# American Institute of Mathematical Sciences

January  2015, 20(1): 189-213. doi: 10.3934/dcdsb.2015.20.189

## On a multiscale model involving cell contractivity and its effects on tumor invasion

 1 Bülent Ecevit University, Faculty of Arts and Sciences, Department of Mathematics, 67100 Zonguldak, Turkey 2 Technische Universität Kaiserslautern, Felix-Klein-Zentrum für Mathematik, Paul-Ehrlich-Str. 31, 67663 Kaiserslautern, Germany, Germany

Received  July 2013 Revised  May 2014 Published  November 2014

Cancer cell migration is an essential feature in the process of tumor spread and establishing of metastasis. It characterizes the invasion observed on the level of the cell population, but it is also tightly connected to the events taking place on the subcellular level. These are conditioning the motile and proliferative behavior of the cells, but are also influenced by it. In this work we propose a multiscale model linking these two levels and aiming to assess their interdependence. On the subcellular, microscopic scale it accounts for integrin binding to soluble and insoluble components present in the peritumoral environment, which is seen as the onset of biochemical events leading to changes in the cell's ability to contract and modify its shape. On the macroscale of the cell population this leads to modifications in the diffusion and haptotaxis performed by the tumor cells and implicitly to changes in the tumor environment. We prove the (local) well posedness of our model and perform numerical simulations in order to illustrate the model predictions.
Citation: Gülnihal Meral, Christian Stinner, Christina Surulescu. On a multiscale model involving cell contractivity and its effects on tumor invasion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 189-213. doi: 10.3934/dcdsb.2015.20.189
##### References:
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Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type,, Translations of Mathematical Monographs, (1968).   Google Scholar [23] K. R. Legate, S. A. Wickström and R. Fässler, Genetic and cell biological analysis of integrin outside-in signaling,, Genes Dev., 23 (2009), 397.  doi: 10.1101/gad.1758709.  Google Scholar [24] B. Lin, W. R. Holmes, C. J. Wang, T. Ueno, A. Harwell, L. Edelstein-Keshet, T. Inoue and A. Levchenko, Synthetic spatially graded Rac activation drives cell polarization and movement,, PNAS, 109 (2012).  doi: 10.1073/pnas.1210295109.  Google Scholar [25] T. Lorenz and C. Surulescu, On a class of multiscale cancer cell migration models: Well-posedness in less regular function spaces,, Math. Models Methods Appl. Sci., 24 (2014), 2383.  doi: 10.1142/S0218202514500249.  Google Scholar [26] G. Meral and C. Surulescu, Mathematical modelling, analysis and numerical simulations for the influence of heat shock proteins on tumour invasion,, J. Math. Anal. 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Surulescu, Some classes of stochastic differential equations as an alternative modeling approach to biomedical problems,, in Nonautonomous Dynamical Systems in the Life Sciences (eds. P. E. Kloeden and C. Pötzsche), (2102), 269.  doi: 10.1007/978-3-319-03080-7_9.  Google Scholar [33] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, Studies in Mathematics and its Applications, (1977).   Google Scholar [34] C. Xue and H. G. Othmer, Multiscale models of taxis-driven patterning in bacterial populations,, SIAM J. Appl. Math., 70 (2009), 133.  doi: 10.1137/070711505.  Google Scholar

show all references

##### References:
 [1] J. C. Adams, Regulation of protrusive and contractile cell-matrix contacts,, J. Cell Sci., 115 (2002), 257.   Google Scholar [2] A. R. A. Anderson, M. A. J. Chaplain, E. L. Newman, R. J. C. Steele and A. M. Thompson, Mathematical modeling of tumor invasion and metastasis,, J. Theoretical Medicine, 2 (2000), 129.   Google Scholar [3] H. T. Banks and C. J. Musante, Well-posedness for a class of abstract nonlinear parabolic systems with time delay,, Nonlinear Anal., 35 (1999), 629.  doi: 10.1016/S0362-546X(98)00053-4.  Google Scholar [4] R. Bellman and K. L. Cooke, Differential-Difference Equations,, Academic Press, (1963).   Google Scholar [5] N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, On the asymptotic theory from microscopic to macroscopic growing tissue models: An overview with perspectives,, Math. Models Methods Appl. Sci., 22 (2012).  doi: 10.1142/S0218202512005885.  Google Scholar [6] H. Berry, Oscillatory behavior of a simple kinetic model for proteolysis during cell invasion,, Biophys. J., 77 (1999), 655.  doi: 10.1016/S0006-3495(99)76921-3.  Google Scholar [7] S. B. Carter, Haptotaxis and the mechanism of cell motility,, Nature, 213 (1967), 256.  doi: 10.1038/213256a0.  Google Scholar [8] F. A. C. C. Chalub, P. A. Markowich, B. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits,, Monatsh. Math., 142 (2004), 123.  doi: 10.1007/s00605-004-0234-7.  Google Scholar [9] M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity,, Netw. Heterog. Media, 1 (2006), 399.  doi: 10.3934/nhm.2006.1.399.  Google Scholar [10] H. J. Eberl and L. Demaret, A finite difference scheme for a degenerated diffusion equation arising in microbial ecology,, Electron. J. Differential Equations, 15 (2007), 77.   Google Scholar [11] A. Eladdadi and D. Isaacson, A mathematical model for the effects of HER2 overexpression on cell proliferation in breast cancer,, Bull. Math. Biol., 70 (2008), 1707.   Google Scholar [12] C. Engwer, T. Hillen, M. Knappitsch and C. Surulescu, Glioma follow white matter tracts: A multiscale DTI-based model,, J. Math. Biol., (2014).  doi: 10.1007/s00285-014-0822-7.  Google Scholar [13] A. van der Flier and A. Sonnenberg, Function and interactions of integrins,, Cell Tissue Res., 305 (2001), 285.   Google Scholar [14] P. Friedl and K. Wolf, Tumour-cell invasion and migration: Diversity and escape mechanisms,, Nature Rev. Cancer, 3 (2003), 362.  doi: 10.1038/nrc1075.  Google Scholar [15] P. Friedl and K. Wolf, Proteolytic and non-proteolytic migration of tumour cells and leucocytes,, Biochem. Soc. Symp., 70 (2003), 277.   Google Scholar [16] R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion,, Cancer Res., 56 (1996), 5745.   Google Scholar [17] T. Hillen, $M^5$ mesoscopic and macroscopic models for mesenchymal motion,, J. Math. Biol., 53 (2006), 585.  doi: 10.1007/s00285-006-0017-y.  Google Scholar [18] J. D. Hood and D. A. Cheresh, Role of integrins in cell invasion and migration,, Nature Rev. Cancer, 2 (2002), 91.  doi: 10.1038/nrc727.  Google Scholar [19] A. Huttenlocher and A. R. Horwitz, Integrins in cell migration,, Cold Spring Harb. Perspect. Biol., 3 (2011).  doi: 10.1101/cshperspect.a005074.  Google Scholar [20] J. Kelkel and C. Surulescu, On some models for cancer cell migration through tissue networks,, Math. Biosci. Eng., 8 (2011), 575.  doi: 10.3934/mbe.2011.8.575.  Google Scholar [21] J. Kelkel and C. Surulescu, A multiscale approach to cell migration in tissue networks,, Math. Models Methods Appl. Sci., 22 (2012).  doi: 10.1142/S0218202511500175.  Google Scholar [22] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type,, Translations of Mathematical Monographs, (1968).   Google Scholar [23] K. R. Legate, S. A. Wickström and R. Fässler, Genetic and cell biological analysis of integrin outside-in signaling,, Genes Dev., 23 (2009), 397.  doi: 10.1101/gad.1758709.  Google Scholar [24] B. Lin, W. R. Holmes, C. J. Wang, T. Ueno, A. Harwell, L. Edelstein-Keshet, T. Inoue and A. Levchenko, Synthetic spatially graded Rac activation drives cell polarization and movement,, PNAS, 109 (2012).  doi: 10.1073/pnas.1210295109.  Google Scholar [25] T. Lorenz and C. Surulescu, On a class of multiscale cancer cell migration models: Well-posedness in less regular function spaces,, Math. Models Methods Appl. Sci., 24 (2014), 2383.  doi: 10.1142/S0218202514500249.  Google Scholar [26] G. Meral and C. Surulescu, Mathematical modelling, analysis and numerical simulations for the influence of heat shock proteins on tumour invasion,, J. Math. Anal. Appl., 408 (2013), 597.  doi: 10.1016/j.jmaa.2013.06.017.  Google Scholar [27] R. E. Mickens, Nonstandard finite difference schemes,, in Applications of Nonstandard Finite Difference Schemes (ed. R. E. Mickens), (2000), 1.  doi: 10.1142/9789812813251_0001.  Google Scholar [28] H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems,, J. Math. Biol., 26 (1988), 263.  doi: 10.1007/BF00277392.  Google Scholar [29] H. G. Othmer and T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations,, SIAM J. Appl. Math., 62 (2002), 1222.  doi: 10.1137/S0036139900382772.  Google Scholar [30] C. Surulescu and N. Surulescu, A nonparametric approach to cell dispersal,, Int. J. Biomath. Biostat., 1 (2010), 109.   Google Scholar [31] C. Surulescu and N. Surulescu, Modeling and simulation of some cell dispersion problems by a nonparametric method,, Math. Biosci. Eng., 8 (2011), 263.  doi: 10.3934/mbe.2011.8.263.  Google Scholar [32] C. Surulescu and N. Surulescu, Some classes of stochastic differential equations as an alternative modeling approach to biomedical problems,, in Nonautonomous Dynamical Systems in the Life Sciences (eds. P. E. Kloeden and C. Pötzsche), (2102), 269.  doi: 10.1007/978-3-319-03080-7_9.  Google Scholar [33] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, Studies in Mathematics and its Applications, (1977).   Google Scholar [34] C. Xue and H. G. Othmer, Multiscale models of taxis-driven patterning in bacterial populations,, SIAM J. Appl. Math., 70 (2009), 133.  doi: 10.1137/070711505.  Google Scholar
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