A numerical model of the fracture healing process that describes tissue development and revascularisation

Abstract

A dynamic model was developed to simulate complex interactions of mechanical stability, revascularisation and tissue differentiation in secondary fracture healing. Unlike previous models, blood perfusion was included as a spatio-temporal state variable to simulate the revascularisation process. A 2D, axisymmetrical finite element model described fracture callus mechanics. Fuzzy logic rules described the following biological processes: angiogenesis, intramembranous ossification, chondrogenesis, cartilage calcification and endochondral ossification, all of which depended on local strain state and local blood perfusion. In order to evaluate how the predicted revascularisation depended on the mechanical environment, we simulated two different healing cases according to two groups of transverse metatarsal osteotomies in sheep with different axial stability. The model predicted slower revascularisation and delayed bony bridging for the less stable case, which corresponded well to the experimental observations. A revascularisation sensitivity analysis demonstrated the potential of the model to account for different conditions regarding the blood supply.

Modelling the fracture-healing process as a moving-interface problem using an interface-capturing approach

“…Independent of the tissue distribution described by the set of domains X i ðtÞ, we also describe the process of revascularisation, which is fundamental for osseous healing (Rhinelander 1974), by tracking the interface of an additional domain X v ðtÞ (Simon et al 2011). We follow the approach of using a coupled LS and VOF (CLSVOF) method (Sussman and Puckett 2000) with minor modifications.…”

“…In contrast to most of the established dynamical fracture-healing models that describe tissue differentiation as the evolution of scalar fields of tissue concentrations or stiffnesses of the fracture callus (Lacroix et al 2002;Garc ıa-Aznar et al 2007;Isaksson et al 2008;Checa and Prendergast 2009;Geris et al 2010;Byrne et al 2011;Simon et al 2011;Repp et al 2015), this procedure is more directly related to the study in which the differentiation hypothesis is postulated on the basis of finite-element (FE) analysis near the healing front (Claes and Heigele 1999). By extension, the interpretation of the parameters inherent to the new model is more intuitive.…”

“…The material parameters, including Young's modulus and Poisson's ratio, for the contributing tissues are listed in Table 1. The chosen parameters are based on the study of Simon et al (2011) Poisson's ratio, which are of the order E ¼ 10 MPa and ¼ 0:4 (Carter and Hayes 1977;Simon et al 2011). For the elements intersected by some tissue interface, a rule of mixture (Simon et al 2011) is applied to determine the material properties.…”

“…The related concentrations are the tissue-volume fractions. Furthermore, we use the standardised callus geometry of Claes and Heigele (1999) of an ovine metatarsus and the non-linear external fixator of Simon et al (2011) (see Figure 1). The long-bone axisymmetric shape and the symmetry of the top and bottom healing domains are utilized to minimise the computational cost for solving the equations.…”

“…More specifically, we want to benchmark such a model with respect to both earlier numerical outcomes (Simon et al 2011) and experimental results (Claes et al 1995).…”