Relationships between geometrical parameters and mechanical properties for a helical braided flow diverter stent

Abstract

BACKGROUND:

Although flow diversion is a promising procedure for aneurysm treatment, the safety and efficacy of this strategy have not been sufficiently characterized. Both mechanical properties and flow reduction effects are important factors in the design of an optimal stent.

OBJECTIVE:

We aimed to clarify the contributions of strut size and pitch to the mechanical properties (radial stiffness and longitudinal flexibility) and geometric characteristics (porosity and pore density) related to flow reduction effects.

METHODS:

Crimping and bending behaviors of the stents were simulated with the finite element method. The relationships between the mechanical properties and geometric characteristics were investigated by changing the strut size and pitch.

RESULTS:

Within the porosity range of 79–82%, the radial stiffness of the stent was similarly influenced by either the strut size or pitch. However, the longitudinal flexibility tended to be influenced more by strut size than by pitch.

CONCLUSIONS:

Adjusting the strut size rather than the pitch can change the mechanical properties while minimizing the change in porosity or pore density related to flow reduction effects.

Keywords

Intracranial aneurysm radial stiffness longitudinal flexibility finite element analysis

1. Introduction

Intracranial aneurysm (IA) is a cerebrovascular disorder comprised of a saccular or fusiform bulging weak area in the wall of a cerebral artery. Approximately 1–5% of the adult population harbors one or more IAs [1, 2], and the annual rupture rate is 1–2% [1]. IA rupture causes subarachnoid hemorrhage in most cases and is associated with a 30 day mortality rate of 45%. Furthermore, 30% of the survivors will have moderate-to-severe disabilities [1].

In recent years, the flow diversion effects of stent placement have received attention, and stent-alone treatment, called “flow diversion”, has been developed as an alternate procedure to surgical clipping or endovascular coil embolization [3]. The stent used in this procedure is called a “flow diverter stent”. The flow diverter stent diverts the flow entering the aneurysm and induces thrombosis inside the aneurysm. Furthermore, it serves as a scaffold for the development of endothelial and neointimal tissue across the aneurysmal neck [4]. Although flow diversion is a promising procedure for aneurysms that are otherwise difficult to treat with coils, the safety and efficacy of this procedure have not been completely characterized. Complications that can occur after this procedure include incomplete occlusion [5], non-occlusion [6], in-stent stenosis [7], in-stent thrombosis [5, 7, 8], side-branch occlusion [4, 9], and rupture [3, 4, 5, 8, 10].

Flow diverter stents have been studied in vivo [11, 12, 13], in vitro [14, 15, 16], and in silico [17, 18, 19]. The porosity and pore density of the stents are important factors for flow diversion efficacy. Porosity is defined as the proportion of the open metal-free area to the total stent area (%). Pore density is defined as the number of pores per area (pores/mm ${}^{2}$ ) [20]. According to previous reports, the reduction effects of velocity in the IA increase with decreasing porosity and with increasing pore density of the stent [21]. To form endothelial and neointimal tissue on the stent, lower porosity is better [4]. However, excessively low porosity or high pore density can result in an embolus to the side branch.

The mechanical properties (e.g., radial stiffness or longitudinal flexibility) of the flow diverter stent affect the technical success of deployment and outcomes. Radial stiffness is defined as how much the diameter of a stent is reduced by the application of external pressure. On the other hand, longitudinal flexibility is defined as flexibility along its longitudinal axis. Although insufficient radial force can cause extra-aneurysmal thrombosis and stroke, excessive radial force can damage the artery [22]. Higher longitudinal flexibility is desirable to conform to tortuous arteries [22].

Typical commercially available flow diverters are constructed of metallic helical wires that are braided. Some stents in other fields, such as Wallstent (Boston Scientific, Marlborough, MA, USA), which also have a helical braided structure, have been studied. Ziao et al. [23] showed that braided wire stents have lower radial stiffness than welded ones. Kim et al. [24] experimentally and numerically investigated the mechanical behavior of fabricated stents and found that, as the pitch decreased or the number of wires increased, the stiffness of the stents to compressive loading increased. They also demonstrated that stents with smaller pitch were likely to retain the openness of the cross-sectional shape. Ni et al. [25] investigated the effects of the structural parameters on the mechanical properties and found that, as wire size increased, radial stiffness increased.

As described above, regarding helical braided stents (HBSs), although the relationships between the specifications and the mechanical properties are active research areas, the problems related to their complications have not yet been solved. To develop a safer and more effective flow diverter stent, further detailed knowledge about them is necessary. In addition, the studies cited above focused only on the mechanical properties. However, flow reduction effects, which affect the outcome, should also be considered in the design of flow diverter stents.

The objective functions of HBSs, including flow reduction (diversion) effects and mechanical properties, are determined by design variables, such as: (1) diameter, (2) length, (3) strut pitch, (4) strut number, (5) strut material, (6) strut size, and (7) braiding pattern. The porosity is determined by (1), (3), (4), and (6), whereas the pore density is determined by (1), (3), and (4). Since some of the objective functions are interdependent, stent design is complicated. Indeed, it is difficult to design a stent that satisfies the best conditions for all objective functions. For example, strut pitch and strut size can be modified to adjust the flow reduction effects, but these changes also affect the mechanical properties of the stent, such as radial stiffness or longitudinal flexibility. Therefore, if the relationships between the flow reduction effects and mechanical properties were clarified, stents could be designed considering both.

The goal of this study was to clarify the contributions of strut size and pitch to the mechanical properties and geometric characteristics (porosity and pore density) related to flow reduction effects.

Table 1
Specifications of fully expanded flow diverters (external diameter of 4.5 mm)

Name	Strut size, $d$ [ $\mu$ m]	Pitch, $p$ [mm]	Porosity [%]	Pore density [pores/mm ${}^{2}$ ]
Pore1-d30	30	4	66.1	20.6
Pore2-d30	30	8	80.3	10.3
Pore3-d30	30	12	84.8	6.9
Pore2-d40	40	8	74.2	10.4
Pore2-d50	50	8	68.3	10.4

Figure 1.

Definitions of design variables ( $D$ : diameter of stent, $d$ : strut size, $p$ : pitch of strut, $L$ : length of stent, $\varphi$ : strut angle) and the length from one strut crossing point to the next strut crossing point along a strut ( $c$ ).

2. Materials and methods

2.1 Specifications of the compared stents

Table 1 shows the specifications of the five stents. All stents were composed of 48 struts [3] and had external diameters of 4.5 mm when fully expanded [26]. The struts were braided in the same patterns used in a previous study [27]. To determine the stent specifications, the base stent (“Pore1-d30”), which was based on the Pipeline Embolization Device (PED; Covidien, Irvine, CA), was created by referring to a previous report [26]. Although the PED is composed of two different strut sizes in a 3:1 ratio [26], in the present study, the stents were composed of only one strut size and one material to simplify the comparison and to clarify the relationships between the specifications and mechanical properties. Next, two different stents were designed with the same strut size “ $d$ ” and different pitches “ $p$ ” (see Fig. 1) of 8 and 12 mm (“Pore2-d30” and “Pore3-d30,” respectively). Finally, two other stents were designed with the same pitch as “Pore2-d30” (8 mm) and strut sizes of 40 and 50 $\mu$ m (“Pore2-d40” and “Pore2-d50”, respectively).

2.2 Numerical simulation

To investigate the radial stiffness and longitudinal flexibility of the stents, crimping tests and bending tests were simulated. The finite element analysis solver ABAQUS/explicit (SIMULIA, Providence, RI) was used because all simulations were unstable owing to their nonlinearity with large-deformation and contact phenomena. A circular beam-element mesh was created for the stent geometries. The nominal element size was set to 0.08–0.10 mm, which corresponded to one-third for Pore1-d30 or one-fourth for the other stents of the length from one strut crossing point to the next strut crossing point along a strut (the length “ $c$ ” shown in Fig. 1). Table 3 shows the number of elements for each nominal element size, and Tables 3 and 5 show the results of grid independence studies with Pore1-d30. The element size was chosen because all relative errors from the result with the next smaller element size were $<$ 5%. The penalty method was chosen for the contact algorithm. A friction coefficient value of 0.15 was assigned for the intra-strut contacts [28]. Co-Cr-Ni material properties were applied to all component struts, as shown in Table 5 [27]. The number of elements for each simulation is shown in Table 7.

Table 2
Nominal element size and the number of elements used in the grid independence study

Nominal element	Crimping test		Bending test
size [mm]	Stent	Crimper	Stent
0.1	3456	31624	17281
0.15	2304	7906	11520
0.3	1152	1947	5760

Table 3

Relative error from the result with the next smaller element size on the grid independence test (crimping) [%]

Inner diameter of crimper [mm]	Element size [mm]
	0.1	0.15	0.3
4.5	$-$	0	0
4.45	$-$	0.89	$-$ 22.97
4.40	$-$	0.46	$-$ 14.98
4.35	$-$	0.38	$-$ 7.68
4.30	$-$	0.20	$-$ 2.98
4.25	$-$	$-$ 0.11	$-$ 2.36
4.20	$-$	2.49	6.78
4.15	$-$	$-$ 0.51	5.32
4.10	$-$	0.11	5.00
4.05	$-$	0.06	2.11
4.00	$-$	0.05	4.01

Table 4

Relative error from the result with the next smaller element size on the grid independence test (bending) [%]

Bending angle [ ${}^{\circ}$ ]	Element size [mm]
	0.1	0.15	0.3
0	$-$	0	0
15	$-$	$-$ 0.55	$-$ 0.82
30	$-$	$-$ 0.63	$-$ 0.87
45	$-$	$-$ 0.72	$-$ 0.90
60	$-$	$-$ 0.79	$-$ 0.90
75	$-$	$-$ 0.85	$-$ 0.87

Table 5

Material properties of Co-Cr-Ni wires

Density [g/cm ${}^{3}$ ]		8
Elastic properties	Young’s Modulus [GPa]	206
Elastic properties	Poisson’s ratio	0.26
Plastic properties	0.2% yield stress [GPa]	2.8
Plastic properties	Isotropic hardening slope [GPa]	8.8

Table 6

The number of elements used in the simulations

Name of stent	Crimping test		Bending test
	Stent	Crimper	Stent
Pore1-d30	3,456	31,624	17,281
Pore2-d30	4,608	31,624	11,521
Pore3-d30	4,608	31,624	7,681
Pore2-d40	4,608	31,624	11,521
Pore2-d50	4,608	31,624	11,521

Table 7

Material properties of steel crimpers

Density [g/cm ${}^{3}$ ]		7.9
Elastic properties	Young’s Modulus [GPa]	202
Elastic properties	Poisson’s ratio	0.3

Figure 2.

Configurations before and after crimping (top: before crimping, $t=$ 0 s, crimper inner diameter, $D=$ 4.5 mm; bottom: after crimping, $t=$ 0.1 s, $D=$ 4.0 mm).

Figure 3.

Configurations before and after bending (top: before bending, $t=$ 0 s, bending angle, $\theta=$ 0 ${}^{\circ}$ ; bottom: after bending, $t=$ 0.45 s, $\theta=$ 75 ${}^{\circ}$ ).

Figure 4.

Schematic of the bending test. (A: The top end nodes are coupled with all degrees of freedom. B: The rotation center is located in the center of the stent. C: The bottom end nodes are fixed.)

2.3 Crimping test

The radial crimping tests were performed by applying a uniform inward radial displacement (0.25 mm) to a cylindrical crimper (see Fig. 2). The crimper dimensions were as follows: inner diameter $=$ 4.5 mm, outer diameter $=$ 4.6 mm, axial length $=$ 8 mm, and wall thickness $=$ 0.05 mm. Steel was chosen as the crimper material for simplification on the basis of its properties, as shown in Table 7 [27]. A shell element mesh was created for the crimper geometries. The mesh size was set to 0.06 mm. A friction coefficient value of 0.01 was assigned for the strut-crimper contacts [24]. The stents having the length equal to half the pitch were crimped. The crimper length was always longer than the stent length during the simulations. Because static behavior was desirable, analysis time was determined such that the kinetic energy of all material bodies was much less than their potential (strain) energy during the analysis [24]. The simulation was performed in an analysis time of 0.1 s; specifically, the stent was crimped in 0.1 s. The average contact pressure ( $P_{r}$ ) was calculated by dividing the force orthogonal to the crimper’s inner surface ( $F_{r}$ ) by the stent length ( $L$ ) and the circumference of crimper’s inner surface.

2.4 Bending test

Based on a previous report [29], cantilever bending tests were performed to investigate longitudinal flexibility. The five stents described above with a length of 10 mm were bent (see Fig. 3). One of the stent ends was fixed; the other was subjected to a rotation angle of $\theta$ (see Fig. 4). The angle $\theta$ was rotated from 0 ${}^{\circ}$ to 75 ${}^{\circ}$ . As with the crimping tests, the analysis time was determined such that the kinetic energy of all material bodies was much less than their potential (strain) energy during the analysis [24]. The stent was bent in 0.45 s.

3. Results

3.1 Radial stiffness

Figure 5(a) shows the relationships between the inner diameter of the crimper and the contact pressure ( $P_{r}$ ) for each stent. First, the stents with the same strut size but with different pitches (Pore1-d30, Pore2-d30, and Pore3-d30) were compared. When the three stents were crimped to 4.0 mm, the contact pressure were 0.613 kPa, 0.220 kPa, and 0.087 kPa for Pore1-d30 (pitch: 4 mm), Pore2-d30 (pitch: 8 mm), and Pore3-d30 (pitch: 12 mm), respectively. The pressure decreased as the pitch increased. The same trend was observed by Kim et al. [24] and Ni et al. [25]. Next, the stents with the same pitch but with different strut sizes (Pore2-d30, Pore2-d40, and Pore2-d50) were compared. When the three stents were crimped to 4.0 mm, the pressure were 0.220 kPa, 0.778 kPa, and 1.926 kPa for Pore2-d30 (strut size: 30 $\mu$ m), Pore2-d40 (strut size: 40 $\mu$ m), and Pore2-d50 (strut size: 50 $\mu$ m), respectively. The pressure increased as the strut size increased. The same trend was observed by Ni et al. [25]. Von Mises stress distributions are shown in Fig. 5(b). Von Mises stress became higher in the central section; similarly, the stents with a larger radial stiffness showed higher von Mises stress distributions.

Figure 5.

Results of the crimping tests. (a) Relationships between the inner diameter of the crimper and the contact pressure ( $P_{r}$ ) for each stent. (b) von Mises stress distributions. (c) Relationships between the contact pressure ( $D=$ 4.0 mm) and the porosity for each stent. (d) Relationships between the contact pressure ( $D=$ 4.0 mm) and the pore density for each stent. The strut size does not affect the pore density in accordance with the definition of pore density.

Figure 6.

Results of the bending tests. (a) Relationships between the bending angle and bending moment for each stent. (b) von Mises stress distributions. (c) Relationships between the bending moment (at 75 ${}^{\circ}$ ) and the porosity for each stent. (d) Relationships between the bending moment (at 75 ${}^{\circ}$ ) and the pore density for each stent for each stent. The strut size does not affect the pore density in accordance with the definition of pore density.

The relationships between contact pressure and geometric characteristics (porosity and pore density) are shown in Figs 5(c) and (d). The porosity and pore density were calculated geometrically by assuming that the external diameter of the stent was 4.0 mm. Figure 5(c) shows the relationship between the porosity and contact pressure. The blue curve indicates the values when the strut size was changed, but the strut pitch was kept constant. The red curve indicates the values when the strut pitch was changed, but the strut size was kept constant. The graph shows that the two values described above tended to be correlated within the porosity range of 79% to 82%.

3.2 Longitudinal flexibility

Figure 6(a) shows the relationships between the bending angle and bending moment for each stent. First, the stents with the same strut size but with different pitches (Pore1-d30, Pore2-d30, and Pore3-d30) were compared. When the three stents were bent to 75 ${}^{\circ}$ , the bending moments were 0.023 N $\cdot$ mm, 0.046 N $\cdot$ mm, and 0.108 N $\cdot$ mm for Pore1-d30 (pitch: 4 mm), Pore2-d30 (pitch: 8 mm), and Pore3-d30 (pitch: 12 mm), respectively. The bending moment increased as the pitch increased. Ni et al. [25] observed the same trend. Next, the stents with the same pitch but with different strut sizes (Pore2-d30, Pore2-d40, and Pore2-d50) were compared. When the three stents were bent to 75 ${}^{\circ}$ , the bending moments were 0.046 N $\cdot$ mm, 0.186 N $\cdot$ mm, and 0.500 N $\cdot$ mm for Pore2-d30 (strut size: 30 $\mu$ m), Pore2-d40 (strut size: 40 $\mu$ m), and Pore2-d50 (strut size: 50 $\mu$ m), respectively. The bending moment increased as the strut size increased.

Figure 6(b) shows the von Mises stress distributions on the bending tests. As the pitch increased or strut size increased, the von Mises stress showed high distribution overall. Further, as the pitch decreased, the stress distributed equally along the axial direction. As the pitch increased, the shape of the center part collapsed and formed a denser strut distribution. Figures 6(c) and (d) show the relationships between the bending moment (at 75 ${}^{\circ}$ ) and the geometric characteristics (porosity and pore density). A stent diameter of 4.0 mm, which is simulating the compressed status of the stents, was used in section 3.1 for the calculation of the porosity and pore density. Instead, a stent diameter of 4.5 mm was used in this section to simulate the non-compressed status of the same stents. In Fig. 6(d), the red curve indicates the values when the strut pitch was changed, but the strut size was kept constant. As the porosity decreased, the bending moment also mildly decreased. On the other hand, the blue curve indicates the values when the strut size was changed, but the strut pitch was kept constant. It is noted that the bending moment dramatically increased when porosity decreased. That is, the graph shows that the bending moment, or the longitudinal flexibility, tended to be more influenced by strut size than by strut pitch.

Figure 7.

Theoretical relationships between (a) the diameter of the stent and length of the stent, (b) the diameter of the stent and porosity, and (c) the diameter of the stent and pore density. The stent length is assumed to be 20 mm when the stent is fully expanded.

4. Discussion

In this study, the relationships between the geometric parameters (strut size and pitch) and the mechanical properties (radial stiffness and longitudinal flexibility) were investigated for typical helical braided flow diverter stents. Similar to the results of previous studies, the present study showed that, as strut size increased, radial stiffness increased and longitudinal flexibility decreased. Both flexural and torsional rigidities of cylindrical beams are proportional to the fourth power of the diameter; thus, the results can be explained on the basis of the mechanics of materials. In addition, as the pitch decreased, radial stiffness increased and longitudinal flexibility increased. When crimped, the stent with smaller pitch deformed more. That is, the strain energy increased as the pitch decreased under the same conditions of crimping, and accordingly, the radial stiffness increased. On the other hand, when bent, as the pitch decreased, even though the length of the strut per stent length increased, the deformation of each part of the strut decreased; thus, longitudinal flexibility increased.

Additionally, as a first attempt, the mechanical properties were associated with the geometric characteristics (porosity and pore density), which are related to flow reduction effects. If adjustment for radial stiffness or longitudinal flexibility is required, the change in strut size will not affect the change in pore density. Additionally, the change in strut size is less sensitive to porosity when adjustment of longitudinal flexibility is required. That is, adjusting the strut size can change the mechanical properties while minimizing the change in flow reduction effects, and adjusting pitch can change the flow reduction effects while minimizing the change in mechanical properties. It seems that the radial stiffness depends on porosity, regardless of the pitch and strut size. However, the data are insufficient, and the range in porosity that enables comparison of the two lines is 79–82% for drawing that conclusion. Therefore, further studies are needed. Figures 7(a)–(c) show the relationships between stent diameter and stent length, porosity, and pore density, respectively. As the “pitch” decreases, there is significant foreshortening (see Fig. 7(a)). Further, as pitch decreases, porosity and pore density change markedly according to the change in the diameter (see Figs 7(b) and (c)). If the strut angle (rather than “pitch,” see Fig. 1) is $>$ 45 ${}^{\circ}$ when the stent is fully expanded, the porosity increases and pore density decreases as the strut angle decreases (i.e., the stent is crimped or the diameter decreases). Once the strut angle reaches 45 ${}^{\circ}$ , that tendency is reversed. Thus, if there is a large discrepancy in diameter between the parent artery and the portion of the stent that is free to expand at the aneurysmal neck area, it has been shown that the porosity at the transition zone (the portion of the stent between where the stent is constrained within the parent artery and the aneurysmal neck in which the stent is fully expanded) is large, and the probability of neointimal formation decreases [30]. On the other hand, if the strut angle is $<$ 45 ${}^{\circ}$ when the stent is fully expanded, the porosity decreases and pore density increases monotonically as the strut angle decreases. In that case, the denser mesh increases the risk of occluding perforators and side branches.

Ma et al. [31] and Xiang et al. [32] showed that the porosity or the pore density could vary according to the deployment technique. Investigation of the deployment technique may lead to the design of optimized stents, but it was not considered in the present study. However, it may take time to master that technique, and deployment in tortuous vascular anatomies is difficult [31]. Therefore, a stent that functions well regardless of deployment skill would be beneficial. Deployment can be improved by modifying the stent or delivery system. For example, the implant of PED Flex is the same as the original device (namely, PED), but the delivery system has been redesigned to enhance device opening and to provide additional safety with a re-sheathing feature [33].

Finally, although the present study was based on the supposition that a typical HBS (e.g., PED) is used, it may be possible to design a safe, effective stent by considering different stent structures [22].

5. Study limitations

The present study has several limitations. No validation of the simulations was performed. However, the same solver, ABAQUS/explicit, and the same computational conditions for friction coefficient, materials, and element model (beam element) used by Ma et al. [27] were used in the present study. Therefore, we believe that the present simulations are valid. The strut size and material were unified in each stent, but actual stents are composed of more than two kinds of struts with different strut sizes and materials. Although fatigue strength was not investigated, it is one of the most important design variables and should thus be considered in future work.

6. Conclusions

Mechanical properties (radial stiffness and longitudinal flexibility) were associated with geometric characteristics (porosity and pore density), which are related to flow reduction effects. Changing the strut size has a greater impact on the radial stiffness and longitudinal flexibility related to porosity or pore density than changing the pitch. Adjusting the strut size can change the mechanical properties while minimizing the change in flow reduction effects, and adjusting the pitch can change the flow reduction effects while minimizing the change in the mechanical properties. These results provide useful information for the design of an optimal HBS.

Conflict of interest

The other authors have no conflicts of interest to disclose.

Footnotes

Acknowledgments

Y.M. and T.I. have received honoraria from Stryker Japan K.K. Y.M. has received honoraria from Asahi Intecc Co., Ltd. Y.M. and H.T. were partially supported by Siemens Healthcare K.K. with a grant provided to our academic institution (Grant No. 35993-00211563). H.T. was supported by NTT Docomo, Inc. with a donation provided to our academic institution that was not related to the present submitted work.

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Relationships between geometrical parameters and mechanical properties for a helical braided flow diverter stent

Abstract

BACKGROUND:

OBJECTIVE:

METHODS:

RESULTS:

CONCLUSIONS:

Keywords

1. Introduction

Table 1 Specifications of fully expanded flow diverters (external diameter of 4.5 mm)

2.1 Specifications of the compared stents

2.2 Numerical simulation

Table 2 Nominal element size and the number of elements used in the grid independence study

2.4 Bending test

3. Results

3.1 Radial stiffness

5. Study limitations

6. Conclusions

Conflict of interest

Footnotes

Acknowledgments

References

Table 1
Specifications of fully expanded flow diverters (external diameter of 4.5 mm)

Table 2
Nominal element size and the number of elements used in the grid independence study