Keywords: mathematical modeling, biomedical engineering, human skin, biomechatronics, fractional calculus, Weyl's fractional derivative, frequency analysis, viscoelasticity.

Human skin tissue, like most other soft tissues, exhibits (electro) viscoelastic behaviour. To obtain complete information about the electroviscoelastic behaviour of human skin, it is necessary to have experimental data over a wide range of time scales. Also, the complex modulus concept is a powerful and widely used tool for characterizing the electroviscoelastic behaviour of materials in the frequency domain. According to this concept bioimpedance moduli can be regarded as complex quantities.

Also, electrical bioimpedance measurement methods have been proposed to study a large number of physiological and bioengineering events. Particularly, a memory function equation, scaling relationships and structural-fractal behaviour of biomaterials were used for the physical interpretation of the Cole-Cole exponents [1]. Also, it is well-known, three expressions for the impedance allow one to describe a wide range of experimental data: the Cole-Cole function, the Cole-Davidson function and the Gavril'yak-Negami function.

Fractional derivatives which describe electroviscoelastic characteristics of materials in this paper are Weyl's fractional derivatives [2]. Such an approach to the description of a non-exponential relaxation is based on the assumption that the relaxing macroscopic system consists of the appropriate number of subsystems, each of which relaxes with its own time constant. The results of many works show that the dielectric characteristics of human skin are typical of semiconductors in the determinate frequency diapason of the applied voltages. The Cole-Cole function, applied in this paper, properly explain impendance in the frequency range 10 Hz to 1 KHz [3].

We can examine the structure and function of human skin categorized into four main layers: (1) the innermost subcutaneous fat layer (hypodermis); (2) the overlying dermis; (3) the viable epidermis; (4) the outermost layer of the tissue (a non-viable epidermal layer) the stratum corneum. Each layer contains the major components with different dielectric and structural characteristics. The behaviour of this polarization admittance can be modeled by means of a frequency dependent capacitive reactance (capacitor) in parallel with a frequency dependent quasi-resistor. We propose a model based upon continuum series of Cole-Cole admittances as structural elements of the skin, described using Weyl's fractional derivatives. These admittances are characterized by continual fractional indexes between 0 and 1. In our model we use approximative discrete fractional indexes. For that purpose, we have used ten index values between 0.1 and 1.0. The measurements of the human skin were carried in vivo. To that effect, the electrical properties of human skin in the range of the applied voltages 0.1 and 1.0 V in the frequency range 0.1 Hz to 100 KHz are modeled theoretically and measured experimentally. At last, the proposed fractional derivative model is fitted to experimental data of impedance tests of human skin to verify behaviour and to obtain the material parameters.

1

R. Nigmatullin, Ya.E. Ryabov, "Cole-Davidson relaxation as a self-similar relaxation process", Physics of the Solid State, 39, 1997. doi:10.1134/1.1129804

2

R. Hilfer, (Editor), "Applications of Fractional Calculus in Physics", World Scientific Pub Co, Singapore, 2000.

3

O.G. Martinsen, S. Grimnes, H. Piltan, "Cutaneous perception of electrical direct current", ITBM-RBM, Volume 25, Issue 4, Pages 240-243, 2004. doi:10.1016/j.rbmret.2004.09.012

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