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Adaptive finite element models have allowed researchers to test hypothetical relationships between the local mechanical environment and the healing of bone fractures. However, their predictive power has not yet been demonstrated by testing hypotheses ahead of experimental testing. In this study, an established mechano-biological scheme was used in an iterative finite element simulation of sheep tibial osteotomy healing under a hypothetical fixation regime, “inverse dynamisation”. Tissue distributions, interfragmentary movement and stiffness across the fracture site were compared between stiff and flexible fixation conditions and scenarios in which fixation stiffness was increased at a discrete time-point. The modelling work was conducted blind to the experimental study to be published subsequently. The simulations predicted the fastest and most direct healing under constant stiff fixation, and the slowest healing under flexible fixation. Although low fixation stiffness promoted more callus formation prior to bridging, this conferred little additional stiffness to the fracture in the first 5 weeks. Thus, while switching to stiffer fixation facilitated rapid subsequent bridging of the fracture, no advantage of inverse dynamisation could be demonstrated. In vivo data remains necessary to conclusively test this treatment protocol and this will, in turn, provide an evaluation of the model’s performance. The publication of both hypotheses and their computational simulation, prior to experimental testing, offers an appealing means to test the predictive power of mechano-biological models.

Stochasticity is a key characteristic of intracellular processes such as gene regulation and chemical signalling. Therefore, characterizing stochastic effects in biochemical systems is essential to understand the complex dynamics of living things. Mathematical idealizations of biochemically reacting systems must be able to capture stochastic phenomena. While robust theory exists to describe such stochastic models, the computational challenges in exploring these models can be a significant burden in practice since realistic models are analytically intractable. Determining the expected behaviour and variability of a stochastic biochemical reaction network requires many probabilistic simulations of its evolution. Using a biochemical reaction network model to assist in the interpretation of time-course data from a biological experiment is an even greater challenge due to the intractability of the likelihood function for determining observation probabilities. These computational challenges have been subjects of active research for over four decades. In this review, we present an accessible discussion of the major historical developments and state-of-the-art computational techniques relevant to simulation and inference problems for stochastic biochemical reaction network models. Detailed algorithms for particularly important methods are described and complemented with Matlab®implementations. As a result, this review provides a practical and accessible introduction to computational methods for stochastic models within the life sciences community.

Iterative computational models have been used to investigate the regulation of bone fracture healing by local mechanical conditions. Although their predictions replicate some mechanical responses and histological features, they do not typically reproduce the predominantly radial hard callus growth pattern observed in larger mammals. We hypothesised that this discrepancy results from an artefact of the models' initial geometry. Using axisymmetric finite element models, we demonstrated that pre-defining a field of soft tissue in which callus may develop introduces high deviatoric strains in the periosteal region adjacent to the fracture. These bone-inhibiting strains are not present when the initial soft tissue is confined to a thin periosteal layer. As observed in previous healing models, tissue differentiation algorithms regulated by deviatoric strain predicted hard callus forming remotely and growing towards the fracture. While dilatational strain regulation allowed early bone formation closer to the fracture, hard callus still formed initially over a broad area, rather than expanding over time. Modelling callus growth from a thin periosteal layer successfully predicted the initiation of hard callus growth close to the fracture site. However, these models were still susceptible to elevated deviatoric strains in the soft tissues at the edge of the hard callus. Our study highlights the importance of the initial soft tissue geometry used for finite element models of fracture healing. If this cannot be defined accurately, alternative mechanisms for the prediction of early callus development should be investigated.

The mechanism by which mechanical stimulus of reparative tissues directs the pattern of healing of a bone fracture is not understood. Several hypotheses have been developed that predict the ossification pattern during healing for a given ambient mechanical environment. These have remained unproved because of the absence of data on stress fields in the reparative tissue of real fractures. The present study examines the predictive performance of the most recent hypothesis that was proposed by Claes and Heigele (J. Biomech. 32 (1999) 255), against measured and calculated data from the clinical fracture reported by Gardner et al. (J. Biomech. 33 (2000) 415). The hypothesis was used to predict ossification from the preceding stress and strain fields present in the FEM of Gardner et al. at four temporal stages during healing. Predictions were then compared with the observed differentiation and maturation. During early healing of the interfragmentary gap region, the hypothesis correctly predicted the formation of connective tissue and fibrocartilage, and during later healing it correctly predicted the beginning of endochondral ossification. At the periphery of the periosteal callus, the hypothesis correctly predicted intramembraneous ossification during early healing, and also its thickening by endochondral ossification during later healing. However, the hypothesis incorrectly predicted intramembraneous ossification during early healing of the main periosteal callus, although in later healing it correctly predicted endochondral ossification.

The exact mechanism by which mechanical stimulus regulates the healing process of a bone fracture is not understood. This has led to the development of several hypotheses that predict the pattern of differentiation of tissue during healing that may arise from characteristic fields of stress or strain at the fracture. These have so far remained unproved because data on stress fields in actual fracture tissue have been unavailable until recently. Thus the present study examines the predictive performance of the hypothesis proposed in J Orthop Res 6 (1988) 736, against measured and calculated data reported in J Biomech 33 (2000) 415, using a 2D FEM of a clinical fracture. The hypothesis was used to predict the influence of stress fields present in the Gardner et al. tissues at four temporal stages during healing. These predictions were then correlated with callus-size, rate of endochondral ossification and ossification pattern subsequently observed by Gardner et al. in the clinical fracture. Results corroborate the hypothesis that high octahedral shear stresses may increase the size of the callus during the initial phase of healing, and they also suggest that this may be true during the later stages of the fracture fixation period. However, compressive dilatational stresses were not found to inhibit endochondral ossification, as suggested by the hypothesis. Although high shear stresses were found in regions indicative of fibrous tissue as postulated by the hypothesis, this was not found to be the case for high tensile dilatational stresses. Also, contour diagrams of Osteogenic index (I) indicated only limited correlation with callus maturation and the pattern of healing. Therefore, the hypothesis was not wholly successful in predicting healing pattern.