A bi-ventricular cardiac atlas built from 1000+ high resolution MR images ofhealthy subjects and an analysis of shape and motion

Wenjia Baia,∗, Wenzhe Shia, Antonio de Marvaob, Timothy J.W. Dawesb, Declan P. O’Reganb, Stuart A. Cookb,c, Daniel Rueckerta

aBiomedical Image Analysis Group, Department of Computing,Imperial College London, UK

bMRC Clinical Sciences Centre, Hammersmith Hospital,Imperial College London, UK

cNational Heart Centre Singapore, Singapore,Duke-NUS Graduate Medical School, Singapore

Abstract

Atlases encode valuable anatomical and functional information from a population. In this work, a bi-ventricular cardiac atlas wasbuilt from a unique data set, which consists of high resolution cardiac MR images of 1000+ normal subjects. Based on the atlas,statistical methods were used to study the variation of cardiac shapes and the distribution of cardiac motion across the spatio-temporal domain. We have shown how statistical parametric mapping (SPM) can be combined with a general linear model tostudy the impact of gender and age on regional myocardial wall thickness. Finally, we have also investigated the influence of thepopulation size on atlas construction and atlas-based analysis. The high resolution atlas, the statistical models and the SPM methodwill benefit more studies on cardiac anatomy and function analysis in the future.

Keywords: cardiac atlas, statistical shape model, statistical motion model, statistical parametric mapping

1. Introduction

Cardiovascular diseases (CVDs) are the leading cause ofdeath globally (Mendis et al., 2011). Consequently, a greateffort has been and is being dedicated to the prevention, di-agnosis, treatment and research of CVDs. Understanding car-diac anatomy and function is essential for determining how theheart adapts to disease. One approach to furthering our exist-ing knowledge is to observe in-vivo cardiac anatomy and func-tion via non-invasive medical imaging techniques and to de-velop computational and mathematical methods to analyse theinformation gathered from the images. In this paper, we builtan atlas of cardiac anatomy from high resolution MR imagesof 1000+ subjects, which enables the alignment of the observa-tions from multiple subjects into a common space so that quan-titative information can be analysed in the common space usingmathematical models.

1.1. Related work

In the past decade, many groups have created cardiac atlasesusing different cohorts. Table 1 gives a brief review of the pre-vious literature. The literature in the table is categorised ac-cording to the objects of interest. Most of the works focus onthe ventricles, whereas some of the works dedicate to all thefour chambers or even the whole heart (including all the fourchambers, aorta, pulmonary veins etc.).

∗Corresponding author.Email address: w.bai@imperial.ac.uk (Wenjia Bai)

In terms of the imaging modality, cine MR and CT are of-ten used to infer shape information (Frangi et al., 2002; Fon-seca et al., 2011; Hoogendoorn et al., 2013), whereas taggedMR, cine MR and US are often used to capture cardiac mo-tion (Chandrashekara et al., 2003; Suinesiaputra et al., 2009;Duchateau et al., 2011). In order to characterise the cardiac my-ocardial fibre architecture, diffusion tensor MRI (DT-MRI) isused for imaging ex vivo human hearts (Lombaert et al., 2012).

In terms of the size of the cohort, most studies include lessthan 100 subjects, whereas the Cardiac Atlas Project (CAP)uses over 3000 subjects, consisting of 2864 asymptomatic vol-unteers from the MESA cohort and 470 patients with myocar-dial infarction from the DETERMINE cohort (Fonseca et al.,2011). The study in (Medrano-Gracia et al., 2014) uses a sub-set of the CAP data set, which consists of 1991 asymptomaticvolunteers.

An atlas normally includes a template image, which is the av-erage intensity image from all the subjects after being aligned toa common space, or a template surface mesh, which describesthe average shape of the objects of interest. A probabilistic atlasmay also be included, which encodes the anatomical label in-formation in the template space. Each voxel of the probabilisticatlas stores a vector value, representing the local probability tobe a certain structure (e.g. the left ventricle, right ventricle ormyocardium) (Lorenzo-Valdes et al., 2004). Sometimes, a sim-ple label map is used, which is a manual segmentation of thetemplate image without any probability information (Zhuanget al., 2010).

Based on the atlas, the cardiac shape or function of a popula-

Preprint submitted to Medical Image Analysis August 26, 2015

Table 1: A brief review of previous literature on cardiac atlases. The literature is being categorised according to the objects of interest.

Literature Modality Subjects Atlas-based analysisLeft ventricleChandrashekara et al. (2003) Tagged MR 17 healthy Statistical motion modelRougon et al. (2004) Tagged MR 11 healthy Statistical motion modelSuinesiaputra et al. (2009) 2D cine MR 44 healthy Statistical contour modelFonseca et al. (2011) 2D cine MR 2864 healthy, 470 patients Statistical shape modelDuchateau et al. (2011) 2D US 21 healthy Statistical motion modelDuchateau et al. (2012) 2D US 71 healthy Statistical motion modelDe Craene et al. (2012) Tagged MR 15 healthy Statistical motion modelMedrano-Gracia et al. (2014) 2D cine MR 1991 healthy Statistical shape model

Both ventriclesFrangi et al. (2002) 2D cine MR 14 healthy Statistical shape modelLorenzo-Valdes et al. (2004) 2D cine MR 14 healthy -Perperidis et al. (2005) 2D cine MR 26 healthy Statistical shape modelLombaert et al. (2012) DT-MRI 10 healthy (ex vivo) Statistical tensor fieldThis work 3D cine MR 1093 healthy Statistical shape/motion models, SPM/GLM

Four chambersLotjonen et al. (2004) 2D cine MR 25 healthy Statistical shape model

Whole heartOrdas et al. (2007) CT 100 healthy/patients Statistical shape modelEcabert et al. (2008) CT 13 patients Statistical shape modelZhuang et al. (2010) Whole-heart MR 10 healthy -Hoogendoorn et al. (2013) CT 138 patients Statistical shape model

tion can be analysed and compared in a common space. The sta-tistical shape model is the most commonly used tool for shapeanalysis. The statistical shape model is essentially a point dis-tribution model (PDM), which describes the probability distri-bution of the vertices of a cardiac surface mesh, for example,the left ventricular endocardial mesh or epicardial mesh. It isoften derived using principal component analysis (PCA) andeach principal component represents a mode of variation ofthe cardiac shape (Perperidis et al., 2005). Apart from PCA,independent component analysis (ICA) has also been used toencode the variation of the cardiac shape (Suinesiaputra et al.,2009). Recently, Hoogendoorn et al. proposed a bilinear shapemodel in which the inter-subject variation and inter-phase vari-ation are encoded separately in two dimensions (Hoogendoornet al., 2009, 2013). To utilise pre-existing statistical shape mod-els, Pereanez et al. proposed a framework of merging multi-modality models by spatial normalisation and eigenspace fu-sion (Pereanez et al., 2014).

Cardiac motion can be encoded either implicitly or explic-itly. Implicit models describe the motion using the variationof the shape across a cardiac cycle (Perperidis et al., 2005;Hoogendoorn et al., 2013) or using the relative position ofthe end-systolic (ES) contour with respect to the end-diastolic(ED) contour (Suinesiaputra et al., 2009; Medrano-Gracia et al.,2012). An alternative way is to describe motion explicitly by es-

timating the displacement field (or the velocity field) via motiontracking for each subject, transporting the displacement field ofall the subjects to the common template space and computingthe statistics (Duchateau et al., 2011; De Craene et al., 2012).We name the explicit model as “statistical motion model” in Ta-ble 1. With the help of a statistical motion model, it becomespossible to detect abnormal motion patterns by using statisticaltesting and computing the statistical significance (Duchateauet al., 2011). To characterise cardiac motion with a compactdescriptor, singular value decomposition (SVD) is used to de-couple and reduce the spatio-temporal dimensionality (McLeodet al., 2013).

The cardiac atlas has many useful applications. The tem-plate image space provides a common reference space forpopulation-based shape analysis (Hoogendoorn et al., 2013).The template surface mesh can be used to generate a per-sonalised patient-specific mesh for biomechanical simulation(Lamata et al., 2011). The statistical shape model can be usedas prior knowledge to guide image segmentation (Heimann andMeinzer, 2009). The fibre orientation atlas has the potential tobe be adapted to a patient image to enable cardiac electrome-chanical modelling (Marchesseau et al., 2013). A review on theapplications of the computational cardiac atlas can be referredto at (Young and Frangi, 2009).

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1.2. Contributions

In this paper, we built a cardiac atlas from the high-resolutioncine MR images of 1093 healthy volunteers. The atlas includesa template image, a template surface mesh and a probabilisticatlas. Based on the atlas, we computed a statistical shape modeland a statistical motion model for cardiac shape and motionanalysis. We also analysed the regional myocardial wall thick-ness using the statistical parametric mapping method. Com-pared to the previous literature, the main contributions of thiswork include:

1. The atlas was built from 3D cine MR images of a largecohort of over 1000 healthy subjects, which were acquiredat Hammersmith Hospital, Imperial College London. Thisforms part of the UK Digital Heart project. As far as weknow, this is one of the largest data sets of healthy subjectsused for building a cardiac atlas. Although it is smallerthan the size of the CAP data set (Fonseca et al., 2011;Medrano-Gracia et al., 2014), the Hammersmith data setis different from it in three aspects:

(a) 2D cine MR (b) 3D cine MR

Figure 1: Improved spatial resolution by moving from 2D MR to 3D MR.

First, the Hammersmith data set was acquired using a3D cine balanced steady-state free precession (b-SSFP) se-quence and has a high resolution of 1.25 × 1.25 × 2 mm,whereas the CAP data set was acquired using a standard2D cine SSFP sequence and the spatial resolution of theimages varies from about 1.4 × 1.4 × 6 mm to 2.5 × 2.5 ×6 mm (Medrano-Gracia et al., 2014). The improved spa-tial resolution, especially the reduction of the slice thick-ness from 6 mm to 2 mm, enables us to characterise thecardiac shape in more detail. (de Marvao et al., 2014)has already demonstrated the improved spatial resolutionby moving from 2D imaging sequence to 3D imaging se-quence. Figure 1 visually demonstrates the improved spa-tial resolution. The high resolution enables more accuratedelineation of the ventricular morphology and forms thefoundation for us to study the subtle change of morpho-logical phenotypes, such as the myocardial wall thickness.In addition, 3D imaging only requires a single breath-hold.Therefore, it does not have the problem of inter-slice shift,which is caused by the inconsistency of the breath-hold po-sitions in 2D imaging (Medrano-Gracia et al., 2014; Suine-siaputra et al., 2014a).

Second, the Hammersmith data set was acquired at thesame hospital using the same scanner system and with thesame imaging protocol. Therefore, the image intensity

range and spatial resolution are consistent. On the con-trary, the CAP data set was contributed by multiple hos-pitals using different MR scanners (Fonseca et al., 2011).Therefore, the image intensity range may be varying.

Third, the ventricular contours for each subject in theHammersmith data set were delineated using a semi-automatic segmentation method, which requires only sixmanual landmarks and all the other steps are automatic(de Marvao et al., 2014). On the contrary, the contourswere manually drawn for the images in the CAP data set(Medrano-Gracia et al., 2014).

However, we have to note that CAP is not only a data setbut also a collaborative platform for researchers to shareresources and to validate their algorithms. It is one of thepioneering work in advancing cardiac data sharing and in-ferring health informatics from a big data set (Suinesia-putra et al., 2014b). The CAP data set has already suc-cessfully lead to fruitful research findings in cardiac im-age analysis and has also been used in many workshops,such as the STACOM workshop series and many MICCAIsegmentation and motion tracking challenges 1. The atlascreated in this paper can be a valuable complement to theprevious atlas works due to its unique properties such ashigh resolution and data consistency.

2. The second contribution and also a novelty of our work isthat we extend the statistical parametric mapping (SPM)method to the domain of cardiac image analysis. SPM iscommonly used used for brain image analysis (Ashburnerand Friston, 2000; Kiebel and Holmes, 2004; Friston et al.,1994) but seldom used on cardiac images. De Craene etal. made one of the first attempts to apply SPM to cardiacimages, using a Gaussian distribution model for motionabnormality detection (De Craene et al., 2012). However,the size of their data set is rather limited (15 healthy and2 patients). Here, we explore more about SPM on cardiacimage analysis and apply it to the Hammersmith data set,which consists of over 1000 subjects and therefore resultsin a higher statistical power. Using statistical testing andgeneralised linear model (GLM), we study the impact ofgender and age on regional myocardial wall thickness ofthe LV and present it on the template surface mesh.

3. The third contribution is that we investigate the impact ofthe population size (the number of subjects) on atlas con-struction and atlas-based analysis. Specifically, the im-pact on the template image, the statistical shape model andstatistical parametric mapping is evaluated and discussed.The results provide some useful insights into the questionof how many subjects are needed for atlas-based analysisof cardiac MR images.

The remainder of this paper is organised as follows. Section2 explains the methods used for building the cardiac atlas andanalysis based on the atlas: statistical shape model, statisticalmotion model and statistical parametric mapping. Section 3presents the results. Section 4 explains limitations. Finally,Section 5 discusses and concludes the work.

1http://www.cardiacatlas.org/web/guest/challenges

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2. Methods

In this section, we will first describe how we construct theatlas. Then, we will carry out analysis on the cardiac shape,motion and myocardial wall thickness based on the atlas.

Figure 2: Flow chart for atlas construction.

Table 2: Descriptive characteristics (mean and standard deviation) of the pop-ulation. EDV: LV end-diastolic volume; ESV: LV end-systolic volume; EF:ejection fraction; LVM: left ventricular mass.

Male (n=493) Female (n=600)Age (yr) 39.9 ± 12.2 40.3 ± 13.1

Height (cm) 177.0 ± 7.3 164.3 ± 6.5Weight (kg) 78.0 ± 11.5 65.3 ± 11.0EDV (mL) 178.2 ± 36.1 138.1 ± 23.9ESV (mL) 64.5 ± 14.6 47.8 ± 9.8

EF (%) 63.8 ± 3.8 65.4 ± 3.8LVM (g) 128.0 ± 24.0 96.0 ± 16.4

2.1. Atlas construction

Figure 2 summarises the flow chart for atlas construction andthe following subsections will explain each step in detail.

2.1.1. Data setThe collection of the single-centre data set was approved by

Hammersmith Hospital’s research ethic committee and all par-ticipants gave written informed consent. The data set used tocreate the atlas in this paper consists of 1093 subjects. Partic-ipants were excluded at screening if they had known cardio-vascular diseases or were being treated for hypertension, dia-betes or hypercholesterolemia. Female subjects were excludedif they were pregnant or breastfeeding but were eligible if theytook oral contraceptives. Standard published safety contraindi-cations to MR images were applied (Shellock, 2003). The maincharacteristics of the population are given in Table 2.

Cardiac MR (CMR) was performed on a 1.5T PhilipsAchieva system (Best, Netherlands). The maximum gradi-ent strength was 33 mT/m and the maximum slew rate 160mT/m/ms. A 32 element cardiac phased-array coil was usedfor signal reception. Scout images were obtained and used toplan a single breath-hold 3D cine balanced steady-state freeprecession (b-SSFP) images in the left ventricular short axis(LVSA) plane from base to apex using the following param-eters (de Marvao et al., 2014): repetition time msec/echo timemsec, 3.0/1.5; flip angle, 50◦; bandwidth, 1250 Hz/pixel; acqui-sition matrix, 160 × 128; native pixel size 2.0 × 2.0 mm; sectionthickness 2 mm overlapping; reconstructed voxel size, 1.25 ×1.25 × 2 mm; number of sections, 50 - 60; cardiac phases, 20;sensitivity encoding (SENSE) factor, 2.0 anterior-posterior and2.0 right-left direction. A typical breath-hold lasts 20-25 sec-onds depending on the heart rate and the number of sectionsacquired.

2.1.2. Template imageThe template image is an average intensity image after all

the subject images have been registered to a common refer-ence space or template space. We perform image registrationin two stages, namely affine registration (accounting for trans-lation, rotation, scaling and shearing) and non-rigid registra-tion (accounting for local deformation). The registrations wereperformed using the IRTK software package (Rueckert et al.,1999).

In the affine registration stage, we first randomly select asubject image as the reference image and perform affine imageregistration between this image and each of the subject image.However, the template image created in this way will be biasedtowards that initial reference image, because image registrationwill force all the subject images to transform and look like theinitial reference image, so will the resulting template image. Inorder to avoid the bias towards the initial reference, we com-pute the mean of all the affine transformations from the initialreference to the subject images. Based on the mean transfor-mation, we determine a new template space, transform all thesubject images to the new template space and compute the av-erage image in the new space. The template image built in thisway can be regarded as an image which represents the averageheart size, position and orientation and therefore avoids or re-duces the bias towards a specific subject (Frangi et al., 2002;Kuklisova-Murgasova et al., 2011; Rueckert et al., 2001).

Figure 3 illustrates the computation of the new templatespace based on the mean transformation. Let I1 be the initialreference image and name it as R(0), N denote the number ofsubject images and AR(0)→i (i = 1, 2, . . . ,N) denote the affinetransformation matrix from initial reference R(0) to image Ii,the average affine transformation matrix is computed as,

A = exp (N∑

i=1

log (AR(0)→i)) (1)

where the exponential and the logarithm of a matrix can becomputed using the Taylor expansion (Alexa, 2002). The mean

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transformation A is also called the geometric mean or the Log-Euclidean mean (Arsigny et al., 2007), since it is the Frechetmean associated with the Log-Euclidean metric.

(a) Initial reference I1

(b) New template space R(1)

Figure 3: Computation of the new template space R(1). AR(0)→i denotes theaffine transformation matrix from the initial reference image R(0) ≡ I1 to thesubject image Ii. The mean transformation A defines where the new templatespace R(1) is.

The mean affine transformation A defines where the new tem-plate space R(1) is with respect to the initial reference R(0). Sub-sequently, the transformation from the template space R(1) tothe subject image Ii can be computed as,

AR(1)→i = AR(0)→i ◦ A−1 (2)

where the symbol “◦” denotes the composition of two affinetransformations. For affine transformations, the compositionis the multiplication of two matrices. With the transformationAR(1)→i, we can transform the subject image Ii to the templatespace R(1) for all the subjects and compute the template image.

In the non-rigid registration stage, we perform B-spline non-rigid image registration between the current template image andeach of the subject images (Rueckert et al., 2001). The param-eters used for non-rigid image registration are: four resolutionlevels from coarse to fine; control point spacing at the finestlevel, 5×5×5 mm; similarity metric, normalised mutual infor-mation (NMI); optimisation method, gradient descent. The rea-son to use B-spline non-rigid registration is that it has demon-strated good performance in previous studies (Klein et al.,2009), but other registration tools can also be used in this step.

Similar to affine registration, we can update the template spaceby computing the mean non-rigid transformation,

T (1) =1N

N∑i=1

TR(1)→i (3)

where T (1) denotes the mean non-rigid transformation, TR(1)→idenotes the non-rigid transformation from the current templateR(1) to the subject image Ii. For non-rigid transformationswhich are modelled using B-splines, computing the mean trans-formation can be performed by computing the mean B-splinecoefficient at each control point. Subsequently, the transforma-tion between the new template R(2) and the subject image Ii canbe computed as,

TR(2)→i = TR(1)→i ◦ (T (1))−1 (4)

With the non-rigid transformation TR(2)→i, we can transform thesubject image Ii to the template space R(2) for all the subjectsand compute the template image.

The update of the reference space and the computation of thetemplate image can be performed iteratively as in (Guimondet al., 2000; Hoogendoorn et al., 2013),

TR(k)→i = TR(k−1)→i ◦ (T (k−1))−1 (5)

R(k) =1N

N∑i=1

Ii(TR(k)→i) (6)

Figure 4 shows the template image without registration, af-ter affine image registration and after iterative non-rigid imageregistrations. If we compare Figure 4(b) and (c), we will seethat non-rigid registration enhances the contrast of the image.In subfigure (b), the boundaries of the left and right ventricularmyocardium look blurry, whereas in subfigure (c), these bound-aries become sharper. The papillary muscle inside the left ven-tricle and the surface of the torso at the top-right corner of thefigure also become sharper. We have found that after two orthree iterations, the change to the template image becomes neg-ligible. In the followings, we will use the template built afterthree iterations as the final template.

2.1.3. Tissue class probabilistic atlasThe tissue class probabilistic atlas is computed by averag-

ing the segmentations of all the subject images after they arealigned to the template space. It would be extremely time-consuming to perform manual segmentation for over 1000 sub-jects. Multi-atlas segmentation has been demonstrated as a ro-bust and accurate segmentation method for brain and cardiacimages in the recent years (Asman and Landman, 2013; Baiet al., 2013; Wang et al., 2013). In this work, we use themulti-atlas PatchMatch algorithm for cardiac image segmenta-tion (de Marvao et al., 2014; Shi et al., 2013a).

Initially, the left ventricular cavity, the myocardium and theright ventricular cavity were manually segmented in 20 sub-jects at the ED and ES phase using the software itkSNAP(Yushkevich et al., 2006) by experienced clinicians in accor-dance with the SCMR recommendations (Schulz-Menger et al.,

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(a) Without registration (b) Affine (c) Non-rigid: Iteration 1 (d) Non-rigid: Iteration 2

Figure 4: The template image without registration, after affine registration and after one and two iterations of non-rigid image registration.

2013). Also, six landmarks were defined at the ED phase foreach subject, including one at the apex, one at the centre ofthe LV cavity on the basal slice and four on the mid-ventricularslice (two for the RV insertion points, one at the RV free walland one at the LV free wall). These images are used as priorknowledge or atlases to guide segmentation2. Given a new im-age (target image), six landmarks are defined on the image toinitialise landmark-based registration between the target imageand the 20 atlas images. Subsequently, non-rigid image reg-istration is performed between the target image and the atlasimages, following which the atlas segmentations can be warpedonto the target image space. The multi-atlas segmentation al-gorithm then goes across each image patch in the input image,looks for similar image patches in the atlas images and assignsthe labels of the atlas image patches onto the input image withconfidence weights, forming the segmentation result. Detailof the multi-atlas segmentation method can be referred to at(de Marvao et al., 2014; Shi et al., 2013a). Except from defin-ing the six landmarks, all the other steps are automatic for seg-mentation. In a validation study on a subset of 138 subjects, theDice overlap metric between the automated segmentation andthe manual segmentation reaches 0.952 for the left ventricularcavity (de Marvao et al., 2014). Figure 5 shows an example ofthe automated segmentation result. The automated segmenta-tion was then visually assessed and screened using the medicalimage viewing software rview which is part of the IRTK soft-ware package3 (Rueckert et al., 1999) by two clinicians withover two years cardiac MR experience. All subjects includedin the data set are images that have passed the manual qualitycontrol, which show good overlay between the segmentationboundary and the intensity image.

Similar to (Lorenzo-Valdes et al., 2004; Kuklisova-Murgasova et al., 2011), we transform the segmentations of thesubjects images to the template space using the non-rigid trans-formation established in the previous section and compute theprobabilistic atlas by averaging all the segmentations. Apartfrom the probabilistic atlas, we also create an atlas label map

2Note that in medical image analysis, generally the term “atlas” means animage averaged from a set of subject images, encoding the information fromthe population. However, for multi-atlas segmentation, an “atlas” means a pairof subject image and corresponding manual segmentation.

3https://www.doc.ic.ac.uk/~dr/software

(a) Segmentation at ED (b) Segmentation at ES

Figure 5: Automated segmentation of a subject image respectively at the EDand ES phase using the multi-atlas PatchMatch algorithm.

by picking up the label with the maximum probability at eachvoxel. An atlas surface mesh is then reconstructed from the la-bel map using the marching cubes algorithm in the VTK library(Kitware Inc.) and smoothed using a Laplacian smoothing fil-ter. Figure 6 displays the template image, maximum probabil-ity label map, surface mesh and tissue class probabilistic atlasat the ED phase.

2.2. Atlas based analysis

Once the atlas is available, we can transform the anatomicaland functional information of the population into the atlas space(the template space) and build statistical shape and motion mod-els of the heart. We can also study the impact of various factorson the cardiac phenotypes using the statistical parametric map-ping method. Here, we will look at two factors as an example,which are gender and age.

2.2.1. Statistical shape modelA statistical shape model describes the variation of the shape

(represented using a surface mesh) across subjects. Principalcomponent analysis (PCA) is often applied for shape analy-sis (Frangi et al., 2002; Lotjonen et al., 2004; Medrano-Graciaet al., 2014). PCA looks for a new coordinate system that ex-plains the input data so that the greatest variance of the data lieson the first coordinate (the first principal component or the first

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(a) Template at ED (b) Label map at ED (c) Surface mesh at ED (d) Volume rendering of prob. atlas

Figure 6: The template image, maximum probability label map, surface mesh and probabilistic atlas at the ED phase.

mode), the second variance of the data on the second coordi-nate and so on. It is accomplished by eigen-decomposition ofthe data covariance matrix.

The input to the PCA is the vertex coordinates of all the sub-ject surface meshes. We propagate the atlas surface mesh toeach subject shape using the deformation fields established inthe previous sections, so that all the subject meshes are repre-sented by the same number of vertices (about 30K vertices inour case for the left and right ventricles combined). Since thesubject meshes may have different positions and orientations,we transform them to the template space using rigid registra-tion to remove the position and orientation differences whileretaining the size and shape information for further analysis.

2.2.2. Statistical motion modelA statistical motion model describes the variation of the mo-

tion across subjects. To build a statistical motion model, firstof all, we perform motion tracking for each subject using thespatio-temporal B-spline image registration with a sparsenessregularisation term (TSFFD) (Shi et al., 2013b). The motionfield estimate is represented by a displacement vector at eachpoint and at each temporal frame t, which measures the dis-placement from the 0-th frame (the ED frame according to theimaging protocol) to the t-th frame. There are in total 20 tem-poral frames in the cine image sequence.

The next step is to transport the motion field of each subjectto the template space. Let x′ = T (x) denote the transformationfrom the template space to the subject space, where x and x′

are respectively the coordinates in the template space and in thesubject space. By considering the spatial transformation as achange of coordinates, we have,

d(x, t) = JT−1 (x′)d′(x′, t) (7)

where d′ denotes an infinitesimal displacement in the subjectspace, d denotes the corresponding infinitesimal displacementin the template space and JT−1 (x′) ≡ dx

dx′ denotes the Jacobianmatrix of the inverse transformation. The Jacobian matrix canalso be computed as,

JT−1 (x′) = [JT (x)]−1 (8)

which facilitates the computation if the inverse transformationis not provided by the image registration programme.

Since the displacement vector is usually non-negligible, wecompute the displacement vector D(x, t) using the following in-tegral,

D(x, t) =

∫du∈D′(x′,t)

[JT (x))]−1du (9)

where du denotes an infinitesimal vector along the displace-ment vector D′(x′, t). This can be computed using numericalintegration, as in (Rao et al., 2004).

There are also other approaches to transport a vector fieldfrom one space to another, such as the parallel transport tech-niques described in (Lorenzi and Pennec, 2013; Qiu et al., 2009;Younes, 2007; Younes et al., 2008). We use the change of co-ordinates method here due to its simplicity, which regards thethe transport of the vector field as a re-orientation process bythe Jacobian matrix. De Craene and Duchateau et al. appliedsimilar techniques in (De Craene et al., 2012; Duchateau et al.,2012).

Once the motion fields are transported to the template spacefor all the subjects, we compute the mean and the covariancematrix of the displacement vector for each point and each tem-poral frame in the template space,

D(x, t) =1N

∑i

Di(x, t) (10)

ΣD(x, t) =1

N − 1

∑i

(Di(x, t) − D(x, t)) · (Di(x, t) − D(x, t))T

(11)

We add the subscript i so that Di(x, t) denotes the displacementvector for subject i. The resulting mean displacement vectorD(x, t) across the template image forms a vector field, whereasthe covariance matrix ΣD(x, t) forms a tensor field. The statis-tics are computed for the displacements which are the directoutput of our motion tracking algorithm. Other measurementssuch as velocity and strain that characterise cardiac function ina different fashion could also be derived and used for statisticalanalysis.

2.3. Statistical parametric mappingStatistical parametric mapping (SPM) is an approach for im-

age analysis which analyses each voxel (or each vertex) usingstatistical test and presents the resulting statistical parameters

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(a) Thickness map on subject mesh (b) Thickness map on atlas mesh

Figure 7: Myocardial wall thickness map represented respectively on the sub-ject surface mesh and on the atlas surface mesh.

on an image or on a mesh. It is widely used in neuroimag-ing for analysis of brain anatomical or functional imaging data(Ashburner and Friston, 2000; Kiebel and Holmes, 2004; Fris-ton et al., 1994). SPM requires a template space in which thestatistical test is performed and the parametric map is presented.In this work, based on the cardiac atlas we have built, we applySPM to cardiac images for analysis of regional myocardial wallthickness.

For each subject, the left ventricular myocardial mesh is re-constructed from the segmentation. For each vertex on the epi-cardium, its minimum distance to the endocardium is measuredas the myocardial wall thickness using the FindClosestPointfunction in the VTK library (Kitware, Inc.). The wall thick-ness is represented on the epicardium of the subject surfacemesh, as illustrated in Figure 7(a). Using the transformationbetween the template and the subject image, the wall thick-ness map in the subject space is transformed to the templatespace and represented on the atlas surface mesh, as illustratedin Figure 7(b). We transform the wall thickness map to the at-las surface mesh for all the subjects so that statistical test can beperformed vertex-wise on the atlas surface mesh. We will thenanalyse the impact of gender and age on regional wall thicknessusing SPM.

A general linear model (GLM) is formulated as follows,

Y = Xβ + ε (12)

where Y denotes the observation, X denotes the design matrix,β denotes the parameters and ε denotes the error term.

Y is a column vector of size N × 1. In our case, the i-thelement Yi denotes the wall thickness value for the i-th subjectat a specific vertex of the surface mesh. Currently, we onlyaccount for gender and age factors in the model and the i-th rowof the design matrix is Xi = [Xi,M , Xi,F , Xi,age]. Xi,M indicatesthe membership to the male group. It has the value 1 if subjecti is male and 0 otherwise. Xi,F indicates the membership tothe female group. It has the value 1 if subject i is female and 0otherwise. Xi,age is the subject age. The matrix X is of size N×3.The parameter vector β is defined as β = [βM , βF , βage]T , whichmodels the impact of gender and age on the observation. Theerror term for each subject is assumed to be an independentlyand identically distributed normal random variable, i.e. εi

i.i.d.∼

N(0, σ2).

If (XT X) is invertible, the least squares estimate to the GLMin Eq. (12) is given by,

β = (XT X)−1XT Y (13)

otherwise, a pseudoinverse (XT X)− is used in place of (XT X)−1

in the equation. The pseudoinverse can be computed using thesingular value decomposition (SVD) of the original matrix X(Golub and Van Loan, 2013).

The parameter estimate is normally distributed: β ∼

N(β, σ2(XT X)−1). It follows that for a column vector c, thelinear compound cT β is also normally distributed: cT β ∼N(cTβ, σ2cT (XT X)−1c). The vector c is also named “contrast”in GLM analysis. The null hypothesis in SPM is that cTβ ≤ 0,whereas the alternative hypothesis is that cTβ > 0. The hypoth-esis can be assessed using,

cT β√σ2cT (XT X)−1c

∼ tN−r (14)

where tN−r is a one-sided Student’s t-distribution with N − r de-grees of freedom and r = rank(X) (Kiebel and Holmes, 2004).Since we need to perform hypothesis tests for all the vertices,we apply the false discovery rate (FDR) method to correct formultiple comparisons (Benjamini and Hochberg, 1995; Ben-jamini and Yekutieli, 2001; Genovese et al., 2002). The p-values of all the vertices are sorted in ascending order and thencorrected using the following equation,

q = p ·Ki

(15)

where p denotes the original p-value for t-test, K denotes thenumber of multiple comparisons (in this case, the number ofvertices of the LV epicardium, around 13,000), i denotes thevertex index after sorting and finally q denotes the correctedp-value.

In our experiment, we use contrasts c1 = [1,−1, 0]T andc2 = [0, 0, 1]T respectively to evaluate the impact of gender andage. The alternative hypothesis for contrast c1 is that the malehas thicker regional wall than the female. The alternative hy-pothesis for contrast c2 is that the old has thicker regional wallthan the young.

3. Results

3.1. Validation of RegistrationA good atlas depends on accurate atlas-to-subject registra-

tion. To validate the registration process, we evaluate the reg-istration accuracy using two metrics, namely the Dice overlapmetric and the mean surface distance. Since automated seg-mentations are available for both atlas and subject, we warpthe atlas segmentation to the subject space using the deforma-tion field resulting from image registration. The Dice overlapmetric between the warped atlas segmentation and the subjectsegmentation is measured. The mean surface distance betweenthe warped atlas mesh and the subject mesh is also measured.Table 3 lists the Dice metric and the mean surface distance. It

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shows that the Dice metric is over 0.9 for the LV/RV cavity andover 0.82 for the myocardium, which demonstrates good over-lap after registration. The mean surface distance is smaller than2 mm, which is similar to or lower than the mean distance errorreported in (Hoogendoorn et al., 2013) on a CT data set.

The constructed atlas is already presented in Figure 6. Inthe following, we will further present the results of atlas-basedanalysis.

Table 3: Evaluation of the registration accuracy.

Dice metricMean surfacedistance (mm)

LV cavity 0.950 ± 0.024 1.31 ± 0.55Myocardium 0.824 ± 0.062 1.38 ± 0.42

RV cavity 0.909 ± 0.031 1.94 ± 0.67

Figure 8: The percentage of variance explained by each mode of PCA for theLV.

3.2. Statistical shape model

PCA is used in our statistical shape model. By inspectingthe eigenvalues of each mode, as shown in Figure 8, we foundthat the first eight modes can explain over 90% of the shapevariation for the LV. The percentage of shape variation encodedin each mode is computed using its eigenvalue divided by thesum of all the eigenvalues.

Figure 9 displays the first three modes of shape variation forthe LV at the ED phase. The first mode accounts for 44% ofthe shape variation and it is mainly associated with the heartsize. The second mode accounts for 17% of the shape variance,which is visually associated with the elongation and oblique-ness of the left ventricle. The third mode accounts for 10%of the shape variation, which is associated with the elongationand roundness. These modes are quite similar to the findings in(Hoogendoorn et al., 2013; Medrano-Gracia et al., 2014), thatthe shape variation mainly includes heart size, elongation androundness.

The first three principal modes of shape variation for the RVare shown in Figure 10. Similar to the LV modes, the firstmode of RV is mainly associated with the size. The secondand the third modes are mainly associated with the elongationand widening of the RV.

Figure 9: The first three modes of shape variation for the LV. For each mode, themean shape and the shape at the +/-3 standard deviation (S.D.) are displayed.The heart at the top-left corner illustrates the orientation (blue: RV, red: LV).

3.3. Statistical motion model

The statistical motion model analyses the mean and the vari-ance of the cardiac motion across the spatio-temporal domain.Figure 11(a) displays the mean displacement field at the 7-th temporal frame for both the LV and RV. The LV displace-ment field clearly shows that during the contraction, the mid-ventricular wall is moving inwards while the basal wall is mov-ing both inwards and downwards. The RV displacement fieldshows similar movement which is inwards and downwards.Figure 11(b) displays the covariance of the displacements, rep-resented by ellipsoids whose length and orientation are deter-mined by the eigenvalues and eigenvectors of the covariancematrices. It shows that the motion at the basal lateral walls ofboth the LV and RV is associated with a higher variability thanthe other parts of the myocardium. On the contrary, the motionat the mid ventricle and at the septum is more certain.

The large variance of the basal lateral wall may come fromseveral sources. First, this is part of the free wall and hence themyocardial motion is less constrained compared to the septum.Second, the basal lateral wall undergoes both inwards contrac-tion and downwards shortening. Its motion is more complexcompared to the mid-ventricular wall and the apex. Third, thebasal lateral wall is relatively thin. The potential subject-to-template registration error may be higher, which also bringssome uncertainty to the statistics.

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Figure 10: The first three modes of shape variation for the RV.

3.4. Statistical parametric mapping

The statistical parametric mapping method performs hypoth-esis tests at each vertex of the surface mesh. Figure 12 displaysthe p-value map of the impact of gender and age on regionalmyocardial wall thickness. The p-values are adjusted by theFDR correction. The smaller the p-value is, the more signifi-cant the impact. Figure 12(a) shows that the gender impact issignificant for most region of the myocardial wall: the male hasa thicker myocardial wall than the female. Figure 12(b) showsthat the age impact is mainly significant at the septum, whichmeans ageing may result in a thicker wall at this region.

We also perform segment-wise comparison of wall thicknessusing the AHA 17-segment model (Cerqueira et al., 2002) andthe statistics are reported in Table 4. It also shows that on aver-age, the males have a thicker wall than the females for all of the17 segments (t-test with p < 0.001 for all segments), which isin line with the SPM findings. Compared to segment-wise com-parison, an advantage of SPM is that it allows detailed vertex-wise characterisation of the impact as shown in Figure 12. Also,the general linear model (GLM) can incorporate multiple fac-tors (either categorical or continuous variables) into the samemodel.

3.5. Impact of Population Size

Two interesting questions are namely (1) how many subjectsare needed for constructing a representative atlas and (2) howthe number of subjects affects the statistics in atlas-based anal-ysis. In the following experiments, we evaluate the impact

(a) Mean displacement field at the 7-th frame

(b) Covariance field at the 7-th frame

Figure 11: Statistical motion model overlaid on the template image and the sur-face mesh. The displacement vectors are represented using arrows. Examples:(1) inwards/downwards movement of the basal region; (2) inwards movementof the mid-ventricular region. The covariance matrices are represented usingellipsoids. Examples: (3) a large ellipsoid at the basal region denoting large co-variance and its orientation denoting the principal direction of the covariance;(4) small covariance at the mid-ventricular region.

of population size (the number of subjects) on the atlas tem-plate image, the statistical shape model and statistical paramet-ric mapping.

An atlas template image is built by averaging 10, 100, 200and 1093 subjects respectively. The top row of Figure 13 showsthe template image when the population size increases. Whenonly 10 subjects are used, Figure 13(a) shows that the templateimage is relatively noisy. When 100 subjects or more are used,the template image becomes much smoother. The bottom rowof the figure shows the difference between each template im-age built using a subpopulation (10, 100 or 200 subjects) andthe one built using the whole population (1093 subjects). Thisshows that when 200 subjects are included, the template imagealready looks very close to the one built using 1093 subjects.

To evaluate the representative power of the statistical shapemodel, we compute the surface reconstruction error. For eachsubject mesh, the distance between the original mesh and thereconstructed mesh using the PCA modes of the shape model iscomputed as,

e =1K

K∑k=1

||Pk − P(recon)k ||2 (16)

where K denotes the number of vertices, Pk denotes the k-thpoint of the original mesh and P(recon)

k denotes the k-th point of

10

(a) Impact of gender

(b) Impact of age

Figure 12: The p-value maps of the impact of gender and age on regional my-ocardial wall thickness and the significance maps. The colour is scaled by thelogarithm of the FDR corrected p-values. The smaller the p-value, the moresignificant the impact is.

the reconstructed mesh. The reconstructed mesh is essentiallythe projection of the original mesh into a subspace spanned bythe PCA modes. We compute PCA modes for the LV or the RVusing 100, 200, 400, 600, 800 and 1093 subjects respectivelyand for all the three cases, we only retain the first 100 modesfor shape reconstruction. The mean reconstruction error overthe 1093 subjects is computed and plotted in Figure 14.

The rationale behind this experiment is that ideally we wouldlike to find a shape model which is representative of all theLV or RV shapes. In practice, however, we are only able touse a subset of the human population to build the shape model.This experiment evaluates how accurate this shape representa-tion can be. Figure 14 shows that using the PCA modes ex-tracted from 100 subjects, the surface reconstruction error isless than 0.3 mm, which is smaller than the voxel spacing. Itdemonstrates that a shape model built from 100 subjects maybe sufficient to represent a much broader population. The fig-ure also shows that when more subjects are included into theshape analysis, we can further reduce the reconstruction errorwithout increasing the number of PCA modes, although the re-duction becomes less pronounced after 400 or 600 subjects areincluded. However, when more subjects are used, we can alsoextract more modes (e.g. 200 modes or more) to reduce thereconstruction error.

Finally, we demonstrate the impact of the population size bytesting the hypothesis that age has an impact on regional wallthickness. Figure 15 shows the FDR corrected p-value mapwhen different population sizes are used. When only 10 sub-jects are used, the statistical power is relatively weak. Thelarger the population, the higher the statistical power. When

Table 4: Comparison of segmental wall thickness (mean and standard devi-ation) between males and females (unit: mm). The segment ID follows thestandard in (Cerqueira et al., 2002).

Segment ID Male Female

Basal

1 6.21 ± 0.99 5.64 ± 0.912 6.38 ± 1.04 5.59 ± 0.903 6.23 ± 0.75 5.55 ± 0.604 5.82 ± 0.84 5.50 ± 0.695 5.38 ± 0.84 5.12 ± 0.686 6.66 ± 0.92 6.08 ± 0.83

Mid

7 6.40 ± 0.90 5.62 ± 0.678 7.06 ± 1.07 6.18 ± 0.969 8.41 ± 1.24 7.03 ± 1.05

10 6.64 ± 0.96 5.94 ± 0.7911 5.94 ± 0.76 5.35 ± 0.6112 6.92 ± 0.93 6.19 ± 0.76

Apical

13 5.47 ± 0.71 4.72 ± 0.5114 6.19 ± 0.83 5.47 ± 0.7415 5.40 ± 0.68 4.83 ± 0.5916 5.87 ± 0.83 5.17 ± 0.61

Apex 17 4.37 ± 0.64 4.00 ± 0.54

200 subjects are used, we can see clearly that age has a differ-ent impact on the septum from on the free wall. This becomesmore salient when 400 subjects are used. After 400 subjects,the change of the p-value map becomes less pronounced.

In hypothesis testing, the required population size also de-pends the distribution of the observed variable. For example,let x1 denote the wall thickness at the septum, x2 denote that atthe free wall and let us assume that both variables follow a nor-mal distribution. To judge whether x1 and x2 are significantlydifferent, one needs to account for their variances. If their vari-ances are low, then a small population size may be sufficientto test the hypothesis. If their variances are high, then moresamples are needed. In our case, statistical parametric mappinginvolves multiple hypothesis tests, which makes the populationsize question even more complex. (Friston et al., 1999) and(Hayasaka et al., 2007) explored this question for brain imagestudies and suggested that random models and statistical poweranalysis can be used to determine the population size.

To summarise, for constructing a good template image, 100subjects would be sufficient but this also depends on the imagequality and the characteristics of the population under study.For constructing a representative shape model, a larger popu-lation results in a better shape model with a higher representa-tive power. For atlas-based hypothesis test, a larger populationresults in a higher statistical power so that subtle factors canbe revealed. The drawback for a larger population is that this

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(a) 10 subjects (b) 100 subjects (c) 200 subjects (d) 1093 subjects

(e) 10 subjects (f) 100 subjects (g) 200 subjects

Figure 13: The top row shows the template image when the population size increases. The bottom shows the difference map between each template image builtusing a subpopulation (10, 100 or 200 subjects) and the one built using the whole population (1093 subjects).

comes with higher image acquisition cost and therefore, a bal-ance needs to be struck.

4. Limitations

The 3D cine MR data set used in this work provides highspatial resolution images that are consistent in 3D and free fromartefacts due to respiration. However, it also comes with somelimitations. The 3D cine image acquisition typically requiresthe subjects to hold their breath for 20-25 seconds depending onthe heart rate and the number of sections acquired (de Marvaoet al., 2014). This can be challenging for patients with seriouscardiovascular diseases to perform such a scan. This is why ourdata set only consists of normal subjects. In contrast, the 2Dcine image acquisition requires 5 or 6 breath holds, each lastingonly 12-15 seconds, which results in potential inter-slice shift.

In addition, the temporal resolution of the 3D cine MR is only20 frames per cardiac cycle. This is relatively low comparedto 2D cine MR in which normally 30-50 frames are acquiredper cardiac cycle. As a result, the statistical motion model thatwe have constructed is of limited temporal resolution. If thepurpose is to detect very fast abnormality motion patterns suchas septal flash, a 2D MR or ultrasound-based motion atlas maybe more appropriate (Duchateau et al., 2012).

5. Discussion and Conclusion

In this paper, we constructed the atlas from the MR imagesof 1000+ subjects. It has to be noted that all the subjects are

healthy volunteers. Therefore, the atlas and the statistical mod-els encode the information of the healthy subjects and representthe normal anatomy and motion. However, this could providea starting point for studying the abnormal anatomy and motion.For example, the image or the surface mesh of a patient canbe aligned to the template space and compared with the nor-mals. Also, the motion of the patient can also be aligned to thetemplate space, so that abnormal motion patterns may be de-tected using the statistical motion model of the normals, suchas the work in (Duchateau et al., 2011). Detection of abnor-mal myocardial wall thickness or abnormal motion pattern isan interesting direction for future research.

As discussed in Section 3.5, the increased population size re-sults in a higher statistical power, which is useful for clinicalstudies, for example, for case control studies in epidemiologi-cal studies. From a technical perspective, a challenge that weface in processing a large data set is the increased amount ofmanual intervention. In this work, manual landmarks are usedto initialise the image registration. Future work is needed toreplace this part with robust and automated image registrationvia landmark detection or organ localisation. Also, manual as-sessment is involved to assess the image segmentation quality,which is important for subsequent statistical analysis since themyocardial wall thickness is computed from the segmentation.Again, an automated quality control method is needed if this orsimilar approaches are to be deployed for even larger data sets,

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(a) LV reconstruction

(b) RV reconstruction

Figure 14: The mean surface reconstruction error using the first 100 PCAmodes extracted from 100, 200, 400, 600, 800 and 1093 subjects respectively.

for example, for projects such as the UK Biobank4, which isplanning to scan up to 100,000 subjects.

In this work, image registration is performed prior to esti-mating the template image. An alternative way for atlas con-struction is to combine image registration and template esti-mation together in a Bayesian framework (Joshi et al., 2004;Ma et al., 2008; Van Leemput, 2009; Zhang et al., 2013). TheBayesian approach iteratively performs group-wise image reg-istration and atlas update. However, group-wise image regis-tration can be computationally expensive for a large data set.In this work, we perform subject-wise image registration andestimate the atlas afterwards.

For building an unbiased atlas, we compute the mean of thenon-rigid transformations using Eq. (3), which is performed bycomputing the mean B-spline coefficient at each control point.This computes the Euclidean mean of the displacement fields.When large displacements are present, however, it would bemore appropriate to compute the mean using the log-Euclideanframework, i.e. the mean of the velocity fields, as suggestedby (Arsigny et al., 2007; Ashburner, 2007; Joshi et al., 2004).Also, registration methods designed for very large deformationsmay be considered (Lombaert et al., 2014). In our data set, allthe images are from healthy subjects. We have found that afterremoving the affine components, the residual non-rigid trans-formations between healthy subjects are relatively small. In thisscenario, computing the Euclidean mean can be a good approx-imation.

4http://imaging.ukbiobank.ac.uk

There are different ways of pre-processing prior to statisti-cal analysis of the surface meshes. For example, in (Frangiet al., 2002), the position, orientation and size differences of thesubject surface meshes are removed. The remaining informa-tion for analysis is local deformations of the surface meshes.(Medrano-Gracia et al., 2014) removes the position, orientationdifferences and only part of the size difference, that is size dueto the subject height. The remaining information for analysiscontains the residual size difference and local deformations. Inthis work, we remove the position and orientation differencesbut retain the size and local deformations for shape analysis. Inthe analysis, we observed very similar modes of shape varia-tions to the results in (Medrano-Gracia et al., 2014).

SPM has many useful applications depending on the designof the experiment. In this work, we use it to study the impact ofgender and age on the myocardial wall thickness. The resultsare in line with clinical findings (Kitzman et al., 1988; Nikitinet al., 2006; Salton et al., 2002). However, we can also applySPM to exploring the unknown covariates, for example, the im-pact of certain genes on cardiac morphology. Since the impactof genetic factors may be very subtle, we need to remove theimpact of the other covariates, such as gender, age, ethnicity,alcohol consumption etc. in order to reveal the influence of thegenetic factors.

It is also common to use global indices of cardiac functionsuch as ventricular volume, myocardial mass or ejection frac-tion to study the influence of genetic or epi-genetic factors.Compared to the analysis of global indices, an advantage ofSPM is that it enables local and detailed evaluation of the im-pact by visualising the distribution across the surface mesh. Thetwo methods are complementary to each other in clinical stud-ies.

In the SPM framework, we use the general linear model(GLM). The limitation of the linear model is that the real in-fluence may not be linear and therefore the model is only an ap-proximation to the real influence. However, the advantages ofthe linear model are that it is simple, able to incorporate multi-ple factors and easy to derive their probability distributions. It isstill one of the most widely used models for statistical analysisof brain images (Kiebel and Holmes, 2004) and for statisticallearning in data mining and inference (Hastie et al., 2009).

To conclude, we have constructed an atlas of the normal car-diac anatomy using high resolution MR images of 1000+ nor-mal subjects. Based on the atlas, we presented the analysis re-sults for cardiac shape, motion and regional wall thickness us-ing statistical methods. We will make the atlas publicly avail-able so that it could benefit future research in the community.We believe that the high resolution atlas, the statistical mod-els and the SPM method will benefit more studies on cardiacanatomy and function analysis in the future.

6. Acknowledgements

The study was supported by the Engineering and Physi-cal Sciences Research Council (EPSRC), UK (EP/K030523/1);the Medical Research Council (MRC), UK; the National In-stitute for Health Research (NIHR) Biomedical Research Cen-

13

(a) 10 subjects (b) 100 subjects (c) 200 subjects (d) 400 subjects

(e) 600 subjects (f) 800 subjects (g) 1093 subjects

Figure 15: The FDR corrected p-value map showing the impact of age on regional wall thickness, when the population size increases.

tre based at Imperial College Healthcare NHS Trust and Impe-rial College London; a British Heart Foundation project grant(PG/12/27/29489) and special grant (SP/10/10/28431); a Well-come Trust-GSK fellowship grant. We would like to thank thestaff at the Hammersmith Hospital and the anonymous volun-teers for performing the MR scans, Dr Andrew King and DrDevis Peressutti for fruitful discussion.

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