The Effect of Surface Curvature on Wound Healing in Bone

The time-independent nonhomogeneous diffusion equation is solved for the equilibrium distribution of wound-induced growth factor over a hemispherical surface. The growth factor is produced at the inner edge of a circular wound and stimulates healing in regions where the concentration exceeds a certain threshold value. An implicit analytic criterion is derived for complete healing of the wound. (C) 2001 Elsevier Science Ltd. All rights reserved

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.Comment: 15 pages, 10 figures, 3 supplementary figure

Surface and bulk stresses drive morphological changes in fibrous microtissues

Engineered fibrous tissues consisting of cells encapsulated within collagen gels are widely used three-dimensional in vitro models of morphogenesis and wound healing. Although cell-mediated matrix remodeling that occurs within these scaffolds has been extensively studied, less is known about the mesoscale physical principles governing the dynamics of tissue shape. Here, we show both experimentally and by using computer simulations how surface contraction through the development of surface stresses (analogous to surface tension in fluids) coordinates with bulk contraction to drive shape evolution in constrained three-dimensional microtissues. We used microelectromechanical systems technology to generate arrays of fibrous microtissues and robot-assisted microsurgery to perform local incisions and implantation. We introduce a technique based on phototoxic activation of a small molecule to selectively kill cells in a spatially controlled manner. The model simulations, which reproduced the experimentally observed shape changes after surgical and photochemical operations, indicate that fitting of only bulk and surface contractile moduli is sufficient for the prediction of the equilibrium shape of the microtissues. The computational and experimental methods we have developed provide a general framework to study and predict the morphogenic states of contractile fibrous tissues under external loading at multiple length scales.Published versio

A Simplified Model for Growth Factor Induced Healing of Wounds

A mathematical model is developed for the rate of healing of a circular or elliptic wound. In this paper the regeneration, decay and transport of a generic \u27growth factor\u27, which induces the healing of the wound, is taken into account. Further, an equation of motion is derived for radial healing of a circular wound. The expressions for the equation of motion and the distribution of the growth factor are related in such a way that no healing occurs if the growth factor concentration at the wound edge is below a threshold value. In this paper we investigate the influence of the behaviour of the thickness of the active layer, in which the growth factor is produced, on the healing process. Also a correction is made to a result in earlier work. © 2006 Elsevier Ltd. All rights reserved

Towards a New Spatial Representation of Bone Remodeling

Irregular bone remodeling is associated with a number of bone diseases such as osteoporosis and multiple myeloma. Computational and mathematical modeling can aid in therapy and treatment as well as understanding fundamental biology. Different approaches to modeling give insight into different aspects of a phenomena so it is useful to have an arsenal of various computational and mathematical models. Here we develop a mathematical representation of bone remodeling that can effectively describe many aspects of the complicated geometries and spatial behavior observed. There is a sharp interface between bone and marrow regions. Also the surface of bone moves in and out, i.e. in the normal direction, due to remodeling. Based on these observations we employ the use of a level-set function to represent the spatial behavior of remodeling. We elaborate on a temporal model for osteoclast and osteoblast population dynamics to determine the change in bone mass which influences how the interface between bone and marrow changes. We exhibit simulations based on our computational model that show the motion of the interface between bone and marrow as a consequence of bone remodeling. The simulations show that it is possible to capture spatial behavior of bone remodeling in complicated geometries as they occur \emph{in vitro} and \emph{in vivo}. By employing the level set approach it is possible to develop computational and mathematical representations of the spatial behavior of bone remodeling. By including in this formalism further details, such as more complex cytokine interactions and accurate parameter values, it is possible to obtain simulations of phenomena related to bone remodeling with spatial behavior much as \emph{in vitro} and \emph{in vivo}. This makes it possible to perform \emph{in silica} experiments more closely resembling experimental observations.Comment: Math. Biosci. Eng., 9(2), 201

The effect of geometry on three-dimensional tissue growth

Tissue formation is determined by uncountable biochemical signals between cells; in addition, physical parameters have been shown to exhibit significant effects on the level of the single cell. Beyond the cell, however, there is still no quantitative understanding of how geometry affects tissue growth, which is of much significance for bone healing and tissue engineering. In this paper, it is shown that the local growth rate of tissue formed by osteoblasts is strongly influenced by the geometrical features of channels in an artificial three-dimensional matrix. Curvature-driven effects and mechanical forces within the tissue may explain the growth patterns as demonstrated by numerical simulation and confocal laser scanning microscopy. This implies that cells within the tissue surface are able to sense and react to radii of curvature much larger than the size of the cells themselves. This has important implications towards the understanding of bone remodelling and defect healing as well as towards scaffold design in bone tissue engineering

Mechanisms of Optical Regression Following Corneal Laser Refractive Surgery: Epithelial and Stromal Responses

Laser vision correction is a safe and effective method of reducing spectacle dependence. Photorefractive Keratectomy (PRK), Laser In Situ Keratomileusis (LASIK), and Small-Incision Lenticule Extraction (SMILE) can accurately correct myopia, hyperopia, and astigmatism. Although these procedures are nearing optimization in terms of their ability to produce a desired refractive target, the long term cellular responses of the cornea to these procedures can cause patients to regress from the their ideal postoperative refraction. In many cases, refractive regression requires follow up enhancement surgeries, presenting additional risks to patients. Although some risk factors underlying refractive regression have been identified, the exact mechanisms have not been elucidated. It is clear that cellular proliferation events are important mediators of optical regression.  This review focused specifically on cellular changes to the corneal epithelium and stroma, which may influence postoperative visual regression following LASIK, PRK, and SMILE procedures

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.Comment: 14 pages, 11 figures, Appendi