Journal of Neuroscience Methods 172 (2008) 122–130

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Journal of Neuroscience Methods

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A robust and accurate algorithm for estimating the complexity of thecortical surface

Jiefeng Jianga, Wanlin Zhub, Feng Shia, Yuanchao Zhangb, Lei Linb, Tianzi Jianga,∗

a National Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Sciences, Beijing 100080, PR Chinab Department of Mathematics, Zhejiang University, Hangzhou 310027, PR China

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Article history:Received 18 December 2007Received in revised form 13 March 2008Accepted 16 April 2008

Keywords:Anatomic MRIBox-countingCortical complexityFractal dimensionReconstructed cortical surface

a b s t r a c t

A fractal dimension (FD) gand has been employed ion the precision of the inpstruction algorithms, theResonance (MR) imaging.by points, as is typical of mused in previous studies dalgorithm that is able toIn this paper, a robust anbox–triangle intersectioncounting method, which hobjects. These two featurevalidated via several expeof these features, the algocerebral cortex.

1. Introduction

The human cerebral cortex plays a crucial role in human intel-ligence. For years, many researchers have dedicated themselvesto investigating its structural features and to comparing the cor-tical folding patterns between groups of individuals. However, dueto the intricate geometry of the cortex, quantifying cortical mor-phology has always been a difficult task. Measurements such asthickness, sulcal depth and curvature only reflect local features ofthe cortex. The Gyrification Index (GI) (Zilles et al., 1988), althoughit gives a global description of cortical complexity, is sensitive tothe direction of slicing (Thompson et al., 2005). Thus, a compactmeasurement which is able to characterize the folding pattern ofthe whole cortex or at least a lobe of the cortex will be of great valuefor researchers.

The fractal, first proposed by Mandlebrot (1982), has beenwidely used to describe self-similar structures to which it is difficultto apply shape analysis in a usual way. Due to its highly convo-luted gyri and sulci, the human brain is a fractal in some spatial

∗ Corresponding author. Tel.: +86 10 8261 4469; fax: +86 10 62551993.E-mail address: jiangtz@nlpr.ia.ac.cn (T. Jiang).

0165-0270/$ – see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.jneumeth.2008.04.018

highly compact description of the shape characteristics of the human brainny studies on brain morphology. The accuracy of FD estimation dependsape description. Facilitated by automatic cerebral cortical surface recon-

e of the cerebral cortex can be more precisely modeled using Magneticthe reconstructed cortical surface is represented by triangles, rather than

s that use voxels, the voxel-based FD estimation algorithms that have beenwork when using the cortical surface as the input. Thus, designing a new

ate the FD from a surface representation becomes of particular interest.urate FD estimation algorithm is proposed. The algorithm is based on aing strategy, which is used for the first time in brain analyses, and a box-en widely used in FD computations of the human brain and other naturalowed the algorithm with robustness. The accuracy of the algorithm wasts using both manually generated datasets and real MR images. As a resultis also suitable for estimating the FD of fractals in addition to that of the

© 2008 Elsevier B.V. All rights reserved.

scales (Kiselev et al., 2003). The shape complexity of a fractal ismeasured by its fractal dimension (FD), a single value that sum-

marizes the variability of an object: the more complex an object(e.g. a brain with more, deeper folds, or more involuted foldingpatterns), the greater its FD value. FD is usually estimated usingthe box-counting method (Gangepain and Roques-Carmes, 1986;Liebovitch and Toth, 1989; Sarraille and Myers, 1994) because ofits robustness in dealing with fractals which do not have strict self-similarity. Since the human brain is not self-similar at all scales, thebox-counting method is especially suitable for computing the FDof the human brain.

Many research studies have adopted fractal analysis to explorethe morphological properties of the human brain using segmentedMagnetic Resonance (MR) images (Bullmore et al., 1994; Cook etal., 1995; Esteban et al., 2007; Free et al., 1996; Kedzia et al., 1997;Kiselev et al., 2003; Li et al., 2007; Takahashi et al., 2004; Zhanget al., 2007). In these studies, the FD of white matter (WM), graymatter (GM), WM/GM surface, and GM/Cerebrospinal fluid (CSF)surface were computed, mostly by the box-counting method. Inorder to eliminate the influence of thickness and give a more com-pact description of the shape, skeletons were employed to studythe cortical folding pattern (Mangin et al., 2004). Some researcherscomputed the skeletons of the cerebral cortex (Ha et al., 2005;

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J. Jiang et al. / Journal of Neuros

Lee et al., 2004) and cerebellum (Liu et al., 2003) from the seg-mented MR images and then used box-counting to obtain the FD.The above cited works calculated the skeletons slice by slice. Zhanget al. (2005, 2007) applied a 3D skeletonization algorithm to WM,and then analyzed the FD of the skeletons. However, voxel-basedmethods have two major drawbacks: they cannot preserve thetopology of cortical surface (e.g. the skeletons and surfaces mayinclude holes); nor can discrete voxels accurately present a contin-uous structure.

Because it overcomes these two drawbacks, the surface of anobject is a better choice for shape analysis, especially in researchon the cerebral cortex because the cortex is thin. The pattern of gyri-fication and fissuration reflects a fractal nature (Luders et al., 2004),which indicates that the FD can be a useful measure when appliedto reconstructed cortical surfaces. The surface-based method wasfirst presented by Thompson et al. (1996), using manually outlinedsulcal surfaces. Over the next several years, a number of automaticcortical surface reconstruction algorithms were invented (Dale etal., 1999; Han et al., 2004; Kim et al., 2005; MacDonald et al., 2000;Xu et al., 2006). An extracted surface provides more accurate detailsof the cerebral cortex than a segmented MRI image, so these recon-struction algorithms have been widely employed in research onabnormal FDs in schizophrenia (Narr et al., 2001, 2004), Williamssyndrome (Thompson et al., 2005) and gender differences of cor-tical complexity (Luders et al., 2004). In addition, Im et al. (2006)did a thorough study of the relationship between FD and other fac-tors like cortical thickness, sulcal depth and folding area. However,the FD estimation method in this work is unstable under differ-ent parameter choices. We will discuss this in greater detail inSection 4.

In this paper, we propose a robust and accurate FD estimationalgorithm which is able to measure cortical folding complexity.This algorithm uses extracted cortical surfaces as input. The box-counting method and a box–surface intersection checking methodare employed to increase robustness. The accuracy was supportedby a series of experiments. In addition, the algorithm is both eas-ily written into code in popular programming languages such as C,C++, Java, C#, etc. and easily used because the user only needs to seta few parameters. After that, it runs without human intervention.

The remainder of this paper is organized as follows: Section2 describes the MR images acquisition and pre-processing, andthe FD estimation algorithm in detail; Section 3 gives the exper-imental results using both artificial data and real MR image data;Section 4 discusses this method, including a comparison between

our algorithm and the algorithm in Im et al. (2006), issues relatingto parameters, a conclusion and some future research directions.

2. Materials and methods

2.1. MR images

57 normal subjects (27 males and 30 females, 23.6 ± 3.9years old) participated in this study. All subjects were right-handed. MR images were scanned on a 3T SIEMENS TrioTimscanner using a magnetization prepared rapid acquisition gradientecho (MP-RAGE) three-dimensional T1-weighted sequence (voxelsize = 1 mm × 1 mm × 1 mm; TR = 2000 ms; TE = 2.6 ms; Nex = 1,slice thickness = 1 mm).

2.2. Pre-processing

Each scan was processed using Freesurfer (http://surfer.nmr.mgh.harvard.edu/)(Dale et al., 1999; Fischl et al., 1999). In brief, thepre-processing stage contained four steps. First, intensity nonuni-

Methods 172 (2008) 122–130 123

formity correction and normalization to stereotaxic space usinglinear transformation were applied to the input image. Second,the voxels of the brain were segmented into GM, WM, CSF andbackground. Third, tessellations of the GM/WM boundary, bound-ary smoothing and automated topological correction (in order toremove holes and genus on the surface) were performed to obtainthe initial surface. Fourth, the obtained surface was used as the ini-tial value for the deformable model to reconstruct the pial surface.Thus, for each scan we got a GM/WM surface and a pial surface foreach hemisphere. We used pial surfaces and their smoothed ver-sion (smoothed using the mris smooth command of FreeSurfer) inthis study, because we are interested in the sulcal and gyral con-volutions of gray matter. The FD of the GM/WM surfaces can beestimated as well for those who are interested in white matterfolding.

A standard brain surface divided into regions of interest (ROI)(Desikan et al., 2006) was mapped back to each subject’s nativeimage space with a high-resolution spherical morphing procedure.ROIs were homologous across subjects. To compute the FD of lobesof a hemisphere, the surface was divided into prefrontal, parietal,temporal and occipital lobes by merging the regions of interest inthe same lobe. The cingulate and insular regions were not includedin the above four regions.

In addition, for each scan, the thickness and curvature at eachvertex were calculated using Freesurfer in the native space.

2.3. Box-counting method

Generally, box-counting of an object is done by placing theobject of interest onto a cubic grid of size r, and counting the num-ber of boxes occupied by the object, namely, N(r). By changing r, aseries of N(r)s are obtained. Then ln N(r) is fitted with ln 1/r. Thefitted slope is the FD estimation of the object. A more detailedintroduction to the box-counting method can be found in Zhang etal. (2005). In this context in which the object was a reconstructedcortical surface which consists of many triangles, we counted theboxes occupied by one or more triangles. Thus, we divided the box-counting on the cortical surface into three sub-procedures: for eachtriangle on the cortical surface, we marked boxes that intersectedit as “occupied boxes”; we counted the number of occupied boxes;and we computed the FD by applying linear fitting. The last twosub-procedures are trivial so we focused on the first step. To checkthe intersection of a box and a triangle, the part of the triangle insidethe box was calculated. If the part was empty, the box was not occu-

pied by a triangle; otherwise, an intersection existed and the boxwas marked. For clarity, we have chosen to illustrate the computa-tion of the interior under a 2D condition; nevertheless, this methodcould be easily extended to 3D. As shown in Fig. 1, a box in 2D is asquare bounded by four lines. The part of a convex polygon (a tri-angle is a special case) which is inside the box can be calculated byiteratively cutting it using the four lines and abandoning the partsoutside the square. A line cutting a convex polygon results in twoparts. It is easy to determine which part to throw away since theposition of the line is known. For example, if the upper boundingline of the square cuts a polygon into two parts, the part above theline is dropped. In a 3D situation, cutting the triangle with the sixfaces of a box gives the interior part of the triangle.

Obviously, testing every triangle–box pair for intersectionrequires too much computation and makes estimation impracti-cal. Thus, bounding-box technique (Parent, 2002) was employedto reduce the numbers of boxes checked for intersection with-out losing precision. In this approach, each triangle is bounded ina bounding cuboid, whose six faces are determined by the min-imum/maximum x, y, z coordinates of the triangle vertices. Thecuboid gives a range of boxes which may be occupied by the

124 J. Jiang et al. / Journal of Neuroscience

Fig. 1. 2D illustration of box–triangle intersection checking method: (A) the triangleand the box. (B) The remaining part (in red) after cutting using the lower bound (inblack). (C) The remaining part after cutting using the upper bound. (D) The remainingpart after cutting using the leftmost bound. (E) The final interior part of the trian-gle (For interpretation of the references to color in this figure legend, the reader isreferred to the web version of the article).

triangle, so only these boxes are required for the intersection test.Because we only need to query the eight vertices of the cuboid toobtain the range, the computation cost is small. Although it usesonly very limited effort, the bounding-box strategy excludes many

Fig. 2. Four typical fractals used to test our algorithm: (A) Cube; (B) Sphere;

Methods 172 (2008) 122–130

boxes which cannot possibly intersect the triangle, so it signifi-cantly improves the efficiency. In practice, it usually reduces thenumber of boxes needed for checking from half a million to lessthan a hundred. In addition, the algorithm can be further accel-erated by labeling a box once intersection checking shows that itintersects a triangle. The box will not be tested again even if it isinside the bounding cuboid of another triangle.

3. Results

3.1. Artificial data results

As was done by Zhang et al. (2005), four typical fractals wereused to test the accuracy of the algorithm (Fig. 2). The first twowere a cubical surface and a spherical surface, whose FDs are well-known to be equal to 2. The third one is a fourth-iteration MengerSponge, and the last one is a fifth-iteration Koch Snowflake. Insteadof creating volumes composed of voxels, we generated surfaces ofthe first three test objects and line segment contours for the KochSnowflake which could be treated as degenerate triangles. The FDresults of our algorithm are shown in Table 1. For comparison pur-poses, we also calculated the FD of the four typical fractals usingmethods in Im et al. (2006) and Thompson et al. (1996). It can beseen that our estimations are closer to the theoretical values thanthose calculated by the other methods. We did not compute the FDof the Menger Sponge and the Koch Snowflake using the methodin Thompson et al. (1996) because the Menger Sponge has holeswhich makes it unsuited for this method and the Koch Snowflake

(C) fourth-iteration Menger sponge; (D) fifth-iteration Koch snowflake.

cience

(2004), the structures in Fig. 5(B–D) should have a higher FD than

J. Jiang et al. / Journal of Neuros

Table 1FD estimation results comparison using four typical fractals

Fractal Cubesurface

Spheresurface

Fourth-iterationMenger Sponge

Fifth-iterationKoch Snowflake

Estimated FD using ourmethod

2.000 2.001 2.671 1.244

Estimated FD usingmethod in Im et al.(2006)

0 1.692 2.649 1.231

Estimated FD usingmethod in Thompsonet al. (1996)

2.000 2.003 - -

Theoretical FD 2 2 2.727 1.262

does not have the area which is needed in this method. In addi-tion, it is interesting to notice that the method in Im et al. (2006)gave surprisingly bad estimations for simple fractals like cubes andspheres. This is due to an inherent flaw, which will be discussed inSection 4.1.

In addition, please note that: first, FD estimations of the MengerSponge and the Koch Snowflake are 2.727 and 1.262 based on frac-tal theory, which are a little larger than our estimations. The reasonis that the fractals we used did not have infinite self-similarity. Inother words, they were not as complex as their ideal versions inmathematical concepts. In order to illustrate this, we constructeda series of Menger Sponges from the second to the fourth-iteration(Fig. 3) and a series of Koch Snowflakes from the third to thefifth-iteration (Fig. 4). The results (Table 2) showed that after eachiteration, the FD estimations went closer to (nevertheless, did notpass) their theoretical values. This convergence actually helps toconfirm that this algorithm is precise. Second, the tests on the KochSnowflakes demonstrated that our algorithm is robust even whenthe surface is degenerate, which means it could also be applied

Fig. 3. Second (A), third (B) and fourth (

Fig. 4. Third (A), fourth (B) and fifth (C)

Methods 172 (2008) 122–130 125

to fractals consisting of line segments. We also estimated the FD ofthese objects using the method in Im et al. (2006) for comparison. Itis clear that our estimations have better convergence and are moreaccurate.

Inspired by Im et al. (2006), we created four artificial objectswhich had folds (like the gyri and the sulci in the human brain) totest whether our algorithm is sensitive to changes in complexity.Fig. 5A was a “base” object; Fig. 5B had deeper folds; Fig. 5C hadincreased frequency of folds; and Fig. 5D had more complex folds.The FDs estimated using our algorithm are 2.025, 2.037, 2.065 and2.060, respectively. According to Im et al. (2006) and Luders et al.

those in Fig. 5A, hence the experiment shows that our algorithm isable to detect changes in complexity caused by a variety of factors.

3.2. Real-world data results

57 MRI scans of normal subjects were used in this study. The datasets were pre-processed using the procedure described in Section2.2. The pre-processing result for one subject is shown in Fig. 6.Four experiments were designed: the first was to compare the FDestimations of pial surfaces with FD estimations of their smoothedversions. Since the original pial surface contains more complexfolds than its smoothed counterpart does, the FD of the formershould be greater than that of the latter. This difference is clearlyobserved in our results (Table 3). The second experiment was toinvestigate the relation between FD estimations and the standarddeviation (S.D.) of the all vertices’ absolute mean curvatures (doCarmo, 1976) (Table 4), which are considered as local spatial vari-ation representations (Luders et al., 2006). The use of the S.D. ofthe absolute curvature was based on three reasons: first, the largerthe absolute curvature, the sharper the neighborhood of the vertex

C) iteration of the Menger sponge.

iteration of the Koch snowflake.

126 J. Jiang et al. / Journal of Neuroscience Methods 172 (2008) 122–130

Table 2FD estimation results of various iteration versions of the Menger sponge and Koch snowflake

Fractal Our FD FD using methodin Im et al. (2006)

Fractal Our FD FD using methodin Im et al. (2006)

Second-iteration MS 2.294 0.656 Third-iteration KS 1.090 0.289Third-iteration MS 2.589 2.223 Fourth-iteration KS 1.204 1.068Fourth-iteration MS 2.671 2.649 Fifth-iteration KS 1.244 1.232Theoretical MS 2.727 Theoretic KS 1.262

MS: Menger sponge; KS: Koch snowflake.

per fo

Fig. 5. Four objects used in sensitivity test. (A) “Base” object. (B) An object with deefolds. The FD estimations are 2.025, 2.037, 2.065 and 2.060, respectively.

changes; whereas the sign of a curvature only shows whether theneighborhood is convex or concave. Second, a difference in aver-

age curvature does not accurately reflect variance in FD. Imaginespheres with different radii. Although their mean curvatures aredifferent, they share the zero S.D. of absolute curvatures. Thus, allspheres have same FD of 2 (Table 4). Third, as shown in Fig. 7,since deeper or more frequent folds results in a higher S.D. anda higher S.D. of absolute curvatures, there may be a positive cor-relation between these two measurements. As shown in Table 5,the four significant results are all positively correlated. This resultsupports our assertion that this algorithm gives a higher FD esti-mation to more complex structures. The last two experiments wereadopted from Im et al. (2006) in order to investigate the relationbetween folding area/average cortical thickness and FD estima-tions. The results are shown in Table 6 and Table 7. Both resultsconfirmed the tendency reported in Im et al. (2006). In addition,we found all ten parts of the FD estimations are positively corre-lated with folding areas, which are larger when the surface is moreconvoluted. This finding again demonstrates that the FD estima-tion of this algorithm is larger when the cortical surface is morecomplex. The degree of correlation displayed in Tables 5–7 varies.This could be the result of the measurements’ differential impactson the shape of the cortical surface. First, the thickness is usually

lds. (C) An object with higher frequency of folds. (D) An object with more complex

between 1.5 and 4 mm; so its influence on shape variability is lim-ited. Thus, it is not surprising that the cortical thickness is poorly

correlated with FD. Second, the relationship between the FD andthe S.D. of curvature is stronger. This may be because the S.D. ofcurvature is an indirect measure of shape variance. Last, the strongrelation between the FD and the folding area may infer that the FDencapsulates the information about the number of folds (Luders etal., 2004).

4. Discussion

4.1. Comparison of our algorithm with the algorithm in Im et al.(2006)

Although our algorithm was suggested by the one used in Imet al. (2006), significant theoretical and practical differences existbetween the two algorithms. Since the cortex surface is composedof many triangles, we need to count the boxes occupied by one ormore triangles. However, as stated in the “fractal dimension” sec-tion in Im et al. (2006), “the number of grid boxes occupied by oneor more vertices of cortical surface, N(r) is counted”. This strategy isflawed in theory. Take Fig. 8 as an example. Just considering thoseboxes occupied by triangle vertices will miss some boxes occupied

J. Jiang et al. / Journal of Neuroscience Methods 172 (2008) 122–130 127

Table 3FD estimation comparison between pial surfaces and a smoothed pial surface

Pial surface FDestimations range

Smoothed pial surfaceFD estimations range

p-Value ofpaired t-test

Left hemisphere 2.195–2.252 2.053–2.087 1.1e−61Right hemisphere 2.185–2.258 2.050–2.082 4.0e−59

Table 4Measurements of five randomly chosen subjects

Subject 1 Subject 2 Subject 3 Subject 4 Subject5

FD of LH hemisphere 2.251 2.206 2.195 2.216 2.233FD of LH prefrontal lobe 2.200 2.158 2.157 2.148 2.204FD of LH parietal lobe 2.170 2.123 2.083 2.127 2.138FD of LH temporal lobe 2.136 2.100 2.088 2.106 2.104FD of LH occipital lobe 2.193 2.145 2.138 2.158 2.147FD of RH hemisphere 2.246 2.200 2.185 2.201 2.211FD of RH prefrontal lobe 2.183 2.137 2.139 2.160 2.182FD of RH parietal lobe 2.132 2.075 2.071 2.093 2.068FD of RH temporal lobe 2.126 2.090 2.063 2.072 2.093FD of RH occipital lobe 2.174 2.149 2.139 2.142 2.141SDAC of LH hemisphere 0.110 0.111 0.111 0.106 0.117SDAC of LH prefrontal lobe 0.149 0.154 0.151 0.146 0.153SDAC of LH parietal lobe 0.153 0.160 0.159 0.163 0.163SDAC of LH temporal lobe 0.149 0.158 0.153 0.149 0.155SDAC of LH occipital lobe 0.162 0.165 0.161 0.168 0.165SDAC of RH hemisphere 0.114 0.108 0.116 0.115 0.116SDAC of RH prefrontal lobe 0.108 0.107 0.109 0.094 0.114

Fig. 6. A sample from data used in a real-world data experiment. (A and B) pial sur-faces of both hemispheres. (C and D) smoothed pial surfaces. (E and F) parcellationof pial surfaces.

by the triangle. This flaw is responsible for the failure of Im etal.’s (2006) method to estimate the FD of a cube and a sphere, asmentioned in Section 3.1.

Missing those boxes is more than a theoretical problem. To showthat Im et al.’s (2006) algorithm sometimes produces inaccurateFD estimations, we reconstructed a cortex surface from the datasetdescribed in Section 2.1 and estimated the FD of its occipital lobeusing both their algorithm and our algorithm. The r-value ranged

SDAC of RH parietal lobe 0.102 0.110 0.107 0.108 0.116SDAC of RH temporal lobe 0.117 0.122 0.127 0.106 0.120SDAC of RH occipital lobe 0.097 0.103 0.100 0.104 0.105FA of LH hemisphere 0.168 0.139 0.137 0.140 0.148FA of LH prefrontal lobe 0.140 0.110 0.116 0.100 0.122FA of LH parietal lobe 0.145 0.108 0.117 0.126 0.126FA of LH temporal lobe 0.149 0.116 0.104 0.115 0.118FA of LH occipital lobe 0.172 0.145 0.150 0.153 0.157FA of RH hemisphere 0.163 0.134 0.138 0.144 0.136FA of RH prefrontal lobe 0.132 0.106 0.113 0.116 0.123FA of RH parietal lobe 0.155 0.098 0.112 0.115 0.105FA of RH temporal lobe 0.157 0.121 0.121 0.124 0.122FA of RH occipital lobe 0.153 0.146 0.138 0.146 0.131AT of LH hemisphere 2.281 2.652 2.533 2.556 2.511AT of LH prefrontal lobe 2.374 2.828 2.603 2.785 2.600AT of LH parietal lobe 2.076 2.512 2.432 2.378 2.320AT of LH temporal lobe 2.603 2.902 2.935 2.792 2.793AT of LH occipital lobe 1.751 2.154 1.965 1.919 2.059AT of RH hemisphere 2.348 2.593 2.588 2.581 2.600AT of RH prefrontal lobe 2.510 2.809 2.717 2.714 2.693AT of RH parietal lobe 2.068 2.406 2.377 2.955 2.110AT of RH temporal lobe 2.593 2.834 2.955 2.916 2.933AT of RH occipital lobe 1.913 2.058 2.110 2.045 2.129

SDAC: standard deviation of absolute curvatures; FA: folding area; AT: average thick-ness, in mm.

Fig. 7. Examples of positive correlation between the FD and S.D. of the absolute curvature. The color indicates the absolute value of the mean curvature at every vertex: (A)‘Base’ object (FD 2.020). (B) An object with deeper folds (FD 2.102). (C) An object with more frequent folds (FD 2.131). Their S.D.s of absolute curvature are 0.176, 0.268 and0.532, respectively.

128 J. Jiang et al. / Journal of Neuroscience Methods 172 (2008) 122–130

Table 5Correlation between the FD estimation and the standard deviation of the absolutevalue of curvature

Correlationcoefficient

p-Value Correlationcoefficient

p-Value

Left hemisphere 0.321 0.015 Right hemisphere 0.238 0.075LH prefrontal lobe 0.442 5.8e−4 RH prefrontal lobe 0.280 0.035LH parietal lobe 0.255 0.055 RH parietal lobe −0.013 0.922LH temporal lobe 0.155 0.251 RH temporal lobe 0.187 0.165LH occipital lobe 0.059 0.680 RH occipital lobe 0.265 0.046

Significance of bold values is p < 0.05.

Table 6Correlation between the FD estimation and the folding area

Correlationcoefficient

p-Value Correlationcoefficient

p-Value

Left hemisphere 0.444 5.4e−4 Right hemisphere 0.522 3.1e−5LH prefrontal lobe 0.468 2.4e−4 RH prefrontal lobe 0.475 1.9e−4LH parietal lobe 0.518 3.6e−5 RH parietal lobe 0.666 1.5e−8LH temporal lobe 0.347 0.008 RH temporal lobe 0.335 0.011LH occipital lobe 0.546 1.1e−5 RH occipital lobe 0.421 0.001

Table 7Correlation between the FD estimation and the average cortical thickness

Correlationcoefficient

p-Value Correlationcoefficient

p-Value

Left hemisphere −0.213 0.112 Right hemisphere −0.330 0.012LH prefrontal lobe −0.044 0.748 RH prefrontal lobe 0.019 0.890LH parietal lobe −0.211 0.115 RH parietal lobe −0.318 0.016LH temporal lobe 0.026 0.849 RH temporal lobe −0.003 0.980

LH occipital lobe −0.147 0.277 RH occipital lobe −0.023 0.867

Significance of bold values is p < 0.05.

from 1.5 to 8 times the average length of the triangle edges. Fig. 9Ashows the number of boxes counted using both algorithms. Notethat even when r is 4 times as large as the average length, the differ-ences between the numbers of boxes counted by the two algorithmsare still obvious. The fitting results (Fig. 9B) show that the FD esti-mated from Im et al.’s algorithm is 1.981, which contradicts thecommonly accepted premise that the FD of any closed surface is noless than 2. Fig. 9(A and B) also demonstrate that, because Im et al.’salgorithm missed more boxes as r became smaller, the estimatedFD decreased until it reached an incorrect value.

To demonstrate that the above example is not coincidental, weused the dataset described in Section 2.1 to test the two algorithms.Comparisons between both algorithms’ FD results of both hemi-spheres’ occipital lobes are shown in Fig. 10(A and B) 38 out of 114of Im et al.’s (2006) estimations were below 2, while all the FDresults using our algorithm were clearly above 2. From the aboveresults and analysis, we assert that due to missing boxes, the ln N(r)

Fig. 8. Comparison between Im et al.’s algorithm

Fig. 9. Comparison of box-counting results of a subject’s left occipital lobe usingboth algorithms: (A) comparison of number of boxes counted. (B) Comparison oflinear fitting result.

counted by Im et al.’s (2006) algorithm exhibits less linearity thanour algorithm, thus their estimations are not stable under variousbox size range settings.

4.2. Parameters

There are only two parameters in this algorithm, namely therange of box sizes and the increment in box size. As shown inSection 4.1, linearity is conserved from small to large values of

(A) and our algorithm (B) in the 2D state.

J. Jiang et al. / Journal of Neuroscience

Fig. 10. Comparison of occipital lobe FD estimations using both algorithms: (A) lefthemisphere. (B) Right hemisphere.

r in our algorithm, thus the choices of r are rather flexible. In

most cases, considering the resolution of the surface and com-putation efficiency, we chose r’s that ranged from 1–1.5 to 8–10times the average length of the triangle edges of the surface. Typ-ically, the algorithm computed the FD of a pial surface whichconsisted of about 120,000 points and 250,000 triangles in about15 min using a Pentium IV 1.8G PC with 512 M RAM. The incre-ment in box size ought to be as small as possible; however, dueto the robustness of our method, a larger incremental step couldbe achieved. Thus, the efficiency was improved. Additionally, witha larger increment, when using the algorithm we were able toput more computational effort on accuracy. For example, as inZhang et al. (2005), we could repeat box-counting several timesfor any fixed box size, using a randomly chosen origin of the gridof boxes each time, and then choose the smallest N(r) to esti-mate the FD. The box–triangle intersection checking criterion couldalso be modified by adding a threshold parameter: in this method,given a box, we calculated the part of the triangle that is insideof it. The area of this part provides a more flexible method forintersection checking. For example, one can set an area thresh-old to filter boxes which only have a small portion intersecting thetriangle.

Methods 172 (2008) 122–130 129

4.3. Summary

In this paper, a robust and accurate algorithm for estimatingthe FD for reconstructed cortical surfaces is presented. It is robustfor the following reasons: it is able to handle fractals that are notmathematically ideal, e.g. the human brain cortical surface; it isable to deal with degenerate surfaces; and it is stable under dif-ferent parameter choices. In addition, its accuracy, sensitivity andpotential for measuring cortical folding complexity and detectingfolding complexity change were verified by tests using both artifi-cial data and real MR images. Furthermore, it is easy to implementusing modern programming languages. Possibilities for furtherwork could include seeking an even more efficient intersectionchecking method to reduce the running time of this method; quan-titatively analyzing the contribution of different folding patterns tothe FD to investigate the underlying anatomical cause of FD change;and applying this method to finding biomarks of brain diseases.

Acknowledgements

This work was partially supported by the Natural Science Foun-dation of China, Grant Nos. 30730035 and 30425004, and theNational Key Basic Research and Development Program (973) GrantNo. 2003CB716100. The authors appreciate the English languageassistance of Dr. Rhoda and Dr. Edmund Perozzi.

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