedness in less regular function spaces, Mathematical Models and Methods in Applied Sciences, 24 (2014), 2383-2436. doi: 10.1142/S0218202514500249.

[35]

Y. Matsukado, C. MacCarty, J. Kernohan, et al., The growth of glioblastoma multiforme (astrocytomas, grades 3 and 4) in neurosurgical practice, Journal of Neurosurgery, 18 (1961), 636-644. doi: 10.3171/jns.1961.18.5.0636.

[36]

G. Meral, C. Stinner and C. Surulescu, On a multiscale model involving cell contractivity and its effects on tumor invasion, Discr. Cont. Dyn. Syst. B, 20 (2015), 189-213. doi: 10.3934/dcdsb.2015.20.189.

[37]

G. Meral and C. Surulescu, Mathematical Modelling, Analysis and numerical simulations for the influence of the heat shock proteins on tumour invasion, Journal of Math. Anal. Appl., 408 (2013), 597-614. doi: 10.1016/j.jmaa.2013.06.017.

[38]

N. Oppenheimer-Marks and P. E. Lipsky, Adhesion molecules and the regulation of the migration of lymphocytes, in Adhesion Molecules and Chemokynes in Lymphocyte Trafficking (ed. A. Hamann), Harwood Acad. Publ. (1997), 55-88.

[39]

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. Theoretical Biol., 323 (2013), 25-39. doi: 10.1016/j.jtbi.2013.01.014.

[40]

M. Sidani, D. Wessels, G. Mouneimne, M. Ghosh, S. Goswami, C. Sarmiento, W. Wang, S. Kuhl, M. El-Sibai, J. M. Backer and R. Eddy, D. Soll and J. Condeelis, Cofilin determines the migration behavior and turning frequency of metastatic cancer cells, The Journal of Cell Biology, 179 (2007), 777-791. doi: 10.1083/jcb.200707009.

[41]

C. Stinner, C. Surulescu and G. Meral, A multiscale model for pH-tactic tumor invasion with time-varying carrying capacities, IMA Journal of Applied Mathematics, 80 (2015), 1300-1321. doi: 10.1093/imamat/hxu055.

[42]

C. Stinner, C. Surulescu and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Analysis, 46 (2014), 1969-2007. doi: 10.1137/13094058X.

[43]

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, Neurocarciology, 46 (2004), 339-350. doi: 10.1007/s00234-003-1114-x.

[44]

Z. Szymanska, J. Urbanski and A. Marciniak-Czochra, Mathematical modelling of the influence of heat shock proteins on cancer invasion of tissue, Journal of Mathematical Biology, 58 (2009), 819-844. doi: 10.1007/s00285-008-0220-0.

[45]

D. S. Tuch, Q-ball imaging, Magnetic resonance in Medicine, 52 (2004), 1358-1372. doi: 10.1002/mrm.20279.

[46]

D. S. Tuch, T. G. Reese, M. R. Wiegell and V. J. Wedeen, Diffusion MRI of complex neural architecture, Neuron, 40 (2003), 885-895. doi: 10.1016/S0896-6273(03)00758-X.

[47]

J. H. Uhm, C. L. Gladson and J. S. Rao, The role of integrins in the malignant phenotype of gliomas, Frontiers in Bioscience, 4 (1999), 188-199.

[48]

S. Wagner, S. M. Rampersad, Ü Aydin, J. Vorwerk, T. F. Oostendorp, T. Neuling, C. S. Herrmann, D. F. Stegeman and C. H. Wolters, Investigation of tDCS volume conduction effects in a highly realistic head model, Journal of neural engineering, 11 (2014), 016002, 14pp. doi: 10.1088/1741-2560/11/1/016002.

[49]

M. F. Wheeler, An elliptic collocation-finite element method with interior penalties, SIAM Journal on Numerical Analysis, 15 (1978), 152-161. doi: 10.1137/0715010.

[50]

M. Wrensch, Y. Minn, T. Chew, M. Bondy and M. S. Berger, Epidemiology of primary brain tumors: Current concepts and review of the literature, Neuro-Oncology, 4 (2002), 278-299.

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