Coupling Chemotaxis and Growth Poromechanics for the Modelling of Feather Primordia Patterning
In this paper we propose a new mathematical model for describing the complex interplay between skin cell populations with fibroblast growth factor and bone morphogenetic protein, occurring within deformable porous media describing feather primordia patterning. Tissue growth, in turn, modifies the tr...
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MDPI AG
2022-11-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/21/4096 |
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| author | Nicolás A. Barnafi Luis Miguel De Oliveira Vilaca Michel C. Milinkovitch Ricardo Ruiz-Baier |
| author_facet | Nicolás A. Barnafi Luis Miguel De Oliveira Vilaca Michel C. Milinkovitch Ricardo Ruiz-Baier |
| author_sort | Nicolás A. Barnafi |
| collection | DOAJ |
| description | In this paper we propose a new mathematical model for describing the complex interplay between skin cell populations with fibroblast growth factor and bone morphogenetic protein, occurring within deformable porous media describing feather primordia patterning. Tissue growth, in turn, modifies the transport of morphogens (described by reaction-diffusion equations) through diverse mechanisms such as advection from the solid velocity generated by mechanical stress, and mass supply. By performing an asymptotic linear stability analysis on the coupled poromechanical-chemotaxis system (assuming rheological properties of the skin cell aggregates that reside in the regime of infinitesimal strains and where the porous structure is fully saturated with interstitial fluid and encoding the coupling mechanisms through active stress) we obtain the conditions on the parameters—especially those encoding coupling mechanisms—under which the system will give rise to spatially heterogeneous solutions. We also extend the mechanical model to the case of incompressible poro-hyperelasticity and include the mechanisms of anisotropic solid growth and feedback by means of standard Lee decompositions of the tensor gradient of deformation. Because the model in question involves the coupling of several nonlinear PDEs, we cannot straightforwardly obtain closed-form solutions. We therefore design a suitable numerical method that employs backward Euler time discretisation, linearisation of the semidiscrete problem through Newton–Raphson’s method, a seven-field finite element formulation for the spatial discretisation, and we also advocate the construction and efficient implementation of tailored robust solvers. We present a few illustrative computational examples in 2D and 3D, briefly discussing different spatio-temporal patterns of growth factors as well as the associated solid response scenario depending on the specific poromechanical regime. Our findings confirm the theoretically predicted behaviour of spatio-temporal patterns, and the produced results reveal a qualitative agreement with respect to the expected experimental behaviour. We stress that the present study provides insight on several biomechanical properties of primordia patterning. |
| first_indexed | 2024-03-09T18:52:03Z |
| format | Article |
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| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T18:52:03Z |
| publishDate | 2022-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-933188ad90744cf695624510007066ec2023-11-24T05:44:50ZengMDPI AGMathematics2227-73902022-11-011021409610.3390/math10214096Coupling Chemotaxis and Growth Poromechanics for the Modelling of Feather Primordia PatterningNicolás A. Barnafi0Luis Miguel De Oliveira Vilaca1Michel C. Milinkovitch2Ricardo Ruiz-Baier3Department of Mathematics, Università di Pavia, Via Ferrata 1, 27100 Pavia, ItalyLaboratory of Artificial & Natural Evolution (LANE), Department of Genetics and Evolution, University of Geneva, 30 Quai Ernest Ansermet, 1211 Geneva, SwitzerlandLaboratory of Artificial & Natural Evolution (LANE), Department of Genetics and Evolution, University of Geneva, 30 Quai Ernest Ansermet, 1211 Geneva, SwitzerlandSchool of Mathematics, Monash University, 9 Rainforest Walk, Melbourne, VIC 3800, AustraliaIn this paper we propose a new mathematical model for describing the complex interplay between skin cell populations with fibroblast growth factor and bone morphogenetic protein, occurring within deformable porous media describing feather primordia patterning. Tissue growth, in turn, modifies the transport of morphogens (described by reaction-diffusion equations) through diverse mechanisms such as advection from the solid velocity generated by mechanical stress, and mass supply. By performing an asymptotic linear stability analysis on the coupled poromechanical-chemotaxis system (assuming rheological properties of the skin cell aggregates that reside in the regime of infinitesimal strains and where the porous structure is fully saturated with interstitial fluid and encoding the coupling mechanisms through active stress) we obtain the conditions on the parameters—especially those encoding coupling mechanisms—under which the system will give rise to spatially heterogeneous solutions. We also extend the mechanical model to the case of incompressible poro-hyperelasticity and include the mechanisms of anisotropic solid growth and feedback by means of standard Lee decompositions of the tensor gradient of deformation. Because the model in question involves the coupling of several nonlinear PDEs, we cannot straightforwardly obtain closed-form solutions. We therefore design a suitable numerical method that employs backward Euler time discretisation, linearisation of the semidiscrete problem through Newton–Raphson’s method, a seven-field finite element formulation for the spatial discretisation, and we also advocate the construction and efficient implementation of tailored robust solvers. We present a few illustrative computational examples in 2D and 3D, briefly discussing different spatio-temporal patterns of growth factors as well as the associated solid response scenario depending on the specific poromechanical regime. Our findings confirm the theoretically predicted behaviour of spatio-temporal patterns, and the produced results reveal a qualitative agreement with respect to the expected experimental behaviour. We stress that the present study provides insight on several biomechanical properties of primordia patterning.https://www.mdpi.com/2227-7390/10/21/4096mathematical modellingnumerical methodslinear and nonlinear poroelasticitychemotaxislinear stability analysistissue growth |
| spellingShingle | Nicolás A. Barnafi Luis Miguel De Oliveira Vilaca Michel C. Milinkovitch Ricardo Ruiz-Baier Coupling Chemotaxis and Growth Poromechanics for the Modelling of Feather Primordia Patterning Mathematics mathematical modelling numerical methods linear and nonlinear poroelasticity chemotaxis linear stability analysis tissue growth |
| title | Coupling Chemotaxis and Growth Poromechanics for the Modelling of Feather Primordia Patterning |
| title_full | Coupling Chemotaxis and Growth Poromechanics for the Modelling of Feather Primordia Patterning |
| title_fullStr | Coupling Chemotaxis and Growth Poromechanics for the Modelling of Feather Primordia Patterning |
| title_full_unstemmed | Coupling Chemotaxis and Growth Poromechanics for the Modelling of Feather Primordia Patterning |
| title_short | Coupling Chemotaxis and Growth Poromechanics for the Modelling of Feather Primordia Patterning |
| title_sort | coupling chemotaxis and growth poromechanics for the modelling of feather primordia patterning |
| topic | mathematical modelling numerical methods linear and nonlinear poroelasticity chemotaxis linear stability analysis tissue growth |
| url | https://www.mdpi.com/2227-7390/10/21/4096 |
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