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. 2018 Nov 24;10(11):e3630.
doi: 10.7759/cureus.3630.

The Goodness-of-fit of the Fractal Dimension as a Diagnostic Factor in Breast Cancer

Sotirios Maipas  1 , Afroditi Nonni  1 , Ekaterini Politi  1 , Helen Sarlanis  1 , Nikolaos G Kavantzas  1
Affiliations

The Goodness-of-fit of the Fractal Dimension as a Diagnostic Factor in Breast Cancer

Sotirios Maipas et al. Cureus. .

Abstract

A large number of studies have found that the fractal dimension increases with the progression towards pathological or more pathological states, but there are also studies that have demonstrated the opposite relationship. In this study, we calculate the nuclear box-counting fractal dimension of 109 malignant, 113 benign, and 80 normal isolated breast cells in order to investigate its possible diagnostic importance. We computed the fractal dimension and its goodness-of-fit (i.e., the r-squared value that describes how well the regression line fits the set of the measurements) for two different sets of box size lengths. The statistical analysis did not confirm an important diagnostic potential of the nuclear fractal dimension of isolated breast cells. However, the goodness-of-fit did display a diagnostic potential. The r-squared value may be able to serve as a complementary diagnostic parameter.

Keywords: box-counting dimension; breast cancer; cancer diagnosis; fractal dimension; goodness-of-fit.

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Conflict of interest statement

The authors have declared that no competing interests exist.

Figures

Figure 1
Figure 1. Breast smear of malignant cells from a case of breast adenocarcinoma at 40x magnification (A) and the same image after the necessary transformations (B)
Figure 2
Figure 2. Breast smear of benign cells from a case of fibroepithelial tumor at 40x magnification (A) and the same image after the necessary transformations (B)
Figure 3
Figure 3. Breast smear of normal epithelial cells at 40x magnification (A) and the same image after the necessary transformations (B)
See this image and copyright information in PMC

References

    1. Weibel ER. Fractals in Biology and Medicine. Vol. 4. Basel: Birkhäuser Verlag; 2005. Mandelbrot’s fractals and the geometry of life: a tribute to Benoît Mandelbrot on his 80th birthday; pp. 3–16.
    1. Mandelbrot B. The Colours Of Infinity. London: Springer; 2010. A geometry able to include mountains and clouds; pp. 38–57.
    1. Karperien AL, Jelinek HF. The Fractal Geometry Of The Brain. New York: Springer; 2016. Box-counting fractal analysis: a primer for the clinician; pp. 13–43. - PubMed
    1. Nuclear fractal dimensions as a tool for prognostication of oral squamous cell carcinoma. Yinti SR, Srikant N, Boaz K, Lewis AJ, Ashokkumar PJ, Kapila SN. J Clin Diagn Res. 2015;9:0–25. - PMC - PubMed
    1. From fractal geometry to fractal analysis. Losa GA, Ristanović D, Ristanović D, Zaletel I, Beltraminelli S. Appl Math. 2016;7:346–354.

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