The exploration of the complex flow patterns within cerebral aneurysms is crucial for risk assessment, therapy planning and medical research. In order to support visual exploration, geometric descriptors are necessary to decompose the complex flow data. An important descriptor is the ostium, the area of inflow into the aneurysm, since other descriptors can be derived from it. We describe an interpolation-based approach to generate a smooth ostium surface that was successfully applied to seven different aneurysm surface models.
A cerebral aneurysm is a pathological artery dilatation and forms an area with a
high risk of rupture. Insight into intravascular flow characteristics is important
for risk assessment, therapy planning and comparison of treatment options (e.g.
different stents and their placement) as well as the development of new, minimal
invasive treatment techniques. A way to retrieve this kind of data is a CFD
simulation, based on 3D reconstructions of the affected vessel. The resulting
flow data is complex and a subdivision into functional unities supports the visual
exploration. Since blood flow is mainly influenced by the surrounding vascular
geometry [
In [
The generation of the ostium surface is a two step approach: Shape features are generated which subsequently form the input for a smooth sampling of the ostium surface by means of interpolation. Some of the shape features are created during the pre-processing whereas others are explicitly computed to meet the requirements of the final ostium surface shape. 2.1
Anatomically the ostium describes the area of inflow into the aneurysm. In
most of the cases, flow data is calculated on a CFD-compliant volume-mesh.
We derive the ostium surface from this mesh as well. A detailed description of
the segmentation process, that is necessary to derive a volume mesh from the
image data, is beyond the scope of this work. We refer to [
Besides the surface of the volume mesh we need two more inputs: the parent
vessel centerline and a contour that describes the location of the ostium with
respect to the surface (Fig. 1a). The parent vessel centerline is also derived from
the surface of the volume mesh, utilizing its Voronoi diagram. A robust
implementation is given as part of the Vascular Modeling Toolkit (VMTK) [
The ostium contour CO and a part of the parent vessel centerline that is projected into the ostium area are the shape features of the ostium surface. These features enable us to generate a closed surface in between. The outer border of the surface is already given by the ostium contour CO through P1 - P4 (Fig. 1a). The projection of the centerline, going through P1 and P2, ensures that the ostium surface adjusts to the location and the bending of the underlying parent-vessel (Fig. 2a). Centerline Correction Beneath the ostium the centerline tends to bulge into the aneurysm (Fig. 2a), since the surface in this area deviates from the tubular shape given in the parent vessel. This part of the parent vessel centerline needs to be corrected. First we identify the last reliable parts of the centerline within the parent vessel, before and after the aneurysm. Each of these parts is defined by two points (a1, a2) on the centerline that have minimal distances to P1 and P2 (b1, b2) respectively. These four points are used as control points for a cubic Lagrange polynomial that will replace the unreliable part of the centerline. The parameterization t for each point is t(a1) = 0:0, t(a2) = !, t(a3) = 1:0 − !, t(a4) = 1:0 where ! is defined by
The Lagrange approximation algorithm is used to sample the polynomial and the original centerline points are replaced with the samples.
Centerline Projection The corrected centerline is projected into to ostium area by interpolating between the vectors −→v1 = P−−1→a2 and −→v2 = P−−2→b1 (Fig. 2a). For each point pi of the n centerline points between a2 and b1 the projected point p′i is defined by p′i = pi + ( 1 − n i ) →−v1 + ( i ) n −→v2
The projected centerline (CP ) starts from P1, resembles the shape of the underlying parent-vessel centerline and ends at P2. 2.3
The projected centerline CP divides the ostium surface, we want to generate, into two parts. Without loss of generality we describe the surface generation process (a) Input data (b) Surface 1 (c) Surface 2 (2) for only one of these parts, since it is a local approach that can be applied to both parts separately.
We subdivide the segment of ostium contour CO that is contained in the chosen part into two segments Ca and Cb of equal geodesic length. The two segments share the division point pd and are uniformly sampled with the same number of samples k. The surface is generated by linear interpolation from Ca to Cb. For this, we need a suitable interpolation path. This path is given by linear segments L1 − Lj that are defined between the j points of the projected centerline CP and pd. Each segment Li is sampled uniformly and has the same number of samples k as the contour segments. For the first linear segment L1, a set of vectors Va is defined by V−→ai = L−−1−iC−→ai, where L1i is the sample on L1 and Cai is the corresponding sample on Ca. The same is done for the last linear segment Lj and Cb with the vector set Vb as result (Fig. 2b).
We now have a set of vectors Va that is related to Ca and a set Vb that is related to Ca respectively. The j linear segments with k samples each define the sampling of our surface. For each sample point Lmn, with 0 ≤ m < j and 0 ≤ n < k, we define a displacement that includes the linear interpolation from Va to Vb
L′mn = Lmn + (
m ) V−−→an + ( m ) V−→bn 1 − j j (3) The regular j ×k structure of the sample point allows us to use simple, vertex Idbased construction rules to define the topology based on the displaced sampling points. The result is a mesh that smoothly interpolates the geometric constraints given by CO (represented by Ca,Cb) and CP (represented by L1 − Lj ) (Fig. 1b). (a) Centerline projection (b) Interpolation scheme
We used MeVisLab 2.1 as prototyping environment, since it wraps the
Visualization Toolkit (VTK) whose mesh processing and visualization capabilities
form the basis of our implementation. As input data seven different polygonal
models were derived from different imaging devices (CTA/MRA) during
clinical standard procedures. The centerlines of the parent vessels were calculated
using VMTK and used as input for the generation of the ostium contours [
The resulting ostium surfaces are smooth and bend natural according to the underlying parent vessel and the aneurysm-vessel transition area (Fig. 1b,c). Medical experts found them to be plausible and correctly related to the particular morphologic situation. Using this ostium surface, the next step is to generate an aneurysm-specific coordinate system that includes the semantic decomposition of the aneurysm (ostium, neck, dome) and special geometric landmarks and axes. This coordinate system will form the basis for adaptable visualizations and support the visual exploration process.
Acknowledgement. This work has been funded by the federal state of SaxonyAnhalt in the scope of the MOBESTAN project (5161AD/0308M).