**5.1 Computation of contact force distribution on vascular wall caused by expansion of stents**

The expansion of a stent in a blood vessel induces pressure on the contact surfaces of the stent and the blood vessel. Analysis of pressure or contact force based on the complicated

Design and Evaluation of Self-Expanding Stents

{*P*(*<sup>m</sup>*)}=(*P*1(*<sup>m</sup>*), *P*2(*<sup>m</sup>*), ..., *P*n(*<sup>m</sup>*))T is given as follows:

The radial displacement {*r*(*m*)

ofcontact, are obtained:

(*s*) and *Ri*

where *Ri*

follows:

follows:

Suitable for Diverse Clinical Manifestation Based on Mechanical Engineering 193

Therefore, the following expressions, namely, the equilibrium equation and the condition

0

Then, by replacing {*P*(*<sup>v</sup>*)} and -{*P*(*<sup>s</sup>*)} with {*P*}, the equation of contact force is obtained as

Equation (11) is solved for {*P*} to obtain the distribution of the contact force between the stent and the blood vessel wall. Since the rigidity of the strut section of the stent is considerably much lower than that of the wire section, *rji*(*s*) is negligible, except for the case in which *i* and *j* are located in the wire section of the stent. Therefore, in the present study, the influence matrix [*C*(*<sup>s</sup>*)] is composed by diagonally placing the partial matrix for the wire section, and the contact force on the strut section of the stent is not evaluated. The Gaussian

Straightening of the blood vessel occurs when a straight stent is inserted into a curved vessel. This insertion causes a straightening force to act on the vascular wall. The straightening effect should be evaluated while considering the interaction process of the expansion of the stent with the curved blood vessel. However, solving this problem is extremely complicated. Therefore, the straightening of the blood vessel is assumed to be independent from the expansion of the vessels by the stent, and it is simplified as shown in Fig. 12. The stent is approximated by a beam with the flexural rigidity obtained in the previous section, and the stent is modeled as a laminated beam by combining with a curved

The beam models are divided into *n* intervals. When a unit force is applied to point *i* of model *m* (*m* = *s*, *v*, which correspond to the stent and the vessel, respectively), the deflection at point *j* is denoted as *dji*(*<sup>m</sup>*). The n-order influence matrix [*D*(*<sup>m</sup>*)] is defined in terms of *dji*(*m*) as

0

, ..., *r*n(*<sup>m</sup>*)

)T due to unknown contact force

(9)

(10)

*mm m CP r* (8)

(contact)

(non contact)

*vs sv CCPRR* (11)

(*v*) denote the initial radii of the stent and the blood vessel, respectively, at *i*.

(*m*)

}=(*r*1(*<sup>m</sup>*), *r*<sup>2</sup>

 

*P P Rr Rr*

 

*P P Rr Rr*

elimination method is used to solve the simultaneous equations.

beam, which is used a model of the blood vessel.

**5.2 Calculation of distribution of straightening force on vascular wall** 

*v s i i vv ss i i ii*

*v s i i vv ss i i ii*

shape of stent is a difficult and time-consuming task. Therefore, a simplified calculation method using the axisymmetrical models is presented as shown in Fig. 11. The stent is modeled by a number of rings, indicated by broken lines in Fig. 11. The wire section is represented by 11 rings. The blood vessel is similarly modeled by rings, where the ring intervals are the same as those of the stent. In addition, it is assumed that the blood vessel is much longer than the stent. It is also assumed that these rings deform axisymmetrically due to the uniformly distributed radial force.

Fig. 11. Model used to compute the contact force between the stent and vascular wall. Thestent and blood vessel are simplified to axisymmetrical models.

Next, let us consider ring *i* (*i* = 1, 2, ..., *n*) in contact along the surfaces of stent and vascular wall. When a unit radial force is applied to ring *i* of model *m* (*m* = *s*, *v*, which correspond to the stent and the blood vessel, respectively), the radial displacement at ring *j*, denoted as *rji*(*<sup>m</sup>*), is calculated using the finite element method. The influence matrix [*C*(*<sup>m</sup>*)] is defined in terms of as *rji*(*m*) as follows:

$$\mathbb{E}\left[\mathbf{C}^{(m)}\right] = \left[ \left\{ r\_1^{(m)} \right\}, \left\{ r\_2^{(m)} \right\}, \dots, \left\{ r\_n^{(m)} \right\} \right] \tag{6}$$

where

$$\mathbb{E}\left\{r\_i^{(m)}\right\} = \left(r\_{1i}^{(m)}, r\_{2i}^{(m)}, \dots, r\_{ni}^{(m)}\right)^{\mathrm{T}}\tag{7}$$

shape of stent is a difficult and time-consuming task. Therefore, a simplified calculation method using the axisymmetrical models is presented as shown in Fig. 11. The stent is modeled by a number of rings, indicated by broken lines in Fig. 11. The wire section is represented by 11 rings. The blood vessel is similarly modeled by rings, where the ring intervals are the same as those of the stent. In addition, it is assumed that the blood vessel is much longer than the stent. It is also assumed that these rings deform axisymmetrically due

Fig. 11. Model used to compute the contact force between the stent and vascular wall.

Next, let us consider ring *i* (*i* = 1, 2, ..., *n*) in contact along the surfaces of stent and vascular wall. When a unit radial force is applied to ring *i* of model *m* (*m* = *s*, *v*, which correspond to the stent and the blood vessel, respectively), the radial displacement at ring *j*, denoted as *rji*(*<sup>m</sup>*), is calculated using the finite element method. The influence matrix [*C*(*<sup>m</sup>*)] is defined in

> 1 2 , , ..., *m m m m C rr rn*

1 2 , , ..., *m mm m*

T

*<sup>i</sup> i i ni r rr r* (7)

(6)

Thestent and blood vessel are simplified to axisymmetrical models.

terms of as *rji*(*m*) as follows:

where

to the uniformly distributed radial force.

The radial displacement {*r*(*<sup>m</sup>*)}=(*r*1(*<sup>m</sup>*) , *r*<sup>2</sup> (*<sup>m</sup>*), ..., *r*n(*<sup>m</sup>*) )T due to unknown contact force {*P*(*m*) }=(*P*1(*<sup>m</sup>*), *P*2(*<sup>m</sup>*), ..., *P*n(*<sup>m</sup>*))T is given as follows:

$$\mathbb{E}\left[\mathbf{C}^{(m)}\right]\left|P^{(m)}\right\rangle = \left\{r^{(m)}\right\}\tag{8}$$

Therefore, the following expressions, namely, the equilibrium equation and the condition ofcontact, are obtained:

$$\begin{aligned} P\_i^{(v)} + P\_i^{(s)} &= 0\\ R\_i^{(v)} + r\_i^{(v)} &= R\_i^{(s)} + r\_i^{(s)} \end{aligned} \quad \text{(contact)}\tag{9}$$

$$\begin{aligned} P\_i^{(v)} &= P\_i^{(s)} = 0\\ R\_i^{(v)} + r\_i^{(v)} &> R\_i^{(s)} + r\_i^{(s)} \end{aligned} \quad \text{(non-contact)}\tag{10}$$

where *Ri* (*s*) and *Ri* (*v*) denote the initial radii of the stent and the blood vessel, respectively, at *i*. Then, by replacing {*P*(*<sup>v</sup>*)} and -{*P*(*s*) } with {*P*}, the equation of contact force is obtained as follows:

$$\mathbb{E}\left[\mathbf{C}^{(v)} + \mathbf{C}^{(s)}\right] \{P\} = \left(R^{(s)} - R^{(v)}\right) \tag{11}$$

Equation (11) is solved for {*P*} to obtain the distribution of the contact force between the stent and the blood vessel wall. Since the rigidity of the strut section of the stent is considerably much lower than that of the wire section, *rji*(*s*) is negligible, except for the case in which *i* and *j* are located in the wire section of the stent. Therefore, in the present study, the influence matrix [*C*(*<sup>s</sup>*)] is composed by diagonally placing the partial matrix for the wire section, and the contact force on the strut section of the stent is not evaluated. The Gaussian elimination method is used to solve the simultaneous equations.

#### **5.2 Calculation of distribution of straightening force on vascular wall**

Straightening of the blood vessel occurs when a straight stent is inserted into a curved vessel. This insertion causes a straightening force to act on the vascular wall. The straightening effect should be evaluated while considering the interaction process of the expansion of the stent with the curved blood vessel. However, solving this problem is extremely complicated. Therefore, the straightening of the blood vessel is assumed to be independent from the expansion of the vessels by the stent, and it is simplified as shown in Fig. 12. The stent is approximated by a beam with the flexural rigidity obtained in the previous section, and the stent is modeled as a laminated beam by combining with a curved beam, which is used a model of the blood vessel.

The beam models are divided into *n* intervals. When a unit force is applied to point *i* of model *m* (*m* = *s*, *v*, which correspond to the stent and the vessel, respectively), the deflection at point *j* is denoted as *dji*(*<sup>m</sup>*). The n-order influence matrix [*D*(*<sup>m</sup>*)] is defined in terms of *dji*(*m*) as follows:

Design and Evaluation of Self-Expanding Stents

chosen only from among commercially available stents.

vessel are simply modeled as a laminated beam.

design method will be described in following sections.

Suitable for Diverse Clinical Manifestation Based on Mechanical Engineering 195

it is more important to design stents suited to each unique symptom of every patient. (Colombo et al., 2002) made evaluations of stent 'deliverability,' 'scaffolding,' 'accurate positioning,' and so on for the average lesion of the coronary artery. They proposed a guideline for use in determining a suitable stent. This guideline was based on clinical trials. Therefore, it is governed by the doctor's sense. One more noteworthy point is that the bestsuited stent cannot be actually available for every specific symptom because a stent must be

Fig. 12. Model for calculation of the straightening force on vascular wall. The stent andblood

In this section, we describe a method for designing a stent that has good mechanical properties to suit diverse clinical manifestation. Figure 13 shows the flow of designing a stent suitable for diverse clinical manifestation. The first step is to determine the radial stiffness of the stent necessary to expand the stenotic part in the blood vessel based on symptom information. Next, based on the determined radial stiffness and the sensitivities of mechanical properties of the stent defined in Section 4, the design variables of a suitable stent are determined. In the second step, the force on the vascular wall by insertion of the designed stent is first evaluated by using the methods described in Section 5. This force is associated with the risk of in-stent restenosis. Next, based on the evaluation result, the designed stent is modified to be more suitable for the symptom. After modification, the force on the vascular wall is evaluated again. The effect of shape modification is confirmed by comparing the forces on the vascular wall between before and after modification. Finally, the modified stent shape is proposed as better suited stent shape. The detail of this proposed

$$\left[\boldsymbol{D}^{(m)}\right] = \left[\left<\boldsymbol{d}\_1^{(m)}\right>, \left<\boldsymbol{d}\_2^{(m)}\right>, \dots, \left<\boldsymbol{d}\_n^{(m)}\right>\right] \tag{12}$$

where

$$\left\{d\_i^{(m)}\right\} = \left(d\_{1i}^{(m)}, d\_{2i}^{(m)}, \dots, d\_{ni}^{(m)}\right)^{\mathrm{T}} \tag{13}$$

The force {*F*(*<sup>m</sup>*)} distributed along the beam is related to the deflection {*d*(*m*) } so as to satisfy the following equation:

$$\left\{ \left[ D^{(m)} \right] \left| F^{(m)} \right> = \left\{ \mathcal{S}^{(m)} \right\} \right. \tag{14}$$

The equilibrium equation and contact condition are given by the following equations:

$$F\_i^{(s)} + F\_i^{(v)} = \mathbf{0} \tag{15}$$

$$
\delta\_i^{(s)} = \delta\_i^{(i)} + \delta\_i^{(v)} \tag{16}
$$

where *di* (*i*) denotes the initial deflection of the blood vessel at point *i*, and *di* (*s*) and *di* (*v*) are the

deflections of the stent and the blood vessel, respectively. Replacing -{*F*(*<sup>v</sup>*)} and {*F*(*<sup>s</sup>*)} with {*F*}, the equation of straightening force is obtained as follows:

$$\left[D^{(\upsilon)} + D^{(s)}\right]\{F\} = \left\{\delta^{(i)}\right\}\tag{17}$$

#### **5.3 Limitations of these calculation methods**

The methods will be useful for improving stent shape in order to reduce the peak force acting on the vascular wall. Although these methods are useful to calculate a contact force and a straightening force on a vascular wall, there are the following limitations:


### **6. Design method of stent suitable for diverse clinical manifestation**

The stent must have the radial stiffness sufficient to expand the stenotic part in the blood vessel outward. Simultaneously, it must be sufficiently flexible to conform to the vascular wall. Neither the symptom nor the blood vessel shape is always in the same state. Therefore,

 1 2 , , ..., *m m m m D dd dn*

The force {*F*(*<sup>m</sup>*)} distributed along the beam is related to the deflection {*d*(*m*)

1 2 , , ..., *m mm m*

*mm m D F*

0 *s v*

 *siv i ii*

deflections of the stent and the blood vessel, respectively. Replacing -{*F*(*<sup>v</sup>*)} and {*F*(*<sup>s</sup>*)} with {*F*},

*vs i DDF*

The methods will be useful for improving stent shape in order to reduce the peak force acting on the vascular wall. Although these methods are useful to calculate a contact force

2. The cross-sectional distortions and thickness changes of the stent and blood vessel were

5. The stent and blood vessel were simplified by using axisymmetrical models in computation of the contact force and using the laminated beam in calculation of the

The stent must have the radial stiffness sufficient to expand the stenotic part in the blood vessel outward. Simultaneously, it must be sufficiently flexible to conform to the vascular wall. Neither the symptom nor the blood vessel shape is always in the same state. Therefore,

3. The friction on the contact surface between the stent and blood vessel was ignored. 4. It was consider that the straightening of the blood vessel and the expansion of the

**6. Design method of stent suitable for diverse clinical manifestation** 

 

The equilibrium equation and contact condition are given by the following equations:

 

and a straightening force on a vascular wall, there are the following limitations:

1. The isotropic material property was assumed for the artery.

vessels by the stent were independent each other.

straightening force, respectively.

the equation of straightening force is obtained as follows:

**5.3 Limitations of these calculation methods** 

(*i*) denotes the initial deflection of the blood vessel at point *i*, and *di*

T

*<sup>i</sup> i i ni d dd d* (13)

(14)

*i i F F* (15)

(17)

(16)

(*s*) and *di*

(*v*) are the

where

where *di*

ignored.

the following equation:

(12)

} so as to satisfy

it is more important to design stents suited to each unique symptom of every patient. (Colombo et al., 2002) made evaluations of stent 'deliverability,' 'scaffolding,' 'accurate positioning,' and so on for the average lesion of the coronary artery. They proposed a guideline for use in determining a suitable stent. This guideline was based on clinical trials. Therefore, it is governed by the doctor's sense. One more noteworthy point is that the bestsuited stent cannot be actually available for every specific symptom because a stent must be chosen only from among commercially available stents.

Fig. 12. Model for calculation of the straightening force on vascular wall. The stent andblood vessel are simply modeled as a laminated beam.

In this section, we describe a method for designing a stent that has good mechanical properties to suit diverse clinical manifestation. Figure 13 shows the flow of designing a stent suitable for diverse clinical manifestation. The first step is to determine the radial stiffness of the stent necessary to expand the stenotic part in the blood vessel based on symptom information. Next, based on the determined radial stiffness and the sensitivities of mechanical properties of the stent defined in Section 4, the design variables of a suitable stent are determined. In the second step, the force on the vascular wall by insertion of the designed stent is first evaluated by using the methods described in Section 5. This force is associated with the risk of in-stent restenosis. Next, based on the evaluation result, the designed stent is modified to be more suitable for the symptom. After modification, the force on the vascular wall is evaluated again. The effect of shape modification is confirmed by comparing the forces on the vascular wall between before and after modification. Finally, the modified stent shape is proposed as better suited stent shape. The detail of this proposed design method will be described in following sections.

Design and Evaluation of Self-Expanding Stents

Fig. 14. Dimension of blood vessel with stenosis

Suitable for Diverse Clinical Manifestation Based on Mechanical Engineering 197

*p v*,

Stratouly et al., 1987; Gow & Hadfield, 1979), with modification.

Table 3. Pressure strain elastic modulus for each type of human artery

stiffness are made as follows.

*<sup>p</sup> <sup>E</sup>*

where *Do* is the blood vessel's outer diameter, and *Do* and *p* are the increase in the blood vessel diameter and the increase in the internal pressure respectively. The pressure strain elastic modulus has been used widely in clinical studies. Therefore, many reports on that subject are available. Table 3 presents value of the pressure strain elastic modulus for each type of human blood vessel, extracted from references (Hayashi et al., 1980; Hayashi, 2005;

> Artery *Ep*,*<sup>v</sup>* Reporter Arteria pulmonalis 0.016 Greenfield and Griggs

Ascending aorta 0.076 Patel et al. (1964) Basilar artery 0.186 (=14) Hayashi et al. (1980) Common carotid artery 0.049 Arndt et al. (1969) Common iliac artery 0.120 Stratouly et al. (1987) Coronary artery 0.602 Gow and Hadfield (1979) Femoral artery 0.433 Patel et al. (1964) Thoracic aorta 0.126 Luchsinger et al. (1962)

**6.1.2 Determination of radial stiffness necessary to expand stenotic part in artery** 

To determine the radial stiffness of the stent necessary to expand the stenotic part in the blood vessel, the requirements are, in addition to the pressure strain elastic modulus *Ep*,*<sup>v</sup>* of the blood vessel, shown as follows. The outer and inner diameters, *Do* and *Di*, of the blood vessel in the normal state, the least diameter *Dl* produced by the stenotic part, and the length of the stenotic part *Ll* are required. The pressure strain elastic modulus of the plaque *Ep*,*<sup>p</sup>* is required. Also required is the inner diameter after the treatment is made, *Dt*, which is an indicator to use as the target setting for the percentage by which to improve the blood flow level there. Given all the values listed above, the calculations for the necessary radial

*o o*

(18)

(1963)

*D D*

Fig. 13. Flow of designing stent suitable for diverse clinical manifestation. Designing a stent suitable for diverse clinical manifestation consists of two steps. In the first step, it is possible to design a stent necessary to expand the stenotic part of a blood vessel. In the second step, the designed stent is modified to suit the patient's symptom better.
