A Multiscale Moving Boundary Model Arising in Cancer Invasion
Abstract
Cancer invasion of tissue is a key aspect of the growth and spread of cancer and is crucial in the process of metastatic spread, i.e., the growth of secondary cancers. Invasion consists in cancer cells secreting various matrix degrading enzymes (MDEs) which destroy the surrounding tissue or extracellular matrix (ECM). Through a combination of proliferation and migration, the cancer cells then actively spread locally into the surrounding tissue. Thus processes occurring at the level of individual cells eventually give rise to processes occurring at the tissue level. In this paper we introduce a new type of multiscale model describing the process of cancer invasion of tissue. Our multiscale model is a two-scale model which focuses on the macroscopic dynamics of the distributions of cancer cells and of the surrounding extracellular matrix, and on the microscale dynamics of the MDEs, produced at the level of the individual cancer cells. These microscale dynamics take place at the interface of the cancer cells and the ECM and give rise to a moving boundary at the macroscale. On the computational side, in order to approximate the newly proposed model, we have developed a novel computational scheme based on a combination of finite elements at the microscale with a new finite difference technique at the macroscale, linking together in a moving boundary formulation of the problem. This two-scale numerical scheme is organized in such a way that it enables us to accurately model all the key processes of cancer invasion at both the macroscale and microscale.
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References
1.
A. Abdulle and C. Schwab, Heterogeneous multiscale FEM for diffusion problems on rough surfaces, Multiscale Model. Simul., 3 (2005), pp. 195--220.
2.
G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), pp. 1482--1518.
3.
S. Alexander, G. E. Koehl, M. Hirschberg, E. K. Geissler, and P. Friedl, Dynamic imaging of cancer growth and invasion: A modified skin-fold chamber model, Histochem. Cell Biol., 130 (2008), pp. 1147--1154.
4.
V. Andasari, A. Gerisch, G. Lolas, A. P. South, and M. A. J. Chaplain, Mathematical modeling of cancer cell invasion of tissue: Biological insight from mathematical analysis and computational simulation, J. Math. Biol., 63 (2011), pp. 141--171.
5.
A. R. A. Anderson, A hybrid mathematical model of solid tumour invasion: The importance of cell adhesion, IMA Math. Med. Biol., 22 (2005), pp. 163--186.
6.
A. R. A. Anderson, M. A. J. Chaplain, E. L. Newman, R. J. C. Steele, and A. M. Thompson, Mathematical modelling of tumour invasion and metastasis, Comput. Math. Methods Med., 2 (2000), pp. 129--154.
7.
A. R. A. Anderson, K. A. Rejniak, P. Gerlee, and V. Quaranta, Microenvironment driven invasion: A multiscale multimodel investigation, J. Math. Biol., 58 (2009), pp. 579--624.
8.
A. H. Baker, D. R. Edwards, and G. Murphy, Metalloproteinase inhibitors: Biological actions and therapeutic opportunities, J. Cell Sci., 115 (2002), pp. 3719--3727.
9.
N. Bellomo, E. De Angelis, and L. Preziosi, Multiscale modelling and mathematical problems related to tumor evolution and medical therapy, J. Theor. Med., 5 (2004), pp. 111--136.
10.
J. Behrens, M. M. Mareel, F. M. Van Roy, and W. Birchmeier, Dissecting tumor cell invasion: Epithelial cells acquire invasive properties after the loss of uvomorulin-mediated cell-cell adhesion, J. Cell Biol., 108 (1989), pp. 2435--2447.
11.
E. B. Brown, R. B. Campbell, Y. Tsuzuki, L. Xu, P. Carmeliet, D. Fukumura, and R. K. Jain, In vivo measurement of gene expression, angiogenesis and physiological function in tumors using multiphoton laser scanning microscopy, Nat. Med., 7 (2001), pp. 864--868.
12.
S. W. Byers, C. L. Sommers, B. Hoxter, A. M. Mercurio, and A. Tozeren, Role of E-cadherin in the response of tumor cell aggregates to lymphatic, venous and arterial flow: Measurement of cell-cell adhesion strength, J. Cell Sci., 108 (1995), pp. 2053--2064.
13.
H. M. Byrne and M. A. J. Chaplain, On the role of cell-cell adhesion in models for solid tumour growth, Math. Comput. Modelling, 24 (1996), pp. 1--17.
14.
H. M. Byrne and M. A. J. Chaplain, Free boundary value problems associated with the growth and development of multicellular spheroids, European J. Appl. Math., 8 (1997), pp. 639--658.
15.
H. M. Byrne, M. A. J. Chaplain, G. J. Pettet, and D. L. S. McElwain, An analysis of a mathematical model of trophoblast invasion, Appl. Math. Lett., 14 (2001), pp. 1005--1010.
16.
H. M. Byrne and L. Preziosi, Modelling solid tumour growth using the theory of mixtures, Math. Med. Biol., 20 (2004), pp. 341--366.
17.
M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system, Math. Models Methods Appl. Sci., 15 (2005), pp. 1685--1734.
18.
M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity, Net. Hetero. Med., 1 (2006), pp. 399--439.
19.
M. A. J. Chaplain, S. R. McDougall, and A. R. A. Anderson, Mathematical modelling of tumor-induced angiogenesis, Annu. Rev. Biomed. Eng., 8 (2006), pp. 233--257.
20.
M. A. J. Chaplain, M. Lachowicz, Z. Szymańska, and D. Wrzosek, Mathematical modelling of cancer invasion: The importance of cell-cell adhesion and cell-matrix adhesion, Math. Models Methods Appl. Sci., 21 (2011), pp. 719--743.
21.
J. J. Christiansen and A. K. Rajasekaran, Reassessing epithelial to mesenchymal transition as a prerequisite for carcinoma invasion and metastasis, Cancer Res., 66 (2006), pp. 8319--8326.
22.
J. Condeelis and J. E. Segall, Intravital imaging of cell movement in tumours, Nat. Rev. Cancer, 3 (2003), pp. 921--930.
23.
M. S. Cooper and L. A. DAmico, A cluster of noninvoluting endocytic cells at the margin of the zebrafish blasto-derm marks the site of embryonic shield formation, Dev. Biol., 180 (1996), pp. 184--198.
24.
T. S. Deisboeck, Z. Wang, P. Macklin, and V. Cristini, Multiscale cancer modeling, Annu. Rev. Biomed. Eng., 13 (2011), pp. 127--155.
25.
M. Egeblad and Z. Werb, New functions for the matrix metalloproteinases in cancer progression, Nature Reviews Cancer, 21 (2002), pp. 161--174.
26.
H. Enderling, M. A. J. Chaplain, A. R. A. Anderson, and J. Vaidya, A mathematical model of breast cancer development, local treatment and recurrence, J. Theor. Biol., 246 (2007), pp. 245--259.
27.
B. Engquist and Y.-H. Tsai, Heterogeneous multiscale methods for stiff ordinary differential equations, Math. Comp., 74 (2005), pp. 1707--1742.
28.
H. B. Frieboes, X. Zheng, C. H. Sun, B. Tromberg, R. Gatenby, and V. Christini, An integrated computational/experimental model of tumor invasion, Cancer Res., 66 (2006), pp. 1597--1604.
29.
P. Friedl, P. B. Noble, P. A. Walton, D. W. Laird, P. J. Chauvin, R. J. Tabah, M. Black, and K. S. Zänker, Migration of coordinated cell clusters in mesenchymal and epithelial cancer explants in vitro, Cancer Res., 55 (1995), pp. 4557--4560.
30.
R. A. Gatenby, Models of tumor-host interaction as competing populations: Implications for tumor biology and treatment, J. Theor. Biol., 176 (1995), pp. 447--455.
31.
R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Res., 56 (1996), pp. 5745--5753.
32.
A. Gerisch and M. A. J. Chaplain, Mathematical modelling of cancer cell invasion of tissue: Local and non-local models and the effect of adhesion, J. Theor. Biol., 250 (2008), pp. 684--704.
33.
P. M. Gresho, D. F. Griffiths, and D. J. Silvester, Adaptive time-stepping for incompressible flow part I: Scalar advection-diffusion, SIAM J. Sci. Comput., 30 (2008), pp. 2018--2054.
34.
V. Guillemin and A. Pollack, Differential Topology, Prentice--Hall, Englewood Cliffs, NJ, 1974.
35.
D. Hanahan and R. A. Weinberg, The hallmarks of cancer, Cell, 100 (2000), pp. 57--70.
36.
D. Hanahan and R. A. Weinberg, Hallmarks of cancer: The next generation, Cell, 144 (2011), pp. 646--674.
37.
F. Helmchen and W. Denk, Deep tissue two-photon microscopy, Nat. Methods, 2 (2005), pp. 932--940.
38.
G. E. Koehl, A. Gaumann, and E. K. Geissler, Intravital microscopy of tumor angiogenesis and regression in the dorsal skin fold chamber: Mechanistic insights and preclinical testing of therapeutic strategies, Clin. Exp. Metastasis, 26 (2009), pp. 329--344.
39.
J. Kolega, The movement of cell clusters in vitro: Morphology and directionality, J. Cell Sci., 49 (1981), pp. 15--32.
40.
M. Lachowicz, Micro and meso scales of description corresponding to a model of tissue invasion by solid tumours, Math. Models Methods Appl. Sci., 15 (2005), pp. 1667--1683.
41.
D. Lagamba, A. Nawshad, and E. D. Hay, Microarray analysis of gene expression during epithelial-mesenchymal transformation, Dev. Dyn., 234 (2005), pp. 132--142.
42.
P. U. Le, T. N. Nguyen, P. Drolet-Savoie, N. Leclerc, and I. R. Nabi, Increased $\beta$-actin expression in an invasive Moloney sarcoma virus-transformed MDCK cell variant concentrates to the tips of multiple pseudopodia, Cancer Res., 58 (1998), pp. 1631--1635.
43.
R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems, SIAM, Philadelphia, 2007.
44.
S. Li, J. L. Guan, and S. Chien, Biochemistry and biomechanics of cell motility, Annu. Rev. Biomed. Eng., 7 (2005), pp. 105--150.
45.
P. Lin, Convergence analysis of a quasi-continuum approximation for a two-dimensional material without defects, SIAM J. Numer. Anal., 45 (2007), pp. 313--332.
46.
P. Macklin and J. S. Lowengrub, A new ghost cell/level set method for moving boundary problems: Application to tumor growth, J. Sci. Comput., 35 (2008), pp. 266--299.
47.
P. Macklin, S. R. McDougall, A. R. A. Anderson, M. A. J. Chaplain, V. Cristini, and J. Lowengrub, Multiscale modelling and nonlinear simulation of vascular tumour growth, J. Math. Biol., 58 (2009), pp. 765--798.
48.
K. J. Painter, N. A. Armstrong, and J. A. Sherratt, The impact of adhesion on cellular invasion processes in cancer and development, J. Theor. Biol., 264 (2010), pp. 1057--1067.
49.
A. J. Perumpanani, J. A. Sherratt, J. Norbury, and H. M. Byrne, Biological inferences from a mathematical model for malignant invasion, Invasion & Metastasis, 16 (1996), pp. 209--221.
50.
A. J. Perumpanani, D. L. Simmons, A. J. H. Gearing, K. M. Miller, G. Ward, J. Norbury, M. Schneemann, and J. A. Sherratt, Extracellular matrix-mediated chemotaxis can impede cell migration, Proc. Roy. Soc. London Ser. B, 265 (1998), pp. 2347--2352.
51.
N. J. Poplawski, U. Agero, J. S. Gens, M. Swat, J. A. Glazier, and A. R. A. Anderson, Front instabilities and invasiveness of simulated avascular tumors, Bull. Math. Biol., 71 (2009), pp. 1189--1227.
52.
L. Preziosi and A. Tosin, Multiphase and multiscale trends in cancer modelling, Math. Model. Nat. Phenom., 4 (2009), pp. 1--11.
53.
L. Preziosi and A. Tosin, Multiphase modelling of tumour growth and extracellular matrix interaction: Mathematical tools and applications, J. Math. Biol., 58 (2009), pp. 625--656.
54.
L. Preziosi and G. Vitale, A multiphase model of tumour and tissue growth including cell adhesion and plastic re-organisation, Math. Models Methods Appl. Sci., 21 (2011), pp. 1901--1932.
55.
I. Ramis-Conde, D. Drasdo, A. R. A. Anderson, and M. A. J. Chaplain, Modeling the influence of the E-cadherin-$\beta$-catenin pathway in cancer cell invasion: A multiscale approach, Biophys. J., 95 (2008), pp. 155--165.
56.
W. Q. Ren and W. E, Heterogeneous multiscale method for the modeling of complex fluids and micro-fluidics, J. Comput. Phys., 204 (2005), pp. 1--26.
57.
W. Rudin, Real and Complex Analysis, WCB/McGraw--Hill, Boston, MA, 1987.
58.
M. Seiki, Membrane-type 1 matrix metalloproteinase: A key enzyme for tumor invasion, Cancer Lett., 194 (2003), pp. 1--11.
59.
P. Soulié, F. Carrozzino, M. S. Pepper, A. Y. Strongin, M. F. Poupon, and R. Montesano, Membrane- type-1 matrix metalloproteinase confers tumorigenicity on nonmalignant epithelial cells, Oncogene, 24 (2005), pp. 1689--1697.
60.
Z. Szymańska, C. M. Rodrigo, M. Lachowicz, and M. A. J. Chaplain, Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interactions, Math. Models Methods Appl. Sci., 19 (2009), pp. 257--281.
61.
A. Tosin and L. Preziosi, Multiphase modeling of tumor growth with matrix remodeling and fibrosis, Math. Comput. Modelling, 52 (2010), pp. 969--976.
62.
D. Trucu, M. A. J. Chaplain, and A. Marciniak-Czochra, Three-scale convergence for processes in heterogeneous media, Appl. Anal., 91 (2012), pp. 1351--1373.
63.
S. Turner and J. A. Sherratt, Intercellular adhesion and cancer invasion: A discrete simulation using the extended Potts model, J. Theor. Biol., 216 (2002), pp. 85--100.
64.
R. A. Weinberg, The Biology of Cancer, Garland Science, New York, 2006.
65.
T. Waerner, M. Alacakaptan, I. Tamir, R. Oberauer, A. Gal, T. Brabletz, M. Schreiber, M. Jechlinger, and H. Beug, ILEI: A cytokine essential for EMT, tumor formation, and late events in metastasis in epithelial cells, Cancer Cell, 10 (2006), pp. 227--239.
66.
A. Wicki, F. Lehembre, N. Wick, B. Hantusch, D. Kerjaschki, and G. Christofori, Tumor invasion in the absence of epithelial-mesenchymal transition: Podoplanin-mediated remodeling of the actin cytoskeleton, Cancer Cell, 9 (2006), pp. 261--272.
67.
K. Wolf, Y. I. Wu, Y. Liu, J. Geiger, E. Tam, C. Overall, M. S. Stack, and P. Friedl, Multi-step pericellular proteolysis controls the transition from individual to collective cancer cell invasion, Nat. Cell Biol., 9 (2007), pp. 893--904.
68.
M. H. Zaman, L. M. Trapani, A. L. Sieminski, D. Mackellar, H. Gong, R. D. Kamm, A. Wells, D. A. Lauffenburger, and P. Matsudaira, Migration of tumor cells in 3D matrices is governed by matrix stiffness along with cellmatrix adhesion and proteolysis, Proc. Natl. Acad. Sci. USA, 103 (2006), pp. 10889--10894.
69.
Z. H. Zheng, X. J. Sun, H. T. Zhou, C. Shang, H. Ji, and K. L. Sun, Analysis of metastasis suppressing function of E-cadherin in gastric cancer cells by RNAi, World J. Gastroenterol., 11 (2005), pp. 2000--2003.
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Submitted: 29 June 2011
Accepted: 4 December 2012
Published online: 5 March 2013
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