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Morphometric analysis of the sacrum using statistical shape modeling

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Title:
Morphometric analysis of the sacrum using statistical shape modeling
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Milligan, Kenneth K. ( author )
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Denver, Colo.
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University of Colorado Denver
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Master's ( Master of science)
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University of Colorado Denver
Degree Divisions:
Department of Bioengineering, CU Denver
Degree Disciplines:
Bioengineering

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Three-dimensional modeling ( lcsh )
Sacrum ( lcsh )
Sacrum ( fast )
Three-dimensional modeling ( fast )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Statistical shape modeling techniques were used to quantify the 3 dimensional anatomic variations of the sacrum and sacroiliac joint surface in the general popu- lation. 25 pelvises were segmented from a from database 223 patients who had CT studies performed for evaluation of abdominal pain at the University of Colorado Hospital. Surface STL meshes were created using Simpleware ScanIP. The statistical shape models was created using the Coherent Point drift algorithm for registration and correspondence. SSM analysis and ordination was performed utilizing procrustes analysis and principal component analysis(PCA). PCA provides dimensionality re- duction of the data for visualization and interpretation of shape features. We found variation in the sacroiliac joint morphology could be quantitatively described using our CPD-SSM technique. We also found the first, second and third PCA modes describe changes in the sacroiliac joint shape.
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Includes bibliographical references.
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Statement of Responsibility:
by Kenneth K. Milligan.

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University of Colorado Denver Collections
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Auraria Library
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982960086 ( OCLC )
ocn982960086
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LD1193.E56 2016m M55 ( lcc )

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MORPHOMETRIC ANALYSIS OF THE SACRUM USING STATISTICAL
SHAPE MODELING by
KENNETH K MILLIGAN B.S., Colorado State University, 2008
A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Bioengineering
2016


This thesis for the Master of Science degree by Kenneth K Milligan has been approved for the Bioengineering Program by
Dana Carpenter, Advisor Kendal Hunter, Chair Vikas Patel Dana Carpenter Stephen Humphries


Milligan, Kenneth K (M.S., Bioengineering)
Morphometric Analysis of the Sacrum using Statistical Shape Modeling Thesis directed by Assistant Professor Dana Carpenter
ABSTRACT
Statistical shape modeling techniques were used to quantify the 3 dimensional anatomic variations of the sacrum and sacroiliac joint surface in the general population. 25 pelvises were segmented from a from database 223 patients who had CT studies performed for evaluation of abdominal pain at the University of Colorado Hospital. Surface STL meshes were created using Simpleware ScanIP. The statistical shape models was created using the Coherent Point drift algorithm for registration and correspondence. SSM analysis and ordination was performed utilizing procrustes analysis and principal component analysis (PC A). PC A provides dimensionality reduction of the data for visualization and interpretation of shape features. We found variation in the sacroiliac joint morphology could be quantitatively described using our CPD-SSM technique. We also found the first, second and third PCA modes describe changes in the sacroiliac joint shape.
The form and content of this abstract are approved. I recommend its publication.
Approved: Dana Carpenter
m


DEDICATION
This thesis is dedicated to Julie, Kayleigh, and Colin.


ACKNOWLEDGMENT
This thesis would not have been possible without the help of Eduardo Novais and Zachary Wuthrich.
v


TABLE OF CONTENTS
Figures .................................................................. vii
1. Introduction............................................................ 1
1.1 Sacroiliac Joint........................................................ 1
1.2 Sacral Anatomy.......................................................... 2
1.3 Morphometric techniques................................................. 3
1.4 Statistical Shape Modeling.............................................. 6
1.5 Shape Parameterization.................................................. 7
1.6 Correspondence.......................................................... 9
1.7 Registration Based Correspondence Methods.............................. 11
1.8 Non-registration based correspondence methods.......................... 13
1.9 Building the Shape model............................................... 13
2. Methods................................................................ 15
3. Result................................................................. 18
4. Discussion............................................................. 26
5. Conclusion............................................................. 30
5.1 Future Work ........................................................... 30
References................................................................ 32
vi


FIGURES
Figure
1.1 Shape classification of the Sacroiliac joint surface based off of a maximum
intensity projection reconstruction........................................... 2
1.2 Frontal and lateral uiew of a 3D surface reconstruction of the sacrum . 3
1.3 Landmarks haue been placed on the points of the stars..................... 5
1.4 Three identical right triangle modified only by translation, rotation, and
scale......................................................................... 6
1.5 Screen capture of the segmentation and 3D reconstruction process. The
software used is Sirnpleware ScanIP suite..................................... 8
1.6 Illustrates the concept of correspondence. Points in the shape rectors must must indexed according to their location on each shape being compared. [37] 10
1.7 Illustration of iteratiue closest points registration. A point mi is automatically corresponded to the closest point is S................................ 12
2.1 Screen capture showing points clouds of the sacrum and alignment of
shapes centroids to the origin............................................... 16
3.1 Three uiews of the of the final shape model using the CPD code............... 18
3.2 Surface reconstruction of the mean shape................................... 19
3.3 Surface reconstruction of the First PC mode. Screen captured from a
coronal perspectiue.......................................................... 20
3.4 Surface reconstruction of the First PC mode. Screen captured from a
lateral perspectiue.......................................................... 21
3.5 Surface reconstruction of the Second PC mode. Screen captured from a
lateral perspectiue.......................................................... 22
3.6 Surface reconstruction of the Third PC mode. Screen capture from a coronal perspectiue ............................................................. 23
vii


3.7 Surface reconstruction of the fifth PC mode. Screen captured from a lateral
perspectiue.................................................................. 24
3.8 Compactness is defined as the number of modes needed to described a giuen
amount of uariation. More compact models can represent greater uariation with few modes. [51, 16]............................................... 25
4.1 Screen capture showing a point cloud SSM of sacrum. This model was
built using a target shape point cloud of 54,700 points................... 27
4.2 The left image is a surface reconstruction from a high density SSM containing 57400 points. The right image is a surface reconstruction from a
low density SSM containing 6993 points.................................... 28
4.3 Illustrates how the compactness of the models change with density of the
point cloud used in the SMM.................................................. 29
viii


1. Introduction
1.1 Sacroiliac Joint
The sacroiliac joint (SI joint) is the interface between the spine and the pelvis. This joint has been implicated as the cause of 10-30% of all low back pain and buttock pain. [10] There are many causes of SI joint pain including inflammatory conditions, trauma, infection, and degenerative osteoarthritis (OA). In the case of degenerative OA the mainstays of treatment are activity modification, NSAIDS, SI joint injections, RF- ablations. Recently there has been an increased use of minimally invasive surgical arthrodesis of the SI joint to relieve pain in cases that fail conservative treatments.
Increasing use of SI joint fusion treatments has created renewed interest in the biomechanical properties of the SI joint. [36] Previous biomechanical studies have looked at motion of the joint, stresses and strains on the articulating surfaces and main supporting ligaments. Even more recent has been an industry supported study comparing two SI joint trajectories and there effects on range of motion in flexion extension, lateral bending, and axial rotations. [49]
All of these previous studies however have not investigated or mentioned the potential effects of SI joint geometry on the biomechanics of the joint. Recent work at the university of Colorado has highlighted the immense variability of the SI joint shape in the general population and attempted to make preliminary classifications of shapes. However these shapes classifications were from 2-D projections of the articular surface. Which prohibited us from commenting on the surface variation from a 3 dimensional perspective.
1


section
Figure 1.1: Shape classification of the Sacroiliac joint surface based off of a maximum intensity projection reconstruction.
1.2 Sacral Anatomy
The Sacrum is one of three bones that make up pelvis. Its general appearance is wedge shaped with the apex pointing inferiorly. Five fused vertebrae, which are oriented dorsally convex, make up the sacrum. There are three joint surfaces of the sacrum. The sacral base is broad circular shape that articulates with the intervertebral disc. The other two joints surfaces are the sacroiliac joints. They are generally described as auricular in shape. [52, 56] However there exist an immense amount of diversity between individuals. The SI joints articulate with the right and left iliac bones. These joints along with the surrounding soft tissue structures transmit forces from the torso to the lower limbs.
2


Figure 1.2: Frontal and lateral view of a 3D surface reconstruction of the sacrum
The SI joint is typically located between the SI and S3 vertebrae.[52] From a coronal perspective the joints appear to be concave with the apex directed towards the midline. As mentioned before it is typically described as auricular shaped but it can also have a more triangular appearance. On average the inferior limb extends 5.6mm and superior limb extends 4.4cm from the center. [56] The average angle between the limbs is 93 degrees where the apex is directed ventrally. [56] The SI joints exhibit a fair amount of inter-subject and intra-subject variability. This variability has been hypothesized to contribute to the development of SI joint pathology. [52] Given the significant variation in the sacrum and SI joints researchers been working on creating detailed quantitative descriptions of sacral morphology. [56, 58, 42]
1.3 Morphometric techniques
Clinical morphological descriptions of the bones have largely been qualitative.
Examples of qualitative descriptions include the auricular shape of the SI joint sur-
3


face and lumbosacral transition variations. Qualitative descriptions are nice when trying to describe geometrical properties that are related to shape. However since there is no way to quantitate the descriptions provided by this analysis it hard to measure difference between shapes. This limits its utility especially when trying to perform statistical analysis between shapes. Other limitations of qualitative descriptions include bias in describing shapes and reproducibility.
Simple quantitative descriptors of sacrum have included linear distance measurements, and angles between features. [56, 58, 42] Although these descriptors are much more amenable to statistical analysis they lack the ability to describe geometrical features of shape. Surface area and gradient measurements of the surface do describe some amount of shape geometry but it still is limited mostly to describing the curvature properties of the shape and global shape. Tools developed originally from anthropology and biology called morphometries helps to provide quantitative descriptions of shape.
Biologist began developing morphometric analysis techniques after the development of several multivariate statistical techniques including principal component analysis and analysis of variance. [22] Original morphometries was based on taking linear distance measurements or angles between landmarks and then using the previous mentioned multivariate statistical techniques for analysis. Landmarks are defined as homologous points between multiple shape that are being compared see figure below. These methods still have trouble describing global geometrical shape properties but when you increase the number of measurements included in the analysis you can extract some global geometrical properties of the shape. However the result of this analysis is difficult to interpret visually and there still remains geometrical information lost. Also linear measurements used in this technique are often affected by size, which is generally thought of as not related to shape.
4


Figure 1.3: Landmarks have been placed on the points of the stars.
A more rigorous treatment of morphometric analysis was created to deal with these limitations and as a result modern day shape theory was created. There are many contributors to the development of shape theory however Kendall and Bookstein are generally regarded as two of the biggest contributors to the held. An important feature of modern day morphometric and shape theory is the rigorous mathematical definition of shape. Which is defined as the geometric information of a structure that is invariant to scale, orientation and translation. [32] Using this definition we can take geometrical structures represented in Euclidean space and transform them to shape space.
Shape space is defined as a space that contains all possible configurations of a
chosen shape. [50] Shape space provides a convenient space for making comparisons
between shapes using metrics such as Procrustes distance. This method of shape
analysis is a very powerful tool for analyzing the shape information provided medical
5


Figure 1.4: Three identical right triangle modified only by translation, rotation, and scale
imaging.
Even though more modern day morphometric tools do provide the ability to quantitate shape differences between geometric structures they are still commonly based on identifying landmarks. This is a very time intensive process that significantly limits its ability to be used on large data sets. It can also be very difficult or even impossible to identify homologues points on complex shapes. This is particularly true for the sacrum where identifying more then three points on the sacroiliac joint surface is nontrival. To address this limitation researchers have been implementing automated tools developed by computer scientist.[10, 12, 47, 16]
1.4 Statistical Shape Modeling
Statistical shape modeling(SSM) is based on combining shape analysis theory with current computer vision techniques developed in the computer science conimu-
6


nity. The techniques allow for semi and fully automatic geometrical information acquisition and analysis. The neuroscience community was one of the first medical research communities to begin applying and developing these techniques for medical applications. In particular they have been using SSM techniques to compare the differences in neuroanatomical structures of diseased populations against the general population. Researchers have recently been using SSM to describe the anatomic variation of bones and soft tissue in efforts to understand how shape leads to diseas.[45, 54, 35, 55, 53, 38, 7, 25] Harris et al. was able to use SSM techniques to quantify the 3-dimensional anatomic variations in patients with femoral acetabular impingement compared to the general population. [25]
There are multiple techniques utilized to build shape models all having various advantages and disadvantages. One of the original techniques was based on active appearance models developed by Cootes et al. [14] Another type of shape modeling is voxel based morphometry. Which is used to analyze shape features that are within the boundary of a shape. This is especially useful in orthopedic applications when analyzing the shape of bone mineralization in the bone. [48] Shape models looking at surface features of shape are typically classified into two groups based on how one chooses to represent the underlying shape. The two categories are known as parametric and non-parametric shape representations. Parametric shape representation are where the boundaries of a shape are mathematically defined by a set of equations. In contrast, non-parametric surface representations are not mathematically defined and are simply defined implicitly. We will expand on both of these representation in the following sections.
1.5 Shape Parameterization
Typically in medical research shape information is originally in the form of medical images. This shape information must first be extracted from the images. The process typically used for this step is called segmentation. This is the process of
7


selecting regions within an image that belong to the shape being study. This process can be time consuming especially for 3 dimension shape analysis of complex shapes. There are a host of semi and fully automated algorithms that can be implemented to speed up the process of segmenting. [46, 27] However many complex shapes are not amenable to these techniques.
Figure 1.5: Screen capture of the segmentation and 3D reconstruction process. The software used is Sirnpleware ScanIP suite.
In the case of three dimensional shape analysis computed tomography (CT) or
magnetic resonance imaging (MRI) are usually used to acquire 3 dimensional shape
information. The segmentation process results in an image mask being created that
represents the region of interest. Image masks are binary images that overlay on the
original image. The pixels in the mask that overlap the region of interest are given a
value of 1. All other pixels in the mask are assigned to zero.It is important to note
that the mask in 3 dimensional segmentation is an image stack with multiple slices
8


that correspond with the slices in the original image stack. This mask representation of the shape defines the shape by the volumetric space it occupies within an image. Typically mask are not used as input into most shape modeling algorithms and must be converted to other representations. This is typically because the lack numerical information of the shape which are used as inputs for SSM algorithms.
Non-parametric shape representations are generally the most widely used representations in orthopedic SSM research. Examples of these types of representations are landmark models, surface meshes and distance transforms. [7, 25] Meshes are probably the most widely used of these examples since there are many robust algorithms that can convert segmentation mask to meshes. [3] There is also increased familiarity in the orthopedic engineering community since mesh representations are also commonly used for computer based biomechanical analysis. The models created for this thesis are based on mesh representations. Distance transform representation is a mathematical function that transforms the pixel information in the mask based on its distance from the boundary. [34] This method of surface representation is utilized in the entropy-based particle system algorithm implemented in the ShapeWorks statistical shape modeling platform. [12] One of the most common parametric surface representations used in shape modeling are based on creating a set spherical harmonic basis functions. This method is employed by SPHARM statistical shape modeling package. [47] Since this method is dependent on representing surfaces on a sphere it is limited to topologically closed shapes. Additionally this technique struggles with complex non-spherical shapes. Since the sacrum is very complex geometrically spherical harmonic based representation have limited utility.
1.6 Correspondence
When using nonparametric shape representations the next step is to create a
configuration or shape vector for each shape. This is vector containing a set of points
on the boundary of the surface. When using mesh based representations it is easiest
9


to use the vertices of the mesh to construct the shape vector. However one important step must be done before you can begin to compare the shape vectors using shape analysis algorithms. This step is known as correspondence which is the process of ordering the points contained in the shape vector based on their location on the surface of shape. This allows for corresponding points on the surface of each shape being analyzed to be compared with one another. This step is quite difficult and is an area of active research. Current methods to deal with this problem can be thought of as registration based or non-registration based. Registration based methods find correspondence of points between shapes while finding a transformation matrix that aligns the shapes. Non-registration based methods find a set of correspondence points between shapes without aligning the shapes.
Figure 1.6: Illustrates the concept of correspondence. Points in the shape vectors must must indexed according to their location on each shape being compared. [37]
10


1.7 Registration Based Correspondence Methods
Registration based method can further be divided into rigid and non-rigid based methods. Rigid based methods utilize rigid registration algorithms to find correspondence. Rigid registration algorithms align and find correspondence between shapes using only translation and rotation transformations. One of the most commonly used rigid registration methods is iterative closest point (ICP) algorithm[5]. Iterative closest point algorithm works by finding a least squares rigid transformation that minimize the distance criterion. The method is performed in an iterative fashion and performs well for similar aligned shapes. However the algorithm finds local minima and can fail to properly align shapes that differ significantly in the starting alignment. Noise within the point sets also can limit the effectiveness of ICP. There are recent updated ICP methods that can overcome the local minima dilemma. [30] However a new class of probabilistic based methods have proven to be more effective in addressing the noise and local minima problems.[30, 44]
In traditional ICP given two point sets the closest pair is assumed to have correspondence. This assumption makes ICP simple and intuitive, however leads to the limitations mentioned above. To overcome this researchers initially used soft correspondence i.e. instead of matching the closet point from one point set to the other. The entire set of points in one point set are used in a weighted fashion to correspond to a point in the other set. [30] Out of this developed probabilistic methods including Gaussian mixture models. This approach uses one point set to represent the gaussian mixture model centroids and the other point set is used as a data set. The GMM centroids are then optimized using maximum likelihood framework to find correspondence. This technique can be utilized for both rigid and non-rigid algorithms.
Coherent Point Drift registration algorithm is a non-rigid registration technique based on the GMM centroid model listed above. [44] CPD extends this concept to allow the GMM centroids move coherently preserving the topologic structure. [44]
11


This allows CPD to perform non-rigid registration between two point sets fast and reliably despite noise in the data. The statistical shape models created in this thesis implemented CPD registration to find correspondence between the shapes.
1.8 Non-registration based correspondence methods
Non-registration methods generally do not find a transformation matrix used to register shapes. Instead one must first align the shapes beforehand. In the case of Shapeworks this is done using ICP. Once the shapes are aligned a new set of statistically optimized correspondence points can be found between the shapes. Entropy based particle system shape analysis developed by Cates is an example of this method. [12] The basic idea of the method is a particle is randomly placed on the surface of a shape and then splits. The resulting two particles are confined to the surface of the shape but free to move apart minimizing the entropy of the system based on a shape cost function. The previous step is then repeated until the desired number of correspondence points are obtained. This method also works well on shapes with a moderate amount of global curvature that typically would require some form of non-rigid registration or warping to find correspondence.
1.9 Building the Shape model
Once correspondence is found between each shape vector we can begin to perform shape analysis. The first step is to align each shape vector in a similar reference frame and scale shape vector so that all non-shape geometric information is removed. This is typically done using Procrustes methods. We can then create a mean shape model using the Procrustes mean which is given by
1 N
(L1)
1
The mean shape for a given set of shapes has many utilities including comparing the distances between two means. However since regions within a shape are highly
12


correlated statistical analysis of mean shapes is not straightforward. [13] One solution is to represent the corresponding shape space by set of de-correlated linear vectors that approximate our shape space. This can be performed using principle component analysis (PC A). PC A also allows you to reduce the dimensionality of shape making it a very powerful tool performing statistical analysis.
There are many statistical tools that can be applied to the PCA loadings obtained. Commonly researchers compare the means of PCA modes between groups. This can be performed using a standard T-Test. Hotelling T-Test is also useful if interested in comparing the entire set of PCA modes from one group are statistical different from another group. Regression techniques are also possible on the PCA loadings which can reveal how shape features change in relation to another variable.
13


2. Methods
IRB approval was obtained to retrospectively collect CT KUBs from 223 patients evaluated at the University of Colorado Hospital for abdominal pain. 71 female patients were randomly selected from this group for segmentation. From this group 25 subjects were chosen for segmentation. Eleven image sets used for analysis were collected using a Siemens SOMATOM Definition Flash CT Scanner(100-140kVp 512x512, 1mm-1.5mm slice thickness, 0.5703-0.98 pixel spacing). Fourteen image sets were collected using Siemens Sensation 64 CT scanner(120kVp 512x512, 1.5mm slice thickness, 0.6875-.976 pixel spacing).
Simpleware ScanIP was used to cubically resample CT images to a cubic voxel size of 1mm using a linear interpolation method. The images were then windowed to enhance bone visualization. Image noise was filtered using a curvature anisotropic diffusion filter. After processing the CT images they were segmented using combination of manual and semi-automatic techniques. Surface meshes were constructed using ScanIP +FE module. Meshing parameters were adjusted in scan IP to achieved surface mesh densities between 10,000-18,000 triangular elements. These hies were then imported into Matlab 2015b for analysis.
14


Figure 2.1: Screen capture showing points clouds of the sacrum and alignment of shapes centroids to the origin
The surfaces were converted into point clouds using the mesh vertices. The
centroid waswas then calculated and the point clouds were transformed to center
their centroids at the origin. The centroid size was calculated for all the shapes and
the shape with the median centroid shape was used as the target for correspondence.
An initial pre-alignment was performed using iterative closest point algorithm on
a randomly selected sampling from each point cloud. The final Registration and
correspondence was performed using the coherent point drift matlab package. [44] The
CPD package performs both rigid and non-rigid registration to find correspondence.
Once the shapes were registered corresponding shape vectors were created. Procrustes
transformation was applied to the shape vectors including Procrustes scaling. The
Procrustes mean was calculated using equation above. Principal component analysis
was then used for dimensionality reduction of the higher dimensional shape space.
15


Reconstructed point clouds were created using the following equation. Where pv is the shape vector for each shape in the set.
N
Pv =P+ (2-1)
i= 1
Visualization was performed varying specihc PC modes from -2sd to +2sd. These point cloud were then exported and surface meshes were created using reverse engineering algorithms.
16


3. Result
The overall shape model built using the CPD registration methodology resulted an overall good quality shape model based on visual appearance. All of the sacral foramen were visual patent. The median sacral crest appeared symmetric and non distorted. The coccyx and sacral facets also appeared to have little distortion.
Figure 3.1: Three views of the of the final shape model using the CPD code.
Reconstruction of the mesh structure from the point clouds appeared to have some distortion around the SI and S2 sacral bodies. The sacral foramen were patent at every level. The distal sacrum and coccyx had minimal distortion. The SI joint surface exhibited little noise and demonstrated a clear auricular shape. Based on the classification system in figure 1.1 we can classify the mean shape as a type II SI joint surface.
17


Figure 3.2: Surface reconstruction of the mean shape
When visualizing the sacral SSM from a coronal view we saw the first PC mode described the sacral width. When moving in the positive direction from the mean the sacrum width increased and conversely decreased when in the negative direction. Visualizing the first PC mode from a lateral view we saw the SI joint surface change shape. This shape varied between a type I and Type 3 depending on the direction you traveled from the mean.
18


Figure 3.3: Surface reconstruction of the First PC mode. Screen captured from a coronal perspective
19


Figure 3.4: Surface reconstruction of the First PC mode. Screen captured from a lateral perspective
Identifying the shape feature described by the second PC mode is much more challenging. Lateral view of the surface reconstructions revealed the superior limb of the SI joint surface varied as we varied the second eigenvector.
20


7 M / /
-2sd Mean +2sd
Figure 3.5: Surface reconstruction of the Second PC mode. Screen captured from a lateral perspective
The third PC mode described variation in the concavity of the SI joint. Variation in the number of sacral vertebral bodies was also described in this mode. Distortion of the sacral vertebral bodies was most significant in this mode.
21


Figure 3.6: Surface reconstruction of the Third PC mode. Screen capture from a coronal perspective
I was unable to interpret any shape features described in the 4th mode PC mode. Similarly the fifth mode also had little visually interpretable shape variation. However median sacral crest prominence does vary in the fifth PC mode.
22


Figure 3.7: Surface reconstruction of the fifth PC mode. Screen captured from a lateral perspective
23


Analyzing the compactness of our SSM. We found the first four eigenmodes captured over 60% of the overall variation in the model. Over 90% of the variation
was captured with the first 16 modes.
Compactness
10 15
Number of Modes
Figure 3.8: Compactness is defined as the number of modes needed to described a given amount of variation. More compact models can represent greater variation with few modes. [51, 16]
24


4. Discussion
The results of this model support that the CPD-SSM registration method can be an effective tool to construct statistical shape models of the sacrum. The mean shape reconstruction appeared to capture the overall structure of the sacrum with little distortion. Given the first four eigenmodes described 60% of the overall variation and 16 modes described 90% of the variation in the model. This demonstrated similar compactness results to other shape modeling studies of bones. [25, 38]
There was some artifact on the ventral surface of the SI and S2 sacral bodies. This distortion was difficult to recognize when just viewing the point cloud of the SSM in figure 3.2. However it was present in all of the point clouds generated suggesting this distortion is inherent in the model itself and not just a limitation of the surface re-meshing software. This is somewhat abnormal since usually poor correspondence occurs in areas of high curvature and overall geometrical complexity. Despite this it is reasonable to conclude that despite the overall lack of local geometrical complexity in this area the correspondence is poor relative to other areas in the model.
Areas of the sacrum with less variation had improved correspondence. This was particularly true for the sacroiliac joint and facet. This is particularly reassuring that models built using the methodology achieve high enough quality correspondence that describing sacroiliac joint variation is a real possibility. This work also demonstrates that this technique is effective at describing additional morphological attributes of the sacrum including the overall curvature of the sacrum, median sacral crest variation.
The most impactful findings of this model are how effective the eigenmodes were at describing areas of variation of the sacrum. Mode 1 captures distinct morphological variations of the sacroiliac joint that were observed in preliminary work done our group. This mode will be very useful in providing an initial classification of the sacroiliac joint morphology.
25


Despite the success in developing workable statistical shape model of the sacrum from point clouds. The quality of the reconstructed meshes is lacking. There are several commonly used methods for reconstructing surface meshes including delaunay, power crust and ball-pivoting algorithms. [23, 2, 4] However these algorithms struggle with complex geometrical shape similarly to the registration algorithms. Increasing the density of the point clouds does help with overall visualization of the SMM. An improvement can even be noticed from just the raw point clouds. As shown in the figure below.
Figure 4.1: Screen capture showing a point cloud SSM of sacrum. This model was built using a target shape point cloud of 54,700 points
This improved point cloud visualization also extends to improved results from surface meshing algorithms. Figure 4.2 highlights the improvement of the distortion on the ventral surface of the sacrum when reconstructing a surface mesh from a high density SSM.
26


Figure 4.2: The left image is a surface reconstruction from a high density SSM containing 57400 points. The right image is a surface reconstruction from a low density SSM containing 6993 points.
Unfortunately there are some negative effects to increasing the number of points. Computational time is greatly affected by the number of points used to build the model. Models built with less then 10,000 points can be computed in under 15 minutes. However, the models built with 54,700 point took over 5 hours to complete. The overall quality of the model also changes as you increase the density of the points in the SSM. This can be seen by a change in the compactness of the model depending on the density of the SSM. In the figure below we see the blue curve corresponding to the lowest density model describes variability at a much quicker rate than higher density models.
27


I 3 1 4 S i T 1 I 1 It U U It K U IT U H H n 11 M
NuniHr tr1 Modn
W'A (dull (Jrii F-oInii TWpdAlL MM pi**n. MMpdm -XaJ fa>n
UiXUjwairYIL lufcM ^i-- 11 4M p^.i-. 1-MLL VO30
Figure 4.3: Illustrates how the compactness of the models change with density of the point cloud used in the SMM
Another hurdle that has presented itself in this initial model is the statistical analysis of the overall model. Principal component analysis is the most widely used method to help describe shape variation in models.[27, 39] Its popularity is due to its ability to reduce the dimensionality of the shape model to a more compact lower dimensional shape space. [27] Principal component analysis is also very effective in detecting and describing global shape changes. [39, 50] However, its utility is limited when trying to describe local variations of shape. [39, 17] Independent component analysis is better suited for local analysis but is limited by its lack of ability to order its shape descriptors making dimensionality reduction difficult.[39] There has been recent development in creating algorithmic techniques to allow for dimensionality reduction using independent component analysis but many algorithms are still early on in there development. [39, 1, 57]
28


5. Conclusion
Despite the limitations of reconstructing surface meshes from the point clouds of our model the results are very encouraging. This work demonstrates that CPD-SSM can be used to determine the mean shape of the sacrum for a given population. This work also revealed principal component analysis is a useful tool for describing sacral morphology. Statistical shape analysis of sacrum proves to be a robust technique that will provide new insights into the morphological variation of the sacroiliac joint in the general population. It also shows promise in detecting morphological difference in patients with sacroiliac joint pain from the general population.
There are many areas where sacral statistical shape models are useful. The models created from this project could easily be used to create 3d printed templates for developing and testing surgical jigs and implants. Combining SSMs with finite element modeling would allow you to systematically describe how complex shape features of the sacrum effect mechanics of the pelvis. It is also possible to extend this application to create personalized finite element models. Typically the hurdles that prevented personalized realtime mechanics simulation revolved around time. Segmenting the medical images and calculating the FE solution are both time intensive processes. However SSMs have the potential to act as a shape atlas speeding up and improving the accuracy automated segmenting techniques. It is also possible to pre calculate FE solutions based on the shape variations described by the SSM. This can then be used as a FE solution database where one can simply interpolate between different solutions.
5.1 Future Work
I am continuing to work on many facets of this project. The next step is to build an SSM of patients with sacroiliac joint pain. This will allow us to investigate the correlation of PC modes with SI joint pain. In addition to this work I am also working on improving the reconstructed meshes from the point clouds. We are currently be
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experimenting with warping the original source mesh to the final mean SSM. This can be accomplished using thin plate splines or reusing the CPD algorithm a second time. However to use this method one will have to implement some form of relaxation function to keep the mesh from self intersecting in places with poor correspondence or high curvature.
The future of statistical shape modeling is especially bright. CPD-SSM code provides a robust tool for registering complex shapes from noisy data. This is especially useful when building SSMs from clinical imaging where data quality is far below the quality of research grade data. It is my hope that this thesis demonstrates how robust this tool is for building high quality shape models.
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[56] Jeffrey T Waldrop, Nabil A Ebraheim, Richard A Yeasting, and W T Jackson. The Location of the Sacroiliac Joint on the Outer Table of the Posterior Ilium. Journal of Orthopaedic Trauma, 7(6), 1993.
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Full Text

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MORPHOMETRICANALYSISOFTHESACRUMUSINGSTATISTICAL SHAPEMODELING by KENNETHKMILLIGAN B.S.,ColoradoStateUniversity,2008 Athesissubmittedtothe FacultyoftheGraduateSchoolofthe UniversityofColoradoinpartialfulllment oftherequirementsforthedegreeof MasterofScience Bioengineering 2016

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ThisthesisfortheMasterofSciencedegreeby KennethKMilligan hasbeenapprovedforthe BioengineeringProgram by DanaCarpenter,Advisor KendalHunter,Chair VikasPatel DanaCarpenter StephenHumphries April29,2016 ii

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Milligan,KennethKM.S.,Bioengineering MorphometricAnalysisoftheSacrumusingStatisticalShapeModeling ThesisdirectedbyAssistantProfessorDanaCarpenter ABSTRACT Statisticalshapemodelingtechniqueswereusedtoquantifythe3dimensional anatomicvariationsofthesacrumandsacroiliacjointsurfaceinthegeneralpopulation.25pelvisesweresegmentedfromafromdatabase223patientswhohadCT studiesperformedforevaluationofabdominalpainattheUniversityofColorado Hospital.SurfaceSTLmesheswerecreatedusingSimplewareScanIP.Thestatistical shapemodelswascreatedusingtheCoherentPointdriftalgorithmforregistration andcorrespondence.SSManalysisandordinationwasperformedutilizingprocrustes analysisandprincipalcomponentanalysisPCA.PCAprovidesdimensionalityreductionofthedataforvisualizationandinterpretationofshapefeatures.Wefound variationinthesacroiliacjointmorphologycouldbequantitativelydescribedusing ourCPD-SSMtechnique.Wealsofoundtherst,secondandthirdPCAmodes describechangesinthesacroiliacjointshape. Theformandcontentofthisabstractareapproved.Irecommenditspublication. Approved:DanaCarpenter iii

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DEDICATION ThisthesisisdedicatedtoJulie,Kayleigh,andColin. iv

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ACKNOWLEDGMENT ThisthesiswouldnothavebeenpossiblewithoutthehelpofEduardoNovaisand ZacharyWuthrich. v

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TABLEOFCONTENTS Figures.......................................vii 1.Introduction...................................1 1.1SacroiliacJoint................................1 1.2SacralAnatomy................................2 1.3Morphometrictechniques...........................3 1.4StatisticalShapeModeling..........................6 1.5ShapeParameterization...........................7 1.6Correspondence................................9 1.7RegistrationBasedCorrespondenceMethods................11 1.8Non-registrationbasedcorrespondencemethods..............13 1.9BuildingtheShapemodel..........................13 2.Methods.....................................15 3.Result......................................18 4.Discussion....................................26 5.Conclusion....................................30 5.1FutureWork.................................30 References ......................................32 vi

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FIGURES Figure 1.1 ShapeclassicationoftheSacroiliacjointsurfacebasedoofamaximum intensityprojectionreconstruction. .....................2 1.2 Frontalandlateralviewofa3Dsurfacereconstructionofthesacrum ..3 1.3 Landmarkshavebeenplacedonthepointsofthestars. ..........5 1.4 Threeidenticalrighttrianglemodiedonlybytranslation,rotation,and scale ......................................6 1.5 Screencaptureofthesegmentationand3Dreconstructionprocess.The softwareusedisSimplewareScanIPsuite. .................8 1.6 Illustratestheconceptofcorrespondence.Pointsintheshapevectorsmust mustindexedaccordingtotheirlocationoneachshapebeingcompared.[37] 10 1.7 Illustrationofiterativeclosestpointsregistration.Apoint m i isautomaticallycorrespondedtotheclosestpointis S .................12 2.1 Screencaptureshowingpointscloudsofthesacrumandalignmentof shapescentroidstotheorigin ........................16 3.1 ThreeviewsoftheofthenalshapemodelusingtheCPDcode. .....18 3.2 Surfacereconstructionofthemeanshape ..................19 3.3 SurfacereconstructionoftheFirstPCmode.Screencapturedfroma coronalperspective ..............................20 3.4 SurfacereconstructionoftheFirstPCmode.Screencapturedfroma lateralperspective ...............................21 3.5 SurfacereconstructionoftheSecondPCmode.Screencapturedfroma lateralperspective ...............................22 3.6 SurfacereconstructionoftheThirdPCmode.Screencapturefromacoronalperspective ................................23 vii

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3.7 SurfacereconstructionofthefthPCmode.Screencapturedfromalateral perspective ...................................24 3.8 Compactnessisdenedasthenumberofmodesneededtodescribedagiven amountofvariation.Morecompactmodelscanrepresentgreatervariation withfewmodes.[51,16] ...........................25 4.1 ScreencaptureshowingapointcloudSSMofsacrum.Thismodelwas builtusingatargetshapepointcloudof54,700points ...........27 4.2 TheleftimageisasurfacereconstructionfromahighdensitySSMcontaining57400points.Therightimageisasurfacereconstructionfroma lowdensitySSMcontaining6993points. ..................28 4.3 Illustrateshowthecompactnessofthemodelschangewithdensityofthe pointcloudusedintheSMM ........................29 viii

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1.Introduction 1.1SacroiliacJoint ThesacroiliacjointSIjointistheinterfacebetweenthespineandthepelvis. Thisjointhasbeenimplicatedasthecauseof10-30%ofalllowbackpainandbuttock pain.[10]TherearemanycausesofSIjointpainincludinginammatoryconditions, trauma,infection,anddegenerativeosteoarthritisOA.Inthecaseofdegenerative OAthemainstaysoftreatmentareactivitymodication,NSAIDS,SIjointinjections, RF-ablations.Recentlytherehasbeenanincreaseduseofminimallyinvasivesurgical arthrodesisoftheSIjointtorelievepainincasesthatfailconservativetreatments. IncreasinguseofSIjointfusiontreatmentshascreatedrenewedinterestinthe biomechanicalpropertiesoftheSIjoint.[36]Previousbiomechanicalstudieshave lookedatmotionofthejoint,stressesandstrainsonthearticulatingsurfacesand mainsupportingligaments.Evenmorerecenthasbeenanindustrysupportedstudy comparingtwoSIjointtrajectoriesandthereeectsonrangeofmotioninexion extension,lateralbending,andaxialrotations.[49] Allofthesepreviousstudieshoweverhavenotinvestigatedormentionedthe potentialeectsofSIjointgeometryonthebiomechanicsofthejoint.Recentwork attheuniversityofColoradohashighlightedtheimmensevariabilityoftheSIjoint shapeinthegeneralpopulationandattemptedtomakepreliminaryclassications ofshapes.Howevertheseshapesclassicationswerefrom2-Dprojectionsofthe articularsurface.Whichprohibitedusfromcommentingonthesurfacevariation froma3dimensionalperspective. 1

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section Figure1.1: ShapeclassicationoftheSacroiliacjointsurfacebasedoofamaximum intensityprojectionreconstruction. 1.2SacralAnatomy TheSacrumisoneofthreebonesthatmakeuppelvis.Itsgeneralappearance iswedgeshapedwiththeapexpointinginferiorly.Fivefusedvertebrae,whichare orienteddorsallyconvex,makeupthesacrum.Therearethreejointsurfacesofthe sacrum.Thesacralbaseisbroadcircularshapethatarticulateswiththeintervertebraldisc.Theothertwojointssurfacesarethesacroiliacjoints.Theyaregenerally describedasauricularinshape.[52,56]Howeverthereexistanimmenseamountof diversitybetweenindividuals.TheSIjointsarticulatewiththerightandleftiliac bones.Thesejointsalongwiththesurroundingsofttissuestructurestransmitforces fromthetorsotothelowerlimbs. 2

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Figure1.2: Frontalandlateralviewofa3Dsurfacereconstructionofthesacrum TheSIjointistypicallylocatedbetweentheS1andS3vertebrae.[52]Froma coronalperspectivethejointsappeartobeconcavewiththeapexdirectedtowardsthe midline.Asmentionedbeforeitistypicallydescribedasauricularshapedbutitcan alsohaveamoretriangularappearance.Onaveragetheinferiorlimbextends5.6mm andsuperiorlimbextends4.4cmfromthecenter.[56]Theaverageanglebetweenthe limbsis93degreeswheretheapexisdirectedventrally.[56]TheSIjointsexhibita fairamountofinter-subjectandintra-subjectvariability.Thisvariabilityhasbeen hypothesizedtocontributetothedevelopmentofSIjointpathology.[52]Giventhe signicantvariationinthesacrumandSIjointsresearchersbeenworkingoncreating detailedquantitativedescriptionsofsacralmorphology.[56,58,42] 1.3Morphometrictechniques Clinicalmorphologicaldescriptionsoftheboneshavelargelybeenqualitative. ExamplesofqualitativedescriptionsincludetheauricularshapeoftheSIjointsur3

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faceandlumbosacraltransitionvariations.Qualitativedescriptionsarenicewhen tryingtodescribegeometricalpropertiesthatarerelatedtoshape.Howeversince thereisnowaytoquantitatethedescriptionsprovidedbythisanalysisithardto measuredierencebetweenshapes.Thislimitsitsutilityespeciallywhentryingto performstatisticalanalysisbetweenshapes.Otherlimitationsofqualitativedescriptionsincludebiasindescribingshapesandreproducibility. Simplequantitativedescriptorsofsacrumhaveincludedlineardistancemeasurements,andanglesbetweenfeatures.[56,58,42]Althoughthesedescriptorsaremuch moreamenabletostatisticalanalysistheylacktheabilitytodescribegeometrical featuresofshape.Surfaceareaandgradientmeasurementsofthesurfacedodescribesomeamountofshapegeometrybutitstillislimitedmostlytodescribing thecurvaturepropertiesoftheshapeandglobalshape.Toolsdevelopedoriginally fromanthropologyandbiologycalledmorphometricshelpstoprovidequantitative descriptionsofshape. Biologistbegandevelopingmorphometricanalysistechniquesafterthedevelopmentofseveralmultivariatestatisticaltechniquesincludingprincipalcomponentanalysisandanalysisofvariance.[22]Originalmorphometricswasbasedontakinglinear distancemeasurementsoranglesbetweenlandmarksandthenusingtheprevious mentionedmultivariatestatisticaltechniquesforanalysis.Landmarksaredenedas homologouspointsbetweenmultipleshapethatarebeingcomparedseegurebelow.Thesemethodsstillhavetroubledescribingglobalgeometricalshapeproperties butwhenyouincreasethenumberofmeasurementsincludedintheanalysisyoucan extractsomeglobalgeometricalpropertiesoftheshape.Howevertheresultofthis analysisisdiculttointerpretvisuallyandtherestillremainsgeometricalinformationlost.Alsolinearmeasurementsusedinthistechniqueareoftenaectedbysize, whichisgenerallythoughtofasnotrelatedtoshape. 4

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Figure1.3: Landmarkshavebeenplacedonthepointsofthestars. Amorerigoroustreatmentofmorphometricanalysiswascreatedtodealwith theselimitationsandasaresultmoderndayshapetheorywascreated.Thereare manycontributorstothedevelopmentofshapetheoryhoweverKendallandBookstein aregenerallyregardedastwoofthebiggestcontributorstotheeld.Animportant featureofmoderndaymorphometricandshapetheoryistherigorousmathematical denitionofshape.Whichisdenedasthegeometricinformationofastructurethat isinvarianttoscale,orientationandtranslation.[32]Usingthisdenitionwecantake geometricalstructuresrepresentedinEuclideanspaceandtransformthemtoshape space. Shapespaceisdenedasaspacethatcontainsallpossiblecongurationsofa chosenshape.[50]Shapespaceprovidesaconvenientspaceformakingcomparisons betweenshapesusingmetricssuchasProcrustesdistance.Thismethodofshape analysisisaverypowerfultoolforanalyzingtheshapeinformationprovidedmedical 5

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Figure1.4: Threeidenticalrighttrianglemodiedonlybytranslation,rotation,and scale imaging. Eventhoughmoremoderndaymorphometrictoolsdoprovidetheabilityto quantitateshapedierencesbetweengeometricstructurestheyarestillcommonly basedonidentifyinglandmarks.Thisisaverytimeintensiveprocessthatsignicantly limitsitsabilitytobeusedonlargedatasets.Itcanalsobeverydicultoreven impossibletoidentifyhomologuespointsoncomplexshapes.Thisisparticularly trueforthesacrumwhereidentifyingmorethenthreepointsonthesacroiliacjoint surfaceisnontrival.Toaddressthislimitationresearchershavebeenimplementing automatedtoolsdevelopedbycomputerscientist.[10,12,47,16] 1.4StatisticalShapeModeling StatisticalshapemodelingSSMisbasedoncombiningshapeanalysistheory withcurrentcomputervisiontechniquesdevelopedinthecomputersciencecommu6

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nity.Thetechniquesallowforsemiandfullyautomaticgeometricalinformation acquisitionandanalysis.Theneurosciencecommunitywasoneoftherstmedicalresearchcommunitiestobeginapplyinganddevelopingthesetechniquesfor medicalapplications.InparticulartheyhavebeenusingSSMtechniquestocomparethedierencesinneuroanatomicalstructuresofdiseasedpopulationsagainst thegeneralpopulation.ResearchershaverecentlybeenusingSSMtodescribethe anatomicvariationofbonesandsofttissueineortstounderstandhowshapeleads todiseas.[45,54,35,55,53,38,7,25]Harrisetal.wasabletouseSSMtechniquesto quantifythe3-dimensionalanatomicvariationsinpatientswithfemoralacetabular impingementcomparedtothegeneralpopulation.[25] Therearemultipletechniquesutilizedtobuildshapemodelsallhavingvarious advantagesanddisadvantages.Oneoftheoriginaltechniqueswasbasedonactive appearancemodelsdevelopedbyCootesetal.[14]Anothertypeofshapemodelingis voxelbasedmorphometry.Whichisusedtoanalyzeshapefeaturesthatarewithin theboundaryofashape.Thisisespeciallyusefulinorthopedicapplicationswhen analyzingtheshapeofbonemineralizationinthebone.[48]Shapemodelslookingat surfacefeaturesofshapearetypicallyclassiedintotwogroupsbasedonhowone choosestorepresenttheunderlyingshape.Thetwocategoriesareknownasparametricandnon-parametricshaperepresentations.Parametricshaperepresentationare wheretheboundariesofashapearemathematicallydenedbyasetofequations.In contrast,non-parametricsurfacerepresentationsarenotmathematicallydenedand aresimplydenedimplicitly.Wewillexpandonbothoftheserepresentationinthe followingsections. 1.5ShapeParameterization Typicallyinmedicalresearchshapeinformationisoriginallyintheformofmedicalimages.Thisshapeinformationmustrstbeextractedfromtheimages.The processtypicallyusedforthisstepiscalledsegmentation.Thisistheprocessof 7

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selectingregionswithinanimagethatbelongtotheshapebeingstudy.Thisprocess canbetimeconsumingespeciallyfor3dimensionshapeanalysisofcomplexshapes. Thereareahostofsemiandfullyautomatedalgorithmsthatcanbeimplementedto speeduptheprocessofsegmenting.[46,27]Howevermanycomplexshapesarenot amenabletothesetechniques. Figure1.5: Screencaptureofthesegmentationand3Dreconstructionprocess.The softwareusedisSimplewareScanIPsuite. InthecaseofthreedimensionalshapeanalysiscomputedtomographyCTor magneticresonanceimagingMRIareusuallyusedtoacquire3dimensionalshape information.Thesegmentationprocessresultsinanimagemaskbeingcreatedthat representstheregionofinterest.Imagemasksarebinaryimagesthatoverlayonthe originalimage.Thepixelsinthemaskthatoverlaptheregionofinterestaregivena valueof1.Allotherpixelsinthemaskareassignedtozero.Itisimportanttonote thatthemaskin3dimensionalsegmentationisanimagestackwithmultipleslices 8

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thatcorrespondwiththeslicesintheoriginalimagestack.Thismaskrepresentation oftheshapedenestheshapebythevolumetricspaceitoccupieswithinanimage. Typicallymaskarenotusedasinputintomostshapemodelingalgorithmsandmust beconvertedtootherrepresentations.Thisistypicallybecausethelacknumerical informationoftheshapewhichareusedasinputsforSSMalgorithms. Non-parametricshaperepresentationsaregenerallythemostwidelyusedrepresentationsinorthopedicSSMresearch.Examplesofthesetypesofrepresentationsare landmarkmodels,surfacemeshesanddistancetransforms.[7,25]Meshesareprobably themostwidelyusedoftheseexamplessincetherearemanyrobustalgorithmsthat canconvertsegmentationmasktomeshes.[3]Thereisalsoincreasedfamiliarityinthe orthopedicengineeringcommunitysincemeshrepresentationsarealsocommonlyused forcomputerbasedbiomechanicalanalysis.Themodelscreatedforthisthesisare basedonmeshrepresentations.Distancetransformrepresentationisamathematical functionthattransformsthepixelinformationinthemaskbasedonitsdistancefrom theboundary.[34]ThismethodofsurfacerepresentationisutilizedintheentropybasedparticlesystemalgorithmimplementedintheShapeWorksstatisticalshape modelingplatform.[12]Oneofthemostcommonparametricsurfacerepresentations usedinshapemodelingarebasedoncreatingasetsphericalharmonicbasisfunctions.ThismethodisemployedbySPHARMstatisticalshapemodelingpackage.[47] Sincethismethodisdependentonrepresentingsurfacesonasphereitislimitedto topologicallyclosedshapes.Additionallythistechniquestruggleswithcomplexnonsphericalshapes.Sincethesacrumisverycomplexgeometricallysphericalharmonic basedrepresentationhavelimitedutility. 1.6Correspondence Whenusingnonparametricshaperepresentationsthenextstepistocreatea congurationorshapevectorforeachshape.Thisisvectorcontainingasetofpoints ontheboundaryofthesurface.Whenusingmeshbasedrepresentationsitiseasiest 9

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tousetheverticesofthemeshtoconstructtheshapevector.Howeveroneimportant stepmustbedonebeforeyoucanbegintocomparetheshapevectorsusingshape analysisalgorithms.Thisstepisknownascorrespondencewhichistheprocessof orderingthepointscontainedintheshapevectorbasedontheirlocationonthe surfaceofshape.Thisallowsforcorrespondingpointsonthesurfaceofeachshape beinganalyzedtobecomparedwithoneanother.Thisstepisquitedicultandis anareaofactiveresearch.Currentmethodstodealwiththisproblemcanbethought ofasregistrationbasedornon-registrationbased.Registrationbasedmethodsnd correspondenceofpointsbetweenshapeswhilendingatransformationmatrixthat alignstheshapes.Non-registrationbasedmethodsndasetofcorrespondencepoints betweenshapeswithoutaligningtheshapes. Figure1.6: Illustratestheconceptofcorrespondence.Pointsintheshapevectorsmust mustindexedaccordingtotheirlocationoneachshapebeingcompared.[37] 10

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1.7RegistrationBasedCorrespondenceMethods Registrationbasedmethodcanfurtherbedividedintorigidandnon-rigidbased methods.Rigidbasedmethodsutilizerigidregistrationalgorithmstondcorrespondence.Rigidregistrationalgorithmsalignandndcorrespondencebetweenshapes usingonlytranslationandrotationtransformations.Oneofthemostcommonly usedrigidregistrationmethodsisiterativeclosestpointICPalgorithm[5].Iterativeclosestpointalgorithmworksbyndingaleastsquaresrigidtransformation thatminimizethedistancecriterion.Themethodisperformedinaniterativefashionandperformswellforsimilaralignedshapes.Howeverthealgorithmndslocal minimaandcanfailtoproperlyalignshapesthatdiersignicantlyinthestarting alignment.NoisewithinthepointsetsalsocanlimittheeectivenessofICP.There arerecentupdatedICPmethodsthatcanovercomethelocalminimadilemma.[30] Howeveranewclassofprobabilisticbasedmethodshaveproventobemoreeective inaddressingthenoiseandlocalminimaproblems.[30,44] IntraditionalICPgiventwopointsetstheclosestpairisassumedtohavecorrespondence.ThisassumptionmakesICPsimpleandintuitive,howeverleadstothe limitationsmentionedabove.Toovercomethisresearchersinitiallyused"softcorrespondence"i.e.insteadofmatchingtheclosetpointfromonepointsettotheother. Theentiresetofpointsinonepointsetareusedinaweightedfashiontocorrespond toapointintheotherset.[30]Outofthisdevelopedprobabilisticmethodsincluding Gaussianmixturemodels.Thisapproachusesonepointsettorepresentthegaussian mixturemodelcentroidsandtheotherpointsetisusedasadataset.TheGMM centroidsarethenoptimizedusingmaximumlikelihoodframeworktondcorrespondence.Thistechniquecanbeutilizedforbothrigidandnon-rigidalgorithms. CoherentPointDriftregistrationalgorithmisanon-rigidregistrationtechnique basedontheGMMcentroidmodellistedabove.[44]CPDextendsthisconceptto allowtheGMMcentroidsmovecoherentlypreservingthetopologicstructure.[44] 11

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ThisallowsCPDtoperformnon-rigidregistrationbetweentwopointsetsfastand reliablydespitenoiseinthedata.Thestatisticalshapemodelscreatedinthisthesis implementedCPDregistrationtondcorrespondencebetweentheshapes. 1.8Non-registrationbasedcorrespondencemethods Non-registrationmethodsgenerallydonotndatransformationmatrixusedto registershapes.Insteadonemustrstaligntheshapesbeforehand.Inthecase ofShapeworksthisisdoneusingICP.Oncetheshapesarealignedanewsetof statisticallyoptimizedcorrespondencepointscanbefoundbetweentheshapes.EntropybasedparticlesystemshapeanalysisdevelopedbyCatesisanexampleofthis method.[12]Thebasicideaofthemethodisaparticleisrandomlyplacedonthesurfaceofashapeandthensplits.Theresultingtwoparticlesareconnedtothesurface oftheshapebutfreetomoveapartminimizingtheentropyofthesystembasedon ashapecostfunction.Thepreviousstepisthenrepeateduntilthedesirednumber ofcorrespondencepointsareobtained.Thismethodalsoworkswellonshapeswith amoderateamountofglobalcurvaturethattypicallywouldrequiresomeformof non-rigidregistrationorwarpingtondcorrespondence. 1.9BuildingtheShapemodel Oncecorrespondenceisfoundbetweeneachshapevectorwecanbegintoperform shapeanalysis.Therststepistoaligneachshapevectorinasimilarreferenceframe andscaleshapevectorsothatallnon-shapegeometricinformationisremoved.This istypicallydoneusingProcrustesmethods.Wecanthencreateameanshapemodel usingtheProcrustesmeanwhichisgivenby x = 1 N N X 1 x i .1 Themeanshapeforagivensetofshapeshasmanyutilitiesincludingcomparing thedistancesbetweentwomeans.Howeversinceregionswithinashapearehighly 12

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correlatedstatisticalanalysisofmeanshapesisnotstraightforward.[13]Onesolution istorepresentthecorrespondingshapespacebysetofde-correlatedlinearvectors thatapproximateourshapespace.Thiscanbeperformedusingprinciplecomponent analysisPCA.PCAalsoallowsyoutoreducethedimensionalityofshapemaking itaverypowerfultoolperformingstatisticalanalysis. TherearemanystatisticaltoolsthatcanbeappliedtothePCAloadingsobtained. CommonlyresearcherscomparethemeansofPCAmodesbetweengroups.Thiscan beperformedusingastandardT-Test.HotellingT-Testisalsousefulifinterestedin comparingtheentiresetofPCAmodesfromonegrouparestatisticaldierentfrom anothergroup.RegressiontechniquesarealsopossibleonthePCAloadingswhich canrevealhowshapefeatureschangeinrelationtoanothervariable. 13

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2.Methods IRBapprovalwasobtainedtoretrospectivelycollectCTKUBsfrom223patientsevaluatedattheUniversityofColoradoHospitalforabdominalpain.71femalepatientswererandomlyselectedfromthisgroupforsegmentation.Fromthis group25subjectswerechosenforsegmentation.Elevenimagesetsusedforanalysis werecollectedusingaSiemensSOMATOMDenitionFlashCTScanner-140kVp 512x512,1mm-1.5mmslicethickness,0.5703-0.98pixelspacing.Fourteenimagesets werecollectedusingSiemensSensation64CTscannerkVp,512x512,1.5mmslice thickness,0.6875-.976pixelspacing. SimplewareScanIPwasusedtocubicallyresampleCTimagestoacubicvoxel sizeof1mmusingalinearinterpolationmethod.Theimageswerethenwindowed toenhancebonevisualization.Imagenoisewaslteredusingacurvatureanisotropic diusionlter.AfterprocessingtheCTimagestheyweresegmentedusingcombinationofmanualandsemi-automatictechniques.Surfacemesheswereconstructed usingScanIP+FEmodule.MeshingparameterswereadjustedinscanIPtoachieved surfacemeshdensitiesbetween10,000-18,000triangularelements.Theseleswere thenimportedintoMatlab2015bforanalysis. 14

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Figure2.1: Screencaptureshowingpointscloudsofthesacrumandalignmentof shapescentroidstotheorigin Thesurfaceswereconvertedintopointcloudsusingthemeshvertices.The centroidwaswasthencalculatedandthepointcloudsweretransformedtocenter theircentroidsattheorigin.Thecentroidsizewascalculatedforalltheshapesand theshapewiththemediancentroidshapewasusedasthetargetforcorrespondence. Aninitialpre-alignmentwasperformedusingiterativeclosestpointalgorithmon arandomlyselectedsamplingfromeachpointcloud.ThenalRegistrationand correspondencewasperformedusingthecoherentpointdriftmatlabpackage.[44]The CPDpackageperformsbothrigidandnon-rigidregistrationtondcorrespondence. Oncetheshapeswereregisteredcorrespondingshapevectorswerecreated.Procrustes transformationwasappliedtotheshapevectorsincludingProcrustesscaling.The Procrustesmeanwascalculatedusingequationabove.Principalcomponentanalysis wasthenusedfordimensionalityreductionofthehigherdimensionalshapespace. 15

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Reconstructedpointcloudswerecreatedusingthefollowingequation.Where p v is theshapevectorforeachshapeintheset. p v = p + N X i =1 c i p q i .1 VisualizationwasperformedvaryingspecicPCmodesfrom-2sdto+2sd.These pointcloudwerethenexportedandsurfacemesheswerecreatedusingreverseengineeringalgorithms. 16

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3.Result TheoverallshapemodelbuiltusingtheCPDregistrationmethodologyresulted anoverallgoodqualityshapemodelbasedonvisualappearance.Allofthesacral foramenwerevisualpatent.Themediansacralcrestappearedsymmetricandnon distorted.Thecoccyxandsacralfacetsalsoappearedtohavelittledistortion. Figure3.1: ThreeviewsoftheofthenalshapemodelusingtheCPDcode. Reconstructionofthemeshstructurefromthepointcloudsappearedtohave somedistortionaroundtheS1andS2sacralbodies.Thesacralforamenwerepatent ateverylevel.Thedistalsacrumandcoccyxhadminimaldistortion.TheSIjoint surfaceexhibitedlittlenoiseanddemonstratedaclearauricularshape.Basedonthe classicationsystemingure1.1wecanclassifythemeanshapeasatypeIISIjoint surface. 17

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Figure3.2: Surfacereconstructionofthemeanshape WhenvisualizingthesacralSSMfromacoronalviewwesawtherstPCmode describedthesacralwidth.Whenmovinginthepositivedirectionfromthemean thesacrumwidthincreasedandconverselydecreasedwheninthenegativedirection. VisualizingtherstPCmodefromalateralviewwesawtheSIjointsurfacechange shape.ThisshapevariedbetweenatypeIandType3dependingonthedirection youtraveledfromthemean. 18

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Figure3.3: SurfacereconstructionoftheFirstPCmode.Screencapturedfroma coronalperspective 19

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Figure3.4: SurfacereconstructionoftheFirstPCmode.Screencapturedfroma lateralperspective IdentifyingtheshapefeaturedescribedbythesecondPCmodeismuchmore challenging.Lateralviewofthesurfacereconstructionsrevealedthesuperiorlimbof theSIjointsurfacevariedaswevariedthesecondeigenvector. 20

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Figure3.5: SurfacereconstructionoftheSecondPCmode.Screencapturedfroma lateralperspective ThethirdPCmodedescribedvariationintheconcavityoftheSIjoint.Variation inthenumberofsacralvertebralbodieswasalsodescribedinthismode.Distortion ofthesacralvertebralbodieswasmostsignicantinthismode. 21

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Figure3.6: SurfacereconstructionoftheThirdPCmode.Screencapturefroma coronalperspective Iwasunabletointerpretanyshapefeaturesdescribedinthe4thmodePCmode. Similarlythefthmodealsohadlittlevisuallyinterpretableshapevariation.However mediansacralcrestprominencedoesvaryinthefthPCmode. 22

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Figure3.7: SurfacereconstructionofthefthPCmode.Screencapturedfromalateral perspective 23

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AnalyzingthecompactnessofourSSM.Wefoundtherstfoureigenmodes capturedover60%oftheoverallvariationinthemodel.Over90%ofthevariation wascapturedwiththerst16modes. Figure3.8: Compactnessisdenedasthenumberofmodesneededtodescribeda givenamountofvariation.Morecompactmodelscanrepresentgreatervariationwith fewmodes.[51,16] 24

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4.Discussion TheresultsofthismodelsupportthattheCPD-SSMregistrationmethodcanbe aneectivetooltoconstructstatisticalshapemodelsofthesacrum.Themeanshape reconstructionappearedtocapturetheoverallstructureofthesacrumwithlittle distortion.Giventherstfoureigenmodesdescribed60%oftheoverallvariationand 16modesdescribed90%ofthevariationinthemodel.Thisdemonstratedsimilar compactnessresultstoothershapemodelingstudiesofbones.[25,38] TherewassomeartifactontheventralsurfaceoftheS1andS2sacralbodies. ThisdistortionwasdiculttorecognizewhenjustviewingthepointcloudoftheSSM ingure3.2.Howeveritwaspresentinallofthepointcloudsgeneratedsuggesting thisdistortionisinherentinthemodelitselfandnotjustalimitationofthesurface re-meshingsoftware.Thisissomewhatabnormalsinceusuallypoorcorrespondence occursinareasofhighcurvatureandoverallgeometricalcomplexity.Despitethisit isreasonabletoconcludethatdespitetheoveralllackoflocalgeometricalcomplexity inthisareathecorrespondenceispoorrelativetootherareasinthemodel. Areasofthesacrumwithlessvariationhadimprovedcorrespondence.Thiswas particularlytrueforthesacroiliacjointandfacet.Thisisparticularlyreassuringthat modelsbuiltusingthemethodologyachievehighenoughqualitycorrespondencethat describingsacroiliacjointvariationisarealpossibility.Thisworkalsodemonstrates thatthistechniqueiseectiveatdescribingadditionalmorphologicalattributesofthe sacrumincludingtheoverallcurvatureofthesacrum,mediansacralcrestvariation. Themostimpactfulndingsofthismodelarehoweectivetheeigenmodeswere atdescribingareasofvariationofthesacrum.Mode1capturesdistinctmorphological variationsofthesacroiliacjointthatwereobservedinpreliminaryworkdoneour group.Thismodewillbeveryusefulinprovidinganinitialclassicationofthe sacroiliacjointmorphology. 25

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Despitethesuccessindevelopingworkablestatisticalshapemodelofthesacrum frompointclouds.Thequalityofthereconstructedmeshesislacking.Thereare severalcommonlyusedmethodsforreconstructingsurfacemeshesincludingdelaunay, powercrustandball-pivotingalgorithms.[23,2,4]Howeverthesealgorithmsstruggle withcomplexgeometricalshapesimilarlytotheregistrationalgorithms.Increasing thedensityofthepointcloudsdoeshelpwithoverallvisualizationoftheSMM.An improvementcanevenbenoticedfromjusttherawpointclouds.Asshowninthe gurebelow. Figure4.1: ScreencaptureshowingapointcloudSSMofsacrum.Thismodelwas builtusingatargetshapepointcloudof54,700points Thisimprovedpointcloudvisualizationalsoextendstoimprovedresultsfrom surfacemeshingalgorithms.Figure4.2highlightstheimprovementofthedistortion ontheventralsurfaceofthesacrumwhenreconstructingasurfacemeshfromahigh densitySSM. 26

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Figure4.2: TheleftimageisasurfacereconstructionfromahighdensitySSMcontaining57400points.Therightimageisasurfacereconstructionfromalowdensity SSMcontaining6993points. Unfortunatelytherearesomenegativeeectstoincreasingthenumberofpoints. Computationaltimeisgreatlyaectedbythenumberofpointsusedtobuildthe model.Modelsbuiltwithlessthen10,000pointscanbecomputedinunder15 minutes.However,themodelsbuiltwith54,700pointtookover5hourstocomplete. Theoverallqualityofthemodelalsochangesasyouincreasethedensityofthepoints intheSSM.Thiscanbeseenbyachangeinthecompactnessofthemodeldepending onthedensityoftheSSM.Inthegurebelowweseethebluecurvecorresponding tothelowestdensitymodeldescribesvariabilityatamuchquickerratethanhigher densitymodels. 27

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Figure4.3: Illustrateshowthecompactnessofthemodelschangewithdensityofthe pointcloudusedintheSMM Anotherhurdlethathaspresenteditselfinthisinitialmodelisthestatistical analysisoftheoverallmodel.Principalcomponentanalysisisthemostwidelyused methodtohelpdescribeshapevariationinmodels.[27,39]Itspopularityisdueto itsabilitytoreducethedimensionalityoftheshapemodeltoamorecompactlower dimensionalshapespace.[27]Principalcomponentanalysisisalsoveryeectivein detectinganddescribingglobalshapechanges.[39,50]However,itsutilityislimited whentryingtodescribelocalvariationsofshape.[39,17]Independentcomponent analysisisbettersuitedforlocalanalysisbutislimitedbyitslackofabilitytoorder itsshapedescriptorsmakingdimensionalityreductiondicult.[39]Therehasbeen recentdevelopmentincreatingalgorithmictechniquestoallowfordimensionality reductionusingindependentcomponentanalysisbutmanyalgorithmsarestillearly onintheredevelopment.[39,1,57] 28

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5.Conclusion Despitethelimitationsofreconstructingsurfacemeshesfromthepointcloudsof ourmodeltheresultsareveryencouraging.ThisworkdemonstratesthatCPD-SSM canbeusedtodeterminethemeanshapeofthesacrumforagivenpopulation.This workalsorevealedprincipalcomponentanalysisisausefultoolfordescribingsacral morphology.Statisticalshapeanalysisofsacrumprovestobearobusttechniquethat willprovidenewinsightsintothemorphologicalvariationofthesacroiliacjointin thegeneralpopulation.Italsoshowspromiseindetectingmorphologicaldierence inpatientswithsacroiliacjointpainfromthegeneralpopulation. Therearemanyareaswheresacralstatisticalshapemodelsareuseful.Themodels createdfromthisprojectcouldeasilybeusedtocreate3dprintedtemplatesfordevelopingandtestingsurgicaljigsandimplants.CombiningSSMswithniteelement modelingwouldallowyoutosystematicallydescribehowcomplexshapefeaturesof thesacrumeectmechanicsofthepelvis.Itisalsopossibletoextendthisapplication tocreatepersonalizedniteelementmodels.Typicallythehurdlesthatprevented personalizedrealtimemechanicssimulationrevolvedaroundtime.Segmentingthe medicalimagesandcalculatingtheFEsolutionarebothtimeintensiveprocesses. HoweverSSMshavethepotentialtoactasashapeatlasspeedingupandimproving theaccuracyautomatedsegmentingtechniques.Itisalsopossibletoprecalculate FEsolutionsbasedontheshapevariationsdescribedbytheSSM.Thiscanthenbe usedasaFEsolutiondatabasewhereonecansimplyinterpolatebetweendierent solutions. 5.1FutureWork Iamcontinuingtoworkonmanyfacetsofthisproject.Thenextstepistobuild anSSMofpatientswithsacroiliacjointpain.Thiswillallowustoinvestigatethe correlationofPCmodeswithSIjointpain.InadditiontothisworkIamalsoworking onimprovingthereconstructedmeshesfromthepointclouds.Wearecurrentlybe 29

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experimentingwithwarpingtheoriginalsourcemeshtothenalmeanSSM.This canbeaccomplishedusingthinplatesplinesorreusingtheCPDalgorithmasecond time.Howevertousethismethodonewillhavetoimplementsomeformofrelaxation functiontokeepthemeshfromselfintersectinginplaceswithpoorcorrespondence orhighcurvature. Thefutureofstatisticalshapemodelingisespeciallybright.CPD-SSMcodeprovidesarobusttoolforregisteringcomplexshapesfromnoisydata.Thisisespecially usefulwhenbuildingSSMsfromclinicalimagingwheredataqualityisfarbelowthe qualityofresearchgradedata.Itismyhopethatthisthesisdemonstrateshowrobust thistoolisforbuildinghighqualityshapemodels. 30

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