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Mathematical modelling of the spatial dispersion of brain MRI lesions in multiple sclerosis Sheikhzadeh, Fahime 2011

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Mathematical Modelling of the Spatial Dispersion of Brain MRI Lesions in Multiple Sclerosis by Fahime Sheikhzadeh B.Sc. Biomedical Engineering, Amirkabir University of Technology, 2007 M.Sc. Biomedical Engineering, Amirkabir University of Technology, 2009 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in The Faculty of Graduate Studies (Biomedical Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April 2012 c Fahime Sheikhzadeh 2011  Abstract Many previous studies in multiple sclerosis (MS) have focused on the relationship between white matter lesion volume and clinical parameters, but few have investigated the independent contribution of the spatial dispersion of lesions to patient disability. In this thesis, we investigate whether a mathematical measure of the 3D spatial dispersion of lesions can reveal clinical significance that is independent of volume. Our hypothesis is that for any two given patients with similar lesion loads, the one with greater lesion dispersion would tend to have a greater disability. We investigate four different approaches for quantifying lesion dispersion and examine the ability of these lesion dispersion measures to act as potential surrogate markers of disability. We propose one connectedness-based measure (compactness), two region-based measures (ratio of minimum bounding spheres and ratio of lesion convex hull to the brain volume), two distance-based measures (Euclidean distance from a fixed point and pair-wise Euclidean distances) and one measure based on network theory (small-worldness). Our data include three sets of MRIs (n = 24, 174, 182) selected from two MS clinical trials. We segment all white matter lesions in each scan with a semi-automatic method to produce binary images of lesion voxels, quantify their spatial dispersion using the defined measures, then perform a statistical analysis to compare the dispersion values to total lesion volume and patient disability. We use linear and rank correlations to investigate the relationships between dispersion, disability, and total lesion volume, and regression analysis to investigate whether there is a potentially meaningful relationship between dispersion and disability, independent of volume. Our main finding is that one distance based measure, Euclidean distance from a fixed ii  Abstract point, consistently correlates with disability score across all three datasets, and has predictive value that is at least partly independent of lesion volume. The results provide support for our hypothesis and suggest that a potentially meaningful relationship exists between patient disability and measurements of lesion dispersion. Finding such relationships can improve the understanding of MS and potentially lead to the discovery of novel surrogate biomarkers for clinical use in designing treatment trials and providing prognostic advice to individual patients.  iii  Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iv  List of Tables  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vi  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vii  Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . .  viii  Dedication  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ix  1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Multiple sclerosis . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Magnetic resonance imaging and monitoring of MS lesions . . 1.3 Thesis motivation . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Thesis contributions . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Lesion dispersion measures . . . . . . . . . . . . . . . 1.4.2 Statistical analysis of the contribution of dispersion to disability . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Thesis organization . . . . . . . . . . . . . . . . . . . . . . .  1 1 2 3 4 4  2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Total white matter lesion volume and disability . . . . . 2.2 Lesion location . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Contribution of lesion location to disability . . .  7 7 9 9  . . . .  . . . .  . . . .  5 6  iv  Table of Contents 2.2.2 2.2.3  Lesion distribution as defined by using lesion probability map . . . . . . . . . . . . . . . . . . . . . . . . . . . . Longitudinal studies on lesion location . . . . . . . . .  10 11  3 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . 3.1 Lesion dispersion measures . . . . . . . . . . . . . . . . . . . 3.1.1 Connectedness-based approach . . . . . . . . . . . . . 3.1.2 Region-based approach . . . . . . . . . . . . . . . . . 3.1.3 Distance-based approach . . . . . . . . . . . . . . . . 3.1.4 Network-based approach . . . . . . . . . . . . . . . . . 3.2 Statistical analysis of the contribution of dispersion to disability  13 15 15 16 17 19 23  4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Total lesion volume normalized by intradural volume . . . . . 4.2 Connectedness-based approach . . . . . . . . . . . . . . . . . 4.2.1 Compactness . . . . . . . . . . . . . . . . . . . . . . . 4.3 Region-based approach . . . . . . . . . . . . . . . . . . . . . 4.3.1 Ratio of minimum bounding sphere and ratio of convex hull . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Distance-based approach . . . . . . . . . . . . . . . . . . . . 4.4.1 Euclidean distance (ED) from a fixed reference point . 4.4.2 Pair-wise Euclidean distances (PWED) analysis . . . . 4.5 Network-based approach . . . . . . . . . . . . . . . . . . . . . 4.5.1 Small-worldness . . . . . . . . . . . . . . . . . . . . . 4.6 Longitudinal study . . . . . . . . . . . . . . . . . . . . . . . .  25 25 26 26 28 28 29 29 32 33 33 34  5 Conclusions . . . . . . 5.1 Discussion . . . . 5.2 Conclusions . . . . 5.3 Future work . . .  . . . .  43 43 45 46  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  48  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  v  List of Tables 3.1  4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8  Mean, standard deviation, ranges, and interquartile ranges (IQR) of EDSS for the first two data sets, and MSFC scores in the third data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Correlation values between EDSS, total lesion volume and lesion dispersion for the first sample of 24 patients . . . . . . . . . . Correlation values between EDSS, total lesion volume and lesion dispersion for the second sample of 174 patients . . . . . . . . Correlation values between MSFC, total lesion volume and lesion dispersion for the third sample of 182 patients . . . . . . Correlation values between leg function component of MSFC, total lesion volume and lesion dispersion for the third data set Correlation values between hand function component of MSFC, total lesion volume and lesion dispersion for the third data set Correlation values between cognitive component of MSFC, total lesion volume and lesion dispersion for the third data set . . . Correlation values between MSFC, total lesion volume and lesion dispersion for the third data set after two years . . . . . . Correlation values between difference of MSFC, difference of total lesion volume and difference of lesion dispersion for the third data set . . . . . . . . . . . . . . . . . . . . . . . . . . .  14  35 36 37 38 39 40 41  42  vi  List of Figures 1.1  Two patients with similar lesion loads but different dispersion  3.1  Examples of different shapes and their corresponding compactness values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Different reference points tested for computing Euclidean distances from a reference point . . . . . . . . . . . . . . . . . . .  3.2  4.1 4.2 4.3  EDSS vs. total lesion volume in the first patient sample . . . . EDSS vs. variance of Euclidean distances from a fixed point and total lesion volume in the first patient sample . . . . . . . EDSS vs. variance of Euclidean distances from a fixed point and total lesion volume in the second patient sample . . . . .  4  16 18 26 30 32  vii  Acknowledgments First and foremost, I would like to express my sincere gratitude to my supervisor, Dr. Roger Tam for all of his support, guidance, patience and motivation. Furthermore, I gratefully acknowledge Dr. Ghassan Hamarneh at the Simon Fraser University (SFU) for his crucial contribution and insightful comments from the early stage of this research. I would also like to thank Andrew Riddehough for proposing the idea of measuring lesion dispersion. Most importantly, heartfelt and deep thanks go to my family for their unwavering love and support. Finally, the financial support of MS/MRI research group at the university of British Columbia is acknowledged.  viii  Dedication  To my parents for always believing in me!  ix  Chapter 1 Introduction 1.1  Multiple sclerosis  Multiple sclerosis (MS) is a disease of the central nervous system (CNS). In MS a chronic inflammatory reaction leads to loss of myelin sheaths around the axons, slowing of nerve conduction and loss of axons. This disease is a common cause of disability in young adults, and especially occurs in young women [8]. There are four subtypes of MS: relapsing remitting (RRMS), secondary progressive (SPMS), primary progressive (PPMS), and progressive relapsing (PRMS). RRMS is defined by relapses and remissions and typically develops into a phase of slowly progressing irreversible disability which is known as SPMS. In PPMS patients, the disease is slowly progressive from the onset. PRMS patients have a steady neurologic decline but also suffer additional attacks. This is the least common of all subtypes [8]. A patient with MS can suffer almost any neurological symptom. The disability progression and symptom severity are available in the form of some clinical measures like expanded disability status scale (EDSS) and multiple sclerosis functional composite (MSFC). EDSS is the most common index for physical disabilities in MS [31]. The score is based upon neurological testing and examination of functional systems in patients, which are areas of the central nervous system that control bodily functions. A higher EDSS value indicates worse clinical status. The values range from 0, normal neurological exam, to 10, death due to MS [3]. An MSFC score contains three main components, each measuring one clinical dimension of MS; 1) quantitive measure of leg function and ambulation, 2) quantitive measure of arm and hand 1  1.2. Magnetic resonance imaging and monitoring of MS lesions function, 3) a measure of cognitive function that assesses auditory information processing speed and ability, as well as calculation ability [13]. To compute the MSFC score, the score for each component is first converted to a z-score based on the pooled data set. The arm/hand component and the cognitive component are both scaled so that higher scores indicate better performance, but the leg function component is reversed. Consequently the formula for combining the three components into an overall MSFC score is: ZMSFC = (Zarm − Zleg + Zcog )/3  (1.1)  A lower MSFC score indicates more severe disability [13] relative to the mean of the reference population.  1.2  Magnetic resonance imaging and monitoring of MS lesions  Magnetic resonance imaging (MRI) is a sensitive imaging modality that can be used for tracking and visualizing MS. MRI can detect areas of inflammation (active lesions) and areas of permanent damage (chronic lesions) in the brain. These images may also include “silent” damage (lesions detected by MRI that do not result in symptoms). One of the MR-based measures that are commonly used to diagnose and monitor MS in the clinical setting is the extent of tissue disruption that is generally visible as brain white matter lesions on T1-weighted, T2-weighted and proton density-weighted (PDweighted) MRI scans [11, 34]. T2-weighted and PD-weighted scans typically show the same lesions, whereas T1-weighted lesions are a subset of the T2weighted/PD-weighted lesions. In this thesis, we focus on lesions that are visible on T2-weighted and PD-weighted scans.  2  1.3. Thesis motivation  1.3  Thesis motivation  Measurement of the total white matter lesion volume on MRI scans is a widely used outcome measure for monitoring the pathological state and progression of multiple sclerosis (MS) [24]. However, previous studies have shown that the relationship between lesion volume and patient disability is generally weak, especially in T2-weighted imaging studies [3]. Specifically, the cross-sectional correlation between T2-weighted lesion volume and the Extended Disability Status Scale (EDSS), which is the most frequently used clinical measure in MS [31], typically ranges between 0.15 to 0.4 for Spearman rank correlation with some studies reporting values as high as 0.6 [3]. In addition to having a limited predictive value, the focus on global lesion volume has left other lesion variables under-explored. In this thesis, we investigate whether mathematical measures of the 3D spatial dispersion of lesion voxels can reveal clinical significance that is independent of lesion volume. As we will describe in more detail in Chapter 2, a number of studies have explored the contribution of lesion location to MS disability, most commonly referred to as representing “distribution” [5, 9], but there has been minimal work done to quantify the spatial extent of MS lesions and its contribution to disability while controlling for volume as a variable. We use the term “dispersion” rather than “distribution” to define spatial extent in order to distinguish our work from studies on lesion location. Our hypothesis is that for any two given patients with similar lesion loads, the one with greater dispersion would tend to have greater disability due to a greater global impact on the brain, thereby reducing its capacity to adapt. Figure 1.1 presents a motivating example, and shows the projection of all lesion voxels onto the largest transverse slice of the brain scans of two patients in our data set. These patients have approximately the same total lesion volume (∼3000 mm3 ), but different spatial dispersion. The EDSS value for the patient with more distributed lesions is higher (6.5 vs. 3.5). Exploring such relationships can improve the understanding of MS and potentially lead to the discovery of novel surrogate biomarkers for clinical use. 3  1.4. Thesis contributions  Figure 1.1: White matter lesions, transverse view; left: a patient with a total lesion volume of 3071 mm3 and EDSS of 3.5, right: a patient with a total lesion volume of 2957 mm3 voxels and EDSS of 6.5. This example illustrates our hypothesis that for any two given patients with similar lesion loads, the one with greater dispersion would tend to have greater disability.  1.4  Thesis contributions  We use a methodology that consists of two main steps to evaluate our hypothesis and assess the relationship between brain lesion dispersion and the disability in MS patients. In the first step we investigate different computational methods for measuring lesion dispersion. In the second step we perform a statistical analysis to determine the strength of the relationship between lesion dispersion and patient disability and compare the contribution of dispersion to that of lesion volume. We use three sets of T2-weighted and proton density-weighted MRIs of 24, 178, and 182 patients selected from two MS clinical trials. To the best of our knowledge, we are the first to study the relationship between spatial lesion dispersion and MS disability.  1.4.1  Lesion dispersion measures  We investigate four different classes of measures for quantifying lesion dispersion. Each class of approaches focuses on a specific concept of spatial dispersion. The first class explores the effect of connectedness between the lesion voxels without incorporating distances; we use one method, termed compactness, from this class. The second class of methods includes region-based  4  1.4. Thesis contributions measures that approximate the region impacted by the lesions. We select two methods as representatives of this class; the ratio of volume of the minimum bounding spheres of the lesion voxels to that of the brain, and the ratio of the volume of the convex hull of the lesion voxels to the brain volume. The third set of measures is distance-based. We propose two measures, one consisting of three components and the other one consisting of four components, for this class. One measure uses the variance, entropy and skewness of the distribution of the Euclidean distances of the lesion voxels from a fixed reference point. The other method in this set uses the mean, variance, entropy and skewness of a set of Euclidean distances, but computed as pair-wise distances between the lesion voxels rather than from a fixed reference point. The last approach, which is derived from network theory, considers the lesions as nodes of a network and then uses the characteristics of the network to find a pattern of lesion dispersion. In this approach, we compute a ratio of small distances between lesion voxels to the large distances. We use a property of networks called small-worldness as a key component of this measure.  1.4.2  Statistical analysis of the contribution of dispersion measures to disability  After computing each measure for our patient samples, we perform a statistical analysis to determine the strength of its relationship to the clinical status, and compare the contribution of dispersion to that of lesion volume. We compute the p-values of both Pearson and Spearman correlations to investigate the statistical significance of relationships between lesion dispersion, the clinical scale, and total lesion volume. In this thesis we make a general assumption of linearity, therefore any discussion of correlation can be taken to mean Pearson correlation unless otherwise specified, and the Spearman values are provided primarily for completeness. We also use regression analysis to investigate whether there is a potentially meaningful relationship between lesion dispersion and clinical score that is independent of total lesion volume. 5  1.5. Thesis organization  Our main finding in this thesis is that one distance-based measure, Euclidean distance from a fixed point, consistently has predictive value that is at least partly independent of lesion volume for all three of our data sets. In addition, a connected-based measure, compactness, has a potentially meaningful correlation with cognitive impairment in the third data set. The results provide support for our hypothesis that for any two given patients with similar lesion loads, the one with greater dispersion would tend to have greater disability, but further investigation is required to determine why some dispersion measures agree with clinical status better than others.  1.5  Thesis organization  The rest of the thesis is organized as follows. In Chapter 2 we present a survey of previous investigations on features of white matter lesions, specifically volume and location, which have been used to monitor MS. In Chapter 3, we describe the data sets used and the detailed methods and algorithms implemented for calculating lesion dispersion and analyzing the results of these calculations. In Chapter 4, we present our results. Finally, Chapter 5 discusses an interpretation of the results, concludes the thesis and presents possible directions for future work.  6  Chapter 2 Related Work In this chapter we review some of common proposed approaches for quantifying lesion characteristics. The previous studies most relevant to this thesis are studies on total lesion volume and lesion location. Lesion volume is used to establish a baseline of clinical relevance for our proposed measures of lesion dispersion. Many studies in MS lesions have focused on the correlation between lesion volume in different locations of the brain and patient disability. Among these studies, some have investigated lesion location using a lesion probability map. This approach is mainly referred as investigating lesion distribution. In contrast, our proposed dispersion measures aim to quantify the spread of the lesions without particular regard for their location.  2.1  Total white matter lesion volume and disability  The total volume of white matter lesions is one of the most used outcome measurements for monitoring the progression of MS [24]. Very early studies like that of Thompson et al. [29] failed to find any correlation between conventional spin echo MRI lesion load and disability. Later on, several studies like those of van Walderveen et al. [33] and Gass et al. [16] which used more accurate measurement methods reported Spearman rank correlation coefficients of r = 0.23 − 0.33 for T2-weighted lesion volumes in mixed MS populations and as high as r = 0.57 [12] for a homogenous cohort of relapsing-remitting patients. Barkhof [3] also reported a selection of studies that have been presented over  7  2.1. Total white matter lesion volume and disability the years looking at correlations between conventional T2-weighted lesion load and EDSS. He observed that Spearman rank correlation coefficients varied between 0.15 and 0.60 [3]. Later on, more studies reported that MRI-derived total lesion load correlates weakly with disability as assessed, for example, by EDSS [7]. In addition to T2-weighted lesion volume, there have been promising studies using T1-weighted lesion volume, which is believed to reflect more permanent brain damage than T2-weighted scans, generally giving better correlations with disability [7]. But T1-weighted lesion volume also has considerable limitations. For example very few automated methods exist for segmenting T1-weighted lesions and the correlation between T1-weighted lesion volume and EDSS can be highly dependent on the intensity threshold used to define the lesions. On the other hand, T2-weighted lesions tend to be much more homogeneous [28]. Longitudinal correlations between changes in lesion volume and clinical status have also been inconsistent across previous studies. In a study by Khaleeli et al. [22], it has been shown that changes in lesion load did not correlate significantly with the progression rate of MS. In their study, changes in T2-weighted lesion load over first two years were studied in a cohort of 101 PPMS patients who were followed-up for 10 years. However some previous studies like that of Fisniku et al. [14] claimed that over a longer time period, changes in T2-weighted lesion load had a significant correlation with changes in disability. In their study, Fisniku et al. [14] have shown that lesion volume and its change at earlier time points were correlated with disability after 20 years. A number of factors and mechanisms are known to negatively affect the strength of the correlation between lesion load and disability, which include: neuroplasticity and cortical reorganization which help the brain adapt to local injury and contribute to functional recovery [4, 7]; in addition to focal lesions, MS is known to cause global white matter and gray matter changes  8  2.2. Lesion location that are not readily detectable on conventional MRI, but contribute to disability [4, 34]; pathological heterogeneity of lesions visualized on PD-weighted and T2-weighted images (edema, demyelination, axonal loss, or gliosis have a similar appearance on T2-weighted imaging) [34]; the fact that a proportion of the lesions visualized on MRI may be clinically silent [34]; and finally, limitations of the EDSS especially its nonlinearity and heavy weighting toward ambulatory deficits [4]. There is evidence that taking cognitive impairment into account may improve correlations with total lesion volume ([1, 15, 26, 27] as reviewed in [7]). However, the Spearman rank correlations between lesion volume and other clinical scores that include cognitive impairment like the multiple sclerosis functional composite (MSFC) are still moderate, ranging from -0.5 to -0.61 (p < 0.001) [14]. Despite inconsistencies in the findings, MRI lesion volume is considered a well established secondary outcome measure in clinical trials of multiple sclerosis. But the focus on global lesion volume has left other lesion variables under-explored. Consequently, researchers have continued to seek alternatives for quantifying the impact of lesions.  2.2 2.2.1  Lesion location Contribution of lesion location to disability  To exploit the fact that the brain function is closely tied to structure, a number of studies have explored the contribution of lesion location to MS disability. Gawne-Cain et al. [17] classified each lesion as either brainstem, cerebellar, subcortical (lesion touching cerebral cortical grey matter), periventricular (touching lateral or third ventricle), or discrete (within cerebral white matter). Then they studied the regional lesion volume in MS and reported that EDSS had a significant correlation with lesion volume in each separate anatomical region, the strongest being with periventricular (r ≥ 0.5 for Spearman correlation) and posterior fossa (brainstem and cerebellum) volumes (r ≥ 0.4 for 9  2.2. Lesion location Spearman correlation) and the weakest with subcortical volumes (r < 0.4 for Spearman correlation). Charil et al. [7] studied the relationship between the site of lesions and type of disability by combining automatic lesion localization with statistical techniques. They reported that in particular, part of the disability contained in the EDSS could be explained by lesions at restricted sites in the white matter, especially within the internal capsule. They showed that lesions at the grey-white junction particularly were implicated in cognitive impairment. Some studies investigated the particular relationship between sites of lesions and cognitive dysfunctions. Tiemann et al. [30] evaluated the significance of total lesion load for predicting general cognitive dysfunction and tested for a correspondence between lesion topography and specific cognitive deficit patterns. Periventricular lesions were significantly related to decreased psychomotor speed, whereas equally distributed cerebral lesion load were not. Their findings supported the idea that periventricular lesions had a determinant impact on cognition in patients with MS.  2.2.2  Lesion distribution as defined by using lesion probability map  A number of studies investigated lesion location using a lesion probability map that is commonly referred to as representing “distribution”. The probability map is produced by computing the probability of each voxel being lesional. In general the probability is defined by the relative voxel intensity. Vellinga et al. [34] investigated the correlations between spatial distribution of lesions and disability (assessed by EDSS and MSFC scores) by using a voxelwise lesion probability map on T2-weighted lesion masks. They found that lesion probability in the periventricular region correlated significantly (p < 0.001) with disability (r = 0.27 for Spearman correlation) and disease duration (r = 0.28 for Spearman correlation), and was higher in progressive than in relapsing disease. However, they found that lesion burden and location were 10  2.2. Lesion location confounding variables in that lesion load influenced relations between disability and lesion probability throughout the brain. In particular, when controlled for lesion load, they found no significant relation between lesion location and disability. In this thesis, we control for lesion volume when assessing the contribution of lesion dispersion. In addition, some studies have been done to evaluate the difference in lesion distribution in groups of patients with specific types of MS. Perri et al. [25] compared the spatial distribution, obtained by probability map, of lesions in patients with PPMS and RRMS. They reported that differences in cerebral pathologic involvement existed between PPMS and RRMS and contributed to variations in clinical disability.  2.2.3  Longitudinal studies on lesion location  Since predicting the progression of MS has important clinical implications, some researchers have studied the lesion location as a potential predictor for the long term clinical outcome in patients. Bodini et al. [5] studied lesion location and topographic distribution of lesions, again employing a lesion probability map to provide a voxel-wise, quantitative description of the topographic distribution of brain lesions. This work investigated whether the location of T2-weighted and T1-weighted lesions at baseline predicted progression over 10 years, performing a retrospective study in the cohort of 80 PPMS patients. Their findings suggested that the location of T2-weighted brain lesions in the motor and associative tracts was an important contributor to the progression of disability in PPMS, and was independent of spinal cord involvement. Dalton et al. [9] investigated associations between the spatial distribution of brain lesions and clinical outcomes in a cohort of MS patients who were followed up 20 years after presenting with a clinically isolated syndrome suggestive of MS. In this study, brain lesion probability maps of T1-weighted and T2-weighted lesions were analyzed adjusting for age and gender using a multiple linear regression model. This study demonstrated that lesion location characteristics 11  2.2. Lesion location were associated with disability after long-term follow-up. Lesions in certain regions of the brain, in particular posterior structures rather than anterior structures, were associated with greater disability as assessed by EDSS. In summary, a review of previous work on exploring white matter lesion properties on brain MRI of MS patients shows that measurement of the total white matter lesion volume is a widely used outcome measure for monitoring the pathological state and progression of MS. However, particularly in T2weighted imaging studies, the relationship between lesion volume and patient disability is generally weak. Consequently, researchers have explored other features of lesions, especially lesion location on T2-weighted scans, and have established some promising biomarkers of disability in MS. These studies have most commonly been done using a lesion probability map that is sometimes referred to as representing “distribution”. But there has been minimal work done to quantify the spatial extent of MS lesions and its contribution to disability while controlling for volume as a variable, which is the main goal in this thesis. In addition, we are proposing measures for lesion dispersion that are not emphasizing lesion location. As a result, our approach minimizes some types of errors (e.g., false positives from multiple comparisons in voxel-based studies and the errors involved in dividing the brain into regions such as the uncertainty of region boundaries).  12  Chapter 3 Materials and Methods We use three sets of T2-weighted and proton density-weighted MRIs to perform our experiments. The first data set is comprised of the MRIs of 24 MS patients, collected from a single selected scanning site participating in an MS clinical trial. The scans were acquired in the axial orientation on a Philips Achieva 3T scanner with a dual-echo sequence with TE1 = 15.0 ms, TE2 = 75.0 ms and TR = 2700.0 ms. The original image dimensions are 256 × 256 × 50 with voxel size 0.937 mm × 0.937 mm × 3.0 mm. For each patient, clinical status is available in the form of an EDSS score. The sample is a mix of 13 RRMS and 11 SPMS patients. The EDSS values are well-distributed in this data set which is not typical in MS studies. Studies of large populations of MS patients have revealed consistently a bimodal EDSS frequency distribution rather than a Gaussian distribution [18], with a large percentage of patients clustered around 6–6.5 and 3–3.5 [35]. The clustered EDSS values can negatively affect the correlation values in our study. We use the first data set to develop our dispersion measures and test the potential of each dispersion measure. We assume that a measure that does not show a positive result in this well-distributed data set is unlikely to show a result in a more realistic patient sample. The second data set is comprised of the MRIs of 174 randomly selected SPMS patients from 33 sites participating in another MS clinical trial. The scans were acquired in the axial orientation using a dual-echo sequence with TE1 = 8.4 - 20.0 ms, TE2 = 60.7 - 98.0 ms and TR = 2000.0 - 3400.0 ms. The original image dimensions are 256 × 256 × 50 with voxel size 0.937 mm × 0.937 mm × 3.0 mm. This patient sample is challenging for use in developing surrogate biomarkers using EDSS correlation because the EDSS values are highly clustered, which is 13  Chapter 3. Materials and Methods Table 3.1: Mean, standard deviation, ranges, and interquartile ranges (IQR) of EDSS for the first two data sets, and MSFC scores in the third data set Dataset 1 2 3  Clinical Score EDSS EDSS MSFC  Number of Patients 24 174 182  Mean 5 5.4 0.03  SD 2.2 1.3 0.76  Range 1.5–8 2.5–8.5 -4.7–1.2  IQR 3.75 2 0.91  typical of the SPMS population, with approximately half of the EDSS scores in this data set having values of 6 or 6.5. The third data set is comprised of the scans of 182 randomly selected SPMS patients enrolled in the same clinical trial as the second data set. We use this third data set to investigate the relationship between lesion dispersion and the MSFC, which includes a cognitive component and is typically better distributed than the EDSS. The second and third data sets overlap by about 50%, and we choose to use two different patient samples for the EDSS and MSFC in order to avoid using the same set of scans repeatedly. Table 3.1 shows the mean, standard deviation, ranges, and interquartile ranges of EDSS in the first two data sets, and MSFC scores in the third data set. The white matter lesions are delineated on each T2-weighted/PD-weighted pair using a semi-automatic method [23] to produce binary images in which the lesion voxels have the value of 1. Briefly, the delineation procedure includes four main steps. In the first step, we use a multi-scale version of the Non-parametric Non-uniform intensity Normalization (N3) method to correct the MR intensity inhomogeneity. In the second step, we use the Brain Extraction Tool (BET) to remove all non-brain tissue. In the third step, two highly experienced radiologists place seed points to mark the location and approximate extent of each T2w lesion. In the last step, we process the seed points by a customized Parzen windows classifier to estimate the intensity distribution of the lesions. We then reduce false positives by using connected component and shape analyses. 14  3.1. Lesion dispersion measures Full details on the automatic segmentation of T2w lesions are provided in [23]. The 3.0 mm slices of the binary images are divided into 1.0 mm slices to remove the effect of voxel anisotropy in the lesion dispersion measurements, resulting in images with dimensions of 256 × 256 × 150. This study has been done in two main steps: 1. Calculating the lesions dispersion for each patient. 2. Analyzing the results to find whether there is a relationship between dispersion and patient status and which descriptor measures the strongest independent contribution to disability.  3.1  Lesion dispersion measures  In this thesis, we investigate four different approaches to finding a descriptor for lesion dispersion.  3.1.1  Connectedness-based approach  Compactness We use a current method for calculating the compactness which was developed by Bribiesca [6]. This measure depends in large part on the sum of the contact surface areas of the face-connected voxels of 3D shapes. To quantify the connectedness of shapes composed of cubic voxels, Bribiesca [6] mathematically defines compactness as follows: C=  n − A/6 √ n − ( 3 n)2  (3.1)  where A corresponds to the total area of the externally visible faces of the solid and n is the total number of voxels. Intuitively, as a shape becomes less compact, there are fewer connections between voxels, and A increases, causing C to decrease. 15  3.1. Lesion dispersion measures  Figure 3.1: Examples of different shapes with 27 voxels and their corresponding compactness [6]; (a) C = 1, (b) C = 0.555, (c) C = 0.444, (d) C = 0. As the number of connected faces decreases, so does the compactness. This measure is independent of the volume of the object and does not change by changing the size of the object. The main advantages of compactness are its ease of computation for voxel data and having a range between 0 and 1, thereby removing the need for any external normalization factor. Its main limitation is that distances between voxels are not modelled. Figure 3.1 shows an example of different shapes and their corresponding compactness and how the compactness decreases when the number of connected faces decreases. We use this measure to quantify how compacted the lesion voxels are, regardless of their size or distances between lesions. From the point of studying lesion dispersion, the assumption is that when lesions are more compacted or the connectedness between lesions is high, they are less dispersed.  3.1.2  Region-based approach  In this approach we estimate the region impacted by computing the lesion minimum bounding sphere for one measure and lesion convex hull for another. The rationale for the measures in this approach is that using the lesions to form a sphere or a convex hull defines a region that is more likely to be impacted by the visible damage than the areas outside of the sphere or the convex hull.  16  3.1. Lesion dispersion measures Ratio of minimum bounding spheres For each patient, we compute the smallest sphere that contains all of the lesion voxels and the smallest sphere containing all of the brain voxels. Then we use the volume ratio of the lesion sphere to the brain sphere as a measurement of lesion dispersion. The volume of the brain sphere acts as a normalization factor. The spherical shape is arbitrary chosen to test the feasibility of the approach. Ratio of convex hull volume to brain volume The motivation for using this measure is to determine if a more accurate regional representation of the lesions and brain would yield stronger results than the ratio of minimum bounding spheres. The convex hull of a set of points, S, is the unique convex polyhedron that contains S and all of whose vertices are points from S. More details on properties of the convex hull and different algorithms for computing the convex hull are provided in [10]. In this study, we use a MATLAB function (convexHull) to calculate the convex hull of lesion voxels. This function is based on the Quickhull algorithm which is provided in [2]. Then we use the ratio of the convex hull volume to the intradural volume (computed using a method based on [20]) as a measurement of lesion dispersion. The intradural volume acts as a normalization factor in this measurement.  3.1.3  Distance-based approach  Euclidean distance from a fixed reference point To quantify lesion dispersion using a distribution of distances, we compute the variance, histogram entropy and skewness of the 3D Euclidean distances between the lesion voxels and a fixed reference point. The variance is computed from the distances directly, whereas the entropy and skewness are computed from a histogram of the distances. The variance of distances indicates how the 17  3.1. Lesion dispersion measures  Figure 3.2: Different reference points used for dispersion measurement. The lowest indicated point in the right image (the center point on the largest slice projected onto the most inferior slice) yields the strongest correlation to EDSS in the first data set. lesions are distributed in terms of distance from the reference point. Skewness measures the asymmetry of the distribution. A negative skew indicates that the bulk of the values lie to the right of the mean and a positive skew shows that the bulk of the values lie to the left of the mean. In our application, negative skewness indicates that the distances tend to be larger than the mean of the distances and positive skewness indicates that the distances tend to be smaller than the mean. We use skewness to determine if asymmetry of the distribution, as a characteristic of dispersion, has a relationship to clinical disability. Entropy is a statistical measure of randomness that can be used to measure the uncertainty of a random variable. In our application, smaller entropy indicates clustered distance values and larger entropy shows that the histogram of distance values has an evenly distributed model. We have tested a number of different reference points for our measurement, including the centroid of the brain, several extremal points, and points inbetween. Figure 3.2 illustrates different reference points. We observe that the results are dependent on the location of the reference point, and that the centre point of the brain defined on the largest slice, but projected onto the most inferior slice, yields the strongest correlations to EDSS in the development data set of 24 patients. This point is located near the brainstem which can be thought of as the trunk of many branching nerve fibres, and has high relevance for motor control. For the other data sets, we also use the same point as the reference point. 18  3.1. Lesion dispersion measures In order to account for natural variations in brain size among different patients, we apply principle component analysis (PCA) to the brain voxels to compute the anterior-posterior, left-right and superior-inferior axes for each patient. The maximum extent along each direction is then used to normalize the lesion distances along the same direction. Mathematically speaking, we use Equation 3.2 to compute the normalized distances:  dir =  xi − xr xb  2  +  yi − yr yb  2  +  zi − zr zb  2  (3.2)  where (xi , yi , zi ) and (xr , yr , zr ) are the coordinates of the lesion voxel and the reference point, expressed in the coordinate system defined by the PCAcomputed axes. xb , yb , and zb are the maximum brain extents in the anteriorposterior, left-right and superior-inferior directions respectively. Pair-wise Euclidean distances analysis To have a distance-based measure that is independent of any reference points, we compute the pair-wise Euclidean distances of lesion voxels in each patient. For pair-wise distances, we also compute the variance, entropy and skewness of the distribution. In addition, we compute the mean of the pair-wise distances, which provides an approximation of how far lesion voxels are located from each other. Similarly to the fixed-point method, the distances are normalized to the brain extents of each patient.  3.1.4  Network-based approach  Small-worldness Another approach that we propose for the topological study of lesion dispersion is derived from network theory. We consider the lesions as nodes and demonstrate that they can form a small-world network. By definition, a small-world network has highly local clustered nodes but also long path lengths between 19  3.1. Lesion dispersion measures elements (clusters). By computing the small-worldness that captures the ratio of small distances to the large distances between nodes, we obtain a measure of dispersion. To study lesion dispersion using network theory, first we define nodes and edges and then form a lesion network in which every lesion area is represented by a node. 1. We define a network in which every lesion area is represented by a node. Due to computing limitations we cannot assign a node to each lesion voxel. Therefore, for each patient, we divide the image into cubic regions of 5 × 5 × 5 voxels. If there is a lesion voxel in a box, the center of that box is considered as a node. The relationships between nodes are represented by connecting edges. Each edge has a weight which indicates the strength of the connection. Here we use the Euclidean distance between two nodes as the weight of the connecting edge. 2. After defining the nodes, we build the minimum spanning tree (MST) of the nodes. In graph theory, a tree is an undirected graph in which one simple path connects any two nodes, which means all nodes are connected in a tree but there are no cycles. A minimum spanning tree is a tree that connects all the nodes together with the total weight (sum of the weights of the edges) smaller than or equal to that of every other possible tree. After computing the MST, we form an edge between every pair of nodes that have an Euclidean distance smaller than a threshold. We have chosen the threshold to maximize the number of networks that fit the small-worldness model in the development data. We use 1/5 of the mean of the maximum edge weights in the MSTs of all patients as the threshold. Using this threshold, 18 networks of the 24 lesion networks we defined for the patients in the first data set, can be considered as smallworld networks. After forming a lesion network for each patient, we use some concepts of 20  3.1. Lesion dispersion measures network analysis to study lesion dispersion. Below, we introduce these concepts including clustering coefficient, characteristic path length, and smallworldness. The clustering coefficient captures the connectivity between lesion voxels. The characteristic path length quantifies how far the lesion clusters are from each other. The small-worldness that is computed using the clustering coefficient along with the path length, can be a measure for dispersion, indicating how clustered the lesion voxels while considering how dispersed the lesion clusters are. Clustering coefficient: The clustering coefficient quantifies how clustered the nodes are in a network. For each node i, the clustering coefficient, ci is: ci =  2Ei ki (ki − 1)  (3.3)  where Ei is the number of existing edges between the neighbors of i, and ki is the number of neighbors of i. The clustering coefficient of the network, c, is the mean of the clustering coefficients of all nodes in the network [19]. By definition, a higher clustering coefficient, indicates more locally clustered nodes. The typical definition of a clustering coefficient does not take into account the distances between nodes. We hypothesize that distances may be informative, therefore we modify the clustering coefficient. In our modification we want to incorporate the Euclidean distances between nodes in quantifying how clustered the network is. For each node, we define cD i as the modified clustering coefficient. Di (3.4) cD i = Diall where Di is the sum of Euclidean distances between every connected pair of neighbors of the node i, and Diall is the sum of the Euclidean distances between all pairs of neighbors of node i, regardless whether they are directly connected to each other or not. Diall is used for normalization since it represents the sum 21  3.1. Lesion dispersion measures of distances if the neighbors of nodes i were fully connected to each other. Characteristic path length: The characteristic path length of the network, L, is the mean of the minimum path length over all node pairs. The minimum path length of two nodes is the minimum number of edges that need to be traversed when going from one node to the other. Small-worldness: To classify a network as small-world, we compare its clustering coefficient and characteristic path length to those of a random network. The random network is usually an Erd¨os-R´enyi random network with the same number of edges and nodes [21]. The clustering coefficient of an Erd¨os-R´enyi random network can be approximated by the edge density. The edge density of a network is the ratio of connections that exists to the number of potential connections of a network. The mean of minimum path length can be approximated by Equation 3.5 [19]. Lrand =  ln(n) ln(< k >)  (3.5)  where n is the network size, or number of nodes, and < k > is the expected value of the degree, across the network. To compare the clustering coefficients and minimum path lengths of a patient network with a random network, we define γp as the ratio of clustering coefficients and λp as the ratio of path lengths of these two networks: γp = and λp =  cp crand Lp Lrand  (3.6)  (3.7)  According to the definition of small-worldness, if γp 1 and λp ≥ 1, the network can be classified as a small-world network. Consequently, a quantita-  22  3.2. Statistical analysis of the contribution of dispersion to disability tive metric of small-worldness is defined as, S=  γp . λp  (3.8)  If S > 1, the network is a small-world network [19]. By computing the modified clustering coefficient and characteristic path length, we obtain small-worldness values for each patient. In our application, a greater small-worldness indicates that lesion voxels are locally clustered and also the distances between lesion clusters are bigger (i.e., the clusters are more dispersed).  3.2  Statistical analysis of the contribution of dispersion to disability  We analyze the results to investigate if there is a statistically significant relationship between lesion dispersion and disability and determine whether such dispersion has the potential to provide information that is additional to and independent from lesion volume. First, we compute Pearson and Spearman correlations to investigate the relationships between lesion dispersion, disability, and total lesion volume (normalized by intradural volume). The Pearson method assumes a linear relationship, whereas the Spearman method is a rank correlation that does not assume any particular type of relationship. The pvalues of both correlations are computed to test for statistical significance. Since we have three variables, in the next step, we use regression analysis to investigate whether there is a potentially meaningful relationship between lesion dispersion and disability, independent of total lesion volume. Linear regression analysis assumes that the dependent variable is a linear combination of the other variables, and it helps us understand how the typical value of the dependent variable (disability) changes when either one of the indepen23  3.2. Statistical analysis of the contribution of dispersion to disability dent variables (lesion dispersion or total lesion volume) is varied, while the other independent variable is held fixed [32]. We compute two multiple regressions: one predicting disability using only volume as the predictive variable and a second regression using both volume and dispersion as the predictive variables. After constructing regression models, the statistical significance of the estimated parameters is checked by an F-test of the overall fit. In addition to studying the cross-sectional correlations in all three data sets, we perform a longitudinal study on the third data set over a period of two years to investigate the change in lesion dispersion across time and its correlation with the change in MSFC. The longitudinal MSFC scores and white matter lesions are provided for 182 patients of the third data set, and therefore we are able to track the changes of lesion dispersion and clinical status of the same patients after two years. We first compute the lesion dispersion measures for the patients after two years, and do a cross-sectional correlation study on obtained values, then we calculate the difference of these measures and baseline measures and compute the correlation values between the changes of lesion dispersion and changes of MSFC scores and normalized total lesion volume.  24  Chapter 4 Results 4.1  Total lesion volume normalized by intradural volume  As discussed in Chapter 2, measurement of the total white matter lesion volume on magnetic resonance images is a widely used outcome measure for monitoring the pathological state and progression of multiple sclerosis. To establish a baseline of clinical significance, we first analyze the relationship between total lesion volume, normalized by intradural volume to minimize the influence of head size, and patient disability in each data sample. In the first patient sample with 24 patients, this measure has a mean of 0.03, standard deviation of 0.03, and range of 2.8 × 10−4 −0.14. Figure 4.1 shows the relationship between total lesion volume and EDSS. Each point represents a patient in this graph and the line is the best fit to the data given by the linear regression of EDSS on total lesion volume. The Pearson and Spearman correlations between EDSS and volume are 0.47 (p = 0.02) and 0.44 (p = 0.02), which are well within the range of published values [3]. The results indicate that the EDSS has a significant linear relationship (p < 0.05) with volume. In the second and larger patient sample with 174 patients, normalized total lesion volume has a mean of 0.008, standard deviation of 0.009, and range of 1 × 10−4 − 0.078. The Pearson and Spearman correlation coefficients between volume and EDSS are 0.14 (p = 0.05) and 0.13 (p = 0.08). The correlation values are not as strong as the values for the smaller data set and only tend toward statistical significance. The normalized total lesion volume in the third  25  4.2. Connectedness-based approach  9 8 7  EDSS  6 5 4 3 2 1  0  0.02  0.04  0.06  0.08  0.1  0.12  0.14  0.16  0.18  Normalized Total Lesion Volume  Figure 4.1: Relationship between total Lesion Volume (Normalized by Intradural Volume) and EDSS in the first patient sample. patient sample with 182 patients, has a mean of 1.4 × 10−2 , standard deviation of 1.4 × 10−2 , and range of 1 × 10−4 − 7 × 10−2 . The Pearson and Spearman correlation coefficients between volume and MSFC are -0.25 (p = 5 × 10−4 ) and -0.39 (p = 3 × 10−8 ) which show that there is a linear relationship between total lesion volume and MSFC (p < 0.01). The correlation values are stronger than the correlation values between total lesion volume in T2-weighted scans and EDSS in the second patient sample. The correlation values in all the first and third patient samples confirm that there is a linear relationship between total lesion volume and clinical status score.  4.2 4.2.1  Connectedness-based approach Compactness  To quantify the strength of the relationships between compactness, EDSS and lesion volume, we calculate the correlations between compactness and clinical status score, and compactness and total lesion volume, and the significance of 26  4.2. Connectedness-based approach adding compactness to the linear regression model of clinical status score on total lesion volume. For the first patient sample, we compute the correlations between compactness and EDSS (r = 0.45 for Pearson correlation, p = 0.02), and compactness and total lesion volume (r = 0.71 for Pearson correlation, p = 7.3 × 10−5 ). Detailed results are provided in Table 4.1. The Pearson correlation between EDSS and compactness is significant and comparable to that between EDSS and volume (r = 0.45 vs. r = 0.47), and shows that patients with lower compactness (i.e., more disconnectedness) tend to have more disability. However, the correlation between compactness and volume is high, and statistically significant, which means that in terms of a linear relationship, these two variables seem to be strongly dependent. Adding compactness to the linear regression model of EDSS on volume is not statistically significant (p = 0.51). Therefore, there does not seem to be a linear relationship between compactness and EDSS that is independent of volume. In the second patient sample, the Pearson correlation value between compactness and EDSS is 0.07 (p = 0.33), and the Pearson correlation value between compactness and total lesion volume is 0.59 (p = 0.25). Table 4.2 contains the correlation values for this data set. Again, it is not statistically significant to add compactness to the linear regression model of EDSS on lesion volume. These results from the larger patient sample also suggest that there is no linear relationship between compactness and EDSS independent of total lesion volume. In the third patient sample, we calculate the correlations between compactness and MSFC (r = −0.27 for Pearson correlation, p = 1 × 10−4 ), and compactness and total lesion volume (r = 0.57 for Pearson correlation, p = 3 × 10−17 ). In this data set, adding compactness to the linear regression model of MSFC on total lesion volume is statistically significant (p < 0.01). The correlation values suggest that lesion compactness and MSFC are related, i.e. patients with less lesion compactness, have lower MSFC which indicates worse disability (Table 4.3). Interestingly, as shown in Table 4.6, compactness is strongly correlated with the cognitive component of MSFC (r = −0.40 for Pearson  27  4.3. Region-based approach correlation, p = 2 × 10−8 ) independent of volume (p = 0.0001). However it is not correlated to the leg function and hand function components of MSFC, independent of volume (Table 4.4 and Table 4.5).  4.3 4.3.1  Region-based approach Ratio of minimum bounding sphere (RMBS) and ratio of convex hull (RCH) volume to the brain volume  As shown in Table 4.1, in the first patient sample, the ratio of minimum bounding spheres (RMBS) is not significantly correlated with EDSS (r = 0.33 for Pearson correlation, p = 0.10), but the ratio of lesion convex hull volume over brain volume (RCH) is correlated with EDSS (r = 0.49 for Pearson correlation, p = 0.01). Again, like compactness, the correlations between total lesion volume and both RMBS and RCH are strong (for Pearson correlation r = 0.42, p = 0.03 for RMBS and r = 0.77, p = 3 × 10−6 for RCH). Adding RCH to the linear model of regression of EDSS on total lesion volume is not statistically significant (p = 0.29). As a result, even though the correlation between RCH and EDSS is significant, adding RCH to the linear regression model of EDSS on volume is not (p = 0.29), and we cannot conclude that RCH is informative about MS disability independent of total lesion volume. For the second patient sample, as shown in Table 4.2, RMBS and RCH are not correlated with EDSS (for Pearson correlation r = 0.03, p = 0.65 for RMBS, and r = 0.09, p = 0.22 for RCH) but are correlated with total lesion volume (for Pearson correlation r = 0.31, p = 2 × 10−5 and r = 0.76, p = 5 × 10−35 ), and it is not statistically significant to add RMBS or RCH to the linear regression of EDSS on total lesion volume (p = 0.97 for RMBS and p = 0.87 for RCH). The results illustrate that there is no linear relationship between RMBS or RCH and EDSS independent of total lesion volume. In 28  4.4. Distance-based approach the third patient sample, Pearson correlation coefficients do not show any relationship between RMBS and MSFC (r = −0.05 , p = 0.42) and RCH and MSFC (r = −0.13 , p = 0.07). But Spearman’s correlation coefficient shows that there is a statistically significant relation between RCH and MSFC (r = −0.21, p = 0.004). The correlation value of RMBS and RCH with total lesion volume (for Pearson correlation r = 0.40, p = 2 × 10−8 for RMBS and r = 0.75, p = 1 × 10−34 for RCH) also confirms the existence of a significant relationship of RMBS and RCH to lesion volume. Adding RMBS or RCH to the linear regression of MSFC on lesion volume is not significant (p = 0.12 for RMBS and p = 0.34 for RCH) which means that neither RMBS nor RCH is providing information about MS patient clinical status, in addition to what total lesion volume is providing.  4.4 4.4.1  Distance-based approach Euclidean distance (ED) from a fixed reference point  Table 4.1 contains the correlation coefficients and p-values that relate EDSS to the variance of Euclidean distance (VED), skewness and entropy of the histogram of Euclidean distance (SHED and EHED) and total lesion volume (V) for the first patient sample. The results show that the EDSS values are significantly correlated with VED (r = 0.57 for Pearson correlation, p = 0.003) and EHED (r = 0.54 for Pearson correlation, p = 0.006), with both correlations being higher than the volume-EDSS correlation (r = 0.47 for Pearson correlation, p = 0.02). In addition, V and VED are not correlated (r = 0.11 for Pearson correlation, p = 0.59) which means these variables are independent for this data set. However, EHED is correlated with V (r = 0.44 for Pearson correlation, p = 0.02). More interestingly, the p-values from the regression analysis show that adding VED to the regression model of EDSS on V  29  4.4. Distance-based approach is statistically significant (p = 0.02), meaning that EDSS and VED are significantly related even after adjusting for V. The same observation can be made for EHED since adding it to the regression model is statistically significant (p < 0.05). The graph in Figure 4.2a illustrates the approximate linear relationship between EDSS and VED in the first patient sample and the diagram in Figure 4.2b shows the relationship between EDSS, volume and VED values using a range of colours (from dark blue, which corresponds to 1.5, to brown, which corresponds to 8). The color of each point represents the EDSS score of that patient. For improved visualization of the overall trends, we interpolate the EDSS values and display them as a color grid in the background. Figure 4.2b shows that for the same volume range, EDSS generally increases with lesion dispersion.  0.18  Normalized Total Lesion Volume  8  7  EDSS  6  5  4  3  2  7.5  0.16  7 0.14  6.5  0.12  6 5.5  0.1  5 0.08  4.5  0.06  4  0.04  3.5 3  0.02 2.5  1  0  50  100  150  200  250  300  Variance of Euclidean Distances from a Fixed Point  (a)  0  0  50  100  150  200  250  300  Variance of Euclidean Distances from a Fixed Point  (b)  Figure 4.2: Relationship between volume, variance of Euclidean distances from a fixed point (VED), and EDSS in the first patient sample (24 patients); a) approximate linear relationship between EDSS and VED. b) EDSS values are shown using a range of colors (dark blue and brown correspond to 1.5 and 8, respectively). For the same volume range, EDSS generally increases as VED increases.  30  4.4. Distance-based approach The correlation values of EDSS with total lesion volume, VED, SHED, and EHED for the second patient sample are provided in Table 4.2. The EDSS values are significantly correlated with VED (r = 0.29 for Pearson correlation, p = 9 × 10−5 ) in this data set as well. On the other hand, VED and total lesion volume are significantly correlated (r = 0.20 for Pearson correlation, p = 0.007). However, adding VED to the linear regression model of EDSS on total lesion volume is also statistically significant (p = 0.0001) which means VED provides new information about the patients, clinical status in addition to what total lesion volume provides. The results show that SHED values are correlated with EDSS (r = −0.17 for Pearson correlation, p = 0.01) and total lesion volume (r = −0.24 for Pearson correlation, p = 0.001) and it is statistically significant to add SHED to the linear regression model of EDSS on total lesion volume (p = 0.02). The EHED and EDSS values do not have a significant correlation (r = 0.11 for Pearson correlation, p = 0.11). In third patient sample, as shown in Table 4.3, the Pearson correlation values indicate a linear relationship between VED and MSFC (r = −0.16, p = 0.03). In addition, there is no linear relationship between VED and total lesion volume (r = 0.09 for Pearson correlation, p = 0.18), and adding VED to the linear regression model of MSFC on total lesion volume is statistically significant (p = 0.02). The correlation values between VED, patient clinical status, and total lesion volume in this data set are consistent with observations from the two previous data sets. SHED and EHED do not have a correlation with MSFC (for Pearson correlation r = 0.03, p = 0.06 for SHED and r = −0.12, p = 0.10 for EHED) but they are correlated with total lesion volume (for Pearson correlation r = −0.19, p = 0.009 for SHED and r = 0.26, p = 1 × 10−4 for EHED). Examining the individual components of the MSFC, VED has a significant correlation with the leg function component (r = −0.17 for Pearson correlation, p = 0.01) and hand function component (r = −0.14 for Pearson correlation, p = 0.04) of MSFC while it does not correlate with the cognitive component (r = −0.01 for Pearson correlation, p = 0.87).  31  4.4. Distance-based approach  9  Normalized Total Lesion Volume  0.035  8  EDSS  7  6  5  4  3  2  0  0.005  0.01  0.015  0.02  0.025  Variance of Euclidean Distances from a Fixed Point  (a)  8 0.03 7 0.025 6 0.02 5  0.015  4  0.01  3  0.005  0  0  0.005  0.01  0.015  0.02  0.025  2  Variances of Euclidean Distances from a Fixed Point  (b)  Figure 4.3: Relationship between volume, variance of Euclidean distances from a fixed point (VED), and EDSS in the second patient sample (174 patients); a) approximate linear relationship between EDSS and VED. b) EDSS values are shown using a range of colors (dark blue and brown correspond to 2.5 and 8.5, respectively). For the same volume range, EDSS generally increases as VED increases.  4.4.2  Pair-wise Euclidean distances (PWED) analysis  Table 4.1 contains the correlation coefficients and p-values that relate EDSS to the mean and variance of pair-wise Euclidean distances (MPWED and VPWED), skewness and entropy of the histogram of the pair-wise Euclidean distances (SHPWED and EHPWED) and total lesion volume (V) in first data set. The EDSS values are correlated with MPWED (r = 0.47 for Pearson correlation, p = 0.01) and EHPWED (r = −0.45 for Pearson correlation, p = 0.02). The MPWED correlation is comparable to the volume-EDSS correlation. However, volume is correlated to MPWED (r = 0.44 for Pearson correlation, p = 0.02) and adding MPWED to the linear regression model of EDSS on volume is not statistically significant (p = 0.08), which means this variable is not independent of volume for this data set.  32  4.5. Network-based approach In the second patient sample, similar to the first sample, according to Pearson correlation values, MPWED and EDSS are linearly correlated (r = 0.20 for Pearson correlation, p = 0.006). MPWED and total lesion volume are correlated as well (r = 0.34 for Pearson correlation, p = 3 × 10−36 ), but the pvalues from the regression analysis confirm that adding MPWED (p = 0.007) or SHPWED (p = 0.003) provides increased predictive value. However the EDSS-MPWED correlation is not as strong as the EDSS-VED correlation (r = 0.29 for Pearson correlation, p = 9 × 10−5 ). The correlation between SHPWED and EDSS (r = −0.21 for Pearson correlation, p = 0.004) is stronger than that between SHED and EDSS (r = −0.17 for Pearson correlation, p = 0.01). Similarly to SHED, SHPWED is also correlated with volume (r = −0.31 for Pearson correlation, p = 2 × 10−5 ). In the third patient sample, there is no significant correlation between MPWED and MSFC (r = −0.07 for Pearson correlation, p = 0.28), but VPWED and MSFC are correlated (r = 0.17 for Pearson correlation, p = 0.01), and adding VPWED to the linear regression model of MSFC on total lesion volume is statistically significant (p = 0.01). However, in this data set MPWED values do not provide meaningful information about patient clinical status, unlike the two previous data sets. The VPWED values yield predictive information about patient clinical status scores, which shows that in general, pair-wise Euclidean distances between lesion voxels hold some promise to describe patient clinical status.  4.5 4.5.1  Network-based approach Small-worldness  Table 4.1 shows the correlation values and p-values that relate EDSS scores to normalized total lesion volume and small-worldness in the first data sample (24 patients). Small-worldness values are correlated with EDSS scores (r = 0.49 for Pearson correlations, p = 0.01), but are also significantly corre33  4.6. Longitudinal study lated with normalized total lesion volume (r = 0.80 for Pearson correlations, 1 × 10−6 ) and it is not statistically significant to add small-worldness to the linear regression of EDSS on total lesion volume.  4.6  Longitudinal study  Table 4.7 contains the cross-sectional correlation values between MSFC, normalized total lesion volume, and lesion dispersion measures of the 182 patients after two years in the third data set which are in general the same as those of the baseline data set (Table 4.3). Table 4.8 illustrates the correlation values between the changes of MSFC scores and changes of total lesion volume and lesion dispersion measures of 182 patients in third data set, over two years. The changes in dispersion measures do not show any significant correlations with the changes in MSFC.  34  Table 4.1: The first sample of 24 patients (13 RRMS, 11 SPMS). Correlation values between EDSS, total lesion volume and lesion dispersion. correlation with EDSS Measures V C RMBS RCH VED SHED EHED MPWED VPWED SHPWED EHPWED S  correlation with V  Pearson  Spearman  Pearson  Spearman  r = 0.47 p = 0.02 r = 0.45 p = 0.02 r = 0.33 p = 0.10 r = 0.49 p = 0.01 r = 0.57 p = 0.003 r = -0.48 p = 0.01 r = 0.54 p = 0.006 r = 0.47 p = 0.01 r = −0.28 p = 0.17 r = −0.29 p = 0.15 r = -0.45 p = 0.02 r = 0.49 p = 0.01  r = 0.44 p = 0.02 r = 0.35 p = 0.08 r = 0.37 p = 0.07 r = 0.44 p = 0.02 r = 0.57 p = 0.003 r = −0.38 p = 0.06 r = 0.39 p = 0.05 r = 0.52 p = 0.008 r = −0.20 p = 0.33 r = −0.25 p = 0.22 r = -0.40 p = 0.04 r = 0.45 p = 0.02  -  -  r = 0.71 p = 7.3 × 10−5 r = 0.42 p = 0.03 r = 0.77 p = 9 × 10−6 r = 0.11 p = 0.59 r = -0.42 p = 0.04 r = 0.44 p = 0.02 r = 0.44 p = 0.02 r = −0.32 p = 0.12 r = −0.31 p = 0.14 r = -0.62 p = 0.001 r = 0.80 p = 1 × 10−6  r = 0.82 p = 2 × 10−6 r = 0.66 p = 5 × 10−4 r = 0.81 p = 2 × 10−6 r = 0.25 p = 0.23 r = −0.33 p = 0.11 r = 0.58 p = 0.02 r = 0.50 p = 0.01 r = -0.43 p = 0.03 r = -0.65 p = 7 × 10−4 r = -0.75 p = 2 × 10−5 r = 0.75 p = 3 × 10−6  Significance of adding the measure to the linear regression of E on V p = 0.45 p = 0.55 p = 0.29 p = 0.0004 p = 0.06 p = 0.02 p = 0.08 p = 0.57 p = 0.50 p = 0.31 p = 0.18  E: EDSS, V: total lesion volume, C: compactness, RMBS: ratio of minimum bounding spheres, RCH: ratio of lesion convex hull to brain volume, VED, SHED, EHED: variance, skewness and entropy of the distribution of Euclidean distances from a fixed point, MPWED, VPWED, SHPWED, EHPWED: mean, variance, skewness and entropy of pair-wise Euclidean distances between lesion voxels, S: small-worldness.  35  Table 4.2: The second patient sample of 174 SPMS patients. Correlation values between EDSS, lesion volume and dispersion. correlation with EDSS Measures V C RMBS RCH VED SHED EHED MPWED VPWED SHPWED EHPWED  correlation with V  Pearson  Spearman  Pearson  Spearman  r = 0.14 p = 0.05 r = 0.07 p = 0.33 r = 0.03 p = 0.65 r = 0.09 p = 0.22 r = 0.29 p = 9 × 10−5 r = -0.17 p = 0.01 r = 0.11 p = 0.11 r = 0.20 p = 0.006 r = 0.01 p = 0.85 r = -0.21 p = 0.004 r = -0.07 p = 0.34  r = 0.13 p = 0.08 r = 0.08 p = 0.25 r = -0.003 p = 0.96 r = 0.06 p = 0.36 r = 0.28 p = 1 × 10−4 r = -0.15 p = 0.04 r = -0.02 p = 0.77 r = 0.11 p = 0.12 r = 0.01 p = 0.84 r = -0.12 p = 0.10 r = -0.04 p = 0.56  -  -  r = 0.59 p = 8 × 10−18 r = 0.31 p = 2 × 10−5 r = 0.76 p = 5 × 10−35 r = 0.20 p = 0.007 r = -0.24 p = 0.001 r = 0.17 p = 0.01 r = 0.34 p = 3 × 10−36 r = -0.10 p = 0.17 r = -0.31 p = 2 × 10−5 r = -0.47 p = 4 × 10−11  r = 0.64 p = 1 × 10−21 r = 0.57 p = 1 × 10−12 r = 0.88 p = 1 × 10−36 r = 0.27 p = 2 × 10−4 r = -0.22 p = 0.003 r = 0.20 p = 0.006 r = 0.38 p = 1 × 10−7 r = -0.13 p = 0.07 r = -0.45 p = 4 × 10−10 r = -0.55 p = 2 × 10−15  Significance of adding the measure to the linear regression of E on V p = 0.96 p = 0.97 p = 0.87 p = 0.0001 p = 0.02 p = 0.20 p = 0.007 p = 0.86 p = 0.003 p = 0.99  E: EDSS, V: total lesion volume, C: compactness, RMBS: ratio of minimum bounding spheres, RCH: ratio of lesion convex hull to brain volume, VED, SHED, EHED: variance, skewness and entropy of the distribution of Euclidean distances from a fixed point, MPWED, VPWED, SHPWED, EHPWED: mean, variance, skewness and entropy of pair-wise Euclidean distances between lesion voxels.  36  Table 4.3: The third patient sample of 182 SPMS patients. Correlation values between MSFC, total lesion volume and lesion dispersion. correlation with MSFC Measures V C RMBS RCH VED SHED EHED MPWED VPWED SHPWED EHPWED  correlation with V  Pearson  Spearman  Pearson  Spearman  r = -0.25 p = 5 × 10−4 r = -0.27 p = 1 × 10−4 r = -0.05 p = 0.42 r = -0.13 p = 0.07 r = -0.16 p = 0.03 r = 0.03 p = 0.65 r = -0.12 p = 0.10 r = -0.07 p = 0.28 r = 0.17 p = 0.01 r = 0.09 p = 0.19 r = 0.09 p = 0.21  r = -0.39 p = 3 × 10−8 r = -0.37 p = 1 × 10−7 r = 0.01 p = 0.83 r = -0.21 p = 0.004 r = -0.13 p = 0.06 r = 0.01 p = 0.83 r = -0.09 p = 0.20 r = -0.08 p = 0.24 r = 0.16 p = 0.02 r = 0.17 p = 0.02 r = 0.10 p = 0.15  -  -  r = 0.57 p = 3 × 10−17 r = 0.40 p = 2 × 10−8 r = 0.75 p = 1 × 10−34 r = 0.09 p = 0.18 r = -0.19 p = 0.009 r = 0.26 p = 3 × 10−4 r = 0.35 p = 1 × 10−6 r = -0.11 p = 0.13 r = -0.31 p = 1 × 10−5 r = -0.49 p = 1 × 10−12  r = 0.6 p = 1 × 10−20 r = 0.64 p = 1 × 10−23 r = 0.84 p = 1 × 10−37 r = 0.24 p = 9 × 10−4 r = -0.20 p = 0.005 r = 0.29 p = 6 × 10−5 r = 0.45 p = 1 × 10−10 r = -0.14 p = 0.04 r = -0.46 p = 5 × 10−11 r = -0.58 p = 6 × 10−18  Significance of adding the measure to the linear regression of M on V p = 0.007 p = 0.12 p = 0.34 p = 0.02 p = 0.96 p = 0.52 p = 0.97 p = 0.01 p = 0.91 p = 0.83  M: MSFC, V: total lesion volume, C: compactness, RMBS: ratio of minimum bounding spheres, RCH: ratio of lesion convex hull to brain volume, VED, SHED, EHED: variance, skewness and entropy of the distribution of Euclidean distances from a fixed point, MPWED, VPWED, SHPWED, EHPWED: mean, variance, skewness and entropy of pair-wise Euclidean distances between lesion voxels.  37  Table 4.4: The third patient sample of 182 SPMS patients. Correlation values between the leg function component of MSFC, total lesion volume and lesion dispersion. Correlation with first component of MSFC Measures V C RMBS RCH VED SHED EHED MPWED VPWED SHPWED EHPWED  Pearson  Spearman  r = -0.24 p = 0.001 r = -0.23 p = 0.001 r = -0.02 p = 0.76 r = -0.16 p = 0.03 r = -0.17 p = 0.01 r = 0.06 p = 0.41 r = -0.12 p = 0.10 r = -0.12 p = 0.09 r = 0.07 p = 0.31 r = 0.13 p = 0.07 r = 0.12 p = 0.10  r = -0.28 p = 8 × 10−5 r = -0.23 p = 0.001 r = -0.14 p = 0.053 r = -0.17 p = 0.02 r = -0.20 p = 0.005 r = 0.05 p = 0.49 r = -0.08 p = 0.24 r = -0.12 p = 0.10 r = 0.03 p = 0.66 r = 0.15 p = 0.04 r = 0.15 p = 0.04  Significance of adding the measure to the linear regression of M1 on V p = 0.05 p = 0.31 p = 0.74 p = 0.01 p = 0.99 p = 0.53 p = 0.71 p = 0.63 p = 0.55 p = 0.99  M1: leg function component of MSFC, V: total lesion volume, C: compactness, RMBS: ratio of minimum bounding spheres, RCH: ratio of lesion convex hull to brain volume, VED, SHED, EHED: variance, skewness and entropy of the distribution of Euclidean distances from a fixed point, MPWED, VPWED, SHPWED, EHPWED: mean, variance, skewness and entropy of pair-wise Euclidean distances between lesion voxels.  38  Table 4.5: The third patient sample of 182 SPMS patients. Correlation values between the hand function component of MSFC, total lesion volume and lesion dispersion. Correlation with second component of MSFC Measures V C RMBS RCH VED SHED EHED MPWED VPWED SHPWED EHPWED  Pearson  Spearman  r = 0.03 p = 0.62 r = 0.02 p = 0.77 r = 0.03 p = 0.66 r = 0.01 p = 0.89 r = -0.14 p = 0.04 r = 0.01 p = 0.82 r = -0.007 p = 0.92 r = -0.02 p = 0.70 r = 0.04 p = 0.57 r = -0.008 p = 0.91 r = 0.03 p = 0.65  r = -0.12 p = 0.09 r = -0.09 p = 0.22 r = -0.05 p = 0.47 r = -0.07 p = 0.29 r = -0.06 p = 0.38 r = 0.05 p = 0.43 r = 0.01 p = 0.83 r = -0.01 p = 0.80 r = 0.01 p = 0.83 r = 0.09 p = 0.19 r = 0.07 p = 0.35  Significance of adding the measure to the linear regression of M2 on V p = 0.99 p = 0.92 p = 0.95 p = 0.01 p = 0.91 p = 0.95 p = 0.71 p = 0.68 p = 0.99 p = 0.57  M2: arm/hand function component of MSFC, V: total lesion volume, C: compactness, RMBS: ratio of minimum bounding spheres, RCH: ratio of lesion convex hull to brain volume, VED, SHED, EHED: variance, skewness and entropy of the distribution of Euclidean distances from a fixed point, MPWED, VPWED, SHPWED, EHPWED: mean, variance, skewness and entropy of pair-wise Euclidean distances between lesion voxels.  39  Table 4.6: The third patient sample of 182 SPMS patients. Correlation values between the cognitive component of MSFC, total lesion volume and lesion dispersion. Correlation with third component of MSFC Measures V C RMBS RCH VED SHED EHED MPWED VPWED SHPWED EHPWED  Pearson  Spearman  r = -0.37 p = 2 × 10−7 r = -0.40 p = 2 × 10−8 r = -0.12 p = 0.08 r = -0.15 p = 0.03 r = -0.01 p = 0.87 r = 0.02 p = 0.74 r = -0.13 p = 0.06 r = -0.02 p = 0.77 r = 0.26 p = 3 × 10−4 r = 0.09 p = 0.22 r = 0.06 p = 0.38  r = -0.39 p = 2 × 10−8 r = -0.40 p = 1 × 10−8 r = -0.21 p = 0.004 r = -0.21 p = 0.003 r = -0.08 p = 0.26 r = 0.01 p = 0.86 r = -0.11 p = 0.12 r = -0.01 p = 0.83 r = 0.29 p = 4 × 10−5 r = 0.12 p = 0.10 r = 0.05 p = 0.47  Significance of adding the measure to the linear regression of M3 on V p = 0.0001 p = 0.96 p = 0.003 p = 0.89 p = 0.58 p = 0.65 p = 0.05 p = 0.0001 p = 0.89 p = 0.02  M3: the cognitive component of MSFC, V: total lesion volume, C: compactness, RMBS: ratio of minimum bounding spheres, RCH: ratio of lesion convex hull to brain volume, VED, SHED, EHED: variance, skewness and entropy of the distribution of Euclidean distances from a fixed point, MPWED, VPWED, SHPWED, EHPWED: mean, variance, skewness and entropy of pair-wise Euclidean distances between lesion voxels.  40  Table 4.7: The third patient sample of 182 SPMS patients after two years. Correlation values between MSFC, total lesion volume and lesion dispersion. Correlation with MSFC Measures V C RMBS RCH VED MPWED VPWED  correlation with V  Pearson  Spearman  Pearson  Spearman  r = -0.23 p = 0.001 r = -0.22 p = 0.002 r = -0.14 p = 0.05 r = -0.12 p = 0.09 r = -0.13 p = 0.07 r = -0.15 p = 0.03 r = 0.06 p = 0.37  r = -0.29 p = 7 × 10−5 r = -0.29 p = 5 × 10−5 r = -0.16 p = 0.02 r = -0.11 p = 0.13 r = -0.16 p = 0.02 r = -0.12 p = 0.10 r = 0.05 p = 0.43  -  -  r = 0.58 p = 1 × 10−17 r = 0.43 p = 1 × 10−9 r = 0.81 p = 9 × 10−43 r = 0.16 p = 0.02 r = 0.32 p = 7 × 10−6 r = -0.12 p = 0.08  r = 0.64 p = 2 × 10−24 r = 0.60 p = 1 × 10−20 r = 0.86 p = 1 × 10−45 r = 0.29 p = 6 × 10−5 r = 0.43 p = 2 × 10−7 r = -0.15 p = 0.04  Significance of adding the measure to the linear regression of M on V p = 0.09 p = 0.70 p = 0.03 p = 0.18 p = 0.24 p = 0.77  M: MSFC, V: total lesion volume, C: compactness, RMBS: ratio of minimum bounding spheres, RCH: ratio of lesion convex hull to brain volume, VED, SHED, EHED: variance, skewness and entropy of the distribution of Euclidean distances from a fixed point, MPWED, VPWED, SHPWED, EHPWED: mean, variance, skewness and entropy of pair-wise Euclidean distances between lesion voxels.  41  Table 4.8: The third patient sample of 182 SPMS patients. Correlation values between difference of MSFC, difference of total lesion volume and difference of lesion dispersion. Correlation with MSFC Measures dV dC dRMBS dRCH dVED dMPWED dVPWED  correlation with V  Pearson  Spearman  Pearson  Spearman  r = -0.12 p = 0.09 r = -0.10 p = 0.14 r = -0.02 p = 0.78 r = 0.04 p = 0.54 r = -0.10 p = 0.18 r = 0.001 p = 0.98 r = 0.04 p = 0.51  r = -0.006 p = 0.93 r = -0.03 p = 0.66 r = 0.002 p = 0.97 r = 0.03 p = 0.66 r = -0.08 p = 0.28 r = -0.02 p = 0.72 r = 0.05 p = 0.46  -  -  r = 0.39 p = 5 × 10−8 r = 0.05 p = 0.48 r = 0.68 p = 5 × 10−26 r = 0.08 p = 0.24 r = -0.03 p = 0.68 r = -0.12 p = 0.08  r = 0.37 p = 4 × 10−7 r = 0.13 p = 0.07 r = 0.75 p = 2 × 10−30 r = 0.11 p = 0.14 r = -0.08 p = 0.26 r = -0.23 p = 0.001  Significance of adding the measure to the linear regression of M on V p = 0.47 p = 0.96 p = 0.97 p = 0.24 p = 0.99 p = 0.82  dM: difference of MSFC, dV: difference of total lesion volume, dC: difference of compactness, dRMBS: difference of ratio of minimum bounding spheres, dRCH: difference of ratio of lesion convex hull to brain volume, dVED, dSHED, dEHED: difference of variance, skewness and entropy of the distribution of Euclidean distances from a fixed point, dMPWED, dVPWED, dSHPWED, dEHPWED: difference of mean, variance, skewness and entropy of pair-wise Euclidean distances between lesion voxels.  42  Chapter 5 Conclusions 5.1  Discussion  In this thesis, we computed the spatial dispersion of lesions in the MRI scans of three sample groups of MS patients using different measures. The patient clinical status was provided in the form of EDSS for the first and the second data sets of 24 and 174 patients, and MSFC for the third data set of 182 patients. First, we developed methods for quantifying lesion dispersion in the first data sample. We used one connectedness-based measure (compactness), two region-based measures (ratio of minimum bounding spheres and ratio of lesion convex hull to brain volume), two distance-based measures (Euclidean distance from a fixed point, and pair-wise Euclidean distance), and one network theory based (small-worldness) measure. In the first patient set, we observed a significant correlation between the connectedness-based measure (compactness) and EDSS (r = 0.45 for Pearson’s correlation, p = 0.02). Also, compactness and total lesion volume correlate strongly (r = 0.71 for Pearson’s correlation, p = 7 × 10−5 ). When the lesion load increases in the constant and limited space of the brain, it increases the probability of lesions connecting, hence, increasing the compactness values. In region-based measures, the convex hull ratio correlates significantly with EDSS as well as total lesion volume. The fact that the lesion convex hull approximates the region impacted by lesions explains the significant correlation between this measure and volume. The strong correlations between compactness and volume and also between the convex hull ratio and volume explain why these two measures do not make an independent contribution to EDSS. 43  5.1. Discussion Among distance-based measures, the variance of Euclidean distances from a reference point correlates strongly with EDSS (r = 0.57 for Pearson’s correlation, p = 0.003) while not correlating with volume (r = 0.11 for Pearson’s correlation, p = 0.59). This indicates the potential of the variance for quantifying the dispersion. The more dispersed lesion voxels yield a greater variance of distances. Finally, we observed a significant correlation between small-worldness and EDSS and a significant correlation between small-worldness and volume that may result from the definition of our lesion network. A greater lesion load results in more connected nodes that increases the clustering coefficient and hence increases small-worldness values. In the next step, we used the connectedness-based, region-based, and distance-based measures to compute lesion dispersion in the second data set in which we selected patients randomly from a clinical trial. The statistical analysis of the results on this data set partially confirms the results from the first data set. The main difference is that the connectedness-based and the region-based measures do not correlate with EDSS in this data set. In addition, the correlation values in the second patient samples are generally weaker than those in the first data set. The difference between the distribution of EDSS scores in the first and the second data sets may explain these observations. Unlike the well-distributed EDSS scores in the first data set, the EDSS scores in the second data set are highly clustered with 43% of values being 6 or 6.5. The lack of correlation between connectedness-based or region-based measures and EDSS indicates that these two approaches may not be applicable for this type of patient population. Finally, we computed the lesion dispersion in a third patient sample using the same measures that we used in the second data set. In this data set, we selected patients randomly from the same clinical trial used for the second data set. The clinical status in the third data set was provided in the form of MSFC scores. We tested this data set to investigate the effect of the clinical status score on our findings from the first and second data sets. The results  44  5.2. Conclusions from the third data set generally confirms the observations from two other data sets. There is no significant correlation between region-based measures and MSFC. But the variance of Euclidean distances from a reference point correlates with MSFC (r = −0.16 for Pearson’s correlation, p = 0.03), the leg function component of MSFC (r = −0.17 for Pearson’s correlation, p = 0.01), and the hand function component of MSFC (r = −0.14 for Pearson’s correlation, p = 0.04). There is a particularly notable result in the third data set that was not found in the first two, and that is the compactness correlates significantly with MSFC (r = −0.27 for Pearson’s correlation, p = 1 × 10−4 ) and particularly with the cognitive component of MSFC (r = −0.40 for Pearson’s correlation, p = 2 × 10−8 ). This observation suggests that the compactness may potentially act as a biomarker for cognitive dysfunction. Overall, we have found that one distance-based measure, variance of Euclidean distances from a fixed point, consistently correlates with the disability score across all three data sets. The variance is only one of three components of the fixed point approach, which may raise the concern of a statistical chance finding, resulting from multiple comparisons. If we apply Bonferroni correction to the third approach which have three related tests, we obtain a new significance value of (0.05/3 = 0.016). The VED yielded a p-value of 0.0004 in the first data set, a p-value of 0.0001 in the second data set, a p-value of 0.02 in the third data set, and a p-value of 0.01 for the leg function and hand function components of MSFC in the third data set. The p-values are well below the corrected threshold in the first two data sets and below the threshold for the physical components of MSFC in the third data set. This may indicate that VED is particularly effective for predicting physical deficits.  5.2  Conclusions  To the best of our knowledge, we are the first to study the contribution of spatial lesion dispersion to MS disability, independent of volume. Up to this  45  5.3. Future work point, we have found and observed that: 1. There exists a potentially meaningful correlation between patient disability and measurements of lesion dispersion that we found by comparing the lesion dispersion values to clinical status scores and total lesion volume. 2. Some distance-based measures are shown to provide new information about the severity of MS that remains independent from and potentially more sensitive than total lesion volume. In particular, VED seems to the yield the strongest and most consistent results when considering all three data set. 3. The distance factor plays a more important role for describing the lesion dispersion in MS patients compared to other approaches, at least for physical disabilities. 4. The lesion dispersion measure based on connectedness may potentially be a biomarker for cognitive impairment in MS patients.  5.3  Future work  The results presented in this thesis provide support for our hypothesis that for any two given patients with similar lesion loads, the one with greater dispersion will tend to have greater disability. However, further investigation will determine why some dispersion measures agree with clinical status better than others. To expand the study of the proposed hypothesis and investigate the existence of a relationship between lesion dispersion and disability, we may benefit from investigating T1 black hole lesion dispersion. We developed our measures with the preliminary data set of 24 patients, and then used these techniques for measuring the dispersion in the other two 46  5.3. Future work data sets. Therefore, the current reference point that we use for computing the variance of distances may be effective for physical disabilities, but potentially less sensitive for cognitive ones. Varying the reference points for this measure may improve the correlation values between this measure and the cognitive component of MSFC in the third data set. In addition, the parameters we used to form the lesion networks for measuring small-worldness are optimized for the first data set. These parameters may not be the best for other data sets. We have found that our preliminary investigations for computing lesion dispersion using a measure derived from network theory to be promising. The main weak point of this approach is the strong correlation with volume. We will do more investigation on this approach to improve the measures. We expect incorporating diffusion tensor imaging to build a network and observing the disruption of the network by the lesions will reveal some information about lesion dispersion and its correlation with disability in MS patients. In our statistical analysis, we only used the three variables of lesion dispersion, lesion volume, and disability. In order to improve the results of this work we will use statistical analysis that controls for other clinically relevant factors such as age, gender and disease duration. In addition, rather than trying to select one optimal measure, we can explore combinations of measures to find a potential composite biomarker for MS.  47  Bibliography [1] P. A. Arnett, S. M. Rao, L. Bernardin, J. Grafman, F. Z. Yetkin, and L. Lobeck. Relationship between frontal lobe lesions and Wisconsin card sorting test performance in patients with multiple sclerosis. Neurology, 44:420–425, 1994. → pages 9 [2] C. B. Barber, D. P. Dobkin, and H. Huhdanpaa. 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