Fractal and stochastic geometry inference for breast cancer: A case study with random fractal models and Quermass-interaction process

Philipp Hermann, Tomáš Mrkvička, Torsten Mattfeldt, Mária Minárová, Kateřina Helisová, Orietta Nicolis, Fabian Wartner, Milan Stehlík

Resultado de la investigación: Article

12 Citas (Scopus)

Resumen

Fractals are models of natural processes with many applications in medicine. The recent studies in medicine show that fractals can be applied for cancer detection and the description of pathological architecture of tumors. This fact is not surprising, as due to the irregular structure, cancerous cells can be interpreted as fractals. Inspired by Sierpinski carpet, we introduce a flexible parametric model of random carpets. Randomization is introduced by usage of binomial random variables. We provide an algorithm for estimation of parameters of the model and illustrate theoretical and practical issues in generation of Sierpinski gaskets and Hausdorff measure calculations. Stochastic geometry models can also serve as models for binary cancer images. Recently, a Boolean model was applied on the 200 images of mammary cancer tissue and 200 images of mastopathic tissue. Here, we describe the Quermass-interaction process, which can handle much more variations in the cancer data, and we apply it to the images. It was found out that mastopathic tissue deviates significantly stronger from Quermass-interaction process, which describes interactions among particles, than mammary cancer tissue does. The Quermass-interaction process serves as a model describing the tissue, which structure is broken to a certain level. However, random fractal model fits well for mastopathic tissue. We provide a novel discrimination method between mastopathic and mammary cancer tissue on the basis of complex wavelet-based self-similarity measure with classification rates more than 80%. Such similarity measure relates to Hurst exponent and fractional Brownian motions. The R package FractalParameterEstimation is developed and introduced in the paper.

Idioma originalEnglish
Páginas (desde-hasta)2636-2661
Número de páginas26
PublicaciónStatistics in Medicine
Volumen34
N.º18
DOI
EstadoPublished - 15 ago 2015

Huella dactilar

Random Fractals
Stochastic Geometry
Fractal Geometry
Fractals
Breast Cancer
Cancer
Breast Neoplasms
Interaction
Fractal
Similarity Measure
Medicine
Neoplasms
Model
Sierpinski Carpet
Boolean Model
Hurst Exponent
Hausdorff Measure
Fractional Brownian Motion
Self-similarity
Randomisation

ASJC Scopus subject areas

  • Epidemiology
  • Statistics and Probability

Citar esto

Hermann, Philipp ; Mrkvička, Tomáš ; Mattfeldt, Torsten ; Minárová, Mária ; Helisová, Kateřina ; Nicolis, Orietta ; Wartner, Fabian ; Stehlík, Milan. / Fractal and stochastic geometry inference for breast cancer : A case study with random fractal models and Quermass-interaction process. En: Statistics in Medicine. 2015 ; Vol. 34, N.º 18. pp. 2636-2661.
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Hermann, P, Mrkvička, T, Mattfeldt, T, Minárová, M, Helisová, K, Nicolis, O, Wartner, F & Stehlík, M 2015, 'Fractal and stochastic geometry inference for breast cancer: A case study with random fractal models and Quermass-interaction process', Statistics in Medicine, vol. 34, n.º 18, pp. 2636-2661. https://doi.org/10.1002/sim.6497

Fractal and stochastic geometry inference for breast cancer : A case study with random fractal models and Quermass-interaction process. / Hermann, Philipp; Mrkvička, Tomáš; Mattfeldt, Torsten; Minárová, Mária; Helisová, Kateřina; Nicolis, Orietta; Wartner, Fabian; Stehlík, Milan.

En: Statistics in Medicine, Vol. 34, N.º 18, 15.08.2015, p. 2636-2661.

Resultado de la investigación: Article

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T2 - A case study with random fractal models and Quermass-interaction process

AU - Hermann, Philipp

AU - Mrkvička, Tomáš

AU - Mattfeldt, Torsten

AU - Minárová, Mária

AU - Helisová, Kateřina

AU - Nicolis, Orietta

AU - Wartner, Fabian

AU - Stehlík, Milan

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N2 - Fractals are models of natural processes with many applications in medicine. The recent studies in medicine show that fractals can be applied for cancer detection and the description of pathological architecture of tumors. This fact is not surprising, as due to the irregular structure, cancerous cells can be interpreted as fractals. Inspired by Sierpinski carpet, we introduce a flexible parametric model of random carpets. Randomization is introduced by usage of binomial random variables. We provide an algorithm for estimation of parameters of the model and illustrate theoretical and practical issues in generation of Sierpinski gaskets and Hausdorff measure calculations. Stochastic geometry models can also serve as models for binary cancer images. Recently, a Boolean model was applied on the 200 images of mammary cancer tissue and 200 images of mastopathic tissue. Here, we describe the Quermass-interaction process, which can handle much more variations in the cancer data, and we apply it to the images. It was found out that mastopathic tissue deviates significantly stronger from Quermass-interaction process, which describes interactions among particles, than mammary cancer tissue does. The Quermass-interaction process serves as a model describing the tissue, which structure is broken to a certain level. However, random fractal model fits well for mastopathic tissue. We provide a novel discrimination method between mastopathic and mammary cancer tissue on the basis of complex wavelet-based self-similarity measure with classification rates more than 80%. Such similarity measure relates to Hurst exponent and fractional Brownian motions. The R package FractalParameterEstimation is developed and introduced in the paper.

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KW - Box-counting dimension

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KW - Hausdorff measure

KW - Pathology

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