Simulation of impedance measurements at human forearm within 1 kHz to 2 MHz

In the last three decades, bioimpedance analysis has become more extensively used to measure clinically important parameters [1]. Non-invasive bioimpedance plethysmography provides waveform information that relates to the distension of blood vessels and can therefore be used for estimating stroke volume [2, 3, 4, 5] and cardiac output [6, 7, 8, 9, 10, 11, 12]. Bioimpedance offers significant advantages over existing techniques such as ultrasound which requires continuous presence of a trained operator and thermodilution which is an invasive method and cannot readily measure beat-to-beat changes. Electrical impedance measurements at the forearm provide a possible way to characterize hemodynamics, and in particular changes in the amount of blood in the arm as a result of vasodilatation and/or the cardiac cycle.

Although the simulation perspective to bioimpedance plethysmography is a rarity there have been several investigations pertaining to the impedance response at forearm section. Some works [13, 14, 15, 16] related to multi-frequency electrical bioimpedance (MF-EBI) for segmental fluid/fat estimation whereas others [17, 18, 19] focused on single frequency measurements to evaluate parameters such as Heart Rate (HR) and Pulse Wave Velocity (PWV). Due to a high variability amongst the impedance values obtained from different works, the application dependence of the measurements can be considered a fair statement. The choice of forearm quite obviously owes to the more approximate cylindrical geometry, which makes it easier to construct and analyze. However, more importantly, the blood dynamics of radial artery can be reflected through bioimpedance considering a forearm section. This can provide an insight to the arm tissue behavior along with the individual contribution of different layers (along with blood flow) to the impedance variations. Also, the forearm provides a simpler and justifiable site for PWV measurement

The objective of this work is to simulate the electrical response of a section of human forearm and obtain the values of impedance over a wide frequency range. These simulation results will be compared to the Cole type response. The simu-lation would be performed at three instances of radial artery diameters to mimic the condition of blood flow. The β dispersion region, as identified by Schwan [20] and Cole theory [21] form the basis of this work.

A 3D finite element model of a simplified human forearm was constructed using Ansys® HFSS, which allows simulation of applied electromagnetic fields in the frequency domain.

Our investigation is limited to the β dispersion region as it is in this frequency range (1 kHz to 100 MHz) that the tissues reflect maximum passive cell membrane capacitance and intracellular organelle capacitance effects [22]. The mathematical consideration of bioimpedance theory is governed by Maxwell’s equations. For most practical situations, the solution to Maxwell’s equations requires a complex matrix approach, which is performed by the finite element modelling platform provided by Ansys® HFSS. Also, the boundary between two dielectrics is automatically specified through Ansys using the defined mesh settings. The forearm was modelled as a structure consisting of three tissue domains – fat, muscle and artery. Even though bone is a major part of the forearm, the overall contribution of bone to the electrical impedance changes is expected to be negligible due to a constant low conductivity over β dispersion frequency range. Hence, bone was not considered to be a part of this model. The fat and muscle regions have been considered as layered, whereas the artery overlaps the domains of both the muscle and fat regions. The tissue domains in the model are therefore not layered. The longitudinal dimension of the section of the arm was taken to be 70 mm, with the fat layer having a thickness of 3 to 6 mm and muscle with a thickness of 10 to 15 mm from the axis. The artery has been modelled to mimic the radial artery blood properties, for which simulation has been performed using three different diameters – 2.3 mm, 2.35 mm, and 2.4 mm as also reported by some previous works [23,24]. The transverse and the longitudinal views of the model are shown in Fig.1 (a) and 1 (b) respectively.

Fig.1 (a)

Fig.1 (b)

A tetra-polar configuration of electrodes was simulated (also shown in Fig 1). The outer set of electrodes supply the current and the inner pair is used for voltage measurements. The electrodes were realized using cylindrical conductors of copper with a diameter of 2 mm and spacing of 24 mm for the outer pair and 11 mm for the inner pair. Additionally, the outer electrodes were connected through a conducting sheet, over which the direction of current flow was specified for simulation purposes. The frequency dependent dielectric properties for each of the tissue materials have been defined using the database developed by Gabriel [25, 26, 27, 28] using the parametric modelling of tissue properties within 10 Hz to 20 GHz (Fig.2). The conductivity and the permittivity values were set for fat, muscle and blood within 1 kHz to 2 MHz.

Fig. 2

The simulation was performed to determine the value of impedance measured across the electrodes at various frequencies within 1 kHz to 2 MHz. The impedance was calculated as the ratio of measured voltage to the applied current at the corresponding electrodes. IEC 60601 technical standards [29] describe safety of medical equipment and prescribe allowable current ranges. The preferred range is between 10 μA to 1 mA for the frequency range of interest [30] and hence 1 mA was chosen for simulation purposes.

Ansys® was used to automatically mesh with maximum element sizes of 10 mm each for fat and muscle and 5 mm for the artery. Fig.3 – (a), (b), and (c) show the mesh distribution over each of the domains.

Fig.3 (a)

Fig.3 (b)

Fig.3 (c)

Simulations were run at frequencies of 1 kHz, 5 kHz, 10 kHz, 50 kHz, 100 kHz, 150 kHz, 200 kHz, 250 kHz, 400 kHz, 500 kHz, 600 kHz, 700 kHz, 800 kHz, 900 kHz, 1 MHz, 1.25 MHz, 1.5 MHz, and 2 MHz. For each of these frequencies, three diameters of the artery were modelled as mentioned previously: 2.3 mm, 2.35 mm, and 2.4 mm.

The simulation was performed to evaluate the dielectric behavior of the compositional properties of fat, muscle and artery in the human forearm. The simulation produced results in the form of electric field distributions (an example shown in Fig. 4 (a) and 4 (b)) throughout the structure at each frequency step, which was used to calculate the voltage drop across the measuring electrodes. The electric field distributions as shown in Fig. 4 (a) and (b) provide a good indication of electric potential gradient throughout the structure volume

Fig.4 (a)

Fig.4 (b)

The voltage was calculated using the line integral of the electric field gradient between the inner electrode pair terminals. The obtained value of the voltage was complex, due to the complex electric field and corresponded to the resistive and reactive behavior of the model. The obtained values have been calculated for impedance (Z) as ratio of measured voltage (V) to current (I) and plotted as a Nyquist plot (as shown in Fig 5) between Re. (Z), i.e. Resistance and Img. (Z), i.e. Reactance for all the frequencies. The three curves correspond to the three diameters of the artery.

Fig.5

The electrical behavior of biological tissues can often be effectively represented using the Cole model. The Cole equation describes the frequency dependence of tissue conductivity and permittivity. The results obtained through the simulations agree with the Cole model in the sense of varying resistance and reactance values with frequency. The geometrical interpretation can be seen as Cole plots for all the three diameters. The semi-circular shape of the plot verifies the agreement of the obtained results with Cole modelling (Fig.5). The major portion of the plot follows the Cole plot trend but the values at lower frequencies were found to be divergent.

The maximum resistance and reactance values were calculated as 445 Ω and 178.5 Ω, 356 Ω and 138 Ω, and 368 Ω and 144.3 Ω for diameters 2.3 mm, 2.35 mm, and 2.4 mm respectively. The expected values at the extremities of the Cole plot (R₀ and R_∞) can be found by extrapolating the Cole trend and observing the values at the resistance axis. The above values of the impedance were found within agreeable limits of some previous results such as [31] (which found average value of forearm resistance of 346 Ω) and [32]) (which found mean±standard deviation values of resistance of 319±21.9 Ω and reactance of 30±2.3 Ω for arm at 50 kHz).

The values obtained at lower frequencies do not follow the Cole trend and reflect the minimum values of reactance for each plot. The simulation results at lower frequencies (1 kHz to 200 kHz) were found to be mainly resistive and possessed a different variation of values than the rest of the plot. The high resistance values and corresponding changes with diameters at lower frequencies are a result of significant change in the conductivity due to change in the arterial diameter. Due to the large conductivity of blood, the major change in the impedance occurs as a resistive change due to change in arterial dimension. Hence at lower frequencies, the response is dominantly resistive. The value of maximum reactance for each Cole plot is defined by a characteristic frequency (F_c) which was found to be different for three diameters of the artery.

Although bone as a biological tissue does possess frequency dependent dielectricity, it has a fairly low and constant conductivity over the frequency range of interest. The low variability of conductance makes it a redundant contender for observing impedance changes with pulsating artery. However, it is believed to offer a significant change in overall resistance value while contributing negligibly to the impedance changes with artery diameter and frequency.

This work aimed to investigate the overall impedance variations caused by pulsatile flow of blood through the radial artery along with other prominent contributing tissue domains. It might be a good idea to study the overall response with bone included in the model even though its electrical properties do not indicate that it introduces significant changes in the overall behavior in the context of this work.

Although the above simulation model lacks blood flow, the consideration of different arterial diameters partly compensates for the hemodynamic effects. The choice of different arterial diameters complies with the characteristics of arterial system. It was reasonable to assume that the diameter changes uniformly at each wavelength flow of blood, in accordance with long wavelength approximation [33] applied to Moens-Korteweg Equation [34] yielding the expression:

where, E is the arterial elastic modulus, h is the arterial wall thickness, r is the radius and ρ is the blood density. Considering c = f λ, the wavelength (λ) can be calculated using frequency (f) as (72 bpm/60) 1.2 Hz (average) [35]. In this case, the pulse wave velocity (c) can be considered as 8 to 10 m/s (based on results of several works [36, 37, 38]) and the maximum fundamental frequency of the blood pressure pulse as 25 Hz [39,40], λ can be evaluated to be 32 to 40 cm which is significantly larger than the dimension of the model. Hence, each wavelength of blood can be assumed ‘longer’ than the proposed arterial dimensions leading to the consideration of blood flow induced arterial diameter changes. The obtained results with different diameters are important in the sense that they reflect the influence of arterial hemodynamics to the impedance values. The obtained variation in the set of values proves that the composition of tissues along with the hemodynamics serve as important contributors to the overall electrical response. The latter, particularly, has been justified by several works [41, 42, 43, 44, 45, 46, 47, 48] analyzing the blood flow effects and contributions to the bioimpedance readings.

The arm structure was enveloped with a radiation boundary and a vacuum region to mimic the prevention of any electromagnetic interference from an outside source. The trade-off between the computing complexity and model efficacy was optimized by selecting the mesh element sizes of 10 and 5 mm for outside layers and artery respectively. Initial simulations with finer mesh sizes produced almost identical results but took longer to converge to a solution. This work neglects the contribution of bone to the overall measurements. Bone is expected to offer higher resistance and capacitance and is predominantly piezoelectric, although dynamic mechanical deformation was not simulated [49].

Also, this study is confined to the compositional properties of blood, fat and muscle tissues and considers dielectric properties to be isotropic. It neglects any dynamic effects, for example, that may cause non-uniform diameter changes along the length of the artery. However, the study provides a useful insight to tissue electrical response through obtained impedance values over the β dispersion frequency range along with changes in impedance variations due to changes in arterial diameter. This can be related to arterial hemodynamics to help estimate the vessel diameter changes through induced impedance variations.

This simulation study aimed at evaluating the impedance values of a section of human forearm Ansys® HFSS was used to construct and simulate a model consisting of three layers of tissues – fat, muscle and artery and the electrical response was simulated over a frequency range of 1 kHz to 2 MHz.

The effect of blood pulsatile flow has been mimicked and analyzed as the electrical response at three different arterial diameters. This approach provides a simulation perspective to analyze impedance contributions from different tissue layers along with blood dynamics. The obtained results clearly assert a good capability of the setup to be quite sensitive towards diameter changes. This may aid future works in investigating the electrical variations due to pulsating artery. The study highlights a basic understanding of the frequency dependent electrical behavior of human forearm tissues and presents an efficient simulation tool to analyze the effects and individual contributions of different tissues to the bioimpedance measurements.