Skoglund, Karl

Stegmann, Mikkel Bille

Corpus callosum analysis is influenced by many factors. The effort in controlling these has previously been incomplete and scattered. This paper sketches a complete pipeline for automated corpus callosum analysis from magnetic resonance images, with focus on measurement standardisation. The presented pipeline deals with i) estimation of the mid-sagittal plane, ii) localisation and registration of the corpus callosum, iii) parameterisation and representation of its contour, and iv) means of standardising the traditional reference area measurements

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On Automating and StandardisingCorpus Callosum Analysis in Brain MRIMikkel B. Stegmann and Karl SkoglundInformatics and Mathematical Modelling, Technical University of DenmarkRichard Petersens Plads, Building 321, DK-2800 Kgs. Lyngby, Denmarkmbs@imm.dtu.dk kas@imm.dtu.dkAbstractCorpus callosum analysis is influenced by manyfactors. The effort in controlling these has pre-viously been incomplete and scattered. This papersketches a complete pipeline for automated corpuscallosum analysis from magnetic resonance images,with focus on measurement standardisation. Thepresented pipeline deals with i) estimation of themid-sagittal plane, ii) localisation and registrationof the corpus callosum, iii) parameterisation andrepresentation of its contour, and iv) means ofstandardising the traditional reference area mea-surements.1 IntroductionBrain morphometry is an important tool for de-tecting and monitoring brain pathologies such asepilepsy, dementia and multiple sclerosis. A com-mon method is to delineate some well-defined areaof the brain to yield a shape for inter- or intra-subject studies. One such structure is the corpuscallosum (CC); a white-matter nervous tissue bun-dle bridging the left and right cerebral hemisphere.This structure is particularly interesting due to themany neurological studies indicating relationshipsbetween the size and shape of the CC, and gen-der, age, handedness, neurodegenerative diseaseset cetera (e.g. [9, 2, 7]). Common to such studiesis that measurements are performed on the two-dimensional cross-section of the CC defined by themid-sagittal plane (MSP), which separates the lefthemisphere from the right. Possibly the most com-mon type of CC measurement consists of partition-ing the CC body into a small set of reference partsand calculating their areas. Such descriptors cansubsequently be used in e.g. group studies betweensubjects with dementia and normal subjects.The literature typically deals with each of thesetopics separately: i) estimation of the mid-sagittalplane (e.g. [1, 8, 15]), ii) corpus callosum locali-sation (e.g. [9, 17, 13]), and iii) corpus callosumpartitioning (e.g. [16]). To bridge these, this pa-per sketches a complete pipeline for automatedcorpus callosum analysis from magnetic resonan-ce images (MRI) taking all of the above elementsinto consideration.2 Methods2.1 Mid-sagittal Plane EstimationThe human brain consists on a coarse level of thecerebrum, the cerebellum and the brainstem. Mostprominent is the cerebrum which is divided intohemispheres connected by a nervous fibre bundle;the corpus callosum. The surface partitioning thisapproximate bilateral symmetry of the cerebrumis typically denoted the mid-sagittal plane (MSP)referring to its relative alignment with the sagittalplane of the human body. Determining the MSPis thus an invariable prerequisite for measurementson structures defined via this plane such as theMSP cross-section of the corpus callosum.Traditionally MSP estimation is based on globalsymmetry which is heavily influenced by global(i.e. head) symmetry. To avoid this, we proposedan approach that optimises a local symmetry mea-sure using the Nelder-Mead simplex method [15];see Figure 1. The MSP is parameterized by az-imuth/elevation angles and the orthogonal distanceto the origin. The objective function is based onprofiles normal to the plane. In addition, [15] in-troduced the use of thin-plate spline fitting us-ing robust statistics for estimating a curved mid-sagittal surface (MSS), e.g. in pathological brains.2.2 Corpus Callosum LocalisationObtaining manual CC tracings is both time-consu-ming, error-prone and operator dependent. In-(a) (b) (c)(d) (e)Figure 1: Automatically estimated mid-sagittalplane (MSP). (a) Initial (above) and optimal (below)MSP. (b) Axial MSP intersection. (c) Coronal MSP in-tersection. (d) Axial and coronal slice with estimatedMSP. (e) Axial and coronal slice with MSP intersec-tion.stead, medical image analysis should aim at re-placing this task. In [13] we introduced the use ofthe class of generative models; Active AppearanceModels (AAMs) [4], for fully automated localisa-tion of the CC (see Figure 2). AAMs establish acompact parameterisation of object variability, aslearned from a representative training set. Objectsare defined by marking up each training exam-ple with points of correspondence, i.e. landmarks.Subsequently, AAMs can be registered rapidly tounseen images. Due to their general nature andability to deal with substantial variability, AAMshave found much use in medical applications [14].The shape models inherent to AAMs rely onlandmarks. Albeit the CC shows an apparent lackof such, papers employing shape models (e.g. us-ing Active Shape Models [5]) have not dealt withthis issue. Thus, we have based our AAMs in[13] on the concept of minimum description length(MDL) as introduced in [6] which led to compactand unique shape descriptors.2.3 Representation and AnalysisWhile the previous sections have dealt with obtain-ing the CC outlines, this will touch upon an outlinerepresentation suitable for extraction of high-levelfeatures. Secondly, extraction of the traditionalCC reference area measurements will be treated.Figure 2: Automated localisation of the corpus cal-losum (cropped). Full MSP search slice (lower-right).2.3.1 Fourier RepresentationClosed, simple curves devoid of strong local curva-ture, such as the CC outline, may be representedby a Fourier series expansion. A 2D curve can berepresented either as a single complex functionf(t) =k=∞∑k=−∞ckeikΩt, (1)or as separate real functions for each axis [12],[x(t)y(t)]=∞∑k=0[pk qkrk sk] [cos kΩtsin kΩt]. (2)Here, t is the arc-length parameter, Ω = 2pi/T isthe angular frequency and T is the period. Thecoefficients ck are calculated using Fourier’s theo-rem [3]: ck = 1T∫ T0e−ikΩtf(t)dt.Usually, shapes come as delineated outlines orannotated landmarks, in which case the input func-tion f(t) is discrete. The coefficients are thenfound using the discrete Fourier transform. TheN -point transform is ck = 1N∑N−1j=0 e−2piijk/Nfj .The Fourier representation has a number ofbenefits. Looking at Equation 2, an intuitive geo-metrical interpretation arises. k = 0 correspondsto a translation in the x and y directions and k = 1represents a general ellipse. Indices greater thanone correspond to other ellipses with increasingfrequency (all are T -periodic) that modulates themain ellipse. As the elliptic components are added,the resulting shape will approximate the originaldata with increasing accuracy (see Figure 3).Another benefit of the Fourier representation isthe analytical nature of the shape-generating equa-tions 1 and 2. It can be shown that a Fouriertransform corresponds to a rotation in RN of theN input landmarks. This means that if all Fouriercoefficients are used, no information is lost in thetransformation. However, to reduce dimensional-ity and regularise the original data points, usually80 100 120 140 1608090100110120Origin + 15 components80 100 120 140 1608090100110120Origin + one component80 100 120 140 1608090100110120Origin + two components80 100 120 140 1608090100110120Origin + five componentsFigure 3: Fourier components of the callosal outline.only the M first coefficients are used. Disregard-ing this approximation, many geometric propertiesand operations can be made in an exact fashion.It is e.g. simple to calculate the curvature at anyposition of the curve. Green’s theorem relates theboundary function to the properties of the wholeenclosed area. This can be used to calculate, forinstance, the area analytically.2.3.2 Partitioning SchemesSince no gross anatomical landmarks exist that de-limit anatomically or functionally distinct callosalregions, several partitioning schemes have been de-vised [16]. These can be roughly divided into threecategories: i) vertical partitioning (i.e. partitionsorthogonal to a line connecting the CC ”endpoints”;the rostrum and the splenium), ii) radial parti-tioning (radial sections emanating from a referencepoints ”between” the rostrum and the splenium),iii) curvilinear partitioning (partitions defined byarc-length along a curvilinear reference line).Due to the lack of landmarks, all approachesneed to be defined uniquely from the contour. Thefirst principal axis of a point-based contour repre-sentation, as shown in Figure 4, can provide suchdefinition for category i). The centroid of the point-based contour provides a reference point for cate-gory ii). For category iii) the midpoints of chordsbetween an arc-length sampling of the inner andouter part of the CC contour may be used [16].Unfortunately, all of the above schemes intro-duce a marked shape-dependent bias of the definedcallosal areas (typically five) [16]. Here we intro-duce two schemes designed to lessen this bias.One solution is to use the landmarks directlyfrom the AAM localisation due to their MDL prop-erties. Figure 5 shows five reference areas deter-Figure 4: Principal axes and one particular set ofcorpus callosum reference areas.Figure 5: Mean corpus callosum area changes forvarying mid-sagittal plane pertubations.mined using this definition, as a function of MSProtation around the 2nd principal axis of the CC.This figure also shows that accurate MSP estima-tion can remove potentially substantial randomeffects, which could be present using the typicalclinical approach of employing in-scanner subjectalignment only. This thereby increases the sen-sitivity of morphometric analyses and allow for areduction of the number of subjects in a given pop-ulation study.Another approach is to further develop cate-gory iii) by eliminating shape-bias using an alter-native definition of the curvilinear reference line.For this we employ the chordal axis transform [11]as a stable and convenient alternative to the me-dial axis transform. All branches of the CAT arepruned away until one axis remains, which sub-sequently parameterises the callosal regions. Anexample showing five uniform partitions of 25 cor-pora callosa is given in Figure 6.3 DiscussionThis paper has described the elements required toestablish a complete pipeline for automated corpusFigure 6: Corpus callosum contours and their chordal axes (left). Corresponding partitions (right).callosum analysis; from MRIs to descriptive fea-tures. Ultimately, such a framework could be im-plemented in-scanner to immediately provide therequired features, e.g. for large-scale data miningof callosal geometry and patient record of everyscanned subject in a radiology department. Eachsubpart of the pipeline aims at standardising thefinal descriptors irrespective of the natural biolog-ical variation.To the best of our knowledge no such systemcurrently exists. However, based on the modulessummarised here, our future work will aim at aunified system producing clinical results from thelarge-scale LADIS study [10] (approx. 600 sub-jects) in collaboration with the Danish ResearchCentre for Magnetic Resonance, DRCMR.AcknowledgmentsCharlotte Ryberg and Egill Rostrup, DRCMR, aregratefully acknowledged for providing MRIs. Allsubjects originated from a subset of the LADISstudy [10]. M. B. Stegmann was supported by TheDanish Medical Research Council, grant no. 52-00-0767, and The Danish Research Agency, grant no.2059-03-0032. K. Skoglund was supported by TheTechnical University of Denmark, DTU.References[1] B. A. Ardekani, J. Kershaw, M. Braun, and I. Kanuo.Automatic detection of the mid-sagittal plane in 3-Dbrain images. IEEE Transactions on Medical Imaging,16(6):947–52, 1997.[2] F. L. Bookstein. Landmark methods for forms withoutlandmarks: localizing group differences in outline shape.Medical Image Analysis, 1(3):225–244, 1997.[3] W. L. Briggs and Van E. Henson. The DFT: An Owner’sManual for the Discrete Fourier Transform. SIAM, 1995.[4] T. F. Cootes, G. J. Edwards, and C. J. Taylor. Activeappearance models. In Proc. of European Conf. on Com-puter Vision 1998, volume 1407 of Lecture Notes in Com-puter Science, pages 484–498. Springer, 1998.[5] T. F. Cootes, C. J. Taylor, D. H. Cooper, and J. Gra-ham. Active shape models – their training and application.Computer Vision and Image Understanding, 61(1):38–59, 1995.[6] R. H. Davies, C. J. Twining, T. F. Cootes, J. C. Waterton,and C. J. Taylor. 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On Automating and Standardising Corpus Callosum Analysis in Brain MRI

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On Automating and Standardising Corpus Callosum Analysis in Brain MRI

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