Three-Dimensional Morphology of the Knee

Data Acquisition

A dataset of 223 male individuals (183 white Americans and 40 black Americans) was scanned using CT. Only normal femurs and tibia were included in this study; femurs or tibia with severe osteophytes and other abnormalities were specifically excluded. Only one femur and tibia was chosen from each individual, with no preference taken to either right or left side.

The bones were CT scanned using 0.625 mm × 0.625 mm × 0.625 mm cubic voxels. The result is high resolution, 3D radiographs in the form of Digital Imaging and Communications in Medicine (DICOM) image slices. This stacked image data were then segmented, and surface models were generated. This process has been found to be reliable with negligible inter- and intra-observer error. These models were then added to the ethnicity-specific statistical bone atlases.

Atlas Creation and Validation

Briefly, a bone atlas is an average mold, or template mesh, that captures the primary shape variation of a bone and allows for the comparison of global shape differences between groups or populations. Bone atlases were developed initially for automatic medical image segmentation ; however, it can be used as a way to digitally recreate a bone and conduct statistical shape analyses. In addition, they have proven useful in biologic anthropology as a means of studying sexual dimorphism and for reconstructing hominid fossils and making shape comparisons among fossil species.

For the ethnicity difference analysis in this study, a previously developed technique for creating a statistical representation of bone shape was used in a novel manner. Two separate statistical atlases of femurs were compiled with one atlas containing only femurs of white Americans and the other only femurs of black Americans. Similarly, two separate atlases were created for the tibia and divided in the same manner (ie, tibiae of white and black Americans). The processes of creating these statistical atlases and adding bones to the atlases are outlined below.

First, all of the bone models in the dataset were compared, and a bone model with average shape characteristics was selected to act as a template mesh. The points in the template mesh are then matched to corresponding points in all of the other training models. This ensures that all of the bones have the same number of vertices and the same triangular connectivity. Next, a series of registration and warping techniques was used to select corresponding points on all the other bone models in the training set. This process of picking point correspondences on new models to be added to the atlas is “nontrivial.” The matching algorithm described here uses several well-known techniques of computer vision, as well as a novel contribution for final surface alignment.

During the first step in the matching algorithm, the centroids of the template mesh and the new mesh were aligned, and the template mesh was prescaled to match the bounding box dimensions of the new mesh. Second, a rigid alignment of the template mesh to the new mesh was performed using a standard vertex-to-vertex iterative closest point (ICP) algorithm. Third, after rigid alignment we performed a general affine transformation without iteration. This method was applied to align the template mesh to the new mesh using 12 degrees of freedom (rotations, translations, scaling, and shear). After the affine transformation step, the template and new model have reached the limits of linear transformation, but local portions of the models still remain significantly distant. Because the goal of final surface-to-surface matching is to create new points on the surface of the new model that will have similar local spatial characteristics as the template model, a novel nonlinear iterative warping approach was developed to reduce misalignment.

To achieve point correspondence ( Fig. 16.1 ), an iterative algorithm is used in which the closest vertex-to-vertex correspondences are found from the template to the new model as before, but now we also find the correspondences from the new model to the template model. Using both of these point correspondences, points on the template mesh are moved toward locations on the new mesh, using a nonsymmetrical weighting of the vectors of correspondence. Next, a subroutine consisting of an iterative smoothing algorithm is applied to the now-deformed template mesh. This smoothing algorithm seeks to average the size of adjacent triangles on the template mesh, thereby eliminating discontinuities. At the beginning of the warping algorithm, the smoothing algorithm uses the actual areas of the surrounding triangles to dictate the smoothing vector applied to each point, which aids in effectively removing outlying points with large triangles. Consequently, at the beginning of the process the template mesh makes large steps, and larger smoothing is required. However, toward the end of the process the smoothing vector is normalized by the total area of the surrounding triangles, which allows for greater expansion of the template mesh into areas of high curvature. After this procedure has been completed on all the femurs and tibiae in their respective atlases, the atlases are ready for morphologic shape analyses and automated metric comparisons.