# Unsupervised Detection of Fibrosis in Microscopy Images Using Fractals and Fuzzy c-Means Clustering

S. K. Tasoulis, Ilias Maglogiannis, V. P. Plagianakos

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S. K. Tasoulis, Ilias Maglogiannis, V. P. Plagianakos. Unsupervised Detection of Fibrosis in Microscopy Images Using Fractals and Fuzzy c-Means Clustering. 8th International Conference on Artificial Intelligence Applications and Innovations (AIAI), Sep 2012, Halkidiki, Greece. pp.385-394, 10.1007/978-3-642-33409-2_40. hal-01521436

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<figure><img src="image_1.png" /></figure>
# Unsupervised Detection of Fibrosis in Microscopy Images Using Fractals and Fuzzy c-means Clustering

S.K. Tasoulis, I. Maglogiannis, and V.P. Plagianakos

Department of Computer Science and Biomedical Informatics,
University of Central Greece, Papassiopoulou 2-4, Lamia, 35100, Greece
{stas,imaglo,vpp}@ucg.gr

**Abstract.** The advances in improved fluorescent probes and better cameras in collaboration with the advent of computers in imaging and image analysis, assist the task of diagnosis in many fields of biologic and medical research. In this paper, we introduce a computer-assisted image characterization tool based on a Fuzzy clustering method for the quantification of degree of Idiopathic Pulmonary Fibrosis (IPF) in medical images. The implementation of this algorithmic strategy is very promising concerning the issue of the automated assessment of microscopic images of lung fibrotic regions.

**Key words:** Image Analysis, Cluster Analysis, Fuzzy Clustering, Fractal Dimension.

## 1 Introduction

Idiopathic pulmonary fibrosis (IPF), also referred to as cryptogenic fibrosing alveolitis, is a chronic, progressive and usually lethal lung disorder of unknown aetiology. Median survival of newly diagnosed patients with IPF is about 3 years, similar to that of clinical non-small cell lung cancer. At present there are no proven therapies for IPF [18]. IPF usually affects patients aged 50-70 years, with a male preponderance, evidenced by a male-to-female ratio of 2:1. The incidence of IPF has been estimated at 10.7 cases per 100,000 men and 7.4 cases per 100,000 women and the incidence appears to be rising [6].

Advances in improved fluorescent probes [5] and better cameras [11, 21] have expanded the capabilities of the light microscope and its usefulness in biologic and medical research. The advent of computers in imaging and image analysis permits quantitative computer analysis. Thus, automatic classification and assessment of microscopic images is possible in an attempt to assist the task of diagnosis.

The field of microscopy image analysis has occupied several research teams and significant research work may be found in the literature of this field. In [14] a tool that classifies biological microscopic images of lung tissue sections with
idiopathic pulmonary fibrosis was presented. Similar tools have also been pro-
posed for the assessment of liver fibrosis [4, 16, 22, 2], the study of micro vascular
circulating leukocytes [9], the assessment of testicular interstitial fibrosis [19, 20],
or that of lung fibrosis [12]. The use of pattern recognition or classification meth-
ods like Support Vector Machines or Neural Networks has enabled the design of
decision-making algorithms, appropriate to microscopic data. Within this con-
text, a method for evaluation of electron microscopic images of serial sections
based on the Gabor wavelets and the construction of a mapping between the
model and the target image has been proposed in [13].

In the present study, we employ one of the most widely used fuzzy clustering
algorithms; namely the Fuzzy c-means [3]. The goal is to identify the presence
and the degree of lung pathology caused by idiopathic pulmonary fibrosis (IPF),
which is a chronic, progressive and usually lethal lung disorder of unknown
etiology [1], whose variability concerning the severity of the lesions it incurs
in the lung is great when assessed by microscopic histological images [12]. Fuzzy
c-means clustering (and more over an improved version of fuzzy c-means) has
also been applied in environmental cases [10]

The Fuzzy c-means clustering method was applied to digital images of sec-
tions, captured using a Nikon ECLIPSE E800 microscope and a Nikon digital
camera DXM1200 at a magnification of 4x. Each image was partitioned into win-
dows of specified size and features were extracted for each window. The main
contribution of the this study is the calculation of the fractal dimension of the
microscopic images and its incorporation to the clustering algorithm.

The rest of the paper is structured as follows. In Section 2 we briefly review
the Fuzzy c-means clustering algorithm and we outline the fractal dimension and
the box-counting method for its computation. Next in Section 3 we introduce
the proposed methodology and finally in Section 4 we present the experimental
analysis and results. The paper ends with concluding results and pointers for
future work.

## 2 Background Information

2.1 Fuzzy Clustering

Data clustering is the process of partitioning a set of data vectors into disjoint
groups (clusters), so that objects of the same clusters are more similar to each
other than objects in different clusters. Different measures of similarity may be
used in the clustering process, where the similarity measure controls how the
clusters are formed. Some examples of measures that can be used in cluster-
ing include distance, connectivity, and intensity. In traditional (hard) clustering,
data are divided into distinct clusters, where each data vector belongs to ex-
actly one cluster. On the other hand, in fuzzy (soft) clustering, data vectors can
belong to more than one cluster, and associated with each element is a set of
membership levels. These indicate the strength of the association between that
data element and a particular cluster. Fuzzy clustering is a process of assigning
these membership levels and then using them to assign data elements to one or more clusters.

One of the most widely used fuzzy clustering algorithms is the Fuzzy C-Means (FCM) algorithm. FCM is a clustering method that assigns each data point to a cluster to some degree that is specified by a membership grade. This technique was originally introduced by Jim Bezdek in [3] as an improvement on earlier clustering methods and attempts to partition a finite collection of data vectors into a collection of fuzzy clusters with respect to some given criterion. A theoretical discussion of FCM can be found in [7].

Given a finite set of data, the algorithm returns a list of cluster centers and a partition matrix indicating the degree to which each element belongs to a given cluster. Like the k-means algorithm, the FCM aims to minimize an objective function, like the following:

$$J_m = \sum_{i=1}^{N} \sum_{j=1}^{C} u_{ij}^{m} \|x_i - c_j\|^2, \quad 1 \le m \le \infty, \quad (1)$$

where m is any real number greater than 1, $u_{ij}$ is the degree of membership of $x_i$ in the cluster j, $x_i$ is the i-th of d-dimensional measured data, $c_j$ is the d-dimension center of the cluster, and $\| \cdot \|$ is any norm expressing the similarity between any measured data and the center. Fuzzy partitioning is carried out through an iterative optimization of the objective function shown above, with the update of membership $u_{ij}$ and the cluster centers $c_j$ by:

$$u_{ij} = \frac{1}{\sum_{k=1}^{C} (\frac{\|x_i - c_k\|}{\|x_i - c_k\|})^{\frac{2}{m-1}}}, \quad (2)$$

where

$$c_j = \frac{\sum_{i=1}^{N} u_{ij}^{m} \cdot x_i}{\sum_{i=1}^{N} u_{ij}^{m}}.$$

This iteration stops when

$$max_{ij}\{|u_{ij}^{k+1} - |u_{ij}^{k}|\} \le \epsilon, \quad (3)$$

where k is the iteration number and $\epsilon$ is a constant between 0 and 1 that controls the termination of algorithm. This procedure converges to a local minimum or a saddle point of $J_m$.

## 2.2 Fractals and Fractal Dimension

It is known that a fractal designates a rough or fragmented geometric shape that can be subdivided into parts, each of which is a reduced-size copy of the whole [8]. Fractals are generally self-similar and independent of scale. In general nature conforms to fractals much more than it does to classical shapes and hence fractals can serve as models for many natural phenomena.
Contrary to classical geometry, fractals are not regular and may have a non-integer dimension. Fractal concepts have provided a new approach for quantifying the geometry of complex or noisy shapes and objects. Fractal geometry has been proven capable of quantifying irregular patterns, such as tortuous lines, crumpled surfaces and intricate shapes, and estimating the ruggedness of systems [15].

In this study, we incorporate the fractal dimension in an attempt to guide the FCM algorithm to construct meaningful clusters. The box-counting approach is one of the frequently used techniques to estimate the FD of an image. Among a number of techniques [15], it was found to be the most appropriate method of fractal dimension estimation.

The box-counting method is easy, automatically computable, and applicable for patterns with or without self-similarity [8, 17]. In the box counting procedure, each input image is covered by a sequence of grids of descending sizes and for each of the grids, two values are recorded. These are the number of square boxes intersected by the image, $N(s)$, and the side length of the squares, $s$.

Then the fractal dimension is calculated by the regression slope $D$ ($1 \le D \le 2$) of the straight line formed by plotting $log(N(s))$ against $log(1/s)$ [15]. An image having a fractal dimension of 1, or 2, is considered as completely differentiable, or very rough and irregular, respectively. The linear regression equation used to estimate the fractal dimension is

$$ \log(N(s)) = \log(K) + D \log(1/s), \qquad (4) $$

where $K$ is a constant and $N(s)$ is proportional to $(1/s)D$.

## 3 Proposed methodology

For this study, Age- and sex- matched, 6-8 week-old mice were used for the induction of pulmonary fibrosis by a single intravenous injection with a dose of 100mg/Kg of body weight (100 mg/kg body weight; 1/3 LD50; Nippon Kayaku Co. Ltd., Tokyo). Bleomycin administration initially induces lung inflammation that is followed by a progressive destruction of the normal lung architecture. To monitor disease initiation and progression, mice were sacrificed at 7, 15 and 23 days after bleomycin injection. Mice injected with saline alone and sacrificed 23 days post injection, served as the control group. For pathology assessment, at each time point, bronchoalveolar lavage (BAL) has been performed (3x 1ml Saline) for the estimation of total and differential cell populations. Finally after perfusion of lungs via the heart ventricle with 10ml Phosphate Buffer Saline, lungs were then removed, weighed dissected and collected, for histology. Sagittal sections from the right lung were used for Hematoxylin and Eosin staining and histopathologic analysis.

When we retrieve the microscopic images the procedure continues as follows. Initially, the input microscopic images are being binarized using the Otsu's method. This method is one of many binarization algorithms, i.e. it automatically performs reduction of a graylevel image to a binary one. The algorithm
assumes that the image contains two classes of pixels (e.g. foreground and back-
ground) and calculates the optimum threshold separating those two classes so
that their combined spread (intra-class variance) is minimal. At the next step
we segment the binary image to n × n windows and then we calculate Fractal
Dimension (FD) of each window. Fractal dimension is a useful feature proposed
to characterize roughness and self-similarity in a picture, which has been used for
texture segmentation, shape classification, and graphic analysis in many fields.
It can be defined as a ratio providing a statistical index of complexity com-
paring how detail in a pattern changes with the scale at which it is measured.
Finally based on the Fractal Dimension and other image properties we perform
clustering using Fuzzy c-means (FCM) clustering algorithm. The fuzzification
parameter *m* that we use in this work is set to the default value 1.25. The com-
plete scheme of the proposed method is shown at Figure 1. Next at Figure 2 an
example of an image is illustrated in various steps of the procedure. From left
to right we can see the input microscopic image, the greyscale image, the binary
image, and the image that shows the clustering result.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Diagram of the proposed methodology.</figcaption></figure>

4 Experiment setup and results

The results that are presented in this section are based on images that correspond
to 7, 15 and 23 days after bleomycin administration in mice (see Figure 3). The
lung images captured using a Nikon ECLIPSE E800 microscope and a Nikon
DXM1200 digital camera, were initially separated to windows of size 100 × 100
<figure><img src="image_3.png" /><figcaption>**Fig. 2.** An example of an image in various steps of the proposed methodology</figcaption></figure>

pixels and the box-counting method was applied at each window to calculate the corresponding fractal dimension. Next, the images were separated into windows of smaller sizes (5, 10, and 20, respectively). Each of this windows corresponds to a data vector, with attributes the fractal dimension of the initial 100 × 100 window that contains it and the mean value of the intensity of the grayscale window image. Finally, to cluster each of the datasets constituted by such data points, we employed the Fuzzy c-Means clustering algorithm, for 3 and 4 clusters, respectively.

At Figures 4 and 5 we can see the clustering results with respect to the window size for all images where windows of the same cluster are colored equally for the 4 and 3 clusters case respectively. In the 4 cluster case the cluster colored in red corresponds to the severe pathology, the orange cluster corresponds to the mild pathology and green colored cluster corresponds to the normal lung. Finally the blue colored cluster denotes the background. In the 3 cluster case the mild pathology class is missing and the colors red, green and blue corresponds to the severe pathology, normal lung and background class respectively. The results are quite encouraging with respect to their medical content. The clustering algorithm seems to work pretty fine in distinguishing pathologic from normal. Thus the majority of the blocks that are clustered according to their fractal dimension into pathological clusters (i.e. severe and mild) belong into the regions indicated by the expert pathologists as fibrotic.

Tables 1 and 2 report the percentage, with respect to the window size for all images (retrieved after 7, 15 and 23 days), of each cluster found. The computed percentages coincide with manual annotation and scoring performed by our collaborating experts pathologists.

<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Original IPF Microscopy Images retrieved after 7, 15 and 23 days respectively</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>**Fig. 4.** Clustering result for the 4 clusters case with respect to the window sizes for the images retrieved after 7, 15 and 23 days respectively</figcaption></figure>

<table><caption>**Table 1.** Percentages of the 4 clusters found by the clustering algorithm.</caption><thead><tr><th>Window Size</th><th>Severe pathology</th><th>Mild pathology</th><th>Normal lung</th><th>Background</th></tr></thead><tbody><tr><td colspan="5">Image retrieved after 7 days</td></tr><tr><td>5</td><td>10.89</td><td>21.59</td><td>23.19</td><td>44.31</td></tr><tr><td>10</td><td>12.59</td><td>26.86</td><td>26.72</td><td>33.81</td></tr><tr><td>20</td><td>16.55</td><td>31.55</td><td>28.48</td><td>23.40</td></tr><tr><td colspan="5">Image retrieved after 15 days</td></tr><tr><td>5</td><td>11.49</td><td>21.99</td><td>22.00</td><td>44.50</td></tr><tr><td>10</td><td>14.17</td><td>27.01</td><td>25.16</td><td>33.64</td></tr><tr><td>20</td><td>14.52</td><td>33.33</td><td>30.56</td><td>21.58</td></tr><tr><td colspan="5">Image retrieved after 23 days</td></tr><tr><td>5</td><td>15.01</td><td>24.74</td><td>25.50</td><td>34.74</td></tr><tr><td>10</td><td>16.83</td><td>27.19</td><td>29.33</td><td>26.64</td></tr><tr><td>20</td><td>16.25</td><td>29.95</td><td>35.82</td><td>17.96</td></tr></tbody></table>
<figure><img src="image_6.png" /><figcaption>**Fig. 5.** Clustering result for the 3 clusters case with respect to the window sizes for the images retrieved after 7, 15 and 23 days respectively</figcaption></figure>

<table><caption>**Table 2.** Percentages of the 3 clusters found by the clustering algorithm.</caption><thead><tr><th>Window Size</th><th>Pathology</th><th>Normal lung</th><th>Background</th></tr></thead><tbody><tr><td colspan="4">Image retrieved after 7 days</td></tr><tr><td>5</td><td>18.68</td><td>31.30</td><td>50.00</td></tr><tr><td>10</td><td>23.07</td><td>38.37</td><td>38.55</td></tr><tr><td>20</td><td>28.76</td><td>45.19</td><td>26.03</td></tr><tr><td colspan="4">Image retrieved after 15 days</td></tr><tr><td>5</td><td>20.52</td><td>30.56</td><td>48.91</td></tr><tr><td>10</td><td>23.79</td><td>37.40</td><td>38.79</td></tr><tr><td>20</td><td>29.64</td><td>44.98</td><td>25.37</td></tr><tr><td colspan="4">Image retrieved after 23 days</td></tr><tr><td>5</td><td>23.61</td><td>33.58</td><td>42.80</td></tr><tr><td>10</td><td>24.68</td><td>39.87</td><td>35.44</td></tr><tr><td>20</td><td>26.72</td><td>44.43</td><td>28.83</td></tr></tbody></table>

# 5 Conclusions

In the present paper, a computer-assisted image characterization tool based on a Fuzzy clustering method was introduced for the quantification of degree of
IPF in medical images. The implementation of this algorithmic strategy is very promising concerning the issue of the automated assessment of microscopic images of lung fibrotic regions. The results obtained so far show that the proposed strategy addresses the vital biological issues concerning their imaging part as this is contained in the specific type of microscopic images.

In a future study, we intent to use the degree of membership to postpro-
cess the clustering results in an attempt to eliminate small isolated regions and
increase the accuracy of the proposed approach.

**Acknowledgments:** The authors would like to thank the European Union (European Social Fund ESF) and Greek national funds through the Operational Program "Education and Lifelong Learning" of the National Strategic Reference Framework (NSRF) - Research Funding Program: "Heracleitus II. Investing in knowledge society through the European Social Fund." for financially supporting this work.

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