1. Introduction
A cerebral aneurysm is a lesion that is presented as a focal bulging in blood vessels, it has a prevalence of about 3% in the population without underlying health conditions [
1] and is typically located at a region of the brain known as the circle of Willis [
2]. The most severe outcome of this lesion is when the aneurysm ruptures leading to a subarachnoid hemorrhage (SAH) [
3,
4], which carries a mortality rate exceeding 40% [
5].
The serious consequences that a cerebral aneurysm may produce, along with the high-complexity treatments the patient can receive, as neurosurgery, create the need for finding a method to predict the rupture risk of a cerebral aneurysm to make a better decision as to whether the patient needs a complex treatment or not. Nowadays, that method does not exist, but it is known that the interaction between the blood flow and artery walls is the key factor in explaining aneurysm formation, as well as its development and rupture [
6].
It is thought that the wall shear stress (WSS) is one of the most important hemodynamic factors to study, where zones with high WSS levels would be more prone to suffer from an aneurysm formation [
7,
8,
9], while zones with low WSS have been related with the development and rupture of an already initiated lesion, finding low-stress levels in those sites where the rupture is produced at the aneurysm dome [
10,
11], although high WSS levels combined with other factors, could also lead to rupture [
12].
Once the aneurysm is initiated, it can have different outcomes, it can rupture young, develop increasing its size to rupture, or be kept in time [
13]. The size of an aneurysm does not evolve at a constant rate [
14], going through periods where it does not change, and other periods where it can grow, even with different growth rates between ruptured and unruptured aneurysms [
15]. It has been shown that aneurysms growth can produce significant morphological changes from at least 23 weeks [
16].
Although it may be intuitive to think that an aneurysm will rupture when it is large, it has also been observed that small aneurysms suffer from ruptures [
17], which suggests that not only its size is a risk factor. In addition to health conditions such as hypertension or trauma to explain why small aneurysms can rupture, it must be considered that the aneurysm is thinner than the artery [
18], and contrary to what could be thought, when the aneurysm grows, it does not necessarily decrease its wall thickness [
19] because it is a living structure that can repair itself.
There are several studies modifying hemodynamic parameters to reflect different health conditions that might affect the rupture risk of an aneurysm, and less common are studies where geometrical factors are modified. Sun et al. [
20] have shown that when an aneurysm grows and its thickness decreases, the rupture risk increases due to a reduction of the WSS and an increment in the Von Mises stress at the aneurysm, and even keeping a constant wall thickness, the maximum Von Mises stress still increases. Nath et al. [
21] studied spherical aneurysm geometries using structural simulations. They found that at the beginning, the aneurysm has a reduction in its maximum Von Mises stress, but then it increases along with the aneurysm size, meaning a higher rupture risk. Other works have used cylindrical/spherical membranes subjected to 1D enlargement to model the development of an aneurysm where the mechanical response is attributed to collagen and elastin [
22].
Those previous works have used a uniform distribution of thickness over the aneurysm dome, however, real aneurysms do not have a uniform thickness distribution, and this can be related to stress concentrations [
23] or aneurysm expansion for abdominal ones [
24]. Nowadays, even with the current medical technology, wall thickness remains a difficult parameter to obtain from patients without performing surgery or removing the aneurysm, especially for cerebral aneurysms. Furthermore, by removing an aneurysm, it has been noted that using a non-uniform thickness distribution in simulations show much higher stress values in the rupture site when compared to a configuration with constant wall thickness [
25].
In the present work, the effect of geometrical parameters is studied, specifically, the effects of the aneurysm wall thickness and its size on its rupture risk, separately varying each parameter for better accounting how each parameter affects the rupture risk of cerebral aneurysms. Six patient-specific geometries are chosen and boundary conditions representing healthy conditions for an adult patient are used. Due to the importance of accounting for blood and wall interaction in simulations for an accurate prediction of patient-specific hemodynamics [
24,
25], a fluid-structure interaction (FSI) approach is used. Besides, as differences in up to 30% in displacement have been observed when comparing two-way, and one-way FSI simulations [
28], the former is used.
Additionally, cases where the aneurysm is virtually removed from lateral geometries are included, with the aim of simulating a state before the lesion initiation.
6. Conclusion
Using computational simulations, this research could study the past evolution of aneurysms and explore potential future developments as well, a task that would otherwise not be possible to perform due to ethical concerns. This study demonstrated that for the studied group, as cerebral aneurysms grow, they trigger significant alterations in hemodynamic and structural parameters which reflect a greater rupture risk environment. For instance, an unruptured geometry can decrease its average WSS by up to 36% when it grows and increase its Von Mises stress by almost 30%. Furthermore, the use of FSI simulations made it possible to assess the effects of aneurysms wall thinning keeping their size unaltered, revealing that even if the aneurysm does not grow, it could be on a high rupture risk state because of a thin wall which is prone to a structural failure since its average Von Mises stress can increase in up to 260%. The use of simulations also enabled the study of the lateral arteries when the aneurysm is removed, finding that for both ruptured aneurysms, the TAWSS was reduced in 75% at the rupture point, whereas for unruptured aneurysms, although they also have a reduction, it was not as high. It is then concluded that even for a healthy patient, either of the geometrical modifications presented in this study implies a significant increment on the rupture risk of the studied cerebral aneurysms, and that there could be an indication of the rupture point of the aneurysms when its TAWSS reduction is obtained, although more samples are needed to give more statistically significant conclusions about this and the geometrical factors effect.
Figure 1.
Selected geometries for the study. With previous rupture, RG-1 to RG-3 at the top, and without previous rupture, UG-1 to UG-3 at the bottom.
Figure 1.
Selected geometries for the study. With previous rupture, RG-1 to RG-3 at the top, and without previous rupture, UG-1 to UG-3 at the bottom.
Figure 2.
Example of the main steps of the scaling process. a) Neck plane, b) aneurysm removal, c) T-spline body, d) scaled body, e) solid body, and f) aneurysm joined to the artery.
Figure 2.
Example of the main steps of the scaling process. a) Neck plane, b) aneurysm removal, c) T-spline body, d) scaled body, e) solid body, and f) aneurysm joined to the artery.
Figure 3.
Unruptured geometries with a) I, b) II and c) III sizes.
Figure 3.
Unruptured geometries with a) I, b) II and c) III sizes.
Figure 4.
Ruptured geometries with a) IV, b) III, c) II, and d) I sizes.
Figure 4.
Ruptured geometries with a) IV, b) III, c) II, and d) I sizes.
Figure 5.
Created surfaces for the artery (in green), with a thickness of 0.35 [mm] and the aneurysm (in gray) with thicknesses of 0.35, 0.2, and 0.10 [mm].
Figure 5.
Created surfaces for the artery (in green), with a thickness of 0.35 [mm] and the aneurysm (in gray) with thicknesses of 0.35, 0.2, and 0.10 [mm].
Figure 6.
Average WSS variation at the aneurysm at diastole (DWSS) and systole (SWSS) in mesh convergence test based on mesh density with linear and quadratic tetrahedral element.
Figure 6.
Average WSS variation at the aneurysm at diastole (DWSS) and systole (SWSS) in mesh convergence test based on mesh density with linear and quadratic tetrahedral element.
Figure 7.
Temporal evolution of the average WSS at the aneurysm for RG1 and UG-1 for each wall thickness.
Figure 7.
Temporal evolution of the average WSS at the aneurysm for RG1 and UG-1 for each wall thickness.
Figure 8.
Evolution of the average displacement at the aneurysm with respect to the aneurysm wall thickness.
Figure 8.
Evolution of the average displacement at the aneurysm with respect to the aneurysm wall thickness.
Figure 9.
Evolution of the average Von Mises stress at the aneurysm with respect to the aneurysm wall thickness.
Figure 9.
Evolution of the average Von Mises stress at the aneurysm with respect to the aneurysm wall thickness.
Figure 10.
Displacement contours at systole for RG-1 using a) thick, b) medium, and c) thin aneurysm wall thickness.
Figure 10.
Displacement contours at systole for RG-1 using a) thick, b) medium, and c) thin aneurysm wall thickness.
Figure 11.
Displacement contours at systole for UG-1 using a) thick, b) medium, and c) thin aneurysm wall thickness.
Figure 11.
Displacement contours at systole for UG-1 using a) thick, b) medium, and c) thin aneurysm wall thickness.
Figure 12.
Temporal evolution of the average WSS at the aneurysm for original aneurysms a) with rupture and b) without rupture, and their size variations.
Figure 12.
Temporal evolution of the average WSS at the aneurysm for original aneurysms a) with rupture and b) without rupture, and their size variations.
Figure 13.
Evolution of the average WSS at the aneurysm with respect to the AR.
Figure 13.
Evolution of the average WSS at the aneurysm with respect to the AR.
Figure 14.
Evolution of the average displacement at the aneurysm with respect to the AR.
Figure 14.
Evolution of the average displacement at the aneurysm with respect to the AR.
Figure 15.
Evolution of the average Von Mises stress at the aneurysm with respect to the AR.
Figure 15.
Evolution of the average Von Mises stress at the aneurysm with respect to the AR.
Figure 16.
Von Mises stress distribution contours during systole for RG-3 geometry and its a) IV, b) III, c) II, and d) I sizes.
Figure 16.
Von Mises stress distribution contours during systole for RG-3 geometry and its a) IV, b) III, c) II, and d) I sizes.
Figure 17.
Von Mises stress distribution contours during systole for UG-3 geometry and its a) III, b) II, and c) I sizes.
Figure 17.
Von Mises stress distribution contours during systole for UG-3 geometry and its a) III, b) II, and c) I sizes.
Figure 18.
TAWSS distribution contours for geometries with the removed aneurysm. In a) RG-1, b) UG-1, c) RG-2, and d) UG-2.
Figure 18.
TAWSS distribution contours for geometries with the removed aneurysm. In a) RG-1, b) UG-1, c) RG-2, and d) UG-2.
Figure 19.
Pressure distribution contours during systole for geometries with the removed aneurysm. In a) RG-1, b) UG-1, c) RG-2, and d) UG-2.
Figure 19.
Pressure distribution contours during systole for geometries with the removed aneurysm. In a) RG-1, b) UG-1, c) RG-2, and d) UG-2.
Figure 20.
Blood flow at the aneurysm region for geometries a) RG-1 and b) UG-1.
Figure 20.
Blood flow at the aneurysm region for geometries a) RG-1 and b) UG-1.
Figure 21.
Blood flow at the aneurysm region for UG-3 a) I, b) II, and c) III sizes.
Figure 21.
Blood flow at the aneurysm region for UG-3 a) I, b) II, and c) III sizes.
Figure 22.
Location of the imported aneurysm on the TAWSS distribution.
Figure 22.
Location of the imported aneurysm on the TAWSS distribution.
Table 1.
Rupture status, type, location and total volume of each geometry.
Table 1.
Rupture status, type, location and total volume of each geometry.
Geometry |
Rupture |
Type |
Location |
Volume [mm3] |
RG-1 |
Yes |
Lateral |
ACA |
57 |
RG-2 |
Yes |
Lateral |
ICA |
583 |
RG-3 |
Yes |
Terminal |
ACA |
121 |
UG-1 |
No |
Lateral |
ICA |
775 |
UG-2 |
No |
Lateral |
ICA |
831 |
UG-3 |
No |
Terminal |
MCA |
1150 |
Table 2.
Displacement and Von Mises stress evolution obtained for the mesh tests and their percentage difference with respect to the 0.15 mm element size.
Table 2.
Displacement and Von Mises stress evolution obtained for the mesh tests and their percentage difference with respect to the 0.15 mm element size.
Element size [mm] |
Displacement [mm] |
D015[%] |
Von Mises stress [MPa] |
D015 [%] |
0.50 |
0.39 |
2.63 |
0.13 |
-7.14 |
0.40 |
0.39 |
2.63 |
0.14 |
0.00 |
0.35 |
0.39 |
2.63 |
0.13 |
-7.14 |
0.30 |
0.39 |
2.63 |
0.14 |
0.00 |
0.25 |
0.39 |
2.63 |
0.14 |
0.00 |
0.20 |
0.39 |
2.63 |
0.14 |
0.00 |
0.15 |
0.38 |
- |
0.14 |
- |
Table 3.
Percentage differences for the Von Mises stress at the aneurysm with respect to the 0.35 [mm] wall thickness.
Table 3.
Percentage differences for the Von Mises stress at the aneurysm with respect to the 0.35 [mm] wall thickness.
Case |
Aneurysm thickness [mm] |
Average Von Mises stress difference [%] |
Diastole |
Systole |
RG-1 |
0.2 |
75.9 |
79.5 |
0.1 |
244.8 |
263.6 |
UG-1 |
0.2 |
82.5 |
87.5 |
0.1 |
200.0 |
212.5 |
Table 4.
Percentage differences for the displacement at the aneurysm with respect to the 0.35 [mm] wall thickness.
Table 4.
Percentage differences for the displacement at the aneurysm with respect to the 0.35 [mm] wall thickness.
Case |
Aneurysm thickness [mm] |
Average displacement difference [%] |
Diastole |
Systole |
RG-1 |
0.2 |
57.1 |
50.0 |
0.1 |
142.9 |
120.0 |
UG-1 |
0.2 |
-4.2 |
-5.4 |
0.1 |
-11.3 |
-11.8 |
Table 5.
Percentage variation of the WSS at the aneurysm dome with respect to the original size.
Table 5.
Percentage variation of the WSS at the aneurysm dome with respect to the original size.
Case |
Size variation |
Area-averaged WSS difference [%] |
Diastole |
Systole |
RG-2 |
III |
57.5 |
49.8 |
II |
123.1 |
116.7 |
I |
275.9 |
261.1 |
RG-3 |
III |
62.7 |
60.4 |
II |
122.7 |
107.8 |
I |
237.7 |
205.2 |
UG-2 |
III |
-35.7 |
-37.2 |
II |
-18.2 |
-17.9 |
UG-3 |
III |
-4.6 |
-0.4 |
II |
-31.4 |
19.0 |
Table 6.
Percentage variation of the average displacement at the aneurysm dome with respect to the original size.
Table 6.
Percentage variation of the average displacement at the aneurysm dome with respect to the original size.
Case |
Size variation |
Average displacement difference [%] |
Diastole |
Systole |
RG-2 |
III |
-21.1 |
-17.9 |
II |
-29.3 |
-21.4 |
I |
-19.0 |
-11.1 |
RG-3 |
III |
-37.1 |
-38.6 |
II |
-54.6 |
-53.6 |
I |
-51.3 |
-48.5 |
UG-2 |
III |
57.1 |
60.0 |
II |
23.8 |
26.7 |
UG-3 |
III |
-21.3 |
-22.8 |
II |
9.6 |
9.6 |
Table 7.
Percentage variation of the average Von Mises stress at the aneurysm dome with respect to the original size.
Table 7.
Percentage variation of the average Von Mises stress at the aneurysm dome with respect to the original size.
Case |
Size variation |
Average Von Mises stress difference [%] |
Diastole |
Systole |
RG-2 |
III |
-19.5 |
-17.2 |
II |
-31.7 |
-31.3 |
I |
-17.1 |
-15.6 |
RG-3 |
III |
-17.1 |
-21.8 |
II |
-22.9 |
-29.1 |
I |
-11.4 |
-10.9 |
UG-2 |
III |
28.0 |
25.0 |
II |
8.0 |
8.7 |
UG-3 |
III |
28.9 |
29.0 |
II |
7.9 |
9.7 |
Table 8.
Comparison between pre-aneurysm and aneurysm states in the TAWSS variation for lateral geometries.
Table 8.
Comparison between pre-aneurysm and aneurysm states in the TAWSS variation for lateral geometries.
Case |
TAWSS [Pa] |
RG-1 |
RG-2 |
UG-1 |
UG-2 |
Aneurysm |
20.7 |
3.6 |
7.8 |
7.7 |
Pre-Aneurysm |
83.2 |
13.9 |
14-2 |
11.5 |
Difference |
-75.1% |
-74.1% |
-45.1% |
-33.0% |