2. The mathematical model of electric cell-substrate impedance sensing (ECIS)

In order to monitor the environments effectively, systematically analyzing the relationship between the electric properties of measured subjects and output of ECIS sensors are needed. In this section, a model related to electric field distribution of ECIS sensing, which can be used in quantifying the ECIS sensor measurements, is created with a partial differential equation. The model of ECIS is established in cylindrical coordinates (r, θ, z) as shown in Figure 1 and simplified into polar coordinates (r, z) due to its axisymmetric property.

#### Figure 1.

necessary. ECIS sensing is one of the techniques among them. The ECIS is becoming an increasingly popular technique, which is able to analyze cell behaviors by measuring the impedance profile spectroscopy [1, 2]. The measured cell impedance provides information about cell morphology and electric properties, including intercellular junction conditions, numbers and densities, attachment, migration, proliferation, invasion, barrier function, membrane capacitance, and cytoplasm conductivity [1–6]. A common ECIS sensor is composed of a working electrode and a counter electrode. Some types of ECIS sensors have a third electrode, the reference electrode, which is used to provide the reference voltage for electrochemical measurements. The traditional ECIS sensors are fabricated on rigid substrate that limits the application in some of dynamically moving environments. Zhang et al. [7, 8] have fabricated the ECIS sensors on stretchable polymer. Such sensors are able to simulate in vitro the dynamic environment of organisms, such as pulsation,

bending, and stretching, which enables investigations on cell behavior that

The cells, attaching and spreading on the ECIS sensors, behave like an insulating medium after seeding. The insulating medium restricts the ion movement between the electrodes [13, 14]. As a result, the measured impedance increases gradually as more cells attach onto the surface. When the cells form a monolayer on the electrodes, the impedance becomes stabilized. The impedance may fluctuate slightly due to cell attachment migration, deformation, and detachment [9, 15–18]. Some chemical, biological, or physical stimuli on measured cells will influence the impedance response due to the changes in cell monolayer caused by cell-cell interactions, cell-substrate interactions, or changing cell electrical properties [2, 9]. Recently, the application of ECIS sensors has been extended to cell-based assays and

The ECIS sensors have different configurations including working electrode dimensions, counter electrode dimensions, and distance between electrodes. However, the relationship between the electrode configuration and detection sensitivity has not been further studied. A study on detection sensitivity of ECIS sensors is

Detection sensitivity is critical in the applications of ECIS sensors, which depends on sensor configuration, such as electrode dimension and the distance between the electrodes [19]. Wang et al. studied the detection sensitivity of ECIS sensors only with interdigital electrodes [20]. Several mathematical models have been introduced to analyze the relationship between measured cell impedance and cell morphology and behaviors [1, 2, 10, 21–28]. In those models, cell membrane and cell cytoplasm were assumed to be capacitors and resistors, respectively, and cell impedance was calculated as a combination of the capacitors and resistors [24–28]. However, the current may switch from one path to another or creating a hybrid path in reality, which was considered by some models [1, 2, 10, 14]. Nevertheless, these models assumed that the current flows radially between the substratum and the ventral surface of the cell, and the electric potential is constant inside the cell. However, the electric potential cannot be assumed to be constant inside the

cell if the current flows through the entirety of the cell. This assumption is

In this study, the influence of ECIS sensor configuration on detection sensitivity and the analysis of paths of current flow of ECIS have been carried out for improving the detection sensitivity, design, and application of ECIS sensors. The ECIS sensors are optimized for water toxicity testing. Several ECIS sensors are used to perform the toxicity testing in detecting the toxic effects from phenol, ammonia, nicotine, and aldicarb, and the impedance response successfully indicate the toxic effect. The gradient of measured impedance qualitatively is related to the concen-

undergoes mechanical stimuli in biological tissue [9–12].

Biosensors for Environmental Monitoring

meaningful for sensor design, fabrication, and applications.

toxicity study [18].

invalidated by Ohm's law.

tration of toxicants.

20

Illustration of cell impedance sensing on a working electrode. The electric potential at the coordinate (r, z) is V (r, z). ρ and ρ1 are the resistivity of the cell culture medium and cytoplasm respectively. Zm1, Zm2 and Zn are the specific impedance of the basal, apical cell membrane, and electrode-electrolyte interface respectively (in Ωm). h1 is the average distance between the ventral surface of cell and electrode-electrolyte interface. h2 is the average thickness of the cell. d is the average horizontal distance between adhesive cells. Vc is the electrical potential on the working electrode.

#### Biosensors for Environmental Monitoring

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

$$\frac{2\pi\mathbf{z}}{-\rho}\left(\frac{\partial V}{\partial r} + r\frac{\partial^2 V}{\partial r^2}\right) - \frac{2\pi r}{-\rho}\left(\frac{\partial V}{\partial \mathbf{z}}\right) = \mathbf{0} \tag{1}$$

$$
\rho \left( \frac{I\_1}{2\pi rz} e\_r + \frac{I\_2}{\pi r^2} e\_x \right) = E \tag{2}
$$

2.2 The calculated impedance of a cell monolayer

<sup>Z</sup> <sup>¼</sup> Zworking <sup>þ</sup> Zcounter <sup>þ</sup> Rs <sup>¼</sup> <sup>1</sup>

DOI: http://dx.doi.org/10.5772/intechopen.81178

Ωm<sup>2</sup>

the cell [19].

tivity of ECIS sensors.

culture medium.

23

electrode Zcounter, and cell culture medium Rs, as shown in Eq. (7).

S1 þ 1 S2 Zn <sup>þ</sup>

3. The design of ECIS sensors for environmental monitoring

3.1 The design guideline of electrode dimensions of ECIS sensors

to guarantee sufficient cell-to-cell contact area.

The design of ECIS sensors includes the dimensions of working electrodes and counter electrodes, and the distance between them is critical in environmental monitoring because those designing parameters will influence the detection sensi-

The radius of working electrode (Ri) and the distance between the edges of the sensing electrodes (dio) can be optimized by using the mathematical model with the parameters related to cell morphology and electric properties and surrounding

3.1.1 The relationship between the radius of working electrode (Ri) and cell impedance

During impedance measurements, ions move through the cell monolayer between the working and counter electrodes which follow many paths. The counter electrodes must have adequate sensing area in order to provide adequate circuit connection. The larger Ri working electrode provides more current paths, which decreases the corresponding impedance. Higher impedance values can improve the data quality of the measured impedance by increasing signal-to-noise ratio, which is useful particularly for sensing small changes in cell behavior. However, the working electrode should not be too small in order to measure adequate number of cells and

In this study, the ECIS sensors with Ri from 100 to 400 μm were fabricated to analyze the relationship between Ri and measured cell impedance. Figure 2 illustrates the cell morphology on those sensors. The simulated cell impedance by using Ri within the same range was also obtained from the mathematical model. The experimental and simulated impedance of cell were shown in Figure 3. The

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

The Modeling, Design, Fabrication, and Application of Biosensor Based on Electric Cell…

where Zn is the specific impedance of the electrode-medium interface (unit

), 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

S Zsingle cell <sup>þ</sup> Zcell\_sub n 

þ Rs (7)

$$I\_1 + I\_2 = I\_1 + dI\_1 + I\_2 - dI\_2 \tag{3}$$

$$
\frac{\partial V}{\partial r}\mathbf{e}\_r + \frac{\partial V}{\partial \mathbf{z}}\mathbf{e}\_\mathbf{z} = -E \tag{4}
$$

$$V(r,z) = AI\_0(2cr)e^{2r^2x^2} + D \tag{5}$$

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 (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 of solution V rð Þ ; z .

#### 2.1 The calculated impedance of a single cell

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

$$Z\_{single\ cell} = \frac{V(r\_c, h\_1) - V(r\_c, h\_1 + h\_2)}{I\_2 + I\_j}$$

$$= \frac{\left(\rho\_1 h\_2 \sqrt{\sigma^2 + \left(2\pi f e e e\_0\right)^2} + 2t\right) (2I\_0 + 2cr\_lc\_1)}{2\pi \left[r\_c^2 I\_0 \sqrt{\sigma^2 + \left(2\pi f e e\_0\right)^2} + \frac{r\_c d}{\rho a h\_2} \left(\rho\_1 h\_2 \sqrt{\sigma^2 + \left(2\pi f e e e\_0\right)^2} + 2t\right) (2I\_0 + 2cr\_lc\_1)\right]}\tag{6}$$

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

The Modeling, Design, Fabrication, and Application of Biosensor Based on Electric Cell… DOI: http://dx.doi.org/10.5772/intechopen.81178
