Cell–Electrode Models for Impedance Analysis of Epithelial and Endothelial Monolayers Cultured on Microelectrodes

Electric cell–substrate impedance sensing has been used to measure transepithelial and transendothelial impedances of cultured cell layers and extract cell parameters such as junctional resistance, cell–substrate separation, and membrane capacitance. Previously, a three-path cell–electrode model comprising two transcellular pathways and one paracellular pathway was developed for the impedance analysis of MDCK cells. By ignoring the resistances of the lateral intercellular spaces, we develop a simplified three-path model for the impedance analysis of epithelial cells and solve the model equations in a closed form. The calculated impedance values obtained from this simplified cell–electrode model at frequencies ranging from 31.25 Hz to 100 kHz agree well with the experimental data obtained from MDCK and OVCA429 cells. We also describe how the change in each model-fitting parameter influences the electrical impedance spectra of MDCK cell layers. By assuming that the junctional resistance is much smaller than the specific impedance through the lateral cell membrane, the simplified three-path model reduces to a two-path model, which can be used for the impedance analysis of endothelial cells and other disk-shaped cells with low junctional resistances. The measured impedance spectra of HUVEC and HaCaT cell monolayers nearly coincide with the impedance data calculated from the two-path model.


Introduction
Electric cell-substrate impedance sensing (ECIS) has been developed as an instrumental method to electrically measure the interaction of adherent cells in culture with their substrate [1].In this method, small gold electrodes serve as substrates for the cells, and the impedances of these electrodes to an alternating current (AC) signal are followed with time.The impedance increases when the cells attach and spread on the electrodes as their insulating plasma membranes restrict the current flow.These impedance values, measured at various AC frequencies, can be used to monitor cell attachment and spreading, cell morphology changes, and cell mobility with great sensitivity [2].In addition, changes in cell morphology are observed as fluctuations in impedance, and these fluctuations can be numerically analyzed to give quantitative estimates of cellular micromotion [3].
One of the essential features of the ECIS method is the use of AC signals at various frequencies [2,4].It is well known that AC techniques can provide information on the actual areas of folded cell membranes and, hence, membrane conductances [5].The main basis for this concept is that a cell membrane can be electrically modeled as a resistor and a capacitor connected in parallel [6].As a result, the amount of transcellular and paracellular currents varies while different frequencies of AC signals are applied.In addition, frequencydependent changes in the measured impedance spectrum are related to cell morphology changes and cell-cell and cell-substrate interactions [4].Previously, Giaever and Keese developed a cell-electrode model (GK model) to study cellular micromotion and analyze the impedance spectrum of fibroblasts measured by ECIS [3,7].This model approximates a confluent cell layer as many connected circular disks on the sensing electrode.Electric currents flow radially in the space underneath the cell and then through basal and apical membranes (transcellular pathway) or around the cell through the intercellular space (paracellular pathway).This original two-path model includes three parameters for data fitting: junctional resistance between adjacent cells over a unit cell area (R b ), a parameter relating to the resistance of the media-filled narrow channels beneath the cell (α), and transmembrane capacitance for a unit area (C m ).By fitting the experimental data of confluent layers of WI-38 and WI-38 VA13 cells with the calculated values obtained from the GK model, model parameters, R b and α, for both fibroblast types were successfully determined [3].
Electric cell-substrate impedance sensing, as the name implies, considers in detail the interaction of the adherent cells with the substrate they have attached to.A cleft, filled with culture media, separates the basal cell membrane from the electrode and thus generates a cleft resistance (R cleft ) [8,9].Unlike most equivalent circuit models used for cell-based impedance assays, the GK model includes the distributed effects of R cleft underneath the cell layer.It is worth noting that R cleft is distributed along with the distributed impedance of the basal cell membrane while the current flows radially in the cell-substrate cleft region and outward into the lateral intercellular space (LIS).This distributed effect is frequencydependent and plays a significant role in the measured impedance spectrum.In ECIS, R cleft can be defined as α 2 = r c 2 (ρ/h), where r c is the cell radius, ρ is the resistivity of the culture medium, and h is the average distance between the cells and the substratum (or the average distance of the cell-electrode cleft) [8].
Lo, Giaever, and Keese extended the GK model for the impedance analysis of epithelial cells [7,10].This model (the LGK model), as illustrated in Figure 1, also describes cells as circular disks adhering to the sensing electrode.In addition to the two pathways defined in the GK model, the LGK model includes a third pathway for the electric currents to flow along the LIS.In through the lateral membrane and out through the apical membrane, this pathway is transcellular and more applicable to most epithelial cells [5,11].This is because epithelial cells have higher junctional resistances than fibroblasts or endothelial cells, and the membrane impedance decreases at high frequencies.This third pathway is necessary for the model calculation of epithelial cells, especially when high-frequency AC signals are used [5].In the LGK model, paracellular resistance consists of a tight junction resistance (R b ) and LIS resistance (R LIS ).Similar to the distributed effects of R cleft and the impedance of the basal cell membrane, distributed R LIS and the distributed impedance of the lateral cell membrane are also considered in the model.The LGK model includes six adjustable parameters: R b , α, C a , (apical membrane capacitance per unit area), C b , (basal membrane capacitance per unit area), C l (lateral membrane capacitance per unit area), and R l (the resistance of the LIS per unit length) [10].By comparing the calculated impedance spectra of the cell-covered electrodes with those measured for confluent layers of MDCK epithelial cells, the LGK model has been demonstrated to fit the experimental data precisely [10].
Although the LGK model is useful for the impedance analysis of epithelial cells, data fitting is challenging if too many fitting parameters are used.Given the purpose of easy use of the cell-electrode model, it is important to limit the number of model parameters to only necessary ones.Among the six parameters in the LGK model, we find it most difficult to adjust the parameter R l (R LIS per unit length) for data fitting.Here, we aim to simplify the LGK model by disregarding the contribution of R l and reducing the fitting parameters from six to five.If R LIS is much lower than R b , R LIS can be considered negligible (i.e., R LIS << R b ), and the distributed effect of the lateral membrane is ignored; therefore, the LGK model can be simplified to only five fitting parameters.The feasibility of LGK model simplification is checked numerically as follows: In the simplest estimation, the resistance of the LIS per unit area can be calculated as R LIS = [ρl/(2πr c )(0.5w)](πr c 2 ), where ρ, l, w, and r c , are the resistivity of culture medium, LIS length, LIS width, and cell radius respectively [5].For confluent MDCK cells, the LIS is approximately 0.25~1.00µm wide (w) and 6 µm long (l) [12], and the cell radius (r c ) is 7 µm [13].With ρ = 60 Ω•cm, the R LIS ranges from 0.25 to 1.00 Ω•cm 2 for an LIS width of 1.00 to 0.25 µm.Yet, the junctional resistance per unit area of MDCK cells (R b ) is around 60~100 Ω•cm 2 (relevant results from this work, which will be described in detail below), which is around 100 times larger than the R LIS .
Sensors 2024, 24, 4214 3 of 14 Although the LGK model is useful for the impedance analysis of epithelial cells, data fitting is challenging if too many fitting parameters are used.Given the purpose of easy use of the cell-electrode model, it is important to limit the number of model parameters to only necessary ones.Among the six parameters in the LGK model, we find it most difficult to adjust the parameter Rl (RLIS per unit length) for data fitting.Here, we aim to simplify the LGK model by disregarding the contribution of Rl and reducing the fitting parameters from six to five.If RLIS is much lower than Rb, RLIS can be considered negligible (i.e., RLIS << Rb), and the distributed effect of the lateral membrane is ignored; therefore, the LGK model can be simplified to only five fitting parameters.The feasibility of LGK model simplification is checked numerically as follows: In the simplest estimation, the resistance of the LIS per unit area can be calculated as RLIS = [ρl/(2πrc)(0.5w)](πrc 2), where ρ, l, w, and rc, are the resistivity of culture medium, LIS length, LIS width, and cell radius respectively [5].For confluent MDCK cells, the LIS is approximately 0.25~1.00µm wide (w) and 6 µm long (l) [12], and the cell radius (rc) is 7 µm [13].With ρ = 60 Ω•cm, the RLIS ranges from 0.25 to 1.00 Ω•cm 2 for an LIS width of 1.00 to 0.25 µm.Yet, the junctional resistance per unit area of MDCK cells (Rb) is around 60~100 Ω•cm 2 (relevant results from this work, which will be described in detail below), which is around 100 times larger than the RLIS.
In this study, we develop a simplified LGK model by ignoring RLIS and treating lateral membranes as lumped impedances.We solve the second-order differential equation in a closed form, and the total impedance only depends on five parameters: Rb, α, Ca, Cb, and Cl.We also evaluate how each model parameter affects the impedance spectra of the cellcovered electrodes and apply this model to analyze the experimental data obtained from MDCK and OVCA429 cells.The third pathway in the LGK model can be ignored if the In this study, we develop a simplified LGK model by ignoring R LIS and treating lateral membranes as lumped impedances.We solve the second-order differential equation in a closed form, and the total impedance only depends on five parameters: R b , α, C a , C b , and C l .We also evaluate how each model parameter affects the impedance spectra of the cell-covered electrodes and apply this model to analyze the experimental data obtained from MDCK and OVCA429 cells.The third pathway in the LGK model can be ignored if the junctional resistance is smaller than the specific impedance of the lateral cell membrane.Based on this assumption, the simplified LGK model is further reduced to a modified GK model, which contains only four fitting parameters: R b , α, C a , and C b .The measured impedance spectra of HUVEC and HaCaT cells are analyzed using the modified GK model and compared with the relevant results obtained from the simplified LGK model.The overall calculated results from the simplified LGK model or modified GK model for all four cell types agree well with the measured impedance spectra.

Impedance Measurements Using ECIS
Electrode arrays (8W1E), an ECIS Zθ instrument, and the acquired software (Version 2.215) for ECIS measurement were obtained from Applied Biophysics (Troy, NY, USA).A lock-in amplifier was used to measure in-and out-of-phase voltages through the sensing electrode at various frequencies.These voltage data were then mathematically converted to resistance and capacitance values considering the cell-electrode system as a series RC circuit.It is noteworthy that this conversion is a standard AC circuit analysis.Treating the cellelectrode system as a parallel RC circuit can obtain equally good results.However, since all components in a series circuit share the same current, applying Ohm's law to determine resistance and capacitive reactance from a series RC circuit is more straightforward than a parallel RC circuit.For ECIS studies, cells were taken from slightly subconfluent cultures 48 h after passage, and a monodisperse cell suspension was prepared using standard tissue culture techniques with trypsin/EDTA.These suspensions were equilibrated at incubator conditions before addition to the electrode-containing wells.Confluent cell layers were formed using 10 5 cell/cm 2 inoculation density.
We monitored cell attachment and spreading using multiple-frequency time series (MFT) measurements with 11 pre-defined frequencies ranging from 62.5 Hz to 64 kHz.These frequencies are 62.5 Hz, 125 Hz, 250 Hz, 500 Hz, 1 kHz, 2 kHz, 4 kHz, 8 kHz, 16 kHz, 32 kHz, and 64 kHz.We also applied a frequency scan measurement with an impedance calculation from the cell-electrode model to determine the morphological parameters of each cell line.In this method, we measured the impedances of the cell-electrode system at 25 different frequencies, ranging from 31.25 Hz to 100 kHz.Each frequency (except the first and last) was generated by multiplying the previous one by 2 1/2 .Because the lock-in amplifier generates single-frequency sine waves with an operating range of 1 mHz to 102 kHz, instead of 128 kHz, we use 100 kHz as the highest frequency for the frequency scan measurement.The calculated values obtained from different cell-electrode models (Z cal ) were fitted with these measured impedance data (Z exp ), including 25 resistance and 25 capacitance reactance points.By using the method of non-linear least squares, we calculated the sum of the squares of the percentage errors (Z cal − Z exp )/Z exp , and refined the model parameters until the minimum sum is attained [14].

Model Derivation 2.3.1. The Simplified LGK Model
The assumptions used for deriving the simplified LGK model are similar to the LGK model assumptions, which have been previously described [3].From Figure 1 and Kirchhoff's Law for AC circuits, we get Equations ( 1)-( 4) can be combined to yield the following differential equation: where and The general solution of Equation ( 5) is where I 0 (γr) and K 0 (γr) are modified Bessel functions of the first and second kind, respectively.Since K 0 (γr) diverges as r goes to zero and the domain of the general solution is [0, r c ], the coefficient B is zero.The general solution of Equation ( 5) becomes We assume that the electrical potential inside the cell, V i , is a constant.For the transcellular current through the apical cell surface, the equation related to the transcellular current can be expressed as Combining Equations ( 1) and ( 9), electrical current through the space between the cell layer and electrode can be expressed as In the simplified LGK model, we assume that the resistance of LIS per unit area (R LIS ) is much lower than the junctional resistance per unit area (R b ) and can be ignored in the model derivation.We also assume that the distributed effect of the lateral membrane can be ignored.Therefore, the equation related to the paracellular current passing through the lateral cell membrane (I l ) can be expressed as Two boundary conditions for determining the two constants, A and V i , are as follows: and We obtain the following matrix equation from the boundary conditions, Equations ( 13) and (14). where Using matrix algebra, we determine the numerical constants A and V i by solving the matrix equation.The total current from the area of the single cell, I ct , can be found by integrating Equation (2): For the simplified LGK Model, the exact solution of the specific impedance of the cell-covered electrode, Z c , is: where and Here, S is the sensing electrode area of 5 × 10 −4 cm 2 .We define the specific impedance of the cell-free electrode, Z n , as a resistance (R n ) and a capacitance (C n ) connected in series as shown in Equation ( 22).R n and C n are obtained respectively from in-phase and outof-phase data measured by the lock-in amplifier at different frequencies.We assume that the specific impedances of the apical, basal, and lateral membranes, Z a , Z b , and Z l , can be calculated as a resistor and a capacitor in parallel, as shown in Equations ( 23)-(25).We also assume that the specific resistance of the cell membrane (R m ) is 10 3 Ω•cm 2 , commonly given for biological membranes [15].The model equation (Equation ( 18)) calculates the specific impedance of the cell-covered electrode, Z c , with the measured Z n and five model parameters, specifically R b , α, C a , C b , and C l .Both resistance and reactance components of the calculated Z c fit the measured Z c at 25 different frequencies, allowing the determination of the five model parameters.

The Modified GK Model
Similar to the simplified LGK model, we use Equations ( 1)- (11) to derive the modified GK model.As shown by most fibroblasts and endothelial cells, the junctional resistance between adjacent cells, R b , is negligible compared with the specific impedance through the lateral cell membrane, Z l .In this case, the third pathway for the electric currents to flow along the LIS and through the lateral membrane no longer exists.Two boundary conditions, Equations ( 13) and ( 14), for determining A and V i can be rewritten as: and We then obtain the following matrix equation from boundary conditions, Equations (26) and ( 27).
For the modified GK Model, the exact solution of the specific impedance of the cellcovered electrode, Z c , is: where As expected, Equations ( 29) and (30) can be easily obtained by plugging (G/Z l ) = 0 into Equations ( 18) and ( 19).The model equation (Equation ( 29)) calculates the specific impedance of the cell-covered electrode, Z c , with the measured Z n and four model parameters, specifically R b , α, C a , and C b .

Measurement of MDCK Cell Attachment and Spreading
To monitor MDCK cell attachment and spreading, we inoculated cells into electrode wells at a 10 5 cells/cm 2 cell density.Electrical impedances of the cell-covered electrodes were followed at various frequencies for 24 h.Three-dimensional illustrations of the changes in measured resistance and capacitance as a function of frequency (spectra) and time are shown in Figure 2a,b.Both resistance and capacitance spectra dramatically change in the first 10 h after cell seeding.In addition, resistance and capacitance time series traced at different frequencies vary in various ways.For example, the measured resistances (Figure 2a) at low frequencies (≤1 kHz) increase quickly in the first 10 h and then slowly reach their final values.However, the measured resistances at high frequencies (>1 kHz) increase in the first 10 h and then drop slightly to their final values.Looking at the measured capacitance time series in the first 10 h (Figure 2b), they significantly decrease at high frequencies (≥4 kHz) but only gradually decrease at low frequencies (<4 kHz).

Parameter Analysis of the Simplified LGK Model
Generally, we use frequency scans before and after cells attach and spread on the electrode to obtain impedances as a function of frequency for both a cell-free electrode and the same electrode confluent with cells.The data curves are similar to the bold purple curves shown in Figure 2a,b, which are the resistance and capacitance spectra at time zero and 24 h after seeding cells.Figure 3 displays the normalized resistance and capacitance spectra of MDCK cells, which are obtained by dividing the resistance and capacitance spectra of a cell-covered electrode by the corresponding values of the same electrode without cells.We used the simplified LGK model equations (Equations ( 18)-( 20)) and the impedance spectrum of the cell-free electrode to calculate and fit both the resistance and capacitance spectra at 25 different frequencies.The best fitting values of the model parameters of MDCK cells, Rb, α, Ca, Cb, and Cl, are 88 Ω•cm 2 , 27 Ω 1/2 •cm, 4.5 µF/cm 2 , 2.0 µF/cm 2 , 1.6 µF/cm 2 , respectively.We estimate the cleft resistance (Rcleft = α 2 ), and the average value of the Rcleft is around 729 Ω•cm 2 .This resistance is surprisingly large, implying the ventral surfaces of MDCK cells are closely in contact with the sensing electrode.
Knowing how changes in each model parameter affect the normalized impedance spectrum is crucial to getting a good model fitting.Figure 4a,b show the normalized resistance and capacitance spectra for different values of Rb: 20, 40, 60, and 80 Ω•cm 2 .When Rb increases, the peak of the normalized resistance spectrum (Figure 4a) shifts leftward and upward.As a result, the values of the normalized resistance increase on the low-frequency side and decrease on the high-frequency side.Figure 4c,d show another normalized resistance and capacitance spectra for different values of α: 20, 30, 40, and 50 Ω 1/2 •cm.When α increases, the left side of the normalized resistance curve (Figure 4c) moves leftward, whereas the peaks of all the curves are almost at the same height.As for normalized capacitance spectra shown in Figure 4b,d, capacitance values at different frequency ranges decrease as Rb or α increases.Finally, as shown in Figure 4e,g, when Ca or Cb increases from 2 µF/cm 2 to 5 µF/cm 2 , the right side of the normalized resistance curve moves We have defined the optimal detection frequency as the frequency at which the seeded cells contribute the maximum relative change in the measured resistance or capacitance [16].For MDCK cell attachment and spreading monitored by MFT runs, the optimal detection frequencies for tracing resistance and capacitance changes are 500 Hz and 64 kHz [16].Figure 2c,d show typical 24 h resistance and capacitance time series data measured at 500 Hz and 64 kHz.Specifically, the measured resistance increases from 5 kΩ to 90 kΩ and then 105 kΩ at 0, 10 h, and 24 h after cell inoculation (Figure 2c).At the same time, the measured capacitance decreases from 3.9 nF to 0.54 nF and then 0.66 nF, as shown in Figure 2d, indicating the formation of a confluent MDCK monolayer [2].Resistance or capacitance tracings fluctuate even after the MDCK cells are confluent.These fluctuations are due to cellular micromotions caused by the rearrangement of cell-cell and cell-substrate adhesion sites.

Parameter Analysis of the Simplified LGK Model
Generally, we use frequency scans before and after cells attach and spread on the electrode to obtain impedances as a function of frequency for both a cell-free electrode and the same electrode confluent with cells.The data curves are similar to the bold purple curves shown in Figure 2a,b, which are the resistance and capacitance spectra at time zero and 24 h after seeding cells.Figure 3 displays the normalized resistance and capacitance spectra of MDCK cells, which are obtained by dividing the resistance and capacitance spectra of a cell-covered electrode by the corresponding values of the same electrode without cells.We used the simplified LGK model equations (Equations ( 18)-( 20)) and the impedance spectrum of the cell-free electrode to calculate and fit both the resistance and capacitance spectra at 25 different frequencies.The best fitting values of the model parameters of MDCK cells, R b , α, C a , C b , and C l , are 88 Ω•cm 2 , 27 Ω 1/2 •cm, 4.5 µF/cm 2 , 2.0 µF/cm 2 , 1.6 µF/cm 2 , respectively.We estimate the cleft resistance (R cleft = α 2 ), and the average value of the R cleft is around 729 Ω•cm 2 .This resistance is surprisingly large, implying the ventral surfaces of MDCK cells are closely in contact with the sensing electrode.
Sensors 2024, 24, 4214 10 of 14 leftward and downward.In general, while changes in Ca or Cb give similar effects to the normalized resistance and capacitance spectra, parameter Ca has more impact than Cb on the calculated spectrum.Notably, both normalized resistance and capacitance spectra change slightly when Cl takes on the values of 1.0, 1.5, 2.0, and 2.5 µF/cm 2 .When R b increases, the peak of the normalized resistance spectrum (Figure 4a) shifts leftward and upward.As a result, the values of the normalized resistance increase on the low-frequency side and decrease on the high-frequency side.Figure 4c,d show another normalized resistance and capacitance spectra for different values of α: 20, 30, 40, and 50 Ω 1/2 •cm.When α increases, the left side of the normalized resistance curve (Figure 4c) moves leftward, whereas the peaks of all the curves are almost at the same height.As for normalized capacitance spectra shown in Figure 4b,d, capacitance values at different frequency ranges decrease as R b or α increases.Finally, as shown in Figure 4e,g, when C a or C b increases from 2 µF/cm 2 to 5 µF/cm 2 , the right side of the normalized resistance curve moves leftward and downward.In general, while changes in C a or C b give similar effects to the normalized resistance and capacitance spectra, parameter C a has more impact than C b on the calculated spectrum.Notably, both normalized resistance and capacitance spectra change slightly when C l takes on the values of 1.0, 1.5, 2.0, and 2.5 µF/cm 2 .

Discussion
We studied cell attachment and spreading on sensing electrodes using ECIS MFT measurement, which is useful for investigating in vitro cell behaviors in real time.Figure 2a,b are examples of three-dimensional plots to demonstrate early dramatic changes in resistance and capacitance spectra (t < 10 h) caused by MDCK cells.The peak value of the normalized resistance spectra, as shown in Figure 3a, represents the maximum relative change in resistance (R cell-covered /R cell-free ), and the peak position is at 707 Hz.Among the 11 frequencies in the MFT runs, 500 Hz is the closest to 707 Hz and can be used to trace time series resistance changes for MDCK cells.In Figure 2c, the early resistance increase (t < 10 h) is mainly due to the cell attachment and spreading.The later resistance increase (t > 10 h) is primarily due to the formation of a barrier function [2].Time series capacitance curves traced at 64 kHz (Figure 2d) also reveal a confluent MDCK cell layer formation within 10 h after cell seeding.These results are consistent with previous reports [2,16].
Cell-electrode models provide theoretical bases for impedance analysis of cell layers measured by ECIS regarding the morphological parameters and cell characteristics in tissue culture [3,10,14,17].Figure 3 demonstrates the ability of the simplified LGK model to fit the measured impedance spectra of confluent MDCK cell layers precisely.Our analytical approach in this paper provides insight into how changes in each cell parameter influence measured impedance spectra.As a result of these modeling studies, one crucial fact is shown in Figure 4a, where the normalized resistance values of MDCK cells increase in the lower frequency region (<1 kHz) but decrease in the higher frequency region (>1 kHz) as the parameter R b increases.This information explains why the measured time series resistances at high frequencies increase in the first 10 h and gradually drop after the barrier function is formed (Figure 2a).In contrast, using resistance tracing at 500 Hz (Figure 2c), we successfully monitor the formation of a barrier function.Similarly, the analysis shown in Figure 4b explains why at high frequencies (≥8 kHz), the measured time series capacitances do not change much even when the barrier function is formed (Figure 2b).Therefore, capacitance tracings at high frequencies such as 64 kHz (Figure 2d) are usually used for monitoring cell coverage on the sensing electrode, implying applications for ECIS wound healing migration assays.
In this study, we have developed simplified LGK and modified GK models and compared the calculated impedance spectra to those measured for confluent layers of MDCK, OVCA429, HaCaT, and HUVEC cell layers.These two models have been demonstrated to fit the experimental data precisely, and the results are shown in Tables 1 and 2. For MDCK and OVCA429 cells, all the parameter values fitted from the simplified LGK model are close to those from the LGK model.This result agrees with our assumption that if R LIS is much lower than R b , then R LIS can be considered negligible in the model derivation, and the distributed effect of the lateral membrane can also be ignored.The simplified LGK model is helpful for the impedance analysis of epithelial cells like MDCK and OVCA429.
The major difference between the modified GK model and the simplified LGK model is the consideration of the third pathway, where the currents pass along the LIS, in through the lateral membrane, and out through the apical membrane.As a result, the junctional resistances (R b ) of MDCK and OVCA429 cells fitted by the modified GK model (two-path model) are overestimated, while the α parameters are somewhat underestimated.Since the modified GK model does not consider the third path, the total current passing through the basal cell membrane is the same as that through the apical cell membrane.Without considering the third-path currents, applying the modified GK model to epithelial cells would overestimate the currents through the basal cell membrane and underestimate the currents through the apical cell membrane.Since capacitive reactance is inversely proportional to the capacitance, this problem causes a slight underestimation of the C a and significantly overestimates the C b in the modified GK model (Table 1).For example, C b values of MDCK and OVCA429 cells are 2.8 µF/cm 2 and 2.6 µF/cm 2 from the simplified LGK model and 3.6 µF/cm 2 and 2.9 µF/cm 2 from the modified GK model.Since the third current path is not considered in the modified GK model, the large C b value from the modified GK model represents the capacitance of the basolateral membrane rather than the basal membrane only.In contrast, considering the third current path for HaCaT and HUVEC cells causes an underestimation of the C b value.As shown in Table 2, C b values of HaCaT and HUVEC cells are 2.1 µF/cm 2 and 2.8 µF/cm 2 from the modified GK model and 1.6 µF/cm 2 and 2.5 µF/cm 2 from the simplified LGK model.
In the GK model, the cell body is characterized as basal and apical membranes stacked together, and the transcellular currents passing through both membranes are the same [3].With the assumption of C a = C b , the specific impedance of the stacked cell membranes is the impedance of the two identical cell membranes connected in series.However, the apical cell membrane usually has more foldings than the basal cell membrane, indicating a higher apical membrane capacitance than the basal membrane capacitance (C a > C b ).In the modified GK model, we consider different current distributions through the basal and apical membranes and successfully estimate C a and C b separately.As shown in Table 2, C a and C b values obtained from the modified GK model are 2.7 µF/cm 2 and 2.1 µF/cm 2 for HaCaT cells and 4.2 µF/cm 2 and 2.8 µF/cm 2 for HUVEC cells.The analyzed data shown in Table 2 also display that the parameter R b obtained from the simplified LGK model or the modified GK model has a similar value, and so does the parameter α.This result indicates the effectiveness of applying the modified GK model (a two-path model) for the impedance analysis of cells with low junctional resistances, such as HaCaT and HUVEC cells.

Conclusions
In this paper, we have developed and validated simplified LGK and modified GK models for the impedance analysis of epithelial and endothelial cells measured by ECIS.Our results provide model parameter and cell characteristic values by computing the specific impedance of the cell-covered electrode (Zc) and fitting the experimental data.Because of the sensitivity of the system and the straightforwardness of the numerical calculation, the impedance analysis of cell layers measured by ECIS will find more applications in

Figure 1 .
Figure 1.A schematic diagram of the simplified LGK model for cell layers cultured on a gold microelectrode.Cells are considered as disk-shaped.The side view (a) displays different current paths, junctional resistance (Rb), LIS resistance (RLIS), and cleft resistance (Rcleft).The side view (b) emphasizes the cell-substratum space and constructs Equations (1)-(4).The cell interior is assumed to be equipotential due to the relatively large cell volume and low intracellular resistivity.

Figure 1 .
Figure 1.A schematic diagram of the simplified LGK model for cell layers cultured on a gold microelectrode.Cells are considered as disk-shaped.The side view (a) displays different current paths, junctional resistance (R b ), LIS resistance (R LIS ), and cleft resistance (R cleft ).The side view (b) emphasizes the cell-substratum space and constructs Equations (1)-(4).The cell interior is assumed to be equipotential due to the relatively large cell volume and low intracellular resistivity.

Figure 2 .
Figure 2. Multiple-frequency time series (MFT) measurements of MDCK cell attachment and spreading.Three-dimensional plots (a,b) are the log of measured resistance and capacitance as a function of log frequency and time.Bold purple curves indicate the impedance spectra at time 0 and 24 h after seeding cells.Plots (c,d) are the time series resistances and capacitances measured at 500 Hz and 64 kHz, respectively.Data were averaged from eight electrode wells and presented as mean ± standard error.

Figure 2 .
Figure 2. Multiple-frequency time series (MFT) measurements of MDCK cell attachment and spreading.Three-dimensional plots (a,b) are the log of measured resistance and capacitance as a function of log frequency and time.Bold purple curves indicate the impedance spectra at time 0 and 24 h after seeding cells.Plots (c,d) are the time series resistances and capacitances measured at 500 Hz and 64 kHz, respectively.Data were averaged from eight electrode wells and presented as mean ± standard error.

Figure 3 .
Figure 3. (a) Normalized resistance and (b) normalized capacitance spectra of an electrode covered with confluent MDCK cells.The curves are obtained from measured resistance and capacitance values at 25 different frequencies by dividing them by the corresponding quantities of the same electrode without cells.The points are calculated values from the simplified LGK model using Equations (18)-(20).Measured values are means ± standard error; n = 14.

Figure 4 .
Figure 4. Normalized resistance (a,c,e,g) and normalized capacitance (b,d,f,h) spectra are calculated from the simplified LGK model using Equations (18)-(20).Here we use Rb = 88 Ω•cm 2 , α = 27 Ω 1/2 •cm, Ca = 4.5 µF/cm 2 , Cb = 2.0 µF/cm 2 , and Cl = 1.6 µF/cm 2 as the basis parameters for model calculations.These values are close to the fitted results of MDCK cells.We change only one parameter in the model calculation, and the parameter values used for model calculation are indicated in each figure.

Figure 3 .
Figure 3. (a) Normalized resistance and (b) normalized capacitance spectra of an electrode covered with confluent MDCK cells.The curves are obtained from measured resistance and capacitance values at 25 different frequencies by dividing them by the corresponding quantities of the same electrode without cells.The points are calculated values from the simplified LGK model using Equations (18)-(20).Measured values are means ± standard error; n = 14.Knowing how changes in each model parameter affect the normalized impedance spectrum is crucial to getting a good model fitting.Figure 4a,b show the normalized resistance and capacitance spectra for different values of R b : 20, 40, 60, and 80 Ω•cm 2 .When R b increases, the peak of the normalized resistance spectrum (Figure4a) shifts leftward and upward.As a result, the values of the normalized resistance increase on the low-frequency side and decrease on the high-frequency side.Figure4c,dshow another normalized resistance and capacitance spectra for different values of α: 20, 30, 40, and 50 Ω 1/2 •cm.When α increases, the left side of the normalized resistance curve (Figure4c) moves leftward, whereas the peaks of all the curves are almost at the same height.As for normalized capacitance spectra shown in Figure4b,d, capacitance values at different frequency ranges decrease as R b or α increases.Finally, as shown in Figure4e,g, when C a or C b increases from 2 µF/cm 2 to 5 µF/cm 2 , the right side of the normalized resistance curve moves leftward and downward.In general, while changes in C a or C b give similar effects to the normalized resistance and capacitance spectra, parameter C a has more impact than C b on the calculated spectrum.Notably, both normalized resistance and capacitance spectra change slightly when C l takes on the values of 1.0, 1.5, 2.0, and 2.5 µF/cm 2 .

Figure 3 .
Figure 3. (a) Normalized resistance and (b) normalized capacitance spectra of an electrode covered with confluent MDCK cells.The curves are obtained from measured resistance and capacitance values at 25 different frequencies by dividing them by the corresponding quantities of the same electrode without cells.The points are calculated values from the simplified LGK model using Equations (18)-(20).Measured values are means ± standard error; n = 14.

Figure 4 .
Figure 4. Normalized resistance (a,c,e,g) and normalized capacitance (b,d,f,h) spectra are calculated from the simplified LGK model using Equations (18)-(20).Here we use Rb = 88 Ω•cm 2 , α = 27 Ω 1/2 •cm, Ca = 4.5 µF/cm 2 , Cb = 2.0 µF/cm 2 , and Cl = 1.6 µF/cm 2 as the basis parameters for model calculations.These values are close to the fitted results of MDCK cells.We change only one parameter in the model calculation, and the parameter values used for model calculation are indicated in each figure.

Figure 4 .
Figure 4. Normalized resistance (a,c,e,g) and normalized capacitance (b,d,f,h) spectra are calculated from the simplified LGK model using Equations (18)-(20).Here we use R b = 88 Ω•cm 2 , α = 27 Ω 1/2 •cm, C a = 4.5 µF/cm 2 , C b = 2.0 µF/cm 2 , and C l = 1.6 µF/cm 2 as the basis parameters for model calculations.These values are close to the fitted results of MDCK cells.We change only one parameter in the model calculation, and the parameter values used for model calculation are indicated in each figure.

Table 1 .
Impedance analysis of MDCK and OVCA429 cells using LGK, simplified LGK, and modified GK models.Values are means ± standard error; n = 14 for MDCK cells; and n = 16 for OVCA429 cells.For the simplified LGK model, we used Equations (18)-(20) and five parameters, R b , α, C a , C b , and C l , to fit the measured resistance and capacitance spectra of cell-covered electrodes.For the modified GK model, we used Equations (29) and (30) and four parameters, R b , α, C a , and C b , to fit the measured impedance spectra.See the list of symbols for the definition of parameters.Percentage errors are less than 15% for LGK and simplified LGK model fitting and less than 10% for modified GK model fitting.

Table 2 .
Impedance analysis of HaCaT and HUVEC cells using simplified LGK and modified GK models.
Values are means ± standard error; n = 4 for HaCaT cells; and n = 16 for HUVEC cells.Percentage errors are less than 15% for simplified LGK model fitting and less than 10% for modified GK model fitting.