Page 157 - Computational Modeling in Biomedical Engineering and Medical Physics

146 Computational Modeling in Biomedical Engineering and Medical Physics

where the real part, R [Ω], is the electric resistance (electro-thermal effect), and
the imaginary part, X [Ω], is the reactance—either inductive (magnetic field) or capaci-
tive (electric field) effects. As mentioned earlier, the body tissues behave like conductive
media when exposed to low frequency electric field; the electrical impedance is thus
represented dominantly by its real part.
Lumped electric circuit models are used in bioimpedance spectroscopy. For example,
the single dispersion Cole model (Cole, 1940; Grimnes and Martinsen, 2008; Ivorra et al.,
2004) has been used in the analysis of blood (Dai and Adler, 2009), body composition
(Buendi et al., 2014), cancer detection (Teixeira et al., 2018), ischemia monitoring
(Guermazi et al., 2014), urea in dialysate measurement (Jensen et al., 2012), tissue analysis
(Guermazi et al., 2014), and hemodialysis (Al-Surkhietal.,2007). Moreover, biological tis-
sues composed of cell clusters and extracellular spaces may be modeled using a simplified
but effective distributed parameters lumped circuit approach and fractional calculus models
(Freeborn, 2013; Vosika et al., 2013), which result in systems of ODEs (Freeborn, 2013;
Ivorra et al., 2004) that may be solved using circuit simulators, for example, SPICE (Nagel
and Pederson, 1973; Nagel, 1975). Other permittivity models—single-dispersion RC
model, extended single-dispersion and double dispersion Cole models, fractional and multi-
scale models—are quoted and referred in the work by Naranjo-Hernández et al. (2019).
The macroscopic, continuous media modelization introduces quantities that aver-
age processes and quantities at cellular scale. At the living matter level, tissues are com-
posed of cells with thin membranes of high electric resistivity that behave as capacitors
(Grimnes and Martinsen, 2008). For harmonic electric excitation of higher frequency,
the electric current flows through tissue and liquids both inside (displacement current)
and outside the cells (conduction current). At low frequencies the current circulates
only through the liquids outside the cells, a conduction current.
The conduction electric current prevails in the electrolyte solutions of the soft tissues
(organs, muscles, etc.) and the body fluids (blood, interstitial fluid, lymph, etc.) (Grimnes
and Martinsen, 2014). The osseous and the adipose tissues, the gases in the breathing paths
and lungs are permeable to the displacement current, which depends on the frequency of
the incident field and the permittivity of the medium (Gabriel et al., 1996)—an electric
property that is explained, at the living matter level, by the dipolar polarization of the bio-
logic medium (Chapter 1: Physical, Mathematical, and Numerical Modeling).
The relaxation effects related to the dipoles depend on the frequency of the incident
electric field. The higher the frequency is, the larger the time lag of the response to the
stimulus is, which leads to an increase in the internal energy. Depending on the fre-
quency of the incident electric field, three groups of processes are observed (Amini
et al., 2018; Bhardwaja et al., 2018; Guermazi et al., 2014): (i) in the range 10 Hz to
10 kHz, named “α dispersion region,” ionic diffusion through the cell membrane and
the counterion processes occur and generate space charge (interfacial) polarization; (ii) in
the range 10 kHz to 100 MHz, labeled “β dispersion region,” the polarization of cell