# Measuring and comparing brain cortical surface area and other areal quantities.

**ABSTRACT** Structural analysis of MRI data on the cortical surface usually focuses on cortical thickness. Cortical surface area, when considered, has been measured only over gross regions or approached indirectly via comparisons with a standard brain. Here we demonstrate that direct measurement and comparison of the surface area of the cerebral cortex at a fine scale is possible using mass conservative interpolation methods. We present a framework for analyses of the cortical surface area, as well as for any other measurement distributed across the cortex that is areal by nature. The method consists of the construction of a mesh representation of the cortex, registration to a common coordinate system and, crucially, interpolation using a pycnophylactic method. Statistical analysis of surface area is done with power-transformed data to address lognormality, and inference is done with permutation methods. We introduce the concept of facewise analysis, discuss its interpretation and potential applications.

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**ABSTRACT:**Permutation methods can provide exact control of false positives and allow the use of non-standard statistics, making only weak assumptions about the data. With the availability of fast and inexpensive computing, their main limitation would be some lack of flexibility to work with arbitrary experimental designs. In this paper we report on results on approximate permutation methods that are more flexible with respect to the experimental design and nuisance variables, and conduct detailed simulations to identify the best method for settings that are typical for imaging research scenarios. We present a generic framework for permutation inference for complex general linear models (glms) when the errors are exchangeable and/or have a symmetric distribution, and show that, even in the presence of nuisance effects, these permutation inferences are powerful while providing excellent control of false positives in a wide range of common and relevant imaging research scenarios. We also demonstrate how the inference on glm parameters, originally intended for independent data, can be used in certain special but useful cases in which independence is violated. Detailed examples of common neuroimaging applications are provided, as well as a complete algorithm - the "randomise" algorithm - for permutation inference with the glm.NeuroImage 02/2014; · 6.25 Impact Factor - SourceAvailable from: David Reese MckayD Reese McKay, Emma E M Knowles, Anderson A M Winkler, Emma Sprooten, Peter Kochunov, Rene L Olvera, Joanne E Curran, Jack W Kent, Melanie A Carless, Harald H H Göring, Thomas D Dyer, Ravi Duggirala, Laura Almasy, Peter T Fox, John Blangero, David C Glahn[Show abstract] [Hide abstract]

**ABSTRACT:**We report effects of age, age(2), sex and additive genetic factors on variability in gray matter thickness, surface area and white matter integrity in 1,010 subjects from the Genetics of Brain Structure and Function Study. Age was more strongly associated with gray matter thickness and fractional anisotropy of water diffusion in white matter tracts, while sex was more strongly associated with gray matter surface area. Widespread heritability of neuroanatomic traits was observed, suggesting that brain structure is under strong genetic control. Furthermore, our findings indicate that neuroimaging-based measurements of cerebral variability are sensitive to genetic mediation. Fundamental studies of genetic influence on the brain will help inform gene discovery initiatives in both clinical and normative samples.Brain Imaging and Behavior 12/2013; · 2.67 Impact Factor - Gregory L Wallace, Stuart F White, Briana Robustelli, Stephen Sinclair, Soonjo Hwang, Alex Martin, R James R Blair[Show abstract] [Hide abstract]

**ABSTRACT:**Although there is growing evidence of brain abnormalities among individuals with conduct disorder (CD), the structural neuroimaging literature is mixed and frequently aggregates cortical volume rather than differentiating cortical thickness from surface area. The current study assesses CD-related differences in cortical thickness, surface area, and gyrification as well as volume differences in subcortical structures critical to neurodevelopmental models of CD (amygdala; striatum) in a carefully characterized sample. We also examined whether group structural differences were related to severity of callous-unemotional (CU) traits in the CD sample. Participants were 49 community adolescents aged 10 to 18 years, 22 with CD and 27 healthy comparison youth. Structural MRI was collected and the FreeSurfer image analysis suite was used to provide measures of cortical thickness, surface area, and local gyrification as well as subcortical (amygdala and striatum) volumes. Youths with CD showed reduced cortical thickness in the superior temporal cortex. There were also indications of reduced gyrification in the ventromedial frontal cortex, particularly for youths with CD without comorbid attention-deficit/hyperactivity disorder. There were no group differences in cortical surface area. However, youths with CD also showed reduced amygdala and striatum (putamen and pallidum) volumes. Right temporal cortical thickness was significantly inversely related to severity of CU traits. Youths with CD show reduced cortical thickness within superior temporal regions, some indication of reduced gyrification within ventromedial frontal cortex and reduced amygdala and striatum (putamen and pallidum) volumes. These results are discussed with reference to neurobiological models of CD.Journal of the American Academy of Child and Adolescent Psychiatry 04/2014; 53(4):456-465.e1. · 6.97 Impact Factor

Page 1

Measuring and comparing brain cortical surface area and other areal quantities

Anderson M. Winklera,b,⁎, Mert R. Sabuncuc,d, B.T. Thomas Yeoe, Bruce Fischlc,d, Douglas N. Greved,

Peter Kochunovf,g, Thomas E. Nicholsh,i, John Blangerog, David C. Glahna,b

aDepartment of Psychiatry, Yale University School of Medicine, New Haven, CT, USA

bOlin Neuropsychiatry Research Center, Institute of Living, Hartford, CT, USA

cComputer Science and Artificial Intelligence Laboratory, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA, USA

dAthinoula A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital, Harvard Medical School, Charlestown, MA, USA

eDepartment of Neuroscience and Behavioral Disorders, Duke-NUS Graduate Medical School, Singapore, Singapore

fMaryland Psychiatric Research Center, Department of Psychiatry, University of Maryland School of Medicine, Baltimore, MD, USA

gDepartment of Genetics, Texas Biomedical Research Institute, San Antonio, TX, USA

hDepartment of Statistics & Warwick Manufacturing Group, University of Warwick, Coventry, UK

iOxford Centre for Functional MRI of the Brain, University of Oxford, UK

a b s t r a c ta r t i c l ei n f o

Article history:

Accepted 6 March 2012

Available online 15 March 2012

Keywords:

Brain surface area

Facewise analysis

Areal interpolation

Pycnophylactic interpolation

Structural analysis of MRI data on the cortical surface usually focuses on cortical thickness. Cortical surface

area, when considered, has been measured only over gross regions or approached indirectly via comparisons

with a standard brain. Here we demonstrate that direct measurement and comparison of the surface area of

the cerebral cortex at a fine scale is possible using mass conservative interpolation methods. We present a

framework for analyses of the cortical surface area, as well as for any other measurement distributed across

the cortex that is areal by nature. The method consists of the construction of a mesh representation of the

cortex, registration to a common coordinate system and, crucially, interpolation using a pycnophylactic

method. Statistical analysis of surface area is done with power-transformed data to address lognormality,

and inference is done with permutation methods. We introduce the concept of facewise analysis, discuss

its interpretation and potential applications.

© 2012 Elsevier Inc. All rights reserved.

Introduction

The surface area of the cerebral cortex greatly differs across species,

whereas the cortical thickness has remained relatively constant during

evolution (Fish et al., 2008; Mountcastle, 1998). At a microanatomic

scale,regional morphology is closely related tofunctional specialization

(RolandandZilles,1998;ZillesandAmunts,2010),contrastingwiththe

columnarorganization of thecortex,in whichcellsfromdifferentlayers

respond to the same stimulus (Buxhoeveden and Casanova, 2002;

Jones, 2000). In addition, Rakic (1988) proposed an ontogenetic

model that explains the processes that lead to cortical arealization and

differentiation of cortical layers according to related, yet independent

mechanisms. Supporting evidence for this model has been found in

studies with both rodent and primates, including humans (Chenn and

Walsh, 2002; Rakic et al., 2009), as well as in pathological states

(Bilgüvar et al., 2010; Rimol et al., 2010).

At least some of the variability of the distinct genetic and develop-

mental processes that seem to determine regional cortical area and

thickness can be captured using polygon mesh (surface-based) repre-

sentations of the cortex derived from T1-weighted magnetic reso-

nance imaging (MRI) (Panizzon et al., 2009; Sanabria-Diaz et al.,

2010; Winkler et al., 2010). In contrast, volumetric (voxel-based)

representations, also derived from MRI, were shown to be unable to

readily disentangle these processes (Winkler et al., 2010).

Mesh representations of the brain allow measurements of the cor-

tical thickness at every point in the cortex, as well as estimation of the

average thickness for pre-specified regions. However, to date, analy-

ses of cortical surface area have been generally limited to two types

of studies: (1) vertexwise comparisons with a standard brain, using

some kind of expansion or contraction measurement, either of the

surface itself (Hill et al., 2010; Joyner et al., 2009; Lyttelton et al.,

2009; Palaniyappan et al., 2011; Rimol et al., 2010), of linear distances

between points in the brain (Sun et al., 2009a,b), or of geometric dis-

tortion (Wisco et al., 2007), or (2) analyses of the area of regions of

interest (ROI) defined from postulated hypotheses or from macro-

scopic morphological landmarks (Dickerson et al., 2009; Durazzo

et al., 2011; Eyler et al., 2011; Kähler et al., 2011; Nopoulos et al.,

2010; Schwarzkopf et al., 2011; Chen et al., 2011). Analyses of expan-

sion, however, do not deal with area directly, depending instead on

non-linear functions associated with the warp to match the standard

brain, such as the Jacobian of the transformation. Moreover, by not

NeuroImage 61 (2012) 1428–1443

⁎ Corresponding author. Fax: +1 203 785 7357.

E-mail address: anderson.winkler@yale.edu (A.M. Winkler).

URL: http://www.glahngroup.org (D.C. Glahn).

1053-8119/$ – see front matter © 2012 Elsevier Inc. All rights reserved.

doi:10.1016/j.neuroimage.2012.03.026

Contents lists available at SciVerse ScienceDirect

NeuroImage

journal homepage: www.elsevier.com/locate/ynimg

Page 2

quantifying the amount of area, these analyses are only interpretable

with respect to the brain used for the comparisons. ROI-based analy-

ses, on the other hand, entail the assumption that each region is ho-

mogeneous with regard to the feature under study, and have

maximum sensitivity only when the effect of interest is present

throughout the ROI.

These difficulties can be obviated by analyzing each point on the

cortical surface of the mesh representation, a method already well

established for cortical thickness (Fischl and Dale, 2000). Pointwise

measurements, such as thickness, are generally taken at and assigned

to each vertex of the mesh representation of the cortex. This kind of

measurement can be transferred to a common grid and subjected to

statistical analysis. Standard interpolation techniques, such as nearest

neighbor, barycentric (Yiu, 2000), spline-based (De Boor, 1962) or

distance-weighted (Shepard, 1968) can be used for this purpose.

The resampled data can be further spatially smoothed to alleviate

residual interpolation errors. However, this approach is not suitable

for areal measurements, since area is not inherently a point feature.

To illustrate this aspect, an example is given in Fig. 1. Methods that

can be used for interpolation of point features do not necessarily com-

pensate for inclusion or removal of datapoints,1unduly increasing or

reducing the global or regional sum of the quantities under study,

precluding them for use with measurements that are, by nature,

areal. The main contribution of this article is to address the technical

difficulties in analyzing the local brain surface area, as well as any

other cortical quantity that is areal by nature. We propose a frame-

work to analyze areal quantities and argue that a mass preserving

interpolation method is a necessary step. We also study different pro-

cessing strategies and characterize the distribution of facewise cortical

surface area.

Method

An overview of the method is presented in Fig. 2. Comparisons of

cortical area between subjects require a surface model for the cortex

Fig. 1. An example demonstrating differences in the nature of measurements. In this analogy, the depth of the soil is similar to brain cortical thickness, whereas the number of trees

is similar to areal quantities distributed across the cortex. These areal quantities can be the surface area itself (in this case, the area of the terrain), but can also be any other

measurement that is areal by nature (such as the number of trees).

1A notable exception is the natural neighbor method (Sibson, 1981). However, the

original method needs modification for use with areal analyses.

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to be constructed. A number of approaches are available (Dale et al.,

1999; Kim et al., 2005; Mangin et al., 1995; van Essen et al., 2001)

and, in principle, any could be used. Here we adopt the method of

Dale et al. (1999) and Fischl et al. (1999a), as implemented in the

FreeSurfer software package (FS).2In this method, the T1-weighted

images are initially corrected for magnetic field inhomogeneities

and skull-stripped (Ségonne et al., 2004). The voxels belonging to

the white matter (WM) are identified based on their locations, on

their intensities, and on the intensities of the neighboring voxels. A

mass of connected WM voxels is produced for each hemisphere,

using a six-neighbors connectivity scheme, and a mesh of triangular

faces is tightly built around this mass, using two triangles per exposed

voxel face. The mesh is smoothed taking into account the local inten-

sity in the original images (Dale and Sereno, 1993), at a subvoxel res-

olution. Topological defects are corrected (Fischl et al., 2001; Ségonne

et al., 2007) ensuring that the surface has the same topological prop-

erties of a sphere. A second iteration of smoothing is applied, result-

ing in a realistic representation of the interface between gray and

white matter (the white surface). The external cortical surface (the

pial surface), which corresponds to the pia mater, is produced by

nudging outwards the white surface towards a point where the tissue

contrast is maximal, maintaining constraints on its smoothness and

on the possibility of self-intersection (Fischl and Dale, 2000). The

white surface is inflated in an area-preserving transformation and

subsequently homeomorphically transformed to a sphere (Fischl

et al., 1999b). After the spherical transformation, there is a one-to-

one mapping between faces and vertices of the surfaces in the native

2Available at http://surfer.nmr.mgh.harvard.edu.

Fig. 2. Diagram of the steps to analyze the cortical surface area. For clarity, the colors represent the convexity of the surface, as measured in the native geometry. (For interpretation

of the references to color in this figure legend, the reader is referred to the web version of this article.)

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A.M. Winkler et al. / NeuroImage 61 (2012) 1428–1443

Page 4

geometry (white and pial) and the sphere. These surfaces are

comprised exclusively of triangular faces.

Area per face and other areal quantities

The surface area for analysis is computed at the interface between

gray and white matter, i.e. at the white surface. Another possible

choice is to use the middle surface, i.e. a surface that runs at the

mid-distance between white and pial. Although this surface is not

guaranteed to match any specific cortical layer, it does not over or

under-represent gyri or sulci (van Essen, 2005), which might be an

useful property. The white surface, on the other hand, matches direct-

ly a morphological feature and also tends to be less sensitive to corti-

cal thinning or thickening than the middle or pial surfaces. Whenever

methods to produce surfaces that represent biologically meaningful

cortical layers are available, these should be preferred.

In contrast to conventional approaches in which the area of all

faces that meet at a given vertex is summed and divided by three,

producing a measure of the area per vertex, for facewise analysis it is

the area per face that is measured and analyzed. Since for each sub-

ject, each face in the native geometry has its corresponding face on

the sphere, the value that represents area per face, as measured

from the native geometry, can be mapped directly to the sphere, de-

spite any areal distortion introduced by the spherical transformation.

Furthermore, since there is a direct mapping that is independent of

theactualareain thenativegeometry,anyotherquantitythatisbiolog-

ically areal can also be mapped to the spherical surface. Examples of

such quantities, that may potentially be better characterized as areal

processes,are theextentof the neuralactivation asobserved with func-

tional MRI, the amount of cortical gray matter, the amount of amyloid

deposited in Alzheimer's disease (Clark et al., 2011; Klunk et al.,

2004), or simply the number of cells counted from optic microscopy

images reconstructed to a tri-dimensional space (Schormann and

Zilles, 1998). Since areal interpolation (described below) conserves

locally, regionally and globally the quantities under study, it allows

accurate comparisons and analyses across subjects for measurements

that are areal by nature, or that require mass conservation on the

surface of the mesh representation.

Registration

Registration to a common coordinate system is necessary to allow

comparisons across subjects (Drury et al., 1996). The registration is

performed by shifting vertex positions along the surface of the sphere

until there is a good alignment between subject and template (target)

spheres with respect to certain specific features, usually, but not

necessarily, the cortical folding patterns. As the vertices move, the

areal quantities assigned to the corresponding faces are also moved

along the surface. The target for registration should be the less biased

as possible in relation to the population under study (Thompson and

Toga, 2002).

A registration method that produces a smooth, i.e. spatially differ-

entiable, warp function enables the smooth transfer of areal quanti-

ties. A possible way to accomplish this is by using registration

methods that are diffeomorphic. A diffeomorphism is an invertible

transformation that has the elegant property that it and its inverse

are both continuously differentiable (Christensen et al., 1996; Miller

et al., 1997), minimizing the risk of vagaries that would be introduced

by the non-differentiability of the warp function.

Diffeomorphic methods are available for spherical meshes (Glaunès

et al., 2004; Yeo et al., 2010a), and here we adopt the Spherical Demons

(SD) algorithm3(Yeo et al., 2010a). SD extends the Diffeomorphic

Demons algorithm (Vercauteren et al., 2009) to spherical surfaces. The

Diffeomorphic Demons algorithm is a diffeomorphic variant of the

efficient, non-parametric Demons registration algorithm (Thirion,

1998). SD exploits spherical vector spline interpolation theory and

efficiently approximates the regularization of the Demons objective

function via spherical iterative smoothing.

Methods that are not diffeomorphic by construction, but in prac-

tice produce invertible and smooth warps could, in principle, be

used for registration for areal analyses. In the Evaluation section we

study the performance of different registration strategies as well as

the impact of the choice of the template.

Areal interpolation

After the registration, the correspondence between each face on

the registered sphere and each face from the native geometry is main-

tained, and the surface area or other areal quantity under study can be

transferred to a common grid, where statistical comparisons between

subjects can be performed. The common grid is a mesh which vertices

lie on the surface of a sphere. A geodesic sphere, which can be con-

structed by iterative subdivision of the faces of a regular icosahedron,

has many advantages for this purpose, namely, ease of computation,

edges of roughly similar sizes and, if the resolution is fine enough,

edge lengths that are much smaller than the diameter of the sphere

(see Appendix A for details). These two spheres, i.e. the registered,

irregular spherical mesh (source), and the common grid (target), typ-

ically have different resolutions. The interpolation method must, nev-

ertheless, conserve the areal quantities, globally, regionally and locally.

In other words, the method has to be pycnophylactic4(Tobler, 1979).

This is accomplished by assigning, to each face in the target sphere,

the areal quantity of all overlapping faces from the source sphere,

weighted by the fraction of overlap between them (Fig. 3).

More specifically, let QiSrepresent the areal quantity on the i-th

face of the registered, source sphere S, i=1, 2, …, I. This areal quantity

can be directly mapped back to the native geometry, and can be the

area per face as measured in the native geometry, or any other quan-

tity of interest that is areal by nature. Let the actual area of the same

face on the source sphere be indicated by Ai

to be transferred to a target sphere T, the common grid, which face

areas are given by AjTfor the j-th face, j=1, 2, …, J, J≠I. Each target

face j overlaps with K faces of the source sphere, being these overlap-

ping faces indicated by indices k=1, 2, …, K, and the area of each

overlap indicated by Ak

assigned to the j-th target face is then given by:

S. The quantities QiShave

O. The interpolated areal quantity to be

QT

j¼

X

K

k¼1

AO

AS

k

k

QS

k

ð1Þ

Similar interpolation schemes have been devised to solve prob-

lems in geographic information systems (GIS) (Flowerdew et al.,

1991; Goodchild and Lam, 1980; Gregory et al., 2010; Markoff and

Shapiro, 1973). Surface models of the brain impose at least one

additional challenge, which we address in the implementation (see

Appendix B). Differently than in other fields, where interpolation is

performed over geographic territories that are small compared to

Earth and, therefore, can be projected to a plane with acceptable

areal distortion, here we have to interpolate across the whole sphere.

Although other conservative interpolation methods exist for this pur-

pose (Jones, 1999; Lauritzen and Nair, 2008; Ullrich et al., 2009),

these methods either use regular latitude-longitude grids, cubed-

spheres, or require a special treatment of points located above a

certain latitude threshold to avoid singularities at the poles. These

3Available at http://sites.google.com/site/yeoyeo02/software/sphericaldemonsrelease.

4From Greek πυκνός (pyknos)=mass, density, and φύλαξις (phylaxis)=guard,

protect, preserve, meaning that the method has to be mass conservative.

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Page 5

disadvantages may render these methods suboptimal for direct use in

brain imaging.

Smoothing

Smoothing can be applied to alleviate residual discontinuities in

the interpolated data due to unfavorable geometric configurations

between faces of source and target spheres. For the purpose of

smoothing, facewise data can be represented either by their barycen-

ters, or converted to vertexwise (see Appendix D for a discussion on

how to convert), and should take into account differences on face

sizes, as larger faces will tend to absorb more areal quantities (see

Appendix A). Smoothing can be applied using the moving weights

method (Lombardi, 2002), defined as

~QT

n¼

∑jQT

jG g xn;xj

?

????

∑jG g xn;xj

???

ð2Þ

where~QT

areal quantity assigned to each of the J faces of the same surface

before smoothing, g(xn, xj) is the scalar-valued distance along the sur-

face between the barycenter xnof the current face and the barycenter

xjof another face, and G(g) is the Gaussian kernel.5

nis the smoothed areal quantity at the n-th face, QjTis the

Statistical analysis

After resampling to a common grid, the facewise data is ready for

statistical analysis. The most straightforward method is to use the

general linear model (GLM). The GLM is based on a number of as-

sumptions, including that the observed values have a linear, additive

structure, that the residuals of the model fit have the same variance

and are normally distributed. When these assumptions are not met,

a non-linear transformation can be applied, as long as the true,

biological or physical meaning that underlies the observed data is

preserved. In the Evaluation section, we show empirically that face-

wise cortical surface area is largely not normal. Instead, the distribu-

tion is skewed and can be better characterized as lognormal. A

generic framework that can accommodate arbitrary areal quantities

with skewed distributions is using a power transformation, such as

the Box–Cox transformation (Box and Cox, 1964), which addresses

possible violations of these specific assumptions, allied with per-

mutation methods for inference (Holmes et al., 1996; Nichols and

Hayasaka, 2003) when the observations can be treated as indepen-

dent, such as in most between-subject analysis.

The application of a statistical test at each face allows the creation

of a statistical map and also introduces the multiple testing problem,

which can also be addressed using permutation methods. These

methods are known to allow exact significance values to be comput-

ed, even when distributional assumptions cannot be guaranteed, and

also to facilitate strong control over family-wise error rate (FWER) if

the distribution of the statistic under the null hypothesis is similar

across tests. If not similar, the result is still valid, yet conservative.

An alternative is to use a relatively assumption-free approach to ad-

dress multiple testing, controlling instead the false discovery rate

(FDR) (Benjamini and Hochberg, 1995; Genovese et al., 2002),

which offers also weak control over FWER. Other approaches for in-

ference, such as the Random Field Theory (RFT) for meshes (Hagler

et al., 2006; Worsley et al., 1999) and the Threshold-Free Cluster

Enhancement (TFCE) (Smith and Nichols, 2009) have potential to be

used, although due to reliance on stringent assumptions or depen-

dence upon specification of certain parameters, these methods need

yet a careful evaluation for facewise areal quantities. Strategies to

present results are discussed in Appendix D.

Evaluation

We illustrate the method using data from the Genetics of Brain

Structure and Function Study, GOBS, a collaborative effort involving

the Texas Biomedical Institute, the University of Texas Health Science

Center at San Antonio (UTHSCSA) and the Yale University School of

Medicine. The participants are members of 42 families, and total sam-

ple size, at the time of the selection for this study, is 868 subjects. We

randomly chose 84 subjects (9.2%), with the sparseness of the selec-

tion minimizing the possibility of drawing related individuals. The

mean age of these subjects was 45.1 years, standard deviation 13.9,

range 18.2–77.5, with 33 males and 51 females. All participants pro-

vided written informed consent on forms approved by each Institu-

tional Review Board. The images were acquired using a Siemens

MAGNETOM Trio 3 T system (Siemens AG, Erlangen, Germany) for

46 participants, or a Siemens MAGNETOM Trio/TIM 3 T system for 38

participants. We used a T1-weighted, MPRAGE sequence with an adia-

batic inversion contrast pulse with the following scan parameters:

TE/TI/TR = 3.04/785/2100 ms, flip angle=13°, voxel size (isotropic)=

0.8 mm. Each subject was scanned 7 (seven) times, consecutively,

using the same protocol, and a single image was obtained by linearly

coregistering these images and computing the average, allowing im-

provement over the signal-to-noise ratio, reduction of motion artifacts

(Kochunov et al., 2006), and ensuring the generation of smooth, accu-

rate meshes with no manual intervention. The image analysis followed

the steps described in the Methods section, with some variation to test

different registration strategies.

Fig. 3. (a) Areal interpolation between a source and a target face uses the overlapping area as a weighting factor. (b) For a given target face, each overlapping source face contributes

an amount of areal quantity. This amount is determined by the proportion between each overlapping area (represented in different colors) and the area of the respective source

face. (c) The interpolation is performed at multiple faces of the target surface, so that the amount of areal quantity assigned to a given source face is conservatively redistributed

across one or more target faces. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

5As with other neuroimaging applications, smoothing after registration implies that

the effective filter width is not spatially constant in native space, neither is the same

across subjects. Smoothing on the sphere also contributes to different filter widths

across space due to the deformation during spherical transformation.

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A.M. Winkler et al. / NeuroImage 61 (2012) 1428–1443

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Registration

To isolate and evaluate the effect of registration, we computed the

area per face after the spherical transformation6and registered each

subject brain hemisphere to a common target using two different reg-

istration methods, the Spherical Demons (Yeo et al., 2010a) and the

FreeSurfer registration algorithm (Fischl et al., 1999b),7each with

and without a study-specific template as the target, resulting in four

different variants. The study-specific targets for each of these

methods were produced using the respective algorithms for registra-

tion, using all the 84 subjects from the sample. The non-specific target

was derived from an independent set of brain images of 40 subjects,

the details of which have been described elsewhere (Desikan et al.,

2006). Areal interpolation was used to resample the areal quantities

to a common grid, a geodesic sphere produced by seven recursive

subdivisions of a regular icosahedron.

The average area per face across subjects was computed after reg-

istration and interpolation to identify eventual systematic patterns of

distortion caused by warping. This can be understood by observing

that, as the vertices are shifted along the surface of the sphere, the

faces that they define, and which carry areal quantities, are also

shifted and distorted. The registration, therefore, causes displacement

of areal quantities across the surface, which may accumulate on cer-

tain regions while other become depleted. Ideally, there should be

no net accumulation when many subjects are considered and the tar-

get is unbiased with respect to the population under study. If pockets

of accumulated or depleted areal quantities are present, this means

that some regions are showing a tendency to systematically “receive”

more areal quantities than others, which “donate” quantities. The av-

erage amount of area after the registration estimates this accumula-

tion and, therefore, can be used as a measure of a specific kind of

bias in the registration process, in which some regions consistently

attract more vertices, resulting in these regions receiving more quan-

tities. The result for this analysis is shown in Fig. 4. Using default set-

tings, SD caused less areal displacement across the surface, with less

regional variation when compared to FS. The pattern was also more

randomly distributed for SD, without spatial trends matching ana-

tomical features, whereas FS showed a structure more influenced by

brain morphology. Using a study specific template further helped to

reduce areal shifts and biases. The subsequent analyses we present

are based on the SD registration with a study-specific template.

6Note that here the area was computed in the sphere with the aim of evaluating the

registration method. For analyses of areal quantities, these quantities should be de-

fined in the native geometry, as previously described.

7The software versions used were FS 5.0.0 and SD 1.5.1.

Fig. 4. A study-specific template (target for the registration) caused less systematic accumulation of areal quantities across the brain when compared with a non-specific template.

Using default parameters, areal accumulation was less pronounced and unrelated to sulcal patterns using Spherical Demons in comparison with FreeSurfer registration. Gains and

losses refer to the area per face that would be expected for areal quantities being redistributed with no bias, i.e. the zero corresponds to the average total surface area of all subjects,

divided by the number of faces.

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Distributional characterization

To evaluate the normality for the cortical area at the white surface

of the native geometry, we used the Shapiro–Wilk normality test

(Shapiro and Wilk, 1965), implemented with the approximations

for samples larger than 50 as described by Royston (1993). The test

was applied after each hemisphere of the brain was registered to a

study-specific template using the Spherical Demons and interpolated

to the geodesic sphere using areal interpolation.

For the vast majority of the faces, the area of the white surface is

not normally distributed (Figs. 5a and 6; see also the Supplemental

Material for maps of skewness and kurtosis). Instead, the lognormal

distribution seems to be more appropriate to describe the data in

most parts of the brain, with the test declaring a much larger number

of faces as normally distributed after a simple logarithmic transfor-

mation. A log-transformation is a particular case of the Box–Cox

transformation (Box and Cox, 1964). For a set of values y={y1, y2,

…, yn}, this transformation uses maximum-likelihood methods to

seek a parameter λ that produces a transformed set

…;~ yng that approximately conforms to a normal distribution. The

transformation is a piecewise function given by:

~ y ¼~ y1;~ y2;

f

~ y ¼

yλ−1

λ

lny

λ≠0

ð

λ ¼ 0

Þ

Þð

8

:

Not surprisingly, the Box–Cox transformation rendered the data

more normally distributed than a simple log-transformation. Howev-

er, an interesting aspect of this transformation is that the parameter λ

is allowed to vary continuously, and it approaches unity when the

data are normally distributed, and zero if lognormally distributed,

serving, therefore, as a summary metric of how normally or lognor-

mally distributed the data are. Throughout most of the brain, λ is

close to zero, although with a relatively wide variation (mode=

−0.057, mean=−0.099, sd=0.493 for the analyzed dataset), indi-

cating that, at the resolution used, the white surface cortical area

can be better characterized across the surface as a gradient of skewed

distributions, with the lognormal being the most common case. The

same was observed for facewise data smoothed in the sphere after

interpolation with FWHM=10 mm (mode=−0.142, mean=−0.080,

sd=0.578).8Maps for the parameter λ are shown in Fig. 5b (see also

the Supplemental Material).

<

ð3Þ

Comparison with expansion/contraction methods

A number of studies have analyzed what has been called ex-

pansion or contraction of the cortical surface when compared to a

reference brain. Different studies adopted different operational defi-

nitions for what these terms would be [e.g. compare Joyner et al.

(2009), Sun et al. (2009b), Hill et al. (2010)], and an unified approach

has not been defined. Notwithstanding, the key difference between

these methods and the proposed areal analysis is that, at the end of

the processing pipeline, areal interpolation ensures the preservation

of the amount (mass) of quantities, whereas these methods do not.

Moreover, in the framework we present, a number of potential prob-

lems that may arise along the pipeline are explicitly addressed. These

problems, along with the solutions we propose, are summarized in

Table 1.

With a variety of expansion/contraction methods available, it is

difficult to identify the best to which areal analysis could be com-

pared. Here we retessellate each subject brain in native space using

the method described by Saad et al. (2004). The expansion/contrac-

tion method was implemented using the following steps: (1) from

the native surface geometry, perform the spherical transformation;

(2) perform the spherical registration to a standard brain; (3) treat

the coordinates x, y and z of the vertices from the native geometry

as three independent scalar fields over the registered sphere, and in-

terpolate these values to the common spherical grid using barycentric

interpolation9; (4) use the interpolated coordinates, together with

the same connectivity scheme between vertices as in the common

grid, to construct a new model of the brain in a subject-specific geom-

etry (Fig. 7); (5) from this new model, compute the area per vertex

and divide it by the area per vertex of the homologous point in the

8For scale comparison, the sphere has radius fixed and set as 100 mm, such that the

Gaussian filter has an HWMH (half width)=1.59% of the geodesic distance between

the barycenter of any face and its antipode.

9The three scalar fields can also be treated as a single vector field and the barycentric

interpolation can be performed in a single step as

xP

yP

zP

2

4

3

5¼

xA

yA

zA

xB

yB

zB

xC

yC

zC

2

4

3

5

δA

δB

δC

2

4

3

5where

x, y, z represent the coordinates of the triangular face ABC and of the interpolated point

P, both in native geometry, and δ are the barycentric coordinates of P with respect to

the same face after the spherical transformation.

Fig. 5. (a) The area of the cortical surface is not normally distributed (upper panels).

Instead, it is lognormally distributed throughout most of the brain (middle panels). A

Box–Cox transformation can further improve normality (lower panels). The same pat-

tern is present without (left) or with (right) smoothing (FWHM=10 mm). (b) Spatial

distribution of the parameter λ across the brain. When λ approaches zero, the distribu-

tion is more lognormal. See the Supplemental Material for the other views of the brain

and histograms for λ.

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Page 8

template. Call this measurement expansion/contraction; (6) optional-

ly, smooth this quantity.

For comparison with the expansion/contraction method, the orig-

inal facewise area was converted to vertexwise, therefore halving the

spatial resolution of the areal data (see Appendix C). In this compar-

ison, we addressed some of the problems presented in Table 1, name-

ly, we registered using Spherical Demons, therefore ensuring smooth

and invertible warps, and as target for registration, we used the

study-specific template that produced the best results in Fig. 4. Fur-

thermore, the measurements were taken at the white surface, rather

than the middle surface, as the last is more prone to be influenced by

the cortical thickness. It is unclear if, when applicable, these aspects

were taken care of in all the different studies that analyzed some

form of expansion/contraction.

After establishing an expansion/contraction procedure, there are

still different ways to compare with areal analysis. The comparison

can be made across subjects or across space, can be global or regional,

and may or may not include smoothing. In Fig. 8 we show that the

average amount of area at each vertex did not produce a similar

spatial map as the average expansion/contraction. Although the two

methods follow remarkably different overall spatial patterns, when

vertices across space were pooled together to produce a global mea-

surement,theyproducedverysimilarresults.Fig.9ashowstherelation-

shipbetweentheglobalcorticalsurfacearea,computedfromthesumof

theareaateachvertex,andaglobalmeasureofexpansioncomputedby

averaging the expansion/contraction at each vertex across space.10The

correlation was very high and helps to validate both methods as a

whole. Likewise, when each vertex was analyzed separately, the corre-

lation across subjectswasalsoveryhigh, asshown in Fig. 10, withanR2

above 0.9 throughout virtually the whole cortex. A spatial comparison

of the average maps, on the other hand, showed a very poor relation-

ship between both approaches, as shown in Fig. 9b. When looking at

each individual subject, rather than at the average, the correlation

across space was still relatively low, albeit not as poor: for the 168

10Note that an exact measurement of expansion/contraction relative to the template

can be produced simply by dividing the global area in native geometry by the area of

the template geometry. In this case, the points in Fig. 9a would lie in a perfectly straight

line, and nothing could be inferred about the relationship between regional variability

on expansion estimates and global measurements.

Fig. 6. Distribution of the uncorrected p-values of the Shapiro–Wilk normality test. For normally distributed data, 5% of these tests are always expected to be declared as not normal

with a significance level of α=0.05. Without transformation or smoothing, near 80% are found as not normal. Logarithmic and Box–Cox transformations render the data more nor-

mally distributed. Observe that the frequencies are shown in logscale. The dashed line (blue) is at the frequency that would be observed for uniformly distributed p-values. (For

interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 1

The proposed framework for areal analyses addresses a number of potential problems that may arise along the processing pipeline.

Processing step Problem Solution

Measurements assigned to vertices at the beginning

of the analysis.

Registration methods that not necessarily produce

smooth and invertible warps.

Interpolation based on points.

Vertices do not hold or convey the same spatial information

as the original faces.

Discontinuities on expansion or contraction that are not

present in the actual brain.

Areal quantities are not preserved at any scale (local,

regional or global).

Results are interpretable only with respect to that same

reference brain.

The local surface area follows a lognormal distribution.

Analyze the faces directly.

Use diffeomorphic registration methods.

Use areal interpolation.

Use of a standard brain to compute the same

measurement that is later analyzed.

Statistical analysis based on assumption of

normality.

Measure and analyze absolute quantities, not relative

to some reference.

Apply a data transformation. Use non-parametric

methods.

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hemispheres analyzed, we found an average linear R2=0.572, sd=

0.044 without smoothing, and R2=0.491, sd=0.065 after smoothing.

These results suggest that, if each vertex is analyzed in isolation,

analysis of surface area and analysis of expansion/contraction tend

to produce similar results. This is the case, for instance, using mass

univariate GLM-based approaches. However, for analysis that involve

spatial information or that combine information across neighboring

vertices,theresultsareexpectedtoberatherdissimilar.Thedifference

stems from the different units of measurement: areal analyses pro-

duce measurements in absolute units of area (e.g. mm2), whereas ex-

pansion/contraction is relative to the given reference. The result

shown in Fig. 10, left panel, also demonstrates, indirectly, that areas

measured in the retessellated brain with the resolution used correlate

reasonably well with the areas obtained using areal interpolation, and

so, have potential to be used as a fast approximation to areal interpo-

lation (Appendix B). Conversely, expansion/contraction measure-

ments can be obtained after areal interpolation simply by dividing

the area per face (or per vertex) by its homologous in the reference

brain.

Validation and stability

Measurements of surface area are valid as long as the surface re-

construction from MR images produces accurate representations of

the cortex. The suggested reconstruction method has been previously

validated (Fischl and Dale, 2000), and is widely used for cortical thick-

ness measurements. Comparison between subjects at the face level

dependsongoodmatchingofhomologiesandtheregistrationmethod

we suggest has, likewise, been previously validated (Klein et al., 2010;

Yeo et al., 2010a). As methods evolve, novel approaches for construct-

ing surface representations of the cortex and for registration have

Fig. 7. After barycentric interpolation of the coordinates in the surface of the sphere, a

new, subject-specific retessellated model is constructed. Areas can be computed

directly from the retessellated model and, once divided by the areas of the homologous

vertices or faces of the reference brain, constitute the measurement of expansion/

contraction.

Fig. 8. Average area (left panels) or expansion/contraction (right panels) per vertex, without (upper panels) and with smoothing (lower panels). Areal analyses and expansion/

contraction differ across space. Smoothing has little global impact.

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A.M. Winkler et al. / NeuroImage 61 (2012) 1428–1443

Page 10

potential to improve the overall quality of areal analyses. The validity

of areal measurements other than surface area itself depends on each

particular measurement technique.

To assess the stability across sessions and scanners, we compared

MR images of the same subject acquired in three different sessions

collected within a 1 year interval. The imaging protocol varied in

terms of acquisition parameters, as well as the number of images

used for averaging and improvements on signal and contrast-to-

noise ratio. The details are summarized in Table 2. The estimated sur-

face area produced by summing the facewise areas over the cortex

after interpolation was very similar across tests, with the largest

difference being 8.2% between Tests A and C (see Table 2), with or

Fig. 9. (a) The sum of the area per vertex correlates well with the average across space of the expansion/contraction at each vertex (i.e. equivalent to a weighed sum considering

each vertex as having the same initial area) for the 168 hemispheres analyzed. For the expansion/contraction, this is not the same as computing the ratio between the global surface

area in native geometry and of the template, in which case, the result would be a perfectly straight line. The high correlation implies that the regional differences in general com-

pensate each other to produce a similar global effect. (b) The correlation between average spatial maps across the 84 subjects, both hemispheres, is very poor between the methods.

[Note that, for (b), attempts to simultaneously plot all the > 300 thousand vertices would not produce meaningful plots in a small space; for this reason only 5% of the vertices were

randomly selected for plotting. The R2were computed from all vertices and, for both (a) and (b), the value corresponds to the goodness of a linear fit.].

Fig. 10. For each isolated vertex, the linear relationship between areal analyses and expansion/contraction is very high across subjects, being above R2= 0.90 virtually across the

whole cortex.

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without smoothing. The mean and standard deviation for facewise

areas were virtually identical across tests, again regardless of smooth-

ing.The pairwise Pearson correlation between thetests forthe facewise

data after registration and interpolation was above 0.80 without

smoothing, and above 0.90 after smoothing with FWHM=10 mm,

showing that the procedure is robust at the face level, even under dif-

ferent scanning conditions and degrees of smoothing.

Discussion

Registration

To be valid, facewise analyses rely on the assumption that micro-

scopic structures can be localized using as reference the features that

are identifiable with MRI and which drive the registration. Features

with such localizing power are important because they help to ensure

goodoverlapofhomologousareasbetweensubjects.Despiteanimplicit

assumption already present in most imaging studies, only recently it

has been demonstrated valid for some cytoarchitetonic areas when

the references are the cortical folding patterns, even though for non-

primary regions, the mismatch may still be substantial (Da Costa et al.,

2011; Fischl et al., 2008, 2009; Hinds et al., 2008, 2009). Other features,

some microscopic and detectable only under ultra-high field strengths

(Augustinack et al., 2005; Bridge and Clare, 2006; Duyn et al., 2007;

Kim et al., 2009), have the potential to be used as the reference as

long as they are demonstrated to be markers of histologically or func-

tionally defined areas, possibly replacing folding patterns altogether,

or used to provide ancillary information. Myeloarchitectural features

may be particularly useful for this application, for being responsible

for most of the contrast observed with MRI (Geyer et al., 2011).

Likewise, areal analyses can be conducted after registration based on

features derived from functional MRI (Sabuncu et al., 2010).

Good matching of homologies, however, depends not only on the

features used to guide the registration, but also on the registration

method itself. For facewise areal analyses, invertibility is necessary

to prevent faces from being folded over others. In addition, methods

that produce smooth warps are necessary to ensure that areal quan-

tities are transferred smoothly, without abrupt variations. Such

abrupt variations would only be acceptable if matching perfectly

with areas where structure and/or function also changes abruptly. A

spatial transformation that allows such perfect matching, however,

cannot be obtained easily in practice, since these borders usually

cannot be observed with current, conventional MRI methods, and im-

portantly, since many of the differences between regions are subtle

and the transitions are gradual. However, invertibility and smooth-

ness, as guaranteed by diffeomorphic methods, albeit important,

may not suffice. Our results show that even methods that produce

smooth varying warps can differ substantially with respect to how

the areal quantities are shifted across the surface. It is possible that

performance differences between these methods might be due to

choices on regularization strategies and associated parameters

(Fischl et al., 1999b; Yeo et al., 2010a), instigating further research

on selections that may produce the most accurate results (Yeo et al.,

2010b). Our experiments also demonstrate that the choice of the tar-

get used for registration affects the distortion in areal measurements.

Areal interpolation

Areal interpolation allows direct analysis of areal quantities in abso-

lutevalues,includingthesurfaceareaitself.Thisisbecauseitistheareal

quantityproperthatisconservativelytransferredbetweengrids.There-

fore, there is no need to apply corrections due to stretches or shrink-

ages, such as using the Jacobian of the transformation (Good et al.,

2001), nor due to the choice of the parametrizable surface (Thompson

and Toga, 1999). Moreover, the results are interpretable directly with

regard to the actual amount of tissue or other measurement under

study, rather than relative to concepts as expansion/contraction,

whicharealwaysrelativetoagivenreference,andcancreatedifficulties

in interpretation and comparison across studies, either due to different

definitions adopted by different authors, or due to the need of a

reference brain. Notwithstanding, after areal interpolation, it continues

to be possible to divide the areas by the areas of the homologous faces

or vertices of a reference brain,and so, obtain an expansion/contraction

measurement. Moreover, areal quantities that are not area itself can

also be divided by the area of each face or vertex in native geometry,

thus converting these quantities to densities if necessary.

It should be emphasized that, as with other interpolation strategies,

areal interpolation is not perfectly reversible, i.e. once the cortical area

of a subject is transferred to a different grid, remapping back to the

subject surface will not produce locally identical results, although the

global areal quantity is always conserved. This is because within each

face, the areal quantity is implicitly assumed to be homogeneously dis-

tributed. This only becomes a problem if low resolution meshes are

used and if several back-and-forth iterations are performed.

Table 2

Stability and robustness of measurements after registration and interpolation were assessed using three test images of the same subject. The measurements were similar across

tests, with similar variability across space and high spatial correlation.

Test ATest B Test C

Manufacturer and model

Sequence

TE/TI/TR (ms)

Flip angle

Voxel size (mm)

Number of acquisitions

Scan date

Cortical surface area (mm2)

Siemens MAGNETOM Trio 3 T

MPRAGE

3.04/785/2100

13°

0.8×0.8×0.8

14

March 2008

176,996

Siemens MAGNETOM Trio/TIM 3 T

MPRAGE

2.83/766/2200

13°

0.8×0.8×0.8

7

March 2008

177,098

Siemens MAGNETOM Allegra 3 T

MPRAGE

2.74/900/2500

8°

1.0×1.0×1.0

1

April 2009

180,949

Not smoothed

Average area per face (mm2)

Standard deviation

Correlation with Test A

Correlation with Test B

Correlation with Test C

0.2937

0.0938

–

0.8218

0.7589

0.2939

0.0910

0.8218

–

0.7863

0.3003

0.0962

0.7589

0.7863

–

Smoothed (FWHM=10 mm)

Average area per face (mm2)

Standard deviation

Correlation with Test A

Correlation with Test B

Correlation with Test C

0.2935

0.0746

–

0.9509

0.9047

0.2936

0.0712

0.9509

–

0.9353

0.3000

0.0748

0.9074

0.9353

–

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Statistical analysis of areal quantities

There are a number of reasons that go beyond purely methodolog-

ical considerations to justify the transformation of the data before sta-

tistical analysis. Measurements related to biological morphology,

such as lengths, areas, volumes or weights, are well known to follow

non-normal distributions. If the diameter of a structure, for instance,

is normally distributed, inevitably both its cross section and its sur-

face area follow skewed distributions, whereas its volume follows

an even more skewed (Gaddum, 1945; Kapteyn and van Uven,

1916). All these related measurements cannot be normally distribut-

ed simultaneously. The skewness is higher when the variability is

relatively large in comparison to the measure of central tendency

that best describes the data, such as the arithmetic or the geometric

mean. If the non-normality is not considered, statistical models are

likely to produce inaccurate results. In this scenario, a power transfor-

mation, such as the Box–Cox transformation, helps to identify subja-

cent, possibly causative, normally distributed effects.

The lognormal distribution, more specifically, is known to arise in

a variety of biological processes. Of particular interest is the autocat-

alytic growth of tissue over time. The number of cells present on a tis-

sue that grows in an unrestricted way can be given by the familiar

formula N=N0ect, where N0is the initial number of cells, and t is

the amount of time in which the cell grows under the circumstances

represented by the constant c, a factor that incorporates a variety of

influences, such as genetic and environmental. N will be lognormally

distributed if either c or t are normally distributed (Koch, 1966;

Limpert et al., 2001). The finding that the facewise cortical surface

area follows mostly lognormal distributions may suggest that the

method may capture these biological effects. Such interpretation

can only subsist under the tenets of accurate and smooth registration.

From a statistical perspective, permutation methods do not rely on

normality, rendering them appropriate in a variety of situations in

which this assumption is not tenable. Nevertheless, the data should,

still, undergo a transformation. As discussed above, the reason is not

merely to conform to normality, although that comes as a bonus,

but also to ensure that underlying biological effects, either multiplica-

tive or proportionally dependent upon an initial value, can be treated

as additive in a linear model (Christensen, 2002). Areal quantities

that are not the cortical surface area itself can, notwithstanding, be

distributed differently, and the framework for statistical analysis out-

lined in the Methods section appears generic enough to accommo-

date a variety cases. The Box–Cox transformation has yet another

advantage when used in combination with permutation methods

under multiple testing conditions: the more stable variance after

the transformation allows the distribution of the statistic under null

hypothesis to become more similar across tests, allowing FWER to

be controlled at a level closer to its nominal value using the distribu-

tion of the maximum statistic.

Further developments and potential applications

Facewise analyses offer the possibility of studying surface area at a

much finer scale than previously. This is a feature of interest in many

research fields across the neurosciences, as well as in medicine. Al-

though the same applies to vertexwise cortical thickness, thickness

and area provide different and complementary insights into process-

es underlying the development of the brain and disorders (Sanabria-

Diaz et al., 2010; Voets et al., 2008; Winkler et al., 2010).

Providedthattheneuronsinthecortex retain largely theirsamerel-

ative positions as the progenitor cells in the embryo (Clowry et al.,

2010; Pierani and Wassef, 2009; Rakic, 1988, 2009), facewise compari-

son of surface area allows one to hypothesize about ontogenetic

processes to the extent that they can be observed and localized with

MRI, even long after the end of phases of massive tangential cellular

proliferation. Until now, this kind of study could not be performed,

either due to lack of methods to analyze cortical surface area without

the constrains imposed by regions of interest, or due to inherent limita-

tions of methods based on expansion or contraction.

The study of local cortical surface area offers, moreover, new possi-

bilities for connectivity analyses, as the need for parcellations based

on macroscopic anatomy is obviated. Indeed, the results of connectivity

analyses are known to be influenced by the choice of the parcellation

that define nodes of putative neuronal networks (Butts, 2009;

Rubinov and Sporns, 2010). Notwithstanding, if a given set of regions

is derived from a different method (Beckmann et al., 2009; Nelson et

al., 2010), these can be directly associated with their corresponding

surface-based areas or areal quantities by means of areal interpolation.

Another potential application is for genetic analyses. Given that

cortical surface area and thickness are both heritable, yet genetically

not correlated (Panizzon et al., 2009; Winkler et al., 2010), these

traits, separately, can be used in a framework similar to voxelwise

genome-wide association studies (vGWAS) (Stein et al., 2010). Iden-

tification of genes that influence surface area has potential to eluci-

date a myriad of developmental, neurologic and psychiatric disorders.

Conclusion

We presented an interpolation method for between-subject anal-

yses of cortical surface area. The method is also suitable for other

quantities that are areal by nature and which require mass conserva-

tion (pycnophylactic property) during interpolation and analysis. We

demonstrated that, when the quantity under study is surface area it-

self, the distribution of the data does not follow a normal distribution,

being instead better characterized as lognormal, and proposed a

framework for statistical analysis and inference. An Octave/MATLAB

implementation of areal interpolation is available from the authors

at http://brainder.org.

Supplementary materials related to this article can be found

online at doi:10.1016/j.neuroimage.2012.03.026.

Acknowledgments

We thank the anonymous reviewers for their helpful remarks. M. R.

Sabuncu received support from a KL2 Medical Research Investigator

Training grant awarded via Harvard Catalyst (NIH grant 1 KL2

RR025757-01 and financial contributions from Harvard University and

its aliations). P. Kochunov received support from the NIBIB grant

EB006395. This study was supported by NIMH grants MH0708143 (PI:

D. C. Glahn), MH078111 (PI: J. Blangero) and MH083824 (PI: D. C.

Glahn). Support for FreeSurfer was provided in part by the National

Center for Research Resources (P41-RR14075, the NCRR BIRN Morpho-

metric Project BIRN002, U24RR021382), the National Institute for Bio-

medical Imaging and Bioengineering (R01EB006758), the National

Institute on Aging (AG022381), the National Center for Alternative

Medicine (RC1AT005728-01), the National Institute for Neurological

Disorders and Stroke (R01-NS052585-01, 1R21-NS072652-01, 1R01-

NS070963) and resources provided by Shared Instrumentation

Grants 1S10RR023401, 1S10RR019307, and 1S10RR023043; additional

support was provided by The Autism & Dyslexia Project funded by the

Ellison Medical Foundation.Noneoftheauthors have financialinterests

todisclose.WearealsothankfultoAndri Tziortziforherscholarlyassis-

tance with terms from Greek.

Appendix A. Geodesic spheres and areal inequalities

The only required feature for the common grid used for the areal

interpolation is that all its vertices must lie on the surface of a sphere.

The algorithm we present in Appendix B requires further that all faces

of the sphere are triangular and that all edges of all faces are much

smaller than the radius, so that areal distortion is minimized when

projecting to a plane.

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A common grid that meet these demands is a sufficiently fine

geodesic sphere. There are different ways to construct such a sphere

(Kenner, 1976). One method is to subdivide each face of a regular

polyhedron with triangular faces, such as the icosahedron, into four

new triangles. The new vertices are projected to the surface of the (vir-

tual) circumscribed sphere along its radius and the process is repeated

recursively a number of times (Lauchner et al., 1969). For the n-th

iteration, the number of faces is given by F=4nF0, the number of

vertices by V=4n(V0−2)+2, and the number of edges by E=4nE0,

where F0, V0and E0are, respectively, the number of faces, vertices and

edges of the polyhedron with triangular faces used for the initial subdi-

vision. For the icosahedron, F0=20, V0=12 and E0=30 (Fig. 11a). For

the analyses in this manuscript, we used n=7, producing geodesic

spheres with 327680 faces and 163842 vertices.

These faces, however, do not have identical edge lengths and areas

(Kenner, 1976), even though the initial icosahedron was perfectly

regular. This is important for areal interpolation, as larger faces on

the target grid do overlap with more faces from the source surfaces,

absorbing larger amounts of areal quantities, possibly causing confu-

sion if one attempts to color-encode the interpolated image according

to the actual areal quantities, in which case, geometric patterns such

as in Fig. 11b will become evident. Moreover, smoothing can cause

quantities that are arbitrarily large or small due to face sizes to be

blurred into the neighbors. Both potential problems can be addressed

by multiplying the areal quantity at each face j, after interpolation, by

a constant given by 4πr2/(Aj

of the geodesic sphere, F is the number of faces, and r is the radius of

the sphere.

TF), where AjTis the area of the same face

Appendix B. Implementation

The areal interpolation for spheres is implemented in two parts. In

the first, we compute inside of which source faces the target vertices

are located, creating a lookup table to be used in the second part. This

is the point-in-polygon problem found in vector graphics applications

(Vince, 2005). Here we calculate the area of each source face, AiS, and

the subsequent steps proceed iteratively for each face in the source.

The barycentric coordinates of each candidate vertex in relation to

the current face i is computed; if their sum equals to unity, the

point is labeled as inside. However, to test if all vertices are inside

every face would needlessly waste computational time. Moreover,

since all points are on the surface of a sphere, the vertices in the target

are never expected to be coplanar to the source triangular faces, so

the test would always fail. The first problem is treated by testing

only the vertices located within a bounding box defined, still in the

3D space, from the source face extreme coordinates. The second

could naïvely be treated by converting the 3D Cartesian coordinates

to 2D spherical coordinates, which allow a fast flattening of the

sphere to the popular plate carrée cylindrical projection. However,

latitude is ill-defined at the poles in cylindrical projections. Moreover,

cylindrical projections introduce a specific type of deformation that is

undesired here: straight lines on the surface (geodesic lines) are dis-

torted. The solution we adopt is to rotate the Cartesian coordinate

system so that the barycenter of the current source face lies at the

point (r, 0, 0), where r is the radius of the source and target spheres.

The barycenter is used for ease of calculation and for being always in-

side the triangle. After rotation, the current face and the nearby can-

didate target vertices are projected to a plane using the azimuthal

gnomonic projection (Snyder, 1987), centered at the barycenter of

the face. The point-in-polygon test can then be applied successfully.

The key advantage of the gnomonic projection is that all geodesics

project as straight lines, rather than loxodromic or other complex

paths as with other projections, which would cause many target ver-

tices to be incorrectly labeled. This projection can be obtained trivially

after the rotation of the 3D Cartesian coordinate system as ϕ=y/x

and θ=z/x, where (x, y, z) are the 3D coordinates of the point being

projected. A potential disadvantage of the gnomonic projection is

the remarkable areal distortion for regions distant from the center

of the projection. Since in typical neuroimaging applications the

source and target spheres are composed of a tessellation of approxi-

mately 3×106faces, Ai

In the second part, the areal interpolation is performed, with the

overlappingareasbeingcalculatedandusedtoweigh thearealquantity

under study. The identification of intersections between two sets of

polygons is also a well studied problem in vector graphics (Chazelle

et al., 1994; Guibas and Seidel, 1987), which solution depends on opti-

mally finding crossings between multiple line segments (Balaban,

1995; Bentley and Ottmann, 1979; Chazelle and Edelsbrunner, 1992).

Most of the efficient available algorithms assume that the polygons

are all coplanar; those that work in the surface of a sphere use

S≪4πr2, and the distortion becomes negligible.

Fig. 11. (a) The common grid can be a geodesic sphere produced from recursive subdivision of a regular icosahedron. At each iteration, the number of faces is quadrupled. (b) After

the first iteration, however, the faces no longer have regular sizes, with the largest face being approximately 1.3 times larger than the smallest as n increases.

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A.M. Winkler et al. / NeuroImage 61 (2012) 1428–1443

Page 14

coordinates expressed in latitude and longitude and require special

treatment of the polar regions. The solution we adopt obviates these

problems by first computing the area of each target face, AjT; the subse-

quent steps are performed iteratively for each face in the target sphere,

using the azimuthal gnomonic projection, similarly as in the first part,

butnowcenteredatthebarycenterofthecurrenttargetfaceateveryit-

eration. The areal quantities assigned to the faces in the target sphere

are initialized as zero before the loop begins. If all three vertices of the

current target face j lie inside the same source face k, as known from

the lookup table produced in the first part, then to the current face

the areal quantity given by QjT=Qk

source faces that surround the target are examined to find overlaps.

This is done by considering the edges of the current target face as vec-

tors organized in counter-clockwise orientation, and testing if theverti-

cesofthecandidatefaceslieontheleft,rightoriftheycoincidewiththe

edge.Ifallthethreeverticesofanycandidatefaceareontherightofany

edge,thereisnooverlapandthecandidatefaceisremovedfromfurther

consideration. If all the three vertices are on the left of all three edges,

then the candidate source face is entirely inside the target, which has

then its areal quantity incremented as QjT←QjT+Qk

faces are those that contain some vertices on the left and some on the

right of the edges of the current, target face. The intersections between

these source and target edges are computed and false intersections be-

tween edge extensions are ignored. A list containing the vertices for

each candidate source face that are inside the target face (known for

being on the left of the three target edges), the target vertices that are

inside each of the source faces (known from the lookup table) and the

coordinates of theintersections between face edges, is used to compute

the convex hull, usingtheQuickhull algorithm(Barberetal., 1996).The

convex hull delimits the overlapping region between the current target

face j and the candidate source face k, which area, Ak

increment the areal quantity assigned to the target face as QjT←QjT+

Qk

The algorithm runs in O n

would be obtained by naïve search. Nevertheless, the current imple-

mentation in Octave/MATLAB, a dynamically typed, interpreted lan-

guage, requires about 24 hours to run in a computer with 2.66 GHz

Intel Xeon processors.

SAjT/Ak

Sis assigned. Otherwise, the

S. The remaining

O, is used to

SAk

O/Ak

S.

ð Þ for n faces, as opposed to O n2

??that

Appendix C. Conversion from facewise to vertexwise

Whenever it is necessary to perform analyses that include mea-

surements taken at each vertex (such as some areal quantity versus

cortical thickness) or when only software that can display vertexwise

data is available (Appendix D), it may be necessary to convert the

areal quantities from facewise to vertexwise. The conversion can be

done by redistributing the quantities at each face to their three con-

stituent vertices. The areal values assigned to the faces that meet at

a given vertex are summed, and divided by three, and reassigned to

this vertex. Importantly, this procedure has to be done after the

areal interpolation, since interpolation methods for vertexwise data

are not appropriate for areal quantities, and before the statistical anal-

ysis, since the average of the results of the statistics of a test is not

necessarily the same as the statistic for the average of the original

data. It should also be observed that conversion from facewise to ver-

texwise data implies a loss of resolution to approximately half of the

original and, therefore, should be performed only if resolution is not a

concern and there is no other way to analyze, visualize, or present

facewise data or results. The conversion does not change the underly-

ing distribution, provided that the resolution of the initial mesh is suf-

ficiently fine.

Appendix D. Presentation of results

To display results, facewise data can be projected from the com-

mon grid to the template geometry, which helps to visually identify

anatomical landmarks and name structures. Projecting data from

one surface to another is trivial as there is a one-to-one mapping be-

tween faces of the grid and the template geometry. The statistics and

associated p-values can be encoded in colors, and a color scale can be

shown along with the surface model.

However, the presentation of facewise data has conceptual dif-

ferences in comparison with the presentation vertexwise data. For

vertexwise data, each vertex cannot be directly colored, for being

dimensionless. Instead, to display data per vertex, typically each

face has its color interpolated according to the colors of its three

defining vertices, forming a linear gradient that covers the whole

face. For facewise data there is no need to perform such interpolation

of colors, since the faces can be shown directly on the 3D space, each

one in the uniform color that represents the underlying data. The

difference is shown in Fig. 12.

Interpolation of colors for vertexwise data should not be confused

with the related, yet different concept of lightning and shading using

interpolation. Both vertexwise and facewise data can be shaded to

produce more realistic images. In Fig. 12 we give an example of

simple flat shading and shading based on linear interpolation of the

lightning at each vertex (Gouraud, 1971).

Currently available software allow the presentation of color-

encoded vertexwise data on the surface of meshes. However, only

very few software applications can handle a large number of colors

per 3D object, being one color per face. One example is Blender

(Blender Foundation, Amsterdam, The Netherlands), which we used

to produce the figures presented in this article. Another option, for in-

stance, is to use low-level mesh commands in MATLAB, such as patch.

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