Studying Anatomy And Disease In Medical Images Using Shape Analysis

DANIEL GOLDBERG-ZIMRING1, DOMINIK S. MEIER3, SYLVAIN BOUIX4 and SIMON K. WARFIELD1'2 Computational Radiology Laboratory, Departments of Radiology 1Brigham & Women's Hospital, Harvard Medical School 75 Francis St. Boston, MA, 02115 USA

2Children's Hospital, Harvard Medical School 300 Longwood Ave. Boston, MA, 02115 USA

3Center for Neurological Imaging, Department of Radiology Brigham & Women's Hospital, Harvard Medical School 221 Longwood Ave. Boston, MA, 02115 USA

4Department of Psychiatry, Harvard Medical School VA Boston Healthcare System, Brockton, MA, 02301 USA [email protected] [email protected] harvard.edu [email protected] harvard. edu [email protected]

The recent development of image analysis and visualization tools allowing explicit 3D depiction of both normal anatomy and pathology provides a powerful means to obtain morphological descriptions and characterizations. Shape analysis offers the possibility of improved sensitivity and specificity for detection and characterization of structural differences and is becoming an important tool for the analysis of medical images. It provides information about anatomical structures and disease that is not always available from a volumetric analysis. In this chapter we present our own shape analysis work directed to the study of anatomy and disease as seen on medical images. Section 2 presents a new comprehensive method to establish correspondences between morphologically different 3D objects. The correspondence mapping itself is performed in a geometry- and orientation-independent parameter space in order to establish a continuous mapping between objects. Section 3 presents a method to compute 3D skeletons robustly and show how they can be used to perform statistical analyses describing the shape changes of the human hippocampus. Section 4 presents a method to approximate individual MS lesions' 3D geometry using spherical harmonics and its application for analyzing their changes over time by quantitatively characterizing the lesion's shape and depicting patterns of shape evolution.

Keywords: Shape analysis; shape correspondence; parameterization; spherical harmonics; skeletons; medial representations.

1. Introduction

The advent of fast three-dimensional (3D) magnetic resonance imaging (MRI) has enabled the routine acquisition of high spatial resolution digital representations of anatomy in vivo. The recent development of image analysis and visualization tools allowing explicit 3D depiction of both normal anatomy and pathology provides a powerful means to obtain morphological descriptions and characterizations.

Shape analysis offers the possibility of improved sensitivity and specificity for detection and characterization of structural differences. A number of approaches for characterizing shape and shape changes have been developed. These methods differ in the underlying representation of shape, in the type of shapes that may be modeled, and in their capacity to differentiate shape changes. Different shape analysis methods have been proposed for a variety of clinical applications. A landmark based shape analysis has been applied to study the effect of different diseases on brain structures. For this landmark-based approach, shape is defined as the information about the landmark configuration that remains unchanged under adjustment of position, orientation or scale.1 Landmarks are manually located on the surface of the studied structures seen on magnetic resonance (MR) images. These landmarks are used to create an average shape of the studied structures, which is later compared to each individual structure deriving displacement maps that are helpful to visualize group differences and to discriminate among clinically meaningful categories.2'3 This approach has been widely used for the study of brain structures in schizophrenia. Buckley et al.2 studied the ventricular dysmorphology in schizophrenia, DeQuardo et al.3 analyzed ventricular enlargement during first-episode schizophrenia, while Tibbo et al.4 studied the corpus callosum shape in male patients with schizophrenia. In a different study landmarks were used to analyze the shape of the corpus callosum and subcortical structures in the fetal-alcohol-affected brain.1

Other shape analysis methods based on MR images have also been applied to the study of different brain structures related studies. All of these shape analysis methods are regularly applied after the corresponding segmentation and registration of the images. Velakoulis et al.5 defined the shape of the hippocampus as the volume of contiguous slices and use it to analyze both, the overall and the behind of the head of the hippocampus volume loss in chronic schizophrenia. Levitt et al.6 studied the correlation between the shape of the caudate nucleus and cognition in neuro-naive schizotypal personality disorder. In that study, they generated a 3D rendering of the caudate nucleus and estimated a shape index using the ratio between the surface area and the volume of the caudate nucleus, which indicate how much a given shape differs from a sphere. Hogan et al.7 studied Hippocampal shape analysis in epilepsy and unilateral mesial temporal sclerosis. They used a color scale showing degrees of outward and inward deviation of each studied hippocampus coregistred to a previously generated average hippocampus. Posener et al.8 also analyzed the shape of the hippocampus but in cases of depression, by superimposing a triangulated mesh of the hippocampus on a hippocampus template and representing the displacement of each graphical point in the mesh by a vector. Sowell et al.9 extracted the cortical surface from the MR images and created a 3D mesh-like render in order to analyze brain surface abnormalities in children, adolescents and young adults with prenatal alcohol exposure. They also assessed the relationship between cortical gray matter density on the brain surface and brain shape. The shape analysis was point-by-point radial distanee estimation between eaeh individual eortieal 3D surfaee and an average 3D eortieal surfaee model. Zilles et al.10 analyzed the inter-subjeet variability of hemispherie shape and defined gender and inter-ethnie differenees between the hemispherie shape of male and female European and Japanese brains. They eonstrueted a mean brain model from all brains and for eaeh sample, and later measured the distanee between eaeh voxel at the surfaee of an individual brain and the eorresponding surfaee voxel of the mean brain. The variability was displayed as a funetion of the standard deviations of the mean absolute distanees between all voxels of the mean brain surfaee and all the eorresponding surfaee voxels of all individual brains. Studholme et al.11 examined a method for the analysis of Jaeobian determinant maps eapturing loeal anatomieal size differ-enees. They applied this method to the study of dementia in Alzheimer's disease and aging by eapturing shape differenees between an individual and a referenee anatomy or between repeated seans of an individual.

Shape analysis has also been used as a tool to develop or further improve some image segmentation and registration algorithms. Pizer et al.12 eonstrueted a stable effieiently ealeulable measure of shape and other geometrie objeet properties and applied it in a uniform method for segmentation and reeognition of image objeets, objeet-based registration, and objeet shape measurement. Tsai et al.13 derived a model-based, implieit parametrie representation of the segmenting eurve and ealeulated the parameters of this implieit model to minimize the region-based energy funetion for medieal image segmentation. Goldberg-Zimring et al.14 proposed an algorithm for the two-dimensional (2D) segmentation of multiple selerosis (MS) lesions based on the assumption that MS lesions have a relatively eireular shape. After an initial global thresholding of the images, for eaeh deteeted segment a shape index and the average intensity inside the region of interest was estimated. This shape index indieating the eloseness of eaeh segment to a eireu-lar shape was used together with the average intensity value as the input for an artifieial neural network in order to diseriminate between MS lesions and deteeted artifaets.

While most of shape analysis teehniques have taken the advantage of MRI and its high resolution and quality images, this kind of analysis ean be applied to studies based on any kind of medieal images. For example, Christodoulou et al.15 developed an algorithm for the automated eharaeterization of earotid plaques reeorded from high-resolution ultrasound images. Carotid plaques were eharaeterized based on extraeted texture features and shape parameters sueh as the X and Y eoordinates maximum length, area, perimeter, and the relation perimeter2 /area.

In this ehapter we present our own shape analysis work direeted to the study of anatomy and disease as seen on medieal images. Seetion 2 presents a new eompre-hensive method to establish eorrespondenees between morphologieally different 3D objeets. Seetion 3 presents a method based on the 3D skeleton to study the effeet of aging on the human Hippoeampus, and See. 4 presents a method to approximate MS lesions' 3D geometry using spherieal harmonies (SH), and to analyze MS lesions ehanges over time.

2. Shape Correspondence

2.1. The correspondence problem

Longitudinal and cross-sectional studies of anatomical shape and structures in clinical images have become a common task in diagnosis, treatment evaluation, surgical planning and exploratory clinical research.

Such morphometric analysis of medical images often requires a one-to-one mapping, or correspondence, between anatomic structures in two or more different but not disparate image sets. So the structures of interest will be similar but morphologically different, such as when a changing structure is imaged over time, e.g., in dynamic cardiac imaging,16'17 the study of disease progression,18'19 or the measurement of local changes in response to therapy.20 Cross-sectional analysis includes the study of anatomic shape variation between normal and pathological structures,21-23 and non-rigid alignment of a standardized anatomic atlas to individual data sets.24'25 In many of these cases, the principal challenge of a correspondence search is that a unique solution for matching points on dissimilar objects does not necessarily exist. Thus, the problem also entails the definition of "allowable" or "probable" correspondences according to basic physical constraints or application-specific knowledge.

This section presents a new comprehensive method to establish correspondences between morphologically different 3D objects. Similarities between shape features guide the correspondence and constraints imposed on a global optimization prevent physically unrealizable matches. The correspondence mapping itself is performed in a geometry- and orientation-independent parameter space in order to establish a continuous mapping between objects. We will present this in three parts: the first describes the concept of shape description by parameterization via SH, the second contains the definition of "Parameter Space Warping" and the third shows example applications of this technique.

2.2. Shape parameterization

The representation of 3D shapes is a well-recognized challenge in computer vision. Parametric models have received particular attention, because they provide concise and robust representations of shape properties well suited to object comparison.

A rough classification of object descriptors distinguishes between local and global representations, with respect to composition, and between implicit and explicit representations with respect to formulation.

Local representations describe the object by a connected series of primitives, such as planar polygons, edges etc., whereas global representations use a single (parametric) function in a global object-oriented coordinate system.

An implicit representation postulates a given relationship, which all points belonging to the object must satisfy. The equation (x/a)2 + (y/b)2 + (z/c)2 = 1, for example, is an implicit representation of an ellipsoid in Cartesian coordinates

[x, y, z], whereas [x,y, z]T = [acos(0)cos(<^),6cos(0)sin(<^),csin(0)]T is an explicit representation of the same ellipsoid, using spherical coordinates (0,

An essential advantage of using a parametric description for shape comparison is that it reduces the dimensionality of the domain, i.e. for a 3D object we step from K3 ^ K2, i.e., a 3D object is represented by a 2D parameter space, a 2D curve is represented by a 1-dimensional (1D) parameter space, etc. Hence we invest extra work in creating an explicit description of the object, but then have the benefit of a much smaller domain, which gives us a reduced search/optimization space.

Obviously, the -mapping of an object geometry onto such a reduced parameter space critically determines the quality of its description. In other words, how we map the geometry will determine how useful the parameter space will be in representing the object's topology. If we choose our parameter space unwisely, the mapping can become very hard to do and we may have to deal with complicated constraints and boundary conditions (BC). It is advantageous, therefore, to choose a map that is topologically equivalent, i.e., a space that complies with the intrinsic BC of the data. BC not addressed by the geometry of the parameter space will have to be formulated explicitly as external constraints, eventually complicating the description and especially the mutual comparison of parameterized objects. For example, the mapping of closed contours and surfaces is useful in a parameter space with an inherent periodicity, like a circle or sphere, respectively, compared to a planar map like the unit square.

2.2.1. Shape parameterization: spherical mapping

Spherical harmonics (SH) are discussed here as basis functions to represent the object surface in parameter space. Because different regions with varying size and shape are to be compared in the correspondence search, it is important that the object representation fulfill the requirement of equidistant sampling, which means that sampled points should all be approximately the same distance from each other. In terms of a 3D object mapping, this is formulated as an equal-area (homologic) mapping of the object onto the unit sphere. Such a mapping demands that the size of an objects representation on the sphere is proportional to its original, size. For example, a region occupying 20% of the object surface should also cover 20% of the map area, in this case 20% of the unit sphere.

Generally, a homologic spherical mapping will require an optimization of some kind. For the examples shown here, a method developed by Brechbuhler et al.26 was used. This method starts from an initial, continuous mapping onto the unit sphere, obtained from solving the Dirichlet problem V20 = 0 and V2^ = 0 for latitude 0 (Fig. 1) and longitude respectively, with boundary conditions 0NP = 0, 0SP = n, where NP and SP are two arbitrary object points that are mapped to the sphere's north pole and south pole, respectively. Finally, longitude must obey an additional periodicity constraint ^>(t + 2n) = ^>(t), which is the equivalent of the geographical "dateline".

Fig. 1. Example of the spherical mapping in two stages: first, an initial mapping is obtained by solving the Dirichlet problem, with two points selected as north an south-pole as the boundary conditions (left). A second step then runs an optimization that seeks a homologic mapping, i.e. distributes all sample points equally over the sphere without overlaps or excessive distortions (right). The latitude ranges from 0 to n. The object shape is one lateral ventricle obtained from brain MRI. NOTE: A color version of this book chapter is available from the authors.

Fig. 1. Example of the spherical mapping in two stages: first, an initial mapping is obtained by solving the Dirichlet problem, with two points selected as north an south-pole as the boundary conditions (left). A second step then runs an optimization that seeks a homologic mapping, i.e. distributes all sample points equally over the sphere without overlaps or excessive distortions (right). The latitude ranges from 0 to n. The object shape is one lateral ventricle obtained from brain MRI. NOTE: A color version of this book chapter is available from the authors.

Thus, there are two sets of partial differential equations to solve, one for latitude 0, and another for longitude This initial spherical projection is then subject to an additional optimization to create an isotropic distribution of the object surface area onto the sphere.26

2.2.2. Parameterization: spherical harmonics

Once a spherical map is obtained, a parameterization of the object surface x is given as a SH series:

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