A penalized spline fitting method to optimize geometric parameters of arterial centerlines extracted from medical images

In order to grasp the spatial and temporal evolution of vascular geometry, three-dimensional (3D) arterial bending structure and geometrical changes of arteries and stent grafts (SG) must be quantified using geometrical parameters such as curvature and torsion along the vasculature centerlines extracted from medical images. Here, we develop a robust method for constructing smooth centerlines based on a spline fitting method (SFM) such that the optimized geometric parameters of curvature and torsion can be obtained independently of digitization noise in the images. Conventional SFM consists of the 3rd degree spline basis function and 2nd derivative penalty term. In contrast, the present SFM uses the 5th degree spline basis function and 3rd and 4th derivative penalty terms, the coefficients of which are derived by the Akaike information criterion. The results show that the developed SFM can reduce the errors of curvature and torsion compared to conventional SFM. We then apply the present SFM to the centerline of the SG in an abdominal aortic aneurysm (AAA), and those of bilateral internal carotid arteries (ICA) in 6 cases: 3 cases with aneurysms and 3 cases without any aneurysm. The SG centerlines were obtained from temporal medical images at three scan times. The strong peak of the curvature could be clearly observed in the distal area of the SG, the inversion of the torsion at 0 months in the middle area of SG disappeared over time, and the torsions around the SG bifurcation at the three time periods were inverted. The curvature-torsion graphs along the ICA centerlines superimposing five aneurysmal positions were useful for investigating the relationship between arterial bending structure and aneurysmal positions. Both ICAs had curvature peak values higher than 0.4 within the ICA syphons. The ICA torsion graphs indicated that left and right ICA tended to be a right- and left-handed helix, respectively. In the left ICA syphon, the biggest aneurysm could be observed downstream of the salient torsion inversion. All aneurysms for 3 cases were positioned at the downstream of the inverted torsion.