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Governing Equations of Tissue Modelling and Remodelling: A Unified Generalised Description of Surface and Bulk Balance
Several biological tissues undergo changes in their geometry and in their bulk material properties by modelling and remodelling processes. Modelling synthesises tissue in some regions and removes tissue in others. Remodelling overwrites old tissue material properties with newly formed, immature tiss...
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Formato: | Online Artículo Texto |
Lenguaje: | English |
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Public Library of Science
2016
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Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4820236/ https://www.ncbi.nlm.nih.gov/pubmed/27043309 http://dx.doi.org/10.1371/journal.pone.0152582 |
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author | Buenzli, Pascal R. |
author_facet | Buenzli, Pascal R. |
author_sort | Buenzli, Pascal R. |
collection | PubMed |
description | Several biological tissues undergo changes in their geometry and in their bulk material properties by modelling and remodelling processes. Modelling synthesises tissue in some regions and removes tissue in others. Remodelling overwrites old tissue material properties with newly formed, immature tissue properties. As a result, tissues are made up of different “patches”, i.e., adjacent tissue regions of different ages and different material properties, within evolving boundaries. In this paper, generalised equations governing the spatio-temporal evolution of such tissues are developed within the continuum model. These equations take into account nonconservative, discontinuous surface mass balance due to creation and destruction of material at moving interfaces, and bulk balance due to tissue maturation. These equations make it possible to model patchy tissue states and their evolution without explicitly maintaining a record of when/where resorption and formation processes occurred. The time evolution of spatially averaged tissue properties is derived systematically by integration. These spatially-averaged equations cannot be written in closed form as they retain traces that tissue destruction is localised at tissue boundaries. The formalism developed in this paper is applied to bone tissues, which exhibit strong material heterogeneities due to their slow mineralisation and remodelling processes. Evolution equations are proposed in particular for osteocyte density and bone mineral density. Effective average equations for bone mineral density (BMD) and tissue mineral density (TMD) are derived using a mean-field approximation. The error made by this approximation when remodelling patchy tissue is investigated. The specific signatures of the time evolution of BMD or TMD during remodelling events are exhibited. These signatures may provide a way to detect remodelling events at lower, unseen spatial resolutions from microCT scans. |
format | Online Article Text |
id | pubmed-4820236 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2016 |
publisher | Public Library of Science |
record_format | MEDLINE/PubMed |
spelling | pubmed-48202362016-04-22 Governing Equations of Tissue Modelling and Remodelling: A Unified Generalised Description of Surface and Bulk Balance Buenzli, Pascal R. PLoS One Research Article Several biological tissues undergo changes in their geometry and in their bulk material properties by modelling and remodelling processes. Modelling synthesises tissue in some regions and removes tissue in others. Remodelling overwrites old tissue material properties with newly formed, immature tissue properties. As a result, tissues are made up of different “patches”, i.e., adjacent tissue regions of different ages and different material properties, within evolving boundaries. In this paper, generalised equations governing the spatio-temporal evolution of such tissues are developed within the continuum model. These equations take into account nonconservative, discontinuous surface mass balance due to creation and destruction of material at moving interfaces, and bulk balance due to tissue maturation. These equations make it possible to model patchy tissue states and their evolution without explicitly maintaining a record of when/where resorption and formation processes occurred. The time evolution of spatially averaged tissue properties is derived systematically by integration. These spatially-averaged equations cannot be written in closed form as they retain traces that tissue destruction is localised at tissue boundaries. The formalism developed in this paper is applied to bone tissues, which exhibit strong material heterogeneities due to their slow mineralisation and remodelling processes. Evolution equations are proposed in particular for osteocyte density and bone mineral density. Effective average equations for bone mineral density (BMD) and tissue mineral density (TMD) are derived using a mean-field approximation. The error made by this approximation when remodelling patchy tissue is investigated. The specific signatures of the time evolution of BMD or TMD during remodelling events are exhibited. These signatures may provide a way to detect remodelling events at lower, unseen spatial resolutions from microCT scans. Public Library of Science 2016-04-04 /pmc/articles/PMC4820236/ /pubmed/27043309 http://dx.doi.org/10.1371/journal.pone.0152582 Text en © 2016 Pascal R. Buenzli http://creativecommons.org/licenses/by/4.0/ This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. |
spellingShingle | Research Article Buenzli, Pascal R. Governing Equations of Tissue Modelling and Remodelling: A Unified Generalised Description of Surface and Bulk Balance |
title | Governing Equations of Tissue Modelling and Remodelling: A Unified Generalised Description of Surface and Bulk Balance |
title_full | Governing Equations of Tissue Modelling and Remodelling: A Unified Generalised Description of Surface and Bulk Balance |
title_fullStr | Governing Equations of Tissue Modelling and Remodelling: A Unified Generalised Description of Surface and Bulk Balance |
title_full_unstemmed | Governing Equations of Tissue Modelling and Remodelling: A Unified Generalised Description of Surface and Bulk Balance |
title_short | Governing Equations of Tissue Modelling and Remodelling: A Unified Generalised Description of Surface and Bulk Balance |
title_sort | governing equations of tissue modelling and remodelling: a unified generalised description of surface and bulk balance |
topic | Research Article |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4820236/ https://www.ncbi.nlm.nih.gov/pubmed/27043309 http://dx.doi.org/10.1371/journal.pone.0152582 |
work_keys_str_mv | AT buenzlipascalr governingequationsoftissuemodellingandremodellingaunifiedgeneraliseddescriptionofsurfaceandbulkbalance |