June  2013, 18(4): 969-1015. doi: 10.3934/dcdsb.2013.18.969

A hybrid model for cell proliferation and migration in glioblastoma

1. 

Department of Mathematics, Konkuk University, Seoul, South Korea

2. 

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, United States

Received  February 2012 Revised  April 2012 Published  February 2013

Glioblastoma is the most aggressive type of brain cancer with the median survival time of one year. A particular microRNA, miR-451, and its counterpart, AMPK complex are known to play a key role in controlling a balance between rapid proliferation and aggressive invasion in response to metabolic stress in the microenvironment. The present paper develops a hybrid model of glioblastoma that identifies a key mechanism behind the molecular switches between proliferative phase and migratory phase in response to metabolic stress and biophysical interaction between cells. We focus on the core miR-451-AMPK control system and show how up- or down-regulation of components in these pathways affects cell proliferation and migration. The model predicts the larger window of bistable systems when there exists a time delay in the inhibitory pathway from CAB39/LKB1/STRAD/AMPK to miR-451. Delayed down-regulation of miR-451 along this pathway would let glioma cells stay longer in the proliferative stage despite relatively low glucose levels, making it a possible therapeutic target. Analysis of the model predicts the existence of a limit cycle with two time delays. We then study a hybrid model for the biomechanical interaction between invasive and proliferative cells, in which all cells are modeled individually, and show how biophysical properties of cells and core miR-451-AMPK control system affect the growth/invasion patterns of glioma spheroids in response to various glucose levels in the microenvironment. The model predicts that cell migration not only depends on glucose availability but also on mechanical constraints between cells. The model suggests that adhesion strength between cells plays an important role in cell shedding from the main core and the disruption of cell-cell adhesion is a pre-requisite for glioma cell invasion. The model also suggests that injection of glucose after surgery will increase visibility of individual migratory cells and the second surgery may eradicate the remaining cancer cells, preventing regrowth of the invisible migratory glioma cells.
Citation: Yangjin Kim, Soyeon Roh. A hybrid model for cell proliferation and migration in glioblastoma. Discrete & Continuous Dynamical Systems - B, 2013, 18 (4) : 969-1015. doi: 10.3934/dcdsb.2013.18.969
References:
[1]

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 (2008), 19678. Google Scholar

[2]

S. Alexander and P. Friedl, Cancer invasion and resistance: Interconnected processes of disease progression and therapy failure,, Trends. Mol. Med., 18 (2012), 13. Google Scholar

[3]

D. Angeli, J. E Ferrell Jr. and E. D. Sontag, Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems,, Proc. Natl. Acad. Sci. USA, 101 (2004), 1822. Google Scholar

[4]

R. P. Araujo and D. L. S. McElwain, A history of the study of solid tumour growth: The contribution of mathematical modelling,, Bull. Math. Biol., 66 (2004), 1039. doi: 10.1016/j.bulm.2003.11.002. Google Scholar

[5]

H. J. Aronen, F. S. Pardo, D. N. Kennedy, J. W. Belliveau, S. D. Packard, D. W. Hsu, F. H. Hochberg, A. J. Fischman and B. R. Rosen, High microvascular blood volume is associated with high glucose uptake and tumor angiogenesis in human gliomas,, Clin. Cancer Res., 6 (2000), 2189. Google Scholar

[6]

K. Asano, C. D. Duntsch, Q. Zhou, J. D. Weimar, D. Bordelon, J. H. Robertson and T. Pourmotabbed, Correlation of n-cadherin expression in high grade gliomas with tissue invasion,, J. Neurooncol., 70 (2004), 3. Google Scholar

[7]

A. F. Baas, J. Kuipers, N. N. Wel, E. Batlle, H. K. Koerten, P. J. Peters and H. C. Clevers, Complete polarization of single intestinal epithelial cells upon activation of lkb1 by strad,, Cell, 116 (2004), 457. Google Scholar

[8]

E. Bandres, N. Bitarte, F. Arias, J. Agorreta, P. Fortes, X. Agirre, R. Zarate, J. A. Diaz-Gonzalez, N. Ramirez and J. J. Sola, microrna-451 regulates macrophage migration inhibitory factor production and proliferation of gastrointestinal cancer cells,, Clin. Cancer Res, 15 (2009), 2281. Google Scholar

[9]

D. P. Bartel, Micrornas: target recognition and regulatory functions,, Cell, 136 (2009), 215. Google Scholar

[10]

J. Boudeau, A. F. Baas, M. Deak, N. A. Morrice, A. Kieloch, M. Schutkowski, A. R. Prescott, H. C. Clevers and D. R. Alessi, Mo25alpha/beta interact with stradalpha/beta enhancing their ability to bind, activate and localize lkb1 in the cytoplasm,, EMBO J., 22 (2003), 5102. Google Scholar

[11]

H. Byrne and L. Preziosi, Modeling solid tumor growth using the theory of mixtures,, Math. Med. Biol., 20 (2004), 341. Google Scholar

[12]

H. M. Byrne, Dissecting cancer through mathematics: From the cell to the animal model,, Nature Reviews, 10 (2010), 221. Google Scholar

[13]

R. Cairns, I. Papandreou and N. Denko, Overcoming physiologic barriers to cancer treatment by molecularly targeting the tumor microenvironment,, Mol. Cancer Res., 4 (2006), 61. Google Scholar

[14]

A. Chauviere, L. Preziosi and H. Byrne, A model of cell migration within the extracellular matrix based on a phenotypic switching mechanism,, Math. Med. Biol., 27 (2010), 255. doi: 10.1093/imammb/dqp021. Google Scholar

[15]

J. D. Cheng and L. M. Weiner, Tumors and their microenvironments: tilling the soil. Commentary re: A. M. Scott et al., A Phase I dose-escalation study of sibrotuzumab in patients with advanced or metastatic fibroblast activation protein-positive cancer,, Clin. Cancer Res. 9 (2003), 9 (2003), 1590. Google Scholar

[16]

G. Cheng, J. Tse, R. K. Jain and L. L. Minn, Micro-environmental mechanical stress controls tumor spheroid size and morphology by supressing proliferation and inducing aopotosis in cancer cells,, PLoS One, (2009). Google Scholar

[17]

S. K. Chintala, J. C. Tonn and J. S. Rao, Matrix metalloproteinases and their biological function in human gliomas,, Int. J. Dev. Neurosci., 17 (1999), 495. Google Scholar

[18]

D. G. Chiro, R. L. DeLaPaz, R. A. Brooks, L. Sokoloff, P. L. Kornblith, B. H. Smith, N. J. Patronas, C. V. Kufta, R. M. Kessler, G. S. Johnston, R. G. Manning and A. P. Wolf, Glucose utilization of cerebral gliomas measured by [18f] fluorodeoxyglucose and positron emission tomography,, Neurology, 32 (1982), 1323. Google Scholar

[19]

A. Cho, Life's patterns: No need to spell it out?, Science, 303 (2004), 782. Google Scholar

[20]

G. Choe, J. K. Park, L. Jouben-Steele, T. J. Kremen, L. M. Liau, H. V. Vinters, T. F. Cloughesy and P. S. Mischel, Active matrix metalloproteinase 9 expression is associated with primary glioblastoma subtype,, Clin. Cancer Res., 8(9) (2002), 2894. Google Scholar

[21]

M. Crawford, E. Brawner, K. Batte, L. Yu, M. G. Hunter, G. A. Otterson, G. Nuovo, C. B. Marsh and S. P. Nana-Sinkam, Microrna-126 inhibits invasion in non-small cell lung carcinoma cell lines,, Biochem. Biophys. Res. Commun., 373 (2008), 607. Google Scholar

[22]

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 (1998), 35347. Google Scholar

[23]

J. C. Dallon and H. G. Othmer, A discrete cell model with adaptive signalling for aggregation of dictyostelium discoideum,, Phil. Trans. Roy. Soc. Lond, B352 (1997), 391. Google Scholar

[24]

J. C. Dallon and H. G. Othmer, How cellular movement determines the collective force generated by the dictyostelium discoideum slug,, J. Theor. Biol., 231 (2004), 203. doi: 10.1016/j.jtbi.2004.06.015. Google Scholar

[25]

F. G. Davis and B. J. McCarthy, Current epidemiological trends and surveillance issues in brain tumors,, Expert Rev. Anticancer Ther., 1 (2001), 395. Google Scholar

[26]

S. J. Day and P. A. Lawrence, Measuring dimensions: The regulation of size and shape,, Development, 127 (2000), 2977. Google Scholar

[27]

T. S. Deisboeck, M. E. Berens, A. R. Kansal, S. Torquato, A. O. Stemmer-Rachamimov and E. A. Chiocca, Pattern of self-organization in tumour systems: Complex growth dynamics in a novel brain tumour spheroid model,, Cell Prolif., 34 (2001), 115. Google Scholar

[28]

T. S. Deisboeck and I. D. Couzin, Collective behavior in cancer cell populations,, Bioessays, 31 (2009), 190. Google Scholar

[29]

T. Demuth and M. E. Berens, Molecular mechanisms of glioma cell migration and invasion,, J. Neurooncol., 70 (2004), 217. Google Scholar

[30]

J. B. Easton and P. J. Houghton, mtor and cancer therapy,, Oncogene, 25 (2006), 6436. Google Scholar

[31]

A. Esquela-Kerscher and F. J. Slack, Oncomirs - micrornas with a role in cancer,, Nat. Rev. Cancer, 6 (2006), 259. Google Scholar

[32]

J. E. Ferrell Jr, Self-perpetuating states in signal transduction: Positive feedback, double-negative feedback and bistability,, Curr. Opin. Cell Biol., 14 (2002), 140. Google Scholar

[33]

P. Friedl and S. Alexander, Cancer invasion and the microenvironment: Plasticity and reciprocity,, Cell, 147 (2011), 992. Google Scholar

[34]

G. Gabriely, T. Wurdinger, S. Kesari, C. C. Esau, J. Burchard, P. S. Linsley and A. M. Krichevsky, Microrna 21 promotes glioma invasion by targeting matrix metalloproteinase regulators,, Mol. Cell Biol., 28 (2008), 5369. Google Scholar

[35]

H. Gal, G. Pandi, A. A. Kanner, Z. Ram, G. Lithwick-Yanai, N. Amariglio, G. Rechavi and D. Givol, Mir-451 and imatinib mesylate inhibit tumor growth of glioblastoma stem cells biochem.,, Biophys. Res. Commun., 376 (2008), 86. Google Scholar

[36]

J. Galle, M. Loeffler and D. Drasdo, Modeling the effect of deregulated proliferation and apoptosis on the growth dynamics of epithelial cell populations in vitro,, Biophysical J., 88 (2005), 62. Google Scholar

[37]

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 (2011), 5692. Google Scholar

[38]

R. A. Gatenby and R. J. Gillies, Why do cancers have high aerobic glycolysis?, Nat. Rev. Cancer, 4 (2004), 891. Google Scholar

[39]

J. Godlewski, M. O. Nowicki, A. Bronisz, S. Williams, A. Otsuki, G. Nuovo, A. Raychaudhury, H. B. Newton, E. A. Chiocca and S. Lawler, Targeting of the bmi-1 oncogene/stem cell renewal factor by microrna-128 inhibits glioma proliferation and self-renewal,, Cancer Res., 68 (2008), 9125. Google Scholar

[40]

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 (2010), 2742. Google Scholar

[41]

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 (2010), 620. Google Scholar

[42]

S. Goldman, M. Levivier, B. Pirotte, J. M. Brucher, D. Wikler, P. Damhaut, E. Stanus, J. Brotchi and J. Hildebrand, Regional glucose metabolism and histopathology of gliomas. A study based on positron emission tomography-guided stereotactic biopsy,, Cancer, 78 (1996), 1098. Google Scholar

[43]

M. Gotte and G. W. Yip, Heparanase, hyaluronan, and CD44 in cancers: A breast carcinoma perspective,, Cancer Research, 66 (2006), 10233. Google Scholar

[44]

R. G. Hahn and T. Nystom, Plasma Volume Expansion Resulting from Intravenous Glucose Tolerance Test,, Comput. Math. Methods Med., (2011). doi: doi:10.1155/2011/965075. Google Scholar

[45]

D. G. Hardie, Amp-activated/snf1 protein kinases: Conserved guardians of cellular energy,, Nat. Rev. Mol. Cell Biol., 8 (2007), 774. Google Scholar

[46]

D. G. Hardie, I. P. Salt, S. A. Hawley and S. P. Davies, Amp-activated protein kinase: An ultrasensitive system for monitoring cellular energy charge,, Biochem. J., 338 (1999), 717. Google Scholar

[47]

H. L. Harpold, J.r. EC and K. R. Swanson, The evolution of mathematical modeling of glioma proliferation and invasion,, J. Neuropathol. Exp. Neurol., 66 (2007), 1. Google Scholar

[48]

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., 27 (2010), 255. doi: 10.1093/imammb/dqq011. Google Scholar

[49]

S. A. Hawley, J. Boudeau, J. L. Reid, K. J. Mustard, L. Udd, T. P. Makela, D. R. Alessi and D. G. Hardie, Complexes between the lkb1 tumor suppressor, strad alpha/beta and mo25 alpha/beta are upstream kinases in the amp-activated protein kinase cascade,, J. Biol., 2 (2003). Google Scholar

[50]

S. A. Hawley, M. A. Selbert, E. G. Goldstein, A. M. Edelman, D. Carling and D. G. Hardie, 5'-amp activates the amp-activated protein kinase cascade, and ca2+/calmodulin activates the calmodulin-dependent protein kinase i cascade, via three independent mechanisms,, J. Biol. Chem., 270 (1995), 27186. Google Scholar

[51]

M. G. Heiden, L. C. Cantley and C. B. Thompson, Understanding the warburg effect: The metabolic requirements of cell proliferation,, Science, 324 (2009), 1029. Google Scholar

[52]

B. Hegedus, A. Czirok, I. Fazekas, T. Babel, E. Madarasz and T. Vicsek, Locomotion and proliferation of glioblastoma cells in vitro: Statistical evaluation of videomicroscopic observations,, J. Neurosurg., 92 (2000), 428. Google Scholar

[53]

G. Helmlinger, P. A. Netti, H. C. Lichtenbeld, R. J. Melder and R. K. Jain, Solid stress inhibits the growth of multicellular tumor spheroids,, Nature Biotechnology, 15(8) (1997), 778. Google Scholar

[54]

A. F. Hezel and N. Bardeesy, Lkb1; linking cell structure and tumor suppression,, Oncogene, 27 (2008), 6908. Google Scholar

[55]

O. Ilina, G. Bakker, A. Vasaturo, R. M. Hofmann and P. Friedl, Two-photon laser-generated microtracks in 3d collagen lattices: principles of mmp-dependent and -independent collective cancer cell invasion,, Phys. Biol., 8 (2011). Google Scholar

[56]

K. Inoki, Y. Li, T. Xu and K. L. Guan, Rheb gtpase is a direct target of tsc2 gap activity and regulates mtor signaling,, Genes. Dev., 17 (2003), 1829. Google Scholar

[57]

K. Inoki, Y. Li, T. Zhu, J. Wu and K. L. Guan, Tsc2 is phosphorylated and inhibited by akt and suppresses mtor signalling,, Nat. Cell Biol., 4 (2002), 648. Google Scholar

[58]

J. Jaalinoja, R. Herva, M. Korpela, M. Hoyhtya and T. Turpeenniemi-Hujanen, Matrix metalloproteinase 2 (mmp-2) immunoreactive protein is associated with poor grade and survival in brain neoplasms,, J. Neurooncol., 46 (2000), 81. Google Scholar

[59]

V. L Jacobs, P. A. Valdes, W. F. Hickey and J. A. De Leo, Current review of in vivo gbm rodent models: emphasis on the cns-1 tumour model,, ASN NEURO, 3 (2011). Google Scholar

[60]

R. K. Jain, Transport of molecules in the tumor interstitium: a review,, Cancer Res., 47 (1987), 3039. Google Scholar

[61]

R. G. Jones and C. B. Thompson, Tumor suppressors and cell metabolism: A recipe for cancer growth,, Genes Dev., 23 (2009), 537. Google Scholar

[62]

L. J. Kaufma, C. P. Brangwynn, K. E. Kasz, E. Filippidi, V. D. Gordon, T. S. Deisboeck and D. A. Weitz, lioma expansion in Collagen I matrices: Analyzing Collagen concentration-dependent growth and motility patterns,, Biophys. J., 89 (2005), 635. Google Scholar

[63]

E. Khain and L. M. Sander, Dynamics and pattern formation in invasive tumor growth,, Phys. Rev. Lett., 96 (2006). Google Scholar

[64]

R. Khanin and V. Vinciotti, Computational modeling of post-transcriptional gene regulation by micrornas,, J. Comput. Biol., 15 (2008), 305. doi: 10.1089/cmb.2007.0184. Google Scholar

[65]

J. W. Kim and C. V. Dang, Cancer's molecular sweet tooth and the warburg effect,, Cancer Res., 66 (2006), 8927. Google Scholar

[66]

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 (2008), 4249. Google Scholar

[67]

Y. Kim and A. Friedman, Interaction of tumor with its microenvironment: A mathematical model,, Bull. Math. Biol., 72 (2010), 1029. doi: 10.1007/s11538-009-9481-z. Google Scholar

[68]

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 (2009), 359. Google Scholar

[69]

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 (2007), 1773. doi: 10.1142/S0218202507002479. Google Scholar

[70]

Y. Kim, S. Roh, S. Lawler and A. Friedman, miR451 and AMPK mutual antagonism in glioma cells migration and proliferation,, PLoS One, 6 (2011). Google Scholar

[71]

Y. Kim, M. Stolarska and H. G. Othmer, The role of the microenvironment in tumor growth and invasion,, Prog. Biophys. Mol. Biol., 106 (2011), 353. Google Scholar

[72]

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 (2010), 401. doi: 10.1007/s00285-009-0307-2. Google Scholar

[73]

W. P. Kloosterman and R. H. Plasterk, The diverse functions of micrornas in animal development and disease,, Dev. Cell, 11 (2006), 441. Google Scholar

[74]

C. Koike, T. D. McKee, A. Pluen, S. Ramanujan, K. Burton, L. L. Munn, Y. Boucher and R. K. Jain, Solid stress facilitates spheroid formation: potential involvement of hyaluronan,, British Journal of Cancer, 86 (2002), 947. Google Scholar

[75]

K. Lamszus, N. O. Schmidt, L. Jin, J. Laterra, D. Zagzag, D. Way, M. Witte, M. Weinand, I. D. Goldberg, M. Westphal and E. M. Rosen, Scatter factor promotes motility of human glioma and neuromicrovascular endothelial cells,, Int. J. Cancer, 75 (1998), 19. Google Scholar

[76]

S. Lawler and E. A. Chiocca, Emerging functions of micrornas in glioblastoma,, J. Neurooncol., 92 (2009), 297. Google Scholar

[77]

J. H. Lee, H. Koh, M. Kim, Y. Kim, S. Y. Lee, R. E. Karess, S. H. Lee, M. Shong, J. M. Kim, J. Kim and J. Chung, Energy-dependent regulation of cell structure by amp-activated protein kinase,, Nature, 447 (2007), 1017. Google Scholar

[78]

C. K. Li, The glucose distribution in 9l rat brain multicell tumor spheroids and its effect on cell necrosis,, Cancer, 50 (1982), 2066. Google Scholar

[79]

J. S. Lowengrub, H. B. Frieboes, F. Jin, Y. L. Chuang, X. Li, P. Macklin, S. M. Wise and V. Cristini, Nonlinear modelling of cancer: Bridging the gap between cells and tumours,, Nonlinearity, 23 (2010). doi: 10.1088/0951-7715/23/1/001. Google Scholar

[80]

A. D. Luca, Niccolo Arena, L. M. Sena and E. Medico, Met overexpression confers hgf-dependent invasive phenotype to human thyroid carcinoma cells in vitro,, Journal of Cellular Physiology, 180 (1999), 365. Google Scholar

[81]

M. Lund-Johansen, R. Bjerkvig, P. A. Humphrey, S. H. Bigner, D. D. Bigner and O. D. Laerum, Effect of epidermal growth factor on glioma cell growth, migration, and invasion in vitro,, Cancer Res., 50 (1990), 6039. Google Scholar

[82]

L. Ma, J. Teruya-Feldstein and R. A. Weinberg, Tumour invasion and metastasis initiated by microrna-10b in breast cancer,, Nature, 449 (2007), 682. Google Scholar

[83]

E. Mandonnet, J. Y. Delattre, M. L. Tanguy, K. R. Swanson, A. F. Carpentier, H. Duffau, P. Cornu, R. Effenterre, J.r. EC and L. Capelle, Continuous growth of mean tumor diameter in a subset of grade ii gliomas,, Ann. Neurol., 53 (2003), 524. Google Scholar

[84]

S. Marino, I. B. Hogue, C. J. Ray and D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology,, J. Theor. Biol., 254 (2008), 178. Google Scholar

[85]

N. I. Markevich, J. B. Hoek and B. N. Kholodenko, Signaling switches and bistability arising from multisite phosphorylation in protein kinase cascades,, J. Cell Biol., 164 (2004), 353. Google Scholar

[86]

M. O. Nowicki, N. Dmitrieva, A. M. Stein, J. L. Cutter, J. Godlewski, Y. Saeki, M. Nita, M. E. Berens, L. M. Sander and H. B. Newton, Lithium inhibits invasion of glioma cells; possible involvement of glycogen synthase kinase-3,, Neuro-oncol, 10 (2008), 690. Google Scholar

[87]

I. Papandreou, R. A. Cairns, L. Fontana, A. L. Lim and N. C. Denko, Hif-1 mediates adaptation to hypoxia by actively downregulating mitochondrial oxygen consumption,, Cell Metab., 3 (2006), 187. Google Scholar

[88]

J. B. Park, H. J. Kwak and S. H. Lee, Role of hyaluronan in glioma invasion,, Cell Adhesion and Migration, 2 (2008), 202. Google Scholar

[89]

M. J. Paszek and V. M. Weaver, The tension mounts: Mechanics meets morphogenesis and malignancy,, J. Mammary Gland Biol. Neoplasia, 9 (2004), 325. Google Scholar

[90]

M. J. Paszek, N. Zahir, K. R. Johnson, J. N. Lakins, G. I. Rozenberg, A. Gefen, C. A. Reinhart-King, S. S. Margulies, M. Dembo, D. Boettiger, D. A. Hammer and V. M. Weaver, Tensional homeostasis and the malignant phenotype,, Cancer cell, 8 (2005), 241. Google Scholar

[91]

C. Perego, C. Vanoni, S. Massari, A. Raimondi, S. Pola, M.G. Cattaneo, M. Francolini, L. M. Vicentini and G. Pietrini, Invasive behaviour of glioblastoma cell lines is associated with altered organisation of the cadherin-catenin adhesion system,, J. Cell Sci., 115 (2002), 3331. Google Scholar

[92]

K. Pham, A. Chauviere, H. Hatzikirou, X. Li, H. M. Byrne, V. Cristini and J. Lowengrub, Density-dependent quiescence in glioma invasion: Instability in a simple reaction-diffusion model for the migration/proliferation dichotomy,, Journal of Biological dynamics, (2011). doi: 10.1080/17513758.2011.590610. Google Scholar

[93]

M. Platten, W. Wick and M. Weller, Malignant glioma biology: role for tgf-beta in growth, motility, angiogenesis, and immune escape,, Microsc. Res. Tech., 52 (2001), 401. Google Scholar

[94]

C. J. Potter, L. G. Pedraza and T. Xu, Akt regulates growth by directly phosphorylating tsc2,, Nat. Cell Biol., 4 (2002), 658. Google Scholar

[95]

L. Preziosi and A. Tosin, Multiphase and multiscale trends in cancer modelling,, Math. Model. Natl. Phenom., 4 (2009), 1. doi: 10.1051/mmnp/20094301. Google Scholar

[96]

L. Preziosi and G. Vitale, A multiphase model of tumor and tissue growth including cell adhesion and plastic reorganization,, Math. Model. Method. Appl. Sci., 21 (2011), 1901. doi: 10.1142/S0218202511005593. Google Scholar

[97]

C. Ragan and M. Zuker amd M. A. Ragan, Quantitative prediction of miRNA-mRNA interaction based on equilibrium concentrations,, PLoS Comput. Biol., 7 (2011). doi: 10.1371/journal.pcbi.1001090. Google Scholar

[98]

K. A. Rejniak and C. J. McCawley, Current trends in mathematical modeling of tumor microenvironment interaction: A survey of tools and applications,, Experimental Biology and Medicine (Maywood), 235 (2010), 411. Google Scholar

[99]

K. A. Rejniak and A. R. A. Anderson, Hybrid models of tumor growth,, WIRES Syst. Biol. Med., 3 (2011), 115. Google Scholar

[100]

K. A. Rejniak and C. J. McCawley, Current trends in mathematical modeling of tumor microenvironment interaction: A survey of tools and applications,, Exp. Biol. Med. (Maywood), 235 (2010), 411. Google Scholar

[101]

A. Ridley, M. Schwartz, K. Burridge, R. Firtel, M. Ginsberg, G. B. Parsons and A. Horwitz, Cell migration: Integrating signals from front to back,, Science, 302 (2003), 1704. Google Scholar

[102]

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 (2006), 816. Google Scholar

[103]

J. M. Rozental, R. L. Levine and R. J. Nickles, Changes in glucose uptake by malignant gliomas: Preliminary study of prognostic significance,, J. Neurooncol., 10 (1991), 75. Google Scholar

[104]

O. Sampetrean, I. Saga, M. Nakanishi, E. Sugihara, R. Fukaya, N. Onishi, S. Osuka, M. Akahata, K. Kai, H. Sugimoto, A. Hirao and H. Saya, Invasion precedes tumor mass formation in a malignant brain tumor model of genetically modified neural stem cells,, Neoplasia, 13 (2011), 784. Google Scholar

[105]

L. M. Sander and T. S. Deisboeck, Growth patterns of microscopic brain tumors,, Phys. Rev. E, 66 (2002). Google Scholar

[106]

M. Scianna, R. M. Merks, L. Preziosi and E. Medico, Individual cell-based models of cell scatter of aro and mlp-29 cells in response to hepatocyte growth factor,, J. Theor. Biol., 260 (2009), 151. Google Scholar

[107]

S. Sen, M. Dong and S. Kumar, Isoform-specific contributions of a-Actinin to Glioma cell mechanobiology,, PLoS One, 4 (2009). Google Scholar

[108]

R. J. Shaw, N. Bardeesy, B. D. Manning, L. Lopez, M. Kosmatka, R. A. DePinho and L. C. Cantley, The lkb1 tumor suppressor negatively regulates mtor signaling,, Cancer Cell, 6 (2004), 91. Google Scholar

[109]

B. I. Shraiman, Mechanicall feedback as a possible regulator of tissue growth,, PNAS, 102 (2005), 3318. Google Scholar

[110]

S. C. Stein, A. Woods, N. A. Jones, M. D. Davison and D. Carling, The regulation of amp-activated protein kinase by phosphorylation,, Biochem. J., 345 (2000), 437. Google Scholar

[111]

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 (2007), 356. Google Scholar

[112]

A. Stein, D. Vader, D. Weitz and L. Sander, The micromechanics of three-dimensional collagen-I gels,, Complexity, 16 (2011), 22. Google Scholar

[113]

M. C. Stella and P. M. Comoglio, HGF: A multifunctional growth factor controlling cell scattering,, Int. J. Biochem. Cell Biol., 31(12) (1999), 1357. Google Scholar

[114]

M. Stolarska, Y. Kim and H. G. Othmer, Multiscale models of cell and tissue dynamics,, Phil. Trans. Roy. Soc. A, 367 (2009), 3525. doi: 10.1098/rsta.2009.0095. Google Scholar

[115]

S. S. Stylli, A. H. Kaye, L. MacGregor, M. Howes and P. Rajendra, Photodynamic therapy of high grade glioma - long term survival,, J. Clin. Neurosci., 12 (2005), 389. Google Scholar

[116]

K. R. Swanson, E. C. Alvord and J. D. Murray, Virtual resection of gliomas: Effect of extent of resection on recurrence,, Math. Comp. Modelling, 37 (2003), 1177. Google Scholar

[117]

K. R. Swanson, J.r. EC and J. D. Murray, A quantitative model for differential motility of gliomas in grey and white matter,, Cell Prolif., 33 (2000), 317. Google Scholar

[118]

L. Tamagnone and P. M. Comoglio, Control of invasive growth by hepatocyte growth factor (hgf) and related scatter factors,, Cytokine Growth Factor Rev., 8(2) (1997), 129. Google Scholar

[119]

L. Trusolino and P. M. Comoglio, Scatter-factor and semaphorin receptors: Cell signalling for invasive growth,, Nat. Rev. Cancer, 2 (2002), 289. Google Scholar

[120]

J. C. Valle-Casuso, A. Gonzalez-Sanchez, J. M. Medina and A. Tabernero, Hif-1 and c-src mediate increased glucose uptake induced by endothelin-1 and connexin43 in astrocytes,, PLoS One, 7 (2012). Google Scholar

[121]

O. Warburg, On the origin of cancer cells,, Science, 123 (1956), 309. Google Scholar

[122]

O. Wartlick, P. Mumcu, A. Kicheva, T. Bittig, C. Seum, F. Julicher and M. Gonzalez-Gaitan, Dynamics of DPP signaling and proliferation control,, Science, 331 (2011), 1154. Google Scholar

[123]

O. Wartlick, P. Mumcu, F. Julicher and M. Gonzalez-Gaitan, Understanding morphogenetic growth control - lessons from flies,, Nat. Rev. Mol. Cell Biol., 12 (2011), 594. Google Scholar

[124]

J. J. Watters, J. M. Schartner and B. Badie, Microglia function in brain tumors,, J. Neurosci. Res., 81 (2005), 447. Google Scholar

[125]

T. Williams and J. E. Brenman, Lkb1 and ampk in cell polarity and division,, Trends. Cell Biol., 18 (2008), 193. Google Scholar

[126]

M. Wiranowska and M. V. Rojiani, "Extracellular Matrix Microenvironment in Glioma Progression,", Glioma - Exploring Its Biology and Practical Relevance, (2011). Google Scholar

[127]

K. Wolf, Y. Wu, Y. Liu, J. Geiger, E. Tam, C. Overall, M. Stack and P. Friedl, Multi-step pericellular proteolysis controls the transition from individual to collective cancer cell invasion,, Nat. Cell Biol., 9(8) (2007), 893. Google Scholar

[128]

K. Wolf, S. Alexander, V. Schacht, L. Coussens, U.H. von Andrian, J. van Rheenen, E. Deryugina and P. Friedl, Collagen-based cell migration models in vitro and in vivo,, Semin. Cell Dev. Biol., 20(8) (2009), 931. Google Scholar

[129]

A. Woods, S. R. Johnstone, K. Dickerson, F. C. Leiper, L. G. Fryer, D. Neumann, U. Schlattner, T. Wallimann, M. Carlson and D. Carling, Lkb1 is the upstream kinase in the amp-activated protein kinase cascade,, Curr. Biol., 13 (2003), 2004. Google Scholar

[130]

W. Xiong and J. E. Ferrell Jr, A positive-feedback-based bistable 'memory module' that governs a cell fate decision,, Nature, 426 (2003), 460. Google Scholar

[131]

R. H. Xu, H. Pelicano, Y. Zhou, J. S. Carew, L. Feng, K. N. Bhalla, M. J. Keating and P. Huang, Inhibition of glycolysis in cancer cells: A novel strategy to overcome drug resistance associated with mitochondrial respiratory defect and hypoxia,, Cancer Res., 65 (2005), 613. Google Scholar

[132]

G. Yao, T. J. Lee, S. Mori, J. R. Nevins and L. You, A bistable rb-e2f switch underlies the restriction point,, Nat. Cell Biol., 10 (2008), 476. Google Scholar

[133]

N. Young and J. R. Brocklyn, Roles of sphingosine-1-phosphate (s1p) receptors in malignant behavior of glioma cells. Differential effects of s1p2 on cell migration and invasiveness,, Exp. Cell Res., 313 (2007), 1615. Google Scholar

[134]

K. Yuan, R. K. Singh, G. Rezonzew and G. P. Siegal, Cell motility in cancer invasion and metastasis, in "Cancer Metastasis - Biology and Treatment,", Springer, (2006), 25. Google Scholar

[135]

Y. Zhou, P. H. Larsen, C. Hao and V. W. Yong, Cxcr4 is a major chemokine receptor on glioma cells and mediates their survival,, J. Biol. Chem., 277 (2002), 49481. Google Scholar

show all references

References:
[1]

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 (2008), 19678. Google Scholar

[2]

S. Alexander and P. Friedl, Cancer invasion and resistance: Interconnected processes of disease progression and therapy failure,, Trends. Mol. Med., 18 (2012), 13. Google Scholar

[3]

D. Angeli, J. E Ferrell Jr. and E. D. Sontag, Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems,, Proc. Natl. Acad. Sci. USA, 101 (2004), 1822. Google Scholar

[4]

R. P. Araujo and D. L. S. McElwain, A history of the study of solid tumour growth: The contribution of mathematical modelling,, Bull. Math. Biol., 66 (2004), 1039. doi: 10.1016/j.bulm.2003.11.002. Google Scholar

[5]

H. J. Aronen, F. S. Pardo, D. N. Kennedy, J. W. Belliveau, S. D. Packard, D. W. Hsu, F. H. Hochberg, A. J. Fischman and B. R. Rosen, High microvascular blood volume is associated with high glucose uptake and tumor angiogenesis in human gliomas,, Clin. Cancer Res., 6 (2000), 2189. Google Scholar

[6]

K. Asano, C. D. Duntsch, Q. Zhou, J. D. Weimar, D. Bordelon, J. H. Robertson and T. Pourmotabbed, Correlation of n-cadherin expression in high grade gliomas with tissue invasion,, J. Neurooncol., 70 (2004), 3. Google Scholar

[7]

A. F. Baas, J. Kuipers, N. N. Wel, E. Batlle, H. K. Koerten, P. J. Peters and H. C. Clevers, Complete polarization of single intestinal epithelial cells upon activation of lkb1 by strad,, Cell, 116 (2004), 457. Google Scholar

[8]

E. Bandres, N. Bitarte, F. Arias, J. Agorreta, P. Fortes, X. Agirre, R. Zarate, J. A. Diaz-Gonzalez, N. Ramirez and J. J. Sola, microrna-451 regulates macrophage migration inhibitory factor production and proliferation of gastrointestinal cancer cells,, Clin. Cancer Res, 15 (2009), 2281. Google Scholar

[9]

D. P. Bartel, Micrornas: target recognition and regulatory functions,, Cell, 136 (2009), 215. Google Scholar

[10]

J. Boudeau, A. F. Baas, M. Deak, N. A. Morrice, A. Kieloch, M. Schutkowski, A. R. Prescott, H. C. Clevers and D. R. Alessi, Mo25alpha/beta interact with stradalpha/beta enhancing their ability to bind, activate and localize lkb1 in the cytoplasm,, EMBO J., 22 (2003), 5102. Google Scholar

[11]

H. Byrne and L. Preziosi, Modeling solid tumor growth using the theory of mixtures,, Math. Med. Biol., 20 (2004), 341. Google Scholar

[12]

H. M. Byrne, Dissecting cancer through mathematics: From the cell to the animal model,, Nature Reviews, 10 (2010), 221. Google Scholar

[13]

R. Cairns, I. Papandreou and N. Denko, Overcoming physiologic barriers to cancer treatment by molecularly targeting the tumor microenvironment,, Mol. Cancer Res., 4 (2006), 61. Google Scholar

[14]

A. Chauviere, L. Preziosi and H. Byrne, A model of cell migration within the extracellular matrix based on a phenotypic switching mechanism,, Math. Med. Biol., 27 (2010), 255. doi: 10.1093/imammb/dqp021. Google Scholar

[15]

J. D. Cheng and L. M. Weiner, Tumors and their microenvironments: tilling the soil. Commentary re: A. M. Scott et al., A Phase I dose-escalation study of sibrotuzumab in patients with advanced or metastatic fibroblast activation protein-positive cancer,, Clin. Cancer Res. 9 (2003), 9 (2003), 1590. Google Scholar

[16]

G. Cheng, J. Tse, R. K. Jain and L. L. Minn, Micro-environmental mechanical stress controls tumor spheroid size and morphology by supressing proliferation and inducing aopotosis in cancer cells,, PLoS One, (2009). Google Scholar

[17]

S. K. Chintala, J. C. Tonn and J. S. Rao, Matrix metalloproteinases and their biological function in human gliomas,, Int. J. Dev. Neurosci., 17 (1999), 495. Google Scholar

[18]

D. G. Chiro, R. L. DeLaPaz, R. A. Brooks, L. Sokoloff, P. L. Kornblith, B. H. Smith, N. J. Patronas, C. V. Kufta, R. M. Kessler, G. S. Johnston, R. G. Manning and A. P. Wolf, Glucose utilization of cerebral gliomas measured by [18f] fluorodeoxyglucose and positron emission tomography,, Neurology, 32 (1982), 1323. Google Scholar

[19]

A. Cho, Life's patterns: No need to spell it out?, Science, 303 (2004), 782. Google Scholar

[20]

G. Choe, J. K. Park, L. Jouben-Steele, T. J. Kremen, L. M. Liau, H. V. Vinters, T. F. Cloughesy and P. S. Mischel, Active matrix metalloproteinase 9 expression is associated with primary glioblastoma subtype,, Clin. Cancer Res., 8(9) (2002), 2894. Google Scholar

[21]

M. Crawford, E. Brawner, K. Batte, L. Yu, M. G. Hunter, G. A. Otterson, G. Nuovo, C. B. Marsh and S. P. Nana-Sinkam, Microrna-126 inhibits invasion in non-small cell lung carcinoma cell lines,, Biochem. Biophys. Res. Commun., 373 (2008), 607. Google Scholar

[22]

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 (1998), 35347. Google Scholar

[23]

J. C. Dallon and H. G. Othmer, A discrete cell model with adaptive signalling for aggregation of dictyostelium discoideum,, Phil. Trans. Roy. Soc. Lond, B352 (1997), 391. Google Scholar

[24]

J. C. Dallon and H. G. Othmer, How cellular movement determines the collective force generated by the dictyostelium discoideum slug,, J. Theor. Biol., 231 (2004), 203. doi: 10.1016/j.jtbi.2004.06.015. Google Scholar

[25]

F. G. Davis and B. J. McCarthy, Current epidemiological trends and surveillance issues in brain tumors,, Expert Rev. Anticancer Ther., 1 (2001), 395. Google Scholar

[26]

S. J. Day and P. A. Lawrence, Measuring dimensions: The regulation of size and shape,, Development, 127 (2000), 2977. Google Scholar

[27]

T. S. Deisboeck, M. E. Berens, A. R. Kansal, S. Torquato, A. O. Stemmer-Rachamimov and E. A. Chiocca, Pattern of self-organization in tumour systems: Complex growth dynamics in a novel brain tumour spheroid model,, Cell Prolif., 34 (2001), 115. Google Scholar

[28]

T. S. Deisboeck and I. D. Couzin, Collective behavior in cancer cell populations,, Bioessays, 31 (2009), 190. Google Scholar

[29]

T. Demuth and M. E. Berens, Molecular mechanisms of glioma cell migration and invasion,, J. Neurooncol., 70 (2004), 217. Google Scholar

[30]

J. B. Easton and P. J. Houghton, mtor and cancer therapy,, Oncogene, 25 (2006), 6436. Google Scholar

[31]

A. Esquela-Kerscher and F. J. Slack, Oncomirs - micrornas with a role in cancer,, Nat. Rev. Cancer, 6 (2006), 259. Google Scholar

[32]

J. E. Ferrell Jr, Self-perpetuating states in signal transduction: Positive feedback, double-negative feedback and bistability,, Curr. Opin. Cell Biol., 14 (2002), 140. Google Scholar

[33]

P. Friedl and S. Alexander, Cancer invasion and the microenvironment: Plasticity and reciprocity,, Cell, 147 (2011), 992. Google Scholar

[34]

G. Gabriely, T. Wurdinger, S. Kesari, C. C. Esau, J. Burchard, P. S. Linsley and A. M. Krichevsky, Microrna 21 promotes glioma invasion by targeting matrix metalloproteinase regulators,, Mol. Cell Biol., 28 (2008), 5369. Google Scholar

[35]

H. Gal, G. Pandi, A. A. Kanner, Z. Ram, G. Lithwick-Yanai, N. Amariglio, G. Rechavi and D. Givol, Mir-451 and imatinib mesylate inhibit tumor growth of glioblastoma stem cells biochem.,, Biophys. Res. Commun., 376 (2008), 86. Google Scholar

[36]

J. Galle, M. Loeffler and D. Drasdo, Modeling the effect of deregulated proliferation and apoptosis on the growth dynamics of epithelial cell populations in vitro,, Biophysical J., 88 (2005), 62. Google Scholar

[37]

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 (2011), 5692. Google Scholar

[38]

R. A. Gatenby and R. J. Gillies, Why do cancers have high aerobic glycolysis?, Nat. Rev. Cancer, 4 (2004), 891. Google Scholar

[39]

J. Godlewski, M. O. Nowicki, A. Bronisz, S. Williams, A. Otsuki, G. Nuovo, A. Raychaudhury, H. B. Newton, E. A. Chiocca and S. Lawler, Targeting of the bmi-1 oncogene/stem cell renewal factor by microrna-128 inhibits glioma proliferation and self-renewal,, Cancer Res., 68 (2008), 9125. Google Scholar

[40]

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 (2010), 2742. Google Scholar

[41]

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 (2010), 620. Google Scholar

[42]

S. Goldman, M. Levivier, B. Pirotte, J. M. Brucher, D. Wikler, P. Damhaut, E. Stanus, J. Brotchi and J. Hildebrand, Regional glucose metabolism and histopathology of gliomas. A study based on positron emission tomography-guided stereotactic biopsy,, Cancer, 78 (1996), 1098. Google Scholar

[43]

M. Gotte and G. W. Yip, Heparanase, hyaluronan, and CD44 in cancers: A breast carcinoma perspective,, Cancer Research, 66 (2006), 10233. Google Scholar

[44]

R. G. Hahn and T. Nystom, Plasma Volume Expansion Resulting from Intravenous Glucose Tolerance Test,, Comput. Math. Methods Med., (2011). doi: doi:10.1155/2011/965075. Google Scholar

[45]

D. G. Hardie, Amp-activated/snf1 protein kinases: Conserved guardians of cellular energy,, Nat. Rev. Mol. Cell Biol., 8 (2007), 774. Google Scholar

[46]

D. G. Hardie, I. P. Salt, S. A. Hawley and S. P. Davies, Amp-activated protein kinase: An ultrasensitive system for monitoring cellular energy charge,, Biochem. J., 338 (1999), 717. Google Scholar

[47]

H. L. Harpold, J.r. EC and K. R. Swanson, The evolution of mathematical modeling of glioma proliferation and invasion,, J. Neuropathol. Exp. Neurol., 66 (2007), 1. Google Scholar

[48]

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., 27 (2010), 255. doi: 10.1093/imammb/dqq011. Google Scholar

[49]

S. A. Hawley, J. Boudeau, J. L. Reid, K. J. Mustard, L. Udd, T. P. Makela, D. R. Alessi and D. G. Hardie, Complexes between the lkb1 tumor suppressor, strad alpha/beta and mo25 alpha/beta are upstream kinases in the amp-activated protein kinase cascade,, J. Biol., 2 (2003). Google Scholar

[50]

S. A. Hawley, M. A. Selbert, E. G. Goldstein, A. M. Edelman, D. Carling and D. G. Hardie, 5'-amp activates the amp-activated protein kinase cascade, and ca2+/calmodulin activates the calmodulin-dependent protein kinase i cascade, via three independent mechanisms,, J. Biol. Chem., 270 (1995), 27186. Google Scholar

[51]

M. G. Heiden, L. C. Cantley and C. B. Thompson, Understanding the warburg effect: The metabolic requirements of cell proliferation,, Science, 324 (2009), 1029. Google Scholar

[52]

B. Hegedus, A. Czirok, I. Fazekas, T. Babel, E. Madarasz and T. Vicsek, Locomotion and proliferation of glioblastoma cells in vitro: Statistical evaluation of videomicroscopic observations,, J. Neurosurg., 92 (2000), 428. Google Scholar

[53]

G. Helmlinger, P. A. Netti, H. C. Lichtenbeld, R. J. Melder and R. K. Jain, Solid stress inhibits the growth of multicellular tumor spheroids,, Nature Biotechnology, 15(8) (1997), 778. Google Scholar

[54]

A. F. Hezel and N. Bardeesy, Lkb1; linking cell structure and tumor suppression,, Oncogene, 27 (2008), 6908. Google Scholar

[55]

O. Ilina, G. Bakker, A. Vasaturo, R. M. Hofmann and P. Friedl, Two-photon laser-generated microtracks in 3d collagen lattices: principles of mmp-dependent and -independent collective cancer cell invasion,, Phys. Biol., 8 (2011). Google Scholar

[56]

K. Inoki, Y. Li, T. Xu and K. L. Guan, Rheb gtpase is a direct target of tsc2 gap activity and regulates mtor signaling,, Genes. Dev., 17 (2003), 1829. Google Scholar

[57]

K. Inoki, Y. Li, T. Zhu, J. Wu and K. L. Guan, Tsc2 is phosphorylated and inhibited by akt and suppresses mtor signalling,, Nat. Cell Biol., 4 (2002), 648. Google Scholar

[58]

J. Jaalinoja, R. Herva, M. Korpela, M. Hoyhtya and T. Turpeenniemi-Hujanen, Matrix metalloproteinase 2 (mmp-2) immunoreactive protein is associated with poor grade and survival in brain neoplasms,, J. Neurooncol., 46 (2000), 81. Google Scholar

[59]

V. L Jacobs, P. A. Valdes, W. F. Hickey and J. A. De Leo, Current review of in vivo gbm rodent models: emphasis on the cns-1 tumour model,, ASN NEURO, 3 (2011). Google Scholar

[60]

R. K. Jain, Transport of molecules in the tumor interstitium: a review,, Cancer Res., 47 (1987), 3039. Google Scholar

[61]

R. G. Jones and C. B. Thompson, Tumor suppressors and cell metabolism: A recipe for cancer growth,, Genes Dev., 23 (2009), 537. Google Scholar

[62]

L. J. Kaufma, C. P. Brangwynn, K. E. Kasz, E. Filippidi, V. D. Gordon, T. S. Deisboeck and D. A. Weitz, lioma expansion in Collagen I matrices: Analyzing Collagen concentration-dependent growth and motility patterns,, Biophys. J., 89 (2005), 635. Google Scholar

[63]

E. Khain and L. M. Sander, Dynamics and pattern formation in invasive tumor growth,, Phys. Rev. Lett., 96 (2006). Google Scholar

[64]

R. Khanin and V. Vinciotti, Computational modeling of post-transcriptional gene regulation by micrornas,, J. Comput. Biol., 15 (2008), 305. doi: 10.1089/cmb.2007.0184. Google Scholar

[65]

J. W. Kim and C. V. Dang, Cancer's molecular sweet tooth and the warburg effect,, Cancer Res., 66 (2006), 8927. Google Scholar

[66]

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 (2008), 4249. Google Scholar

[67]

Y. Kim and A. Friedman, Interaction of tumor with its microenvironment: A mathematical model,, Bull. Math. Biol., 72 (2010), 1029. doi: 10.1007/s11538-009-9481-z. Google Scholar

[68]

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 (2009), 359. Google Scholar

[69]

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 (2007), 1773. doi: 10.1142/S0218202507002479. Google Scholar

[70]

Y. Kim, S. Roh, S. Lawler and A. Friedman, miR451 and AMPK mutual antagonism in glioma cells migration and proliferation,, PLoS One, 6 (2011). Google Scholar

[71]

Y. Kim, M. Stolarska and H. G. Othmer, The role of the microenvironment in tumor growth and invasion,, Prog. Biophys. Mol. Biol., 106 (2011), 353. Google Scholar

[72]

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 (2010), 401. doi: 10.1007/s00285-009-0307-2. Google Scholar

[73]

W. P. Kloosterman and R. H. Plasterk, The diverse functions of micrornas in animal development and disease,, Dev. Cell, 11 (2006), 441. Google Scholar

[74]

C. Koike, T. D. McKee, A. Pluen, S. Ramanujan, K. Burton, L. L. Munn, Y. Boucher and R. K. Jain, Solid stress facilitates spheroid formation: potential involvement of hyaluronan,, British Journal of Cancer, 86 (2002), 947. Google Scholar

[75]

K. Lamszus, N. O. Schmidt, L. Jin, J. Laterra, D. Zagzag, D. Way, M. Witte, M. Weinand, I. D. Goldberg, M. Westphal and E. M. Rosen, Scatter factor promotes motility of human glioma and neuromicrovascular endothelial cells,, Int. J. Cancer, 75 (1998), 19. Google Scholar

[76]

S. Lawler and E. A. Chiocca, Emerging functions of micrornas in glioblastoma,, J. Neurooncol., 92 (2009), 297. Google Scholar

[77]

J. H. Lee, H. Koh, M. Kim, Y. Kim, S. Y. Lee, R. E. Karess, S. H. Lee, M. Shong, J. M. Kim, J. Kim and J. Chung, Energy-dependent regulation of cell structure by amp-activated protein kinase,, Nature, 447 (2007), 1017. Google Scholar

[78]

C. K. Li, The glucose distribution in 9l rat brain multicell tumor spheroids and its effect on cell necrosis,, Cancer, 50 (1982), 2066. Google Scholar

[79]

J. S. Lowengrub, H. B. Frieboes, F. Jin, Y. L. Chuang, X. Li, P. Macklin, S. M. Wise and V. Cristini, Nonlinear modelling of cancer: Bridging the gap between cells and tumours,, Nonlinearity, 23 (2010). doi: 10.1088/0951-7715/23/1/001. Google Scholar

[80]

A. D. Luca, Niccolo Arena, L. M. Sena and E. Medico, Met overexpression confers hgf-dependent invasive phenotype to human thyroid carcinoma cells in vitro,, Journal of Cellular Physiology, 180 (1999), 365. Google Scholar

[81]

M. Lund-Johansen, R. Bjerkvig, P. A. Humphrey, S. H. Bigner, D. D. Bigner and O. D. Laerum, Effect of epidermal growth factor on glioma cell growth, migration, and invasion in vitro,, Cancer Res., 50 (1990), 6039. Google Scholar

[82]

L. Ma, J. Teruya-Feldstein and R. A. Weinberg, Tumour invasion and metastasis initiated by microrna-10b in breast cancer,, Nature, 449 (2007), 682. Google Scholar

[83]

E. Mandonnet, J. Y. Delattre, M. L. Tanguy, K. R. Swanson, A. F. Carpentier, H. Duffau, P. Cornu, R. Effenterre, J.r. EC and L. Capelle, Continuous growth of mean tumor diameter in a subset of grade ii gliomas,, Ann. Neurol., 53 (2003), 524. Google Scholar

[84]

S. Marino, I. B. Hogue, C. J. Ray and D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology,, J. Theor. Biol., 254 (2008), 178. Google Scholar

[85]

N. I. Markevich, J. B. Hoek and B. N. Kholodenko, Signaling switches and bistability arising from multisite phosphorylation in protein kinase cascades,, J. Cell Biol., 164 (2004), 353. Google Scholar

[86]

M. O. Nowicki, N. Dmitrieva, A. M. Stein, J. L. Cutter, J. Godlewski, Y. Saeki, M. Nita, M. E. Berens, L. M. Sander and H. B. Newton, Lithium inhibits invasion of glioma cells; possible involvement of glycogen synthase kinase-3,, Neuro-oncol, 10 (2008), 690. Google Scholar

[87]

I. Papandreou, R. A. Cairns, L. Fontana, A. L. Lim and N. C. Denko, Hif-1 mediates adaptation to hypoxia by actively downregulating mitochondrial oxygen consumption,, Cell Metab., 3 (2006), 187. Google Scholar

[88]

J. B. Park, H. J. Kwak and S. H. Lee, Role of hyaluronan in glioma invasion,, Cell Adhesion and Migration, 2 (2008), 202. Google Scholar

[89]

M. J. Paszek and V. M. Weaver, The tension mounts: Mechanics meets morphogenesis and malignancy,, J. Mammary Gland Biol. Neoplasia, 9 (2004), 325. Google Scholar

[90]

M. J. Paszek, N. Zahir, K. R. Johnson, J. N. Lakins, G. I. Rozenberg, A. Gefen, C. A. Reinhart-King, S. S. Margulies, M. Dembo, D. Boettiger, D. A. Hammer and V. M. Weaver, Tensional homeostasis and the malignant phenotype,, Cancer cell, 8 (2005), 241. Google Scholar

[91]

C. Perego, C. Vanoni, S. Massari, A. Raimondi, S. Pola, M.G. Cattaneo, M. Francolini, L. M. Vicentini and G. Pietrini, Invasive behaviour of glioblastoma cell lines is associated with altered organisation of the cadherin-catenin adhesion system,, J. Cell Sci., 115 (2002), 3331. Google Scholar

[92]

K. Pham, A. Chauviere, H. Hatzikirou, X. Li, H. M. Byrne, V. Cristini and J. Lowengrub, Density-dependent quiescence in glioma invasion: Instability in a simple reaction-diffusion model for the migration/proliferation dichotomy,, Journal of Biological dynamics, (2011). doi: 10.1080/17513758.2011.590610. Google Scholar

[93]

M. Platten, W. Wick and M. Weller, Malignant glioma biology: role for tgf-beta in growth, motility, angiogenesis, and immune escape,, Microsc. Res. Tech., 52 (2001), 401. Google Scholar

[94]

C. J. Potter, L. G. Pedraza and T. Xu, Akt regulates growth by directly phosphorylating tsc2,, Nat. Cell Biol., 4 (2002), 658. Google Scholar

[95]

L. Preziosi and A. Tosin, Multiphase and multiscale trends in cancer modelling,, Math. Model. Natl. Phenom., 4 (2009), 1. doi: 10.1051/mmnp/20094301. Google Scholar

[96]

L. Preziosi and G. Vitale, A multiphase model of tumor and tissue growth including cell adhesion and plastic reorganization,, Math. Model. Method. Appl. Sci., 21 (2011), 1901. doi: 10.1142/S0218202511005593. Google Scholar

[97]

C. Ragan and M. Zuker amd M. A. Ragan, Quantitative prediction of miRNA-mRNA interaction based on equilibrium concentrations,, PLoS Comput. Biol., 7 (2011). doi: 10.1371/journal.pcbi.1001090. Google Scholar

[98]

K. A. Rejniak and C. J. McCawley, Current trends in mathematical modeling of tumor microenvironment interaction: A survey of tools and applications,, Experimental Biology and Medicine (Maywood), 235 (2010), 411. Google Scholar

[99]

K. A. Rejniak and A. R. A. Anderson, Hybrid models of tumor growth,, WIRES Syst. Biol. Med., 3 (2011), 115. Google Scholar

[100]

K. A. Rejniak and C. J. McCawley, Current trends in mathematical modeling of tumor microenvironment interaction: A survey of tools and applications,, Exp. Biol. Med. (Maywood), 235 (2010), 411. Google Scholar

[101]

A. Ridley, M. Schwartz, K. Burridge, R. Firtel, M. Ginsberg, G. B. Parsons and A. Horwitz, Cell migration: Integrating signals from front to back,, Science, 302 (2003), 1704. Google Scholar

[102]

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 (2006), 816. Google Scholar

[103]

J. M. Rozental, R. L. Levine and R. J. Nickles, Changes in glucose uptake by malignant gliomas: Preliminary study of prognostic significance,, J. Neurooncol., 10 (1991), 75. Google Scholar

[104]

O. Sampetrean, I. Saga, M. Nakanishi, E. Sugihara, R. Fukaya, N. Onishi, S. Osuka, M. Akahata, K. Kai, H. Sugimoto, A. Hirao and H. Saya, Invasion precedes tumor mass formation in a malignant brain tumor model of genetically modified neural stem cells,, Neoplasia, 13 (2011), 784. Google Scholar

[105]

L. M. Sander and T. S. Deisboeck, Growth patterns of microscopic brain tumors,, Phys. Rev. E, 66 (2002). Google Scholar

[106]

M. Scianna, R. M. Merks, L. Preziosi and E. Medico, Individual cell-based models of cell scatter of aro and mlp-29 cells in response to hepatocyte growth factor,, J. Theor. Biol., 260 (2009), 151. Google Scholar

[107]

S. Sen, M. Dong and S. Kumar, Isoform-specific contributions of a-Actinin to Glioma cell mechanobiology,, PLoS One, 4 (2009). Google Scholar

[108]

R. J. Shaw, N. Bardeesy, B. D. Manning, L. Lopez, M. Kosmatka, R. A. DePinho and L. C. Cantley, The lkb1 tumor suppressor negatively regulates mtor signaling,, Cancer Cell, 6 (2004), 91. Google Scholar

[109]

B. I. Shraiman, Mechanicall feedback as a possible regulator of tissue growth,, PNAS, 102 (2005), 3318. Google Scholar

[110]

S. C. Stein, A. Woods, N. A. Jones, M. D. Davison and D. Carling, The regulation of amp-activated protein kinase by phosphorylation,, Biochem. J., 345 (2000), 437. Google Scholar

[111]

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 (2007), 356. Google Scholar

[112]

A. Stein, D. Vader, D. Weitz and L. Sander, The micromechanics of three-dimensional collagen-I gels,, Complexity, 16 (2011), 22. Google Scholar

[113]

M. C. Stella and P. M. Comoglio, HGF: A multifunctional growth factor controlling cell scattering,, Int. J. Biochem. Cell Biol., 31(12) (1999), 1357. Google Scholar

[114]

M. Stolarska, Y. Kim and H. G. Othmer, Multiscale models of cell and tissue dynamics,, Phil. Trans. Roy. Soc. A, 367 (2009), 3525. doi: 10.1098/rsta.2009.0095. Google Scholar

[115]

S. S. Stylli, A. H. Kaye, L. MacGregor, M. Howes and P. Rajendra, Photodynamic therapy of high grade glioma - long term survival,, J. Clin. Neurosci., 12 (2005), 389. Google Scholar

[116]

K. R. Swanson, E. C. Alvord and J. D. Murray, Virtual resection of gliomas: Effect of extent of resection on recurrence,, Math. Comp. Modelling, 37 (2003), 1177. Google Scholar

[117]

K. R. Swanson, J.r. EC and J. D. Murray, A quantitative model for differential motility of gliomas in grey and white matter,, Cell Prolif., 33 (2000), 317. Google Scholar

[118]

L. Tamagnone and P. M. Comoglio, Control of invasive growth by hepatocyte growth factor (hgf) and related scatter factors,, Cytokine Growth Factor Rev., 8(2) (1997), 129. Google Scholar

[119]

L. Trusolino and P. M. Comoglio, Scatter-factor and semaphorin receptors: Cell signalling for invasive growth,, Nat. Rev. Cancer, 2 (2002), 289. Google Scholar

[120]

J. C. Valle-Casuso, A. Gonzalez-Sanchez, J. M. Medina and A. Tabernero, Hif-1 and c-src mediate increased glucose uptake induced by endothelin-1 and connexin43 in astrocytes,, PLoS One, 7 (2012). Google Scholar

[121]

O. Warburg, On the origin of cancer cells,, Science, 123 (1956), 309. Google Scholar

[122]

O. Wartlick, P. Mumcu, A. Kicheva, T. Bittig, C. Seum, F. Julicher and M. Gonzalez-Gaitan, Dynamics of DPP signaling and proliferation control,, Science, 331 (2011), 1154. Google Scholar

[123]

O. Wartlick, P. Mumcu, F. Julicher and M. Gonzalez-Gaitan, Understanding morphogenetic growth control - lessons from flies,, Nat. Rev. Mol. Cell Biol., 12 (2011), 594. Google Scholar

[124]

J. J. Watters, J. M. Schartner and B. Badie, Microglia function in brain tumors,, J. Neurosci. Res., 81 (2005), 447. Google Scholar

[125]

T. Williams and J. E. Brenman, Lkb1 and ampk in cell polarity and division,, Trends. Cell Biol., 18 (2008), 193. Google Scholar

[126]

M. Wiranowska and M. V. Rojiani, "Extracellular Matrix Microenvironment in Glioma Progression,", Glioma - Exploring Its Biology and Practical Relevance, (2011). Google Scholar

[127]

K. Wolf, Y. Wu, Y. Liu, J. Geiger, E. Tam, C. Overall, M. Stack and P. Friedl, Multi-step pericellular proteolysis controls the transition from individual to collective cancer cell invasion,, Nat. Cell Biol., 9(8) (2007), 893. Google Scholar

[128]

K. Wolf, S. Alexander, V. Schacht, L. Coussens, U.H. von Andrian, J. van Rheenen, E. Deryugina and P. Friedl, Collagen-based cell migration models in vitro and in vivo,, Semin. Cell Dev. Biol., 20(8) (2009), 931. Google Scholar

[129]

A. Woods, S. R. Johnstone, K. Dickerson, F. C. Leiper, L. G. Fryer, D. Neumann, U. Schlattner, T. Wallimann, M. Carlson and D. Carling, Lkb1 is the upstream kinase in the amp-activated protein kinase cascade,, Curr. Biol., 13 (2003), 2004. Google Scholar

[130]

W. Xiong and J. E. Ferrell Jr, A positive-feedback-based bistable 'memory module' that governs a cell fate decision,, Nature, 426 (2003), 460. Google Scholar

[131]

R. H. Xu, H. Pelicano, Y. Zhou, J. S. Carew, L. Feng, K. N. Bhalla, M. J. Keating and P. Huang, Inhibition of glycolysis in cancer cells: A novel strategy to overcome drug resistance associated with mitochondrial respiratory defect and hypoxia,, Cancer Res., 65 (2005), 613. Google Scholar

[132]

G. Yao, T. J. Lee, S. Mori, J. R. Nevins and L. You, A bistable rb-e2f switch underlies the restriction point,, Nat. Cell Biol., 10 (2008), 476. Google Scholar

[133]

N. Young and J. R. Brocklyn, Roles of sphingosine-1-phosphate (s1p) receptors in malignant behavior of glioma cells. Differential effects of s1p2 on cell migration and invasiveness,, Exp. Cell Res., 313 (2007), 1615. Google Scholar

[134]

K. Yuan, R. K. Singh, G. Rezonzew and G. P. Siegal, Cell motility in cancer invasion and metastasis, in "Cancer Metastasis - Biology and Treatment,", Springer, (2006), 25. Google Scholar

[135]

Y. Zhou, P. H. Larsen, C. Hao and V. W. Yong, Cxcr4 is a major chemokine receptor on glioma cells and mediates their survival,, J. Biol. Chem., 277 (2002), 49481. Google Scholar

[1]

Eugene Kashdan, Svetlana Bunimovich-Mendrazitsky. Hybrid discrete-continuous model of invasive bladder cancer. Mathematical Biosciences & Engineering, 2013, 10 (3) : 729-742. doi: 10.3934/mbe.2013.10.729

[2]

Sebastián Ferrer, Francisco Crespo. Parametric quartic Hamiltonian model. A unified treatment of classic integrable systems. Journal of Geometric Mechanics, 2014, 6 (4) : 479-502. doi: 10.3934/jgm.2014.6.479

[3]

Ghendrih Philippe, Hauray Maxime, Anne Nouri. Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solution. Kinetic & Related Models, 2009, 2 (4) : 707-725. doi: 10.3934/krm.2009.2.707

[4]

Hongbin Guo, Michael Yi Li. Global dynamics of a staged progression model for infectious diseases. Mathematical Biosciences & Engineering, 2006, 3 (3) : 513-525. doi: 10.3934/mbe.2006.3.513

[5]

Margherita Carletti, Matteo Montani, Valentina Meschini, Marzia Bianchi, Lucia Radici. Stochastic modelling of PTEN regulation in brain tumors: A model for glioblastoma multiforme. Mathematical Biosciences & Engineering, 2015, 12 (5) : 965-981. doi: 10.3934/mbe.2015.12.965

[6]

Danilo T. Pérez-Rivera, Verónica L. Torres-Torres, Abraham E. Torres-Colón, Mayteé Cruz-Aponte. Development of a computational model of glucose toxicity in the progression of diabetes mellitus. Mathematical Biosciences & Engineering, 2016, 13 (5) : 1043-1058. doi: 10.3934/mbe.2016029

[7]

Faker Ben Belgacem. Uniqueness for an ill-posed reaction-dispersion model. Application to organic pollution in stream-waters. Inverse Problems & Imaging, 2012, 6 (2) : 163-181. doi: 10.3934/ipi.2012.6.163

[8]

Christoph Sadée, Eugene Kashdan. A model of thermotherapy treatment for bladder cancer. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1169-1183. doi: 10.3934/mbe.2016037

[9]

Zijuan Wen, Meng Fan, Asim M. Asiri, Ebraheem O. Alzahrani, Mohamed M. El-Dessoky, Yang Kuang. Global existence and uniqueness of classical solutions for a generalized quasilinear parabolic equation with application to a glioblastoma growth model. Mathematical Biosciences & Engineering, 2017, 14 (2) : 407-420. doi: 10.3934/mbe.2017025

[10]

Tracy L. Stepien, Erica M. Rutter, Yang Kuang. A data-motivated density-dependent diffusion model of in vitro glioblastoma growth. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1157-1172. doi: 10.3934/mbe.2015.12.1157

[11]

Tiberiu Harko, Man Kwong Mak. Travelling wave solutions of the reaction-diffusion mathematical model of glioblastoma growth: An Abel equation based approach. Mathematical Biosciences & Engineering, 2015, 12 (1) : 41-69. doi: 10.3934/mbe.2015.12.41

[12]

Urszula Ledzewicz, Heinz Schättler. Controlling a model for bone marrow dynamics in cancer chemotherapy. Mathematical Biosciences & Engineering, 2004, 1 (1) : 95-110. doi: 10.3934/mbe.2004.1.95

[13]

Eugene Kashdan, Svetlana Bunimovich-Mendrazitsky. Multi-scale model of bladder cancer development. Conference Publications, 2011, 2011 (Special) : 803-812. doi: 10.3934/proc.2011.2011.803

[14]

Avner Friedman, Harsh Vardhan Jain. A partial differential equation model of metastasized prostatic cancer. Mathematical Biosciences & Engineering, 2013, 10 (3) : 591-608. doi: 10.3934/mbe.2013.10.591

[15]

Esther Chigidi, Edward M. Lungu. HIV model incorporating differential progression for treatment-naive and treatment-experienced infectives. Mathematical Biosciences & Engineering, 2009, 6 (3) : 427-450. doi: 10.3934/mbe.2009.6.427

[16]

Wei Feng, Shuhua Hu, Xin Lu. Optimal controls for a 3-compartment model for cancer chemotherapy with quadratic objective. Conference Publications, 2003, 2003 (Special) : 544-553. doi: 10.3934/proc.2003.2003.544

[17]

Hsiu-Chuan Wei. Mathematical and numerical analysis of a mathematical model of mixed immunotherapy and chemotherapy of cancer. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1279-1295. doi: 10.3934/dcdsb.2016.21.1279

[18]

J. Ignacio Tello. On a mathematical model of tumor growth based on cancer stem cells. Mathematical Biosciences & Engineering, 2013, 10 (1) : 263-278. doi: 10.3934/mbe.2013.10.263

[19]

Marcello Delitala, Tommaso Lorenzi. Recognition and learning in a mathematical model for immune response against cancer. Discrete & Continuous Dynamical Systems - B, 2013, 18 (4) : 891-914. doi: 10.3934/dcdsb.2013.18.891

[20]

Ben Sheller, Domenico D'Alessandro. Analysis of a cancer dormancy model and control of immuno-therapy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1037-1053. doi: 10.3934/mbe.2015.12.1037

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (47)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]