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

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.

Curvature dependences of wave propagation in reaction–diffusion models

Reaction–diffusion waves in multiple spatial dimensions advance at a rate that strongly depends on the curvature of the wavefronts. These waves have important applications in many physical, ecological and biological systems. In this work, we analyse curvature dependences of travelling fronts in a single reaction–diffusion equation with general reaction term. We derive an exact, non-perturbative curvature dependence of the speed of travelling fronts that arises from transverse diffusion occurring parallel to the wavefront. Inward-propagating waves are characterized by three phases: an establishment phase dominated by initial and boundary conditions, a travelling-wave-like phase in which normal velocity matches standard results from singular perturbation theory and a dip-filling phase where the collision and interaction of fronts create additional curvature dependences to their progression rate. We analyse these behaviours and additional curvature dependences using a combination of asymptotic analyses and numerical simulations.

3D printed scaffold design for bone defects with improved mechanical and biological properties

Bone defect treatment is still a challenge in clinics, and synthetic bone scaffolds with adequate mechanical and biological properties are highly needed. Adequate waste and nutrient exchange of the implanted scaffold with the surrounded tissue is a major concern. Moreover, the risk of mechanical instability in the defect area during regular activity increases as the defect size increases. Thus, scaffolds with better mass transportation and mechanical properties are desired. This study introduces 3D printed polymeric scaffolds with a continuous pattern, ZigZag-Spiral pattern, for bone defects treatments. This pattern has a uniform distribution of pore size, which leads to uniform distribution of wall shear stress which is crucial for uniform differentiation of cells attached to the scaffolds. The mechanical, mass transportation, and biological properties of the 3D printed scaffolds are evaluated. The results show that the presented scaffolds have permeability similar to natural bone and, with the same porosity level, have higher mechanical properties than scaffolds with conventional lay-down patterns 0-90° and 0-45°. Finally, human mesenchymal stem cells are seeded on the scaffolds to determine the effects of geometrical microstructure on cell attachment and morphology. The results show that cells in scaffold with ZigZag-Spiral pattern infilled pores gradually, while the other patterns need more time to fill the pores. Considering mechanical, transportation, and biological properties of the considered patterns, scaffolds with ZigZag-Spiral patterns can mimic the properties of cancellous bones and be a better choice for treatments of bone defects.

Analysis of Mechanical Properties and Permeability of Trabecular-Like Porous Scaffold by Additive Manufacturing

Human bone cells live in a complex environment, and the biomimetic design of porous structures attached to implants is in high demand. Porous structures based on Voronoi tessellation with biomimetic potential are gradually used in bone repair scaffolds. In this study, the mechanical properties and permeability of trabecular-like porous scaffolds with different porosity levels and average apertures were analyzed. The mechanical properties of bone-implant scaffolds were evaluated using finite element analysis and a mechanical compression experiment, and the permeability was studied by computational fluid dynamics. Finally, the attachment of cells was observed by confocal fluorescence microscope. The results show that the performance of porous structures can be controlled by the initial design of the microstructure and tissue morphology. A good structural design can accurately match the performance of the natural bone. The study of mechanical properties and permeability of the porous structure can help address several problems, including stress shielding and bone ingrowth in existing biomimetic bone structures, and will also promotes cell adhesion, migration, and eventual new bone attachment.

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.

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.

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.

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.

Substrate curvature as a cue to guide spatiotemporal cell and tissue organization

Recent evidence clearly shows that cells respond to various physical cues in their environments, guiding many cellular processes and tissue morphogenesis, pathology, and repair. One aspect that is gaining significant traction is the role of local geometry as an extracellular cue. Elucidating how geometry affects cell and tissue behavior is, indeed, crucial to design artificial scaffolds and understand tissue growth and remodeling. Perhaps the most fundamental descriptor of local geometry is surface curvature, and a growing body of evidence confirms that surface curvature affects the spatiotemporal organization of cells and tissues. While well-defined in differential geometry, curvature remains somewhat ambiguously treated in biological studies. Here, we provide a more formal curvature framework, based on the notions of mean and Gaussian curvature, and summarize the available evidence on curvature guidance at the cell and tissue levels. We discuss the involved mechanisms, highlighting the interplay between tensile forces and substrate curvature that forms the foundation of curvature guidance. Moreover, we show that relatively simple computational models, based on some application of curvature flow, are able to capture experimental tissue growth remarkably well. Since curvature guidance principles could be leveraged for tissue regeneration, the implications for geometrical scaffold design are also discussed. Finally, perspectives on future research opportunities are provided.

3D analysis of the osteonal and interstitial tissue in human radii cortical bone

Human cortical bone has a complex hierarchical structure that is periodically remodelled throughout a lifetime. This microstructure dictates the mechanical response of the tissue under a critical load. If only some structural features, such as the different porosities observed in bone, are primarily studied, then investigations may not fully consider the osteonal systems in three-dimensions (3D). Currently, it is difficult to differentiate osteons from interstitial tissue using standard 3D characterization methods. Synchrotron radiation micro-computed tomography (SR-μCT) in the phase contrast mode is a promising method for the investigation of osteons. In the current study, SR-μCT imaging was performed on cortical bone samples harvested from eight human radii (female, 50-91 y.o.). The images were segmented to identify Haversian canals, osteocyte lacunae, micro-cracks, as well as osteons. The significant correlation between osteonal and Haversian canal volume fraction highlights the role of the canals as sites where bone remodelling is initiated. The results showed that osteocyte lacunae morphometric parameters depend on their distance to cement lines, strongly suggesting the evolution of biological activity from the beginning to the end of the remodelling process. Thus, the current study provides new data on 3D osteonal morphometric parameters and their relationships with other structural features in humans.