A principal component analysis-based framework for statistical modeling of bone displacement during wrist maneuvers

Abstract

We present a method for the statistical modeling of the displacements of wrist bones during the performance of coordinated maneuvers, such as radial-ulnar deviation (RUD). In our approach, we decompose bone displacement via a set of basis functions, identified via principal component analysis (PCA). We utilized MRI wrist scans acquired at multiple static positions for deriving these basis functions. We then utilized these basis functions to compare the displacements undergone by the bones of the left versus right wrist in the same individual, and between bones of the wrists of men and women, during the performance of the coordinated RUD maneuver. Our results show that the complex displacements of the wrist bones during RUD can be modeled with high reliability with just 5 basis functions, that captured over 91% of variation across individuals. The basis functions were able to predict intermediate wrist bone poses with an overall high accuracy (mean error of 0.26 mm). Our proposed approach found statistically significant differences between bone displacement trajectories in women versus men, however, did not find significant differences in those of the left versus right wrist in the same individual. Our proposed method has the potential to enable detailed analysis of wrist kinematics for each sex, and provide a robust framework for characterizing the normal and pathologic displacement of the wrist bones, such as in the context of wrist instability.

Keywords: Principal component analysis; Sex differences; Statistical modeling; Wrist bone displacement.

Figure 1:

Representative T₁-weighted images of one volunteer in the various static positions. All volunteers (N=18, 36 wrists) were imaged in the extreme RD, neutral, and extreme UD positions while a subset (N=8, 16 wrists) was also imaged in halfway positions, outlined in red above.

Figure 2:

Flowchart for constructing the PCA-based basis functions and creating the motion model.

Figure 3:

Representative surfaces of the wrist bones used for creating the statistical model in the (A) UD, (B) neutral, and (C) RD positions for a study participant. PCA was employed to construct basis functions based on bone displacement between the positions.

Figure 4:

Rendering of the first two principal components for the RUD motion. The α term refers to the model coefficient, and the first and second principal components were varied separately, i.e. all other model coefficients were set to zero. The first component was in the expected RUD direction while the second component was a combination of flexion-extension and bone spacing scaling. The dotted lines are intended to aid in visually comparing the bone positions between each row.

Figure 5:

Percent variance explained by each eigenvector (N=18 right wrists). The total number of eigenvectors was the number of right wrist positions minus 1, i.e., (18 × 3)−1.

Figure 6:

Rendering of the individual kinematic model of one of the volunteers. Subtle flexion-extension was observed during the RUD maneuver in some volunteers, such as the one shown above. The positions rendered are (A) extreme RD, (B) halfway between extreme RD and neutral, (C) neutral, (D) halfway between neutral and extreme UD, and (E) extreme UD. Top row is a volar view of the wrist, while the bottom row is a top down view. The arrows indicate subtle flexion-extension.

Figure 7:

Example prediction of the bone displacement model to the intermediate positions from one volunteer mid-way between neutral and UD (top row) and mid-way between neutral and RD (bottom row) positions. “Original” (far left column) refers to surfaces of the intermediate positions, after bone atlas alignment. “Model Prediction” (middle column) are the bone positions predicted by the model. “Overlap” (right column) refers to the overlay of the “Original” and “Model Prediction” bone surfaces. The mean Euclidean distance in this case was 0.24 ± 0.20 mm.

Figure 8:

The model coefficients between the right wrist (plotted in red) and the left wrist (plotted in blue) versus the RUD angle for 4 representative volunteers. Lower RUD angles refer to more radial deviation while higher RUD angles refer to more ulnar deviation.

Figure 9:

Men are represented in blue while women are represented in red. A second order model was fitted for each volunteer to estimate how each wrist moved uniquely, and these models were sampled at eight RUD angles for comparison. Lower RUD angles refer to more radial deviation while higher RUD angles refer to more ulnar deviation.

Figure 10:

The fitted models for men and women were sampled at five RUD angles and rendered above. The angles chosen were approximately the same as the five static positions a subset of the volunteers did during scanning. The main difference appears to be overall wrist scaling with men having larger wrists then women.