1. Università degli Studi di Napoli Federico II
FACOLTA’ DI INGEGNERIA
CORSO DI LAUREA IN
INGEGNERIA BIOMEDICA
(CLASSE DELLE LAUREE IN INGEGNERIA DELL’INFORMAZIONE n.9)
“Homogenized trigonal models
for biomechanical applications”
Relatore: Candidato:
Ch.mo Prof. Ing. CIERVO MARCO
MASSIMILIANO FRALDI
Matr. 691/939
Correlatore:
GIANPAOLO PERRELLA

2. Description
Hierarchical structures: importance in biomechanics
Ligaments: Anterior Cruciate Ligament (ACL)
Healing ACL ruptures: 6-cord wire-rope scaffold
Analytical model: George A. Costello’s theory of the wire rope
Homogenized Trigonal model: example of development

3. Anterior Cruciate Ligament
Ligaments hierarchical structure
Viscoelastic behaviour

4. ACL Kinematics
Flexion: 46.3 deg. Flexion: 44.8 deg.
Abduction: 0 deg. Abduction: 10.0 deg.
External rotation: 0 deg. External rotation: 29.1 deg.
A. Guadagno,Relatore Ch.mo Prof. Ing. A. Pepino e Correlatore Ing. A. Ranavolo J Biomech. 2010 July 20; 43(10): 2039–2042. doi:10.1016/j.jbiomech.2010.03.015.
Napoli, 2004/2005. A Knee-Specific Finite Element Analysis of the Human Anterior
VALUTAZIONE DELLE FORZE COMPRESSIVE E DI TAGLIO ALL’ARTICOLAZIONE DEL GINOCCHIO Cruciate Ligament Impingement against the Femoral
CON SISTEMA DI ANALISI COMPUTERIZZATA MULTIFATTORIALE DEL MOVIMENTO, Intercondylar Notch
Hyung-Soon Park1,†, Chulhyun Ahn,†, David T. Fung, Yupeng Ren, and Li-Qun
Zhang

5. ACL Kinematics
Journal of Biomechanics 38 (2005) 293–298
Interactions between kinematics and loading during
walking for the
normal and ACL deficient knee
Thomas P. Andriacchia, Chris O. Dyrby

6. Altman’s Scaffold
Biomaterials 23 (2002) 4131–4141
Silk matrix for tissue engineered anterior cruciate
ligaments
Gregory H. Altmana, Rebecca L. Horana, Helen H.
Lua, Jodie Moreaua, Ivan Martinb,
John C. Richmondc, David L. Kaplana

7. Analytical Model
core wires h1 h2 h3

8. Kinematics of a wire
T axial tension in the wire
H twisting moment in the wire
N, N’ shear force in the wire, along the local cooordinates system directions
G, G’ bending moment in the wire in x and y directions, along the local coordina-
tes system
κ, κ’ curvature of the wire in x and y directions, along the local coordinates
system
т twist per unit length of the wire
Mechanical Engineering Series, Springer - Theory of Wke Rope, 2nd ed. - George A. Costello

9. Simple straigth strand
Rc core radius Geometrical characteristics
Rw wire radius
α wire helix angle
R = Rc+2Rw strand total radius
r = Rc+Rw strand helical radius
E Young Modulus Material properties
v Poisson’s ratio
ξc = ε core axial strain Deformations
Δα
ξw = ξc - Tg α wire axial strain
βr = Tg α - Δα + ν Rc ξc + Rw ξw helical rototional strain
ξw
r Tg α
β total rototional strain
т = r βr = R β angle of twist per unit lenght
Costitutive assumptions
Δт’ = 1 - 2 Sin α Δα + ν Rc ξc +2 Rw ξw Sin α Cos α
2
r r  F / AE   kεε kεβ   ε 
M / ER 3  = k k ββ   β 
Δκ’ = - 2 Sin α Cos α Δα + ν Rc ξc +2 Rw ξw Cos2 α curve variation  t   βε  
r r
Core Loads Hipothesis of small displacements:
Fc = � E Rc2 ξc Mc = E Rc4 т’ � Δα<<1, ξ<<1
4 (1 + v)
Hw = E Rw4 Δт’ � Wire Loads
Nw’ = Hw Cos α - Gw’ Sin α Cos α
2
Tw = � E Rw2 ξw 4 (1 + v)
r r
Gw’ = E Rw4 Δκ’ � Fw = mw (Tw Sin α + Nw’ Cos α) Mw = mw (Hw Sin α + Gw’ Cos α + r Tw Cos α + r Hw Sin α )
4

10. 30 Fibers
R1 = [19 ± 2.8]10-6 m
1 Bundle
Silk fibroin average radius
Rf = 10.4067 10 m
Biomaterials 24 (2003) 401–416
-5 Silk-based biomaterials
Gregory H. Altman,Frank Diaz,Caroline Jakuba,Tara
Equivalent fiber radius Calabro,Rebecca L. Horan,
Jingsong Chen,Helen Lu,John Richmond, David L.
Kaplan
On the right pilot-scale manu-
facturing equipment for the
fabrication of silk wire-rope
matrices
Two particluars (B) and (C) of
(i) and (iii), showing the
extraction of the fibroins
On the left a close-up view of
(i): the twisting machines
showing the motor controlled
spring-loaded clamps.

11. 6 Bundles
Rch3 = R2 Geometrical characteristics
Rwh3 = R3
αh3
E R34 Sin αh3 � Loads
Ach1wh2wh3 =
2(2 + ν)Cos2 αh3
Gych1wh2wh3’ = Ach1wh2wh3 Δκch1wh2wh3’
ξch1wh2wh3 Deformations
ξch1wh2ch3 = ξch1wh2
βrch1wh2h3 = Δтch1wh2’ Rh3
βrch1wh2wh3 = ξch1wh2wh3 - Δαch1wh2wh3 + rh3
ν R3 ξch1wh2wh3 + R2 ξch1wh2ch3
Tg αh3 Tg αh3
Δтch1wh2wh3’ = 1 - 2 Sin2αh3 Δαch1wh2wh3 + ν R3 ξch1wh2wh3 + R2 ξch1wh2ch3 Sin αh3 Cos αh3
rh3 rh32
Δκch1wh2wh3’ = - 2 Sin αh3 Cos αh3 Δαch1wh2wh3 + ν R3 ξch1wh2wh3 + R2 ξch1wh2ch3 Cos2 αh3
rh3 rh32

12. 6 Bundles
Rch3 = R2 Geometrical characteristics
Rwh3 = R3
αh3
E R34 Sin αh3 � Loads
Awh1wh2wh3 =
2(2 + ν)Cos2 αh3
Gywh1wh2wh3’ = Awh1wh2wh3 Δκwh1wh2wh3’
ξwh1wh2wh3 Deformations
ξwh1wh2ch3 = ξwh1wh2
βrwh1wh2h3 = Δтwh1wh2’ Rh3
βrwh1wh2h3 = ξwh1wh2wh3 - Δαwh1wh2wh3 + ν R3 ξwh1wh2wh3 + R2 ξwh1wh2ch3
Tg αh3 rh3 Tg αh3
Δтwh1wh2wh3’ = 1 - 2 Sin αh3 Δαwh1wh2wh3 + ν
2
R3 ξwh1wh2wh3 + R2 ξwh1wh2ch3 Sin αh3 Cos αh3
rh3 rh32
Δκwh1wh2wh3’ = - 2 Sin αh3 Cos αh3 Δαwh1wh2wh3 + ν
R3 ξwh1wh2wh3 + R2 ξwh1wh2ch3Cos2 αh3
rh3 rh32

13. 3 Strands
Rch2 = 0 Geometrical characteristics
Rwh2
1
rh2 =Rwh2 1+
3 Sin2 αh2
αh2
Tch1wh2=Fch1wh2h3 Loads
Hch1wh2=Mch1wh2h3
mh3 E R34 Sin αh3 � E R24 �
Ach1wh2 = +
2(2 + ν)Cos2 αh3 4
Mch1ch2=0
Fch1ch2=0
ξch1wh2 Deformations
ξch1ch2 = ξch1
βrch1h2 = Δтch1’ Rh2
ξch1wh2 - Δαch1wh2 + ν 2 R3 ξch1wh2wh3 + R2 ξch1wh2ch3
βrch1wh2 = rh2
Tg αh2 Tg αh2
Δтch1wh2’ = 1 - 2 Sin2αh2 Δαch1wh2+ ν 2 R3 ξch1wh2wh3 + R2 ξch1wh2ch3 Sin αh2 Cos αh2
rh2 rh22
Δκch1wh2’ = - 2 Sin αh2 Cos αh2 Δαch1wh2 + ν 2 R3 ξch1wh2wh3 + R2 ξch1wh2ch3 Cos2 αh2
rh2 rh22

14. 3 Strands
Rch2 = 0 Geometrical characteristics
Rwh2
1
rh2 =Rwh2 1+
3 Sin2 αh2
αh2
Twh1wh2=Fwh1wh2h3 Loads
Hwh1wh2=Mwh1wh2h3
mh3 E R34 Sin αh3 � E R24 �
Awh1wh2 = +
2(2 + ν)Cos2 αh3 4
Mwh1ch2=0
ξwh1wh2
Fwh1ch2=0
ξwh1wh2 Deformations
ξwh1ch2 = ξwh1
βrwh1h2 = Δтwh1’ Rh2
ξwh1wh2 - Δαwh1wh2 + ν 2 R3 ξwh1wh2wh3 + R2 ξwh1wh2ch3
βrwh1wh2 = rh2
Tg αh2 Tg αh2
Δтwh1wh2’ = 1 - 2 Sin2αh2 Δαwh1wh2+ ν 2 R3 ξwh1wh2wh3 + R2 ξwh1wh2ch3Sin αh2 Cos αh2
rh2 rh22
Δκwh1wh2’ = - 2 Sin αh2 Cos αh2 Δαwh1wh2 + ν 2 R3 ξwh1wh2wh3 + R2 ξwh1wh2ch3Cos2 αh2
rh2 rh22

15. 6 Cords
Rch1 Geometrical characteristics
Rwh1
αh1
Twh1 = Fh1 Loads
Hwh1 = Mh1
Awh1 = mh2 Awh1wh2
Gywh1 = Awh1 Δκh1’
Fwh1 = mh1 (Twh1 Sin αh1 + Nwh1’ Cos αh1)
Mwh1 = mh1 (Hwh1 Sin αh1 + Gwh1’ Cos αh1 + rh1 Twh1 Cos αh1 + rh1 Hwh1 Sin αh1)
ξch1 = ε Deformations
ξwh1
βh1
т = rh1 βrh1 = Rh1 βh1
βrh1 = ξwh1 - Δαh1 + ν 4 R3 ξwh1wh2wh3 + 2 R2 ξwh1wh2ch3 + 4 R3 ξch1wh2wh3 + 2 R2 ξch1wh2ch3
Tg αh1 rh1 Tg αh1
Δтh1’ = 1 - 2 Sin αh1 Δαh1 + ν 4 R3 ξwh1wh2wh3 + 2 R2 ξwh1wh2ch3 + 4 R3 ξch1wh2wh3 + 2 R2 ξch1wh2ch3 Sin αh1 Cos αh1
2
rh1 rh12
Δκh1’ = - 2 Sin α Cos α Δα + ν 4 R3 ξwh1wh2wh3 + 2 R2 ξwh1wh2ch3 + 4 R3 ξch1wh2wh3 + 2 R2 ξch1wh2ch3 Cos2 α
r r2

16. Analytical model
Results and validation
600
Calculated
500 G. Vunjak-Novakovic et al.
Root Mean Square Error Percentage
2

 xi − xi 
n
400
∑ x 

i 

RMSEP =
i  = 0.023
Load (N)
n
300
Annu. Rev. Biomed. Eng. 2004. 6:131–56
TISSUE ENGINEERING OF LIGAMENTS
G. Vunjak-Novakovic1, Gregory Altman2,3, Rebecca Horan2 and David L.
Kaplan2
200
1 Massachusetts Institute of Technology, Harvard-MIT Division of Health
Sciences and Technology, Cambridge, Massachusetts 02139; email: gorda-
na@mit.edu
2 Department of Biomedical Engineering, Tufts University, Medford,Massa-
chusetts 02155; email: gregory.altman@tufts.edu, david.kaplan@tufts.edu,
0.05 0.10 0.15 0.20 0.25 rebecca.horan@tufts.edu
3 Tissue Regeneration, Inc., Medford, Massachusetts 02155
Displacement (%)

17. Homogenized model
 σ rr   S11 S12 S13 S14 0 0   ε rr  Costello’s constants Trigonal constants [Pa]
σ  S kεε=0.52972 S11=5.02232*1010
 θθ   12 S22 S23 S24 0 0   εθθ 
   kεβ=0.0566074 S12=3.34821*1010
 F / AE   kεε kεβ   ε  σ 33  S13 S23 S33 S34 0 0   ε 33 
 3
=  ⇔  = ⋅ . kβε=0.122818 S22=5.02232x1010
Mt / ER  k βε k ββ   β 
  σ 3θ  S14 S24 S34 S44 0 0  2ε 3θ 
σ 3 r   0 0 0 0 S55 S56   2ε 3 r  kββ=0.0160126 S33=1.06738x109
      S34=1.68011*109
σ θ r   0
   0 0 0 S56 S66   2εθ r 
  
S44=2.86704*107
S55=2.86704*107
S66=8.37054*109
ΔL
Displacements along the z-axis [m] Rotations around the z-axis [deg]

18. Homogenized model
Results and validation
600 Analytical model 40
Homogenized model
500
Rotation (deg)
60
Load (N)
400
80
300
Analytical model
100
Homogenized model
200
0.10 0.15 0.20 0.25 0.10 0.15 0.20 0.25
Displacement (%) Displacement (%)
Root Mean Square Error Percentage Root Mean Square Error Percentage
2 2
n 
 xi − xi  n 
 xi − xi 
∑ x 

i 
 ∑ x 
 
RMSEP =
i  = 0.028 i  i  = 0.086
RMSEP =
n n

19. Conclusions and
future developments
NON LINEAR BEHAVIOUR DEVELOPMENT
3D FEM JOINT IMPLEMENTATION WITH ON-SITE TESTS
IMPROVEMENT OF CURRENT SCAFFOLDS
DESIGN OF NEW STRUCTURES

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