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Fractal analysis of MRI data for the characterization of patients with schizophrenia and bipolar disorder

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Institute of Physics and Engineering in Medicine Phys. Med. Biol. 60 (2015) 1697–1716

Physics in Medicine & Biology doi:10.1088/0031-9155/60/4/1697

Fractal analysis of MRI data for the characterization of patients with schizophrenia and bipolar disorder Letizia Squarcina1, Alberto De Luca2,3, Marcella Bellani1, Paolo Brambilla4,5, Federico E Turkheimer6 and Alessandra Bertoldo2 1

  Department of Public Health and Community Medicine, Section of Psychiatry and Section of Clinical Psychology, InterUniversity Centre for Behavioural Neurosciences, University of Verona, Verona, Italy 2   Department of Information Engineering (DEI), University of Padova, Padova, Italy 3   Department of Neuroimaging, Scientific Institute, IRCCS ‘Eugenio Medea’, Bosisio Parini, LC, Italy 4   Department of Experimental and Clinical Medical Sciences (DISM), InterUniversity Center for Behavioral Neurosciences, University of Udine, Udine, Italy 5   Department of Psychiatry and Behavioral Sciences, UTHouston Medical School, Houston, TX, USA 6   Department of Neuroimaging, Institute of Psychiatry, King’s College London, UK E-mail: [email protected] Received 23 October 2014, revised 18 December 2014 Accepted for publication 30 December 2014 Published 29 January 2015 Abstract

Fractal geometry can be used to analyze shape and patterns in brain images. With this study we use fractals to analyze T1 data of patients affected by schizophrenia or bipolar disorder, with the aim of distinguishing between healthy and pathological brains using the complexity of brain structure, in particular of grey matter, as a marker of disease. 39 healthy volunteers, 25 subjects affected by schizophrenia and 11 patients affected by bipolar disorder underwent an MRI session. We evaluated fractal dimension of the brain cortex and its substructures, calculated with an algorithm based on the box-count algorithm. We modified this algorithm, with the aim of avoiding the segmentation processing step and using all the information stored in the image grey levels. Moreover, to increase sensitivity to local structural changes, we computed a value of fractal dimension for each slice of the brain or of the particular structure. To have reference values in comparing healthy subjects with patients, we built a template by averaging fractal dimension values of the healthy volunteers data. Standard deviation was evaluated and used to create a confidence interval. We also performed a slice by slice t-test to assess the 0031-9155/15/041697+20$33.00  © 2015 Institute of Physics and Engineering in Medicine  Printed in the UK

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difference at slice level between the three groups. Consistent average fractal dimension values were found across all the structures in healthy controls, while in the pathological groups we found consistent differences, indicating a change in brain and structures complexity induced by these disorders. Keywords: automatic data processing, neuroimaging, brain morphology, magnetic resonance imaging, fractals (Some figures may appear in colour only in the online journal) 1. Introduction Bipolar disorder (BD) and schizophrenia (SCZ) are severe psychiatric disorders with partially overlapping symptoms (Craddock et al 2010). They also share modifications to brain structure in respect to healthy subjects: for example, in both diseases alterations have been found in thalamus, frontal and temporal cortex in magnetic resonance images (MRI) along with reduction in grey matter (GM) volume (Yu et al 2010, Anderson et al 2013), ventricular enlargement, (Shenton et al 2001, Cecil et al 2002, Yuksel et al 2012) and damage to corpus callosum (Brambilla et al 2003, 2005, Bellani et al 2009, Li et al 2013). Fractals are geometric objects introduced by Mandelbrot (1983), characterized by self-similarity (structure repetition at different scales) and internal homothety, which is a generalization of an affine space transformation which includes more alterations on vectors than the pure scaling of the modules of vectors, as rotations, translations and axial symmetries. Moreover, while objects defined in the Euclidean geometry have an integer topological dimension, fractals have a fractional dimension (FD), which is an index of structural complexity. To evaluate the FD of a fractal object, Mandelbrot (1983) introduced the box-count algorithm. The calculation of the FD of a 3D object consists, when applied to images, in covering the object of interest with squares (e.g. boxes) of increasing sizes and for each size, counting the number of needed boxes. Then the FD is obtained as the logarithm of the ratio between the number of boxes and size. In the latest decade the application of fractal methods has been increasing in the biomedical field, in particular in the analysis of anatomical images. Self-similarity and a fractal dimension are common findings in natural objects and there is evidence that, across a finite number of scales, the brain and its observable functional activity exhibits fractal behaviour. Power law scaling of synchronization metrics has been observed in normal volunteers under resting conditions in EEG, MEG and fMRI studies (LinkenkaerHansen et al 2001, Fraiman et al 2009, Benayoun et al 2010, Chialvo 2010, Expert et al 2011). Interestingly, the scaling of brain functional activity is correlated to behavioural scaling laws, an observation that links the ultimate emergence of brain complex output to the fractal dynamic of elementary neuronal assemblies (Palva et al 2013). Given the intrinsic link between brain function and structure and the fact that functional activity drives structural plasticity it is not surprising that cortical structure may present, to a degree, fractal properties. Kiselev et al (2003) analyzed brain MRI data of six healthy subjects and demonstrated fractal-like structure of grey matter (GM) in the range of millimetres. This result was confirmed also by Jiang et al (2008), who applied the box-count algorithm to the reconstructed 3D brain cortex of 57 healthy subjects and demonstrated fractal properties for the structure of the cortex, as well as its substructures, lobes and hemispheres; the computed FD was consistent across subjects and structures. Liu et al have demonstrated fractal geometry in cerebellum (Liu et al 2003). 1698

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An emerging view stemming from the observed self-similar nature of the central nervous system structure and activity is that the brain is a self-organized critical system (SOC, Fraiman et al 2009, Petermann et al 2009, Shew et al 2009) Systems in a critical state are poised on the cusp of a transition between ordered and random behaviour and demonstrate complex patterning of fluctuations at all scales of space and time. Critical brain dynamics exhibit an optimal dynamical range for information processing (Kinouchi and Copelli 2006, Beggs 2008) and are efficient in learning complex rules via plastic adaptation (De Arcangelis et al 2010). If the normal brain exhibits fractal like properties one should expect brain disorders to break self-similarity and be reflected in distorted fractal metrics. Indeed, fractal metrics have been demonstrated to be affected in neurological disorders such as multiple sclerosis (Esteban et al 2007, 2009) where 3D box-count method revealed differences in WM and GM of the pathologic population at early stages of disease. In post-stroke patients (Zhang et al 2008) those with impaired motor abilities were found to have lower FD values in the motor cortex. In the field of neurodegenerative disorders, King et al (2010) found significantly lower FD in 3D rendering from T1 images of the cortical ribbon of Alzheimer’s disease patients when compared with healthy subjects. Interestingly, fractal brain properties have also been demonstrated in psychiatric cohorts. The analysis of images of brain of patients affected by attention-deficit disorder (Li et al 2007), revealed lower complexity of the left prefrontal cortex respect to healthy controls. Sandu and colleagues (2008) found that whole brain FD of patients affected by schizophrenia was different from that computed in healthy volunteers. In particular, they showed how FD calculated at the boundary between segmented white and grey matter results higher in patients. Higher FD in schizophrenia patients was found also by Narr et al (2004), who considered the 3D surfaces of superior frontal, inferior frontal, temporal, parietal and occipital lobes of patients with a first episode of psychosis. Nonetheless, results in schizophrenia are not always in agreement: reduced FD was found in patients affected by either schizophrenia or obsessive–compulsive disorder (Ha et al 2005), when considering the cortical surface of the whole brain or of the two hemispheres. Even if the direction of change in FD values is not clear, these disorders functional disabilities are undoubtedly reflected into changes in shape and complexity of grey matter. In this work we wanted to assess the fractality of the brain of two psychiatric conditions, schizophrenia and bipolar disorder. The rationale for a change in the geometric properties of the brain in these two pathological populations stems from the current hypothesis on the aetiology of these two disorders. Both these disorders are strongly associated with the dysfunction of GABAergic neurons and a particular class of these, the Parvalbumin interneurons (Brambilla et al 2003, Benes et al 2007, 2009, Benes 2010, Marin 2012, Jiang et al 2013). Together with pyramidal neurons, interneurons constitute the basic cellular motif of the brain, entrain Gamma activity hence are key to brain processing (Borgers and Kopell 2003, Lord et al 2013); at the same time they have an equally relevant role in plasticity and development (Le Magueresse and Monyer 2013) Hence we posited that in these two disorders, GABA dysfunction would have altered the natural complexity of brain structure and cause a loss of fractality. Given the contrasting results in previous applications of fractal geometry to these cohorts, we paid particular attention to the validation of the analytics. It is to be noted that FD algorithms so far have produced a single FD value as an output for each subject. Since structural changes in the brain could be localized to specific anatomical areas, we considered that it would be more suitable to obtain more location-specific information, thus we focused on obtaining a different FD for each brain slice. Moreover, building on the work of Zook et al (2005) and Lv et al (2009), which combined the information stored in the grey levels of MR images with the fractal approach to improve the characterization of tissues, we developed a 1699

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technique based on box-counting to detect structural modifications in the brain without the need of segmenting the T1-weighted images into white matter (WM) and grey matter (GM). 2.  Materials and methods 2.1. Dataset

39 healthy volunteers (all Caucasians, mean age 28.4, ± 5,4 17 males, 22 females), 25 subjects affected by schizophrenia (all Caucasians, mean age 34,96 +/ −  7,69 years, 15 males, 10 females) and 11 patients affected by bipolar disorder (all Caucasians, mean age 41 +/ −  8,91, 5 males, 6 females) were recruited and gave their consent to the study. Patients were recruited from the South-verona psychiatric care register (PCR) (Tansella and Burti 2003), a community-based mental health register. The clinical diagnosis of bipolar disorder was established using the Italian version of the Structured clinical interview for DSM-IV (SCID I) (First et al 2000) and confirmed with the clinical consensus of two staff members. The studies were conducted according to the Declaration of Helsinki. 2.2.  MRI acquisition

MRI data were obtained using a Siemens 3.0 Tesla Magnetom Allegra MRI scanner (Siemens Ag). T1-weighted MPRAGE images were acquired with the following acquisition parameters: matrix size 256  ×  256; slice thickness = 1.0 mm, 0.94  ×  0.94 mm in plane resolution, 150 slices; TR = 2.3 ms; TE = 3.93 ms. 2.3.  Data processing 2.3.1. T1-weighted pre-processing.  T1-weighted data were pre-processed using FSL (Jenkinson et al 2012, Smith et al 2004, Woolrich et al 2009). Images were skull-stripped, segmented into GM, WM and cerebro-spinal fluid (CSF) and registered to the MNI-152 T1weighted 1 mm3 template (Grabner et al 2006). Brain extraction was performed using FSL BET (Smith 2002). The resulting volumes were segmented using FSL FAST (Zhang et al 2001). T1-weighted images and their segmentations were then registered to the MNI 152 1 mm template using FSL FLIRT (Jenkinson et al 2002). The previously obtained WM and GM maps were also aligned to the MNI space using the linear transformation calculated for the T1-weighted map. This operation was performed to align all the subjects to the same spatial coordinates, allowing comparisons of slice-by-slice values. Alignment to the MNI template also allowed the consideration of some major brain structures, i.e. frontal lobe, occipital lobe, parietal lobe and temporal lobe obtained from the MNI structural atlas (Collins et al 1995, Mazziotta et al 2001). 2.3.2.  Box-count algorithm.  The standard box-count algorithm (Mandelbrot, 1983) defines a

kernel function H, called box, ⎧  1  if  (x − x 0 )2 + (y − y )2 + (z − z 0 )2 ≤ r 2 0 H (x, y, z, r ) = ⎨ ⎩ 0 elsewhere, 0 < x 0< w, 0 < y0 < h, 0 < z 0 0 ∼ . I (x, y, z ) = ⎨ ⎩ 0 if I (x, y, z ) = 0

(4)

A linear regression of the logarithm of the count on the inverse of the logarithm of the box size allows the estimation of FD: logN (r ) FD .  = rlim  → 0 − log (r )

(5)

We customized the algorithm with the aim of keeping the information stored in the intensity values of the image: the contribution of each voxel to the count of occurrences is weighted ∼ according to its intensity. Thus, the image I (i, j, k ) is re-defined as: ∼ I (x, y, z ) = max (I (x, y, z )) . (6) The original algorithm, for every box size r, adds 1 to the count of occurrences when a box contains at least one non-negative pixel (non-empty box), 0 otherwise. In our modified algorithm, the value 1 is replaced with the maximum intensity value inside the considered box. 2.4.  FD of T1-weighted images 2.4.1.  FD of grey matter.  FD was computed on the grey matter of whole brain and on some

major structures. In each case we performed the analysis computing a FD value for each slice of the considered volume. Two approaches have been used to compute the FD. The first approach, which we refer to as global, consists in computing one FD value for each slice by directly applying the box-count algorithm to the entire slice. The second, which we named local, consists in computing multiple FD values for every slice, then averaging them. FD values are computed inside square blocks, of size n × n voxels with n ∈ [15, 20, 25, 30, 35, 40, 50, 65] . The FD value is then assigned to the inner 5  ×  5 block. We introduced this subdivision and averaging procedure to increase sensitivity to local structural changes. A graphical representation of this method is depicted in figure 1. The parameter n is a key value for the algorithm. To assess the best value to be used in in our study, we decided to use the value that maximized the distance between a subset of 5 healthy subjects and 5 subjects affected by schizophrenia (see section 2.6). We defined the distance index d (a, b ) as: 1701

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Figure 1.  In our local box-count algorithm, FD is computed in boxes of size n  ×  n voxels, where the box-count algorithm is applied. The value is then assigned to the inner 5  ×  5 block. This is done to increase sensitivity to local structure changes.

∑ a (s) − b (s)

d (a, b ) = 

(7)

s

where a, b are the vectors containing the FD values of the slices of two selected subjects; and s is the slice number. We compared results obtained using the standard 2D box-count (Liu et al 2003, Zhang et al 2006) against those obtained using our customized (both local and global) versions. We binarized images before applying the standard box-count algorithm, while for our modified algorithm we obtained grey matter images multiplying the T1-weighted (T1w) images with ROIs grey matter masks. 2.4.2. FD of unsegmented images.  In order to verify the possibility of analyzing the T1-weighted images using fractal geometry without differentiating between white and grey matter, we considered FD values obtained with the log–log plot as in the work of Kiselev and colleagues. An object can be considered a fractal if the log–log plot of the occurrences versus the box-side size results in a straight line, whose slope is the FD of the object. We applied our customized box-count on each slice of the T1-weighted volumes of each healthy subject, then analyzed the log–log plot to demonstrate the fractality of the non-segmented brain cortex. We computed Pearson’s r correlation coefficient between the obtained line and the ideal straight line that connected the first two points. Moreover, we computed the linear regression line, extracted the residuals and averaged them; we expected their mean value to be near zero. Then, we applied both the local and the global algorithms, as described in section 2.4.1, to the selected structures. Also in this case, we evaluated the influence of the parameter n on the FD calculation, with the same procedure described above. 1702

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2.5.  Template creation

FD is not an absolute value and depends on many factors, as box size, size of the analyzed structure, image brightness: these factors may vary across subjects. Thus, only results obtained using the same approach and algorithm can be compared. An absolute value of FD for healthy structures does not exist: to overcome this problem we built a template (TMP) by averaging FD values of an age and sex-matched subset of the healthy volunteers data. Standard deviation (SD) was evaluated and used to create a confidence interval of 2SD around mean values. To evaluate the robustness of the method we used to create the template, we implemented a bootstrapped leave n out t-test, using n = 15, 25. In particular we generated 100.000 unique vectors dividing our healthy controls in training set and testing set, then for each axial slice and for each tested structure we performed a two sample t-test between the two groups and counted the number of statistically different slices. After this phase, we created a template using data from all subjects and used it as a reference against FD values of patients. 2.6.  Evaluation of FD in relation to pathology 2.6.1.  Analysis of distance index.  After the healthy template was calculated, we used it as a

reference for comparison with patients, taken both as a group and individually. We computed the distance index within groups to test the sensitivity of the proposed methods, while a comparison between the healthy template and individual patients was used to address the capabilities and limitations of the methods when dealing with single subjects. We used the previously defined distance index (equation (7)) to evaluate our results and in particular, we computed the average distance index within the healthy group and between the pathological groups and the healthy. We calculated for each group the average FD value, standard deviation, coefficient of variation (CV) and finally the increment of distance of the pathological groups in respect to the healthy group, using the operator ‘distance increment’: diff  (H,X) = 

d(TMP,X) − d(TMP,TMP) d(TMP,TMP)

(8)

where TMP is the healthy template group, X is one of the pathological groups and d(H, X) is the distance index. 2.6.2.  Slice by slice t-test.  We performed a slice by slice t-test to assess the difference at slice

level between the three groups. For each slice, we built a vector formed by the average FD value of the slice of all the subjects and performed a t-test between the template, the healthy group and each of the two pathological groups. Then, we counted the number of slices that resulted significantly different between groups. The test was formulated as: FD TMP ≠ FD X, α = 0.05

where X is SCZ or BD. 3. Results 3.1.  Results on healthy subjects 3.1.1.  FD of segmented images.  We computed FD of GM, in terms of both binary images and

of grey levels, first considering the whole brain and then considering four separate structures. 1703

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Table 1.  Results

on healthy subjects on binary GM images. d is the distance within the healthy group the number of different slices between the groups. FD values are similar across the lobes and higher on the whole brain. Grey matter -Standard box-count

Whole brain   Local 2D   Global 2D Frontal Lobe   Local 2D   Global 2D Occipital Lobe   Local 2D   Global 2D Parietal Lobe   Local 2D   Global 2D Temporal Lobe   Local 2D   Global 2D

Mean FD,TMP (SD)

Mean d (SD)

1.46 (0.01) 1.66 (0.01)

2.32 (0.93) 2.14 (0.89)

1.29 (0.04) 1.49 (0.05)

1.11 (1.04) 4.73 (1.09)

1.30 (0.01) 1.45 (0.02)

1.48 (0.56) 2.11 (0.39)

1.32 (0.004) 1.51 (0.01)

1.25 (0.50) 1.64 (0.39)

1.34 (0.008) 1.60 (0.04)

1.39 (0.32) 2.69 (0.48)

Tables 1 and 2 report the average FD values computed on the healthy subjects using the local and global approaches, with the custom and the standard box-count algorithm, with n = 35. The selection of n is a crucial step of the algorithm and it is discussed in section 3.2.1. Figure 2 shows how FD varies across the brain and how the two methods yield similar results in terms of FD behaviour, even if FD values are higher when computed with the global approach. The average distances of both approaches within the healthy group are similar and CVs are in the order of 2%. FD values computed with the global approach are generally higher (between 12 and 20%) than those computed with the local approach. This difference is increased in the temporal lobe in respect to the whole volume. Using the box-count algorithm on grey matter images with both global and local approach results in higher FD values than our customized box-count, even if only of the order of 3 to 4%. 3.1.2. Analysis of non-segmented T1-weighted images.  We verified the applicability of our approaches to T1-weighted images without differentiating between grey and white matter using the customized box-count algorithm. Analysis of these data was performed only with the customized box-count, as the standard is not applicable to non-binary data. The number of boxes needed to cover the image versus the size of the boxes can be approximated in the log–log plot with a straight line (figure 3(a) for a single subject and (b) considering all subjects together). We computed the residuals of the fit to a straight line for each subject and slice. Their average value and standard deviations were always very close to 0 (in the order of magnitude of 10−15). We computed Pearson’s r coefficient between the obtained regression line and the ideal line obtained interpolating the first two points only. Mean whole brain r value was 0.991 (0.03). A similar procedure has been applied to the frontal lobe, r = 0.988 (0.028), temporal lobe, r = 0.981 (0.036), occipital lobe, r = 0.985 (0.020), parietal lobe, r = 0.982 (0.036). Table 3 reports results on every analyzed structure. Consistent average FD values were found across all the structures. The last column of the table reports the average distance 1704

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Table 2.  Results on healthy subjects on GM T1w images. Images were obtained by multiplying ROIs GM masks with the T1w images, keeping the grey levels values. d is the distance within the healthy group. FD values are similar across the lobes and higher on the whole brain.

Grey matter - Customized box-count Whole brain   Local 2D   Global 2D Frontal Lobe   Local 2D   Global 2D Occipital Lobe   Local 2D   Global 2D Parietal Lobe   Local 2D   Global 2D Temporal Lobe   Local 2D   Global 2D

Mean FD,TMP (SD)

Mean d (SD)

1.40 (0.01) 1.59 (0.01)

2.18 (0.84) 1.95 (0.77)

1.24 (0.04) 1.43 (0.05)

3.89 (0.96) 4.40 (0.96)

1.24 (0.01) 1.39 (0.02)

1.39 (0.50) 1.94 (0.35)

1.27 (0.004) 1.44 (0.01)

1.16 (0.43) 1.35 (0.30)

1.28 (0.05) 1.54 (0.04)

1.32 (0.31) 2.90 (0.44)

Figure 2.  Average whole brain slice by slice FD values in grey matter ±2SD (dashed

lines) computed on all healthy subjects, computed with the local (a) and the global (b) approaches. Only values for slices between 40–140 are represented.

within the healthy group, computed by averaging the distance of FD of each healthy subject from the mean FD value of the template. This analysis shows that the considered lobes have similar average FD values. CVs on NS data are higher than those on segmented data, in the order of 3–7%. 3.1.3.  Creation of the template.  We used a leave-n-out procedure with n = 15 and 25 to test if the template performances depended on the number of subjects used and on the selected subjects themselves, using a t-test (p = 0.05) slice by slice to count the number of different slices between the training ant test groups. For healthy, the number of different slices between the training and test set was around 5% (from 4.9% for the temporal lobe and the local method with the test set of size 25, to 5.1% for

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Figure 3.  Regression of the number of boxes needed to cover the image versus the size

of the considered box on slice 55 of a representative healthy subject (taken as example) (a) and average regression line of slice 55 for all subjects ± 2SD (dashed lines) (b). Latest points, which exhibit a different behavior, correspond to a box-side greater than 100 pixels, over half of the image. Similar results were found for all the remaining slices and subjects.

Table 3.  Results

on healthy subjects, on T1-w without distinction into white and grey matter. d is the distance within the healthy controls group. On T1weighted images, FD values are very similar across the lobes and higher for the whole brain. Local approach

Region

Mean FD,TMP (SD)

Mean d (SD)

Whole brain Frontal lobe Occipital lobe Temporal lobe Parietal Lobe

1.63 (0.003) 1.42 (0.18) 1.42 (0.06) 1.41 (0.01) 1.44 (0.01)

1.62 (0.08) 1.5 (0.37) 0.62 (0.24) 0.78 (0.22) 0.41 (0.18)

Global approach Region

Mean FD,TMP (SD)

Mean d (SD)

Whole brain Frontal lobe Occipital lobe Temporal lobe Parietal Lobe

1.84 (0.03) 1.63 (0.05) 1.52 (0.04) 1.76 (0.07) 1.59 (0.02)

2.91 (0.57) 3.40 (0.70) 2.18 (0.42C) 2.54 (0.54) 1.36 (0.31)

the whole brain with the local method and test set of size 15) as expected once selecting the confidence level of 2SD around the average values. 3.2.  Results on patients 3.2.1. Dependence of FD from neighbourhood size n.  Table 4 reports the results of our method on 10 subjects, 5 healthy and 5 schizophrenia affected subjects with n ranging from 15 to 65. We chose to adopt n = 35 for T1-weighted images through all the study, as this value, in most structures, lead to the highest separation between the groups. Compared to a neighbourhood size of 15 (fastest computation, used here as a reference value), this value 1706

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Table 4.  Results

on different structures and data for different neighborhood size, n = 35 (in bold) was chosen due to the higher discriminative power. d(X, X) = distance index respect to n = 15 (base value), H = healthy, P = schizophrenia affected subject. Whole brain, GM, subjects 1–5 40–45, custom box-count n

d(H,P)

d(H,H)

diff d(H,P) (%)

15 20 25 30 35 40 50 65

5,13 5,05 5,19 5,01 5,86 5,44 5,2 4,41

1,67 1,52 1,58 1,61 1,58 1,56 1,64 1,65

207 232 228 211 271 249 217 167

Frontal lobe, GM, subjects 1–5 40–45, custom box-count n

d(H,P)

d(H,H)

15 20 25 30 35 40 50 65

7,7 8,21 8,67 8,94 8,37 8,01 8,24 6,82

4,8 4,99 4,96 5,33 4,86 4,91 5,14 4,62

diff d(H,P) (%) 60 65 75 68 72 63 60 48

Whole brain, T1w, subjects 1–5 40–45, custom box-count n

d(H,P)

d(H,H)

diff d(H,P) (%)

15 20 25 30 35 40 50 65

1,66 1,21 1,6 1,83 2,05 1,64 1,65 1,9

0,94 0,75 0,88 0,93 1,07 1,1 1,21 0,94

77 61 82 97 92 49 36 102

performed a better separation of these schizophrenic subjects from healthy and in many cases reduced internal differences of the healthy group. The choice of n only affects the magnitude of the separation: differences between pathological and healthy subjects were revealed also with other values of n. 3.2.2.  FD values on grey matter.  Tables 5 and 6 report results obtained on GM images of

patients. We expected FD values on some structures, as frontal and parietal lobes, to show higher distances from healthy values, because GM of these regions is described in literature as most exposed to mutations induced by the considered pathologies. Results obtained with the local approach on the whole brain agreed with our expectations, with distance increment of 58.3% for SCZ and 98.6% for BD. Global approach showed distance increment of 50.5 and 102% for SCZ and BD respectively (table 5). When considering the image grey levels, the 1707

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Table 5. 

Results on patients using on GM T1w images using the standard boxcount algorithm. Images were obtained by multiplying ROIs GM masks to the T1w images, therefore keeping the grey levels values. Grey matter - Standard box-count

Whole brain   Local 2D   Global 2D Frontal Lobe   Local 2D   Global 2D Occipital Lobe   Local 2D   Global 2D Parietal Lobe   Local 2D   Global 2D Temporal Lobe   Local 2D   Global 2D

diff (SCZ)(%)

diff (BD) (%)

Mean FD,SCZ (SD)

Mean FD,BD (SD)

58.3 50.5

98.6 101.7

1.43 (0.06) 1.63 (0.04)

1.42 (0.06) 1.61 (0.04)

40.6 27.9

66.0 52.7

1.25 (0.16) 1.45 (0.25)

1.25 (0.17) 1.46 (0.16)

25.2 15.7

67.8 38.3

1,28 (0.07) 1.43 (0.06)

1.27 (0.07) 1.42 (0.06)

52.7 24.9

95.5 56.9

1.29 (0.05) 1.47 (0.07)

1.28 (0.05) 1.46 /(0.07)

30.61 6.19

75.4 33.0

1,31 (0.05) 1.57 (0.07)

1.30 (0.05) 1.56 (0.07)

Table 6. Results

on pathological subjects on GM T1w images using the customised box-count algorithm, obtained by multiplying ROIs GM masks to the T1w images, keeping the grey levels values. diff expresses the increased percentage of distance from the template. Grey matter - Customized box-count

Whole brain   Local 2D   Global 2D Frontal Lobe   Local 2D   Global 2D Occipital Lobe   Local 2D   Global 2D Parietal Lobe   Local 2D   Global 2D Temporal Lobe   Local 2D   Global 2D

diff (SCZ) (%)

diff (BD) (%)

Mean FD,SCZ (SD)

Mean FD,BD (SD)

65.9 65.7

95.1 108.2

1.37 (0.05) 1.56 (0.04)

1.36 (0.05) 1.56 (0.04)

42.7 33.3

63.0 51.0

1.20 (0.15) 1.39 (0.24)

1.19 (0.20) 1.41 (0.16)

27.8 18.6

66.7 46.2

1.22 (0.07) 1.37 (0.05)

1.22 (0.07) 1.37 (0.05)

58.7 39.5

92.9 67.9

1.23 (0.05) 1.41 (0.06)

1.22 (0.05) 1.40 (0.06)

32.8 21.2

69.6 39.4

1.26 (0.05) 1.52 (0.06)

1.25 (0.05) 1.51 (0.08)

difference in the distance index reached, for the parietal lobe, the values of 58% for the group of schizophrenia-affected patients. For BD patients the ROI resulting with most increased distance index was also parietal lobe with 98.06%. Figure 4 shows a graphical representation of the comparison between the healthy controls, the schizophrenia affected group and the bipolar disorder affected group average FD. Analysis 1708

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Figure 4.  Average (bold lines) ± 2SD (dashed line) whole brain FD of grey matter for

healthy controls (HC, circles), schizophrenia patients (SCZ, solid) and bipolar disorder patients (BD, crosses) (a) and boxplots of distance indices (d) of the same groups compared with template values (b). Both pathological inter-groups’ distance indices are statistically different from the healthy intra-group distance index.

was performed on GM of the whole brain using the local approach with the customized boxcount using, n = 35. We compared the pathologic group to the healthy using the distance index. Panel B of figure 4 depicts the boxplots of the distance indices of the three groups. Subjects affected either by bipolar disorder or schizophrenia show a distance from the group of healthy subjects which is higher than the intra-group distance of healthy. We compared slice-by-slice FD values of patients and healthy using t-tests (p = 0.05). In the frontal lobe, with local-custom approach, the comparison resulted in 89 significantly different slices out of 114 in the case of SCZ and 83 for BD. In the parietal lobe different slices were 45 for SCZ group and 45 for BD group, out of 95 slices. Results on the other two considered lobes show the advantages offered by the slice by slice approach: some slices show very little difference between the groups while for other slices the difference becomes relevant. For SCZ group different were 38 out of 76 and 46 out of 77 on occipital and temporal lobe respectively, while for BD different slices were 29 and 42 respectively. We are thus able to localize which slices show differences. As can be seen in figure 5 in the case of temporal lobe, although globally the groups aren’t well distinct, in some intervals of slices, as 65–70 the distinction becomes clear. No relevant differences in term of statistically different slices were found between the two groups of patients, with a maximum of 9 slices in the occipital lobe, using the global approach and the non-modified box-count algorithm. 3.2.3.  FD values on T1w data.  Even if results are less stable than those obtained in the grey

matter, when dealing with the undistinguished brain tissue the local approach detects differences between groups when analyzing T1w data, without segmentation, thus considering grey and white matter with no distinction (table 7), with distance indices related to the pathological groups higher than those of healthy from 6.3% for schizophrenia patients in the temporal lobe to 92.6% for BD in the parietal lobe. Bipolar disorder group shows more difference in FD in respect to the template than the group of schizophrenia, when considering the distance indices, particularly in the whole brain and in the frontal, temporal and occipital lobes (local approach). Average FD values of bipolar disorder and SCZ subjects are lower than those of healthy. Whole brain t-test resulted, using the global approach, in 50 and 46 statistically different slices out of 143 respectively. The same t-test on the other lobes reported 63 (SCZ) and 50 1709

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Figure 5.  Average FD values ± 2SD (dashed lines) slice by slice on frontal lobe (a), temporal lobe (b), parietal lobe (c) and occipital lobe (d), computed using the local approach with the customized box-count for healthy controls (circles), schizophrenia patients (solid line) and bipolar disorder patients (crosses). Frontal lobe and parietal lobe show great separation between the average lines of the groups through all the slices. Differences are clearly observable also in occipital and temporal lobe, but are localized in a limited number of slices.

(BD) out of 108 different slices for the frontal lobe, 5 (SCZ) and 5 (BD) out of 73 different slices for the occipital lobe, 7 (SCZ) and 4 (BD) out of 95 for the parietal lobe and 13 (SCZ) and 14 (BD) out of 76 different slices for the temporal lobe. The local approach detected, considering the whole brain, 71 different slices for SCZ and 55 for BD. In the frontal lobes the different slices were 63 for SCZ and 50 for BD, in the temporal lobe 19 for SCZ and 13 for BD, in the parietal lobe 44 for SCZ and 41 for BD, in the occipital lobe 38 for SCZ and 34 for BD. No remarkable differences were found between SCZ and BD in terms of statistically different slices: 4. Discussion Aim of this work was to 1) test the hypothesis that self-similarity is reduced in schizophrenic and bipolar cohorts and 2) test the sensitivity and reliability of fractal analytics for the analysis of brain structural data. Quantitative indices that discriminate brain images of subjects affected by psychiatric diseases from those of healthy are of great interest, because they could potentially help to improve the diagnostic processes in the context of significant diagnostic instability (Salvatore et al 2009). 1710

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Table 7.  Results on patients using local and global approaches on T1w images. The distance increment is higher for the bipolar group, when considering the local approach.

Local approach Region

diff (SCZ) (%)

diff (BD) (%)

Mean FD,SCZ (SD)

Mean FD,BD (SD)

Whole brain Frontal lobe Occipital lobe Temporal lobe Parietal lobe

43.1 28.3 23.7 6.3 40.7

58.7 39.0 43.6 33.2 92.6

1.62 (0.08) 1.41 (0.11) 1.41 (0.06) 1.41 (0.05) 1.44 (0.05)

1.62 (0.08) 1.41 (0.11) 1.41 (0.06) 1.40 (0.05) 1.43 (0.05)

Global approach Region

diff (SCZ) (%)

diff (BD) (%)

Mean FD,SCZ (SD)

Mean FD,BD (SD)

Whole brain Frontal lobe Occipital lobe Temporal lobe Parietal lobe

2.52 12.2 −3.67 −4.06 −3.24

2.81 23.7 −3.44 2.46 3.67

1.83 (0.05) 1.61 (0.22) 1.51 (0.09) 1.76 (0.09) 1.58 (0.11)

1.83 (0.05) 1.61 (0.22) 1.51 (0.10) 1.76 (0.11) 1.57 (0.10)

Alterations in brain morphology in psychiatric illnesses have already been proven, using MRI imaging techniques. For example, it has been shown that abnormalities in grey matter are frequent in schizophrenia: voxel-based morphometry studies on T1-weighted images demonstrate a diffuse loss of GM volume especially in frontal and temporal lobes (Mathalon et al 2001 Honea et al 2005, Bora et al 2011) which has also been related with clinical outcome (Ubukata et al 2013). Also, bipolar disorder is associated with loss of GM volume (Bora et al 2010, Yu et al 2010). Differently from these state-of-the art techniques, we wanted a method able to evaluate specific structures and over all, the changes happening in individual patients. Group-analysis as VBM or GM thickness analysis do not, in fact, allow the consideration of single patients. We believe that this is an important feature of our method and that the evaluation of individuals could greatly help in the diagnostic process. Other studies integrated texture analysis with fractal geometry (Gonçalves and Bruno 2013, Lin et al 2013). This could be subject of future work building on our results: with our method, we aimed at obtaining a direct measure of complexity of brain areas with fractal dimension. The combination of our method with texture detection could improve the performances of the algorithm, especially with more challenging aims as classification, as authors did in Lin et al 2013 in the field of pulmonary oncology. In this work we showed that brain images of patients affected either by BD or SCZ have lower FD than controls, suggesting less structural complexity, especially in frontal lobe, which is known to be heavily affected in these disorders (Davidson et al 2003, Lyoo et al 2004, Lin et al 2011, Quan et al 2013). Previous work of our group (Altamura et al 2013) demonstrated that BD and SCZ share, in addition to a number of clinical, a common neurobiological signature in that prefrontal cortex resulted to be hypoactive in both diseases but differed in terms of white matter metabolism. Since in this work the two groups do not differ in terms of structural complexity, the data suggest that the developmental trajectory due to interneuronal dysfunction might be similar in the two disorders while the metabolic substrate differs. Most studies that employ fractal geometry compute a single FD for each considered structure: Esteban et al (2007), as well as Zhang et al (2008) obtained a single FD value for each 1711

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subject. It is important to note that many neurological pathologies can produce lesions and mutations at local level in well-defined, specific regions. Computations of FD on the whole brain may lead to underestimation of these modifications due to the high percentage of tissue included, with misleading results. Instead, we computed one FD value for each slice, increasing the sensitivity of the method to local modifications of the structures. The possibility of obtaining a FD value for each slice allows the identification of brain portions which show heavier structure modification in respect to the whole structure. We decided to calculate a FD value for every slice where each slice is handled as a 2D object. Other works (Esteban et al 2007, Zhang et al 2008, Sandu et al 2008) proved that the usage of 3D box-count improved the evaluation of FD in the brain in respect to previously used 2D approaches. We also developed a 3D version of our technique: instead of dividing 2D images in 5  ×  5 blocks, we divided 3D input data in 5  ×  5 × 5 blocks and performed 3D box-count (customized and standard) on the n × n × n voxels surrounding each cube. Values were then averaged in each slice. This 3D version dramatically increased computation time and did not provide significant improvements to the results, thus we discarded it favouring the two 2D methods we implemented. We found marked differences between controls and patients both with the global and the local approach. FD values resulted slightly different for each considered method, but it is to be noted that they were always consistent, both in terms of mean FD and of distance within/ between groups, with all methods separately. We found unstable results when evaluating T1 images, when we did not differentiate grey from white matter, especially using the global approach in the temporal, parietal and occipital lobes. These are the brain areas where differences between patients and healthy are less definite. Moreover, the global approach is susceptible to uncertainties when used in conjunction with a non-segmented image, due to its less sensitivity to local changes. This could also be the reason why with this method we found small negative distance indices in the occipital, parietal and temporal lobes of patients. When computing the distance indices, the global approach seems to suit bigger structures better than smaller portions of image. Since the global approach, by definition, computes a single FD value for each slice, it could be less sensitive to smaller modifications which could be identified with the use of a local approach. In the latter case, the size of the neighbourhood, i.e. the parameter n of the algorithm, is a key value which affects the performance of the process. We found no single value to use in all cases, and usage of non-optimal values may lead to unstable results. Small values of n increase the sensitiveness of the algorithm to local variations, but decrease the performance of the box-count algorithm. Large values increase the accuracy of the box-count algorithm in distinguishing between the groups, but decrease global local sensitivity. We chose to use the value which maximizes the distance index between the groups as a criterion. Obviously, this is a subjective choice which could constitute a point of weakness of this work although it was optimized on an independent cohort. It is to be noted, though, that the value of n = 35 that we chose, is also conservative regarding the distance index within the healthy group. Moreover, the size of the portion of the image we consider when using this local approach corresponds to approximately 12 cm2.which we assume is a reasonable size, i.e. fractal characteristics of the underlying tissue, which is a non-ideal fractal, should not be affected by our taking into account just a portion of the image. This is also the reason because FD values calculated with this approach result, in general, lower than those calculated with the global approach. In fact, since the slice FD calculated with the local method is the mean of the FD values for the different portions, areas with a regular structure contribute to a decrease in the value of FD. This does not have an impact on the sensitivity of the method to the disease, nor on the applicability of a fractal method itself, which we validated using a log–log plot as in (Kiselev et al 2003). 1712

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Figure 6.  FD of a patient with BD (solid line) compared with template average FD (solid line with circles) +/ −  2SD (shaded lines with circles) in the whole brain.

We also demonstrated that fractal analysis can be done on the whole brain and in the considered structures, without distinguishing between grey and white matter, thus without the need of a segmentation pre-processing step. We found differences in the whole brain, in the frontal and occipital lobe, particularly when using the local approach with the custom boxcount algorithm. Discrimination between patients and controls is in this case weaker than after segmentation and also unstable in some areas, but the analysis becomes noticeably faster. Moreover, the need to manually check results of segmentation is a process which must not be underestimated in presence of large datasets, as it is necessary due to the fact that the segmentation process is not immune to errors in the estimation of tissues and some voxels or structures could be spuriously identified as belonging to the wrong class. With this algorithm, the need of assessing the goodness of segmentation is overcome. One way to combine the speed of the analysis on T1w images and the better accuracy of that on segmented volumes could be applying first the algorithm on T1-weighted images analysis to rapidly reveal the presence of abnormal FD values then, only on subjects of interest, data could be segmented to have a more detailed characterization of these subject using fractal analysis on the output of segmentation. Finally, we want to emphasize how this method is suitable to the analysis of single subject data. In figure 6 FD values of a subject affected by BD are shown. As can be seen, FD values are below the mean FD—2SD template values in a large range of slices. This demonstrates that it is possible to identify changes in brain complexity not only in groups of patients, but also in single subjects. Further studies are needed to understand if FD behaviour can be linked to clinical information as length of disease, medication or symptoms severity and more importantly, in the prediction of conversion for subjects at risk. A limitation of this study is that subjects are not well age-matched. Unfortunately the use of multivariate statistical approach for the comparison of our different groups leads to an overestimation of group-specific differences (Dukart et al 2011). 5. Conclusion In summary, we show that fractal geometry can provide new insight in the evaluation of psychiatric diseases, in particular schizophrenia and bipolar disorder. Structural brain modification can be identified by means of a purely geometric analysis of the 2D slice-by-slice structure of the brain. We demonstrated that with this analysis differences in brain geometry are relevant both considering only grey matter and without any segmentation. Fractal 1713

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