Investigating the Effects of Demographics on Shoulder Morphology and Density Using Statistical Shape and Density Modeling

Abstract

A better understanding of how the shape and density of the shoulder vary among members of a population can help design more effective population-based orthopedic implants. The main objective of this study was to develop statistical shape models (SSMs) and statistical density models (SDMs) of the shoulder to describe the main modes of variability in the shape and density distributions of shoulder bones within a population in terms of principal components (PCs). These PC scores were analyzed, and significant correlations were observed between the shape and density distributions of the shoulder and demographics of the population, such as sex and age. Our results demonstrated that when the overall body sizes of male and female donors were matched, males still had, on average, larger scapulae and thicker humeral cortical bones. Moreover, we concluded that age has a weak but significant inverse effect on the density within the entire shoulder. Weak and moderate, but significant, correlations were also found between many modes of shape and density variations in the shoulder. Our results suggested that donors with bigger humeri have bigger scapulae and higher bone density of humeri corresponds with higher bone density in the scapulae. Finally, asymmetry, to some extent, was noted in the shape and density distributions of the contralateral bones of the shoulder. These results can be used to help guide the designs of population-based prosthesis components and pre-operative surgical planning.

Introduction

Total shoulder arthroplasty is widely regarded as a clinically successful surgery used to relieve pain or restore movement to an arthritic shoulder joint [1–3]. Revision rates, however, are growing at a faster pace than for knee and hip arthroplasty implants [4]. Poor bone quality around the implants or suboptimal load transfer between implant components and surrounding bone continues to affect long-term fixation and ultimately leads to implant loosening and the need for revision surgeries [5,6]. Finite element (FE) analyses have proven effective for a better understanding of implant/bone mechanics while studying the effect of varying implant design [7–10]. However, these models are generally developed from small cohorts of shoulders and their results may not be representative of the broader population. The key to improving this aspect of implant design is developing models that can incorporate the normal variations in shoulder bone morphology and density distributions within populations and differences associated with patient demographics such as sex and age.

The effects of sex [11–15] and age [14–18] on the morphology and density distribution of the shoulder have been previously assessed through radiographic measurements or assessment of anatomic parameters. However, with the emergence of statistical shape models (SSM) [19] and statistical density models (SDM) [20], it has become possible to systematically investigate the correlations of demographics with the main modes of three-dimensional shape and density variations of the shoulder and incorporate this variability into finite element analyses.

Statistical shape models and SDMs are capable of systematically describing the morphology and complex density distribution of bones in terms of a relatively small set of uncorrelated variables called principal components (PCs). PCs can efficiently capture the main modes along which the three-dimensional shape and spatial distribution of bone density can vary among the members of a population. Previously, SSMs have successfully been applied to describe the main modes of variation in the shape of different human organs such as the liver [21,22], heart [23,24], and brain [25,26]. Examples of bones and joints include the femur [27], hip [28,29] and knee joint [30]. Furthermore, SDMs have been applied to the femur [20,31,32] and the mandibular condyle [33].

Recently published studies have demonstrated a variety of SSM approaches to modeling the shoulder, including general modeling pipeline development [34], predicting muscle attachment sites [35], reconstructing glenoid defects [36] and analyses of demographics-related (sex, ethnicity) variability in proximal humerus morphologies [37]. A statistical model based on 58 proximal humeri, described by Kamer et al. [38], reported typical modes of variability in bone morphology. In terms of bone density, they focused on characterizing typical densities in probed regions rather than characterizing predominate forms of variability. A SSM/SDM based on 53 scapulae, described by Burton et al. [39], was used to explore trends and variability of scapular bone density distributions and correlations with PC scores of shape variability. They leveraged cluster analysis techniques to identify unique subgroups that maximally differed in terms of bone density distributions.

Statistical shape models and SDMs of shoulder bones are potentially valuable tools for designing orthopedic implants. Using SSMs and SDMs to characterize the shape and density distribution variability, between sexes and across age groups, would allow the creation of more representative SSM/SDM-based FE models that could be used to fine-tune implants for specific patients cohorts. None of the aforementioned shoulder SDMs, however, were used to characterize the effects of sex as a biological variable, nor the effect of age on humerus and scapula bone density distributions. Furthermore, advances in additive manufacturing allow, to some extent, a decoupling of implant shape and stiffness properties (by using compliant or porous structures). Thus, decoupling shape and density variability in SSMs/SDMs is advantageous as it permits separate characterization. It is still important, however, to identify correlations between shape and density distributions to avoid designing implants for unrealistic or rare shape and density combinations. However, only Burton et al. [39] measured correlations between shape and density distributions, and only for the scapula. Moreover, the relationship of the humerus shape or density to that of the scapula could be rigorously investigated using SSMs/SDMs which can have implications for the design of shoulder implants.

Finally, shoulder morphological and density distribution symmetry, if present, could permit the use of the contralateral shoulder as an intrasubject control for clinical assessment and research studies [40]. As demonstrated previously by others [41], bilateral morphological symmetry has enabled the contralateral shoulder bones to be used as templates to guide shoulder surgical reconstruction, but density distribution similarities have not been rigorously studied. Bilateral morphology and density distribution comparisons can be facilitated using SSMs and SDMs.

Thus, the objectives of this study were: (1) to develop SSMs and SDMs for the humerus and scapula to systematically explore the main modes of shape and density variation in a population. (2) to investigate the correlations of sex and age with the shape and density distribution of the shoulder using the developed statistical models. (3) to explore correlations between shape and density distribution of the shoulder using separately created SSMs and SDMs for both the humerus and scapula. (4) to evaluate correlations between the humerus and scapula in terms of shape and density distribution using SSMs and SDMs and (5) to assess the symmetry of contralateral shoulders in terms of both shape and density using our statistical models.

Methods

Development of Statistical Shape and Density Models

Specimens.

Separate SSMs and SDMs were created for the humerus and scapula using computed tomography (CT) scans of 75 human cadaveric shoulder joints. This set includes 57 male (20 pairs) and 18 female shoulders (1 pair) from 54 donors (Table 1). CT scans were performed using a GE Healthcare Discovery CT750 HD CT scanner (GE Healthcare, Waukesha, WI) with a slice thickness of 0.625 mm, pixel spacing of 0.59 mm, tube voltage of 120 kVp and tube current of 200 mA. The actual bone-related disease diagnoses in our overall sample were not known and CT-based diagnoses were not performed.

Table 1.

The demographic data for the population

	Mean ± standard deviation	Range (minimum to maximum)
Age (years)	73 ± 12	21–94
Height (cm)	172 ± 10	147–191
Weight (kg)	63 ± 18	30–116
BMI ($\text{kg}/{\mathrm{m}}^2$)	21 ± 5	12–35

Three-Dimensional Model Reconstruction.

Outer surface geometries of each scapula and humerus were segmented from CT scan data using the three-dimensional medical image processing software 3dslicer [42] (Fig. 1(a)). A threshold-based segmentation protocol, based on previously described techniques, was employed for each CT scan to label, reconstruct and smooth triangle-tessellated models of each bone (Fig. 1(b)) [43]. The procedure was repeated for each scapula and humerus, and triangular meshes were extracted for all the segmented three-dimensional models. Right-side scapulae and humeri data were reflected in the midsagittal plane to be left-sided, to simplify the development of the SSMs.

Fig. 1.

Steps taken toward developing SSMs from CT images: (a) image segmentation, (b) surface modeling, (c)co-registration, (d) mesh morphing, and (e) statistical shape modeling

Co-Registration and Surface Mapping.

A baseline scapula and humerus shape from one donor were randomly chosen from the specimen pool, and both were remeshed in meshlab [44] to obtain a smooth and uniform topology with a mean edge-length of 0.6 mm. Resulting surface meshes comprised approximately 110,000 vertices and 230,000 faces for the scapula and 90,000 vertices and 170,000 faces for the humerus. The three-dimensional meshes of the remaining scapulae and humeri were imported into meshlab and aligned to the baseline shapes using a two-step registration procedure. First, a landmark-based co-registration was performed based on manually identified homologous points on the scapulae (acromion tip, coracoid tip, and superior, inferior and anterior glenoid rim points) and humeri (central, inferior and anterior humeral head border points, and a single mid-diaphyseal medial surface point). These coarse alignments were refined using meshlab's iterative closest point algorithm (Fig. 1(c)). Subsequently, the two baseline meshes were mapped/morphed to each of the aligned meshes in the model dataset using R3DS Wrap 3.2 (R3DS, Voronezh, Russia) (Fig. 1(d)). Mesh morphing control points varied from model to model and were manually selected on extremums upon the need for such points due to a misalignment. On the humerus, two points (most proximal and most distal) were identified while in most cases there was no need for control points. On the scapula, costal and dorsal points were chosen to ensure curvature was properly accounted for. This process resulted in a mesh for each specimen with similar topology and identical vertex numbering but customized to the individual shape of each specimen (Fig. 2). Having the same topology simplified point-to-point comparisons between the models. Commonly used measures of proximal humerus and glenoid morphology were extracted from each model using custom scripts in matlab R2017b (Mathworks Inc., Natick, MA) (Table 2).

Fig. 2.

Baseline mesh mapped to each of the aligned meshes to obtain similar topology across all the specimens

Table 2.

Commonly used measures of proximal humerus and glenoid morphology summarized across the entire study set

Proximal humeral morphology
Head radius (mm)	25.2 ± 2.2
Head offset (mm)	7.2 ± 1.8
Retroversion (deg)	32.6 ± 8.4
Inclination (deg)	53.8 ± 2.5
Shaft diameter (mm)	21.5 ± 2.1
Glenoid morphology
Coronal radius of curvature (mm)	34.1 ± 4.7
Transverse radius of curvature (mm)	33.7 ± 10.6
Version (deg)	−5.8 ± 5.4
Inclination (deg)	4.7 ± 5.4
Torsion (deg)	6.8 ± 5.4

Creating Homologous Volumetric Meshes.

For creating SDMs with consistent sampling points, a baseline volumetric mesh was morphed to the individual surface meshes of each specimen. First, baseline humerus and scapula tetrahedral element meshes were generated in tetgen [45] from the same baseline surface models described above. The mesh elements had mean edge-lengths of 0.7 mm, resulting in approximately 810,000 nodes/4,800,000 elements for the humerus and 460,000 nodes/2,500,000 elements for the scapula. These base meshes were morphed to individual specimens via displacements applied directly to the surface nodes of the baseline mesh model to match the surface to any other specimen; the inner volumetric mesh deformed accordingly via the element shape functions and mesh connectivity. This process was performed using abaqus/cae (Dassault Systèmes Simulia Corp, Johnston, RI) and resulted in a set of 75 corresponding shoulder mesh models. All the volumetric meshes were qualitatively assessed to confirm none of the default shape error criteria in abaqus/cae were violated.

Assigning Nodal Density Properties.

Each humerus and scapula mesh was transformed back into its original CT coordinates and imported into the open-source mesh preprocessing software mitk-gem [46]. Here, the CT image intensity (in HU) at each node location was assigned by sampling from the original corresponding CT data. As a result, the set of 75 three-dimensional humerus and scapula meshes with homologous mesh topologies each had specimen-specific CT image intensity data assigned at each node. Due to the homologous mesh topologies, the CT image intensity of any given node within a model could be compared directly with the CT image intensity of the same node (at the same relative position) within any other model.

Principal Component Analyses.

The vertex coordinate data for each specimen surface model were imported into matlab, where the coordinates for all n vertices were concatenated to one vector $\mathbb{X}$ that described the shape, i.e., $\mathbb{X}\hspace{0.17em}=(x_1,{\hspace{0.17em}y}_1,\hspace{0.17em}{z}_1,\hspace{0.17em}\dots ,\hspace{0.17em}{x}_n,{\mathrm{ y}}_n,{\mathrm{ z}}_n);$ these individual vectors were assembled into a matrix comprising all point coordinate data for all specimens. The mean of each surface model point across specimens was calculated, and then principal component analyses (PCAs) were performed separately for the humerus and scapula surface coordinate matrices to compute PCs of their shapes. These PCs described the main modes of shape variation with respect to the average shapes of the humerus and scapula (Fig. 1(e)). For facilitating interpretation, the positive directions of some PCs were inverted, such that increasing a PC generally increased the volume of a bone.

For the SDM, bone densities at each sampling point of a specimen were expressed as a vector that was assembled into a matrix containing data for all specimens. The mean density at each node across specimens was calculated, and PCA was performed to identify the main modes of density variation within the humeri and scapulae. As above, the positive directions of some PCs were inverted such that increasing a PC generally increased the average density of a bone. All PC scores were normalized by dividing each PC score by the standard deviation of the corresponding PC calculated over the entire population.

Evaluating Compactness and Robustness of Statistical Models.

The SSMs and SDMs for the humerus and scapula were analyzed to identify the number of significant modes of variation (as opposed to modes that are spurious noise) [47].

To evaluate the compactness of each of the SSMs and SDMs, the percentage of interspecimen variability explained by each PC was calculated. Furthermore, for each specimen, the error in reconstructing the shape and density distribution of the humerus and scapula using a compact SSM/SDM was evaluated (results available in Appendix A). The robustness of the model development process to interoperator and specimen variability was also investigated and included as supplementary materials (Appendices B and C).

Data Analysis.

The shapes and density distributions of the humerus and scapula, as described using principal components, were compared for males versus females. To reduce the bias of the larger average male subject sizes, a more comparable subgroup of males and females in the height range of 157–170 cm was chosen, which included 7 male and 11 female donors. The age, height, weight and BMI of these donor sets were not significantly different, although the proximal humerus and glenoid shapes based on clinical measures were (Table 3). Student's t-tests were used to compare the shape and density PC scores of the humeri/scapulae of these well-matched subsets (excluding contralateral bones) and statistically significant differences were determined (p ≤ 0.05). To evaluate for relationships between bone density distributions and age, Pearson correlation coefficients were calculated for age with the PC scores of all the specimens in the SDM (excluding contralateral bones). Moreover, correlation coefficients between PC scores of the SSM and PC scores of the SDM were calculated for the humerus and scapula separately. Furthermore, correlation coefficients between PC scores of the humerus and PC scores of the scapula were computed for each of the SSM and SDM. Finally, the shape and density PC scores of another subgroup containing 21 pairs of contralateral humeri and scapulae were analyzed using paired t-tests in order to identify asymmetry in right- versus left-sided bone shapes or densities. Each statistical analysis using SSM/SDM data only included significant principal components, as determined by the results of our analyses.

Table 3.

The demographic data for male and female subgroups, including differences in proximal humerus and glenoid morphologies

	Male	Female	p
Age (years)	79 ± 9	73 ± 10	0.24
Height (cm)	166.1 ± 3.6	163.3 ± 4.3	0.19
Weight (kg)	55.2 ± 10.3	51.5 ± 7.8	0.46
BMI ($\text{kg}/{\mathrm{m}}^2$)	20 ± 3	19 ± 3	0.54
Proximal humeral morphology
Head radius (mm)	25.2 ± 1.1	22.6 ± 1.5	0.001
Head offset (mm)	7.9 ± 0.8	6.0 ± 1.8	0.017
Retroversion (deg)	30.9 ± 4.7	27.7 ± 8.5	0.37
Inclination (deg)	53.7 ± 1.9	53.9 ± 3.5	0.90
Shaft diameter (mm)	21.2 ± 0.9	18.8 ± 0.6	<0.001
Glenoid morphology
Coronal radius of curvature (mm)	35.4 ± 4.9	32.9 ± 3.7	0.25
Transverse radius of curvature (mm)	37.9 ± 10.7	35.2 ± 13.7	0.67
Version (deg)	−2.3 ± 4.4	−7.1 ± 4.5	0.040
Inclination (deg)	0.8 ± 5.6	4.9 ± 5.4	0.15
Torsion (deg)	4.5 ± 2.3	10.5 ± 4.3	0.004

Results

The Compactness of the Statistical Shape Models and Statistical Density Models and Identification of Significant Modes.

The highly correlated nodal coordinates of the shapes were reduced into a relatively small set of 74 uncorrelated and independent shape variables (SSM PC scores), and nodal densities were reduced to 74 uncorrelated SDM PC scores. The effects of each PC are described in Tables 4–7. The percentage of variability accounted for by each PC for both the humerus and scapula are shown in Fig. 3. For the scapula SSM and SDM, 9 and 17 PCs were identified as significant, which cumulatively explained 90.5% and 55.9% of the variability in those models, respectively. For the humerus SSM and SDM, 5 and 10 PCs were identified as significant, which cumulatively explained 94.4% and 58.7% of the variability in those models, respectively. Using only these significant PCs, the SDMs were still able to reproduce the density distribution patterns for each specimen (see Appendix A).

Table 4.

Description of the effects of increasing each humerus SSM PC score shown to be significantly influenced by sex^a (of well-matched subjects), age^b, side^c (when pairs), correlated with the humerus SDM^d or the scapula SSM^e

		Effect of increasing PC by 1 standard deviation on proximal humeral morphology
PC #	Description	Head radius (mm)	Head offset (mm)	Retroversion (deg)	Inclination (deg)	Shaft diameter (mm)
1	Overall scaling; increase in sized,e	1.5	0.1	1.1	−0.4	1.1
2	Axial elongation and medial rotation of the humeral headc	0.2	0.0	1.5	−0.3	0.3
3	Generally, an increase of overall girtha	0.8	0.3	2.6	0.2	0.9
4	Increase of humeral retroversionb,c	−0.2	0.4	7.3	0.0	0.3
5	Lateral bowing of the humeral shafta,c	0.4	1.2	−0.4	1.1	0.5

Table 7.

Description of the effects of increasing each scapula SDM PC score shown to be significantly influenced by sex^a (of well-matched subjects), age^b, side^c (when pairs), correlated with the scapula SSM^d or the humerus SDM^e

PC #	Description
1	Increasing the density over the entire boneb,e
2	Thinning of the cortical shell at the inferior edge of the glenoid and increasing the density of the central and superior subchondral bone in the glenoid regiona,c,e
3	Thinning of the cortical shell at the superior and inferior edge of the glenoid and increasing the trabecular bone density over the entire glenoidd,e
4	Thickening of the superior and central subchondral bone in the glenoid regione
9	Increasing the trabecular bone density in the superior region of the glenoid and thinning of the inferior subchondral bone in the glenoid regionb
13	Thinning of the inferior subchondral bone in the glenoid region and increasing the trabecular bone density in the superior region of the glenoidc

Fig. 3.

The cumulative sum of the variability percentage explained by the respective number of PCs: (a) for the SSM and (b) for the SDM (significant modes of variation are separated using dashed lines)

Sex Analysis.

The mean shape of the scapula and humerus across the entire model set, as well as the averages from the well-matched male and female subsets, are shown in Fig. 4. Qualitatively compared to average female bone shapes, the average male scapula had a longer medial/inferior border and a longer coracoid, and the average male humerus had a larger humeral head. The SSM PC scores for male versus female humeri and scapulae are compared in Figs. 5 and 6. For the humerus, statistically significant differences were identified in the scores of PC 3 and 5 of the SSM, with average PC scores differing by +1.4 ± 0.3 and +1.0 ± 0.3 standard deviations, respectively, for males relative to the females (all p ≤ 0.01). Statistically significant differences were identified in the scores of PC 1, 2, and 9 for the scapula SSM, with average PC scores differing by +1.3 ± 0.2, −1.1 ± 0.4 and −0.9 ± 0.4 standard deviations, respectively, for males relative to the females (all p ≤ 0.04). The effects of increasing each significant SSM PC score on the shape of the humerus and scapula are described in Tables 4 and 5, both qualitatively and in terms of typical radiographic measures of proximal humerus and glenoid morphology.

Fig. 4.

The shape of the humerus and scapula averaged over the entire population and averaged over male/female subgroup (these subgroups were well-matched for age, height, weight, and BMI as explained in Table 3): (a) population mean, (b) female subgroup mean, (c) male subgroup mean, and (d) superposition of male and female means

Fig. 5.

The boxplot of the PC scores of male and female humeri in the well-matched set for shape (green/darker boxes: PC scores of male humeri, yellow/lighter boxes: PC scores of female humeri). The boxes represent the first to the third quartiles, whiskers represent the range, lines in the box represent the median values, and squares represent the mean values of the PC scores. The blue/lighter icons show the average humerus shape deviated along the corresponding PC by +3σ, while the red/darker icons show the same for deviations by −3σ. The arrows show the effect of each PC on the shape of the humerus along the positive direction of that PC (Arrows with a vane at the end show the directions of out-of-plane rotations).

Fig. 6.

The boxplot of the PC scores of male and female scapulae in the well-matched set for shape (green/darker boxes: PC scores of male scapulae, yellow/lighter boxes: PC scores of female scapulae). The boxes represent the first to the third quartiles, whiskers represent the range, lines in the box represent the median values, and squares represent the mean values of the PC scores. The blue/lighter icons show the average scapula shape deviated along the corresponding PC by +3σ, while the red/darker icons show the same for deviations by −3σ. The arrows show the effect of each PC on the shape of the scapula along the positive direction of that PC (Arrows with a vane at the end show the directions of out-of-plane rotations).

Table 5.

Description of the effects of increasing each scapula SSM PC score shown to be significantly influenced by sex^a (of well-matched subjects), age^b, side^c (when pairs), correlated with the scapula SDM^d or the humerus SSM^e

		Effect of increasing PC by 1 standard deviation on glenoid morphology
PC #	Description	Coronal radius of curvature (mm)	Transverse radius of curvature (mm)	Version (deg) (retroversion negative)	Inclination (deg)	Torsion (deg)
1	Overall scaling; increase in sizea,e	0.6	−7.6	1.3	−0.3	−2.0
2	External rotation of the medial border of the scapula and medial rotation of the inferior regiona	0.8	−0.3	−1.7	1.1	1.2
3	Scaling along the superior-inferior axis and elongation of the acromion and coracoid	−0.3	−1.9	0.1	2.1	−1.0
4	Inferolateral elongation of the bone, posterior tilting of the medial scapular border and anterior tilting of the acromion and coracoidc	−0.4	0.8	−1.8	0.1	1.9
5	Elongation of the coracoid and anterior tilting of the acromion	0.2	−0.5	−0.1	−0.3	−0.3
6	Posterior tilting of the medial scapular border and anterior tilting of the coracoidb	−0.2	−2.7	0.7	2.5	1.6
7	Anterior tilting of the medial scapular border and posterior tilting of the coracoidd	−0.1	1.3	−2.3	−0.9	−0.3
8	Inferolateral elongation of the bone, anterior tilting and elongation of the acromionc	−0.9	−0.3	−1.2	0.5	−1.2
9	Greater glenoid inclination and posterior tilting of the acromiona	−0.2	−1.1	0.1	1.3	−0.2

Figure 7 depicts the mean density distribution of the proximal humerus and glenoid across the entire model set (mapped to the mean shapes), using all the PCs. Qualitatively comparing bone density distributions between sexes (Fig. 8), male bones appeared to have a denser trabecular bone in the subarticular zone and tuberosities of the humeral head, and the superior glenoid region. Comparing SDM PC scores for the humerus (Fig. 9), statistically significant differences were identified in the scores of PC 2, 3, and 5 with average PC scores differing by −1.1 ± 0.5, +1.2 ± 0.4, and −1.2 ± 0.4 standard deviations, respectively, for males relative to females (all p ≤ 0.04). Comparing PC scores for the scapula (Fig. 10), statistically significant differences were identified in the score of PC 2, with average PC scores differing by −1.2 ± 0.5 standard deviations, for males relative to the females (p = 0.03). The effects of increasing significant SDM PC scores on the density distribution of the proximal humerus and glenoid are described in Tables 6 and 7 (see Appendix D for even more detail).

Fig. 7.

The bone density distribution averaged over the entire population (pairs were averaged): (a) for the humerus and (b) for the glenoid

Fig. 8.

Comparing the male with female bone density distribution (Left: male, right: female). They were averaged over the well-matched set and mapped to the overall average bone shape: (a) for the humerus and (b) for the glenoid.

Fig. 9.

The boxplot of the PC scores of male and female humeri in the well-matched set for density (green/darker boxes: PC scores of male humeri, yellow/lighter boxes: PC scores of female humeri). The boxes represent the first to the third quartiles, whiskers represent the range, lines in the box represent the median values, and squares represent the mean values of the PC scores. Only the first five PCs, of the ten used for statistical analyses, are shown for brevity.

Fig. 10.

The Boxplot of the PC scores of male and female scapulae in the well-matched set for density (green/darker boxes: PC scores of male scapulae, yellow/lighter boxes: PC scores of female scapulae). The boxes represent the first to the third quartiles, whiskers represent the range, lines in the box represent the median values, and squares represent the mean values of the PC scores. Only the first nine PCs, of the 17 used for statistical analyses, are shown for brevity.

Table 6.

Description of the effects of increasing each humerus SDM PC score shown to be significantly influenced by sex^a (of well-matched subjects), age^b, side^c (when pairs), correlated with the humerus SSM^d or the scapula SDM^e

PC #	Description
1	Increasing the density over the entire bone and thickening the cortical shell, with a more marked effect below the anatomical neckb,c,e
2	Thinning of the cortical shell, especially medially below the surgical neck of the humerus and increasing cortical bone density at the proximal humerusa,e
3	Increasing the subarticular trabecular bone density and thickening cortical bone at the tuberosities, and thinning of the cortical shell below the surgical necka,e
5	Thickening of the cortical shell all over the proximal humerusa
6	Increasing the subarticular trabecular bone density, thinning the medial cortex inferior to the surgical neck and thickening the cortical shell on the lateral aspect of the proximal humerusb
7	Increasing the subarticular trabecular bone density, thinning the subchondral bone and cortical shell on the medial aspect and thickening the cortical shell on the lateral cortex inferior to the surgical neck of the humerusd
8	Thickening the cortical shell inferior to the surgical neck and thinning the subchondral bone of the humeruse
10	Decreasing the density of the subchondral bone and thickening the cortical shell inferior to the surgical neck of the humerusc

Age Analysis.

For the humerus SDM, the scores of PC 1 and 6 demonstrated significant but “weak” and “moderate” inverse correlations [48] with age (ρ = −0.29, and ρ = −0.40, both p ≤ 0.03), respectively. For the scapula, the scores of PC 1 and 9 showed similar weak, but significant, inverse correlations with age (ρ = −0.31, and ρ = −0.32, both p ≤ 0.02). No significant correlations with age were observed for any of the other significant PC scores.

Correlation Between Statistical Shape Models and Statistical Density Models.

The score of PC 1 of the SSM and SDM for the humerus showed a weak but significant correlation (ρ = 0.31, p = 0.007). Also, the score of PC 1 of the SSM and the score of PC 7 of the SDM for the humerus showed a moderate but significant correlation (ρ = 0.41, p = 0.0002). Interestingly, the score of PC 1 of the SSM also showed a weak but significant inverse correlation with the score of PC 2 of the SDM for both the humerus and scapula (ρ = −0.25, p = 0.03 for the humerus, and ρ = −0.39, p = 0.0006 for the scapula). Moreover, the score of PC 7 of the SSM and the score of PC 3 of the SDM for the scapula showed a moderate but significant inverse correlation (ρ = −0.42, p = 0.0002).

Correlation Between Humerus and Scapula.

For both the SSM and SDM, the score of PC 1 of the humerus and scapula showed a “strong” and significant correlation (ρ = 0.77, p < 0.0001 for the SSM, and ρ = 0.75, p < 0.0001 for the SDM). Interestingly, for the SDM, there were moderate but significant correlations between the score of PC 2 of the humerus and scapula (ρ = 0.51, p < 0.0001). Also, for the SDM, moderate but significant inverse correlations were noted between the score of PC 3 of the humerus and scapula (ρ = −0.52, p < 0.0001), and the score of PC 8 of the humerus and the score of PC 4 of the scapula (ρ = −0.40, p = 0.0004).

Contralateral Pairs Analysis.

For the humerus SSM, statistically significant differences were observed in the scores of PC 2, 4 and 5, with average PC scores differing by +0.4 ± 0.9, +0.5 ± 1.0, and −0.5 ± 0.9 standard deviations, respectively, for right humeri relative to the left (all p ≤ 0.04). For the scapula, statistically significant differences were identified in the scores of PC 4 and 8 of the SSM, with average PC scores differing by +0.4 ± 0.6, and +0.3 ± 0.5 standard deviations, respectively, for right scapulae relative to the left (all p ≤ 0.01).

For the humerus SDM, statistically significant differences were observed in the scores of PC 1 and 10, with average PC scores differing by −0.3 ± 0.7, and +0.4 ± 0.7 standard deviations, respectively, for right bone relative to the left (all p ≤ 0.03). For the scapula, statistically significant differences were observed in the scores of PC 2 and 13 of the SDM, with average PC scores differing by −0.4 ± 0.8, and −0.3 ± 0.7 standard deviations, respectively, for right bone relative to the left (all p ≤ 0.03).

Discussion

The primary objective of this study was to develop shoulder bone (humerus and scapula) SSMs and SDMs, which could have applications in population-based modeling of bone-implant mechanics, population-based implant design and surgical planning. Statistical analyses were performed on SSM/SDM PC scores to find correlations with demographics of the given population, between the shape and density distributions, between the humerus and scapula, and between sides of paired specimens.

Our findings related to sex-based differences in humeral bone morphology and density distributions share similarities and differences with previously reported work; and can be most closely compared with the shoulder bone morphology studies by Jacobson et al. [11], Robertson et al. [12], the humerus SSM described by Sintini et al. [37], and bone density analyses by Barvencik et al. [13], Yamada et al. [15] and Tingart et al. [14]. Most notably the first PC score of the SSM (size scaling) in our study was not significantly different between well-matched male and female subgroups. This deviates from the findings of all three previous morphological studies. However, it should be noted that none of these previous studies had height-matched male and female subgroups. Thus, some of the differences they reported may be attributed to overall differences in height between sexes. We did identify statistically significant increases in the third PC score of the SSM for males, which increases retroversion, shaft diameter, head radius and overall girth; similar findings were reported by both Jacobson et al. [11] and Sintini et al. [37]. Significant differences in the fifth PC score of the SSM indicated that male humeri have greater lateral bowing, also manifesting as an increase in head offset and inclination. This interesting finding was not reported in any of the aforementioned studies, may have implications for the design of long stems in sex-specific implants. However, the percentage of variability explained by this PC (2.1%) is relatively small and perhaps does not warrant the inclusion of additional designs.

Considering humeral bone density distributions, we did not identify significant differences in the first PC score of the SDM (overall density scaling), suggesting similar overall bone density between sexes. This differs from the findings of Barvencik et al. [13], who concluded that the bone volume to total volume ratio is significantly greater for males than females in various regions throughout the humeral head. Yamada et al. [15] also found an approximately 15% higher trabecular bone tissue percentage in the humeral head of males versus females. Our study did, however, identify significant differences in the second PC score of the SDM (males have thicker cortical bone, especially medially below the surgical neck of the humeral head). Interestingly, although Tingart et al. [14] found no significant differences in mean cortical thickness of the proximal humeral diaphysis between females and males (3.9 ± 0.44 mm versus 4.6 ± 1.02 mm, p = 0.08); the differences were close to being statistically significant. They also demonstrated that males have a significantly higher bone mineral density (BMD) in the surgical neck than females. Male and female sets were not matched for height, age or BMI in their study. Overall, when combined with the sex-based differences in bone morphology observed in our study (larger shaft diameter implying larger area moment of inertia), male humeri are mechanically capable of withstanding greater bending loads, which may have implications for sex-specific humeral stem design.

Considering that we did not observe sex-based differences in the overall size of the humerus, it was surprising that the first PC score of the scapula SSM (size scaling) was significantly greater for males than the females (for size-matched donors). In agreement with our findings, Jacobson et al. [11] showed that male scapulae are significantly larger than the female scapulae in approximately half of their anatomic parameters. However, they did not have height-matched male and female comparison groups. Furthermore, the first PC in our study accounted for more than simple scaling, as subtle differences in glenoid curvature and orientation are also observed. Statistically significant sex-based differences were also found for the second PC score of our scapula SSM, which describes subtle differences in glenoid orientation and curvature, and also indicates that female scapulae have a greater external rotation of the medial border and higher medial rotation of the inferior region of the scapula. This finding was not reported in the study of Jacobson et al. [11]. They also reported no significant difference between males and females in terms of their glenoid inclinations, whereas the significant sex-based differences for the second and ninth PC score of the scapula SSM both indicate that female scapulae have a greater glenoid inclination. This is interesting because previous studies have reported that glenoid inclination is significantly greater in patients with full-thickness rotator cuff tears versus those without [49], and women are at greater risk for rotator cuff pathologies [50]. That said, women are more likely to have partial rather than the full cuff tears associated with glenoid inclination [49,51]. Differences in glenoid inclination may have implications for implant designs, indicating that perhaps sex-specific shoulder implants should compensate or accommodate these differences. However, the percentage of shape variability explained by the second and ninth PCs are only 11.1% and 1.4%, respectively.

Considering scapular bone density distributions, we did not identify significant differences in the first PC score of the SDM (overall density scaling). Significant differences were identified in the second PC score of the SDM, which implies that compared to males, female scapulae have a thinner cortical shell at the inferior edge of their glenoid and higher density at the central and superior glenoid subchondral bone. This deviates from the findings of Mahaffy et al. [52], who concluded that there is no significant difference between sexes regarding density distribution in the glenoid (with only slightly higher densities for females). However, their samples were only focused on type E2 eroded glenoids. The differences identified in our study may have implications regarding optimal screw patterns and trajectories for glenoid implants in males versus females.

Considering correlations between humeral bone density distributions and age, our study findings imply that age has a significant but weak (ρ = −0.29) inverse relationship with the first PC score of the humerus SDM, suggesting bone density loss with age and cortical thinning, especially below the anatomical neck. Generally, these results are in accordance with that of previously published anatomical studies. A model developed by Roosa et al. [16] indicated that the areal BMD of the proximal humerus declined by 29% between ages 30 and 80 (p < 0.001) in addition to declines in cortical bone mass, area, and thickness with aging (all p < 0.01). Kirchhoff et al. [17] noticed a strong inverse correlation between age and bone volume to total volume ratio of the humeral head, being more marked in females (ρ = −0.72, p < 0.0001). By analyzing CT values in HU, Yamada et al. [15] found significant differences in mean trabecular bone tissue percentage of the humeral head for patients in three different age groups. They noticed that the average percentage of bone tissue reduced from 60.4% ± 12.1% in patients aged between 40 and 60 years to 48.2% ± 14.6% in patients aged over 60 years. In another study by Laval-Jeantet et al. [18], it was shown that the mean cortical porosity grows from 4.6% in men and 4% in women at the age of 40 to more than 10% at the age of 80. Apparent BMD, which is inversely linked to porosity, was shown to decrease with age in both males and females. Tingart et al. [14] demonstrated that the mean cortical thickness of the proximal humerus is significantly lower in donors aged over 70 versus younger donors (3.8 ± 0.86 mm versus 4.8 ± 0.96 mm, p < 0.05). Our relatively weak correlation of the first PC score for the humerus with age could be due to the relatively small numbers of younger subjects in our donor pool. We also found a moderate significant inverse correlation with the sixth PC score of the humerus SDM. In accordance with the results of Yamada et al. [15], its effect implies that the density of the subarticular trabecular bone of the humeral head decreases with age. Surprisingly, it also implies that the cortical bone of the proximal humerus below the surgical neck tends to become thicker on the medial side compared to the lateral part with age, possibly to compensate for the reduced subarticular density. This effect was not reported in any of the aforementioned studies.

Our findings related to correlations between scapular bone density distributions and age suggest that age also has a significant but weak inverse influence on the first PC score of the scapula SDM, suggesting a natural bone density loss of both shoulder bones with age.

A strength of our study was developing separate statistical models for shape and density instead of combining them into a single model, as in some previous works [20,31,32]. This allowed us to separate the correlated main modes of shape and density variation of the shoulder. Interestingly, the first PC score of the SSMs had weak but significant correlations with the second PC score of the SDMs, for both the humerus and the scapula. This finding suggests that there is a direct but weak significant correlation between the size of a bone and the thickness of its cortical shell, being more marked for the scapula. Furthermore, the first PC score of the humerus SSM and the SDM showed a weak but significant correlation, which implies that the larger humeri are also, on average, denser. In agreement with our results, Sintini et al. [37] found that the cortical thickness of the proximal humerus had a weak, but statistically significant, correlation with the first PC score of their proximal humerus SSM, describing the size of the bone (ρ = 0.25). Furthermore, Burton et al. [39] indicated that the first PC score of their SSM had a moderate but significant correlation with the first PC score of their SDM (ρ = 0.41).

Another interesting finding of this study was identifying a strong and significant correlation between the first PC score of the humerus and scapula, for both the SSM and SDM. These results suggest that donors with bigger humeri have bigger scapulae and higher bone density of humeri corresponds with higher bone density in the scapulae/glenoids.

Considering bilateral symmetry of shoulder morphology, statistically significant differences were observed for contralateral humeri and scapulae. Differences in the second, fourth and fifth PC scores of the humerus SSM and the fourth and eighth PC scores of the scapula SSM were observed. Taken all together, these results suggest asymmetries in contralateral humeri in terms of the humeral length, medial rotation of the humeral head, retroversion, inclination, head offset and lateral bowing of the humeral shaft. They also imply asymmetries in contralateral scapulae regarding glenoid orientation, the size of the inferolateral side of the bone, the orientation of the medial scapular border, the orientation of the coracoid, length and orientation of the acromion. However, these PCs combined only explain 8.9% and 5.8% of the total variability in shapes of the humeri and scapulae in our set, respectively. Therefore, these differences in shape may be inconsequential from a clinical perspective where one side may be used as a template for reconstructing the other. Verhaegen et al. [41] found an average difference of 2 mm in scapular offset (equal to 0.25 standard deviations of this parameter across their population), 2 $\text{deg}$ in glenoid version (equal to 0.5 standard deviations) and 2 $\text{deg}$ in glenoid inclination (equal to 0.5 standard deviations) between pairs in their set of 50 pairs of contralateral scapulae using CT images and three-dimensional model reconstruction. From these results, they concluded that contralateral scapulae are quite symmetrical in terms of scapular offset, glenoid version, and inclination. Also, Shi et al. [53] indicated that there is no significant difference in contralateral glenoids' length (p = 0.53), width (p = 0.42), area (p = 0.36), or circumference (p = 0.73) and concluded that contralateral glenoids are strongly symmetric in shape and size.

In terms of the density, the results of our study indicated that the first PC score of the humerus SDM, which mostly scales the density over the entire bone and thickens the cortical shell, especially below the anatomical neck, is significantly different between contralateral humeri. This implies that there is, to some extent, asymmetry in the density of the paired humeri and the thickness of their cortical shells. Diederichs et al. [54] concluded that there is a strong correlation between contralateral humeri in terms of BMD for both the distal (ρ = 0.90) and the proximal humerus (ρ = 0.74) (all p < 0.01), but did not do point-by-point comparisons of sites which our SDM is built upon. Also, in our study, statistically significant differences in the second PC score of the SDM for contralateral scapulae were observed, suggesting a difference in the thickness of the cortical shell at the inferior edge of the glenoid and in the density of the central and superior glenoid subchondral bone, between paired scapulae. Our observations regarding the asymmetry in the density distributions of paired bones may be attributed to the effect of the dominant hand. However, as the information regarding the dominant hand of the donors was not available, we were not able to further investigate this effect.

Our study is limited by the number of specimens and their age range (mean 73 ± 13). However, the sensitivity of our results to the number of included specimens was investigated through a robustness study (Appendix C). In the future, by including younger specimens, we will be able to further investigate the effect of age on the density distribution of the shoulder. Another limitation of our study is that 71 out of 75 specimens were from donors of Caucasian ethnicity. Including specimens from other ethnicities in future studies would allow us to investigate the effect of this factor on the shape and density distribution of the bones. The lack of compactness of the SDMs, compared to our SSMs (where only a few PCs were required to describe 95% of shape variations of the bones), is a limitation of our model; however, the SDMs could still successfully be used to reconstruct all the specimens in our set and capture the patterns of their density distributions effectively (Appendix A). This lack of compactness can be attributed to the high three-dimensional variability in density distribution across specimens (compared to shape variations), possibly due to adaptations of bones according to their in vivo mechanical loadings. Furthermore, the SSMs only considered the external cortex of the bones. In future studies, including the inner cortical boundary (and therefore cortical bone thickness) as a shape parameter may be more effective than incorporating it as a bone density distribution parameter. This may allow more compact SDMs; however, it will likely increase variability in the SSMs and requires delicate segmentation near articular surfaces where the shell is thinner. Moreover, the density distribution for the humerus is more uniform compared to the scapula, which has more complex geometry and density distribution. In future work, it may be advantageous to develop scapula SDMs that focus more on a certain area of interest (e.g., the glenoid), as the actual structural distribution of bone in different regions may not be of use in all models. For the purposes of surgical planning, however, a full-bone SSM/SDM offers the opportunity to predict the predisease shape of the articulation based on surrounding bones, which could be used to identify the optimal placement of the center of rotation of the prosthesis. This capability has not been demonstrated in this study, but will be a focus of future work. Finally, bones were not normalized to subject sizes, which increases the amount of variability in the dataset (which is mostly accounted for in PC 1 of the SSMs). This could have made it harder to appreciate nuanced shape changes occuring in PC 1 in combination with scaling.

Statistical shape and density models are tools capable of describing the main modes of variation in the shapes and density distributions of shoulder bones within a population. This study further confirmed that the demographics of a population, such as sex and age, have a significant influence on the shape and density distribution of the shoulder, and demonstrated that there are weak and moderate, but significant, correlations between many modes of shape and density variations in the shoulder. It also noted that donors with bigger humeri have bigger scapulae and higher bone density of humeri corresponds with higher bone density in the scapulae. Finally, an asymmetry of contralateral bones was identified in terms of density distributions. The humerus and scapula SSMs and SDMs created from this research were freely shared in a public online repository in the website link,1 allowing open access for other researchers to use and build on these models. Thus, the results of this study can be used to help guide the designs of population-based prosthesis components and pre-operative surgical planning.

Appendix A. Specimen Reconstruction Using a Compact Statistical Shape Models/Statistical Density Models

For each specimen, the root-mean-square error (RMSE) in reconstructing the surface of the humerus using the first five PCs of the SSM, and the RMSE in reconstructing the surface of the scapula using the first nine PCs were computed. The average RMSE across all the specimens and nodes in reconstructing the surface of the humerus using the compact SSM was 2.1 ± 0.6 mm. The average RMSE in reconstructing the surface of the scapula using the compact SSM was 2.2 ± 0.4 mm.

The density distribution of each specimen was also reconstructed using the first ten PCs of the SDM for the humerus and the first 17 PCs for the scapula and then was compared with its original CT. The average RMSE across all the specimens and nodes in reconstructing the density distribution of the humerus using the compact SDM was 169.2 ± 12.9 HU. The average RMSE in reconstructing the density distribution of the scapula using the compact SDM was 195.7 ± 15.9 HU. However, these reduced models were able to capture the pattern of density distribution for each specimen for both the humerus and scapula (Fig. 11).

Fig. 11.

Comparing the original CT image of the specimen with the maximum reconstruction error (left) with its reconstructed model (right): (a) for the humerus and (b) for the glenoid

Appendix B. Statistical Shape Models Robustness Analysis

A subset of ten scapulae was chosen for intra/inter-observer error analysis. The average absolute surface to surface distance over the entire scapula was quantified between the mean shape and meshes transformed +1 standard deviation (SD) along each PC (Fig. 12). Using the same CT-based models, a second SSM was generated by a second observer, and comparisons of the resulting mean shape and PCs were performed (in terms of the average absolute surface to surface distances). Finally, the PC values required to reproduce the shape of a specific scapula from the training set were compared for the two SSMs.

Fig. 12.

Example of the absolute surface to surface distance between the mean and +1 SD along a PC

The first five (of nine) PCs accounted for 95.2% of the variation in this subset (Fig. 13). A similar trend was observed in the average surface to surface distances associated with transforming along each PC (Fig. 13). The average surface to surface distance between the mean shapes of the two different SSMs was 0.002 mm, and the PCs computed for either SSM corresponded in terms of their influence on the model shape. Finally, similar scaling of PCs from either SSM was required to reproduce the shape of two of the training set specimens (Fig. 14), further suggesting the robustness of the SSMs.

Fig. 13.

The percentage of variability in the training set explained by each PC and cumulative percentage of variability (read from the left side) as well as the surface to surface distance between the mean shape and +1 SD of each PC (read from the right scale)

Fig. 14.

The SD values of each PC required to produce two scapulae in training set with two SSMs generated by different observers

Appendix C. Statistical Density Models Sensitivity Analysis

To assess the sensitivity of the SDMs to contributions of individual specimens, we used the add-one-in-approach. One additional specimen was included in the sets of humeri and scapulae, after which PCA was performed again to determine the sensitivity of the spatial distribution of the densities to the particular specimens used in the study. The additional bone instances were excluded from the original SSMs and SDMs, as the CT image of the extra humerus was missing its corresponding scapula and vice versa. In the evaluation of the robustness of the model, the average and the maximum of the absolute differences in the densities across all the nodes of the SDM based on 75 specimens and the one including the extra specimen, by a deviation of one σ along the first several PCs, were calculated (Fig. 15).

Fig. 15.

The average and the maximum of the absolute differences in densities in HU across all the nodes by a deviation of σ along the first few PCs: (a) for the humerus and (b) for the scapula

Appendix D. The Effects of Statistical Density Models Principal Components

The effects of some significant modes of density variation in the population, referred to in the text, on the density distribution of the proximal humerus (Fig. 16) and glenoid (Fig. 17) are visualized.

Fig. 16.

The colormaps show the effects of perturbing the corresponding modes of variation by one standard deviation, on the density distribution of the proximal humerus

Fig. 17.

The colormaps show the effects of perturbing the corresponding modes of variation by one standard deviation, on the density distribution of the glenoid

Funding Data

• National Institute of Arthritis and Musculoskeletal and Skin Diseases of the National Institutes of Health (Award No. R03AR066841; Funder ID: 10.13039/100000069).

Nomenclature

cm =

centimeter

HU =

Hounsfield units

kg =

kilogram

kVp =

kilovoltage peak

m =

meter

mA =

milliampere

mm =

millimeter

p =

p-value in a t-test

ρ =

Pearson correlation coefficient

σ =

standard deviation

* =

statistically significant difference

$\mathbb{X}$ =

shape vector representing coordinates of the vertices for a single specimen