Symmetric Proliferation: An Examination of the Fractal Geometry of Tumors

Abstract

S ymmetric P roliferation : A n E xamination of the F ractal G eometry of T umors B S J Rachel Lew Sierpinski carpet model of plane fractals From flowers to faces, nature is abound with symmetry. Most natural objects tend to form according to patterns, a tendency which mathematicians readily exploit in order to create theoretical models of the world around us. The height of a cliff, for instance, is modeled by a one-dimensional line; a snake’s path through the sand is modeled across a two- dimensional surface; a block of ice is modeled as a mass extending into three dimensions. But what about objects in between dimensions? In fact, between the 1-D and the 2-D there exist objects known as fractals. These mathematical objects are infinitely self-similar, which means that upon magnification of a certain part of a fractal, one sees the figure of the overall fractal, and so on unto infinity. Self-symmetry “Picture, for example, water trickling through only the most loosely packed areas of soil in a pot; in the same way, the blood vessels of a tumor grow into the weakest areas of the tissue around it.” allows a fractal to have fractional dimensions because it is not purely linear--the border of a fractal cannot be traced--but this lack of boundedness also means that the fractal never encircles a defined area. Imagine a tree whose branches branch out infinitely, or a snowflake with six tips, each of which looks like the original snowflake, and so on and so forth. Clearly, in fractal models, as in all models of nature, there is a difference between the mathematical and the natural. Natural fractal objects are composed of discrete units and are not infinitely divisible--the fractal pattern must end somewhere, or else, for instance, one might find tiny branches at the cellular level of a tree branch. To correct for this quality, scientists define natural fractal objects as having statistical self-similarity, or when “the statistical properties of the pieces [of an object] are proportional to the statistical properties of the whole” (Grizzi et al., 2008). In the human body, statistically self-similar models are most commonly applied to branching structures in the lung and in networks of blood vessels. The latter application has had particular importance in medical studies of cancer, as there is evidence that understanding the fractal geometry of tumor vasculature may aid in identification and targeted treatment of cancers. Tumor vasculature can in fact be described by a fractal model, and is often distinguished from normal vasculature by either an abnormally high or abnormally low fractal dimension (Zook and Iftekharuddin, 2005). An object’s fractal dimension is a constant between the integers 1 and 2, and might be described as how ‘proliferative’ the object looks; i.e., an object with a higher fractal dimension looks more like an object with true area than like a curve. Baish and Jain observed that blood vessels in the tumors of mice had higher fractal dimensions than the mice’s normal arteries and veins, claiming that “the fractal dimension quantified the degree of randomness to the vascular distribution, a characteristic not easily captured by the vascular density” (Baish and Jain, 2000). Moreover, the researchers noted that tumor vessels tended to be more twisted than normal vessels, having “many smaller bends upon each larger bend” (Baish and Jain, 2000). They also found that the way tumor vessels grew and branched closely matched a type of statistical growth called invasive percolation. In invasive percolation, a substance moves through a medium which has varying degrees of strength, penetrating the weakest areas of the medium and thus branching out to form a network. Picture, for example, water trickling through only the most loosely packed areas of a pot of soil; in the same way, the blood vessels of a tumor grow into the weakest areas of the tissue around it. On the other hand, normal capillaries are traditionally modeled by the Krogh cylinder model, which assumes that the capillaries are straight, relatively spaced, and only reach a cylindrical volume of tissue immediately 18 • B erkeley S cientific J ournal • S ymmetry • F all 2015 • V olume 20 • I ssue 1