## Info

the sampled specimens. We slice each model in two perpendicular directions: first along the sagittal direction, termed the longitudinal direction, and then along the direction that is perpendicular to the sagittal direction, which has been described previously, termed the latitudinal direction.

The PP head has a circular profile in the sagittal plane, varying across its width. The radii of the best-fit circles could be found for the PP to see if the joint is a conforming joint.

Take the PP of the ring finger as an example. Eleven paralleled auxiliary planes are first inserted into the PP head model, with equal distance between each other throughout the width of the PP head. Intersection curves are obtained (Figure 7.32) and fitted into 11 circles in the sagittal plane sections. Since the leftmost and the rightmost curves are not fitted into satisfactory circles according to data fitting verification, they are removed from this analysis (Figure 7.33); the other nine curves are subject to further analysis.

Ideally, the centres of the best-fit circles would be aligned on the same straight line. However, in practice they are not. This can be seen in Figure 7.34. The best-fit circles are obtained using the least-squares fitting method, developed to automate this process.

Data extracted from the best-fit circles are exported to Excel for further analysis. The x, y, z coordinate values and the radii for the best-fit circles with the mean and standard deviations were calculated. Figure 7.35 shows the values for the y, z coordinates and the radii of the best-fit circles from all section curves.

Figure 7.32 Feature curves in the sagittal planes of the PP head (left) and curves fitted to circles (right)

Figure 7.33 Leftmost and rightmost fitted circles in the sagittal plane

Figure 7.34 Centres of best-fit circles in front view and left view

Figure 7.34 Centres of best-fit circles in front view and left view

Figure 7.35 Circle profiles in the sagittal plane, varying across the PP head width

From these data we can find that, although the centres of the best-fit circles are not ideally on a straight line, the standard deviation is rather small: for the y, z coordinates the standard deviations are 0.463 and 0.248; for the radius of the best-fit circles the standard deviation is 0.255. The lowest centre point among all these circles tends to be on the left of the PP trough.

Figure 7.36 shows the values for the y, z coordinates and the radii of the best-fit circles for all sampled ring fingers and could be regarded as the generalized model.

It can be found that, for the generalized ring finger models, the circular profiles of the PP head in the sagittal plane vary across its width. The model tends to have the same maximum best-fit circle radius for the two condyles. The centre points are not strictly lying on a straight line. The lowest centre point tends to be coincident with the PP trough, with other centre points varying towards the bearing surface of the proximal interphalangeal joint (PIPJ).

The cartilaginous surface of the MP base is concave and bi-condylar to articulate against the PP head. Tessellated curves are first obtained on the surface of the MP base using two sets of auxiliary planes, perpendicular to each other. The tessellated curves then trim each

x-value of best-fit circles

Figure 7.36 Averaged profiles in the plane, varying with PP head width x-value of best-fit circles

Figure 7.36 Averaged profiles in the plane, varying with PP head width

Figure 7.37 Feature curves fitted to circles in sagittal planes for the MP base

other to construct the best-fit circles for the MP base. The trimmed curves are then used to fit corresponding circles in the sagittal plane (Figure 7.37).

From the data on the values of the x, y, z coordinates and the radii for the best-fit circles in the sagittal planes, it was found that:

• The circular profiles do not vary so much across the width of the MP base, except near the edge area.
• The centres of the circles tend to have the same y value, except near the edge area.
• The Z coordinate does not vary so much across the base width.
• The same properties of the circular profiles are also applicable to the averaged circular profiles of the MP base.

The bearing surfaces of the existing finger joint prostheses were designed to be conforming to increase joint stability. However, the above analysis shows that the MP base has a larger radius of curvature than the PP head, indicating that the sides of the joint do not conform to each other.

Figure 7.38 Corresponding radii of circular profiles in the sagittal planes for the PP and MP

The averaged radii of the circular profiles in the sagittal planes for the ring fingers of the PP and MP are shown in Figure 7.38. The MP base has a larger radius of curvature than that of the PP head. Therefore, the PIPJ does not match well in essence. However, the radius of curvature does not change much for both phalanges in the middle region of the bearing surfaces. It can be calculated that PP radii of the circular profiles measured from sagittal plane sections are on average 1.67 mm smaller than the MP radii.

Eleven NURBS curves are extracted from the PP head of the finger joint bone in order to find the geometrical features of the bearing surface. The PP head consists of two sagittally circular condyles, which are merged together to form a bi-condylar articular surface and are broader anteriorly than posteriorly. It could be found that the two condyles are not identical, although they are similar in profile shapes and sizes (Figures 7.39 and 7.40).

To analyse the geometrical features of the generalized bone model, finger joint surface feature curves were obtained from all the bone sample specimens. The curve dimensions vary among different curves of the same bone model and the same curve for different bone models. Figure 7.41 shows the fifth curve extracted from all the bone specimens for the PP head. Similar results were also found for the PP base and MP base respectively.

From the above figures we can see that the finger joint surface feature curves follow the same geometrical style for different finger joint specimens. We can categorize the bone samples into several groups in the finger joint surface replacement surgery.

Theoretically, any of the dimensions R, L, Wpph, Wppb and Wmpb can be used to define the sizes of the finger joint prostheses. From the above analysis it can be found that the PP lengths

Figure 7.39 Surface curve profiles for the PP head of specimen 2

Figure 7.40 PP head shape: broader anteriorly than posteriorly in the transverse plane