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A Continuum Mathematical Model of Substrate-Mediated Tissue Growth
We consider a continuum mathematical model of biological tissue formation inspired by recent experiments describing thin tissue growth in 3D-printed bioscaffolds. The continuum model, which we call the substrate model, involves a partial differential equation describing the density of tissue, [Formu...
Autores principales: | , , |
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Formato: | Online Artículo Texto |
Lenguaje: | English |
Publicado: |
Springer US
2022
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Materias: | |
Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8891221/ https://www.ncbi.nlm.nih.gov/pubmed/35237899 http://dx.doi.org/10.1007/s11538-022-01005-7 |
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author | El-Hachem, Maud McCue, Scott W. Simpson, Matthew J. |
author_facet | El-Hachem, Maud McCue, Scott W. Simpson, Matthew J. |
author_sort | El-Hachem, Maud |
collection | PubMed |
description | We consider a continuum mathematical model of biological tissue formation inspired by recent experiments describing thin tissue growth in 3D-printed bioscaffolds. The continuum model, which we call the substrate model, involves a partial differential equation describing the density of tissue, [Formula: see text] that is coupled to the concentration of an immobile extracellular substrate, [Formula: see text] . Cell migration is modelled with a nonlinear diffusion term, where the diffusive flux is proportional to [Formula: see text] , while a logistic growth term models cell proliferation. The extracellular substrate [Formula: see text] is produced by cells and undergoes linear decay. Preliminary numerical simulations show that this mathematical model is able to recapitulate key features of recent tissue growth experiments, including the formation of sharp fronts. To provide a deeper understanding of the model we analyse travelling wave solutions of the substrate model, showing that the model supports both sharp-fronted travelling wave solutions that move with a minimum wave speed, [Formula: see text] , as well as smooth-fronted travelling wave solutions that move with a faster travelling wave speed, [Formula: see text] . We provide a geometric interpretation that explains the difference between smooth and sharp-fronted travelling wave solutions that is based on a slow manifold reduction of the desingularised three-dimensional phase space. In addition, we also develop and test a series of useful approximations that describe the shape of the travelling wave solutions in various limits. These approximations apply to both the sharp-fronted and smooth-fronted travelling wave solutions. Software to implement all calculations is available at GitHub. SUPPLEMENTARY INFORMATION: The online version contains supplementary material available at 10.1007/s11538-022-01005-7. |
format | Online Article Text |
id | pubmed-8891221 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2022 |
publisher | Springer US |
record_format | MEDLINE/PubMed |
spelling | pubmed-88912212022-03-08 A Continuum Mathematical Model of Substrate-Mediated Tissue Growth El-Hachem, Maud McCue, Scott W. Simpson, Matthew J. Bull Math Biol Original Article We consider a continuum mathematical model of biological tissue formation inspired by recent experiments describing thin tissue growth in 3D-printed bioscaffolds. The continuum model, which we call the substrate model, involves a partial differential equation describing the density of tissue, [Formula: see text] that is coupled to the concentration of an immobile extracellular substrate, [Formula: see text] . Cell migration is modelled with a nonlinear diffusion term, where the diffusive flux is proportional to [Formula: see text] , while a logistic growth term models cell proliferation. The extracellular substrate [Formula: see text] is produced by cells and undergoes linear decay. Preliminary numerical simulations show that this mathematical model is able to recapitulate key features of recent tissue growth experiments, including the formation of sharp fronts. To provide a deeper understanding of the model we analyse travelling wave solutions of the substrate model, showing that the model supports both sharp-fronted travelling wave solutions that move with a minimum wave speed, [Formula: see text] , as well as smooth-fronted travelling wave solutions that move with a faster travelling wave speed, [Formula: see text] . We provide a geometric interpretation that explains the difference between smooth and sharp-fronted travelling wave solutions that is based on a slow manifold reduction of the desingularised three-dimensional phase space. In addition, we also develop and test a series of useful approximations that describe the shape of the travelling wave solutions in various limits. These approximations apply to both the sharp-fronted and smooth-fronted travelling wave solutions. Software to implement all calculations is available at GitHub. SUPPLEMENTARY INFORMATION: The online version contains supplementary material available at 10.1007/s11538-022-01005-7. Springer US 2022-03-02 2022 /pmc/articles/PMC8891221/ /pubmed/35237899 http://dx.doi.org/10.1007/s11538-022-01005-7 Text en © The Author(s) 2022 https://creativecommons.org/licenses/by/4.0/Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ (https://creativecommons.org/licenses/by/4.0/) . |
spellingShingle | Original Article El-Hachem, Maud McCue, Scott W. Simpson, Matthew J. A Continuum Mathematical Model of Substrate-Mediated Tissue Growth |
title | A Continuum Mathematical Model of Substrate-Mediated Tissue Growth |
title_full | A Continuum Mathematical Model of Substrate-Mediated Tissue Growth |
title_fullStr | A Continuum Mathematical Model of Substrate-Mediated Tissue Growth |
title_full_unstemmed | A Continuum Mathematical Model of Substrate-Mediated Tissue Growth |
title_short | A Continuum Mathematical Model of Substrate-Mediated Tissue Growth |
title_sort | continuum mathematical model of substrate-mediated tissue growth |
topic | Original Article |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8891221/ https://www.ncbi.nlm.nih.gov/pubmed/35237899 http://dx.doi.org/10.1007/s11538-022-01005-7 |
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