How Cerebral Vessel Tortuosity Affects Development and Recurrence of Aneurysm: Outer Curvature versus Bifurcation Type

Background and Purpose Previous studies have assessed the relationship between cerebral vessel tortuosity and intracranial aneurysm (IA) based on two-dimensional brain image analysis. We evaluated the relationship between cerebral vessel tortuosity and IA according to the hemodynamic location using three-dimensional (3D) analysis and studied the effect of tortuosity on the recurrence of treated IA.Methods We collected clinical and imaging data from patients with IA and disease-free controls. IAs were categorized into outer curvature and bifurcation types. Computerized analysis of the images provided information on the length of the arterial segment and tortuosity of the cerebral arteries in 3D space.Results Data from 95 patients with IA and 95 controls were analyzed. Regarding parent vessel tortuosity index (TI; <i>P</i><0.01), average TI (<i>P</i><0.01), basilar artery (BA; <i>P</i>=0.02), left posterior cerebral artery (<i>P</i>=0.03), both vertebral arteries (VAs; <i>P</i><0.01), and right internal carotid artery (<i>P</i><0.01), there was a significant difference only in the outer curvature type compared with the control group. The outer curvature type was analyzed, and the occurrence of an IA was associated with increased TI of the parent vessel, average, BA, right middle cerebral artery, and both VAs in the logistic regression analysis. However, in all aneurysm cases, recanalization of the treated aneurysm was inversely associated with increased TI of the parent vessels.Conclusions TIs of intracranial arteries are associated with the occurrence of IA, especially in the outer curvature type. IAs with a high TI in the parent vessel showed good outcomes with endovascular treatment.

A Note on the Strong Maximum Principle for Fully Nonlinear Equations on Riemannian Manifolds

We investigate strong maximum (and minimum) principles for fully nonlinear second-order equations on Riemannian manifolds that are non-totally degenerate and satisfy appropriate scaling conditions. Our results apply to a large class of nonlinear operators, among which Pucci’s extremal operators, some singular operators such as those modeled on the p- and $$\infty $$ ∞ -Laplacian, and mean curvature-type problems. As a byproduct, we establish new strong comparison principles for some second-order uniformly elliptic problems when the manifold has nonnegative sectional curvature.

Boundary properties of fractional objects: Flexibility of linear equations and rigidity of minimal graphs

The main goal of this article is to understand the trace properties of nonlocal minimal graphs in {\mathbb{R}^{3}}, i.e. nonlocal minimal surfaces with a graphical structure.We establish that at any boundary points at which the trace from inside happens to coincide with the exterior datum, also the tangent planes of the traces necessarily coincide with those of the exterior datum.This very rigid geometric constraint is in sharp contrast with the case of the solutions of the linear equations driven by the fractional Laplacian, since we also show that, in this case, the fractional normal derivative can be prescribed arbitrarily, up to a small error.We remark that, at a formal level, the linearization of the trace of a nonlocal minimal graph is given by the fractional normal derivative of a fractional Laplace problem, therefore the two problems are formally related. Nevertheless, the nonlinear equations of fractional mean curvature type present very specific properties which are strikingly different from those of other problems of fractional type which are apparently similar, but diverse in structure, and the nonlinear case given by the nonlocal minimal graphs turns out to be significantly more rigid than its linear counterpart.