Abstract
Glioblastoma multiforme (GBM) is one of the deadliest human cancers and is characterized by fast growth and aggressive invasion. GBM also communicates with microglia and macrophages which are recruited by tumor cells to facilitate growth and invasion. In this study we investigate the biochemical and cell-mechanical interactions between the glioma cells and the microenvironment including resident glial cells and M1/M2 microglia that enhance tumor invasion. We develop various types of mathematical models that involve reaction-diffusion equations or multi-scale hybrid models for the important components in this mutual interaction. In particular, we investigate the dynamics of intracellular signaling including miR-451 and AMPK, biochemical interaction of a glioma with M1/M2 microglia via CSF-1-EGF-TGF-β signaling, and glioma cell infiltration through the narrow intercellular space via the regulation of myosin II. We show that these models can replicate the key features of the experimental findings and make novel predictions to guide future experiments aimed at the development of new anti-invasive strategies.
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References
R. Abe, S.C. Donnelly, T. Peng, R. Bucala, and C.N. Metz. Peripheral blood fibrocytes: differentiation pathway and migration to wound sites. J. Immunol., 166(12):7556–62, 2001.
B.D. Aguda, Y. Kim, M.G. Hunter, A. Friedman, and C.B. Marsh. MicroRNA regulation of a cancer network: Consequences of the feedback loops involving miR-17-92, E2F, and Myc. PNAS, 105(50):19678–19683, 2008.
Judit Anido, Andrea Sáez-Borderías, Alba Gonzàlez-Juncà, Laura Rodón, Gerard Folch, Maria A Carmona, Rosa M Prieto-Sánchez, Ignasi Barba, Elena Martínez-Sáez, Ludmila Prudkin, et al. Tgf-β receptor inhibitors target the cd44 high/id1 high glioma-initiating cell population in human glioblastoma. Cancer cell, 18(6):655–668, 2010.
A. Bartocci, D.S. Mastrogiannis, G. Migliorati, R.J. Stockert, A.W. Wolkoff, and E.R. Stanley. Macrophages specifically regulate the concentration of their own growth factor in the circulation. Natl Acad Sci U S A, 84(17):6179–83, 1987.
C. Beadle, M.C. Assanah, P. Monzo, R. Vallee, S.S. Rosenfield, and P. Canoll. The role of myosin II in glioma invasion of the brain. Mol. Biol. Cell, 19:3357–3368, 2008.
L.T. Bemis and P. Schedin. Reproductive state of rat mammary gland stroma modulates human breast cancer cell migration and invasion. Cancer Res, 60(13):3414–8, 2000.
D. Bray. Cell movements. New York: Gerland, 1992.
D.R. Brown. Dependence of neurones on astrocytes in a coculture system renders neurones sensitive to transforming growth factor beta1-induced glutamate toxicity. J Neurochem, 72(3):943–53, 1999.
J.J. Casciari, S.V. Sotirchos, and R.M. Sutherland. Glucose diffusivity in multicellular tumor spheroids. Cancer Research, 48(14):3905–3909, 1988.
M. A. J. Chaplain. Avascular growth, angiogenesis and vascular growth in solid tumours: The mathematical modeling of the stages of tumour development. Mathl. Comput. Modeling, 23(6):47–87, 1996.
N.A. Charles, E.C. Holland, R. Gilbertson, R. Glass, and H. Kettenmann. The brain tumor microenvironment. Glia, 59:1169–1180, 2011.
D. Chen, J.M. Roda, C.B. Marsh, T.D. Eubank, and A. Friedman. Hypoxia inducible factors-mediated inhibition of cancer by GM-CSF: a mathematical model. Bull Math Biol, 74(11):2752–2777, 2012.
M.B Chen, M.X. Wei, J.Y. Han, X.Y. Wu, C. Li, J. Wang, W. Shen, and P.H. Lu. MicroRNA-451 regulates AMPK/mTORC1 signaling and fascin1 expression in HT-29 colorectal cancer. Cellular signalling, 26(1):102–109, 2014.
A. Claes, A.J. Idema, and P. Wesseling. Diffuse glioma growth: a guerilla war. Acta Neuropathologica, 114(5):443–458, 2007.
Salvatore J Coniglio, Eliseo Eugenin, Kostantin Dobrenis, E Richard Stanley, Brian L West, Marc H Symons, and Jeffrey E Segall. Microglial stimulation of glioblastoma invasion involves epidermal growth factor receptor (EGFR) and colony stimulating factor 1 receptor (CSF-1R) signaling. Molecular medicine, 18(3):519, 2012.
B.E. Crute, K. Seefeld, J. Gamble, B.E. Kemp, and L.A. Witters. Functional domains of the alpha1 catalytic subunit of the AMP-activated protein kinase. J Biol Chem, 273(52):35347–54, 1998.
J. C. Dallon and H. G. Othmer. How cellular movement determines the collective force generated by the dictyostelium discoideum slug. J. Theor. Biol., 231:203–222, 2004.
J.F. de Groot and M.R. Gilbert. New molecular targets in malignant gliomas. Curr Opin Neurol, 20:712–718, 2007.
Micah Dembo and Francis Harlow. Cell motion, contractile networks, and the physics of interpenetrating reactive flow. Biophys. J, 50:109–121, 1986.
R.B. Dickinson and R.T. Tranquillo. A stochastic model for adhesion-mediated cell random motility and haptotaxis. J Math Biol, 31(6):563–600, 1993.
Azadeh Farin, Satoshi O. Suzuki, Michael Weiker, James E. Goldman, Jeffrey N. Bruce, and Peter Canoll. Transplanted glioma cells migrate and proliferate on host brain vasculature: a dynamic analysis. Glia, 53(8):799–808, June 2006.
A. Friedman, C. Huang, and J. Yong. Effective permeability of the boundary of a domain. Commun. In Partial differential equations, 20(1&2):59–102, 1995.
E.A. Gaffney, K. Pugh, P.K. Maini, and F. Arnold. Investigating a simple model of cutaneous wound healing angiogenesis. J Math Biol, 45(4):337–74, 2002.
M.P. Gantier, C.E. McCoy, I. Rusinova, D. Saulep, D. Wang, D. Xu, A.T. Irving, M.A. Behlke, P.J. Hertzog, F. Mackay, and B.R. Williams. Analysis of microRNA turnover in mammalian cells following Dicer1 ablation. Nucleic Acids Res, 39(13):5692–703, 2011.
R.A. Gatenby and R.J. Gillies. Why do cancers have high aerobic glycolysis? Nat Rev Cancer, 4(11):891–9, 2004.
J. Godlewski, A. Bronisz, M.O. Nowicki, E.A. Chiocca, and S. Lawler. microRNA-451: A conditional switch controlling glioma cell proliferation and migration. Cell Cycle, 9(14):2742–8, 2010.
J. Godlewski, H.B. Newton, E.A. Chiocca, and S.E. Lawler. MicroRNAs and glioblastoma; the stem cell connection. Cell Death Differ, 17(2):221–8, 2010.
J. Godlewski, M.O. Nowicki, A. Bronisz, G. Nuovo J. Palatini, M. D. Lay, J.V. Brocklyn, M.C. Ostrowski, E.A. Chiocca, and S.E. Lawler. MircroRNA-451 regulates LKB1/AMPK signaling and allows adaptation to metabolic stress in glioma cells. Molecular Cell, 37:620–632, 2010.
K. Gordon. Mathematical modelling of cell-cycle-dependent chemotherapy drugs-implications for cancer treatment. Ph.D. Thesis, University of Dundee, Dundee., pages –, 2006.
D.J. Gow, K.A. Sauter, C. Pridans, L. Moffat, A. Sehgal, B.M. Stutchfield, S. Raza, P.M. Beard, Y.T. Tsai, G. Bainbridge, P.L. Boner, G. Fici, D. Garcia-Tapia, R.A. Martin, T. Oliphant, J.A. Shelly, R. Tiwari, T.L. Wilson, L.B. Smith, N.A. Mabbott, and D.A. Hume. Characterisation of a novel fc conjugate of macrophage colony-stimulating factor. Molecular Therapy, 22(9):1580–1592, 2014.
M.E. Gracheva and H.G. Othmer. A continuum model of motility in ameboid cells. Bull. Math. Biol., 66:167–194, 2004.
M.G. Vander Heiden, L.C. Cantley, and C.B. Thompson. Understanding the Warburg effect: the metabolic requirements of cell proliferation. Science, 324(5930):1029–33, 2009.
B. Hong, B. Wiese, M. Bremer, H.E. Heissler, F. Heidenreich, J.K. Krauss, and M. Nakamura. Multiple microsurgical resections for repeated recurrence of glioblastoma multiforme. Am J Clin Oncol, 36(3):261–268, 2013.
S. Ivkovic, C. Beadle, S. Noticewal, S.C. Massey, K.R. Swanson, L.N. Toro, A.R. Bresnick, P. Canoll, and S.S. Rosenfelda. Direct inhibition of myosin II effectively blocks glioma invasion in the presence of multiple motogens. Mol Biol Cell, 23(4):533–542, 2012.
L.J. Kaufman, C.P. Brangwynne, K.E. Kasza, E. Filippidi, V.D. Gordon, T.S. Deisboeck, and D.A. Weitz. Glioma expansion in collagen i matrices: Analyzing collagen concentration-dependent growth and motility patterns. Biophys. J. BioFAST, 89:635–650, 2005.
H.D. Kim, T.W. Guo, A.P. Wu, A. Wells, F.B. Gertler, and D.A. Lauffenburger. Epidermal growth factor-induced enhancement of glioblastoma cell migration in 3D arises from an intrinsic increase in speed but an extrinsic matrix and proteolysis-dependent increase in persistence. Mol Biol Cell, 19:4249–4259, 2008.
J.W. Kim and C.V. Dang. Cancer’s molecular sweet tooth and the Warburg effect. Cancer Res, 66(18):8927–30, 2006.
Y. Kim. Regulation of cell proliferation and migration in glioblastoma: New therapeutic approach. Frontiers in Molecular and Cellular Oncology, 3:53, 2013.
Y. Kim and A. Friedman. Interaction of tumor with its microenvironment : A mathematical model. Bull. Math. Biol., 72(5):1029–1068, 2010.
Y. Kim, H. Jeon, and H.G. Othmer. The role of the tumor microenvironment in glioblastoma: A mathematical model. IEEE Trans Biomed Eng, 64(3):519–527, 2017.
Y. Kim, S. Lawler, M.O. Nowicki, E.A Chiocca, and A. Friedman. A mathematical model of brain tumor : pattern formation of glioma cells outside the tumor spheroid core. J. Theo. Biol., 260:359–371, 2009.
Y. Kim and H.G. Othmer. A hybrid model of tumor-stromal interactions in breast cancer. Bull Math Biol, 75:1304–1350, 2013.
Y. Kim, G. Powathil, H. Kang, D. Trucu, H. Kim, S. Lawler, and M. Chaplain. Strategies of eradicating glioma cells: A multi-scale mathematical model with miR-451-AMPK-mTOR control. PLoS One, 10(1):e0114370, 2015.
Y. Kim and S. Roh. A hybrid model for cell proliferation and migration in glioblastoma. Discrete and Continuous Dynamical Systems-B, 18(4):969–1015, 2013.
Y. Kim, S. Roh, S. Lawler, and A. Friedman. miR451 and AMPK/MARK mutual antagonism in glioma cells migration and proliferation. PLoS One, 6(12):e28293, 2011.
Y. Kim, M. Stolarska, and H.G. Othmer. A hybrid model for tumor spheroid growth in vitro I: Theoretical development and early results. Math. Models Methods in Appl Scis, 17:1773–1798, 2007.
Y. Kim, M. Stolarska, and H.G. Othmer. The role of the microenvironment in tumor growth and invasion. Prog Biophys Mol Biol, 106:353–379, 2011.
Y. Kim, J. Wallace, F. Li, M. Ostrowski, and A. Friedman. Transformed epithelial cells and fibroblasts/myofibroblasts interaction in breast tumor: a mathematical model and experiments. J. Math. Biol., 61(3):401–421, 2010.
S. Koka, J.B. Vance, and G.I. Maze. Bone growth factors: potential for use as an osseointegration enhancement technique (oet). J West Soc Periodontol Periodontal Abstr, 43(3):97–104, 1995.
Yoshihiro Komohara, Yukio Fujiwara, Koji Ohnishi, and Motohiro Takeya. Tumor-associated macrophages: Potential therapeutic targets for anti-cancer therapy. Advanced drug delivery reviews, 2015.
M. Kovacs, J. Toth, C. Hetenyi, A. Malnasi-Csizmadia, and J.R. Sellers. Mechanism of blebbistatin inhibition of myosin II. JOURNAL OF BIOLOGICAL CHEMISTRY, 279(34):35557–35563, 2004.
M. Kretzschmar. Transforming growth factor-beta and breast cancer: Transforming growth factor-beta/smad signaling defects and cancer. Breast Cancer Res, 2(2):107–115, 2000.
J.E. Kudlow, C.Y. Cheung, and J.D. Bjorge. Epidermal growth factor stimulates the synthesis of its own receptor in a human breast cancer cell line. J Biol Chem, 261(9):4134–8, 1986.
L.A. Kunz-Schughart, S. Wenninger, T. Neumeier, P. Seidl, and R. Knuechel. Three-dimensional tissue structure affects sensitivity of fibroblasts to tgf-beta 1. Am J Physiol Cell Physiol, 284(1):C209–19, 2003.
H.G. Lee and Y. Kim. The role of the microenvironment in regulation of cspg-driven invasive and non-invasive tumor growth in glioblastoma. Japan J. Indust. Appl. Math., 32(3):771–805, 2016.
W. Lee, S. Lim, and Y. Kim. The role of myosin II in glioma invasion: A mathematical model. PLoS One, 12(2):e0171312, 2017.
C.K. Li. The glucose distribution in 9l rat brain multicell tumor spheroids and its effect on cell necrosis. Cancer, 50(10):2066–73, 1982.
H.Y. Li, Y. Zhang, J.H Cai, and H.L. Bian. MicroRNA-451 inhibits growth of human colorectal carcinoma cells via downregulation of PI3K/AKT pathway. Asian Pacific Journal of Cancer Prevention, 14(6):3631–3634, 2013.
W. Li and M.B. Graeber. The molecular profile of microglia under the influence of glioma. Neuro Oncol, 14:958–978, 2012.
S.C. Lin and D.G. Hardie. AMPK: Sensing Glucose as well as Cellular Energy Status. Cell Metab, 27(2):299–313, 2018.
D.S. Markovic, R. Glass, M. Synowitz, N. Van Rooijen, and H. Kettenmann. Microglia stimulate the invasiveness of glioma cells by increasing the activity of metalloprotease-2. J Neuropathol Exp Neurol, 64:754–762, 2005.
J. Massague. Tgf-beta signal transduction. Annual Review of Biochemistry, 67(1):753, 1998.
J. Massagué. TGF [beta] in cancer. Cell, 134(2):215–230, 2008.
J. Mercapide, .R. Cicco, J.S. Castresana, and A.J. Klein-Szanto. Stromelysin-1/matrix metalloproteinase-3 (mmp-3) expression accounts for invasive properties of human astrocytoma cell lines. Int J Cancer, 106(5):676–82, 2003.
H.G. Moller, A.P. Rasmussen, H.H. Andersen, K.B. Johnsen, M. Henriksen, and M. Duroux. A systematic review of microrna in glioblastoma multiforme: micro-modulators in the mesenchymal mode of migration and invasion. Mol Neurobiol, 47(1):131–44, 2013.
P. Namy, J. Ohayon, and P. Tracqui. Critical conditions for pattern formation and in vitro tubulogenesis driven by cellular traction fields. J Theor Biol, 227(1):103–20, 2004.
A.K. Newell-Litwa, R. Horwitz, and M.L. Lamers. Non-muscle myosin II in disease: mechanisms and therapeutic opportunities. Dis Model Mech, 8(12):1495–1515, 2015.
H. Oren, N. Duman, H. Abacioglu, H. Ozkan, and G. Irken. Association between serum macrophage colony-stimulating factor levels and monocyte and thrombocyte counts in healthy, hypoxic, and septic term neonates. Pediatrics, 108(2):329–332, 2001.
M. R. Owen and J. A. Sherratt. Pattern formation and spatiotemporal irregularity in a model for macrophage-tumour interaction. J. Theor. Biol., 189:63–80, 1997.
M.R. Owen, H.M. Byrne, and C.E. Lewis. Mathematical modelling of the use of macrophages as vehicles for drug delivery to hypoxic tumour sites. Journal of Theoretical Biology, 226(4):377–391, 2004.
A.J. Perumpanani and H.M. Byrne. Extracellular matrix concentration exerts selection pressure on invasive cells. Eur J Cancer, 35(8):1274–80, 1999.
C. Peskin. The immersed boundary method. Acta Numerica, 11:479–517, 2002.
G. J. Pettet, H. M. Byrne, D. L. S. McElwain, and J. Norbury. A model of wound healing angiogenesis in soft-tissue. Math. Bios, 136:35–64, 1996.
J. Pinkas and B.A. Teicher. Tgf-beta in cancer and as a therapeutic target. Biochem Pharmacol, 72(5):523–9, 2006.
G.G. Powathil, K.E. Gordon, L.A. Hill, and M.A. Chaplain. Modelling the effects of cell-cycle heterogeneity on the response of a solid tumour to chemotherapy: Biological insights from a hybrid multiscale cellular automaton model. J Theor Biol, 308:1–19, 2012.
S.M. Pyonteck, L. Akkari, A.J. Schuhmacher, R.L. Bowman, L. Sevenich, D.F. Quail, O.C. Olson, M.L. Quick, J.T. Huse, V. Teijeiro, M. Setty, C.S. Leslie, Y. Oei, A. Pedraza, J. Zhang, C.W. Brennan, J.C. Sutton, E.C. Holland, D. Daniel, and J.A. Joyce. CSF-1R inhibition alters macrophage polarization and blocks glioma progression. Nat Med, 19(10):1264–72, 2013.
D.A. Reardon, K.L. Fink, T. Mikkelsen, T.F. Cloughesy, A. O’Neill, S. Plotkin, M. Glantz, P. Ravin, J.J. Raizer, K.M. Rich, D. Schiff, W.R. Shapiro, S. Burdette-Radoux, E.J. Dropcho, S.M. Wittemer, J. Nippgen, M. Picard, and L.B. Nabors. Randomized phase II study of cilengitide, and integrin-targeting arginine-glycine-aspartic acid peptide, in recurrent glioblastoma multiforme. J Clin Oncol, 26:5610–5617, 2008.
K.A. Rejniak and R.H. Dillon. A single cell-based model of the ductal tumour microarchitecture. Computational and Mathematical Methods in Medicine, 8(1):51–69, 2007.
A.A.V. Los Reyes, E. Jung, and Y. Kim. Optimal control strategies of eradicating invisible glioblastoma cells after conventional surgery. J. Roy Soc Interface, 12:20141392, 2015.
Z. Rong, U. Cheema, and P. Vadgama. Needle enzyme electrode based glucose diffusive transport measurement in a collagen gel and validation of a simulation model. Analyst, 131(7):816–21, 2006.
S. Saffarian, I.E. Collier, B.L. Marmer, E.L. Elson, and G. Goldberg. Interstitial collagenase is a brownian ratchet driven by proteolysis of collagen. Science, 306(5693):108–11, 2004.
Y. Sakamoto, S. Prudhomme, and M. H. Zaman. Viscoelastic gel-strip model for the simulation of migrating cells. Annals of Biomedical Engineering, 39(11):2735–2749, 2011.
L.M. Sander and T.S. Deisboeck. Growth patterns of microscopic brain tumors. Phys. Rev. E, 66:051901, 2002.
H. Schattler, Y. Kim, U. Ledzewicz, A.A.d. los Reyes V, and E. Jung. On the control of cell migration and proliferation in glioblastoma. Proceeding of the IEEE Conference on Decision and Control,, 978-1-4673-5716-6/13:1810–1815, 2013.
S. Sen, M. Dong, and S. Kumar. Isoform-specific contributions of a-cctinin to glioma cell mechanobiology. PLoS One, 4(12):e8427, 2009.
J. A. Sherratt and J. D. Murray. Models of epidermal wound healing. Proc.R. Soc.Lond., B241:29–36, 1990.
D.L. Silbergeld and M.R. Chicoine. Isolation and characterization of human malignant glioma cells from histologically normal brain. J Neurosurg, 86(3):525–31, 1997.
Daniel J Silver, Florian A Siebzehnrubl, Michela J Schildts, Anthony T Yachnis, George M Smith, Amy A Smith, Bjorn Scheffler, Brent A Reynolds, Jerry Silver, and Dennis A Steindler. Chondroitin sulfate proteoglycans potently inhibit invasion and serve as a central organizer of the brain tumor microenvironment. The Journal of Neuroscience, 33(39):15603–15617, 2013.
A.M. Stein, T. Demuth, D. Mobley, M. Berens, and L.M. Sander. A mathematical model of glioblastoma tumor spheroid invasion in a three-dimensional in vitro experiment. Biophys J, 92(1):356–65, 2007.
C. L. Stokes and D. A. Lauffenburger. Analysis of the roles of microvessel endothelial cell random motility and chemotaxis in angiogenesis. J. theor. Biol, 152:377–403, 1991.
L.M. Swift, H. Asfour, N.G. Posnack, A. Arutunyan, M.W. Kay, and N. Sarvazyan. Properties of blebbistatin for cardiac optical mapping and other imaging applications. Pflugers Arch., 464(5):503–512, 2012.
B. Szomolay, T.D. Eubank, R.D. Roberts, C.B. Marsh, and A. Friedman. Modeling the inhibition of breast cancer growth by GM-CSF. J. Theor Biol, 303:141–151, 2012.
S. Tang, H. Liu, G. Chen, Q. Roa, and Y. Geng. Internalization and half-life of membrane-bound macrophage colony-stimulating factor. Chinese Sc Bull, 45:1697–1703, 2000.
R.G. Thorne, S. Hrabetov, and C. Nicholson. Diffusion of epidermal growth factor in rat brain extracellular space measured by integrative optical imaging. J Neurophysiol, 92(6):3471–81, 2004.
J-W. Tsai, K. H. Bremner, and R. B. Vallee. Dual subcellular roles for lis1 and dynein in radial neuronal migration in live brain tissue. Nat. Neurosci., 10:970–979, 2007.
J.C. Utting, A.M. Flanagan, A. Brandao-Burch, I.R. Orriss, and T.R. Arnett. Hypoxia stimulates osteoclast formation from human peripheral blood. Cell Biochem Funct, 28(5):374–380, 2010.
M.M. Valter, O.D. Wiestler, and T. Pietsch. Differential control of vegf synthesis and secretion in human glioma cells by il-1 and egf. International Journal of Developmental Neuroscience, 17(5–6):565–577, 1999.
L.M. Wakefield, D.M. Smith, T. Masui, C.C. Harris, and M.B. Sporn. Distribution and modulation of the cellular receptor for transforming growth factor-beta. J Cell Biol, 105(2):965–75, 1987.
O. Warburg. On the origin of cancer cells. Science, 123(3191):309–14, 1956.
A. Wesolowska, A. Kwiatkowska, L. Slomnicki, M. Dembinski, A. Master, M. Sliwa, K. Franciszkiewicz, S. Chouaib, and B. Kaminska. Microglia-derived TGF-beta as an important regulator of glioblastoma invasion–an inhibition of TGF-beta-dependent effects by shRNA against human TGF-beta type ii receptor. Oncogene, 27(7):918–30, 2008.
E.A. Woodcock, S.L. Land, and R.K. Andrews. A low affinity, low molecular weight endothelin-a receptor present in neonatal rat heart. Clin Exp Pharmacol Physiol, 20(5):331–4, 1993.
Xian-zong Ye, Sen-lin Xu, Yan-hong Xin, Shi-cang Yu, Yi-fang Ping, Lu Chen, Hua-liang Xiao, Bin Wang, Liang Yi, Qing-liang Wang, et al. Tumor-associated microglia/macrophages enhance the invasion of glioma stem-like cells via tgf-β1 signaling pathway. The Journal of Immunology, 189(1):444–453, 2012.
B.B. Zhang, G. Zhou, and C. Li. AMPK: an emerging drug target for diabetes and the metabolic syndrome. Cell metabolism, 9(5):407–416, 2009.
Z. Zi, Z. Feng, D. A. Chapnick, M. Dahl, D. Deng, E. Klipp, A. Moustakas, and Xuedong Liu. Quantitative analysis of transient and sustained transforming growth factor-beta signaling dynamics. Molecular Systems Biology, 7(492):1–12, 2011.
Acknowledgements
YJK was supported by the Basic Science Research Program through the National Research Foundation of Korea by the Ministry of Education (NRF-2015R1D1A1A0 1058702). WHL was supported by the National Institute for Mathematical Sciences (NIMS) Grant funded by the Korean government (Grant No. B18130000).
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Kim, Y. et al. (2018). The Role of Microenvironment in Regulation of Cell Infiltration in Glioblastoma. In: Stolarska, M., Tarfulea, N. (eds) Cell Movement. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-96842-1_2
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