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Identifier 000425140
Title Modelling analysis and computation of cell-induced phase transitions in fibrous biomaterials
Alternative Title Μοντελοποίηση, ανάλυση και υπολογισμοί φάσεων που προκαλούνται από κύτταρα σε ινώδη βιουλικά
Author Γρέκας, Γεώργιος
Thesis advisor Ροζάκης, Φοίβος
Μακριδάκης, Χαράλαμπος
Abstract Modelling, Analysis and Computation of Cell-Induced PhaseTransitions in Fibrous BiomaterialsbyGeorgios GrekasSubmitted to the Department of Mathematics and Applied Mathematicson July, 2019, in partial fulfillment of therequirements for the degree ofDoctor of PhilosophyAbstractBy exerting mechanical forces, biological cells cause striking spatial patterns of lo-calised deformation in the surrounding fibrous collagen matrix. Tether-like pathsof high densification and fiber alignment form between cells, and radial hair-likebands emanate from cell clusters. While tethers may facilitate cell communication,the mechanism for their formation is unclear. In this study, modeling, numericalmethods and computations are combined to show that tether formation is a densifi-cation phase transition of the fibrous extracellular matrix, caused by microbucklinginstability of network fibers under compression. The mechanical behaviour of the ex-tracellular matrix (ECM) caused by cell contraction is modelled and analysed froma macroscopic perspective employing the theory of nonlinear elasticity for phasetransitions. It is assumed that individual collagen fibers can sustain tension butbuckle and collapse under compression. Averaging over fiber orientations yields atwo-phase bistable strain energy density for fibrous collagen, with a densified sec-ond phase. Simulated energy-minimizing deformations exhibit strain discontinuitiesbetween the low- and the high-density phase, which localizes within intercellulartethers and radial emanations from cell clusters, as experimentally observed. Theformation of the localized deformations is due to the fact that the resulting strainenergy function fails to be rank one convex, and essentially equivalent to a multi-wellpotential. This failure of rank-one convexity implies a loss of ellipticity of the Euler-Lagrange PDEs of the corresponding energy functional. For a wide class of similarproblems it is known that there exist oscillatory minimizing sequences with finer microstructures involving increasing numbers of strain jumps. Similar behaviour isobserved in computed solutions as the mesh size decreases. In order to show thatthis mesh dependence is not a numerical artefact, a higher gradient term is added tothe model. This term also introduces an internal length scale, which is an additionalmaterial parameter related to characteristic fiber length, bending stiffness and otherparameters of the fiber network.To approximate a variational problem involving a non rank-one convex strain-energyfunction, regularized by a higher gradient term, a non conforming finine elementmethod is used. It is shown that a suitable numerical scheme is utilized in the sensethat the numerical approximations indeed converge in the limit to minimisers ofthe continuous problem. This is done by employing the theory ofΓ-convergence ofthe approximate energy minimisation functionals to the continuous model when thediscretisation parameter tends to zero. This is a rather involved task due to thestructure of numerical approximations which are defined in spaces with lower regu-larity than the space where the minimisers of the continuous variational problem aresought. Furthermore, when the energy functional contains terms with exponentialgrowth, additional embedding theorems are required. For this purpose, an embed-ding has been proved of piecewise polynomial spaces admitting discontinuities in thegradient, into appropriate Orlicz spaces.
Language English
Subject Noncinear elasticity
Αλλαγές φάσεων
Γ- σύγκλιση
Μη γραμμική ελαστικότητα
Issue date 2019-11-22
Collection   Faculty/Department--Faculty of Sciences and Engineering--Department of Applied Mathematics--Doctoral theses
  Type of Work--Doctoral theses
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