PhD LaraVigneron

7/21/2019 PhD LaraVigneron

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st

nd

rd

rd


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Ae

eaji j iai ib

C0

ci

iclk l l k D

d

di i

E

E

en

e


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Fint

Fext

{Fl

}4l=1

f

g

gj jth

H

H

H(A, B)

A

Bh(A, B)

A

B

I

I

J

H

K

K

Fl n S V

nEi

R3

r

S

S

S0 S

St S t Su S u s

t

t

U

U0

u

ui iuFEM

uXFEM

u

uI

uII

V

V0 V

V

Ve

e


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X

Xs

Xs,t Xs tx

xi ixs

xs,t xs txs,t1 xs,t

u Wint Wext

i iei ei

i i

s

x (x,y,z)T

f

fi i= x, y,zg gi i= x, y,zF Fij i, j =x, y,z

G Gij i, j=x, y,zH Hijkl i ,j,k,l= x, y,z

xs ( sx , sy , sz )T

xf

fxx

fxy

fxz

fyx

fyy

fyz

fzx

fzy

fzz


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x.f fxx + fyy + fzz

x.F

Fxxx

+ Fxyy

+ Fxzz

Fyxx

+ Fyyy

+ Fyzz

Fzxx

+ Fzyy

+ Fzzz

g.f x,y,zi gifi

G: F x,y,zi,j GijFij

H :F x,y,zk,l HijklFkl

Ff ith x,y,zj Fijfj

FG

(i, k)th

x,y,z

j

FijGjk


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st

nd

nd

rd

rd


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1 2mm


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20 mm

6 mm

7mm

37.6 14mm


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st

st

st

st

st


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1st iMR image

Landmark displacementcomputation

Biomechanical modelupdate

Biomechanicalmodel building

Rigidly registered

preoperative images

Rigid transformationcomputation

Image warping

Nonrigid registration

2nd iMR image 3rd iMR image

Constitutive

laws

Rigid biomechanicalmodel transformation

Rigid imagetransformation

Rigid registration

Updatedpreoperative images


Biomechanical modelupdate

Image warping

Nonrigid registration


Landmark displacementcomputation

Initialbiomechanical model


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st


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kth iMR image

Discontinuity definition

FEM- orXFEM-based

biomechanical model update

(k+1)th iMR image

Segmentation

Segmentation

Rigid registration toreference iMR image

Volumedisplacement field

Initial biomechanical model

Biomechanical model

building

Volume meshing

Segmentation

Constitutivelaws


(a) (b)

Rigidly-registeredpreoperative images


Surface landmarkdisplacement computation

Surface meshing

Rigidly registered

kth iMR image

Rigidly registered

(k+1)th iMR image

Updated 1st iMR imagenonrigidly registered to

kth iMR image

Volume

displacement field


Image warping

Image warping

Image warping

Updated preoperative imagesnonrigidly registered to

(k+1)th iMR image

Edge extractionEdge extractionEdge extraction

Edge distancecomputation

Edge distance

computation

(c)

Distance for validation ofserial nonrigid registration

Updated 1st iMR image

nonrigidly registered to(k+1)th iMR image

Updated kth iMR image

nonrigidly registered to(k+1)th iMR image

Updated preoperative imagesnonrigidly registered to

kth iMR image

Distance for validation ofnonrigid registration

Rigidly registeredkth iMR image

Rigidly registered(k+1)th iMR image

st


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st

st

st

st

st

st

st


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st

st

0.9375mm

0.9375 0.9375 2.5mm


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0.93750.93752.5mm

0.9375mm


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Xs= X(s), s= (s, r) : s,r [0, 1] Xs


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Fint(Xs) + Fext(Xs) = 0,

Fint

Fext

Fint

Fint(Xs) =

s(

Xss

) 2

s2(

2Xss2

),

= 0

Fext

Fext

Xs,t = X(s, t)

Xs,t

t =Fint(Xs,t) + Fext(Xs,t),

Xs,t

Xs

Fext(x)

Fext(x) =SminD(x),

D(x)

Smin

+1 1

Smin=

1 if D(x)> D(x + D(x))1 if D(x)< D(x + D(x)).


40/150

st

(Ik, Ik+1)

Ik Ik+1 Iwk Ik+1 I

wk

Ik

Ik

I1

Iw

1,k

I1 Ik+1 I

w1,k Ik+1

H(A, B)

A

B


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st

st


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Wint= Wext,

Wint Wext

Wint =

V

:xudV

Wext =

V

b.udV +

St

t.udS.

V

dV V0 dV0,

V0 V

x X.

E=12

[(Xu)T + (Xu) + (Xu)T.Xu].

E 12

((xu)T + xu) =,


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= H :,

H

H

E

Wint =

V0

:x(u)dV0=V0

: (xu)dV0 =V0

:dV0.

V0

: dV0 =

V0

b.udV0+

S0t

t.udS0.

b t u

(1

2 V0 : dV0) =(V0 b.udV0) +(S0t t.udS0),

V0

: dV0 =

V0

(H :) :dV0 =1

2(

V0

(H :) : dV0) =1

2(

V0

: dV0).

(1

2

V0

: dV0 V0

b.udV0 S0t

t.udS0) = 0,

(E) = 0,

E

E=1

2

V0

: dV0 V0

b.udV0 S0t

t.udS0,


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u(x)

uFEM(x)

uFEM(x) =

iIi(x)ui,

I

i(x) ui

i(x) i

i

i(xj) =

1 if i= j0 if i=j on i,

xj j

u

FEM

(x)

xi

i

uFEM(xi) =Ni=1

i(xi)ui=ui.

ui i

uFEM(x)

i(x) i ei (x)

e

ei (x) =

1

xi

0 xj=i,je.

ei (x) =aei + b

eix + c

eiy

2Ae ,

aei =xjym

xmyjbei =yj ymcei =xm xj ,


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Ae

(xi, yi) i

ei (x) =aei + beix+ ceiy+ deiz

6Ve ,

aei =det

xj yj zjxm ym zm

xp yp zp

, bei =det

1 yj zj1 ym zm

1 yp zp

,

cei =det

xj 1 zjxm 1 zmxp 1 zp

, dei =det

xj yj 1xm ym 1xp yp 1

,

Ve

(xi, yi, zi) i

Ku= f or iI

Kijui= f

i i, j = 1,...,n,

Kij =

V

BTi HBjdV with i =

ix

0 0

0 iy

0

0 0 iz

iy

ix

0

0 iz

iy

iz

0 ix

,

fi= V idV + St tidS.

u K

f

u uFEM(x) x

u

2 n

3 n

n u

u 2 3

x


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BTi HBj

Ae

Ve

uXFEM(x) =iI

i(x)ui+iJ

i(x)nEij=1

gj(x)aji.

I

i(x) ui

nEi aji i

J

I

i(x)

gj(x)

uXFEM(x) xk k


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kJ

uXFEM(xk) =iI

i(xk)ui+iJ

i(xk)

nEij=1

gj(xk)aji

=uk+nEij=1

gj(xk)ajk .

uXFEM(xk)=uk.

uk

k uXFEM(x)

uk k uFEM(x)

uk

gj(x)

iJ

H(x) = 1 f or (x x).en > 0

1 f or (x

x).en < 0,

x

x

x en x

H(x)

{Fl(r, )}4l=1 ={rsin2 , rcos2 , rsin2sin , rcos2sin },


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s1

es2(x*)

en(x*)

x

x*

Crack

surface

es1(x*)

s2

s

es(x*)

en(x*)

x

x*

(a)

Crack

(b)

r

x

es(xtip), en(xtip))

es1(xtip),en(xtip))

rsin

2

H(x)

Fl(x) = Fl(r, ), l = 1, ..., 4

H(x)

Fl(r, )

s1

x

Crack

surface

s2

rxtip

es1(xtip)

en(xtip)

es2(xtip)

(b)

r

(a)

s

Crack

x

xtip

es(xtip)

en(xtip)

Crack

front

uXFEM(x) =iJ

i(x)ui+jJ

j(x)H(x)aj+kK

k(x)(4

l=1

Fl(x)clk),

J

K aj


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clk H(x) Fl(x)

c J K

K={k I : xc k} J={j I :j c=, j /K},

xc k k k

V(x) =

1 f or (x x).en > 00 f or (x x).en < 0,

J(x) =

HII(x) f or H I(x)< 00 f or H I(x)> 0,

HI(x) HII(x)


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Crack

Crack

-1 1

-1

1

-1

1

0

H

(x) H

(x) J(x)

Kd= g oriI

Kijdi=gi i, j = 1,...,n,

di =

uiai

ci

,

Kij =

Kuuij Kuaij KucijKauij Kaaij Kacij

Kcuij Kcaij K

ccij

,

gi= gui gai gc1i gc2i gc3i gc4i

T .


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Krs

ij =

V

(Bri

)THBsj

dV (r, s= u, a, c),

gui =

V

idV +

St

tidS

gai =

V

iH dV +

St

tiH dS

gcli =

V

iFldV +

St

tiFldS l= 1, 2, 3, 4.

ui =

i

x 0 00 iy

0

0 0 iz

iy

ix

0

0 iz

iy

iz

0 ix

, ai =

(iH)

x 0 00 (iH)y

0

0 0 (iH)z

(iH)y

(iH)x

0

0 (iH)z

(iH)y

(iH)z

0 (iH)x

,

ci =

Bc1i B

c2i B

c3i B

c4i

,

cli =

(iFl)x

0 0

0 (iFl)y

0

0 0 (iFl)z

(iFl)y

(iFl)x

0

0 (iFl)z

(iFl)y

(iFl)z

0 (iFl)x

.

J

K

J

K


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r

J

J

J

A

A

above

A

below

Aabove/A

Abelow/A

= 104

J

Ae

Ve


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Kuuij

Kaaij

Ae

Kuaij K

auij

AeaboveAebelow Kucij

Kucij Kucij K

ucij K

ucij

H(x) = 0

Aeabove A

ebelow

r

Aeabove

Aebelow

rx,y

(a) (c)(b)

FE FE FE

A

E

D

CB

r


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h O(h)

O(

h)

O(

h)

h

O(h)

h

h

O(h)

gj(x)

gj(x)


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uI(x) = 12

r2

cos( 2)[ 1 + 2 sin2( 2)]sin(

2)[ + 1 2cos2(

2)]

uII(x) = 1

2

r

2

sin(

2)[ + 1 + 2 cos2(

2)]

cos( 2

)[ 1 2sin2( 2

)]

,

r

= E

2(1 +), =

3 4

31+

E

Fl(x) =Fl(r, ), l= 1, ..., 4

(a) Mode I (b) Mode II (c) Mode III


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(1a)

(3a)

(1b)

(2a)

(3b)

(2b)

(1c)

(2c)


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2mm

0.80.4mm

1.60.5mm


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st

st

nd

st

st

st


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1st iMR image

FEM-basedbiomechanical model update

2nd iMR image

Segmentation

Segmentation

Rigid registration to1st iMR image

Volume displacement field

Linear elastic law


(a)

Whole-brain region boundarydisplacement field computation

Image warping

Edge extractionEdge extraction


Biomechanical modelbuilding

Volume meshing

Surface meshing

Internal tumor region boundary

displacement field computation

1st iMR image 2nd iMR imageVolume displacement field

(b)


2nd iMR image


st

nd

st


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st

25mm

25 mm

3, 247

K

K= 6

6 VK

4

i=1

a(fi) maxi=1,...,6

l(ei)

,


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(a) (d)(c)(b)

25mm

N


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(a) (b)

3.84

2.89

1.94

0.99

0.03

Displ. mag.mm

max: 3.84min: 0.03

max: 2.62min: 0.02

25 mm


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(a) (b)

3.80

2.35

0.90

-0.54

-1.99

Disp. ymm

x

yz z

xy

st


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(a) (b) (c)

K 0.24

K

0.45

K

0.24


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(a)

(e)

(c)(b)

(d)

K 0.45

st

nd

nd


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(a) (b) (c)

4.98

3.75

2.52

1.29

0.06(f)

Displ. mag.mm

(d) (e)

st

nd

st

st

nd


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(a) (b)

5.01

3.77

2.53

1.28

0.04max: 4.98min: 0.06

Displ. mag.mm

max: 5.01min: 0.04


83/150

4.47

2.97

1.46

-0.41

-1.55

Displ. ymm

(b)(a) (c)

max: 3.80min: -1.99

max: 4.47min: -1.55

x

yz z

xy

z

xy

0.05

0.95

0.06

0.95

0.18mm

0.02


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(b)

5.01

3.77

2.53

1.28

0.04

(a)

Displ. mag.mm

max: 6.42min: 0.13

max: 5.01min: 0.04

st

st

nd

st

nd

1.24 mm

1.08 mm


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(c)

max: 5.71min: -2.50

(b)

Displ. y

mm4.47

2.97

1.46

-0.41

-1.55max: 4.47min: -1.55

x

yz z

xy

z

xy

(a)

(c)(b)

0.06

0.02

0.00

-0.07

-0.11

Strain yy

max: 0.95min: -0.27

max: 0.06min: -0.11

(a)

x

yz z

xy

z

xy


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st

nd

st

nd

st

nd

st

st


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E= 3000P a

= 0.45

E= 10 P a

= 0.05

1.08mm

2 mm

st

2.5 mm


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1.24mm

1.10mm

1.08mm


89/150


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st

nd


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0.9 0.7 mm

3.7 mm


94/150

st

nd


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st

st

st

st

st

nd

rd

st

nd

rd

st

rd

st

rd

nd

st

nd

st


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2nd iMR image

FEM-basedbiomechanical model update

3rd iMR image

SegmentationSegmentation




Healthy-brain region boundarydisplacement field computation

(b)


2nd iMR image 3rd iMR imageVolume displacement field

Image warpingwith resection





(a)


2nd iMR image


3rd iMR image

Updated 2nd iMR imagenonrigidly registered to

3rd iMR image



st

nd

rd

st

nd


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rd

rd

3rd

nd

nd

nd

rd

th

nd

th

st

th

rd

st

st

nd

st

st

st

rd

rd

st

st

st

nd

st

3, 317


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3rd iMR image

XFEM-basedbiomechanical model update

4th iMR image





Healthy-brain region boundarydisplacement field computation

(b)


3rd iMR image 4th iMR imageVolume displacement field






(a)


3rd iMR image


4th iMR image

Updated 3rd iMR imagenonrigidly registered to

4th iMR image



1st resectiondiscontinuity definition

nd

rd

th

st

rd


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9, 951

873

st

rd

th

st

st

nd

rd

th

rd

nd

rd

nd

nd

th

th

th

rd

th

th

th


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(a)

(c)

(b)

x

y

z x

yz

(d)

nd

3rd

st

3rd

st

2nd

st


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1st

3rd

2nd

4th

3rd


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1st

3rd

2nd

4th

3rd

5th


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rd

nd

rd

rd

nd

rd

nd

nd

st

rd

nd

2nd

2nd


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(a) (b) (c)

rd

2nd

2nd

3rd

rd

th

rd

th

th

th


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th

th

rd

Ik

(k = 1,...5)

(Ik, Ik+1)

Ik Ik+1 I

wk Ik+1 I

wk

Ik

Ik Ik+1 I1

Ik+1 I

w1,k Ik+1 I

w1,k

I1

(I

k, I

k+1)

H(Ik, Ik+1) H(Iwk, Ik+1)

(Ik, Ik+1) (Iwk, Ik+1)

1 mm


106/150

rd

4th

5th

th


107/150

H(Ik,

Ik

+1)

H(Iwk

,Ik+1)

k

=

1,.

..

4

(Ik,

Ik+1)

Ik

1st

2nd

3rd

H(I1,

I2

)

H(Iw1

,I2

)

H

(I2,

I3

)

H(Iw2

,I3

)

H(I3,

I4

)

H(Iw3

,I4

)

H(I4,

I5

)

H(Iw4

,I5

)

H(I1,

Ik

+1)

H(Iw1

,k,

Ik+1)

k

=

1,...

4

(Ik,

Ik+1)

I

1

Ik

Ik

1st

2nd

3rd

H(I1,

I2

)

H(Iw1

,I2

)

H

(I1,

I3

)

H(Iw1

,2,

I3

)

H(I1,

I4

)

H(Iw1

,3,

I4

)

H(I1,

I5

)

H(Iw1

,4,

I5

)


108/150

3rd

25mm

I1

H(I4, I5) = 1.70mm

H(Iw4, I5) = 1.37 mm H(I1, I2) = 1.36 mm

H(Iw1, I2) = 1.28 mm

I1

(Ik, Ik+1)

H(I1, Ik+1) H(Iw1,k, Ik+1)

H(I1, Ik+1)

H(I1, Ik+1) H(Iw1,k, Ik+1)

H(I1, I2) = 1.24mm

H(I1, I5) = 1.78 mm H(I1, I2) = 1.01 mm

H(I1, I5) = 1.68mm


109/150

st

st


110/150


111/150

3 mm

10mm


112/150

85%

75%


113/150

256 256

0.859 0.859mm

2nd

st

nd

1st

st

828

1, 656

30

24


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nd

nd

2nd

(a) (b)(a) (b) (c)

st

st


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2nd

d retr,leftn dretr,rightn

n

An(xretrn )uAn+ H(x

retrn )An(x

retrn )aAn

+ Bn(xretrn )uBn

+ H(xretrn )Bn(xretrn )aBn

=An(xretrn )uAn+ An(x

retrn )aAn

+ Bn(xretrn )uBn

+ Bn(xretrn )aBn

=dretr,leftn

H(x) +1

An(xretrn )uAn+ H(x

retrn )An(x

retrn )aAn

+ Bn(xretrn )uBn

+ H(xretrn )Bn(xretrn )aBn

=An(xretrn )uAn An(xretrn )aAn

+ Bn(xretrn )uBn

Bn(xretrn )aBn=dretr,rightn

H(x)

1

+1 1 xretrn

n An Bn


116/150

n

Am(x) Bm(x) Am Bm

uAn uBn An Bn aAn aBn

An Bn

An Bn

st

2.17 mm

1.78mm

2.17mm

1.76 mm

2.17 mm 1.78 mm


117/150

st

2nd

st

nd

st

nd


118/150

(a)

(d)(c)

(b)

nd

rd


119/150

2nd iMR image

3rd iMR image


Rigid registration to2nd iMR image


Linear elastic law


(a)

Image warping

Edge extractionEdge extraction


Biomechanical modelbuilding

Volume meshing

Surface meshing

2nd iMR image 3rd iMR imageVolume displacement field

(b)

Updated 2nd iMR imagenonrigidly registered to

3rd iMR image


XFEM-basedbiomechanical model update

Left hemisphere region boundarydisplacement field computation

Right hemisphere region boundary

displacement field computation

Cut lipsdisplacement field computation

Retraction cutdiscontinuity definition

nd

rd

nd


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st

nd

nd

st

nd

nd

nd

256 256 60

0.859 0.859 2.5mm

rd

nd

nd

K 0.42

E= 3000P a

= 0.45


121/150

nd

rd


122/150

2, 198

6, 594

561

7, 155

nd

rd


123/150

nd

rd

(a) (b)

max: 3.44min: 0.00

9.33

7.00

4.67

2.33

0.00

Displ. mag.mm

max: 9.33min: 0.00


124/150

rd

(a) (b) (c)

10.7

8.07

5.43

2.78

0.14

Displ. mag.mm

rd

3rd


125/150

(1a) (1c)(1b)

(2a) (2c)(2b)

displacement field of

brain hemispheredisplacement field of

cut lip


brain hemisphere


cut lip


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(a) (b)

12.0

9.03

6.02

3.02

0.02

Displ. mag.mm

xy

z

(c)

x y

z

nd

nd

rd

nd


127/150

nd

rd

1.36 mm

1.10mm

nd

nd


128/150

nd

rd

nd

rd

nd

rd


129/150


130/150


131/150


132/150


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st


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Documents

PhD LaraVigneron