Fractional Calculus Model of Electrical Impedance Applied to Human Skin

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 (beta) related to remnant memory and corrected four essential parameters (R-0, R-infinity, alpha, tau(alpha)). We further generalized single generalized element by introducing specific partial sum of Maclaurin series determined by parameters (beta(*), gamma,delta ...). 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 beta* = gamma = delta = ... = 0. 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

Electrical characteristics of female and male human skin

Bioimpedance spectroscopy (BIS) is a popular method for characterizing the electrical properties of biological tissues. In this study, BIS measurement data of female and male human skin were analyzed and compared. The electrical characteristics of tissue were followed according to four-parameters of the Cole-Cole model: low frequency resistance R0; high frequency resistance R∞; relaxation time t and parameter a. Individual electrical characteristics of human skin were determined for 30 women and 30 men. The distribution and one-way analysis of variance (one-way ANOVA) of the Cole-Cole parameters R0, R∞, t, a within the human population indicated their different dependence on gender. Parameter a, which is higher in the female subjects (a =0.83±0.03) than in the male subjects (a=0.7±0.05), is strongly dependent on gender (p=0). Parameter R∞ also significantly depends on gender (p=0.002), while t and R0 seem to be slightly related to gender (p>0.05). [Acknowledgments - This work was supported by the Ministry of Education and Science of the Republic of Serbia (41006).

Modeling of bioimpedance for human skin based on fractional distributed-order modified cole model

Eksperimentalni podaci otpornosti i računa necelobrojnog reda koriste se za modeliranje bioimpedansnih osobina ljudske kože. Uveli smo i predložili modifikovani Kole model koristeći pri tom operator distribuiranog necelog reda koji je zasnovan na Caputo-Weyl-ovim izvodima necelog reda.Naš predloženi model predstavlja izmenjen jedno-disperzijski Kole model, jer uvodi nove parametre k i σ u jedno-disperzijskoj Kole impedansnoj jednačini. Ovi parametri karakterišu širinu intervala oko frakcionog indeksa α i oni su važni za precizniji opis bioimpedansnih osobina ljudske kože. Predloženi modifikovani Kole model mnogo bolje daje fitovanje date eksperimentalne krive u datom frekventnom opsegu u poređenju sa sa postojećim Kole modelima. Fitovanje je urađeno primenom Levenberg-Marquardt algoritma nelinearnih najmanjih kvadrata.Electrical impedance measurement data and fractional calculus have been utilized for modeling bioimpedance properties of human skin. We introduced and proposed revisited Cole model using modified distributed order operator based on the Caputo-Weyl fractional derivatives. Our proposed model presents essentially modified single-dispersion Cole model, since it introduces a new parameters k and σ in single-dispersion Cole impedance equation. These parameters characterize the width of interval around fractional index α and they are important for more accurate describing bioimpedance properties of human skin. The impedance spectrum was measured in a finite frequency range up to 100 kHz. Our proposed modified Cole model fits much better to experimental curve in a given frequency range compared to existing Cole models. The fitting is done using the Levenberg-Marquardt nonlinear least squares

Electrical characteristics of female and male human skin

Bioimpedance spectroscopy (BIS) is a popular method for characterizing the electrical properties of biological tissues. In this study, BIS measurement data of female and male human skin were analyzed and compared. The electrical characteristics of tissue were followed according to four-parameters of the Cole-Cole model: low frequency resistance R0; high frequency resistance R∞; relaxation time t and parameter a. Individual electrical characteristics of human skin were determined for 30 women and 30 men. The distribution and one-way analysis of variance (one-way ANOVA) of the Cole-Cole parameters R0, R∞, t, a within the human population indicated their different dependence on gender. Parameter a, which is higher in the female subjects (a =0.83±0.03) than in the male subjects (a=0.7±0.05), is strongly dependent on gender (p=0). Parameter R∞ also significantly depends on gender (p=0.002), while t and R0 seem to be slightly related to gender (p>0.05)

Modeling of bioimpedance for human skin based on fractional distributed-order modified cole model

Eksperimentalni podaci otpornosti i računa necelobrojnog reda koriste se za modeliranje bioimpedansnih osobina ljudske kože. Uveli smo i predložili modifikovani Kole model koristeći pri tom operator distribuiranog necelog reda koji je zasnovan na Caputo-Weyl-ovim izvodima necelog reda.Naš predloženi model predstavlja izmenjen jedno-disperzijski Kole model, jer uvodi nove parametre k i σ u jedno-disperzijskoj Kole impedansnoj jednačini. Ovi parametri karakterišu širinu intervala oko frakcionog indeksa α i oni su važni za precizniji opis bioimpedansnih osobina ljudske kože. Predloženi modifikovani Kole model mnogo bolje daje fitovanje date eksperimentalne krive u datom frekventnom opsegu u poređenju sa sa postojećim Kole modelima. Fitovanje je urađeno primenom Levenberg-Marquardt algoritma nelinearnih najmanjih kvadrata.Electrical impedance measurement data and fractional calculus have been utilized for modeling bioimpedance properties of human skin. We introduced and proposed revisited Cole model using modified distributed order operator based on the Caputo-Weyl fractional derivatives. Our proposed model presents essentially modified single-dispersion Cole model, since it introduces a new parameters k and σ in single-dispersion Cole impedance equation. These parameters characterize the width of interval around fractional index α and they are important for more accurate describing bioimpedance properties of human skin. The impedance spectrum was measured in a finite frequency range up to 100 kHz. Our proposed modified Cole model fits much better to experimental curve in a given frequency range compared to existing Cole models. The fitting is done using the Levenberg-Marquardt nonlinear least squares

Discrimination of mineral waters using near-infrared spectroscopy and aquaphotomics

Voda je jedan od najčešće proučavanih materijala danas, ali uprkos tome mnoga njena svojstva i dalje ostaju nerazjašnjena i neiskorišćena. Voda je neophodna za normalno funkcionisanje ljudskog organizma, između ostalog zbog toga, poremećaji homeostaze vode u ljudskom telu leže u osnovi mnogih bolesti. Analiza vode i njene ispravnosti za upotrebu u ljudskoj ishrani uglavnom se bavi onim što je prisutno u vodi - koncentracijama prisutnih anjona i katjona, prisustvu mikroorganizama i tome slično. Različite vrste voda se uglavnom i klasifikuju upravo prema vrsti elemenata koje sadrže, koncentraciji prisutnih elemenata, ili pak odnosu između koncentracije pojedinih jona i njihov efekat na ljudski organizam razmatra se isključivo sa stanovišta elemenata koji su prisutni u njoj. Međutim, iako je poznato da voda formira različite tipove klastera i može da se organizuje oko prisutnih elemenata na različite načine, klasifikacija voda na osnovu organizacije vodenih molekula, kao i efekti različito klasterizovanih voda na ljudski organizam, za sada ne postoje u literaturi. Predmet ovog rada je diskriminacija različitih tipova voda na osnovu njihovog spektra u bliskoj infracrvenoj oblasti, primenom multivarijacione analize i novog pristupa za tumačenje spektara vode u ovoj oblasti, poznatog pod nazivom Akvafotomika. Akvafotomika interpretira spektar vode u bliskoj infracrvenoj oblasti preko posebno definisanih koordinata vodene mreže (water matrix coordinates - WAMACS) kojima su pripisani tačno određeni vibracioni modovi molekula vode preko kojih se može zaključiti kako se molekuli vode organizuju. Na taj način, primenom saznanja akvafotomike, voda se može opisati i sa aspekta njene organizacije u klastere, i time se omogućiti i diskriminacija voda na osnovu prisutnih tipova klastera što je prikazano u ovom radu.Despite that water is one of the most studied materials today its dynamic properties are still not well understood. Water state in human organism is of high importance for normal healthy functioning of human body. Different kinds of water are usually classified according to their present solutes and concentrations of these solutes, but though it is known that water molecules can form clusters around present solutes, the classification of waters based on types of water molecular organization and present clusters is not present in current literature. In this study the multivariate analysis is used for classification of commercial mineral waters based on their near-infrared spectra (NIR). Further, the aquaphotomics has been applied, a new approach for interpretation of near-infrared spectra of water, that gives insight into organization of water molecules in each of these waters

Discrimination of mineral waters using near-infrared spectroscopy and aquaphotomics

Voda je jedan od najčešće proučavanih materijala danas, ali uprkos tome mnoga njena svojstva i dalje ostaju nerazjašnjena i neiskorišćena. Voda je neophodna za normalno funkcionisanje ljudskog organizma, između ostalog zbog toga, poremećaji homeostaze vode u ljudskom telu leže u osnovi mnogih bolesti. Analiza vode i njene ispravnosti za upotrebu u ljudskoj ishrani uglavnom se bavi onim što je prisutno u vodi - koncentracijama prisutnih anjona i katjona, prisustvu mikroorganizama i tome slično. Različite vrste voda se uglavnom i klasifikuju upravo prema vrsti elemenata koje sadrže, koncentraciji prisutnih elemenata, ili pak odnosu između koncentracije pojedinih jona i njihov efekat na ljudski organizam razmatra se isključivo sa stanovišta elemenata koji su prisutni u njoj. Međutim, iako je poznato da voda formira različite tipove klastera i može da se organizuje oko prisutnih elemenata na različite načine, klasifikacija voda na osnovu organizacije vodenih molekula, kao i efekti različito klasterizovanih voda na ljudski organizam, za sada ne postoje u literaturi. Predmet ovog rada je diskriminacija različitih tipova voda na osnovu njihovog spektra u bliskoj infracrvenoj oblasti, primenom multivarijacione analize i novog pristupa za tumačenje spektara vode u ovoj oblasti, poznatog pod nazivom Akvafotomika. Akvafotomika interpretira spektar vode u bliskoj infracrvenoj oblasti preko posebno definisanih koordinata vodene mreže (water matrix coordinates - WAMACS) kojima su pripisani tačno određeni vibracioni modovi molekula vode preko kojih se može zaključiti kako se molekuli vode organizuju. Na taj način, primenom saznanja akvafotomike, voda se može opisati i sa aspekta njene organizacije u klastere, i time se omogućiti i diskriminacija voda na osnovu prisutnih tipova klastera što je prikazano u ovom radu.Despite that water is one of the most studied materials today its dynamic properties are still not well understood. Water state in human organism is of high importance for normal healthy functioning of human body. Different kinds of water are usually classified according to their present solutes and concentrations of these solutes, but though it is known that water molecules can form clusters around present solutes, the classification of waters based on types of water molecular organization and present clusters is not present in current literature. In this study the multivariate analysis is used for classification of commercial mineral waters based on their near-infrared spectra (NIR). Further, the aquaphotomics has been applied, a new approach for interpretation of near-infrared spectra of water, that gives insight into organization of water molecules in each of these waters

Advancement of the Process for Extraction, Chromatography and Characterization of Fullerenes

In the first phase of this work, the basic C60 and C70 as well as the higher fullerenes, mainly C76 and C84, were Soxhlet extracted with p-xylene and n-heptane from the samples of carbon soot, produced in electrical arc, yielding 5.9% and 0.7% of the extract, respectively. The remaining, p-xylene insoluble soot was then extracted with pyridine, by original advanced method, increasing the entire extract yield for an additional 5.9% to 11.8%. The procedures for increases of the basic and higher fullerenes yields, as well as for additional, selective extraction of the higher fullerenes were found. Chromatographic separation of the basic and the higher C76 and C84 fullerenes from p-xylene extract was performed by continual elution, in one phase of the process, under atmospheric pressure, with original, defined, gradient of solvents from 5% toluene in hexane to pure 100% toluene, on active Al2O3 column, by a new, improved method Identifications of the basic and the higher fullerenes in chromatographically purified fractions, as well as in the obtained extracts, were performed using determined techniques of IR and UV/VIS spectroscopy that have not been presented previously for the higher fullerenes

Fractional order Cole model of bioimpedance of the human skin: new results

In this paper, electrical impedance measurement data and fractional calculus have been utilized for modeling bioimpedance properties of human skin. In relation to our experimental in vivo conditions, structure and complexity of the human skin, we suggested that bio-electrical behavior of the human skin can be described by the series layer Cole model based on modif ied fractional distributed-order based on the Caputo-Weyl fractional derivatives. The equivalent total impedance Zc(w) of this new electric circuit is given by the next equation. The impedance spectrum was measured in a nite frequency range up to 100kHz. Our proposed modi ed Cole model much better t to experimentally curve in given frequency range in compare to existing Cole models. The tting is done using Levenberg-Marquardt nonlinear least squares.http://www.biomath.bg/2012/ConferenceBook.pd

Further results on modeling of bioimpedance of the human skin: calculus of non-integer order approach

The fractional integro-differential operators are a generalization of integration and derivation to non-integer order (fractional) operators [1]. Fractional calculus is the essential analytical approach for modeling any kind of complex systems. Further, bioelectro-physical properties of human skin tissue, like most other soft tissues, exhibit electrical behavior [2,3] where human skin consists of several layers with distinct dielectric properties. Today, bioimpedance measurements provide an important method for the noninvasive investigation of tissue structure and properties or for monitoring physiological change (i.e., ‘‘static’’ or ‘‘dynamic’’ human organism properties). Moreover,the complex modulus concept is a powerful and widely used tool for characterizing the electrical behavior of materials in the frequency domain. In this case, according to the proposed concept, bioimpendance moduli can be regarded as complex quantities. In the field of bioimpedance measurements, the Cole (Cole model) impedance model [4] is widely used for characterizing biological tissues because of its simplicity and good fit with measured data, illustrating the behavior of tissue impedance as a function of frequency.In this study, we apply fractional calculus to modeling of electrical properties of biological systems and derive a new class of models for electrical impedance of human skin. According to literature data, the human skin is usually observed as a relative simple structure, and equivalent electrical model of skin doesn’t include tissue lamination. Such relaxation processes occur because the epidermis is a mosaic in which layers of laminated, inhomogeneous cell structure pile up on top of one another. Frequency-dependent components such as CPE (constant phase element), that exists in the single-dispersion Cole model, can be considered composed of an infinite number of lumped components. Recently, authors [5] suggested using the three-layer skin numerical model in the MHz frequency range where each layer of skin is represented by the corresponding Cole–Cole model. So, we proposed the skin structure as a more complex system, consisting of several layers(Fig.1). In relation to our experimental in vivo conditions, structure and complexity of the human skin, we suggested that bio-electrical behavior of the human skin can be described by the series layer Cole model using modified fractional distributed-order based on the Caputo-Weyl fractional derivatives. Our proposed model presents essentially modified single-dispersion Cole model, since it introduces a new parameters k and σ in single-dispersion Cole impedance equation. These parameters characterize the width of interval around fractional index α. Comparing our model to well-known Cole models, we conclude that these parameters are important for more accurate describing bioimpedance properties of human skin. Our modified Cole model much better fit to experimentally curve in given frequency range in compare to existing Cole models. The fitting is done using Levenberg-Marquardt nonlinear least squares. In that way, one may conclude that the electrical properties of skin can be modeled using a more discrete Cole impedance element rather than one discrete Cole impedance element.Last, some our results are related to generalized Cole element as well as constant phase element (CPE). These generalizations is described by the novel equation which presented parameter (β ) and corrected four essential parameters ( R0 , R oo, α , tau α ). Using serial combinations of the primary model elements we may obtain two new models and defined them by the appropriate equations and electrical schemes.http://www.mi.sanu.ac.rs/projects/booklet_of_abstracts.pd