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BIOINFORMATICS

Vol. 28 ECCB 2012, pages i619–i625

doi:10.1093/bioinformatics/bts411

From phenotype to genotype: an association study of

longitudinal phenotypic markers to Alzheimer’s disease

relevant SNPs

Hua Wang1, Feiping Nie1, Heng Huang1,∗, Jingwen Yan2, Sungeun Kim2,

Kwangsik Nho2, Shannon L. Risacher2, Andrew J. Saykin2, Li Shen2,∗, for

the Alzheimer’s Disease Neuroimaging Initiative†

1Department of Computer Science and Engineering, University of Texas at Arlington, TX 76019, USA and

2Department of Radiology and Imaging Sciences, Indiana University School of Medicine, Indianapolis, IN 46202, USA

ABSTRACT

Motivation: Imaging genetic studies typically focus on identifying

single-nucleotide polymorphism (SNP) markers associated with

imaging phenotypes. Few studies perform regression of SNP values

on phenotypic measures for examining how the SNP values change

when phenotypic measures are varied. This alternative approach

may have a potential to help us discover important imaging genetic

associations from a different perspective. In addition, the imaging

markers are often measured over time, and this longitudinal profile

may provide increased power for differentiating genotype groups.

How to identify the longitudinal phenotypic markers associated to

disease sensitive SNPs is an important and challenging research

topic.

Results:

Taking into account the temporal structure of the

longitudinal imaging data and the interrelatedness among the SNPs,

we propose a novel ‘task-correlated longitudinal sparse regression’

model to study the association between the phenotypic imaging

markersandthegenotypesencodedbySNPs.Inournewassociation

model, we extend the widely used ?2,1-norm for matrices to tensors

to jointly select imaging markers that have common effects across

all the regression tasks and time points, and meanwhile impose

the trace-norm regularization onto the unfolded coefficient tensor to

achieve low rank such that the interrelationship among SNPs can be

addressed. The effectiveness of our method is demonstrated by both

clearly improved prediction performance in empirical evaluations and

a compact set of selected imaging predictors relevant to disease

sensitive SNPs.

Availability: Software is publicly available at:

http://ranger.uta.edu/%7eheng/Longitudinal/

Contact: heng@uta.edu or shenli@inpui.edu

∗To whom correspondence should be addressed.

†Data used in preparation of this article were obtained from theAlzheimer’s

Disease Neuroimaging Initiative (ADNI) database (adni.loni.ucla.edu).

As such, the investigators within the ADNI contributed to the design

and implementation of ADNI and/or provided data but did not

participate in analysis or writing of this report. A complete listing

of ADNI investigators can be found at: http://adni.loni.ucla.edu/wp-

content/uploads/how_to_apply/ADNI_Acknowledgement_List.pdf.

1

Neuroimaging genetics is an emerging research field, where brain

imaging is used as quantitative phenotypes to investigate the role

of genetic variation in brain structure and function. It holds great

promise for a systems biology of the brain to better understand

complex neurobiological systems, from genetic determinants to

cellular processes to the complex interplay of brain structure,

function, behavior and cognition. Disorders of the nervous system

are associated with complex neurobiological changes, which may

lead to profound alterations at all levels of organization.

Genome-wide associationstudies

increasingly performed to correlate high-throughput single-

nucleotide polymorphism (SNP) data to large-scale imaging

data. To facilitate such association analysis, many studies used

a hypothesis-driven approach (Glahn et al., 2007) by making

significant reduction in one or both data types. For example,

some whole-brain studies focused on a small number of genetic

variables (e.g. Brun et al., 2009; Filippini et al., 2009; Hariri et al.,

2006; Nichols and Inkster, 2009), and some whole-genome studies

examined a limited number of imaging variables (e.g. Baranzini et

al., 2008; Potkin et al., 2009; Seshadri et al., 2007). Many SNPs

have been identified as risk factors for Alzheimer’s disease (AD),

see those in the AlzGene database (www.alzgene.org).

So far most studies focus on selecting and associating SNPs to

AD status or imaging phenotypes. Very few studies have been done

to directly examine how the SNP values change when phenotypic

measures are varied, i.e. via regression of SNPvalues on phenotypic

measures. This alternative approach may have a potential to help us

discover important imaging genetic associations from a different

perspective. In this study, we perform such an initial analysis for

finding phenotypic imaging markers that are related to SNPs from

or proximal to AlzGene candidates.

Neuroimaging measures have been widely studied to predict

disease status and/or cognitive performance (Batmanghelich et al.,

2009;Shenetal.,2010a).However,whetherthesemeasurescoupled

withtheirlongitudinalprofileshavesufficientpowertoinferrelevant

genotype groups is still an under-explored yet important topic inAD

research. A simple strategy typically used in longitudinal studies

(e.g. Risacher et al., 2010) is to analyze a single summarized value

such as average change rate of change or slope. This approach may

be inadequate to distinguish the complete dynamics of cognitive

trajectories and thus become unable to identify the underlying

genetic structure.

INTRODUCTION

(GWAS) have been

© The Author(s) 2012. Published by Oxford University Press.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which

permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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H.Wang et al.

With these observations, in this work, we propose a new task-

correlated longitudinal sparse regression framework to effectively

identifythelongitudinalphenotypicmarkersrelatedtocandidateAD

SNPs. Based on the emerging structured sparse learning techniques,

which has been effectively applied in imaging genetics studies

(Wangetal.,2011a,b,2012a,b),thenewcombinedstructuredsparse

regularizations are introduced to tackle the longitudinal phenotypic

patterns and biological genotypic correlations. The proposed new

computational biology model consists of three major components.

First,duetotheserialmeasuresoftheimagingphenotypesovertime,

we propose a novel longitudinal regression analysis method. As a

result, the regression coefficients assess the relationships between

longitudinal phenotypes and their genetic makeups. Second, certain

SNPs are naturally correlated via different ways, e.g. multiple

SNPs from one single gene often jointly carry out similar genetic

functionalities, SNPs in high linkage disequilibrium (LD) are

linked together in meiosis. To incorporate such SNP correlations

in our association studies, we propose to use the trace/nuclear

norm regularization (Candès and Recht, 2009; Nie et al., 2012) to

approximately minimize the rank of regression coefficient matrix,

such that the coefficients of phenotypes associated to correlated

SNPs are linearly dependent. Finally, through enforcing the ?2,1-

normregularization,theimagingfeatureselectionacrossmostSNPs

are coupled (Argyriou et al., 2007; Obozinski et al., 2006), so that

the identified imaging phenotypes have common influence on all the

SNPs.

We apply the proposed method to the Alzheimer’s Disease

Neuroimaging Initiative (ADNI) cohort (Mueller et al., 2005) for

identifying longitudinal phenotypes using a set of SNPs based on

the AlzGene database. Our empirical results yield not only clearly

improved prediction performance in all test cases but also a compact

set of associations between phenotypes and genotypes that are in

accordance with prior research findings.

2

Both SNP and structural magnetic resonance imaging (MRI) data

used in the preparation of this article were obtained from the ADNI

database (adni.loni.ucla.edu). One goal of ADNI has been to test

whether serial MRI, positron emission tomography (PET), other

biological markers and clinical and neuropsychological assessment

can be combined to measure the progression of mild cognitive

impairment (MCI) and early AD. For up-to-date information, we

refer interested readers to www.adni-info.org.

MATERIALS AND DATA SOURCES

2.1

TheSNPdatausedinthisstudy(Saykinetal.,2010)weregenotyped

using the Human 610-Quad BeadChip (Illumina, Inc., San Diego,

CA, USA).Among all SNPs, only SNPs, belonging to the top 40AD

candidate genes listed on the AlzGene database (www.alzgene.org)

as of June 10, 2010, were selected after the standard quality control

(QC) and imputation steps.The QC criteria for the SNPdata include

(i) call rate check per subject and per SNPmarker, (ii) gender check,

(iii) sibling pair identification, (iv) the Hardy–Weinberg equilibrium

test, (v) marker removal by the minor allele frequency and (vi)

population stratification. As the second pre-processing step, the

quality-controlled SNPs were imputed using the MaCH software

(Li et al., 2010) to estimate the missing genotypes. After that, the

SNP genotypes

Illumina annotation information based on the Genome build 36.2

was used to select a subset of SNPs, belonging to the top 40 AD

candidate genes (Bertram et al., 2007).The above procedure yielded

1224 SNPs from 37 genes. For the remaining three genes, no SNPs

were available on the genotyping chip.

2.2MRI analysis and extraction of imaging

phenotypes

Two widely used automated MRI analysis techniques were used to

process and extract imaging genotypes across the brain from all

the MRI scans of ADNI participants as previously described (Shen

et al., 2010b). First, voxel-based morphometry (VBM) (Ashburner

and Friston, 2000) was performed to define modulated gray matter

(GM) maps and extract local GM values for target regions. Second,

automated parcellation via FreeSurfer V4 (Fischl et al., 1999, 2002)

was conducted to define volumetric and cortical thickness values for

regions of interest (ROIs) and to extract total intracranial volume

(ICV). Further information is available in (Shen et al., 2010b). The

time points examined in this study for imaging markers included

baseline (BL), Month 6 (M6), Month 12 (M12) and Month 24

(M24).All the participants with no missing BL/M6/M12/M24 MRI

measurementswereincludedinthisstudy.Figure2showsthenames

oftheseROIsinthebrainspace.Allthesemeasureswereadjustedfor

baseline ICV using the regression weights derived from the healthy

control (HC) participants.

3 TASK-CORRELATED LONGITUDINAL SPARSE

REGRESSION

For the association study of longitudinal imaging phenotypes to the

genotypes, the input imaging features are a set of matrices X =

{X1,X2,...,XT}∈Rd×n×Tcorresponding to the measurements at

T consecutive time points, where Xtis the imaging measurements

for a certain type of imaging markers, such as VBM or FreeSurfer

markers used in this study, at time t?1≤t≤T?. Obviously, X is a

points. The output genetic variations described by c SNPs for the n

subject samples forms a matrix Y =?y1,...,yn

learn from {X,Y} a model that can reveal the associations between

the longitudinal imaging phenotypes X and the genotypes Y.

A straightforward method for relating imaging phenotypes and

SNPs is to perform regression at each time point separately,

which, though, does not take into account the valuable information

conveyed by the longitudinal patterns of the phenotypic inputs. To

overcomethislimitation,differentfrompreviousstudiesthatlearned

the regression coefficient matrix for each time point individually,

we aim to learn a unified longitudinal regression model to find

the genetic features that are associated to the longitudinal imaging

patterns over all the measurement time points. To this end, we

expect to learn a coefficient tensor (a stack of coefficient matrices)

B={B1,···,BT}∈Rd×c×Tto reveal the temporal changes of the

coefficient matrices. In this article, we propose to use the low-rank

structured sparse regularizations to explore the temporal patterns

and the interrelatedness between SNPs in a new task-correlated

longitudinal sparse regression model.

tensor data with d imaging features, n subject samples and T time

?T∈Rn×c, where the

yi∈Rcis the SNP values of the ith subject sample. Our goal is to

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Association study of longitudinal phenotypic markers to AD relevant SNPs

3.1 Task-correlated longitudinal sparse regression

using low-rank structured sparse regularizations

The simplest model to associate the the phenotypic markers to the

genotypes is the multivariate regression model, which solves the

following optimization problem:

min

B

J0=L?B?+γ||B||2

2=L?B?+γ

T

?

t=1

d

?

k=1

||bk

t||2

2,

(1)

where bktdenotes the kth row of coefficient matrix Btat time t, and

L?B?is the proposed longitudinal loss and defined as

L?B?=||B⊗1XT−Y||2

Because the objective J0in Equation (1) can be decoupled for

each individual time point and does not consider the longitudinal

correlations between the imaging features and the SNPs, it is

not suitable for longitudinal data analysis and feature selection.

Because the selected imaging markers with temporal changes are

desired to connect all the SNPs, the T groups of regression tasks

at different time points should not be decoupled and have to

be performed simultaneously. Thus, we introduce the structured

sparse regularization (Argyriou et al., 2007; Nie et al., 2010;

Obozinski et al., 2006) into the longitudinal data regression and

feature selection model as following:

F=

T

?

t=1

||XT

tBt−Y||2

F.

(2)

min

B

J1=L?B?+γ

d

?

k=1

?

?

?

?

T

?

t=1

||bkt||2

2.

(3)

Apparently, J1in Equation (3) can no longer be decoupled over

time dimension. Upon solution, the imaging features with common

influences to all the SNPs across all the time points will be identified

out due to the second term in Equation (3), which essentially is a

tensor extension of the widely used ?2,1-norm for matrices.

To further take into account that many SNPs are interrelated

together and their effects on brain structure or disease progression

could overlap, we expect to further develop J1in Equation (3) to

leverage the useful information conveyed by the SNP correlations.

Mathematically speaking, due to the interrelatedness among the

SNPs, the learning vector

jshould have certain correlations,

where?bt

n-mode tensor T ∈RI1×I2×···×In, we denote unfoldk

RIk×?I1...Ik−1Ik+1...In

Then we can achieve our goal by minimizing the rank of B?1/=

following optimization problem:

?

?bt

?

?

jdenotes the jth column of Bt. Namely, the coefficient

matrices Bt

should be of low rank. Given a general

?1≤t≤T?

?T?=T?k?∈

?

as the unfolding operation along its kth mode.

[B1,B2,...,BT]∈Rd×?c×T/induced from B, which leads to the

min

B

J2=L?B?+γ1

d

?

k=1

?

?

?

T

?

t=1

||bkt||2

2+γ2?B?∗,

(4)

where??∗denotesthetrace-normofamatrix,andwithoutambiguity

we drop the subscript of the matrix B?1/for notation brevity. Given

trace-norm of M is defined as ?M?∗=?min?n,m/

amatrixM∈Rn×manditssingularvaluesσi

?1≤i≤min?n,m??,the

i=1

σi=Tr?MMT?1

2.

It has been shown that (Candes and Tao, 2010; Candès and

Recht, 2009) the trace-norm is the best convex approximation of

the rank-norm. Therefore, the third term of J2in Equation (4)

indeed minimizes the rank of the unfolded learning model B, such

that the correlations among the SNPs are captured. Due to its

both capabilities for imaging marker selection and task correlation

integration, we call J2defined in Equation (4) as the proposed

‘task-correlated longitudinal sparse regression model’.

3.2A new optimization algorithm and its global

convergence

Because our new objective J2is non-smooth, the problem in

Equation (4) is difficult to solve in general. Some existing methods,

such as LARS (Efron et al., 2004), linear gradient search (Liu et al.,

2009), proximal (Beck and Teboulle., 2009) methods, can solve it,

but not efficiently. Thus, in this subsection we derive a new efficient

algorithm to solve J2with rigorous proof of its global convergence.

Taking the derivative of J2w.r.t Btand set it to zeros, we have:

2XtXT

tBt−2XtY+2γ1DBt+2γ2¯DBt=0,

(5)

where D is a diagonal matrix with D(k,k)=

?BBT?−1/2/2. Thus, we can derive

Bt=(XtXT

When the time t changes from 1 to T, we can compute Bt

by Equation (6). Because D and ¯D depend on B and can be seen

as latent variables, we propose an iterative algorithm to obtain the

global optimum solutions of B in Algorithm 1.

Algorithm 1: A new algorithm to minimize J2in Equation (4).

Data: X ∈Rd×n×T, Y ∈Rn×c.

1. Initialize B(0)∈Rd×c×Tusing the regression results at each

individual time point.;

repeat

2. Calculate the diagonal matrix D, where the k-th diagonal

element is computed as

2

?BBT?−1

until Converges;

Result: B={B1,B2,...,BT}∈Rd×c×T.

1

2

??T

t=1||bk

t||2

2

and¯D=

t+γ1D+γ2¯D)−1XtY.

(6)

?1≤t≤T?

1

??T

2.;

t+γ1D+γ2¯D)−1XtY.;

t=1||bkt||2

2

.;

3. Calculate¯D=1

4. Update Btby Bt=(XtXT

2

We summarize the convergence of Algorithm 1 as following.

Theorem 1. Algorithm 1 monotonically decreases J2in Equation

(4) in each iteration, and converges to the globally optimal solution.

Proof: In Algorithm 1, in each iteration we denote the updated

Btas˜ Btand the updated L as ˜ L. From step 4 we know that:

T

?

T

?

˜ L+γ1

t=1

Tr(˜ BT

tD˜ Bt)+γ2

T

?

T

?

t=1

Tr(˜ BT

t¯D˜ Bt)≤

L+γ1

t=1

Tr(BT

tDBt)+γ2

t=1

Tr(BT

t¯DBt).

(7)

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H.Wang et al.

In each iteration, denote the updated B as ˜ B and the updated bkt

as˜bkt, according to the definitions of D and¯D, we can write

||?T

2

||?T

2

˜ L+γ1

2

d

?

d

?

k=1

t=1˜bkt||2

t=1||bkt||2

t=1bkt||2

??T

?

t=1

?

t=1

2

??T

+γ2

2Tr

?

˜ B˜ BT?

?

BBT?−1

2?

≤

L+γ1

2

k=1

2

t=1||bkt||2

+γ2

2Tr

BBT?

BBT?−1

2?

.

(8)

Following (Nie et al., 2010, 2012), it can be verified that

?

?

?

?

?

T

?

||˜bkt||2

2−

?T

t=1||˜bkt||2

??T

?T

2

2

t=1||bkt||2

t=1||bkt||2

??T

2−Tr˜ B˜ BT?

BBT?1

?

t=1

d

?

2

≤

?

T

?

||bkt||2

2−

2

2

t=1||bkt||2

2

.

(9)

Tr

?

?

˜ B˜ BT?1

BBT?−1

BBT?−1

2≤

Tr

2−TrBBT?

2.

(10)

Adding the both sides of Equations (8–10) together, we obtain

˜ L+γ1

d

?

k=1

?

?

?

?

?

?

T

?

T

?

||˜bkt||2

2+γ2Tr

?

˜ B˜ BT?1

2≤

L+γ1

k=1

?

t=1

||bkt||2

2+γ2Tr

?

BBT?1

2.

(11)

Thus, our algorithm decreases the objective value of Equation (4)

in each iteration. When the objective value keeps unchange,

Equation (5) is satisfied, i.e. the KKT condition of the objective is

satisfied.Ouralgorithmreachesoneoftheoptimalsolution.Because

our objective in Equation (4) is a convex problem, our Algorithm 1

will converge to one of the globally optimal solution.

Computational analysis. In the iteration loop of Algorithm 1,

Step 2 is computationally trivial. Step 3 solves a singular value

decomposition (SVD) problem, and Step 4 solves a system of linear

equations, both of which, thereby the whole algorithm, are well

studied in literature and can be solved very efficiently by existing

numerical packages.

2

4

In this section, we evaluate the proposed method by applying it to

theADNI cohort, where a wide range of imaging markers measured

overaperiodof2yearsareexaminedandassociatedtoSNPsthatare

relevant toAD. The goal is to discover a compact set of phenotypic

imaging markers that are closely related to AD-sensitive genotypes

encoded by SNPs.

EXPERIMENTAL RESULTS AND DISCUSSIONS

4.1 Improved prediction of SNPs from longitudinal

phenotypic imaging markers

We first evaluate the proposed method by applying it to the ADNI

cohort to predict the SNPs of the participants from each of their

Table 1. Numbers of participants in the experiments using two different

types of imaging markers

Imaging phenotypes # Total# AD # MCI# HC

VBM

FreeSurfer

424

474

86

100

194

216

144

158

two types of imaging phenotypes, i.e. VBM markers and FreeSurfer

markers, tracked over four different time points, including BL

and M06/M12/M24. Because some subjects of the ADNI cohort

do not have complete imaging marker measurements over all the

four time points, in our experiments we use the subject samples

that have both SNPs data and complete imaging measurements.

As a result, two subsets of ADNI subjects are included in our

experiments, one for each type of imaging phenotypes, as detailed in

Table 1.

We compare the proposed method against its three close

counterparts including multivariate linear regression (LR) method,

ridge regression (RR) method and least absolute shrinkage and

selection operator (Lasso) (Tibshirani, 1996) method. LR method

is the most broadly used association model in both statistical

learning and imaging genetics. RR method is the regularized

version of LR model to avoid over-fitting. Lasso method replaces

the squared ?2-norm regularization in RR method by the ?1-

norm regularization, from which sparse results can be achieved

(Tibshirani, 1996). Different to these compared methods, our

new association model imposes structured sparsity via the tensor

?2,1-norm regularization for phenotypic marker selection and the

trace-norm regularization for capturing the interrelationships among

different SNPs. We implement two versions of the proposed method

as follows. First, we implement our method by only imposing the

trace-norm regularization, denoted as ‘Ours (Trace-norm only)’,

which only makes use of the SNPs’ correlations, but does not

select longitudinal imaging markers. Second, we implement the

full version of the proposed method, denoted as ‘Ours’, which

solves the problem in Equation (4). For measuring the regression

performance of the five compared association models, we use a 5-

foldcross-validationstrategybycomputingthePearson’scorrelation

coefficient(CORR)andtherootmeansquareerror(RMSE)between

the predicted and the actual SNP values, which are reported in

Figure 1.

As can be seen from Figure 1, if we only use the baseline

data, the proposed method is reduced into a conventional multi-

task regression model, which appears as a matrix but not a tensor

and achieves only the slightly better performance than the RR and

Lasso methods. On the other hand, by using the longitudinal data,

the performance of the proposed method is significantly improved,

e.g. for predicting SNPs using the longitudinal data over all the four

time points, the proposed (BL∼M24) method achieves the CORR

of 0.793 and 0.812 and the RMSE of 0.314 and 0.301, respectively,

which are much better than the case of using only the baseline

data.

Inaddition,Figure1alsoshowsthattheusageoflongitudinaldata

can improve the performances of all the LR, RR and Lasso methods,

althoughtheimprovementsaremuchlessthantheproposedmethod.

These results demonstrate the effectiveness of using longitudinal

dataforimprovedregressionfromimagingphenotypestogenotypes,

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Association study of longitudinal phenotypic markers to AD relevant SNPs

BLBL~M06 BL~M12BL~M24

0.5

0.6

0.7

0.8

Correlation coefficient

Linear Regression

Ridge Regression

Lasso

Ours (Trace−norm only)

Ours

VBM

BLBL~M06BL~M12 BL~M24

0.5

1

1.5

2

RMSE

Linear Regression

Ridge Regression

Lasso

Ours (Trace−norm only)

Ours

VBM

BLBL~M06 BL~M12 BL~M24

0.5

0.6

0.7

0.8

Correlation coefficient

Linear Regression

Ridge Regression

Lasso

Ours (Trace−norm only)

Ours

FreeSurfer

BLBL~M06 BL~M12BL~M24

0.5

1

1.5

2

RMSE

Linear Regression

Ridge Regression

Lasso

Ours (Trace−norm only)

Ours

FreeSurfer

(a)

(b)

(c)

(d)

Fig. 1. Regression performance with respect to the use of different number

of longitudinal time points by three different methods

especiallybytheproposedmethod,whichhasthecapabilitytomake

use of the input data through longitudinal feature selection; and the

integration of the interrelatedness among the SNPs.

4.2

One primary goal of this study is to identify a subset of imaging

phenotypes that are highly correlated to certain SNPs to capture

important imaging genomic associations in AD research. Thus, we

examine the phenotypic imaging markers identified by the proposed

methods, which are relevant to the genotypes encoded by SNPs.

Identification of longitudinal imaging markers

4.2.1 Identified imaging markers with high AD risks

Figure 2 are the overall regression coefficients for all the VBM

and FreeSurfer measures with respect to the 1224 SNPs used in

this study. Because these SNPs areAlzGene candidates or proximal

to the candidates, the results in Figure 2 can help identify SNP-

relevant imaging phenotypes and have a potential to gain biological

insightsfromgenetobraintosymptoms.Besides,thetop10selected

VBM imaging features, as well as their association coefficients,

are visualized in Figure 3 by mapping them onto the human

brain.

A first glance at the association weigh maps shows that

the selected imaging markers have clear patterns that span

across all the four studied time points, which demonstrates

that these phenotypic markers are longitudinally stable thus can

serve as screening target over the course of AD progression.

We also observe that hippocampal measures (LHippocampus,

RHippocampus, LHippVol and RHippVol) are identified, which

is in accordance with the fact that in the pathological pathway

of AD, medial temporal lobe including hippocampus is firstly

affected, followed by progressive neocortical damage (Braak and

Braak, 1991; Delacourte et al., 1999). The thickness measures

of isthmus cingulate (LIsthmCing and RIsthmCing), frontal pole

(LFrontalPole and RFrontalPole) and posterior cingulate gyrus

(LPostCingulate and RPostCingulate) are also selected, which,

again, is accordance with the fact that the GM atrophy of these

regions is high inAD (Lehmann et al., 2010; McEvoy et al., 2009).

In summary, the identified longitudinally stable markers strongly

agree with the existing findings, which warrants the correctness

of the discovered phenotype–genotype associations, and reveals

Shown in

the complex relationships among MRI measures, genetic variations

and diagnosis status. This is of clear importance for theoretical

research and clinical practices for a better understanding of AD

mechanism.

4.2.2 Case studies: markers identified for rs423958-APOE and

rs11136000-CLU

We provide two case studies to show the top

10 FreeSurfer markers associated with two major AD risk SNPs:

rs423958-APOE and rs11136000-CLU. We explore the associations

between the FreeSurfer markers and the two SNPs in four different

subject groups induced from the ADNI data, i.e. the groups of All,

AD, MCI and HC participants, respectively. The number of the

subjects in each group is available in Table 1. We select the imaging

markers by their average regression coefficients over all the four

time points. The top 10 FreeSurfer markers relevant to rs423958-

APOE and their regression coefficients are shown in Figure 4 and

those relevant to rs11136000-CLU are shown in Figure 5. From

Figure 4 we can see that most of the top 10 FreeSurfer markers for

rs423958-APOE in the four different testing groups are well-known

AD-sensitive phenotypes, such as hippocampal volume in All, AD,

MCI and HC groups; amygdala volume in All, AD, MCI and HC

groups; accumbens volume in All and MCI groups and entorhinal

cortex thickness in AD and HC groups; Similar patterns are also

observed for rs11136000-CLU, as shown in Figure 5.Although data

are not shown due to space limit, our VBM analyses have also

yielded similar results. The complete imaging marker identification

results by our method for both VBM and FreeSurfer markers on

the top 10 identified SNPs are available at the author’s website

at http://ranger.uta.edu/%7eheng/imgsnp/. These results have again

demonstrated the promise of the proposed method in terms of its

capability to identify imaging markers relevant to AD-sensitive

SNPs.

5

Elucidating the associations between longitudinal phenotypic

imaging markers and AD sensitive SNPs is of important value

for both scientific research and clinical practice. In this article,

we presented a new task-correlated longitudinal sparse regression

method to identify longitudinal imaging markers to AD-relevant

SNPs. In our newly proposed regression model, we imposed a

tensor ?2,1-norm regularization extended from the standard matrix

?2,1-norm to capture the temporal patterns in the longitudinal data

over all the tasks of interest, and meanwhile imposed the trace-

norm regularization onto the unfolded coefficient tensor such that

the interrelatedness among the SNPs during the progression of

AD conversion is addressed. Due to the additional time dimension

of the input data and the non-smoothness of the tensor ?2,1-

norm and trace-norm, solving the formulated objective of our

new method was very challenging. Therefore, we presented an

efficient iterative algorithm and rigorously proved its convergence

to the global optimum. We applied the proposed method to

the ADNI cohort and evaluated it in both SNPs prediction and

longitudinal imaging marker identification. The clearly improved

regression performance in the prediction and highly suggestive

imaging markers selected by our new method have validated its

effectiveness.

CONCLUSIONS

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H.Wang et al.

VBM

LAmygdala

RAmygdala

LAngular

RAngular

LCalcarine

RCalcarine

LCaudate

RCaudate

LAntCingulate

RAntCingulate

LMidCingulate

RMidCingulate

LPostCingulate

RPostCingulate

LCuneus

RCuneus

LInfFrontal_Oper

RInfFrontal_Oper

LInfOrbFrontal

RInfOrbFrontal

LInfFrontal_Triang

RInfFrontal_Triang

LMedOrbFrontal

RMedOrbFrontal

LMidFrontal

RMidFrontal

LMidOrbFrontal

RMidOrbFrontal

LSupFrontal

RSupFrontal

LMedSupFrontal

RMedSupFrontal

LSupOrbFrontal

RSupOrbFrontal

LFusiform

RFusiform

LHeschl

RHeschl

LHippocampus

RHippocampus

LInsula

RInsula

LLingual

RLingual

LInfOccipital

RInfOccipital

LMidOccipital

RMidOccipital

LSupOccipital

RSupOccipital

LOlfactory

ROlfactory

LPallidum

RPallidum

LParahipp

RParahipp

LParacentral

RParacentral

LInfParietal

RInfParietal

LSupParietal

RSupParietal

LPostcentral

RPostcentral

LPrecentral

RPrecentral

LPrecuneus

RPrecuneus

LPutamen

RPutamen

LRectus

RRectus

LRolandic_Oper

RRolandic_Oper

LSuppMotorArea

RSuppMotorArea

LSupramarg

RSupramarg

LInfTemporal

RInfTemporal

LMidTemporal

RMidTemporal

LMidTempPole

RMidTempPole

LSupTempPole

RSupTempPole

LSupTemporal

RSupTemporal

LThalamus

RThalamus

BL

M6

M12

M24

1

2

3

4

5

FreeSurfer

LCerebWM

RCerebWM

LCerebCtx

RCerebCtx

LLatVent

RLatVent

LInfLatVent

RInfLatVent

LCerebellWM

RCerebellWM

LCerebellCtx

RCerebellCtx

LThalVol

RThalVol

LCaudVol

RCaudVol

LPutamVol

RPutamVol

LPallVol

RPallVOl

LHippVol

RHippVol

LAmygVol

RAmygVol

LAccumVol

RAccumVol

LBanksSTS

RBanksSTS

LCaudAntCing

RCaudAntCing

LCaudMidFrontal

RCaudMidFrontal

LCuneus

RCuneus

LEntCtx

REntCtx

LFusiform

RFusiform

LInfParietal

RInfParietal

LInfTemporal

RInfTemporal

LIsthmCing

RIsthmCing

LLatOccipital

RLatOccipital

LLatOrbFrontal

RLatOrbFrontal

LLingual

RLingual

LMedOrbFrontal

RMedOrbFrontal

LMidTemporal

RMidTemporal

LParahipp

RParahipp

LParacentral

RParacentral

LParsOper

RParsOper

LParsOrb

RParsOrb

LParsTriang

RParsTriang

LPostCent

RPostCent

LPostCing

RPostCing

LPrecent

RPrecent

LPrecuneus

RPrecuneus

LRostAntCing

RRostAntCing

LRostMidFrontal

RRostMidFrontal

LSupFrontal

RSupFrontal

LSupParietal

RSupParietal

LSupTemporal

RSupTemporal

LSupramarg

RSupramarg

LFrontalPole

RFrontalPole

LTemporalPole

RTemporalPole

LTransvTemporal

RTransvTemporal

BL

M6

M12

M24

5

10

15

20

Fig. 2. Weight maps of the association between imaging markers and the SNPs learned by the proposed method

Fig. 3. Visualization of top 10 VBM features selected by the proposed method at four different time points. The colors of the selected brain regions show the

regression coefficients of the corresponding VBM markers

(a)

(b)(c)(d)

Fig. 4. Top 10 FreeSurfer markers identified for rs423958-APOE.

(a)

(b)(c)

(d)

Fig. 5. Top 10 FreeSurfer markers identified for rs11136000-CLU

Funding: This research was supported by the National Science

Foundation Grant (NSF) CCF-0830780, CCF-0917274, DMS-

0915228 and IIS-1117965 at UTA; and by NSF IIS-1117335, NIH

UL1 RR025761, U01 AG024904, NIA RC2 AG036535, NIA R01

AG19771 and NIA P30 AG10133-18S1 at IU.

Data collection and sharing for this project was funded by

the ADNI (National Institutes of Health Grant U01 AG024904).

ADNI is funded by the National Institute on Aging, the National

Institute of Biomedical Imaging and Bioengineering, and through

generous contributions from the following: Abbott; Alzheimer’s

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Page 7

Association study of longitudinal phenotypic markers to AD relevant SNPs

Association; Alzheimer’s Drug Discovery Foundation; Amorfix

Life Sciences Ltd.; AstraZeneca; Bayer HealthCare; BioClinica,

Inc.; Biogen Idec Inc.; Bristol-Myers Squibb Company; Eisai

Inc.; Elan Pharmaceuticals Inc.; Eli Lilly and Company; F.

Hoffmann-La Roche Ltd and its affiliated company Genentech,

Inc.; GE Healthcare; Innogenetics, N.V.; Janssen Alzheimer

Immunotherapy Research & Development, LLC.; Johnson &

Johnson Pharmaceutical Research & Development LLC.; Medpace,

Inc.; Merck & Co., Inc.; Meso Scale Diagnostics, LLC.; Novartis

Pharmaceuticals Corporation; Pfizer Inc.; Servier; Synarc Inc.

and Takeda Pharmaceutical Company. The Canadian Institutes

of Health Research is providing funds to support ADNI clinical

sites in Canada. Private sector contributions are facilitated by the

FoundationfortheNationalInstitutesofHealth(www.fnih.org).The

granteeorganizationistheNorthernCaliforniaInstituteforResearch

and Education and the study is coordinated by the Alzheimer’s

Disease Cooperative Study at the University of California, San

Diego. ADNI data are disseminated by the Laboratory for Neuro

Imaging at the University of California, Los Angeles. This research

was also supported by NIH grants P30 AG010129, K01 AG030514

and the Dana Foundation.

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