# Multivariate tensor-based morphometry on surfaces: application to mapping ventricular abnormalities in HIV/AIDS.

**ABSTRACT** Here we developed a new method, called multivariate tensor-based surface morphometry (TBM), and applied it to study lateral ventricular surface differences associated with HIV/AIDS. Using concepts from differential geometry and the theory of differential forms, we created mathematical structures known as holomorphic one-forms, to obtain an efficient and accurate conformal parameterization of the lateral ventricular surfaces in the brain. The new meshing approach also provides a natural way to register anatomical surfaces across subjects, and improves on prior methods as it handles surfaces that branch and join at complex 3D junctions. To analyze anatomical differences, we computed new statistics from the Riemannian surface metrics-these retain multivariate information on local surface geometry. We applied this framework to analyze lateral ventricular surface morphometry in 3D MRI data from 11 subjects with HIV/AIDS and 8 healthy controls. Our method detected a 3D profile of surface abnormalities even in this small sample. Multivariate statistics on the local tensors gave better effect sizes for detecting group differences, relative to other TBM-based methods including analysis of the Jacobian determinant, the largest and smallest eigenvalues of the surface metric, and the pair of eigenvalues of the Jacobian matrix. The resulting analysis pipeline may improve the power of surface-based morphometry studies of the brain.

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- Jie Shi, Olivier Collignon, Liang Xu, Gang Wang, Yue Kang, Franco Leporé, Yi Lao, Anand A. Joshi, Natasha Leporé, Yalin Wang[Show abstract] [Hide abstract]

**ABSTRACT:**Blindness represents a unique model to study how visual experience may shape the development of brain organization. Exploring how the structure of the corpus callosum (CC) reorganizes ensuing visual deprivation is of particular interest due to its important functional implication in vision (e.g., via the splenium of the CC). Moreover, comparing early versus late visually deprived individuals has the potential to unravel the existence of a sensitive period for reshaping the CC structure. Here, we develop a novel framework to capture a complete set of shape differences in the CC between congenitally blind (CB), late blind (LB) and sighted control (SC) groups. The CCs were manually segmented from T1-weighted brain MRI and modeled by 3D tetrahedral meshes. We statistically compared the combination of local area and thickness at each point between subject groups. Differences in area are found using surface tensor-based morphometry; thickness is estimated by tracing the streamlines in the volumetric harmonic field. Group differences were assessed on this combined measure using Hotelling’s T 2 test. Interestingly, we observed that the total callosal volume did not differ between the groups. However, our fine-grained analysis reveals significant differences mostly localized around the splenium areas between both blind groups and the sighted group (general effects of blindness) and, importantly, specific dissimilarities between the LB and CB groups, illustrating the existence of a sensitive period for reorganization. The new multivariate statistics also gave better effect sizes for detecting morphometric differences, relative to other statistics. They may boost statistical power for CC morphometric analyses.Neuroinformatics 02/2015; · 3.10 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Finding the neuroanatomical correlates of prematurity is vital to understanding which structures are affected, and to designing efficient prevention and treatment strategies. Converging results reveal that thalamic abnormalities are important indicators of prematurity. However, little is known about the localization of the abnormalities within the subnuclei of the thalamus, or on the association of altered thalamic development with other deep gray matter disturbances. Here, we aim to investigate the effect of prematurity on the thalamus and the putamen in the neonatal brain, and further investigate the associated abnormalities between these two structures. Using brain structural magnetic resonance imaging, we perform a novel combined shape and pose analysis of the thalamus and putamen between 17 preterm (41.12 ± 5.08 weeks) and 19 term-born (45.51 ± 5.40 weeks) neonates at term equivalent age. We also perform a set of correlation analyses between the thalamus and the putamen, based on the surface and pose results. We locate significant alterations on specific surface regions such as the anterior and ventral anterior (VA) thalamic nuclei, and significant relative pose changes of the left thalamus and the right putamen. In addition, we detect significant association between the thalamus and the putamen for both surface and pose parameters. The regions that are significantly associated include the VA, and the anterior and inferior putamen. We detect statistically significant surface deformations and pose changes on the thalamus and putamen, and for the first time, demonstrate the feasibility of using relative pose parameters as indicators for prematurity in neonates. Our methods show that regional abnormalities of the thalamus are associated with alterations of the putamen, possibly due to disturbed development of shared pre-frontal connectivity. More specifically, the significantly correlated regions in these two structures point to frontal-subcortical pathways including the dorsolateral prefrontal-subcortical circuit, the lateral orbitofrontal-subcortical circuit, the motor circuit, and the oculomotor circuit. These findings reveal new insight into potential subcortical structural covariates for poor neurodevelopmental outcomes in the preterm population.Brain Structure and Function 11/2014; · 7.84 Impact Factor -
##### Conference Paper: 3D pre- versus post-season comparisons of surface and relative pose of the corpus callosum in contact sport athletes

Yi Lao, Niharika Gajawelli, Lauren Haas, Bryce Wilkins, Darryl Hwang, Sinchai Tsao, Yalin Wang, Meng Law, Natasha Leporé[Show abstract] [Hide abstract]

**ABSTRACT:**Mild traumatic brain injury (MTBI) or concussive injury affects 1.7 million Americans annually, of which 300,000 are due to recreational activities and contact sports, such as football, rugby, and boxing[1]. Finding the neuroanatomical correlates of brain TBI non-invasively and precisely is crucial for diagnosis and prognosis. Several studies have shown the in influence of traumatic brain injury (TBI) on the integrity of brain WM [2-4]. The vast majority of these works focus on athletes with diagnosed concussions. However, in contact sports, athletes are subjected to repeated hits to the head throughout the season, and we hypothesize that these have an influence on white matter integrity. In particular, the corpus callosum (CC), as a small structure connecting the brain hemispheres, may be particularly affected by torques generated by collisions, even in the absence of full blown concussions. Here, we use a combined surface-based morphometry and relative pose analyses, applying on the point distribution model (PDM) of the CC, to investigate TBI related brain structural changes between 9 pre-season and 9 post-season contact sport athlete MRIs. All the data are fed into surface based morphometry analysis and relative pose analysis. The former looks at surface area and thickness changes between the two groups, while the latter consists of detecting the relative translation, rotation and scale between them.SPIE Medical Imaging; 03/2014

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Multivariate Tensor-based Morphometry on Surfaces:

Application to Mapping Ventricular Abnormalities in HIV/AIDS

Yalin Wang, PhD*,a,b,1, Jie Zhangc, Boris Gutmana, Tony F. Chan, PhDb, James T. Becker,

PhDd,e,f, Howard J. Aizenstein, MD, PhDe, Oscar L. Lopez, MDd, Robert J. Tamburo, PhDe,

Arthur W. Toga, PhDa, and Paul M. Thompson, PhDa

a Lab. of Neuro Imaging, Dept. of Neurology, UCLA School of Medicine, Los Angeles, CA 90095,

USA

b Dept. of Mathematics, UCLA, Los Angeles, CA 90095, USA

c Statistics Department, Univ. of Wisconsin-Madison, Madison, WI 53706, USA

d Dept. of Neurology, Univ. of Pittsburgh, Pittsburgh, PA 15260, USA

e Dept. of Psychiatry, Univ. of Pittsburgh, Pittsburgh, PA 15260, USA

f Dept. of Psychology, Univ. of Pittsburgh, Pittsburgh, PA 15260, USA

Abstract

Here we developed a new method, called multivariate tensor-based surface morphometry (TBM),

and applied it to study lateral ventricular surface differences associated with HIV/AIDS. Using

concepts from differential geometry and the theory of differential forms, we created mathematical

structures known as holomorphic one-forms, to obtain an efficient and accurate conformal

parameterization of the lateral ventricular surfaces in the brain. The new meshing approach also

provides a natural way to register anatomical surfaces across subjects, and improves on prior

methods as it handles surfaces that branch and join at complex 3D junctions. To analyze

anatomical differences, we computed new statistics from the Riemannian surface metrics - these

retain multivariate information on local surface geometry. We applied this framework to analyze

lateral ventricular surface morphometry in 3D MRI data from 11 subjects with HIV/AIDS and 8

healthy controls. Our method detected a 3D profile of surface abnormalities even in this small

sample. Multivariate statistics on the local tensors gave better effect sizes for detecting group

differences, relative to other TBM-based methods including analysis of the Jacobian determinant,

the largest and smallest eigenvalues of the surface metric, and the pair of eigenvalues of the

Jacobian matrix. The resulting analysis pipeline may improve the power of surface-based

morphometry studies of the brain.

Please address correspondence to: Dr. Yalin Wang, Laboratory of Neuro Imaging, Dept. of Neurology, UCLA School of Medicine,

635 Charles E. Young Drive South, Suite 225, Los Angeles, CA 90095-7332, USA, Phone: (310) 206-2101, Fax: (310) 206-5518,

ylwang@loni.ucla.edu.

*Submitted to Neuroimage

1Acknowledgements: This work was funded by the National Institutes of Health through the NIH Roadmap for Medical Research,

Grant U54 RR021813 entitled Center for Computational Biology (CCB). Additional support was provided by the National Institute on

Aging (AG021431 to JTB, AG05133 to OLL, and AG016570 to PMT), the National Library of Medicine, the National Institute for

Biomedical Imaging and Bioengineering, and the National Center for Research Resources (LM05639, EB01651, RR019771 to PMT,

AI035041 and DA025986 to JTB).

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Published in final edited form as:

Neuroimage. 2010 February 1; 49(3): 2141–2157. doi:10.1016/j.neuroimage.2009.10.086.

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1. Introduction

Surface-based analysis methods have been extensively used to study structural features of

the brain, such as cortical gray matter thickness, complexity, and patterns of brain change

over time due to disease or developmental processes (Thompson and Toga, 1996; Dale et al.,

1999; Thompson et al., 2003; Chung et al., 2005). Cortical mapping methods have revealed

the 3D profile of structural brain abnormalities in Alzheimer’s disease, HIV/AIDS, Williams

syndrome, epilepsy, schizophrenia, and bipolar disorder (Thompson et al., 2003, 2004b,

2005a,b, 2009). Surface models have also proven useful for studying the shape of

subcortical structures such as the hippocampus, basal ganglia, and ventricles (Thompson et

al., 2004a; Styner et al., 2004, 2005; Yushkevich et al., 2006; Morra et al., 2009).

One fruitful area of research combines surface-based modeling with deformation-based

methods that measure systematic differences in structure volumes and shapes. Deformation-

based morphometry (DBM) (Ashburner et al., 1998; Chung et al., 2001; Wang et al., 2003;

Chung et al., 2003b), for example, uses deformations obtained from the nonlinear

registration of brain images to a common anatomical template, to infer 3D patterns of

statistical differences in brain volume or shape. Tensor-based morphometry (TBM)

(Davatzikos et al., 1996; Thompson et al., 2000a; Chung et al., 2003a; Ashburner, 2007;

Leporé et al., 2008; Chung et al., 2008) is a related method, that examines spatial derivatives

of the deformation maps that register brains to common template. Morphological tensor

maps are used to derive local measures of shape characteristics such as the Jacobian

determinant, torsion or vorticity. DBM, by contrast, analyzes 3D displacement vector fields

encoding relative positional differences in anatomical structures across subjects, after

mapping all brain images to a common stereotaxic space (Thompson et al., 1997; Cao et al.,

1997). One advantage of TBM for studying brain structure is that it also derives local

derivatives and tensors from the deformation for further analysis. When applied to surface

models, TBM can even make use of the Riemannian surface metric to characterize local

surface abnormalities. In this paper, we extend tensor-based morphometry to the

multivariate analysis of surface tensors. We illustrate the approach by applying it to analyze

lateral ventricular surface abnormalities in patients with HIV/AIDS. The overall goal of the

work is to find new and informative descriptors of local shape differences that can pick up

disease effects with greater statistical power than standard methods.

The lateral ventricles – fluid-filled structures deep in the brain – are often enlarged in

disease, and can provide sensitive measures of disease progression. Surface-based analysis

approaches have been applied in many studies to examine ventricular surface morphometry

(Thompson et al., 2004a; Styner et al., 2005; Thompson et al., 2006; Carmichael et al., 2006;

Ferrarini et al., 2006; Carmichael et al., 2007a,b,c; Ferrarini et al., 2008a,b; Chou et al.,

2008, 2009a,b). Ventricular changes typically reflect atrophy in surrounding structures, and

ventricular measures and surface-based maps often provide sensitive (albeit indirect)

assessments of tissue reduction that correlate with cognitive deterioration in illnesses.

Ventricular measures have also recently garnered interest as good biomarkers of progressive

brain change in dementia. They can usually be extracted from brain MRI scans with greater

precision than hippocampal surfaces or other models (Weiner, 2008).

Thompson et al. (2004a, 2006) analyzed ventricular shape with a parametric surface-based

anatomical modeling approach originally proposed in (Thompson and Toga, 1996). In one

type of analysis, a medial axis is derived passing down the center of each ventricular horn.

The local radial size – an intuitive local measure of thickness – can then be defined as the

radial distance between each boundary point and its closest point on the associated medial

axis (see Thompson et al. (2004a) for details; see work by Styner and Yushkevich for related

‘m-rep’ approaches). Based on the local radial size, multiple regression, or structural

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equation models (Chou et al., 2009b), may be used to assess the simultaneous effects of

multiple factors or covariates of interest, on surface morphology. Given maps of surface-

based statistics, false discovery rate (FDR) methods or permutation methods may be used to

assign overall (corrected) p-values for effects seen in surface based statistical maps. In the

largest ventricular mapping study to date (N = 339), Carmichael et al. (2006) applied the

radial distance method together with an automated single-atlas segmentation method to

analyze localized ventricular expansion in Alzheimer’s disease (AD) and mild cognitive

impairment. This method was also applied in a series of ventricular expansion studies

(Carmichael et al., 2007a,b,c). More recently, the same method was also extended to

combine multiple segmentations (using an approach called “multi-atlas fluid image

alignment”) to create more accurate segmentations of the ventricular surface. These methods

have been used to study genetic effects in AD (Chou et al., 2008), genetic influences on

ventricular structure in normal adult twins (Chou et al., 2009b). These methods found

correlations between ventricular expansion and CSF biomarkers of pathology, and with

baseline and future clinical decline (Chou et al., 2009a).

Styner et al. (2005) also modeled the lateral ventricles using geometrical surfaces, by

transforming each ventricle into a spherical harmonic-based shape description. They applied

this method to explore the effects of heritability and genetic risk for schizophrenia on

ventricular volume and shape. Extending this work to a diagnostic classification problem,

Ferrarini et al. (2007) used an unsupervised clustering algorithm, generating a control

average surface and a cloud of corresponding nodes across a dataset, to study ventricular

shape variations in healthy elderly and AD subjects (Ferrarini et al., 2006, 2008a,b).

As an illustrative application, we studied ventricular surface abnormalities associated with

HIV/AIDS. Our proposed multivariate TBM method detected areas of statistically

significant deformation even in a relatively small test dataset – from 11 subjects with HIV/

AIDS and 8 matched healthy controls2. For comparison, we also compared our multivariate

TBM method with simpler, more standard, Jacobian matrix based statistics. In a comparison

of overall effect sizes for different surface-based statistics, our multivariate TBM method

detected areas of abnormality that were generally consistent with simpler approaches, but

gave greater effect sizes (and therefore greater statistical power) than all other Jacobian

matrix based statistics including the Jacobian determinant, largest and least eigenvalue, or

the pair of eigenvalues of the local Jacobian matrix. Our method to compute the Jacobian

matrix and multivariate TBM is also quite general and can be used with other surface

models and triangulated meshes from other analysis programs used in neuroimaging (Fischl

et al., 1999;Van Essen et al., 2001;Thompson et al., 2004b).

Figure 1 summarizes our overall sequence of steps used to analyze lateral ventricular surface

morphometry. We used lateral ventricular surface models from our previously published

study (Thompson et al., 2006). We deliberately chose a small set of surfaces, to see if group

differences were detectable in a small sample, and if so, we aimed to find out which surface-

based statistics gave greatest effect sizes for detecting these differences. Constrained

harmonic map (Joshi et al., 2007;Shi et al., 2007) was used to match ventricular surfaces and

multivariate statistics were applied to identify regions with significant differences between

the two groups. Based on this, we created statistical maps of group differences.

2A study results on the full data set used in the previous study (Thompson et al., 2006) is reported in Appendix.

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2. Automatic Surface Registration with Holomorphic One Forms

2.1. Theoretical Background

Differential forms are used here as the basis for surface modeling and parameterization.

They belong to a branch of differential geometry known as the exterior calculus. Basic

principles of these mathematical constructs are reviewed here, assuming some knowledge of

differential geometry. For a more detailed introduction to differential forms, the reader is

referred to a differential geometry text such as Bachman (2006).

Suppose S is a surface embedded in ℝ3, with induced Euclidean metric g. In the terminology

of differential geometry, S is considered to be covered by an atlas {(Uα, φα)}. Suppose (xα,

yα) is the local parameter on the chart (Uα, φα). We say that a coordinate system (xα, yα) for

the surface is isothermal, if the metric has the representation

λ(xα, yα) is the conformal factor, which is a positive scalar function defined on each point on

the surface.

, where

The Laplace-Beltrami operator is defined as

This operator may be used to measure the regularity (smoothness) of signals that are defined

on a surface, and it is the extension of the standard Laplacian operator to general manifolds,

such as curved surfaces. A function f: S → ℝ is harmonic, if Δgf ≡ 0.

Suppose ω is a differential one-form with the representation fαdxα+gαdyα in the local

parameters (xα, yα), and fβdxβ + gβdyβ in the local parameters (xβ, yβ). Then

ω is a closed one-form, if on each chart (xα, yα),

equals the gradient of some function. An exact one-form is also a closed one-form. The de

Rham cohomology group H1(S, ℝ) is the quotient group between closed one-forms and

exact one-forms. If a closed one-form ω satisfies

Hodge theory claims that in each cohomology class, there exists a unique harmonic one-

form. The gradient of a harmonic function is an exact harmonic one-form.

. ω is an exact one-form, if it

, then ω is a harmonic one-form.

The Hodge star operator turns a one-form ω into its conjugate *ω, *ω = −gαdxα + fαdyα. If

we rewrite the isothermal coordinates (xα, yα) in the complex format zα = xα + iyα, then the

isothermal coordinate charts form a conformal structure on the surface. A topological

surface with a conformal structure is called a Riemann surface.

Holomorphic differential forms may be generalized to Riemann surfaces by using the notion

of conformal structure. A holomorphic one-form is a complex differential form, such that on

each chart, it has the form

, where ω is a harmonic one-form, and f(z) is a

holomorphic function. All holomorphic one-forms form a linear space, which is isomorphic

to the first cohomology group H1(S, ℝ).

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For a genus 0 open boundary surface with s boundaries, all holomorphic one-forms form a

real s-dimensional linear space. At a zero point p ∈ M of a holomorphic one-form ω, any

local parametric representation has

In general there are s − 1 zero points for a holomorphic one-form defined on a genus 0 open

boundary surface with s boundaries.

A holomorphic one-form induces a special system of curves on a surface, the so-called

conformal net. Horizontal trajectories are the curves that are mapped to iso-v lines in the

parameter domain. Similarly, vertical trajectories are the curves that are mapped to iso-u

lines in the parameter domain. The horizontal and vertical trajectories form a web on the

surface. The trajectories that connect zero points, or a zero point with the boundary, are

called critical trajectories. The critical horizontal trajectories form a graph, called the

critical graph.

2.2. Illustration in the 2D Planar Case

Suppose we have an analytic function φ: z → w, z, w ∈ ℂ, that maps the complex z-plane to

the complex w-plane. We will demonstrate the concepts of the holomorphic one-form, a

conformal net, and our proposed algorithms, by using three simple planar cases.

For simplicity, we simply choose φ as the function that squares a complex number, i.e., w =

z2; the mapping caused by this function may be considered to be a deformed grid in the

complex plane, which is visualized in Figure 2(a) and (b). The holomorphic one-form is the

complex differential dw = 2zdz. The red and blue curves in the w-plane are the horizontal

trajectories and vertical trajectories of dw. They are mapped to the red and blue lines in the

w-plane. The conformal net is formed by horizontal and vertical trajectories. We examine a

horizontal trajectory γ; φ maps γ to a horizontal line, namely, along γ the imaginary part of

the differential form dw is always zero. Similarly, along a vertical trajectory, the real part of

dw is always zero.

Therefore, in order to trace horizontal trajectories, we only need to find a direction, along

which the value of the differential form is real. Similarly, the vertical trajectories may be

traced by following directions in which dw is always imaginary.

The center of the z-plane is a zero point, which is mapped to the origin of the w-plane. In the

neighborhood of the zero point, the horizontal trajectories may be represented as the level

sets

and the vertical trajectories may be represented, by the implicit function

– both of these are hyperbolic curves.

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At the zero point, more than one of the horizontal trajectories intersect with each other, and

more than one of the vertical trajectories intersect with each other. The trajectories through

the zero point are called critical trajectories. The horizontal critical trajectories partition the

z-plane into 4 patches. Each patch is mapped by φ to a half plane in the w-plane.

Suppose z0 is the zero point on the z-plane, the map φ may be recovered by integrating the

differential form, ω, namely

(1)

Now, suppose instead that we have an analytic map from a 2-hole annulus (a simple type of

surface) to the complex plane. In Figure 2(c), we visualize the map in the same way as in the

previous discussion.

We denote the differential form of the map as ω. The horizontal and vertical trajectories in

the z-plane are represented as red and blue curves, which are mapped to the horizontal and

vertical lines in w-plane.

It is still true that along horizontal trajectories, the imaginary part ω is zero; along vertical

trajectories, the real component of ω is zero. The zero point is the intersection of two

horizontal trajectories and also of two vertical trajectories. The horizontal and vertical

trajectories form the conformal net. Similarly, we illustrate an analytic map from a 4-hole

annulus to the complex plane in Figure 2(d). There are three zero points.

The conformal net has a simple global structure. The critical graph partitions the surface into

a set of non-overlapping patches that jointly cover the surface, and each patch is either a

topological disk or a topological cylinder (Strebel, 1984). This is important because it allows

a general surface to be converted into a set of non-overlapping parametric meshes, even if

the surface has holes or branches in it. Each patch Ω ⊂ M may be mapped to the complex

plane by integration of the holomorphic one-form on it (Equation 1). We will use this

property to automatically partition the ventricular surface (Section 4.1). After being cut open

at three extreme points, the surface becomes topologically equivalent to a genus 0 with 2

open boundary surface (see Figure 2(c)).

The structure of the critical graph and the parameterizations of the patches are determined

by the conformal structure of the surface. If two surfaces are topologically homeomorphic to

each other and have similar geometrical structures, they can support consistent critical

graphs and segmentations (i.e., surface partitions), and their parameterizations are consistent

as well. Therefore, by matching their parameter domains, the entire surfaces can be directly

matched in 3D (this could be considered an induced mapping, or a push-forward mapping,

in the terminology of differential geometry). This generalizes prior work in medical imaging

that has matched surfaces by computing a smooth bijection to a single canonical surface,

such as a sphere or disk (Fischl et al., 1999; Thompson et al., 2000b).

2.3. Algorithm to Compute Canonical Holomorphic One-forms

Suppose that the input mesh has n+1 boundaries, ∂M = γ0–γ1–···–γn. Without loss of

generality, we map γ0 to the outer boundary and the others to the inner boundaries in the

parameter domain.

The following is the algorithm pipeline to compute the canonical holomorphic one-forms:

1.

Compute the basis for all exact harmonic one-forms;

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2.

Compute the basis for all harmonic one-forms;

3.

Compute the basis for all holomorphic one-forms;

4.

Construct the canonical conformal parameterization.

2.3.1. Basis for Exact Harmonic One-forms—The first step of the algorithm is to

compute the basis for exact harmonic one-forms. Let γk, k = 1, …, n, be an internal

boundary, we compute a harmonic function fk: S → ℝ by solving the following Dirichlet

problem on the mesh M:

where δkj is the Kronecker function, and Δ is the discrete Laplace-Beltrami operator

implemented using the co-tangent formula proposed in (Pinkall and Polthier, 1993).

The exact harmonic one-form ηk can be computed as the gradient of the harmonic function

fk, ηk = dfk, and {η1, η2, ···, ηn} form the basis for all the exact harmonic one-forms.

2.3.2. Basis for Harmonic One-forms—After getting the exact harmonic one-forms,

we will compute the closed one-form basis. Let γk (k > 0) be an inner boundary. Compute a

path from γk to γ0, denote it by ζk. ζk cuts the mesh open to Mk, while ζk itself is split into

two boundary segments and in Mk. Define a function gk: Mk → ℝ by solving a

Dirichlet problem,

Compute the gradient of gk and let τk = dgk, then map τk back to M, where τk becomes a

closed one-form. Then we need to find a function hk: M → ℝ, by solving the following

linear system: Δ(τk + dhk) ≡ 0.

Updating τk to τk + dhk, we now have {τ1, τ2, …, τn} as a basis set for all the closed but non-

exact harmonic one-forms.

With both the exact harmonic one-form basis and the closed non-exact harmonic one-form

basis computed, we can construct the harmonic one-form basis by taking the union of them:

{η1, η2, ···, ηn; τ1, τ2, ···, τn}.

2.3.3. Basis for Holomorphic One-forms—In Step 1, we computed the basis for exact

harmonic one-forms {η1, ···, ηn}. Now we compute their conjugate one-forms {*η1, ···, *ηn},

so that we can combine all of them together into a holomorphic one-form basis set.

First of all, for ηk we compute an initial approximation by a brute-force method using the

Hodge star. That is, rotating ηk by 90° about the surface normal to obtain (Wang et al.,

2007). In practice such an initial approximation is usually not accurate enough because of

the digitization errors introduced in the image segmentation and surface reconstruction

steps. To improve accuracy, we employ a technique that uses the harmonic one-form basis

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that we just computed. From the fact that ηk is harmonic, we can conclude that its

conjugate *ηk should also be harmonic. Therefore, *ηk may be represented as a linear

combination of the base harmonic one-forms:

Using the wedge product ∧ (Heinbockel, 2001), we can construct the following linear

system,

We solve this linear system to obtain the coefficients ai and bi (i = 1, 2, ···, n) for the

conjugate one-form *ηk. Pairing each base exact harmonic one-form in the basis with its

conjugate, we get a basis set for the holomorphic one-form group on M:

2.3.4. Canonical Conformal Parameterization—Given a Riemann surface M, there

are infinitely many holomorphic one-forms, but each of them may be expressed as a linear

combination of the basis elements. Figure 3 illustrates four different holomorphic one-forms

on a left lateral ventricular surface. Its induced conformal parameterization is visualized by

the texture mapping of a checkerboard onto the surface. We define a canonical conformal

parametrization as a linear combination of the holomorphic bases

n. A canonical conformal holomorphic one-form can be computed by

, i = 1, …,

(2)

The conformal parametrization induced by Equation 1 from the canonical conformal

holomorphic one-form is called the canonical conformal parametrization, as visualized in

Figure 3(a). The selected canonical conformal parameterization has a more uniform

parameterization on each horn, which is very useful as it can be used to study the surface

morphology of each horn. Compared with our prior work (Wang et al., 2007), the current

canonical conformal parameterization directly works on open boundary surfaces and does

not require us to compute a double covering of the surface and a homology basis. So, it is

more computationally efficient. The obtained conformal parameterization is unique, so it

provides a consistent parameterization across surfaces. Another benefit of this specific

parameterization is that it maximizes the uniformity of the induced grid over the entire

domain, in the sense of optimizing a conformal energy.

2.4. Surface Registration by Constrained Harmonic Map

The canonical conformal parameterization, obtained using this method, is an intrinsic

property of the overall surface structure (i.e., depends on its boundary number, and how the

components of the surface join in 3D). As a result, the surface partitions that are obtained

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from modeling different subjects’ anatomy are very similar, and the surface of the ventricles

is split up into horns in a logical and consistent way. Thus, we can also match entire surfaces

through the parameter domain. There are numerous nonrigid surface registration algorithms

(Holden, 2008) that can be applied to register each segmented surface via the parameter

domain. To register these surfaces to each other, here we apply a constrained harmonic map

through the parameter domain. The constrained harmonic map can be computed as follows.

Given two surfaces S1 and S2, whose conformal parameterizations are τ1: S1 → ℝ2 and τ2: S2

→ ℝ2, we want to compute a map, φ: S1 → S2. Instead of directly computing φ, we can

easily find a harmonic map between the parameter domains. We look for a harmonic map, τ:

ℝ2 → ℝ2, such that

Then the map φ can be obtained by

mapping relationship.

. The below diagram illustrates the

Since τ is a harmonic map, and τ1 and τ2 are conformal maps, the resulting φ is a harmonic

map. When landmark curves needs to be matched, such as the boundaries of each

component of the ventricles, we guarantee the matching of both ends of the curves. We also

match the rest of the curves in 3D based on unit speed parameterizations of both curves.

3. Multivariate Tensor-Based Morphometry

3.1. Derivative Map

Suppose φ: S1 → S2 is a map from the surface S1 to the surface S2. To simplify the

formulation, we use the isothermal coordinates of both surfaces as the arguments. Let (u1,

v1); (u2, v2) be the isothermal coordinates of S1 and S2, respectively. The Riemannian metric

of Si is represented as , i = 1,2.

In the local parameters, the map φ can be represented as φ(u1, v1) = (φ1(u1, v1), φ2(u1, v1)).

The derivative map of φ is the linear map between the tangent spaces, dφ: TM(p) →

TM(φ(p)), induced by the map φ. In the local parameter domain, the derivative map is the

Jacobian of φ,

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Let the position vector of S1 be r(u1, v1). Denote the tangent vector fields as

Because (u1, v1) are isothermal coordinates, and only differ by a rotation of π/2.

Therefore, we can construct an orthonormal frame on the tangent plane on S1 as

{

}. Similarly, we can construct an orthonormal frame on S2 as

{ }.

.

The derivative map under the orthonormal frames is represented as

In practice, smooth surfaces are approximated by triangle meshes. The map φ is

approximated by a simplicial map, which maps vertices to vertices, edges to edges and faces

to faces. The derivative map dφ is approximated by the linear map from one face [v1, v2, v3]

to another one [w1, w2, w3]. First, we isometrically embed the triangle [v1, v2, v3],[w1, w2,

w3] onto the plane ℝ2; the planar coordinates of the vertices of vi, wj are denoted using the

same symbols vi, wj. Then we explicitly compute the linear matrix for the derivative map dφ,

(3)

3.2. Multivariate Deformation Tensor-Based Statistics

In our work, we use multivariate statistics on deformation tensors (Leporé et al., 2008) and

here we adapt these concepts to be applicable to surface tensors. In related work, Chung et

al. (2003a, 2008) have also proposed to use the surface metrics as the basis for morphometry

in cortical studies. Let J be the derivative map and define the deformation tensors as S =

(JTJ)1/2. Instead of analyzing shape differences based on the eigenvalues of the deformation

tensor, e.g. Cai (2001), we use a recently introduced family of metrics, the “Log-Euclidean

metrics” (Arsigny et al., 2006). These metrics make computations on tensors easier to

perform, as they are chosen such that the transformed values form a vector space, and

statistical parameters can then be computed easily using standard formulae for Euclidean

spaces (Wang et al., 2008b).

We apply Hotelling’s T2 test (Hotelling, 1931) on sets of values in the Log-Euclidean space

of the deformation tensors. Given two groups of n-dimensional vectors Si, i = 1, …, p, Tj, j =

1, …, q, we use the Mahalanobis distance M to measure the difference between the mean

vectors for two different groups of subjects,

where S̄ and T̄ are the means of the two groups and Σ is the combined covariance matrix of

the two groups. In this formula, the 3 distinct components of the deformation tensor, which

is a 2 × 2 symmetric matrix and has two duplicate off-diagonal terms, are converted to a

vector of the same dimension.

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4. Results

4.1. Automatic Lateral Ventricular Surface Registration via Holomorphic One-Forms

The concave shape, complex branching topology and extreme narrowness of the inferior and

posterior horns of the lateral ventricles have made it difficult for surface parameterization

approaches to impose a grid on the entire lateral ventricular structure without introducing

significant area distortion. For example, many papers in the engineering literature that claim

to have extracted the lateral ventricles tend to show the inferior horn omitted or too short;

the occipital horn is often also truncated at the calcar avis, but there is usually an additional

small volume of CSF at the occipital tip of the occipital horns, and this is hard for most

segmentation algorithms to properly connect with the rest of the ventricles. To correctly

model the entire lateral ventricular surface, we automatically locate and introduce three cuts

on each ventricle. The cuts are motivated by examining the topology of the lateral ventricles,

in which several horns are joined together at the ventricular “atrium” or “trigone”.

Meanwhile, we keep the cutting curves consistent across surfaces. After the topology is

modeled in this way, a lateral ventricular surface, in each hemisphere, becomes an open

boundary surface with 3 boundaries(see Figure 4(a)). We computed the exact harmonic one-

form (Figure 4(b)), its conjugate one-form (Figure 4(c)), and canonical holomorphic one-

form (Figure 4(d). With the conformal net introduced in this way (Figure 4(e)), each lateral

ventricular surface may be divided into 3 pieces (Figure 4(f)). Although surface geometry is

widely variable across subjects, the zero point locations are intrinsically determined by the

surface conformal structures, and the partitioning of the surface into component meshes is

highly consistent across subjects. This topological decomposition, to model the structure as

a set of connected surfaces, may be considered to be a topology optimization operation. The

topological optimization also helps to enable a uniform parametrization on some areas that

otherwise are very difficult for usual parametrization methods to capture (e.g., the tips of a

pointed structure). Figure 4(f) illustrates the automatic surface segmentation result for the

same pair of lateral ventricular surfaces, which is similar to the manual surface

segmentations used in prior research (Thompson et al., 2006). Even so, the new automated

partitioning method improves on past work as it avoids arbitrarily chopping the surface into

3 parts using a fixed coronal plane. Figure 5 illustrates a pair of segmentation results from

two groups. In the Figure, part (a) shows a pair of ventricular surfaces extracted from a 3D

MRI image of a healthy control subject and (b) is from a patient with HIV/AIDS. The

ventricular surfaces in (b) are dilated due to the disease. The two surfaces are similar overall,

so the surface segmentation results are consistent. Each ventricular surface was

automatically segmented into 3 pieces. We labeled each piece with a different color. Figure

5 demonstrates that our algorithm can successfully identify matching parts on surfaces that

are significantly different.

After the surface segmentation, each lateral ventricular surface is divided up into three

surfaces, each topologically equivalent to a cylinder. For each piece, we computed a new set

of holomorphic one-forms on it and conformally mapped it to a rectangle. Figure 6

illustrates a left lateral ventricular surface example. After segmentation, each piece of

overall surface is topologically equivalent to a surface with one open boundary, i.e., a

cylinder. This can then be conformally mapped to a rectangle without any singularities. We

compute the new conformal parameterization and visualize the conformal coordinates using

the red and blue lines on the parameterization domain. Then we register the two components

of the surfaces via the constrained harmonic map as explained in Section 2.4.

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4.2. Multivariate Tensor-Based Morphometry Study of the Lateral Ventricular Surface in

HIV/AIDS

In our experiments, we compared ventricular surface models extracted from 3D brain MRI

scans of 11 HIV/AIDS individuals and 8 matched control subjects, using data previously

examined in (Thompson et al., 2006). After registration of these surfaces to a common

space, and across subjects using a harmonic mapping, we computed the surface Jacobian

matrix and applied multivariate tensor-based statistics to study differences in ventricular

surface morphometry. We ran a permutation test with 5, 000 random assignments of subjects

to groups to estimate the statistical significance of the areas with group differences in

surface morphometry. We also used a pre-defined statistical threshold of p=0.05 at each

surface point to estimate the overall significance of the group difference maps by non-

parametric permutation testing (Thompson et al. 2005). In each case, the covariate (group

membership) was permuted 5,000 times and a null distribution was developed for the area of

the average surface with group-difference statistics above the pre-defined threshold in the

significance maps. The overall significance is defined of the map is defined as the

probability of finding, by chance alone, a statistical map with at least as large a surface area

beating the pre-defined statistical threshold of p=0.05. This omnibus p-value is commonly

referred to as the overall significance of the map (or the features in the map), corrected for

multiple comparisons. It basically quantifies the level of surprise in seeing a map with this

amount of the surface exceeding a pre-defined threshold, under the null hypothesis of no

systematic group differences. The experimental results are shown in Figure 10. After fixing

the template parametrization, we used Log-Euclidean metrics to establish a metric on the

surface deformation tensors at each point. We conducted a permutation test on the

suprathreshold area of the resulting Hotelling’s T2 statistics (Hotelling, 1931). The statistical

maps are shown in Figure 10. The threshold for significance at each surface point was

chosen to be p = 0.05. Although sample sizes are small, we still detected large statistically

significant areas, consistent with prior findings (Thompson et al., 2006). The permutation-

based overall significance of the group difference maps, corrected for multiple comparisons,

were p = 0.0066 for the right ventricle and 0.0028 for the left ventricle, respectively.

To explore whether our multivariate statistics provided extra power when running TBM on

the surface data, we conducted four additional statistical tests using different tensor-based

statistics derived from the Jacobian matrix. The other statistics we studied were: (1) the pair

of eigenvalues of the Jacobian matrix, treated as a 2-vector; (2) the determinant of Jacobian

matrix; (3) the largest eigenvalue of Jacobian matrix; and (4) the smallest eigenvalue of

Jacobian matrix. For statistics (1) we used Hotelling’s T2 statistics to compute the group

mean difference. In cases of (2), (3) and (4), we applied a Student’s t test to compute the

group mean difference at each surface point. For these four new statistics, their calculated

statistical maps are shown in Figure 11(a)–(d), respectively. For each statistic, we also

computed the overall p-values (see Table 1). Areas of surface abnormalities detected by

different tensor-based surface statistics were relatively consistent. The experiments also

strongly suggested that the newly proposed multivariate TBM method has more detection

power in terms of effect size (and the area with suprathreshold statistics), probably because

it captures more directional and rotational information when measuring geometric

differences.

In these maps, the statistics based on the multivariate analysis of the surface metric do in

fact give the highest effect sizes for detecting differences between the two groups. The left

ventricle has a very broad region that shows very low p-values, and the extent of the group

differences is readily apparent. A more common TBM-derived statistic would to look at the

Jacobian of the surface metric, which in this context essentially encodes local differences in

surface area, across subjects. Figure 11(b) shows that the Jacobian of the surface metric is

not in fact a particularly strong descriptor of group differences, in the sense that only very

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small and localized regions pass the voxel-level significance threshold. As in Leporé et al.

(2008), we also examined several other tensor-derived statistics to see which ones were

really carrying the majority of the relevant information for picking up group differences.

The maps based on the two eigenvalues of the surface metric (Figure 11(a)) are really quite

effective in picking up group differences, although they do not quite have the same effect

size as the analysis of the full surface metric tensor. This is in line with intuition, as the

analysis of the full surface metric tensor conserves a substantial amount of information on

local surface geometry, and has 3 independent components (as it is equivalent to a 2 × 2

symmetric matrix). The two eigenvalues of the surface metric (Figure 11(a)) still retain

much relevant information, giving intermediate effect sizes, and the least information is

retained by analyzing only the single scalar value at each point provided by the determinant

of the surface metric tensor (which is the square of the determinant of the Jacobian of the

deformation mapping; Figure 11(b)).

As Figures 11(c) and (d) show, the information in the pair of eigenvalues is present to some

extent when analyzing each of the eigenvalues separately, but the multivariate test on the

pair of eigenvalues considered together gives better (lower) effect sizes. This is again to be

expected as the multivariate tests can draw upon information on the observed sample

covariance between the scalar components being analyzed, making tests for group

differences substantially more powerful.

In Figure 7, the cumulative distribution function of the p-values observed for the contrast of

patients versus controls is plotted against the corresponding p-value that would be expected,

under the null hypothesis of no group difference, for the different scalar and multivariate

statistics. For null distributions, the cumulative distribution of p-values is expected to fall

approximately along the line (represented by the dotted line); large deviations from that

curve are associated with significant signal, and greater effect sizes represented by larger

deviations (the theory of false discovery rates gives formulae for thresholds that control false

positives at a known rate). We note that the deviation of the statistics from the null

distribution generally increases with the number of parameters included in the multivariate

statistics, with statistics on the full tensor typically outperforming scalar summaries of the

deformation based on the eigenvalues.

As such, it makes sense to use the multivariate analysis for surface-based TBM studies, so

long as other component maps are also created to better interpret the geometrical origin of

any detected differences.

5. Discussion

The current study has two main findings. First, it is possible to analyze differences in surface

morphometry by building a set of parametric surfaces using concepts from exterior calculus,

such as differential one-forms and conformal nets. This is a high-level branch of

mathematics that has not been extensively used in brain imaging before, but it provides a

rigorous framework for representing, splitting, parameterizing, matching, and measuring

surfaces. Second, the analysis of parametric meshes in computational studies of brain

structure can be made more powerful by analyzing the full multivariate information on

surface morphology. In this context, we are using multivariate to mean a multi-component

description of surface shape at each and every point, rather than treating the observations at

many different points on a surface as a multivariate vector (which can also be done, for

example, in statistical shape analysis that use PCA on shapes). Work on tensor-based

analysis of surfaces was proposed initially by Chung et al. (2003a), who noted that cortical

surface data in children could be analyzed for patterns of growth over time by considering

the local contraction or expansion of parametric grids adapted to the cortex (see Thompson

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et al. (1997, 2000a) for related work). A key barrier to getting these methods to work in

general on subcortical surfaces is the complex topology of many subcortical surfaces, which

this paper provides a method to understand.

In terms of validating the method, at least two types of validation are necessary. It is

important to first show that the methods do in fact create conformal mappings (as checked

by the assessment of angles in the resulting grids) and, second, show by formal proof that

the patches form a total cover of the surface and are guaranteed not to overlap. For

generating non-overlapping patches that are each proven to be conformal maps, the

holomorphic one-form is a well-studied mathematical tool. According to Hodge theory, the

holomorphic one-form induces a conformal mapping (i.e., parameterization) between a

surface and the complex plane. The conformal parameterization induces a simple global

structure, the conformal net. There are formal proofs that the resulting surface partition

consists of non-overlapping patches that jointly cover the surface, and each patch is either a

topological disk or a topological cylinder (Strebel, 1984; Luo, 2006). Basically, our

segmentation algorithm just traces a critical graph so that each of the surface segments

obtained is either a topological disk or a topological cylinder, and there is no overlap

between patches.

In prior work, we also verified some of the formal properties of these maps, using

artificially-generated, synthetic surfaces. In Wang et al. (2005b), we built a two-hole torus

surface (essentially like a 3D solid version of a figure 8), and found that the holomorphic

one-form method was able to correctly split it into two patches with a rectangular

parameterization. Via texture mapping, we also showed that the hippocampus can be

represented as an open-boundary genus-one surface (a cylinder), and we built one-form

based single-patch parameterizations. These correctly mapped the patch boundaries to the

boundaries of the hippocampal surface, without causing the extremely dense gridding that is

evident at the poles of a spherical parameterization. In Wang et al. (2005c), we used

empirical plots to verify that the resulting maps were very close to conformal, even when

internal landmark constraints were added to enforce landmark correspondences inside the

conformal maps. Histograms of the angle difference from a conformal (angle-preserving)

mapping were tightly clustered around zero even when a large number of landmark

constraints were enforced (Wang et al., 2005c; Lui et al., 2006a,b). In Wang et al. (2005a),

we also showed texture-mapped data verifying the surface registrations. In that paper, we

used an additional term to improve the mutual information between hippocampal surface

features (scalar fields defined on the surfaces) in two different subjects. Plots of the

conformal factor and mean curvature were shown, before and after surface registration, to

verify that the registration improved the alignment of corresponding features in parameter

space, and, via the pull-back mapping, on the 3D surfaces. In the current work, we did not

use mutual information to align curvature maps within the surface patches. This is because

there is no reason to think that curvature is a reliable guide to homology on the ventricular

surface instead, it seems reasonable to set up a computed correspondence between the two

surfaces, that is smooth and conformal, and matches all the horns at once. If features on the

ventricular surfaces were identified that could be used to independently validate the

registration mappings, they could be used for verification, or included in the mapping

themselves. However, there is no ground truth surface correspondence for ventricles across

subjects, except for the logical requirement that each horn maps to its homolog (as is seen in

our maps).

Finally, there may be rare pathological cases in which the patches in the critical graph may

not correspond anatomically across different subjects, but we did not see any such examples

in tests on large numbers of subjects. As the three horns on each side of the lateral ventricles

are quite elongated, we found that in practice the patch boundaries always generated a cross

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on the surface (see Figure 6) in which the temporal and occipital horns met at a point, and

the frontal horn wrapped around them in such a way that two points in the frontal horns

parameter space met at the same 3D point. So long as the three tips of the ventricular horns

can be isolated, such a decomposition is enforced, because the corners of the patches in

parameter space contain right angles, so they still do when mapped into 3D via an angle-

preserving (conformal) map.

We also assessed the sensitivity of the algorithm to simulated differences, by introducing

spatially varying deformations of into the left ventricular surfaces of the control group. For

each left ventricular surface, we selected a rectangular area in the parameter space of the

frontal horn, as shown in Figure 8(a). We slightly adjusted each surface point’s geometry

coordinates by the following equation,

where r and k are tunable parameters that allowed us to vary the geometric deformation.

Using surface registration, we introduced this geometric deformation to all the surfaces in

the control group, to generate a new group of surfaces, matched to the original ones. We

applied our multivariate tensor-based morphometry method to assess the geometric

differences between these two groups. Using this scheme, abnormalities are deliberately

created and mathematically defined, and the nature and context of the deformations can be

systematically varied to determine the conditions that affect detection sensitivity.

Figure 8 illustrates the results of the experiments on these synthetic datasets. (a) illustrates

the selected rectangle (labeled in yellow) on a specific left ventricular surface. We

conducted 3 sets of experiments with different parameters: r = 0.005, k = 1 (Figure 8(b)), r =

0.005, k = 7 (Figure 8(c)) and r = 0.01, k = 7 (Figure 8(d)). Although the imposed

deformations were relatively small scale geometric perturbations involving only one

coordinate, our algorithm still successfully picked up the areas that were significantly

different by construction. For the frontal horn regions that involved the simulated

deformation, the overall (corrected) significance of the group difference maps were p =

0.1642, 0.0248 and 0.0004, respectively, for the deformations of increasing severity.

These results illustrate the graded response of the multivariate tensor-based morphometry

algorithm, in assessing deformations of spatially varying magnitude across the ventricular

surfaces. This also demonstrates the effectiveness of our algorithm for picking up subtle

morphometric deformations across surfaces.

In our experiment, an important step is the automatic partitioning of a complex 3D surface,

i.e., we cut open a ventricular surface at its three extreme points. This turns the surface into a

genus 0 surface with 2 open boundaries that is topologically equivalent to the surface in

Figure 2(c). Obviously, the original ventricular surface is a genus 0 surface. Theoretically it

is topologically equivalent to a sphere. However, the concave shape, complex branching

topology and extreme narrowness of the inferior and posterior horns make it extremely

difficult to compute a meaningful regular mapping from a ventricular surface to a sphere

(Friedel et al., 2005)3.

3One public domain implementation is available in a software package, ShapeAtlasMaker, developed by the UCLA Center for

Computational Biology – http://www.loni.ucla.edu/twiki/bin/view/CCB/TaoSphericalMap.

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