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

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. A cancer risk can utilize determination of fractal dimension. Inspired by Sierpinski-Carpet, we introduce a flexible parametrical 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 of the cancer data and we apply it to the images. It was found out that mastopathic tissue deviates significantly more strongly from Quermass-interaction process, which describes interactions among the particles, than mammary cancer tissue does. It was shown that Quermass-interaction process serves as a model describing the tissue, which structure is broken to a certain level.However, random fractals model fits well for mastopathic tissue. Impact of ramification level is addressed as well as Hausdorff measure of both mastopatic and mammary cancer tissues is calculated. The R package FractalParameterEstimation is developed and introduced in the paper.