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Fraktale Geometrie zur Objektivierung des Gradings beim Prostatakarzinom

Fractal geometry in the objective grading of prostate carcinoma

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Zusammenfassung

Hintergrund

Ein möglicher Ansatz komplexe Muster im Tumorgewebe objektiv klassifizieren zu können, ist die mathematische Erfassung der Verteilung von Tumorzellkernen, die als geometrische Repräsentation der Krebszellen dienen, durch fraktale Dimensionen. Die Existenz, sowie die Veränderungen der fraktalen Struktur der Verteilung der Zellkerne haben wichtige Konsequenzen für eine objektive Klassifizierung der Tumoren. Weiterhin kann auch die Komplexität des Tumorwachstums in verschiedenen Karzinomen sowie die interzellulären Interaktionen im Gewebesystem dadurch verglichen werden.

Ergebnisse

In dieser Arbeit stellen wir eine theoretische Einführung in die fraktale Geometrie sowie in die Algorithmen, die auf der Rényi-Familie der fraktalen Dimensionen basieren, dar. Wir führen ein geometrisches Modell für die Bewertung von Prostatakarzinomgeweben ein und erklären den Zusammenhang zwischen dem geometrischen Tumormuster und den fraktalen Dimensionen der Rényi-Familie.

Abstract

Background

A possible approach to objectively classify complex patterns in tumor tissue is a mathematical and statistical investigation of the distribution of cell nuclei as a geometric representation of cancer cells by fractal dimensions. Both the existence and changes in the fractal structure of tumor tissue have important consequences for the objective system of tumor grading. In addition, the complexity of growth in different carcinomas or their intercellular interactions can be compared to each other.

Results

We present a theoretical introduction into fractal geometry as well as in the computer algorithms based upon the Rényi family of fractal dimensions. Finally, a geometric model of prostate cancer is introduced and the relationship between geometric patterns of prostate tumor and the fractal dimensions of the Rényi family are explained.

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Literatur

  1. Altunbay D, Cigir C, Sokmensuer C, Gunduz-Demir C (2010) Color graphs for automated cancer diagnosis and grading. IEEE Trans Biomed Eng 57(3):665–674

    Article  PubMed  Google Scholar 

  2. Baish JW, Jain RK (2000) Fractals and cancer. Cancer Res 60(14):3683–3688

    CAS  PubMed  Google Scholar 

  3. Dey P (2005) Fractal geometry: basic principles and applications in pathology. Anal Quant Cytol Histol 27:284–290

    PubMed  Google Scholar 

  4. Goutelle S, Maurin M, Rougier F et al (2008) The Hill equation: a review of ist capabilities in pharmacological modelling. Fundam Clin Pharmacol 22:633–648

    Article  CAS  PubMed  Google Scholar 

  5. Gil J, Gimeno M, Laborda J et al (2006) Tangential algorithm for calculation of the fractal dimension of kidney tubuli sections. Int J Morphol 24(1):31–34

    Google Scholar 

  6. Grassberger P, Procaccia I (1983) Characterization of strange attractors. Phys Rev Lett 50(5):346–349

    Article  Google Scholar 

  7. Hill AV (1910) The possible efects of the aggregation of the molecules of hemoglobin on the dissociation curves. Proc Physiol Soc 40:4

    Google Scholar 

  8. Karperien A, Jelinek HF, Milosevic NT (2013) Reviewing lacunarity analysis and classification of microglia in neuroscience. In: Waliszewski P (ed) Fractals and complexity. Format, Wrocław, pp 50–56, ISBN 978-83-936598-0-7

  9. Landini G, Murray PI, Misson GP (1995) Local connected fractal dimensions and lacunarity analyses of 60° fluorescein angiograms. Invest Ophthalmol Vis Sci 36:2749–2755

    CAS  PubMed  Google Scholar 

  10. Loeffler M, Greulich L, Scheibe P et al (2012) Classifying prostate cancer malignancy by quantitative histomorphometry. J Urol 187(5):1867–1875

    Article  PubMed  Google Scholar 

  11. Mandelbrot B (1967) How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science 156:636–638

    Article  CAS  PubMed  Google Scholar 

  12. Mitchell M (2009) Complexity. A guided tour. Oxford University Press, New York

  13. Page SE (2011) Diversity and complexity. Princeton University Press, Princeton

  14. Papaiannou VE, Pneumatikos I (2010) Fractals and power law in pulmonary medicine. Implications for the clinician. Pneumon 23(3):250–259

    Google Scholar 

  15. Peitgen HO, Jürgens H, Saupe D (2004) Chaos and fractals: new frontiers of science, 2. Aufl. Springer, Berlin Heidelberg New York

  16. Schuster HG (1988) Deterministic chaos. VCH Verlagsgesellschaft, Weinheim

  17. Talu S (2011) Fractal analysis of normal retinal vascular network. Oftalmologia 55(4):11–16

    PubMed  Google Scholar 

  18. Taverna G, Colombo P, Grizzi F et al (2009) Fractal analysis of two-dimensional vascularity in primary prostate cancer and surrounding non-tumoral parenchyma. Pathol Res Pract 205(7):438–444

    Article  PubMed  Google Scholar 

  19. Waliszewski P (1994) Mathematical methods in molecular biology. Biotechnologia 3(26):7–14

    Google Scholar 

  20. Waliszewski P, Molski M, Konarski J (1998) On the holistic approach in cellular and cancer biology: nonlinearity, complexity, and quasi-determinism of the dynamic cellular network. J Surg Oncol 68:70–78

    Article  CAS  PubMed  Google Scholar 

  21. Waliszewski P, Molski M, Konarski J (1999) Self-similarity, collectivity, and evolution of fractal dynamics during retinoid-induced differentiation of cancer cell population. Fractals 7(2):1–11

    Article  Google Scholar 

  22. Waliszewski P, Konarski J, Molski M (2000) On the modification of fractal self-space during tumor progression. Fractals 8(2):195–203

    Article  Google Scholar 

  23. Waliszewski P, Konarski J (2001) Tissue as a self-organizing system with fractal dynamics. Adv Space Res 28(4):545–548

    Article  CAS  PubMed  Google Scholar 

  24. Waliszewski P, Konarski J (2003) Gompertzian curve reveals fractal properties of tumor growth. Chaos, Solitons, and Fractals 16:665–674

  25. Waliszewski P, Konarski J (2005) On time-space of nonlinear phenomena with Gompertzian dynamics. Biosystems 80:91–97

    Article  PubMed  Google Scholar 

  26. Waliszewski P (2005) A principle of fractal-stochastic dualism and Gompertzian dynamics of growth and self organization. Biosystems 82(1):61–73

    Article  PubMed  Google Scholar 

  27. Waliszewski P (2009) A principle of fractal-stochastic dualism, couplings, complementarity and growth. CEAI 11(4):45–52

    Google Scholar 

  28. Waliszewski P (2013) Towards the objective grading of prostate cancer. In: Waliszewski P (ed) Fractals and complexity. Format, Wrocław, pp 16–24, ISBN 978-83-936598-0-7

  29. Waliszewski P, Dominik A, Wagenlehner F et al (2014) Complexity measures and classifiers in objective prostate cancer grading, 29th EAU Congress, Stockholm, Poster 741

  30. West BJ, Bhargava V, Goldberger L (1986) Beyond the principle of similitude: renormalization in the bronchial tree. J Appl Physiol 60:1089–1097

    CAS  PubMed  Google Scholar 

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Danksagung

Wir danken Herrn Dr. Martin Obert, einem erfahrenen Forscher auf dem Gebiet der fraktalen Geometrie, für hilfreiche Diskussionen.

Einhaltung ethischer Richtlinien

Interessenkonflikt. P. Waliszewski, F. Wagenlehner, S. Gattenlöhner und W. Weidner geben an, dass kein Interessenkonflikt besteht. Dieser Beitrag beinhaltet keine Studien an Menschen oder Tieren.

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Authors and Affiliations

  1. Klinik für Urologie, Andrologie und Kinderurologie, Justus-Liebig-Universität, Rudolf-Buchheim-Straße 7, 35392, Gießen, Deutschland

    P. Waliszewski FEBU, F. Wagenlehner & W. Weidner

  2. Institut für Pathologie, Justus-Liebig-Universität, Gießen, Deutschland

    S. Gattenlöhner

Authors
  1. P. Waliszewski FEBU
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  2. F. Wagenlehner
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  3. S. Gattenlöhner
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  4. W. Weidner
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Correspondence to P. Waliszewski FEBU.

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Waliszewski, P., Wagenlehner, F., Gattenlöhner, S. et al. Fraktale Geometrie zur Objektivierung des Gradings beim Prostatakarzinom. Urologe 53, 1186–1194 (2014). https://doi.org/10.1007/s00120-014-3472-x

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  • DOI: https://doi.org/10.1007/s00120-014-3472-x

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