Governing equations of tissue modelling and remodelling: A unified generalised description of surface and bulk balance
arXiv:1512.07644 · doi:10.1371/journal.pone.0152582
Abstract
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 time signatures of BMD or TMD during remodelling events may provide a way to detect these events occurring at lower, unseen spatial resolutions from microCT scans.
14 pages, 8 figures. V2: minor stylistic changes, more detailed derivation of Eqs (30)-(31), additional comments on implication of BMD and TMD signatures for microCT scans
References in corpus (6)
- A multiscale mechanobiological model of bone remodelling predicts site-specific bone loss in the femur during osteoporosis and mechanical disuse
- Spatio-temporal structure of cell distribution in cortical Bone Multicellular Units: a mathematical model
- Bone refilling in cortical bone multicellular units: Insights into tetracycline double labelling from a computational model
- Osteocytes as a record of bone formation dynamics: A mathematical model of osteocyte generation in bone matrix
- Endocortical bone loss in osteoporosis: The role of bone surface availability
- Potential theory, path integrals and the Laplacian of the indicator