Table 1.
Foot bone volume, surface area and bone density (Mean ±SD).
Figure 1.
Positional relationship between a bone's COM and COS.
Fig. 1a Positional relationship between COM and COS on x-y plane; Fig. 1b Positional relationship between COM and COS on x-z plane; Fig. 1c Positional relationship between COM and COS on y-z plane; Fig. 1d Distance between bone's COM and COS. The bones' COS and COM are derived from the calculation of Eqs. (2) and (3). When choosing coordinate system with origin at COM, the coordinates of COS relative to COM can be derived as . By using
,
and
, 384 pieces' bone coordinates of COS with respect to COM can be located on x-y, y-z and x-z planes. See Fig. 1a, 1b and 1c (unit is mm). Through Eq. (4), the distance of these 384 pieces of bones' COS to the COM can be calculated, resulting in Fig. 1d.
Figure 2.
Relationship between the COM and COS of the cross-sectional image.
Fig. 2a Positional relationship between the COM and COS of the cross-sectional image through the coordinate origin; Fig. 2b Positional relationship between the COS of the cross section and the COS of the bone. When the position value of the cross-sectional VE at z-axis is approximately equal to the bone's COS, i.e. , the cross section is the tomography that goes through the bone's COS. Calculate the bone's cross-sectional COM and COS, and then calculate the distance between the two points by using the plane distance formula. See Fig. 2a. Fig. 2b is the distance between the cross-sectional COS and the COS of bone
on x-y plane calculated by the plane distance formula.
Figure 3.
Fig. 3a and 3e Structure by constraints and loads; Fig. 3b and 3f Axial force distribution in the structure; Fig. 3c and 3g Shear distribution in the structure; Fig. 3d and 3h Bending moment distribution in the structure. Fig. 3i Relationship between internal square position and strength. The rods in the structure are all rigid and the connections between the rods are rigid also. Two squares are drawn with a side length of 1 and 0.2 respectively. Connect the vertices of the two squares and a simple structural mechanics model is forged. Set the two bottom vertices of the bigger square to connect with the hinge bearing on the ground. The top of the bigger square is subjected to distributed load (size is 1). The vertical coordinate of the smaller square COS superposes the bigger square. Change the horizontal coordinate from −0.3 to +0.3. By using the software of SMSolver, the calculation results are shown in Fig. 3a–3i.
Figure 4.
Application of the superposition principle of the bone's COM and COS.
Fig. 4a Positional relationship between the COS of the compact bone and the COS of the bone; Fig. 4b–m Relationship between the bone tissue's density and distribution radius, where axis x stands for the tissue's density and axis y for the standardized mean distribution radius of the tissue. The data were collected from 192 pieces of foot bone of the wrestlers and 192 ones of the footballers. *p<0.05, **p<0.01. When 1.65, the bone tissue is defined as compact bone. Eqs (2) and (3) are used to calculate the compact bone's COM and COS while Eq. (4) the distance between the two points and Eq. (5) the distribution radius of bone tissue. Fig. 4a is the result of the distance between the compact bone's COM and COS standardized by the bone tissue's radius. Eq. (6) is applied to calculate same density tissue radius. Then standardize it by the bone tissue's radius. See Fig. 4b–4m.
Figure 5.
COM and COS of non-bone tissues.
Fig. 5a Positional relationship between the whole foot's COM and COS; Fig. 5b Positional relationship between the COM and COS of ankle skin; Fig. 5c Positional relationship between the COM and COS of non-bone tissues (a group of cross sections selected around the ankle joint); Fig. 5d Positional relationship between the COM and COS when a ROI (region of interest) of 1 cm3 is established around the COS of the talus. The grey ball stands for the COM position and the red ball for the COS position. The radius of the ball is 0.5 mm. According to the definition of non-bone tissue density, Eqs. (2) and (3) are used to calculate COM and COS of non-bone tissue. The three-dimensional model is constructed by the software of Mimics.