Applied Mathematical Sciences, Vol. 5, 2011, no. 30, 1449 - 1458

Local Fuzzy c-Means Clustering

for Medical Spectroscopy Images

Andres Barrea

CIEM - Universidad Nacional de Cordobaabarrea@mate.uncor.edu

Abstract

In this paper we use a local fuzzy c-means clustering for analysis ofdata from spectroscopy which allows validate the hypothesis of the ac-tion clioquinol (CQ), a new drug for prostate tumors. But even throughthe algorithms proposed, it is possible to see that the action of the drugdepends on the concentration of copper in tissue, which is known byprevious studies to be higher in tumor tissue than in healthy subjects.

Mathematics Subject Classification: 92-08, 69Uxx

Keywords: Prostate cancer - Fuzzy c-means clustering (FCM) - Imagesegmentation

1 Introduction

Recently, organic copper complexes has been report as proteasome inhibitorsand apoptosis inducers [2].

Apoptosis is a highly conserved cellular suicide program in multicellularorganism from worms to human. This cellular death program serves as ameans to maintain multicellular organisms by discarding and damaged andundesirable cells [5].

It has been suggested that cancer cells are more sensitive to several apoptosis-inducing stimuli than normal cells, including proteasome inhibitors [4].

The proteasome is a intracellular protease, and its functions in cells area variety of important intracellular events, including cell cycle progression,antigen-presenting pathway and apoptosis [1].

Copper stimulates proliferation and migration of human endothelias cellsand is requered for the secretion of several angiogenic factors by tumor cells([3], [10]).

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In addition , there is increasing evidence suggesting a correlation betweenapoptosis and angiogenesis. For example, some angiogenesis inhibitors, in-cluding angiostatin and endostatin, induce dormancy of primary tumors andmetastases by indirectly increasing apoptosis in tumor cells.

There is evidence that copper chelation acts through inhibition of NF-κBactivity in cancer cell lines although the molecular mechanism has not beenshow [12].

Clioquinol (CQ) is a lipophilic compound of quinoline class that is capableof forming stable complexes with copper(II) ions [14]. CQ is a well toleratedcopper binding compound and CQ-copper complexes is a potent proteasomeinhibitor and apoptosis inducer in tested prostate and breast cancer.

In this work we applied an algorithm, Fuzzy C-means Clustering (FCM),for analysis of data from spectroscopy which allows validate the hypothesis ofthe action Cu and CQ in prostate tumors.

FCM clustering is an unsupervised technique that has been successfullyapplied to feature analysis, clustering, and classifier designs in fields such asastronomy, geology, medical imaging, target recognition and image segmenta-tion.

Fuzzy C-means (FCM) is one of the most used methods for image segmen-tation and its success chiefly attributes to the introduction of fuzziness for themembership of each image pixels.However, one disadvantage of standard FCMis not to consider any spatial information in image context.

Recently, many researchers have incorporated local spatial information intothe original FCM algorithm to improve the performance of image segmenta-tion. [19, 20, 21, 22, 23].

Different algorithms have been proposed in the literature, all with limita-tions in different areas such as computing time, precision and convergence. Werefer to the bibliography for a more detailed explanation of the various algo-rithms and their applications. Nevertheless the application of FCM to imagesfrom spectroscopic studies of this type of work are not found in the literature.

The main objective of this work to propose a methodology that allowssimple and fast spectroscopic analysis of images to check clinical hypothesis towhich it has been through more sophisticated methods.

2 Data Description

Tissue samples were obtained from male mice. The samples are divided intohealthy tissue (C4 Normal), tissue untreated injected with C4 line cells (C4Tumor) and tissue treated with CQ (C4 CQ Tumor). Four samples weremeasured under the following conditions. X-ray focused Beam (5 mx5 m beamsize using a KB mirrors system , 10.0 keV ) was used for fluorescence excitationof the samples. The geometrical arrangement was the standard, i.e., the sample

Local fuzzy c-means... 1451

is mounted at 45 degree angle relative to the beam path and the detectionsystem is at 90 degrees. A Ketek silicon drift detector 100 mm2 active areawas used for fluorescence detection. The intensity of the micro x-ray beam wasmonitored using a small ion chamber mounted downstream the KB mirrors infront of the sample. The samples were mounted in an XYZ positioner, whichallows moving the sample with 0.1 m accuracy in any direction. The scanswere performed in both vertical and horizontal directions to cover the entiresize of the sample by the excitation beam, using a step size of 5 microns inboth directions. The acquisition time was 1 seconds per point. The data wasanalyzed using a Matlab code written for this purpose. The fluorescence countswere normalized by the intensity of the incoming beam.

Figure (1) represents the elemental distribution of Ca, Fe, Cu and Zn fromsample normal (C4 Normal). The numbers on each axis represent the actualpoint position in millimeters. The fluorescence intensities are plotted usinga color code that is shown on the left of each picture. The minimum andmaximum values were selected to enhance the contrast between backgroundsignal and fluorescence signal. The spots having the maximum value or higherare shown in red. The color code range for the elements was the same in allpictures for comparison purposes.

Figure (2) represents the distribution of metals from sample with injectedC4 cell without treatment (C4 Tumor), and figure (3) sample treated with CQ(C4 CQ Tumor).

Figure 1: Distribution Normal Tissue

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Figure 2: Distribution Untreated Tumor

Figure 3: Distribution Treated Tumor

3 Algorithm Description

In this paper we use a similar version of the an algorithm presented in [16] andwe refer to it for a detailed analysis of the algorithm.

Given a set of data {x1, ..xN}, one can raise the problem of separating theminto K sets, so that each group representing a particular condition among thedata that they belong to, this problem is called clustering. There are manyalgorithms to do this, in this case we consider a generalization of the well-known C-means algorithm.

In the algorithm C-means we consider prototypes vectors (centers) C1, ..CK

which are the representatives of the K groups.

The components of these vectors are variables and they minimized the

Local fuzzy c-means... 1453

following functional:

J(C) =K∑i=1

N∑j=1

d2(xj, Ci).

Now, we consider the following generalized functional:

Jm(C) =K∑i=1

N∑j=1

(uji)md2(xj, Ci),

where the element uji represents the probability of data xj to belong to thegroup i. Of course,we need the following condition:

K∑i=1

uji = 1,

and the parameter m is called the parameter of fuzziness, chosen equal totwo in this case.

The algorithm earlier is not designed initially for images, as it does notconsider the local distribution (spatial) of data, two vectors equal will be equalregardless of their location space for resolve this the function is penalized witha term that takes into account this distribution as follows:

Jm(C) =K∑i=1

N∑j=1

(uji)md2(xj, Ci)

+ αK∑i=1

N∑j=1

(uji)md2(xj, Ci).

where xj is the average in a window determined by the user.The idea is to implement this algorithm to the maps of distribution of Ca,

Cu, Fe and Zn obtained by spectroscopy of normal tissue, untreated tumortissue and tumor tissue treated with the drug. This will form the vectors

xk = (Ca,Cu, Fe, Zn),

where the image is read from top to bottom and from left to right for theposition k and placed in the vector concentrations given by the image at thatpoint.

4 Results and Discussion

First, we project the data on the coordinates Cu, Fe and Zn and to observea possible separation of these two groups; which is not observable.

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Figure 4: Projection Data

By applying the algorithm to data to separate them into two groups andtaking for each concentration of the highest probability of copper being in agroup (matrix U) there is a separation almost disjoint value in 0.022 for thetreated tissue. In the region intersection probability of being in a group orthe other this at [0.49, 0.51]. In all figures, y-axis is the copper concentration(normalized) and x-axis represent the membership of each group of vectors.

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 20

0.05

0.1

0.15

Figure 5: Separation of Cu into 2 groups - Treated Tissue

For the untreated tissue the separation is almost the same but the thresholdis less than 0.022.

For the normal tissue there is no evidence of the separation in two groupshave a bearing on the amount of copper, i.e. the intersection is great, moreoverthe group 2 included in almost group 1. Put another way data with the sameamount of copper may belong to any group with high probability.

In the following three graphics viewing the separation of other metals

Local fuzzy c-means... 1455

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 20

0.02

0.04

0.06

Figure 6: Separation of Cu into 2 groups - Untreated Tissue

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 20.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0.022

Figure 7: Separation of Cu into 2 groups - Normal Tissue

1 1.2 1.4 1.6 1.8 20

0.02

0.04

0.06Ca

1 1.2 1.4 1.6 1.8 20

0.05

0.1

0.15Cu

1 1.2 1.4 1.6 1.8 20

0.02

0.04

0.06

0.08Fe

1 1.2 1.4 1.6 1.8 20

0.05

0.1

0.15Zn

Figure 8: Separation of the four metals into 2 groups- Treated Tissue

Despite low sample data, in the case of the treated tissue and untreatedtissue the separation into two groups is apparently related to the concentrationof copper, which in principle was not obvious, and gives indications of existenceof a concentration threshold that separates two types of processes and two

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1 1.2 1.4 1.6 1.8 20

0.01

0.02

0.03

0.04Ca

1 1.2 1.4 1.6 1.8 20

0.02

0.04

0.06Cu

1 1.2 1.4 1.6 1.8 20

0.05

0.1

0.15

0.2

0.25

Fe

1 1.2 1.4 1.6 1.8 20

0.02

0.04

0.06

0.08

0.1

0.12

Zn

Figure 9: Separation of the four metals into 2 groups- Untreated Tissue

1 1.2 1.4 1.6 1.8 20

0.02

0.04

0.06

0.08Ca

1 1.2 1.4 1.6 1.8 20.005

0.01

0.015

0.02

0.025

0.03

Cu

1 1.2 1.4 1.6 1.8 20

0.05

0.1

0.15Fe

1 1.2 1.4 1.6 1.8 20

0.02

0.04

0.06Zn

Figure 10: Separation of the four metals into 2 groups - Normal Tissue

kinds of cells.

5 Summary

In this paper a local fuzzy clustering method is proposed for obtain a charac-terization of the action of a new drug in prostate cancer. The method allowsto observe the action of the drug depends on the concentration of copper in tis-sues, a hypothesis that emerged from previous clinical trials. This algorithm issimple and allows to analyze images obtained by spectroscopy of tissues treatedwith the drug, the results are easy to interpret and allow them to support theclinical hypothesis.

The proposed methodology is simple and fast, and we want to present it asa tool in the analysis of complex chemical processes combined with standardqualitative methods used by researchers.

ACKNOWLEDGEMENTS. The author is thankful to Dr. Raul Barrea

Local fuzzy c-means... 1457

because without their data and ideas that work would not have been possible.

References

[1] Oikawa, T. et al. 1998. The proteasome is involved in angiogenesis. Bio-chemical and Biophysical Research Communications 246, 243-248.

[2] Daniel, K. et al. 2004. Organic copper complexes as a new class of protea-some inhibitors and apoptosis inducers in human cancer cells. BiochemicalPharmacology 67, 1139-1151.

[3] Hu, G. 1998.Copper stimulates proliferation of human endothelias cellsunder culture. J Cell Bioche. 69, 326-335.

[4] Adams, J. 2003. Potential for proteasome inhibition in the treatment ofcancer. Drug Discov Today. 8, 307-315.

[5] Song, Z., Steller, H. 1999. Death by design: mechanism and control ofapoptosis. Trends Cell Biol 9, 49-52.

[6] Theophanides, T. Anastassopoulou, J. 2002. Copper and carcinogenesis.Crit Rev Oncol Hematol. 42, 57-64.

[7] Twombly, R. 2003. First proteasome inhibitor approved for multiplemyeloma. J Natl Cancer Inst 95:845.

[8] Kumar, A. et al 2004. Nuclear factor κB: its role in health and disease. JMol Med. 82, 434-448.

[9] Adams, J. et al 1999. Proteasome inhibitors: A novel class of potent andeffective antitumor agents. Cancer Research 2615-2622.

[10] Lowndes, S. Harris, A. 2005. The role of copper in tumor angiogenesis. JMammary Gland Biol Neoplasia. 10, 299-310.

[11] Folkman, J. 1971. Tumor angiogenesis: therapeutic implications. N EnglJ Med. 285, 1182-1186.

[12] Pan, Q. et al 2003. Tetrathiomolybdate inhibits angiogenesis and metas-tasis through suppression of the NFκB signaling cascade. Mol Cancer Res.10, 701-706.

[13] Ding. W. et al 2005. Anticancer activity of the antibiotic clioquinol. Can-cer Res. 8, 3389-3395

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[14] Lipniacki, T. et al 2004. Mathematical model of NFκB regulatory module.J Theo Biol. 228, 195-215.

[15] Olivier, S. et al 2006. Can NFκB be a target for novel and efficient anti-cancer agents? Bioch Pharmacology. 72, 1054-1068.

[16] Weiling, C. et al. 2007. Fast and robust fuzzy c-means clustering algo-rithms incorporating local information for image segmentation PatternRecognition, 40, 3, 825-838.

[17] M.N. Ahmed, et al.,2002 A modified fuzzy c-means algorithm for bias fieldestimation and segmentation of MRI data, IEEE Trans. Med. Imaging 21,pp. 193-199

[18] S.C. Chen,et al. 2004 Robust image segmentation using FCM with spatialconstraints based on new kernel-induced distance measure, IEEE Trans.Systems Man Cybernet. B 34 (4), pp. 1907-1916.

[19] D. L. Pham,et al. 1999 An adaptive fuzzy c-means algorithm for imagesegmentation in the presence of intensity inhomogeneities, Pattern Recog-nition Lett. 20, pp. 57-68.

[20] Y. A. Tolias,et al. 1998 On applying spatial constraints in fuzzy imageclustering using a fuzzy rule based system, IEEE Signal Process Letters(5), pp. 57-68.

[21] Y. A. Tolias,et al. 1998 Robust image segmentation using FCM withspatial constraints based on new kernel-induced distance measure, IEEETrans. Systems Man Cybernet. A 28, pp. 359-369.

[22] A. W. C. Liew,et al. 2000 Fuzzy image clustering incorporting spatialcontinuity, Inst. Elec. Eng. Vis. Image Signal Process 147, pp. 185-192.

[23] D. L. Pham,et al. 2002 , Fuzzy clustering with spatial constraints IEEEProceedings of the International Conference Image Processing, pp. 65-68.

Received: October, 2010