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Mathematical model and its fast numerical method for the tumor growth
Synergistic effect of blocking cancer cell invasion revealed by computer simulations
1.  Division of Mathematical Oncology, Institute of Medical Science, University of Tokyo, 461 Shirokanedai Minatoku, Tokyo, 1088639, Japan 
References:
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V. V. Artym, Y. Zhang, F. SeillierMoiseiwitsch, K. M. Yamada and S. C. Mueller, Dynamic interactions of cortactin and membrane type 1 matrix metalloproteinase at invadopodia: defining the stages of invadopodia formation and function, Cancer Res, 66 (2006), 30343043. doi: 10.1158/00085472.CAN052177. 
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H. F. Bigg, C. J. Morrison, G. S. Butler, M. A. Bogoyevitch and Z. Wang, et al., Tissue inhibitor of metalloproteinases4 inhibits but does not support the activation of gelatinase A via efficient inhibition of membrane type 1matrix metalloproteinase, Cancer Res, 61 (2001), 36103618. 
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M. Egeblad and Z. Werb, New functions for the matrix metalloproteinases in cancer progression, NatRevCancer, 2 (2002), 161174. doi: 10.1038/nrc745. 
[4] 
D. Hoshino, N. Koshikawa, T. Suzuki, V. Quaranta and A. M. Weaver, et al., Establishment and validation of computational model for MT1MMP dependent ECM degradation and intervention strategies, PLoS Comput Biol, 8 (2012), e1002479. doi: 10.1371/journal.pcbi.1002479. 
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K. Ichikawa, ACell: graphical user interface for the construction of biochemical reaction models, Bioinformatics, 17 (2001), 483484. doi: 10.1093/bioinformatics/17.5.483. 
[6] 
K. Ichikawa, A modeling environment with threedimensional morphology, ACell3D, and Ca2+ dynamics in a spine, Neuroinformatics, 3 (2005), 4964. 
[7] 
E. Maquoi, D. Assent, J. Detilleux, C. Pequeux and J. M. Foidart, et al., MT1MMP protects breast carcinoma cells against type I collageninduced apoptosis, Oncogene, 31 (2012), 480493. doi: 10.1038/onc.2011.249. 
[8] 
H. Nagase, R. Visse and G. Murphy, Structure and function of matrix metalloproteinases and TIMPs, Cardiovasc Res, 69 (2006), 562573. doi: 10.1016/j.cardiores.2005.12.002. 
[9] 
T. Nonaka, K. Nishibashi, Y. Itoh, I. Yana and M. Seiki, Competitive disruption of the tumorpromoting function of membrane type 1 matrix metalloproteinase/matrix metalloproteinase14 in vivo, MolCancer Ther, 4 (2005), 11571166. 
[10] 
M. Schoumacher, R. D. Goldman, D. Louvard and D. M. Vignjevic, Actin, microtubules, and vimentin intermediate filaments cooperate for elongation of invadopodia, J Cell Biol, 189 (2010), 541556. doi: 10.1083/jcb.200909113. 
[11] 
K. Taniwaki, H. Fukamachi, K. Komori, Y. Ohtake and T. Nonaka, et al., Stromaderived matrix metalloproteinase (MMP)2 promotes membrane type 1MMPdependent tumor growth in mice, Cancer Res, 67 (2007), 43114319. doi: 10.1158/00085472.CAN064761. 
[12] 
A. Watanabe, D. Hosino, N. Koshikawa, M. Seiki and T. Suzuki, et al., Critical role of transient activity of MT1MMP for ECM degradation in invadopodia, PLoS Comput Biol, 9 (2013), e1003086. 
[13] 
A. M. Weaver, Invadopodia: Specialized cell structures for cancer invasion, ClinExpMetastasis, 23 (2006), 97105. doi: 10.1007/s1058500690141. 
show all references
References:
[1] 
V. V. Artym, Y. Zhang, F. SeillierMoiseiwitsch, K. M. Yamada and S. C. Mueller, Dynamic interactions of cortactin and membrane type 1 matrix metalloproteinase at invadopodia: defining the stages of invadopodia formation and function, Cancer Res, 66 (2006), 30343043. doi: 10.1158/00085472.CAN052177. 
[2] 
H. F. Bigg, C. J. Morrison, G. S. Butler, M. A. Bogoyevitch and Z. Wang, et al., Tissue inhibitor of metalloproteinases4 inhibits but does not support the activation of gelatinase A via efficient inhibition of membrane type 1matrix metalloproteinase, Cancer Res, 61 (2001), 36103618. 
[3] 
M. Egeblad and Z. Werb, New functions for the matrix metalloproteinases in cancer progression, NatRevCancer, 2 (2002), 161174. doi: 10.1038/nrc745. 
[4] 
D. Hoshino, N. Koshikawa, T. Suzuki, V. Quaranta and A. M. Weaver, et al., Establishment and validation of computational model for MT1MMP dependent ECM degradation and intervention strategies, PLoS Comput Biol, 8 (2012), e1002479. doi: 10.1371/journal.pcbi.1002479. 
[5] 
K. Ichikawa, ACell: graphical user interface for the construction of biochemical reaction models, Bioinformatics, 17 (2001), 483484. doi: 10.1093/bioinformatics/17.5.483. 
[6] 
K. Ichikawa, A modeling environment with threedimensional morphology, ACell3D, and Ca2+ dynamics in a spine, Neuroinformatics, 3 (2005), 4964. 
[7] 
E. Maquoi, D. Assent, J. Detilleux, C. Pequeux and J. M. Foidart, et al., MT1MMP protects breast carcinoma cells against type I collageninduced apoptosis, Oncogene, 31 (2012), 480493. doi: 10.1038/onc.2011.249. 
[8] 
H. Nagase, R. Visse and G. Murphy, Structure and function of matrix metalloproteinases and TIMPs, Cardiovasc Res, 69 (2006), 562573. doi: 10.1016/j.cardiores.2005.12.002. 
[9] 
T. Nonaka, K. Nishibashi, Y. Itoh, I. Yana and M. Seiki, Competitive disruption of the tumorpromoting function of membrane type 1 matrix metalloproteinase/matrix metalloproteinase14 in vivo, MolCancer Ther, 4 (2005), 11571166. 
[10] 
M. Schoumacher, R. D. Goldman, D. Louvard and D. M. Vignjevic, Actin, microtubules, and vimentin intermediate filaments cooperate for elongation of invadopodia, J Cell Biol, 189 (2010), 541556. doi: 10.1083/jcb.200909113. 
[11] 
K. Taniwaki, H. Fukamachi, K. Komori, Y. Ohtake and T. Nonaka, et al., Stromaderived matrix metalloproteinase (MMP)2 promotes membrane type 1MMPdependent tumor growth in mice, Cancer Res, 67 (2007), 43114319. doi: 10.1158/00085472.CAN064761. 
[12] 
A. Watanabe, D. Hosino, N. Koshikawa, M. Seiki and T. Suzuki, et al., Critical role of transient activity of MT1MMP for ECM degradation in invadopodia, PLoS Comput Biol, 9 (2013), e1003086. 
[13] 
A. M. Weaver, Invadopodia: Specialized cell structures for cancer invasion, ClinExpMetastasis, 23 (2006), 97105. doi: 10.1007/s1058500690141. 
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