**6. Simulations and results**

### **6.1. Blood flow simulation**

We have performed comparative tests against FLUENT software <sup>2</sup> both in 2D and 3D space. In this case, we mostly aim at validating numerical accuracy of the DEC method rather than the ability to precisely describe the actual blood flow features near aneurysm, due to the present difficulties in *in vivo* analysis of velocity, particularly in the small cerebral vessels. Each group of the comparison between DEC and FLUENT consists of blood flow simulation on several identical meshes with the same geometry but different resolution.

The first group of simulations using a 2D aneurysm model (generated from the profile of a patient-specific geometry) is performed on three meshes composed of 210177, 19753 and 2160 triangles respectively. The contours of velocity magnitude and streamlines computed by DEC and FLUENT are shown in Figure 10. The unit of velocity displayed in all the figures is *mm*/*s*

<sup>2</sup> FLUENT is a commercial product of ANSYS (http://ansys.com), widely used in industries.

#### 18 Will-be-set-by-IN-TECH 240 Aneurysm

**Figure 10.** Comparison of velocity field on 2D aneurysm model. The contours of velocity magnitude and streamlines in the whole region (the first two rows) and in the sac (the last two rows) are computed by DEC and FLUENT on the identical meshes of three different resolution.

by default. Besides, we plot the profiles of the velocity magnitude over two lines across the inlet and the aneurysm respectively in Figure 11. When the mesh resolution decreases by 10 times, the contours and profiles are almost the same, and show the similarity between the two methods. In addition, the streamlines show a strong agreement for flow patterns and vortex structures in terms of the positions of the vortex centers. Only when we reduce the mesh resolution by 100 times, the result computed by DEC loses some detail information inside the sac; less fluid flows into the sac, and the vortex inside the sac is missing. But the main features of the velocity field still remain the same, such as the flow pattern (characterized by the streamlines) and the variation of the velocity field in space (characterized by the profiles).

**Figure 11.** Comparison of velocity profiles over two lines across the inlet and aneurysm

The second group of simulations is performed similarly on two meshes of a 3D patient-specific aneurysm with a wide neck. In Figure 12, the streamlines show the similar movement of blood. In both cases there is a low-speed region surrounded by high-speed flows observed on slice 1, which represents a local vortex in this region. There are two obvious vortices in the FLUENT result, reflected both by the low-speed regions observed on slice 2 and the swirls of streamlines, but in the DEC result on the coarser mesh, the smaller vortex is not that obvious. These results show that the DEC method can capture large-scale flow structures, while small-scale differences exist between the two approaches. Considering both accuracy and computational efficiency, we choose the lower-resolution mesh of 34029 tetrahedra for the simulation of coil embolization, and the computational time of simulating 1*s* real time is 44*s* on an Intel i7 3.33*GHz* processor.

#### **6.2. Real-time simulation of coil embolization**

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**Figure 10.** Comparison of velocity field on 2D aneurysm model. The contours of velocity magnitude and streamlines in the whole region (the first two rows) and in the sac (the last two rows) are computed

by default. Besides, we plot the profiles of the velocity magnitude over two lines across the inlet and the aneurysm respectively in Figure 11. When the mesh resolution decreases by 10 times, the contours and profiles are almost the same, and show the similarity between the two methods. In addition, the streamlines show a strong agreement for flow patterns and vortex structures in terms of the positions of the vortex centers. Only when we reduce the mesh resolution by 100 times, the result computed by DEC loses some detail information inside the sac; less fluid flows into the sac, and the vortex inside the sac is missing. But the main features of the velocity field still remain the same, such as the flow pattern (characterized by the streamlines) and the variation of the velocity field in space (characterized by the profiles).

by DEC and FLUENT on the identical meshes of three different resolution.

Triangles 210*K* 20*K* 2*K*

DEC

FLUENT

DEC

FLUENT

We predict the embolization outcome for an intracranial aneurysm with a small sac of volume 132.1*mm*<sup>3</sup> (usually the sac volume of an intracranial aneurysm varies from 100 to 1000*mm*3) and a wide neck of dimension 7.0*mm*. From the clinical experience, the final coil packing density is around 25%, *i.e.*, the porosity *ϕ* is around 75%. In Figure 13, we show the blood flow without coils, with 10% and 25% volume filled with coils. The velocity magnitude contours are compared on two sections, crossing the neck and the sac respectively. From the comparison on the neck section, we can see that every incremental increase in coil packing density is accompanied by a decrease in cross-neck flow rate. Additionally on the sac section, after inserting the first coil, the velocity magnitude over the whole sac region has been reduced, which creates a favorable hemodynamic environment for the deployment of the following coils.

Then we show the inverse influence of the blood flow on the coil deployment. Figure 14 displays the simulated process of placing the first coil into the small aneurysm. After the catheter is advanced in the vascular network to reach the position of the aneurysm, the coil is delivered through the catheter and inserted into the aneurysm. The contact among the catheter, the coil and the vessel wall, as well as the interaction of the blood flow are all included in this simulation. The pre-computation of blood velocity covering one cardiac cycle at 5 levels of coil density is accomplished in less than 5 minutes, and then using the pre-computed velocity field, the simulation of coil deployment is performed in real time.

**Figure 12.** Comparison of velocity field on the 3D patient-specific aneurysm model. The first row displays the geometry of the model and the position of two slices chosen for comparing velocity magnitude contours. In the following rows, streamlines and contours of velocity magnitude are computed by DEC and FLUENT on the identical meshes, which are mesh 1 of 55711 tetrahedra and mesh 2 of 34029 tetrahedra.

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**Figure 12.** Comparison of velocity field on the 3D patient-specific aneurysm model. The first row displays the geometry of the model and the position of two slices chosen for comparing velocity magnitude contours. In the following rows, streamlines and contours of velocity magnitude are computed by DEC and FLUENT on the identical meshes, which are mesh 1 of 55711 tetrahedra and

DEC mesh 1

FLUENT mesh 1

> DEC mesh 2

FLUENT mesh 2

mesh 2 of 34029 tetrahedra.

streamlines slice 1 slice 2

**Figure 13.** Coil embolization of a small aneurysm. The velocity magnitude on two sections (a) before the embolization (b), after the first coil deployed (c), and after the final coil deployed (d).

**Figure 14.** Simulation of coil embolization: (a) The catheter (black) reaches at the aneurysm neck through vessels. (b)-(f) The first coil (silver) is delivered by the catheter and inserted into the aneurysm. The colorful volume displays the periodically varying velocity field.
