**San
Francisco Bay Area Statistical Association**

**Meeting**

**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**

** ** https://campus-map.stanford.edu/
provides interactive campus map, search

for
ŇSequoia HallÓ to location the building.

Please note parking is free on campus after 4pm.

**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

Email: Milan.Stehlik@jku.at

**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.

**References**

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.