SMST-2010: Open Source Stent Calculator by NDC

Open Source Stent Calculator SMST 2010

Craig Bonsignore NDC craig.bonsignore@nitinol.com Monday, May 17, 2010

Acknowledgments Deborah Tolomeo Tom Duerig Aaron Kueck Payman Saffari

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• Introduction to Open Source • Introduction to Stent Designs • Structural Mechanics • Stent Calculator Applications • Community and Resources

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ASPECT

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• Introduction to Open Source • Introduction to Stent Designs • Structural Mechanics • Stent Calculator Applications • Community and Resources

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Stent Designs

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• Introduction to Open Source • Introduction to Stent Designs • Structural Mechanics • Stent Calculator Applications • Community and Resources

A simple stent pattern

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Array of “Z” shaped elements

Array of “V” shaped elements

Array of Simple BEAMS

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Beam, fixed at one end, free but guided at the other

Stress = σ =

FL 2Z

FL 2

12EIδ Force = F = L3

FL 2

E = modulus of elasticity

I = moment of inertia, beam cross section Z = section modulus, beam cross section =

W = load on beam

I distance from neutral axis to extreme fiber

Solve for STRAIN and FORCE 3w Strain = ε = 2 ⋅ δ L

FL 2

12EI Force = F = 3 ⋅ δ L

FL 2

E = modulus of elasticity

I = moment of inertia, beam cross section w = Strut Width L = strut length

Strain and Force for a Simple Beam Stent

Etw F = 3 !" ! L

3w ! = 2 "# ! L

w !

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Stent Calculator Relating design INPUTS and OUTPUTS tube diameter modulus of elasticity density mass absorption coefficient number of struts strut width strut length strut thickness material removal transition temperature vessel diameter vessel compliance systolic pressure diastolic pressure

strength / stiffness scaffolding constrained profile expansion angles foreshortening surface area contact area percent open area contact pressure mean strain strain amplitude safety factor radiopacity

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mean strain strain amplitude safety factor

Understanding: Radial Pulsatile Fatigue

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Service History and Pulsatile Fatigue Durability ➊ Stent is manufactured at a small diameter ➋ Stent is expanded to a large diameter and shape set ➌ Stent is constrained to a small diameter in a delivery sheath ➍ Stent is deployed into a vessel with diameter X ➎ Outward force of the stent causes the vessel to expand to X+ ➏ At diastolic blood pressure, the stented vessel contracts slightly ➐ At systolic blood pressure, the stented vessel expands slightly ➑ The stent experiences fatigue loading as the pressure cycles

diameter

➊

➋

➌

➍

➎

➏

➐

➑

Systolic Diameter Diastolic Diameter Balanced Diameter Vessel Diameter

➑

Service History and Pulsatile Fatigue Durability ➋

➌

➍

diameter

➊

➎

➏

➐

➑

Systolic Diameter Diastolic Diameter Balanced Diameter Vessel Diameter

Systolic Pressure / Diameter / Strain

Strain Amplitude

Mean Strain 1

-3

-2

-1

-2

Diastolic Pressure / Diameter / Strain

• Introduction to Open Source • Introduction to Stent Designs • Structural Mechanics • Stent Calculator Applications • Community and Resources

Stent Calculator Worksheet: 34 Design Inputs !"# !"4 !"9 !"! !"= !"C !"A !"I !"< !"#3

!"#$"%&#'()$%*$+,"' $%&'( )*+,-./'0/&'(*+)1 $%15.*51 15.*51/6.'*)7/&8.&*+0-.-)&:%5*,'*5-./786+5-./'0/5*,8); 5%.6> >6((/5?8&@)-11/'0/.6>/5*,8); B%15.*5%8))-. 15.*5/(-);5?/5'/8))-./56);-)51 >%6D-E%.6> 6D-E/>875?F/61G&*5 H%,.87;6E86(/;6D/,-5>--)/'*5-./56);-)51 J%,.87;&8.&*+0-.-)K6(/1D6)/'0/,.87;>%,.87;-%.6> >875?/'0/,.87;$%,.87;-1 )*+,-./'0/,.87;-1/6.'*)7/&8.&"

-$("' 2 2 ++ ++ ++ ++ ++ ++ ++ 2

./0,# #3 !4 #"<#= 3"#A3 #"433 3"#93 3"#=3 3"333 3"#4= A

!"## !"#4 !"#9 !"#!

1234#''%1/2/5#"#2' >%@-.0 +8)8+*+/-L-&85M-/@-.0/>875? +%>875? >875?/.-+'M6(/8)/N)81?8); +%5?8&@)-11 >6((/5?8&@)-11/.-+'M6( O0 O0/'0/N)81?-7/&'+D')-)5

++ ++ ++ 7-;P

3"34= 3"39C 3"3=< 4A

!"#= !"#C !"#A !"#I !"#< !"43 !"4# !"44 !"49 !"4! !"4= !"4C

6/"#2(/0%123+#27#' Q%O0%('> +'7*(*1/'0/-(61K&85R/65/B'>/O0 O0%('> B'>/O0/0'./7-N)8);/Q Q%O0%?8;? +'7*(*1/'0/-(61K&85R/65/O0/T8;? O0%?8;? T8;?/O0/0'./7-N)8);/Q O0%8)U-&K') V)U-&K')/D'8)5/8)/Q/M1/O0 Q%&61-# Q/0'./O0/W/O0%('> Q%&61-4 Q/0'./O0%('>/W/O0/W/O0%8)U-&K') Q%&61-9 Q/0'./O0%8)U-&K')/W/O0/W/O0%?8;? Q%&61-# Q/0'./O0/Y/O0%?8;? Q +'7*(*1/'0/-(61K&85R/65/1D-&Z-7/O0 7-)185R%)8K 7-)185R/'0/$8K)'( 15.68)%-)7*.6)&- -)7*.6)&-/(8+85

SD6 7-;P SD6 7-;P 7-;P SX6 SX6 SX6 SX6 SX6 +;[++\9 ]

<!333 G= 9!333 9A #< <!333 9#A!# 9!3=< 9!333 9!3=< C"A 3"!3]

!"4A !"4I !"4< !"93 !"9# !"94 !"99 !"9!

!#28(4#%1/2/5#"#2' :%1-5 -ED6)7-7/8))-./786+-5-./'0/15-)5 :%M-1 786+-5-./'0/M-11-( : 6)6(R181/786+-5-. P%D-.&-)5 &'+D(86)&-^/]/&?6);-/8)/786+-5-. P%D.-11*.&'+D(86)&-^/D.-11*.-/0'./]/&?6);X%1R15'(8& 1R15'(8&/D.-11*.-/65/8+D(6)5/185X%78615'(8& 78615'(8&/D.-11*.-/65/8+D(6)5/185X%+-6) +-6)/D.-11*.-/65/8+D(6)5/185-

++ ++ ++ ] ++T; ++T; ++T; ++T;

I"33 C"=3 C"=3 C] #33 #=3 =3 #33

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Stent Calculator Worksheet: 78 Design Outputs !"#$ !"#9 !"#? !"#G !"#> !"!F !"!8 !"!7

!"#$"%&'(#$)'*$%+,-.) %&'())* +,-.(/0120'())*03/1,+40'5/' :&'/5-; 2,))<0'1+*=/35+(401,=(/0453-(=(/ @&*=/,= (A('BC(0)(+D=E0120*=/,= H&*=/,=&/3H H54=E0120*=/,=I03*J',= H&*=/,= H54=E0120*=/,=I0K+5*E(4 H&./54D( H54=E0120./54D(I0K+5*E(4 H&3;(L H54=E01203;(LI0K+5*E(4 = H3))0=E5'M+(**I0K+5*E(4

6 --------

78 8">8$ 8"##F F"88G F"FG7 F"FG> F"F>! F"888

!"!# !"!! !"!$ !"!9 !"!? !"!G !"!> !"$F

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4(D 4(D 4(D 4(D -4(D 4(D --

7F"F 8!"> $"8 !F"F F"88G 89"9 #"! F"F?>

!"$8 !"$7 !"$# !"$! !"$$

6#$2"7%+,-.) P&'())&'/5-; P&=1=3)&'/5-; P&'()) P&=1=3) QR

----S

8"98F 8$">$F 8"$9> 8$"$!! 7"$!S

!"$9 !"$? !"$G !"$> !"9F !"98 !"97 !"9# !"9! !"9$ !"99 !"9? !"9G !"9> !"?F !"?8 !"?7

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--U7 ---U7 --U7 --U7 --U7 S S ---U7 --U7 --U7 --U# --U# --U# --U# -D

F"F>? F"88# F"F78 F"F8# $F"# #8?"! 8$">S G!"8S F"F99 F"F?? F"F8? F"F88 F"F8F F"FF7 F"FF8 $"F78 ##"9!F

!"?# !"?! !"?$ !"?9 !"??

;*(#$"%*8%=$#/5, V = H 3);E3 [

---/34 --U!

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% % S S

G"G7\JF7 $">8\JF7 8"9!S 8"8FS

)(+D=E0120'())05+0'/5-;(40*=3=( )(+D=E0120*=(+=05+0'/5-;(40*=3=( )(+D=E0120'())03=03+3)<*5*0453-(=(/ )(+D=E0120*=(+=03=03+3)<*5*0453-(=(/ 21/(*E1/=(+5+D03=03+3)<*5*0453-(=(/

1,=(/0/345,*0120=,.( H3))0=E5'M+(** *=/,=0H54=E 3);E30;3/3-(=(/ -1-(+=01205+(/B3

!"#$ !"#> !"#! !"#I !"#K

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;<9;<9--EF JB. ;<--

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!"#A !"## !"#H !"H= !"HG

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OB. --;<-;<--

="=G> K"I== K"#H= ="=A$ ="G#I

!"H$ !"H> !"H! !"HI !"HK !"HA

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-------

K"I= K">G K"A= K"#K K"AG A"=G

(/"%/'9#;#45.+$'0/'7020+4#8'9*0:#/#"$ !"H# ?(2?.'-2.+ 5?,@?4.+F/24.?4S./.+923430.-2?2, !"HH 32/?.'?(2?.'-2.+ 9(.+F240+45?,@?4.+F/24.?4S./.+923430.!"G== 32/?.'-2.+ 5?,@?432U296)+4.?4S./.+923430.-2?2,

32F 32F --

GK"GI! >"#I# ="=H=

!"G=G ?(2?.'30.5?)/09 5?,@?4.+F/24.?430.5?)/094*,255@,2 !"G=$ 32/?.'?(2?.'30.5?)/099(.+F240+45?,@?4.+F24.?430.5?)/094 !"G=> 32/?.'30.5?)/09 5?,@?432U296)+4.?430.5?)/09

32F 32F --

GI"KIH !">I! ="G=G

!"G=! ?(2?.'5R5?)/09 5?,@?4.+F/24.?45R5?)/094*,255@,2 !"G=I 32/?.'?(2?.'5R5?)/099(.+F240+45?,@?4.+F24.?45R5?)/09 !"G=K 32/?.'5R5?)/09 5?,@?432U296)+4.?45R5?)/09

32F 32F --

GK"KIG >">KG ="=A#

V V V V

G"K!V G"$IV G"!GV G"=HV

V V W

G"$IV ="GKV $"!H

!"G=A !"G=# !"G=H !"GG=

(/"0*+'102%#$ 5?,.0+'72552/ 5?,.0+'S./.+923 5?,.0+'30.5?)/09 5?,.0+'5R5?)/09

5?,.0+4.?472552/430.-2?2, 5?,.0+4.?4-2.+4S./.+923430.-2?2, 5?,.0+4.?430.5?)/094*,255@,2 5?,.0+4.?45R5?)/094*,255@,2

305,%#'(0<#/='304/." !"GGG 5?,.0+'-2.+ -2.+45?,.0+ !"GG$ 5?,.0+'.-*/0?@32 5?,.0+4.-*/0?@32 !"GG> ;'58 8.6F@245.82?R48.9?),

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Stent Calculator Worksheet: What-If Analysis

!"# !"4 !"9 !"! !"= !"C !"A !"I !"< !"#3

-$("' 2 2 ++ ++ ++ ++ ++ ++ ++ 2

./0,# #3 !4 #"<#= 3"#A3 12311 3"#93 3"#=3 3"333 3"#4= A

./0,# #3 !4 #"<#= 3"#A3 12411 3"#93 3"#=3 3"333 3"#4= A

./0,# #3 !4 #"<#= 3"#A3 12511 3"#93 3"#=3 3"333 3"#4= A

./0,# #3 !4 #"<#= 3"#A3 62111 3"#93 3"#=3 3"333 3"#4= A

./0,# #3 !4 #"<#= 3"#A3 62611 3"#93 3"#=3 3"333 3"#4= A

./0,# #3 !4 #"<#= 3"#A3 62711 3"#93 3"#=3 3"333 3"#4= A

./0,# #3 !4 #"<#= 3"#A3 62811 3"#93 3"#=3 3"333 3"#4= A

./0,# #3 !4 #"<#= 3"#A3 62911 3"#93 3"#=3 3"333 3"#4= A

./0,# #3 !4 #"<#= 3"#A3 62:11 3"#93 3"#=3 3"333 3"#4= A

./0,# #3 !4 #"<#= 3"#A3 62;11 3"#93 3"#=3 3"333 3"#4= A

./0,# #3 !4 #"<#= 3"#A3 62311 3"#93 3"#=3 3"333 3"#4= A

!"## !"#4 !"#9 !"#!

<=>?#''%</=/@#"#=' >%@-.0 +8)8+*+/-L-&85M-/@-.0/>875? +%>875? >875?/.-+'M6(/8)/N)81?8); +%5?8&@)-11 >6((/5?8&@)-11/.-+'M6( O0 O0/'0/N)81?-7/&'+D')-)5

++ ++ ++ 7-;P

3"34= 3"39C 3"3=< 4A

!"#= !"#C !"#A !"#I !"#< !"43 !"4# !"44 !"49 !"4! !"4= !"4C

A/"#=(/0%<=>+#=B#' Q%O0%('> +'7*(*1/'0/-(61K&85R/65/B'>/O0 O0%('> B'>/O0/0'./7-N)8);/Q Q%O0%?8;? +'7*(*1/'0/-(61K&85R/65/O0/T8;? O0%?8;? T8;?/O0/0'./7-N)8);/Q O0%8)U-&K') V)U-&K')/D'8)5/8)/Q/M1/O0 Q%&61-# Q/0'./O0/W/O0%('> Q%&61-4 Q/0'./O0%('>/W/O0/W/O0%8)U-&K') Q%&61-9 Q/0'./O0%8)U-&K')/W/O0/W/O0%?8;? Q%&61-# Q/0'./O0/Y/O0%?8;? Q +'7*(*1/'0/-(61K&85R/65/1D-&Z-7/O0 7-)185R%)8K 7-)185R/'0/$8K)'( 15.68)%-)7*.6)&- -)7*.6)&-/(8+85

SD6 7-;P SD6 7-;P 7-;P SX6 SX6 SX6 SX6 SX6 +;[++\9 ]

<!333 G= 9!333 9A #< <!333 9#A!# 9!3=< 9!333 9!3=< C"A 3"!3]

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Trend Analysis: Diameter

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Trend Analysis: Strain v. Diameter

1.8% 1.6% 1.4%

Strain

1.2% 1.0% 0.8% 0.6% 0.4% 0.2% 0.0% 5.9

6.1

6.3

6.5

6.7

6.9

7.1

Vessel Diameter (mm) mean strain strain amplitude

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Trend Analysis: Fatigue Safety Factor v. Diameter

3.0 2.5

Nsf

2.0 1.5 1.0 0.5 0.0 5.9

6.1

6.3

6.5

6.7

6.9

7.1

Vessel Diameter (mm) Fatigue Safety Factor

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Trend Analysis: Wall Thickness

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Trend Analysis: Radial Force v. Wall Thickness 0.6 0.5

RRF (N/cm)

0.4 0.3 0.2 0.1 0.0 0.10

0.12

0.14

0.16

0.18

0.20

0.22

0.24

Starting Wall Thickness (mm) Radial Resistive Force

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Trend Analysis: Strain v. Wall Thickness

1.6% 1.4% 1.2%

Strain

1.0% 0.8% 0.6% 0.4% 0.2% 0.0% 0.10

0.12

0.14

0.16

0.18

0.20

0.22

0.24

Starting Wall Thickness (mm) mean strain strain amplitude

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Trend Analysis: Fatigue Safety Factor v. Wall Thickness

3.0 2.5

Nsf

2.0 1.5 1.0 0.5 0.0 0.10

0.12

0.14

0.16

0.18

0.20

0.22

0.24

Starting Wall Thickness (mm) Fatigue Safety Factor

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Trend Analysis: Strut Length

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Trend Analysis: Balanced Diameter v. Strut Length

7.8 7.6

Diameter (mm)

7.4 7.2 7.0 6.8 6.6 6.4 6.2 0.5

0.7

0.9

1.1

1.3

1.5

1.7

1.9

Strut Length (mm) Balanced Diameter, Diastolic Pressure Balanced Diameter, Systolic Pressure

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Trend Analysis: Strain v. Strut Length

1.8% 1.6% 1.4%

Strain

1.2% 1.0% 0.8% 0.6% 0.4% 0.2% 0.0% 0.5

0.7

0.9

1.1

1.3

1.5

1.7

1.9

Strut Length (mm) Mean Strain Strain Amplitude

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Trend Analysis: Stiffness v. Strut Length

0.30 0.25

k (N/mm)

0.20 0.15 0.10 0.05 0.00 0.5

0.7

0.9

1.1

1.3

1.5

1.7

1.9

Strut Length (mm) stent stiffness vessel stiffness

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Trend Analysis: Radial Force v. Strut Length 3.0 2.5

RRF (N/cm)

2.0 1.5 1.0 0.5 0.0 0.5

0.7

0.9

1.1

1.3

1.5

1.7

1.9

Strut Length (mm) Radial Resistive Force

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Trend Analysis: Fatigue Safety Factor v. Strut Length 4.5 4.0 3.5

Nsf

3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.5

0.7

0.9

1.1

1.3

1.5

1.7

1.9

Strut Length (mm) Fatigue Safety Factor

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Beyond Excel

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Stent Calculator Python Script

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Input Parameter Variation Distributions w_strut

Af 0.13

D_ves 29

0.1

7.4 7.2 0.12

0.09

6.8 6.6 0.11

6.4

0.08

6.2 0.1

0.07

5.8 5.6

Quantiles 100.0% maximum 99.5% 97.5% 90.0% 75.0% quartile 50.0% median 25.0% quartile 10.0% 2.5% 0.5% 0.0% minimum

Quantiles 0.10188 0.09478 0.09224 0.08877 0.08571 0.0822 0.07887 0.07568 0.07245 0.06979 0.0654

Moments Mean Std Dev Std Err Mean Upper 95% Mean Lower 95% Mean N

100.0% maximum 99.5% 97.5% 90.0% 75.0% quartile 50.0% median 25.0% quartile 10.0% 2.5% 0.5% 0.0% minimum

Quantiles 0.12942 0.12357 0.12064 0.11727 0.11436 0.11103 0.10764 0.10451 0.10118 0.0979 0.09246

Moments 0.0822706 0.0050412 0.0000713 0.0824104 0.0821309 5000

Mean Std Dev Std Err Mean Upper 95% Mean Lower 95% Mean N

100.0% maximum 99.5% 97.5% 90.0% 75.0% quartile 50.0% median 25.0% quartile 10.0% 2.5% 0.5% 0.0% minimum

Quantiles 28.9812 28.3039 27.9808 27.6372 27.3482 27.0077 26.6784 26.3731 26.0336 25.7423 25.2261

Moments 0.1109817 0.0049797 7.0423e-5 0.1111197 0.1108436 5000

Mean Std Dev Std Err Mean Upper 95% Mean Lower 95% Mean N

100.0% maximum 99.5% 97.5% 90.0% 75.0% quartile 50.0% median 25.0% quartile 10.0% 2.5% 0.5% 0.0% minimum

Moments 27.010197 0.497228 0.0070319 27.023983 26.996412 5000

Mean Std Dev Std Err Mean Upper 95% Mean Lower 95% Mean N

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7.47098 7.12231 6.98744 6.81438 6.67221 6.50646 6.33404 6.18413 6.02162 5.85584 5.58439 6.5035834 0.2462738 0.0034828 6.5104113 6.4967555 5000

Performance Output Variation

Distributions mass

RF_hoop

P_contact 240

220

0.8

200

0.7

180

37 0.6

35 33

160 140

0.5

120

31 0.4

29 27

100 80

0.3

60 0.2

Quantiles 43.711 40.6728 38.9594 37.0779 35.4285 33.5771 31.8344 30.245 28.5801 27.189 23.0055

Moments

100.0% maximum 99.5% 97.5% 90.0% 75.0% quartile 50.0% median 25.0% quartile 10.0% 2.5% 0.5% 0.0% minimum

Quantiles 0.86631 0.69305 0.62551 0.56028 0.49934 0.44051 0.38689 0.34204 0.29576 0.26879 0.19739

Moments 33.641953 2.6485309 0.0374559 33.715383 33.568523 5000

Mean Std Dev Std Err Mean Upper 95% Mean Lower 95% Mean N

100.0% maximum 99.5% 97.5% 90.0% 75.0% quartile 50.0% median 25.0% quartile 10.0% 2.5% 0.5% 0.0% minimum Mean Std Dev Std Err Mean Upper 95% Mean Lower 95% Mean N

0.019 0.018 0.017

0.0018

0.016 0.015 0.014

0.0017

0.013 0.012

0.0016

0.0015

0.008 0.007 0.006

0.0014

Mean Std Dev Std Err Mean Upper 95% Mean Lower 95% Mean N

2.7 2.6 2.5 2.4 2.3

0.0013

2.2

Quantiles 0.01929 0.01716 0.01582 0.01447 0.01327 0.01188 0.01058 0.00942 0.00804 0.00693 0.00429

Moments 129.27845 26.814249 0.3792107 130.02187 128.53503 5000

2.8

0.011 0.01 0.009

100.0% maximum 99.5% 97.5% 90.0% 75.0% quartile 50.0% median 25.0% quartile 10.0% 2.5% 0.5% 0.0% minimum

N_sf

2.9

Quantiles 239.065 210.871 185.473 163.718 146.462 127.811 110.679 96.0923 81.1837 67.4174 44.0718

Moments 0.446724 0.0846035 0.0011965 0.4490696 0.4443784 5000

strain_amplitude

0.005 0.004

Quantiles

100.0% maximum 99.5% 97.5% 90.0% 75.0% quartile 50.0% median 25.0% quartile 10.0% 2.5% 0.5% 0.0% minimum Mean Std Dev Std Err Mean Upper 95% Mean Lower 95% Mean N

strain_mean

100.0% maximum 99.5% 97.5% 90.0% 75.0% quartile 50.0% median 25.0% quartile 10.0% 2.5% 0.5% 0.0% minimum

Quantiles 0.00181 0.00171 0.00167 0.00162 0.00158 0.00154 0.00149 0.00145 0.0014 0.00137 0.00131

Moments 0.0119248 0.0019716 2.7882e-5 0.0119794 0.0118701 5000

Mean Std Dev Std Err Mean Upper 95% Mean Lower 95% Mean N

3.04766 2.92413 2.84917 2.75368 2.67807 2.59938 2.52502 2.46235 2.39731 2.33963 2.20963

Moments 0.001539 0.0000669 9.4597e-7 0.0015409 0.0015372 5000

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100.0% maximum 99.5% 97.5% 90.0% 75.0% quartile 50.0% median 25.0% quartile 10.0% 2.5% 0.5% 0.0% minimum Mean Std Dev Std Err Mean Upper 95% Mean Lower 95% Mean N

2.6040063 0.1137918 0.0016093 2.6071611 2.6008514 5000

Predicting Relationships: Radial Force vs. Mass Bivariate Fit of RF_hoop By mass

0.8

0.7

RF_hoop

0.6

0.5

0.4

0.3

0.2 23

mass

Polynomial Fit Degree=2

Polynomial Fit Degree=2 RF_hoop = -0.55894 + 0.0297866*mass + 0.0005111*(mass-33.642)^2

Summary of Fit RSquare RSquare Adj Root Mean Square Error Mean of Response Observations (or Sum Wgts)

0.88079 0.880742 0.029217 0.446724 5000

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Fatigue Performance: Constant Life Diagram Bivariate Fit of strain_amplitude By strain_mean

0.0018

strain_amplitude

0.0017

0.0016

0.0015

0.0014

0.0013 0.004

0.006 0.007 0.008 0.009 0.01 0.011 0.012 0.013 0.014 0.015 0.016 0.017 0.018 0.019 strain_mean

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• Introduction to Open Source • Introduction to Stent Designs • Structural Mechanics • Stent Calculator Applications • Community and Resources

Open Stent Design: The Book

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Open Stent Design: SolidWorks

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Open Stent Design: Calculator

CHAPTER 4. STENT CALCULATOR FORMULAS

4.10

Force and Strain Calculations

The relationships between stress, load, deflection, and strain have been thoroughly documented for a variety of beam loading conditions. Force and strain related to a specified strut deflection are based on the formulation for a beam fixed at one end, and free but guided at the other as documented in Machinery’s Handbook [1]. Strain = ! =

Force = F =

3w "# L2

FL 2

CHAPTER 4. STENT CALCULATOR FORMULAS

�d is the maximum strain experienced within the strut when the stent is constrained from the fully expanded state to the analysis diameter. This is equal to epsilon in Figure 4.5 by the definition of the ”free but guided” beam as described in Machinery’s Handbook [1]. 3wstrut · δd (Lstrut )2 �d = 1.64 % �d =

12EI !" L3

I = moment of inertia, beam cross section w = strut width

�1mm

L = strut length

Figure 4.5: Beam fixed at one end, and free but guided at the other.

4.11

Fhoop is the hoop component of the force exerted by a single strut when the stent is constrained from the fully expanded state to the analysis diameter. This is equal to F in Figure 4.5 by the definition of the ”free but guided” beam as described in Machinery’s Handbook [1].

Fhoop

3wstrut · δ1mm (Lstrut )2 = 1.10 %

�1mm =

12 · E · I · δd (Lstrut )3 = 1.03 · 10−1 N

(4.78)

Fhoop 1mm is the hoop component of the force exerted by a single strut when the stent is constrained from the fully expanded state to a diameter one millimeter less than the analysis diameter. This allows for later calculation of stent forces normalized per millimeter diameter constraint.

12 · E · I · δ1mm (Lstrut )3 = 6.92 · 10−2 N

Fhoop 1mm = Fhoop 1mm

(4.79)

(4.81)

Pressure and Stiﬀness Calculations

In this section, the forces and other calculations derived above are used to estimate radial resistive force in terms that are common for bench testing. RFhoop is the hoop component of the force exerted when the stent is constrained from the fully expanded state to 1mm less than the expansion diameter, normalized by length in centimeters. This value is consistent with radial resistive force type measurement (RRF) generated from a collar type fixture. By convention, it is expressed in terms of Newtons per centimeter length, and is thus normalized by length. Fhoop 1mm � mm � · 10 · Xcell cm = 0.44 N/cm

RFhoop = RFhoop

(4.82)

RFtrf is the true radial component of the force exerted when the stent is constrained from the fully expanded state to 1mm less than the expanded diameter, normalized by length in centimeters. This value is consistent with radial resistive force type measurement (RRF) generated from a Blockwise or MSI type testing fixture. This is also expressed in terms of newtons per centimeter length, and is thus also normalized by length, and evaluated for a 1mm diameter constraint.

(4.80)

�1mm is the maximum strain experienced within the strut when the stent is constrained from the fully expanded state to one millimeter less than the analysis diameter.

FL 2

E = modulus of elasticity

Fhoop =

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