2 Anatomical models in medical image analysis

2 Anatomical models in medical image analysis

Anatomical modeling is a rapidly growing field of research, where three main application areas have evolved over the last decades: visualization and education, functional analysis and segmentation. Since the focus of attention in this thesis is directed to anatomical models for segmentation purposes, this section will briefly discuss the first two applica- tions, whereas models aimed at medical image segmentation are discussed in more detail.

2.1 Anatomical models for visualization and educational purposes Recent advances in computer technology have enabled the development of digital ana- tomical models for education and training purposes. For example, highly detailed ana- tomical atlas models are digitally represented in the form of a set of labeled voxels [1-9].

By combining photorealistic renderings of segmented voxel data with background infor- mation about the visualized structures, different aspects of human anatomy can be explored three-dimensionally. The added value of such atlases over plain paper atlases is the facility to interactively explore the spatial structure of an organ in three dimensions, where background information about the function of an organ can be retrieved on demand. A well-known example of such an atlas model for visualization and education purposes is the visible human atlas [4-9], where human anatomy can be viewed interac- tively in combination with its appearance in a number of radiological imaging modali- ties.

A novel educational application of anatomical atlas models is surgical simulation and

training in a virtual reality environment. Again, the rapid progression in computer hard-

ware has triggered the development of a broad spectrum of such applications, where real-

istic atlas models are visualized three-dimensionally, and response to user interaction is

provided by visual and in some cases tactile feedback [10-12]. The actual clinical applica-

tion of virtual- and augmented reality methods in medicine is slowly increasing, though

currently such techniques are mainly applied experimentally in a research setting. Clini-

cal application examples such as augmented reality displays have been described in e.g.

[13, 14]. For a recent overview of the state-of-the-art applications of anatomical models in surgical simulation applications the reader is referred to [15].

A third promising application of anatomical atlas models in combination with visual- ization techniques lies in radiological image simulation. In this application, the appear- ance of different organs in a radiological image is simulated by modeling the entire imaging chain from physical tissue characteristics to the underlying physical principles and transfer functions of an imaging modality. This way, the role of specific parameters in the imaging chain on the resulting image conditions can be investigated. Examples of this application of anatomical atlas models have been described in [16-19] for nuclear imaging and in [20, 21] for magnetic resonance images.

2.2 Anatomical models for functional analysis

A second important application field of anatomical models in medicine is the analysis and simulation of physical processes as they occur throughout the human body. By cou- pling physiological knowledge with image data, a better insight can be obtained in the features that distinguish normal from pathological conditions. An excellent example of this application is provided by Kaye [22], who modeled cardiothoracic interactions dur- ing respiration and the changes therein as a result of a pneumothorax based on an ana- tomical model of the thoracic contents. Other examples of the applications of anatomical models for functional analysis are given in [23, 24].

A specialized application area of analytical anatomical models is formed by the dynamic cardiac deformation models [25-63], which are aimed at modeling the contrac- tion pattern of left [27-62] and right [63] ventricles over the cardiac cycle. These defor- mation models are dedicated to the tracking and analysis of wall motion, often by means of a set of characteristic parameter functions, from which wall motion abnormalities can be recognized as deviations from a normal set of functions (e.g.[27, 42, 43]). Further- more, regional parameters for cardiac function like wall stress and strain can be estimated from the deformations mapping the model to an image set.

A third class of analytical shape models is aimed at quantitatively analyzing anatomi-

cal variations and group differences in organ shapes over a population [64-76]. The idea

behind such models is the definition of a non-linear transformation (B-spline basis func-

tions [64], thin-plate spline interpolants [65-69], elastic-[70] or viscous fluid deforma-

tions ([71, 72, 74, 76])), which allow the mapping of a set of training samples in a

standardized space. The deformation effort required to map a shape sample onto the

standardized shape template is a measure for the shape difference between a sample and

the template model. With these models insight can be obtained in anatomical variabili-

ties, where pathological organ shapes can be distinguished from normal shapes [69, 74-

76]. Many of such statistical shape models have been applied for segmentation purposes

as well, as will be discussed later in this section.

2.3 Anatomical models for medical image segmentation

The need to incorporate prior knowledge into image segmentation methods is nowadays widely recognized. Especially in medical imaging, where many aspects of the imaging conditions are difficult to control, the incorporation of knowledge about the shape, loca- tion, appearance and spatial context of an organ is essential. In this section, different classes of anatomical models are discussed in the context of a commonly applied subdivi- sion of segmentation methods into five abstraction levels. From high-level down to low- level operations, one can distinguish the scene level, the single object level, the image entity level, the low-level segmentation level and the preprocessing level (see Figure 2.1).

Each stage in the image interpretation hierarchy is classified according to the degree of prior knowledge involved in operations at that level. The different anatomical modeling methods described in the literature can be placed mainly at the top three levels of the image interpretation pyramid, and are discussed in a bottom-up order.

Physically-based deformable models (snakes):

A widely acknowledged object representation applied for segmentation of medical imag- ery is the deformable model, which has been recently surveyed in Singh [77]. Terzopou- los [78] introduced the deformable model concept to create realistic computer

animations by viewing an object surface as an elastic sheet and deforming the object by mimicking the physical deformation behavior of the sheet. The first application to image

Preprocessing Low-level image analysis Lower image interpretation

Raw images Filtered images Image features: edges,

regions, texture Scene elements: 3D-surfaces,

volumes, contours Objects

Object recognition High level scene interpretation Scene

Figure 2.1: General classification of segmentation methods.

segmentation was described by Kass [79], who introduced the nowadays widely applied active contours, also referred to as snakes. Roughly, three types of snakes have evolved since then: parametric snakes, implicit snakes and probabilistic snakes.

Parametric snakes

In brief, a parametric snake is a curve expressed in coordinate functions (x(s) and y(s)), where s represents the parametric domain [0,1]. The shape of the contour is governed by an energy functional:

(2.1) where the first integral term represents an internal deformation energy of the model, which is balanced with an external scalar field P(v), typically defined from an image fea- ture such as the local image gradient. Parameter functions w

(s) and w

(s) represent two physical properties of the contour, i.e. the ability to stretch and bend respectively. These functions can be used to impose a preferred shape on the model and to locally control shape characteristics like the object smoothness of the resulting segmentation. The third term E

represents shape constraints introduced by the user, e.g. by fixating a point of the contour to an image point. The final shape of an active contour in an image corre- sponds to a minimum in E(v), which can be found by numerically solving the Euler- Lagrange equation of Equation 2.1.

Extensions of the two-dimensional snake model to three dimensions (deformable bal- loons) have since been reported [49, 80-94], as well as many modifications of the origi- nal energy formulation to improve robustness of the snake- and balloon methods with respect to spurious feature points and initial positioning [81, 95, 96], transitions in topology [97, 98] and simultaneous detection of multiple objects [99]. Applications to left-ventricular segmentation in dynamic cardiac MR-images are given in e.g. [53, 62, 84, 100-104]

Implicit snakes

One of the practical limitations of parametric snakes is the requirement of an initial guess, which is reasonably close to the desired shape. Furthermore, these snakes are not suitable to describe shape protrusions or extrusions that a shape may posses. A different class of physically-based deformable models designed to circumvent these shortcomings is the level-set approach [105-111], which simulates the propagation of wave fronts with curvature dependent speeds. In [108, 109], the original formulation of Cassalles [105, 106] is modified to an energy minimization, whereas in [111, 112] an extension for simultaneous segmentation of multiple objects is described. An advantage of these implicit snakes over the parametric snake formulation is the lack of assumptions made about topological structure of an object. Therefore topologically adaptable snakes have shown to be useful for segmentation of complexly shaped objects of which little prior shape knowledge is available, for instance branching vessel structures. However, in the

E v w s v

s w s v

s ds P v s ds E_userds

( ) = ( ) ∂ ( ) ( ( ))

∂ + ∂

∂ Ê

Ë Á ˆ

¯ ˜ + +

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absence of geometric shape constraints other than connectivity, the ways to incorporate shape knowledge are limited in case prior knowledge is available.

Probabilistic snakes

Due to the locally distributed nature of prior shape knowledge of the original parametric and implicit snake formulations, the facilities to incorporate shape knowledge other than smoothness constraints to restrict the allowable shape domain are limited. Therefore these snakes are less suitable for object recognition purposes and can be generally classi- fied to the low-level image interpretation stage in Figure 2.1. As a result of this, paramet- ric and implicit snakes are mainly suitable for application in a highly interactive setting.

By selecting a shape parametrization expressed on an orthonormal basis, i.e. a repre- sentation that allows the definition of an object shape as a weighted sum of known basis functions, the shape parameters become physically interpretable. A preferred shape can thus be imposed, based on the parameter distributions over a set of training samples, where the snake is allowed to deform following population-based shape deformations.

The probabilistic snake is preferentially attracted towards feature patterns in the image data, which are consistent with its trained shape. This makes the model matching more robust with respect to initial position and noise and therefore these snakes can be classi- fied to the object recognition level in Figure 2.1.

Applications of probabilistic snakes have been described in Vemuri et al. [113-115], who developed a model based on a deformable superquadric in combination with a locally superimposed deformation field expressed on an orthonormal wavelet basis. Staib et al. describe a two-dimensional probabilistic snake [116] and a three-dimensional [103] probabilistic balloon model applicable to deformations of four surface topologies constructed on a Fourier basis, whereas orthonormal Fourier parametrizations have been described in [117-119] that are applicable to segmentation and recognition problems of free-form closed surfaces.

Statistical shape models

Probabilistic snakes describe a shape and its natural variations by means of the parameter distributions of a shape parametrization and have shown to be a powerful representation for population based shape knowledge. The necessary prerequisite of a parametrization on an orthonormal basis however introduces limitations to the shape topology of these models. In contrast, statistical shape models do not require a shape parametrization, and are therefore not subjected to the topological or shape constraints intrinsic to a lumped parameter model. This makes these statistical models suitable to describe free-form shapes consisting of multiple objects simultaneously.

A widely acknowledged statistical shape model is the Point Distribution Model

(PDM) as introduced by Cootes et al. [120-125]. A Point Distribution Model (PDM)

describes the average shape and characteristic shape variations of a set of training sam-

ples, which are given in the form of a set of points on the sample boundaries. By apply-

ing an affine registration of the shape samples, the most characteristic local shape

variations around a shape average can be extracted by means of a principal component analysis on the sample point distributions. The only necessary condition for the calcula- tion of a point distribution model is the definition of a point-correspondence between points in successive training samples, which ensures a compact and specific model. In two dimensions this point correspondence is typically defined in application specific assumptions, which are difficult to generalize in three dimensions. The development of more generic methods to define point correspondence for two [126-128] and three [129, 130] dimensions is currently an active field of research.

The application of point distribution models to image segmentation is known as Active Shape Model (ASM). A key difference between the ASM matching method and the matching mechanism for parametric and implicit snake models is the absence of an energy functional based on elastic material properties. For ASM’s, the image matching is performed by calculating a suggested boundary location for each point in the PDM based on image information. This allows an elegant coupling between high-level knowl- edge about object shapes and low-level image features. The suggested boundary hypoth- eses can be generated either using a simple edge filter, but also using custom edge filters for each point in the shape samples, a statistical gray-value model for each sample point or other forms of prior knowledge about an organ’s image appearance [122, 131-133].

The model pose- and shape parameters are iteratively updated to optimally fit the hypothesized shape, where the model is only allowed to deform along the most charac- teristic eigendeformations. A final solution is reached when the generated candidate boundary points coincide with the model boundaries.

A second important statistical shape modeling method for segmentation purposes is based on spatial normalization of a set of training images (2D) or image volumes (3D).

By optimally registering a set of segmented voxel volumes into a standardized space by applying affine [134-140] or higher dimensional transformations such as thin-plate spline interpolants [141-143], an image scene can be expressed as a probability map or as an average shape with a locally defined variance measure respectively. These models are generally applied to segmentation problems by weighting a feature-based probability density function with a spatial probability distribution of an organ shape [138, 139, 141-143] in a Bayesian formulation.

Because statistical shape models allow a coupling between low-level image data and higher level knowledge about individual organ shapes and their spatial context in a scene, they can be classified to the scene interpretation level in the image interpretation hierarchy in Figure 2.1.

Boundary template models

Boundary template models (boundary = 2D contour or 3D surface) can be seen as pre-

shaped deformable models of an organ boundary which are only allowed to deform

within restricted modes of deformation. Generally, boundary templates represent one

shape instance of a particular organ or set of organs in the form of one 2D contour [144-

150], a set of contours forming a 3D surface[151] or a set of coupled analytical primi-

tives, as has been demonstrated in 2D by Yuille[152], who describes the eye and the mouth as a combination of circles and parabolic arcs. In 3D this analytical approach has been exemplified by Delibasis [153], who modeled part of the brain stem as a combina- tion of globally deformable superquadrics. Furthermore, in Chapters 4, 5 and 6 of this thesis [154-156], a novel boundary template model is presented that describes the tho- racic anatomy as a set of analytical primitives combined by means of Constructive Solid Geometry.

The matching of boundary templates is performed by either balancing an internal energy term with an external energy similar to snake-based approaches [144, 151], using global cost optimization strategies based on dynamic programming [144, 146-150], or by optimizing an energy function directly on explicit model parameters, thereby omit- ting an internal energy function [152-156]. During the matching, the allowed deforma- tion modes are restricted by prior knowledge, which is not necessarily population-based.

Such prior knowledge can be for instance an allowed shape interval on a set of radials along the template boundaries [145], assumptions about small deformations from the template shape [151], deformations along cascaded affine transforms [154-156], a restriction of the template deformations along orthogonal curves [144], or a restriction on the parameter bounds in analytical templates [153].

Due to the restricted degrees of deformation in boundary template matching, explicit prior knowledge can be imposed on the shape and its deformations resulting in more robust matching behavior, although boundary templates are less flexible than other snake approaches. Due to their relative rigidity, the boundary templates are applicable to single objects [144, 145] as well as to scenes consisting of multiple objects [151, 152, 154- 156], where an optimum is sought for the scene model as a whole. Boundary templates can therefore be classified to the object level and the top (scene) level in Figure 2.1.

Volumetric templates

Volumetric templates consist of a segmented voxel set, which is often constructed by

manually delineating a number of organs in a representative image set, and are therefore

derived from a single shape instance. Such models are matched to image data from a dif-

ferent subject by deforming the model on the basis of attraction forces, which are derived

from local similarity measures. The dimensionality of the defined transformations deter-

mines the matching accuracy that can be achieved with these deformations. Several

approaches have been described based on affine [137] , piecewise affine [157], non-rigid

[158], elastic [159, 160], thin-plate spline interpolants spanned by landmarks [161] and

viscous fluid deformations [71, 162-167], where such template models are most com-

monly applied to segmentation of brain structures. In [166, 167] examples of LV seg-

mentation from SPECT [166] and MR [167] image data have been described. Due to

the locally distributed nature of shape knowledge in these models, the model-image

matching is computationally an order of magnitude more expensive than for boundary

template approaches.

Since volumetric templates are matched as a whole, the topological structure of the model is preserved throughout the matching procedure, which makes it suitable to seg- ment multiple objects in a scene simultaneously. Therefore, these models can be placed on the scene interpretation level in the image interpretation pyramid in Figure 2.1.

High level scene models

On the highest level of abstraction in the image interpretation pyramid the high-level scene models can be found. These models explicitly describe knowledge about the scene domain under investigation. This can be knowledge about a typical size or volume of an organ, the typical image features for a set of organs in a particular image modality [168], the spatial relations of different organs with respect to each other [169-171] or the spa- tial embedding of objects in the image scene by means of a Voronoi diagram ([172], Chapter 3 [173] in this thesis). Two common representations for such high level knowl- edge are explicit rules [174, 175] and semantic- or frame networks [168-171, 176].

Explicit rules store knowledge in the form of ‘if ... then’ rules. This representation is often chosen to represent procedural knowledge consisting of a large number of discrete facts, and is highly flexible in its application, easy to extend and allows combination of multiple rules in a straighforward manner. Examples of the application of explicit rules to trace the intrathoracic airway trees in CT images and to semantically interpret MR brain scans are given in [175, 177] respectively.

Semantic- and frame networks are structured object descriptions, which describe a set of objects and their mutual relations as an attributed graph. Each object is described with a record data structure containing slots, which can be attributed a symbolic or numeric value. Relationships between different records are described by links, which characterize inheritance relations, neighborhood relations and part-subpart hierarchies. Frame- and semantic net representations are commonly applied in a framework for hypothesis gener- ation and verification such as a blackboard system [168, 171]. In [170, 171, 175, 178], detailed implementations of frame representations have been described for knowledge driven segmentation of MR images of the brain, whereas applications to segmentation of thoracic- and abdominal CT scans have been described in [168, 169] respectively.

2.4 Summary

This chapter provides an overview of the applications of anatomical models in medical

image analysis. Anatomical models have rapidly become valuable tools for visualization,

educational, functional and shape analysis and segmentation purposes. In particular for

segmentation applications, anatomical models can be utilized to extend segmentation

algorithms with prior knowledge about the scene under investigation. This application is

discussed in detail following a common subdivision of image segmentation into five

abstraction levels in an image interpretation pyramid: preprocessing, low-level segmenta-

tion, lower image interpretation, object recognition and scene interpretation (see Figure

2.1). Anatomical models for segmentation purposes can be classified mainly to the high- est three levels in this image interpretation hierarchy.

Anatomical models on the lower image interpretation level are mainly limited in application to the formation of coherent scene elements such as contours and regions from low-level image information. Parametric and implicit snakes are examples of such models at this level, because the facilities to impose prior shape knowledge other than smoothness and connectivity constraints are limited. On the object recognition level, trainable models such as probabilistic snakes, point distribution models and object tem- plates can be distinguished, which combine prior knowledge about an average object shape and its characteristic shape deformations with domain specific low-level feature knowledge. At the highest level of abstraction, which involves the simultaneous process- ing of multiple objects within a scene, point-distribution models, volumetric template models and boundary template models can be utilized to describe the shapes of multiple organs in the scene within their spatial context. These models allow a coupling between high-level anatomical knowledge and low-level image features by defining different types of transforms, which enable a data driven deformation of the model to a feature pattern in the image data. A second class of anatomical models on the scene interpretation level are the high level scene models, which explicitly describe knowledge about the scene domain under investigation in a set of rules, a frame representation or a semantic net- work. These knowledge representations are commonly applied in a framework for hypothesis generation and verification such as a blackboard system.

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