With the increase of precision and speed of DNA sequencing, it has become clear that just standard evolu- tionary genetic model of cancer may be not enough to understand the nature of can- cer . A recently observed sharp increase in the complexity and variability of genetic signatures of activated/mutated genes asso- ciated with cancer has considerably slowed the advancement in this direction. There is a hope that physical sciences can provide a missing link to understand and eventually eradicate cancer.
Fractal geometry is one of the intrigu- ing mathematical constructs. If a surface is fractal, its geometry repeats itself periodi- cally at different scales .
In 1997, Sedivy and Mader proposed a connection between cancer and fractal . It was justified by the observation that cancer tissues look rather random, chaotic. Fractal, on the other hand, typically occurs when the geom- etry is formed from chaos (or far-from equilibrium processes, which are quite similar to chaos as well). Indeed, cancer- specific fractal geometry of tumors was found at the tissue scale when analyzing tumor perimeters [4,5]. Fractal geometry was also found in the structure of tumor angiogenesis [6,7].
The search for appearance of fractals at the single-cell level is of particular inter- est. For example, it is still unclear if can- cer develops from an individual cell. In other words, if there is a clear cancer marker for individual cells rather than the entire tumor. A strong correlation between cancer and fractal at the cellular level could be such a biophysical marker. Nevertheless, high-resolution (electron) images of cells did not show the expected transition to fractal geometry when cells become can- cerous. Cells derived from cancer and nor- mal tissues demonstrated almost ideal frac- tal geometry (although having different
“It has been found that the fractal geometry on the cell surface (and conceivably, chaos) is reached only at a particular stage of progression to cancer, when premalignant (immortal) cells are transformed into cancerous.”
fractal dimension, in other words, the degree of ‘roughness’ of the fractal surface) [8,9]. Imaging of cells by means of atomic force microscopy (AFM) [10,11] demonstrated higher than SEM resolution of cell surface. It also showed a clear segregation between cancer, immortal (precancerous), and normal cells when using the frac- tal dimension parameter. However, the use of fractal dimension does not imply presence of fractal geometry. Technically, fractal dimension can be calculated for any surface which is not necessarily fractal.
It is here worth mentioning one important feature of the definition of fractal. Rigorously speaking, fractals are defined for the infinite range of scales, from infinitely small to infi- nitely large. In reality, it is obviously impossible to obtain an image with infinite resolution and unlimited in size. For example, when imaging cells, the geometric scale is limited by the cell size (~10 μm) and the resolution of the used imaging technique (1–20 nm in the case of the AFM imaging). Therefore, it is plausible to limit the fractal definition to these scales when study- ing individual cells. This has been done when studying cell fractals.
The emergence of fractal geometry on cell surface during progression toward cancer has been finally found in , in which human cer- vical epithelial cells were imaged by a high-res- olution AFM technique. The maps of adhesion between the AFM probe and cell surface were analyzed, which can be approximately treated as high-resolution topography images (see  for detail). The cervical cell model was chosen as a well-developed system to study development of an epithelial cancer. To better understand what was discovered in , let us describe a few technical details. A self-correlation function is used to define fractal and calculate the fractal dimension. The definition of fractal implies that the self-correlation function obeys a power-law dependence on the geometric scale. It can easily be seen as a straight line in the log–log scale. The tilt of that straight line is proportional to the fractal dimension. Divergence of the self- correlation function from this behavior means deviation from being fractal. This deviation was studied in .
The results reported in  were quite unex- pected. It has been found that the fractal geome- try on the cell surface (and conceivably, chaos) is reached only at a particular stage of progression to cancer, when premalignant (immortal) cells are transformed into cancerous. Before and after that, geometry of the cell surface deviates from fractal. Normal and cancer cells diverged from fractal in different directions. The parameter of deviation from fractal (called ‘multifractality’; multifractality is zero for fractal ) was posi- tive for normal and immortal cells (approaching zero for immortal cells), and turn to negative for cancer, crossing zero between immortal and can- cerous cells. Contrary to the initial expectations, these results vote in favor of considering cancer (at least, at the cellular level) as rather determin- istic development, and not a chaotic disbalance of biochemical reactions (note: this can only be concluded for the reactions which shape the cell surface). At the same time, transition from immortal to cancer cells can be considered as a switch associated with a chaotic disbalance. Later development of cancer is associated with further and further deviation from fractal.
The study of fractal geometry of cell surface performed in  was purely physical, nonspecific from biochemical point of view. It is interesting to discuss biomolecular nature of the changes observed. As was shown, the pericellular brush changed substantially when normal cells turn into cancerous [13,14]. The term ‘(peri)cellular brush’ was introduced in [15,16] as the pericellular layer detected by AFM. It consists of glycoca- lyx (or pericellular coat), microvilli, membrane ‘wrinkles’ (microridges). When preparing cells for AFM imaging, the cells were freeze-dried. When exposed to ambient humidity, the pericel- lular brush was collapsed. Thus, the observed surface geometry is topography created by microvilli, microridges and glycocalyx.
It is also interesting to note on geometrical size of surface features which contributes the most to the change of fractal during progression to can- cer. Comparing the difference in self-correlation function, which was measured to study fractal as described above, one can see in  that the largest changes can be observed at the scale of surface features ranging between 1 and 300 nm. This is quite large scale to speak about difference in particular molecules. It should be clusters of molecules and microvilli’s/microridges. Thus, the change in fractal geometry on the cell surface during progression to cancer is presumably asso- ciated with aggregates or clusters of microvilli/ microridges and glycocalyx molecules.
How these results may be useful? Besides straightforward utilization of the parameter describing deviation from fractal as a new physi- cal biomarker, these results can give some insight for modeling cancer. Next, these results can give some insight for modeling cancer. A nonchaotic behavior of cancer cells votes in favor of models which consider cancer as a switch to another cell behavior rather than chaotic disbalance , for example, a switch to an evolutionary regression to the single cell organisms. These results may also bring a new strategy to attack on cancer. For example, it is known that chaotic behavior is typically associated with the existence of points of instability , the points which influence cha- otic development the most. By disturbing these instability points, it is possible in principle to prevent development of the chaotic behavior. This may open a new avenue to fight cancer. Scientists can start identifying those instabili- ties in the pathways related to the formation of cell surface. Targeting those points with specific chemicals might prevent the development of the chaotic behavior, and conceivably, to turn the switch to cancer off.
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Fractals and cancer.
Recent studies have shown that fractal geometry, a vocabulary of irregular shapes, can be useful for describing the pathological architecture of tumors and, perhaps more surprisingly, for yielding insights into the mechanisms of tumor growth and angiogenesis that complement those obtained by modern molecular methods. This article outlines the basic methods of fractal geometry and discusses the value and limitations of applying this new tool to cancer research.
- Correspondence re: J. W. Baish and R. K. Jain, Fractals and Cancer. Cancer Res., 60: 3683-3688, 2000. [Cancer Res. 2001]
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