QUANTIFYING BRAIN WHITE MATTER STRUCTURAL CHANGES in NORMAL AGING USING FRACTAL DIMENSION by LUDUAN ZHANG Submitted in Partial

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QUANTIFYING BRAIN WHITE MATTER STRUCTURAL CHANGES IN

NORMAL AGING USING FRACTAL DIMENSION

LUDUAN ZHANG

Submitted in partial fulfillment of the requirements

For the degree of Doctor of Philosophy

Dissertation Advisor: Guang H. Yue, Ph.D.

Department of Biomedical Engineering

CASE WESTERN RESERVE UNIVERSITY

January, 2006

CASE WESTERN RESERVE UNIVERSITY

SCHOOL OF GRADUATE STUDIES

We hereby approve the dissertation of

______

candidate for the Ph.D. degree *.

(signed)______(chair of the committee)

______

(date) ______

*We also certify that written approval has been obtained for any proprietary material contained therein.

I grant to Case Western Reserve University the right to use this work, irrespective of any copyright, for the University’s own purposes without cost to the University or to its students, agents and employees. I further agree that the University may reproduce and provide single copies of the work, in any format other than in or from microforms, to the public for the cost of reproduction.

Luduan Zhang

To my loving parents,

my brother, Yuanheng, and my sister, Hongying.

Table of Contents

Table of Contents ...... vi

List of Tables ...... x

List of Figures...... xi

Acknowledgements ...... xiv

List of Abbreviations ...... xvi

Abstract………………………………………………………………………………..xvii

1. Introduction and Background ...... 1

1.1. Introduction...... 1

1.2. Brain Anatomy and Histology ...... 3

1.2.1. Nerve Cells and Function...... 3

1.2.2. White Matter and Gray Matter...... 5

1.3. Aging of White Matter...... 6

1.3.1. Overview...... 6

1.3.2. Imaging of White Matter Changes...... 7

1.3.3. Histopathologic Correlates of White Matter Changes on MR...... 8

1.3.4. Clinical Correlates of White Matter Changes...... 8

1.4. Skeleton – Shape Medial Representation ...... 9

1.5. Fractal Dimension – Shape Complexity Descriptor ...... 12

1.5.1. The Concept of the “Fractal” ...... 12

1.5.2. Fractal Dimension...... 13

1.5.2.1. Fractal Dimension Definition ...... 13

1.5.2.2. Measurement of Fractal Dimension...... 15

1.6. Dissertation Objectives and Organization ...... 16

1.6.1. Objectives ...... 16

1.6.2. Specific Aims...... 16

1.6.3. Dissertation Organization ...... 18

1.7. References...... 21

2. Fractal Dimension in Human Cerebellum Measured by Magnetic Resonance

Imaging ...... 25

2.1. Abstract...... 25

2.2. Introduction...... 26

2.3. Materials and Methods...... 29

2.3.1. Subjects...... 29

2.3.2. Collection of MR Head Images ...... 29

2.3.3. Segmentation and Resampling of CB Images ...... 30

2.3.4. Segmentation of WM and GM...... 31

2.3.5. Extraction of WM Skeleton ...... 31

2.3.6. Measurement of CB Fractal Dimension ...... 32

2.3.7. Statistical Analysis...... 33

2.4. Results...... 33

2.5. Discussion...... 35

2.6. References...... 40

3. A Three-dimensional Fractal Analysis Method for Quantifying White Matter

Structure in Human Brain ...... 44

3.1. Abstract...... 44

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3.2. Introduction...... 45

3.3. Materials and Methods...... 48

3.3.1. MR Images...... 48

3.3.2. Image Processing ...... 49

3.3.3. Fractal Analysis Methods ...... 51

3.3.3.1. Box-counting Method in HarFA...... 52

3.3.3.2. Fractal Analysis Methods of the Present Study ...... 53

3.3.4. Accuracy and Sensitivity Assessment ...... 56

3.4. Results...... 58

3.4.1. Phantom Results...... 58

3.4.2. MR Results...... 59

3.4.3. Box Size...... 61

3.5. Discussion...... 63

3.5.1. Importance of Shape Descriptor (Skeleton) in FD Analysis ...... 66

3.5.2. Comparison with HarFA and Pseudo-3D Method...... 66

3.5.3. Accuracy Assessment ...... 68

3.5.4. Sensitivity Assessment...... 70

3.5.5. Summary...... 71

3.6. References...... 72

4. Quantifying Brain White Matter Changes in Normal Aging using Fractal

Dimension ...... 76

4.1. Abstract...... 76

4.2. Introduction...... 77

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4.3. Materials and Methods...... 79

4.3.1. Subjects...... 79

4.3.2. Head Images Acquisition...... 79

4.3.3. Image Processing ...... 80

4.3.4. Measurement of Fractal Dimension...... 80

4.3.5. Volume Measurement...... 80

4.3.6. Statistical Analysis...... 82

4.4. Results...... 83

4.4.1. Age and Gender Effects on WM Structural Changes ...... 83

4.4.1.1. Age Effect ...... 83

4.4.1.2. Gender Effect...... 84

4.4.1.3. Age by Gender Effect ...... 84

4.4.2. Asymmetry of WM...... 85

4.5. Discussion...... 99

4.5.1. WM Structural Changes with Aging...... 99

4.5.2. Gender Differences of WM ...... 101

4.5.3. Asymmetry of WM...... 102

4.5.4. Summary...... 103

4.6. References...... 105

5. Conclusions and Future Directions ...... 111

5.1. Conclusions...... 111

5.2. Future Directions ...... 114

List of Tables

Table 1.1 Summary of MR-data acquisition, data set and investigated structures ...... 20

Table 2.1 Results of fractal dimension (FD) in cerebellum ...... 36

Table 3.1 Results of fractal dimension/topological dimension of phantoms ...... 58

Table 3.2 Results of fractal dimension (mean ± standard error of the mean) of WM skeleton, surface and general structure ...... 61

Table 3.3 Results of box size range and length of data segments (dr) of phantoms...... 63

Table 3.4 Results of box size range of WM...... 64

Table 3.5 Summary of used software packages and my contributions to the data process methods...... 68

Table 4.1 Results of white matter fractal dimension...... 86

Table 4.2 Results of white matter volume...... 87

Table 4.3 Post hoc analysis P value table for age effect on white matter fractal dimension...... 95

Table 4.4 Post hoc analysis P value table for gender effect on white matter fractal dimension...... 96

Table 4.5 Post hoc analysis P value table for hemispheric asymmetry of white matter fractal dimension...... 96

Table 4.6 Post hoc analysis P value table for age effect on white matter volume...... 97

Table 4.7 Post hoc analysis P value table for gender effect on white matter volume.... 98

Table 4.8 Post hoc analysis P value table for hemispheric asymmetry of white matter volume...... 98

List of Figures

Figure 1.1 Cells in the central nervous system ...... 4

Figure 1.2 Brain anatomy...... 5

Figure 1.3 Skeleton illustration...... 10

Figure 1.4 Illustration of dimension calculation ...... 14

Figure 1.5 FD of fractals...... 14

Figure 1.6 Traditional 2D box-counting method ...... 16

Figure 2.1 A sample slice of CB MR image and its resampling to different resolutions 30

Figure 2.2 Histogram of the CB image set of a subject ...... 32

Figure 2.3 Illustration of the steps to extract the CB skeleton in a sample slice ...... 33

Figure 2.4 Illustration of the generated CB skeletons in two sample slices from two subjects...... 34

Figure 2.5 Illustration of the linear fit to obtain the fractal dimension in a single subject

...... 35

Figure 3.1 Flowchart of image processing steps...... 49

Figure 3.2 Brain tissue segmentation in a sagittal, coronal and transverse slice from one subject ...... 50

Figure 3.3 2D box-counting method in HarFA...... 53

Figure 3.4 Linear regression analysis to obtain the initial FD of the entire WM skeleton from one subject...... 55

Figure 3.5 Single slope analysis...... 56

Figure 3.6 Generated phantoms ...... 57

Figure 3.7 Illustration of the generated WM skeletons in two sample slices and in three

dimensions from one young and one old subjects ...... 60

Figure 3.8 95% confidence interval plot of FD results of young and old groups for the whole brain (Whole), left hemisphere (Left), right hemisphere (Right) and cerebellum

(CB)...... 62

Figure 3.9 Linear regression analysis of chosen box sizes of phantoms ...... 65

Figure 4.1 Image processing results based on one sample coronal slice from one young subject (A-C) and one sample coronal slice from one old subject (D-F) ...... 81

Figure 4.2 Box plot of WM skeleton FD results of young (males and females) and old groups (males and females) for the (A) whole brain, (B) left hemisphere, and (C) right hemisphere ...... 88

Figure 4.3 Box plot of WM surface FD results of young (males and females) and old groups (males and females) for the (A) whole brain, (B) left hemisphere, and (C) right hemisphere ...... 89

Figure 4.4 Box plot of WM general structure FD results of young (males and females) and old groups (males and females) for the (A) whole brain, (B) left hemisphere, and (C) right hemisphere...... 90

Figure 4.5 Box plot of WM absolute volume results of young (males and females) and old groups (males and females) for the (A) whole brain, (B) left hemisphere, and (C) right hemisphere...... 91

Figure 4.6 Box plot of WM relative volume results of young (males and females) and old groups (males and females) for the (A) whole brain, (B) left hemisphere, and (C) right hemisphere...... 92

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Figure 4.7 Box plot of FD results of left hemisphere (Left) and right hemispheres

(Right) in male (young and old) and female participants (young and old)...... 93

Figure 4.8 Box plot of volume results of left hemisphere (Left) and right hemispheres

(Right) in male (young and old) and female participants (young and old)...... 94

Figure 5.1 One sample T1-weighted slice of a stroke patient...... 115

Figure 5.2 Image processing results of one sample T1-weighted MR slice ...... 116

Figure 5.3 Fractal dimension (FD) results ...... 118

Figure 5.4 Eight sample slices of a stroke patient and fractal dimension (FD) values corresponding to the eight slices...... 119

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Acknowledgements

Foremost, I would like to express my deepest appreciation to Dr. Guang H. Yue of the Cleveland Clinic Foundation (CCF) for his guidance over the last five years. As my mentor and research advisor, he is the one who has taught me much about being a good scientist. His continued support, valuable expertise and encouragement have helped me overcome the obstacles during these years. He was always on my side!

I am sincerely grateful to my academic advisor, Dr. Marty Pagel of Case Western

Reserve University (CWRU), for having kept me posted on various departmental deadlines to graduate smoothly. In addition, he provided me valuable suggestions not only for this project, but also for my career planning.

I am deeply indebted to Dr. David Dean of CWRU who has brought me great ideas about my project. His expertise in neuroanatomy and morphometrics has helped me keep my research focused. He was extremely responsible, not only supervising the research part of my manuscripts or proposals, but also spotting every typo-mistake.

I also express my sincere gratitude to Drs. David L. Wilson and Lee White of

CWRU, for joining my guidance committee. Dr. Wilson’s considerable expertise in medical imaging has greatly extended my knowledge in the field of image processing.

Dr. White has provided me with opinions about computer algorithms.

In addition, I would like to thank Drs. Jingzhi Liu and Vinod Sahgal of CCF, and

Drs. Igor Efimov and Miklos Gratzl of CWRU. Dr. Liu helped me create the idea, and gave me many good suggestions for this project. Dr. Sahgal has provided me with great opinions about the clinical applications of this research. Dr. Efimov, my former academic

xiv

advisor, helped me shape the program of study and solve academic problems. Thank you,

Dr. Gratzl, for offering me a lot of support during my adaptation to the different culture.

Many thanks to everyone from the CCF neural control lab who worked with me.

Among others, this includes Dr. Yin Fang, Dr. Zuyao Shan, Dr. Vlodek Siemionow, Bing

Yao, Haibin Huang, Vinoth Ranganathan, Qi Yang and our new member Dr. Didier

Allexandre. Thank you all for providing me an inspiring environment and a place full of fun.

I also thank all my friends who helped and supported me during the past few years.

I am grateful to Ms. Christine Kassuba, the editorial assistant of CCF, for her last minute, sweet help on writing corrections.

Finally, and most importantly, my acknowledgements go to my parents, my brother, Yuanheng, my sister, Hongying, and Quanwei Lu. Their unconditional love and unfailing faith give me the strength to go through all kinds of challenges in life. My Ph.D. would not have been possible without their incredible support and encouragement. Thank you, my family!

List of Abbreviations

2D Two-dimensional

3D Three-dimensional

CB Cerebellum

C-R Crown-rump

DTI Diffusion Tensor Imaging

FD Fractal Dimension

GM Gray Matter

MR Magnetic Resonance

MRI Magnetic Resonance Imaging

WM White Matter

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Quantifying Brain White Matter Structural Changes

in Normal Aging Using Fractal Dimension

Abstract

LUDUAN ZHANG

It has been reported that brain white matter (WM), which conducts information to and from the cerebral cortex, experiences degeneration in normal aging. Age-related WM changes have increasingly been explored in brain research due to its possible prognostic significance for diseases such as motor function impairment, cognitive deficits, and depression. Current clinical diagnosis of WM degeneration still depends primarily on visual rating scales due to the lack of objective measurements. Although volume analysis based on magnetic resonance (MR) images, a conventional technique in WM quantification research, is appropriate in assessing brain atrophy, it only captures one of multiple aspects of a full structural characterization of WM and tells little about WM shape adaptations. The goal of this research was to develop a shape analysis method

(fractal dimension [FD] analysis) to study the structurally complex changes of WM as a result of aging in three parts. First, the FDs of human cerebellum (CB) WM interior structure (skeleton) in MR images were measured in healthy young subjects using traditional FD methods, which served as a feasibility study for this research. The results showed that the CB skeleton was a highly fractal structure and no differences of CB FDs were detected between men and women. Second, a three-dimensional (3D) FD method

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was developed and validated to measure the FD of human brain WM interior structure,

WM surface and WM general structure simultaneously. The results indicated that the method was accurate in quantifying 3D brain WM structures and sensitive in detecting age-related degeneration of these structures. Finally, a cross-sectional study was conducted to investigate WM complexity changes in normal aging. WM FDs of old people were found to be significantly smaller than those of young people, suggesting that

WM structural complexity declined with normal aging. These findings provided new information in describing brain WM structural complexity, which might in the future serve as an objective diagnostic index or as a predictive parameter for neurological diseases. This method might be used for longitudinal studies to evaluate the effect of disease or aging on WM.

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Chapter 1

Introduction and Background

1.1. Introduction

Human brain white matter (WM) has been reported to experience degeneration as a consequence of normal aging and neurological disease. Although the clinical or functional consequences of WM degeneration are not well understood, correlation studies have shown a close association between WM changes and motor function impairment

(Breteler et al., 1994; Baloh et al., 2003), cognitive deficits (de Groot et al., 1998;

Gunning-Dixon and Raz, 2000), and depression (Pantoni and Garcia, 1995; Inzitari et al.,

2000). Because WM changes could be a predictive parameter for disease, WM degeneration has increasingly been of interest in brain research.

Magnetic resonance imaging (MRI) provides a good tool to assess WM degeneration because of its relatively high sensitivity in detecting WM shape. In clinical diagnosis based on MR (magnetic resonance) images, there is no single best scale but 13 visual rating scales (Mantyla et al., 1997) are commonly used to rate WM changes. The

1 Chapter 1 Introduction and Background 2 variety of rating scales may be one of the reasons why observations of WM changes in previous studies have been inconsistent (Mantyla et al., 1997). Quantitative techniques can be expected to provide more consistent rating scales for WM adaptations. Volumetric

WM analysis is a conventional technique of quantification. It is sensitive in detecting

WM atrophy but only captures one of multiple aspects of the structure and tells little about WM shape and internal structural changes. Because of this limitation, current diagnosis of WM degeneration still depends primarily on visual rating scales.

Shape analysis has been reported to provide new information on WM shape changes versus conventional volumetric measurements (Gerig et al., 2001). In this technique, appropriate shape representations (Costa and Cesar 2001) are extracted before the shape is characterized by shape descriptors. The criteria of an appropriate shape representation of a biological object are that it adequately represents shape variation and is biologically meaningful. Three WM shape representations are described in the three- dimensional (3D) studies reported in Chapters 3 and 4: general, surface, and interior

(skeleton) structures, whereas only the skeleton was analyzed in the two-dimensional

(2D) study discussed in Chapter 2. The set of WM voxels in a segmented image is a general structural representation of WM shape. The external surface of those voxels identifies a set of boundary voxels. The central skeleton of those voxels represents the essential structure of the WM object, and it is used to represent the interior shape of the

WM object.

In this project I investigate WM shape changes due to normal aging using a complexity descriptor – fractal dimension (FD). Our research extended the knowledge of age-related WM structural changes by complementing size information (provided by

Chapter 1 Introduction and Background 3 volume measurements) with complexity information. We expect that the FD will serve as a quantitative index for estimating age- or disease-related WM degeneration.

In this chapter I first discuss the general biological background of WM tissue, including brain anatomy and histology, imaging of WM, histopathology, and clinical correlates of WM changes with aging. I then provide methodological background, including WM skeletonization (definition and computational methods) and the FD

(fractal concept and measurements of FD). My dissertation’s objectives and organization are finally described.

1.2. Brain Anatomy and Histology

The human brain processes and integrates information from the external environment and generates appropriate behavioral responses. Accomplishing these tasks requires an anatomical configuration with considerable complexity. The human brain is a network of hundreds of billions of interconnected nerve cells. Knowledge of the brain’s anatomical structure and the pathways of information flow in the brain is important not only for understanding its normal function but also for identifying specific regions that are disturbed by neurological diseases (Kandel et al., 2000). This section overviews brain anatomy from the microscopic (Section 1.2.1) to the macroscopic level (Section 1.2.2).

1.2.1. Nerve Cells and Function

At the microscopic level, brain function is accomplished by nerve cells and their connections. There are two classes of cells in the central nervous system: nerve cells

(neurons) and glial cells.

Chapter 1 Introduction and Background 4

Neurons are the main signaling units of the nervous system (Kandel et al., 2000).

A neuron consists primarily of a cell body, dendrites, and an axon (Figure 1.1A). The cell body is the metabolic center of the cell and processes the neural information. Dendrites receive incoming information from other neurons. The axon conveys information to other neurons. To increase the signal conduction speed, axons are wrapped in a sheath of insulating lipoprotein called myelin.

(A) Dendrites Cell body Myelin

Axon Axon

Presynaptic neuron Postsynaptic neurons

(B) (C) Capillary Oligodendrocyte End-foot

End-foot Astrocyte

Myelin Axon Neuron

Figure 1.1. Cells in the central nervous system. (A) Neuron. (B) Oligodendrocyte in white matter. (C) Astrocyte. (Figures taken from Kandel et al., 2000 and modified.)

Glial cells provide support for neurons and are not directly involved in information processing. There are two major types of glial cells in the central nervous system (brain and spinal cord): oligodendrocytes and astrocytes. Oligodendrocytes are a

Chapter 1 Introduction and Background 5

major portion of the WM produce myelin used to insulate neuron axons (Figure 1.1B)

(WM is discussed in the next section). Astrocytes are thought to bring nutrients to neurons, forming end-feet on the surface of neuron. They also help form the blood-brain barrier by forming end-feet on the brain’s capillaries (Figure 1.1C).

1.2.2. White Matter and Gray Matter

The brain is composed of three major structures: cerebrum, cerebellum, and brainstem

(Figure 1.2A). The cerebrum, the largest part of the brain, has perceptual, motor, and

cognitive functions, including memory and emotion. The cerebellum, located at the base

of the brain, modulates motor coordination, maintains equilibrium, and modulates

sensory and cognitive functions. The brainstem, connecting the brain and the spinal cord, acts as a relay station and plays many other important functions.

(A) (B) (C)

Figure 1.2. Brain anatomy. (A) Brain divisions (Image taken from www.brainexplorer.org). (B) Brain tissue (T1-weighted MR coronal image). (C) Structure of a nerve bundle. (Image taken from Purves et al., Life: The Science of Biology, 4th Edition, by Sinauer Associates and WH Freeman.)

Both the cerebrum and cerebellum consist of an outer layer, known as the gray matter (GM), overlaying a central core of WM (Figure 1.2B). The GM is also called

Chapter 1 Introduction and Background 6

cortex, cerebral cortex (approximately 3mm thick) and cerebellar cortex (approximately

2mm thick), for cerebrum and cerebellum, respectively. Histologically, neuronal cell

bodies constitute the GM. The WM, located in the central and subcortical regions of the

cerebral and cerebellar hemispheres, plays an essential function of conducting

information to and from the cerebral cortex. Histologically, the WM contains nerve fibers

(bundles of axons that wrapped in a sheath of myelin), supporting cells (oligodendrocytes and astrocytes), interstitial space, and vascular structures (Figure 1.2C).

1.3. Aging of White Matter

Aging is an inevitable phenomena of life. Normal brain undergoes morphological

changes with increasing age. In this section we focus on the aging of WM. Section 1.3.1

overviews aging of the brain. Section 1.3.2 discusses imaging of WM, including normal

WM and aged WM. Section 1.3.3 discusses histopathology of WM changes. Section

1.3.4 discusses clinical correlates of age-related WM changes.

1.3.1. Overview

Conspicuous age-related changes in the brain are decreased brain weight and volume,

dilation of ventricular space, increases of cerebrospinal fluid volume and WM

degeneration (Albert and Knoefel, 1994). At a microscopic level, normal aging is

demonstrated by a decrease in the number of neurons while an increase in the size and

number of glial cells. The process includes dendrite and dendritic spine loss, cell body

loss, and shrinkage of axons and myelin (Boss, 1991; Mrak et. al., 1997). The

macroscopic changes are indeed reflective of the microscopic changes. For example,

deep WM changes correspond to gliosis, demyeliniztion, small vessel disease, widened

Chapter 1 Introduction and Background 7

Virchov-Robin spaces, and atrophy and shrinkage of axons and myelin around blood vessels (Challa and Moody, 1987; Kirkpatrick and Hayman, 1987; de Groot et al, 1998).

Periventricular WM changes correspond to demyelination and reactive gliosis (Scheltens et al., 1995).

The loss of dendrites reduces neuronal interaction. Axonal and myelin thinning decrease conduction velocity, which reduces the speed of information processing and increases response time. On the system level, aging results in reduced rate and amount of motor activity, slowed information processing, recent memory, retrieval and response time (Boss, 1991).

1.3.2. Imaging of White Matter Changes

The introduction of neuroimaging techniques has brought many reports regarding WM changes in elderly people, particularly in those with vascular risk factors, cerebrovascular diseases, and cognitive impairments (Pantoni and Garcia, 1995). Compared with other imaging techniques, MR imaging has been found sensitive in detecting brain WM changes due to normal aging and a variety of diseases. Normal WM appears white macroscopically because of a large amount of myelin. It is hypointense on T2-weighted images and hyperintense on T1-weighted images (Figure 1.2B), primarily due to the lipid content of myelin. WM changes that reduce myelin content result in increased signal on

T2-weighted images and reduced signal on T1-weighted images (Barkhof and Scheltens,

2002), and are described as bilateral, punctuate, patchy, or diffuse. MR signal characteristics of WM changes are similar and relatively nonspecific, and the foci of WM changes include periventricular, subcortical, and deep WM signal alterations (Ketonen,

1998). Although it is hard to distinguish tissue changes related to normal aging from

Chapter 1 Introduction and Background 8

neurodegenerative disorders, other features could be used to assist in diagnosis, such as

the pattern of the lesions (shape, size, distribution) and location (Barkhof and Scheltens,

2002).

1.3.3. Histopathologic Correlates of White Matter Changes on MR

Age-related WM adaptation is a heterogeneous process, ranging from normal to

pathological (Pantoni, 2002), probably because of different degenerative mechanisms.

Although histopathologic correlates of WM on MR are not well understood, ischemia is

the most commonly cited case. Other possible contributing mechanisms include blood-

brain barrier alterations, edema, loss of autoregulation (Pantoni, 2002). Pantoni and

colleagues (1997, 2002) reviewed these hypotheses on the etiology of WM changes and concluded that the most accepted hypothesis is that alterations of small parenchymal

blood vessels play a central role. Because of its peculiar terminal type of blood supply,

WM is vulnerable to ischemia (Pantoni and Garcia, 1997). Different locations of WM

changes may also have different pathological mechanisms. de Groot and colleagues

(1998) suggest that WM alterations at a deep parenchymal locations are primarily due to

ischemic origins whereas periventricular WM changes are the result of long-term

augmentation of periventricular fluid concentrations, perhaps due to breakdown of the

ventricular ependym.

1.3.4. Clinical Correlates of White Matter Changes

Schmidt and colleagues carried out a three-year (1999, 2000) and six-year (2003) follow-

up Stroke Prevention Study in Austrian to provide information on the rate of progression

of MRI WM lesions in elderly individuals without neuropsychiatric disease. Based on

Chapter 1 Introduction and Background 9

MR images, the authors measured total volume of the WM and found that lesion grade at the baseline was the only significant predictor of lesion progression. Punctate WM lesions are not ischemic, not progressive and benign while early confluent lesions are ischemic, progressive, or malignant.

Despite many existing discrepancies, WM changes have been consistently reported in association with motor and gait impairment (Breteler et al., 1994; Baloh et al.,

2003), global and selective cognitive deficit (de Groot et al., 1998; Gunning-Dixon et al.,

2000), and depression (Pantoni and Garcia, 1995; Inzitari et al, 2000). Gunning-Dixon and colleagues (2000) concluded that WM changes exert detrimental effects on cognitive function by influencing speed of neural transmission and interneuronal connectivity.

These changes, in turn, bring about cumulative generalized slowing, eventually resulting in a variety of cognitive deficits. Inzitari and colleagues (2000) reviewed the clinical consequences of WM changes in the aging population and found that the functional expression of WM changes is distributed from normal to severely disabled populations.

Based on this fact, they concluded that WM changes could be one of the age-related adaptive processes that cause the transition to disability in the elderly. WM changes have been postulated to have predictive potential for ischemic stroke, intracerebral hemorrhage, cognitive decline, and dementia (Kapeller et al., 1998).

1.4. Skeleton – Shape Medial Representation

The notion of image object’s “skeleton” was introduced by Blum (1967). It is the best known form of the medial axis representation. One of very illustrative definitions of the medical axis is given by grassfire analogy. Imagine that a fire started at the boundary of a shape that spreads inward at a constant speed (Figure 1.3A). When the fire wavefronts

Chapter 1 Introduction and Background 10

(A) (B)

Figure 1.3. Skeleton illustration. (A) The grassfire process. (B) The skeleton result. meet, quench points (equidistant from two different parts of the boundary) form and the fire is extinguished. The locus of these quench points is the medial axis skeleton (e.g., dashed lines in Figure 1.3A and solid lines in Figure 1.3B). The medial axis skeleton can be thought of as the essence of an object’s shape. It is a thin (one-voxel width in 3D and one-pixel width in 2D case) and centered line figure. It represents the local object symmetries. The medial axis skeleton has three major properties: topology preservation

(i.e., retaining the topology of the original object), geometry preservation (i.e., skeleton being in the middle of the object and stable under the most important geometrical transformation including translation, rotation, and scaling), and connectivity preservation

(i.e., no connected object is split away or completely deleted and no hole in an object is created or eliminated).

A variety of methods have been proposed to approximate the medial axis skeleton in 2D and 3D in the literature. They can be classified into three categories: topological thinning, distance transform, and Voronoi methods. The topological thinning method

(Lam et al., 1992; Ma and Sonka, 1996) is an iterative process that peels off as many boundary voxels (pixels in 2D case) as possible until only one-voxel (or one-pixel in 2D case) thick skeleton is left. This process is analogous to the grassfire definition of the

Chapter 1 Introduction and Background 11 medial axis skeleton. The procedure of thinning preserves topology and connectivity of the object. This algorithm is relatively easy to implement in 3D and in parallel. The major disadvantage of the topological thinning is that it is limited in geometry preservation. The best medial axis skeleton is not always located and the results are geometric- transformation sensitive. The distance transformation method (Kimmel et al., 1995; Zhou et al., 1998) detects ridges (i.e., local curvature extrema) in distance maps of the boundary points and considers the ridges as skeletal points. The advantages of the method are that it preserves geometry but the obtained skeleton does not represent the true topology of the object. The Voronoi method (Brandt and Algazi, 1992) calculates the

Voronoi diagram of the boundary points and the skeleton is a subgraph of the Voronoi

Diagram. The method preserves both geometry and topology of the object and is mainly designed for extraction of a skeleton of continuous objects. Unfortunately, the Voronoi calculations are a computationally expensive process, especially when analyzing large, complex objects. The method is relatively difficult to implement in 3D because Voronoi skeletons are often excessively hairy (i.e., too much branch noise in the medial axis skeleton domain) and pruning algorithms are needed to carry out in these procedures

(Shaked and Bruckstein, 1998).

In this research, the complexity of WM shape has been quantified. The major concerns in extracting medial axis skeletons are topology preservation and connectivity preservation. Whether all of the medial axis skeletons are located in the perfect medial place (geometry preservation) is not strictly required. For this reason, the thinning method is adopted to obtain skeleton of the WM. In Chapter 2, a 2D regular thinning

Chapter 1 Introduction and Background 12

method is used to extract the skeleton of 2D cerebellar WM images. In Chapters 3 and 4,

a 3D thinning method (Ma and Sonka, 1996) is employed to a yield 3D WM skeleton.

1.5. Fractal Dimension – Shape Complexity Descriptor

1.5.1. The Concept of the “Fractal”

The fractal concept, first developed by Mandelbrot (1982), who coined the term from the

Latin adjective “fractus”, provides a useful tool to quantify the inherent irregularity of some images. A fractal is a rough and irregular object made of parts that are in some way similar to the whole. It is mathematically defined as any set for which the dimension, a continuous function, exceeds the discrete topological dimension. Fractals have three properties:

• Self-similar – Fractals are composed of smaller versions of itself. When

magnified, they turn out to be identical to the entire object.

• Infinitely detailed – When magnified, fractals do not become simple, but remain

as complex as they were without magnification.

• Fractal dimension (FD)

FD is the most unique property of fractals. Compared with topological dimension in conventional geometry, the FD, termed in fractal geometry, is fractional. It serves as an index of the morphometric variability and complexity of the object. Fractals are frequently seen in nature, such as clouds, coastlines, snowflakes, as well as the human body. A wide range of fractals in characterizing the shape of the human body have been

Chapter 1 Introduction and Background 13 studied, such as the lung (Prakash et al., 2002; Kido and Sasaki, 2003) and brain (Kedzia et al., 2002; Kiselev et al., 2003).

1.5.2. Fractal Dimension

1.5.2.1. Fractal Dimension Definition

Dimension is a term used to measure the size of a set. We are all familiar with geometric objects such as a line segment, a square, and a cube. We know they have topological dimension 1, 2, and 3, respectively. Fractals have dimensions (FD), too.

Mathematically, FD is frequently called Hausdorff dimension. The Hausdorff dimension accurately measures dimension of an arbitrary metric space, including complicated sets such as fractals. Here we explain the FD definition intuitively and omit formal mathematical derivations and proofs (see (Mandelbrot, 1982) for mathematical details).

For geometric objects, first, if a 1-dimensional line segment with the magnification of 2 is examined, 2 identical line segments are seen. For a 2D square, 4 identical shapes are obtained with a magnification of 2. Finally, if a 3D cube is magnified by 2 times, 8 identical cubes are produced. Figure 1.4 demonstrates the magnification procedure.

According to these three examples, we can get a clear pattern of relationship between magnification and dimension and written as Eq. 1.1.

eD = N (1.1)

Chapter 1 Introduction and Background 14

Figure Dimension No. of identical shapes

Line 1 2 = 21

Square 2 2 = 22

Cube 3 2 = 23 D 2 = 2D

Figure 1.4. Illustration of dimension calculation. where e is magnification, N is number of copies, and D is dimension. The inserted table in Figure 1.4. also illustrates the power-law relationship. The dimension is then obtained by the logarithmic operation on Eq. 1.1 as following (Eq. 1.2).

DN= ln / ln e (1.2)

D=ln4/ln3=1.26

(A)

Koch Snowflake Sierpinski Triangle Menger Sponge

(B) D = 1.26 D = 1.58 D = 2.73

Figure 1.5. FD of fractals. (A) Illustration of FD calculation of snowflake. (B) FDs of some famous fractals.

Chapter 1 Introduction and Background 15

Now the FD can be calculated using Eq. 1.2. Take one of the famous fractals, a Koch

snowflake as an example. If part of a snowflake is magnified by 3 times (Figure 1.5A), 4

identical copies are produced. The FD of the snowflake is therefore 1.26

( D =ln 4 / ln 3 =1.26 ). Figure 1.5B lists FD values of some famous fractals, including

Koch Snowflake, Sierpinski Triangle, and Menger Sponge.

1.5.2.2. Measurement of Fractal Dimension

There are various methods commonly used for estimating FD, including box-counting,

correlation dimension, and surface model methods (Kenkel and Walker, 1996). This

research adopted a box-counting method. Details of the other methods can be found in

Kenkel and Walker’s review (1996).

The box-counting method (Mandelbrot, 1982; Kenkel and Walker, 1996) is the

most desirable method in for image FD estimation because it can apply fractal patterns

with or without self-similarity, such as brain (i.e., the brain is self-similar only in a

certain range of scales and thus not strictly self-similar). It works by repeatedly covering

the fractal image with different–sized boxes (r) and then evaluating the number of boxes

(N ) needed to cover the fractal completely (Figure 1.6). Figure 1.6 demonstrates two

meshes with different-sized boxes (5 pixels and 15 pixels) overlaid on a 2D WM slice

(binary image). For the two meshes, a total of 676 boxes (green boxes in Figure 1.6A)

and 115 boxes (green boxes in Figure 1.6B) were found to cover the WM pixels

completely.

The FD is defined in the power-law relationship (Eq. 1.3):

Nk= r−FD (1.3)

Chapter 1 Introduction and Background 16

(A) (B)

Figure 1.6. Traditional 2D box-counting method. Mesh with different sizes (r) was put in the 2D sample WM slice (binary image) and number of boxes (N) that cover the WM completely was counted (green boxes in the images). (A) r = 5 pixels, N = 676. (B) r = 15 pixels, N = 115. The images were generated by box-counting package from HarFA (http://www.fch.vutbr.cz/lectures/imagesci/ ) using data from one subject.

where k is a constant. The FD was obtained by linearly fitting the equation (Eq. 1.4):

ln NF=Dln ()1/ r+ln k (1.4)

1.6. Dissertation Objectives and Organization

1.6.1. Objectives

The overall objective of the project is to develop a fractal analysis method for a better

understanding of the morphological adaptations of brain WM as a result of aging. The

potential unique information provided by the study is that the method might eventually

lead to an objective diagnosis scale for age- or disease-related brain WM degeneration.

The method may also be useful in longitudinal studies to evaluate effects of disease

development, progression of aging, or medical intervention on brain WM structural

adaptations. Correspondingly, this project had three specific aims.

1.6.2. Specific Aims

(1) To measure FD of 2D human cerebellar WM skeletons. The medial axis skeleton is a major feature of WM structure. It is hypothesized that these skeletons represent shape

Chapter 1 Introduction and Background 17 of the WM image object, and box-counting analysis can adequately describe complexity of the WM skeleton configuration. To test this hypothesis, 2D WM skeletons of human cerebellum will be extracted from MR images using a thinning method (Lam et al., 1992) and the skeleton FD (box-counting dimension) will be calculated. MR brain and cerebellum images from elderly and young subjects will be collected. By calculating the number of pixels belonging to the WM skeleton images at different scales, the box- counting dimension can be deduced from the relationship between the size and number of pixels.

(2) To develop a method for fractal dimension measurements of three-dimensional cerebral and cerebellar skeletons, surfaces, and general structures. It is hypothesized that the 3D box-counting method will provide a more robust tool for detecting WM structural changes than the 2D method. For this reason, a 3D volumetric fractal analysis method that quantifies not only interior and surface brain structures, but also the WM general structure will be developed. Compared with the existing procedures that use only one shape descriptor (surface or general structure), a combination of multi-shape descriptors (skeleton, surface, and general structure) may provide a more robust detection tool than a single one. It is also hypothesized that the 3D method will accurately measure the WM FD. To test this hypothesis, mathematical fractal phantoms will be generated, and the computed FDs will be compared with their theoretical values. It is further hypothesized that the 3D method will be sensitive in detecting WM structural differences.

To test this hypothesis, statistical analysis will be performed on the computed FD values in a number of young and old subjects.

Chapter 1 Introduction and Background 18

(3) To investigate WM structural changes in normal aging. It is hypothesized that the

FD of old subjects will be smaller than that of young people as a consequence of age-

related WM degeneration. This hypothesis will be tested by comparing the difference in

FD values of the young and old groups. The WM FDs will be computed using the newly-

developed 3D method. Statistical analysis will be performed to investigate the effects of

aging on whole-brain and regional WM structures.

1.6.3. Dissertation Organization

Chapter 1: Introduction and Background

This chapter introduces the general background, including an overview of the

WM anatomy and pathology, shape analysis and fractal geometry. It also describes the overall objective for this project, and the significance and specific aims to achieve the objectives.

The next three chapters (Chapters 2, 3, and 4) correspond to the three specific aims. Each chapter approximately matches a manuscript that has been published in, accepted to publish in, or submitted to a peer-reviewed journal for publication.

Chapter 2: Fractal Dimension in Human Cerebellum Measured by Magnetic

Resonance Imaging

This chapter (Liu et al., 2003) describes a pilot study. FD of human cerebellar

WM skeleton of young adults is measured using traditional box-counting method. FD values of men and women are also compared.

Chapter 3: A Three-dimensional Fractal Analysis Method for Quantifying White

Matter Structure in Human Brain

Chapter 1 Introduction and Background 19

This chapter (Zhang et al., 2005a) describes a 3D fractal analysis method. The accuracy of the method is validated by phantom images with theoretical dimensions. The sensitivity of the method in detecting WM changes in aging is examined and discussed by applying the method to a number of young and old MRI brain images.

Chapter 4: Quantifying Brain White Matter Changes in Normal Aging Using

Fractal Dimension

This chapter (Zhang et al., 2005b) provides a cross-sectional study of WM changes in normal aging. FDs of WM skeleton, surface and general structures are measured in young and old adults. Results of age-related WM FD changes are presented and gender differences in WM FD changes and WM hemispheric asymmetry information are also provided.

Chapter 5: Conclusions and Future Directions

This chapter contains the overall conclusions and the future directions of this research.

Table 1.1 summarizes the MR-data acquisition activities, data set, and investigated structures of this project.

Chapter 1 Introduction and Background 20

Table 1.1. Summary of MR-data acquisition, data set and investigated structures

Chapter Head Images Acquisition Subjects FD measurements Chapter 2 • 1.5 T Siemens Vision • 24 young healthy • Pseudo-3D scanner subjects (17-35 years cerebellar WM • 3D Turboflash imaging old, mean age ± standard skeletons sequence, T1-weighted deviation = 27.7 ± 4.4 • Contiguous coronal years) slices (n = 128, each 2 • 12 males (20-35 years mm thick) old, 28.8 ± 3.7 years) • In-plane resolution, 1 × 1 • 12 females (17-35 years mm2 old, 26.7 ± 5.0 years) Chapter 3 • 1.5 T Siemens Vision • 6 healthy subjects • Left hemisphere scanner • 3 young subjects (17, 32 of cerebrum (3D • 3D Turboflash imaging and 32 years) WM skeletons, sequence, T1-weighted • 3 old subjects (72, 75 surfaces, general • Contiguous coronal and 76 years) structures) slices (n = 128, each 2 • Right hemisphere mm thick) of cerebrum (3D • In-plane resolution, 1 × 1 WM skeletons, mm2 surfaces, general structures) • Cerebellum (3D WM skeletons, surfaces, general structures) Chapter 4 • 1.5 T Siemens Vision • 24 young subjects (17-35 • Left hemisphere scanner years old, mean age ± of cerebrum (3D • 3D Turboflash imaging standard deviation = WM skeletons, sequence, T1-weighted 27.7 ± 4.4 years) surfaces, general • Contiguous coronal • 12 old subjects (72-80 structures) slices (n = 128, each 2 years old, 74.8 ± 2.6 • Right hemisphere mm thick) years) of cerebrum (3D • In-plane resolution, 1 × 1 • Young group: 12 males WM skeletons, mm2 (28.8 ± 3.7 years) and 12 surfaces, general females (26.7 ± 5.0 structures) years) • Old group: 5 males (75.0 ± 3.3 years) and 7 females (74.7 ± 2.2 years) Chapter 5 • 1.5 T Philips scanner • One stroke patient • 2D WM skeletons • MPRAGE sequence, T1- (left and right weighted hemispheres) • Contiguous transverse slices (n = 28, each 5mm thick) • In-plane resolution, 1 × 1 mm2

Chapter 1 Introduction and Background 21

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Kido, S., and S. Sasaki. Fractal analysis for quantitative evaluation of diffuse lung

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after Binswanger's report. A review. Stroke 1995; 26:1293-301.

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correlates of white matter changes on MRI in Alzheimer’s disease and normal

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hyperintensities: three-year follow-up of the Austrian Stroke Prevention Study.

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Chapter 2

Fractal Dimension in Human Cerebellum

Measured by Magnetic Resonance Imaging

2.1. Abstract

Fractal dimension has been used to quantify the structures of a wide range of objects in biology and medicine. We measured the fractal dimension of human cerebellum (CB) in magnetic resonance images of 24 healthy young subjects (12 men and 12 women). CB images were resampled to a series of image sets with different 3D resolutions. At each resolution, the skeleton of the CB white matter was obtained and the number of pixels belonging to the skeleton was determined. Fractal dimension of the CB skeleton was calculated using the box-counting method. The results indicated that the CB skeleton is a highly fractal structure, with a fractal dimension of 2.57 ± 0.01. No significant difference in the CB fractal dimension was observed between men and women.

*This chapter was published in Biophysical Journal 2003; 85: 4041-4046. 25 Chapter 2 Fractal Dimension in Human Cerebellum Measured by Magnetic Resonance Imaging 26

2.2. Introduction

Fractal analysis has recently been applied to study a wide range of objects in biology and medicine (Kenkel and Walker, 1996; Cross, 1997; Heymans et al., 2000; Losa, 2000), especially in brain structures and processes. It has been used successfully in quantifying brain cell morphologies (Porter et al., 1991; Smith et al., 1991, 1993; Takeda et al., 1992;

Smith and Behar, 1994; Soltys et al., 2001) and the shape of the brain (Bullmore et al.,

1994; Cook et al., 1995; Free et al., 1996; Thompson et al., 1996; Rybaczuk et al., 1996;

Rybaczuk and Kedzia, 1996; Kedzia et al., 1997, 2002; Pereira et al., 2000; Iftekharuddin et al., 2000; Blanton et al., 2001). Bullmore et al. (1994) measured the complexity of the boundaries between the white matter (WM) and gray matter (GM) in human brain magnetic resonance (MR) images and applied the results to a controlled study of schizophrenic and manic-depressive patients. They determined the mean FD of all subjects to be 1.402, and found that the manic-depressive patients had higher FD than controls, whereas the schizophrenic patients had lower FD than controls. Thompson et al. (1996) quantified morphometric variance of the surfaces of the supracallosal sulcus, the cingulated and maginal sulci, the anterior and posterior rami of the calcarine sulcus, and the parieto-occipital sulcus in cryoplaned specimen photographs of human brain, and found that the FDs of these structures were in a narrow range around 2.10. The investigators concluded, based on their observation of low variance in the FD, that there was a stable FD value for normal brain, which could be used as a control for pathological studies. Blanton et al. (2001) investigated the influence of age and gender on structural complexity and cortical sulcus developmental trends in children and adolescents by measuring FD of the surface of the brain sulcal/gyral convolution in MR images using the

Chapter 2 Fractal Dimension in Human Cerebellum Measured by Magnetic Resonance Imaging 27 algorithm of Thompson et al. (1996). Their results showed significant age-related FD changes in the frontal regions and gender-related FD alterations in the left superior frontal and right inferior frontal regions. Free et al. (1996) studied the convolution of cerebral cortex by measuring variance in the WM surface in human brain MR images, and obtained the FD in a narrow range from 2.24 to 2.41. These investigators found a high degree of symmetry in FD between the right and left hemispheres in the subjects aged from 23 to 53, whereas no gender-related difference was observed. Differences in

FD between the normal gyral patterns and those in epilepsy and gyral abnormality were also compared. Kedzia et al. (2002) studied the microangioarchitecture of human brain vessels during the fetal period and found that FD increased from 1.26 in the fourth month to 1.53 in the fifth month and 1.6 in the sixth and seventh months. The results showed a close correspondence between structure and function. Pereira et al. (2000) quantified the edge irregularities of brain lesions to evaluate the degree of tumor malignancy in human brain MR images. Iftekharuddin et al. (2000) used FD to detect and locate brain tumor.

Their method may be applied to extract progressive tumor development information by comparing to normal brain data. Kedzia et al. (1997) studied senile brain atrophy by comparing the young brain (NMR data) and the older brain with visible senile changes

(photo data). They found that the FDs of the gray matter surfaces were 2.35 for the younger brain and 2.29 for the senile brain. Cook et al. (1995) estimated the FD of the cortical WM boundary in MR images of normal human brain and compared it with those of patients with frontal lobe epilepsy, temporal lobe epilepsy, and dementia of Alzheimer type. They concluded that patients with frontal lobe epilepsy had decreased FD (FD <

Chapter 2 Fractal Dimension in Human Cerebellum Measured by Magnetic Resonance Imaging 28

1.27) whereas patients with temporal lobe epilepsy and dementia had FDs in the normal range (1.32-1.48 and 1.37-1.50, respectively).

Although fractal research on the brain has been ongoing, little work has been done on the human cerebellum (CB). Most studies in the literature were dealing with CB cell morphometrics (Takeda et al., 1992; Smith et al., 1993). Only one group applied FD to evaluate CB surface complexity (Rybaczuk et al., 1996; Rybaczuk and Kedzia, 1996).

They quantified the CB growth process by measuring the FD from 120 crown-rump (C-

R) length to 200 C-R length in the fetal period (Rybaczuk et al., 1996). They observed that the FD increased to 2.25 from 120 to 130 C-R, decreased to 2.12 from 130 to 180 C-

R, then increased to 2.26 from 180 to 200 C-R, resulting from corresponding bulk growth

(reducing FD) and surface growth (augmenting FD) at the different developmental stages.

The investigators then measured the FD of adult CB surface and compared it to the FD of the fetal CB (Rybaczuk and Kedzia, 1996). They found that the adult FD, which was in the range of 2.25-2.30, slightly exceeded the FD of fetal CB at the final growth period.

Despite the fact that the skeleton is a very important morphometric character of an image, there has been no report on the fractal property of the skeleton of human CB WM.

As is apparent from the MR CB images, the CB WM bears a tree-like pattern, which resembles a fractal structure. The purpose of this study was to measure the FD of human

CB skeleton using T1-weighted MR head images. Since the shape of WM is well represented by its skeleton, it is expected that morphometric changes in CB WM due to

WM diseases may be evaluated using the WM skeleton. Moreover, interior CB lesions may be studied by skeleton, whereas CB surface evaluation in the previous studies can only used to examine lesions on the surface. Therefore, it is desirable to develop a

Chapter 2 Fractal Dimension in Human Cerebellum Measured by Magnetic Resonance Imaging 29

method to extract the CB WM skeleton and explore its usefulness in analyzing FD of CB.

In this study, the skeleton of human CB WM was first created from the MR CB images at

different spatial resolutions, and the FD was then calculated by applying the box-

counting definition. The significance and potential applications of the FD are addressed

and possible improvements of the method are suggested in the Discussion.

2.3. Materials and Methods

2.3.1. Subjects

Twenty-four young healthy subjects (17-35 years old, mean age ± standard deviation =

27.7 ± 4.4) participated in the study, including 12 men (20-35 years old, mean age ± standard deviation = 28.8 ± 3.7 ) and 12 women (17-35 years old, mean age ± standard

deviation = 26.7 ± 5.0 ). The experimental procedures were approved by the Institutional

Review Board at the Cleveland Clinic Foundation. All subjects signed informed consent

prior to the participation.

2.3.2. Collection of MR Head Images

Coronal MRI brain images covering the whole cerebellum were collected on a 1.5-T

Siemens Vision scanner (Erlangen, Germany) using a circularly polarized head coil. A

total of 128 contiguous coronal brain slices (each 2 mm thick) were imaged with the 3D

Turboflash imaging sequence (TR (repetition time) / TE (echo time) = 11.4 ms / 4.4 ms).

The flip angle was 10° and the in-plane resolution was 1 x 1 mm2.

Chapter 2 Fractal Dimension in Human Cerebellum Measured by Magnetic Resonance Imaging 30

2.3.3. Segmentation and Resampling of CB Images

The CB images were segmented out manually from the collected head images.

Background noise in each slice was removed by deleting the pixels with intensities lower than a certain threshold. The images were then standardized to the same dimension

(matrix size = 128 × 64, pixel size = 1 × 1 mm2, slice thickness = 2 mm). The number of

CB slices was 23 - 29, depending on the individual subject. The segmented CB images were resampled to be 1 mm in thickness to keep the isotropy (Figure 2.1).

The resultant images were further resampled to a series of image sets with different 3-D resolutions (∆), i.e., 1/4, 1/2, 2, and 4 mm (Figure 2.1). The resampling was performed using the subvoxel interpolation algorithm, i.e., the so-called point spread function (PSF) resampling, in the MEDx 3.2 software (Sensor Systems, Inc., Sterling,

VA). The algorithm considered each image value as a weighted integral, and the

1 × 1 × 2 mm3 1 × 1 × 1 mm3

0.25 × 0.25 × 0.25 mm3 0.5 × 0.5 × 0.5 mm3

2 × 2 × 2 mm3 4 × 4 × 4 mm3

Figure 2.1. A sample slice of CB MR image and its resampling to different resolutions. The spatial resolution (voxel size) is indicated below each image.

Chapter 2 Fractal Dimension in Human Cerebellum Measured by Magnetic Resonance Imaging 31 weighting function was the PSF of the imaging sensor. The cubic convolution approach, specified in a kernel file, was used to approximate the PSF (Boult and Wolberg, 1993). In most cases, this algorithm provided excellent results compared with the nearest-neighbor or trilinear resampling approaches. All the image analyses were performed on a Sun

UltraSPARC 333 MHz workstation with 640 MB RAM.

2.3.4. Segmentation of WM and GM

The interfaces between WM and GM in each image set were extracted using a user- modified contour algorithm in Matlab 6.1 (The MathWorks, Natick, MA). The histogram

(Figure 2.2) was used to determine the threshold between WM and GM. Denoting the intensity of the highest peak (A, corresponding to GM); and the intensity of the valley after the GM peak (B), the threshold was taken as A+(B-A)/3. This selection was based on our experience after comparing results obtained using different thresholds. After thresholding, GM was removed, and the contour image of WM was obtained in each slice

(Figure 2.3A). The rendered contour images were then converted to binary images, i.e., intensities of pixels inside the contours were assigned the value 1, otherwise 0 (Figure

2.3B).

2.3.5. Extraction of WM Skeleton

The thinning method in the Matlab 6.1 image processing toolbox was applied to the binary images. CB skeletons were obtained from each image slice by iteratively deleting successive layers of pixels on the boundary of WM images until only a skeleton remained

(Figure 2.3C) (Lam et al., 1992).

Chapter 2 Fractal Dimension in Human Cerebellum Measured by Magnetic Resonance Imaging 32

GM A 5000

A + (B - A) / 3 4000 ls Pixe 3000 of r e b WM m

u 2000 N

B 1000 CSF

0 0 50 100 150 200 250 Intensity

Figure 2.2. Histogram of the CB image set of a subject. The x axis indicates the intensity of the pixels; the y axis indicates the number of pixels at each intensity value. The threshold for separating the white matter (WM) and gray matter (GM) is indicated by the vertical dashed line between the GM and WM peaks.

The resultant skeletons in each resolution were color- washed and overlaid onto the corresponding CB images so we could inspect the accuracy of the skeletonization

(Figures 2.3D and 2.4).

2.3.6. Measurement of CB Fractal Dimension

The box-counting method (Mandelbrot, 1982; Kenkel and Walker, 1996) was adopted to compute the FD (D) of the CB skeleton. This method is suitable for structures that lack strict self-similarity. At each resolution ∆ (i.e., 1/4, 1/2, 1, 2, and 4 mm), the number of pixels (N) belonging to the CB skeleton was counted. The counting of the skeleton pixels was performed using the MEDx 3.2 software. The FD is defined in the power-law relationship:

NC=∆−D (2.1)

Chapter 2 Fractal Dimension in Human Cerebellum Measured by Magnetic Resonance Imaging 33

(A) Contour image

(B) Binarizing

(D) Overlay

Figure 2.3. Illustration of the steps to extract the CB skeleton in a sample slice. (A) A contour image was created by thresholding. (B) A binary image was generated by assigning value 1 to the pixels inside the contour. (C) The skeleton of the CB image was produced by thinning the binary image. (D) The CB skeleton was color-washed and overlaid onto the original CB image.

where C is a constant. The FD (D) was obtained by linearly fitting ln(N) = ln(C) –

Dln(∆) (Figure 2.5).

2.3.7. Statistical Analysis

The FDs of the men and women groups were compared using the independent t test (at

95% confidence level) to determine if significant difference existed between genders.

2.4. Results

In Figure 2.4, two sample slices of the CB from two subjects in different resolutions are shown. In each subject, the images became more blurred from the top images to the

Chapter 2 Fractal Dimension in Human Cerebellum Measured by Magnetic Resonance Imaging 34

Figure 2.4. Illustration of the generated CB skeletons in two sample slices from two subjects. The skeletons were overlaid onto the original CB images in different resolutions. Visual inspection showed that skeletons of the CB WM were reproduced with reasonable accuracy. bottom ones due to reduced spatial resolution. The extracted WM skeletons, being overlaid onto the original CB images, are in yellow color. These images show that WM skeletons were generated at satisfactory accuracy using the methods described previously.

The computed FDs for the 24 subjects are listed in Table 2.1. The FD was 2.571 ±

0.006 and 2.569 ± 0.007 (mean ± SE of the mean (SEM)) for the men and women groups, respectively. No significant difference was detected between genders by the independent t test at 95% confidence level in our examined age range (t = 0.28, P = 0.78). The correlation coefficients (R2) of the regression fits are also listed in Table 2.1, which show that the data were fitted by linear functions excellently (R2 = 0.9949 ± 0.0002 for men,

R2 = 0.9946 ± 0.0002 for women).

Chapter 2 Fractal Dimension in Human Cerebellum Measured by Magnetic Resonance Imaging 35

10 lnN

0 -2 -1 0 1 2 3

ln∆ Figure 2.5. Illustration of the linear fit to obtain the fractal dimension in a single subject. The x axis is image resolution (∆) and the y axis is number of pixels in the skeleton (N) in logarithmic scale. The data (open circles) were fitted using linear regression. The negative slope of the fitted curve (solid line) represents the fractal dimension. The figure and the high correlation coefficient (R2) show that the data can be fitted by a linear function excellently.

2.5. Discussion

We measured the FD of human CB WM skeleton at 2.57 ± 0.01. This indicates that human CB is a highly fractal structure, consistent with conclusions drawn from studies of the CB surface (Rybaczuk et al., 1996; Rybaczuk and Kedzia, 1996). Our results are notably higher than those of the former studies as mentioned in the introduction. This is because different structures were evaluated in their and our studies. Most of the other studies measured FDs of the brain or cerebellum surfaces/boundaries whereas we measured that of the CB WM skeleton. Therefore, it is not strange to see the differences between their results and our results because different structures are expected, in general, to have different FD values. Meanwhile, the differences between the FDs evaluated from different aspects of the same structure (e.g., surface, boundary, and skeleton of the CB) suggest that a single structure may bear different FDs depending on the angles of view.

Chapter 2 Fractal Dimension in Human Cerebellum Measured by Magnetic Resonance Imaging 36

Table 2.1. Results of fractal dimension (FD) in cerebellum

Men Women R2 Subject Age FD Age FD Men Women 1 20 2.540 17 2.557 0.9953 0.9948 2 26 2.547 20 2.578 0.9948 0.9950 3 27 2.596 23 2.562 0.9961 0.9945 4 28 2.563 25 2.622 0.9954 0.9934 5 28 2.591 26 2.562 0.9949 0.9943 6 28 2.583 27 2.562 0.9943 0.9941 7 29 2.580 27 2.558 0.9949 0.9953 8 30 2.587 29 2.581 0.9949 0.9948 9 30 2.589 29 2.541 0.9936 0.9955 10 32 2.541 30 2.551 0.9950 0.9953 11 32 2.564 32 2.584 0.9950 0.9948 12 35 2.576 35 2.568 0.9943 0.9936 Average 28.8 2.571 26.7 2.569 0.9949 0.9946 SE 1.1 0.006 0.8 0.007 0.0002 0.0002

SE is standard error of the mean. R2 is the correlation coefficient of the linear fit.

Therefore, it may be necessary to evaluate the fractal properties of a biological structure,

especially those of high complexity such as the brain and CB, from different aspects to

make a more comprehensive understanding.

The FDs in our study were in a small range (2.540-2.596) and did not change

significantly with age (17 – 35 years), which is consistent with the studies on cerebrum

sulcal surface (Thompson et al., 1996) and white matter surface (Free et al., 1996).

However, the FD of the CB changes significantly during the fetal period, as demonstrated by Rybaczuk et al. (1996). Age-related changes in FD of sulcal surface in frontal regions of the cerebrum were also found in children and adolescents (Blanton et al., 2001). For the elderly people, decreases in brain gray matter surface FD were observed (Kedzia et

Chapter 2 Fractal Dimension in Human Cerebellum Measured by Magnetic Resonance Imaging 37

al., 1997). These findings seem to suggest that the FD of human brain may be relatively

invariable in some specific age ranges. Once developed, the human brain and its FD may remain relatively stable for a certain period of time. Furthermore, FD changes of the brain with age may not be in a progressive and continuous pattern but may occur abruptly at some particular age ranges. It is also possible that significant changes may be found in certain specific areas of the brain while the global FD of the whole brain remains relatively constant.

We found no significant difference between the FDs of the men and women, which is consistent with the observation by Free et al. (1996). However, a recent study reported gender-related differences when quantifying specific regions of the cerebrum

(Blanton et al., 2001). These findings may suggest that there are no gender-related differences in FD of the whole brain in men and women of the same age range, even though differences may indeed exist in certain localized regions. Based on the present data, the same may hold true not only for the whole brain, but also for the CB.

Another important aspect is regarding the comparison of FD between patients and normal individuals. As suggested by previous studies, some patients may have FD similar to those of normal subjects, especially when the brain was quantified as a whole (Cook et al., 1995; Free et al., 1996). For example, Cook and his co-investigators (1995) found no differences in the FDs of the cortical WM boundary between normal subjects and patients with temporal lobe epilepsy and dementia (whereas patients with frontal lobe epilepsy did have lower FD). These findings suggest that either the tested patients had no structural degeneration in the measured cortical regions or the FD measure is not sensitive enough to identify adaptive changes in these particular patient populations. The sensitivity may

Chapter 2 Fractal Dimension in Human Cerebellum Measured by Magnetic Resonance Imaging 38

be improved by quantifying FD separately in specific cortical or CB regions, such as the

cerebrocerebellum, spinocerebellum, and vestibulocerebellum (Ghez and Thach, 2000),

especially when the quantified regions suffered damage or pathological changes. It is

reasonable to expect that structural changes after brain or CB damage may lead

detectable FD changes, as also suggested in the previous studies (Bullmore et al., 1994;

Cook et al., 1995; Free et al., 1996; Pereira et al., 2000; Iftekharuddin et al., 2000). In

our view, quantification of brain structures by FD, either in normal or diseased

conditions, will be a major application in future FD-related biomedical research.

The accuracy of the skeletonization method directly affects the accuracy of the

FD computation. The images in Figure 2.4 show that the skeleton generated by our

method represented the true CB skeletal structure with satisfactory accuracy. However,

the generated skeleton was not connected in some areas due to thresholding. This may

have little influence on the FD calculation because the box-counting dimension does not

require branch connection. The method may need to be improved if other definitions of

FD are to be used. In the data aspect, the best way is to collect raw imaging data at

different resolutions directly rather than resampling the images. However, this is not

practicable at this time for us due to the limit of the MR imaging technique. That’s to say,

using a 1.5 T MR image scanner, the spatial resolution cannot reach the values require by

our image processing method, i.e., 0.25 × 0.25 × 0.25 mm3 or 0.5 × 0.5 × 0.5 mm3. If we choose to do it anyway, the image quality will be deteriorated. However, when MR imaging advances (e.g., with higher magnetic field, faster imaging speed, more sensitive signal detection, etc.) in the future, it may become possible to do so and we may consider doing it. Another concern is that the skeleton extraction method used in the study is not a

Chapter 2 Fractal Dimension in Human Cerebellum Measured by Magnetic Resonance Imaging 39 true 3D algorithm. The skeletons were obtained in 2D slices rather than in a 3D environment. This may introduce certain errors to the FD computation due to the lack of information on the connection relationship between the slices. An accurate 3D skeletonization algorithm is certainly useful in improving the accuracy of the FD calculation as well as in other brain studies. Additionally, a quantitative method to validate the skeletonization is also demanded even though there has not been an effective one in the field due to various difficulties. Investigation of the accuracy of the reconstructed original CB shape from skeletons would be a good approach and may be explored in the future.

As a final remark, fractal geometry is a very useful tool to describe and understand various biological systems with fractal properties. FD can serve as an index for quantifying structural or functional complexity of the neural system during the stages of development, degeneration, reorganization, or evolution because in these processes the

FD may be dynamically changing (Rybaczuk et al., 1996; Rybaczuk and Kedzia, 1996;

Kedzia et al., 2002). The method we developed in this project and its future improvements might potentially be applied to study CB or the brain in normal and diseased conditions. It would be useful for characterizing the property changes of the brain during the developmental processes of certain diseases that affect the WM structures, such as multiple sclerosis (Graves et al., 1986; Davie et al., 1995) and autism

(Cook, 1990; Courchesne et al., 2001; Lee et al., 2002). We hope that the obtained information may eventually be useful to the improvement of diagnosis accuracy of such diseases.

Chapter 2 Fractal Dimension in Human Cerebellum Measured by Magnetic Resonance Imaging 40

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Chapter 3

A Three-dimensional Fractal Analysis

Method for Quantifying White Matter

Structure in Human Brain

3.1. Abstract

Fractal dimension (FD) is increasingly used to quantify complexity of brain structures.

Previous research that analyzed FD of human brain mainly focused on two-dimensional measurements. In this study, we developed a three-dimensional (3D) box-counting method to measure FD of human brain white matter (WM) interior structure, WM surface and WM general structure simultaneously. This method, which firstly incorporates a shape descriptor (3D medial axis skeleton) representing interior structure and combines the three features, provides a more comprehensive characterization of WM structure.

WM FD of different brain segments was computed to test robustness of the method. FDs of fractal phantoms were computed to test the accuracy of the method. The consistency

*This chapter is in press in Journal of Neuroscience Methods, 2005. 44 Chapter 3 A Three-dimensional Fractal Analysis Method for Quantifying White Matter Structure in Human Brain 45 of the computed and theoretical FD values suggests that our method is accurate in measuring FDs of fractals. Statistical analysis was performed to examine sensitivity of the method in detecting WM structure differences in a number of young and old subjects.

FD values of the WM skeleton and surface were significantly greater in young than old individuals, indicating more complex WM structures in young people. These results suggest that our method is accurate in quantifying three-dimensional brain WM structures and sensitive in detecting age-related degeneration of the structures.

3.2. Introduction

Age-related white matter (WM) degeneration has increasingly been of interest due to its possible prognostic significance for diseases, such as motor function impairment

(Breteler et al., 1994; Baloh et al., 2003), cognitive deficit (de Groot et al., 1998;

Gunning-Dixon and Raz, 2000), depression (Pantoni and Garcia, 1995; Inzitari et al.,

2000), or dementia (Kapeller and Schmidt, 1998). Current clinical diagnosis of WM degeneration still depends on visual rating scales due to the lack of appropriate measurement tools. Although volumetric analysis, a conventional technique in WM structural quantification research, is an appropriate assessment of WM atrophy, characterization by volume only captures one of multiple aspects of the structure.

Moreover, shape analysis of brain structures has been reported to provide new information which is not accessible by volumetric measurements (Gerig et al., 2001).

New morphometric analysis techniques are therefore needed for better descriptions of

WM shape changes in aging and diseases.

Fractal dimension (FD) (Mandelbrot, 1982), a shape complexity descriptor, has increasingly been applied to brain research, including determining neural cell

Chapter 3 A Three-dimensional Fractal Analysis Method for Quantifying White Matter Structure in Human Brain 46 morphometrics (Smith and Behar, 1994; Soltys et al., 2001), time series (Preissl et al.,

1997; Shimizu et al., 2004), and macroscopic structures of gray matter (GM) and white matter (WM) (Bullmore et al., 1994; Cook et al., 1995; Free et al., 1996; Rybaczuk et al.,

1996; Rybaczuk and Kedzia, 1996; Kedzia et al., 1997; Sato et al., 1996; Thompson et al., 1996; Blanton et al., 2001; Kiselev et al., 2003; Liu et al., 2003; Luders et al., 2004).

So far, the brain’s cortical surface has been the most frequently investigated feature because of the obvious fractal characteristics of the brain’s sulci and gyri convolutions. Investigators have reported two-dimensional (2D) cortical WM contours

(Bullmore et al., 1994; Cook et al., 1995), 2D brain tissue (cerebrospinal fluid, WM, GM) contours (Sato et al., 1996), sulci surfaces (Thompson et al., 1996; Blanton et al., 2001), cortical surfaces (Luders et al., 2004), WM surface (Free et al., 1996), and cerebellar surface of the human fetus (Rybaczuk et al., 1996) and adults (Rybaczuk and Kedzia,

1996). Although surface analysis provides folding patterns of the brain, its ability to detect structural abnormalities or lesions for biomedical research purposes is limited to the surface only. Significant structural changes may occur beneath the surface (i.e., within the parenchyma). To the best of our knowledge, only one study has examined the entire shape of the brain (Kiselev et al., 2003), and our earlier study evaluated the interior structure of cerebellar WM (Liu et al., 2003). Kiselev et al. reported the FD of the entire geometry of the human cerebral cortex. Our study reported results from quantifying the interior 2D “skeletonized” structure of the cerebellar WM.

In the field of pattern recognition, the image medial axis “skeleton” is an important shape descriptor for distinguishing one object from another. The term skeleton has been used to describe center-line of an object structure that summarizes its shape,

Chapter 3 A Three-dimensional Fractal Analysis Method for Quantifying White Matter Structure in Human Brain 47 size, and orientation. It retains the shape information of the original object and is a reduced representation of it. Skeletons have successfully been applied to FD analysis of

2D mathematical fractals (Daya Sagar, 1996; Foroutan-pour et al., 1999). Although

Kiselev et al. (2003) and we (Liu et al., 2003) attempted a new form of structural feature analysis (other than surface measurements), all of these studies only touched one view.

This may be partly due to the fact that it is difficult to fit one FD method to all views. In the current study, a three-dimensional (3D) FD analysis tool that is able to simultaneously evaluate the entire shape (general structure), interior shape (skeleton), and surface shape of human brain WM in 3D magnetic resonance (MR) volumetric images was developed.

Simultaneous analysis of the whole objects provides a more complete evaluation of the

WM structure and lowers the possibility of missing important information.

Choices of FD method are critical in analyzing the whole object. Current widely used methods in brain structure feature quantification include box-counting (Bullmore et al., 1994; Cook et al., 1995; Sato et al., 1996; Rybaczuk et al., 1996; Rybaczuk and

Kedzia, 1996; Liu et al., 2003), surface-based algorithm (Thompson et al., 1996; Blanton et al., 2001; Luders et al., 2004), and fast Fourier transform-based methods (Kiselev et al., 2003). Among them, box-counting is the most appropriate and desirable method in brain structural FD estimation because it can apply fractal patterns with or without self- similarity. For this reason, we chose the box-counting method for the current study. The traditional box-counting method has been applied in 2D cases (Bullmore et al., 1994;

Cook et al., 1995; Sato et al., 1996). Despite the fact that 3D analysis provides more information with greater accuracy, there has been no detailed report on 3D volumetric box-counting analysis of the brain. Two studies may have been involved in 3D brain

Chapter 3 A Three-dimensional Fractal Analysis Method for Quantifying White Matter Structure in Human Brain 48

structure (Rybaczuk et al., 1996; Rybaczuk and Kedzia, 1996), and one performed

pseudo-3D structural analysis of cerebellar WM (Liu et al., 2003). Although Rybaczuk

and colleagues stated that they performed a 3D FD evaluation, they did not provide

details of their FD calculations. Therefore, it is not known if the method was pseudo-3D

box-counting (calculation slice by slice) or true 3D box-counting. Our previous study

(Liu et al., 2003) simulated a 3D FD structural analysis of cerebellar WM based on slice- by-slice 2D skeleton image analysis. The purpose of the study reported here was to

develop a 3D box-counting analysis method to simultaneously quantify three features

(general structure, skeleton and surface) in images of 3D human brain WM structure.

Significance and potential applications of this method are addressed in Section 3.5.

3.3. Materials and Methods

3.3.1. MR Images

Coronal MR human head images covering the entire cerebrum and cerebellum were

collected on a 1.5-T Siemens Vision scanner (Erlangen, Germany) using a circularly

polarized head coil. Contiguous coronal slices (n = 128, each 2 mm thick) were acquired

with a 3-D Turboflash imaging sequence (TR [repetition time] / TE [echo time] = 11.4

ms / 4.4 ms). The flip angle was 10° and the in-plane resolution was 1 x 1 mm2. Six subjects participated in this study, including three young (17, 32 and 32 years) and three old (72, 75 and 76 years) healthy subjects. The experimental procedures were approved by the Institutional Review Board at the Cleveland Clinic Foundation. All subjects gave informed consent prior to participation.

Chapter 3 A Three-dimensional Fractal Analysis Method for Quantifying White Matter Structure in Human Brain 49

3.3.2. Image Processing

A flowchart of the processing steps is shown in Figure 3.1. The MR head images were resampled via trilinear interpolation, yielding 1-mm slice thickness to keep the isotropy using MEDx 3.41 software (Sensor Systems, Inc., Sterling, VA). The brain was segmented from the resampled head images using the brain extraction tool (BET) (Smith,

2002) in the FSL package (FMRIB Software Library from the Oxford Centre for

Functional Magnetic Resonance Imaging of the Brain) of MEDx 3.41. This algorithm used a tessellated surface model that deformed itself to fit the brain’s surface by

Figure 3.1. Flowchart of image processing steps.

Chapter 3 A Three-dimensional Fractal Analysis Method for Quantifying White Matter Structure in Human Brain 50 application of a set of locally adaptive model forces. A triangular tessellation of a sphere’s surface was initialized inside the brain and slowly evolved toward its edge by iteratively splitting each triangle and adjusting each vertex’s position to keep the surface well spaced and smooth, until achieving the required complexity. The left and right hemispheres and cerebellum were segmented manually from the coronal brain images using the MEDx tracing tool, and mask images corresponding to each segmentation were generated. The manual segmentation is illustrated in Figure 3.2, in which segmented WM

(discussed below) has been masked and WM of the left and right hemispheres and cerebellum WM are shown.

(A) (B) (C)

(D) (E) (F)

Cerebrospinal fluid White matter of left cerebrum Gray matter White matter of right cerebrum Cerebellar white matter

Figure 3.2. Brain tissue segmentation in a sagittal, coronal and transverse slice from one subject . Top Panel: T1-weighted sagittal image (A), coronal image (B) and transverse image (C). Bottom panel: segmentation results for a sagittal image (D), coronal image (E) and transverse image (F).

Chapter 3 A Three-dimensional Fractal Analysis Method for Quantifying White Matter Structure in Human Brain 51

The WM was segmented from the brain images using FSL FAST (FMRIB's

Automated Segmentation Tool) (Zhang et al., 2001) and recorded as binary (black-and- white) images. FSL FAST is one of the parametric statistical segmentation approaches, which labels pixels according to probability values determined by image intensity distributions and spatial neighborhood information coded by a hidden Markov random field model. During the segmentation, FAST corrected spatial intensity variations (or radiofrequency inhomogeneities) to improve segmentation performance using

Guillemaud and Brady’s algorithm (Guillemaud and Brady, 1997). The segmentation of gray matter, WM and cerebrospinal fluid is illustrated in Figure 3.2, in which the results on a sagittal, coronal, and transverse slice are shown.

A 3D thinning method was then applied to the binary images to obtain skeletons of the WM (Ma and Sonka, 1996). The method peeled off as many boundary voxels as possible without affecting the general shape of the WM. A set of deleting templates

(position configuration of object voxels and background voxels in a 3 × 3 × 3 window)

was built up to determine boundary voxels to be removed. This algorithm deleted in

parallel every object voxel that satisfied at least one deleting template until no voxel

could be deleted. The thinning procedure was fully parallel and applied only once in each

iteration. The method was implemented by the C++ language. Finally, the mask images

were applied to the WM skeleton and WM general structure images to obtain WM

skeletons and WM general structure images of the left and right hemispheres and the

cerebellum.

3.3.3. Fractal Analysis Methods

Chapter 3 A Three-dimensional Fractal Analysis Method for Quantifying White Matter Structure in Human Brain 52

The traditional 2D box-counting method is discussed in Chapter 1. In previous research

which used the traditional method, 5-10 meshes were usually taken and box sizes (r)

were recommended to be power of two: 2, 4, 8, 16, …, 2n (Penn et al., 2000; Liu et al.,

2003). The traditional method used in previous research was limited by small numbers of fractal coverage (choice of 5-10 meshes), most probably because of computational load.

In addition, the “power of 2” box size choice may mar the FD results because of the seemingly small errors in the linear regression analysis. In this study we developed a 3D box-counting method which provides a more accurate FD measurement. It was derived from HarFA 2D box-counting method and implemented by a custom-designed software written in Matlab 6.5 (The MathWorks, Natick, MA).

3.3.3.1. Box-counting Method in HarFA

Nezadal and colleagues modified the box-counting method and implemented it in HarFA software to perform fractal and harmonic analysis of 2D digitized images (Buchnicek et al., 2000; Nezadal et al., 2001). The modifications include: (i) box-counting mechanism,

(ii) choice of box sizes, and (iii) single slope analysis.

Different from the traditional method, which counts boxes needed to cover the

object completely, HarFA counts three categories of boxes: NB , which contains only the

black background; NW , which covers only the white object; and NBW , which covers the

border of the white object [e.g., those boxes which contain at least part of the white

object (Figure 3.3)]. According to this counting mechanism, it obtains not one, but three

FDs ( FDB , FDW , FDBW ) that characterize the properties of black background, white

Chapter 3 A Three-dimensional Fractal Analysis Method for Quantifying White Matter Structure in Human Brain 53

(A) (B)

Figure 3.3. 2D box-counting method in HarFA. Mesh with different sizes (r) was put in the 2D sample WM slice (black-and-white image). Number of boxes (NW) which cover the white object WM (specified with green boxes), number of boxes (NB) which cover the black background (specified with blue boxes) and number of boxes (NBW) which cover the WM borders (other boxes) were counted: (A) r = 5 pixels, NW = 245.0, NBW = 431.0, NB =1052.0 and (B) r = 15 pixels, NW = 4.0, NBW = 111.0, NB = 77.0. The images were generated by box-counting package from HarFA (http://www.fch.vutbr.cz/lectures/imagesci/) using data from the same subject as in Figure 1.6.

object and white-black border of the object. Another two FDs ( FDWBW , FDBBW ) can also

be computed from NWBW ( the sum of NW and NBW ) and NBBW (the sum of NB and

NBW ). FDWBW is the traditional box-counting dimension.

To improve the method’s performance, HarFA uses 2 to 1/3 of the smallest

dimension of an image as the box size range. A new single slope analysis tool was also

implemented in HarFA to determine the range of box sizes.

3.3.3.2. Fractal Analysis Methods of the Present Study

The box-counting method used in this study was derived from HarFA. We expanded the

2D method to 3D to analyze our 3D MR data. The entire procedure includes three major

steps: (i) counting the number of boxes needed to cover the skeletons and WM, (ii)

performing linear regression analysis to obtain the initial FD, and (iii) performing single

slope analysis to obtain the precision FD.

A 3D mesh was overlaid onto the 3D WM skeleton and WM volume images. The

origin of the mesh was chosen randomly in the images to improve the box-counting

Chapter 3 A Three-dimensional Fractal Analysis Method for Quantifying White Matter Structure in Human Brain 54

performance. For the skeleton images, NBW , the number of boxes covering the skeletons,

was counted. For the WM volume images, NBW and NW were counted. The number of

white boxes NW was always zero for skeleton images because skeleton has the one-voxel image pattern. Eq. 3.1 describes the counting of number of boxes,

3 NNii+1 =+(lx×ly×lz)r (3.1)

where Ni+1 is the newly counted number of boxes, Ni is the already counted number of

boxes, lx , ly , lz are the sizes of the processing box in x , y , z direction, and r is the chosen box size. The counting stopped after all images had been processed. The box size ranged from 2 to 1/3 of the smallest dimension of the image for both images (WM skeleton images and WM volume images).

The FD was subsequently determined by the relationship between the number of voxels and box size (Eq. 1.3). Linear regression analysis was performed to obtain the

initial FDBW (Figure 3.4), FDWBW , which were the slopes of the functions (Eq. 3.2-3.3):

ln ()NFBW =+DBW ln (1/ r) ln (KBW ) (3.2)

ln ()NFWBW =+DWBW ln (1/ r) ln (KWBW ) (3.3)

FDBW (or FDWBW ) of skeletons characterizes the complexity of the skeletons. FDBW of

WM images represents the complexity of the WM surface, whereas FDWBW shows the complexity of the WM general structure.

The range of box sizes is important in the box-counting method since the images of the fractals we studied are not pure fractals. When the size of an overlaid box is too small, no fractal but only a mosaic from black-and-white voxels can be seen. On the other

Chapter 3 A Three-dimensional Fractal Analysis Method for Quantifying White Matter Structure in Human Brain 55

Fractal dimension FD BW 11

y = 2.2002x + 12.2218 10 R2 = 0.99399 9

) 7 BW N (

ln 6

2 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 ln(1/r)

Figure 3.4. Linear regression analysis to obtain the initial FD of the entire WM skeleton from one subject. The x-axis is inverse box size (1/r) and the y-axis is number of voxels in the WM skeleton (NBW) in logarithmic scale. The data (empty circles) were fitted using linear regression. The slope of the fitted curve (solid line) represents the FD. The arrow points to the non-linear part of the line. hand, when the box size is too large, the fractal will disappear into the background

(Nezadal et al., 2001). The distortion is shown in the linear regression analysis as disturbance of linearity (arrow in Figure 3.4). Accurate determination of FD requires the linear part of the functions (Eqs. 3.2-3.3). Single slope analysis was performed to find the linear portion of the function and obtain the precise FD. In the initial FD analysis, linear regression analysis was applied to the entire data set to obtain one FD value (initial FD).

In the single slope analysis, the data points were divided into several segments.

Linear regression was sequentially applied to each segment to obtain FD of that segment data (data 1 ~ data dr+1, data 2 ~ data dr+2, data 3 ~ data dr+3,…, where dr was the length of the analyzed data point segment). Thus, there were several FD values corresponding to the entire data set. The length of the data segment was specified prior to the analysis (10 points for cerebellum analysis, 25 points for other brain regions in this

Chapter 3 A Three-dimensional Fractal Analysis Method for Quantifying White Matter Structure in Human Brain 56 study). A new set of FDs was then obtained and displayed on Figure 3.5. In this figure, each point was colorized according to the correlation coefficient of the linear regression.

The linear portion of the original function was extracted from the constant portion of the curve (similar FDs with standard deviation less than 0.01) with high correlation. The FD was measured as the mean of the FDs in the linear portion.

3.3.4. Accuracy and Sensitivity Assessment

To determine whether our method was accurate in measuring FD, we simulated three regular geometrical objects and two famous fractals in 3D images (Figure 3.6) using

Matlab 6.5 and measured their dimensions. (i) A 100 × 100 × 100 cube (image size: 120

× 120 × 120, Figure 3.6A). Voxels inside the cube were set to 1 and those outside the cube set to 0. We calculated FDWBW, FDW, FDBW with the first two being dimensions of the volume and the third being dimension of the surface. (ii) A 120 × 120 × 120 filled

FD:2.4078±0.0062 3 0.95

0.9 2.5 0.85 Linear Portion 2 0.8

0.75 1.5 FD 0.7

1 0.65

0.6 0.5 0.55

0 -4 -3.5 -3 -2.5 -2 -1.5 -1 R2 ln (1 / r)

Figure 3.5. Single slope analysis. The same data as in Fig. 3.4 were used in this demonstration. The color of each square represents correlation coefficient of the linear regression. The finalized FD was the mean of the linear portion of the curve. The chosen grid size was from 6 to 36.

Chapter 3 A Three-dimensional Fractal Analysis Method for Quantifying White Matter Structure in Human Brain 57

(A) (B)

Figure 3.6. Generated phantoms: (A) 3D solid cube (size, 100 × 100 × 100). (B) 3D solid sphere (diameter, 100). (C) third-iteration Koch Snowflake. All the blue “lines” were solid tubes with diameter 6. The red line indiates the skeletons of the Koch curves and has one voxel width. (D) Second-iteration Menger Sponge (size, 100 × 100 × 100).

cube (image size: 120 × 120 × 120). All voxels in the images were set to 1. We calculated

FDWBW and FDW as dimensions of the volume. The images (not shown) were similar to the cube (Figure 3.6A). (iii) A sphere with diameter 100 (image size: 120 × 120 × 120,

Figure 3.6B). We calculated FDWBW, FDW, FDBW with the first two being dimensions of the volume and the third being dimension of the surface. (iv) A 3rd iteration Koch

Snowflake (image size: 305 × 305 × 15, Figure 3.6C). All the “lines” in the images were

solid tubes with diameter 6. Skeletons of the Koch curves were generated (red lines in

Figure 3.6C). FDWBW of Koch curves general structure and skeletons were measured. (v)

A 2nd iteration 100 × 100 × 100 Menger Sponge (image size: 102 x 102 x102, Figure

3.6D). FDWBW of general structure was measured.

Chapter 3 A Three-dimensional Fractal Analysis Method for Quantifying White Matter Structure in Human Brain 58

To determine whether our method was sensitive in detecting age-related WM

structure changes, we analyzed and compared image data of three young and three old

healthy subjects. The FDs of the three young and three old subjects were compared using

an independent t test to determine whether age affected the WM structures. Significant differences were detected at p<0.05.

3.4. Results

3.4.1. Phantom Results

FD results of the phantoms are shown in Table 3.1. When measuring dimensions of non-

fractals (cube, filled cube and sphere) using our method, the FDs became topological

dimensions. Both FDWBW and FDW represented dimensions of the object volume but

differed in that FDWBW incorporated the surface information into volume information, whereas FDW represented only volume information. FDW could not provide any useful

information for fractals. Here we used it to test the dimension of non-fractals. For

Table 3.1. Results of fractal dimension/topological dimension of phantoms

Object FDWBW FDW FDBW FDSkel FDT DVolume DSurface Cube 2.893 3.127 2.011 - - 3 2 Filled Cube 3 3 - - - 3 - Sphere 2.777 3.209 2.009 - - 3 2 Koch Snowflake 2.311 - - 1.256 1.262 - - Menger Sponge 2.796 - - - 2.727 - -

FDWBW, FD of general structure (black-and-white boundary and interior white object); FDW, FD of interior white object; FDBW, FD of surface; FDSkel, FD of skeletons; FDT, theoretical FD; DVolume, theoretical topological dimension of volume; DSurface, theoretical topological dimension of surface.

Chapter 3 A Three-dimensional Fractal Analysis Method for Quantifying White Matter Structure in Human Brain 59

fractals, only FDWBW was taken as a dimension of their volume structure (general

structure).

FDWBW of phantoms are very close to their theoretical dimensions (cube: 2.893 vs.

3; filled cube: 3 vs. 3; sphere: 2.777 vs. 3; Menger Sponge: 2.796 vs. 2.727). This indicates that FDWBW is an accurate measurement of dimension of general structure.

FDWBW and theoretical dimension (Hausdorff dimension) of the Koch Snowflake are noncomparable because the Hausdorff dimension is a measurement of the Koch

Snowflake curve shape, whereas FDWBW describes the general structure. As expected, our

computed FD of the skeleton of the Koch Snowflake closely matched its Hausdorff

dimension (1.256 vs. 1.262). This indicates that the FD of the skeleton is an accurate

measurement of shape. The similarity between the FDBW (2.011 for the cube and 2.009

for the sphere) and surface dimension (DSurface = 2) suggests that our method is accurate to measure surface dimension. We did not measure FDBW of phantom fractals because we

did not know the theoretical FD of the surface fractals.

3.4.2. MR Results

A WM skeleton is shown in a 2D sample slice of one young (Figure 3.7A) and one old

subject (Figure 3.7B). The extracted 2D WM skeletons, being overlaid onto the original head images, are shown by the yellow lines. The 3D skeletons of the young and old subjects are shown in Figure 3.7C and Figure 3.7D, respectively. These images show that the WM shape was well represented by the skeleton. The skeleton of the old subject appears to be looser than that of the young subject by visual inspection, which seems to indicate that the WM of the old subject had a less complicated pattern. The same conclusion was drawn at a quantification level by examining quantitative FD data (Table

Chapter 3 A Three-dimensional Fractal Analysis Method for Quantifying White Matter Structure in Human Brain 60

(A) (B)

Figure 3.7. Illustration of the generated WM skeletons in two sample slices and in three dimensions from one young and one old subjects: (A) 2D skeleton of a young subject; (B) 2D skeleton of an old subject; (C) 3D skeleton of the young subject; (D) 3D skeleton of the old subject. Visual inspection showed that skeletons of the young subject have a more complicated pattern than that of the old subject.

3.2), in which means of FD of the old subjects were smaller than those of the young

subjects for all the structures.

The FDs of the entire brain, left and right hemispheres and cerebellum were

computed in three features – WM skeletons, WM surfaces and WM general structures.

Table 3.2 shows the FD means and standard errors of the means for young and old

groups. Individual data for each subject is given in Figure 3.8. Most FD values of the old

subjects were smaller than those of the young subjects with a few exceptions for the WM

surface (Figure 3.8B) and general structure (Figure 3.8C) measurements. Statistical

results by independent t tests at 95% confidence level are also shown in Figure 3.8.

Significant differences were detected between young and old individuals in entire WM skeleton ( P = 0.008), left WM skeleton ( P = 0.046 ), right WM skeleton ( P = 0.007 ) and cerebellar WM skeleton ( P = 0.040 ), albeit with only three subjects in each group. With

Chapter 3 A Three-dimensional Fractal Analysis Method for Quantifying White Matter Structure in Human Brain 61

Table 3.2. Results of fractal dimension (mean ± standard error of the mean) of WM skeleton, surface and general structure

Feature Group Region Whole brain Left cerebrum Right cerebrum Cerebellum Skeleton Young 2.397 ± 0.009 2.262 ± 0.010 2.301 ± 0.009 2.231 ± 0.013 Old 2.325 ± 0.012 2.166 ± 0.032 2.213 ± 0.015 2.124 ± 0.033 Surface Young 2.507 ± 0.008 2.414 ± 0.014 2.425 ± 0.004 2.124 ± 0.010 Old 2.470 ± 0.009 2.387 ± 0.013 2.368 ± 0.013 2.097 ± 0.027 General structure Young 2.565 ± 0.008 2.275 ± 0.027 2.338 ± 0.063 2.130 ± 0.016

Old 2.526 ± 0.013 2.221 ± 0.013 2.323 ± 0.050 2.091 ± 0.034

respect to FDs of WM surfaces, significant differences were also found in the entire WM

surface ( P = 0.037 ) and right WM surface ( P = 0.01). No differences were found in the left WM surface and cerebellar surface. No significant differences were observed in FDs of WM general structures.

3.4.3. Box Size

Box size ranges of phantoms are listed in Table 3.3. Choice of box size range is critical

since it directly affects the resultant FD. Regression lines for the chosen box size ranges

are shown in Figure 3.9. Most ranges stay in smaller box sizes except those for the

Menger Sponge, whose range is in the middle part of the box sizes (11~18). This is

because we excluded choice of box sizes 1~10. Our tested Menger Sponge was a 2nd- iteration sponge, not the ideal infinite-iteration one. The size of smaller holes of the

Menger Sponge was 10 × 10 × 10. If we included box sizes smaller than 10, our method would not detect the existing smaller holes and the resultant FD would be larger than the

Hausdorff dimension and would not be accurate. We performed linear regression analysis among box size ranges of 2~10 and 19~34 and obtained FDs of 2.84 and 2.90,

Chapter 3 A Three-dimensional Fractal Analysis Method for Quantifying White Matter Structure in Human Brain 62

Gro up 2.5 Young ** Old 2.4 D D D D D ** F F F F F n n n n n 2.3 * * o o o o o et et et et et el el el el el k k k k k 2.2 s s s s s M M M M M W W W W W 2.1

2.0 (A) Whole Left Right CB Brain region Group 2.6 Young * Old 2.5 D D D D * 2.4 e F e F e F e F c c c c a a a a f f f f 2.3 r r r r u u u u s s s s 2.2 WM WM WM WM 2.1 2.0 (B) Whole Left Right CB Brain region Group 2.6 Young D D D D Old F F F F 2.5 e e e e ur ur ur ur t t t t c c c c 2.4 u u u u r r r r t t t t 2.3 s s s s l l l l a a a a r r r r 2.2 ne ne ne ne ge ge ge ge 2.1 M M M M 2.0 W W W W (C) 1.9 Whole Left Right CB Brain region

Figure 3.8. 95% confidence interval plot of FD results of young and old groups for the whole brain (Whole), left hemisphere (Left), right hemisphere (Right) and cerebellum (CB). (A) FDs of WM skeletons. (B) FDs of WM surface. (C) FDs of WM general structure. Data for each young subject (black circle) and old subject (red triangle) are given. ( ⊕ ) Group mean, * P < 0.05 , ** P < 0.01.

Chapter 3 A Three-dimensional Fractal Analysis Method for Quantifying White Matter Structure in Human Brain 63

respectively. Both values were larger than the Hausdorff dimension (2.727). Lengths of

data segments in the slope analysis are also listed in Table 3.3. They were determined by

testing several values and choosing the one that would give the highest correlation

coefficient and the most constant portion in the slope analysis.

The box size ranges of different brain structures are listed in Table 3.4. They may

be considered as references when studying FDs of the brain using the box-counting

method in the future. The box size range of the cerebellum is obviously smaller than that

of the cerebrum because the cerebellum is smaller.

3.5. Discussion

Almost all previous brain FD studies investigated the FD of the brain surfaces. The limitation of the surface research is that it cannot explore structural features of the interior of the brain. Our previous study (Liu et al., 2003) provided a pseudo-3D method to quantify the interior structure of the cerebellum but was not able to examine the surface pattern. In this study, we developed a 3D volumetric fractal analysis method that

Table 3.3 Results of box size range and length of data segments (dr) of phantoms

Object FDWBW FDW FDBW FDSkel Cube 2-11 (6) 2-8 (5) 2-13 (6) - Filled Cube - - - - Sphere 2-11 (6) 2-8 (5) 2-17 (6) - Koch Snowflake 2-15 (6) - - 2-14 (11)

Menger Sponge 11-18 (4) - - -

FDWBW, FD of general structure (black-and-white boundary and interior white object). FDW, FD of interior white object. FDBW, FD of surface. FDSkel, FD of skeletons. Number in the parentheses is length of data segments (dr).

Chapter 3 A Three-dimensional Fractal Analysis Method for Quantifying White Matter Structure in Human Brain 64

Chapter 3 A Three-dimensional Fractal Analysis Method for Quantifying White Matter Structure in Human Brain 65

12 12 (A) (B) 10 10

8 8 ) ) ) N N WBW data N 6 6 ln( ln( W data ln( BW data WBW data 4 4 BW LR W data W LR BW data 2 WBW LR 2 BW LR WBW(or W) data W LR WBW (or W) LR WBW LR 0 0 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 ln(1 / r) ln(1 / r)

10 12 (C) (D) 11 8 10

6 9 ) ) W W 8 WB WB 4 (N

(N 7 ln ln 2 6

Skeleton data 5 0 General structure data General structure LR 4 WBW data Skeleton LR WBW LR -2 3 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 ln(1 / r) ln(1 / r)

Figure 3.9. Linear regression analysis of chosen box sizes of phantoms. (A) Cube results. Red square, white and black-and-white (WBW) box-counting data in different box sizes. Red line, linear regression line (LR) of WBW box-counting data. Green circle, white (W) box-counting data in different box sizes. Green line, regression line of white box-counting data. Blue triangle, black-and-white (BW) box- counting data in different box sizes. Blue line, regression line of BW box-counting data. Filled cube results. Black diamond, WBW (or white) box-counting data. Black line, regression line of WBW (or white) box-counting data. (B) Sphere results. The illustrations of color symbols and lines are the same as in (A). (C) Koch Snowflake results. Red square, WBW (general structure) box-counting data in different box sizes. Red line, regression line of WBW (general structure) box-counting data. Green circle, WBW (skeleton) box-counting data in different box sizes. Green line, regression line of WBW (skeleton) box-counting data. (D) Menger Sponge results. Red square, WBW box-counting data in different box sizes. Red line, regression line of WBW box-counting data.

quantified not only interior and surface brain WM structures, but also the WM general structure. Compared with the existing procedures, our method has the following advantages: (i) It quantifies different shape features all at once and provides more information. Each different shape descriptor has different detection abilities. A combination of these shape descriptors would provide a more robust detection tool than a single one. (ii) It is a true 3D fractal analysis method that is suitable for complicated 3D

Chapter 3 A Three-dimensional Fractal Analysis Method for Quantifying White Matter Structure in Human Brain 66

brain structure descriptions, whereas a 2D or pseudo-3D method only characterizes part

of the brain structure. (iii) Introduction of skeleton in 3D fractal analysis improves sensitivity of the FD method.

3.5.1. Importance of Shape Descriptor (Skeleton) in FD Analysis

Our results also showed the importance of incorporating skeletons in 3D FD analysis.

This procedure made our method more sensitive at detecting structural variations.

Skeletonized images have been proven to be able to provide a better material for box-

counting method compared with unskeletonized images on 2D mathematical fractals

(Foroutan-pour et al., 1999). The thickness of the skeleton affects the choice of box sizes and, therefore, the value of FD. Our phantom study of the Koch Snowflake confirmed that an accurate FD can be obtained once applying our FD analysis on skeletons. In our study of MR images of young and old individuals, we did not observe significant differences in the WM general structures (Figure 3.8C), but did detect significant differences in the FDs of the skeletons between the two groups. This suggests that the skeleton is also a better material in 3D case and the FD analysis of skeletonized images is more sensitive than that of unskeletonized images. Introduction of the WM general structure may only cause the true shape information to disappear into the unnecessary information or noise. The observation of greater standard deviation of box size boundary

(Table 3.4) for a majority of areas also illustrates this point.

3.5.2. Comparison with HarFA and Pseudo-3D Method

Our method was also a successful 3D extension of the 2D box-counting method in HarFA

(Buchnicek et al., 2000; Nezadal et al., 2001). By adopting the counting mechanism of

Chapter 3 A Three-dimensional Fractal Analysis Method for Quantifying White Matter Structure in Human Brain 67 white boxes and white and black boxes after overlaying different-sized 3D meshes, we obtained FDs of the brain WM skeleton, WM surface and WM general structure all at once. Furthermore, our method has other advantages: (i) A new shape descriptor – skeleton – was proposed in the box-counting calculation, which provided more sensitive measurements. (ii) In the single slope analysis, different lengths of the data segments corresponding to the different neural structures were specified. This step was important since the length of a data segment for a bigger fractal in the brain may exceed the size of a smaller fractal. The choice of the same values might severely distort the box size range of some structures. In our study, the length of the data segment was chosen to be 10 for the cerebellum, different from 25 for other areas, to reflect the smaller size of the cerebellum.

Compared with 2D method and pseudo-3D method (Liu et al., 2003), our method is better because it provides a more comprehensive characterization of 3D structure.

However, direct comparison between pseudo-3D (or 2D) method and our method is not practical and necessary because different image processing procedures generate different ranges of FD values. The pseudo-3D cerebella skeleton FD study (Liu et al., 2003) reported much higher FD values (2.540~2.596) in young subjects than our 3D study

(2.205~2.247). This large difference between FD values of the pseudo- and true-3D methods was most probably contributed by the fact that that the pseudo-3D analysis was based on a structure not representing real biological shape variation (lack of connection relationship information between the brain slices).

Table 3.5 summarizes the used software packages and my contributions to the data process methods, including image processing and fractal analysis methods.

Chapter 3 A Three-dimensional Fractal Analysis Method for Quantifying White Matter Structure in Human Brain 68

Table 3.5. Summary of used software packages and my contributions to the data process methods

Data processing Software/literature My contributions Extracting brain BET (brain extraction tool), FSL package of Medx software Segmenting white matter FAST (FMRIB’s Image Automated Segmentation processing Tool), FSL package, Medx software Extracting skeleton 3D thinning (Ma and C++ coding Sonka, 1996) Putting 3D meshes Matlab coding Box counting 2D box-counting in • 3D box counting HarFA (Harmonic and derived from HarFA 2D Fractal Analysis) software box-counting • Two categories of boxes: NBW, NWBW • Matlab coding Single slope analysis Single slope analysis in • Data segments Fractal HarFA determined by object analysis size • Matlab coding

Others • Combining two shape descriptors: fractal dimension (FD) and skeleton • Measuring FD of skeleton, surface, and general structures of object simultaneously • Matlab coding

3.5.3. Accuracy Assessment

Our phantom results showed that our method can accurately measure topological

dimension of non-fractals and FD of fractals. The FDWBW of phantoms are very close to their theoretical dimensions, the computed FD of the Koch Snowflake skeleton closely

matched its Hausdorff dimension, and the FDBW of the cube and sphere was very similar to the surface dimension (see Table 3.1 and Section 3.4.1). Although there were small differences between computed FDs and theoretical dimensions, they were expected.

Chapter 3 A Three-dimensional Fractal Analysis Method for Quantifying White Matter Structure in Human Brain 69

FDWBW of the cube was 2.893, slightly smaller than its topological dimension 3. This was expected since the cube did not fill in the whole 3D space and black background has been taken into consideration. The smaller the box sizes were, the bigger difference between the number of counting boxes in the cube (red squares in Figure 3.9A) and number of boxes in the filled cube (black diamonds in Figure 3.9A). Such counting differences made the slope of regression line (red line in Figure 3.9A) for the cube smaller than that of filled cube (black line in Figure 3.9A). FDW of the cube was 3.127, slightly greater

than its topological dimension 3. This was also expected because the number of white

boxes (green circles in Figure 3.9A) related to large box sizes was smaller than that of the

filled cube (black diamonds in Figure 3.9A), which made the slope in the linear

regression analysis bigger. Both FDW and FDWBW of the filled cube were 3, identical to

the theoretical value. This result indicates that our method was accurate to measure the

dimension of the volume. FDWBW, FDW and FDBW of the sphere were 2.777, 3.209 and

2.009, respectively. The jumping trends of its counting data (Figure 3.9B) were the same as those of the cube, and the same conclusions may be drawn.

The box-counting dimension (1.256) of Koch skeleton was slightly smaller than the Hausdorff dimension (1.262). This was expected because we only measured 3rd- iteration Koch curve, whereas the Hausdorff dimension is a measurement of the infinite- iteration Koch curve. Figure 3.9C shows our computed FD based on the linear part of the counting data. FDWBW of the Menger Sponge was 2.796, slightly larger than its Hausdorff

dimension 2.727. This might also be due to the measurement of only the 2nd-iteration sponge. Figure 3.9D shows its linear regression results.

Chapter 3 A Three-dimensional Fractal Analysis Method for Quantifying White Matter Structure in Human Brain 70

3.5.4. Sensitivity Assessment

The results of significant differences in WM skeletons and WM surfaces between only

three young and three old subjects (Figure 3.8A-B) suggest that our method is sensitive to

detect complexity changes of interior WM represented by skeletons and WM surface

changes as a result of aging (Guttmann et al., 1998; Schmidt et al., 2003). The method

even detected asymmetrical WM skeletons between the left and right hemispheres within

the three young subjects ( P = 0.045).

The statistically based inference showing that sensitivity of the general structure box-counting analysis is low may not be true because we had a limited number of subjects. If we had studied more subjects, more significant results may have been obtained. A more likely alternative explanation is that the box-counting method may not be appropriate for examining the WM general structure. Other FD analysis methods need to be explored for more sensitive quantification of brain WM general structures.

The accuracy and sensitivity of the FD analysis method can be influenced by

accuracy of the image processing procedure (Figure 3.1). Among these processing steps,

WM segmentation and thinning are relatively more important. Misclassification of WM

would affect FD of surface and general structure directly and FD of skeleton indirectly.

Compared with other statistical segmentation methods, in which only intensity

information is taken into account, FAST incorporates the spatial information encoded

through the mutual influences of neighboring sites. This makes the method less sensitive

to noise and more robust (Zhang et al., 2001). The segmentation results (Figure 3.2) show

that FAST provides a good WM classification for our data. Geometry preservation and

connectivity preservation are two major concerns of thinning algorithms. Thinning

Chapter 3 A Three-dimensional Fractal Analysis Method for Quantifying White Matter Structure in Human Brain 71 methods without satisfactorily meeting these two requirements would mar FD assessment and even worse, make FD values calculated from skeletons useless because skeletons do not represent the real shape well. Our results (Figure 3.7) show that the thinning method adopted in this study obtained satisfactory results, in which the WM shape was well represented by the skeleton.

3.5.5. Summary

We have developed a novel 3D FD method and tested the method’s feasibility using geometric phantoms and MR data of healthy young and old subjects. The study shows that the method is accurate and sensitive in detecting age-related WM structural changes.

More solid conclusion will be drawn based on larger sample sizes. Systematic analysis of

WM structural changes in aging and diseases, which requires a large sample size, is beyond the scope of this methodological paper and will be performed in the future. This method might eventually lead to an objective diagnostic scale for age- or disease-related

WM degeneration. It may also be useful in longitudinal studies to evaluate effects of disease development, progression of aging, or medical intervention on brain WM structural adaptations.

Chapter 3 A Three-dimensional Fractal Analysis Method for Quantifying White Matter Structure in Human Brain 72

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Chapter 4

Quantifying Brain White Matter Changes in

Normal Aging using Fractal Dimension

4.1. Abstract

Although brain white matter (WM) degeneration in aging is a well recognized problem, its quantification has mainly relied on volumetric measurements, which lack details in describing the degenerative adaptation. In this study, brain WM structural complexity was evaluated in healthy old and young adults by analyzing three-dimensional fractal dimension (FD) of the WM segmented from magnetic resonance brain images. WM volumes were also measured for the whole, left, and right hemispheres. No significant volume decline was found in the old group. Significantly smaller WM FD values were detected in the old compared to young subjects. Specifically, interior structure complexity of the WM degenerated in the left hemisphere in old men but that was found in the right hemisphere in old women. Men showed more complex WM patterns (greater

*This chapter has submitted to Neurobiology of Aging for publication. 76 Chapter 4 Quantifying Brain White Matter Changes in Normal Aging using Fractal Dimension 77

FD values) than women. An asymmetrical right-greater-than-left-hemisphere complexity pattern was observed in the interior and general structures of the WM while the surface complexity was symmetrical across the WM structures of the two hemispheres. These results suggest that brain WM structural complexity deteriorates with age but the degeneration is not uniformly distributed across the genders and hemispheres.

4.2. Introduction

Aging is accompanied by anatomical and functional degenerative adaptations in the nervous system (Albert and Knoefel, 1994). Among which, age-related brain white matter (WM) degeneration has long been recognized as it interferes normal communications within the nervous system and disrupts regulatory functions of the nervous system towards various body systems. It has been suggested that age-related

WM changes could potentially act as disease-prediction parameters, such as motor function impairments (Breteler et al., 1994; Baloh et al., 2003), cognitive deficits (de

Groot et al., 1998;Gunning-Dixon and Raz, 2000), depression (Pantoni and Garcia, 1995;

Inzitari et al., 2000), or dementia (Kapeller and Schmidt, 1998). Despite the importance of the WM structural integrity in maintaining normal body function and its adaptive information in predicting/diagnosing various disorders, reports of accurate assessments of multi-feature brain WM structure in vivo are scarce. This is largely due to the lack of appropriate methods for the quantification of different aspects of the WM structure.

Brain WM structure has most frequently been studied by magnetic resonance imaging (MRI) using various methods, among which volumetric analysis has most often been performed. Volumetric analysis is an appropriate assessment for brain atrophy but it might not be sensitive in characterizing WM structural features and their adaptations.

Chapter 4 Quantifying Brain White Matter Changes in Normal Aging using Fractal Dimension 78

So far, there have been no consistent conclusions regarding whether age or neurological disorders induce brain WM volume reduction. While some investigators have observed a significant decrease in WM volume with normal aging (Guttmann et al., 1998; Jernigan et al., 2001; Ge et al., 2002), others did not see the degenerative change (Good et al.,

2001a; Sato et al., 2003). Moreover, the volume measurement only captures one of multiple features of the WM structural characterization and reveals very little about adaptations of the features because of limitations of volumetric evaluation in describing nonlinear structures, such as complexity and variability of the WM structural organization.

Shape analysis of brain structures has been suggested to provide new information that is not accessible by conventional volumetric measurements (Gerig et al., 2001).

Structural fractal analysis (Mandelbrot, 1982) provides one of such shape descriptors and is a prominent method in quantifying morphometric complexity and variability of nonlinear structures. Although a relatively large number of fractal studies on brain gray matter (GM) have been reported (Sato et al., 1996; Thompson et al., 1996; Kedzia et al.,

1997; Blanton et al., 2001; Kiselev et al., 2003; Lee et al., 2004; Luders et al., 2004), few such studies have investigated brain WM (Bullmore et al., 1994; Cook et al., 1995; Free et al., 1996; Liu et al., 2003) and they all only examined surface (or contour) features of the structure, and none has performed three-dimensional (3D) volumetric fractal analysis.

Furthermore, there has been no report of fractal analysis on WM structural changes in normal aging.

Recently, we have developed a method that performs 3D fractal analysis of three features (interior structure, surface [interface between GM and WM], and general

Chapter 4 Quantifying Brain White Matter Changes in Normal Aging using Fractal Dimension 79 structure) of brain WM structures with high accuracy and sensitivity (Zhang et al., 2005).

The combination of the three-shape representations provides a more comprehensive characterization of the WM structures in detecting age- or disease-related human brain

WM degenerative and regenerative adaptations. The purpose of this study was to evaluate complexity and variability alterations of brain WM in normal aging using the 3D and multi-feature fractal analysis method, and to compare the volume results measured using the conventional volumetric technique.

4.3. Materials and Methods

4.3.1. Subjects

Twenty-four young (17 – 35 years old, mean age ± standard deviation = 27.7 ± 4.4 years) and 12 old (72 – 80 years old, 74.8 ± 2.6 years) subjects participated in the study. Within the young group, there were 12 males (28.8 ± 3.7 years) and 12 females (26.7 ± 5.0 years). The old-subject group consisted of 5 males (75.0 ± 3.3 years) and 7 females (74.7

± 2.2 years). All the subjects were healthy without known neurological or psychological disorders at the time of the study. The experimental procedures were approved by the

Institutional Review Board at the Cleveland Clinic Foundation. All subjects signed informed consent prior to the participation.

4.3.2. Head Images Acquisition

Coronal magnetic resonance (MR) human head images covering the entire brain were collected on a Siemens 1.5-T Vision scanner (Erlangen, Germany) using a circularly polarized head coil. Contiguous coronal brain slices (n = 128, each 2 mm thick) were

Chapter 4 Quantifying Brain White Matter Changes in Normal Aging using Fractal Dimension 80 acquired with a 3-D Turboflash imaging sequence (TR [repetition time] / TE [echo time]

= 11.4 ms / 4.4 ms). The flip angle was 10° and the in-plane resolution was 1 x 1 mm2.

4.3.3. Image Processing

The image processing procedures were the same as those used in our recently study of developing a 3D fractal analysis method (Chapter 3 and Zhang et al., 2005). Briefly, there were five steps: (i) resampling of the MR head images, (ii) brain extraction from the head images, (iii) manual segmentation of individual structures (see below), (iv) automated

WM segmentation, and (v) automated WM skeleton extraction. Details of image processing methods are discussed in Chapter 3. Figure 4.1 demonstrates the image processing results based on one sample slice from a young subject (Figure 4.1A-C) and one sample slice from an old subject (Figure 4.1D-F). The WM segmentation results

(Figures 4.1C and 4.1F) indicated that in general, the FAST method (Zhang et al., 2001) classified WM of the cerebrum well.

4.3.4. Measurement of Fractal Dimension

The recently developed 3D fractal analysis method, discussed in Chapter 3, was used to measure FDs of the WM (Zhang et al., 2005). The FD values of the WM skeleton, WM surface and WM general structure were computed for the whole brain, left, and right hemispheres.

4.3.5. Volume Measurement

The absolute volumes (ml) and relative volumes (%) of the WM were measured for the whole brain, left, and right hemispheres. The absolute volumes were determined by multiplying the number of voxels by unit volume of the voxel. The relative volumes were

Chapter 4 Quantifying Brain White Matter Changes in Normal Aging using Fractal Dimension 81

(A) (D)

(B) (E)

Cerebrospinal fluid White matter of left cerebrum Gray matter White matter of right cerebrum White matter skeleton

Figure 4.1. Image processing results based on one sample coronal slice from one young subject (A-C) and one sample coronal slice from one old subject (D-F). Left panel: image processing results for the young subject. (A) A T1-weighted coronal head image. (B) Brain segmentation result. (C) Brain tissues segmentation and skeleton results overlaid on the original head image. Right panel: image processing results for the old subject. (D) A T1-weighted coronal head image. (E) Brain segmentation result. (F) Brain tissues segmentation and skeleton results overlaid on the original head image.

Chapter 4 Quantifying Brain White Matter Changes in Normal Aging using Fractal Dimension 82

referred as relative percentage of WM within the intracranial cavity (GM + WM +

cerebrospinal fluid).

4.3.6. Statistical Analysis

A two-way analysis of variance (ANOVA) (age [young vs. old], gender [male vs.

female]) was conducted separately to analyze the WM structure of the whole cerebrum,

left hemisphere, and right hemisphere to determine the age and gender effects on the WM

FDs and volumes. A post hoc test was performed to analyze the age effect in males and

females and the gender effect in young group and old group. A three-way repeated

measures ANOVA (within, hemisphere of the brain [left vs. right]; between, age [young

vs. old], gender [male vs. female]) was performed to detect the left-right hemispheric

asymmetry of WM. A post hoc test was performed to analyze the asymmetry of a

specific group. Significant differences were accepted at P < 0.05.

There were two difficulties to model the data: unbalance design (24 young people vs. 12 old people) and missing values which were due to image collection errors (brains were not scanned completely for some subjects). The conventional ANOVA model that is based on the least-squares approach (Fox, 1997) can be misestimated when the two problems exist. Likelihood-based estimation scheme to fit the two-way or three-way

ANOVA model (Davis, 2002) was, therefore, used in this study. The type III F-statistics and p-values were calculated based on a general Wald-type F statistics (Wolfinger and

Chang, 1995).

Chapter 4 Quantifying Brain White Matter Changes in Normal Aging using Fractal Dimension 83

4.4. Results

4.4.1. Age and Gender Effects on WM Structural Changes

The FD means and standard deviations by age group and gender are shown in Table 4.1.

FD differences between young and old individuals in male participants and female participants, and the differences between men and women in the young and old group by post hoc tests are shown in Figures 4.2-4.4 (corresponding P values are shown in Tables

4.3 and 4.4).

The volume means and standard deviations by age group and gender are shown in

Table 4.2. Volume differences between young and old individuals in male participants and female participants, and the differences between men and women in the young and old group by post hoc tests are shown in Figures 4.5 and 4.6 (corresponding P values are shown in Tables 4.6 and 4.7). Outliners were found in volumetric analysis (Figures 4.5,

4.6 and 4.8). Excluding these outliners did not affect statistical conclusions.

4.4.1.1. Age Effect

The FD value was significantly smaller in the old than young group in the WM skeleton for the whole brain and left hemisphere (Table 4.1). The markedly smaller WM skeleton

FD values in the old individuals suggest that the WM complexity decreases with age.

This is consistent with the visual inspection (Figure 4.1), in which the brain of the older person showed a shrinking WM structure and less complex WM skeleton pattern compared with the brain of the young individual. With respect to general structure, the

FD value was significantly smaller in the old than young group for the whole brain.

Chapter 4 Quantifying Brain White Matter Changes in Normal Aging using Fractal Dimension 84

No significant volume differences (absolute volume and relative volume) were observed between young and old groups (Table 4.2 and Figures 4.5 and 4.6).

4.4.1.2. Gender Effect

The FD was significantly greater in men than in women in the WM skeleton of the whole brain and right hemisphere, the WM surface of the whole brain and right hemisphere, and the WM general structure of the whole brain (Table 4.1). The results suggest that men have a more complex WM pattern than women. Post hoc analysis revealed that the men- greater-than-women WM complexity pattern was more obvious in the old than young group, including WM skeleton of the right hemisphere (Figure 4.2C), WM surface of the whole brain and right hemisphere (Figure 4.3A, C) and WM general structure of the whole brain (Figure 4.4A).

The absolute volume and relative volumes were significantly greater in men than in women (Table 4.2). Post hoc analysis revealed that the men-greater-than-women WM volume pattern was more obvious in the young than old group (Figures 4.5 and 4.6).

4.4.1.3. Age by Gender Effect

Although no significant age-gender interaction was found on FD (Table 4.1), Post hoc analysis showed that complexity pattern of the WM interior structure experienced significant age-related degeneration in the left hemisphere in men (Figure 4.2B) and right hemisphere in women (Figure 4.2C). These results suggest that age had asymmetric effects on the hemispheres and such effects were gender dependent.

With respect to surface and general structure, although no significant age-gender interaction was detected, post hoc analysis reported women had greater age-related WM

Chapter 4 Quantifying Brain White Matter Changes in Normal Aging using Fractal Dimension 85

changes than men in the whole brain WM surface (Figure 4.3A) and whole brain WM

general structure (Figure 4.4A).

With respect to absolute and relative volumes, no significant age-gender

interaction was found in two-way ANOVA analysis (Table 4.2) and post hoc analysis

(Tables 4.6 and 4.7).

4.4.2. Asymmetry of WM

The repeated measures ANOVA revealed a significant smaller FD of the left than right

WM skeleton ( P = 0.015), which indicates that interior structure of the WM in the left

hemisphere has a less complex pattern than the right hemisphere. A significant smaller

FD of the left than right WM general structure was also observed ( P = 0.0011), showing that the entire shape of the left hemisphere WM has a less complex pattern than the right hemisphere. No significant FD differences were found between the left and right WM surfaces, indicating that convolution pattern of the WM surface is symmetrical across the two hemispheres measured by the fractal technique. WM FD asymmetry results for young and old men, and young and old women are shown in Figure 4.7 (P values are given in Table 4.5). The WM skeleton and WM general structure showed obvious right- greater-than-left (rightward) asymmetry in young (men and women) and old (men and women) group, although significant hemisphere differences were detected only in WM skeletons in old men, and in WM general structure in young men.

No significant volume differences were detected between left and right WM hemispheres, suggesting that WM volume was symmetrical across the two hemispheres

(Figure 4.8 and Table 4.8).

Chapter 4 Quantifying Brain White Matter Changes in Normal Aging using Fractal Dimension 86

Chapter 4 Quantifying Brain White Matter Changes in Normal Aging using Fractal Dimension 87

Chapter 4 Quantifying Brain White Matter Changes in Normal Aging using Fractal Dimension 88

2.44 (A) ** ** 2.42

2.40

2.38 FD FD FD FD FD 2.36

2.34

2.32 Group * Young * 2.30 Old Male Female All

* ** 2.30 (B)

2.25

FD 2.20

2.15

2.10 Male Female All

2.325 (C) * 2.300

2.275 FD FD FD FD 2.250

2.225

2.200 **

Male Female All

Figure 4.2. Box plot of WM skeleton FD results of young (males and females) and old groups (males and females) for the (A) whole brain, (B) left hemisphere, and (C) right hemisphere. ( ⊕ ) Group mean, * P < 0.05 , ** P < 0.01.

Chapter 4 Quantifying Brain White Matter Changes in Normal Aging using Fractal Dimension 89

2.53 (A) * 2.52

2.51 2.50 FD FD FD FD 2.49

2.48

2.47 * Group 2.46 Young ** 2.45 Old Male Female All

2.450 (B)

2.425

FD 2.400

2.375

2.350 Male Female All

2.450 (C)

2.425

2.400 FD FD FD

2.375

2.350 *

Male Female All

Figure 4.3. Box plot of WM surface FD results of young (males and females) and old groups (males and females) for the (A) whole brain, (B) left hemisphere, and (C) right hemisphere. ( ⊕ ) Group mean, * P < 0.05 , ** P < 0.01.

Chapter 4 Quantifying Brain White Matter Changes in Normal Aging using Fractal Dimension 90

2.58 (A) *** 2.57 *** 2.56 2.55 2.54 FD FD FD FD 2.53

2.52 * 2.51 Group 2.50 Young Old *** 2.49 Male Female All

2.35 (B)

2.30

FD 2.25

2.20

2.15 Male Female All

2.5 (C)

2.4

2.3 FD FD 2.2

2.1

2.0 Male Female All

Figure 4.4. Box plot of WM general structure FD results of young (males and females) and old groups (males and females) for the (A) whole brain, (B) left hemisphere, and (C) right hemisphere. ( ⊕ ) Group mean, * P < 0.05 , ** P < 0.01, *** P < 0.001.

Chapter 4 Quantifying Brain White Matter Changes in Normal Aging using Fractal Dimension 91

650 (A) ) ) ) l l l 600 m m m ( ( (

e e e 550 m m m u u u l l l 500 Vo Vo Vo e e e t t t u u u l l l 450 so so so Group ** Ab Ab Ab 400 Young 350 Old Male Female All

320 (B) 300 ) ) l l m m 280 ( ( e e 260 m m u u l l 240 Vo Vo e e t t 220 u u l l so so 200 Ab Ab 180 * 160 Male Female All

320 (C) 300 l) l) m m 280 ( (

e e m m 260 lu lu o o 240 V V e e t t 220 lu lu o o s s b b 200 A A 180 * 160 Male Female All

Figure 4.5. Box plot of WM absolute volume results of young (males and females) and old groups (males and females) for the (A) whole brain, (B) left hemisphere, and (C) right hemisphere. ( ⊕ ) Group mean, ▲, outliner, * P < 0.05 , ** P < 0.01.

Chapter 4 Quantifying Brain White Matter Changes in Normal Aging using Fractal Dimension 92

42 (A)

) 40 % (

e 38 m u l 36 Vo e v i

t 34 a l Group Re 32 * Young 30 Old Male Female All

18.0 (B) 17.5 ) % (

17.0 e m u l 16.5 Vo

e 16.0 v i t a l 15.5 Re 15.0 **

Male Female All

17.5 (C)

) 17.0 % (

e 16.5 lum o

V 16.0 e v i t

la 15.5 e R 15.0 *

Male Female All

Figure 4.6. Box plot of WM relative volume results of young (males and females) and old groups (males and females) for the (A) whole brain, (B) left hemisphere, and (C) right hemisphere. ( ⊕ ) Group mean, ▲, outliner, * P < 0.05 , ** P < 0.01.

Chapter 4 Quantifying Brain White Matter Changes in Normal Aging using Fractal Dimension 93

Hemispher 2.35 Left ** Right 2.30

2.25 FD 2.20

2.15 (A) 2.10 Young Old Young Old Male Female

2.450

2.425

2.400 FD

2.375

2.350 (B) Young Old Young Old Male Female

2.5 **

2.4

2.3 FD FD FD 2.2

2.1 (C) 2.0 Young Old Young Old Male Female

Figure 4.7. Box plot of FD results of left hemisphere (Left) and right hemispheres (Right) in male (young and old) and female participants (young and old). (A) WM skeleton FD results, (B) WM surface results, (C) WM general structure results. ( ⊕ ) Group mean, ** P < 0.01.

Chapter 4 Quantifying Brain White Matter Changes in Normal Aging using Fractal Dimension 94

Area 320 (A) Left 300

) Right l

(m 280

m 260 u l 240 Vo

te 220 u l o s 200 b A 180 160 Young Old Young Old Male Female

18.0 (B)

) 17.5 %

( 17.0 e m u

l 16.5

Vo 16.0 e v i t

a 15.5 l

Re 15.0 14.5 Young Old Young Old Male Female

Figure 4.8. Box plot of volume results of left hemisphere (Left) and right hemispheres (Right) in male (young and old) and female participants (young and old). (A) WM absolute volume results, (B) WM relative volume results. ( ⊕ ) Group mean, ▲, outliner.

Chapter 4 Quantifying Brain White Matter Changes in Normal Aging using Fractal Dimension 95

Table 4.3 Post hoc analysis P value table for age effect on white matter fractal dimension

Region Male Female Skeleton Whole 0.097 0.003** Left 0.025* 0.109 Right 0.920 0.013* Surface Whole 0.660 0.022* Left 0.238 0.312 Right 0.672 0.204 General structure Whole 0.139 <0.001*** Left 0.588 0.832 Right 0.124 0.701

* P < 0.05 , ** P < 0.01, *** P < 0.001. Whole, whole white matter (WM). Left, WM of left cerebrum. Right, WM of right cerebrum.

Chapter 4 Quantifying Brain White Matter Changes in Normal Aging using Fractal Dimension 96

Table 4.4 Post hoc analysis P value table for gender effect on white matter fractal dimension

Region Young Old Skeleton Whole 0.026* 0.010* Left 0.308 0.944 Right 0.259 0.008** Surface Whole 0.034* 0.004**

Left 0.068 0.373 Right 0.347 0.043* General structure Whole 0.032* <0.001*** Left 0.142 0.527 Right 0.116 0.536

* P < 0.05 , ** P < 0.01, *** P < 0.001. Whole, whole white matter (WM). Left, WM of left cerebrum. Right, WM of right cerebrum.

Table 4.5 Post hoc analysis P value table for hemispheric asymmetry of white matte fractal dimension

Young Old Men Women Men Women Skeleton 0.296 0.243 0.007** 0.666

Surface 0.411 0.938 0.414 0.668

General structure 0.005** 0.137 0.733 0.095

* P < 0.05 , ** P < 0.01.

Chapter 4 Quantifying Brain White Matter Changes in Normal Aging using Fractal Dimension 97

Table 4. 6 Post hoc analysis P value table for age effect on white matter volume

Region Male Female Absolute volume (ml) Whole 0.282 0.157 Left 0.120 0.355 Right 0.141 0.429 Relative volume (%) Whole 0.615 0.661 Left 0.592 0.491 Right 0.537 0.305

Whole, whole white matter (WM). Left, WM of left cerebrum. Right, WM of right cerebrum.

Chapter 4 Quantifying Brain White Matter Changes in Normal Aging using Fractal Dimension 98

Table 4.7 Post hoc analysis P value table for gender effect on white matter volume

Region Young Old Absolute volume (ml) Whole 0.008** 0.060 Left 0.012* 0.247 Right 0.023* 0.355 Relative volume (%) Whole 0.049* 0.238

Left 0.003** 0.035* Right 0.018* 0.153

* P < 0.05 , ** P < 0.01. Whole, whole white matter (WM). Left, WM of left cerebrum. Right, WM of right cerebrum.

Table 4.8 Post hoc analysis P value table for hemispheric asymmetry of white matte volume

Young Old Men Women Men Women Absolute volume 0.795 0.959 0.889 0.956

Relative volume 1.000 1.000 1.000 1.000

Chapter 4 Quantifying Brain White Matter Changes in Normal Aging using Fractal Dimension 99

4.5. Discussion

This study is the first to apply FD analysis to MR data for detection of WM structural adaptations in normal aging. Our methods allowed us to simultaneously evaluate three

WM structures: the interior (represented by the skeletons), surface (interface between the

GM and WM), and general structures of WM of the cerebrum. The study also, for the first time, comprehensively addressed the left-right hemispheric asymmetry of the WM structural complexity of the three structures.

4.5.1. WM Structural Changes with Aging

We observed WM skeleton FD decreases in the old group in the whole brain and left hemisphere. This finding suggests that complexity of the interior WM structure of each of the organizations decreased markedly with normal aging. Because the FD measures of

WM skeletons might reflect complexity of WM fiber bundle connectivity network (e.g., fibers crossings and bifurcations), the decrease in FD of WM skeletons could be an indication of reduced connectivity in the WM network. Postmortem studies have reported losses of thinner fibers with preservation of thicker ones in WM of aging brain, which confirmed the loss of higher-order fiber bifurcations (Marner et al., 2003).

Our findings also demonstrated that age effects on WM interior structural complexity was not uniform, but rather asymmetrical across the hemispheres. WM interior structure complexity of the left hemisphere in men and right hemisphere in women experienced significant age-related reductions. This leftward age effect was also observed in a regional WM volumetric study (Resnick et al., 2003) and WM signal

(regional GM-WM contrast ratio) study (Davatzikos and Resnick, 2002). In a

Chapter 4 Quantifying Brain White Matter Changes in Normal Aging using Fractal Dimension 100 longitudinal study, Resnick and colleagues detected significant age-related WM losses in all the brain regions examined, with a greater loss in the left than right temporal lobe

(Resnick et al., 2003). Davatzikos and coworkers reported that age-related WM signal weakening (reflection of WM demyelination and changes in water, protein and mineral content of the tissue) occurred primarily in the left hemisphere (Davatzikos and Resnick,

2002). Although gender differences regarding asymmetrical age-related WM changes across the hemispheres have not been discussed in the literature (Davatzikos and Resnick,

2002; Resnick et al., 2003), several studies have explored this point in aging brain. Shan et al. (2005) found significantly greater asymmetrical volume lost in both the left hemisphere and left frontal lobe in elderly men than women. Xu and colleagues (2000) found that brain atrophy with aging was more significant in male than female subjects in the posterior part of the right frontal lobe. Moreover, they observed that only male subjects had atrophy in the middle portion of the right temporal lobe, the left basal ganglia, the parietal lobe, and the cerebellum. Our observations of asymmetrical hemispheric age-related WM changes in men and women may provide neuroanatomical substrates for support of lateral asymmetry of human brain functions. Lateralized brain functions have been recognized for more than a century, in which the left hemisphere plays a special role on language function and the right hemisphere is linked to spatial manipulation. Our finding that elderly men have marked WM complexity changes in the left hemisphere is consistent with one behavior study that suggested that in men spontaneous language ability decreases rapidly with age but this function is well- preserved in elderly women (Ardila and Rosselli, 1996). In general, women outperform men on language tasks and underperform men on spatial tasks. There are many other

Chapter 4 Quantifying Brain White Matter Changes in Normal Aging using Fractal Dimension 101 differences in cognitive, sensorimotor and behavioral functions between men and women and they may be attributed to gender-biased asymmetrical organization of the brain structures. Age-related asymmetrical neural degeneration (such as WM) between genders is likely a consequence of gender-selective function deterioration.

Our results showed that FD values of WM surface (whole brain and right hemisphere) and WM general structure (whole brain) decreased in the elderly people as well. The FD decrease in the WM surface could reflect indirectly the sulcal widening in aging, which is consistent with a recent cortical surface study, in which the FD measurement of skeletonized cortical surface (medial surface between the interface of the

GM and WM and the pial surface) decreased in old people (Lee et al., 2004). The diminished WM general structure FD could possibly reflect WM volumetric atrophy in aging that has repeated been observed previously (Guttmann et al., 1998; Jernigan et al,

2001; Ge et al., 2002).

In the volumetric analysis, no significant WM volume reduction was found in aging. Our finding is consistent with some previous studies (Good et al., 2001a; Sato et al., 2003) whereas does not agree with others (Guttmann et al., 1998; Jernigan et al.,

2001; Ge et al., 2002). Such conflict may be due to the differences of image processing procedures and sample sizes. It may also suggest that volumetric measurement is not sensitive in detecting WM structural changes in aging. FD measurement is, therefore, more appropriate to detect WM structural changes for a small sample size study.

4.5.2. Gender Differences of WM

Our results showed that men had significant higher complexity pattern (higher FD) than women for the WM interior (whole brain and right hemisphere), surface (whole brain and

Chapter 4 Quantifying Brain White Matter Changes in Normal Aging using Fractal Dimension 102

right hemisphere), and general structures (whole brain). Although no studies have

examined gender differences in WM complexity, other cortical structural complexity has

been reported with significantly greater values in women compared to men in the

superior-frontal and parietal lobes (Luders et al., 2004). We found that men had a larger

volume of WM than women, which is consistent with previous studies (Gur et al., 1999).

It is not clear if such size differences are associated with WM complexity discrepancies

between men and women, especially the discrepancy in the WM general structure that, to some degree, may reflect volume alterations.

4.5.3. Asymmetry of WM

WM anatomical asymmetry in healthy people has been explored in volumetric studies

using structural MRI (Resnick et al., 2000; Good et al., 2001b; Pujol et al., 2002) and in

anisotropic measurement using diffusion tensor (magnetic resonance) imaging (DTI)

(Buchel et al., 2004; Park et al., 2004). A left-greater-than-right (leftward) volumetric

asymmetry was reported in WM of the whole brain and regional WM (frontal and

temporal regions) (Resnick et al., 2000; Good et al., 2001b; Pujol et al., 2002). The

asymmetrical pattern of WM anisotropy (neuronal fiber bundle quality, such as

myelination, density, coherence of the fibers) was found to be different among regions

within the WM structure (Buchel et al., 2004; Park et al., 2004). For example, a leftward

asymmetry was observed in the arcuate fascicle, whereas a rightward asymmetry was

found in WM of the inferior parietal lobe (Buchel et al., 2004).

The WM complexity asymmetry could be functional-related and reflect influences

evolution and development. Our results revealed that WM had a rightward complexity

asymmetry pattern in the interior and general structures. Such rightward asymmetries

Chapter 4 Quantifying Brain White Matter Changes in Normal Aging using Fractal Dimension 103

were not age-dependent. With respect to gender effect, we found that men had a more

pronounced complexity asymmetry of the interior and general WM structures than

women. This finding is consistent with studies that have report results of volumetric

asymmetry (Good et al., 2001b; Pujol et al., 2002) and with other investigations that have

shown more asymmetrical brain structures in male than the female individuals (Toga and

Thompson, 2003). The complexity of the WM surface convolution was found to be

symmetrical between the right and left hemispheres, which was consistent with findings

of a previous study (Free et al., 1996). Such symmetrical composition is likely to reflect

not only the shape of the WM, but also the inner boundary of cortical surface. The

complexity asymmetry of inner surface of the cerebral cortex is, therefore, possibly

different from the skeletonized cortical surface shape (Lee et al., 2004) and outer surface

of cerebral cortex (Luders et al., 2004). Based on FD analysis, Lee and colleagues

reported that cortical surface shape had a rightward complexity asymmetry (Lee et al.,

2004). Luders and colleagues found that complexity asymmetry of cortical surface (outer

boundary) differed among regions, with higher complexity in the left parietal and left

occipital cortices, as well as in the right inferior-frontal regions (Luder et al., 2004).

In our volumetric analysis, we found that WM volume was left-right symmetrical,

which does not agree with previous studies (Resnick et al., 2000; Good et al., 2001b;

Pujol et al., 2002). This may be also due to differences of image processing procedures

and sample sizes.

4.5.4. Summary

This study explored cerebral WM structural complexity changes in normal aging and hemispheric WM complexity asymmetry using a recently developed 3D fractal analysis

Chapter 4 Quantifying Brain White Matter Changes in Normal Aging using Fractal Dimension 104 method (Zhang et al., 2005). WM structural complexity declined with age in all the analyzed anatomical organizations. Aged-related asymmetrical changes were found in the WM interior structure, which showed that complexity of the WM interior structure diminished in the left hemisphere in men and right hemisphere in women. WM structures in men were more complex (higher FD) than in women. Finally, a right- greater-than-left complexity asymmetry pattern in the WM interior structure and general structure was observed, whereas complexity of the WM surface was symmetrical. With respect to volumetric analysis, no significant volume reductions of WM were observed.

Men had a larger WM volume than women. WM volume was found to be left-right symmetrical. Our findings have added new information that complements previous age- related WM studies and provide anatomical evidence that might be helpful to explain lateralized brain functions. The FD measurement of WM structural complexity may serve as an objective diagnostic index for age- or disease-related WM degeneration and perhaps treatment-related regeneration. It may also be useful in longitudinal studies to estimate onset of neurodegenerative diseases, evaluate effects of disease development, progression of aging, as well as the rehabilitation interventions. This may be the first noninvasive objective way to demonstrate the morphologic basis of cerebral plasticity.

Chapter 4 Quantifying Brain White Matter Changes in Normal Aging using Fractal Dimension 105

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Chapter 5

Conclusions and Future Directions

5.1. Conclusions

The main purpose of this study was to determine whether human aging disrupts normal

white matter (WM) structure of the brain by measuring MR (magnetic resonance) image-

based WM fractal dimension (FD). I first performed a pilot study to test the feasibility of

the application of FD theory to human brain WM. I then developed a three-dimensional

(3D) FD method, which is more relevant, accurate and sensitive in detecting age-related

brain WM structural changes compared with conventional FD methods. Finally, I

investigated the WM shape complexity alterations in normal aging using my developed

3D FD method. The main findings and conclusions are summarized below.

Chapter 2 presented measurements of the FD of 2D human cerebellar WM

skeletons in MR images of 24 healthy young subjects. The medial axis “skeleton” is a major feature and represents essence of the WM structure. It was hypothesized that the

111 Chapter 5 Conclusions and Future Directions 112

skeleton represents the shape of the WM and that box-counting FD analysis is

appropriate to evaluate complexity of the WM skeleton. To test this hypothesis, skeletons were extracted from the cerebellar WM, and the FD was calculated using pseudo-3D box- counting method derived from the conventional 2D box-counting method. A 3D box- counting procedure was simulated by slice-by-slice 2D skeleton image analysis. Because cerebellar WM bears an obvious “tree-like” fractal pattern, it is the most appropriate region in the brain to test the applicability of skeleton and fractal theory on the WM. The findings indicate that cerebellum (CB) skeleton is a highly fractal structure with a FD of

2.57 ± 0.01. No significant CB FD difference was observed between men and women.

The results suggest that human brain WM bears a fractal structure and FD might potentially be applied to study cerebellar and/or the cerebral normal and pathological

WM changes.

Chapter 3 described procedures of developing a 3D FD (box-counting) method to quantify FD of human brain WM interior, surface, and general structures simultaneously.

This method, which firstly incorporated a shape descriptor (3D skeleton) representing the

interior structure and later combined the three features together, provided a more

comprehensive characterization of the WM structure. It was hypothesized that the 3D

box-counting method would accurately measure FD of the fractals and be sensitive in

detecting age-related WM adaptations. To test this hypothesis, fractal phantoms with

known FDs were generated, and the computed FD was compared with its theoretical

value to determine the accuracy of the measurement. The consistency of the computed

and theoretical FD values indicates that the method is accurate in measuring FDs of the

fractals. The WM FDs of a number of young and old healthy subjects were computed to

Chapter 5 Conclusions and Future Directions 113 find if there are any differences between the two groups to determine the sensitivity of the technology. FD values of the WM skeleton and surface were significantly greater in young than old individuals, suggesting more complex WM structures in young than old people. These results suggest that the new method is accurate in quantifying 3D brain

WM structural complexity and sensitive in detecting age-related WM degeneration.

In Chapter 4, I systematically evaluated age-related WM structural complexity changes in normal aging by analyzing 3D FDs of the WM images segmented from high- resolution MR brain images using our developed 3D FD method (Chapter 3). It was hypothesized that the FD values of old subjects would be smaller than those of young people as a consequence of age-related WM degeneration. To test this hypothesis, the FD values of young and old individuals were computed and differences between the two groups were compared. I found that the WM structural complexity declined with age in all the analyzed anatomical organizations (lower FDs in old than young people). Age- related asymmetrical changes were observed in the WM interior structure, which showed that complexity of the WM interior structure diminished in the left hemisphere in men and right hemisphere in women. Overall, WM structures in men were more complex

(higher FD) than in women. Finally, we found that the WM complexity asymmetry pattern was different among the shape features. A right-greater-than-left complexity asymmetry pattern in the WM interior structure and general structure was observed, whereas complexity of the WM surface was symmetrical. My findings have supplied new information in addition to previous age-related WM studies (a majority of which are volumetric investigations) and provided anatomical evidence that might be helpful to explain lateralized brain functions.

Chapter 5 Conclusions and Future Directions 114

In conclusion, I have developed a method that accurately detected age-related

brain WM structural degeneration. I am working to improve the method so that it can

serve as an objective diagnosis index for age- or disease-related WM degeneration and

perhaps treatment-related regeneration.

5.2. Future Directions

This dissertation has described a newly-developed 3D FD method that is accurate and

sensitive in detecting WM structural degeneration in normal aging. Building on the work presented in this dissertation, I propose future clinical applications in estimating onset and development of neurodegenerative diseases by detecting the structural complexity changes of the brain.

Because in-vivo imaging signal characteristics of WM adaptation is similar and relatively nonspecific, it is hard to distinguish changes of the tissue related to normal aging from neurodegenerative disorders. Current clinical diagnosis of WM degeneration relies primarily on visual rating scales, which uses other features, such as patterns or locations (shape, size, distribution) of lesions, as auxiliary parameters that are unable to separate WM diseases from aging development. Because of robust ability of the FD analysis in assessing shape/structure variability and complexity, we expect that the FD measurement to differentiate patterns of WM changes between normal aging and neurological disorders or from one type of neurological disease to another. We also expect regional FD analysis to detect locations of lesions.

In a recent preliminary experiment, we measured WM skeleton FD of one stroke patient based on his MR brain images. The transverse T1-weighed MR head images

Chapter 5 Conclusions and Future Directions 115 were collected on a Philips 1.5-T scanner. Contiguous transverse brain slices (n = 28, each 5mm thick) were acquired with a MPRAGE sequence. The in-plane resolution was

1 x 1 mm2. Figure 5.1 shows one sample slice of this stroke patient, with lesion observable in the right hemisphere.

Figure 5.1. One sample T1-weighted slice of a stroke patient. The arrow points to the lesion area.

Because the MR slice was too thick (5 mm), it was not practical to apply our 3D FD method. We therefore, measured FD of 2D WM skeleton of the left and right hemispheres. Figure 5.2 illustrates the image processing results using the sample MR slice (Figure 5.2A) to extract the brain image (Figure 5.2B), segment the WM, and identify the skeletons (Figure 5.2C). The results suggest that the FD method can successfully recognize degenerative WM adaptation after stroke. Among the 28 slices, we excluded the first and last few slices to avoid image processing errors and calculated the FDs of the WM skeletons from slice #8 to slice #24. The FD values of left and right hemispheres in each slice are shown in Figure 5.3A. The FD difference between left and right hemispheres (FDL - FDR) in each slice is shown in Figure 5.3B.

Chapter 5 Conclusions and Future Directions 116

(A) (B) (C)

White matter of left brain White matter of right brain Skeleton

Figure 5.2. Image processing results of one sample T1-weighted MR slice (A) from a stroke patient. (B) Brain extraction results. (C) White matter segmentation and skeleton extraction results.

As previously discussed, brain WM structure has a right-greater-than-left asymmetrical pattern in healthy adults. An appropriate negative value of FD difference

(FDL - FDR) might therefore, suggest that the complexity pattern of WM in the slice is

normal. Such normal range of FD is to be determined in the future. Since the right

hemisphere represents the lesioned hemisphere and the left hemisphere represents the

normal hemisphere in this patient, a positive value of the FD difference between WMs of

the two hemispheres (FDL - FDR) might indicate WM structural degeneration in the

lesioned areas. We found relatively large differences (positive values) in slices #14 - #17

(Figure 5.3B). The results suggest that the method is able to accurately identify

degenerated WM in the lesioned brain regions. A paired two-tailed t test comparison of

the four FD values of the left side with those on the right side (corresponding to the four slices (slices #14 - #17) in Figure 5.4) showed a statistical significant difference between

the lesioned and healthy sides (P = 0.006). One limitation of this case study is that the

analysis was based on poor-resolution (1 × 1 × 5 mm3) MR images. We will, therefore,

Chapter 5 Conclusions and Future Directions 117 collect higher resolution MR images so that we can evaluate the WM structural complexity more accurately using the 3D method.

In summary, this case-study indicates the ability of the FD method to quantify degenerative developments following stroke. The goal of this research in the immediate future is to improve sensitivity of the method so that even minute changes can be recognized. This technique will be applied to clinical and aging populations to determine whether unique characteristics of WM alterations could be identified within given normal and pathological populations. In the long term, attempts will be made to develop one or more parameters of the method into objective diagnostic indices. These indices are expected to assist clinicians in estimating the stages of a disease, evaluating progression of a disease or aging, and assessing effectiveness of the treatments.

Chapter 5 Conclusions and Future Directions 118

(A) Left brain Right brain 1.45

1.4

1.35 FD FD 1.3

1.25

1.2 8 1012141618202224 Slice #

(B) 0.08

0.06

0.04 ) ) 0.02 (R (R D D F F 0 ) - ) - (L (L 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 D D F F -0.02

-0.04

-0.06

-0.08 Slice #

Figure 5.3. Fractal dimension (FD) results. (A) Scatter plot of individual FD data. (B) FD differences between the left and right brain slices.

Chapter 5 Conclusions and Future Directions 119

Slice #12 Slice #13 Slice #14 Slice #15

FDL 1.380 1.351 1.379 1.370 FDR 1.417 1.419 1.337 1.318

Slice #16 Slice #17 Slice #20 Slice #22

FDL 1.410 1.399 1.341 1.305 FDR 1.339 1.360 1.393 1.342

Figure 5.4. Eight sample slices of a stroke patient and fractal dimension (FD) values corresponding to the eight slices. FDL, FD of left hemisphere. FDR , FD of right hemisphere.