How osteons form: A quantitative hypothesis-testing analysis of cortical pore filling and wall asymmetry.

Osteon morphology provides valuable information about the interplay between different processes involved in bone remodelling. The correct quantitative interpretation of these morphological features is challenging due to the complexity of interactions between osteoblast behaviour, and the evolving geometry of cortical pores during pore closing. We present a combined experimental and mathematical modelling study to provide insights into bone formation mechanisms during cortical bone remodelling based on histological cross-sections of quiescent human osteons and hypothesis-testing analyses. We introduce wall thickness asymmetry as a measure of the local asymmetry of bone formation within an osteon and examine the frequency distribution of wall thickness asymmetry in cortical osteons from human iliac crest bone samples from women 16-78 years old. Our measurements show that most osteons possess some degree of asymmetry, and that the average degree of osteon asymmetry in cortical bone evolves with age. We then propose a comprehensive mathematical model of cortical pore filling that includes osteoblast secretory activity, osteoblast elimination, osteoblast embedment as osteocytes, and osteoblast crowding and redistribution along the bone surface. The mathematical model is first calibrated to symmetric osteon data, and then used to test three mechanisms of asymmetric wall formation against osteon data: (i) delays in the onset of infilling around the cement line; (ii) heterogeneous osteoblastogenesis around the bone perimeter; and (iii) heterogeneous osteoblast secretory rate around the bone perimeter. Our results suggest that wall thickness asymmetry due to off-centred Haversian pores within osteons, and that nonuniform lamellar thicknesses within osteons are important morphological features that can indicate the prevalence of specific asymmetry-generating mechanisms. This has significant implications for the study of disruptions of bone formation as it could indicate what biological bone formation processes may become disrupted with age or disease.

Reliable and efficient parameter estimation using approximate continuum limit descriptions of stochastic models.

Stochastic individual-based mathematical models are attractive for modelling biological phenomena because they naturally capture the stochasticity and variability that is often evident in biological data. Such models also allow us to track the motion of individuals within the population of interest. Unfortunately, capturing this microscopic detail means that simulation and parameter inference can become computationally expensive. One approach for overcoming this computational limitation is to coarse-grain the stochastic model to provide an approximate continuum model that can be solved using far less computational effort. However, coarse-grained continuum models can be biased or inaccurate, particularly for certain parameter regimes. In this work, we combine stochastic and continuum mathematical models in the context of lattice-based models of two-dimensional cell biology experiments by demonstrating how to simulate two commonly used experiments: cell proliferation assays and barrier assays. Our approach involves building a simple statistical model of the discrepancy between the expensive stochastic model and the associated computationally inexpensive coarse-grained continuum model. We form this statistical model based on a limited number of expensive stochastic model evaluations at design points sampled from a user-chosen distribution, corresponding to a computer experiment design problem. With straightforward design point selection schemes, we show that using the statistical model of the discrepancy in tandem with the computationally inexpensive continuum model allows us to carry out prediction and inference while correcting for biases and inaccuracies due to the continuum approximation. We demonstrate this approach by simulating a proliferation assay, where the continuum limit model is the well-known logistic ordinary differential equation, as well as a barrier assay where the continuum limit model is closely related to the well-known Fisher-KPP partial differential equation. We construct an approximate likelihood function for parameter inference, both with and without discrepancy correction terms. Using maximum likelihood estimation, we provide point estimates of the unknown parameters, and use the profile likelihood to characterise the uncertainty in these estimates and form approximate confidence intervals. For the range of inference problems considered, working with the continuum limit model alone leads to biased parameter estimation and confidence intervals with poor coverage. In contrast, incorporating correction terms arising from the statistical model of the model discrepancy allows us to recover the parameters accurately with minimal computational overhead. The main tradeoff is that the associated confidence intervals are typically broader, reflecting the additional uncertainty introduced by the approximation process. All algorithms required to replicate the results in this work are written in the open source Julia language and are available at GitHub.

Interpreting how nonlinear diffusion affects the fate of bistable populations using a discrete modelling framework.

Understanding whether a population will survive or become extinct is a central question in population biology. One way of exploring this question is to study population dynamics using reaction-diffusion equations, where migration is usually represented as a linear diffusion term, and birth-death is represented with a nonlinear source term. While linear diffusion is most commonly employed to study migration, there are several limitations of this approach, such as the inability of linear diffusion-based models to predict a well-defined population front. One way to overcome this is to generalize the constant diffusivity, D , to a nonlinear diffusivity function D ( C ) , where C > 0 is the population density. While the choice of D ( C ) affects long-term survival or extinction of a bistable population, working solely in a continuum framework makes it difficult to understand how the choice of D ( C ) affects survival or extinction. We address this question by working with a discrete simulation model that is easy to interpret. This approach provides clear insight into how the choice of D ( C ) either encourages or suppresses population extinction relative to the classical linear diffusion model.

Extinction of Bistable Populations is Affected by the Shape of their Initial Spatial Distribution.

The question of whether biological populations survive or are eventually driven to extinction has long been examined using mathematical models. In this work, we study population survival or extinction using a stochastic, discrete lattice-based random walk model where individuals undergo movement, birth and death events. The discrete model is defined on a two-dimensional hexagonal lattice with periodic boundary conditions. A key feature of the discrete model is that crowding effects are introduced by specifying two different crowding functions that govern how local agent density influences movement events and birth/death events. The continuum limit description of the discrete model is a nonlinear reaction-diffusion equation, and we focus on crowding functions that lead to linear diffusion and a bistable source term that is often associated with the strong Allee effect. Using both the discrete and continuum modelling tools, we explore the complicated relationship between the long-term survival or extinction of the population and the initial spatial arrangement of the population. In particular, we study different spatial arrangements of initial distributions: (i) a well-mixed initial distribution where the initial density is independent of position in the domain; (ii) a vertical strip initial distribution where the initial density is independent of vertical position in the domain; and, (iii) several forms of two-dimensional initial distributions where the initial population is distributed in regions with different shapes. Our results indicate that the shape of the initial spatial distribution of the population affects extinction of bistable populations. All software required to solve the discrete and continuum models used in this work are available on GitHub .

A quantitative analysis of cell bridging kinetics on a scaffold using computer vision algorithms.

Tissue engineering involves the seeding of cells into a structural scaffolding to regenerate the architecture of damaged or diseased tissue. To effectively design a scaffold, an understanding of how cells collectively sense and react to the geometry of their local environment is needed. Advances in the development of melt electro-writing have allowed micron and submicron polymeric fibres to be accurately printed into porous, complex and three-dimensional structures. By using melt electrowriting, we created a geometrically relevant in vitro scaffold model to study cellular spatial-temporal kinetics. These scaffolds were paired with custom computer vision algorithms to investigate cell nuclei, cell membrane actin and scaffold fibres over different pore sizes (200-600 µm) and time points (28 days). We find that cells proliferated much faster in the smaller (200 µm) pores which halved the time until confluence versus larger (500 and 600 µm) pores. Our analysis of stained actin fibres revealed that cells were highly aligned to the fibres and the leading edge of the pore filling front, and we found that cells behind the leading edge were not aligned in any particular direction. This study provides a systematic understanding of cellular spatial temporal kinetics within a 3D in vitro model to inform the design of more effective synthetic tissue engineering scaffolds for tissue regeneration. STATEMENT OF SIGNIFICANCE: Advances in the development of melt electro-writing have allowed micron and submicron polymeric fibres to be accurately printed into porous, complex and three-dimensional structures. By using melt electrowriting, we created a geometrically relevant in vitro model to study cellular spatial-temporal kinetics to provide a systematic understanding of cellular spatial temporal kinetics within a 3D in vitro model. The insights presented in this work help to inform the design of more effective synthetic tissue engineering scaffolds by reducing cell culture time; which is valuable information for the implant or lab-grown-meat industries.

Model-based data analysis of tissue growth in thin 3D printed scaffolds.

Tissue growth in three-dimensional (3D) printed scaffolds enables exploration and control of cell behaviour in more biologically realistic geometries than that allowed by traditional 2D cell culture. Cell proliferation and migration in these experiments have yet to be explicitly characterised, limiting the ability of experimentalists to determine the effects of various experimental conditions, such as scaffold geometry, on cell behaviour. We consider tissue growth by osteoblastic cells in melt electro-written scaffolds that comprise thin square pores with sizes that were deliberately increased between experiments. We collect highly detailed temporal measurements of the average cell density, tissue coverage, and tissue geometry. To quantify tissue growth in terms of the underlying cell proliferation and migration processes, we introduce and calibrate a mechanistic mathematical model based on the Porous-Fisher reaction-diffusion equation. Parameter estimates and uncertainty quantification through profile likelihood analysis reveal consistency in the rate of cell proliferation and steady-state cell density between pore sizes. This analysis also serves as an important model verification tool: while the use of reaction-diffusion models in biology is widespread, the appropriateness of these models to describe tissue growth in 3D scaffolds has yet to be explored. We find that the Porous-Fisher model is able to capture features relating to the cell density and tissue coverage, but is not able to capture geometric features relating to the circularity of the tissue interface. Our analysis identifies two distinct stages of tissue growth, suggests several areas for model refinement, and provides guidance for future experimental work that explores tissue growth in 3D printed scaffolds.

Modelling cell guidance and curvature control in evolving biological tissues.

Tissue geometry is an important influence on the evolution of many biological tissues. The local curvature of an evolving tissue induces tissue crowding or spreading, which leads to differential tissue growth rates, and to changes in cellular tension, which can influence cell behaviour. Here, we investigate how directed cell motion interacts with curvature control in evolving biological tissues. Directed cell motion is involved in the generation of angled tissue growth and anisotropic tissue material properties, such as tissue fibre orientation. We develop a new cell-based mathematical model of tissue growth that includes both curvature control and cell guidance mechanisms to investigate their interplay. The model is based on conservation principles applied to the density of tissue synthesising cells at or near the tissue's moving boundary. The resulting mathematical model is a partial differential equation for cell density on a moving boundary, which is solved numerically using a hybrid front-tracking method called the cell-based particle method. The inclusion of directed cell motion allows us to model new types of biological growth, where tangential cell motion is important for the evolution of the interface, or for the generation of anisotropic tissue properties. We illustrate such situations by applying the model to simulate both the resorption and infilling components of the bone remodelling process, and to simulate root hair growth. We also provide user-friendly MATLAB code to implement the algorithms.

The role of mechanical interactions in EMT.

The detachment of cells from the boundary of an epithelial tissue and the subsequent invasion of these cells into surrounding tissues is important for cancer development and wound healing, and is strongly associated with the epithelial-mesenchymal transition (EMT). Chemical signals, such as TGF-ß, produced by surrounding tissue can be uptaken by cells and induce EMT. In this work, we present a novel cell-based discrete mathematical model of mechanical cellular relaxation, cell proliferation, and cell detachment driven by chemically-dependent EMT in an epithelial tissue. A continuum description of the model is then derived in the form of a novel nonlinear free boundary problem. Using the discrete and continuum models we explore how the coupling of chemical transport and mechanical interactions influences EMT, and postulate how this could be used to help control EMT in pathological situations.

A free boundary mechanobiological model of epithelial tissues.

In this study, we couple intracellular signalling and cell-based mechanical properties to develop a novel free boundary mechanobiological model of epithelial tissue dynamics. Mechanobiological coupling is introduced at the cell level in a discrete modelling framework, and new reaction-diffusion equations are derived to describe tissue-level outcomes. The free boundary evolves as a result of the underlying biological mechanisms included in the discrete model. To demonstrate the accuracy of the continuum model, we compare numerical solutions of the discrete and continuum models for two different signalling pathways. First, we study the Rac-Rho pathway where cell- and tissue-level mechanics are directly related to intracellular signalling. Second, we study an activator-inhibitor system which gives rise to spatial and temporal patterning related to Turing patterns. In all cases, the continuum model and free boundary condition accurately reflect the cell-level processes included in the discrete model.

Mechanical Cell Competition in Heterogeneous Epithelial Tissues.

Mechanical cell competition is important during tissue development, cancer invasion, and tissue ageing. Heterogeneity plays a key role in practical applications since cancer cells can have different cell stiffness and different proliferation rates than normal cells. To study this phenomenon, we propose a one-dimensional mechanical model of heterogeneous epithelial tissue dynamics that includes cell-length-dependent proliferation and death mechanisms. Proliferation and death are incorporated into the discrete model stochastically and arise as source/sink terms in the corresponding continuum model that we derive. Using the new discrete model and continuum description, we explore several applications including the evolution of homogeneous tissues experiencing proliferation and death, and competition in a heterogeneous setting with a cancerous tissue competing for space with an adjacent normal tissue. This framework allows us to postulate new mechanisms that explain the ability of cancer cells to outcompete healthy cells through mechanical differences rather than an intrinsic proliferative advantage. We advise when the continuum model is beneficial and demonstrate why naively adding source/sink terms to a continuum model without considering the underlying discrete model may lead to incorrect results.

Modeling Osteocyte Network Formation: Healthy and Cancerous Environments.

Advanced cancers, such as prostate and breast cancers, commonly metastasize to bone. In the bone matrix, dendritic osteocytes form a spatial network allowing communication between osteocytes and the osteoblasts located on the bone surface. This communication network facilitates coordinated bone remodeling. In the presence of a cancerous microenvironment, the topology of this network changes. In those situations, osteocytes often appear to be either overdifferentiated (i.e., there are more dendrites than healthy bone) or underdeveloped (i.e., dendrites do not fully form). In addition to structural changes, histological sections from metastatic breast cancer xenografted mice show that number of osteocytes per unit area is different between healthy bone and cancerous bone. We present a stochastic agent-based model for bone formation incorporating osteoblasts and osteocytes that allows us to probe both network structure and density of osteocytes in bone. Our model both allows for the simulation of our spatial network model and analysis of mean-field equations in the form of integro-partial differential equations. We considered variations of our model to study specific physiological hypotheses related to osteoblast differentiation; for example predicting how changing biological parameters, such as rates of bone secretion, rates of cancer formation, and rates of osteoblast differentiation can allow for qualitatively different network topologies. We then used our model to explore how commonly applied therapies such as bisphosphonates (e.g., zoledronic acid) impact osteocyte network formation.

Cell proliferation and migration explain pore bridging dynamics in 3D printed scaffolds of different pore size.

Tissue growth in bioscaffolds is influenced significantly by pore geometry, but how this geometric dependence emerges from dynamic cellular processes such as cell proliferation and cell migration remains poorly understood. Here we investigate the influence of pore size on the time required to bridge pores in thin 3D-printed scaffolds. Experimentally, new tissue infills the pores continually from their perimeter under strong curvature control, which leads the tissue front to round off with time. Despite the varied shapes assumed by the tissue during this evolution, we find that time to bridge a pore simply increases linearly with the overall pore size. To disentangle the biological influence of cell behaviour and the mechanistic influence of geometry in this experimental observation, we propose a simple reaction-diffusion model of tissue growth based on Porous-Fisher invasion of cells into the pores. First, this model provides a good qualitative representation of the evolution of the tissue; new tissue in the model grows at an effective rate that depends on the local curvature of the tissue substrate. Second, the model suggests that a linear dependence of bridging time with pore size arises due to geometric reasons alone, not to differences in cell behaviours across pores of different sizes. Our analysis suggests that tissue growth dynamics in these experimental constructs is dominated by mechanistic crowding effects that influence collective cell proliferation and migration processes, and that can be predicted by simple reaction-diffusion models of cells that have robust, consistent behaviours.

Mineral density differences between femoral cortical bone and trabecular bone are not explained by turnover rate alone.

Bone mineral density distributions (BMDDs) are a measurable property of bone tissues that depends strongly on bone remodelling and mineralisation processes. These processes can vary significantly in health and disease and across skeletal sites, so there is high interest in analysing these processes from experimental BMDDs. Here, we propose a rigorous hypothesis-testing approach based on a mathematical model of mineral heterogeneity in bone due to remodelling and mineralisation, to help explain differences observed between the BMDD of human femoral cortical bone and the BMDD of human trabecular bone. Recent BMDD measurements show that femoral cortical bone possesses a higher bone mineral density, but a similar mineral heterogeneity around the mean compared to trabecular bone. By combining this data with the mathematical model, we are able to test whether this difference in BMDD can be explained by (i) differences in turnover rate; (ii) differences in osteoclast resorption behaviour; and (iii) differences in mineralisation kinetics between the two bone types. We find that accounting only for differences in turnover rate is inconsistent with the fact that both BMDDs have a similar spread around the mean, and that accounting for differences in osteoclast resorption behaviour leads to biologically inconsistent bone remodelling patterns. We conclude that the kinetics of mineral accumulation in bone matrix must therefore be different in femoral cortical bone and trabecular bone. Although both cortical and trabecular bone are made up of lamellar bone, the different mineralisation kinetics in the two types of bone point towards more profound structural differences than usually assumed.

A level-set method for the evolution of cells and tissue during curvature-controlled growth.

Most biological tissues grow by the synthesis of new material close to the tissue's interface, where spatial interactions can exert strong geometric influences on the local rate of growth. These geometric influences may be mechanistic or cell behavioural in nature. The control of geometry on tissue growth has been evidenced in many in vivo and in vitro experiments, including bone remodelling, wound healing, and tissue engineering scaffolds. In this paper, we propose a generalisation of a mathematical model that captures the mechanistic influence of curvature on the joint evolution of cell density and tissue shape during tissue growth. This generalisation allows us to simulate abrupt topological changes such as tissue fragmentation and tissue fusion, as well as three dimensional cases, through a level-set-based method. The level-set method developed introduces another Eulerian field than the level-set function. This additional field represents the surface density of tissue-synthesising cells, anticipated at future locations of the interface. Numerical tests performed with this level-set-based method show that numerical conservation of cells is a good indicator of simulation accuracy, particularly when cusps develop in the tissue's interface. We apply this new model to several situations of curvature-controlled tissue evolutions that include fragmentation and fusion.

Differing Effects of Parathyroid Hormone, Alendronate, and Odanacatib on Bone Formation and on the Mineralization Process in Intracortical and Endocortical Bone of Ovariectomized Rabbits.

Bone is formed by deposition of a collagen-containing matrix (osteoid) that hardens over time as mineral crystals accrue and are modified; this continues until bone remodeling renews that site. Pharmacological agents for osteoporosis differ in their effects on bone remodeling, and we hypothesized that they may differently modify bone mineral accrual. We, therefore, assessed newly formed bone in mature ovariectomized rabbits treated with the anti-resorptive bisphosphonate alendronate (ALN-100µ g/kg/2×/week), the anabolic parathyroid hormone (PTH (1-34)-15µ g/kg/5×/week), or the experimental anti-resorptive odanacatib (ODN 7.5 µM/day), which suppresses bone resorption without suppressing bone formation. Treatments were administered for 10 months commencing 6 months after ovariectomy (OVX). Strength testing, histomorphometry, and synchrotron Fourier-transform infrared microspectroscopy were used to measure bone strength, bone formation, and mineral accrual, respectively, in newly formed endocortical and intracortical bone. In Sham and OVX endocortical and intracortical bone, three modifications occurred as the bone matrix aged: mineral accrual (increase in mineral:matrix ratio), carbonate substitution (increase in carbonate:mineral ratio), and collagen molecular compaction (decrease in amide I:II ratio). ALN suppressed bone formation but mineral accrued normally at those sites where bone formation occurred. PTH stimulated bone formation on endocortical, periosteal, and intracortical bone surfaces, but mineral accrual and carbonate substitution were suppressed, particularly in intracortical bone. ODN treatment did not suppress bone formation, but newly deposited endocortical bone matured more slowly with ODN, and ODN-treated intracortical bone had less carbonate substitution than controls. In conclusion, these agents differ in their effects on the bone matrix. While ALN suppresses bone formation, it does not modify bone mineral accrual in endocortical or intracortical bone. While ODN does not suppress bone formation, it slows matrix maturation. PTH stimulates modelling-based bone formation not only on endocortical and trabecular surfaces, but may also do so in intracortical bone; at this site, new bone deposited contains less mineral than normal.

Osteoblasts infill irregular pores under curvature and porosity controls: a hypothesis-testing analysis of cell behaviours.

The geometric control of bone tissue growth plays a significant role in bone remodelling, age-related bone loss, and tissue engineering. However, how exactly geometry influences the behaviour of bone-forming cells remains elusive. Geometry modulates cell populations collectively through the evolving space available to the cells, but it may also modulate the individual behaviours of cells. To factor out the collective influence of geometry and gain access to the geometric regulation of individual cell behaviours, we develop a mathematical model of the infilling of cortical bone pores and use it with available experimental data on cortical infilling rates. Testing different possible modes of geometric controls of individual cell behaviours consistent with the experimental data, we find that efficient smoothing of irregular pores only occurs when cell secretory rate is controlled by porosity rather than curvature. This porosity control suggests the convergence of a large scale of intercellular signalling to single bone-forming cells, consistent with that provided by the osteocyte network in response to mechanical stimulus. After validating the mathematical model with the histological record of a real cortical pore infilling, we explore the infilling of a population of randomly generated initial pore shapes. We find that amongst all the geometric regulations considered, the collective influence of curvature on cell crowding is a dominant factor for how fast cortical bone pores infill, and we suggest that the irregularity of cement lines thereby explains some of the variability in double labelling data as well as the overall speed of osteon infilling.

Late stages of mineralization and their signature on the bone mineral density distribution.

PURPOSE: Experimental measurements of bone mineral density distributions (BMDDs) enable a determination of secondary mineralization kinetics in bone, but the maximum degree of mineralization and how this maximum is approached remain uncertain. We thus test computationally different hypotheses on late stages of bone mineralization by simulating BMDDs in low-turnover conditions. MATERIALS AND METHODS: An established computational model of the BMDD that accounts for mineralization and remodeling processes was extended to limit mineralization to various maximum calcium capacities of bone. Simulated BMDDs obtained by reducing turnover rate from the reference trabecular BMDD under different assumptions on late stage mineralization kinetics were compared with experimental BMDDs of low-turnover bone. RESULTS: Simulations show that an abrupt stopping of mineralization near a maximum calcium capacity induces a pile-up of minerals in the BMDD statistics that is not observed experimentally. With a smooth decrease of mineralization rate, imposing low maximum calcium capacities helps to match peak location and width of simulated low-turnover BMDDs with peak location and width of experimental BMDDs, but results in a distinctive asymmetric peak shape. No tuning of turnover rate and maximum calcium capacity was able to explain the differences found in experimental BMDDs between trabecular bone (high turnover) and femoral cortical bone (low turnover). CONCLUSIONS: Secondary mineralization in human bone does not stop abruptly, but continues slowly up to a calcium content greater than 30 wt% Ca. The similar mineral heterogeneity seen in trabecular and femoral cortical bones at different peak locations was unexplained by the turnover differences tested.

Modeling the Effect of Curvature on the Collective Behavior of Cells Growing New Tissue.

The growth of several biological tissues is known to be controlled in part by local geometrical features, such as the curvature of the tissue interface. This control leads to changes in tissue shape that in turn can affect the tissue's evolution. Understanding the cellular basis of this control is highly significant for bioscaffold tissue engineering, the evolution of bone microarchitecture, wound healing, and tumor growth. Although previous models have proposed geometrical relationships between tissue growth and curvature, the role of cell density and cell vigor remains poorly understood. We propose a cell-based mathematical model of tissue growth to investigate the systematic influence of curvature on the collective crowding or spreading of tissue-synthesizing cells induced by changes in local tissue surface area during the motion of the interface. Depending on the strength of diffusive damping, the model exhibits complex growth patterns such as undulating motion, efficient smoothing of irregularities, and the generation of cusps. We compare this model with in vitro experiments of tissue deposition in bioscaffolds of different geometries. By including the depletion of active cells, the model is able to capture both smoothing of initial substrate geometry and tissue deposition slowdown as observed experimentally.

Towards a cell-based mechanostat theory of bone: the need to account for osteocyte desensitisation and osteocyte replacement.

Bone׳s mechanostat theory describes the adaptation of bone tissues to their mechanical environment. Many experiments have investigated and observed such structural adaptation. However, there is still much uncertainty about how to define the reference mechanical state at which bone structure is adapted and stable. Clinical and experimental observations show that this reference state varies both in space and in time, over a wide range of timescales. We propose here an osteocyte-based mechanostat theory that encodes the mechanical reference state in osteocyte properties. This theory assumes that osteocytes are initially formed adapted to their current local mechanical environment through modulation of their properties. We distinguish two main types of physiological processes by which osteocytes subsequently modify the reference mechanical state at different timescales. One is cell desensitisation, which occurs rapidly and reversibly during an osteocyte׳s lifetime. The other is the replacement of osteocytes during bone remodelling, which occurs over the long timescales of bone turnover. The novelty of this theory is to propose that long-lasting morphological and genotypic osteocyte properties provide a material basis for a long-term mechanical memory of bone that is gradually reset by bone remodelling. We test this theory by simulating long-term mechanical disuse (modelling spinal cord injury), and short-term mechanical loadings (modelling daily exercises) with a mathematical model. The consideration of osteocyte desensitisation and of osteocyte replacement by remodelling is able to capture a number of phenomena and timescales observed during the mechanical adaptation of bone tissues, lending support to this theory.

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 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.