title: Fractal and stochastic geometry inference for breast cancer: a case study with random fractal models and Quermass-interaction process.
abstract: 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.
authors.affiliation: Department of Applied Statistics, Johannes Kepler University Linz, Linz, Austria.
authors.forename: Philipp
authors.initials: P
authors.lastname: Hermann
authors.affiliation: Department of Applied Mathematics and Informatics, University of South Bohemia in České Budějovice, České Budějovice, Czech Republic.
authors.forename: Tomáš
authors.initials: T
authors.lastname: Mrkvička
authors.affiliation: Institute of Pathology, Universitătsklinikum Ulm, Ulm, Germany.
authors.forename: Torsten
authors.initials: T
authors.lastname: Mattfeldt
authors.affiliation: Department of Mathematics, Slovak University of Technology, Bratislava, Slovak Republic.
authors.forename: Mária
authors.initials: M
authors.lastname: Minárová
authors.affiliation: Department of Mathematics, Czech Technical University in Prague, Prague, Czech Republic.
authors.forename: Kateřina
authors.initials: K
authors.lastname: Helisová
authors.affiliation: Institute of Statistics, University of Valparaíso, Valparaíso, Chile.
authors.forename: Orietta
authors.initials: O
authors.lastname: Nicolis
authors.affiliation: Department of Applied Statistics, Johannes Kepler University Linz, Linz, Austria.
authors.forename: Fabian
authors.initials: F
authors.lastname: Wartner
authors.affiliation: Department of Applied Statistics, Johannes Kepler University Linz, Linz, Austria.|Departamento de Matemática, Universidad Técnica Federico Santa María, Valparaíso, Chile.
authors.forename: Milan
authors.initials: M
authors.lastname: Stehlík
mesh_terms: D000465:Algorithms; D001943:Breast Neoplasms / Q000175:diagnosis / Q000473:pathology*; D018270:Carcinoma, Ductal, Breast; D003198:Computer Simulation; D003936:Diagnosis, Computer-Assisted / Q000379:methods*; D005260:Female; D017709:Fractals; D006801:Humans; D010336:Pathology / Q000379:methods*; D018570:Risk Assessment / Q000379:methods*; D013269:Stochastic Processes
keywords: Hausdorff measure; Quermass-interaction process; box-counting dimension; breast cancer; pathology