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Fractals, Vol. 17, No. 2 (2009) 1–9 c World Scientific Publishing Company
FRACTAL DIMENSION AND LACUNARITY OF TRACTOGRAPHY IMAGES OF THE HUMAN BRAIN P. KATSALOULIS∗ , D. A. VERGANELAKIS† and A. PROVATA∗,‡ ∗Institute of Physical Chemistry, National Center for Scientific Research “Demokritos”, 15310 Athens, Greece †Medical Diagnostic Center Encephalos-Euromedica 15233 Halandri, Greece ‡
[email protected] Received February 27, 2008 Revised October 29, 2008 Accepted December 9, 2008
Abstract Tractography images produced by Magnetic Resonance Imaging scans have been used to calculate the topology of the neuron tracts in the human brain. This technique gives neuroanatomical details, limited by the system resolution properties. In the observed scales the images demonstrated the statistical self-similar structure of the neuron axons and its fractal dimensions were estimated using the classic Box Counting technique. To assess the degree of clustering in the neural tracts network, lacunarity was calculated using the Gliding Box method. The twodimensional tractography images were taken from four subjects using various angles and different parts in the brain. The results demonstrated that the average estimated fractal dimension of tractography images is approximately Df = 1.60 with standard deviation 0.11 for healthy human-brain tissues, and it presents statistical self-similarity features similar to many other biological root-like structures. Keywords:
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1. INTRODUCTION The use of Magnetic Resonance Imaging (MRI) in medicine has greatly improved the accurate visualization of internal structures of the human body.1,2 The simple and non-intrusive nature of MRI made it an important tool in diagnosis in various situations, where a detailed view is required. In contrast with Computed Tomography (CT), MRI can image soft tissues with increased and variable image contrast without using ionizing radiation.3 For this reason, in situations where tumor detection or frequent periodical tests are required, MRI is superior to CT and is commonly used.4–6 MRI actually detects proton signals from water molecules which are abundant in the human body and brain. Therefore, each image represents a map of water distribution as a function of the position in the body. The conventional MRIs are twodimensional images (planes) of human body parts and organs, which represent the three-dimensional structure of the tissue. Three major planes are used: Axial, Sagittal and Coronal. In specific processing, three-dimensional or volume models of the subject are used. On top of this technology, other techniques have been developed which answer more specialized questions, such as Functional MRI, Diffusion MRI and Magnetic Resonance Spectroscopy. All techniques are enhanced with post-processing based on computer analysis. The human brain in particular, consists of white and gray matter that can be clearly seen in conventional MRI. The former consists of neurons which interconnect various areas of brain together, in the form of bundles. These pathways form the brain structure and are responsible for the various neuro-cognitive aspects. There are about 100 billion
Fig. 1
neurons within a human brain interconnected in complex manners. They vary in sizes (µm – cm) and in shapes. Although, some of those neuron bundles can be seen with conventional MRI techniques, the majority of those (due to their small size and similar relaxation — T1 and T2 — parameters) are missed. A recently developed technique, the so-called Diffusion Tensor Imaging (DTI) Tractography,7,8 allows the in vivo study of white matter neuronal fibers with a resolution of the orders of mm. DTI actually measures the anisotropic diffusion of water molecules within neuronal tracts. Since the structure of the brain is not homogeneous, the diffusion is anisotropic in all three axes.9 The local properties of water diffusion in the tracts can be processed to create two or three dimensional images and structures,10,11 where the shape represents the fibers and the color the diffusion direction. DTI measures water diffusion within the neuronal tracts in many directions (15 in this study) and characterizes diffusion on a voxel-by-voxel basis. After processing, the neuron structure is illustrated with a two-dimensional image using three colors which correspond to the different diffusion directions: red (left-to-right), green (posterior-toanterior) and blue (superior-to-inferior), see Fig. 1. Selecting a Region Of Interest (ROI) that covers a certain colored tract of neuronal bundles achieves the minimization of contamination with other neuronal tracts. In this work we opted to estimate structural parameters related to the distribution of neuron tracts in the brain, based on the information obtained from the Diffusion Tensor Images. These DTI structures seem to be primarily selfsimilar in the statistical sense and they present
DTI images of the same area in the brain. The two images are snapshots taken from a different angle.
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scale-invariance over 2–3 orders of magnitude. In the sequel we will use the term ‘self-similarity’ to represent ‘statistical self-similarity’, since we deal with natural objects. DTI structures of the neurons optically resemble the arrangements of networks of tree roots, blood vessels, ganglia, river networks and other biological and physical systems. One of the main properties of these systems is branching in many levels, which is an indication of self-similarity in their structures.12,13 The degree of (statistical) self-similarity which characterizes a particular network of neurons emanating from certain part of the human brain can be determined using fractal analysis. By comparing the degree of self-similarity (given quantitatively by the fractal dimension Df ) from healthy tissues and damaged tissues, it might be possible to quantitatively detect the presence of structural problems. These problems can originate from brain hematomas, tumors or other neurological disorders. In this case, Df could serve as a primary index related to the extent of damage.14 By introducing such quantitative measures of statistical selfsimilarity, it will become possible to numerically characterize tissues, which is also the main direction of this study. Fractality has already been detected in various biological structures, including blood vessels, bronchial tree, neural connections, dendritic arbors of ganglion cells and many others.14–24 Various studies exist which explore the self-similarity and fractality of these structures and various models have been proposed. In particular, the dendritic arbor structures of retinal ganglion cells in animals (such as ferret, cat and goldfish) have been investigated by analysing the retrograde transport of injected fluorescent microspheres.16–18 Also the same method was used for the analysis of the morphology of the Lateral Geniculate Nucleus neurons of the cat.19 The analysis of the images obtained using the retrograde tracers showed that the dendritic morphology of the neurons could be quantitatively described using, among other measures, the fractal dimension. The dimensionality attributed to the different type of dendritic arbors was shown to depend on the type of neuron cell, the organism and the development stage. Values of Df in the range 1.2–1.6 were frequently encountered. Following this path, we focus on diffusion tensor tractography images and calculate the fractal dimension and the lacunarity of fiber tracts as presented in DTI images, in the human brain.
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The need for interconnection between the different areas of the brain has led to an optimal network of neural pathways, in many scales. The existence of many scales in the network of neural axons, and the overall self-similar topology in their distribution, can be quantitatively addressed through the corresponding fractal dimension Df . Since different fractals with the same fractal dimension may have different topology, the lacunarity Λ(r) is a function that characterises the coarseness of the fractal in the different length scales r.25–27 Lacunarity distinguishes structures which present accumulation of mass as opposed to scattered and non-connected objects, with holes in their spatial distribution. Fractal dimension and lacunarity address different aspects of the spatial distribution of neuron pathways in the brain and both can be “tracked” by DTIs originating from the tractography technique. In the next section the features of the DTIs used in this study are described. In Secs. 3 and 4 the results of the fractal box-counting and the lacunarity analysis are presented and discussed. In the concluding section the main conclusions are drawn and open problems are presented.
2. METHODS 2.1. Data Four different subjects have been used in this analysis, termed S.1, . . . , S.4, in Table 1. Two of the subjects had intracranial hematoma (subjects S.1 and S.2) and two were healthy. Although only two of the subjects were healthy, all areas are selected from the healthy hemisphere parts. In all cases, two symmetrically located areas placed in the two brain hemispheres were considered. In some cases, clustered together in Table 1, DTIs from the same Region Of Interest (ROI) but viewed from different angles are considered. The areas studied in this work varied in shape and size; the size of the chosen areas was between ∼200– 500 mm2 . The location and the size of the selected ROIs reflects the local branching of the neurons and the total area covered by them, and is thus expected to influence the locally calculated fractal dimension.
2.2. Diffusion Tensor Images The MRI measurements were performed on a 1.5 Tesla General Electric CVi system (Milwaukee Wisconsin USA). The parameters of the DTI single-shot spin-echo Echo Planar Imaging (EPI)
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P. Katsaloulis et al. Table 1 DTI Fractal Dimension of the Four Subjects (S.1, . . . , S.4). The First Column of all Subjects Presents the Results of Every DTI. In the Second Column DTIs from the Same Area are Averaged Together. S. 1
S.1
S.2
S. 2
S.3
S.3
S.4
S.4
1.444 1.539 1.571 1.686 1.718 1.758 1.645 1.622 1.596 1.686∗ 1.659 1.590 1.611 1.597∗∗ 1.620 1.465 1.440 1.547 1.512 1.533 1.595 1.560 1.485 1.534
1.444
1.534 1.337 1.390 1.590 1.540 1.468 1.533 1.596 1.541 1.568 1.686 1.742 1.771 1.706 1.711
1.534 1.337 1.390 1.590 1.540 1.468 1.533 1.596 1.541 1.568
1.445 1.621 1.734 1.793 1.840 1.829 1.776
1.445 1.621 1.734
1.697 1.551 1.648 1.647 1.696 1.719 1.489 1.495 1.527
1.697 1.551
1.555
1.721 1.634 1.641 1.659 1.600 1.609
1.809
1.678 1.489 1.495 1.527
1.733 1.706 1.711
1.452
1.531 1.578 1.509
pulse sequence were: flip angle 900, Echo Time (TE): 85 ms, Repetition Time (TR): 10700 ms, slice thickness: 3 mm, spacing between slices: 0 mm, Field Of View (FOV): 26 cm, matrix: 128 × 128, and No. Of Excitations (NEX): 1. The diffusion gradients were applied along 15 non-collinear directions. The scan time for each subject was approximately 3 minutes. The coil that was used to transmit the radiofrequency pulses and detect the MRI signal was an eight-element head coil. The processing of the data was performed off-line at a workstation using the Functool software (GE). After determining the lower signal threshold, circular ROIs were placed on the predetermined areas of interest, bilaterally. The area of the ROIs between opposite regions of the same subject was kept constant (a few mm2 ). (Note that, it is critical to place the ROIs within the actual nerve bundles under investigation without contaminating the measurements by generated signals of adjacent structures.) For each subject a series of images was taken from various areas of the brain. In some cases multiple snapshots of the same area with different viewing angle were used. All images can be
downloaded from our website (http://limnos.chem. demokritos.gr/dti) or are available upon request. A typical example of DTI can be seen in Fig. 1. All the images have been transformed into black and white bitmaps in order to calculate the various parameters (see Fig. 2). Color values have been ignored and treated equally. The resolution of each image was 512 × 512 pixels. This is far inferior to the actual dimensions of the neuron axons. Since the diameter of axons is usually in the order of a few µm, this resolution is unable to demonstrate the actual network complexity. Still, even with this low resolution representation, a scale invariance of the structure is prominent. With methods of finer resolution it would be possible to detect neuron details in even lower length scales and to determine the lower cutoff scale of self-similarity.
3. FRACTAL BOX-COUNTING ANALYSIS There are many ways to calculate the scaling properties of natural objects and many empirical
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(a)
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(b)
methods and algorithms have been suggested in the international literature. The most common one is the Box Counting method, which approximates the Hausdorff fractal dimension. Other scaling indices frequently used are the information dimension and correlation dimension, which are also measures of the self-similarity of the structure.28,29 The box counting algorithm gives reliable indications on the existence of self-similarity, can be easily implemented in digitalised data and is used in the sequel for the analysis of the DTI images. This method divides the space (in our case the 2D image) in equal partitions of length r. In this study squares of length r, ranging between 1 ≤ r ≤ 64 pixels, have been used. If N (r) is the number of boxes of size r which contain, partly or fully, the original (tract) structure, the fractal dimension Df of this structure is given as: log N (r) log N (r) = − lim . r→0 log(1/r) r→0 log r
Df = lim
Number of neuron-containing boxes N(r)
Fig. 2 Representation of the same DTI neuronal tracts. On the left (a) the original image is shown, with colors indicating the diffusion direction along the neuron axon. On the right (b) is the black and white representation of (a) image, used to calculate fractality and lacunarity of this image. Black color (in b) is used to define the presence of the neuron tracts. White color indicates the space where no neuron tracts are detected, associated with the neurons forming the black part.
experimental DTI data box counting fit
10000
1000
100
10 1
10
Box size r
Fig. 3 Example of box counting analysis of a DTI (subject 1, image indicated by asterisk * in Table 1). Symbols X represent the experimental data, while the solid line corresponds to the fit of the values. The calculated fractal dimension of this image is 1.6856.
(1)
In order to estimate Df , N (r) over r is plotted in a double logarithmic scale and the slope of the produced graph is calculated. Figure 3 presents an example of fractal dimension calculation using the proposed box counting method. For this figure, image taken from subject S.1, indicated with asterisk * in Table 1 was used. The data points form a straight line on a double logarithmic scale, whose slope represents the fractal dimension. Since the resolution of the images is 512 × 512 pixels, windows
of sizes ranging from 1 to 64, were used, for all subjects. As seen in Fig. 3, scaling extends in the two observed orders of magnitude in the r scale. The value of Df = 1.6856 < 2 (R2 = 0.9988), indicates that the axons of the neurons crossing this specific area of the brain do not homogeneously cover the space around them, but they are self-similarly distributed covering a subspace of dimension ∼ 1.69. Using the box counting method the Hausdorff fractal dimension Df of a series of MRI images were calculated. Since multiple DTIs of the same area
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P. Katsaloulis et al. Table 2 The Average Values and Standard Deviation of the Fractal Dimension for Each of the Four Subjects. The Last Column Shows the Average Fractal Dimension and Standard Deviation, with all DTIs Taken into Account.
Average Std. Dev.
Subject 1
Subject 2
Subject 3
Subject 4
1.584 0.084
1.581 0.126
1.720 0.142
1.608 0.092
have been calculated, the average value of multiple DTIs is also provided. Results are presented in Table 1 where both actual and averaged data is displayed. In this table the calculated fractal dimension Df is given for all DTI images belonging to all subjects S.i where i = 1, . . . , 4. For every image, the average Df value taken from different angles is calculated for all subjects and is also provided in the Table 1. It has been found that the smallest value of Df is 1.34, found for subject 2, while the largest value of Df is 1.84 (or 1.81 if we take into account the averaged values), found for subject 3. The structure of the tractography, and therefore the calculated fractal dimension, is directly affected by the originally selected ROIs in the brain. We have also calculated the averages of Df , as well as the standard deviation, in order to compare between the different subjects. The average results are presented in Table 2. It has been found that the average fractal dimension of DTIs has a value of around 1.6. The structure of neural tract networks and the fractal dimension reminds of dendritic structures like the arbors of retinal ganglion cells which present fractal dimensions around 1.25– 1.6, or even the plant roots, which have fractal dimensions around 1.3–1.5, depending on the type of plant. A histogram of all DTIs is presented in Fig. 4. The averaged values of each area has been used, in order not to include artifacts caused by double representation of duplicate areas. The most common values are found to be between the values of 1.57 and 1.62.
4. LACUNARITY ANALYSIS Although two objects might have the same fractal dimension, their structure might be statistically very different. Lacunarity Λ is a measure of clustering of structures as opposed to scattered ones. Instead of only defining the percentage of space not occupied by the structure, lacunarity is more
Total 1.604 0.112
Fig. 4 Histogram of DTIs of all four subjects. The horizontal axis represents the fractal dimension Df of the DTI, while the vertical axis represents the frequency of appearance of a given Df .
interested in the holes and their distribution on the object under observation. In the case of fractals, even if two fractal objects have the same dimension Df , they could have different lacunarity if the fractal “pattern” is different. In the current study lacunarity is used to describe the complex interconnectivity of neurons as opposed to a random distribution of them and to detect differences in holes or clustering in the DTIs.25,26 The method used to calculate lacunarity Λ is based on the Gliding-box algorithm.25,27 Squares of various size (r×r) were considered and for every size r the number m(r) of sites which belong to neurons [black sites in Fig. 3(b)] were calculated. Lacunarity is defined as: Λ(r) =
m2 (r) + m(r)2 , m(r)2
(2)
where m2 (r) is the variance and m(r) is the average density of neurons. A 512 × 512 grid of the DTI images was used and values of r between 1 and 64 were considered. The calculated lacunarity of a typical DTI (image taken from subject S.1 and denoted by double asterisk ** in Table 1) is presented in Fig. 5. The data
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DTI image Fractal image Random image
Lacunarity Λ(r)
10
1 1
10
Box size r Fig. 5 Calculated lacunarity of 3 images: a) symbol X represents data of DTI image, subject S.1, image denoted by double asterisk (**); b) open circles represent a fractal image with almost the same fractal dimension as the data set; and c) triangles represent a random image with same density as the data set.
set used, indicated with Symbol-X in Fig. 5, originates from subject S.1 and corresponds to fractal dimension Df = 1.597. Along with the DTI image, two additional simulated images have been used for demonstration purposes. The triangles represent a simulated Random structure that has the same density (density d = 0.075) as the DTI image, but the structure is completely random and uncorrelated. It is noted here, that for large values of r, the lacunarity of the random structure approaches the value of 2, which is precisely the lacunarity of a Poisson process where the value of the 2nd moment equals the square of the mean, m2 = m2 . The open circles represent a Sierpinski carpet which has the same fractal dimension (Df = 1.58) as image (a). The density of 9th iteration of the Sierpinski carpet used is 0.088, similar to the DTI image used. A first observation is that the DTI image has higher lacunarity than the depicted Random image which present the lowest lacunarity of all. This is not surprising because the random structure is expected to be completely scattered, given the low neural density of the DTI image (d = 0.075) imposed on the simulated random structure. Using similar arguments, the DTI image has also higher lacunarity than the Sierpinski fractal and this is also due to the scattered nature of this fractal. (Note that the fractal dimension of the Sierpinski carpet is DfS = ln 3/ ln 2.) On the other hand the human brain DTI images examined here represent all the neuron tracts crossing a certain part of the brain, thus all neuron tracts are coming together in
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this “central” part forming thus a single cluster, an interconnected structure, giving for this reason the highest values to the lacunarity. The lacunarity of all other DTI data sets shows similar characteristics to the one presented in Fig. 5. Lacunarity analysis can not attribute a mere characteristic value to each DTI image but provides a finer or secondary analysis, which can help to further classify the distribution of (fractal) neuron tracts from one length scale to another. Different lacunarity of two DTI images may (a) indicate that the represented neurons originate from different parts of the brain, or (b) indicate damaged versus healthy tissue. Except for the lacunarity analysis, the evaluation of higher moments of the neuron tracts distribution could help in the detection of possible finer deformations in their structure.
5. CONCLUSIONS The aim of this work was to measure the structural complexity of brain neuronal axons, using tractography DTI images and to compare the complexity of the textures originating from healthy parts of the brain. It was found that no real difference is present even for the subjects which suffer from intracranial hematoma (subjects S.1 and S.2), since all areas studied were healthy. The average fractal dimension of subjects S.1, S.2 and S.4 is the same, around 1.58–1.6, while For subject S.3 the number of provided DTIs is not enough to define whether the small difference by 0.14 is significant or not. The similar values of Df found in all subjects could be explained, since all selected areas originate from healthy brain areas, which still retain the original structure and functionality. Although four subjects are not enough to generalize, this work provides good indications of the presence of scale invariance in the topology of the neuron axons. This work can be further extended in the future, with a larger number of subjects, to obtain more precise values of the fractal dimensions as a function also of the part of the brain, the age and/or the gender of the subjects. Another interesting, although expected, observation is that tractographs of the same area but viewed from different observation angles have similar fractal dimension. For example DTIs of the first subject have values of Df 1.54 and 1.57 for the same brain area (e.g. 2nd and 3rd image of S.1). The facts that (a) the statistical characteristics (including the fractal dimension) of the neuron axons network do
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not change much when viewed from different angles, and (b) the structure does not cover completely the 2D space when viewed from any angle, indicate that the true fractal dimension is closed to the one obtained from the 2D images and it is certainly less than 2. The calculated Df values are then expected to be close to the actual true fractal dimension, even if the original neural structure is 3D, while the available data is given by 2D images. Lacunarity confirmed the difference of tractography images, compared to Random images and Fractal objects with the same density. In both case the lacunarity values were higher, demonstrating in the former case the interconnection in the neuron distribution and in the latter case the clustering of the structure and the connectivity of the neuron ensemble. It would be interesting to compare healthy with damaged tissue in the human brain, since all results in this study are based on healthy tissues. Preliminary results indicate that the fractal dimension of tractography images should be different, since axons of damaged neurons appear to be shorter. This might be due to the axonal withdrawal or even neuron death. In this case the axons will not be functional, and thus the activity will decrease or even vanish. The activity reduction is expected to start at the most distal part of the axon. The type of damage (brain injury versus brain tumor and other diseases) may also demonstrate different distributions. In a future study DTI images from known damaged tissue could be used, to quantitatively explore the possible differences compared to healthy tissue. Although tractography could be in 2D or 3D space, the analysis performed in this work deals with 2D images of neuron tracts. Another approach of fractal analysis of DTI is to use the 3D model of the water diffusion. Using 3D tractography information will improve the fractal dimension estimation of the neuron tract distribution, since these objects actually live in the 3D space. Still, tracts consist of neuron fibers, which roughly can be modeled as one dimensional threads. The distribution of them is rarefied and thus even the 2D estimation is close enough to the actual fractal dimension. A clear indication of it is the close values in the fractal dimensions obtained, when the same image is captured from different angles. Therefore we expect that the conclusions drawn here will not be significantly altered by the proposed 3D analysis.
As a conclusion, it is clear that tractography images are fractals and the neuron tracts do not cover homogeneously the whole brain area. The structure of the neuron tract networks is statistically scale-invariant and their length and the space which they occupy seem to follow scaling laws. These results are in accordance with the expectation that the brain neuron tracts depicted with this method will have similar fractal structure with other biological tissues, such as neuron networks in the body, retinal ganglion cells, bronchial tree and vessels.
ACKNOWLEDGMENTS This work was partially financed by the European Union and the General Secretariat for Research and Technology, Greece, within the Framework of the Regional Operational Program of Attiki 2004–2008.
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