Fractional Calculus Model of Electrical Impedance Applied to Human Skin

Abstract

Fractional calculus is a mathematical approach dealing with derivatives and integrals of arbitrary and complex orders. Therefore, it adds a new dimension to understand and describe basic nature and behavior of complex systems in an improved way. Here we use the fractional calculus for modeling electrical properties of biological systems. We derived a new class of generalized models for electrical impedance and applied them to human skin by experimental data fitting. The primary model introduces new generalizations of: 1) Weyl fractional derivative operator, 2) Cole equation, and 3) Constant Phase Element (CPE). These generalizations were described by the novel equation which presented parameter related to remnant memory and corrected four essential parameters We further generalized single generalized element by introducing specific partial sum of Maclaurin series determined by parameters We defined individual primary model elements and their serial combination models by the appropriate equations and electrical schemes. Cole equation is a special case of our generalized class of models for Previous bioimpedance data analyses of living systems using basic Cole and serial Cole models show significant imprecisions. Our new class of models considerably improves the quality of fitting, evaluated by mean square errors, for bioimpedance data obtained from human skin. Our models with new parameters presented in specific partial sum of Maclaurin series also extend representation, understanding and description of complex systems electrical properties in terms of remnant memory effects.

Fractional Dynamics in Bioscience and Biomedicine and the Physics of Cancer

“…Recently it is showed that the fractional model perfectly fits the test data of memory phenomena in different disciplines [46] they have found that a possible physical meaning of the fractional order is an index of memory. From this viewpoint fractional calculus has found many applications in new research on physics of biological structures and living organisms, from DNA dynamics [47][48][49] to protein folding [50], cancer cells [51], tumor-immune system [52], modeling of some human autoimmune diseases such as psoriasis [53], bioimpedance [54], spiking neurons [55], and also the transport of drugs across biological materials and human skin [56] and electrical impedance applied to human skin [57] and even modeling of HIV dynamics [58].…”

Application of the Caputo‐Fabrizio and Atangana‐Baleanu fractional derivatives to mathematical model of cancer chemotherapy effect

“…[20][21][22][23][24][25][26][27] The fractional-order differential equations are naturally related to systems with multiple timescale dynamics (related to the complexity of the medium) and memory effects, which exist in most biological systems. [28][29][30][31][32][33][34][35][36][37] Mathematical models involving fractional differential equations have been proven valuable in understanding the dynamics of tumor-immune system and how host immune cells and cancerous cells evolve and interact. Considering the diffusion of drugs in cancer cells and fractality of DNA walks, in Namazi et al, 38 the authors considered the fractional diffusion equation to model and predict the effect of chemotherapy on cancer cells.…”

A method based on the Jacobi tau approximation for solving multi-term time–space fractional partial differential equations

“…These applications cross diverse disciplines, such as chemical physics [1], viscoelasticity [2], electricity [3], finance [4], control theory [5], biomedical engineering [1], fluid mechanics [6] and other sciences (see [6,7,8] and references therein). In fact, it has been found that the fractional-order models are more adequate than the previously used integer-order models [9,10,11], because fractional-order derivatives and integrals enable the description of the memory and hereditary properties of different substances.…”