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ORIGINAL ARTICLE
Shape and curvedness analysis of brain morphology
using human fetal magnetic resonance images in utero
Hui-Hsin Hu
•
Hui-Yun Chen
•
Chih-I Hung
•
Wan-Yuo Guo
•
Yu-Te Wu
Received: 13 June 2012 / Accepted: 20 October 2012 / Published online: 8 November 2012
Ó Springer-Verlag Berlin Heidelberg 2012
Abstract The 3-D morphological change has gained
increasing significance in recent investigations on human
fetal brains. This study uses a pair of new indices, the shape
index (SI) and curvedness index (CVD), to quantify 3-D
morphological changes in developing brains from 22 to
33 weeks of gestation. The SI was used to automatically
locate the gyral nodes and sulcal pits, and the CVD was
used to measure the degree of deviation of cortical shapes
from a flat plane. The CVD values of classified regions were
compared with two traditional biomarkers: cerebral volume
and cortical surface area. Because the fetal brains dramat-
ically deform with age, the age effect was controlled during
the comparison between morphological changes and
volume and surface area. The results show that cerebral
volume, the cortical surface area, and the CVD values of
gyral nodes and sulcal pits increased with gestational age.
However, with age controlled, the CVD values of gyral
nodes and sulcal pits did not correlate with cerebral volume,
but the CVD of gyral nodes increased slightly with the
cortical surface area. These findings suggest that the SI, in
conjunction with the CVD, provides developmental infor-
mation distinct from the brain volumetry. This approach
provides additional insight into 3-D cortical morphology in
the assessment of fetal brain development.
Keywords Shape Curvedness Fetus MR Volume
Area Brain
Introduction
In utero magnetic resonance image (MRI) has become a
crucial instrument for the study of early brain development
in human fetuses. In utero MRIs provide a higher quality of
tissue contrast than ultrasound, and are without the limitation
of postmortem data such as tissue deformation after death
and fixation. Recent neuroimaging studies have successfully
applied image reconstruction to in utero MRIs (Habas et al.
2010a, b; Kim et al. 2010; Rousseau et al. 2006), pinpointing
the 3-D morphology of early gyrification (Clouchoux et al.
2012; Hu et al. 2011; Rajagopalan et al. 2011).
An efficient method for quantifying 3-D morphology is
to examine surface curvature and its derivatives (Batchelor
et al. 2002; Luders et al. 2006). A recent study used
principal curvatures to extract the skeleton of the cortical
surface from the reconstructed fetal MRIs (Clouchoux et al.
2012). The skeleton facilitated the visual inspection of the
evolution of sulcal fundi in fetuses, but did not provide
quantitative information on early gyrification. The shape
index (SI) and curvedness index (CVD), which derive from
the polar system of principal curvatures, demonstrate an
exceptional ability to quantify 3-D morphology (Koenderink
and van Doorn 1992). As shown in Fig. 1, the SI can locate
outward cylindrical surfaces (indicated by black arrows in
H.-H. Hu H.-Y. Chen C.-I. Hung Y.-T. Wu (&)
Department of Biomedical Imaging and of Radiological
Sciences, National Yang-Ming University, No. 155,
Li-Nong Street, Section 2, Pei-Tou, Taipei 112, Taiwan, ROC
e-mail: yute.wu@msa.hinet.net
H.-Y. Chen W.-Y. Guo (&)
Department of Radiology, Taipei Veterans General Hospital,
201, Section II, Shih-Pai Road, Taipei 11217, Taiwan, ROC
e-mail: wyguo@vghtpe.gov.tw
W.-Y. Guo
Faculty of Medicine, School of Medicine,
National Yang-Ming University, Taipei, Taiwan, ROC
Y.-T. Wu
Brain Research Center, National Yang-Ming University,
Taipei, Taiwan, ROC
123
Brain Struct Funct (2013) 218:1451–1462
DOI 10.1007/s00429-012-0469-3
Fig. 1c) with a value of 0.5, whereas at corresponding
locations (indicated by black arrows in Fig. 1d), the CVD
values increased and showed that these surfaces became
sharper along the x-axis. In other words, the SI discriminates
shapes surrounding surface points, and the CVD quantifies
the degree of deviation of the shape from the flat plane.
The SI and CVD have been applied to neuroimages of
developing human brains. Use of the SI has been extended
to preterm neonates (Rodriguez-Carranza et al. 2008) and
to fetal brains (Hu et al. 2011). The joint probability map of
SI and CVD was used to extract cortical morphology in
neonates with congenital heart disease (Awate et al. 2010).
The map delineated the distribution of changes in the SI
and in the CVD, but did not describe changes between the
different shapes that are consistent with anatomical fea-
tures such as gyral crowns and sulcal fundi.
Recent studies have documented the significance of the
separation between gyral crowns and sulcal fundi in
investigating brain morphology. An MR study of normal
adult brains reported that ‘‘T’’-shaped gyral crowns
appeared to have more fibers than other cortical shapes (Li
et al. 2010). A postmortem study of fetuses also showed
more axonal connections in gyral crowns than in sulcal
fundi (Takahashi et al. 2012). Our MR study on fetuses
showed that the gyral surface became smoother, whereas
the sulcal surface becomes more angular with gestational
age (Hu et al. 2011). However, a further quantification of
morphological changes between gyral crowns and sulcal
fundi need to be provided. Moreover, comparisons between
quantitative changes in fetal brain morphology and other
traditional biometric indices in the assessment of brain
development are also lacking.
Cerebral volume and cortical surface area are two tra-
ditional and significant biometric measures in the research
and assessment of fetal brain development. Cerebral vol-
ume reflects the proliferation rate of brain tissue during the
fetal stage. Previous research has documented abnormali-
ties in both cerebral volume and cortical morphology in
certain neurodevelopmental disorders. For example, a fetus
with microcephaly shows a cerebral volume that is below
the third percentile, and a wider and shallower gyral shape
than normal brains at the same term (Garel 2004). This
implies a correlation between brain morphology and cere-
bral volume. The cortical surface area is another biometric
measure that might be related to changes in cortical mor-
phology. This is based on a well-known explanation in
which, to avoid a substantial increase in cerebral volume,
the cortical surface folds to accommodate the expansion of
the cerebral cortex (Bradley et al. 1991; Le Gros Clark
1945). Accordingly, the cortical surface area expands with
the emergence of the sulci and gyri, or with the increase in
surface complexity caused by the deforming sulci and gyri.
Fig. 1 Shape index (SI) and
curvedness index (CVD).
Although the SI and CVD are
combinations of the maximum
and minimum curvature,
namely principal curvatures (a),
they represent distinct
properties of a surface. Based on
the polar system of principal
curvatures (b), the SI is defined
as the angle of the system,
whereas the CVD is defined as
the distance in the system. As
shown in an example of a
simulated cosine surface where
the SI and CVD values are
transformed into different
colors, the SI (c) discriminates
and locates the inward (blue)
and outward (gray) shapes; the
CVD (d) quantifies the
curvature of the surface. Once
the SI locates all the outward
surfaces (arrows in c), the CVD
show that these outward
surfaces become increasingly
angular along the x-axis (arrows
in d)
1452 Brain Struct Funct (2013) 218:1451–1462
123
However, Dubois et al. found that preterm babies with
intrauterine growth restriction showed a smaller cortical
surface area than normal preterm babies, but a similar
surface complexity to those of normal controls at the same
term (Dubois et al. 2008). This implies that, for brains with
certain developmental disorders, the cortical surface area
could not expand with the emergence of sulci and gyri.
These observations prompt us to investigate brain mor-
phology and its relationship to cerebral volume and the
cortical surface area, in normal fetuses.
This study uses new curvature derivatives, the SI and the
CVD, to quantify 3-D morphological changes in gyral
nodes and sulcal pits in developing human fetal brains, and
explores the relationship between morphological change
and two traditional biometric measures: cerebral volume
and cortical surface area. We used the term of gyral nodes
and sulcal pits, rather than gyral crown and sulcal fundus,
because the crest line of gyri and sulci have not been well-
developed for young fetuses. The SI was used to locate, on
the brain surface of each fetus, the gyral nodes, sulcal pits,
and two shapes that connect the nodes and pits, namely, the
gyral saddle and the sulcal saddle (Fig. 2). The CVD was
then used to measure the curvature of each cortical shape.
Finally, the quantitative relationship between the CVD
values of classified regions and the cerebral volume/corti-
cal surface area was investigated under the control of
gestational age. All the steps of this experiment are sum-
marized in Fig. 3.
Cerebral volume has been one of the most important
clinical biomarkers in assessing the development of fetal
brains. Better understanding of the correlation between
brain morphology and cerebral volume for normal brains
may help identifying the clinical value of measuring brain
morphology in fetal development. The normal trajectories
of morphological changes between gyral nodes and sulcal
pits in this study may provide useful information in further
assessment of neurodevelopmental disorders such as Cor-
nelia de Lange Syndrome (Spaggiari et al. 2012) and
microcephaly (Garel 2004).
Methods and materials
Data acquisition
This study used the MR brain images of 27 fetuses between
22 and 33 weeks of gestational age (GA). Each fetus was
from a single pregnancy. The Institutional Review Board of
Taipei Veterans General Hospital (VGHIRB No. 96-12-
26A) approved this study, and informed consent for MR
examination was obtained from each mother. Fetuses that
had been examined for non-central nervous system (CNS)
abnormalities were retrospectively selected. The fetuses
had undergone an additional MR examination of the brain
and/or had shown suspicious ventricular dilatation on
ultrasound, but were regarded as normal fetuses after fur-
ther MR examination.
A half-Fourier acquisition single-shot fast spin-echo
(SSFSE by GE, Milwaukee, WI, USA or HASTE by Sie-
mens, Erlangen, Germany) scanning sequence with body
phased-array coils was used. The system used was a 1.5-
Tesla MR scanner (GE or Siemens). Three orthogonal
image views of the fetal brain (the coronal, axial, and
sagittal planes) were acquired.
Slice thicknesses of 4 and 5 mm were used for pre-
serving sufficient SNR and reducing fetal movement, dur-
ing ultrafast MR imaging. Of 27 fetuses, 7 were scanned
with a 4 mm slice thickness and 20 with a 5 mm slice
thickness. The GAs of those scanned with a 4 mm slice
thickness were 167, 170, 190, 193,197, 204, and 232 days,
Fig. 2 Several local shapes and their corresponding SI values
(courtesy of Koenderink and van Doorn 1992). The SI was used to
locate four cortical shapes, including gyral nodes, sulcal pits, and two
transitional shapes connecting the gyral nodes and sulcal pits: the
gyral saddle and the sulcal saddle. In this study, the sulcal pits are
located by the surface points with SI \ -0.5. The gyral nodes are
located by surface points with SI [0.5. Because the concavities
(SI \ 0) and convexities (SI [ 0) are separated by the saddle shape
with SI = 0, the surface points of 0 \ SI \0.5 were defined as the
gyral saddle, and the surface points of -0.5 \ SI \ 0 as the sulcal
saddle
Brain Struct Funct (2013) 218:1451–1462 1453
123
1454 Brain Struct Funct (2013) 218:1451–1462
123
and the GAs of those scanned with a 5 mm slice thickness
were 154, 159, 162, 163 165, 175, 182, 188, 191, 193, 208,
210, 211, 211, 212, 218, 222, 225, 231, and 234 days. The
GA difference between the 4-mm and 5-mm groups was
statistically insignificant (two-sample t test: t =-0.1796,
P = 0.8601). Two matrix sizes 256 9 56 and 512 9 512
were obtained, with two fields of view, 280 9 280 mm
2
and 400 9 400 mm
2
, respectively. Image resolutions were
1.09 9 1.09 and 0.78 9 0.78 mm
2
. To obtain a consistent
in-plane resolution of the images, each MR slice was
resampled to 1 9 1mm
2
by linearly interpolating the
image intensity without losing or adding detailed cerebral
contour features. When this was done, the image resolution
became 1 9 1 9 4 and 1 9 1 9 5mm
3
.
Image reconstruction
Three-dimensional fetal brain MRIs were obtained through
image reconstruction by merging three original brain
images orthogonal to each other. This ensured that the SI
and CVD values were calculated from continuous
and complete 3-D brain surfaces. The reconstruction was
iteratively executed in two steps: co-registration and
reconstruction (Rousseau et al. 2006).
During co-registration, the three original images sepa-
rately underwent two rigid transformations: (1) a trans-
formation between each 2-D slice of the original image
and a 3-D reconstructed image; and (2) a transformation
between the original 3-D image and the reconstructed 3-D
image. Here, the reconstructed image was initially set as
the coronal image that had been linearly interpolated to a
resolution of 1 9 1 9 1mm
3
. An intensity-based simi-
larity measure [normalized mutual information (NMI)]
was computed and reached its maximum, while two slices
or two images were optimally aligned. During recon-
struction, the rigid transformations obtained during
co-registration were used to renew the reconstructed 3-D
image.
The programming language used was Matlab7.0 code
(MathWorks, Inc., Natick, MA, USA). The resolution of
the resultant reconstructed images became 1 9 1 9
1mm
3
. Two cases were excluded from this study because
of image reconstruction failure.
Brain image segmentation
Fetal brain images were manually segmented from the
reconstructed images. This study included two brain con-
tours: the outer contour between cerebrospinal fluid (CSF)
and the cerebral cortex, and the inner contour between
cerebral cortex and white matter. These two brain contours,
Fig. 4 a An example of manual segmentation on coronal slice.
b Brain image of outer surface between cerebrospinal fluid and
cerebral cortex. c Brain image of inner surface between cerebral
cortex and white matter (light gray area). Difference between outer
and inner surface were also shown (dark gray area)
Fig. 3 Flowchart of data analysis. This study used the SI and CVD to
quantify morphological changes in the developing cerebral cortex,
and then compared the morphological changes to the cerebral volume
and the cortical surface area. Step 1: original MR fetal brain images
show excellent in-plane resolution but thick slice thicknesses (4 and
5 mm). For example, Row A presents three orthogonal views of a
sagittal image. The coronal view (left) and axial view (middle) were
more blurred than the sagittal view (right). Step 2: three original
images are merged into an image with 3D coherence. This
reconstructed image shows high quality along three views. Step 3:
brain contour between the cerebral cortex and cerebrospinal fluid (red
curve in a) is manually segmented. Each segmented brain image was
then binarized to standardize the image intensity among subjects. The
SI and CVD values were computed on the binarized image. Step 4:
the SI value was used to locate four cortical shapes, including gyral
nodes, sulcal pits, and two transitional shapes between gyral nodes
and sulcal pits: the gyral saddle and the sulcal saddle. Step 5: the
CVD value was used to quantify the sharpness of the cortical surface.
Higher CVD values represent a more angular cortical surface. Step 6:
the CVD value of each shape was computed for each fetus. Step 7:
cerebral volume and the cortical surface area were computed for each
fetus. Step 8: the partial correlation coefficient was calculated to
compare the CVD value of each cortical shape with the cerebral
volume and the cortical surface area
b
Brain Struct Funct (2013) 218:1451–1462 1455
123
excluding the cerebellum and the spine, was traced with
MIPAV (Center for Information Technology, National
Institutes of Health) which provided a user interface and
semiautomatic methods for facilitating brain segmentation.
The intra-rater error was 3 % for one rater, and 8 % for two
independent raters. Accordingly, only a single rater (HH
Hu) segmented all brain images. The rater simultaneously
inspected three orthogonal views to ensure the 3-D topo-
logical correctness of the brain contours. This was because
sulci orientations often allow them to be only identified
from one viewing angle. Then, all segmented images were
reviewed by a radiologic physician (WY Guo). As shown
in Fig. 3, Step 3(a), the contour (red curve) was finally
segmented on the coronal view. Figure 4c presents the
difference between outer surface and inner surface of the
same brain slice.
Calculation of shape index (SI) and curvedness
index (CVD)
The definitions of the SI and the CVD are described in the
Appendix. Figure 5 shows the distribution of the SI and
CVD values on a simulated 3-D surface, formulated as
z ¼ cos yðÞsin xðÞexp 0: 2yðÞexp 0:1xðÞ ð1Þ
where x 2 C 2p : 2p
fg
and y 2 C
1
2
p :
7
2
p
:
Fig. 5 Distribution of SI and CVD values on a simulated surface.
The SI and CVD values were transformed into the colors on the
surface. Top-left panel: distribution of SI values on the surface. Top-
right panel: distribution of CVD values on the surface. Middle panel:
we focused on the SI and CVD values of the points along the diagonal
line of the x–y plane for the surface. Bottom-left panel: SI values of
the points along the diagonal line from left to right. The periodic SI
values encode the shape of the hills on the surface. The peaks were
located by an SI value of 1. Bottom-right panel: the CVD values of
the points along the diagonal line from left to right. Decreasing CVD
values show that the hills become increasingly smooth
1456 Brain Struct Funct (2013) 218:1451–1462
123
We acquired k
1
and k
2
by differentiating implicit func-
tions of the surfaces (Eq. 1), and then the SI and CVD
values by inserting k
1
and k
2
into the SI and CVD equations
(Eqs. A1 and A2).
Several ‘‘hills’’ and ‘‘valleys’’ on the surface decreased
in height and became smoother along the x-axis (Fig. 5).
On the diagonal points (middle panel), the SI values
showed a trend in the shape of ‘‘hills’’ and ‘‘valleys’’
Fig. 6 Distribution of SI (upper row) and CVD values (lower row) over outer brain surfaces with increasing age. The SI values located four
cortical shapes based on the division in Fig. 2
Fig. 7 Partial correlation (r) of CVD values of each cortical shape to
the cerebral volume and to the cortical surface area, for outer cortical
surfaces. Positive correlations with gestational age were found in the
CVD values of each cortical shape (top row), cerebral volume (left,
bottom row), and the cortical surface area (middle, bottom row).
However, with gestational age controlled, only the correlation
between the CVD value of gyral nodes and outer cortical surface
area reached statistical significance (P \ 0.05) (right, bottom row)
Brain Struct Funct (2013) 218:1451–1462 1457
123
(bottom-left panel), whereas the CVD values showed a
decreasing trend that was consistent with the smoothening
along the x-axis (bottom-right panel). Therefore, we can
use the SI value of the surface point to locate the shape of
interest, and the CVD value to quantify the degree of a
specific shape deviated from the flat plane.
Estimating CVD and SI on brain images
We focused on the SI and CVD of surface voxels in an
image. Surface voxels (white voxels in Step 3c of Fig. 3)
were acquired from each segmented slice of a brain image
volume. All surface voxels comprise a digital 3-D surface.
The following describes the calculation of the SI and CVD
of a voxel.
First, to eliminate the possibility of inconsistent intensity
among images, which can lead to incomparable results
among subjects, all brain images were binarized with a
value of 100 to the inside of the brain contour and a value of
0 to the background (Fig. 3, Step 3d). Second, the intensity-
based method proposed by Thirion and Gourdon (1993) was
used to obtain k
1
and k
2
. This method avoided the para-
metrical modeling of brain surface. Let a 3-D digital image
be denoted by f ðx; y; zÞ, and the partial derivatives
of ðx; y; zÞ=ox be denoted by f
x
, of ðx; y; zÞ= oy by f
y
,
of ðx; y; zÞ=oz by f
z
, o
2
f ðx; y; zÞ=ox
2
by f
xx
, o
2
f ðx; y; zÞ=oy
2
by f
yy
, o
2
f ðx; y; zÞ=oz
2
by f
zz
, o
2
f ðx; y; zÞ=oxoy by f
xy
,
o
2
f ðx; y; zÞ=oyoz by f
yz
, and o
2
f ðx; y; zÞ=oxoz by f
xz
for
simplicity. These partial derivatives represent the intensity
gradient among adjacent voxels. The mean curvature (S)
and Gaussian curvature (K) at a voxel location ðx; y; zÞ were
given by Thirion and Gourdon (1993).
S ¼
1
2h
3=2
f
2
x
ðf
yy
þ f
zz
Þ2f
y
f
z
f
yz
þ
f
2
y
ðf
xx
þ f
zz
Þ2f
x
f
z
f
xz
þ f
2
z
ðf
xx
þ f
yy
Þ2f
x
f
y
f
xy
K ¼
1
h
2
h
f
2
x
ðf
yy
f
zz
f
2
yz
Þþ2:f
y
f
z
ðf
xz
f
xy
f
xx
f
yz
Þ
þ f
2
y
ðf
xx
f
zz
f
2
xz
Þþ2f
x
f
z
ðf
yz
f
xy
f
yy
f
xz
Þ
þf
2
z
ðf
xx
f
yy
f
2
xy
Þþ2f
x
f
y
ðf
xz
f
yz
f
zz
f
xy
Þ
i
h ¼ f
2
x
þ f
2
y
þ f
2
z
The principal curvatures (k
1
and k
2
) at the voxel location
ðx; y; zÞ in the images are the solutions of the following
equation (Thirion and Gourdon 1993):
k
i
ðx; y; zÞ¼S
ffiffiffiffi
D
p
with D ¼ S
2
Kði ¼ 1; 2 and k
1
jj
k
2
jjÞ
Finally, the SI and CVD values of each surface voxel
were obtained by inserting k
1
and k
2
into Eqs. A1, A2.
Based on the SI value, all surface voxels in an image
were divided into gyral nodes, sulcal pits, and two transi-
tional cortical shapes: gyral saddles and sulcal saddles. The
division of these four shapes is shown in the bottom line in
Fig. 2. The gyral node was fully convex (0.5 \ SI B 1),
and the sulcal pit was fully concave (-0.5 [ SI C-1).
Cylindrical shapes (SI = 0.5 or -0.5) were used to separate
the concavities and convexities from the saddle-like shapes.
The saddle-like shapes were divided into gyral saddles and
sulcal saddles using the symmetric saddle (SI = 0, in
Fig. 2). The gyral saddle was composed of outward saddle-
like shapes (0 \ SI \ 0.5), and the sulcal saddle was
composed of inward saddle-like shapes (-0.5 \ SI \ 0).
Fig. 8 Distribution of SI (upper row) and CVD values (lower row) over inner brain surfaces with increasing age
1458 Brain Struct Funct (2013) 218:1451–1462
123
After the SI located one of the cortical shapes, the CVD
values of voxels belonging to the same shape were aver-
aged for each fetus (Fig. 3, Step 6). Finally, the CVD
changes were compared to the cerebral volume and cortical
surface area.
Measuring cerebral volume and cortical surface area
The cerebral volume was calculated in the reconstructed
image by summing the size of all voxels inside the brain
contour. The unit of the cerebral volume was 1 9 1 9 1mm
3
.
The cerebral surface area was obtained using the Lind-
blad approach (2005), particularly for estimating the sur-
face area of a 3-D binarized image volume. Each voxel on
the brain surface was assigned a weighting value based on
its local configuration; then the cerebral surface area was
equal to the summation of the value of all these weightings.
Because our reconstructed brain volume was isotropic,
with a resolution of 1 9 1 9 1mm
3
, the unit of the brain
surface area was mm
2
.
Statistics
The partial correlation coefficient (r) was calculated to
examine the relationship of average CVD values to cere-
bral volume and the cortical surface area. GA was set as the
control variable. The critical value (P value) of the statis-
tics was set to 0.05.
Results
Figure 6 shows the distribution of the SI and CVD on a
series of brain outer surfaces with an increasing GA. Results
from the study showed that the SI successfully discrimi-
nated the sulcal pits (red) and the gyral nodes (dark blue), as
well as the two saddle-like shapes (light blue and yellow)
that connected the sulcal pits and gyral nodes. Furthermore,
the largest CVD values (the red area in the bottom row of
Fig. 6) emerged on the ridgeline of cortical folds and
grooves, and increased with increasing age. These results
show that the SI efficiently located the four cortical shapes
for fetuses at different ages, and that the CVD reflected the
sharpness and smoothness of the surface.
Figure 7 shows that GA correlated positively with the
CVD values of four cortical shapes (top row), the cerebral
volume (left panel, bottom row), and the outer cortical
surface area (middle panel, bottom row). However, with
GA as the control, only the correlation between the CVD
values of gyral nodes and cortical surface area reached a
significant value (r = 0.44,P= 0.03) (right panel, bottom
row). All the CVD values were unrelated to cerebral vol-
ume (P [ 0.05) (right panel, bottom row).
This study also examined the partial correlation between
cerebral volume and the outer cortical surface area. The
result showed that cerebral volume increased significantly
with the outer cortical surface area with GA as the control
(r = 0.86, P \ 0.0001).
Fig. 9 Partial correlation (r) of CVD values of each cortical shape to
the cerebral volume and to the cortical surface area, for inner cortical
surfaces. Positive correlations with gestational age were found in the
CVD values of each cortical shape (top row), cerebral volume (left,
bottom row), and the cortical surface area (middle, bottom row).
However, with gestational age controlled, only the CVD values of
gyral nodes and gyral saddle significantly correlated with inner corital
surface area (P \ 0.05) (right, bottom row)
Brain Struct Funct (2013) 218:1451–1462 1459
123
Figures 8 and 9 show the results of inner cortical surface.
Comparing Figs. 6, 7 to Figs. 8, 9, we found that the results
between outer and inner cortical surface were similar.
However, the results of inner surface presented that CVD
values both in gyral node and gyral saddle are positively
correlated with inner surface area, whereas the results of
outer surface presented that CVD value only in gyral node is
positively correlated with outer surface area. Besides, the
r became larger and p value became smaller in the results of
inner surface than in that of outer surface, r value changed
from 0.44 to 0.61 and p value changed from 0.03 to 0.002
(CVD of gyral node with surface area in Figs. 7g, 9g).
Discussion
This study demonstrated that the concurrent use of the SI
and CVD is feasible for quantifying 3-D morphological
changes in developing fetal brains. The mean curvature
(S) and Gaussian curvature (K) are well-known curvature
derivatives that have been used extensively for quantifying
3-D morphological changes in adult (Luders et al. 2006)
and fetal brains (Batchelor et al. 2002; Clouchoux et al.
2012). The S is defined as the arithmetic average of prin-
cipal curvature (S = (k
1
? k
2
)/2); the K as the geometric
average (K = square(k
1
9 k
2
)). Although the S and K, as
well as the SI and CVD, are all combinations of principal
curvatures; the SI and CVD can describe surface shape
changes in more detail. As shown in Fig. 10, the S and K
identify only several shapes by comparing the sign of the S
and K, and cannot subdivide these shapes. The SI values
can identify continuous surface shapes toward convexity or
concavity (Fig. 2).
The morphology of fetal brains varies substantially with
gestational age, especially during the formation of the
primary cortical folding (20–36 weeks). To evaluate such
morphological changes, several studies used age-dependent
reference images (Rajagopalan et al. 2011; Habas et al.
2010a, b). Their approaches were useful in detecting local
morphological changes, and developing a specific spatio-
temporal pattern for fetal brain in vivo (Habas et al. 2012).
However, the shape measurement, such as the displace-
ment between each voxel and the reference image (Habas
et al. 2010a, b), varied with each chosen reference image.
That is, the baseline of shape changes varied by age. This is
because the number of reference images to be used remains
unknown. For example, Rajagopalan et al. (2011) used two
brain reference images within 20–28 weeks of gestation,
whereas Clouchoux et al. (2012) used four brain reference
images within 25–35 weeks of gestation. Our study showed
that the SI and CVD can capture shape changes without
any reference image, and were calculated on each surface
voxel. This suggests that the SI and CVD can be further
used in reference-based analysis.
Our finding showed that the CVD values of each cortical
shape did not vary with cerebral volume with age con-
trolled (Fig. 7, bottom-right panel). This suggests that in
fetal development the CVD values of cortical shapes pro-
vide different information from brain volumetry. This also
shows that two measures that correlate highly with age may
or may not correlate with each other. As the scatter plots
show in Fig. 7, all measures were highly correlated with
GA. With GA controlled, cerebral volume is still highly
correlated with the cortical surface area (r = 0.86,
P \ 0.001). However, cerebral volume is not correlated
with CVD values (Fig. 7, bottom-right panel). In past
studies, two biometric measures such as the gyrification
index and cortical surface area (Clouchoux et al. 2012) and
local volume and shape using the template-dependent
manner (Rajagopalan et al. 2011) showed similar growth
Fig. 10 Surface shapes
identified by the sign of the
mean curvature (S) and
Gaussian curvature (K).
Contrary to the H and K, which
identify shapes by their sign, the
SI value can discriminate
surface shapes
1460 Brain Struct Funct (2013) 218:1451–1462
123
patterns over identical developmental periods. Results from
our study suggest that further quantitative examination
with controlled GA should be conducted to ensure corre-
lation between two biometric measures in the fetal stage.
This surface-based study used two curvature-based
indices, the shape index (SI) and curvedness (CVD), to
quantify brain morphology for fetuses. A recent study by
Habas et al. (2012) used mean curvature to detect the timing
of formation of primary cortical folding. They focused on
12 selected sulci and gyri forming between 20 and 28 weeks
of gestation. Their results showed that the absolute value of
mean curvature increased with age for all the selected sulci
and gyri. These regional results are consistent with our
results, in which the CVD value increased with age for four
cortical shapes. Besides, they normalized the volume of
fetal brains to averaged reference image, and found the
volume-normalization surface area still increasing with age.
This increase in surface area is due to the growth of cortical
folding (Habas et al. 2012). Our results show that while
controlling age, the CVD value increase with surface area
for gyri but is unrelated with brain volume (Fig. 9). That is,
the CVD value appears to reflect the growth of surface area
from the development of cortical folding, rather from the
development of brain volume.
Our results show that the cortical surface area expanded
with the curvature of the gyral nodes, rather than that of
sulcal pits (Figs. 7, 9). Previous studies also observed such
a difference in relation to the amount of axonal connec-
tions. A postmortem MR tractography study visually
inspected the radial organization of fetal brains, and then
found more axonal connections in gyral crowns than in
sulcal fundi (Takahashi et al. 2012). Similar results were
obtained in the study of an in vivo adult brain, in which
more axonal connections emerged in the ‘‘T-shape’’ gyral
crowns (Li et al. 2010). However, these methods neither
provided quantitative analyses, nor probed morphological
changes in sulcal fundi. Our approach introduced automatic
quantification of morphological changes separately in gyral
crowns and in sulcal fundi. This can provide additional
insight into the formation of gyri and sulci in the fetal stage
by further comparing the morphological changes to bio-
metric measures such as fiber density from in utero MR
tractography (Kasprian et al. 2008) and the thickness of the
subplate zone (Corbett-Detig et al. 2011).
In summary, we demonstrated the effectiveness of the SI
in separating the gyral nodes and sulcal pits, and the ability
of the CVD to quantify brain morphology during the
essential period of primary cortical folding formation,
without the need to establish age-associated templates. The
results suggest that, for fetuses, the SI and CVD provide
developmental information distinct from the cerebral vol-
ume. The proposed method can be useful for further
investigations on the formation of cortical convolutions,
and for establishing the baseline to discriminate normal and
pathological brains in prenatal MR imaging.
Acknowledgments This study was funded in part by the National
Science Council and NSC 99-2314-B-075-033-MY2, NSC 100-2221-
E-010-009, Taipei Veterans General Hospital V97C1-025 and the
Aim for the Top University Plan from Ministry of Education for
National Yang-Ming University. NSC support for the Center for
Dynamical Biomarkers and Translational Medicine, National Central
University, Taiwan (NSC 100-2911-I-008-001).
Appendix: Definition of SI and CVD
The SI and the CVD were firstly proposed by Koenderink
and van Doorn (1992). They modified the polar system of
principal curvatures\k
1
,k
2
[, the angle and distance as (see
Fig. 1b).
Shape index SIðÞ¼
2
p
arctan
k
2
þ k
1
k
2
k
1
and ðA1Þ
Curvedness CVDðÞ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k
2
2
þ k
2
1
2
r
; respectively: ðA2Þ
Several properties of the SI and CVD are listed as
follows:
1. The SI value can be positive and negative. The CVD
value is always positive.
2. The SI value ranges between -1 and 1. Figure 2
displays several shapes and their corresponding SI
values such as the spherical dome (k
1
= k
2
= 0,
SI = 1), spherical cup (k
1
= k
2
= 0, SI =-1), cylin-
drical (|SI| = 0.5), and saddle-like (|SI| \ 0.5) shapes.
3. A flat plane (k
1
= k
2
= 0) makes the value of SI = 0
and of CVD = 0.
4. A symmetric saddle with k
1
=-k
2
(middle plot in
Fig. 2) makes a value of SI = 0.
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