Modelling the early phases of bone regeneration around an endosseous oral implant

The objective of this study was to see whether a mathematical model of fracture healing was able to mimic bone formation around an unloaded screw-shaped titanium implant as it is well-believed that both processes exhibit many biological similarities. This model describes the spatio-temporal evolution of cellular activities, ranging from mesenchymal stem cell migration, proliferation, differentiation to bone formation, which are initiated and regulated by the growth factors present at the peri-implant site. For the simulations, a finite volume code was used and adequate initial and boundary conditions were applied. Two sets of analyses have been performed, in which either initial and boundary condition or model parameter values were changed with respect to the fracture healing model parameter values. For a number of combinations, the spatio-temporal evolution of bone density was well-predicted. However reducing cell proliferation rate and increasing osteoblast differentiation and osteogenic growth factor synthesis rates, the simulation results were in agreement with the experimental data.     

Turing instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner model

In this paper we consider a modified spatiotemporal ecological system originating from the temporal Holling-Tanner model, by incorporating diffusion terms. The original ODE system is studied for the stability of coexisting homogeneous steady-states. The modified PDE system is investigated in detail with both numerical and analytical approaches. Both the Turing and non-Turing patterns are examined for some fixed parametric values and some interesting results have been obtained for the prey and predator populations. Numerical simulation shows that either prey or predator population do not converge to any stationary state at any future time when parameter values are taken in the Turing-Hopf domain. Prey and predator populations exhibit spatiotemporal chaos resulting from temporal oscillation of both the population and spatial instability. With help of numerical simulations we have shown that Turing-Hopf bifurcation leads to onset of spatio-temporal chaos when predator's diffusivity is much higher compared to prey population. Our investigation reveals the fact that Hopf-bifurcation is essential for the onset of spatiotemporal chaos.     

Expression of bone morphogenetic proteins during membranous bone healing

For the reconstructive plastic surgeon, knowledge of the molecular biology underlying membranous fracture healing is becoming increasingly vital. Understanding the complex patterns of gene expression manifested during the course of membranous fracture repair will be crucial to designing therapies that augment poor fracture healing or that expedite normal osseous repair by strategic manipulation of the normal course of gene expression. In the current study, we present a rat model of membranous bone repair. This model has great utility because of its technical simplicity, reproducibility, and relatively low cost. Furthermore, it is a powerful tool for analysis of the molecular regulation of membranous bone repair by immunolocalization and/or in situ hybridization techniques. In this study, an osteotomy was made within the caudal half of the hemimandible, thus producing a stable bone defect without the need for external or internal fixation. The healing process was then catalogued histologically in 28 Sprague-Dawley rats that were serially killed at 1, 2, 3, 4, 5, 6, and 8 weeks after operation. Furthermore, using this novel model, we analyzed, within the context of membranous bone healing, the temporal and spatial expression patterns of several members of the bone morphogenetic protein (BMP) family, known to be critical regulators of cells of osteoblast lineage. Our data suggest that BMP-2/-4 and BMP-7, also known as osteogenic protein-1 (OP-1), are expressed by osteoblasts, osteoclasts, and other more primitive mesenchymal cells within the fracture callus during the early stages of membranous fracture healing. These proteins continue to be expressed during the process of bone remodeling, albeit less prominently. The return of BMP-2/-4 and OP-1 immunostaining to baseline intensity coincides with the histological appearance of mature lamellar bone. Taken together, these data underscore the potentially important regulatory role played by the bone morphogenetic proteins in the process of membranous bone repair.     

A nonlinear compartmental model of Sr metabolism. I. Non-steady-state kinetics and model building

A model of Sr metabolism was developed by using plasma and urinary Sr kinetic data obtained in groups of postmenopausal women who received four different oral doses of Sr and collected during the Sr administration period (25 days) and for 28 days after cessation of treatment. A nonlinear compartmental formalism that is appropriate for study of non-steady-state kinetics and allows dissociation of variables pertaining to Sr metabolism (system 1) from those indirectly operating on it (system 2) was used. At each stage of model development, the dose-dependent model response was fitted to the four sets of data considered simultaneously (1 set per dose). A seven-compartment model with internal Sr distribution and intestinal, urinary, and bone metabolic pathways was selected. It includes two kinds of nonlinearities: those accounting for saturable intestinal and bone processes, which behave as intrinsic nonlinearities because they are directly dependent on Sr, and extrinsic nonlinearities (dependent on system 2), which suggest the cooperative involvement of plasma Sr changes in modulating some intestinal and bone mineral metabolic pathways. With the set of identified parameter values, the initial steady-state model predictions are relevant to known physiology, and some peculiarities of model behavior for long-term Sr administration were simulated.     

Time delay can enhance spatio-temporal chaos in a prey–predator model

In this paper we explore how the time delay induced Hopf-bifurcation interacts with Turing instability to determine the resulting spatial patterns. For this study, we consider a delayed prey–predator model with Holling type-II functional response and intra-specific competition among the predators. Analytical criteria for the delay induced Hopf-bifurcation and for the delayed spatio-temporal model are provided with numerical example to validate the analytical results. Exhaustive numerical simulation reveals the appearance of three types of stationary patterns, cold spot, labyrinthine, mixture of stripe-spot and two non-stationary patterns, quasi-periodic and spatio-temporal chaotic patterns. The qualitative features of the patterns for the non-delayed and the delayed spatio-temporal model are the same but their occurrence is solely controlled by the temporal parameters, rate of diffusivity and magnitude of the time delay. It is evident that the magnitude of time delay parameter beyond the Hopf-bifurcation threshold mostly produces spatio-temporal chaotic patterns.     

Host-parasitoid spatial dynamics in heterogeneous landscapes

This paper explores the effect of spatial processes in a heterogeneous environment on the dynamics of a host-parasitoid interaction. The environment consists of a lattice of favourable (habitat) and hostile (matrix) hexagonal cells, whose spatial distribution is measured by habitat proportion and spatial autocorrelation (inverse of fragmentation). At each time step, a fixed fraction of both populations disperses to the adjacent cells where it reproduces following the Nicholson-Bailey model. Aspects of the dynamics analysed include extinction, stability, cycle period and amplitude, and the spatial patterns emerging from the dynamics.
We find that, depending primarily on the fraction of the host population that disperses in each generation and on the landscape geometry, five classes of spatio-temporal dynamics can be objectively distinguished: spatial chaos, spirals, metapopulation, mainland-island and spiral fragments. The first two are commonly found in theoretical studies of homogeneous landscapes. The other three are direct consequences of the heterogeneity and have strong similarities to dynamic patterns observed in real systems (e.g. extinction-recolonisation, source-sink, outbreaks, spreading waves).
We discuss the processes that generate these patterns and allow the system to persist. The importance of these results is threefold: first, our model merges into a same theoretical framework dynamics commonly observed in the field that are usually modelled independently. Second, these dynamics and patterns are explained by dispersal rate and common landscape statistics, thus linking in a practical way population ecology to landscape ecology. Third, we show that the landscape geometry has a qualitative effect on the length of the cycles and, in particular, we demonstrate how very long periods can be produced by spatial processes.     

Orthotropic properties of cancellous bone modelled as parameterized cellular material

Analysis of stresses and strains in bone tissues and simulation of their adaptive remodelling require exhaustive information about distribution of constitutive properties of cancellous bone and their relationships to microstructural parameters. Homogenization of "equivalent" trabecular microstructures appears to be an advantageous tool for this task. In this study, parameterized orthotropic constitutive models of cancellous bone are derived from finite element analysis of repeatable microstructure cells. The models, based on a space-filling dodecahedron, are fully three-dimensional and are parameterized with four shape parameters. Variation of the parameters allows to imitate most of typical microstructure patterns observed in real bones, along with a variety of intermediate geometries. Finite element models of cells are generated by a special-purpose structured mesh generator for any arbitrary set of shape parameter values. Static numerical tests are performed for an exhaustive number of parameter value sets (microstructure instances). Coefficients of elastic orthotropic stiffness matrix are determined as tabularized functions of elastic constants versus the shape parameters. Additionally, they are correlated to apparent density and principal fabric tensor values. Comparison of the results with micro-FE data obtained for a large set of cancellous bone specimens proves a good agreement.     

Orthotropic properties of cancellous bone modelled as parameterized cellular material

Analysis of stresses and strains in bone tissues and simulation of their adaptive remodelling require exhaustive information about distribution of constitutive properties of cancellous bone and their relationships to microstructural parameters. Homogenization of “equivalent” trabecular microstructures appears to be an advantageous tool for this task. In this study, parameterized orthotropic constitutive models of cancellous bone are derived from finite element analysis of repeatable microstructure cells. The models, based on a space-filling dodecahedron, are fully three-dimensional and are parameterized with four shape parameters. Variation of the parameters allows to imitate most of typical microstructure patterns observed in real bones, along with a variety of intermediate geometries. Finite element models of cells are generated by a special-purpose structured mesh generator for any arbitrary set of shape parameter values. Static numerical tests are performed for an exhaustive number of parameter value sets (microstructure instances). Coefficients of elastic orthotropic stiffness matrix are determined as tabularized functions of elastic constants versus the shape parameters. Additionally, they are correlated to apparent density and principal fabric tensor values. Comparison of the results with micro-FE data obtained for a large set of cancellous bone specimens proves a good agreement.     

The spatial temporal response of the Limulus eye neural set

The Limulus lateral eye provides an example of a neural set showing lateral inhibition. Experimental evidence suggests the use of a linear, time-invariant, spatio-temporal, differential-integral equation as an approximation describing the set. A general solution is obtained by the Fourier and Laplace transform methods. The temporal pattern of the neural response depends both on the subset interactions and the intrinsic dynamics of component subsets. Since subset interaction strengths depend on the spatial frequencies (pattern) of the input light intensity, the output time response from the whole set depends on the input spatial frequencies. A possible temporal code for spatial patterns is discussed.     

Instabilities in spatially extended predator-prey systems: spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations

We investigate the emergence of spatio-temporal patterns in ecological systems. In particular, we study a generalized predator-prey system on a spatial domain. On this domain diffusion is considered as the principal process of motion. We derive the conditions for Hopf and Turing instabilities without specifying the predator-prey functional responses and discuss their biological implications. Furthermore, we identify the codimension-2 Turing-Hopf bifurcation and the codimension-3 Turing-Takens-Bogdanov bifurcation. These bifurcations give rise to complex pattern formation processes in their neighborhood. Our theoretical findings are illustrated with a specific model. In simulations a large variety of different types of long-term behavior, including homogenous distributions, stationary spatial patterns and complex spatio-temporal patterns, are observed.     

Spatial and spatio-temporal patterns in a cell-haptotaxis model

We investigate a cell-haptotaxis model for the generation of spatial and spatio-temporal patterns in one dimension. We analyse the steady state problem for specific boundary conditions and show the existence of spatially hetero-geneous steady states. A linear analysis shows that stability is lost through a Hopf bifurcation. We carry out a nonlinear multi-time scale perturbation procedure to study the evolution of the resulting spatio-temporal patterns. We also analyse the model in a parameter domain wherein it exhibits a singular dispersion relation.     

A multiscale mechanobiological model of bone remodelling predicts site-specific bone loss in the femur during osteoporosis and mechanical disuse

We propose a multiscale mechanobiological model of bone remodelling to investigate the site-specific evolution of bone volume fraction across the midshaft of a femur. The model includes hormonal regulation and biochemical coupling of bone cell populations, the influence of the microstructure on bone turnover rate, and mechanical adaptation of the tissue. Both microscopic and tissue-scale stress/strain states of the tissue are calculated from macroscopic loads by a combination of beam theory and micromechanical homogenisation. This model is applied to simulate the spatio-temporal evolution of a human midshaft femur scan subjected to two deregulating circumstances: (i) osteoporosis and (ii) mechanical disuse. Both simulated deregulations led to endocortical bone loss, cortical wall thinning and expansion of the medullary cavity, in accordance with experimental findings. Our model suggests that these observations are attributable to a large extent to the influence of the microstructure on bone turnover rate. Mechanical adaptation is found to help preserve intracortical bone matrix near the periosteum. Moreover, it leads to non-uniform cortical wall thickness due to the asymmetry of macroscopic loads introduced by the bending moment. The effect of mechanical adaptation near the endosteum can be greatly affected by whether the mechanical stimulus includes stress concentration effects or not.     

Spiral and Rotor Patterns Produced by Fairy Ring Fungi

A broad class of soil fungi form the annular patterns known as ‘fairy rings’ and provide one of the only means to observe spatio-temporal dynamics of otherwise cryptic fungal growth processes in natural environments. We present observations of novel spiral and rotor patterns produced by fairy ring fungi and explain these behaviors mathematically by first showing that a well known model of fairy ring fungal growth and the Gray-Scott reaction-diffusion model are mathematically equivalent. We then use bifurcation analysis and numerical simulations to identify the conditions under which spiral waves and rotors can arise. We demonstrate that the region of dimensionless parameter space supporting these more complex dynamics is adjacent to that which produces the more familiar fairy rings, and identify experimental manipulations to test the transitions between these spatial modes. These same manipulations could also feasibly induce fungal colonies to transition from rotor/spiral formation to a set of richer, as yet unobserved, spatial patterns.     

Patterns of spread of influenza A in Canada

Understanding spatial patterns of influenza transmission is important for designing control measures. We investigate spatial patterns of laboratory-confirmed influenza A across Canada from October 1999 to August 2012. A statistical analysis (generalized linear model) of the seasonal epidemics in this time period establishes a clear spatio-temporal pattern, with influenza emerging earlier in western provinces. Early emergence is also correlated with low temperature and low absolute humidity in the autumn. For the richer data from the 2009 pandemic, a mechanistic mathematical analysis, based on a transmission model, shows that both school terms and weather had important effects on pandemic influenza transmission.     

Adaptive pattern classification and universal recoding: I. Parallel development and coding of neural feature detectors

This paper analyses a model for the parallel development and adult coding of neural feature detectors. The model was introduced in Grossberg (1976). We show how experience can retune feature detectors to respond to a prescribed convex set of spatial patterns. In particular, the detectors automatically respond to average features chosen from the set even if the average features have never been experienced. Using this procedure, any set of arbitrary spatial patterns can be recoded, or transformed, into any other spatial patterns (universal recoding), if there are sufficiently many cells in the network's cortex. The network is built from short term memory (STM) and long term memory (LTM) mechanisms, including mechanisms of adaptation, filtering, contrast enhancement, tuning, and nonspecific arousal. These mechanisms capture some experimental properties of plasticity in the kitten visual cortex. The model also suggests a classification of adult feature detector properties in terms of a small number of functional principles. In particular, experiments on retinal dynamics, including amarcrine cell function, are suggested.Supported in part by the Advanced Research Projects Agency under ONR Contract No. N00014-76-C-0185     

Effect of bed rest immobilization on metabolic turnover of bone and bone mineral density

Immobilization induces abnormal bone metabolism and severe decalcification of bone. To investigate the effect of middle-term immobilization on bone metabolism, we studied 10 young healthy males and females during bed rest for 20 days. Bone mineral density (BMD) rapidly decreased in both lumbar and metacarpal bones. No bone showed consistent BMD alterations, partial increase and partial decrease, and both lumbar and metacarpal bone showed similar rapid BMD change. Urinary excretion of pyridinoline tended to slightly increase by day 10, and to decline by day 20 (mean +/-SE: 34.2 +/-7.4, 26.3+/-4.6 nmol day-1, respectively). Neither alkaline phosphatase (isoform III) nor tartrate-resistant acid phosphatase, changed, suggesting that in the early stage of immobilization bone matrix in some part might increase or be resorbed without any activation of osteoblast or osteoclast, resulting in rapid calcification or decalcification, respectively.     

Elastic properties of cancellous bone derived from finite element models of parameterized microstructure cells

Evaluation of constitutive properties of cancellous bone and their relationships to microstructural parameters is a crucial issue in analysis of stresses and strains in bone tissues and simulation of their remodelling. Known limitations of experimental methods as well as of the micro-FE techniques make the analysis and homogenization of 'equivalent' trabecular microstructures an advantageous tool for this task. In this study, parameterized orthotropic constitutive models of cancellous bone are derived from finite element analysis of repeatable microstructure cells. Two cell types are analysed: cube- and prism-based. The models are fully three-dimensional, have realistic curvilinear shapes and are parameterized with three shape parameters. Variation of the parameters allows to imitate most of the typical microstructure patterns observed in real bones, along with variety of intermediate geometries. Finite element models of cells are generated by a special-purpose structured mesh generator for any arbitrary set of shape parameter values. Six static numerical tests are performed for an exhaustive number of parameter value sets (microstructure instances). Multi-point boundary conditions imposed on the models ensure mutual fitting of deformed neighbouring cells. Values of computed stresses allow to determine all coefficients of elastic orthotropic stiffness matrix. Results have a form of tabularized functions of elastic constants versus the shape parameters. Comparison of the results with micro-FE data obtained for a large set of cancellous bone specimens proves a good agreement, though evidently better in the case of the prism-based cell model.     

The effect of muscle loading on the simulation of bone remodelling in the proximal femur

A large number of finite element analyses of the proximal femur rely on a simplified set of muscle and joint contact loads to represent the boundary conditions of the model. In the context of bone remodelling analysis around hip implants, muscle loading affects directly the spatial distribution of the remodelling signal. In the present study we performed a sensitivity analysis on the effect of different muscle loading configurations on the outcome of the bone remodelling simulation. An anatomical model of the femur with the implanted stem in place was constructed using the CT data of the Visible Human Project dataset of the National Institute of Health. The model was loaded with three muscle force configurations with increasing level of complexity. A strain adaptive remodelling rule was employed to simulate the post-operative bone changes around the implant stem and the results of the simulation were assessed quantitatively in terms of the bone mineral content changes in 18 periprosthetic regions of interest. The results showed considerable differences in the amount of bone loss predicted between the three cases. The simplified models generally predicted more pronounced bone loss. Although the overall remodelling patterns observed were similar, the bone conserving effect of additional muscle forces in the vicinity of their areas of attachment was clear. The results of this study suggest that the loading configuration of the FE model does play an important role in the outcome of the remodelling simulation.     

Comparison of histomorphometric values in iliac trabecular bone of beagle dogs raised under different breeding systems.

In order to clarify individual differences in bone metabolism among colony-raised beagle dogs, histomorphometric values of iliac trabecular bone and values of serum biochemical constituents related to bone were examined in 10 and 17 beagle dogs raised, respectively, under our two breeding systems in which differences in factors such as exercise, ultraviolet rays, and mineral content of the diet affect bone metabolism. At the age of 14 months, all dogs were injected with tetracycline hydrochloride and calcein twice for double bone labeling in order to measure dynamic as well as static parameters by bone histomorphometry and the ilium was later biopsied. The measurement on cancellous bone areas of undecalcified iliac sections was performed with a semiautomatic image analyser. Values of total calcium, phosphorus, alkalinephosphatase activity, parathyroid hormone and calcitonin in serum were also determined. The results showed that there were no significant differences between the two groups in histomorphometric values, except for the osteoid volume (p less than 0.05) and osteoid surface/trabecular surface ratio (p less than 0.01) in females, or in serum biochemical constituents, except for alkalinephosphatase activity (p less than 0.001) in males, indicating there were virtually no individual differences in bone metabolism in normal colony raised beagle dogs.     

Effects of dietary bread crust Maillard reaction products on calcium and bone metabolism in rats

Maillard reaction products (MRP) consumption has been related with the development of bone degenerative disorders, probably linked to changes in calcium metabolism. We aimed to investigate the effects of MRP intake from bread crust on calcium balance and its distribution, and bone metabolism. During 88 days, rats were fed control diet or diets containing bread crust as source of MRP, or its soluble high molecular weight, soluble low molecular weight or insoluble fractions (bread crust, HMW, LMW and insoluble diets, respectively). In the final week, a calcium balance was performed, then animals were sacrified and some organs removed to analyse calcium levels. A second balance was carried out throughout the experimental period to calculate global calcium retention. Biochemical parameters and bone metabolism markers were measured in serum or urine. Global calcium bioavailability was unmodified by consumption of bread crust or its isolate fractions, corroborating the previously described low affinity of MRP to bind calcium. Despite this, a higher calcium concentration was found in femur due to smaller bones having a lower relative density. The isolate consumption of the fractions altered some bone markers, reflecting a situation of increased bone resorption or higher turnover; this did not take place in the animals fed the bread crust diet. Thus, the bread crust intake does not affect negatively calcium bioavailability and bone metabolism.