Abstract
Introduction
Trabecular bone transmits loads to the cortical shell and is therefore most active in bone remodeling. This remodeling alters trabecular material strength thereby changing the bending stiffness. Accounting for trabecular material heterogeneity has been shown to improve empirical-µFEM correlations by allowing for more realistic trabecular bending stiffness. In µFEMs to reduce computation time, region averaging is often used to scale image resolution. However, region averaging not only alters trabecular architecture, but inherently alters the CT-intensity of each trabeculae. The effect of CT-intensity variations on computationally derived apparent modulus (Eapp) in heterogenous µFEMs has not been discussed. The objectives of this study were to compare trabecular Eapp among i) hexahedral and tetrahedral µFEMs, ii) µFEMs generated from 32 µm, 64 µm, and 64 µm down-sampled from 32 µm µ-CT scans, and iii) µFEMs with homogeneous and heterogeneous tissue moduli.
Methods
Fourteen cadaveric scapulae (7 male; 7 female) were micro-CT scanned at two spatial resolutions (32 µm & 64 µm). Virtual bone cores were extracted from the glenoid vault, maintaining a 2:1 aspect ratio, to create µFEMs from the 32 µm, 64 µm, and down-sampled 64 µm scans. Custom code was used to generate µFEMs with 8-node hexahedral elements (HEX8), while maintaining the bone volume fraction (BV/TV) of each HEX8 32 µm model (BV/TV=0.24±0.10). Each virtual core was also generated as a 10-node tetrahedral (TET10) µFEM. All µFEMs were given either a homogeneous tissue modulus of 20 GPa, or a heterogeneous tissue modulus scaled by CT-intensity. All FEMs were constrained with identical boundary conditions and compressed to 0.5% apparent strain. The apparent modulus of each model was compared.
Results
Comparing error in mean Eapp, TET10 32 µm µFEMs with a homogeneous tissue modulus had an error of 7%, and a heterogeneous tissue modulus an error of 1%. Larger errors occurred for both down-sampled and scanned 64 µm µFEMs with both homogeneous and heterogeneous tissue moduli. The error in Eapp as a function of trabecular thickness (Tb.Th*) was larger for µFEMs generated from 64 µm scans, than the down-sampled 64 µm µFEMs. The errors were lowest for Tb.Th* greater than 0.225 mm and for µFEMs generated with heterogeneous tissue moduli. The error in Eapp as a function of volume fraction (BV/TV) was lowest above 0.225 for µFEMs with both homogeneous and heterogeneous tissue moduli and hexahedral and tetrahedral elements. Error was lower for the down-sampled 64 µm µFEMs versus scanned 64 µm µFEMs.
DISCUSSION
This study compared the Eapp of linear isotropic µFEMs generated with hexahedral or tetrahedral elements from 32 µm, 64 µm, or down-sampled 64 µm µ-CT scans, with a homogeneous or heterogeneous tissue modulus. It was found that except at the highest spatial resolution, tetrahedral elements underestimate Eapp. Down-sampling to half the original scan spatial resolution is not equivalent in Eapp to FEMs generated from scans at that spatial resolution, and both models underestimate the Eapp of the highest spatial resolution models. In general, accounting for trabecular material heterogeneity decreased errors in Eapp.
