Cellular Electrical Micro-Impedance Parameter Artifacts Produced by Passive and Active Current Regulation

Abstract

This study analyzes the cellular microelectrode voltage measurement errors produced by active and passive current regulation, and the propagation of these errors into cellular barrier function parameter estimates. The propagation of random and systematic errors into these parameters is accounted for within a Riemannian manifold framework consistent with information geometry. As a result, the full non-linearity of the model parameter state dependence, the instrumental noise distribution, and the systematic errors associated with the voltage to impedance conversion, are accounted for. Specifically, cellular model parameters are treated as the coordinates of a model space manifold that inherits a Riemannian metric from the data space. The model space metric is defined in terms of the pull back of an instrumental noise-dependent Fisher information metric. Additional noise sources produced by the evaluation of the cell-covered electrode model that is a function of a naked electrode random variable are also included in the analysis. Based on a circular cellular micro-impedance model in widespread use, this study shows that cellular barrier function parameter estimates are highly model state dependent. Systematic errors produced by coaxial lead capacitances and circuit loading can also lead to significant and model state-dependent parameter errors and should, therefore, be either reduced or corrected for analytically.

Introduction

Micro-impedance sensors have applications ranging from toxicology screening to cellular motility and adhesion interaction studies. One of the most common micro-impedance sensors currently available consists of a gold micro-electrode with a large counter electrode.12,13 Although such sensors have been increasingly used to probe and study time- and frequency-dependent cellular impedances,1,4,6–8,11,18,20–23,25,27–29 several forms of random and systematic errors may potentially corrupt these impedance measurements. Despite the importance of this technology, few studies have attempted to identify and reduce these errors and their propagation into cellular barrier function parameter estimates based on cell–cell, cell–matrix, and membrane impedance components.30

Several forms of time- and frequency-dependent artifacts can potentially corrupt cellular micro-impedance measurements.10 Systems based on phase sensitive detection are particularly susceptible to synchronous and 60 Hz noise. Following phase sensitive detection,19 this noise can appear at different frequencies depending on the lock-in frequency. The sampling rate can also produce complicated noise components as a result of aliasing. Gaussian noise exists in most circuits and even with very efficient filtering, analog to digital noise is always present and sets a lower limit to the achievable resolution. The large frequency-dependent changes in the gold electrode impedance can also introduce frequency-dependent loading artifacts. Coaxial cable capacitances are another source of frequency-dependent systematic errors.

The nonlinearity of the functions used to model cellular barrier function are coupled to both time-dependent and systematic instrumental noise, and present a significant obstacle to cellular barrier function analysis. The sensitivity of each of the parameters can vary significantly from one model state to another.5 A recent study has quantified this sensitivity with respect to instrumental noise fluctuations using information geometry.9 Geometric methods can account for both systematic and random error propagation. Random error propagation can be accounted for by transformations of the Fisher metric. Systematic errors can be quantified using the displacements in parameter values from their true values based on the Fisher metric.

The objective of this study is to quantify the time-dependent and systematic error propagation into cellular impedance parameter estimates using both passive and active current sources. Using a consistent geometric framework, a data space metric is defined in terms of time-dependent instrumental noise levels, and model states are mapped into this data space. Systematic errors are quantified by calculating the distance between displaced parameter states that arise from the voltage to impedance conversion.

Methods and Materials

To meet this study’s objective, impedance estimates of a cell-covered industry standard gold micro-electrode and an electrode circuit model, with known impedance, were obtained using passive and active current source circuit configurations. The passive current source circuit configuration, commonly implemented in these types of measurements, uses a large resistor in series with the electrode to give an approximately constant 1 μA current source. The active current source circuit uses a transconductance amplifier to provide a more constant 1 μA current source. Using an electrode circuit model with a known impedance, within an error defined by the component resistor and capacitor tolerances, the voltage to impedance conversion models were tested. This system’s random and systematic errors were then used to define a data metric for a parameter precision and displacement analysis. The cell-covered electrode model, instrumental circuit configuration, and noise levels were then unified within a consistent Riemannian manifold framework of information geometry.

Cell Culture

Endothelial cells were isolated from porcine pulmonary arteries obtained from a local abattoir. The endothelial cells were cultivated at 37 °C and 5% CO₂. Cell culture media consisted of M199 (GibcoBRL) and 10% fetal bovine serum (Hyclone) supplemented with vitamins (Sigma), glutamine (GibcoBRL), penicillin and streptomycin (GibcoBRL), and amino acids (Sigma). Endothelial cells were inoculated onto a series of gold micro-electrodes (Applied Biophysics) coated with fibronectin (BD Biosciences) to facilitate cellular adhesion.

Cellular Impedance Circuit Electronics and Analysis

Figure 1 shows a detailed schematic of the active current source. The voltage-dependent current source was constructed using a modified Howland current pump with a precision FET input high common mode rejection ratio operational amplifier to produce a transimpedance amplifier with an extremely high output impedance.16 If the resistors are perfectly matched, the output impedance using an AD845 operational amplifier is over 500 MΩ; this degrades to 25 MΩ when using 0.1% resistors, a 25-fold improvement over using a passive current source. The circuit was composed of an inverting amplifier, U1, connected to a modified Howland current pump, U2. The inverting amplifier reduced the signal amplitude by a factor of 100, corrected the signal phase inversion introduced by the second stage inverting Howland current pump, and provided the ability to adjust the transconductance gain, via R3, and correct for any small voltage offset, via R11, that would have otherwise caused a small DC bias current to flow through the electrode array.

Figure 1

Constant 1 μA current source based on a Howland pump. A precision field effect transistor with a high common mode rejection ratio operational amplifier produces a transimpedance amplifier with a very high output impedance. The inverting amplifier, U1, connects to a modified Howland current pump, U2. The inverting amplifier reduces the signal amplitude by a factor of 100 and corrects the signal phase inversion introduced by the second stage inverting Howland current pump. The inverting amplifier also provides the ability to adjust the transconductance gain, via R3, and corrects for any small voltage offset, via R11

A lock-in amplifier (Stanford Research SR830) provided 1 volt AC reference signals between 56 Hz and 56 kHz to the electrode via either a series 1 MΩ resistor or a voltage-controlled Howland pump current source as shown in Figs. 2a and 2b. In the first configuration, shown in Fig. 2a, a 1 MΩ resistor was used in series with the 1v_rms AC reference signal. In the second configuration, shown in Fig. 2b, the voltage-controlled current source, shown in Fig. 1, maintained a constant 1 μA current. In each circuit, the conversion of the measured voltage to an equivalent impedance follows from basic circuit analysis, the details of which are outlined in Appendix A.

Figure 2

Passive and active circuit configurations for cellular impedance measurements. The circuit parameters in both cases are: V _AC = 1 ∠ 0^o, R _s = 50 Ω, C _ps = 86 pF, R _cc = 1 MΩ, C _pv = 90 pF, R _v = 10 MΩ, and C _v = 25 pF. (a) In the passive current source configuration, a large 1 MΩ resistor, R _cc, provides an approximately constant current provided that it is much greater than the electrode load impedance Z _e. The parasitic capacitance of the coaxial cable connecting R _s to R _cc produces a negligible contribution as a result of the low source impedance. (b) In the active current source configuration, a voltage to current source converter provides an almost constant 1 μA current source to the remainder of the circuit

Electrode Model and Calibration

An electrode circuit model consisting of a parallel resistor, R _p, and capacitor, C _p, combination in series with another resistor, R _c, was used to evaluate the system.26 This circuit provided a known frequency-dependent impedance standard, within the propagated tolerance of the circuit components, to evaluate the system. The manufacturer’s labeled values were R _p = 1.00 ± 0.01 MΩ, C _p = 10.0 ± 0.2 nF, and R _c = 2.20 ± 0.02 kΩ. The measured values using a BK Precision 889 A Bench LCR/ESR Meter were R _p = 0.975 ± 0.001 MΩ, C _p = 8.656 ± 0.001 nF, and R _c = 2.165 ± 0.001 kΩ. The component values were chosen to produce a similar voltage response to the naked electrode over the frequency range of interest. The circuit model impedance, z _mod, is

$$ z_{{\bmod }} = \frac{{R_{{\text{p}}} }} {{1 + j\omega R_{{\text{p}}} C_{{\text{p}}} }} + R_{{\text{c}}} . $$

(1)

A BK Precision 889 A Bench LCR/ESR Meter also provided coaxial cable, electrode circuit model, and cellular micro-electrode impedances measurements.

Cell-Covered Electrode Model

Figure 3 outlines a quantitative description of an endothelial cell monolayer layer cultivated on an electrode. Current can flow between cells, underneath the cells, and through the membrane via capacitive and resistive coupling. A closed form solution for circular cells based on the continuity arguments outlined in Fig. 3b has been derived in previous studies,14,15 and is given by

$$ z_{{\text{c}}} = z_{{\text{n}}} {\left[ {\frac{{z_{{\text{n}}} }} {{z_{{\text{n}}} + z_{{\text{m}}} }} + \frac{{\frac{{z_{{\text{m}}} }} {{z_{{\text{n}}} + z_{{\text{m}}} }}}} {{\frac{{\gamma r_{{\text{c}}} }} {2}\frac{{I_{0} {\left( {\gamma r_{{\text{c}}} } \right)}}} {{I_{1} {\left( {\gamma r_{{\text{c}}} } \right)}}} + R_{{\text{b}}} {\left( {\frac{1} {{z_{{\text{n}}} }} + \frac{1} {{z_{{\text{m}}} }}} \right)}}}} \right]}^{{ - 1}} , $$

(2)

where z _c is the cell-covered electrode impedance, z _n the naked electrode impedance, z _m the cell membrane impedance, R _b the cell–cell junction impedance, r _c is the radius of a single cell, and I ₀(γr _c) and I ₁(γr _c) are modified Bessel functions of the first kind of zero and first order, respectively.

Figure 3

Model system definition. (a) The total measured impedance is a function of the naked electrode resistance, Z _n, the impedance underneath the cells, α, the trans-cellular impedance, Z _m, and the resistance between adjacent cell, R _b. (b) A micro-continuum description of the cell-covered electrode can be formulated by defining the basal, I, electrode, I _n, and membrane, I _m, currents as a function of the radial coordinate r. The electrode voltage is V _n and the surrounding electrolyte has a voltage V ₀

The variable γ is defined as

$$ \gamma = {\sqrt {\frac{\rho } {h}{\left( {\frac{1} {{z_{{\text{n}}} }} + \frac{1} {{z_{{\text{m}}} }}} \right)}} }, $$

(3)

where ρ is the media resistivity and h is the cell substrate separation distance. The cell membrane impedance, z _m, can be considered a series combination of an apical and basal membrane impedance that consists of a parallel resistor and capacitor combinations, i.e.,

$$ z_{{\text{m}}} = \frac{{2R_{{\text{m}}} }} {{1 + j2\pi fR_{{\text{m}}} C_{{\text{m}}} }}, $$

(4)

where each membrane consists of a parallel resistor, R _m, and capacitor, C _m, combination. A parameter, α, can be defined as

$$ \alpha = r_{{\text{c}}} {\sqrt {\frac{\rho } {h}} }, $$

(5)

that represents the cell–matrix impedance. Based on the model given by Eq. (2), this study will consider the two parameters α and R _b, to be optimized based on voltage measurements, as the local coordinates of a model space and the impedance function maps these points into an impedance data space.

Geometric Error Analysis

Associated with each measured impedance, z, is a statistical distribution of voltage values. If these voltage measurements are normally distributed with mean μ and covariance matrix Σ, then

$$ p{\left( {v;{\left\{ {\mu ,\Sigma } \right\}}} \right)} = {\left( {2\pi \det \Sigma } \right)}^{{ - n}} \exp {\left( { - \frac{1} {2}{\left( {{\mathbf{v}} - \varvec{\upmu}} \right)}^{T} \Sigma {\left( {{\mathbf{v}} - \varvec{\upmu}} \right)}} \right)}, $$

(6)

where n represents the number of measured frequencies, v is a random voltage variable, and T represents the transpose. The population mean, μ, and covariance are assumed to vary smoothly with respect to z. As described in more detail in Appendix B, if θ parameterizes the set of distributions then the Fisher metric follows from the relation

$$ g_{{ab}} (\theta ) = E{\left[ {\frac{\partial } {{\partial \theta ^{a} }}\ln p(v;\theta )\frac{\partial } {{\partial \theta ^{b} }}\ln p(v;\theta )} \right]}, $$

(7)

where E denotes the expectation value and the indices are defined over the range 1 ≤ a,b ≤ q. In the case of a normal distribution with covariance matrix, Σ, that does not vary significantly with respect to change in the data state

$$ g_{{ab}} = \frac{{\partial \varvec{\upmu}^{T} }} {{\partial z^{a} }}\Sigma ^{{ - 1}} \frac{{\partial \varvec{\upmu}}} {{\partial z^{b} }} = \Sigma ^{{ - 1}}_{{ab}} . $$

(8)

The parameter precision is based on the pull-back of this metric and systematic errors are quantified in terms of distances based on this metric as described in Appendix C.

Experimental Probability Estimation

To apply the above theory in practice, it is necessary to use experimental measurements to estimate the underlying, or population, distribution and propagate the errors in terms of the Fisher information metric. The Fisher metric is estimated from the measured naked and cell-covered electrode noise statistics. Average voltage estimates of the population average, ${\varvec{\upmu}}$, were calculated from the N data samples, or observations, at each frequency, i.e.,

$$ \ifmmode\expandafter\bar\else\expandafter\=\fi{v}_{ek} = \frac{1} {N}{\sum\limits_{i = 1}^N {v^{i}_{ek} } }, $$

(9)

where $\ifmmode\expandafter\bar\else\expandafter\=\fi{v}_{ek}$ represents the data sample average at the kth frequency and v ⁱ _ek represents the ith voltage sample at the kth frequency. The data variance–covariance matrix at each frequency,

$$ S_{k} = {\left[ {\begin{array}{*{20}c} {{S^{{{\Re }{\Re }}}_{k} }} & {{S^{{{\Re }{\Im }}}_{k} }} \\ {{S^{{{\Im }{\Re }}}_{k} }} & {{S^{{{\Im }{\Im }}}_{k} }} \\ \end{array} } \right]} $$

(10)

was calculated using the relations

$$ \begin{aligned}{} & S^{{{\Re }{\Re }}}_{{k}} = {\sum\limits_{i = 1}^N {\frac{{{\left( {v^{i}_{{{e\Re }{k} }} - \ifmmode\expandafter\bar\else\expandafter\=\fi{v}_{{{e\Re }{k} }} } \right)}{\left( {v^{i}_{{{e\Re }{k} }} - \ifmmode\expandafter\bar\else\expandafter\=\fi{v}_{{{e\Re }{k} }} } \right)}}} {{{\left( {N - 1} \right)}}}} },\quad S^{{{\Re }{\Im }}}_{{{k} }} = {\sum\limits_{i = 1}^N {\frac{{{\left( {v^{i}_{{e{\Re }{k} }} - \ifmmode\expandafter\bar\else\expandafter\=\fi{v}_{{e{\Re }{k} }} } \right)}{\left( {v^{i}_{{e{\Im }{k} }} - \ifmmode\expandafter\bar\else\expandafter\=\fi{v}_{{e{\Im }{k} }} } \right)}}} {{{\left( {N - 1} \right)}}}} }, \\ & S^{{{\Im }{\Re }}}_{{{k} }} = {\sum\limits_{i = 1}^N {\frac{{{\left( {v^{i}_{{e{\Im }{k} }} - \ifmmode\expandafter\bar\else\expandafter\=\fi{v}_{{e{\Im }{k} }} } \right)}{\left( {v^{i}_{{e{\Re }{k} }} - \ifmmode\expandafter\bar\else\expandafter\=\fi{v}_{{e{\Re }{k} }} } \right)}}} {{{\left( {N - 1} \right)}}}} },\quad {\text{and }}\quad S^{{{\Im }{\Im }}}_{{{k} }} = {\sum\limits_{i = 1}^N {\frac{{{\left( {v^{i}_{{e{\Im }{k} }} - \ifmmode\expandafter\bar\else\expandafter\=\fi{v}_{{e{\Im }{k} }} } \right)}{\left( {v^{i}_{{e{\Im }{k} }} - \ifmmode\expandafter\bar\else\expandafter\=\fi{v}_{{e{\Im }{k} }} } \right)}}} {{{\left( {N - 1} \right)}}}} }, \\ \end{aligned} $$

(11)

where the real and imaginary impedance component standard deviations were defined as

$$ s^{{{\Re }{\Re }}}_{k} = {\sqrt {S^{{{\Re }{\Re }}}_{k} } }\quad {\text{and}}\quad {\text{ }}s^{{{\Im }{\Im }}}_{k} = {\sqrt {S^{{{\Im }{\Im }}}_{k} } }, $$

(12)

respectively. The correlation coefficients at frequency f _k were calculated from the relation

$$ r^{{{\Re }{\Im }}}_{k} = r^{{{\Im }{\Re }}}_{k} = \frac{{S^{{{\Re }{\Im }}}_{k} }} {{{\sqrt {S^{{{\Re }{\Re }}}_{k} } }{\sqrt {S^{{{\Im }{\Im }}}_{k} } }}}. $$

(13)

The unbiased variance–covariance matrix, $S_k,$ at the kth frequency provides an unbiased estimate of the population variance–covariance matrix $\Sigma_k.$

Errors in the statistical estimators of the mean and variance of the underlying probability density also contribute to the uncertainty. The population variance of the sample mean and sample variance based on N samples can be used to justify ignoring this effect. The analysis presented in this study assumes that these contributions are negligible. The uncertainties in the circuit elements produce another Fisher metric on the control spaces involved in the voltage to impedance conversion.

Results

The results of this study are organized in such a way as to illustrate the different error sources and their contribution to the precision and accuracy of the cellular impedance and model parameters. Using the Fisher information matrix, random noise produces a decrease in parameter precision whereas systematic errors decrease the accuracy by producing displacements from the true value. The first two subsections quantify the random and systematic errors in this system and their effect on voltage measurements and impedance estimates. The last two subsections illustrate the propagation of these errors into the model parameter estimates.

Random Sample Space Voltage Estimates

Figure 4 provides a statistical summary of the measured voltages from a naked gold electrode, a cell-covered gold electrode, and the electrode circuit model. At each frequency, the statistics of 64 voltage measurements sampled at a rate of 32 Hz for 2 s were evaluated using Eqs. (9–11). The electrode circuit model frequency-dependent average voltages are in qualitative agreement with those of the gold electrode. Assuming a constant 1 μA electrode current, the equivalent impedances can be estimated by scaling the measured voltages by a factor of 10⁶. Figure 4 also illustrates the real and imaginary frequency dependence of the noise variance of the naked electrode, the cell-covered electrode, and the resistor and capacitor electrode model. At lower frequencies, the resistor and capacitor model has a lower noise level than the naked and cell-covered electrodes. A variance peak, consistent with 60 Hz noise, is also present in the resistor and capacitor circuit model.

Figure 4

Circuit model, naked electrode, and cell-covered electrode voltage statistics as a function of frequency using the passive current source. (a) The real noise variances of the cell-covered, ${\left\langle {{\Re }^{{\text{2}}}_{{\text{c}}} } \right\rangle }$, and naked, ${\left\langle {{\Re }^{{\text{2}}}_{{\text{n}}} } \right\rangle }$, electrodes are similar but the circuit model, ${\left\langle {{\Re }^{{\text{2}}}_{{\text{m}}} } \right\rangle }$, is significantly less at lower frequencies. (b) The imaginary cell-covered, ${\left\langle {{\Im }^{{\text{2}}}_{{\text{c}}} } \right\rangle }$, naked, ${\left\langle {{\Im }^{{\text{2}}}_{{\text{n}}} } \right\rangle }$, and model, ${\left\langle {{\Im }^{{\text{2}}}_{{\text{m}}} } \right\rangle }$, variance components show a similar pattern to the real variance measurements. (c) The average values of the cell-covered, ${\left\langle {{\Re }_{{\text{c}}} } \right\rangle }$, naked, ${\left\langle {{\Re }_{{\text{n}}} } \right\rangle }$, and circuit model, ${\left\langle {{\Re }_{{\text{m}}} } \right\rangle }$, resistive mean components decrease with increasing frequency and are in qualitative agreement with each other. (d) The average reactive components of the cell-covered, ${\left\langle {{\Im }_{{\text{c}}} } \right\rangle }$, naked, ${\left\langle {{\Im }_{{\text{n}}} } \right\rangle }$, and circuit model, ${\left\langle {{\Im }_{{\text{m}}} } \right\rangle }$, components show a similar pattern to the real values. The electrode circuit model noise levels are significantly lower than the electrode noise levels at lower frequencies

Systematic Voltage Data Space Displacements

Figure 5 shows the measured and predicted normalized and random impedance deviations of the electrode circuit model using the passive and active circuit configurations. Measurements obtained from an LCR meter are included for comparison. In Figs. 5a and 5b, the voltage measurements for both the passive and Howland current sources were scaled by a factor of 10⁶ to produce impedance estimates assuming a constant 1 μA electrode current. Figures 5c and 5d show the data space components of the deviations to the ideal 1 μA current source evaluated using the Fisher metric. Using the frequency-dependent noise levels results in a different pattern than the normalized data. During a numerical optimization, these will be the regions that contribute the most to systematic errors. The high frequency regions contribute significantly to the optimized fit error if they are not corrected. It is important to realize that these quantified shifts based on the Fisher information metric are more important than the normalized curves shown in Figs. 5a and 5b. It is clear from Figs. 5c and 5d that the parasitic capacitive elements will have a more severe effect on the parameter optimization if they are not corrected. The components of the displacement between states assuming a constant 1 μA current source and corrections produced by the circuit model show at what frequencies the most significant errors contribute to the parameter uncertainty. The total error, or variance, will be a sum of these components.

Figure 5

The normalized and statistical systematic deviations of frequency-dependent real and imaginary impedance components. (a) The normalized deviation of the resistance, (R _c  − R _n)/R _n, following a 10⁶ scaling, shows the voltage divider effect that occurs at low frequencies when a passive current source is used and a constant 1 μA electrode current is assumed. Using an active current source partially corrects this. (b) The normalized reactance shows a similar voltage divider effect at low frequencies and large normalized deviations at high frequencies produced by circuit capacitances and other artifacts. (c) The geodesic displacements of the real impedance show the significance of the circuit capacitive effects at higher frequencies. Although the normalized deviations show the voltage divider effects, the circuit capacitive elements produce statistically more significant deviations at higher frequencies. Using a constant current source does not correct this. (d) A similarly increasing geodesic separation distance is observed at increasing frequencies with the imaginary impedance component

Model Parameter Precision Analysis

In the absence of systematic errors, random noise fluctuations can enter into the parameter optimization through both the naked and cell-covered electrode measurements. For this particular study, Fig. 4 summarizes the statistics of a representative naked and cell-covered electrode that the following results will be based on. Figure 6 shows a precision analysis resulting from pulling the cell-covered electrode and naked electrode noise metrics back to the model space. The left column of subplots, (a–c), summarizes a precision analysis produced by cell-covered electrode fluctuations while the right column, (d–f), shows the contribution produced by the naked electrode. Plots (a) and (d) refer to the parameter α while plots (b) and (e) refer to the parameter R _b. Plots (c) and (f) summarize the case when both parameters are considered simultaneously.

Figure 6

Parameter precision analysis of α, R _b and joint α-R _b using the Fisher metric induced when naked electrode covered fluctuations (a–c) or naked electrode fluctuations (d–f) are used to define the Fisher metric. Plots (a) and (d) refer to the parameter α, while plots (b) and (e) refer to the parameter R _b. Plots (c) and (f) summarize the case when both parameters are considered simultaneously

Systematic Model Space Displacements

Systematic errors in the measured voltages enter into the data analysis through both the naked and the cell-covered electrode impedance estimates. Figure 7 shows the weighted separation distance in the data space to the model submanifold using both a passive current source and an ideal constant current source. The displacement from each submanifold model state produced by the measured coaxial capacitance C _p, R _cc, and load impedance is a function of the model state. The displacement is equivalent to the increase in χ ² produced by the systematic error components.

Figure 7

Systematic errors produced by naked and cell-covered impedance estimates using a passive and constant current source. (a–c) Passive systematic errors. Plot (a) shows the systematic errors produced by the naked electrode only, (b) the cell-covered electrode, and (c) both naked and cell-covered electrode impedance systematic errors. (d–f) Constant current systematic error summary shows the systematic errors produced by the (d) naked electrode, (e) cell-covered, and (f) a combination of both naked and cell-covered electrode errors. A constant current source significantly reduces the systematic errors. The cell-covered systematic errors make a more significant contribution to the systematic errors. The combination of naked and cell-covered systematic errors partially compensate each other

Parameter systematic errors can enter through systematic errors in either the naked, cell-covered, or a combination of both the naked and cell-covered electrode impedance measurements. In both the passive and active current sources, systematic errors produced by the naked electrode are less than those produced by the cell-covered electrode alone. When both sources of systematic errors are included, the overall systematic errors are less than those produced by the cell-covered electrode alone. Some compensation, therefore, occurs when the naked and cell-covered electrode systematic errors occur together. The systematic errors produced by the cell-covered electrode dominate the overall systematic errors. Increasing values of R _b and decreasing α values are more sensitive to naked electrode systematic errors. Increasing values of both the R _b and α model states are more susceptible to cell-covered impedance systematic errors. In the absence of any circuit model corrections, a constant current source produces a significant reduction in systematic errors compared to those produced by the passive current source. Although an active current source reduces the systematic errors compared to the passive source, large values of χ ² still remain for most states.

Random and Systematic Corrections to Optimized Model Parameters

To correct the systematic errors indicated in Fig. 7, a calibration correction at each frequency point was carried out using known series resistance and capacitor combinations. Based on the frequency-dependent impedance estimates of the naked and cell-covered electrodes, a range of capacitors and resistors were chosen at each frequency to provide a calibration of known impedances. The quoted error was based on the propagated voltage following the interpolation transformation and the calibration standard errors. Figure 8 shows the impedance estimates based on the calibration and corrections at each frequency.

Figure 8

Improved data fit using random and systematic error corrections. The two model fits illustrate the case of (a) including random noise without any systematic error corrections, (b) including both random and systematic errors in the optimization. Not including any noise measurements in the optimization produced very poor quality fits that often do not converge to meaningful results. Including noise measurements stabilized the convergence but still produced poor quality fits (χ ²_ν  = 8.417 × 10⁶). Including systematic errors significantly improved the fit quality (χ ²_ν  = 9.889 × 10⁵)

Discussion

The quantification of cellular barrier function parameters optimized from electrical impedance measurements is complicated by the model non-linearity, frequency-dependent noise levels, and systematic errors associated with the voltage to impedance conversion. Noise estimates play a critical role in optimizing and quantifying these parameters. By constructing a Fisher metric using measured noise levels, it is possible to examine the contributions of naked and cell-covered electrode measurement errors to the parameter precision. In addition, the Fisher metric can quantify systematic errors as the distances separating actual impedance values from their values when shifted by the circuit parameter component values. Noise measurements also play an important role in stabilizing the parameter analysis. If noise measurements are not included in the optimization, the data rarely converge to meaningful results. The large errors that occur at low frequencies produce instabilities in the optimization if these data points are equally weighted with the rest of the data.

The conceptual images that one should have while reading the statistical results presented in Fig. 4 are the following. An estimate of the location of the naked, cell-covered, and model electrode states in their respective probability manifolds are first obtained by calculating the frequency-dependent averages and covariance components. Although n different frequencies are measured, one should still think of this as a single point in a neighborhood of ℜ²ⁿ × ℜ²ⁿⁿ⁺ⁿ on the naked, S _n, cell-covered, S _c, and electrode model, S _m, probability manifolds. The uncertainty associated with this point, based on the number of samples in the statistical estimator, defines neighborhoods of the naked, U _n, cell-covered, U _c, and model, U _m, probability manifolds to examine. The number of data samples, N, in this study is assumed to be large enough to justify using a single point of the naked electrode probability manifold with respect to the uncertainties in this system.

Complications in the error propagation arise because some of the parameters in the model function are, themselves, random variables. The fact that the naked electrode impedance must be input to the model function, therefore, introduces an additional noise source in the analysis. If the model function inverse exists, random and systematic errors associated with these additional variables can be included in the model parameter error analysis by using fiber bundle mappings. As a result, this geometric framework has the freedom to include multiple forms of measurement errors in the parameter analysis.

The systematic errors produced by uncorrected loading, which in turn are produced by the circuit parameter R _cc, coaxial capacitances C _p, and the lock-in amplifier input impedance, have two important effects on the parameter optimization. Systematic errors shift the measured voltage away from the true voltage and, thereby, increase the χ ²_ν value. Systematic errors, however, limited this minimum separation distance, as indicated by the large χ ²_ν values. The maximum likelihood point can also be shifted to another model state. These are important considerations when trying to numerically discriminate among different possible models.

Increasing random noise levels decrease the model state precision. In this sensor and model, however, systematic errors appear to have a much more detrimental effect on the parameter estimates if they are not corrected. Relatively small amounts of parasitic capacitances produce large shifts in the location of the model state. This study also partially explains the very large χ ²_ν values observed in previous studies.10 In particular, coaxial cable capacitances can produce large shifts in the impedance value.

The statistical approximation to the population covariance–variance matrix can potentially introduce an additional form of error into the Fisher information matrix that should be considered in this analysis. Careful consideration of the required number of data points to reduce these contributions to negligible values is, therefore, important. The number of data samples was assumed to be large enough to justify taking the estimated states as a single point. In reality, however, the finite sampling of data points introduces yet another source of uncertainty in the system. The Cramer–Rao theorem2,24 states that the covariance of any unbiased estimator $\hat{\theta } = (\hat{\theta }^{i} )$ is bounded by the inverse of the Fisher information matrix, i.e.,

$$ Cov{\left[ {\hat{\theta }^{i} ,,\hat{\theta }^{j} } \right]} \ge g^{{ij}} $$

(14)

where a ^ij ≥ b ^ij implies that a ^ij − b ^ij forms a positive semi-definite matrix. This limit is attained asymptotically in the sense that if $\hat{\theta }_{N}$ is an estimator based on the N independent observations $v^{1} ,v^{2} , \ldots ,v^{N}$ from the distribution p(v, θ), then the covariance of $\hat{\theta }_{N}$ tends to g ^ij/N as N tends to infinity,

$$ Cov{\left[ {\hat{\theta }^{i} ,\hat{\theta }^{j} } \right]} \to \frac{1} {N}g^{{ij}} $$

(15)

The distribution of the estimator $\hat{\theta }_{N}$ tends to the normal distribution N (θ, g ^ij/N).

When used with a lock-in amplifier, a Howland current source significantly reduced systematic errors associated with impedance estimates below 1 kHz, as compared to a passive current source. At frequencies above 1 kHz, however, both the passive and active current source configurations produced significant errors. Carefully circuit model corrections are, therefore, necessary to convert the measured voltage into an equivalent impedance in both cases.

Appendices

Appendix A

Voltage and Impedance Conversion Circuit Analysis

Using basic circuit analysis,17 the measured electrode voltage based on the passive current source shown in Fig. 2a is

$$ v_{{\text{e}}} {\left( {z_{{\text{e}}} } \right)} = {\left\{ {\frac{{z_{{\text{L}}} {\left( {z_{{\text{e}}} } \right)}}} {{z_{{{\text{cc}}}} + z_{{\text{L}}} {\left( {z_{{\text{e}}} } \right)} + z_{{\text{s}}} + \frac{{z_{{\text{s}}} }} {{z_{{{\text{ps}}}} }}z_{{{\text{cc}}}} + \frac{{z_{{\text{s}}} }} {{z_{{{\text{ps}}}} }}z_{{\text{L}}} {\left( {z_{{\text{e}}} } \right)}}}} \right\}}v_{{\text{s}}} , $$

(A.1)

where v _e is the measured electrode voltage, z _e represents either the naked, cell-covered, or electrode model impedance, z _ps = 1/jωC _ps, z _pc = 1/jωC _pc and z _pv = 1/jωC _pv represent coaxial lead impedances, R _s = 50 Ω the current source impedance, R _cc = 1 MΩ, z _v = R _v/(1 + jωR _v C _v) the lock in amplifier input impedance, and the term

$$ z_{{\text{L}}} {\left( {z_{{\text{e}}} } \right)} = \frac{{z_{{{\text{pc}}}} zz_{{{\text{pv}}}} z_{{\text{v}}} }} {{z_{{\text{e}}} z_{{{\text{pv}}}} z_{{\text{v}}} + z_{{{\text{pc}}}} z_{{{\text{pv}}}} z_{{\text{v}}} + z_{{{\text{pc}}}} z_{{\text{e}}} z_{{\text{v}}} + z_{{{\text{pc}}}} z_{{\text{e}}} z_{{{\text{pv}}}} }}, $$

(A.2)

represents the parallel combination of the circuit elements to the right of R _cc. When the source resistance, z _s = R _s, is very small compared to z _ps, this result simplifies to the familiar voltage divider law, i.e.,

$$ v_{{\text{e}}} {\left( {z_{{\text{e}}} } \right)} = {\left\{ {\frac{{z_{{\text{L}}} {\left( {z_{{\text{e}}} } \right)}}} {{z_{{{\text{cc}}}} + z_{{\text{L}}} {\left( {z_{{\text{e}}} } \right)}}}} \right\}}v_{{\text{s}}} . $$

(A.3)

Converting the measured voltage to an equivalent impedance, therefore, follows from the inverse relation

$$ z_{{\text{e}}} {\left( {v_{{\text{e}}} } \right)} = \frac{{z_{{\text{v}}} {\left( {z_{{{\text{cc}}}} + z_{{\text{s}}} + \frac{{z_{{\text{s}}} }} {{z_{{{\text{ps}}}} }}z_{{{\text{cc}}}} } \right)}}} {{z_{{\text{v}}} {\left( {\frac{{v_{{\text{s}}} }} {{v_{{\text{e}}} }} - 1 - \frac{{z_{{\text{s}}} }} {{z_{{{\text{ps}}}} }}} \right)} - {\left( {\frac{{z_{{\text{v}}} }} {{z_{{{\text{pc}}}} }} + \frac{{z_{{\text{v}}} }} {{z_{{{\text{pv}}}} }} + 1} \right)}{\left( {z_{{{\text{cc}}}} + z_{{\text{s}}} + \frac{{z_{{\text{s}}} }} {{z_{{{\text{ps}}}} }}z_{{{\text{cc}}}} } \right)}}}. $$

(A.4)

In the limit that the source voltage resistance, R _s, is small

$$ z_{{\text{e}}} {\left( {v_{{\text{e}}} } \right)} = \frac{{z_{{\text{v}}} z_{{{\text{cc}}}} }} {{z_{{\text{v}}} {\left( {\frac{{v_{{\text{s}}} }} {{v_{{\text{e}}} }} - 1} \right)} - {\left( {\frac{{z_{{\text{v}}} }} {{z_{{{\text{pc}}}} }} + \frac{{z_{{\text{v}}} }} {{z_{{{\text{pv}}}} }} + 1} \right)}z_{{{\text{cc}}}} }}. $$

(A.5)

If, in addition, the lock-in amplifier input impedance, z _v, is very large

$$ z_{{\text{e}}} {\left( {v_{{\text{e}}} } \right)} = \frac{{z_{{{\text{cc}}}} }} {{{\left( {\frac{{v_{{\text{s}}} }} {{v_{{\text{e}}} }} - 1} \right)} - {\left( {\frac{1} {{z_{{{\text{pc}}}} }} + \frac{1} {{z_{{{\text{pv}}}} }}} \right)}z_{{{\text{cc}}}} }}. $$

(A.6)

Notice the inverted parallel combination of z _pc and z _pv. In the limit that the parallel combination of z _pc and z _pv is very large

$$ z_{{\text{e}}} {\left( {v_{{\text{e}}} } \right)} = \frac{{z_{{{\text{cc}}}} }} {{{\left( {\frac{{v_{{\text{s}}} }} {{v_{{\text{e}}} }} - 1} \right)}}}\quad {\text{or}}\quad z_{{\text{e}}} = \frac{{z_{{{\text{cc}}}} v_{{\text{e}}} }} {{v_{{\text{s}}} - v_{{\text{e}}} }}. $$

(A.7)

When z _cc is much smaller than z _c, then $v_{{\text{e}}} \ll v_{{\text{s}}}$ and we get

$$ z_{{\text{e}}} {\left( {v_{{\text{e}}} } \right)} = \frac{{v_{{\text{e}}} z_{{{\text{cc}}}} }} {{v_{{\text{s}}} }}. $$

(A.8)

The measured voltage using the constant current source shown in Fig. 2b is

$$ v_{{\text{e}}} {\left( {z_{{\text{e}}} } \right)} = \frac{1} {{\frac{1} {{z_{{\text{e}}} }} + \frac{1} {{z_{{{\text{pa}}}} }} + \frac{1} {{z_{{{\text{pc}}}} }} + \frac{1} {{z_{{{\text{pv}}}} }} + \frac{1} {{z_{{\text{v}}} }}}}i, $$

(A.9)

where z _c is the electrode impedance, z _pa = 1/{jω C _pa}, z _pc = 1/{jωC _pc} and z _pv = 1/{jωC _pv} represent coaxial lead impedances, and z _v = R _v/(1 + jωR _v C _v) is the lock-in amplifier impedance. The corresponding impedance, given the measured voltage is

$$ z_{{\text{e}}} {\left( {v_{{\text{e}}} } \right)} = \frac{1} {{\frac{1} {{v_{{\text{e}}} }}i - {\left( {\frac{1} {{z_{{{\text{pa}}}} }} + \frac{1} {{z_{{{\text{pc}}}} }} + \frac{1} {{z_{{{\text{pv}}}} }} + \frac{1} {{z_{{\text{v}}} }}} \right)}}}. $$

(A.10)

In the limit that the lock-in amplifier input impedance is very large, the cell-covered impedance reduces to

$$ z_{{\text{e}}} {\left( {v_{{\text{e}}} } \right)} = \frac{1} {{\frac{1} {{v_{{\text{e}}} }}i - {\left( {\frac{1} {{z_{{{\text{pa}}}} }} + \frac{1} {{z_{{{\text{pc}}}} }} + \frac{1} {{z_{{{\text{pv}}}} }}} \right)}}}. $$

(A.11)

Furthermore, if the coaxial cable capacitance is small, then

$$ z_{{\text{e}}} {\left( {v_{{\text{e}}} } \right)} = \frac{{v_{{\text{e}}} }} {{i_{{\text{s}}} }}. $$

(A.12)

Appendix B

Statistical Manifold Analysis

The measurement of each physical impedance state $\zeta \in Z$ produces a statistical distribution of measured voltages depending on the experimental configuration, ρ, defined by the circuit parameters and the instrumental data acquisition settings. If C represents the space of all experimental configurations, such that $\rho \in C$, then define a map, $h:Z \times C \to S:\zeta \times \rho \mapsto p{\left( {v;\theta } \right)}$, from the physical space Z to a manifold, $S = \{ p(v;\theta ):\theta = \{ \theta ^{1} , \ldots ,\theta ^{q} \} \in \Theta \}$ 2,3, of parameterized voltage probability densities. The term v represents a random voltage variable belonging to the sample space V = ℜ²ⁿ, and p(v;θ) is the probability density function of v, parameterized by θ. The experimental configuration or control space, $C \subset {\Re }^{{\text{m}}}$, is parameterized by the set of variables c having components that represent the circuit element variables C _p, R _cc, R _v, etc., illustrated in Fig. 2. The mapping h(ζ, ρ) from the measurement state (ζ, ρ) to the population probability distribution, p(v;θ) defines the error associated with a given measurement. The mean, variance, and θ dependence of p(v;θ) play an important role in defining systematic and random errors associated with a particular measurement. In practice, statistical estimators of the mean and variance can be applied to actual measurements and used in the analysis.

The subscripts n and c are used to denote the naked electrode or cell-covered electrode cases. For example, the measurement of the naked electrode impedance, Z _n, and the cell-covered impedance, Z _c, have associated spaces S _n and S _c, respectively. A similar convention is applied to the other spaces, maps and points. Since the same circuit configuration is used to make both naked and cell-covered electrode measurements, the same set of experimental configuration or control parameters defines the maps h _n(ζ _n, ρ _n) and h _c(ζ _c, ρ _c).

The set of parameters m = (α, R _b) represents the cellular model space coordinates of some open subset M of Euclidean space. The fact that the naked electrode impedance must be included in the domain of the model function introduces an additional complication in the analysis. The cellular impedance model function given by Eq. (2), $\psi :M \times Z_{{\text{n}}} \to Z_{{\text{c}}} \subset {\Re }^{{2n}} ,$ maps each model state $\pi \in M$ and naked electrode impedance state, $\zeta _{{\text{n}}} \in Z_{{\text{n}}} ,$ into a cell-covered impedance physical space element, $\zeta _{{\text{c}}} \in Z_{{\text{c}}} \subset {\Re }^{{2n}} .$ The two sets of 2n coordinate components, $z_{{\text{n}}} = (z^{1}_{{\text{n}}} , \ldots ,z^{{2n}}_{{\text{n}}} )$ and $z_{{\text{c}}} = (z^{1}_{{\text{c}}} , \ldots ,z^{{2n}}_{{\text{c}}} ),$ of the naked and cell-covered electrode physical spaces, Z _n and Z _c, respectively, represent the real and imaginary parts of the n frequencies $\{ f_{1} , \ldots ,f_{n} \} .$

There exist two forms of error associated with the optimization of model states using cellular micro-impedance measurements. The first arises because measurements of a particular cell-covered impedance state produce a distribution of measured voltages governed by the population probability density p _c(v _c;θ _c). The second arises because naked electrode impedance estimates are required to evaluate the model state mapping $\psi :M \times Z_{{\text{n}}} \to Z_{{\text{c}}} \subset {\Re }^{{2n}} .$ The mapping h ⁻¹_n : S _n × C _n→Z_n, therefore, implies that z _n is a random variable as is z_c under the transformation ψ. This contribution can be considered as the probability density p _n(v _n;θ _n(h _n·ψ ⁻¹)) as a function of the cell-covered impedance state z _c. Hence, when both the naked and cell-covered noise distributions are considered, the sample space can be defined as V = V _n × V _c, where V _n is the sample space for the naked electrode random variable, v _n, and V _c is the cell-covered sample space for the cell-covered random variable, v _c. The joint distribution on V is p(v;θ). If the probabilities are independent, then p(v;θ) = p _n(v _n;θ _n) · p _c (v _c;θ _c). Although a probability density p _n(v _n;θ _n(h _n ·ψ ⁻¹)) is a function of the cell-covered impedance state z _c, it is more convenient to include this contribution using push forwards and pull backs of an equivalent metric to be defined shortly.

For the purpose of this study, we will assume that each physical impedance value, z, produces a normal distribution of voltage values when it is measured. The manifold S, therefore, consists of all normal probability distribution functions on the sample space V = ℜ²ⁿ parameterized by a single coordinate chart (Θ, θ) consisting of the components of the mean, μ, and the components of the upper triangle, Δ, of the population covariance matrix, Σ, i.e.,

$$ p{\left( {v;{\left\{ {\mu ,\Sigma } \right\}}} \right)} = {\left( {2\pi \det \Sigma } \right)}^{{ - n}} \exp {\left( { - \frac{1} {2}{\left( {{\mathbf{v}} - \varvec{\upmu}} \right)}^{T} \Sigma {\left( {{\mathbf{v}} - \varvec{\upmu}} \right)}} \right)}, $$

(B.1)

where T is the transpose, and the coordinate chart maps p(v;θ) to the ordered pair (μ, Δ) in ℜ²ⁿ × ℜ^2n×n+n. Associated with each set of parameters z = (z ¹, z ², ..., z ²ⁿ) is a mean, μ(z), and population covariance matrix, Σ(z), that are assumed to vary smoothly with respect to z.

We may define the composite smooth functions k _c = h _c · ψ: M × Z _n→S _c by

$$ k_{{\text{c}}} {\left( \pi \right)} = p_{{\text{c}}} {\left( {v_{{\text{c}}} ;{\left\{ {\mu _{{\text{c}}} {\left\{ {h_{{\text{c}}} {\left[ {\psi {\left( {\pi ,\varsigma _{{\text{n}}} } \right)}} \right]}} \right\}},\Sigma _{{\text{c}}} {\left\{ {h_{{\text{n}}} {\left[ {\psi {\left( {\pi ,\varsigma _{{\text{n}}} } \right)}} \right]}} \right\}}} \right\}}} \right)}. $$

(B.2)

Similarly, the composite smooth functions k _n = h _n: M × Z _n→S _n are defined by

$$ k_{{\text{n}}} {\left( \pi \right)} = p_{{\text{n}}} {\left( {v_{{\text{n}}} ;{\left\{ {\mu _{{\text{n}}} {\left\{ {h_{{\text{n}}} {\left( {\pi ,\varsigma _{{\text{n}}} } \right)}} \right\}},\Sigma _{{\text{n}}} {\left\{ {h_{{\text{n}}} {\left( {\pi ,\varsigma _{{\text{n}}} } \right)}} \right\}}} \right\}}} \right)}. $$

(B.3)

The maps, k _c and k _n, therefore, assign to each model state, π, the normal distributions such that the mean and covariance matrix are associated with π, both of which may be estimated by experimental data. Assuming that the function k _c is regular, (i.e., the differentials ψ _* and h _c*. of the composite map, k _c = h _c·ψ, have maximal rank), the image D = k _c(M × Z _n ) of M × Z _n in S _c is a, possibly immersed, submanifold of S _c. As such, it locally satisfies the definition of a statistical manifold.2,3 If k _c is one-to-one then by definition D will be an embedded manifold.

The Fisher metric on the manifold S is defined as

$$ g_{{ab}} (\theta ) = E{\left[ {\frac{\partial } {{\partial \theta ^{a} }}\ln p{\left( {v;\theta } \right)}\frac{\partial } {{\partial \theta ^{b} }}\ln p{\left( {v;\theta } \right)}} \right]}, $$

(B.4)

where E denotes the expectation value and the indices 1 ≤ a,b ≤ q. In the case of a normal distribution, the Fisher information matrix becomes

$$ g_{{ab}} = \frac{{\partial \varvec{\upmu}^{T} }} {{\partial z^{a} }}\Sigma ^{{ - 1}} \frac{{\partial \varvec{\upmu}}} {{\partial z^{b} }} + \frac{1} {2}tr{\left( {\Sigma ^{{ - 1}} \frac{{\partial \Sigma }} {{\partial z^{a} }}\Sigma ^{{ - 1}} \frac{{\partial \Sigma }} {{\partial z^{b} }}} \right)}, $$

(B.5)

where

$$ \frac{{\partial \varvec{\upmu}^{T} }} {{\partial z^{a} }} = {\left\{ {\frac{{\partial \mu ^{1} }} {{\partial z^{a} }},\frac{{\partial \mu ^{2} }} {{\partial z^{a} }}, \ldots ,\frac{{\partial \mu ^{{2n}} }} {{\partial z^{a} }}} \right\}} $$

(B.6)

and

$$ \frac{{\partial \Sigma }} {{\partial z^{a} }} = {\left[ {\begin{array}{*{20}c} {{\frac{{\partial \Sigma ^{{1,1}} }} {{\partial z^{a} }}}} & {{\frac{{\partial \Sigma ^{{1,2}} }} {{\partial z^{a} }}}} & { \ldots } & {{\frac{{\partial \Sigma ^{{1,2n}} }} {{\partial z^{a} }}}} \\ {{\frac{{\partial \Sigma ^{{2,1}} }} {{\partial z^{a} }}}} & {{\frac{{\partial \Sigma ^{{2,2}} }} {{\partial z^{a} }}}} & { \cdots } & {{\frac{{\partial \Sigma ^{{2,2n}} }} {{\partial z^{a} }}}} \\ { \vdots } & { \vdots } & { \ddots } & { \vdots } \\ {{\frac{{\partial \Sigma ^{{2n,1}} }} {{\partial z^{a} }}}} & {{\frac{{\partial \Sigma ^{{2n,2}} }} {{\partial z^{a} }}}} & { \cdots } & {{\frac{{\partial \Sigma ^{{2n,2n}} }} {{\partial z^{a} }}}} \\ \end{array} } \right]}. $$

(B.7)

If the covariance matrix, Σ, does not vary significantly with respect to change in the data state, we can use the fact that

$$ g_{{ab}} = \frac{{\partial \varvec{\upmu}^{T} }} {{\partial z^{a} }}\Sigma ^{{ - 1}} \frac{{\partial \varvec{\upmu}}} {{\partial z^{b} }} = \Sigma ^{{ - 1}}_{{ab}} $$

(B.8)

to greatly simplify the analysis.

To quantify shifts produced by systematic errors, let [a, b] be a closed interval in R, and γ: [a, b]→D a smooth curve. The length of γ is then defined as using the relation

$$ L(\gamma ): = {\int\limits_a^b {{\left\| {\frac{{d\gamma }} {{dt}}(t)} \right\|}} }dt. $$

Working with the coordinates (z ¹(γ(t)),..., z ²ⁿ(γ(t))) and using the abbreviation

$$ \ifmmode\expandafter\dot\else\expandafter\.\fi{z}^{i} (t): = \frac{d} {{dt}}{\left( {z^{i} (\gamma (t))} \right)}, $$

gives us

$$ L(\gamma ) = {\int\limits_a^b {{\sqrt {g_{{ij}} {\left( {z(\gamma (t))} \right)}\ifmmode\expandafter\dot\else\expandafter\.\fi{z}^{i} (t)\ifmmode\expandafter\dot\else\expandafter\.\fi{z}^{j} (t)} }} }dt. $$

Systematic errors can be defined in terms of the minimum, or geodesic, distance between the corrected and uncorrected points or as the path lengths produced by continuous shifts in control parameters.

Appendix C

Geometric Precision Analysis

Two important contributions to the parameter uncertainty need to be considered. First, for a given naked electrode impedance state, ζ _n ∈  Z _n, each model state, π ∈ M, is mapped into a cell-covered impedance state, ζ _c ∈  Z _c, such that $\psi :M \times Z_{{\text{n}}} \to Z_{{\text{c}}} :\pi \times \zeta _{{\text{n}}} \mapsto \zeta _{{\text{c}}}$ has an associated statistical distribution, p _c(v _c;θ _c), via the experimental configuration map $h_{{\text{c}}} :Z_{{\text{c}}} \times C_{{\text{c}}} \to S_{{\text{c}}} :\zeta _{{\text{c}}} \times \rho _{{\text{c}}} \mapsto p_{{\text{c}}} {\left( {v_{{\text{c}}} ;\theta _{{\text{c}}} } \right)}$ of measured cell-covered voltage states. The pull back of the Fisher metric associated with the measured cell-covered voltage noise distribution sets an upper bound on the attainable precision produced by this noise source. The second, and subtler, contribution arises because naked electrode impedance estimates, z _n, are required to evaluate the model function. The transformation, or push forward, of the naked electrode error distribution to the cell-covered electrode impedance value must, therefore, also be included. These two sources of error can be included by evaluating the push forward of the model function map ψ: M × Z _n→Z _c, i.e.,

$$ D\psi = {\left[ {\begin{array}{*{20}c} {{D_{{\text{m}}} \psi }} & {{D_{{z_{{\text{n}}} }} \psi }} \\ \end{array} } \right]} = {\left[ {\begin{array}{*{20}c} {{\frac{{\partial z_{{\text{c}}} }} {{\partial m}}}} & {{\frac{{\partial z_{{\text{c}}} }} {{\partial z_{{\text{n}}} }}}} \\ \end{array} } \right]} $$

(C.1)

where

$$ D_{{\text{m}}} \psi = \frac{{\partial z_{{\text{c}}} }} {{\partial m}} = {\left[ {\begin{array}{*{20}c} {{\frac{{\partial z^{1}_{{\text{c}}} }} {{\partial \alpha }}}} & {{\frac{{\partial z^{1}_{{\text{c}}} }} {{\partial R_{{\text{b}}} }}}} \\ {{\frac{{\partial z^{2}_{{\text{c}}} }} {{\partial \alpha }}}} & {{\frac{{\partial z^{2}_{{\text{c}}} }} {{\partial R_{{\text{b}}} }}}} \\ { \vdots } & { \vdots } \\ {{\frac{{\partial z^{{2n}}_{{\text{c}}} }} {{\partial \alpha }}}} & {{\frac{{\partial z^{{2n}}_{{\text{c}}} }} {{\partial R_{{\text{b}}} }}}} \\ \end{array} } \right]} $$

(C.2)

and

$$ D_{{z_{{\text{n}}} }} \psi = \frac{{\partial z_{{\text{c}}} }} {{\partial z_{{\text{n}}} }} = {\left[ {\begin{array}{*{20}c} {{\frac{{\partial z^{1}_{{\text{c}}} }} {{\partial z^{1}_{{\text{n}}} }}}} & {{\frac{{\partial z^{1}_{{\text{c}}} }} {{\partial z^{2}_{{\text{n}}} }}}} & { \cdots } & {{\frac{{\partial z^{1}_{{\text{c}}} }} {{\partial z^{{2n}}_{{\text{n}}} }}}} \\ {{\frac{{\partial z^{2}_{{\text{c}}} }} {{\partial z^{1}_{{\text{n}}} }}}} & {{\frac{{\partial z^{2}_{{\text{c}}} }} {{\partial z^{2}_{{\text{n}}} }}}} & { \cdots } & {{\frac{{\partial z^{2}_{{\text{c}}} }} {{\partial z^{{2n}}_{{\text{n}}} }}}} \\ { \vdots } & { \vdots } & { \ddots } & { \vdots } \\ {{\frac{{\partial z^{{2n}}_{{\text{c}}} }} {{\partial z^{1}_{{\text{n}}} }}}} & {{\frac{{\partial z^{{2n}}_{{\text{c}}} }} {{\partial z^{2}_{{\text{n}}} }}}} & { \cdots } & {{\frac{{\partial z^{{2n}}_{{\text{c}}} }} {{\partial z^{{2n}}_{{\text{n}}} }}}} \\ \end{array} } \right]}. $$

(C.3)

Similarly, the push forward of the cell-covered electrode impedance to cell-covered electrode voltage map, h _c: Z _c × C _c→S _c, is given by

$$ Dh_{{\text{c}}} = {\left[ {\begin{array}{*{20}c} {{D_{{z_{{\text{c}}} }} h_{{\text{c}}} }} & {{D_{{c_{{\text{c}}} }} h_{{\text{c}}} }} \\ \end{array} } \right]} = {\left[ {\begin{array}{*{20}c} {{\frac{{\partial \theta _{{\text{c}}} }} {{\partial z_{{\text{c}}} }}}} & {{\frac{{\partial \theta _{{\text{c}}} }} {{\partial c_{{\text{c}}} }}}} \\ \end{array} } \right]} $$

(C.4)

and the push forward of the naked electrode impedance to naked electrode voltage map, h _n: Z _n × C _n→S _n, is given by

$$ Dh_{{\text{n}}} = {\left[ {\begin{array}{*{20}c} {{D_{{z_{{\text{n}}} }} h_{{\text{n}}} }} & {{D_{{c_{{\text{n}}} }} h_{{\text{n}}} }} \\ \end{array} } \right]} = {\left[ {\begin{array}{*{20}c} {{\frac{{\partial \theta _{{\text{n}}} }} {{\partial z_{{\text{n}}} }}}} & {{\frac{{\partial \theta _{{\text{n}}} }} {{\partial c_{{\text{n}}} }}}} \\ \end{array} } \right]}. $$

(C.5)

The pull back of the cell-covered noise Fisher metric gives rise to the model space metric G _c defined as

$$ G_{{\text{c}}} {\left( \pi \right)}{\left( {\begin{array}{*{20}c} {{\frac{\partial } {{\partial m^{a} }}}} & {{\frac{\partial } {{\partial m^{b} }}}} \\ \end{array} } \right)} = {\left( {h_{{\text{c}}} \cdot \psi } \right)}^{*} g_{{S_{{\text{c}}} }} = g_{{S_{{\text{c}}} }} {\left( {\begin{array}{*{20}c} {{D_{{z_{{\text{c}}} }} h_{{\text{c}}} \cdot D_{{\text{m}}} \psi \frac{\partial } {{\partial m^{a} }}}} & {{D_{{z_{{\text{c}}} }} h_{{\text{c}}} \cdot D_{{\text{m}}} \psi \frac{\partial } {{\partial m^{b} }}}} \\ \end{array} } \right)}, $$

(C.6)

while the pull back of the naked electrode noise contribution, G _n, is

$$ G_{{\text{n}}} {\left( \pi \right)}{\left( {\begin{array}{*{20}c} {{\frac{\partial } {{\partial m^{a} }}}} & {{\frac{\partial } {{\partial m^{b} }}}} \\ \end{array} } \right)} = g_{{S_{{\text{n}}} }} {\left( {\begin{array}{*{20}c} {{D_{{z_{{\text{n}}} }} h_{{\text{n}}} \cdot D_{{z_{{\text{n}}} }} \psi ^{{ - 1}} \cdot D_{{\text{m}}} \psi \frac{\partial } {{\partial m^{a} }}}} & {{Dz_{{\text{n}}} h_{{\text{n}}} \cdot D_{{z_{{\text{n}}} }} \psi ^{{ - 1}} \cdot D_{{\text{m}}} \psi \frac{\partial } {{\partial m^{b} }}}} \\ \end{array} } \right)}. $$

(C.7)

An additional complication that needs to be addressed is the propagation of errors in the circuit element values that are used to convert the measured voltages into impedances. The measurement and electrode circuit model component values have uncertainties given by the manufacturer’s specifications and/or the multi-meter precision. To ensure that these errors are not contributing significantly to the parameter precision analysis, a Fisher matrix that embodies their assumed Gaussian errors is defined on the control spaces used to define the impedance to voltage conversion. The naked and cell-covered control spaces are defined in terms of the Cartesian product C _n = C _c = R _cc × C _p and the electrode circuit model C _m = R _p × C _p × R _c. The naked and cell-covered to impedance to voltage conversions are, therefore, defined as h _n: C _n × Z _n→S _n and h _c: C _c × Z _c→S _c, respectively. The sensitivity to the circuit component uncertainties can be tested using the block diagonal Fisher metric defined on C _n × Z _n and C _c × Z _c. To include these contributions, a similar set of pull backs is defined to describe the model parameter precision in terms of the circuit component tolerances, i.e.,

$$ G_{{C_{{\text{c}}} }} (\pi ){\left( {\begin{array}{*{20}c} {{\frac{\partial } {{\partial m^{a} }}}} & {{\frac{\partial } {{\partial m^{b} }}}} \\ \end{array} } \right)} = {\left( {\begin{array}{*{20}c} {{D_{{c_{{\text{c}}} }} h_{{\text{c}}} \cdot D_{{\text{m}}} \psi \frac{\partial } {{\partial m^{a} }}}} & {{D_{{c_{{\text{c}}} }} h_{{\text{c}}} \cdot D_{{\text{m}}} \psi \frac{\partial } {{\partial m^{b} }}}} \\ \end{array} } \right)}, $$

(C.8)

and

$$ G_{{C_{{\text{n}}} }} (\pi ){\left( {\begin{array}{*{20}c} {{\frac{\partial } {{\partial m^{a} }}}} & {{\frac{\partial } {{\partial m^{b} }}}} \\ \end{array} } \right)} = {\left( {\begin{array}{*{20}c} {{D_{{c_{{\text{n}}} }} h_{{\text{n}}} \cdot D_{{z_{{\text{n}}} }} \psi ^{{ - 1}} \cdot D_{{\text{m}}} \psi \frac{\partial } {{\partial m^{a} }}}} & {{D_{{c_{{\text{n}}} }} h_{{\text{n}}} \cdot D_{{z_{{\text{n}}} }} \psi ^{{ - 1}} \cdot D_{{\text{m}}} \psi \frac{\partial } {{\partial m^{b} }}}} \\ \end{array} } \right)}. $$

(C.9)

The precision associated with a parameter or group of parameters follows from the determinant components defined by these metrics, i.e.,

$$ \Lambda ^{1}_{\alpha } = {\sqrt {G_{{\alpha \alpha }} } }, $$

(C.10)

$$ \Lambda ^{1}_{{R_{{\text{b}}} }} = {\sqrt {G_{{R_{{\text{b}}} R_{{\text{b}}} }} } }, $$

(C.11)

and

$$ \Lambda ^{2}_{{\alpha R_{{\text{b}}} }} = {\sqrt {\det {\left( {\begin{array}{*{20}c} {{G_{{\alpha \alpha }} }} & {{G_{{\alpha R_{{\text{b}}} }} }} \\ {{G_{{R_{{\text{b}}} \alpha }} }} & {{G_{{R_{{\text{b}}} R_{{\text{b}}} }} }} \\ \end{array} } \right)}} }. $$

(C.12)