Fractal study of tumors

Abstract

Fractal geometry was first invented by mathematicians of the late 19th and early 20th century to study problems as abstract as the existence of continuous functions non derivable at any point, the continuum hypothesis, or the existence of topological spaces with strange properties. It was not until the second half of the 20th century that fractal geometry became popular, and it was applied to the study of natural phenomena by the hand of B. Mandelbrot. The study of tumor growth using fractal geometry is relatively recent. This study is based on the fractal character of tumor contours. One of their most important features is roughness, since its increase is related to the tumor growth. Tumor growth dynamic can be described by a function that depends on both time and position, known as local width function, as well as by some power law exponents, called critical exponents. The aim of this final degree project is to study all these concepts in connection with fractals. Elementary concepts of the theory of fractals will also be studied, as the iterated function systems, the fractal dimension and the fractal interpolation functions. Since fractal dimension can only be computed accurately for some specific mathematical objects, we will study some methods that estimate fractal dimension of tumors based on its fractal characteristics. Finally, we will develop some Matlab programs which will compute the fractal dimension of tumors, as well as the roughness of their contours. Furthermore, we will compare our results with the experimental data.