Alternative General Fitting Methods for Real-Time Cell-Count Experimental Data Processing

Alternative General Fitting Methods for RealTime Cell-Count Experimental Data Processing Juan Alfonso Serrano, Pablo Pérez, Gloria Huertas, and Alberto Yúfera, Member, IEEE Abstract —This paper reports two general methods for extraction of cell-electrode electrical model parameters in cell culture (CC) assays. The presented approaches can be applied to CC assays based on electrical cell-substrate impedance spectroscopy (ECIS) technique for real-time supervision, providing the cell number per square centimeter, i.e., the cell density, as main result. Both of the proposedmethods - minimization of system equations and data predictive model - search, during the experiment, the optimum values of the electrical model parameters employed for the electrodecell model. The results of this search enable a fast and efficient calculation of the involved cell-electrode model parameters and supply real-time information on the cell number. For the sake of experimental validation, we applied the proposed methods to practical CCs in cell growth assays with a cell line of AA8 Chinese hamster ovarian fibroblasts and the Oscillation Based Test technique for bioimpedance measurements. These methods can be easily extrapolated to any general cell lines and/or other bioimpedance test methodologies. Index Terms —Bioimpedance, cell-culture, ECIS, electrical electrode model, OBT. I. INTRODUCTION MANY biological and medical assays are based on the preparation of a cell culture (CC) for a given cell line and then observation of the cell progress over time after applying a given product such as a toxin or a drug, external conditions, a protocol, etc. [1]. This is a common and useful procedure employed in biomedical labs and exploited in fields such as cell biology [2], genetics [3], biochemistry [4], pharmacology [5], tissue engineering [6], immunology [7], and food control [8]. A CC assay requires seeding a number of cells on a substrate, commonly a Petri dish, and applying This work was supported in part by the Spanish Government’s, Ministerio de Ciencia, Innovacióny Universidades and in part by the Plan Estatal 2017–2020 Retos: Real Time Monitoring of Hemodynamic Variables Using Intelligent Stents (iSTENT) with Capaci-tive Sensors and Bioimpedance, co-financed with FEDER under Project RTI2018-093512-B-C21. The associate editor coordinating the review of this article and approving it for publication was Dr. Shanhong Xia. (Corresponding author: Juan Alfonso Serrano.) Juan Alfonso Serrano, Pablo Pérez, and Alberto Yúfera are with the Departamento de Tecnología Electrónica, ETSII, Universidad de Sevilla, 41012 Seville, Spain, and also with the Instituto de Microelectrónica de Sevilla (IMSE-CNM-CSIC), 41092 Sevilla, Spain (e-mail: serrano@ imse-cnm.csic.es; [email protected]; [email protected]). Gloria Huertas is with the Departamento de Electrónica y Electromagnetismo, Facultad de Física, Universidad de Sevilla, 41012 Seville, Spain, and also with the Instituto de Microelectrónica de Sevilla (IMSE-CNMCSIC), 41092 Sevilla, Spain (e-mail: [email protected]). several conditions or protocols, defined in the context of the assay. To obtain information at a given time instant about the cell status, e.g., their size, number, type, and health, it is generally required to “kill” the cells and fix them [1] for microscope observation. This is why these assays are commonly called “end-point protocols”: the CC finishes when its observation is decided. For this reason, CC assays need many parallel preparations in terms of observation points specified in the assay description, requiring subsequent statistical analysis, significant human workload, and many material resources. To overcome the drawbacks of end-point protocols, Giaever and Keese [9] proposed the so-called Electrical Cell-Substrate Sensing (ECIS) technique as an alternative procedure to perform real-time monitoring of CCs, thereby avoiding multiple samples. This study suggested testing the impedance of a CC when it is positioned between two electrodes (when 4wires setups are used, the current lines of the applied electric field do not correctly sense the cells). Given that cells are attached to the CC substrate (electrodes), it is possible to have an estimation of the cell number by measuring the bioimpedance between the two electrodes. There results were developed by leveraging prior studies [10]–[13], which demonstrated the feasibility of the impedance-based technique. Practical information reported in most previous ECIS studies includes the growth evolution over time by monitoring the bioimpedance measured between the two electrodes (resistive and capacitive components) from cell seeding to confluence (electrode surface full of cells) phases. These data are reliable for CC characterization and have been useful for biomedical teams. However, the evolution of the cell numbers over time has not been reported. The key to access this information relies on the knowledge of the electrical model for the electrodecell system, which is mandatory for decoding the electrical bioimpedance measurements. The first proposal for an electrode-cell electrical model was described in [10]. This approach considered the cell culture as a monolayer and obtained the solution of Maxwell’s equations for the electrical field in the electrode-cell interface. To find the solution, parameter values such as Rb(barrier resistance between cells), h(electrode-cell interface distance), and r(cell radius) were proposed and selected to fulfill the equations derived. In a subsequent study, Huang et al. [14], considered a single cell on top of an electrode and extracted the electrical performance of the system by simulating its electrical response to an AC signal with finiteelement simulation tools. A new resistance Rgap –thesocalled gap resistance –was incorporated to the electrodecell electrical model that represented the obstruction to the electrical current flowing along the electrode-cell interface, which usually extends several nanometers. They also included a geometrical parameter ff – the so-called fill factor –that described the percentage of electrode area covered by the cells. Other approaches [15] employs a RC model to evaluate the load response associated to cells, as suggested in [10], without considering their size or any other cell properties. Recently, a novel approach [16] was proposed to employ the model in [14] with empirical parameter fitting, modifying the dependence of Rs– the so-called spreading resistance – on ff. This dependence may be due to changes in extracellular composition during the experiment. The main conclusions in this study were that electrode mismatch precluded the use of the same parameter values for the electrode-electrical model, requiring evaluation for each sample or CC. The electrode impedance was observed to evolve over time throughout the assay (several days in this case). In addition, it was necessary to wait until the confluence phase of the CC to apply the proposed fitting model. The present study proposes the use of empirical data for electrode-cell electrical model calculations as in [16], but incorporating only the initial data measurements of the CC assay and calculating the electrode-cell parameter values for calibration. Herein, we propose two approaches to find the values of the parameters in the electrical model in real time. The first one is based on the analytical solution of the electrical response obtained by minimization of system equations (MSE). The parameter values are provided from the equations that describe the system solution. As an example, we applied the Oscillation Based Test (OBT) method to measure the system bioimpedance. Machine Learning (ML) algorithms are increasingly relevant to data science and experimental data from bioimpedance sensing is not an exception. The application of ML algorithms to analyze bioimpedance cell measurements has been previously evaluated by the authors [17]. Following this idea, the second method searches for the best-fitting Fig. 1. 8E10E commercial electrodes from Applied Biophysics (AB). This is the substrate employed in the CC assays evaluated in this study. Each well contains 10 circular microelectrodes (diameter of 250 μm) connected in parallel and 1 reference electrode (GND). parameter values from a design space generated by moving the corresponding cell-electrode electrical parameter within a given range while evaluating the parameter proposal using the OBT equations. These parameter values are employed for training a k-nearest neighbor (KNN) algorithm [18], which will be applied for data decoding after the calibration phase. The article is organized as follows. Section II describes the material and methods, the CC protocol, the electrode-cell electrical model, and the main OBT equations employed. Both of the proposed experimental fitting models, i.e., MSE and data predictive model (DPM), are described in detail in Section III. The main results from the same CC data are reported and compared in Section IV. Finally, conclusions are highlighted in Section V. II. MATERIALS AND METHODS A. Cell-Culture Assay The experimental assays were based on commercial electrodes provided by AB [2] (Fig. 1). These devices contain eight separated wells, each with ten circular biocompatible gold microelctrodes of 250-μm diameter in parallel for input current, and a large ground electrode for output current. This study was performed using a cell line described next. The biological sample under test (SUT) was formed by Chinese hamster ovarian fibroblasts. This cell line is identified as AA8 (American Type Culture Collection). This sample was immersed in a McCoy’s medium supplemented with 10 % (v/v) foetal calf serum, 2 mM L-glutamine, 50 μg/mL streptomycin, and 50 U/mL penicillin. The growing environment was established at 37 ◦C and 5 % CO2 in a humid atmosphere. Different initial cell numbers were seeded for our experiments, namely 2.500, 5.000, and 10.000, obtained by dilution of an initially high cell density. Petri-dish cultures were also made with the same cell density for further matching with the proposed bioimpedance (BioZ)test. B. Cell-Electrode Electrical Model A CC assay was initiated in the aforementioned commercial devices. As pointed out above, the system had eight wells, each with a two-electrode configuration: input and output (Fig. 1). Cells were deployed on the electrodes along with the medium solution. The cell-electrode electrical model describes the behavior of a CC and corresponding electrodes. Fig. 2 shows the electrode model in a saline solution, while Fig. 3 shows Fig. 2. Electrode electrical model immersed in an ionic solution. A double layer is formed near the electrode; an Inner Helmholtz Plane is defined by the distance at which water dipoles are aligned with the electrode surface. An Outer Helmholtz Plane is defined as the distance at which solvated ions are aligned near the electrode. Fig. 3. BioZ electrical model: electrical model of an electrode in a saline solution considering the influence of the cells covering an ff percentage of the electrode area. its evolution considering the influence of the cells covering an ff percentage of the electrode area. This model has been extensively explored in the literature, [9]–[14] and [19]. To test the cell growth over the electrodes, the OBT measurement technique was employed [20]. The sensor system in the OBT architecture forms an oscillator where the biological interface is connected as a load (Z)into the main loop. As a result, a biological sensor is created. The CC assay growth is related to changes in bioimpedance, which generate variations over the amplitude and frequency of the signal generated in the oscillator. The OBT technique is just an example. In OBT the central frequency can be tuned by the band pass filter [13], the reactive impedance due to the cell membrane capacitance is very high for the lower frequency spectrum (up to several kHz). The sensor frequency selection ensures that electrical current lines do not pass through the cell. The proposed fitting methods can be easily extrapolated to other measurement techniques. A previous study [16] described the changes on electrical model parameters and their importance to predict the values of ff or cell density in real time, as well as the equations and parameters of the sensor and its configuration to collect data. The BioZ electrical model shown in Fig. 3 is described by Rgap,Rs, and the following parameters: R1=Rct (1−ff)R2=Rct ff C1=Cdl ·(1−ff)C2=Cdl ·ff (1) where Rct and Cdl are the charge transfer resistance and the double-layer capacitor, respectively [19], modeled in Fig. 2. The parameters R1,R2,C1,andC2model the electrical behavior of the microelectrodes covered by the cells. Nodal analysis provides the transfer function that models the BioZ: Hz(s)= k2·s2+k1·ω0z Qz·s+k0·ω2 0z s2+ω0z Qz·s+ω2 0z (2) where, k2=Rsk1=Rs+Rgap ·R1 2·Rgap +R1+R2 k0=Rs+R1·(Rgap +R2) Rgap +R1+R2 ω0z=Rgap +R1+R2 Rgap ·(R·C)2Qz=ω0z Rgap ·R·C 2·Rgap +R1+R2 (3) C. OBT System As mentioned in Section II.A, the measurement method we applied for transforming BioZ into a measurable variable was OBT. This is simply a possible technique; any other could be used. The electronic system oscillates at a given amplitude and frequency. These two parameters are defined by the BioZ value of the CC and can provide the cell growth information along with a proper knowledge of the sensor electrical model. The oscillation system used to measure BioZ was reported in [16]; the same block diagram and parameter values were applied in the present study. The Barkhausen stability criterion – see below Equation (4) – is the mathematical condition that the closed-loop feedback system must fulfill to obtain sustained oscillations. Thus, the oscillation parameters aosc and fosc can be derived by setting the real and imaginary parts of Equation (4) to 0: 1+H(s=j)·N(aosc, osc)=0(4) where ωosc =2πfosc,andH(s=jω) is the transfer function with no linear element of the OBT, which includes the BioZ, a band-pass filter, and other electronic elements; N(aosc, ωosc)is the descriptive function of the comparator, that is, the linearized mathematical description of the comparator, whose calculation was described in [20]. Equation (4) will be used as a demonstrative vehicle to show how both fitting methods work; any other system equations could be employed. III. ESTIMATION METHODS OF REAL-TIME EXPERIMENTAL CELL-CULTURE GROWTH In this section, the two methods proposed for analysis of the measurements provided by the sensor are compared. Firstly, Fig. 4. Flowchart of the MSE method. A cost function is used to estimate Rsi and fp from the initials measurements, and Rgap ,Δ Rs , and ff for each measurement. minimization of the module of the function in Equation (4), denoted as MSE. Secondly, DPM, which is a method based on electrical parameter fitting from initial experimental measures. Both methods rely on the numerical solution of the dynamic models for the system and the oscillation condition expressed in Equation (4). In particular, the oscillation condition is given by Equation (4), where the real and imaginary parts, denoted as eq1 and eq2, respectively, from now on (defined in the Appendix as Equations (1) and (2), respectively), must be zero. A. Minimization of System Equations This method is based on a previous study [16]. Joining eq1 and eq2, derived by setting the real and imaginary parts of Equation (4) to zero, and three parameters derived from the electrochemical interface and the cells under test, namely Rgap,Rs,andff, we have 2 equations and 3 variables. The solution for these equations, as previously reported in [16], is found by setting the values of Rgap and Rsthrough a calibration experiment using a previous cell line, given that these values seem to be specific to each cell line and well type. However, herein, the method is focused on a different approach for the estimation of cell growth; the major advantage of this approach is that it enables estimation with no previous experiments by minimizing a multivariable function with constraints. This function must contain the three unknown variables and meet the oscillation condition defined in Equation (4). Fig. 4 shows the flow diagram of the on-line estimation process. It minimizes a cost function, which is explained below, after each CC measurement to obtain Rgap,Rs,and ff. We waited for a few hours at the beginning of the each experiment until Rsi and fp, which is one of the poles in Equation (2) used to estimate initial values of Rct and Cdl, could be estimated. Matlab was the environment used to conduct all testing. The Matlab function used to minimize the function was fmincon, which finds the minimum of a constrained nonlinear multivariable function. The fmincon algorithm used was sqp [21]. The function to be minimized must ensure that the Barkhausen stability criterion is fulfilled. Therefore, we employed the following function: f(Rgap,Rs,ff)=(eq12+eq22)(5) This function implies that the module of the real and imaginary parts in Equation (4) must be 0. Given that both terms are squared, it is ensured that eq1 and eq2 cannot compensate each other to minimize f(Rgap,Rs,ff)to0. In this way, it is ensured that the Barkhausen stabilization criterion is fulfilled, i.e., eq1 and eq2 are 0. Note that fmincon needs a starting value per execution for each variable, as expressed in Equation (6), in addition of possible restrictions (or bounds) of such variables, expressed in Equation (7). ff n=ff n−1Rgap =500Rs=0(6) ff bounds =0.05 0.999 %1 Rgap.bounds =100 2e3 Rs.bounds =−Rsi 2·Rsi (7) The initial value of ff for each fmincon iteration is the value of ff at the previous point of the experiment. The ff limits its value to the range [0,1]. The initial value of Rgap for each fmincon iteration is 500 . The limits for Rgap precludes that it becomes negative and excessively large, that is, its variation is allowed for positive values (negative ones do not make physical sense). The initial value of Rsfor each fmincon iteration is 0 . The limits imposed on Rsprevent Rsfrom becoming negative, or too large, but still allow for a wide variation range. The estimation process is described next. First, it is required to wait for the CC to begin to grow. Then, using the initial values of fosc and aosc, the position of the pole, fp,aswell as the initial values of Rsand Rsi, are calculated as in [16]. This is achieved by the fmincon minimization procedure of the function expressed in Equation (5), using fpand Rsas unknown variables together with ff (the values of Rgap and Rsare the same as in Equation (6), and the initial value of ff is estimated from the initial cell number in each well). This estimation is only computed at the beginning of the experiment, with the initial conditions and bounds provided in Equation (8) below. ff n=ff ini fp=90Hz Rs=500 fbounds =0.05 0.999 %1 fp.bounds =0.11e3Hz Rs.bounds =0.12e3(8) At this point, it is already possible to estimate the values of Rgap,Rs,andff at any arbitrary point of the experiment through fmincon. Interestingly, note that Rgap and Rsdo not affect the electrode-cell model when ff →0, that is, at the beginning of the experiment. B. Dataset Predictive Model Note that the first proposed method, i.e., MSE, finds the most suitable operation point for the sensor, which corresponds to a certain oscillatory solution. This requires a certain amount of computational power at every iteration. TABLE I PARAMETER BOUNDS An alternative approach is presented in this section. The DPM method relies on the definition of a full dynamic model for the whole growing process in which the cell-electrode interface is evolving through the experiment; the values for Rsand Rgap are not constant in this case. A certain degree of variance across the different electrodes involved in the experimental setup is assumed. According to Fig. 2, values for Cdl,Rct ,andRscan be computed to provide an electrical model from the electrode material and conductive ion concentration in the solution [19]. Nominal values for such parameters are Cdl =32nF,Rct =1.3M,andRs=500. Fig. 5 shows a general diagram of the DPM process. This method requires the computation of a multivariate dataset accounting for the variation of the electrode-model electrical parameters over a certain range (Input Range). Then, a KNN algorithm [18] network is trained using this multivariate dataset to provide the parameters that better fit our experimental data during the calibration step (Electrical Parameter Determination). Using the electrical parameters computed in the calibration step, a prediction of ff values is obtained through a linear regression from the expected ff curves for such electrical values. The algorithm steps are described in detail next. 1) Dataset Generation: As described above, the electrode model is a source of significant error in terms of how the electrical parameters contribute to the ff estimation. The fact that real electrodes introduce some uncertainty and dispersion across different devices cannot be neglected. To generate a multivariate dataset establishing the relation between ff and the electrical model, a range for the electrical parameters is defined. TABLE I reports the ranges of Cdl,Rct ,andRs obtained from the MSE method. A numerical simulation was performed using the aforementioned parameters and setting ff to zero (initial status). The results are depicted in Fig. 6 and Fig. 7, where the amplitude and frequency sensor output are described for the whole multivariate range. The contribution arising from Rct to the frequency and amplitude curves is almost negligible. This is why we present Rct and Cdl as the dependent variables. 2) KNN Training and Prediction: Training a KNN Network using the previously described dataset enables the prediction of the closest point to a given output value from the sensor (aosc and fosc)over the early stage of the experiment. The KNN algorithm provides the most suitable fitting for the electrical model that corresponds to the experimental data acquired over the initial period (ff →0). 3) Fill Factor (Ff) Estimation: At this point, the electrical model parameters are known, thus enabling the calculation of ff from the electrical model. The analysis of the frequency and amplitude curves in terms of ff already revealed a nearly linear behavior [11]. Hence, to estimate the value of ff, it is sufficient to compute the maximum frequency ( fmax)(corresponding to ff →1) from the electrical model to describe a linear function using all the known elements: fosc =fosc.max −fosc.min ff max −fmin ·ff +fosc.min (9) where ffmax =1andffmin =0. Then, we have that: fosc =(fosc.max −fosc.min)·ff +fosc.min (10) which can be rewritten and solved for ff: ff =fosc −fosc.min fosc.max −fosc.min (11) where fosc corresponds to the frequency experimentally measured and ff is the estimated fill factor. IV. RESULTS In this section, we report the results for each estimation method. As a comparison metric, we obtained the cell concentration, i.e., Ncell /cm2, obtained by traditional methods, optically counting the number of cells with the Leica DMI1 inverted microscopy every 24h until the confluence phase was reached. The cell concentration (Ncon)was obtained by using ff in Equation (12) below. Traditional cell counting was obtained as the mean of 3 counting samples. This traditional counting method provides an error margin. We estimated the cell area for the upper and lower margins of this traditional counting method to obtain the cell-density error margin in the MSE and DPM methods. Ncon =Ncell Awell [cm2]where Ncell =Awell Acell ×ff (12) where Ncell is the cell number in the well at any arbitrary moment, Awell is the well area, and Acell is the mean cell area. In our case, we considered that rcell =13.3μm. A. Minimization of System Equations Results related to the MSE method are depicted in Fig. 9. Note that for each initial cell number, the error was very small. Indeed, for the greatest initial cell number, the error was minimum (10.000 is a very small cell number). The initial error was not acknowledged given that it was caused by some perturbations resulting from the adaptation of the cells to the wells and deviations of temperature, humidity, and CO2from their ideal values. These deviations were due to the opening of the incubator; they were corrected after a few hours. Note that, for initial cell numbers of 5.000 and 10.000, the error ranges of the traditional counting method and MSE were similar after 48 hours from the starting point of the experiment until the end. B. Dataset Predictive Model For DPM,Fig. 9 illustrates the prediction for the same experimental values as in the previous method. The resulting curves provide good matching with the experimental ones. There exists some dispersion probably due to the electrode shape and sensing surface. For most of the results, the mean Fig. 5. Flowchart of the DPM method. This flowchart depicts the steps needed to estimate the value of ff using the DPM method. The algorithm generates a dataset for potential experimental results using the analytical sensor model. This dataset encodes a function of fosc and aosc with respect to variations on electrical parameters (such as those introduced by the electrodes). An estimate of such electrical parameters is performed from the initial measurements acquired to enable real-time prediction of different ff values. Fig. 6. Frequency dataset for variation of the electrical parameters Cdl and Rs . value of the cell radius estimation (blue line) is evaluated within statistical significance with respect to traditional counting methods (which may have a certain dispersion in some measurements). C. Comparison Both methods provided a good enough estimation of the cell concentration. By comparing Fig. 9 and Fig. 8, we can see that DPM was better for a low initial cell number and for the first hours of the experiment, while the MSE method was Fig. 7. Amplitude dataset for variation of the electrical parameters Cdl and Rs . Fig. 8. Comparison of cell concentration between traditional methods and the DPM method for three initial cell numbers (2500, 5000, and 10000 cells), where the red line represents the traditional counting method with its error range (vertical red line), and the blue line represents the estimation of cell concentration using the DPM method with its error range (blue dashed lines). better for a higher initial cell number and from the middle to the end of the experiment. TABLE II shows the mean relative errors of both methods with respect to traditional counting in row er.The relative Fig. 9. Comparison of cell concentration between traditional methods and the MSE method for three initial cell numbers (2500, 5000, and 10000 cells), where the red line represents the traditional counting method with its error range (vertical red line), and the blue line represents the estimation of cell concentration using the MSE method with its error range (blue dashed lines). TABLE II RELATIVE ERROR error was estimated through the next equation: er=1 n n  i=1|Ncon.method(i)−Ncon.trad(i)| Ncon.trad(i)(13) where eris the mean relative error, Ncon.method and Ncon.trad denote the cell concentration vectors estimated by each method and by traditional counting, respectively, and nis the number of samples employed to estimate er. The first row shows the mean relative error of the whole experiment. The next rows show the mean relative error when ff was less than 0.5 and greater or equal than 0.5, respectively. The MSE method provided better estimates of cell concentration when ff was greater than 0.5 and for experiments whose initial cell numbers were high. By contrast, the DPM method provided better estimates of cell concentration when ff was lower than 0.5 and for experiments whose initial cell numbers were low. V. CONCLUSIONS Two analytical methods were proposed in this study to obtain the evolution of cell concentration in a growth experiment of CC assay. Previous methods reported by the authors [16] apply an estimation algorithm which requires the full experiment information to be able to perform estimation. The proposed methods allow to estimate the cell concentration during the experiment (in real-time) without having to wait until the end of the experiment. The ECIS technique, along with the electrical model of an electrode in a saline solution, were applied to estimate the bioimpedance of a CC assay in each well and the ff parameter, thereby achieving real-time prediction of the cell density present in the assay. Each method has a working zone, defined by the experiment progress and initial cell concentration, where it is more accurate. A comparison of the ranges of both methods (Fig. 9 and Fig. 8, in dashed blue line) with the ranges of traditional counting (Fig. 9 and Fig. 8, in dashed red line) shows that the proposed methods achieve positive results. In addition TABLE II reports the error and the best working zone of each method in order to evaluate their accuracy. The parameters Rgap and Rsdo not affect when ff is low [16]. The MSE method was less accurate in this part of the experiment given that it estimates these parameters for each measurement. By contrast, when ff is high, Rgap and Rs affect the most (their estimations are more accurate) and thus the estimation of ff (and cell concentration) is more accurate. The DPM method is based on a previous estimation of the cell-electrode model parameters. This estimation provides a very accurate estimation of the ff when this parameter is low, given that Rgap and Rsdo not affect in the interval ranging from the beginning to the middle of the experiment. When the ff was high, small errors in the value of Rgap and Rscould cause a decrease in accuracy. For this reason, we conclude that the combination of both methods can probably result in a general procedure for more accurate estimates of the cell concentration at any point of the experiment without prior calibration. According to the results obtained, we consider that using the DPM method when the estimated ff is low and the MSE method for ff greater than a certain value, in particular when ff >0.5, can give rise to a better fitting method. For future experiments, both methods will be combined and their accuracy and operation will be verified. APPENDIX Below are the equations obtained by calculating the real and imaginary parts in Equation (4). Note that eq1 and eq2 are Equations (1) and (2) in this appendix. Re(1+HBP(s)·HZ(s)·HCMP.F(s)·N(aosc, osc))=0(1) Im(1+HBP(s)·HZ(s)·HCMP.F(s)·N(aosc, osc))=0(2) where s=j·ωosc,HZis the transfer function of the cellelectrode electrical model; HBP and HCMP.Fare the transfer functions of the band-pass filter and the pre-comparator filter of the OBT system, respectively; and N(aosc,ωosc)is the comparator describing function of the OBT system [20]. Estimates of N(aosc,ωosc), HBP,andHCMP.Fare reported in [16]. 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