A New Continuum Theory for Incompressible Swelling Materials
Authors: 
Pierre Degond, Marina A. Ferreira, Sara Merino-Aceituno, Mickaël NahonAuthors Info & Claims
Pages 163 - 197
Abstract
Swelling media (e.g., gels, tumors) are usually described by mechanical constitutive laws (e.g., Hooke or Darcy laws). However, constitutive relations of real swelling media are not well-known. Here, we take an opposite route and consider a simple packing heuristics, i.e., the particles can't overlap. We deduce a formula for the equilibrium density under a confining potential. We then consider its evolution when the average particle volume and confining potential depend on time under two additional heuristics: (i) any two particles can't swap their position; (ii) motion should obey some energy minimization principle. These heuristics determine the medium velocity consistently with the continuity equation. In the direction normal to the potential level sets the velocity is related with that of the level sets, while in the parallel direction, it is determined by a Laplace--Beltrami operator on these sets. This complex geometrical feature cannot be recovered using a simple Darcy law.
References
[1]
D. Ambrosi and L. Preziosi, On the closure of mass balance models for tumor growth, Math. Models Methods Appl. Sci., 12 (2002), pp. 737--754.
[2]
I. S. Aranson and L. S. Tsimring, Patterns and collective behavior in granular media: Theoretical concepts, Rev. Modern Phys., 78 (2006), pp. 641--692.
[3]
M. Ben Amar and P. Ciarletta, Swelling instability of surface-attached gels as a model of soft tissue growth under geometric constraints, J. Mech. Phys. Solids, 58 (2010), pp. 935--954.
[4]
M. Bertsch, R. Dal Passo, and M. Mimura, A free boundary problem arising in a simplified tumour growth model of contact inhibition, Interfaces Free Bound., 12 (2010), pp. 235--250.
[5]
M. Bertsch, D. Hilhorst, H. Izuhara, and M. Mimura, A nonlinear parabolic-hyperbolic system for contact inhibition of cell-growth, Differ. Equ. Appl, 4 (2012), pp. 137--157.
[6]
H. Byrne and D. Drasdo, Individual-based and continuum models of growing cell populations: a comparison, J. Math. Biol., 58 (2009), pp. 657--687.
[7]
H. Byrne and L. Preziosi, Modelling solid tumour growth using the theory of mixtures, Math. Med. Biol., 20 (2003), pp. 341--366.
[8]
M. A. Chaplain and B. Sleeman, A mathematical model for the growth and classification of a solid tumor: A new approach via nonlinear elasticity theory using strain-energy functions, Math. Biosci., 111 (1992), pp. 169--215.
[9]
F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, Grad. Texts in Math. 264, Springer, New York, 2013.
[10]
P. Colli, G. Gilardi, E. Rocca, and J. Sprekels, Vanishing viscosities and error estimate for a Cahn--Hilliard type phase field system related to tumor growth, Nonlinear Anal. Real World Appl., 26 (2015), pp. 93--108.
[11]
V. Cristini, J. Lowengrub, and Q. Nie, Nonlinear simulation of tumor growth, J. Math. Biol., 46 (2003), pp. 191--224.
[12]
P. Degond, M. A. Ferreira, and S. Motsch, Damped Arrow--Hurwicz algorithm for sphere packing, J. Comput. Phys., 332 (2017), pp. 47--65.
[13]
D. Drasdo and S. Höhme, A single-cell-based model of tumor growth in vitro: Monolayers and spheroids, Phys. Biol., 2 (2005), pp. 133--147.
[14]
H. Federer, Geometric Measure Theory, Springer, New York, 2014.
[15]
A. Friedman, A free boundary problem for a coupled system of elliptic, hyperbolic, and Stokes equations modeling tumor growth, Interfaces Free Bound., 8 (2006), pp. 247--261.
[16]
S. Gallot, D. Hulin, and J. Lafontaine, Riemannian Geometry, Universitext 3, Springer, New York, 1990.
[17]
M. Goodman and S. Cowin, A continuum theory for granular materials, Arch. Ration. Mech. Anal., 44 (1972), pp. 249--266.
[18]
A. Hawkins-Daarud, K. G. van der Zee, and J. Tinsley Oden, Numerical simulation of a thermodynamically consistent four-species tumor growth model, Int. J. Numer. Methods Biomed. Eng., 28 (2012), pp. 3--24.
[19]
S. Hecht and N. Vauchelet, Incompressible limit of a mechanical model for tissue growth with non-overlapping constraint, Commun. Math. Sci., 15 (2017), pp. 1913--1932.
[20]
D. Hilhorst, J. Kampmann, T. N. Nguyen, and K. G. Van Der Zee, Formal asymptotic limit of a diffuse-interface tumor-growth model, Math. Models Methods Appl. Sci., 25 (2015), pp. 1011--1043.
[21]
M. Leroy-Lerêtre, G. Dimarco, M. Cazales, M.-L. Boizeau, B. Ducommun, V. Lobjois, and P. Degond, Are tumor cell lineages solely shaped by mechanical forces?, Bull. Math. Biol., 79 (2017), pp. 2356--2393.
[22]
E. H. Lieb and M. Loss, Analysis, Grad. Stud. Math. 14, AMS, Providence, RI, 2001.
[23]
B. Maury, A time-stepping scheme for inelastic collisions, Numer. Math., 102 (2006), pp. 649--679.
[24]
B. Maury, A. Roudneff-Chupin, F. Santambrogio, and J. Venel, Handling congestion in crowd motion models, Netw. Heterog. Media, 6 (2011), pp. 485--519.
[25]
S. Motsch and D. Peurichard, From short-range repulsion to Hele-Shaw problem in a model of tumor growth, J. Math. Biol., 76 (2018), pp. 205--234.
[26]
B. Perthame, F. Quirós, M. Tang, and N. Vauchelet, Derivation of a Hele-Shaw type system from a cell model with active motion, Interfaces Free Bound., 16 (2014), pp. 489--508.
[27]
B. Perthame, F. Quirós, and J. L. Vázquez, The Hele--Shaw asymptotics for mechanical models of tumor growth, Arch. Ration. Mech. Anal., 212 (2014), pp. 93--127.
[28]
B. Perthame and N. Vauchelet, Incompressible limit of a mechanical model of tumour growth with viscosity, Philos. Trans. A, 373 (2015), 20140283.
[29]
T. Roose, S. J. Chapman, and P. K. Maini, Mathematical models of avascular tumor growth, SIAM Rev., 49 (2007), pp. 179--208.
[30]
C. Walker, E. Mojares, and A. del Río Hernández, Role of extracellular matrix in development and cancer progression, Internat. J. Molecular Sci., 19 (2018), E3028.
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© 2020, Society for Industrial and Applied Mathematics.
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Society for Industrial and Applied Mathematics
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Published: 01 January 2020
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