Positive phase error from parallel conductance in tetrapolar bio-impedance measurements and its compensation

The basic set-up for a tetrapolar, or four-electrode, impedance measurement consists of two current carrying electrodes used to inject alternating current, I, into the sample and two additional electrodes to pick-up the voltage that is produced across the sample, V_PU. The measured impedance of the sample, Z_M, is then calculated as the complex ratio of the pick-up voltage to the injected current: Z_M = V_PU / I.

If a parallel conductive pathway exists the measuring circuit is modified as in figure 1. In this case the current injected into the sample is reduced to the new value i_S and there will be a concomitant drop in the voltage measured across the sample. The discrepancy between the actual sample impedance and the measured impedance results from the fact that the full injected current is used to calculate the measured impedance. The existence of a parallel current pathway means that the current flowing through the sample is over-estimated such that the measured impedance is less than the true sample impedance. It will be shown that this artefact has the appearance of an inductance having increasing impedance magnitude with frequency and positive phase at low to intermediate frequencies.

Fig. 1

Circuit analysis of the set-up depicted in figure 1 shows that, in the presence of a parallel conductive pathway, the current reaching a sample with impedance Z_S is reduced by a factor given by the ratio of the total impedance of the circuit, Z_TOT, to the impedance of the pathway containing the sample:

Here Z_CC is the total impedance of the current carrying electrodes. The voltage measured by the pick-up electrodes is given by:

These expressions can be combined to give an expression for the measured impedance:

Simulations were performed in MatLab (The Mathworks, UK) to examine the role played by the various components present in this expression and to characterise the distortion to the impedance spectra that results.

Experimental data was acquired using an EndOhm-12 device (World Precision Instruments, UK) designed for trans-epithelial electrical resistance measurements of cell samples grown on transwell membranes. Figure 2 shows a schematic of the measurement rig containing solution but no transwell membrane. The corresponding cross-sectional geometry, axially symmetric about r = 0, is also shown. The pick-up (PU) electrodes are symmetrically apposed Ag/AgCl sintered pellets 2mm in diameter. One is centred in the base of the well chamber. The second is centred in a pillar in the lid of the device, which can be inserted into the well chamber. The current-carrying (CC) electrodes are circular strip sterling silver electrodes 4.2mm wide in the base and 1mm tall in the pillar. Upon fabrication, the CC electrodes are coated with a bleach-generated chloride layer (Clorox, California, USA).

Fig.2

The EndOhm device is designed for single frequency resistance measurements but was interfaced with a freshly calibrated Solartron 1260 impedance analyser (Advanced Measurement Technology, UK) for spectral analysis. Electrical impedance spectroscopy measurements were made at logarithmically spaced frequencies in the range from 100Hz to 1MHz with ten points per decade. A 1V signal was applied. The current carrying and pick-up circuitry were connected by a common ground.

Artefact seen in the spectra was confirmed as being independent of the Solartron analyzer by using a separate current source and monitoring the voltage on the pick up electrodes with voltage probes and an oscilloscope.

The chloride layer initially coating the CC electrodes suffers thinning over time, which does not recondition fully upon re-chloriding. To examine the effect of contact impedance on the acquired spectrum, spectra with and without (one month later) the chloride layer being present on the CC electrodes were compared. The data were acquired after filling the main well chamber of the device with 3ml of phosphate buffered saline (PBS) (Sigma, UK); inserting a blank transwell membrane (Corning Inc., NY, USA) and adding 0.5ml PBS to the transwell centre.

To validate the fitting procedure the EndOhm device was filled with 5ml of commercially available known-conductivity calibration solutions (Hanna Instruments, UK). No transwell membranes were used in these cases. Conductivities ranging from 0.14S/m to 1.28S/m were used to correspond with the physiological range. The temperature was recorded and the corresponding conductivity was determined from the solution’s technical sheet. The lid was seated on the device allowing full submergence of the upper electrodes.

The finite element model (FEM) of the EndOhm device was constructed in Comsol Multiphysics (Comsol AB, Sweden) using the AC/DC module under quasi-static conditions. The cylindrical geometry of the device was simplified to a 2D axially symmetric model (figure 2, right) The upper electrode being used for current injection was set to the port boundary condition with a fixed current input, while the lower CC electrode was set to ground. The pick-up electrodes were set to the floating potential boundary condition. To match the experimental conditions used for the calibration solutions, the well chamber was modelled as being filled with 5ml of solution and with the upper pillar separated from the base of the chamber by 4.3mm. The sub-domain corresponding to the solution was set to the appropriate temperature-specific conductivity. No permittivity information was available from the manufacturers so the relative permittivity was set to 80, a typical value for liquids such as water and saline. The initial model consisted of 344 triangular elements. Adaptive mesh refinement was used when solving the model to maximise accuracy. The final mesh consisted of 2,734 elements.

When making tetrapolar impedance measurements, it is the transfer impedance between the CC and PU electrodes that is measured meaning that the sensitivity of the measuring device should be considered (5; 7; 8). The sensitivity, S, of the device measures the contribution of each element within the volume to the overall impedance measured and was calculated as:

Here J_CC is the current density that results by using the CC electrodes for current injection and J_PU is the current density produced by reversing the set-up and using the PU electrodes for current injection. I is the integral of the normal current density injected through the surface, δΩ, of the particular electrode being used for current injection. For example:

Positive sensitivity means that an increase in the resistivity of the sample will result in a corresponding increase in the measured resistance. Conversely, negative sensitivity results in a decrease in the measured resistance for the same increase in resistivity. Since the sensitivity is normalised to the current level, this distribution is in fact independent of the current level used.

The geometric factor or cell constant of a measurement rig is the coefficient of proportionality between the resistivity, ρ, of a homogeneous sample and the measured transfer resistance, R. The calculated sensitivity from the FEM can be used to calculate the pseudo geometric factor of the measurement rig being modelled. This factor, g, is calculated by summing the contribution from all volume elements, dv, throughout the measurement volume V:

This expression was used to calculate the expected transfer resistance of each known conductivity solution.

A second measure of the expected transfer resistance was calculated from the FEM by numerically implementing the measuring procedure. First, the voltage on the upper and lower PU electrodes was calculated as:

The expected transfer resistance was calculated by dividing the voltage difference, corresponding to V_PU, by the current injected through the CC electrode (as calculated above):

Fitting to the equivalent electrical circuit model of parallel conductance depicted in figure 1 was performed in MatLab using the Neldr-Mead simplex algorithm (12). The fitting was carried out with nine free parameters as follows. First, a resistance, R_S, and capacitance, C_S are used to represent the solution impedance:

Here ω is the angular frequency and j = $-1. Each of the CC electrodes is represented with a charge transfer resistance, R_CT, and a constant phase element (13; 14) with double layer capacitance, C_dl, and exponent, α, to capture the inhomogeneous nature of the electrode surface:

A resistance, R_P, represents the parallel conductive pathway. The total impedance incorporating the solution, the CC electrodes and the parallel pathway is calculated as:

Finally the fitted impedance spectrum, Z_F, is:

Given the complex nature of the acquired data, optimisation was achieved by minimising the sum of squared differences between the measured, Z_M, and fitted, Z_F, data for both the real and imaginary components as per the following cost function:

Here N is the total number of spectral data points acquired. The fitting procedure is looped with the result of the fit being used as the starting point for subsequent iterations to minimise the risk of the fitting procedure stopping prematurely due to a local minimum. To ensure a physically realistic fit, only positive values were allowed for each of the fitting parameters and the alpha parameter was restricted to the range from 0.6 to 1.

Impedance spectra of cell cultures were acquired using a human colorectal tumour cell line (HCT116). These were grown until confluent on collagen treated polytetrafluoroethylene (PTFE) membranes (Corning Inc., NY, USA) that promote cell attachment and spreading. The transwell insert diameter is 12mm. The insert’s membrane has a growth area of 1.12cm2 with 0.4μm pores and a nominal thickness of 30μm. It rests on the upper lip of the well chamber and extends approximately 1.6cm into the device with the membrane of the transwell positioned between the upper and lower electrodes. Cell layers of different density were grown by seeding the transwell membranes at four different levels: approximately confluent, 33%, 11% and 4% coverage. Two wells were used for each seeding level. Two blank transwells were also incubated to provide a baseline measurement under equivalent conditions. After four days growth, the cells were removed from the incubator and allowed to reach equilibrium with room temperature for a period of one hour. Each transwell was then washed twice with PBS. Care was taken to ensure that the cell layers did not lift off the transwell membrane during washing. The transwells were then inserted into the EndOhm device containing 3ml of PBS (replenished for each transwell membrane). A further 0.5ml of PBS was added to the transwell to cover the cells and create a contact between the upper and lower electrodes. With this procedure, the solution inside and outside of the transwell was approximately level. Micrographs were acquired before and after the impedance measurements using a Nikon Diaphot microscope running LUCIA G software and equipped with a DXM1200F camera. The cells were then lysed and the absorbance of the samples at 280nm was measured using a Helios Zeta spectrophotometer (Thermo Scientific, UK). The optical density was converted to cell number based on a calibration curve. The calibration curve was determined by performing linear regression on optical density measurements acquired using cell extracts for which the cell numbers had been established using conventional cell counting with a hemocytometer. Figure 3 shows an example micrograph from a well seeded at 11% confluency and the equivalent electrical circuit used to model the transwell membranes.

Fig. 3

From the micrograph of figure 3 it can be seen that there are areas of the transwell membrane to which no cells have adhered. In the equivalent electrical circuit of the transwell membrane these areas are collectively represented by a resistance, R_GAP. As the foci merge the area of blank transwell membrane is reduced. Where cells have grown, the cell density is non-uniform. These regions are represented by the conventional cell model, in which the extracellular space is represented by a resistance denoted R_E; the intracellular space is represented by a resistance denoted R_I; and the capacitive nature of the cell membrane is represented by a reactance denoted X_M. The solution above and below the transwell membrane is represented by the resistance, R_SOL, in series with the transwell membrane layer. The sample impedance, Z_S, introduced above for situations where the device is filled only with solution was replaced by this model in order to extract the electrical properties of the transwell membranes from the spectra.

As there are a large number of parameters in this model, the result is sensitive to the initial conditions. Fitting was begun with a blank transwell membrane and a gap resistance of 0# since this would be the condition to bypass the cellular compartment of the model. Each subsequent transwell was initialised with the result from the previous fit, with the exception that the gap resistance was incremented by 50# for the first seeding level to reflect the fact that the resistance of these zones will increase with the existence of cell foci. The extracellular pathway is preserved via the R_E parameter of the conventional cellular model.

Figure 4 shows a simulation to contrast the appearance of artefact due to electrode polarisation effects with artefact caused by the presence of a parallel conductive pathway. Electrode polarisation effects (+) in the impedance spectrum diminish with increasing frequency as the double layer capacitance has greater bearing on the coupling to the sample. At sufficiently high frequency the measured impedance converges on the true sample impedance, simulated as 500# and zero phase in this instance (□).

Fig. 4

Conversely, in the case of a parallel conductance (o), the measured impedance does not converge on the true sample impedance. The measured impedance is inversely dependent on the polarisation impedance of the CC electrodes introducing a frequency dependence, even for the case of a purely conductive parallel path and a purely conductive sample. As the frequency increases, the electrode impedance will decrease and the current flowing through the sample will increase. This results in the measured impedance increasing in magnitude and having positive phase. As the frequency increases further, the current flowing in the sample will approach the actual injected current level and the measurement error will reduce. The measured impedance would only converge on the true sample impedance when the resistance of the parallel pathway increases sufficiently to satisfy the open circuit condition, i.e. infinite impedance.

Further simulations in which a parallel conductance is present are shown in figure 5. These simulations examine the effect of the CC electrode contact impedance and the resistance of the parallel pathway on the impedance spectrum. For each of the scenarios presented, the true sample impedance (again 500# and no phase) is never measured accurately.

Fig. 5

For the cases of CC electrodes with high contact impedance, the measurement error is immediately apparent because of the increase in impedance magnitude and peaking phase profile above 100kHz. The error is accentuated when the resistance of the parallel pathway is low. In this case the variability is over 80% and the mean error across the spectrum is over 70%. When the CC electrodes have low contact impedance, the phase profile is much flatter, being less than three degrees across the full spectrum in these examples. The magnitude of the measured impedance spectrum is correspondingly more uniform and the variability drops to below 10%. However, the measurement error is still considerable and reaches over 50% when the parallel pathway is of low resistance. The variability and mean error for each case is quantified in table 1.

Figure 6 shows typical data acquired from the EndOhm device with a transwell membrane in situ. In the time that lapsed between measurements the chloride coating was stripped from the lower CC electrode. In keeping with simulated data, the variability in the spectrum increased from 6.84% for the chlorided case to 29.86% for the dechlorided case. The solid lines show the result of fitting this data to the model of parallel conductance. The fitting results show a dramatic rise in contact impedance for one of the electrodes. This is consistent with the fact that the chloride layer has been stripped from the lower CC electrode.

Fig. 6

Figure 7 shows the impedance spectra for three known conductivity calibration solutions. Increasing impedance magnitude with frequency and positive phase are seen in each of the spectra. The level of artefact is inversely correlated with the conductivity of the solution. These characteristics are consistent with the presence of a parallel conductive pathway. The solid lines are the result of fitting these data to the equivalent electrical circuit model of parallel conductance.

Fig. 7

A good fit between the model and the data is achieved for each solution. The fitted solution resistance (values given in table 2) increases linearly with resistivity. Regression analysis through the origin gives a value of 15.7m^-1 for the pseudo geometric factor of the rig.

Figure 8 shows the sensitivity map of the EndOhm device. The sensitivity is highly non-linear, with distinct positive and negative regions. The sensitivity is large and positive between the upper and lower PU electrodes. In the insulated regions between the PU and CC electrodes are two regions of large negative sensitivity. Away from these dominant regions, the sensitivity goes to zero.

Fig. 8

Since this model is axially symmetric, the pseudo geometric factor (or cell constant) can be calculated for the device by integrating over the full 3D volume. The sensitivity based pseudo geometric factor for the device is 13.3m^-1. The numerical implementation of the measuring procedure gives a higher pseudo geometric factor of 16.0m^-1.

The values estimated for the solution resistance, R_S, by fitting to the equivalent circuit model of parallel conductance are shown in table 2 along with the finite element modelling results of expected transfer resistance. There is good agreement between the fitted solution resistance and the numerical estimate, evidenced by the fact that the geometric factors calculated by these two approaches are within 2% of each other. The sensitivity based measurement gives a lower estimate for the geometric factor.

The optical density measurements confirmed that the number of cells grown on the transwell membranes was proportional to the seeding level used. Micrographs of three transwells depicting different levels of cell growth are shown in figure 9a In the impedance spectra acquired from the transwells, the impedance magnitude is increased with respect to the blank membrane for all transwells on which cells have been cultured. However, the characteristic !dispersion expected for such measurements is not present and their ordering does not always reflect the seeding levels used. Figure 9b shows sample spectra corresonding to the micrographs in the same figure along with the spectra fitted to the equivalent electrical circuit model of parallel conductance. The corresponding reconstructed spectra depicting the impedance from within the measurement rig are shown in figure 9c

Fig. 9

The blank transwell shows no dispersion with frequency meaning that the impedance of the rig is purely resistive, as expected. Dispersions are seen for the other transwells. The magnitude of the dispersion is seen to increase with increasing seeding level. Figure 10 quantifies the dispersion as the drop in impedance from the low frequency limit, i.e. no intracellular flow, to the high frequency limit, i.e. no reactance due to the cell membranes. This drop in impedance is expressed as a percentage of the low frequency limit. No change corresponds to no dispersion and therefore no cellular component contributing whereas a decrease denotes a dispersion and therefore the presence of a cellular comonent. At low cell levels the gaps on the transwell membrane on which no cells have grown dominate. As the confluency of the cell layer increases, the cellular components of the model become more dominant and the magnitude of dispersion increases. The magnitude of the dispersion reaches a maximum as the transwell membrane becomes completely covered by the cell layer.

Fig. 10