Geometric structures in biology and medicine. Configurations and topology change in physiological and pathological processes
Keywords:
biological structure, anisotropy, annulus fibrosus, material homogenization, mastopatic and cancer tissue, random fractalsAbstract
Life is change. Each living organism undergoes permanent change during its lifespan. Since a living organism is open system, it interchanges mass and energy with the ambient, it has external interactions. Many physiological processes of various kinds (mechanical, chemical, electrochemical) run permanently in the organism changing its inner structure, architecture and geometry. Components are permanently rebuilt, some of them fade out, others originate. Everything is in an unceasing movement, all in a wonderful and perfect harmony. Indeed, geometry and topology of the organs, tissues, cells and organelles develop during the turbulent physiological process of life. Moreover, pathological changes caused by injuries or diseases run in the human body as well. They are often devastating.
This paper deals with the geometry of the human body and its parts, tackling topological and non-topological changes within. Two previous papers of the author, describing a biomechanical and a biomedical investigation are regarded from topological point of view; two main parts of the paper are devoted to the homogenization of material properties of intervertebral disc of human spine, and the fractal analysis modelling the cancer tissue growth.
References
Baish, J.W., Jain, R.K.: Fractals and cancer, Perspectives in Cancer Research, Vol. 6 (2000), 83-95.
Hearmon, R.F.S.: Introduction to applied anisotropic elasticity, The Clarendon Press, Oxford, 1961.
Hermann, P., Mrkvička, T., Mattfeldt, T., Minárová, M., Helisová, K., Nicolis, O., Wartner,F., Stehlík, M.: Fractal and stochastic geometry inference for breast cancer: a case study with random fractal models and Quermass-interaction process, Statistics in Medicine Vol 34, Issue 18 (2015), 2636-2661.
S. G. Lechnickij: Theory of elasticity of the anisotropic body, Gostechizdat, Moscow - Leningrad, 1957.
Lindstrom, J.: A Method for Estimating the Fractal Dimension of Digital Color Images, " Umea University, 2008.
Mandelbrot, B.: The Fractal Geometry of Nature, publisher: W.H. Freeman, New York, 1982.
Minárová, M. and Sumec, J.: Stress - Strain Response of the Human Spine Intervertebral Disc as an Anisotropic Body Mathematical Modelling and Computation, Information Technology and Computational Physic, Special Issue of Open Physics, Vol. 14, No. 1 (2016), 426-435.
Pompilio, A.: Topology of Fractals, Honors thesis, University of Dayton, 2019.
Porchon, H.: Fractal topology foundations, Topology and its Applications, Vol. 159, Issue 14 (2012), 3156-3170.
Sinelnikov, R. D.: Atlas of Human anatomy, Part I., Publishing house AVICENUM - Prague, 1980.
Sokolnikoff, J.S.: Mathematical theory of elasticity, MacGraw-Hill, New York, 1950.
Stehlík, M., Giebel, S.M., Prostakova, J., Schenk, J.P.: Statistical inference on fractals for cancer risk assessment, Pakistan Journal of Statistic, Vol. 30, Issue 4 (2014), 439-454.
Stehlík, M., Hermann, P., Giebel, S., Schenk, J.P. : Multifractal Analysis on Cancer Risk, Recent Studies on Risk Analysis and Statistical Modeling. Contributions to Statistics. Springer (2018), 17-33.
Stehlík, M., Mrkvička, T., Filus, J., Filus, L.: Recent developments on testing in cancer risk: a fractal and stochastic geometry, Journal of Reliability and Statistical Studies, Vol 5 (2012), 83-95.
Stoyan, D., Kendall, W.S., Mecke, J.: Stochastic Geometry and Its Applications, Second Edition. J. Wiley: Chichester, 1995.
Valenta, J.: Biomechanics, (in Czech), Academia, Prague, 1985.
