. 2015 Mar:72:71-80.

doi: 10.1016/j.bone.2014.11.006. Epub 2014 Nov 15.

Trabecular plates and rods determine elastic modulus and yield strength of human trabecular bone

Ji Wang¹, Bin Zhou², X Sherry Liu³, Aaron J Fields⁴, Arnav Sanyal⁵, Xiutao Shi⁶, Mark Adams⁷, Tony M Keaveny⁸, X Edward Guo⁹

Affiliations

¹ Bone Bioengineering Laboratory, Department of Biomedical Engineering, Columbia University, New York, NY, USA. Electronic address: jw2857@columbia.edu.
² Bone Bioengineering Laboratory, Department of Biomedical Engineering, Columbia University, New York, NY, USA. Electronic address: bz2159@columbia.edu.
³ Bone Bioengineering Laboratory, Department of Biomedical Engineering, Columbia University, New York, NY, USA; McKay Orthopaedic Research Laboratory, Department of Orthopaedic Surgery, University of Pennsylvania, Philadelphia, PA, USA. Electronic address: xiaoweil@mail.med.upenn.edu.
⁴ Department of Orthopaedic Surgery, University of California San Francisco, San Francisco, CA, USA; Department of Mechanical Engineering, University of California Berkeley, Berkeley, CA, USA. Electronic address: FieldsA@orthosurg.ucsf.edu.
⁵ Department of Mechanical Engineering, University of California Berkeley, Berkeley, CA, USA. Electronic address: arnavsanyal@berkeley.edu.
⁶ Bone Bioengineering Laboratory, Department of Biomedical Engineering, Columbia University, New York, NY, USA. Electronic address: xs2163@columbia.edu.
⁷ Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY, USA. Electronic address: mark.adams@columbia.edu.
⁸ Department of Mechanical Engineering, University of California Berkeley, Berkeley, CA, USA. Electronic address: tmk@me.berkeley.edu.
⁹ Bone Bioengineering Laboratory, Department of Biomedical Engineering, Columbia University, New York, NY, USA. Electronic address: exg1@columbia.edu.

PMID: 25460571
PMCID: PMC4282941
DOI: 10.1016/j.bone.2014.11.006

Trabecular plates and rods determine elastic modulus and yield strength of human trabecular bone

Ji Wang et al. Bone. 2015 Mar.

. 2015 Mar:72:71-80.

doi: 10.1016/j.bone.2014.11.006. Epub 2014 Nov 15.

Authors

Ji Wang¹, Bin Zhou², X Sherry Liu³, Aaron J Fields⁴, Arnav Sanyal⁵, Xiutao Shi⁶, Mark Adams⁷, Tony M Keaveny⁸, X Edward Guo⁹

Affiliations

¹ Bone Bioengineering Laboratory, Department of Biomedical Engineering, Columbia University, New York, NY, USA. Electronic address: jw2857@columbia.edu.
² Bone Bioengineering Laboratory, Department of Biomedical Engineering, Columbia University, New York, NY, USA. Electronic address: bz2159@columbia.edu.
³ Bone Bioengineering Laboratory, Department of Biomedical Engineering, Columbia University, New York, NY, USA; McKay Orthopaedic Research Laboratory, Department of Orthopaedic Surgery, University of Pennsylvania, Philadelphia, PA, USA. Electronic address: xiaoweil@mail.med.upenn.edu.
⁴ Department of Orthopaedic Surgery, University of California San Francisco, San Francisco, CA, USA; Department of Mechanical Engineering, University of California Berkeley, Berkeley, CA, USA. Electronic address: FieldsA@orthosurg.ucsf.edu.
⁵ Department of Mechanical Engineering, University of California Berkeley, Berkeley, CA, USA. Electronic address: arnavsanyal@berkeley.edu.
⁶ Bone Bioengineering Laboratory, Department of Biomedical Engineering, Columbia University, New York, NY, USA. Electronic address: xs2163@columbia.edu.
⁷ Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY, USA. Electronic address: mark.adams@columbia.edu.
⁸ Department of Mechanical Engineering, University of California Berkeley, Berkeley, CA, USA. Electronic address: tmk@me.berkeley.edu.
⁹ Bone Bioengineering Laboratory, Department of Biomedical Engineering, Columbia University, New York, NY, USA. Electronic address: exg1@columbia.edu.

PMID: 25460571
PMCID: PMC4282941
DOI: 10.1016/j.bone.2014.11.006

Abstract

The microstructure of trabecular bone is usually perceived as a collection of plate-like and rod-like trabeculae, which can be determined from the emerging high-resolution skeletal imaging modalities such as micro-computed tomography (μCT) or clinical high-resolution peripheral quantitative CT (HR-pQCT) using the individual trabecula segmentation (ITS) technique. It has been shown that the ITS-based plate and rod parameters are highly correlated with elastic modulus and yield strength of human trabecular bone. In the current study, plate-rod (PR) finite element (FE) models were constructed completely based on ITS-identified individual trabecular plates and rods. We hypothesized that PR FE can accurately and efficiently predict elastic modulus and yield strength of human trabecular bone. Human trabecular bone cores from proximal tibia (PT), femoral neck (FN) and greater trochanter (GT) were scanned by μCT. Specimen-specific ITS-based PR FE models were generated for each μCT image and corresponding voxel-based FE models were also generated in comparison. Both types of specimen-specific models were subjected to nonlinear FE analysis to predict the apparent elastic modulus and yield strength using the same trabecular bone tissue properties. Then, mechanical tests were performed to experimentally measure the apparent modulus and yield strength. Strong linear correlations for both elastic modulus (r(2) = 0.97) and yield strength (r(2) = 0.96) were found between the PR FE model predictions and experimental measures, suggesting that trabecular plate and rod morphology adequately captures three-dimensional (3D) microarchitecture of human trabecular bone. In addition, the PR FE model predictions in both elastic modulus and yield strength were highly correlated with the voxel-based FE models (r(2) = 0.99, r(2) = 0.98, respectively), resulted from the original 3D images without the PR segmentation. In conclusion, the ITS-based PR models predicted accurately both elastic modulus and yield strength determined experimentally across three distinct anatomic sites. Trabecular plates and rods accurately determine elastic modulus and yield strength of human trabecular bone.

Keywords: Elastic modulus; Finite element; Individual trabecula segmentation; Microarchitecture; Plate and rod; Yield strength.

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Figures

**Figure 1**
Illustration of ITS-based PR modeling on a cubical trabecular bone specimen. (A) the original 3D volume of the trabecular bone. (B) Microstructural skeleton with the trabecular type labeled for each voxel. Plate skeleton voxels are shown in red, surface edge voxels in green, rod skeleton voxels in blue. (C) Segmented microstructural skeleton with individual trabeculae labeled by color for each skeleton voxel. (D) Recovered trabecular bone with individual trabeculae labeled by color for each voxel. (E) PR model with shell and beam elements and color indicating different trabeculae. (F) Details of the beam-shell connection.

**Figure 2**
Meshing trabecular rods into beam elements. (A) Original microarchitecture of a trabecular rod; (B) Rod-rod junction or plate-rod junction at both ends of the trabecular rod skeleton; (C) Shape refining nodes divide the rod into three beam elements.

**Figure 3**
Meshing trabecular plates into shell elements. (A) Original microarchitecture of the trabecular plate; (B) Plate-rod junctions connecting plate and rod skeletons; (C) Plate-plate junctions connecting plate-arc skeletons; (D) Plate edge junctions and shape refining nodes are added to construct triangular shell elements.

**Figure 4**
μCT image of human trabecular bone from (A) PT, (B) FN, and (C) GT; PR models for (D) PT, (E) FN, and (F) GT; corresponding voxel models for (G) PT, (H) FN, and (I) GT.

**Figure 5**
A randomly chosen typical set of strain-stress curves acquired from the mechanical testing experiment, voxel model FEA and PR model FEA.

**Figure 6**
Linear regressions of the elastic modulus (A, C) and yield strength (B, D) between PR model prediction and voxel model prediction and experimental measurements (data pooled from three sites).

**Figure 7**
Bland-Altman plots of the prediction error of PR model compared to voxel model and mechanical testing experiment. Error = (PR model - voxel model or experiment) / mean.

**Figure 8**
(A∼C) Linear regressions between bone volume fraction and the elastic modulus along x, y and z axes determined by voxel models; (D∼F) linear regressions between the elastic modulus along x, y and z axes predicted by PR models and voxel models.

**Figure 9**
(A∼C) Linear regressions between bone volume fraction and the yield strength along x, y and z axes determined by voxel models; (D∼F) linear regressions between the yield strength along x, y and z axes predicted by PR models and voxel models.

**Figure 10**
Comparison between PR model and voxel model for the test set of trabecular bone specimens at distal tibia and distal radius.

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Trabecular plates and rods determine elastic modulus and yield strength of human trabecular bone

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Trabecular plates and rods determine elastic modulus and yield strength of human trabecular bone

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