A Geometric Method for Automatic Extraction of Sulcal Fundi

A Geometric Method

for Automatic Extraction of Sulcal Fundi

Michael Hofer*

C.-Y. Kao, G. Sapiro, J. Stern, K. Rehm and D.A. Rottenberg

*Geometric Modeling and Industrial Geometry Vienna University of Technology

University of Minnesota, Minneapolis VA Medical Center

FSP Meeting Graz March 23, 2007

What are sulcal fundi?

I Sulci = crevices of convoluted human brain surface

I Sulcal fundi = 3D curves lying in the depths of the cerebral cortex.

Why are sulcal fundi important?

I Intrinsic value in brain research

I Fundi are often used as

landmarks for deformation fields warping cortical surfaces of different brains onto each other

I Longitudinal and cross-sectional studies of brain structure and function, cortical thickness, gray-matter density

What is the state of the art?

Fig. from [Lohmann 1998]

I Manual labeling of voxels in MRI brain volume using GUI which displays three orthogonal 2D brain slices

I Process is extremely tedious, time consuming and prone to human error

I Expert anatomist needs 1 day for manually marking 6 fundi per hemisphere

Overview of our algorithm

I Our goal: Automatic sulcal fundi extraction

I Main steps are:

Segmentation into brain/non-brain matter

I Input: T 1-weighted MRI brain volume (1 mm isotropic voxels) acquired at Montreal Neurologic Institute, provided by Dr. Alan C. Evans.

I Output: stripped skull using Brain Extraction Tool (BET)^a.

ahttp://www.fmrib.ox.ac.uk/fsl/bet/

Surface extraction

I Input: Segmented MRI brain volume

I Output: Topologically correct triangular mesh representing the pial (GM-CSF) surface of cerebral cortex extracted by publicly available software FreeSurfer^a.

ahttp://surfer.nmr.mgh.harvard.edu/

Outer hull surface computation

Morphological closing using the level set function:

I Move surface Φ = 0 outward by a time parameter T

I Move surface inward by same amount of time

I Governing equation:

 Φt+ V (t)|∇Φ| = 0 Φ(x , 0) = Φ, where

V (t) =

 1 for t ≤ T

−1 for T < t ≤ 2T

 .

Pial surfaces and outer hulls for different brains

Outer hulls overlaid on MR images

Axial, coronal, and sagittal slices of the MRI brain volume and the outer hull surface of one hemisphere.

Sulcal depth computation

a. Euclidean

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B B B BBBBBBBBBBBBBBBBBBB B B B B B B B B B B B B BB B B B BB B B BBBBBBBB C C C CCCCCCCCCCCCCCCCCCC C C C C C C C C CC C C C CC C C C CC CCCCCCCCC

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b. on mesh

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c. our measure

A A AAAAAAAAAAAAAAAAAAAAA A A A A A A A A A AA A A A AA A A AAAAAAAAAA

BBBBBBBBBBBBBBBBBBBBBBBBB B B B B B B B B B BB B B B BB B B B BBBBBBB CCCCCCCCCCCCCCCCCCCCCCCCC C C C C C CC C C C CC C C C CC C C CCCCCCC

DDDDDDDDDDDDDDDDDDDDDDDDD D D D D D D D D D DD D D D DD D D D DDDDDDD

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I Propose a new geodesic depth measure of pial surface s to outer hull surface h: d (C ) > d (B) > d (A) ∼= d (D)

I Calculation on a rectangular grid using a fast sweeping method restricted to the (CSF) region between h and s

I Propagate depth onto mesh using e.g. trilinear interpolation

Computed sulcal depth — illustrated on surface

We slice through the pial and hull surface at height indicated in previous figure with a black line. (Left) hull surface. (Middle) intersection curve with the pial surface. (Right) pial surface. The color corresponds to the computed depth (in mm) from the hull.

Computed sulcal depth — overlaid onto MRI

0 5 10 15 20 25 30 35 38.7

Color coded geodesic depth (see color bar) displayed on axial, coronal, and sagittal slices of the pial surface overlaid onto the MRI brain volume.

Sulcal regions extraction

I Define the sulcal regions of the pial surface as those with a depth d larger than a depth threshold D

I In the literature, D is usually chosen 2 − 3 mm

I We use D = 2.5 mm

I Results in approximately 50 components per hemisphere

Sulcal fundus extraction — endpoint computation

C C C C CCCCCCCCCCCCCCCCC C C C C C C C C C C C C CC C C C CC C C C C CCCCCCC

p₁

p2

p3

p4

C p₁

B p1

N

I Fundi endpoints p1, . . . , p4 of a component C are those points of the boundary B of C that are extremal according to the principal component direction in a local neighborhood N.

Sulcal fundus extraction — thinning step

For each component C take triangles of boundary strip B that correspond to the computed endpoints p_j; add them to an initial skeleton list S ; repeat until all triangles of C are processed:

1. Find the triangle ∆ of B with the least depth.

2. If removing ∆ would change the connectivity of the mesh, then add ∆ to the skeleton list S . Else, we remove ∆ from the list of boundary triangles B and add the edge neighbor triangles of ∆ to B.

c

c

p₂

p1 I The longest non-branching path of a sulcal region runs from p₁ to p2.

Sulcal fundus extraction — results

Automatically extracted sulcal fundi illustrated as thick 3D curves.

Michael Hofer A Geometric Method for Automatic Sulcal Fundi Extraction

Sulcal fundi smoothing

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I Extracted sulcal fundi are represented as polylines

I Smooth polylines using weighted splines in manifolds x (u), Z _un

u1

w (x (u))k¨x (u)k²du → min

Experiments, data analysis and evaluation

Manual extraction is considered as ”gold standard”. Need to compare to it.

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Comparison automatic to manual — central sulci

dots = manual, curves = automatic

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Comparison automatic to manual — central sulci

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rater 2

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Problems with manual labeling

Manual labeling of the central sulcus fundi in 6 consecutive slices of the original MRI volume: If the sulcus is parallel to the plane of section, manual-labeling may miss points in the indicated areas.

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Measure distances handmarked to automatic

p_m p_s

p_a r

S

I Compute closest points ps to manually-labeled voxels p_m on the pial surface S .

I To compensate for errors in the surface extraction: measure the distance r = kps− p_ak to the automatically extracted points pa.

Major sulcal fundi — recap

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Distances handmarked to automatic — histograms

The histograms of r = kps − p_ak for three different brains. The horizontal axis is r and the vertical axis is the number of fundus points. The two rows show the results for the same brains for two different human raters.

0 2 4 6 8 10 12

0 100 200 300 400 500

0 2 4 6 8 10 12

0 50 100 150 200 250 300

0 2 4 6 8 10 12

0 50 100 150 200 250 300 350

0 2 4 6 8 10 12

0 100 200 300 400 500

0 2 4 6 8 10 12

0 50 100 150 200 250

0 2 4 6 8 10 12

0 50 100 150 200 250 300 350 400

Distances handmarked to automatic — numbers

The mean values per major sulci are taken over six brains for two different raters.

I n . . . mean of number n of handmarked voxels per fundi¯

I ¯r . . . mean of distance value r

I m . . . mean of total percentage of voxels with r < 2 mm.¯

I σ_(·) . . . corresponding standard deviation

calc. cent. olfa. prec. supe. temp.

¯

n 54.5 63.5 45.7 15.5 48.4 82.0

σ_n 4.2 3.8 4.3 4.0 10.9 10.0

¯

r 1.0 1.1 1.0 1.8 2.5 2.4

σ_r 0.4 0.4 0.4 0.9 0.8 0.6

¯

m 86% 86% 87% 63% 52% 55%

σm 9% 9% 9% 28% 19% 11%

Summary of evaluation results

Possible reasons for divergence of automatic and manual results:

I Shortcomings in our definition of sulcal depth and fundal location

I Errors in the extracted mesh surface

I Errors in the manual labeling

Advantages of automatic extraction

I Objective extraction of sulcal fundi

I Improved quality and reproducibility of process

I Considerable time savings

I Automatically process large number of high-resolution MRI data sets

I Automatic extraction also returns secondary and tertiary fundi

Comparison to previous work

I Both voxel-based and surface mesh-based representations of cortical surface have been used.

I Two distinct approaches: curvature; distance functions.

Previous work: Curvature based approaches

Fig. from [Bartesaghi et al. 2001]

Fig. from [M´emoli et al. 2004]

I Extract WM-GM boundary surface and compute mean surface curvature

I Fundi are curves lying within areas of extremal mean surface curvature.

I Manually mark two endpoints of a fundi

I Use dynamic programming [Khaneja et al. 1998] to find a connecting curve

I Use fast marching algorithm on triangle meshes [Bartesaghi et al.

2001] or implicit surfaces [M´emoli et al. 2004] to connect two points with a weighted geodesic.

Previous work: Distance based approaches

Figs. from [Le Goualher et al. 1999]

I Distance based approaches compute medial sulcal surfaces (”sulcal ribbons”) from volumetric data.

I Fundi are inferior margins of these surfaces [Lohmann 1998, Le Goualher et al. 1999, Cachia et al. 2003]

I Combination of curvature and distance based computations [Tao et al. 2004]

Open questions

I Experts only agree on nomenclature for the major sulci (e.g.

central and superior frontal)

I Secondary and tertiary sulcal patterns vary greatly from individual to individual

I Classification of our results into primary, secondary, tertiary sulcal fundi

I Automatic labeling of sulcal fundi

Conclusion

I Sequence of geometric algorithms for automatic extraction of sulcal fundi from MR images.

I Novel depth measure, high quality polyline representation of sulcal fundi.

I Results are useful for downstream applications in computational neuroanatomy.

References

I C.-Y. Kao, M. Hofer, G. Sapiro, J. Stern, K. Rehm, and D.A.

Rottenberg: A geometric method for automatic extraction of sulcal fundi. Transactions on Medical Imaging 26, 2007, to appear.

I C.-Y. Kao, M. Hofer, G. Sapiro, J. Stern, and D.A.

Rottenberg: A geometric method for automatic extraction of sulcal fundi. Proc. IEEE International Symposium on

Biomedical Imaging, (electronic), 1168-1171, 2006.