Abstract
We give a description of fibroblast cell diffusion in a soft tissue, paying special attention to the coupling of force, matter and microforce balance laws through a suitable dissipation principle. To this end we cast our framework into a multi-level schematics, comprising both kinematics and kinetics, which is based on a characterization of the free energy. This way we lay down first a force balance law, where force and stress fields are defined as power conjugate quantities to velocity fields and their gradients, then we give a species molar balance law, with chemical potential test fields, as power conjugate quantities to the rate of change of species concentration, and finally a microforce balance law. The main feature of this framework is the constitutive expression for the chemical potential which turns out to be split in a natural way into a term derived from the homogeneous convex part of the free energy and an active external chemical potential giving rise to the spinodal decomposition. The active part of the chemical potential is given an expression depending on the cell density and resembling the one defined in [29], where it is meant to characterize an upward cell diffusion induced by cell motility.
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Notes
- 1.
Throughout the paper we will consistently denote test fields by underlying the corresponding symbol.
References
Allen, S.M., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1085–1095 (1979)
Anand, L.: A Cahn-Hilliard-type phase-field theory for species diffusion coupled with large elastic deformations. J. Mech. Phys. Solids 60, 1983–2002 (2012)
Bates, P.W., Fife, P.C.: Spectral comparison principles for the Cahn-Hilliard and phase-field equations, and time scales for coarsening. Phys. D 43, 335–348 (1990)
Bower, A.F., Guduru, P.R., Sethuraman, V.A.: A finite strain model of stress, diffusion, plastic flow, and electrochemical reactions in a lithium-ion half-cell. J. Mech. Phys. Solids 59, 804–828 (2011)
Cahn, J.W.: On spinodal decomposition. Acta Metall. 9, 795–801 (1961)
Cahn, J.W.: Spinodal decomposition. Trans. Metall. Soc. AIME 242, 89–103 (1968)
Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258–267 (1958)
Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. III. Nucleation in a two-component incompressible fluid. J. Chem. Phys. 31, 688–699 (1959)
Carter, S.B.: Principles of cell motility: the direction of cell movement and cancer invasion. Nature 208, 1183–1187 (1965)
Carter, S.B.: Haptotaxis and the mechanism of cell motility. Nature 213, 256–260 (1967)
Cates, M.E., Tailleur, J.: Motility-induced phase separation. Ann. Rev. Condens. Matt. Phys. 6, 219–244 (2015)
Chatelain, C., Balois, T., Ciarletta, P., BenAmar, M.: Emergence of microstructural patterns in skin cancer: a phase separation analysis in a binary mixture. New J. Phys. 13, 115013 (2011)
Chen, L., Fan, F., Hong, L., Chen, J., Ji, Y.Z., Zhang, S.L., Zhu, T., Chen, L.Q.: A phase-field model coupled with large elasto-plastic deformation: application to lithiated silicon electrodes. J. Electrochem. Soc. 161, F3164–F3172 (2014)
Coleman, B.D., Noll, W.: The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Rational. Mech. Anal. 13, 167–178 (1963)
COMSOL, Inc.: COMSOL Multiphysics® Reference Manual, version 5.3
Cui, Z.W., Gao, F., Qu, J.M.: A finite deformation stress-dependent chemical potential and its applications to lithium ion batteries. J. Mech. Phys. Solids 60, 1280–1295 (2012)
Di Leo, C.V., Rejovitzky, E., Anand, L.: A Cahn-Hilliard-type phase-field theory for species diffusion coupled with large elastic deformations: application to phase-separating Li-ion electrode materials. J. Mech. Phys. Solids 70, 1–29 (2014)
Eshelby, J.D.: Elastic energy-momentum tensor. J. Elast. 5, 321–335 (1975)
Fife, P.C., Penrose, O.: Interfacial dynamics for thermodynamically consistent phase-field models with nonconserved order parameter. Electron. J. Differ. Equ. 1–49, 1995 (1995)
Foty, R.A., Steinberg, M.S.: The differential adhesion hypothesis: a direct evaluation. Dev. Biol. 278, 255–263 (2005)
Gierer, A., Meinhardt, H.: A theory of biological pattern formation. Kybernetik 12, 30–39 (1972)
Gurtin, M.E.: Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance. Phys. D 92, 178–192 (1996)
Gurtin, M.E., Fried, E., Anand, L.: The Mechanics and Thermodynamics of Continua. Cambridge University Press, Cambridge (2010)
Larché, F., Cahn, J.W.: Linear theory of thermomechanical equilibrium of solids under stress. Acta Metall. 21, 1051–1063 (1973)
Larché, F., Cahn, J.W.: Overview no. 41 The interactions of composition and stress in crystalline solids. Acta Metallurgica 33, 331–357 (1985)
Meinhardt, H.: Models of biological pattern formation: from elementary steps to the organization of embryonic axes. Curr. Top. Dev. Biol. 81, 1–63 (2008)
Murray, J.D., Oster, G.F.: Cell traction models for generating pattern and form in morphogenesis. J. Math. Biol. 19, 265–279 (1984)
Murray, J.D., Oster, G.F.: Generation of biological pattern and form. IMA J. Math. Appl. Med. Biol. 1, 51–75 (1984)
Oster, G.F., Murray, J.D., Harris, A.K.: Mechanical aspects of mesenchymal morphogenesis. J. Embryol. exp. Morph. 78, 83–125 (1983)
Podio-Guidugli, P.: Models of phase segregation and diffusion of atomic species on a lattice. Ricerche mat. 55, 105–118 (2006)
Preziosi, L., Scianna, M.: Mathematical models of the interaction of cells and cell aggregates with the extracellular matrix. In: Preziosi, L., Chaplain, M., Pugliese, A. (eds.) Mathematical Models and Methods for Living Systems, pp. 131–210. Springer (2016)
Steinberg, M.S.: Differential adhesion in morphogenesis: a modern view. Curr. Opin. Genet. Dev. 17, 281–286 (2007)
Tatone, A., Recrosi, F., Repetto, R., Guidoboni, G.: From species diffusion to poroelasticity and the modeling of lamina cribrosa. J. Mech. Phys. Solids 124, 849–870 (2019)
Tiribocchi, A., Wittkowski, R., Marenduzzo, D., Cates, M.E.: Active model H: scalar active matter in a momentum-conserving fluid. Phys. Rev. Lett. 115, 188302–1–5 (2015)
Turing, A.M.: The chemical basis of morphogenesis. Philos. Trans. Roy. Soc. London Ser. B Biol. Sci. 237, 37–72 (1952)
Wittkowski, R., Tiribocchi, A., Stenhammar, J., Allen, R.J., Marenduzzo, D., Cates, M.E.: Scalar \(\varphi ^4\) field theory for active-particle phase separation. Nat. Commun. 5, 4351 (2014)
Wu, C.H.: The role of Eshelby stress in composition-generated and stress-assisted diffusion. J. Mech. Phys. Solids 49, 1771–1794 (2001)
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Recrosi, F., Repetto, R., Tatone, A., Tomassetti, G. (2020). Mechanical Model of Fiber Morphogenesis in the Liver. In: Carcaterra, A., Paolone, A., Graziani, G. (eds) Proceedings of XXIV AIMETA Conference 2019. AIMETA 2019. Lecture Notes in Mechanical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-030-41057-5_55
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