A Comprehensive Numerical Approach to Coil Placement in Cerebral Aneurysms: Mathematical Modeling and In Silico Occlusion Classification

TL;DR

This work tackles sub-optimal coil placement in endovascular aneurysm treatment by developing a comprehensive discrete elastic rod (DER) model that represents coils as a reduced 1D system with bending, twisting, and inextensibility. It combines an octree-based collision detection, a symplectic semi-implicit Euler time integration, and explicit contact/friction models to simulate realistic coil deployments within patient-specific aneurysm geometries. An in silico Raymond-Roy occlusion classification (RROC) is introduced, using voxelized coil distributions to quantify local packing densities in the neck, boundary, and core regions, enabling statistical analyses and sensitivity studies of coil parameters. The framework enables large-scale, preclinical exploration of placement strategies and materials, informing catheter positions, coil shapes, and expected occlusion outcomes prior to clinical intervention. Overall, the method provides a physics-grounded, computationally efficient tool for predicting occlusion quality and guiding embolization planning.

Abstract

Endovascular coil embolization is one of the primary treatment techniques for cerebral aneurysms. Although it is a well established and minimally invasive method, it bears the risk of sub-optimal coil placement which can lead to incomplete occlusion of the aneurysm possibly causing recurrence. One of the key features of coils is that they have an imprinted natural shape supporting the fixation within the aneurysm. For the spatial discretization our mathematical coil model is based on the Discrete Elastic Rod model which results in a dimension-reduced 1D system of differential equations. We include bending and twisting responses to account for the coils natural curvature. Collisions between coil segments and the aneurysm-wall are handled by an efficient contact algorithm that relies on an octree based collision detection. The numerical solution of the model is obtained by a symplectic semi-implicit Euler time stepping method. Our model can be easily incorporated into blood flow simulations of embolized aneurysms. In order to differentiate optimal from sub-optimal placements, we employ a suitable in silico Raymond-Roy type occlusion classification and measure the local packing density in the aneurysm at its neck, wall-region and core. We investigate the impact of uncertainties in the coil parameters and embolization procedure. To this end, we vary the position and the angle of insertion of the microcatheter, and approximate the local packing density distributions by evaluating sample statistics.

Figures (20)

• Figure 1: Coiling of an aneurysm at different packing densities. (\ref{['fig:coiling_photo1']}) Incomplete occlusion (low packing density), with only framing coil inserted. (\ref{['fig:coiling_photo2']}) Completed (high volumetric occlusion) coiling procedure with additional filling coil in place.
• Figure 2: Morphological features of the three aneurysms considered in this study. (\ref{['fig:Geom_Small_1']})--(\ref{['fig:Geom_Small_2']}) is denoted as small aneurysm, (\ref{['fig:Geom_Bir_1']})--(\ref{['fig:Geom_Bir_2']}) is a bifurcation aneurysm and (\ref{['fig:Geom_Gamm_1']})--(\ref{['fig:Geom_Bir_2']}) a narrow neck aneurysm. From the left to the middle column the perspective is changed by a rotation of 90 degrees. The neck width, dome width and height are given. In the right column, at (\ref{['fig:Geom_Small_3']}), (\ref{['fig:Geom_Gamm_3']}), (\ref{['fig:Geom_Bir_3']}), the insertion direction of the catheter is sketched in green and the position of the catheter is shown in red.
• Figure 3: A space curve $\boldsymbol{x}$ parameterized by its arc-length $s$ together with the material directors $\boldsymbol{D}_{1}(s), \boldsymbol{D}_{2}(s), \boldsymbol{D}_{3}(s)$ and their corresponding Darboux-vector $\boldsymbol{\Omega}(s)$.
• Figure 4: A discrete space curve with material points $\boldsymbol{x}_i$ for $i \in \{1,...,N\}$ and edges $\boldsymbol{e}^j=\boldsymbol{x}_{j+1}-\boldsymbol{x}_{j}$ for $j \in \{1,...,N-1\}$ on which the material frames $\boldsymbol{D}_{1}^j, \boldsymbol{D}_{2}^j, \boldsymbol{D}_{3}^j$ are attached.
• Figure 5: Framing of a Helix where $\boldsymbol{D}_1$ is represented in green and $\boldsymbol{D}_2$ in red (\ref{['fig:Frenet']}) An example for a material Frame with the director $\boldsymbol{D}_1$ in green and $\boldsymbol{D}_2$ in red (\ref{['fig:Bishop']}) Bishop Frame with directors $\boldsymbol{U}_1$ in green and $\boldsymbol{V}_2$ in red. (\ref{['fig:Combined_Frames']}) In red, we show a material Frame of directors $\boldsymbol{D}_1,\boldsymbol{D}_2$ and in green a Bishop frame for the curve $\boldsymbol{U}_1,\boldsymbol{V}_2$. Since the Bishop frame is twist free we can rotate it on each edge into the material frame to get the absolute reference angle with respect to the Bishop frame. On two consecutive edges $(j,j+1)$, the difference in this angle is the twist of the material frame $\tau^j$.