RELATIONSHIPS BETWEEN IMAGE GREY-SCALE, ELASTIC MODULUS AND ELEMENT SIZE FOR FINITE ELEMENT MODELLING OF BONE-LIKE MATERIALS

Abstract

Purpose: Subject-specific computational models of anatomical components can now be generated from image data and used in the assessment of orthopaedic interventions. Whilst such models are becoming more commonplace, there are few studies that have investigated the effects of element size or the level of morphological detail that is required to model complex hierarchical structures such as trabecular bone [1,2]. The purpose of this study was to investigate the modelling of the mechanical properties of a trabecular-like material under compression, and in particular to evaluate the relationship between the grey-scale dependent density, determined from microCT, and the elastic modulus at different levels of mesh density.

Materials and Methods: A specific open cell rigid foam (pcf 7.5, Sawbone, Sweden), with a porosity over 95% and cell size between 1.5 and 2.5 mm, was chosen to represent human cancellous bone. Five cylindrical specimens (24 mm diameter, 19 mm height) were compressed under a maximum displacement of 0.35 mm in a materials testing machine. All the specimens were imaged by microCT (ƒÝCT80, Scanco Medical, Switzerland) before the test to provide the geometry and greyscale distribution. Corresponding finite element models of each specimen were generated using proprietary software (ScanFE, Simpleware, UK) with an element size of 1.5 mm. The elastic modulus of each element was based on the image greyscale using a conversion factor that was determined, through trial-and-error, by matching the predicted displacement with the experimental measurement in the linear region (> 0.2 mm). For each specimen, models of higher and lower mesh densities were constructed by down-sampling the microCT output to different levels. The smallest element size was limited to 0.5 mm due to computational restrictions, however a smaller 8 mm cube of the material was also analysed, with element sizes down to 0.25 mm.

Results: From the experimental tests, the mean apparent elastic modulus of the specimens was found to be 15 MPa, as compared with 18 MPa specified by the manufacturer. For the computational models, the predicted elastic modulus of the whole specimen models was found to decrease with decreasing element size. In particular, there was a large drop in the predicted values when the element size was of the same order as the cell size. With larger elements, the results indicated some convergence and there was reasonable agreement with the experimental results. For the cube model with smaller element sizes, the predicted moduli values again appeared to converge as the element size was reduced to 0.25 mm.

Conclusion: The results of this study show that the optimum conversion factor from an image greyscale value to an elastic modulus varies with element size. If the finite element method is to be used effectively to model bone, then the mesh size must either be sufficiently large to use a continuum approach or sufficiently small to capture the behaviour of the individual trabeculae. Different conversion factors will be required to determine the material properties from the image greyscale in each case.