Study of Fractal Behavior of Cell Surface Gives a Hint to a New Way of Attack on Cancer

Igor Sokolov (left) and Maxim E. Dokukin

Authors: Igor Sokolov^1,2.3, Maxim E. Dokukin¹

Affiliation:
¹Department of Mechanical Engineering, Tufts University, Medford, MA, USA
²Department of Biomedical Engineering, Tufts University, Medford, MA, USA
³Department of Physics, Tufts University, Medford, MA, USA.

Fractal is one of the most intriguing geometry in nature. Fractal surface keeps repeating itself through different scales. Fractal typically occurs from chaos or far from equilibrium processes (which are actually quite similar to chaos as well) [1]. Sedivy and Mader were two of the first scientists who published -- back in 1997 -- a hypothesis about possible connection between cancer and fractal [2]. The reason behind this idea was in the observation that cancer typically behaves rather randomly and chaotically. Cancer-specific fractal behavior of tumors at the macroscale was found when analyzing the tumor perimeters [3, 4]. Similar behavior was found in the analysis of geometry of vascular system in tumors (antiangiogenesis) [5, 6]. However, when going down to the smaller scale of single cells, the analysis did not show the expected transition to fractal behavior when cells become cancerous.

Both cancer and normal cells demonstrated almost ideal fractal behavior [7, 8], although with different fractal dimension (the degree of roughness of the fractal surface). More precise measurements done with the help of atomic force microscopy (AFM) [9] did show a better segregation between cancer and normal cells based on the use of fractal dimension. The issue is that fractal dimension can be calculated for any surface, no matter whether or not it’s fractal. Although we noticed that cancer cells seemed to be closer to ideal fractals than normal, there was no actual analysis done of how close the cell surfaces were to fractal.

In our recently published paper [10], we finally did study the question of how good the fractal approximation is for cancer and normal cells. Specifically, we looked at human cervical epithelial cells. The cells were so-called primary cells, which were derived from cervical tissue of either healthy humans or tumors of cancer patients. In addition to cancer and normal cells, we added an intermediate stage of cell progression towards cancer, immortal or precancerous cells (these cells were genetically mutated normal cells). Furthermore, we looked at cells at different number of cell population doubling (cell passage, or cell age). This is important because there are various evidence that immortal cells start demonstrating malignant behavior with the increase of population doubling. Similarly, cancerous cells increase their aggressiveness with the number of cell divisions. So, we assumed that the cells are aligned towards cancer with the number of their population doubling.

To see how close the cell surface is to fractal, we need to describe how fractal is defined mathematically. To find if the surface is fractal or not, one needs to calculate so-called self-correlation function, and see how this function depends on the size of the surface features [11]. Sometimes it is easier to calculate the Fourier transform of the surface as a function of the inverse (reciprocal) feature size. A simple power dependence of these functions is the definitive property of fractal. The deviation from that simple power dependence is the deviation from fractal.

Figure 1 shows typical examples of fractal surfaces as well as the image of a cell of study; its zoomed image is recorded with AFM. The magnitude of the Fourier transform as a function of the reciprocal feature size is shown in Fig.1 at the top right column. The power law in this log-log scale should be a straight line. One can see that normal and cancer cells can be approximated by two straight lines rather than one. The immortal cells of this example demonstrate the behavior closest to a straight line, i.e., fractal. To characterize the deviation from fractal, we introduced a new surface parameter, multi-fractality. The multi-fractality is the difference between the two slopes of the Fourier magnitude shown in Fig. 1. If multi-fractality is zero the cell surface is fractal.

Figure 1 (To view with higher resolution, click on the image): Examples of fractal objects (right column). The middle column: An actual image of a cell obtained by means of scanning electron microscope (SEM) and a high-resolution image obtained by an atomic force microscope (AFM). The right column: (Top) The definition of fractal as the straight line of the magnitude versus reciprocal size plotted in log-log scale; (Bottom) Dependence of the multi-fractality parameter on the stage of progression towards cancer. Fractal is reached when the multi-fractality parameter is zero.

Figure 1 (the bottom right column) shows essentially the main result of the work, the dependence of the multi-fractality parameter on the stage of progression towards cancer, from normal through immortal to malignant behavior. Within each cell type, the results are ordered by the number of population doubling. One can see that contrary to the previous expectations, cancer is not a pure fractal. Moreover, its behavior deviates from fractal further with the cancer progression. It seems that the ideal fractal (zero multi-fractality) is reached at a particular moment of transition between precancerous (immortal) cells to cancer.

Based on these results, we can speculate that it votes in favor of theories which consider cancerous cells as another state of cell functioning which is pretty deterministic rather than chaotic. Moreover, it is known that the development of chaotic behavior is typically associated with some bifurcation points. Having effective influences on such points, one could thus prevent chaos, and maybe cancer, from development. Thus, the search of such bifurcation points in the biochemical pathways responsible for the morphology of the cell surface, and effectively influencing those bifurcation points might be a new way to attack on cancer.

We gratefully acknowledge partial funding for this work by Tufts Collaborates! Grant an NSF CMMI-1435655 (IS).

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