I'm not sure how much value there is in arguing about what I meant when I wrote the post. It seems to me, that I should be trusted to express my intentions. Let's not devolve into that OK?
I believe I may revive the thread on the question of randomness if you are interested
OK. Thank you. I see the part at the bottom which says: "Whether fractals will ultimately find a place in the oncologist’s toolbox awaits more controlled comparisons with conventional pathological procedures."
You missed the important point I cited. The question of randomness is not addressed in 'conventional pathological procedures,' which is a different issue. The issue in the article is that the apparent observed randomness is explained by fractal properties of Cancer at the molecular level.
That was written over 20 years ago. Do you have something more current?
I'm not saying you're wrong. But it seems like you are in a rush to correct me. Let's talk about it, OK?
If I grant that it's not random, it's fractal, what does that change about what I wrote; what are the implications on my assertion if the word chaos is replaced by fractal?
The age of the publication is not an issue. This is just an example of the proper use of fractals in cause-and-effect outcomes that appear random and are explained by Fractal relationships. If you replace fractal with chaos it changes the meaning. Fractal properties throughout nature are explained by 'Chaos Theory,' which is a different meaning to chaos as you use it, which is more like a synonym for randomness. Contemporary computer programs for predictable patterns in weather prediction and genetic mutation research.
the following article is another concerning cancer.
Skip main navigation
FUTURE ONCOLOGYVOL. 11, NO. 22GENERAL CONTENT – EDITORIAL
Fractals: a possible new path to diagnose and cure cancer?
Published Online:15 Oct 2015
https://doi.org/10.2217/fon.15.211
About
The war on cancer is rather far from being victorious. The number of oncology patients has been increasing. In part, it can probably be explained by general aging of human population. There is multiple evidence of correlation between cancer development and polluted environment, genetic predisposition to cancer and exposure to some hazardous chemicals. Nevertheless, the general nature of cancer is not known as of yet. Traditional biochemical studies of cancer seem to be running out of steam. With the increase of precision and speed of DNA sequencing, it has become clear that just standard evolutionary genetic model of cancer may be not enough to understand the nature of cancer [
1]. A recently observed sharp increase in the complexity and variability of genetic signatures of activated/mutated genes associated 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 intriguing mathematical constructs. If a surface is fractal, its geometry repeats itself periodically at different scales [
2]. In 1997, Sedivy and Mader proposed a connection between cancer and fractal [
3]. It was justified by the observation that cancer tissues look rather random, chaotic. Fractal, on the other hand, typically occurs when the geometry 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 antiangiogenesis [
6,
7].
The search for appearance of fractals at the single-cell level is of particular interest. For example, it is still unclear if cancer develops from an individual cell. In the 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 cancerous. Cells derived from cancer and normal tissues demonstrated almost ideal fractal geometry (although having different 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 fractal 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 infinitely 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 studying 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 [
12], in which human cervical epithelial cells were imaged by a high-resolution 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 [
12] 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 [
12], 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 [
12].