Clinical case
Patient-specific image data (Fig. 1) were acquired by 3D rotational angiography performed on a biplane Philips Allura Xper FD20 system (Philips Medical Systems B.V., Best, the Netherlands). The data were segmented with open-source software 3DSlicer 4.8.1 (http://www.slicer.org) and further post-processed with the Vascular Modeling Toolkit 1.3 (http://www.vmtk.org) [19]. Aneurysm shape and dimensions are presented in Fig. 2.
The site of the aneurysm is a special anatomical site in the human body, as it is the only region where two vessels (vertebral arteries) merge to become one vessel (basilar artery). This causes disturbed flow that may play a role in aneurysm development. From a clinical point of view, only one vessel is needed (and there are norm variants where only one vertebral artery is present). To reduce the probability of disturbed flow, it is a common clinical practice to occlude the smaller vertebral artery just before this merging point. Stenting of both arteries would result in the two stents touching each other in the central part of the basilar artery as they cannot be woven into each other. Such a scenario showed to result in major thromboembolic complications as it usually prevents a re-endothelialization of these touching stent parts.
Since the studied giant fusiform aneurysm could not be treated with a single flow-diverter stent, three flow-diverter SILK stents (Balt Extrusion, Montmorency, France) of 4.5 × 40, 5 × 40 and 5.5 × 40 mm, were deployed in series along the basilar and right vertebral arteries in a standard telescopic technique with an overlap of about 30%, i.e., at least 1 cm between the three single stents. In detail, first the distal stent was deployed using a conventional technique. However, after deployment, the central wire was not removed, but a microcatheter again advanced through the first flow-diverter stent, and a second flow-diverter stent was deployed in an overlapping manner, with an overlap of about 1 cm. This procedure was repeated again to cover the full length of the aneurysm. Additionally, the left vertebral artery was occluded endovascularly.
As our experiments were conducted retrospectively with permanently anonymized patient data, our local ethics committee deemed the study exempt from the requirement for approval.
Virtual stenting
Geometrical models of the clinically used flow-diverter stents were reconstructed according to manufacturer specifications and deployed using a fast virtual stenting technique [20], which was previously validated on a set of real clinical cases [21, 22]. The deployment procedure started from computing a centerline from the right vertebral artery along the basilar artery. Then, the flow-diverter models were deployed one by one in series, starting from the largest one to the least. The overlapping between two neighboring stents was about 30% of the stent length. To estimate malapposition, for each overlapping region between two neighboring stents a set of five cross sections was used. The cross sections were equally distributed along the overlapping region. For each cross section, an area between the outer and inner stents was measured. Then the measured area was divided by the lumen area of the inner stent. The observed malapposition between d = 5.5 mm and d = 5 mm flow-diverter stents ranges between 10 and 12%. A range of 11 to 12% occurred between d = 5 mm and d = 4.5 mm flow-diverters. The result of the virtual deployment is presented in Fig. 2 (at the right). The final deployment was carefully inspected by three experienced neurointerventionalists and showed a sufficient agreement with the real deployment.
Numerical simulations
Since patient-specific flow data were not available for the studied case, a realistic velocity curve was scaled to match an average flow rate of 100 ml/min, which is in physiological range of the vertebral arteries [23]. For the pretreatment setting, the flow rate in the left and right vertebral arteries was assumed to be equal. The maximum Reynolds numbers (Re) for the left and right vertebral arteries were 227 and 251, respectively; the average Re numbers were 151 and 160, respectively. A plug velocity profile was imposed at both inlets. Since inlet segments of the left and right vertebral arteries had sufficient lengths, realistic velocity profiles were developed in the aneurysm inflow zone. The free outflow condition was imposed at the outlet. The vessel wall was assumed rigid. Additionally, for the treated setting a zero inlet flow rate was imposed at the inlet of the left vertebral artery, since the left vertebral artery was occluded during the treatment.
Blood was considered as a Newtonian fluid with a dynamic viscosity \(\mu\) of 3.5 mPas and density \(\rho\) of 1050 kg/m3. A computational mesh was generated using snappyHexMesh tool from the OpenFOAM CFD Toolbox version 5.0 (OpenFOAM Foundation Ltd, London, UK). The mesh consisted primarily of hexahedral elements with a base size of 0.125 mm both for the untreated case and for the case with telescopic stenting. Additional refinement procedure was conducted to ensure a precise evaluation of the velocity gradients near the vessel wall and near the flow-diverters braiding [24]. To ensure grid independency, a mesh doubling test was carried out for both cases. The simulations were conducted with different base element sizes: 1 mm, 0.5 mm, 0.25 mm and 0.125 mm. The simulation results for the base element sizes of 0.25 and 0.125 mm differ only on about 1–2%. No significant change in the results was found, proving the numerical solution was accurate. The final mesh size was 5 million elements for the untreated case and 27 million elements for the case with flow-diverters.
Duration of the cardiac cycle, \(T\) was set to 1 s. The Navier–Stokes equations for an incompressible fluid and continuity equation were solved using PIMPLE (merged PISO-SIMPLE) algorithm with OpenFOAM CFD Toolbox version 5.0. PIMPLE algorithm ensures numerical stability at higher time discretization steps and allows computation even in the cases where Courant number is greater than 1. A Gauss linear scheme was used for spatial discretization, whereas a Crank–Nicolson scheme was used for a temporal discretization. Transient CFD simulations before and after the treatment were performed under physiologically relevant pulsatile flow conditions. A time period of seven cardiac cycles was sufficient to reach a converged periodic solution with a time discretization step of 1 ms. The numerical results for the last (seventh) cardiac cycle were used for post-processing. An open-source package for scientific visualization ParaView 5.0.1 (Kitware, New York, USA) was used for visualization of the velocity field, whereas in-house developed software was employed for the calculation of hemodynamic parameters.
Hemodynamic parameters
For correct comparison between the cases, only the aneurysm volume itself was considered excluding the flow-diverters. The following hemodynamic parameters were analyzed before and after the treatment.
A flow reduction \(R\):
$$R = \frac{{U_{\text{avg}}^{\text{pre}} - U_{\text{avg}}^{\text{post}} }}{{U_{\text{avg}}^{\text{pre}} }} \cdot 100\% ,$$
(1)
where \(U_{\text{avg}}^{\text{pre}}\) and \(U_{\text{avg}}^{\text{post}}\) are space-averaged velocities in the aneurysm before and after the treatment, respectively.
Wall shear stress (WSS) distribution:
$$\overrightarrow {\text{WSS}} = \mu \frac{{{\text{d}}u}}{{{\text{d}}n}},$$
(2)
where \(\frac{{{\text{d}}u}}{{{\text{d}}n}}\)—velocity gradient along the normal to the vessel wall.
Since WSS changes during the cardiac cycle, the additional parameters were employed to characterize the shear stress acting on the vessel wall-TAWSS (time-averaged WSS) and TAWSSV (time-averaged WSS vector), which are defined as follows:
$${\text{TAWSS}} = \frac{1}{T}\mathop \smallint \limits_{0}^{T}\ \left| {\overrightarrow {\text{WSS}} } \right|{\text{d}}t,$$
(3)
$${\text{TAWSSV}} = \frac{1}{T}\left| {\mathop \smallint \limits_{0}^{T} \overrightarrow {\text{WSS}} \;{\text{d}}t} \right|.$$
(4)
TAWSS shows the average magnitude of WSS during the cardiac cycle, whereas TAWSSV characterizes the magnitude of the resulting shear stress vector.
Also, the oscillatory shear index (OSI) was used for evaluation of rate of change in shear stress vector:
$${\text{OSI}} = \frac{1}{2}\left\{ {\frac{{\left| {\mathop \smallint \nolimits_{0}^{T} \overrightarrow {\text{WSS}} \;{\text{d}}t} \right|}}{{\mathop \smallint \nolimits_{0}^{T} \left| {\overrightarrow {\text{WSS}} } \right|{\text{d}}t}}} \right\},$$
(5)
where OSI of 0.5 corresponds to completely oscillatory WSS, whereas OSI of 0 corresponds to unidirectional. Additionally, to characterize thrombogenic conditions in the aneurysm, the relative residence time (RRT) was calculated as follows [25, 26]:
$${\text{RRT}} = \frac{1}{\text{TAWSSV}}.$$
(6)
RRT represents a residence time of blood near the vessel wall.
