Evaluation of single cell electrical parameters from bioimpedance of a cell suspension

Abstract

The present study introduces a simple and detailed analysis technique to extract the electrical properties of a single cell from the impedance spectroscopy data from a group of cells in suspension, leading to a more reliable and cost effective diagnosis process for disease detection. The existing method for bioimpedance measurement, by trapping a single cell in a microchannel, is quite a complex process and suffers from localized joule heating. Considering that biological cells show their natural characteristics and functionality in a colony of similar cells rather than in an individual environment, the extraction of single cell electrical parameters from the impedance measurement of a group of suspended cells may provide more reliable and effective information. Experimental and theoretical analyses were performed to extract single cell permittivity, conductivity, membrane capacitance and cytoplasm resistance, utilizing the established Maxwell's mixture theory. The bioimpedance of the suspended HeLa cells was characterized with a controlled volume fraction of cells in the suspension, and the measurement was performed by varying the voltage to investigate the change in permittivity and conductivity of the HeLa cells. The proposed technique showed the membrane capacitance and cytoplasm resistance of a single HeLa cell to be in the 1.8 nF cm⁻² and 35 kΩ cm² ranges, respectively. Analysis of the measured impedance data also reveals that the relative permittivity and conductivity of a single HeLa cell is a function of the applied potential and frequency.

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Introduction

Traditional disease detection techniques require an expensive and complex labeling process and an extensive biochemical assay. However, through various experimental results, phenomenal progress has been made which has established that non-biological properties, such as the mechanical, electrical and optical parameters of cells, are also modified during disease progression and undergo pathological changes.^1–3 Sometimes the signal from the non-biological parameters may show an early sign of the disease before significant changes are observed in the biological signal. Therefore, the current trend observed in the medical research community is for the exploration of an alternative, label-free, non-invasive detection technique for the identification of cancers,^4–7 bacteria,^8,9 the status of tissues,^10–12 cytotoxicity assessments,^13,14 drug screening and toxin detection¹⁵ etc. by quantitative investigation of the passive non-biological parameters of cells to enrich clinical processes.

Electrical bioimpedance spectroscopy (EBIS) has been used to characterize the complex biological system at the individual cellular level for the early diagnosis, prevention and treatment of complex diseases like cancer, malaria, AIDS etc. EBIS is able to produce a signature to distinguish the level of disease progress and abnormality of cells, and consequently establish a relationship between the electrical properties, biological functions and pathological attributes of cells. Although the EBIS technique is continuously exploited for the analysis of complex biological systems, its efficacy may further be enhanced by detailed and accurate information of single-cell parameters for the reliable diagnosis of diseases. Therefore, analysis of the electrical properties of single cells has become a new trend to enable the correlation of cellular events to aid the understanding of complex physiological processes. To this end, several experimental and numerical studies for single cell impedance characterization are available in the literature. Jang and Wang¹⁶ performed electrical impedance analysis of a single HeLa cell by capturing it inside a fabricated three-pillar microstructure in a microchannel. Hua and Pennell¹⁷ fabricated a chevron-like structure of electrodes in a microfluidic channel to capture single cells and measured the volume changes using impedance. Malleo et al. demonstrated a hydrodynamic cell trap system for the continuous differential impedance analysis of a single cell.¹⁸ Using the micropipette technique with impedance spectroscopy enables the direct measurement of the impedance of an individual cell membrane, however, the technique is invasive since the pipette punctures the cell.^19,20 Further, new techniques have evolved to measure the information from an individual single cell in a non-destructive way by using the microfluidic channel with integrated microelectrodes to interface with the cells directly.^21–23 A planar microhole-based structure has been explored to measure the impedance of a single cell without disturbance of the electrode polarization.^24,25 However, the interpretation of data gained using the microhole-based method was limited, due to the difficulty in observing the exact cellular morphology.

Electrical attributes of single-cell analysis provide information about the bio-physiological properties of the cell, which are sensitive to the bio-physical changes in the cell. However, this technique suffers from several challenges which need to be overcome. The handling of a single cell in a microchannel is complicated, and requires trapping mechanisms in the micro-channel for a single cell impedance measurement.^16,26 The throughput of the cell capturing devices is limited unless a large number of traps are available in the micro-channel.^27,28 The integration of electrodes, together with the multiplexed impedance measurements increase the complexity of the system. For large arrays of traps, complex active matrix methods are needed to measure the signals from multiple electrodes. Both the large size of cell, and its variation in shape, together with the difficulty of handling a single cell would introduce significant errors into the results. Moreover, microelectrodes only enabled for single cell measurements suffer from localized joule heating, produced by highly confined current pathways. Polarization of microelectrodes is also significant, and special attention is needed to extenuate the effect.^29,30 Furthermore, cells from the same cell line may have a different biological status, such as in different cell division cycles, in different stage of apoptosis etc. Due to the heterogeneity of a biological system, it is expected that cells show their natural characteristics in a colony of similar cells, rather than an individual environment. These shortcomings result in major difficulties when electrical properties are utilized to distinguish between normal and cancerous cells through single cell analysis. Alternatively, the electrical properties of a single cell can be estimated through the EBIS measurement of cells in suspensions, using the well established Maxwell's mixture theory³¹ without interference from the above issues. The analysis relates the complex permittivity of the suspension to the complex permittivity of the particle, the suspending medium and the volume fraction. Therefore, this technique provides a comparatively easy alternative way to extract the single cell parameters without involving the complex technology required for single cell analysis in a microchannel.

In the present study, the electrical properties of a single HeLa cell have been analyzed from the impedance spectroscopy data acquired from its group of cells in suspension. A detailed experimental and theoretical analysis has been performed to extract the single cell permittivity, conductivity, membrane capacitance and cytoplasm resistance using Maxwell's mixture theory. EBIS of suspended HeLa cells are measured using an impedance analyzer in the frequency range of 100 Hz to 10 MHz with a controlled volume fraction of cells in the suspension. The experiments were performed for different applied potentials to evaluate the permittivity and conductivity of HeLa cells, and then analyzed to extract the single cell parameters. The present study demonstrates a simple and detailed analysis technique to extract single cell parameter values from the impedance measurement of a group of suspended cells, in comparison to other existing techniques involving Maxwell's mixture theory and equivalent electrical circuit models.

Theoretical modelling

Maxwell's mixture theory

The basic principle of bioimpedance is based on Ohm's law, where the potential is measured by applying a small current across a group of cells, and subsequently impedance values are determined with a frequency sweep. The recorded bioimpedance is a measure of the complex dielectric properties of the cells, which are characterized in terms of permittivity and conductivity. The suspended biological cells in media form a heterogeneous system and their dielectric properties are generally described by Maxwell's mixture theory.³¹ For a spherical particle dispersed in a suspending medium at a low volume fraction, Maxwell's mixture theory provides an equivalent complex dielectric permittivity of the cells in the frequency domain, according to eqn (1).where f̃_cm is the complex Clausius–Mossotti factor:

The subscripts “mix”, “p” and “m” refer to mixture, particle and medium, respectively, and [small epsilon, Greek, tilde] is the complex permittivity represented as: , where , and ε, σ, ω and φ are the permittivity, conductivity, angular frequency, and volume fraction of the cells in suspension, respectively. Although Maxwell's theory is only valid for low volume fraction (φ < 10%), Hanai and Koizumi^32,33 later extended the theory for all volume fractions, which is depicted in eqn (3).

In terms of a single shelled spherical cell model in suspension as shown in Fig. 1, the complex permittivity of the cell is:³¹

with γ = R + d/R, where [small epsilon, Greek, tilde]_mem and [small epsilon, Greek, tilde]_i are the complex permittivity of cell membrane and cytoplasm, respectively, R is the radius of the cell and d is the membrane thickness. Thus, the complex permittivity of the cell is a function of the dielectric properties of the membrane, its internal properties (mainly cytoplasm), and size of the individual cell. The complex bioimpedance (Z̃_mix) of the cells in suspension is related to the equivalent complex permittivity of the mixture and the geometrical parameters of the measurement system given by:³¹where G is a geometric constant, which is the ratio of the electrode area (A) to the gap (g) – A/g between the electrodes. In this study, impedance spectroscopy of cells suspended in media is carried out using an electric cell-substrate impedance sensing (ECIS) based device with known geometric constant. Subsequently, the complex impedance and equivalent complex permittivity of the mixture of suspended cells are found out using eqn (5).

Fig. 1 Schematic diagram of a single-shell model of a cell in suspension.

Equivalent electrical model of single cell

In general, an equivalent electrical circuit model analogous to the physical model is utilized to describe the electrical properties of the suspended cells. Although it is quite complex to analyze the equivalent circuit model, the conductivity of the membrane and the permittivity of the cytoplasm were considered to be very low in the present study, giving the simple mathematical expressions as stated in eqn (6) and (7).

[small epsilon, Greek, tilde]_i = −jσ_i/ω (6)

[small epsilon, Greek, tilde]_mem = ε (7)

where σ_i is the conductivity of the cytoplasm and ε is the permittivity of the membrane. Using the above assumptions, eqn (4) can be simplified to eqn (8) and (9).where and with

On dividing the real and imaginary parts of [small epsilon, Greek, tilde]_p in eqn (9), a quadratic equation is obtained:

where

On assuming that l = σ_i/ε and equating the imaginary part of eqn (9), the permittivity of the single cell is obtained:

Using eqn (10), a value of l is calculated, the relative permittivity (ε) is obtained from eqn (11), and finally the conductivity (σ_i) is incurred through σ_i = l × ε.

The impedance of the cell suspension consists of the impedance of the medium, represented by a combination of resistance and capacitance, and impedance of the cells. According to Foster and Schwan,³⁴ a single cell is analogous to a cytoplasm resistor (R_i) in series with a membrane capacitor (C_mem) as represented in Fig. 2. The cell membrane consists of a thin phospholipid bilayer with very low conductivity, and acts as a dielectric material offering a capacitive pathway to the system.

Fig. 2 Simplified equivalent circuit model of a single cell in suspension.

The cell cytoplasm can be considered as a highly conducting ionic solution with a large concentration of dissolved organic material, which is considered to be a resistive pathway to the electrical signal in the electrical equivalent of the system. The values of the simplified, frequency dependent cell parameters are determined by the dielectric and conductivity properties of the cell, and the medium, cell size, volume fraction and geometric constant of the EBIS system:³¹

The frequency dependent relative permittivity (ε) and conductivity (σ_i) of the single cell obtained in eqn (10) and (11) are used in eqn (12) and (13) to extract the single cell membrane capacitance and cytoplasm resistance.

Materials and methods

Cell suspension preparation

In the present study, bioimpedance spectroscopy was performed on a HeLa cell line suspended in PBS buffer solution. HeLa cells are the cell line of human cervical carcinoma, which is one of the leading causes of death in women all over the world. HeLa cells were cultured in MEM media supplemented with 10% heat-inactivated fetal bovine serum, L-glutamine, pyruvic acid, and 1% antimycotic antibiotic. The cells were grown in a humidified atmosphere containing 5% carbon dioxide at 37 °C. The confluent cell population was removed by treament with 0.25% trypsin and 0.02% EDTA for 5 min. Subsequently, the cells with a population size of 10⁶ were re-suspended in 400 μL of fresh PBS buffer media, with pH value 7.4 and conductivity of 1.56 S m⁻¹. Bioimpedance spectroscopy of the suspended cells was performed using an ECIS based device.

Electrical bioimpedance spectroscopy

The impedance measurement of the cell suspensions was carried out in an ECIS-8W1E DD (Applied BioPhysics, USA) cell culture well. An Agilent precision impedance analyzer 4294-A was used to record the impedance data from the ECIS device. The ECIS-8W1E DD culture well consists of eight separate mini-culture wells having an individual circular working electrode (WE) and common counter electrode (CE) made of thin gold film. However, a portion of the individual WE is coated with a bio-compatible polymer, with a circular exposed part 250 μm in diameter at the centre of WE to minimize the electrode–media interface. The ECIS device ECIS-8W1E DD was mounted on a printed circuit board (PCB) and the necessary electrical connections were used to connect it with the impedance analyzer. Initially, 400 μL of the PBS media was added to one of the wells of the ECIS device, and the impedance-phase angle values were measured. This serves as a control and a baseline for the impedance measurement system in the absence of HeLa cells. Subsequently, HeLa cells suspended in 400 μL of PBS media was added to the wells of the ECIS device for impedance spectroscopy. The impedance magnitude and phase angle were recorded using an Agilent precision impedance analyzer 4294-A, in the frequency range of 100 Hz to 10 MHz, for different excitation voltages from 10 mV to 1 V peak–peak. Each measurement was also repeated three times without disturbing the ECIS system. There are 167 data points measured across the entire frequency range. The average recorded data were then analyzed using Maxwell's mixture theory and the equivalent circuit model to extract the electrical parameters of the single HeLa cell.

Algorithm to calculate single cell electrical parameters

In the present parameter extraction technique, the complex bioimpedance of the group of suspended cells with a known volume fraction was measured, then, the following steps were performed to extract electrical properties of single cell:

• The complex permittivity of the cell mixture ([small epsilon, Greek, tilde]_mix) was calculated using eqn (5) with a given geometric constant (G).

• The complex permittivity of the single cell ([small epsilon, Greek, tilde]_p) was estimated using the already calculated [small epsilon, Greek, tilde]_mix and the known values of [small epsilon, Greek, tilde]_m and the volume fraction (φ), according to eqn (3).

• The relative permittivity of the membrane (ε) and conductivity of the cytoplasm (σ_i) of a single cell was determined using eqn (10) and (11).

• The membrane capacitance and cytoplasm resistance of a single cell were extracted using eqn (12) and (13).

Results and discussion

Impedance of HeLa cells in suspension

Fig. 3 shows the Bode diagram of the impedance magnitude and phase angle variation with frequency for both of the PBS media, without and with suspended HeLa cells, at the operating voltage of 10 mV. The Bode magnitude of only PBS media without cells decreases with increasing frequency, representing a transition from a capacitive to resistive behavior as the applied signal moves from low to high.³⁵ The phase plot shows that the phase angle of only PBS media without cells moves towards zero degrees with increasing frequency. Fig. 3 also shows the effects of adding HeLa cells to the PBS media on the impedance data. Since the media is more conductive than the suspended HeLa cells, the impedance of the media is lower, and the phase angle of the media is less negative than that of the cells with suspended media. The Bode diagram depicts that upon adding the cells to the media, the impedance value increases abruptly at low frequencies (<100 kHz), whilst both of the curves coincide at higher frequencies. Similarly, the phase angle decreases at low frequencies for media with cells, whilst the two phase curves coincide at higher frequencies. This experimental observation indicates that the HeLa cells influence the impedance data at lower frequencies. In the EBIS system, a coating capacitance is formed due to the thin polymer layer on the electrode surface, and is in the nF range, resulting in its dominance in higher frequency zones. In addition, the solution resistance also plays a dominant role at higher frequencies. Therefore, the coating capacitance and solution resistance behave as an RC circuit at higher frequencies, leading to all of the curves coinciding without changing the impedance spectrum.

Fig. 3 Bode diagram of ECIS system with and without cells, suspended in PBS solution.

Fig. 4 shows the variation in the magnitude and phase of impedance at different voltage levels varying from 10 mV to 1 V. At all of the operating potentials, the impedance magnitude of the suspended HeLa cells decreases, and the phase angle increases with increasing frequency. The impedance values decrease rapidly in the lower frequency range (<100 kHz), whilst all of the curves coincide at the higher frequency range. Fig. 4a also indicates that the impedance magnitude decreases with increasing applied potential, e.g. at 1 kHz, the impedance of the suspended HeLa cells reduces from 2.2 kΩ to 702 Ω with increased potential from 10 mV to 1 V. Similarly, the phase angle value decreases with higher applied voltage, as observed in Fig. 4b. This observation is attributed to the opening of more ionic channels in the cell membrane at higher voltage, influencing the permittivity of the cell membrane and the conductivity of the cell cytoplasm.^16,36 It is expected that a higher electric field will greatly influence the ion exchange process between the intra- and extra-cellular solution, which leads to lower impedance and higher (more positive) phase angle of the cells.³⁷ At 1 V, the initial impedance value was quite low compared to the lower operating voltage, however, it followed the same trend in variation with an increase in frequency as observed in Fig. 4a. The lower impedance value is attributed to the breakdown of the dielectric membrane, due to a high electric field which is maintained throughout the entire frequency range. The coincidence of all of the impedance curves above 10 kHz, thereby maintaining a near equal impedance value at a higher frequency range, represents the coating capacitance of the ECIS system which does not alter with applied voltage.

Fig. 4 (a) Impedance magnitude spectroscopy of suspended HeLa cells at different operating potentials, (b) variation of the phase of HeLa cells in suspension at various voltages.

Extraction of single cell parameters

Maxwell's mixture theory was explored for single cell analysis, utilizing the impedance data measured at various voltages. In this approach, the volume fraction of 1 million HeLa cells suspended in a total volume of 400 μL is represented as:

The radius of a HeLa cell (R) is 10 μm, while the typical thickness of the membrane (d) is 5 nm.³⁷ The geometric constant (G = A/g) of ECIS, as mentioned in eqn (5), is not easy to determine because of the non-linear electric field. However, the approximate value of G is found by assuming that the area of the electrode (A) is equal to 1 cm² and the average gap between the electrodes (g) is 3 mm. The measured relative permittivity of the PBS medium used in the experiment is 136, and the conductivity of the medium is 1.56 S m⁻¹. Hence, the complex permittivity of PBS may be expressed as eqn (15).

where ε₀ = 8.854 × 10⁻¹² F m⁻¹ and ω is angular frequency. Using impedance spectroscopy data obtained from the experiment, the complex permittivity of the mixture ([small epsilon, Greek, tilde]_mix) was calculated using eqn (5), as given in the Theoretical Modelling section. Subsequently, [small epsilon, Greek, tilde]_mix, [small epsilon, Greek, tilde]_m and φ were substituted into eqn (3) to estimate the complex permittivity ([small epsilon, Greek, tilde]_p) of a single HeLa cell. Fig. 5a and b illustrate the variation in the conductivity and permittivity of the single HeLa cell, respectively, with frequency sweep from 100 Hz to 10 MHz, for various operational voltages in the range of 1 mV to 1 V. The results show that both the conductivity and relative permittivity of the equivalent single HeLa cell increase with a higher operational voltage at a lower frequency range.

Fig. 5 Variation in the (a) conductivity and (b) relative permittivity of a single HeLa cell with frequency at different voltages.

From Fig. 5a, it may be observed that at low frequency, the conductivity of the single HeLa cell increases from 0.13 S m⁻¹ to 0.23 S m⁻¹ as the voltage increases from 10 mV to 1 V. The results indicate that the higher electric field opens more ionic channels in the cell membrane, and thus enhances the charge exchange process between the cytoplasm and extracellular solution. This phenomenon allows a higher current to flow through the cell membrane and cytoplasm, leading to an increase in conductivity. The availability of ionic channels enhances the charge exchange between the cytoplasm and the extra-cellular solution. Wang and Jang³⁷ showed similar variations in permittivity and conductivity for different voltages, measured by trapping a single HeLa cell inside a microchannel. Fig. 5b shows that the measured relative permittivity of the single HeLa cell is nearly the same for different operating voltages up to 300 mV. At higher applied voltages above 300 mV, the slope of the relative permittivity of the cell is sharper than the low voltage data up to 4 kHz, and thereafter, all the permittivity data remains nearly same for the entire operating voltage. Under higher electric fields, the capacitance of the cell membrane may be fully charged at lower operating frequencies, whereas in higher frequency zones it cannot be fully charged within one cycle.^37,38 This demonstrates the decrease in the relative permittivity of the HeLa cell with increasing frequency at higher operating potentials. The above experimental facts depict that the relative permittivity and conductivity of the single HeLa cell are a function of the applied potential and frequency. This information may be useful for the electroporation of cell membranes and the characterization of different diseased cells.

Additionally, parameters such as the cell membrane capacitance and cytoplasm resistance of a cell may be utilized for the identification of cell type, and may also be explored for diagnostic and prognostic applications correlating with disease progression. Therefore, the extraction of the membrane capacitance and cytoplasm resistance of a single cell has emerged as a major requirement to gain detailed insight into diseases and their characterization. The obtained relative permittivity (ε) and conductivity (σ_i) of the single cell are further used to obtain the membrane capacitance and cytoplasm resistance of the single HeLa cell using eqn (12) and (13), and the variations with frequency are shown in Fig. 6a and b, respectively.

Fig. 6 Variation of the (a) membrane capacitance and (b) cytoplasm resistance with respect to frequency.

Since the permittivity of the membrane decreases continuously with the frequency, the membrane capacitance shows a similar variation, as observed in Fig. 6a. Its value decreases from 50 nF cm⁻² at 100 Hz, to 9 pF cm⁻² at 1 MHz. In the β-dispersion range (above 100 kHz), the membrane capacitance becomes almost constant, in the range of a few nF cm⁻², which matches closely with the generally accepted value for the single HeLa cell membrane capacitance obtained by cell trapping.¹⁶ As depicted in Fig. 6b, the cytoplasm resistance decreases with frequency in the low frequency range, while it becomes almost constant (35 kΩ cm²) in the β-dispersion range, which is also similar to the generally accepted value for single HeLa cell cytoplasm resistance obtained by the method of cell trapping.¹⁶ Table 1 compares the extracted cell membrane capacitance and cytoplasm resistance using the single cell trapping method by Wang and Jang³⁷, and our method of impedance measurement of a colony of suspended cells. Therefore, the experimental and theoretical analysis presented in this paper shows that the electrical properties of a single cell may be evaluated through the EBIS measurement of a colony of cells in a suspension, using the well-established Maxwell's mixture theory, and avoids the use of complicated single cell trapping for an impedance study. It is expected that the extracted parameters will provide more realistic and practical information about the cell because the measurements were conducted in conditions closely resembling ambient conditions. Although the present study has demonstrated the feasibility of this technique for one type of cell, the validity and reproducibility of this approach using a variety of cells requires confirmation.

Table 1 Comparison of extracted single cell parameters

Single cell parameters	By single cell trapping method (Wang)	By cell suspension method
Resistance of cytoplasm (Ω cm^2)	6.0 × 10^4	3.5 × 10^4
Membrane capacitance (F cm^−2)	2.5 × 10^−9	1.8 × 10^−9

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Conclusion

In this study, the electrical properties of a single HeLa cell were extracted through the bioimpedance analysis of a colony of its cells in suspension using the Maxwell's mixture theory. Using the bioimpedance spectroscopy data of the HeLa cells in PBS medium at different voltages, the complex permittivity of the mixture was calculated, which was further analyzed to extract the permittivity and conductivity of a single HeLa cell. Experimental observation revealed that the relative permittivity and conductivity of a single HeLa cell is a function of the applied potential and frequency. At low frequency, the conductivity of a single cell increases from 0.13 S m⁻¹ to 0.23 S m⁻¹ as the voltage is increased from 10 mV to 1 V, which is attributed to the opening of more membrane ion channels at higher electric fields, allowing a higher current to flow through the cell membrane and cytoplasm, leading to an increase in conductivity. At lower frequencies, the relative permittivity of the cell membrane decreases with the voltage. The relative permittivity of a single HeLa cell is almost constant over the entire frequency range at a lower operating voltage of 10–300 mV, whilst at higher applied voltage between 500 mV and 1 V, it reduces rapidly with increasing frequency. This indicates that under a higher electric field, the capacitance of the cell membrane may be fully charged at a lower operating frequency, whereas in higher frequency zones, it may not be able to be fully charged within one cycle. In the β-dispersion range, the membrane capacitance and cytoplasm resistance were calculated to be in the ranges of 1.8 nF cm⁻² and 35 kΩ cm², respectively, which matches closely with the values obtained from a cell trapping method found in the literature. Therefore, the present study provides an alternative technique to extract the electrical properties of a single cell from the bioimpedance spectroscopy data from a colony of cells in suspension, by avoiding the complexities of single cell capture and making impedance measurements in microchannels.

Acknowledgements

The authors would like to acknowledge the staff members of the MEMS & Microelectronics Laboratory of ATDC for their help. The authors would also like to thank the staff members of the Cell Culture Group of SMST for providing the cell line to conduct the experiment.

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