Mechanical Response of a Metallic Stent

K. Ravi-Chandar and Renjun Wang

Department of Aerospace Engineering and Engineering Mechanics

Center for Mechanics of Solids, Structures and Materials

Collaborators: Prof. Suncica Canic, UH, Dr. Zvonko Krajcer, St. Luke’s

Outline

Stents in vascular applications Failure modes Mechanics problem

Experimental characterization Analysis of the deformation

Coupled stent-artery deformation

Outlook

Arteries

J Humphrey, Cardiovascular Solid Mechanics, Springer, 2002

Abdominal Aortic Aneurysm

C.E. Ruiz, et al, Circulation, 1997;96:2438-2448

Structural changes in the artery

Dramatic decrease in elastin and smooth muscle cell content

Increase in collagen Degradation of arterial resistance

to the blood pressure

Treatment

Surgical placement of stent-grafts Extensive surgery – not all patients

are suitable for this procedure Endovascular placement of stents

and stent-grafts Quick, simple procedure – currently

still experimental Long-term consequences are not well

characterized

AnueRx Bifurcated Stent

WallStent

Villareal, Howell, and KrajcerTex Heart Inst J 2000;27:146-9

Abdominal Aortic Aneurysm

Stent-graft Stent

C.E. Ruiz, et al, Circulation, 1995;91:2470-2477

Abdominal Aortic Aneurysm

Short term

Reduction in the size of aneurysm

Long termDilation of proximal side of artery

Wever et al., (2000), European Journal of Vascular and Endovascular Surgery, 19: 197–201.

Endoleaks Chuter et al. (2001), Journal of Vascular Surgery, 34, 98–105.

Migration and other forms of failure Shames, Sanchez, Rubin and Sicard, (2002) Vascular and Endovascular Surgery, 36, 77-83

Bell (2002), Editorial, Vascular Medicine 7, 253–255

Our Objectives

Evaluate the mechanical response of the stent in appropriate configurations – Experimental

Develop the appropriate structural mechanics description - Analytical

Evaluate the coupled response of the stent and the blood vessel - Analytical

Experimental apparatus

Stent

Balloon

Pressure Meter

Air I n

Pressure Meter

Solid Tube

Air In

Balloon

Stent

Internal Pressure External Pressure

Pressure-diameter relationship

-100

-50

0

50

100

150

0 5 10 15 20 25

Diameter - mm

Pre

ssu

re -

mm

of

Hg

Internal Pressure

External Pressure

No friction

with friction

Normal range of the aorta

Pressure-length relationship

-100

-50

0

50

100

150

0 10 20 30 40 50 60 70 80

Length - mm

Pre

ssu

re -

mm

of

Hg

Internal Pressure

External Pressure

No friction

with friction

Helical spring model – Kirchhoff-Love theory

a

r0, 0 – initial radius and pitch angle

r, – radius and pitch angle

Pa – axial force; Ps – transverse shear force

MB – bending moment; Mt – twisting moment

q – effect of pressure loading on the wire

Fz – external axial force

Equilibrium equations

0cos

sincos

qrPP sa

zsa FPP cossin

0cos

sincossin)cossin(

2

2

qr

MMPPr tBsa

(1)

(2)

(3)

Pressure loading

pn

rp

nl

rq

sin22

The internal pressure loading is distributed uniformly over the n wires, resulting in the load distribution q:

Curvature and twist evolution

0

02

0

cos

r

0

000

cossin

r

0

022

0

coscos)(

rrEIEIM B

0

000

sincossincos)(

rrGIGIM ppt

r

2cos

r

cossin

Curvature:

Twist:

Bernoulli-Euler Beam Theory:

Coulomb Torsion Theory:

(4)

(5)

Geometrical constraint

The braiding of the n wires results in contact at the cross-over points; these are constrained frictionally and therefore the wire is not allowed to unwind helically as the stent expands. This can be expressed as a constraint between the radius and the pitch angle of the helix:

0

0

coscos rr

(6)

Pressure-diameter relationship

0

00

0

022

22

2

sincossincoscos

coscossin

sin2

cos

rrr

GI

rrr

EI

r

np

p

This is an exact relationship within the restrictions of the Kirchhoff-Love slender rod theory, without any adjustable parameters.

Parameters of the stent

 

n Number of wires 36E Modulus of elasticity 200 GPaG Shear modulus 77 GPaa Radius of the stent wire 0.170 mPitch angle of the helix at zero pressure 34r Radius of the stent at zero pressure 0.01 m

L Length of the stent 0.08 m

Effect of friction

Friction acts on the cylindrical surface of the stent in the axial direction, and is given by:

0

00

22200

03

sinsincos

coscossin

cossin2

cos

rGI

rEI

rnrr

np

p

rpF f 2

Pressure-diameter relationship

-100

-50

0

50

100

150

0 5 10 15 20 25

Diameter - mm

Pre

ssu

re -

mm

of

Hg

Internal Pressure

External Pressure

No friction

with friction

Pressure-length relationship

-100

-50

0

50

100

150

0 10 20 30 40 50 60 70 80

Length - mm

Pre

ssu

re -

mm

of

Hg

Internal Pressure

External Pressure

No friction

with friction

Axial force measurements

Balloon

StentString

Pressure Meter

Air I n

Rigid Post

Weight

0

00

0

022

2

22 sincossincoscoscoscossin

cos

sin2

rrr

GI

rrr

EIp

n

rF p

z

Axial force measurement

0

1

2

3

4

5

6

7

0 20 40 60 80 100 120 140 160 180

p - mm of Hg

Fz -

N

Theory

Experiment

Spatially varying pressure – A beam-on-elastic-foundation model

x2r0

v(x)

r(x)

(a)

aeff prfxd

rdEI )(

4

4

~f(r)

Governing differential equation

a

p

p

rrr

GI

rrr

EI

r

n

xd

rd

r

naE

0

00

0

022

22

2

4

44

cossincossincoscoscossin

sin2

cos

sin8

0 , dx

drRrFixed boundary:

Free boundary:

Compliant boundary:

0 ,02

2

3

3

dx

rd

dx

rd

vesselstentvesselstent dx

dr

dx

drrr ,

Example 1 –Fixed ends

String

Clamp

Plastic Block

Air I n

Stent

Balloon

Metallic Tube

Comparison to experiments

0.8

0.9

1

1.1

1.2

0 0.2 0.4 0.6 0.8 1

x/L

r/R

ExperimentalData

Beam on elastic foundation modeln

Example 2 – stent exiting a catheter

0

5

10

15

0 10 20 30 40 50 60 70 80

x - mm

r -

mm

Experimental Data

Beam on elastic foundation model

Coupled response of the aorta and stent

Response of the aorta

Curve fit to data from:Länne T et al 1992, Noninvasive measurement of diameter changes in the distal abdominal aorta in man, Ultrasound in Med & Biol,18:451-457.

Coupled response - results

5

7

9

11

13

-15 0 15 30 45 60 75

Axial Position(mm)

Rad

ius

(mm

)

AortaAorta

Stent

Aneurysm RLa

RHa

Aorta

PH=150mmHgPL=65mmHgRH

a=8.73mmRL

a=7.75mm

Summary

Experimental methods developed to evaluate the mechanical response of stentsAnalytical models were developed to characterize the response of the stent by itself and coupled with the aortaThe procedures established should enable design of AAA stentsProf Canic is working on embedding these models with fluid flow simulations

Dilation of the aorta

Source:Wever et al., (2000), European Journal of Vascular and Endovascular Surgery, 19: 197–201.