Group-wise analysis on myelination profiles of cerebral cortex using the second eigenvector of Laplace-Beltrami operator

G R O U P - W I S E A N A LY S I S O N M Y E L I N AT I O N
P R O F I L E S O F C E R E B R A L C O R T E X U S I N G T H E
S E C O N D E I G E N V E C T O R O F L A P L A C E - B E LT R A M I
O P E R AT O R
S E U N G - G O O K I M , J O H A N N E S S T E L Z E R , P I E R R E - L O U I S B A Z I N
A D R I A N V I E H W E G E R , T H O M A S K N Ö S C H E
ISBI 2014, 1st of May, Beijing, China

O V E R V I E W
• Aim: Intersubject correspondence for group-wise
analysis on myelination profiles from the high-field MRI

O V E R V I E W
• Method: Parametrization using the second Laplace-
Beltrami eigenvector

O V E R V I E W
• Method: Parametrization using the second Laplace-
Beltrami eigenvector
• Application: Statistical inference using random field
theory on Heschl’s gyrus in auditory cortex

M Y E L I N AT I O N P R O F I L E
A N D I N - V I V O I M A G I N G
I N T R O D U C T I O N & M O T I VA T I O N

M Y E L O A R C H I T E C T U R E O F C O R T E X
(cc) Quasar Jarosz

Beck (1928) G M
W M
Nieuwenhuys (2013) Ed. Geyer & Turner
(cc) Quasar Jarosz

Vogt (1903)
C Y T O - M Y E L O -
Beck (1928) G M
W M
(cc) Quasar Jarosz

Vogt (1903)
C Y T O - M Y E L O -
Beck (1928) G M
W M
(cc) Quasar Jarosz
Hopf (1954)

I N - V I V O I M A G I N G O F I N T R A C O R T I C A L
M Y E L O A R C H I T E C T U R E U S I N G 7 T M R I
• Quantitative T1
mapping (qT1)
• T1 inversely
correlates with
myelination
• Myelin is the main
contribution to the T1
contrast [1]
Geyer et al. (2011) Front Hum Neurosci
ex-vivo
in-vivo
[1] Eickhoff et al. (2005) Hum Brain Mapp

I N T E R S U B J E C T C O R R E S P O N D E N C E
C S F G M W M

W H O L E B R A I N R E G I S T R AT I O N ?
• Not yet fully developed for
high-field MRIs
• Signal loss in ventral
regions
• Different image contrast
(quantitative T1 mapping)
• Dimensionality from  
sub-mm resolution

M O T I VAT I O N : C O R R E S P O N D E N C E
• To construct correspondence between myelination
profiles in order to infer group-wise differences
• Circumventing whole-brain registration issues
• More precise than averaging within a ROI

M O T I VAT I O N : C O R R E S P O N D E N C E
• To construct correspondence between myelination
profiles in order to infer group-wise differences
• Circumventing whole-brain registration issues
• More precise than averaging within a ROI
• Parametrization using the second Laplace-Beltrami
eigenvector
Lévy (2006) SMI

T H E S E C O N D L A P L A C E - B E LT R A M I
E I G E N V E C T O R
• Monotonous increase along the longest
geodesic distance

T H E S E C O N D L A P L A C E - B E LT R A M I
E I G E N V E C T O R
• Monotonous increase along the longest
geodesic distance
• Used to construct medial axes & Reeb
graph for arbitrarily shaped structures
Seo et al. (2011) SPIE
Shi et al. (2008) MICCAIShi et al. (2008) IEEE CVPR
Reuter et al. (2009) CAD

A P P L I C AT I O N : H E S C H L’ S G Y R U S ( H G )

A P P L I C AT I O N : H E S C H L’ S G Y R U S ( H G )
A I L P
Wallace et al. (2002) Exp Brain Res
A I
L P
S TA

M Y E L I N AT I O N P R O F I L E
E S T I M AT I O N
P R E P R O C E S S I N G

S U B J E C T S & I N - V I V O I M A G I N G
• Six healthy participants: all male, age=25 ± 2 y.o.
• MP2RAGE (magnetization-prepared rapid gradient
echo with two inversion times) at 0.7 mm isovoxel
using a 7 T scanner (Siemens)
Marques et al. (2010) NeuroImage
qT1T1w

T1w at 1mm isovoxel
qT1 at 0.7 mm isovoxel

T1w at 1mm isovoxel
qT1 at 0.7 mm isovoxel
Regenerated surfaces

M A N U A L D E L I N E AT I O N O F H G
• Duplication/sulcus (10) • Single HG (LH:1, RH:1)

R E A L I S T I C C O R T I C A L L AY E R I N G [ 1 ]
[1] Waehnert et al. (2013) NeuroImage

R E A L I S T I C C O R T I C A L L AY E R I N G [ 1 ]
http://www.cbs.mpg.de/institute/software/cbs-hrt[1] Waehnert et al. (2013) NeuroImage

T H E S E C O N D L A P L A C E -
B E LT R A M I E I G E N V E C T O R
PA R A M E T E R I Z A T I O N & I N F E R E N C E

• Laplace-Beltrami (LB) operator 𝚫 of function 𝒇 defined on an
arbitrary manifold is given by:
L A P L A C E - B E LT R A M I E I G E N V E C T O R S
D f := div(grad f)
M 2 R2
⇢ R3

• To find eigenvector Ѱj and eigenvalue 𝝀j of LB, solve:
0 = l0 < l1  l2  ···
y0,y1,y2 ···
D f := div(grad f)
M 2 R2
⇢ R3
Dyj = ljyj

0 = l0 < l1  l2  ···
y0,y1,y2 ···
D f := div(grad f)
[1] Qui et al., (2006) TMI; [2] Aubry et al. (2011) ICCV
CY = lAY C Cotagent matrix
A Mass matrix
• Discretization of LB using FEM, then the eigenvectors can be
computed from generalized eigenvalue problem [1,2]:
M 2 R2
⇢ R3
Dyj = ljyj

0 = l0 < l1  l2  ···
y0,y1,y2 ···
D f := div(grad f)
[1] Qui et al., (2006) TMI; [2] Aubry et al. (2011) ICCV
CY = lAY C Cotagent matrix
A Mass matrix
• Discretization of LB using FEM, then the eigenvectors can be
computed from generalized eigenvalue problem [1,2]:
http://www.di.ens.fr/~aubry/wks.html
*smoothed for visualization
M 2 R2
⇢ R3
Dyj = ljyj

L E V E L S E T S B A S E D O N L B 2

PA R A M E T E R I Z E D M Y E L I N P R O F I L E S
C S F
G M
W M

AV E R A G E D M Y E L I N I M A G E S
Left Right
1st
2nd
2315
2827
Corticaldepth
AL PM
0
0.5
1
2435
2773
Corticaldepth
AL PM
0
0.5
1
2493
2762
Corticaldepth
AL PM
0
0.5
1
2543
2833
Corticaldepth
AL PM
0
0.5
1
T1 (ms)

S TAT I S T I C A L I N F E R E N C E
dIhemi = Ileft Iright dIorder = I1st I2nd
• Paired differences matching order or hemisphere

dIorder = b0 +edIhemi = b0 +e
• Paired t-test (left vs. right; 1st vs. 2nd)

• Paired t-test covarying the other variables & interaction
dIhemi = b0 +b1 ⇥order+e dIorder = b0 +b1 ⇥hemi+e

• Paired t-test covarying the other variables & interaction
dIhemi = b0 +b1 ⇥order+e dIorder = b0 +b1 ⇥hemi+e
http://www.math.mcgill.ca/keith/surfstat/Worsely et al. (2009) NeuroImage
• RFT for multiple comparisons correction; FWHM= 2 pixels

Left - Right
L-R controlling
order
Effect of order
in L-R diff
1st - 2nd
1st-2nd covarying
hemisphere
Effect of hemi
in 1st-2nd diff

R E S U LT
• Greater T1 in the left HG 
(Higher myelin in the right HG)
[1] Warrier et al., 2009, J Neurosci

R E S U LT
• Lateralization of HG?
• Structural difference of HG between hemispheres
and specialized sensitivity to temporal/spectral
information [1]

R E S U LT
• Lateralization of HG?
• Structural difference of HG between hemispheres
and specialized sensitivity to temporal/spectral
information [1]
• The application is for demonstration of inter-structure
comparison of myelination profiles

F U R T H E R A P P L I C AT I O N S
• Other regions: primary
somatosensory/motor areas

F U R T H E R A P P L I C AT I O N S
5 10 15 20
0
20
40
60
inx−coordinate
Order of eigenvector
5 10 15 20
0
2
4
6
8
iny−coordinate
5 10 15 20
0
5
10
15
inz−coordinate
data1
data2
data3
data4
data5
data6
Y(p) = q(p)+e(p),
q(p) =
k
Â
i=0
bjyj
ˆb = (y0
y) 1
y0
Y
[1] Kim et al. (2012) MMBIA
• Other regions: primary
somatosensory/motor areas
• Shape descriptor for group
differentiation (e.g. musicians):
Fourier coefficients [1] or the
eigenvalues of LB operator

T h i s w o r k i s f u n d e d b y t h e I n t e r n a t i o n a l M a x P l a n c k R e s e a rc h
S c h o o l o n N e u ro s c i e n c e o f C o m m u n i c a t i o n ( I M P R S - N e u ro c o m )
T H A N K Y O U F O R AT T E N T I O N !
© Hans-Joachim Krumnow

Group-wise analysis on myelination profiles of cerebral cortex using the second eigenvector of Laplace-Beltrami operator

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Group-wise analysis on myelination profiles of cerebral cortex using the second eigenvector of Laplace-Beltrami operator