**6.3.2 Modification method of the stent shape to suit the clinical manifestation**

The stent designed in the previous section has a uniform radial stiffness *Kp* along its axial direction. The radial stiffness *Kp* causes excessive expansion at the normal part of the artery because *Kp* was calculated for expansion of the stenotic part, which is generally stiffer than the normal part. The objective for the mechanical stimulus can be attained by decreasing the radial stiffness to a value corresponding to that of a normal artery. At the same time, the

in-stent restenosis occurred primarily at both ends of the stent during the period one to three months after stenting. In the study of (Lal et al., 2007), the majority of patients in the target patient population showed in-stent restenosis at the stent ends. (Yazdani and Berry, 2009) cultured a stented native porcine carotid artery under physiologic pulsatile flow and pressure conditions for a week. They confirmed that the proliferation of smooth muscle cells occurred significantly at both ends of the stent. By summarizing these reports and evaluation results of the forces on the vascular wall, it is concluded that the force concentration provokes neointimal thickening due to the hyperplasia of smooth muscle

Next, the expansion of the vascular wall is examined. The normal part of the artery is expanded to become much larger than the target diameter. This excessive expansion causes expansion of the entire stented part. As a result, stagnation and vortices of blood flow are

The designed stent has mechanical properties sufficient to expand the stenotic part of theartery, but is not suitable for the normal part of the artery. Thus, it requires

**6.3 Effective method to modify the stent shape in consideration of the risk of in-stent** 

In the previous section, it was shown that although a stent suitable for the assumed symptoms could be designed using the proposed design method, further modifications

*Objective for the mechanical stimulus*: as shown in Fig. 18, the contact force on the normal part of the artery is concentrated at both ends of the stent. This force concentration provokes neointimal thickening from the hyperplasia of smooth muscle cells. Therefore, the contact force must be reduced at both ends of the stent. However, this force should be larger than a

*Objective for the blood flow*: as stated above, the normal part of the artery is expanded to become much larger than the target diameter. This expansion state of the artery causes stagnation and vortices of blood flow. As a result, stent thrombosis may be induced. It is very important to reduce stagnation and vortex creation to decrease the risk of stent thrombosis. Therefore, the vascular wall should be expanded flatly by the insertion of the stent. The flat expansion of the artery can prevent the generation of stagnation and vortices

causes excessive expansion at the normal part of the artery

was calculated for expansion of the stenotic part, which is generally stiffer than

the normal part. The objective for the mechanical stimulus can be attained by decreasing the radial stiffness to a value corresponding to that of a normal artery. At the same time, the

along its axial

were still required. Two kinds of design objectives are set up for shape modification.

**6.3.2 Modification method of the stent shape to suit the clinical manifestation** 

The stent designed in the previous section has a uniform radial stiffness *Kp*

induced, further increasing the potential for stent thrombosis.

**6.3.1 Design objective for the shape modification** 

certain limit so that stent functions on the lesion.

cells, i.e., in-stent restenosis.

modification.

**restenosis** 

in the stented artery.

because *Kp*

direction. The radial stiffness *Kp*

objective for blood flow must be attained. Therefore, the stent should have a radial stiffness *Kp* to expand the vascular wall at normal part only for the stent thickness *t*. This can achieve the flat expansion of the artery. By changing the stent radial stiffness at the normal part from *Kp* to *Kp* , the contact force can also decrease. In this study, it was decided to adopt the radial stiffness *Kp* as a compromise between the two design objectives.

The methods using the influence matrix described in Section 5 are used for the shape modification. Figure 19 shows the concept for modification of the stent shape. The distributed contact force *Pi* on the stenotic part can be calculated based on the radial stiffness *Kp* , as follows:

$$P\_i^\* = \frac{\pi D\_t K\_p^\* \Delta r\_t l\_w^\*}{n\_{\rm CP}} \tag{22}$$

where *Dt* is the inner diameter of the blood vessel after treatment, *rt* is the increase in the vascular radius of the target, *lw* is the wire length of the stent designed based on the radial stiffness *Kp* . Also, *n*CP represents the number of calculation points on the wire section. For simplicity, it was assumed that a uniform contact force was exerted on the vascular wall.

Fig. 19. Concept for modification of the stent shape. The distributed contact force is obtained from the radial stiffness of the stent on the stenotic part by using the design method. The force that should be applied on the normal artery is calculated by using the obtained distributed contact force on the stenotic part.

As described in Section 5, the radial displacement of the vascular wall {*r*(*<sup>v</sup>*)} = (*r*<sup>1</sup> (*<sup>v</sup>*), *r*2(*<sup>v</sup>*), ..., *rn*(*<sup>v</sup>*))T due to the unknown contact force {*P*} = (*P*1, *P*2, ..., *Pn*)T is given as follows:

$$\mathbb{E}\left[\mathbf{C}^{\left(\upsilon\right)}\right](P) = \left\{ r^{\left(\upsilon\right)} \right\}\tag{23}$$

where [*C*(*<sup>v</sup>*)] is the influence matrix of the blood vessel, defined by the radial displacement of the vascular wall due to the unit radial force. The calculated contact force *Pi* is substituted

Design and Evaluation of Self-Expanding Stents

is frequently reported.

Suitable for Diverse Clinical Manifestation Based on Mechanical Engineering 205

On the other hand, it is recognized that straightening force increases at the stenotic-healthy tissue interface (the axial location is 5 mm in Fig. 21(b)). The concentration of straightening force also occurs at the axial location of 8 mm. After modification of the stent shape, the flexural and shear rigidities of the stent vary with the axial location. The bending state of the stent changes at the changing point of the rigidities, which corresponds to the turn of the stent shape. Therefore, straightening force increases due to changing of the stent bending state based on rigidities changing. Although the risk of the vessel rupture slightly increases at the stenotic-healthy tissue interface, it is achieved that the straightening force is significantly reduced at both ends of the stent, where the hyperplasia of smooth muscle cells

(a)

(b) Fig. 20. The SDCO stent 6 mm in diameter is modified to suit the assumed symptom of the coronary artery. (a) Design variables after modification are determined by using the map of the radial stiffness. (b) The modified SDCO has a nonuniform shape along its axial direction.

into equation (23), and the influence matrix [*C*(*<sup>v</sup>*)] is downsized to the matrix [*C*normal(*<sup>v</sup>*)] of the normal blood vessel.

$$\left\{ \mathbf{C}\_{\text{normal}}^{\left(\mathbf{v}\right)} \right\} \left\{ P^{\*\*} \right\} = \left\{ r\_{\text{normal}}^{\left(\mathbf{v}\right)} \right\} + \left[ \mathbf{C}\_{\text{stensosis}}^{\left(\mathbf{v}\right)} \right] \left\{ P^{\*} \right\} \tag{24}$$

[*C*stenosis(*<sup>v</sup>*)] is the influence matrix of the stenotic part, and {*r*normal(*<sup>v</sup>*)} is the radial displacement of the normal part of the blood vessel wall. Equation (24) is solved for {*P*to obtain the distributed force *Pi* that can expand the normal part to the stent thickness *t*.

Here, it is assumed that the wire length after modification is *lw* and the calculation points from *k* to *l* are included in the modified wire section. The required radial stiffness *Kp* is defined as follows:

$$\left(K\_p^{\ast \ast} = \frac{\sum\_{i=k}^{l} P\_i^{\ast \ast}}{\pi D\_t l\_w^{\ast \ast} t} \right) \tag{25}$$

The *Kp* value calculated by equation (25) is plotted on the radial stiffness map. As a result, the wire width *tw* after modification is determined from the *lw* value and the curve of *Kp* on the map. Therefore, the designed stent has to be modified to the shape of *lw* and *tw* at the normal part of the blood vessel. Considering that an increase in the stent length is undesirable, the designer should keep the wire length *lw* equal to *lw* after the modification. However, it is possible that the wire width *tw* after the modification cannot be obtained because of the mismatch between the assumed *lw\*\** value and the curve of *Kp* . In this case, the designer can increase the wire length *lw\*\** , perform the same procedures, and determine the *lw* and *tw* values.

The required radial stiffness *Kp*<sup>1</sup> at both ends of the SDCO is 50.3 MPa/m. The radial stiffness *Kp*<sup>2</sup> at the normal part of the artery except for both stent ends is also calculated as 155.2 MPa/m. From the dot-dashed curves of *Kp*<sup>1</sup> and *Kp*<sup>2</sup> shown in Fig. 20(a), it is determined that *lw*<sup>1</sup> is 2.52 mm, *tw*<sup>1</sup> is 0.087 mm, *lw*<sup>2</sup> is 2.01 mm, and *tw*<sup>2</sup> is 0.094 mm. Figure 20(b) shows the modified shape of the SDCO.

The flexural rigidity *Kb*<sup>1</sup> at both ends of the SDCO is 5.75×10*-*6 Nm2, and the shear rigidity *Ks*<sup>1</sup> is 0.20 N. At the normal part of the artery except for both stent ends, the flexural rigidity *Kb*<sup>2</sup> of the SDCO is 6.00×10*-*6 Nm2, and the shear rigidity *Ks*<sup>2</sup> is 0.42 N.

#### **6.3.3 Confirmation of the effect of the shape modification**

Figure 21 shows a comparison of the force on the vascular wall by insertion of the stent before and after the modifications. After the modifications of the SDCO, an approximately 80 % reduction in the concentrated contact force was attained (Fig. 21(a)). Furthermore. the concentrated straightening force at the stent ends after modification was reduced to approximately 35 % of that before modification of the SDCO (Fig. 21(b)).

into equation (23), and the influence matrix [*C*(*<sup>v</sup>*)] is downsized to the matrix [*C*normal(*<sup>v</sup>*)] of the

[*C*stenosis(*<sup>v</sup>*)] is the influence matrix of the stenotic part, and {*r*normal(*<sup>v</sup>*)} is the radial displacement of the normal part of the blood vessel wall. Equation (24) is solved for {*P*to

from *k* to *l* are included in the modified wire section. The required radial stiffness *Kp*

\*\*

*p*

*K*

after modification is determined from the *lw*

on the map. Therefore, the designed stent has to be modified to the shape of *lw*

\*\*

*l i i k*

*Dl t* 

the normal part of the blood vessel. Considering that an increase in the stent length is

be obtained because of the mismatch between the assumed *lw\*\** value and the curve of *Kp*

In this case, the designer can increase the wire length *lw\*\** , perform the same procedures,

is 0.087 mm, *lw*<sup>2</sup>

is 0.20 N. At the normal part of the artery except for both stent ends, the flexural

Figure 21 shows a comparison of the force on the vascular wall by insertion of the stent before and after the modifications. After the modifications of the SDCO, an approximately 80 % reduction in the concentrated contact force was attained (Fig. 21(a)). Furthermore. the concentrated straightening force at the stent ends after modification was reduced to

of the SDCO is 6.00×10*-*6 Nm2, and the shear rigidity *Ks*<sup>2</sup>

approximately 35 % of that before modification of the SDCO (Fig. 21(b)).

at the normal part of the artery except for both stent ends is also calculated as

and *Kp*<sup>2</sup>

at both ends of the SDCO is 5.75×10*-*6 Nm2, and the shear rigidity

\*\*

*t w*

value calculated by equation (25) is plotted on the radial stiffness map. As a result,

*P*

Here, it is assumed that the wire length after modification is *lw*

undesirable, the designer should keep the wire length *lw*

values.

modification. However, it is possible that the wire width *tw*

and *tw*

155.2 MPa/m. From the dot-dashed curves of *Kp*<sup>1</sup>

Figure 20(b) shows the modified shape of the SDCO.

is 2.52 mm, *tw*<sup>1</sup>

**6.3.3 Confirmation of the effect of the shape modification** 

v vv \*\* \* *C Pr C P* normal normal stenosis (24)

that can expand the normal part to the stent thickness *t*.

and the calculation points

(25)

equal to *lw*

at both ends of the SDCO is 50.3 MPa/m. The radial

is 2.01 mm, and *tw*<sup>2</sup>

value and the curve of *Kp*

after the modification cannot

shown in Fig. 20(a), it is

is 0.42 N.

and *tw*

after the

is 0.094 mm.

is

.

at

normal blood vessel.

obtain the distributed force *Pi*

defined as follows:

the wire width *tw*

and determine the *lw*

determined that *lw*<sup>1</sup>

The flexural rigidity *Kb*<sup>1</sup>

stiffness *Kp*<sup>2</sup>

rigidity *Kb*<sup>2</sup>

*Ks*<sup>1</sup>

The required radial stiffness *Kp*<sup>1</sup>

The *Kp*

On the other hand, it is recognized that straightening force increases at the stenotic-healthy tissue interface (the axial location is 5 mm in Fig. 21(b)). The concentration of straightening force also occurs at the axial location of 8 mm. After modification of the stent shape, the flexural and shear rigidities of the stent vary with the axial location. The bending state of the stent changes at the changing point of the rigidities, which corresponds to the turn of the stent shape. Therefore, straightening force increases due to changing of the stent bending state based on rigidities changing. Although the risk of the vessel rupture slightly increases at the stenotic-healthy tissue interface, it is achieved that the straightening force is significantly reduced at both ends of the stent, where the hyperplasia of smooth muscle cells is frequently reported.

(b)

Fig. 20. The SDCO stent 6 mm in diameter is modified to suit the assumed symptom of the coronary artery. (a) Design variables after modification are determined by using the map of the radial stiffness. (b) The modified SDCO has a nonuniform shape along its axial direction.

Design and Evaluation of Self-Expanding Stents

ISSN 0028-4793 (print version).

0028-4793 (print version).

833-840, ISSN 0741-5214.

0195-668X (print version).

Information Center, Fort Belvoir.

(print version).

1042-3931.

28, No. 7, (July 2009) 1126-1137, ISSN 0278-0062.

(June 2002) 1773-1780, ISSN 0028-4793 (print version).

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The modified stent can expand the vascular wall in a more flat manner. Therefore, modification of the designed stent by using the proposed method can relax the force concentration at both ends of the stent by attaining flat expansion of the vascular wall.
