**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 stiffness are made as follows.

Design and Evaluation of Self-Expanding Stents

Table 4. Assumed patient symptom information

designer can proceed to work with some design propositions.

The required radial stiffness *Kp*

Suitable for Diverse Clinical Manifestation Based on Mechanical Engineering 199

in Fig. 16. From this line, it is possible to determine the design variables for a stent with sufficient radial stiffness to expand the stenotic part of the blood vessel. Based on these, the

is plotted as a broken line on the radial stiffness map shown

s

Coronary artery Parameter

Outer diameter, *Do* (mm) 4.90 Inner diameter, *Di* (mm) 4.06 Least diameter of lesion, *Dl* (mm) 2.5 Length of lesion, *Ll* (mm) 10

Total flexion angle (deg.) (Flexion angle (deg.)) 90 (45) Rate of stenosis by ECST method (%) 38.4 Pressure strain elastic modulus of artery, *Ep*,*<sup>v</sup>* (MPa) 0.602 Pressure strain elastic modulus of diseased artery, *Ep*,*vl* (MPa) 0.628

Inner diameter after treatment, *Dt* (mm) 4.56

Schematic view of assumed symptom

Fig. 16. Proposed designs having the radial stiffness necessary to expand the stenotic part of a blood vessel. By assuming that a stent is inserted into a coronary artery, a value of 0.602 MPa was used for the pressure strain elastic modulus *Ep*,*<sup>v</sup>* of the normal part of the coronary artery, and a value of 0.628 MPa for the pressure strain elastic modulus *Ep*,*vl* of the diseased artery. For the coronary artery model, *Do* = 4.90 mm, *Di* = 4.06 mm, and *Dl* = 2.5 mm were

**6.1.3 Range of selectable flexural rigidity and the dilemma of selecting the design**

After inserting an originally straight stent into a curved blood vessel and leaving it there, the stent generally conforms to the blood vessel shape. Nevertheless, because the flexural

assumed. The corresponding part of the map is magnified and displayed.

By knowing that the increase in the blood vessel diameter is obtainable from circumferential strain of a cylinder, the blood vessel with stenosis can be modeled by simply using springs connected in parallel, as presented in Fig. 15. Thus, the internal pressure in blood vessel *p* ,

which is necessary to expand the stenotic part, can be obtained by

Fig. 15. Simplified modeling of expansion of stenosis in artery by insertion of stent

When measuring the pressure strain elastic modulus of the plaque *Ep*,*<sup>p</sup>* is difficult, it is possible to replace the sum of the elastic moduli, *Ep*,*<sup>v</sup>* + *Ep*,*<sup>p</sup>*, by the pressure strain elastic modulus of the diseased blood vessel *Ep*,*vl*, which can be easily measured.

As a matter of fact, the diameter of the stent, which is inserted into the stenotic part, is greater than the target vascular diameter after treatment. In conclusion, from the stent diameter *ds* and the target vascular diameter *Dt* obtained after treatment, the radius reduction *r* in stent after inserting can be calculated by means of the following equation:

$$
\Delta r = \frac{d\_s - D\_t}{2} \tag{20}
$$

By substituting equations (19) and (20) into *Kp* = *p*/*r*, which was defined by (Yoshino et al., 2008) to obtain the radial stiffness, the radial stiffness *Kp* necessary to expand the stenotic part can be obtained from

$$\mathcal{K}\_p^\* = \mathcal{Z} \left( E\_{p,v} + E\_{p,p} \right) \frac{D\_t - D\_l}{D\_o \left( d\_s - D\_t \right)} = \mathcal{Z} E\_{p,vl} \frac{D\_t - D\_l}{D\_o \left( d\_s - D\_t \right)} \tag{21}$$

The obtained *Kp* is the least required stiffness to expand the stenotic part in the blood vessel. Therefore, when designing a stent, the chosen stiffness should be greater than this *Kp* value.

Now consider the case where a stent with a diameter of 6 mm is inserted into a coronary artery. The symptoms shown in Table 4 are examples based on references (Gow & Hadfield, 1979; Le Floc'h et al., 2009). Based on this data, the radial stiffness *Kp* necessary to expand the stenotic part in the blood vessel is calculated by equation (21) to be *Kp* = 366.7 MPa/m.

By knowing that the increase in the blood vessel diameter is obtainable from circumferential strain of a cylinder, the blood vessel with stenosis can be modeled by simply using springs connected in parallel, as presented in Fig. 15. Thus, the internal pressure in blood vessel *p*

> , , , *tl tl pv pp p vl*

*DD DD pE E E D D*

Fig. 15. Simplified modeling of expansion of stenosis in artery by insertion of stent

modulus of the diseased blood vessel *Ep*,*vl*, which can be easily measured.

part can be obtained from

The obtained *Kp*

When measuring the pressure strain elastic modulus of the plaque *Ep*,*<sup>p</sup>* is difficult, it is possible to replace the sum of the elastic moduli, *Ep*,*<sup>v</sup>* + *Ep*,*<sup>p</sup>*, by the pressure strain elastic

As a matter of fact, the diameter of the stent, which is inserted into the stenotic part, is greater than the target vascular diameter after treatment. In conclusion, from the stent diameter *ds* and the target vascular diameter *Dt* obtained after treatment, the radius reduction *r* in stent after inserting can be calculated by means of the following equation:

> 2 *s t d D*

By substituting equations (19) and (20) into *Kp* = *p*/*r*, which was defined by (Yoshino et al., 2008) to obtain the radial stiffness, the radial stiffness *Kp* necessary to expand the stenotic

> \* , , , 2 2 *t l t l*

Now consider the case where a stent with a diameter of 6 mm is inserted into a coronary artery. The symptoms shown in Table 4 are examples based on references (Gow & Hadfield,

*DD DD K EE <sup>E</sup>*

*os t os t*

(21)

*Dd D Dd D*

is the least required stiffness to expand the stenotic part in the blood vessel.

*p pv pp p vl*

Therefore, when designing a stent, the chosen stiffness should be greater than this *Kp*

1979; Le Floc'h et al., 2009). Based on this data, the radial stiffness *Kp*

the stenotic part in the blood vessel is calculated by equation (21) to be *Kp*

*<sup>r</sup>* (20)

value.

necessary to expand

= 366.7 MPa/m.

*o o*

(19)

which is necessary to expand the stenotic part, can be obtained by

\*

,

The required radial stiffness *Kp* is plotted as a broken line on the radial stiffness map shown in Fig. 16. From this line, it is possible to determine the design variables for a stent with sufficient radial stiffness to expand the stenotic part of the blood vessel. Based on these, the designer can proceed to work with some design propositions.


Table 4. Assumed patient symptom information

Fig. 16. Proposed designs having the radial stiffness necessary to expand the stenotic part of a blood vessel. By assuming that a stent is inserted into a coronary artery, a value of 0.602 MPa was used for the pressure strain elastic modulus *Ep*,*<sup>v</sup>* of the normal part of the coronary artery, and a value of 0.628 MPa for the pressure strain elastic modulus *Ep*,*vl* of the diseased artery. For the coronary artery model, *Do* = 4.90 mm, *Di* = 4.06 mm, and *Dl* = 2.5 mm were assumed. The corresponding part of the map is magnified and displayed.
