2.2 The calculated impedance of a cell monolayer

The governing equation of electric field distribution of ECIS sensing (as shown in Eq. (1)) can be obtained from the differential form of Ohm's law between electric potential and current (as shown in Eq. (2)), Kirchhoff's circuit law at a point of interest (r, z) (as shown in Eq. (3)), and the gradient of electric potential (as shown in Eq. (4)). The solution of the governing equation is shown in Eq. (5), which is the same as the solution in Giaever et al. ECIS model when the variable z is held as constant [1, 2, 23, 29]. The detailed information about the mode can be referred to in [19]. These three coefficients A, D, and c are calculated as A = �2.3, D = 3.3, and

> � <sup>2</sup>π<sup>r</sup> �ρ

∂V ∂z � �

2c2z<sup>2</sup>

I<sup>1</sup> þ I<sup>2</sup> ¼ I<sup>1</sup> þ dI<sup>1</sup> þ I<sup>2</sup> � dI<sup>2</sup> (3)

¼ 0 (1)

¼ E (2)

þ D (5)

ez ¼ �E (4)

c = 4749.83 by using the parameters listed in [19, 30–36].

∂V ∂r þ r ∂2 V ∂r2

ρ I1 2πrz

> ∂V ∂r er þ ∂V ∂z

� �

er þ I2 <sup>π</sup>r<sup>2</sup> ez

V rð Þ¼ ; z AI0ð Þ 2cr e

(r,z); V is the electric potential at the point (r, z); and dI1 and dI2 are the

infinitesimally small currents of I1 and I2. dI1 and dI2 have the same sign; I0ð Þ 2cr is the modified Bessel function of the first kind; A, D, and c are the coefficients

In this model, the impedance of a single cell (Zsingle cell) is able to be calculated by dividing the electric potential difference between the apical V(rc, h1) and ventral surfaces of a single cell V(rc, h1 + h2)0 by the total current flowing through and

Zsingle cell <sup>¼</sup> V rc ð Þ� ; <sup>h</sup><sup>1</sup> V rc ð Þ ; <sup>h</sup><sup>1</sup> <sup>þ</sup> <sup>h</sup><sup>2</sup>

ρ1h<sup>2</sup>

where I2 is the current flowing through a single cell, Ij is the current flowing through the intercellular junction gap; h1 is the average distance between the ventral surface of cell and electrode-electrolyte interface; h2 is the average thickness of the cell layer; rc is the average radius of a single cell; f is the measurement frequency; ρ<sup>1</sup> is the resistivity of cell cytoplasm; ε is the relative permittivity of the cell membrane; <sup>ε</sup><sup>0</sup> is the vacuum permittivity, which is 8.85 � <sup>10</sup>�<sup>12</sup> F/m; and <sup>t</sup> and

2

q

� �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ<sup>2</sup> þ ð Þ 2πf εε<sup>0</sup>

� �

σ are the thickness and conductivity of the cell membrane, respectively.

I<sup>2</sup> þ Ij

þ 2t

ð Þ 2I<sup>0</sup> þ 2crcI<sup>1</sup>

2

þ 2t

ð Þ 2I<sup>0</sup> þ 2crcI<sup>1</sup>

(6)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ<sup>2</sup> þ ð Þ 2πf εε<sup>0</sup>

� �

where ρ is the resistivity of the cell culture medium (electrolyte); I1 and I2 are the current flowing through the point (r,z) in r and z directions, respectively; er and ez are the unit vectors of the r and z directions; E is the electric field at any point

� �

2πz �ρ

Biosensors for Environmental Monitoring

2.1 The calculated impedance of a single cell

ρ1h<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ<sup>2</sup> þ ð Þ 2πf εε<sup>0</sup>

q

2

þ rcd ραh<sup>2</sup>

around the cell, as shown in Eq. (6).

of solution V rð Þ ; z .

¼

22

2π rc <sup>2</sup>I<sup>0</sup>

q

The impedance of a cell monolayer (Z) is calculated as the sum of the impedance on current path, including the impedance from working electrode Zworking, counter electrode Zcounter, and cell culture medium Rs, as shown in Eq. (7).

$$Z = Z\_{\text{working}} + Z\_{\text{counter}} + R\_\text{s} = \left(\frac{\mathbf{1}}{\mathbf{S}\_1} + \frac{\mathbf{1}}{\mathbf{S}\_2}\right) \left[Z\_n + \frac{\mathbf{S}\left(Z\_{\text{single cell}} + Z\_{\text{cell,sub}}\right)}{n}\right] + R\_\text{s} \tag{7}$$

where Zn is the specific impedance of the electrode-medium interface (unit Ωm<sup>2</sup> ), which can be calculated according to the parameters referred to [19, 37–41]; S1 and S2 are the surface areas of the working and counter electrodes, respectively; S is the total surface area of the ECIS sensor, which contains the working electrode, counter electrode, and nonelectrode area; n is the number of cells seeded on the ECIS sensor; Rs is the impedance of the culture medium, which can be calculated according to the parameters referred to [19, 42–47]; Zcell-sub is the impedance of the culture medium between the electrode-electrolyte interface and ventral surface of cell, which can be calculated by dividing the electric potential difference between the edge and center of a single cell by the total current flowing through and around the cell [19].
