Abstract:
Systems, apparatus and methods for designing and operating a cell-electrode impedance sensor to detect chemical and biological samples, including biological cells. In one implementation, a method of designing a cell-electrode impedance sensor includes determining a cell free cell-electrode impedance and a cell covered cell-electrode impedance based on a design model for the cell-electrode impedance sensor, wherein the design model is based on one or more factors, the factors including properties and elements of a cell-electrode impedance measurement system, using the cell free cell-electrode impedance and the cell covered cell-electrode impedance to obtain a sensor sensitivity of the cell-electrode impedance measurement system, and choosing one or more design parameters of the cell-electrode impedance sensor in the cell-electrode impedance measurement system to maximize the sensor sensitivity.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
       [0001]    This patent document is a divisional application of and claims priority to U.S. patent application Ser. No. 12/532,992, filed on Sep. 24, 2009, which is a U.S. National Stage under 35 U.S.C. §371 of International Patent Application No. PCT/CN2007/002260, filed on Jul. 25, 2007, which claims benefit of priority of Chinese Patent Application No. 200710098717.4, filed on Apr. 25, 2007. The entire contents of the before-mentioned patent applications are incorporated by reference as part of the disclosure of this application. 
     
    
     TECHNICAL FIELD 
       [0002]    The present disclosure relates to detecting chemical and biological samples, including biological cells. 
       BACKGROUND 
       [0003]    Living cells are surrounded by an outer cell membrane that restricts the movement of ions and solutes between the cell interior and the exterior of the cell. Changes in the electrical activity of cell membranes can reflect the biophysical state of the cells and further reflect underlying physiological and biochemical processes occurring within the cell, as well as biophysical changes occurring at the surface of the cell or in the cell membrane. Electrical devices and apparatus for the measurement of electrical impedance can be used to detect the state of the electrophysiological activity of living cells and their cell membranes. 
         [0004]    Cell-electrode impedance sensing and cell substrate electrical impedance sensing are two related electrical measurements that are based on the application of a small alternating current (AC) electrical signal to probe the value of the impedance of sensor electrodes immersed in a conductive medium. Living cells can attach and grow on the surface of the sensor electrodes and can alter the electric field between electrodes causing a change in the electrical impedance that can be detected by the sensor electrodes. The measurement of impedance by the sensor can reflect the electrophysiological state of the cell and can allow the biophysical properties of the cell to be monitored. 
       SUMMARY 
       [0005]    In one example, a method to design a cell-electrode impedance measurement system is described where a range of dimensions of a cell-electrode impedance sensor in the system can be determined based on a model for cell-electrode impedance to maximize sensor sensitivity. In addition, a frequency range to optimize the parameters of the sensor (e.g., sensitivity, dimensions, etc.) can also be determined. 
         [0006]    In one aspect, a method of designing a cell-electrode impedance sensor is described. The method includes determining a cell free cell-electrode impedance and a cell covered cell-electrode impedance based on a design model for the cell-electrode impedance sensor, wherein the design model is based on one or more factors, the factors including properties and elements of a cell-electrode impedance measurement system, using the cell free cell-electrode impedance and the cell covered cell-electrode impedance to obtain a sensor sensitivity of the cell-electrode impedance measurement system, and choosing one or more design parameters of the cell-electrode impedance sensor in the cell-electrode impedance measurement system to maximize the sensor sensitivity. 
         [0007]    This and other aspects can include one or more of the following features. The cell-electrode impedance sensor can include an interdigitated electrode array comprising a plurality of branch elements. The design parameters can include one or more of a width of the branch elements, a length of the branch elements, a surface area of the branch elements, and a number of the branch elements. The method can further include choosing an effective width lesser than the width chosen based on the design model, to account for a non-uniform electric field distribution over each branch element. The elements of the cell-electrode impedance measurement system can include one or more of a cell culture vessel, a cell culture substrate, and circuitry. The properties of the cell-electrode impedance measurement system can include one or more of a property of a cell culture medium and a property of the cell. Determining the cell free cell-electrode impedance and the cell covered cell-electrode impedance can include identifying values for the properties of the cell-electrode impedance measurement system, substituting the values in a first equation associated with the cell free cell-electrode impedance to determine the cell free cell-electrode impedance, and substituting the values in a second equation associated with the cell covered cell-electrode impedance to determine the cell covered cell-electrode impedance. The cell free cell-electrode impedance can be represented by a Helmholtz interfacial capacitance and a spreading resistance in the design model. The spreading resistance can be associated with a proportionality coefficient. The cell covered cell-electrode impedance can be represented by factors including one or more of a resistance between a plurality of cells, a capacitance between the plurality of cells, a resistance between a gap between a surface of a cell and a cell culture vessel substrate, and a capacitance between the surface of the cell and the cell culture vessel substrate. The resistance and capacitance between the plurality of cells, and the resistance and capacitance between the gap between the surface of the cell and the cell culture vessel substrate can each be associated with a proportionality coefficient. The resistance and capacitance between the plurality of cells, and the resistance and capacitance between the gap between the surface of the cell and the cell culture vessel substrate can depend on an extent of attachment between a surface of the cells and a surface of the sensor. The surface of the cells can be completely attached to the surface of the sensor. The surface of the cells can be partially attached to the surface of the sensor. Obtaining the sensor sensitivity of the system can further include determining the cell density of the system. The method can further include determining a frequency to deliver an alternating current (AC) signal to the sensor, wherein delivering the AC signal at the determined frequency can further improve sensor sensitivity. Determining the frequency can include determining a derivative of an equation for sensor sensitivity with respect to frequency and solving for the frequency by equating the derivative to zero. 
         [0008]    In another aspect, an apparatus for measuring cell-electrode impedance in biological cells is described. The apparatus includes a cell-culture vessel that includes electrodes and circuitry electrically coupled to the electrodes. The electrodes are structured in an interdigitated array of electrode branch elements and the cell-culture vessel is operable to hold a cell-culture medium having biological cells attachable to the electrodes. The circuitry includes a stimulator to apply an alternating current (AC) signal to the electrodes and configured to control a frequency of the AC signal to be between 10 kHz and 40 kHz. 
         [0009]    This and other aspects can include one or more of the following features. The electrodes can be located in the cell-culture vessel at a location to be immersed in the cell culture medium in the cell-culture vessel when the cell culture medium is present. The electrodes arranged in an interdigitated array of electrode branch elements include first electrode branch elements connected to a first electrical terminal of the circuitry and second electrode branch elements connected to a second electrical terminal of the circuitry, wherein the first and second electrode branch elements are interleaved and the circuitry applies the AC signal at the first and second electrical terminals. Each branch element can have a width. Adjacent branch elements can be separated by a distance. The branch element can have a width from 10 μm to 100 μm. A surface of the cells can be attached to a surface of each branch element. The cell-electrode impedance can depend on an extent of the attachment. A surface of the cells can be completely attached to the surface of the electrodes. A surface of the cells can be partially attached to the surface of the electrodes. A width of the branch elements can be altered to account for the extent of the attachment. The altered width can equal the sum of a width of the branch element and twice a radius of the biological cells. 
         [0010]    In another aspect, a method for measuring cell-electrode impedance is described. The method can include immersing an interdigitated electrode array having electrode branch elements with a width from 10 μm to 100 μm in a cell culture medium in a cell-culture vessel to allow cells in the cell culture medium to attach to the electrode branch elements, and applying an alternating current (AC) signal to the interdigitated electrode array in a frequency range from 10 kHz to 40 kHz to measure a cell-electrode impedance for detecting cells. 
         [0011]    This and other aspects can include one or more of the following features. The method can further include predetermining a number of branch elements, the branch element width, and a space between branch elements using a design model. The method can further include collecting an output from the electrode using a data acquisition module. The method can further include processing the collected output using a computer system. The method can further include providing input to the electrode using the computer system. The frequency range can be chosen to maximize a sensitivity of the electrode when employed in a cell-electrode impedance measurement system. 
         [0012]    The system and techniques described can provide one or more advantages. Firstly, the cell-electrode impedance measurement system can be designed to maximize sensitivity of the system and to maximize the number of cells that can be studied. Further, a frequency of input AC signals can be determined which can further maximize sensitivity of the system. In addition, cell-electrode impedance sensors can be designed based on a design model. 
         [0013]    The details of one or more implementations are set forth in the accompanying drawings and the description below. Other features and advantages will be apparent from the description and drawings, and from the claims. 
     
    
     
       DESCRIPTION OF DRAWINGS 
         [0014]      FIG. 1  is an example of a schematic of a cell-electrode impedance measurement system. 
           [0015]      FIG. 2  is an example of an equivalent circuit for cell free cell-electrode impedance. 
           [0016]      FIG. 3A  is an example of an equivalent circuit for cell covered cell-electrode impedance. 
           [0017]      FIG. 3B  is an example of an equivalent circuit for cell covered cell-electrode impedance when in low frequency range. 
           [0018]      FIG. 3C  is an example of an equivalent circuit for cell covered cell-electrode impedance when in high frequency range. 
           [0019]      FIG. 4  is an example of a cell-electrode impedance sensor. 
           [0020]      FIG. 5A  is an example of cell-electrode impedance sensor and an equivalent circuit for cell-electrode impedance. 
           [0021]      FIG. 5B  is an example of cell-electrode impedance sensor and an equivalent circuit for cell-electrode impedance. 
           [0022]      FIG. 6  is an example of equivalent circuits of sensor electrodes with different lengths. 
           [0023]      FIG. 7  is a plot of sensor sensitivity in response to changing frequency. 
           [0024]      FIG. 8  is a plot of sensor sensitivity in response to changing frequency. 
           [0025]      FIG. 9  is a plot of sensor sensitivity in response to changing frequency. 
           [0026]      FIG. 10  is an example of a schematic of cells attached on an electrode surface. 
           [0027]      FIG. 11  is a plot validating the design model for cell-electrode impedance. 
           [0028]      FIG. 12  is a plot of normalized impedance measured over time in response to varied frequency. 
           [0029]      FIG. 13  is a plot of sensor sensitivity in response to changing frequency. 
           [0030]      FIG. 14  is a flow chart of an example of designing a cell-electrode impedance measurement system. 
           [0031]      FIG. 15  is a flow chart of an example of maximizing sensitivity of a cell-electrode impedance measurement system. 
       
    
    
       [0032]    Like reference symbols in the various drawings indicate like elements. 
       DETAILED DESCRIPTION 
       [0033]      FIG. 1  depicts an example of a cell-electrode impedance measurement system  100 . The system  100  can include a cell culture vessel  105  and circuitry  110 . The cell culture vessel  105  can include elements to culture biological cells as well as elements to detect electrical properties of the biological cells, such as electrodes. The circuitry  110  can include elements to measure the electrical output produced by the physiological changes undergone by the cells in the cell culture vessel  105 . The circuitry  110  can include an AC stimulator  115 , a detection resistor  120 , a data acquisition module  125 , and a digital processor  130  (e.g., a computer). The AC stimulator can provide AC test signals at different frequencies for cell-electrode impedance sensing. The data acquisition module  125  can be used to collect output signals from the system  100 . In some implementations, the output can include the voltage between two electrodes in the system  100 . In other implementations, the output can include voltages between one or more pairs of electrodes in the system. In addition, the output can include current measurements, resistance measurements, etc. The rate of collection, for example, the number of voltage measurements per second, can be altered based on user input and the data acquisition module  125 . The digital processor  130  can be used to store the data collected by the data acquisition module  125 , process the data and perform additional operations related to the processed data. For example, the digital processor  130  may be configured to prepare spread sheets, perform statistical operations such as averaging, perform calculations such as calculating impedance from measured voltages, currents, and resistances, etc., as well as transferring the data by methods including storing on a portable storage device (e.g., compact disc, USB flash drive, etc.), transmitting over a network (e.g., wired, wireless, the internet), etc. The digital processor  130  can also be configured to provide input signals to the system  100  to alter experimental conditions, e.g., frequency of AC signals from the AC stimulator  115 . In addition, the digital processor  130  can be configured to receive input from a user. Such input can include constants for use in data processing, input to alter experimental conditions, etc. 
         [0034]    The system  100  can include a cell culture vessel  105 . The cell culture vessel  105  can include a vessel substrate  135 , a cell culture medium  140 . Electrodes  145  can be positioned in the cell culture vessel  105 . The electrodes  145  can be immersed in the cell culture medium  140 . In some implementations, the electrodes  145  can be arranged as an interdigitated array and positioned such that a large surface area of the electrodes are available upon which the cells in the cell culture vessel can rest. The cell culture vessel  105  can contain the biological cells  105  under study. The cell culture vessel  105  can be manufactured from a bio-compatible insulating material, e.g., plastic, glass, polydimethylsiloxane (PDMS), etc. The vessel substrate  135  can be manufactured from the same material as the cell culture vessel  105 . The cell culture medium  140  can include proteins, sugar, salt, amino acids, antibiotics, and other nutrition that cells need. Cells  150  growing in the cell culture vessel  105  can form a monolayer on the surface of the cell culture vessel  105  as well as the surface of the electrodes  145 . Examples of cells that can be studied include HeLa, HepG2, NIN-3T3, VERO, etc. Properties of the cells that can be studied include cell proliferation, cell apoptosis, cell differentiation, cell migration, cell micromotion, cell attachment, etc. The cells  150  can be of several shapes (e.g., spherical, hemispherical, etc.). A cell  150  can be attached either completely or partially to the surface of the electrode  145 . When a cell  150  is partially attached, only a portion of the cell  150  can be in contact with the surface of the electrode  145 . In addition, a cell can be attached to the surface of the cell culture vessel  105  and can have no contact with the surface of the electrode  145 . 
         [0035]    In some implementations, the AC stimulator  115  can be designed and operated to supply an input AC voltage at a pre-determined frequency. The frequency can be altered based on user input. The cell culture medium  140  and the electrode  145  effectuate a part of a circuit through which a probe current  155  is flowed. The presence of cells on the electrode surface  145  can cause impedance which can affect the probe current  155 . The probe current  155  can be affected by cells that are attached completely or partially attached to the surface of the electrodes  145 . The voltage between two electrode surfaces  145  can be collected by the data acquisition module  125  and stored in the digital processor  130 . Alternatively, or in addition, the probe current can be collected and stored in the digital processor  130 . The digital processor  130  can be configured to calculate the impedance due to the cells  150  for every instance that the voltage and/or the probe current between the electrode surfaces  145  were recorded. In this manner, the cell-impedance of the system  100  can be determined. 
         [0036]    The impedance of the system described in  FIG. 1  can be attributed to at least two sources, namely the cells  150  in the cell culture vessel  105  and the cell culture vessel  105  itself. In the absence of cells  150 , the system  100  is a cell-free cell-electrode impedance sensor.  FIG. 2  depicts an example of a schematic of an equivalent circuit for cell-free cell-electrode impedance for the system  100  in  FIG. 1 . The equivalent circuit depicts representations of the Helmholtz double layer interfacial capacitance, C I  ( 205 ) and the spreading resistance of the cell culture media, R S  ( 210 ). 
         [0037]      FIG. 3A  depicts an example of an equivalent circuit for cell covered cell-electrode impedance for system  100 . When cells attach and grow on the electrode surface  145 , the impedance of the cells  150  to the flow of probe current  155  can be modeled by the equivalent circuit shown in  FIG. 3A . The impedance can be a function of the electrical resistance between growing cells, R cell  ( 310 ). In addition, the cells which can be attached on the electrode surface  145  can be treated as a capacitance, C cell  ( 320 ), which can be a biophysical property of the insulating cell membrane. A gap can be present between a cell  150  growing on the substrate surface and the underside of the attached cell  150 . The gap between the underside of the cell  150  and the surface of the substrate can be composed of the cell culture medium  140  and can cause a resistance, R gap  ( 305 ) and a capacitance, C gap ( 315 ). The values of R cell  ( 310 ), C cell /( 320 ), R gap  ( 305 ), and C gap  ( 315 ) for a cell  150  can depend on factors including one or more of the extent of attachment of the cell  150  to the substrate surface, the biological nature of the cell, the cell culture medium  140 , etc. 
         [0038]    In some implementations, in measuring cell-electrode impedance, the sensitivity of the system  100  can be defined as the smallest impedance, caused by presence of cells in the cell culture vessel  105 , that can be measured. The total impedance measured by the system  100  can be a function of the impedance due to the cells  150 , the impedance due to the cell culture medium  140 , and the density of cells, Q cell . In some implementations, the sensitivity of the system  100  can be a function of the cell free cell-electrode impedance (Z cell free-total ), the cell covered cell-electrode impedance (Z cell-covered-total ), and the cell density, Q cell , and can be depicted by equation (1). 
         [0000]    
       
         
           
             
               
                 
                   
                     Sensitivity 
                      
                     
                       ( 
                       f 
                       ) 
                     
                   
                   = 
                   
                     
                       
                          
                         
                           
                             Z 
                             
                               cell 
                                
                               
                                 - 
                               
                                
                               covered 
                                
                               
                                 - 
                               
                                
                               total 
                             
                           
                            
                           
                             ( 
                             f 
                             ) 
                           
                         
                          
                       
                       - 
                       
                          
                         
                           
                             Z 
                             
                               cell 
                                
                               
                                 - 
                               
                                
                               free 
                                
                               
                                 - 
                               
                                
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                            
                           
                             ( 
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                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The impedance of the system  100  can be affected by the frequency, f, of the signal from the AC stimulator  115 . 
         [0039]      FIG. 3B  depicts an equivalent circuit model for cell impedance when the applied AC signal to the electrodes of the sensor is in a selected low frequency range. For example, this selected low frequency can range between [(1/10)×(π·R cell ·C cell )] and [(1/10)×(π·R gap ·C gap )]. In the low frequency range, the resistance of the cells, R cell  ( 310 ), and the resistance of the gaps, R gap  ( 305 ), dominate over the capacitance of the cells, C cell  ( 320 ), and the capacitance of the gaps, C gap  ( 315 ). 
         [0040]      FIG. 3C  depicts an equivalent circuit model for cell impedance when the applied AC signal to the electrodes of the sensor is in a selected high frequency range. For example, this high frequency can range between [(5/2)×(π·R cell ·C cell ] and [(5/2)×(π·R gap ·C gap )]. In the high frequency range, the capacitance of the cells, C cell  ( 320 ), and the capacitance of the gaps, C gap  ( 315 ), dominate over the resistance of the cells, R cell  ( 310 ), and the resistance of the gaps, R gap  ( 305 ). The impedance of the system  100  can also be affected by the geometry of the cell-electrode impedance sensor. 
         [0041]      FIG. 4  depicts an example of a schematic of a cell-electrode impedance sensor  400 . The sensor  400  can include two electrodes  403  and  405 . The two electrodes  403  and  405  can be configured to form an interdigitated array  410  formed of interleaved branch elements  415  where each branch element  415  is connected to one sensor electrode  403  or  405  and the interleaved branch elements  415  include one group of branch elements  415  connected to the sensor electrode  403  and another group of branch elements  415  connected to the other sensor electrode  405 . Referring to both  FIGS. 1 and 4 , the first group of branch elements is connected to a first electrical terminal of the circuitry  110  and the second group of branch elements is connected to a second electrical terminal of the circuitry  110 . The branch elements of the first and the second group can be interleaved and the circuitry  110  applies the AC signal at the first and second electrical terminals. In some implementations, the branch elements  415  of each electrode can be arranged in an alternating manner. Alternatively, the branch elements  415  can be arranged in any order, for example, sequentially, alternatively, staggered, etc. In some implementations, the branch elements  415  can also be positioned on both sides of each electrode. 
         [0042]      FIG. 4  also depicts a magnified view  420  of a portion of the sensor  400 . The magnified view  420  illustrates branch elements  415  of the sensor. In the example illustrated in  FIG. 4 , the branch elements  415  of each electrode  405  are arranged in an alternating manner. In some implementations, the branch elements  415  can have a width, W ( 425 ). Two adjacent branch elements  15  can be separated by a distance, D ( 430 ). In some implementations, the branch elements  415  of both electrodes  405  can have a uniform width, W ( 425 ). In other implementations, the width, W ( 425 ), of the branch elements  415  of the same electrode  405  can be uniform. The width, W ( 425 ), of branch elements  415  of different electrodes can be non-uniform. In other implementations, the width, W ( 425 ), of each branch element  415  in the same electrode  405  can be non-uniform. In some implementations, the distance, D ( 430 ), between two branch elements  415  can be uniform. In other implementations, the branch elements  415  can be staggered with a non-uniform distance, D ( 430 ), separating the branch elements  415 . 
         [0043]    The sensitivity of a sensor can depend on factors including the design parameters of the sensor  400 . The design parameters of the sensor  400  can include the material used to fabricate the sensor  400 , the width, W ( 425 ), of each branch element  415 , the distance between two branch elements, D ( 430 ), the length of the branch element  415 , etc. The total area of the electrode can be represented by A total . The total length of the sensor  400  can be represented by L total . In some implementations, the length of each branch element  415  can be uniform. In such implementations, the total length of the electrode, L total , can be represented by the length of each branch element, L, and the number of branch elements, N. In some implementations, the branch element  415  can have a width between 10 μm and 100 μm, or between 10 μm and 80 μm, such as one of 20 μm, 40 μm, and 60 μm for a range of sensing applications. The distance between two branch elements can be 20 μm in some sensor designs. The sensor  400  can be fabricated using various materials such as glass, silicon, certain plastics, etc. by methods including standard lift-off fabrication method. One method to determine optimal design parameters of a sensor is to fabricate sensors of different design parameters and employ each sensor in a cell-electrode impedance measurement system. 
         [0044]    In some implementations, a design model based on fundamental principles can be developed to predict the cell free cell-electrode impedance and cell covered cell-electrode impedance of a cell-electrode impedance system  100 . Based on the predicted impedance and given the conditions of the cell impedance measurement system  100 , the design parameters of the sensor, such as branch element  415  dimensions, can be determined to optimize the system  100  to increase sensitivity and increase number of cells detected. In addition, the design model can be employed to determine an optimal frequency to maximize sensitivity of the system  100 . 
         [0045]    The total impedance measured by the system  100  can be a function of the impedance due to the cells and the impedance due to the electrode. The impedance due to the electrode, represented by Z cell-free-total , can be determined using equation (2). 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
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         [0046]    When cells attach and grow on the surface of the sensor  400 , the impedance due to the cells, represented by Z cell-covered-total , can be determined using equation (3). 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
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                                                 π 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   fC 
                                                   gap 
                                                 
                                               
                                               ) 
                                             
                                             
                                               - 
                                               1 
                                             
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           N 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               
                                 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     
                                       R 
                                       S 
                                     
                                   
                                   + 
                                   
                                     2 
                                      
                                     
                                       
                                         ( 
                                         
                                           j 
                                            
                                           
                                               
                                           
                                            
                                           2 
                                            
                                           
                                               
                                           
                                            
                                           π 
                                            
                                           
                                               
                                           
                                            
                                           
                                             fC 
                                             I 
                                           
                                         
                                         ) 
                                       
                                       
                                         - 
                                         1 
                                       
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   2 
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           
                                             R 
                                             cell 
                                           
                                           × 
                                           
                                             
                                               ( 
                                               
                                                 j 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 2 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 π 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   fC 
                                                   cell 
                                                 
                                               
                                               ) 
                                             
                                             
                                               - 
                                               1 
                                             
                                           
                                         
                                         
                                           
                                             R 
                                             cell 
                                           
                                           + 
                                           
                                             
                                               ( 
                                               
                                                 j 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 2 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 π 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   fC 
                                                   cell 
                                                 
                                               
                                               ) 
                                             
                                             
                                               - 
                                               1 
                                             
                                           
                                         
                                       
                                       + 
                                       
                                         
                                           
                                             R 
                                             gap 
                                           
                                           × 
                                           
                                             
                                               ( 
                                               
                                                 j 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 2 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 π 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   fC 
                                                   gap 
                                                 
                                               
                                               ) 
                                             
                                             
                                               - 
                                               1 
                                             
                                           
                                         
                                         
                                           
                                             R 
                                             gapl 
                                           
                                           + 
                                           
                                             
                                               ( 
                                               
                                                 j 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 2 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 π 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   fC 
                                                   gap 
                                                 
                                               
                                               ) 
                                             
                                             
                                               - 
                                               1 
                                             
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 L 
                                 total 
                               
                               / 
                               2 
                             
                              
                             
                                 
                             
                              
                             L 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0047]    In the equivalent circuit equations (1) and (2), the interfacial capacitance, C I  ( 205 ) can be a function of the permittivity of free space, ∈ 0 , the effective dielectric constant of the double layer separating the ionic charges and the electrode, ∈ ρ , area of a single electrode branch element  415 , A, and the thickness of the double layer, d dl . The interfacial capacitance, C I  ( 205 ) can be determined using equation (4). 
         [0000]    
       
         
           
             
               
                 
                   
                     C 
                     I 
                   
                   = 
                   
                     
                       
                         ɛ 
                         0 
                       
                        
                       
                         ɛ 
                         ρ 
                       
                        
                       A 
                     
                     
                       d 
                       dl 
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0048]    In the equivalent circuit equations (1) and (2), the spreading resistance, R S  ( 210 ) can be a function of the resistivity of the cell culture medium  140 , ρ, the area of a single electrode branch element  415 , A, and a proportionality coefficient, K. The spreading resistance, R S  ( 210 ) can be determined using equation (5) below: 
         [0000]    
       
         
           
             
               
                 
                   
                     R 
                     S 
                   
                   = 
                   
                     
                       ρ 
                        
                       
                           
                       
                        
                       K 
                     
                     
                       π 
                        
                       
                           
                       
                        
                       A 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0049]    In the equivalent circuit equation (2), the resistance of the gaps between cells, R cell    310 , and the resistance of the cells to the gap between the cell and the substrate, R gap    305 , can be inversely proportional to the electrode area, A. The capacitance of the cells that are attached to the electrode surfaces, C cell    320 , and the capacitance of the gap between the cell and the substrate, C gap    315 , can be directly proportional to the electrode area, A. The resistances and the capacitances can be represented by equation (6) below: 
         [0000]        R   cell   =K   1   A   −1   ; R   gap   =K   z   A   −1   ; C   cell   =K   3   A; C   gap   =K   4   A   (6)
 
         [0050]    The proportionality coefficients, K, K 1 , K 2 , K 3 , and K 4  can be determined empirically. In one example, experimental data can be fitted to the design model using the least square fitting or other fitting algorithm. In some implementations, the values of the proportionality coefficients can be 85 ohm·mm 2  and 48 ohm·mm 2  for K 1  and K 2 , respectively, and 60 nF/mm 2  and 6 nF/mm 2  for K 3  and K 4 , respectively. 
         [0051]    Substituting equation (6) in equation (2) and equation (3), the impedance due to the electrode, Z cell-free-total , and the impedance due to the cells, Z cell-covered-total , can be represented by equation (7) and equation (8), respectively: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             Z 
                             
                               cell 
                                
                               
                                 - 
                               
                                
                               free 
                                
                               
                                 - 
                               
                                
                               total 
                             
                           
                            
                           
                             ( 
                             f 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             2 
                              
                             
                               ( 
                               
                                 
                                   
                                     ρ 
                                      
                                     
                                         
                                     
                                      
                                     K 
                                   
                                   π 
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       j 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       f 
                                        
                                       
                                         
                                           
                                             ɛ 
                                             0 
                                           
                                            
                                           
                                             ɛ 
                                             ρ 
                                           
                                         
                                         
                                           d 
                                           dl 
                                         
                                       
                                     
                                     ) 
                                   
                                   
                                     - 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                           
                             A 
                             . 
                             N 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             2 
                              
                             
                               ( 
                               
                                 
                                   
                                     ρ 
                                      
                                     
                                         
                                     
                                      
                                     K 
                                   
                                   π 
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       j 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       f 
                                        
                                       
                                         
                                           
                                             ɛ 
                                             0 
                                           
                                            
                                           
                                             ɛ 
                                             ρ 
                                           
                                         
                                         
                                           d 
                                           dl 
                                         
                                       
                                     
                                     ) 
                                   
                                   
                                     - 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                           
                             
                               
                                 A 
                                 . 
                                 
                                   L 
                                   total 
                                 
                               
                               / 
                               2 
                             
                              
                             
                                 
                             
                              
                             L 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             Z 
                             
                               cell 
                                
                               
                                 - 
                               
                                
                               covered 
                                
                               
                                 - 
                               
                                
                               total 
                             
                           
                            
                           
                             ( 
                             f 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               
                                 
                                   
                                     2 
                                      
                                     
                                       ( 
                                       
                                         
                                           
                                             ρ 
                                              
                                             
                                                 
                                             
                                              
                                             K 
                                           
                                           π 
                                         
                                         + 
                                         
                                           
                                             ( 
                                             
                                               j 
                                                
                                               
                                                   
                                               
                                                
                                               π 
                                                
                                               
                                                   
                                               
                                                
                                               f 
                                                
                                               
                                                 
                                                   
                                                     ɛ 
                                                     0 
                                                   
                                                    
                                                   
                                                     ɛ 
                                                     ρ 
                                                   
                                                 
                                                 
                                                   d 
                                                   dl 
                                                 
                                               
                                             
                                             ) 
                                           
                                           
                                             - 
                                             1 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   2 
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           
                                             K 
                                             1 
                                           
                                           × 
                                           
                                             
                                               ( 
                                               
                                                 j 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 2 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 π 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   fK 
                                                   3 
                                                 
                                               
                                               ) 
                                             
                                             
                                               - 
                                               1 
                                             
                                           
                                         
                                         
                                           
                                             K 
                                             1 
                                           
                                           × 
                                           
                                             
                                               ( 
                                               
                                                 j 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 2 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 π 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   fK 
                                                   3 
                                                 
                                               
                                               ) 
                                             
                                             
                                               - 
                                               1 
                                             
                                           
                                         
                                       
                                       + 
                                       
                                         
                                           
                                             K 
                                             2 
                                           
                                           × 
                                           
                                             
                                               ( 
                                               
                                                 j 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 2 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 π 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   fK 
                                                   4 
                                                 
                                               
                                               ) 
                                             
                                             
                                               - 
                                               1 
                                             
                                           
                                         
                                         
                                           
                                             K 
                                             2 
                                           
                                           + 
                                           
                                             
                                               ( 
                                               
                                                 j 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 2 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 π 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   fK 
                                                   4 
                                                 
                                               
                                               ) 
                                             
                                             
                                               - 
                                               1 
                                             
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             A 
                             . 
                             N 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               
                                 
                                   
                                     2 
                                      
                                     
                                       ( 
                                       
                                         
                                           
                                             ρ 
                                              
                                             
                                                 
                                             
                                              
                                             K 
                                           
                                           π 
                                         
                                         + 
                                         
                                           
                                             ( 
                                             
                                               j 
                                                
                                               
                                                   
                                               
                                                
                                               π 
                                                
                                               
                                                   
                                               
                                                
                                               f 
                                                
                                               
                                                 
                                                   
                                                     ɛ 
                                                     0 
                                                   
                                                    
                                                   
                                                     ɛ 
                                                     ρ 
                                                   
                                                 
                                                 
                                                   d 
                                                   dl 
                                                 
                                               
                                             
                                             ) 
                                           
                                           
                                             - 
                                             1 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   2 
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           
                                             K 
                                             1 
                                           
                                           × 
                                           
                                             
                                               ( 
                                               
                                                 j 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 2 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 π 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   fK 
                                                   3 
                                                 
                                               
                                               ) 
                                             
                                             
                                               - 
                                               1 
                                             
                                           
                                         
                                         
                                           
                                             K 
                                             1 
                                           
                                           × 
                                           
                                             
                                               ( 
                                               
                                                 j 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 2 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 π 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   fK 
                                                   3 
                                                 
                                               
                                               ) 
                                             
                                             
                                               - 
                                               1 
                                             
                                           
                                         
                                       
                                       + 
                                       
                                         
                                           
                                             K 
                                             2 
                                           
                                           × 
                                           
                                             
                                               ( 
                                               
                                                 j 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 2 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 π 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   fK 
                                                   4 
                                                 
                                               
                                               ) 
                                             
                                             
                                               - 
                                               1 
                                             
                                           
                                         
                                         
                                           
                                             K 
                                             2 
                                           
                                           + 
                                           
                                             
                                               ( 
                                               
                                                 j 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 2 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 π 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   fK 
                                                   4 
                                                 
                                               
                                               ) 
                                             
                                             
                                               - 
                                               1 
                                             
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 A 
                                 . 
                                 
                                   L 
                                   total 
                                 
                               
                               / 
                               2 
                             
                              
                             
                                 
                             
                              
                             L 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0052]    According to equations (7) and (8), the cell free cell-electrode impedance and the cell covered cell-electrode impedance are inversely proportional to the electrode area, A, and the total length of the electrode, L total . The total length of the electrode, L total , can be expressed as a function of the length of each branch element  415 , L, and the number of branch elements, N. The total length of the electrode, L total , can be expressed by equation (9). 
         [0000]        L   total =2 LN   (9)
 
         [0000]    Equations (7) and (8) show that the cell free cell-impedance (Z cell-free-total (f)) and the cell covered cell-impedance (Z cell-covered-total (f)) are inversely proportional to the length of each branch element  415 , L, and the number of branch elements, N. From equation (1), it can be determined that the sensitivity of the cell-impedance sensor is inversely proportional to the length of each branch element  415  and the number of branch elements. Therefore, in the design of a cell-electrode impedance sensor, the reduction of electrode branch number or a reduction in the electrode width can improve the sensitivity of the sensor, but can decrease the number of cells which could be monitored by the sensor. 
         [0053]      FIG. 5A  and  FIG. 5B  show examples of cell-electrode impedance sensors including branch elements of different areas. The cell-electrode impedance sensor depicted in  FIG. 5A  includes two branch elements  505  and  510 . The cell-electrode impedance sensor depicted in  FIG. 5B  includes two branch elements  520  and  525 . In the examples shown, the areas of  505  and  510  are equal to each other while the areas of  520  and  525  are equal to each other. The area of the branch elements  505  and  510  is twice the area of the branch elements  520  and  525  in  FIG. 5B . The equivalent circuit of cell-electrode impedance in  FIG. 5A  and  FIG. 5B  are represented by  515  and  530 , respectively. In  FIGS. 5A and 5B , the resistance represents the sum of R cell  ( 310 ) and R gap  ( 305 ) and the capacitance represents the sum of C cell  ( 320 ) and C gap  ( 315 ). In the example shown, each branch element  505  and  510  contain 7 cells attached to their surface. The equivalent circuit of each cell represents R cell  ( 310 ), R gap  ( 305 ), C cell  ( 320 ), and C gap  ( 315 ). Therefore, the total resistance and capacitance of cell culture vessel  505  is associated a factor of 1/7. In the example shown, the surface of each electrode of cell culture vessel  510  contains twice the number of cells attached as the surface of electrodes of cell culture vessel  515 . The equivalent circuit of each cell represents R cell  ( 310 ), R gap  ( 305 ), C cell  ( 320 ), and C gap  ( 315 ). Therefore, the total resistance and capacitance of cell culture vessel  510  is associated a factor of 1/14. An electric field is generated between the electrodes  505  and  510  and the electrodes  520  and  525  when a voltage is applied. 
         [0054]      FIG. 6  depicts an example of equivalent circuits of a cell-electrode impedance measurement systems of different lengths. The number of branch elements  415  in cell culture vessel  610  is twice the number of branch elements  415  in cell culture vessel  605 . Thus, in the example shown, the total length of the electrodes in cell culture vessel  610  is twice that of the electrodes in the cell culture vessel  605 . The equivalent circuit of each cell represents R cell  ( 310 ), R gap  ( 305 ), C cell  ( 320 ), and C gap  ( 315 ). In the example shown, the surface of each electrode in both culture vessels contains the same number of cells attached. Since the number of branch elements  415  in cell culture vessel  610  is twice the number of branch elements  415  in cell culture vessel  605 , the number of cells attached to the branch elements  415  of cell culture vessel  610  is twice the number of cells attached to the branch elements  415  of cell culture vessel  605 . Therefore, the total resistance and capacitance of cell culture vessel  605  are associated with a factor of 1/6. The total resistance and capacitance of cell culture vessel  610  are associated with a factor of 1/12. 
         [0055]      FIG. 7  depicts a plot of sensor sensitivity in response to frequency for three different values for the total electrode length, L total . The total electrode length, L total , is altered by increasing the number of branch elements  415 , N. The number of branch elements in the sensors tested were 40, 80, and 160. The width of the branch element, W, and the inter-electrode spacing, D, were maintained constant at 20 μm and 20 μm, respectively. As depicted in  FIG. 7 , the sensitivity (in ohms) increased at the frequency range of 0-40 kHz and decreased for frequencies greater than 40 kHz. 
         [0056]      FIG. 8  depicts a plot of sensor sensitivity in response to frequency for different values for the inter-electrode distance, D. In one example, the number of branch elements  415  was 60 and the width of the electrode, W, was 20 μm. The inter-electrode spacing, D, was 20 μm, 40 μm, and 80 μm. In a second example, the number of branch elements  415  was 60 and the width of the electrode, W, was 40 μm. The inter-electrode spacing, D, was 20 μm, 40 μm, and 80 μm. In a third example, the number of branch elements  415  was 60 and the width of the electrode, W, was 80 μm. The inter-electrode spacing, D, was 20 μm, 40 μm, and 80 μm. As depicted in  FIG. 8 , the sensitivity (in ohms) was not affected by the change in inter-electrode spacing, D. 
         [0057]      FIG. 9  depicts a plot of sensor sensitivity in response to frequency for different values of the electrode width, W. In one example, the number of branch elements  415  was 60 and the inter-electrode distance, D, was 20 μm. The width, W, was 20 μm, 40 μm, and 80 μm. In a second example, the number of branch elements was 60 and the inter-electrode distance, D, was 40 μm. The width, W, was 20 μm, 40 μm, and 80 μm. In a third example, the number of branch elements  415  was 60 and the inter-electrode distance, D, was 80 μm. The width, W, was 20 μm, 40 μm, and 80 μm. As depicted in  FIG. 7 , the sensitivity (in ohms) decreased with increasing width, W. 
         [0058]      FIG. 10  depicts an example of cells  150  totally or partially attached to the surface of a branch element  415 . In  FIG. 7 , cells  150 , depicted by 1, are positioned on the edge of the branch element  415  such that half the surface of the cell is attached to the branch element  415 , while half of the surface of the cell is not. Cells  150 , depicted by 2, are positioned on the edge of the branch element  415  such that more than half of the surface of the cell is attached to the edge of the branch element  415  while the remainder of the surface of the cell is not. Cells  150 , depicted by 3, are positioned on the edge of the branch element  415  such that less than half of the surface of the cell is attached to the edge of the branch element  415 , while the remainder of the cell is not. Cells  150 , depicted by 4, are positioned on the branch element  415  such that the total surface of the cell is attached to the branch element  415 . Cells under category 4 can be positioned anywhere on the branch element  415 . In some implementations, only the cells that are attached entirely to the branch element  415  can be used to determine cell covered cell-electrode impedance. In other implementations, the contribution to the measured signals from cells that are partially attached to the branch elements  415  can be included. 
         [0059]    Both, the cells attached totally and the cells attached partially, can be defined as effective cells that contribute to the impedance. Numerous cells can occupy positions along the edges of the branch elements  415 . A branch element  415  can be represented by a length, L, and a width, W. The individual cells on the branch element  415  surface can fall under one of the categories 1, 2, 3, and 4 depicted in  FIG. 7 . The average area of any cell occupying a position on the edge of the branch element  415  can be considered as equivalent to half the area of a cell  150 . The area of a cell  150 , in turn, can be represented as the distance of one radius of a cell tangential to the long edge of the branch element  415  for the total length of the edges of the branch elements  415 . An effective branch element  415  width, W eff , can be defined by equation (10), to account for effective cells along both of the long edges of the branch element  415 . 
         [0000]        W   eff   =W+ 2 r   cell   (10)
 
         [0000]    In addition, the effect of sharp turns of electrodes on the electric field generated by the AC signal, known as the edge effect, can also be accounted for by using the effective branch element width, W eff . The sharp turns on the electrode can increase the strength of the local electric field due to electric charge accumulation at the location of the sharp turn. This can alter the total electric field strength of the electrode. The effect of the altered electric field near the sharp turns can be accounted for by at least assuming the width of the branch element, W, to be an effective width, W eff  which is set to be greater than the actual width W of the branch element. The relationship regarding the effective electrode width and the effect of the electric field can be complex when the electric field is a non-uniform field caused by the edge effect where the strength of electric field in the edge is higher than that in the middle of the electrode. Experience and various tests have demonstrated that using a properly selected effective electrode width W eff &gt;W can compensate for the non-uniformity of the electric field distribution from the edge to the center of the electrode during the design process of the impedance sensors described in this application. As a result of using this effective electrode width, W eff , the design model based on the uniform field distribution over the electrode can be used to design actual sensors where the field distribution is not uniform due to the edge effect. An effective area of the branch element  415 , A eff , can be defined and represented by equation (11): 
         [0000]        A   eff   =W   eff   ×L   (11)
 
         [0000]    In some implementations, the length of the branch element can also be replaced by an effective branch element length, which includes the radius of the cell near the short edge of the branch element  415 . 
         [0060]    The cell radius, r cell , is an intrinsic parameter of the type of the living cell attached to the cell-electrode impedance sensor. Both, the edge effect and r cell , can affect the sensitivity of the sensor. As the width of the branch element  415  decreases, the cells which cover the edge of the branch element  415  can occupy a larger portion of the surface are of the branch element  415 . Both, A eff  and the edge effect, can be manipulated to improve the sensitivity of the sensor and capturing the impedance from a larger number of cells as the width of the branch element  415  decreases. 
         [0061]    The accuracy of equations (7) and (8), in predicting the cell free and cell covered cell-electrode impedance, respectively, can be determined by experimentally measuring cell free and cell covered cell-electrode impedances in a cell-electrode impedance measurement system  100  and comparing the measured values to the design prediction. For example, HeLa cell cultures with a cellular radius of about 9.5 μm were studied. Branch elements of widths, W, 20 μm, 40 μm, and 80 μm were chosen. This resulted in effective widths, W eff , of 39 μm (20+2*9.5), 59 μm (40+2*9.5), and 99 μm (80+2*9.5), respectively. 
         [0062]      FIG. 11  depicts a comparison between cell-electrode impedance calculated using equations (7) and (8) and experimentally measured cell-electrode impedance using the HeLa cell cultures. The sensitivity of the cell-electrode impedance measurement system  100  is plotted against the effective widths, W eff , of 3 branch elements  415 . The experimentally measured sensitivity data, shown in  FIG. 11 , shows that the measured sensitivity (represented by points in  FIG. 11 ) lies on the computationally obtained fitted curve (represented by the line in  FIG. 11 ). 
         [0063]      FIG. 12  depicts plots of normalized impedance measured over time in response to changing frequency. In the example shown, the cell-electrode impedance was measured during cell proliferation.  FIG. 12  shows that during cell proliferation, as time increases, the frequency-dependent normalized impedance increases ( FIG. 12 , (a)). It can also be seen from  FIG. 12  that, during cell proliferation, as time decreases, the frequency-dependent normalized impedance decreases as time increases ( FIG. 12 , (b)). 
         [0064]      FIG. 13  depicts the frequency characteristics of a cell-electrode impedance sensor. A frequency range to maximize the sensitivity of the cell-electrode impedance measurement system can be determined using the design model. In measuring cell-electrode impedance, AC signals from the AC stimulator  115  can be delivered at different frequencies. Three important frequencies can be deduced from the equivalent circuit proposed in equation (7) and (8), namely, f low , f middle , and f high . In the low frequency range, interface capacitance, C I , dominates resistance and the equivalent impedance circuit can be represented by  FIG. 3B . The equivalent circuit which is effective in low frequency range can be a equated to a high-pass circuit and the cut-off frequency at which the impedance of C I  is equal to the sum of R S , R cell , and R gap  can be defined by equation (12). 
         [0000]    
       
         
           
             
               
                 
                   
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                   12 
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         [0000]    A frequency, f low , can be defined as lower than one fifth one-fifth of f cut-off-low . At a frequency lower than f low , the impedance of cell covered cell-electrode sensor can be dominated by interface capacitance, C I . In this range, the impedance of cell free cell-electrode sensor is also dominated by C I  and the sensitivity of the sensor can be low. Further, as the frequency decreases, the sensitivity can decrease and approach zero. The frequency, f low , can be represented by equation (13). 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
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         [0065]    In the high frequency range, R cell  can dominate C cell , and R gap  can dominate C gap . The equivalent impedance circuit can be represented by  FIG. 3C . The equivalent circuit can be equated to a high-pass circuit and the cut off frequency can be represented by equation (14). 
         [0000]    
       
         
           
             
               
                 
                   
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                   14 
                   ) 
                 
               
             
           
         
       
     
         [0000]    A frequency, f high , can be defined as five times f cut-off-high . At a frequency higher than f cut-off-high , the impedance of both cell free and cell covered cell-electrode impedance can be dominated by R S . Further, the sensitivity can be low and can approach zero as the frequency increases. The frequency, f high , can be represented by equation (15). 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
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                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
         [0066]    A third frequency, f middle , can be defined as the frequency at which the sensitivity can be maximum. This frequency can be determined by solving the derivative of the equation for sensitivity (1). The derivative can be represented by equation (16). 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
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                               ( 
                               
                                 Sensitivity 
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                                                
                                               
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                         = 
                           
                          
                         0 
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
         [0000]    As illustrated in  FIG. 13(   a ), the cell free cell-electrode impedance and the cell covered cell-electrode impedance are nearly equal at frequencies closer to f low  and f high . The difference between the cell free cell-electrode impedance and the cell covered cell-covered impedance is greatest at frequencies near f middle . Since the sensitivity is directly proportional to the difference between the cell covered cell-electrode impedance and the cell free cell-electrode impedance, sensitivity of the impedance measurement system can be maximized at and near a frequency equal to f middle . The dimensions of a sensor are independent of the frequency of the AC signal. In some implementations, the sensitivity can be maximized when the frequency range of the input signal is between 10 kHz and 40 kHz. This operating frequency range from 10 kHz to 40 kHz can be determined based on experimental data of the sensors. Equivalent circuit parameters described in this specification can be used to fit to measured experimental data to determine equivalent circuit parameters and the resultant equivalent circuit parameters are then applied to the circuit equations to calculate the operating frequency range for the sensor. 
         [0067]      FIG. 14  depicts a flow chart of an example of designing a cell-electrode impedance measurement system. The values of parameters of the system can be determined at  1405 . The parameters can include one or more of the properties of the cell culture vessel, the culture medium, the culture cells, the proportion coefficients to represent R gap , R cell , C gap , and C cell , etc. The cell free cell-electrode impedance can be determined from a design model at  1410 . The cell free cell-electrode impedance can be determined using equation (7). The cell covered cell-electrode impedance can be determined from the design model at  1415 . The computational cell covered cell-electrode impedance can be determined using equation (8). Using the computationally determined cell free and cell covered cell-electrode impedances, the sensitivity of a system can also be determined at  1420 . The computational sensitivity can be determined using equation (1). The computational cell free and cell covered cell-electrode impedances can be used to predict parameters of a cell-electrode impedance sensor in the system at  1425 . The cell-electrode impedance sensor can be an interdigitated array. The predicted parameters can include the area of the branch elements of the interdigitated array, the width of the branch elements, and the number of branch elements. Based on equations (7) and (8), the sensitivity of the sensor is inversely proportional to the area of the branch elements and the number of branch elements. In addition, the area of the branch elements and the number of elements is directly proportional to the number of cells that can be studied. By predicting the cell-electrode impedances, a sensor can be designed that can maximize sensitivity while also maximizing the number of cells that can be studied. 
         [0068]      FIG. 15  depicts a flow chart of an example of optimizing the frequency of AC signals applied to a cell-electrode impedance measurement system. Based on the sensitivity of the cell-electrode impedance measurement system, determined from equation (1), and the cell free and cell covered cell-electrode impedances determined from equations (7) and (8), the optimum frequency of AC signals can be determined at  1505 . In some implementations, the equations representing cell free cell-electrode impedance (7) and cell covered cell-electrode impedance (8) can be substituted into the equation representing sensitivity equation (1). The equation representing sensitivity can be differentiated with respect to frequency and the resulting expression can be set to zero. A solution to the equation can yield an optimum frequency value. AC signals can be applied to the cell-electrode impedance measurement system at the determined optimum frequencies at  1510 . The cell free and cell covered cell-electrode impedances of the system can be measured at  1515 . 
         [0069]    Hence, one design model for designing a cell-electrode impedance sensor system is based on an equivalent circuit model for interdigitated array structure cell-electrode impedance sensor, including a first equivalent circuit model the cell free sensor and a second equivalent circuit model for the cell covered sensor. The equivalent circuit model in this design model can include parameters Rcell, Ccell, Rgap and Cgap. Rcell is the resistance of the mean gap distance between the cells which are attached on the electrodes of the sensor. This parameter can be used to monitor the mean gap distance between cells in real time. Ccell is the capacitance of cells created by their insulating cell membranes. This parameter can reflect changes in the biophysical properties of the cell membranes. Rgap is the resistance of the gap between the attached cells and the electrode surfaces. This parameter can be used in real time monitor the mean gap between the attached cells and electrode surface substrate. Cgap is the capacitance between the attached cells and the electrode surface substrate. This parameter can be used to reflect the gap between attached cell membranes and electrode surface substrate. 
         [0070]    Part or entirety of the design processes described in this specification may be implemented as computer-implemented processes. The design parameters obtained from the described design processes can be used to fabricate sensor devices in a wide range of applications including sensing biological cells. 
         [0071]    While this specification contains many specifics, these should not be construed as limitations on the scope of any invention or of what may be claimed, but rather as descriptions of features specific to particular embodiments. Certain features that are described in this specification in the context of separate embodiments can also be implemented in combination in a single embodiment. Conversely, various features that are described in the context of a single embodiment can also be implemented in multiple embodiments separately or in any suitable subcombination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a subcombination or variation of a subcombination. Only a few implementations and examples are described. However, other variations, modifications, and enhancements are possible and are within the scope of the following claims.