Medical Image Analysis 18 (2014) 635–646

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Medical Image Analysis

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

Personalized assessment of craniosynostosis via statistical shapemodeling

http://dx.doi.org/10.1016/j.media.2014.02.0081361-8415/� 2014 Elsevier B.V. All rights reserved.

⇑ Corresponding author. Address: Sheikh Zayed Institute for Pediatric SurgicalInnovation, Children’s National Medical Center, 111 Michigan Avenue NW,Washington, DC, USA.

E-mail address: carlos.sanchez.mendoza@gmail.com (C.S. Mendoza).

Carlos S. Mendoza a,b,⇑, Nabile Safdar a, Kazunori Okada c, Emmarie Myers a, Gary F. Rogers d,Marius George Linguraru a,e

a Sheikh Zayed Institute for Pediatric Surgical Innovation, Children’s National Medical Center, Washington, DC, USAb Signal Processing Department, University of Sevilla, Sevilla, Spainc Computer Science Department, San Francisco State University, San Francisco, CA, USAd Division of Plastic and Reconstructive Surgery, Children’s National Medical Center, Washington, DC, USAe Departments of Radiology and Pediatrics, School of Medicine and Health Sciences, George Washington University, Washington, DC, USA

a r t i c l e i n f o

Article history:Received 20 January 2013Received in revised form 11 February 2014Accepted 22 February 2014Available online 12 March 2014

Keywords:CraniosynostosisComputational anatomyShape analysisComputer-assisted diagnosisGraph-cut segmentation

a b s t r a c t

We present a technique for the computational analysis of craniosynostosis from CT images. Our fullyautomatic methodology uses a statistical shape model to produce diagnostic features tailored to the anat-omy of the subject. We propose a computational anatomy approach for measuring shape abnormality interms of the closest case from a multi-atlas of normal cases. Although other authors have tackled malfor-mation characterization for craniosynostosis in the past, our approach involves several novel contribu-tions (automatic labeling of cranial regions via graph cuts, identification of the closest morphology to asubject using a multi-atlas of normal anatomy, detection of suture fusion, registration using maskedregions and diagnosis via classification using quantitative measures of local shape and malformation).Using our automatic technique we obtained for each subject an index of cranial suture fusion, anddeformation and curvature discrepancy averages across five cranial bones and six suture regions.Significant differences between normal and craniosynostotic cases were obtained using these character-istics. Machine learning achieved a 92.7% sensitivity and 98.9% specificity for diagnosing craniosynostosisautomatically, values comparable to those achieved by trained radiologists. The probability of correctlyclassifying a new subject is 95.7%.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

1.1. Craniosynostosis

Craniosynostosis is a congenital condition characterized by pre-mature fusion of the cranial sutures; the incidence is 1 in 2100–2500 live births (Lajeunie et al., 1995). It is usually detected earlyin life, both due to its cosmetic manifestations and functionalconsequences, as it can result in limited brain growth, elevated in-tra-cranial pressure, and respiratory and visual impairment. Earlydiagnosis is crucial for management, prevention of complications,and consideration for early surgical correction (Kirmi et al., 2009;Panchal and Uttchin, 2003).

Although some authors have claimed the risk of submittinginfants to ionizing radiation (Raschle et al., 2012), three-dimensional reconstruction from computed tomography (CT) isthe de facto imaging standard for investigating potential cranio-synostosis, and 3D CT imaging is essential for preoperative diagno-sis to allow surgical planning as well as postsurgical assessment(Branson and Shroff, 2011; Xia et al., 2000; Cohen et al., 2008;Tartaro et al., 1998; Ursitti et al., 2011). Findings from CT includeclosure and possible ridging of sutures (Branson and Shroff,2011). Furthermore, as the infant brain increases in size rapidlyduring the first year of life, volume expansion results in compensa-tory areas of cranial overgrowth and abnormal morphology, whichare also best evaluated on three-dimensional reconstructions fromCT data.

Craniosynostosis has been classified in terms of the prema-turely fused suture and the resulting cranial malformation: sagittal(scaphocephaly), coronal (anterior plagiocephaly), metopic (trigo-nocephaly) and lambdoid (posterior plagiocephaly) (Kirmi et al.,2009). See Fig. 1 for an indication of the cranial bones and sutures

Fig. 1. Reconstructions of a cranium with labeled bones and sutures. Left: Anterior view. Right: Posterior view.

636 C.S. Mendoza et al. / Medical Image Analysis 18 (2014) 635–646

involved in craniosynostosis (notice the large open region of theanterior fontanelle that is present in newborns). Note that shapeabnormality can also occur without pathological suture fusionand it is often addressed by behavioral adjustments or mechanicaldevices, without surgical correction (Park and Yoon, 2012).Although diagnosis rests on the presence of a suture fusion (theexception being metopic craniosynostosis, which depends on theseverity of trigonocephaly), assessing the need for surgical inter-vention may require consideration of several clinical factors,including the subjective evaluation of cranial shape and the degreeof malformation. The surgical procedure itself is conditioned on theseverity of shape abnormality in the different bones which com-prise the cranial vault (Kirmi et al., 2009). Surgery can involve sig-nificant morbidity since the scalp is retracted, much of the craniumis resected, and the bone segments are reshaped and repositionedto achieve the desired configuration. Thus, it is desirable to rely onobjective quantitative descriptions of shape abnormality to assistin the decision making process.

1.2. Statistical shape models

Shape analysis is a mature sub-field of medical image comput-ing and analysis. Pioneering work by D’Arcy Thompson in 1917first proposed the study of biological shapes by characterizing spa-tial transformations mapping different biological shapes into eachother (Thompson, 1917). Biological shape data combine two sortsof information: geometric location and biological homology(matching semantics). The link between these two are the land-marks, a set of semantically corresponding points located in allthe forms of a shape series. Analysis of shapes as configurationsof landmarks is nuclear to Thompson’s approach. As the locationsof homologic landmarks are matched across shape forms, spatialtransformations providing such correspondences can be obtainedat the landmarks, and interpolated elsewhere.

Other classic shape statistics are based on ratios of distances be-tween landmarks or angles submitted to multivariate analysis,such that the full geometry is often lost. It was not until 1978 thatThompson’s approach was re-visited by Bookstein in his book ‘‘TheMeasure of Biological Shape and Shape Change’’ (Bookstein, 1978).In the 80s and 90s several authors contributed to the developmentof the current body of knowledge in shape statistics (Bookstein,1991; Small, 1996; Kendall, 1999). Statistical shape descriptionsin this context are defined in terms of landmarks (anatomicallymeaningful points) and pseudo-landmarks (secondary landmarksobtained heuristically from primary landmarks). Shape is consid-ered independently of location, orientation and scale, such thatmain differences in the aforementioned approaches relate to themethod for correction of these features. Once the landmarks are

thus transformed, their coordinates are combined into high-dimensional vectors, each representing a particular shape instance.

For the use of landmarks as basis of a statistical shape model(SSM), Cootes et al. (1992) have coined the name Point DistributionModels (PDMs), which has become quite popular in literature andis often used synonymously to landmark-based SSMs. The majorityof current shape models are based on PDMs (Cootes et al., 1992).PDMs involve the construction of typical shapes and typical vari-ability based on a set of training shapes via principal componentanalysis (PCA), and a large percentage of shape variability can oftenbe captured by the main modes in PCA-transformed shape space.As with preceding landmark-based techniques, an essentialrequirement for building shape models with PDMs is that land-marks on all training samples are located at corresponding loca-tions. In many medical image applications shape instances arerepresented as labeled objects in 3D space, and often it is difficultto establish meaningful landmark correspondences without te-dious and time-consuming expert work. In any case, establishingdense point correspondences between all shapes of the trainingset is generally the most challenging part of 3D model construc-tion, and at the same time one of the major factors influencingmodel quality (Heimann and Meinzer, 2009). Although a numberof strategies to automatically establish landmark correspondencevia registration have been proposed (Frangi et al., 2001; Rueckertet al., 2003), landmark approaches suffer from further limitationssuch as numerical instability, inability to accurately capture highcurvature locations and difficulty in handling topological changes(Tsai et al., 2003).

Alternatively, landmark-free SSMs have been proposed in theliterature by representing the shape of labeled regions over thewhole domain of the labeled image, which can be regarded asdense PDMs in which all pixels in the image domain implicitly be-come landmarks. An important trend in landmark-free SSMs is torepresent shape by deformation fields (Soatto and Yezzi, 2002;Cootes et al., 2004). For each shape instance a similarity(rigid + isotropic scaling) registration procedure corrects for poseand scale, and subsequent non-rigid diffeomorphic (smooth andinvertible) registration provides a deformation field that repre-sents the shape over the entire image domain. Such fields can beoperated upon linearly to construct a mean deformation field andits principal modes of variation via PCA. Registration of all shapeinstances is performed either with respect to a template shape,or iteratively via group-wise registration (Heimann and Meinzer,2009).

Additionally, landmark-free SSMs can be constructed fromsigned distance functions (SDFs) (Danielsson, 1980) or level setsof shapes. As first proposed by Leventon et al. (2000), a labeled re-gion in an image can be represented over the entire image domain

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by assigning to each pixel the negative or positive distance to thecontour of the shape, depending on whether the pixel is located in-side or outside the shape contour, respectively. This approach, fur-ther popularized by Tsai et al. (2003) and Rousson et al. (2004),requires previous similarity registration, to correct for pose andscale, which can be performed with respect to a template, or bygroup-wise iterative techniques. In general, this approach hastwo main drawbacks. First, SDFs are treated as elements of a linearvector space, and operations such as averaging are performed. Yet,the space of SDFs is a nonlinear Riemannian manifold and is notclosed under linear operations (Srivastava et al., 2012). Therefore,the use of linear analysis tools such as PCA gives rise to an incon-sistent framework for shape modeling (Tsai et al., 2003; Kim et al.,2007; Cremers et al., 2007). This drawback has limited effect if theSDFs are sufficiently similar. In such case, the part of the manifoldsupporting the shapes is approximately linear provided that themanifold does not have too much curvature (Kim et al., 2007). Thisis why methods based on linear manipulation of signed distancefunctions (Leventon et al., 2000; Tsai et al., 2003; Rousson et al.,2004) work well when there is small shape variation. Second, whilethe first few principal components capture most of the variation inthe space of SDFs, they will not necessarily capture the variation inthe space of the embedded contours. As a consequence, one mayneed to include a larger number of eigen-modes (compared toPCA on landmark-based representations) in order to capture cer-tain details of the modeled shape (Cremers et al., 2007).

Therefore, using SDF representations is appropriate when deal-ing with limited shape variation and when no particular restrictionapplies to the number of eigen-modes to consider after PCA. Theconsistency of linear methods under such conditions is enoughfor most applications.

1.3. Shape analysis for craniosynostosis

Craniosynostosis diagnosis and surgical planning constitute anideal problem domain for the application of formal statistical shapecharacterization. Despite the fact that suture fusion is a clear indi-cation of craniosynostosis (except for metopic phenotypes), assess-ing the need for surgical correction relies heavily on subjectiveassessment of shape abnormality. The surgical procedure, whetherresponding to aesthetical or functional needs, attempts to obtainan adequate cranial morphology, by correcting both the deforma-tion of the cranial bones due to compensatory over-growth andthe possible ridging (i.e. increased curvature) of ossified sutural re-gions. The definition of a desired morphological reference, and thedegree of bone reshaping to perform in order to achieve it, both re-spond to mental constructions by experienced plastic surgeons orneurosurgeons, and are thus highly subjective. In the case ofmetopic craniosynostosis, diagnosis relies solely on shapecharacterization.

Ideally, shape analysis in craniosynostosis should provide adense description of deformation and ridging over the differentbones/sutures. Furthermore, such description should help guidethe surgical correction of the cranium, so need to be expressed interms of physical discrepancies (e.g. in millimeters) with respectto a proper desirable shape reference, tailored to the anatomy ofthe subject, to compensate for the effect of healthy variations ofthe cranial shape.

In the rest of this Section we will frame the contributions of ourfully automatic methodology to formalizing these features via sta-tistical shape modeling and computational geometry.

1.3.1. Statistical shape models of cranial anatomyPrevious approaches are based on PDMs (Cootes et al., 1992) on

pseudo-landmarks due to the difficulty of defining shape homolo-gies on the cranial vault. Pseudo-landmarks are typically obtained

by systematic sampling with respect to structures in the base ofthe skull after rigid registration for pose-correction (Marcuset al., 2009; Saber et al., 2012) or by obtaining surface correspon-dence using the method of consistent patch decomposition andparameterization (Lamecker et al., 2006). These approaches areage-variant and require age stratification to achieve consistency.Furthermore, they suffer from the limitations of defining landmarkcorrespondence, as outlined in Section 1.2.

In the present approach we circumvent these limitations byusing a landmark-free SDF shape representation of volumetric cra-nial shapes to build an SSM of normal anatomy. Age-invariance isachieved using a registration procedure that maximizes the over-lap of anatomical structures at the base of the skull allowing forisotropic scale variations. The constructed SDF-based SSM is usedto provide references of normal cranial anatomy for each subjectunder study, via analysis of the PCA-transformed shape space.The underlying assumption is that the morphological variabilityof an anatomical structure within different subjects, even acrosspopulations, is sufficiently small such that the high-dimensionalpoints that represent the shape set lie in close proximity to eachother on the shape manifold. Such analysis involves projectingthe subject under study into the PCA shape space. Then, the refer-ence is obtained either constraining the projection to a normal var-iation range, or by selecting the most similar available normalshape in the training set (i.e. using the model as a multi-atlas).The similarity is computed using a metric in the PCA space. Thisis a significant departure from previous approaches that performshape characterization in terms of mean shape (Marcus et al.,2009; Saber et al., 2012), and allows accounting for normal varia-tions in healthy anatomy (e.g. due to ethnicity (Dean et al., 1998)).

1.3.2. Computational geometry for local shape characterizationGiven mesh representations of the surface of the calvarium for a

subject under study and for a tailored cranial shape reference, ourmethodology is capable of computing dense malformation descrip-tions over the subject’s surface mesh. Craniosynostosis is charac-terized by local deformations over cranial bones and ridging ofsutures. Previous work by Marcus et al. (2009) represented defor-mation at each landmark in terms of a number of standard devia-tions from the mean shape of the obtained PDM, assuming aGaussian multi-variate shape distribution. In the present approach,we compute deformations and curvature discrepancies in terms ofphysical dimensions (millimeters) with respect to the closest pointin the shape reference. This allows for more meaningful interpreta-tion during surgical planning and follow up, in a manner closely re-lated to the surgeon’s mental process in subjective shapeassessment.

Furthermore, local shape malformation can be quantified overanatomically meaningful regions (bone segments and sutures) asin clinical practice, thanks to a novel automatic graph-cut methodbased on labeling priors and low contact interface detection toautomatically segment and label the cranial bones and sutures.As a result, automatic diagnosis becomes available, and the malfor-mation measurements are more useful for planning surgery involv-ing the reshaping of the individual bone segments of the subject.

2. Materials and methods

This Section provides a detailed description of the methods thatallow a fully automatic diagnosis of different types of non-syn-dromic craniosynostosis, and the computation of local malforma-tions over the cranial surface in terms of a tailored reference ofnormal anatomy. Fig. 2 shows a schematic representation of theimage processing pipeline which produces a shape descriptionfrom a CT image of the head of the subject, and ultimately a

Fig. 2. Schematic of the proposed shape analysis technique. ROI: Region of interest. SDF: Danielsson’s signed distance function.

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diagnosis via classification. What follows is a description of the dif-ferent tasks represented by blocks in the schematic in Fig. 2.

2.1. Data

After approval of the institutional review board (IRB), from theimage repository system in our institution we obtained scansfor subjects with non-syndromic craniosynostosis with ages0–12 months between the years 2005–2012. We retrieved allavailable CT scans containing the words ‘‘synostosis’’, ‘‘craniosyn-ostosis’’, ‘‘trigonocephaly’’,‘‘scaphocephaly’’ and ‘‘plagiocephaly’’in the radiological reports. Controls were selected from subjects re-ported to the emergency room for trauma, and were manuallyscreened to exclude hydrocephalus, intra-cranial tumor, intra-cra-nial hemorrhage, hardware (e.g. shunts), craniofacial trauma andprior craniofacial surgery. CT scans of healthy infants are retro-spective data and can be used without any exposure to radiation.Improper protocol studies or poor-quality images were criteriafor exclusion from subsequent analysis, and particularly those withan axial spacing above 5 mm. This resolution is consistent withcommon clinical practice for craniosynostosis (Vannier et al.,1994; Craven et al., 1995; Park et al., 1996). The images were ac-quired with the following scanners: General Electrics LightSpeedUltra, General Electrics LightSpeed Discovery 690, Philips Brilliance40 and Philips Brilliance 64. The axial in-plane pixel size ranged be-tween 0.26 and 0.49 mm, and the axial spacing ranged between0.33 and 5 mm. We gathered a total of 90 normal subjects, 27 withsagittal synostosis, 16 with metopic synostosis, 3 with right coro-nal synostosis and 5 with left coronal synostosis (total 141 cases).The analysis for coronal synostoses was performed by inverting thepolarity according to the diagnosis of right versus left coronal. Suchcohort of CT scans obtained with different resolutions and differentscanners, can be considered as representative of clinical practicefor craniosynostosis and will illustrate realistically the propertiesof our proposed techniques.

2.2. Cranial bone segmentation

To obtain a 3D representation of the cranium of a subject, thevoxels with intensity above 100 Hounsfield units (HU) are labeled,as to avoid all soft tissues (<100 HU) and preserve all cranial tissueswhich are still in ossification in infants (and thus less dense thancranial tissue in adults). Then the largest connected componentof voxels is selected. To obtain a volumetric representation of thebony structures in the cranium but excluding the open sutures(which exhibit a lower density) each subject is subsequently thres-holded so that the denser 50% of the samples of bony tissue are

kept. These threshold values were evaluated on cases not used inthe analysis described in this paper. As a result we obtain a volumecontaining the cranium, and another containing the cranial boneswith open sutures (except at ossified sutures).

2.3. Spatial normalization via registration

As outlined in Section 1.2, shape analysis requires a previouscorrection of the pose (location, orientation and scale) of eachshape instance. For our application we resort to a template-basedregistration scheme. We selected a head CT image of a healthy sub-ject (the template) in order to define the reference pose. On thistemplate, a set of manual landmarks was selected on structuresat the base of the cranium (the nasion (N), the opisthion (O) andthe two clinoid processes of the dorsum sellae (D1,D2)). Equivalentanatomical landmarks have been proposed in previous literature asbeing able to separate the calvarium and the base of the skull(Lamecker et al., 2006; Marcus et al., 2009). N, D1 and D2 togetherdefine a horizontal plane that dissects the cranium approximatelyabove the supra-orbital notches (P1). O, D1, and D2 define an obli-que plane passing through the lower occipital bone (P2). The inter-section of the regions above P1 and P2 describe the region ofinterest (ROI) for cranial morphology assessment (approximatelycoinciding with the cranial vault). See Fig. 3.

According to the defined template, we proceed to align all sub-jects in order to neutralize their initial pose differences. For eachsubject, the center of mass of the binary volume is obtained. Then,the registration procedure is initialized with a translation equal tothe vector connecting the center of mass for the subject and thetemplate. This approach assumes that the moments of mass ofthe structures are similar for both images, which represent thesame body part in different patients. We then implement a gradi-ent descent optimization scheme on translation, orientation andscale parameters. At each iteration the employed optimizationmetric is equivalent to the sum of squared differences (SSD), typi-cally used for registration of binary volumes (Petti et al., 1994),which provides a measure of overlap (exclusive-or of the binaryvoxel sets):

SSDðA;BÞ ¼XM

i¼1

ðAðiÞ � bBði;~hÞÞ2; ð1Þ

where AðiÞ is the template (fixed image) defined over a volume withM elements and evaluated at location i, and bBði;~hÞ is the floating im-age transformed by the set of translation, rotation and scale param-eters ~h, linearly interpolated at i. The metric is computedconsidering structures in the template that lie inferior to the P1and P2 planes described above (see Fig. 3), as to prevent abnormal

Fig. 3. Illustration of landmark and plane locations. The figure on the right allows to identify the cranial structures used in the registration as those shaded in red (lying underthe planes). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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anatomy in the cranial region to interfere in the alignment proce-dure. The computed transformation was allowed seven degrees offreedom (rotation, translation and isotropic scale), in order to ac-count for pose and scale differences (e.g. due to age and growth)across subjects.

As a result of this procedure, we obtain a pose-neutral represen-tation of every subject in a common physical frame. Furthermore,since the overlap of structures at the cranial base is maximized,P1 and P2 allow the delineation of the region of the interest (cra-nial vault) for all the subjects. The accuracy of ROI definition is thusdetermined by the accuracy of the registration procedure itself,even when the landmarks may not exactly overlap due to malfor-mation. However, pathological deformations from craniosynostosisare negligible at the base of the skull (Kotrikova et al., 2007; de Oli-veira et al., 2011) and they should not affect the accuracy of land-mark detection. The overall accuracy of the registration was alsoconfirmed by the authors via visual inspection of all results.

2.4. Surface model generation

Each volume containing the cranium of a subject can be repre-sented as a surface model or mesh of the outer interface of the cra-nial bones. Using a marching cubes procedure (Lorensen and Cline,1987) we can obtain a mesh representation of the cranial surface.The resulting surface model contains both the outer and inner sur-faces of the cranium. Holes on the cranial surface due to normalanatomy (e.g. the fontanelle, see Fig. 1) will also be representedin the resulting mesh.

To obtain a single-layered, genus-zero surface over which wecan represent local malformations, we need to obtain an equivalentwatertight model. This is a common problem in computationalgeometry for which a number of methods have been proposed,with higher or lower degrees of computational complexity (seefor example Carr et al. (2001), Fleishman et al. (2005), Jun(2005)). For the problem at hand a simpler approach has beenadopted: constrained relaxation of an embedding sphere. Thismethodology is known as ShrinkWrap and was first proposed byPope (2009). The triangular mesh of the embedding sphere alreadyexhibits several of the desirable traits we seek: the triangles havean average size which can be customized, they all obey the right-hand rule and are outward facing. Then, this mesh needs to bemodified so it approximates the surface of the model. This isaccomplished by collapsing the triangular mesh of the embeddingsphere onto the surface of the model as defined by the model’s ver-tex array and connectivity list. This is achieved by iterative appli-cation of windowed cardinal sine smoothing filter (Taubin et al.,1996). Please see Pope (2009) for details.

The resulting mesh is watertight, regular (homogeneous trian-gle sizes), and the desired level of detail can be controlled by defin-ing embedding spheres with varying numbers of vertices. To obtainthe final desired surface representation of the cranial vault we

dissect the obtained genus-zero mesh with planes P1 and P2 regis-tered to the subject under study. The portion of the mesh lyingabove the planes is kept for subsequent analysis.

2.5. Statistical shape model of normal cranial anatomy

As outlined in Section 1.3.1 our methodology to obtain localdeformation and abnormal curvature is obtained in physical unitsto facilitate a representation that mimics mental processes by sur-geons in planning cranial shape correction interventions. Suchmalformation features are obtained in terms of a subject-specificmodel of normal anatomy obtained from a statistical shape modelof normal cranial shapes. We used a total of 90 normal subjects in a5-fold scheme, resulting in 5 groups of 72 (4 out of 5 normal sub-jects) instances for training, and using the remaining 18 (1 out of 5normal subjects) for testing.

We adopt the SDF representation (Leventon et al., 2000; Tsaiet al., 2003; Rousson et al., 2004) for binary volumes of cranialshapes. As a result, each normal subject is turned into a high-dimensional vector (as many components as voxels in the volume).The set of all subjects lie on a Riemannian manifold roughly assim-ilable to a hyperplane. Then PCA can be performed on the set ofvectors to obtain a 71-dimensional PCA shape space. As discussedin Section 1.2, the first few principal components may capturemost of the variation in the space of SDFs, but they will not neces-sarily capture the variation in the space of the embedded surfaces.Therefore, we employ the full 71-dimensional PCA space in all sub-sequent analysis.

For every test subject we project its SDF into the PCA shapespace. To constrain the resulting projection to lie in the subspaceof allowed shapes, several techniques are possible, depending onthe model chosen for the distribution of shapes in the shape space.Some authors have proposed Gaussian mixture models (Cootes andTaylor, 1999), non-parametric distribution estimation (Kim et al.,2007), or more commonly Gaussian multi-variates (Stegmannet al., 2006). But the most common approach is to consider eachmode as an independent Gaussian distribution (Heimann andMeinzer, 2009; Cerrolaza et al., 2011), due to its simplicity and re-duced computational cost. This then translates in bounding eachcomponent of the projection to lie in a range extending �3raround the mean shape (origin of the shape space), with r beingthe standard deviation of each component as computed by thePCA (the square root of the corresponding eigenvalue). This is theusual choice when the Gaussian distribution assumption is in place(Heimann and Meinzer, 2009; Cerrolaza et al., 2011), and guaran-tees 99.7% of the distribution samples are included. Then, the con-strained projection can be used to reconstruct the SDF and surfacemodel generation (see Section 2.4) can be applied to produce a sur-face model of tailored normal anatomy for each subject. We callthis technique the constrained projection (CP) approach.

640 C.S. Mendoza et al. / Medical Image Analysis 18 (2014) 635–646

Alternatively to reconstructing from constrained projections,we also produce for each subject a reference normal shape by usingthe shape space as a multi-atlas, i.e. choosing as shape referencethe closest normal shape from the training set of normal cases(closest normal (CN) approach). The closest normal can be ob-tained by computing the Mahalanobis distance from the projectionto all the projections of all the cases employed for the constructionof the shape model, and choosing the closest normal case as ana-tomical reference.

Finally, we also use the mean cranial shape as anatomical refer-ence (the mean shape (MS) approach). This will allow us to com-pare our methodology with pre-existing approaches (Marcuset al., 2009; Saber et al., 2012), both using average cranial shapes.See Fig. 4 for a comparison of shape references for one particularsubject using the three different methods (MS, PR and CN).

2.6. Bone labeling and detection of fused sutures

From the output of the cranial bone segmentation described inSection 2.2 we obtain a volume representation of the cranial boneswith open sutures. This representation can be used to separate thedifferent cranial bones. To the best of our knowledge, we are thefirst authors to propose a method to analyze/label cranial bonesand sutures.

A graph-cut approach is adopted for the separation of bones atlow contact degree interfaces (Liu et al., 2008). The term contactdegree refers to the number of voxels that belong to a certain bonesegment and touch a voxel that belongs to an adjacent bonesegment. It serves as a surrogate of suture fusion when the voxelsbelong to bone segments that are separated by a suture in healthysubjects.

Graph-cut techniques are capable of minimizing many sorts ofcustom energies defined on a graph analogue of an image, and they

Fig. 4. Subject (solid green) and shape reference (translucid red) derived from the shape mrow: Anterior view. First column: Mean shape (MS) approach. Second column: Constrainterpretation of the references to colour in this figure legend, the reader is referred to

exhibit good performance and guaranteed convergence (for thebinary case) via min-cut/max-flow computation. Since we must as-sign multiple labels, a-expansion and a-b-swap variations(Kolmogorov and Zabih, 2004) of the basic algorithm (Boykovet al., 2001; Boykov and Kolmogorov, 2004) are employed instead.

Our node system is constructed from all the voxels in the su-ture-free bone volume. The subject and the template are aligned.Then, bones are identified based on a spatial relation betweenthe subject’s data and the labels in the template. To achieve thiswe introduce a novel regional cost term that, for each label, variesaccording to the quotient between the distance to the bone withthat label in the template, and the distance to the furthest othertemplate bone. Our edge cost is fixed and equal for all edgesinvolving different labels. Analytically, energy E is defined as thesum over the nodes P of the graph, of a unary and a binary term

Eðf Þ ¼ a �Xp2P

DpðfpÞ þXðp;qÞ2K

Vp;qðfp; fqÞ; ð2Þ

where a is a tuning constant parameter, f is a labeling scheme thatassigns label fp to the node p;Dpð�Þ is a data penalty function that as-signs a cost DpðfpÞ to having label fp at node p;Vp;qð�; �Þ is an interac-tion potential that assigns cost Vp;qðfp; fqÞ to having labels fp and fq atnodes p and q, and K is the set of all pairs of neighboring nodes. Inthe classic method, the regional term is formulated in a Bayesianframework as the log-likelihood of obtaining the intensity valueof the node from the intensity distribution of a given label,

DpðfpÞ ¼ � ln PðIpjfpÞ; ð3Þ

with Ip the intensity at p. Instead, as we are labeling a binary vol-ume, we derive the label energy in terms of a spatial prior. As a con-sequence, the label energy depends on the anatomical location ofthe node, while the classical graph-cut methodology assigns label

odel by different techniques. First row: Superior view. Second row: Left view. Thirdined projection (CP) approach. Third column: Closest normal (CN) approach. (For

the web version of this article.)

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energies according to node intensity. If the distance to the cranialbone with label f in the template is df ,

DpðfpÞ ¼ dfp dfp þmaxfðdf jf – f pÞ

� ��; ð4Þ

and the edge term as in the classic method:

Vp;qðfp; fqÞ ¼1 if f p – f q;

0 otherwise:

�ð5Þ

Using this energy, the cut with minimum cost will encourage a sep-aration that assigns labels in accordance to the bone distribution inthe template, but at the same time favoring cuts involving minimalnumbers of nodes (i.e. at the open sutures).

On the resulting labeled volume, each cranial suture is posi-tioned between two adjacent labeled cranial bones, appearing asa gap between the bones, and it can be open or fused. There shouldnot be adjacent voxels with different labels unless one of the su-tures is fused. In such case the graph-cut labeling would be drivenby the template-based term Dp, since no low contact interfacewould be available at the expected location of the fused suture.Thus, the resulting labeling for the suture region would be basedmostly on the template prior. In the resulting labeling, detectionof fused sutures can be achieved by counting voxels at the contactbetween different bone labels (e.g. if the left frontal and right fron-tal bone segment labels meet, then the metopic suture is fused).Thus, a fusion index (FI) can be computed for each suture as thenumber of two-neighbor cliques that exhibit a particular combina-tion of the corresponding two different labels. To guarantee consis-tent FIs, we compute these adjacencies on spatially (pose andresolution) normalized versions of the volumes.

For sutures that are closed (i.e. with high FI), the graph-cutlabeling of the two fused bone segments is subsequently refinedwith null edge cost in Eq. (5), which produces a separation whichignores the contact degree of the interface and performs based so-lely on the template prior.

Once a labeled cranial volume is produced, a derived mesh isobtained as in Section 2.4, and suture regions are identified at ver-tices lying simultaneously closer than 7.5 mm. to the two bonesegments shared by the suture. All remaining vertices are assignedto the closest bone segment.

2.7. Diagnostic features

We compute several features on each subject’s cranial vaultwith respect to the normal shape reference model (generated viaCP, CN or MS, see Section 2.5). First we obtain a fusion index (FI)to discriminate between open and close sutures as described inSection 2.6. Secondly, for every point on the subject’s surface mod-el, we compute the Euclidean distance to the closest point in theshape reference surface model. This provides a local estimate ofdeformation. Finally, we compute the absolute difference betweenthe curvature at each point on the subject’s surface and the curva-ture at the closest point in the shape reference surface model,which will highlight ridged sutures. We estimate local curvaturevalues following the method in Gumhold et al. (2001), by defininga neighborhood of points at a certain distance from the point ofinterest p, fitting a plane minimizing the squared distance to thesepoints and computing the quotient between the distance from theneighborhood center to the plane and the average distance ofthe neighbors to the plane (Gumhold et al., 2001; Doria, 2011).The curvature estimate ji at point pi is then defined as

ji ¼2di

l2i

; ð6Þ

where di is the distance from point pi to the fitted plane, and li isthe average distance from the points in the neighborhood of pi to

the fitted plane. This methodology was chosen for its simplicityand because it provides an intuitive way to set the scale for curva-ture computation (i.e. the size of the sphere containing the neigh-bors employed for plane fitting). We empirically found goodresults for a sphere of radius equal to 1 cm and used it for all sub-sequent analysis.

For deformations as well as for curvature discrepancies, wecompute the mean, standard deviation, maximum and minimumacross five cranial bones (left/right parietal (LP/RP), left/right fron-tal (LF/RF), occipital (Oc)), and six suture-adjacent regions (meto-pic suture (M), left/right coronal (LC/RC), sagittal (S), and left/right lambdoid (LL/RL)).

2.8. Validation

An operator manually labeled the five cranial bones (LP, RP, LF,RF, Oc) in the suture-free bone volumes for seven normal and se-ven abnormal randomly-picked subjects. Using this gold standardwe can evaluate the accuracy of our bone labeling technique. Fur-thermore, the operator placed four anatomical landmarks (N, D1,D2, O) on each of the 14 subjects, which allow us to validate theregistration procedure and the assumption that differences in thecranial base for different age groups can be corrected for by isotro-pic scaling to a good accuracy level. To validate the alignmentusing the structures at the base of the skull and the anatomicallocations of planes P1 and P2 defining the ROI, we computed theerror/distance between the locations of N, D1, D2 and O obtainedfrom manual tracing and automatic detection.

For all the remaining validation we used the complete set of 141subjects (90 normal, 18 metopic, 27 sagittal, 8 coronal) as de-scribed in Section 2.1. To assess the ability of FI to detect fused su-tures we compared the FI values with a gold standard of the fusionstatus of the suture, for each of the six sutures. This gold standardwas obtained from the clinical diagnosis of craniosynostosis for allbut the metopic suture, which closes during the first year of lifeeven in healthy subjects. For the metopic suture thus, an operatorvisually inspected the patency of the suture using 3D reconstruc-tions in accordance to the usual clinical protocol for assessingthe fusion of the other sutures in routine diagnosis of craniosynos-tosis. The significance of differences in FI was assessed usingMann–Whitney U-tests (Gibbons and Chakraborti, 2003).

Subsequently, we computed the mean, standard deviation,maximum and minimum of both shape assessment features(deformation and curvature discrepancy) across each of the cranialbones (LF, RF, LP, RP, Oc) and each of the regions adjacent to the su-tures (M, LC, RC, S, LL, RL). This produced a total of 88 features foreach subject. For the coronal subjects, as mentioned in Section 2.1,we reversed the left/right polarity in the right coronal cases, so thatthe values on the left side (LF, LP, LC) can be considered ipsilateral,and the values on the right side (RF, RP, RC) can be considered con-tralateral to the fused suture. These values were compared be-tween craniosynostosis cases (sagittal, metopic, coronal) andnormal cases using Mann–Whitney U-tests. Significance wasestablished at a p-value of 0.05.

2.9. Classification

To further determine the discriminative power of our shapedescription, we created for each subject a feature vector containingthe 6 fusion indices (FI Only), the 88 shape feature statistics fromall bones and sutures (Shape Only), and the 94-dimensional com-bination of both (FI & Shape). We normalized the ranges of varia-tion for each component to ½0;1�, and performed PCA on thefeature vectors and selected the most significant components cov-ering 95% of the variance. This percentage results in 37, 35 and 31components for the CN, MS and CP, respectively, when including FI

Table 1Cranial bone labeling validation parameters, for each cranial bone (top) and averagedall/fused/non-fused bones.

Cranial bone Sensitivity Specificity Dice Jaccard

Left parietal 0.973 ± 0.037 0.997 ± 0.002 0.983 ± 0.020 0.967 ± 0.039Right parietal 0.990 ± 0.017 0.997 ± 0.005 0.990 ± 0.012 0.981 ± 0.023Left frontal 0.971 ± 0.027 0.993 ± 0.009 0.964 ± 0.035 0.932 ± 0.063Right frontal 0.988 ± 0.022 0.994 ± 0.006 0.974 ± 0.023 0.949 ± 0.043Occipital 0.974 ± 0.063 0.999 ± 0.001 0.984 ± 0.035 0.971 ± 0.064All average 0.979 ± 0.038 0.996 ± 0.006 0.979 ± 0.028 0.960 ± 0.052Fused average 0.970 ± 0.030 0.993 ± 0.007 0.964 ± 0.026 0.932 ± 0.047Non-fused

average0.987 ± 0.040 0.999 ± 0.001 0.991 ± 0.022 0.984 ± 0.040

Table 2Average FI values for different sutures and results for Mann–Whitney U-tests forsuture fusion status versus fusion indices.

Sagittal suture Metopic suture Coronal suture

Fused sutures FI 366.78 ± 177.35 364.09 ± 125.30 282.00 ± 57.03Non-fused sutures FI 3.96 ± 11.97 13.19 ± 22.48 31.26 ± 51.36p-Value <0.001 <0.001 <0.001

642 C.S. Mendoza et al. / Medical Image Analysis 18 (2014) 635–646

& Shape (94-dimensional features). It results in 34, 33 and 28 com-ponents for CN, MS and CP, respectively, when including ShapeOnly (88-dimensional features). PCA was not performed for FI Only(6-dimensional vectors).

With the transformed feature vectors we performed leave-one-out classification of all the subjects using three different tech-niques: regularized linear discriminant analysis (Fisher, 1936;Friedman, 1989), random forests (Breiman, 2001; Ho, 1998) andsupport vector machines (Vapnik and Kotz, 1982; Cortes andVapnik, 1995).

Linear discriminant analysis (LDA) assumes that the conditionalprobability density function of features for each class can be mod-eled as a multivariate Gaussian. We used regularized discriminantanalysis (Friedman, 1989) to mitigate the effects of a small numberof samples in a high-dimensional space, with parameters opti-mized using also leave-one-out classification.

Random forests (RF) use an ensemble classifier that consists ofmany decision trees and outputs the class that is the mode of theclasses output by individual trees. RF method has been shown tobe accurate, to automatically deal with redundancy and skewedpopulations, and to lend itself naturally to multi-class classificationproblems (Breiman, 2001). In our implementation we used 200trees, 4 component subset at each node, all samples are input ateach tree, and the Gini impurity index as node splitting criterion.

Support vector machines (SVM) are supervised learning modelswith associated learning algorithms that analyze data and recog-nize patterns, used for classification and regression analysis. Mul-ti-class SVM aims to assign labels to instances by using supportvector machines, where the labels are drawn from a finite set ofseveral elements, like in our problem. We used the one-versus-one approach, in which classification is done by a max-wins votingstrategy. For our implementation we used regularized support vec-tor classification, and a linear kernel. The soft margin parameterwas optimized using leave-one-out classification.

3. Results

We performed a comparison between anatomical landmarksobtained automatically and those manually selected. The error(mm) was 6.44 ± 3.36 for N, 2.65 ± 1.13 for D1, 2.56 ± 1.31 for D2,and 4.60 ± 1.97 for O. The error is low for D1, D2 and O, as thesethree landmarks are located on the base of the cranium. It is pos-sible that the error for N is higher due to its location right belowthe forehead, which might be affected by malformation in somecraniosynostosis cases.

In Table 1 we present numerical results for the validation of thebone labeling algorithm in Section 2.6. We present sensitivity,specificity, Dice’s coefficient and Jaccard index, averaged acrossall subjects for every cranial bone. Subsequently, we present theaverage of these parameters computed either globally, for thosebones which share a closed suture, and for those which share anopen suture. The accuracy is worse for closed sutures, since the ac-tual location of the closed suture is determined by proximity to thecorresponding suture in the template. Nevertheless, the average

Table 3Correct classification rate for leave-one-out classification of 141 cases. Results are shownnormal shape reference (MS, CP and CN) and three different subsets of features (FI Only, Shasupport vector machines. MS: mean shape approach, CP: constrained projection approach

Features FI Only Shape only

Method LDA RF SVM LDA

MS 0.879 0.908 0.879 0.872CP 0.723CN 0.936

Dice/Jaccard overlaps from bones sharing a fused suture are stillhigh (0.96,0.93).

In Table 2 we present the mean values of the fusion indices inopen and closed sutures, and the p-values computed for the differ-ences between the two at each suture. Note how the automaticallycomputed fusion index exhibits significantly different values foropen and closed sutures.

In Table 3 we present the average correct classification rate forthe three shape reference methods (mean shape (MS), constrainedprojections (CP) and closest normal (CN)); three different classifi-cation schemes (LDA, RF, linear SVM); and for features includingthe fusion indices (FI), the malformation/curvature statistics, andthe combination of both. For shape-only we obtained 34 compo-nents after PCA, and for the combination of FI and shape we ob-tained 37 components. The best results were obtained for LDAafter PCA using CN, with a correct classification rate of 0.957.

In Fig. 5 we present some examples of malformation and curva-ture discrepancy (using the CN approach) computed locally overthe surface of the subject’s cranium. We present results for a nor-mal subject, another with sagittal craniosynostosis, and anotherwith metopic craniosynostosis. Note that the higher malformationvalues are reached on the occipital bone in the sagittal case, andthe highest curvature discrepancy happens on the frontal bonesin the metopic case.

In Table 4 we highlight the features for which the mean valuevaries significantly between craniosynostosis and normal subjects,for each of the cranial regions. Note that most of the features exhi-bit significant differences. This suggests the ability of our methodto identify relevant anatomical areas for diagnosing craniosynosto-sis. Subsequently, in Table 5 we show the mean values for malfor-mations and curvature discrepancies in the different cranial bones,and also in the fused suture representative for each type of

for three classification methods (LDA, RF and SVM), three methods for establishing ape Only and FI and Shape). LDA: linear discriminant analysis, RF: random forests, SVM:, CN: closest normal approach. Best result highlighted with boldface.

FI and Shape

RF SVM LDA RF SVM

0.808 0.830 0.922 0.851 0.8790.787 0.794 0.874 0.823 0.9150.872 0.872 0.957 0.879 0.908

Fig. 5. Subject surface models with labeled bones/sutures (first column), deformation from closest normal subject (second column) and curvature discrepancy with closestnormal subject (third column). First row: Normal subject. Second row: Sagittal synostosis. Thirs row: Metopic synostosis.

Table 4Significance (p < 0.05) for mean malformation and curvature discrepancy for subjects with different types of craniosynostosis against normal subjects. Results are shown for fivecranial bones and six suture regions. �: significant, �: non-significant. LP/RP: left/right parietal bone, LF/RF: left/right frontal bone, Oc: occipital bone, M: metopic suture, LC/RC:left/right coronal suture, S sagittal suture, and LL/RL: left/right lambdoid suture. Mf: malformation, Cv: curvature discrepancy.

Type LPMf.

LPCv.

RPMf.

RPCv.

LFMf.

LFCv.

RFMf.

RFCv.

OcMf.

OcCv.

SMf.

SCv.

LCMf.

LCCv.

LLMf.

LLCv.

RCMf.

RCCv.

RLMf.

RLCv.

MMf.

MCv.

SagittalC.

� � � � � � � � � � � � � � � � � � � � � �

MetopicC.

� � � � � � � � � � � � � � � � � � � � � �

CoronalC.

� � � � � � � � � � � � � � � � � � � � � �

Table 5Average malformation (mm) and curvature discrepancy (mm�1) for all subjectclasses. We show results on all cranial bones for each class. In the last two rows weshow the malformation and curvature discrepancy in the fused suture for cranio-synostosis subjects, and the average across all sutures in the case of normal subjects.LP/RP: left/right parietal, LF/RF: left/right frontal, Oc: occipital, Sut: fused suture(average of all sutures for normal cases). Mf: malformation, Cv: curvaturediscrepancy.

Type Normal Sagittal Metopic Coronal

LP Mf. 1.50 ± 0.66 4.99 ± 1.98 2.49 ± 1.88 3.28 ± 1.33LP Cv. 0.51 ± 0.08 0.69 ± 0.12 0.58 ± 0.13 0.60 ± 0.08RP Mf. 1.44 ± 0.62 4.40 ± 2.06 2.96 ± 2.42 2.90 ± 0.91RP Cv. 0.52 ± 0.09 0.75 ± 0.14 0.55 ± 0.09 0.57 ± 0.06LF Mf. 1.13 ± 0.50 2.89 ± 1.52 2.57 ± 1.71 2.46 ± 1.24LF Cv. 0.59 ± 0.16 0.78 ± 0.20 0.92 ± 0.27 0.75 ± 0.15RF Mf. 1.11 ± 0.63 2.93 ± 1.52 3.20 ± 2.07 1.68 ± 0.74RF Cv. 0.53 ± 0.12 0.71 ± 0.20 0.89 ± 0.26 0.68 ± 0.15Oc Mf. 1.78 ± 1.10 5.59 ± 3.07 2.24 ± 1.21 4.14 ± 3.43Oc Cv. 0.54 ± 0.11 0.94 ± 0.31 0.57 ± 0.14 0.71 ± 0.21Sut. Mf. 1.44 ± 0.22 4.11 ± 2.46 3.43 ± 2.26 2.14 ± 1.94Sut. Cv. 0.53 ± 0.02 0.78 ± 0.24 0.89 ± 0.38 0.74 ± 0.14

C.S. Mendoza et al. / Medical Image Analysis 18 (2014) 635–646 643

craniosynostosis. For normal cases we averaged the values for allsutures to provide a comparison benchmark for fused sutures incraniosynostosis subjects. For coronal subjects, the left cranialbones are taken as ipsilateral, and the right as contralateral dueto the polarity inversion mentioned in Section 2.1. Note how thehigh malformation and curvature values appear in different cranialregions depending on the type of craniosynostosis. See for examplehow sagittal cases have the highest curvature discrepancies in theoccipital bone, as opposed to metopic cases with higher curvaturediscrepancies in the frontal bones. Note how the malformation(mean and standard deviation) on coronal subjects stays lowerthan for other types of craniosynostosis for the frontal bone onthe contralateral side, but relatively high in the ipsilateral parietalbone.

Finally in Table 6 we show in more detail the results of the clas-sification procedure for the best method, LDA and CN. We showsensitivity and specificity for each class, and for craniosynostosisin general. Note that our fully automatic methods performsimilarly to the trained human expert (sensitivity = 0.96 and

Table 6Classification results for LDA and CN using FI and shape features. N: normal, S:sagittal, M: metopic, C: coronal. T: true class, A: assigned class.

T A

N S M C Sensitivity per class Specificity per class

N 89 0 1 0 0.989 0.927S 1 25 1 0 0.926 1.000M 2 0 14 0 0.875 0.984C 1 0 0 7 0.875 1.000

Sensitivity for Craniosynostosis = 0.927Specificity for Craniosynostosis = 0.989Posterior Success Probability = 0.957

644 C.S. Mendoza et al. / Medical Image Analysis 18 (2014) 635–646

specificity = 1.00 according to Vannier et al. (1994)), supporting thehypothesis that computer-assisted diagnosis can support the clin-ical management of craniosynostosis. Finally we show that theposterior probability of correctly classifying a new subject usingour technique rises to 95.7%.

4. Discussion

In the previous sections we presented a novel set of technolo-gies to provide a personalized morphological description of cranialanatomy which allowed systematic characterization of malforma-tion in craniosynostosis. By expressing anatomy with respect to anormalized spatial frame that corrects for pose and isotropic cra-nial growth via registration, we compared the shape of subjectsfrom a large age range. Our definition of normality is key to thesuccess of our technique. Normal anatomy on the cranial vaultwas modeled as the structures at the cranial base were registeredto show maximal overlap. By comparing the morphology of eachsubject with the closest normal shape from our anatomicalmulti-atlas of cranial anatomy we detected malformations moreprecisely, as opposed to previous approaches that define abnor-mality in relation to the average cranial shape.

We showed that establishing a cranial shape reference fromaverage does not produce the best results. This is possibly due tothe inability of such reference to adapt to the normal shape varia-tion across subjects, which may be due to various factors such asethnicity (Dean et al., 1998). On the other hand, using constrainedprojections allowed the synthesis of normal shape references tai-lored to the subject under study. The results for this approach werenot the best either. One possible reason is that these reconstruc-tions from constrained projections sometimes can be very similarto abnormal shapes due to craniosynostosis, and thus the mea-sured malformation was too low. We obtained the best resultsby comparing abnormal subjects with the closest normal subjectfrom the multi-atlas in PCA-transformed shape space. A cleardrawback of this approach is the dependence of the analysis withthe particular set of images included in the multi-atlas.

As our ultimate goal is to assist surgical planning, it is crucial tocharacterize dysmorphology in a per-bone basis, since the surgicalcorrection of the calvarium can be performed separately on the

Fig. 6. Comparison of a metopic subject (red) that was wrongly classified as a normal subreferences to colour in this figure legend, the reader is referred to the web version of th

individual bones. Using our graph-cut based excision technique,we circumscribed our analysis to the different anatomical regionsof the cranium. This allowed us to discriminate the different typesof craniosynostosis by detecting distinct patterns of malformationand curvature abnormality, mimicking clinical practice. Also, it al-lowed us to identify the location of the different sutures. Then, bycoupling shape features with our novel fusion index for the identi-fied sutures, our system detected the different types of craniosyn-ostosis with a degree of accuracy comparable with that achievedby trained radiologists (sensitivity: 92.3%, specificity: 98.9%). Wetested the classification using only FI and only the shape features.FI is a good discriminant for craniosynostosis, but not for metopiccases in which the fusion of the suture is not necessarily patholog-ical, and there are also cases with partially fused sutures which arenot clearly identified by FI. Partial volume effects and noise add tothe uncertainty. On the other hand, shape features are great atpicking up abnormal morphologies, but are not indicative of crani-osynostosis for some of the milder cases in the absence of suturefusion cues. Furthermore, shape features are unable to correctlydiscriminate non-synostotic cases with positional plagiocephaly/brachycephaly from craniosynostosis cases with comparable mal-formations. The classifier performing best was LDA, as opposedto typically more robust alternatives like RF and SVM. The reasonmight be that variations among the same class cases are very largefor each class. In such a case, any hyperplane learning methods(SVM) or piecewise hyperplane learning methods (RF) would notwork well.

One possible drawback of our methodology is the loss of infor-mation that occurs as a consequence of representing local shapeabnormality by a few measures (mean, standard deviation, maxi-mum, minimum). This approach was chosen because dealing withhigh-dimensional descriptions seems less intuitive, particularly forinterpretation by clinicians. Side-by-side with a good visualizationof the variation of local shape features, a description in terms of afew parameters computed on anatomically-meaningful regions ismore suitable for routine interpretation by radiologists orsurgeons.

In Fig. 6 we provide a comparison between one of the subjectsclinically diagnosed with metopic craniosynostosis and classifiedby our system as being normal, and its closest normal cranial shapefrom the cranial multi-atlas. Although the ridging of the metopicsuture is clear, the degree of malformation in this case is limitedin comparison to the closest normal shape. In cases like this itmight be preferable to recommend clinical observation, since thecosmetic implications are negligible. A shape quantification systemlike the one described herein might help in detecting cases inwhich diagnosis of premature synostosis may not require surgicalintervention, complementing the judgment of a surgeon. This isparticularly true for metopic craniosynostosis, for which diagnosisis more subjective due to the difficulty of establishing if the fusionof the suture was premature or not.

It is well know that localized abnormalities are commonly ob-served in craniosynostosis patients. What is not common knowl-edge is how these dysmorphologies contribute to the diagnosis of

ject, and the closest normal case in the multi-atlas (blue). (For interpretation of theis article.)

C.S. Mendoza et al. / Medical Image Analysis 18 (2014) 635–646 645

craniosynostosis in general and to the classification of differenttypes of craniosynostosis. The automated diagnosis of craniosynos-tosis has many potential applications. It could reduce variability oftraditional diagnosis using non-automated methods. Additionally,it could enable the analysis of a large number of subjects, whichcould aid in elucidating the epidemiological characteristics of thedisease or enable correlation with genetic factors. Furthermore,our methods can be applied to compare subject’s anatomy beforeand after surgical correction in terms of a desirable morphology,providing an objective assessment of the success of the procedure.Ultimately, our representation of malformation in terms of localphysical distances from a preferred shape reference that is tailoredto the anatomy of each subject and our quantification across ana-tomically meaningful regions could be of great value in planningthe surgical correction of cranial shape.

The presented methodology has been proven to discriminatedifferent types of craniosynostosis (sagittal, metopic, coronal) withan accuracy comparable to that achieved by trained radiologists(sensitivity: 92.3% versus 96%, specificity: 98.9% versus 100%), ona realistic cohort of CT scans with varying resolutions and acquiredwith different scanners. The probability of correctly classifying asubject using our automatic system is 95.7%.

Acknowledgments

The authors would like to thank Tanakorn Kittisarapong for hishelp with data processing. This project was supported by a philan-thropic gift from the Government of Abu Dhabi to Children’sNational Medical Center. Its contents are solely the responsibilityof the authors and do not necessarily represent the official viewsof the donor.

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