San Francisco Bay Area Statistical Association



Time: 4:30 – 6:00 pm, January 26, 2017

          (4:30-5:00 networking and light refreshment, 5:00-6:00 seminar)

Location: Department of Statistics, Stanford University, Sequoia Hall Room 200

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Speaker: Professor Milan Stehlik

                Department of Applied Statistics, Johannes Kepler University in Linz, Austria

                 Linz Institute of Technology (LIT), Johannes Kepler University in Linz, Austria

                 Institute of Statistics, Universidad de Valpara ́őso, Valpara ́őso, Chile


Title: Multi-fractal and Multi-criteria aggregation for cancer risk assessment

Abstract: There exists a big need for discrimination between mammary cancer and mastopathy tissues (see Hermann et al. (2015)). Non invasive techniques generally may produce inverse problems, e.g. estimating a Hausdorff fractal dimension from boundary of examined tissue (see Kiselak et al. (2013)). We will discuss these issues in the context of our recent results (see e.g. Filus, Filus and Stehl ́ők (2009)). During the talk we will discuss several issues which bring light into both fractal based cancer modelling and more general stochastic geometry models and their comparisons.

When we consider fractal based cancer diagnostic, many times a statistical procedure to assess the fractal dimension is needed. We shall look for some analytical tools for discrimination between cancer and healthy ranges of fractal dimensions of tissues. Baish and Jain (2000) discussed planar tissue preparations in mice which has a remarkably consistent scaling exponents (fractal dimensions) for tumor vasculature even among tumor lines that have quite different vascular densities and growth characteristics. In Stehl ́ők et al. (2015) we provide extensive study of cancer risk assessment on simulated and real data and fractal based cancer. Both non-random and random carpets have been validated for modelling of the cancer growth, and it was shown that only random carpets can be used. We constructed a statistical test, which is able to distinguish between the two groups, masthopathy and mammary cancer (see Stehl ́ők et al. (2012)).

The inter-patient variability of fractal dimension for mammary cancer is high (see Stehl ́ők et al. (2016)) and therefore multifractality is a better concept (see Nicolis et al. (2016)). This is a feasible and parsimous solution for a more delicate problem of multi-objective aggregation of information. This touches bases for general topological approach for aggregation Stehl ́ők (2016), which links to Sugeno integral as a way to aggregation in bornological spaces.

The algebraic and topologic properties of cancer growth are available via appropriate set structures, e.g. bornology (see Paseka, Solovyov, and Stehl ́ők (2015), Paseka, Solovyov, and Stehl ́ők (2016)). Such structures can be very useful for defining fractal cancer hypothesis. Nephroblastoma (given by Wilms tumour) is the typical tumour of the kidneys appearing in childhood, which does not satisfy fractal cancer hypothesis. We illustrate on recent pre/post clinical study the effect on chemotherapy to Euclidean volumes of such tumors (see Hermann et al. (2015)b.)

Author acknowledges support of FONDECYT Regular N 1151441.


Baish J.W. and Jain R.K. (2000). Fractals and cancer. Cancer Research, 60, 3683-3688.

Filus J., Filus L. and Stehl ́ők M. (2009). Pseudoexponential modelling of cancer diagnostic testing.

Biometrie und Medizinische Informatik, Greifswalder Seminarberichte Heft 15, 41-54.

Kiselak, J., Pardasani,K. R., Adlakha,N. Agrawal, M. and Stehl ́ők, M. (2013) On some probabilistic

     aspects of diffusion models for tissue growth, The Open Statistics and Probability Journal

Hermann, P.; Mrkvicka, T.; Mattfeldt, T., Minarova, M.; Helisova, K.; Nicolis, O.; Wartner, F.;

Stehl ́ők M., (2015) Fractal and stochastic geometry inference for breast cancer: a case study with

    random fractal models and Quermass-interaction processÓ Statistics in Medicine,34, 2636-2661

Hermann, P., Giebel, S.M., Schenk, J.-P., and Stehl ́ők, M. (2015). Dynamic Shape Analysis - before

    and after chemotherapy. Proc. International Conference on Risk Analysis (ICRA6), 339-346.

Nicolis, O., Kiselak, J. Porro, F. and Stehl ́ők, M. Multi-fractal cancer risk assessment, Stochastic

    Analysis and Applications, DOI: 10.1080/07362994.2016.1238766

Paseka, J., Solovyov, S. A. and Stehl ́ők, M. (2015) ÓLattice-valued bornological systemsÓ, Fuzzy Sets

    and Systems, 259, 15, 68-88

Paseka, J., Solovyov, S. A. and Stehl ́ők, M. (2016), On a topological universe of L-bornological

   spaces, Soft Computing 20:2503-2512

Stehl ́ők M., Mrkvi˙cka T., Filus J. and Filus L. (2012), Recent development on testing in cancer risk:

   a fractal and stochastic geometry, Journal of Reliability and Statistical Studies Vol. 5, 83-95.

Stehl ́ők,M., Giebel, S. M., Prostakova,J. Schenk J.P. (2014), Statistical inference on fractals for

   cancer risk assessmentÓ Pakistan Journal of Statistic, Vol. 30(4), 439-454.

Stehl ́ők M., Hermann, P., and Nicolis, O.(2016) Fractal based cancer modelling, Revstat, 14, 2, 139-155

Stehl ́ők, M. (2016). On convergence of topological aggregation functions. Fuzzy Sets and Systems, 287:4856.