# Some Methods for Modeling Cortical Surface Yuai Hua 2010.10.29.

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Some Methods for Modeling Cortical Surface Yuai Hua 2010.10.29

What is a cortical surface like?

Gyral region, sulcal region, gyral crest, sulcal fundi Sulcal Region Gyrus Crest Line Gyral Region Sulcal Fundi

Sulcal basin & Gyral basin Sulcal Basin Gyral Basin (cross-section)

What will I present ? Cortical sulcal parcellation Cortical fundi extraction Cortical gyral parcellation Cortical sulcal bank segmentation Gyral folding pattern analysis

Part A. Cortical sulcal parcellation Key Technique: Principal direction flow field tracking method ( proposed by Gang Li et al. ) Goal: Finding sulcal region sulcal basin &

Some Basic Concepts ----- the maximum and minimum values of curvatures at a point p on a surface. ----- the vectors along which the curvatures are principal. Principal curvatures Principal directions

principal directions

Triangulated cortical surface

sulcal regions & sulcal basins? How to find Stages: (1) Estimate principal curvatures and principal directions at each point; (2) Finding sulcal regions; (3) Finding sulcal basins.

Step1. Calculate the normal vectors of each triangle face. X X1X1 X2X2 X5X5 X4X4 X3X3 X6X6 (1) Estimate principal curvatures and principal directions at each point.

Weingarten Matrix is a symmetric matrix. Its eigenvalues are the principal curvatures. Its eigenvectors are the principal directions. Weingarten Matrix Step2. Calculate Weingarten Matrix in each triangle face.

Step4. Calculate the eigenvalues and eigenvectors of each Weingarten Matrix. Those are the principal curvatures and principal directions. Step3. Calculate Weingarten Matrix at each vertex by weighted averaging its adjacent faces. X X1X1 X2X2 X5X5 X4X4 X3X3 X6X6

Only the maximum principal curvatures and its corresponding principal directions are adopted.

Calculating the directional derivative of maximum principal curvature along the corresponding principal direction and ensuring it decreases by choosing appropriate principal direction. Keeping in mind…… fundi Principal direction

The principal direction points towards the sulcal fundus from the gyral crest; The principal curvatures are large positive and negative values at gyral crown and sulcal fundi. Thus

(2) Finding sulcal regions Let n = the number of the total vertices on the cortical surface

In order to segment cortical surface into sulcal regions and gyral regions, we should solve Problem : is unknown Knowing: is a normal random variable & can be estimated.

Hence, according to the Bayes theory and a special method (proposed by Zhang et al., 2010), we can estimate X by solving During this process, hidden Markov random field model, expectation maximization method and iterated algorithms are used. So far, a cortical surface is segmented into a series of sulcal regions and gyral regions.

(3) Finding sulcal basins Idea: following the maximum principal directions from a gyral crown region until to the sulcal fundus. The vertices that converge to the same fundus are grouped together, and these vertices form a sulcal basin. Thus the cortical surface are segmented into different sulcal basins. fundi Principal direction

Idea: at a sulcal fundus and gyral crown, the produced flow field should be close to the original direction field, and at flat areas, it should vary smoothly. Step1 Estimating flow field Problem: at a flat cortical region, the two principal curvatures might be very small, so we may not find the exactly maximum principal direction. Method: Principal direction flow field diffusion

n : normal vector Estimating flow field V(X) maximum principal curvature maximum principal direction weighting parameter gradient operator where Using calculus of variation to solve the equation

Step2 Sulcal basin segmentation Method: Principal direction flow field tracking —Searching for flow trajectories Given a vertex on a flow trajectory with is calculated as the principal direction the one-ring adjacent vertex of. the next vertex

X6X6 V(X) X5X5 X4X4 X1X1 X3X3 X2X2 X =x′

V(X) X V(X′) V(X) X V(X′) x′

The region at which the flow field tracking procedure stops should be a fundus. Thus every vertex flows to a fundus. Those vertices flow to the same fundus are grouped together as a sulcul basin. Cortical sulcal parcellation is over

Part B. A pipeline for cortical fundi extraction (Proposed by Gang Li et al)

Step1 Estimating curvatures and curvature derivatives Step2 Detecting sulcal fundi segments The maximum principal curvature The minimum principal curvature, The principal directions. Directional derivative

Criterion for fundus point : Fundus point

Procedure for fundi segmenting (1) Procedure for finding fundi points: in the cortical surface For each triangle (three vertices are )

(2) Connect the adjacent fundi points to form fundi segments:

Candidate fundi segment: if there is any candidate fundi point in it; Strict fundi segment: if there is no candidate fundi point in it; Two types of Provisional fundi segments

b) Expanding again For every vertex with negative maximum curvature in a fundi curve, connect itself with its adjacent vertex which is in another fundi curve, obtain a new fundi curve. (3) Linking sulcal fundi segments a) Starting from a strict fundi segment, adding the adjacent segments to it, and go on, a fundi curve will be obtained. There may be more than one fundi curve.

c) Pruning the fundi curves less than three segments. The remain fundi curves may include some very short ones. How to tell the different kinds of the short fundi curve? Two types of short fundi curves: interrupted and inherent. Next, it is necessary to connect those interrupted short fundi curves to the long ones.

d) From each endpoint of the extracted fundi curve, searching the geodesic region to find whether there exists another fundi curve in the region. If any, connect the endpoints to the newly found sulcal fundi curve. e) Smoothing the extracted sulcal fundi curves. Due to the numerical error, the extracted sulcal fundi curves may exist sharp bumps, it should be smoothed. Method : minimizing the geodesic distance between the endpoints or junction points of each piece of the extracted sulcal fundi.

Part C. Cortical Gyral Parcellation Technique: Using probabilistic atlas and graph cuts ( proposed by Gang Li et al. )

Each gyral patch is a part of gyral basin Each gyral patch is bounded by adjacent sulcal fundi and interrupted at junctions of gyral basins Each gyral patch belongs to only one gyral basin Each gyral basin is composed of one or more gyral patches. Characteristics of a gyrus:

Every kinds of gyral patchs have been labeled by experts in the form of Probabilistic Atlas. What we know: Probabilistic atlas is a series of maps of human brain anatomic regions. These maps were produced from a set of whole-head MRI. Each MRI was manually delineated to identify a set of 56 structures in the brain, most of which are within the cortex. Each type of region has a label.

n: number of gyral patches p-th gyral patch in sulcal surface k-th gyral structure in the Probabilitic Atlas Area of k-th gyral structure in the P-Atlas Area of the intersection Likelihood of belonging to. Area of p-th gyral patch in sulcal surface The label which the p-th gyral patch in sulcal surface be assigned corresponding to the gyral basin in the P-Atlas

Maximum principal curvature at vertex in gyral patch. Set of all neighboring vertices between two neighboring gyral patches. represents the weight between two neighboring gyral patches.

. N: the set of neighboring gyral patch pairs an adjust parameter. by using graph cuts method. The cortex is segmented into different gyral basins

Part D. Cortical sulcal bank segmentation Sulcal bank: Each side of a sulcal basin. A sulcal basin has two opposite sulcal banks. Goal: Segment a sulcal basin into two sulcal banks. Technique: Graph partition (proposed by Gang Li et al.)

Step1: Segment a cortical surface into sulcal basins. The triangular mesh of a sulcal basin Procedure Step2: Rough sulcal bank segmentation Angular similarity between two vertices : unit normal direction) will be large if otherwise, small. ( are in the same sulcal bank;

Distance similarity between two vertices : Euclidean distance; geodesic distance (the shortest path will be large if sulcal bank; otherwise, small. connecting two vertices along the triangular mesh of a sulcal basin) are in the same

Similarity between two vertices will be large if otherwise, small. ( is a weight parameter) are in the same sulcal bank;

, Rough graph partition: Then set Firstly, using Normal cuts method to divide the sulcal baisn into two opposing sulcal banks A and B

Technique: construct a energy function and minimize it. Step3: Fine sulcal bank segmentation Goal: making the boundary clearer. Cortical sulcal bank segmentation is over

(proposed by Kaiming Li, et al) Part E. Gyral folding pattern analysis

Gyral folding patterns:

Gyral crown Gyrus Sulcus Hinge Line Hinge Different parts of cortex

Five different parts of cortex Blue: sulcus basin Red: gyrus crown Yellow: sub gyrus crown Green ： Central area Light blue: sub sulcus basin

Goal: 1. segment cortex into five different classes 2. detect the hinge. Firstly, build a coordinate system as follows: O: Any vertex on a sulcal surface N: The normal direction at O R o : A randomly selected vector on tangent plane Then build a polar coordinate system in tangent Plane, then build a Cartesian-Polar Coordinate System.

Cartesian-Polar Coordinate System N P RαRα RORO C(α,x, y) α y x o A profile

gyrus crown: 1; sub gyrus crown: 2 central area: 3; sub sulcus basin: 4 sulcus basin: 5 f i represents the value of i-th point, then calculate the value of the profile: Set some threshold, the gyral surface can be divided into five regions and the hinge points can be detected. Assign each class a value: For each profile, suppose there are N points on it, Procedure

References Gang Li et al.,Automatic cortical sulcal parcellation based on surface principal direction flow field tracking. ( http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WNP-4VXDTX7- 1&_user=130907&_coverDate=07%2F15%2F2009&_rdoc=1&_fmt=high&_orig=search&_origin=search& _sort=d&_docanchor=&view=c&_searchStrId=1522441145&_rerunOrigin=scholar.google&_acct=C00000 4198&_version=1&_urlVersion=0&_userid=130907&md5=6abccc308c9b83e504db8c4f51819d29&searcht ype=a) Gang Li et al.,An automated pipeline for cortical fundi extraction. ( http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6W6Y-4Y9XM1S- 1&_user=130907&_coverDate=06%2F30%2F2010&_rdoc=1&_fmt=high&_orig=search&_origin=search& _sort=d&_docanchor=&view=c&_searchStrId=1522445802&_rerunOrigin=google&_acct=C000004198&_ version=1&_urlVersion=0&_userid=130907&md5=74a63ac5ec24e920f55e874ba48d64ea&searchtype=a ) Gang Li et al., Automatic cortical gyral parcellation using probabilistic atlas and graph cuts. ( http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WNP-4VXDTX7- 1&_user=130907&_coverDate=07%2F15%2F2009&_rdoc=1&_fmt=high&_orig=search&_ori gin=search&_sort=d&_docanchor=&view=c&_searchStrId=1522455825&_rerunOrigin=schol ar.google&_acct=C000004198&_version=1&_urlVersion=0&_userid=130907&md5=1d5aa28 7fe1024dddbf3c85e02287e05&searchtype=a )

http://en.wikipedia.org/wiki/Principal_curvature http://planetmath.org/encyclopedia/WeingartenMatrix.html http://www.loni.ucla.edu/Atlases/LPBA40 http://en.wikipedia.org/wiki/Cut_(graph_theory) Gang Li et al., Cortical sulcal bank segmentation via geometric similarity based graph partition. (http://www.springerlink.com/content/l5213t641r587l83/)http://www.springerlink.com/content/l5213t641r587l83/ Kaiming Li et al., Gyral folding pattern analysis via surface profiling. (http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2915584/)

Thank you