A comparison of density–modulus relationships used in finite element modeling of the shoulder
Introduction
Clinical-computed tomography (CT) scans are commonly performed for diagnostics and surgical planning of upper limb orthopaedic surgical procedures. Improvements in surgical procedures, implant designs, understanding of joint biomechanics, and pathologic conditions can be elucidated using clinical-resolution-derived computational finite element models (FEMs). As initial input to these models a constitutive relationship must be chosen that relates the CT-intensity to the bones’ mechanical properties, to ensure that the resulting model is an accurate representation of the bone being modeled.
These density–modulus relationships have been shown to result in clinical-resolution-derived whole-bone FEMs that are highly correlated with experimental results (R2 > 0.90) [1], [2]. These relationships are thought to be site-specific, with anatomic site- and subject-specific modeling parameters shown to greatly improve the accuracy of clinical-resolution-derived FEMs [3], [4], [5], [6]. However, it is common for relationships developed for one anatomic site, such as the hip, to be used in another, due to a paucity of established relationships. Pooling relationships from multiple anatomic sites to improve the modeling of mechanical properties in alternative sites is one approach to cover a greater density range. However, this method neglects site-specific trabecular architecture, the local distribution of bone, and the geometric contributions from the cortical structure of whole-bones.
Anatomic site-specific linear isotropic density–modulus relationships are commonly used in biomechanics research for accurate material mapping in FEMs derived from commercially available (Mimics, Materialise, Leuven, BE.; Simpleware, Synopsys, Mountain View, CA, USA) and open source (BoneMat) software, making these relationships essential to FEM development. A large number of density–modulus relationships exist within the literature [7], with relationships primarily developed by testing physical trabecular and/or cortical bone specimens. However, when mechanically testing specimens, variations in experimental testing protocols have resulted in large systematic errors due to end-artifacts, specimen geometry, misrepresented boundary conditions, and the loss of load-carrying capacity of outer trabeculae due to coring [7], [8], [9]. To improve model accuracy and reduce these errors, computational µ-FEMs that account for mineral heterogeneity [10], [11], [12], [13], [14], [15] may provide a robust method of density–modulus development.
A computational methodology has recently been reported that uses µ-FEMs and co-registered QCT-FEMs to compare the loading of trabecular bone cores [16]. This methodology eliminates some of the errors associated with traditional experimental mechanical testing of trabecular bone cores [17] and allows for the use of identical boundary conditions across models. Consistent with previous work, this methodology uses apparent strain energy density (SEDapp) to compare multi-resolution modeling of trabecular bone [16], [18]. Accounting for trabecular tissue heterogeneity at the micro-level has been shown to improve µ-FEM accuracy by allowing for a more accurate representation of trabecular bending stiffness [10], [11], [12], [13], [15]. Computationally, this is represented as a heterogeneous distribution of varying tissue modulus and is consistent with studies that have illustrated variations in trabecular tissue density superficially and at the core due to trabecular bone remodeling [19], [20].
Six relationships are commonly used in shoulder FE studies [21], [22], [23], [24], [25], [26], with only a single study having used scapular trabecular bone samples [22] for development. Shoulder FE studies lack experimental validation of the FE results, limiting the ability to translate outcomes and compare studies. This study compares these six relationships on the ability to predict SEDapp in µ-FEMs derived from glenoid trabecular bone.
Section snippets
Micro finite element model generation
Fourteen full-arm cadaveric specimens were obtained (7 male; 7 female; mean age 67 ± 8 years). The scapula was removed and denuded of all soft tissues. The glenoid fossa of each scapula was scanned with a micro-computed tomography scanner (Nikon XT H 225 ST, Nikon Metrology, NV, 95 kVp, 64 µA, 3141 projections, 1000 ms exposure). To include the entire glenoid structure in all scans from the largest to the smallest specimen, a fixed spatial resolution of 32 µm was used. As recommended for
Results
When considering comparisons between QCT-FEMs and µ-FEMs with a homogeneous tissue modulus, near absolute statistical agreement (Y = X) was observed between the µ-FEMs and the QCT-FEMs using the Morgan et al. [21] pooled relationship (Table 2). Not surprisingly, due to the similarity between the two relationships (Table 2), the Gupta and Dan [22] and Carter and Hayes [24] models showed near identical REML linear regression fit parameters. All relationships other than the Morgan et al. [21]
Discussion
This study compared the six most commonly used density–modulus relationships used in finite element (FE) modeling of the shoulder using a computational methodology with co-registered µ-FEMs [16]. When a homogeneous effective tissue modulus is used in µ-FEMs the results suggest that density–modulus relationships mapped to co-registered QCT-FEMs pooled from multiple anatomic sites, may accurately predict the apparent strain energy density (SEDapp) of glenoid trabecular bone. When considering a
Acknowledgments
The authors would like to thank Shruthi Poolacherla for her assistance with data collection. Funding for this work was provided in part by a Lawson Health Research Internal Research Fund Grant. Nikolas K. Knowles is supported in part by the Natural Sciences and Engineering Research Council of Canada and by a Transdisciplinary Bone & Joint Training Award from the Collaborative Training Program in Musculoskeletal Health Research at The University of Western Ontario. Ethics approval was obtained
Conflict of interest
The authors have no conflict of interest with this work.
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