Surface geometric analysis of anatomic structures using biquintic finite element interpolation

The surface geometry of anatomic structures can have a direct impact upon their mechanical behavior in health and disease. Thus, mechanical analysis requires the accurate quantification of three-dimensional in vivo surface geometry. We present a fully generalized surface fitting method for surface geometric analysis that uses finite element based hermite biquintic polynomial interpolation functions. The method generates a contiguous surface of C2 continuity, allowing computation of the finite strain and curvature tensors over the entire surface with respect to a single in-surface coordinate system. The Sobolev norm, which restricts element length and curvature, was utilized to stabilize the interpolating polynomial at boundaries and in regions of sparse data. A major advantage of the current method is its ability to fully quantify surface deformation from an unstructured grid of data points using a single interpolation scheme. The method was validated by computing both the principal curvature distributions for phantoms of known curvatures and the principal stretch and principal change of curvature distributions for a synthetic spherical patch warping into an ellipsoidal shape. To demonstrate the applicability to biomedical problems, the method was applied to quantify surface curvatures of an abdominal aortic aneurysm and the principal strains and change of curvatures of a deforming bioprosthetic heart valve leaflet. The method proved accurate for the computation of surface curvatures, as well as for strains and curvature change for a surface undergoing large deformations.