Curvilinear Three-Dimensional Modeling of Spinal Curves with Dual Kriging

Abstract

In spinal deformation studies, three-dimensional reconstruction of the spine is frequently represented as a curve in space fitted to the vertebral centroids. Conventional interpolation techniques such as splines. Bezier and the least squares method are limited since they cannot describe precisely the great variety of spinal morphologies. This article presents a more general technique called dual kriging, which includes two mathematical constituents (drift and covariance) to adjust the interpolated functions to spinal deformity better. The cross-validation technique was used to compare the parametric representations of spinal curves with different combinations of drift and covariance functions. Model validation was performed from a series of analytic curves reflecting typical scoliotic spines. Calculation of geometric torsion, a sensitive parameter, was done to evaluate the accuracy of the kriging models. The best model showed an absolute mean difference of 1.2 x 10∼5 (±7·1 × l()∼ ⁵)  mm⁻¹ between the analytical and estimated geometric torsions compared to 5·25 × 10∼ (±3.7 × ³10∼²) mm* ¹ for the commonly used least-squares Fourier series method, a significant improvement in spinal torsion evaluation.

Keywords:

• Dual kriging
• spine
• geometric torsion
• cross-validation
• least squares
• interpolation