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Fuzzy Fractals in Cervical Cancer

T. Sudha and G. Jayalalitha*

Department of Mathematics, VISTAS, Chennai, India

Abstract

To describe complicated systems often requires a mathematical model because it is designed for precise description. To model dynamic processes in biology mathematics is used. Using mathematical and biological data to model the growth of cancer cell is a boom area of cancer disease.

Cancer cells keeps developing and apparently unpredictable behavior arising in a deterministic system because of great sensitivity to initial conditions. This chapter represents the images of cancer based on cell formation in the tissue. It explained the complexity of Cancer Cell (CC) images using Fuzzy matrix method, and Sausage method with Maximum Modulus theorem.

Matrices are used in representing the cancer cell images of different stage of Cancer patients. The value computed from a square matrix of numbers by a rule of combining products of the matrix entries And from Eigen Vectors shows the cancer cells extension by stretching or compressing and also Eigen values shows the factor of compressing. Eigen Vectors of Normal Cancer cells to Abnormal Cells a re-irregular. The method to find the dimension of irregular figures in Fractal Geometry is called Box-Counting method. Using Box Counting method the Area, perimeter and radius of the Cancer cells are used in Sausage method to find out the invasiveness of cancer cells and also the irregularity growth of cancer cells. Fuzzy Fractals is a noval mathematical combination. This approach is led to proper results which will improve the determination and the stage of the cancer.

Keywords: Fractals, fuzzy, fuzzy matrix, cervical cancer, sausage method

1.1 Introduction

1.1.1 Fuzzy Mathematics

Fuzzy Mathematics is a branch of mathematics which contains Fuzzy Set theory and Fuzzy Logic. In 1965 Lotfi A. Zadeh introduced Fuzzy concepts.

1.1.1.1 Fuzzy Set

Fuzzy set is a generalization of a classical set and the membership function a generalization of the characteristic function [23]. Since we refer to a universal (crisp) set X, some elements of a fuzzy set may have the degree of membership zero [23]. Often it is appropriate to consider those elements of the universe that have a nonzero degree of membership in a fuzzy set [1, 23]. Example: Is Ganesh is honest? Answer is; Extremely honest, very honest, slightly honest, sometime honest ......etc.

For a fuzzy set, the characteristic function allows various degrees of membership for the elements of a given set [24]. If X is a collection of objects denoted generically by x, then fuzzy set à in X is a set of ordered pairs: à = {(x, µÃ (x))| x e X}, µ à (X) is called the membership function or grade of membership (also degree of compatibility or degree of truth) of x in A that maps X to the membership space M (When M contains only the two points 0 and 1, à is non fuzzy and µÃ(X) is identical to the characteristic function of a non fuzzy set) [24, 27].

The range of the membership function is a subset of the nonnegative real numbers whose superimum is finite [24]. Elements with a zero degree of membership are normally not listed [1, 24].

The application of Fuzzy set theory is a high-yielding and interesting area in the Medical field. For decision making problems Fuzzy set theory plays a very important role [7]. Fuzzy set theory has already been used in some medical expert systems [7]. Sanchez [8] formulated the diagnostic models involving fuzzy matrices representing the medical knowledge.

1.1.1.2 Fuzzy Logic

Fuzzy logic deals with reasoning with inexact or fuzzy concepts. Hence the well established isomorphism's between Boolean algebra, set theory and propositional logic can be extended in a natural way between fuzzy algebra, fuzzy set theory and fuzzy logic [2].

1.1.1.3 Fuzzy Matrix

Fuzzy matrices were introduced for the first time by Thomson [6] who discussed the convergence of powers of fuzzy matrix [2, 26, 28]. Fuzzy matrices play a vital role in scientific development. By a fuzzy matrix, we mean a matrix over a fuzzy algebra [26]. A Boolean matrix is a special case of a fuzzy matrix with entries from the set {0, 1} [28]. In practice, fuzzy matrices have proposed to represent fuzzy relations in a system based on fuzzy set theory [14, 15]. A fuzzy matrix can be interpreted as a binary fuzzy relation [2, 26].

Cagman et al. [13] defined fuzzy soft matrix theory and its application in decision making. A fuzzy associative matrix express fuzzy logic rules in matrix form this rules usually takes to variables as input, mapping clearly to a 2-dimensional matrix [10, 11], although theoretically a matrix of any number of dimensions is possible [12, 20].

1.1.2 Fractals

Mandelbrot defined Fractal as a special class of subsets of a complete metric space for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension. In other words, a given set of randomized, hyperbolic, Iterated Function System (IFS) is able to generate a particular image as a fractal set [29]. Fractals are important because it CHANGE the most basic ways to analyze and understand experimental data. The term Fractals was coined by Benoit Mandelbrot in 1975. Fractals are kind of shapes that can be seen in nature. Lightning, Clouds are the examples of Fractals as shown in Figure 1.1. Cantor set, Sierpinski Triangle, Von Koch curve are also examples of fractals. One of the characteristic of Fractals is self-similarity. Three types of self-similarity found in fractals such as

• Exact Self-Similarity [17, 30]
  According to Perfect self-similarity each of the fractals is composed of smaller versions of itself.
• Quasi-Self-Similarity [17, 30]
  This is a loose form of self-similarity the fractal appears approximately (but not exactly) identical at different scales.
• Statistical Self-Similarity [17, 30]
  Random fractals are examples of fractals which are statistically self-similar, but neither exact nor quasi-self-similar.

Figure 1.1 Examples of fractals.

1.1.2.1 Fractal Geometry

The shapes comes out from the Fractals geometry are rough and infinitely complex. However Fractals geometry is still about making shapes, measuring the shapes and defining the shapes [9].

Fractal geometry characterizes the complexity of the image analysis [9, 25]. It is helpful to qualifying the morphologies that are considered random or irregular. It also provides of pattern formation in diffusion and percolation [9]. It usually has a non-integer dimension and greater than topological dimension and less than Euclidean dimension [9].

The box-counting dimension is calculated as follows:

where F is a non-empty bounded set in Rn, Nd(F) is the minimum number of the sets covering F and their radii are no larger than d [25]. Sample data can be obtained for (-log d, log Nd(F)) [25]. Slope of the line can be estimated of the box-counting dimension by the least-squares method [21, 22]. Images of cancer can be treated as 2D [25].

1.1.3 Fuzzy Fractals

In Fuzzy Fractals, the algorithm for a specific problem can be coined by incorporating the fuzzy set and fractals together and create a new dimension by considering the data as a time series problem. The Fuzzy Fractals are used to find the relationship between the crisp values and their properties for an investigation [29].

1.1.4 Cervical Cancer

Cancer is an abnormal growth of cells and is uncontrolled in a way that leads to metastasize shown in Figure 1.2 [5]. Cancer can be defined as a disease in which a group of abnormal cell grows uncontrollably by disregarding the normal rules of cell division [3]. Normal cells are constantly subject to signals that dictate whether the cell should divide, differentiate into another cell or die [3]. Cancer cells develop a degree of autonomy from these signals resulting in uncontrolled growth and proliferation. If this proliferation is allowed to continue and spreads, shown in Figure 1.7, it can be fatal [3, 5].

Initiation and progression of cancer depend on both external factors in the environment (tobacco, chemicals, radiation and infectious organisms) and factors within the cell (inherited mutations, hormones, immune conditions and mutations that occur from metabolism) [3]. Pap Smear Test shown in Figure 1.3 is a tool, to locate the cancer disease explains in the cervical part [4, 5]. Human Papilloma Virus [HPV] shown in Figure 1.7 infection is a necessary factor in the development of nearly all cases of Cervical Cancer [18, 19].

In this chapter, Section 1.2.1 explained Fuzzy method that gives uncertainty growth of Cancer cells and Section 1.2.2 shows the Sausage methods that shows the Invasiveness of the Cervix Cancer and also explained Maximum Modulus theorem in Section 1.4. In Section 1.5 Results are discussed.

Figure 1.2 Cells of cervix.

Figure 1.3 Pap smear test of the cervical cancer.

1.2 Methods

1.2.1 Fuzzy Method

Here it explained uncertainty growth of cervix cells, for this many patients Cervix cancer cells images of different...