Fractal and stochastic geometry inference for breast cancer: a case study with random fractal models and Quermass-interaction process
Language English Country England, Great Britain Media print-electronic
Document type Comparative Study, Journal Article, Research Support, Non-U.S. Gov't
PubMed
25847279
DOI
10.1002/sim.6497
Knihovny.cz E-resources
- Keywords
- Hausdorff measure, Quermass-interaction process, box-counting dimension, breast cancer, pathology,
- MeSH
- Algorithms MeSH
- Diagnosis, Computer-Assisted methods MeSH
- Carcinoma, Ductal, Breast MeSH
- Fractals MeSH
- Risk Assessment methods MeSH
- Humans MeSH
- Breast Neoplasms diagnosis pathology MeSH
- Pathology methods MeSH
- Computer Simulation MeSH
- Stochastic Processes MeSH
- Check Tag
- Humans MeSH
- Female MeSH
- Publication type
- Journal Article MeSH
- Research Support, Non-U.S. Gov't MeSH
- Comparative Study MeSH
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.
Departamento de Matemática Universidad Técnica Federico Santa María Valparaíso Chile
Department of Applied Statistics Johannes Kepler University Linz Linz Austria
Department of Mathematics Czech Technical University Prague Prague Czech Republic
Department of Mathematics Slovak University of Technology Bratislava Slovak Republic
Institute of Pathology Universitătsklinikum Ulm Ulm Germany
Institute of Statistics University of Valparaíso Valparaíso Chile
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