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**|**HHS Author Manuscripts**|**PMC2970518

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- Abstract
- 1 Introduction
- 2 Atlas Construction
- 3 Automated Identification of Major Sulci
- 4 Experimental Results
- 5 Conclusion
- References

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Med Image Comput Comput Assist Interv. Author manuscript; available in PMC 2011 January 1.

Published in final edited form as:

Med Image Comput Comput Assist Interv. 2010; 13(Pt 3): 49–56.

PMCID: PMC2970518

NIHMSID: NIHMS242596

Yonggang Shi: ude.alcu.inol@ihsy

See the article "A randomized placebo-controlled clinical trial of five smoking cessation pharmacotherapies" in *Arch Gen Psychiatry* on page 1253.

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In this paper we propose a novel and robust system for the automated identification of major sulci on cortical surfaces. Using multiscale representation and intrinsic surface mapping, our system encodes anatomical priors in manually traced sulcal lines with an intrinsic atlas of major sulci. This allows the computation of both individual and joint likelihood of sulcal lines for their automatic identification on cortical surfaces. By modeling sulcal anatomy with intrinsic geometry, our system is invariant to pose differences and robust across populations and surface extraction methods. In our experiments, we present quantitative validations on twelve major sulci to show the excellent agreement of our results with manually traced curves. We also demonstrate the robustness of our system by successfully applying an atlas of Chinese population to identify sulci on Caucasian brains of different age groups, and surfaces extracted by three popular software tools.

The automated identification of major sulci on the human cortex is a challenging problem with important applications in brain mapping[1]. While their form and location can vary quite significantly, there is no difficulty for an anatomist to observe the regularity of major sulci based simply on the *geometry* of cortical surfaces regardless of their size, orientation, and the software used to extract them from MRI images. From an engineering point of view, this simplicity is critical for a computational system to achieve the same level of robustness, which essentially lies in its ability of modeling sulcal anatomy with the geometry of cortical surfaces. To this end, we propose in this work a novel system for automated sulci identification by integrating local geometry, i.e., curvature, with global descriptors derived from the Laplace-Beltrami eigen-system [2,3,4] and intrinsic surface mapping [5].

At the core of our system is an intrinsic atlas of major sulci that represents prior knowledge in training data and enables the integration of curvature information into major sulci. This atlas-based approach is most related to learning-based methods for sulci identification in previous work [6,7,8,9,10]. Principal component analysis were used to model sulcal basins [6] and sulcal lines on the sphere [7]. Boosting methods were used in [9,10]. The Markovian relation of multiple sulci was incorporated with graphical models [8,10]. The main novelty in our system is that the modeling of cortical anatomy is based entirely on intrinsic geometry, which eliminates the use of coordinates in conventional MRI atlases such as the Talairach atlas. This makes our system robust to pose differences and variations across populations. We demonstrate this robustness by using an atlas constructed from a Chinese population to detect twelve major sulci on Caucasian brains of different age groups and surfaces extracted by different software tools.

The rest of the paper is organized as follows. In section 2, we describe the construction of the intrinsic atlas for the modeling of sulcal anatomy. The sulci identification algorithm based on this atlas is developed in section 3. Experimental results are presented in section 4. Finally conclusions are made in section 5.

As illustrated in Fig. 1, there are two main steps in the construction of the atlas. In the first step, a multiscale representation of the cortical surface is constructed. In the second step, surface maps at a selected scale are computed to bring manually traced sulcal lines to an atlas surface. We describe the two steps in detail next.

Let = (, ) denote the triangular mesh representation of a cortical surface, where and are the set of vertices and triangles. We construct the multiscale representation of by using the eigen-system of its Laplace-Beltrami operator , which is defined as

$${\mathrm{\Delta}}_{\mathcal{M}}f=-\lambda f.$$

(1)

The spectra of is discrete and we denote the eigenvalues as *λ*_{0} ≤ *λ*_{1} ≤ · · · and the corresponding eigenfunctions as *f*_{0}, *f*_{1}, · · ·. To numerically compute the eigenfunctions, we use the finite element method and solve a generalized matrix eigenvalue problem[2,3,4].

Let *X*(·, 0) : → ^{3} denote the coordinate function on , i.e., *X*(**p**, 0) = **p** for **p** . Using the eigen-system, we can express the heat diffusion of the coordinate function as

$$X(\mathbf{p},t)=\sum _{n=0}^{\infty}{e}^{-{\lambda}_{n}t}{f}_{n}(\mathbf{p}){\int}_{\mathcal{M}}{f}_{n}(\mathbf{q})X(\mathbf{q},0)d\mathcal{M}$$

(2)

By replacing the coordinates of the vertices on with *X*(·, *t*), we have a multiscale representation of . For numerical implementation, we approximate the diffusion with the first 300 eigenfunctions that are computed efficiently with the spectrum shift technique [11]. As an illustration, we show in Fig. 2 the multi-scale representation of a cortical surface at the scale *t* = 0, 10, 100, 1000. With the increase of the scale, we can see the surface exhibits more regularity that is common across population. However, sulcal landmarks on the original surface might be overly distorted in terms of length and angle if *t* is too large. In our work, we usually choose *t* = 100 as a tradeoff between surface regularity and landmark distortion.

To construct the atlas at a selected scale, we extend the intrinsic surface mapping technique developed for sub-cortical surfaces in [5] to cortical surfaces. Given a pair of surfaces
${\mathcal{M}}_{1}^{t}$ and
${\mathcal{M}}_{2}^{t}$ at a scale *t*, we compute two maps
${u}_{1}:{\mathcal{M}}_{1}^{t}\to {\mathcal{M}}_{2}^{t}$ and
${u}_{2}:{\mathcal{M}}_{2}^{t}\to {\mathcal{M}}_{1}^{t}$ by minimizing the following energy:

$$E={\int}_{{\mathcal{M}}_{1}^{t}}[\sum _{j=1}^{3}{\alpha}_{D}^{j}{({\xi}_{1}^{j}-{\xi}_{2}^{j}{u}_{1})2}^{+}+{\int}_{{\mathcal{M}}_{2}^{t}}[\sum _{j=1}^{3}{\alpha}_{D}^{j}{({\xi}_{2}^{j}-{\xi}_{1}^{j}{u}_{2})2}^{+}$$

(3)

where
${\xi}_{i}^{j}(i=1,2;j=1,2,3)$ are the feature functions defined on the surfaces that characterize their intrinsic geometry. In each integral, the first term penalizes the difference of feature functions, the second term encourages inverse consistency, and the third term uses the harmonic energy [12] for smoothness regularization, where *J*_{u1} and *J*_{u2} are the Jacobian of the maps. The regularization parameters *α _{D}*,

To model the sulcal anatomy intrinsically, we define three feature functions on cortical surfaces as shown in Fig. 3(a) by using the Reeb graph of the first and second eigen-function of its Laplace-Beltrami operator [5]. We can see these functions characterize the frontal-posterior, superior-inferior, and medial-lateral profile of the surface intrinsically. By projecting each point $\mathbf{p}{\mathcal{M}}_{i}^{t}$ to ( ${\xi}_{i}^{1}(\mathbf{p})\phantom{\rule{0.16667em}{0ex}}{\xi}_{i}^{2}(\mathbf{p})\phantom{\rule{0.16667em}{0ex}}{\xi}_{i}^{3}(\mathbf{p})$), we construct an embedding of each surface in the feature space. As shown in Fig. 3(b), the two cortical surfaces align much better in the embedding space, which enables us to find a good initial map between them by simply looking for the nearest point on the other surface. Starting from the initial maps, we iteratively evolve them by solving a pair of PDEs on the surfaces in the gradient descent directions [5].

By choosing one of the surface in the training set as the atlas surface, we compute the maps from all other surfaces in the training data to this atlas surface . All the manually traced sulcal lines can then be projected onto the atlas surface as shown in the third row of Fig. 1. Since both the multiscale representation and surface mapping are established via intrinsic geometry, we denote the collection of sulcal lines projected onto the atlas surface as the intrinsic atlas of major sulci.

Using the atlas of major sulci, we develop in this section an automated system for their identification on cortical surfaces. Given a new surface , the skeletal representation of its sulcal region is first computed based on the mean curvature of [13] as shown in Fig 4(a). The skeleton is decomposed into a set of branches *B* = {*B*_{1}, *B*_{2}, · · ·}, where each branch is a polyline on . The multiscale representation of and the map *u* : → is then computed. With the map *u*, the branches are projected onto the atlas surface and denoted as = {_{1}, _{2}, · · ·,}. Our goal is to extract anatomically consistent sulcal lines from these skeletal branches.

(a) Skeletal branches of the sulcal region. (b) Candidate branches (black) together with the training curves of the superior frontal sulcus. (c) Sample paths (black) and the most likely path (red). (d) The detected curve on the original surface.

Let
${S}_{i}=\{{S}_{i}^{1},{S}_{i}^{2},,{S}_{i}^{J}\}$ denote the set of training curves for the *i*-th sulcus on the atlas surface. Given the prior model *S _{i}*, the challenge is how to model the likelihood of skeletal branches in . The difficulty arises from the fact that

$${P}_{{S}_{i}}(C)=\{y(\mathbf{x}){S}_{i}^{j},xC\left|\right|x-y\left|\right|={min}_{z{S}_{i}^{j}}$$

(4)

where
${S}_{i}^{j}=arg{min}_{j}{d}_{H}(C,{S}_{i}^{j})$ and *d _{H}* is the Hausdorff distance. Given a branch

Using the candidate branches in , we construct a directed graph to generate a set of sample paths on as candidate curves for the sulcus. Given two nodes * _{p}* and

Let
${C}_{i}=\{{C}_{i}^{1},{C}_{i}^{2},\}$ be the set of sample paths for the *i*-th sulcus. For a candidate curve, its distance to the training data is defined as:

$$d({C}_{i}^{k};{S}_{i})=\underset{j}{min}(\overline{d}({C}_{i}^{k},{S}_{i}^{j})+\overline{d}({S}_{i}^{j},{C}_{i}^{k}))$$

(5)

where (·, ·) is the average distance from points on a curve to the other curve. We also define the “sulcality” of each curve as:

$$\mathit{Sulcality}({C}_{i}^{k})={\sum}_{{\widehat{B}}_{p}{C}_{i}^{k}}$$

(6)

which measures how good the path follows the sulcal regions. We then define the likelihood of each curve as

$$\mathcal{L}({C}_{i}^{k}{S}_{i})={e}^{-d({C}_{i}^{k};{S}_{i})/\mathit{Sulcality}({C}_{i}^{k})}$$

(7)

and choose the detection result as the sample curve in *C _{i}* with the maximum likelihood. As an illustration, we show on the set of sample paths and the path with the maximum likelihood for the superior frontal sulcus in Fig. 4(c). The branches in this path are then connected via a curvature-weighted geodesic on the original surface to obtain the detected sulcus as plotted in Fig. 4(d).

Let *C _{i}*

$$d({C}_{i1}^{p},{C}_{i2}^{q};{S}_{i1},{S}_{i2})=\underset{j}{min}(\overline{d}({C}_{i1}^{p},{S}_{i1}^{j})+\overline{d}({S}_{i1}^{j},{C}_{i1}^{p})+\overline{d}({C}_{i2}^{q},{S}_{i2}^{j})+\overline{d}({S}_{i2}^{j},{C}_{i2}^{q}))$$

(8)

The joint likelihood of the two curves is then defined as:

$$\mathcal{L}({C}_{i1}^{p},{C}_{i2}^{q}{S}_{i1},{S}_{i2})={e}^{-d({C}_{i1}^{p},{C}_{i2}^{q};{S}_{i1},{S}_{i2})/(\mathit{Sulcality}({C}_{i1}^{p})\mathit{Sulcality}({C}_{i2}^{q}))}$$

(9)

The joint detection results are the pair of curves in the sample space *C _{i}*

In this section, we present experimental results on four different datasets, including surfaces extracted from 3 software tools, to demonstrate our sulci detection system.

In the first experiment, we applied our method to pial surfaces extracted from the 3T MRI images of 65 Chinese young subjects (18 ~ 27 years) by Freesurfer [14]. The left hemispheres were used in this work. Twelve major sulci as listed in Table 1 were manually traced on each surface. We used 40 surfaces as training data to construct the intrinsic atlas of major sulci, which is shown on the third row of Fig. 1, and the algorithm developed in section 3 was used to identify the twelve major sulci on the other 25 surfaces for testing and quantitative validation. For robustness, the central and post-central sulcus were detected jointly by maximizing the joint likelihood in (9). The other 10 sulci were detected separately by maximizing the likelihood in (7).

Quantile statistics in testing data. (*S*_{1}:central; *S*_{2}:post-central; *S*_{3}:pre-central; *S*_{4}:superior-temporal; *S*_{5}:intraparietal; *S*_{6}:inferior-frontal; *S*_{7}:superior-frontal; *S*_{8}:olfactory; *S*_{9}:collateral; *S*_{10}:parietal-occipital; *S*_{11}:cingulate; *S*_{12}:calcarine.)

As an illustration, the results from 4 subjects in the testing data are plotted in Fig. 5. For better visualization, we plotted the detected curves on the surfaces at the scale *t* = 10 in Fig. 5(a) and (b). As shown in Fig. 5(c), the automatically detected sulcal lines align very well with manually traced sulci plotted in black. Quantile statistics of two distances *d _{am}* and

In the second experiment, we applied the atlas built from the Chinese population in the first experiment to detect sulci on three different datasets of Caucasian brains. The first dataset consists of the left pial surfaces of eight elderly subjects (63 ~ 85 years) extracted by Freesurfer. The second dataset is composed of the left pial surfaces of two young adults extracted by BrainSuite [15]. The third dataset consists of white matter surfaces of two young adults extracted by BrainVisa [16]. For better visualization, we also plot the detected curves on all surfaces at the scale *t* = 10 in Fig. 6. These results demonstrate the robustness of our method across ethnic and age groups, and different software tools for surface extraction.

In summary, we have developed a novel system for the automated detection of major sulci on cortical surfaces. Quantitative validations on twelve major sulci showed excellent agreement between our system and manual tracing. We also demonstrated the robustness of our system across different populations and surface extraction tools. For future work, we will augment our system with the intrinsic modeling of gyral landmarks to further improve its performance.

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