An electrochemical impedance spectroscopy parameter extraction method of a heuristic global optimization algorithm

Abstract

This invention discloses a heuristic global optimization algorithm for extracting electrochemical impedance spectroscopy (EIS) parameters. The method first collects E. coli impedance spectroscopy data using an impedance testing instrument and preprocesses it to obtain effective impedance spectroscopy data. Next, an equivalent circuit model is selected from the effective impedance spectroscopy data, and a transfer function is used to describe it, resulting in the equivalent circuit model transfer function. Then, the equivalent circuit model transfer function is rearranged to obtain its real and imaginary parts, which are then transformed to obtain a multi-objective optimization model. Finally, the multi-objective optimization model is solved using a heuristic global optimization algorithm to obtain the optimal estimated parameters of the circuit model, i.e., the EIS parameters. This invention has good accuracy and convergence, enabling accurate EIS parameter estimation, reducing operational burden, and accelerating experimental progress.

Description

Technical Field

[0001] This invention belongs to the field of bioelectrochemistry research and mainly relates to a method for extracting electrochemical impedance spectroscopy parameters using a heuristic global optimization algorithm. Background Technology

[0002] Electrochemical impedance spectroscopy (EIS) is a technique used to characterize internal electrochemical processes. It obtains complex information about an electrochemical system by measuring its electrical properties under an alternating current field. EIS is widely used in batteries, sensors, and biomedicine. In EIS data processing, parameter estimation is an effective and practical method for fully understanding the internal electrochemical processes, as it provides a quantitative description of the electrochemical reactions. However, parameter extraction in EIS has always been a challenge and bottleneck in this field. In recent years, many new algorithms have been developed to extract impedance spectroscopy parameters more quickly and accurately, such as neural network-based methods, Bayesian statistical methods, and genetic algorithms. Traditional EIS parameter extraction methods are mainly based on model fitting or traditional mathematical solutions. However, model fitting and traditional mathematics, when solving complex multi-parameter nonlinear problems, rely heavily on initial values ​​for convergence results and speed, easily getting trapped in local optima and experiencing excessively long execution times. Incorrect initial values ​​sometimes compromise accuracy and reliability, resulting in high complexity, poor accuracy, and weak generalization ability. This also indirectly increases the learning cost and barrier for experimenters. Meanwhile, before interpreting any data extracted from equivalent circuit modeling, the system needs to be well characterized to verify whether it meets the three necessary conditions of stability, linearity, and causality. However, the verification of the validity and reliability of EIS data is often overlooked by many experimenters, resulting in unreliable or incorrect experimental data. Summary of the Invention

[0003] The purpose of this invention is to provide a heuristic global optimization algorithm for extracting electrochemical impedance spectroscopy parameters. This addresses the technical problems in existing technologies where data validity screening and evaluation indicators, convergence results, and convergence speed rely too heavily on initial values, easily leading to local optima, excessively long execution times, and low stability and accuracy.

[0004] The specific steps of the electrochemical impedance spectroscopy parameter extraction method based on this heuristic global optimization algorithm are as follows:

[0005] Step 1: Collect E. coli impedance spectrum data using an impedance testing instrument.

[0006] Step 2: Preprocess the impedance spectrum data Z(ω) of E. coli to obtain the preprocessed effective impedance spectrum data.

[0007] Step 3: Select an equivalent circuit model from the effective impedance spectrum data obtained in Step 2 to obtain the equivalent circuit model.

[0008] Step 4: Describe the equivalent circuit model using a transfer function to obtain the equivalent circuit model transfer function.

[0009] Step 5: Rearrange the transfer function of the equivalent circuit model obtained in Step 4 to separate the real and imaginary parts, thus obtaining the real and imaginary parts of the transfer function of the equivalent circuit model.

[0010] Step 6: Transform the real and imaginary parts of the transfer function of the equivalent circuit model to obtain the multi-objective optimization model.

[0011] Step 7: Solve the multi-objective optimization model using a heuristic global optimization algorithm to obtain the optimal estimated parameters of the circuit model.

[0012] Step 8: Repeat steps 1 to 7 every preset time interval to obtain the electrochemical impedance spectroscopy parameter extraction data of Escherichia coli at different time periods.

[0013] Preferably, the specific process for collecting E. coli impedance spectroscopy data in step 1 is as follows:

[0014] 1-1. Culture Escherichia coli at different growth stages and place them in petri dishes.

[0015] 1-2. In a petri dish, the E. coli obtained in step 1-1 is modified onto the electrode surface and added to the test cell using potassium ferricyanide / potassium ferrocyanide solution as a redox probe.

[0016] 1-3. Electrochemical impedance spectroscopy was performed on the cell under test using an impedance testing instrument to obtain the impedance spectrum data of Escherichia coli.

[0017] 1-4. Transfer the E. coli impedance spectrum data Z(ω) obtained in step 1-3 to the computer.

[0018] Preferably, the preprocessing in step 2 is performed as follows:

[0019] 2-1. Using the real and imaginary parts of the E. coli impedance spectrum data Z(ω) obtained in step 1 as the dependent variable and the frequency as the independent variable, plot them separately to obtain (real part - frequency)logZ real -logf graph, (imaginary part - frequency)logZ imag -logf graph.

[0020] 2-2. The logZ obtained in step 2-2 real -logf graph, logZ imagThe -logf plots were smoothed using the Savitzky-Golay filter to obtain the smoothed E. coli impedance spectrum data Z′(ω).

[0021] 2-3. Perform Hilbert transform (HT) verification on the smoothed and filtered E. coli impedance spectrum data Z′(ω) obtained in step 2-3 to obtain the effective impedance spectrum data of E. coli. The specific process of the HT verification method is as follows:

[0022] 2-4-1. Separate the real and imaginary parts of the smoothed and filtered E. coli impedance spectrum obtained in step 2-3 to obtain the real part Z at each corresponding frequency measurement point. Re (ω), imaginary part Z Im (ω).

[0023] 2-4-2. According to the Hilbert transformation formula below, for the real part Z obtained in step 2-4-1... Re (ω), Imaginary part Z Im (ω) are transformed respectively:

[0024]

[0025] 2-4-3. Simplify the transformation obtained in step 2-4-2 to obtain the transformed real part. virtual part The complex form of the Hilbert transform Z HT (ω).

[0026] 2-4-4. Based on the Hilbert transform Z obtained in step 2-4-3 HT Using the impedance spectrum data Z(ω) and E. coli, calculate the residual ε(ω) before and after the Hilbert transform; calculate the overall quality index ε of the input impedance spectrum according to the following formula:

[0027]

[0028] Where j is the frequency point index, ω j Let ω be the angular frequency at the j-th position, where ω = 2πf is the angular frequency, f is the frequency, and N1 is the total number of frequency points.

[0029] 2-4-5. Set the overall quality index threshold for the impedance spectrum. The impedance spectrum is screened for validity and reliability. Data exceeding a threshold is considered invalid, while data less than or equal to the threshold is considered valid, i.e., valid impedance spectrum data.

[0030] As a preferred option, the specific process for selecting the equivalent circuit model in step 3 is as follows:

[0031] 3-1. The effective impedance spectrum data of Escherichia coli is visualized using the Nyquist plot. If the intersection of the Nyquist plot of the effective impedance spectrum data with the real axis is not at zero, it is determined that there is solution impedance R1; otherwise, there is no solution impedance R1.

[0032] 3-2. Based on the Nyquist plot and relaxation time distribution (DRT) transformation of the effective impedance spectrum data obtained after preprocessing in step 2, determine the number of parallel resistors R and capacitors C. The specific process for determining the number of parallel RC resistors is as follows:

[0033] 3-2-1. Manually observe the Nyquist plot of the effective impedance spectrum data obtained after preprocessing in step 2 to obtain the number of semicircles on the Nyquist plot.

[0034] 3-2-2. Perform DRT transformation on the effective impedance spectrum data obtained after preprocessing in step 2 to obtain the number of peaks in the DRT transformation graph, and exclude the overlapping semicircles caused by similar electrochemical processes in step 3-2-1.

[0035] 3-2-3. Determine the number of parallel RC circuits in the impedance spectrum model by finding the maximum values ​​of the number of semicircles and peaks on the Nyquist plot.

[0036] 3-3. If the Nyquist plot of the effective impedance data obtained after preprocessing in step 2 has a 45° inclined straight line, it is determined that diffusion impedance is involved; otherwise, it is determined that the equivalent circuit model obtained in step 3 does not have diffusion impedance.

[0037] Preferably, the specific process of using a transfer function to describe the equivalent circuit model in step 4 is as follows:

[0038] 4-1. Using constant phase angle element impedance The RC obtained in step 3-2 is described, and the impedance of the constant phase angle element is given. The expression is shown in the following formula:

[0039]

[0040] Where i is the index of the (RC) numbers; j represents the unit imaginary number; and Q is the constant phase angle coefficient value. α is the constant phase angle element exponent factor; f is the frequency.

[0041] 4-2. If the equivalent circuit model obtained in step 3-3 has diffusion impedance, then describe it according to the Warburg impedance formula below; otherwise, do not describe it:

[0042]

[0043] Where Z0 is the diffusion impedance coefficient; denoted as the diffusion frequency, L as the diffusion thickness, and D as the diffusion coefficient; coth is the hyperbolic cotangent function.

[0044] 4-3. Based on the descriptions of the circuit model elements obtained in steps 4-1 to 4-2 and the series-parallel theorem, calculate the total impedance of the equivalent circuit model to obtain the following EIS true dynamic description of the equivalent circuit model, i.e., the transfer function of the equivalent circuit model:

[0045]

[0046] Where R1 is the solution impedance; N2 is the number of parallel RC circuits; n is the sequence number of the parallel R and C circuits; R 1+n Z represents the resistance value of R in the nth parallel R and C connection. Qn Let C be the impedance value of the nth parallel R-C.

[0047] Preferably, the specific process of rearranging the equivalent circuit model transfer function in step 5 to separate the real and imaginary parts is as follows:

[0048] 5-1. Expand the total impedance obtained in step 4-3 according to the complex number operation theorem to obtain the real part. and the virtual part

[0049]

[0050]

[0051] Where R1 is the solution impedance; Q n α is the impedance coefficient of C in the nth parallel R and C; α is the exponential coefficient of C in the nth parallel R and C; ω is the angular frequency.

[0052] Preferably, the specific steps for performing multi-objective optimization model transformation on the equivalent circuit model transfer function in step 6 are as follows:

[0053] 6-1. Calculate the optimization objective function F at the j-th frequency point using the formula below. j :

[0054]

[0055]

[0056] Where j is the frequency point index; Z(ω) j Z Re (ω j Z Im (ω j) are the E. coli impedance spectrum data, the real part of the E. coli impedance spectrum data, and the imaginary part of the E. coli impedance spectrum data at the j-th frequency point obtained in step 2, respectively. These represent the equivalent circuit model impedance, real part of the impedance, and imaginary part of the impedance at the j-th frequency point obtained in step 4-4, respectively; f j (R1,R2,Q1,α) represents the residual between the equivalent circuit model impedance and the original effective impedance at the j-th frequency point.

[0057] 6-2. Optimize the objective function F at a total of p frequency points obtained in step 6-1. j By combining these equations, we obtain a set of p-columns of objective function equations, which are the multi-objective optimization functions for the equivalent circuit model.

[0058] 6-3. Based on the multi-objective optimization functions obtained in steps 6-1 and 6-2, two objective functions can be obtained at each frequency point, namely, the objective function F with the optimal real part. Re,j The optimal objective function F with imaginary part Im,j .

[0059] 6-4. Add the following constraints to the control parameters of the multi-objective optimization function obtained in step 6-2: The upper and lower limits of the resistance value are constrained to [1e-07, 1e07]; the upper and lower limits of the capacitance coefficient are constrained to [1e-12, 1e-03]; and the upper and lower limits of the exponent coefficient are constrained to [0, 1].

[0060] Preferably, the specific process for using a heuristic global optimization algorithm to find the optimal solution for the multi-objective optimization model in step 7 is as follows:

[0061] 7-1. Pass the multi-objective function, control parameters, and parameter constraints obtained in step 6 into the heuristic global optimization algorithm.

[0062] 7-2. Set the population size using the following formula. and Get the initial population size X i (i = 1, 2, ..., n) and

[0063]

[0064] Where t represents the current iteration, and It is a coefficient vector. It is the position vector of the prey. This represents the position vector of the search agent. To reduce linearly from 2 to 0 during the iteration process, It is a random vector in [0,1].

[0065] 7-3. Determine whether the control parameters input in step 7-1 meet the constraints input in step 7-1. If they do, proceed to step 7-4; otherwise, return to step 7-1 and randomly select data within the constraints to modify the control parameters.

[0066] 7-4. Using the multi-objective function obtained in step 7-1 as the multi-objective fitness function, calculate the fitness function for each individual, and save the three individuals with the best fitness to obtain the three individuals X with the best fitness. a X β X δ .

[0067] 7-5. Update the current position of the search agent according to the following formula to obtain the updated position of the search agent.

[0068]

[0069] in, These are the top three individuals with the best fitness; These are the top three individuals with the best fitness after the update; This indicates the updated position of the current search agent.

[0070] 7-6. Based on the formula in step 7-2 and the updated position of the search agent obtained in step 7-5, update... and

[0071] 7-7. Based on the multi-objective fitness function obtained in step 7-4, update the fitness of each individual, and obtain the top three individuals X with the best updated fitness. a X β X δ .

[0072] 7-8. Determine whether the maximum number of iterations has been reached or the optimization stopping condition has been met. If the conditions are met, exit, return the optimal solution, and exit the algorithm. Otherwise, repeat steps 7-5 to 7-8.

[0073] The beneficial effects of this invention are:

[0074] 1. This invention proposes an overall quality evaluation index for EIS data based on the Hilbert transform. A complete overall quality evaluation index algorithm is specifically designed for the input EIS data, effectively eliminating the influence of experimental errors or improper operation, improving the accuracy and reliability of EIS, and providing effective data support for subsequent global optimization algorithms. Simultaneously, a method based on the transfer function concept of equivalent circuits is designed to determine the objective function, and the mathematical equations are rearranged to separate the real and imaginary parts. The problem is transformed into a multi-objective optimization problem, and then the equations are solved under a heuristic global optimization algorithm to extract the parameters of the equivalent circuit elements. Accurate parameter estimation can be achieved.

[0075] 2. This invention presents an automated method for extracting electrochemical impedance spectroscopy (EIS) parameters based on equivalent circuits, eliminating the need for manual, repetitive, line-by-line data processing using commercial or open-source software. When dealing with large volumes of EIS data, automation significantly reduces the user's workload and accelerates the experimental process.

[0076] 3. This invention relies on heuristic intelligent algorithms and optimization problem theory methods to design a set of heuristic global optimization algorithms for extracting equivalent circuit parameters based on EIS. Due to the fact that traditional mathematics and highly mathematical techniques rely too much on the initial value when solving complex multi-parameter nonlinear problems, they are prone to getting trapped in local optima and have excessively long execution times. Incorrect initial values ​​sometimes lead to a lack of accuracy and reliability. Heuristic global optimization algorithms have an advantage in this regard because they can handle incorrect guesses or limits and avoid local optima.

[0077] 4. This invention designs a heuristic global optimization algorithm. This method is based on the optimization of genes that satisfy the equation under certain constraints. It is relatively simple to use and easy to understand. It does not require derivatives or initial values. It can process the global optimum in a shorter computation time and shows good accuracy and convergence, indicating that it is a candidate for simulation of complex equivalent circuit models in EIS.

[0078] 5. This invention designs a method to extract equivalent circuit parameters and transform them into a multi-objective model, providing a more accurate fitness function for EIS curves. Compared with other studies, it employs a heuristic crossover operator to solve the optimization problem, improving accuracy and robustness. It is also highly versatile and suitable for parallel processing, improving the accuracy of estimated parameters with less computation time. It can handle large lower and upper bounds, obtaining more practical results in a shorter computation time, thus making it a powerful tool for EIS parameter estimation. Attached Figure Description

[0079] Figure 1 This is a flowchart illustrating the implementation steps of the present invention;

[0080] Figure 2 This is a flowchart of the heuristic global optimization algorithm of the present invention;

[0081] Figure 3 This is a comparison of the original and smoothed Nyquist impedance spectra of Escherichia coli obtained in this invention;

[0082] Figure 4 This is a comparison of the original and smoothed logz-logf impedance spectra of Escherichia coli obtained in this invention;

[0083] Figure 5 The first derivative plots of the real and imaginary parts of the original and smoothed Escherichia coli impedance spectra obtained in this invention are shown.

[0084] Figure 6 The residual plot of the E. coli impedance spectrum obtained in this invention after Hilbert transformation and the original data is shown.

[0085] Figure 7 The quality evaluation index diagram is obtained from ten experiments on the impedance spectrum of Escherichia coli obtained in this invention.

[0086] Figure 8 The equivalent circuit model diagram of the impedance spectrum of Escherichia coli obtained in this invention is shown.

[0087] Figure 9 This is the DRT transformation diagram of the impedance spectrum of Escherichia coli obtained in this invention;

[0088] Figure 10 The Nyquist comparison diagram of the Escherichia coli impedance spectrum obtained in this invention, fitted based on a global optimization algorithm;

[0089] Figure 11 The image shows a comparison of logz-logf impedance spectra of Escherichia coli obtained in this invention, fitted using a global optimization algorithm.

[0090] Figure 12 The impedance spectrum of Escherichia coli obtained in this invention is a graph showing the change in fitness with the number of iterations in a global optimization algorithm. Detailed Implementation

[0091] The present invention will be further described below with reference to the accompanying drawings.

[0092] like Figure 1 As shown, a heuristic global optimization algorithm for extracting electrochemical impedance spectroscopy parameters is used to extract and fit the equivalent circuit element parameters of the impedance spectrum of Escherichia coli. Based on the fact that the obtained E. coli impedance spectrum can be replaced by an equivalent circuit model and that the heuristic global optimization algorithm can simply and efficiently solve nonlinear multi-objective optimization model problems, the analysis experiment of extracting and fitting E. coli impedance spectrum parameters is carried out.

[0093] like Figure 1 The diagram shown is a flowchart illustrating the implementation steps of the present invention. The specific steps of the extraction and fitting method are as follows:

[0094] Step 1: Use an impedance testing instrument to collect impedance spectrum data of E. coli at different growth stages.

[0095] 1-1. Culture Escherichia coli at different growth stages and place them in petri dishes.

[0096] 1-2. The Escherichia coli obtained in step 1-1 is modified onto the electrode surface and added to the test cell using potassium ferricyanide / potassium ferrocyanide solution as a redox probe.

[0097] 1-3. The test cell obtained in step 1-2 is subjected to electrochemical impedance spectroscopy using an impedance testing instrument to obtain Escherichia coli impedance spectroscopy data.

[0098] 1-4. Transfer the E. coli impedance spectrum data obtained in step 1-3 to the computer.

[0099] Step 2: Preprocess the E. coli impedance spectrum data Z(ω) obtained in Step 1 to obtain the preprocessed effective impedance spectrum data. The specific preprocessing process is as follows:

[0100] 2-1. Using the real and imaginary parts of the E. coli impedance spectrum data Z(ω) obtained in step 1 as dependent variables and the frequency as the independent variable, plot the data to obtain logZ. real -logf graph, logZ imag -logf graph, such as Figure 4 As shown in the diagram.

[0101] 2-2. The logZ obtained in step 2-2 real -logf graph, logZ imag The -logf plots were smoothed using a Savitzky-Golay filter to obtain the smoothed E. coli impedance data, as shown below. Figure 3 The solid line indicates the logz-logf graph, as shown below. Figure 4 As shown by the solid line, the specific process of smoothing filtering is as follows:

[0102] 2-3-1. Set the filter window length to 2m+1, where m is the frequency point in the window; each measurement point is f=(-m,-m+1,···,0,1,···,m-1,m+1), and use a k-1 degree polynomial to fit the data points within the window, as shown in the following formula:

[0103] z = a0 + a1f + a2f 2 +...+a k-1 f k-1

[0104] In the formula, a0 is the polynomial coefficient, and z is the fitted value at the measurement point.

[0105] 2-3-2. By arranging the 2m+1 polynomials obtained in step 2-2-1 side by side, we obtain a system of linear equations:

[0106]

[0107] 2-3-3. Representing the system of equations obtained in step 2-2-2 in matrix form, we obtain the following simplified matrix form:

[0108] Z (2m+1)×1 =F (2m+1)×k ·A k +E (2m+1)×1

[0109] In the formula, Z (2m+1)×1 F represents a 2m+1 row, 1 column fitted matrix. (2m+1)×k Let A be a 2m+1 row, k column matrix. k Let E represent a k-row, 1-column coefficient matrix. (2m+1)×1 This represents a 2m+1 row, 1 column fitting error matrix.

[0110] 2-3-4. Solve the matrix equation obtained in step 2-2-3 using the least squares method to obtain the least squares solution of A.

[0111]

[0112] In the formula, F T This represents the transpose of the frequency measurement point matrix.

[0113] 2-3-5. The least squares solution obtained in step 2-2-4 Multiply by the frequency measurement point matrix Z to obtain the filtered value of the Z matrix. for:

[0114]

[0115] 2-3-6. Based on the formula in step 2-3-5, obtain the real part of the impedance after smoothing and filtering. virtual part and complex form

[0116] To evaluate the smoothing filtering effect, the first derivatives of the real and imaginary parts of the original impedance obtained in step 2-2 and the smoothed impedance obtained in step 2-3-6 are calculated respectively. The first derivatives of the real and imaginary parts of the original and smoothed impedance spectra of *E. coli* are shown below. Figure 5 As shown.

[0117] 2-3. Perform Hilbert transform (HT) verification on the smoothed and filtered E. coli impedance spectrum data Z′(ω) obtained in step 2-3 to obtain the effective impedance spectrum data of E. coli. The specific process of the HT verification method is as follows:

[0118] 2-4-1. Separate the real and imaginary parts of the smoothed and filtered E. coli impedance spectrum obtained in step 2-3 to obtain the real part Z at each corresponding frequency measurement point. Re (ω), imaginary part Z Im (ω).

[0119] 2-4-2. Based on the HT transformation formula below, calculate the real part Z obtained in step 2-4-1. Re (ω), Imaginary part Z Im (ω) are transformed respectively:

[0120]

[0121] 2-4-3. Simplify the transformation obtained in step 2-4-2 to obtain the real part after transformation. virtual part The complex form Z of the HT transform HT (ω):

[0122]

[0123]

[0124]

[0125] In the formula, ω=2πf is the angular frequency, and f is the frequency.

[0126] 2-4-4. Based on the formula in step 2-4-3 and the formula below, calculate the residual ε(ω) before and after the Hilbert transform; Figure 6 As shown. Calculate the overall quality index ε of the input impedance spectrum using the following formula:

[0127] ε(ω j )=|Z(ω j )-Z HT (ω j )|

[0128]

[0129] In the formula, |·| represents the modulo operation; j is the frequency point index; ω j Let ω be the angular frequency at the j-th position; ω = 2πf is the angular frequency, f is the frequency, and N is the total number of frequency points.

[0130] 2-4-5. Set the overall quality index threshold for the impedance spectrum. To screen for the effectiveness and reliability of impedance spectroscopy, this algorithm selects a threshold of [threshold value missing]. Data exceeding the threshold is considered invalid, while data less than or equal to the threshold is considered valid.

[0131] 2-4-6. Ten impedance spectroscopy measurements were performed on Escherichia coli. The overall quality evaluation index of the impedance spectrum in each experiment is as follows: Figure 7 As shown.

[0132] Step 3: Select an equivalent circuit model from the effective impedance spectrum data obtained in Step 2 to obtain the equivalent circuit model, such as... Figure 8 As shown, the specific process for selecting the equivalent circuit model is as follows:

[0133] 3-1. The effective impedance spectral data of *E. coli* were visualized using a Nyquist plot. Since the intersection of the Nyquist plot with the real axis is not at zero, the solution impedance R was determined. s .

[0134] 3-2. Based on the Nyquist plot and DRT transformation obtained after preprocessing in step 2, determine the number of parallel charge transfer resistors (R) and double-layer capacitors (C). The specific process for determining the number of R and C is as follows:

[0135] 3-2-1. Manually observe the Nyquist plot obtained after preprocessing in step 2 to find that there is 1 semicircle on the upper part of the Nyquist plot.

[0136] 3-2-2. Perform DRT transformation on the effective impedance spectrum data obtained after preprocessing in step 2. The resulting DRT transformation graph has only one peak. Figure 9 As shown. Exclude the overlapping semicircles caused by similar electrochemical processes in step 3-2-1.

[0137] 3-2-3. Based on the maximum number of semicircles and peaks on the Nyquist plot, determine that the number of R and C in the impedance spectral model is 1.

[0138] 3-3. Based on the fact that there is no 45° inclined straight line on the Nyquist plot obtained after preprocessing in step 2, it is determined that the model does not have diffusion impedance.

[0139] Step 4: Describe the equivalent circuit model obtained in Step 3 using a transfer function to obtain the equivalent circuit model transfer function. The specific process of describing the equivalent circuit model using a transfer function is as follows:

[0140] 4-1. Due to reasons such as the roughness of the electrode surface, a capacitor cannot be replaced by a pure capacitor element. Therefore, a constant phase angle is used instead. The impedance of a constant phase angle element is... The RC obtained in step 3-2 is described using R2 to describe the charge transfer resistance R, and the double-layer capacitance C is described using a constant-phase-angle element Q1. The expression for the impedance of the constant-phase-angle element is as follows:

[0141]

[0142] In the formula, Q is the constant phase angle coefficient value; j represents the unit imaginary number; α is the constant phase angle element exponent factor; and f is the frequency.

[0143] 4-2. The equivalent circuit model obtained in step 3-3 does not have diffusion impedance. Diffusion impedance is not described by a transfer function.

[0144] 4-3. Based on the descriptions of the circuit model components obtained in steps 4-1 to 4-3, calculate the total impedance of the equivalent circuit model to obtain the true dynamic description of the equivalent circuit model in EIS, i.e., the transfer function of the equivalent circuit model. The formula is shown below:

[0145]

[0146] In the formula, R1 is the solution resistance.

[0147] Step 5: Rearrange the transfer function of the equivalent circuit model from Step 4 to separate the real and imaginary parts, obtaining the real and imaginary parts of the transfer function of the equivalent circuit model. The specific process of separating the real and imaginary parts is as follows:

[0148] 5-1. Expand the total impedance obtained in step 4-4 according to the following formula to obtain the real part Z. Re And the imaginary part Z Im :

[0149]

[0150]

[0151]

[0152] In the formula, j is the unit imaginary number.

[0153] Step 6: Transform the real and imaginary parts of the transfer function of the equivalent circuit model obtained in Step 5 to obtain the multi-objective optimization model. The specific process of transforming the real and imaginary part equations into the multi-objective optimization model is as follows:

[0154] 6-1. Calculate the optimization objective function F at the j-th frequency point using the formula below. j :

[0155]

[0156]

[0157] In the formula, j is the frequency point index; Z(ω) j Z Re (ω j Z Im (ω j ) are the effective impedance data of E. coli at the j-th frequency point obtained in step 2, the real part of the effective impedance data of E. coli, and the imaginary part of the effective impedance data of E. coli; is the original effective impedance at the j-th frequency point obtained in step 2; These represent the equivalent circuit model impedance, real part of the impedance, and imaginary part of the impedance at the j-th frequency point obtained in step 4-4, respectively; f j (R1,R2,Q1,α) represents the residual between the equivalent circuit model impedance and the original effective impedance at the j-th frequency point.

[0158] 6-2. Optimize the objective function F at a total of p frequency points obtained in step 6-1. j By combining these equations, we obtain a set of p-columns of optimization objective function equations. The equivalent circuit model formula obtained in step 4 will then be rewritten as a multi-objective function, as shown in the following formula:

[0159]

[0160] 6-3. Based on the multi-objective optimization functions obtained in steps 6-1 and 6-2, two objective functions can be obtained at each frequency point, namely, the objective function F with the optimal real part. Re,j The optimal objective function F with imaginary part Im,j For example, the experimental results for the frequency at position j=10, with the multi-objective function shown below:

[0161]

[0162] In the formula, 5.999 × 10 2 7.4983 and 7.4983 are the real part and imaginary part of the effective impedance data of E. coli at the j-th frequency point obtained in step 2, respectively.

[0163] 6-4. Add constraints to the parameters of the multi-objective optimization model obtained in step 6-2, as shown in Table 1 below, to obtain the parameter constraint list.

[0164] Table 1 Parameter Constraints

[0165]

[0166] The heuristic global optimization algorithm is used to solve the multi-objective optimization model obtained in step 6. The flow of the heuristic global optimization algorithm of this invention is as follows: Figure 2 As shown.

[0167] Step 7: Obtain the optimal estimated parameters of the circuit model. The fitted Nyquist plot is shown below. Figure 10 As shown, the logz-logf graph is as follows: Figure 11 As shown. The specific process of solving the multi-objective optimization model using the heuristic global optimization algorithm is as follows:

[0168] 7-1. Pass the multi-objective function, control parameters, and parameter constraints obtained in step 6 into the heuristic global optimization algorithm.

[0169] 7-2. Set the population size to 1000, and calculate using the following formula. and Get the initial population size X i (i = 1, 2, ..., n) and

[0170]

[0171]

[0172] In the formula, t represents the current iteration. and It is a coefficient vector. It is the position vector of the prey. This represents the position vector of the search agent. To reduce linearly from 2 to 0 during the iteration process, It is a random vector in [0,1].

[0173] 7-3. Determine whether the input parameters meet the constraints. If they do, proceed to step 7-4; otherwise, return to step 7-2 to modify the initialization conditions.

[0174] 7-4. Using the multi-objective function obtained in step 7-1 as the multi-objective fitness function, calculate the fitness function for each individual, and save the three individuals with the best fitness to obtain the three individuals with the best fitness: X a X β X δ .

[0175] 7-5. Update the current position of the search agent according to the following formula to obtain the updated position of the search agent.

[0176]

[0177] in, These are the top three individuals with the best fitness; These are the top three individuals with the best fitness after the update; This indicates the updated position of the current search agent.

[0178] 7-6. Based on the formula in step 7-2 and the updated position of the search agent obtained in step 7-5, update... and

[0179] 7-7. Based on the multi-objective fitness function obtained in step 7-4, update the fitness of each individual, and obtain the top three individuals X with the best updated fitness. a X β X δ .

[0180] 7-8. Determine whether the maximum number of iterations has been reached or the optimization stopping condition has been met. If the conditions are met, exit and return the optimal solution to end the algorithm. Otherwise, repeat steps 7-5 to 7-8.

[0181] 7-9. Plot the fitness results obtained in step 7-8 to obtain a curve showing the change in fitness with the number of iterations, as shown below. Figure 12 As shown.

[0182] 7-10. The extraction errors of each parameter obtained in step 7-8 are plotted in a table, as shown in Table 2 below.

[0183] Table 2 Parameter Extraction Error

[0184]

Claims

1. A method for extracting electrochemical impedance spectroscopy parameters using a heuristic global optimization algorithm, characterized in that, Includes the following steps: Step 1: Modify the surface of the electrode with E. coli from the culture dish, then add it to the test cell using potassium ferricyanide / potassium ferrocyanide solution as a redox probe, and collect the impedance spectrum data of E. coli using an impedance testing instrument. And transmit it to the computer; Step 2: Analysis of E. coli impedance spectroscopy data Preprocessing is performed to obtain effective impedance spectrum data; Step 3: Select the equivalent circuit model from the effective impedance spectrum data to obtain the equivalent circuit model; Step 4: Describe the equivalent circuit model using a transfer function to obtain the equivalent circuit model transfer function; Step 5: Rearrange the transfer function of the equivalent circuit model to separate the real and imaginary parts, obtaining the real and imaginary parts of the transfer function of the equivalent circuit model. The specific process is as follows: The transfer function of the equivalent circuit model is expanded using the complex number operation theorem to obtain the real part. and the virtual part : ； ； in, The solution impedance; Let C be the impedance coefficient of the nth parallel resistor R and capacitor C. Let C be the exponential coefficient of the nth parallel resistor R and capacitor C; Angular frequency; Step 6: Transform the real and imaginary parts of the transfer function of the equivalent circuit model to obtain the multi-objective optimization model; Step 7: Solve the multi-objective optimization model using a heuristic global optimization algorithm to obtain the optimal estimated parameters of the circuit model, i.e., the electrochemical impedance spectroscopy parameters. Step 8: Repeat steps 1 to 7 at preset intervals to obtain the electrochemical impedance spectroscopy parameters of Escherichia coli at different time intervals.

2. The method for extracting electrochemical impedance spectroscopy parameters using a heuristic global optimization algorithm according to claim 1, characterized in that, In step 2, the specific process of preprocessing is as follows: 2-1. Escherichia coli impedance spectroscopy data Using the real and imaginary parts of the variable as the dependent variable and the frequency as the independent variable, plot them to obtain the real-frequency ratio. Figure, Imaginary part - frequency picture; 2-2. Regarding picture, The figures show E. coli impedance spectral data obtained by applying a Savitzky-Golay filter for smoothing. ; 2-3. Impedance spectral data of E. coli after smoothing and filtering Hilbert transform verification was performed to obtain the effective impedance spectrum data of Escherichia coli.

3. The method for extracting electrochemical impedance spectroscopy parameters using a heuristic global optimization algorithm according to claim 2, characterized in that, In step 3, the specific process for selecting the equivalent circuit model is as follows: 3-1. The effective impedance spectral data of *E. coli* are visualized using Nyquist plots. If the intersection of the Nyquist plot with the real axis is not at zero, solution impedance is determined. Otherwise, it does not have solution resistance. ; 3-2. Based on the Nyquist plot and DRT transformation of the effective impedance spectrum data, determine the number of parallel resistors R and capacitors C; 3-3. If the Nyquist plot of the effective impedance data has a straight line with a 45° inclination, it indicates the participation of diffusion impedance; otherwise, the equivalent circuit model does not have diffusion impedance.

4. The method for extracting electrochemical impedance spectroscopy parameters using a heuristic global optimization algorithm according to claim 3, characterized in that, The specific process for determining the number of parallel resistors R and capacitors C, as described in section 3-2, is as follows: 3-2-1. Based on the Nyquist plot of the effective impedance spectrum data, obtain the number of semicircles on the upper part of the Nyquist plot; 3-2-2. Perform DRT transformation on the effective impedance spectrum data to obtain the number of peaks in the DRT transformation graph; exclude the overlapping semicircles caused by similar electrochemical processes in step 3-2-1; 3-2-3. The maximum number of semicircles and peaks on the Nyquist plot is the number of parallel resistors R and capacitors C in the impedance spectrum model.

5. The method for extracting electrochemical impedance spectroscopy parameters using a heuristic global optimization algorithm according to claim 4, characterized in that, The specific process of step 4 is as follows: 4-1. Using constant phase angle element impedance Describe the resistor R and capacitor C obtained in step 3-2; 4-2. If the equivalent circuit model obtained in step 3-3 has diffusion impedance, it shall be described according to the Warburg impedance formula; otherwise, it shall not be described. 4-3. Based on the descriptions of the circuit model elements and the series-parallel theorem obtained in steps 4-1 to 4-2, calculate the total impedance of the equivalent circuit model, and obtain the following true dynamic description of the electrochemical impedance spectroscopy (EIS) of the equivalent circuit model, i.e., the transfer function of the equivalent circuit model: ； in, N is the solution impedance; N2 is the number of parallel resistors R and capacitors C; n is the serial number of the parallel resistors R and capacitors C. Let R be the resistance value of the nth parallel resistor R and capacitor C. Let C be the impedance value of the nth parallel resistor R and capacitor C.

6. The method for extracting electrochemical impedance spectroscopy parameters using a heuristic global optimization algorithm according to claim 5, characterized in that, Step 6 is as follows: 6-1. Calculate the optimization objective function at the j-th frequency point using the formula below. : ； ； in, Frequency point number; The first E. coli impedance spectrum data at the frequency point, real part of E. coli impedance spectrum data, and imaginary part of E. coli impedance spectrum data; The first The equivalent circuit model impedance, real part of impedance, and imaginary part of impedance at the frequency point; For the first The residual between the equivalent circuit model impedance and the original effective impedance at the frequency point; 6-2. Optimize the objective function at the p frequency points obtained in step 6-1. By combining these equations, we obtain a set of p-columns of objective function equations, which are the multi-objective optimization functions for the equivalent circuit model.

7. The method for extracting electrochemical impedance spectroscopy parameters using a heuristic global optimization algorithm according to claim 6, characterized in that, Step 6 also includes: in the multi-objective optimization function, two objective functions can be obtained at each frequency point, namely the real-optimal objective function. The optimal objective function of the imaginary part ; Add constraints to the control parameters of the multi-objective optimization function obtained in step 6-2: Limit the upper and lower resistance values ​​within [...]. , In; the upper and lower limits of the capacitance coefficient are constrained within [ , In; the upper and lower limits of the exponential coefficient are constrained within [ , 1] in.

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