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.  doi: 10.1137/S0036142901384162.  Google Scholar

[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.  doi: 10.1007/s00607-008-0003-x.  Google Scholar

[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.  doi: 10.1007/s00607-008-0004-9.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.mcm.2009.12.002.  Google Scholar

[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).  doi: 10.1142/S0218202512005885.  Google Scholar

[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.  doi: 10.1142/S021820251350053X.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1051/mmnp/20127106.  Google Scholar

[10]

S. Coons, Anatomy and growth patterns of diffuse gliomas,, in The gliomas (eds. M. Berger and C. Wilson), (1999), 210.   Google Scholar

[11]

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

[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.   Google Scholar

[13]

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

[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.   Google Scholar

[15]

M. Descoteaux, High Angular Resolution Diffusion MRI: From Local Estimation to Segmentation and Tractography,, Ph.D. thesis, (2008).   Google Scholar

[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.  doi: 10.1007/s00285-014-0822-7.  Google Scholar

[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.  Google Scholar

[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.  doi: 10.1137/040603565.  Google Scholar

[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.  doi: 10.1093/imanum/drm050.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[22]

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

[23]

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

[24]

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

[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.  doi: 10.1093/imammb/dqq011.  Google Scholar

[26]

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

[27]

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

[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.  doi: 10.1137/S0036139999358167.  Google Scholar

[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.  doi: 10.1158/0008-5472.CAN-07-2491.  Google Scholar

[30]

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

[31]

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

[32]

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

[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.  doi: 10.1101/gad.1758709.  Google Scholar

[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.  doi: 10.1142/S0218202514500249.  Google Scholar

[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.  doi: 10.3171/jns.1961.18.5.0636.  Google Scholar

[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.  doi: 10.3934/dcdsb.2015.20.189.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2013.06.017.  Google Scholar

[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), (1997), 55.   Google Scholar

[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.  doi: 10.1016/j.jtbi.2013.01.014.  Google Scholar

[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.  doi: 10.1083/jcb.200707009.  Google Scholar

[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.  doi: 10.1093/imamat/hxu055.  Google Scholar

[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.  doi: 10.1137/13094058X.  Google Scholar

[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.  doi: 10.1007/s00234-003-1114-x.  Google Scholar

[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.  doi: 10.1007/s00285-008-0220-0.  Google Scholar

[45]

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

[46]

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

[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.   Google Scholar

[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).  doi: 10.1088/1741-2560/11/1/016002.  Google Scholar

[49]

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

[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.   Google Scholar

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.  doi: 10.1137/S0036142901384162.  Google Scholar

[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.  doi: 10.1007/s00607-008-0003-x.  Google Scholar

[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.  doi: 10.1007/s00607-008-0004-9.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.mcm.2009.12.002.  Google Scholar

[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).  doi: 10.1142/S0218202512005885.  Google Scholar

[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.  doi: 10.1142/S021820251350053X.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1051/mmnp/20127106.  Google Scholar

[10]

S. Coons, Anatomy and growth patterns of diffuse gliomas,, in The gliomas (eds. M. Berger and C. Wilson), (1999), 210.   Google Scholar

[11]

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

[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.   Google Scholar

[13]

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

[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.   Google Scholar

[15]

M. Descoteaux, High Angular Resolution Diffusion MRI: From Local Estimation to Segmentation and Tractography,, Ph.D. thesis, (2008).   Google Scholar

[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.  doi: 10.1007/s00285-014-0822-7.  Google Scholar

[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.  Google Scholar

[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.  doi: 10.1137/040603565.  Google Scholar

[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.  doi: 10.1093/imanum/drm050.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[22]

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

[23]

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

[24]

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

[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.  doi: 10.1093/imammb/dqq011.  Google Scholar

[26]

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

[27]

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

[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.  doi: 10.1137/S0036139999358167.  Google Scholar

[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.  doi: 10.1158/0008-5472.CAN-07-2491.  Google Scholar

[30]

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

[31]

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

[32]

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

[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.  doi: 10.1101/gad.1758709.  Google Scholar

[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.  doi: 10.1142/S0218202514500249.  Google Scholar

[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.  doi: 10.3171/jns.1961.18.5.0636.  Google Scholar

[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.  doi: 10.3934/dcdsb.2015.20.189.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2013.06.017.  Google Scholar

[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), (1997), 55.   Google Scholar

[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.  doi: 10.1016/j.jtbi.2013.01.014.  Google Scholar

[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.  doi: 10.1083/jcb.200707009.  Google Scholar

[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.  doi: 10.1093/imamat/hxu055.  Google Scholar

[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.  doi: 10.1137/13094058X.  Google Scholar

[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.  doi: 10.1007/s00234-003-1114-x.  Google Scholar

[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.  doi: 10.1007/s00285-008-0220-0.  Google Scholar

[45]

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

[46]

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

[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.   Google Scholar

[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).  doi: 10.1088/1741-2560/11/1/016002.  Google Scholar

[49]

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

[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.   Google Scholar

[1]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242

[2]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316

[3]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433

[4]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321

[5]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051

[6]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[7]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118

[8]

Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457

[9]

Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137

[10]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458

[11]

S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020435

[12]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[13]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073

[14]

Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297

[15]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[16]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

[17]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050

[18]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323

[19]

Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076

[20]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (28)
  • HTML views (0)
  • Cited by (17)

[Back to Top]