The tali of the Japanese subjects in a previous analysis of the calcaneus morphology  were analyzed in the present study. Briefly, CT scans of subjects without talar and calcaneal injuries or disorders were retrospectively identified in the database at our institution. Fifty-six feet were enrolled in this study (men, 31 feet; women, 25 feet); mean age of men was 49.2 [SD 17.5] years (range, 20–82 years) and that of women was 52.6 [SD 21.6] years (range, 17–87 years). The institutional review board and appropriate ethics committee approved this retrospective review. CT images were acquired using an Aquilion Multi-detector CT scanner (Toshiba Medical Systems, Otawara, Japan). Detailed information of the CT image parameters was described in a previous study . The images were imported into a commercial software (Mimics Version 9.0; Materialise Inc., Leuven, Belgium) and a 3-D bone surface of the talus was generated. Mirror image models of left-side specimens were subsequently created using the Geomagic Design X (3-D Systems Inc., Rock Hill, SC, USA) and were treated as right-side specimens.
Articular Surface Analysis
To quantify the orientations of the articular surfaces, a body-fixed coordinate system was defined for the talus. For this, we used a typical human talus as a standard specimen and defined a body-fixed orthogonal coordinate system with the X-, Y-, and Z-axes the representing mediolateral, anteroposterior, and dorsoplantar axes, respectively, (Fig. 1) as in a previous report . The origin of the coordinate system was defined as the centroid of the 37 landmarks on the surface of the talus . All specimens were superimposed on this standard specimen based on the 37 landmark coordinates  using the Generalized Procrustes Analysis method [22–24] to place them in the same coordinate system.
Three regions of the articular surfaces in the talus, i.e., the superior surface of the trochlea, calcaneal articular surfaces, and navicular articular surface, were manually extracted by outlining the visible borders of the corresponding articular surfaces. To quantify the orientation of the three articular surfaces, the planes were fitted to each articular surface using the least-squares method, and the normal vector of each plane was calculated (Fig. 2). Furthermore, the first principal axis was calculated (Fig. 2).
For the trochlea and subtalar articular surfaces (Fig. 2a, b), the angle between the normal vector and the Y-axis projected on the sagittal (YZ-) plane was defined as the superoinferior angle, whereas the angle between the normal vector and the Z-axis projected on the frontal (XZ-) plane was defined as the mediolateral angle. The angle between the principal axis and the Y-axis projected on the horizontal (XY-) plane was defined as the rotational angle. For the navicular surface (Fig. 2c), the angles between the normal vector and the Y-axis projected on the sagittal plane, between the normal vector and the Y-axis projected on the horizontal plane, and between the principal axis and the Z-axis projected on the frontal plane were defined as the superoinferior, mediolateral, and rotational angles, respectively. The angles were positive if the surfaces were oriented superiorly, laterally, and internally rotating directions, respectively.
To assess the curvatures of the articular surfaces, the superior trochlea, the posterior calcaneal articular surface, and the navicular surface were approximated by cylindrical and spherical surfaces, using the least-squares minimization (Solid Primitive command in Geomagic Design X, 3D Systems, Rock Hill, SC, USA) (Fig. 3). The subtalar articular facet consisting of the anterior, middle, and posterior calcaneal articular surfaces is essentially a plane joint on which the calcaneus slightly translates and rotates . However, the posterior calcaneal surface is cylindrical and thus approximated by a cylinder. The calculated radii of curvature were normalized by the talar length, defined as the distance between the most distomedial point on the posterior calcaneal articular surface and the center of the navicular articular surface (landmark points of 23 and 36 described in Nozaki et. al. ) for comparisons.
Measurement Reproducibility Assessment
The intra-rater reproducibility for the measurements in each of the 56 tali (two independent measurements per talus) was assessed using the intraclass correlation coefficient with one-way random effects, absolute agreement, single rater/single measurement (ICC, model 1, 1), and 95% confidence intervals. The standard error of measurement (SEM) was calculated to represent the consistency of the results within individuals in the same unit as the original measurement. The 95% minimal detectable change (MDC95) was also determined for assuming the amount of change required to represent a true change (i.e., exceeding measurement error) . Repeated measurements were conducted by one observer with a 4-week interval. Calculation of the ICC was performed using the SPSS statistics (version 25.0, IBM, Armonk, NY, USA).
Multivariate analysis of variance (MANOVA) was individually conducted for each three articular surfaces to investigate sex-related differences in the orientation angles and curvatures as well. Age-related differences in the orientation angles and curvatures of the articular surfaces were also analyzed using the Pearson’s correlation coefficient. The statistical significance level was set at P < 0.05. All statistical analyses were performed in the open source R software, version 3.5.2 (R Core Team, 2016).