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Fractal Case Study for Mammary Cancer: Analysis of Interobserver Variability

  • Philipp Hermann
  • Sarah Piza
  • Sandra Ruderstorfer
  • Sabine Spreitzer
  • Milan StehlíkEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 136)

Abstract

This paper discusses some features of the distribution of box-counting fractal dimension measured on a real data set from mammary cancer and masthopathy patients. During the study we found several reasons why mammary cancer and its following distribution cannot be easily represented by single box-counting dimension. The main problem is that without a histopathological examination of the tumor a simple algorithm based only on single box-counting dimension is difficult to be constructed. We have tried to understand the distribution underlying the real data, especially its departures from normality. Both normal and gamma distributions are related to the Tweedy distributions, which are given by multi-fractal dimension spectra present in histopathological images. Without having a histological examination of the data multifractality is unavoidable as can be seen from several analysis in this paper. We have seen a fair differentiation between cancer and masthopathy. Finally we studied the depths of the data based on the information divergence. Some practical conclusions are also given.

Keywords

Depth Discrimination Mammary cancer Multifractality Skewness 

Notes

Acknowledgments

The research received partial support from the WTZ Project No. IN 11/2011 “Thermal modelling of cancer”. The corresponding author acknowledges Proyecto Interno 2015, REGUL. MAT 12.15.33, Modelaciòn del crecimientode tejidos con aplicaciones a la Investigaciòn del càncer. First author thanks to the support of ANR project Desire FWF I 833-N18. We also thank the editor and reviewers, whose insightful comments helped us to sharpen the paper considerably.

References

  1. 1.
    Baish, J.W., Jain, R.K.: Fractals and cancer. Perspect. Cancer Res. 60, 3683–3688 (2000)Google Scholar
  2. 2.
    Breslow, N.E., Day, N.E.: Statistical Methods in Cancer Research, Volume 1: The Analysis of Case Controls Studies. IARC, Lyon (1980)Google Scholar
  3. 3.
    Chakravarthi, S., Choo, Z.W., Nagaraja, H.S.: Susceptibility to renal candidiasis due to immunosuppression induced by breast cancer cell lines. Sci. World J. 5(1), 5–10 (2010)Google Scholar
  4. 4.
    Enby, E.: A breast cancer tumor consisted of a spore-sac fungus (Ascomycotina). 3rd Millennium Health Care Sci. 18(1), 8–10 (2013)Google Scholar
  5. 5.
    George L.E., Kamal H.S.: Breast cancer diagnosis using multi-fractal dimension spectra. In: 2007 IEEE International Conference on Signal Processing and Communications (ICSPC’07) (2007)Google Scholar
  6. 6.
    Hermann, P., Mrkvička, T., Mattfeldt, T., Minárová, M., Helisová, K., Nicolis, O., Wartner, F., and Stehlík, M.: Fractal and stochastic geometry inference for breast cancer: a case study with random fractal models and Quermass-interaction process. Statistics in Medicine (2015) doi:  10.1002/sim.6497
  7. 7.
    Kitsos, C.P.: Cancer Bioassays: A Statistical Approach, p. 110. LAMBERT Academic Publisher, Saarbrucken. ISBN 978-3-659-29451-8 (2012)Google Scholar
  8. 8.
    Kitsos, C.P.: Estimating the relative risk for the breast cancer. Biom. Lett. 47(2), 133–146 (2010)Google Scholar
  9. 9.
    Mandelbrot, B.: The Fractal Geometry of Nature. W.H. Freeman and Co., New York (1982)zbMATHGoogle Scholar
  10. 10.
    Mrkvička, T., Mattfeldt, T.: Testing histological images of mammary tissues on compatibility with the Boolean model of random sets. Image Anal. Stereol. 30, 11–18 (2011)Google Scholar
  11. 11.
    Pázman, A.: Nonlinear statistical Models (chapters 9.1 and 9.2). Kluwer Academic Publication, Dordrecht (1993)Google Scholar
  12. 12.
    R Development Core Team. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing: Vienna, 2010. http://www.R-project.org [12 June 2013]
  13. 13.
    Stehlík, M., Giebel, S.M., Prostakova, J., Schenk, J.P.: Statistical inference on fractals for cancer risk assessment. Pakistan J. Statist. 30(4), 439–454 (2014)Google Scholar
  14. 14.
    Stehlík, M.: Distributions of exact tests in the exponential family. Metrika 57(2), 145–164 (2003)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Stehlík, M., Fabián, Z., Střelec, L.: Small sample robust testing for normality against Pareto tails. Commun. Stat.—Simul. Comput. 41(7), 1167–1194 (2012b)Google Scholar
  16. 16.
    Stehlík, M., Wartner, F., Minárova, M.: Fractal analysis for cancer research: case study and simulation of fractals. PLISKA—Studia Mathematica Bulgarica 22, 195–206 (2013)MathSciNetGoogle Scholar
  17. 17.
    Wosniok, W., Kitsos, C., Watanabe, K.: Statistical issues in the application of multistage and biologically based models. In: Prospective on Biologically Based Cancer Risk Assessment, pp. 243–273. Plenum Publication (NATO Pilot Study Publication by Cogliano, Luebeck, Zapponi (eds.)) (1998)Google Scholar
  18. 18.
    Zuo, Y., Serfling, R.: Nonparametric notions of multivariate “scatter measure” and “more scattered” based on statistical depth functions. J. Multivar. Anal. 75(1), 62–78 (2000)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Philipp Hermann
    • 1
  • Sarah Piza
    • 1
  • Sandra Ruderstorfer
    • 1
  • Sabine Spreitzer
    • 1
  • Milan Stehlík
    • 1
    • 2
    Email author
  1. 1.Department of Applied StatisticsJohannes-Kepler-University LinzLinz a. D.Austria
  2. 2.Departamento de MatemáticaUniversidad Técnica Federico Santa MaríaValparaísoChile

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