Med Biol Eng Comput DOI 10.1007/s11517-014-1175-5
Original Article
Extracting the parameters of the double‑dispersion Cole bioimpedance model from magnitude response measurements Todd J. Freeborn · Brent Maundy · Ahmed S. Elwakil
Received: 7 May 2013 / Accepted: 25 June 2014 © International Federation for Medical and Biological Engineering 2014
Abstract In the field of bioimpedance measurements, the Cole impedance model is widely used for characterizing biological tissues and biochemical materials. In this work, a nonlinear least squares fitting is applied to extract the double-dispersion Cole impedance parameters from simulated magnitude response datasets without requiring the direct impedance data or phase information. The technique is applied to extract the impedance parameters from MATLAB simulated noisy magnitude datasets showing less than 1.2 % relative error when 60 dB SNR Gaussian white noise is present. This extraction is verified experimentally using apples as the Cole impedances showing less than 3 % relative error between simulated responses (using the extracted impedance parameters) and the experimental results over the entire dataset. Keywords Bioimpedance · Cole impedance model · Magnitude/modulus retrieval
1 Introduction The field of bioimpedance, which measures the passive electrical properties of biological materials, uses these measurements to give information about the electrochemical processes in tissues and can be used to characterize tissues or monitor physiological changes [1, 2, 16]. One T. J. Freeborn (*) · B. Maundy Department of Electrical and Computer Engineering, University of Calgary, 2500 University Dr. N.W., Calgary, Canada e-mail:
[email protected] A. S. Elwakil Department of Electrical and Computer Engineering, University of Sharjah, P.O. Box 27272, Sharjah, United Arab Emirates
of the most widely used models in the field of bioimpedance is the Cole impedance model introduced by Kenneth Cole in 1940 [5] and is widely used for characterizing biological tissues and biochemical materials. In literature, this model has also been commonly referred to as the Cole– Cole model or Cole–Cole impedance model. However, the Cole–Cole model is actually a similar model introduced by the Cole brothers in 1941 regarding dielectric permittivity [6]. Therefore, care needs to be taken when describing the model to prevent confusion between work with dielectric permittivity and impedance. The single-dispersion Cole model, shown in Fig. 1a, is composed of three hypothetical circuit elements. A high-frequency resistor R∞, a resistor R1 and a constant phase element (CPE), with impedance ZCPE = 1/(jω)α C or 1/sα C in the s-domain, where C is the pseudo-capacitance and α is its order. The units of the pseudo-capacitance are F/s1−α and were originally proposed in [27] where [s] is the unit of time not to be confused with s, the Laplace transform operator. In this work, the units of the pseudo-capacitance are described as [F] for simplicity. The name of the CPE is in reference to the phase angle, φCPE, which is independent of frequency and dependent only on the order, α, given as φCPE = απ/2. While α ∈ R is mathematically possible, the values from experimentally collected data are typically in the range of 0 < α < 1. These devices have also been called fractional capacitors, in reference to their order which takes a value between the traditional circuit elements of a resistor and capacitor. It is for this reason that we use a capacitor as the schematic representation of CPEs in this work. The impedance of the single-dispersion Cole model is then given by
Z(s) = R∞ +
R1 = Z ′ + jZ ′′ 1 + sα1 R1 C1
(1)
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• Investigating the age-related changes of human dentine with potential to create nondestructive test methods applied to early caries and micro-leakage identification [8];
(a)
(b) Fig. 1 Theoretical a single-dispersion Cole impedance model and b double-dispersion Cole impedance model
Noting that sα = (jω)α = ωα cos (απ/2) + j sin (απ/2) . This model has become very popular because of its simplicity and good fit with measured data, illustrating the behavior of impedance as a function of frequency. An expanded model, the double-dispersion Cole model, is used to accurately represent the impedance over a larger frequency range or for more complex materials. This model, shown in Fig. 1b, is composed of an additional parallel combination of a resistor (R2) and CPE in series with the single-dispersion Cole model with total impedance given by Z(s) = R∞ +
R2 R1 + 1 + sα1 R1 C1 1 + sα2 R2 C2
(2)
Physiologically, the resistances in this model are contributed by the numerous intracellular, extracellular, and cellular membrane resistances within the tissue, with capacitance contributed by the membrane capacitances of the numerous tissue cells. The parameters α1,2 are dimensionless quantities known as dispersion coefficients. It is possible to regard them in several ways, including, as distributions of relaxation times caused by the heterogeneity of cell sizes and shapes, measures of the deviation from ideal capacitors in the equivalent circuit, or as measures of physical processes like the Warburg diffusion [16]. Now, while these models do not provide an explanation of the underlying mechanisms, there has been a large and expanding body of research regarding their use [12]. Specifically, the double-dispersion Cole model and its parameters have been investigated for applications in: • Characterizing intestinal tissue excised from sheep [24]; • Monitoring necrosis of human tumor xenografts during and/or after hyperthermia treatment [21];
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• To characterize a particular tissue or material using the double-dispersion Cole model requires the determination of the seven impedance parameters (R∞ , R1 , R2 , α1 , α2 , C1 , C2). Early methods extracted the parameters graphically from an impedance plot relating the imaginary impedance, Z ′′, to the real impedance, Z ′. With the rise of computers and very powerful numerical fitting software, the majority of parameters are now estimated using nonlinear least squares routines fitting experimental data to the desired model. Parameters are selected such that the least squares error between the experimental data and estimated response are minimized. While these fitting processes were initially applied to impedance data [19], research has been expanded to extract the parameters without requiring direct measurement of the impedance. Instead, parameters are extracted only from the real part of the impedance (Z ′ ) [3, 26], the imaginary (Z ′′ ) part, or the modulus [3, 4, 13, 22] component of the impedance and even from the time domain step response [14, 15]. Methods without requiring fitting routines have also been investigated to extract the parameters from the magnitude response [9, 20] and the time domain response to a triangle-wave current input [10]. A significant motivation in the research of alternative methods for extracting the impedance parameters is to reduce the amount of hardware and cost of instruments for these measurements [4]. Traditionally, to collect the impedance data requires an impedance analyzer which is expensive and not portable. Low-cost hardware to accomplish this same task was reported in [17, 25]. Lowering the cost of instruments that extract the Cole parameters without requiring direct measurement of the impedance has the potential to significantly reduce the barriers to conduct research, especially with regard to real-time monitoring with portable devices [11]. In the following sections, a method of nonlinear least squares fitting (NLSF) is applied to extract the double-dispersion Cole impedance parameters from MATLAB simulated noisy magnitude response datasets without requiring impedance data. The impact of the noise on the accuracy of the NLSF is analyzed with respect to the relative error of the extracted impedance parameters. The NLSF is further verified with PSPICE simulations using approximated CPEs in the realization of the double-dispersion Cole impedance and experimental results collected when apples are used as the Cole impedances.
Med Biol Eng Comput
where a1 = 1/(C2 R2 ), a2 = 1/(C1 R1 ), a3 = 1/(C1 C2 R1 R2 ), b1 = (R∞ + R2 + RL )/(C2 R2 (R∞ + RL )), b2 = (R∞ + R1 + RL ) /(C1 R1 (R∞ + RL )), b3 = (R∞ + R1 + R2 + RL )/(C1 C2 R1 R2 (R∞ + RL )). The magnitude response of (3) is given by
X02 + X12 RL · |T (jω)| = R0 + RL X22 + X32
Cole Impedance Model
where the values for the coefficients X0 , X1 , X2, and X3 are given in Table 1. From the magnitude response, we note that there are three distinct regions: (1) at very low frequencies, when the CPEs behave as open circuits with very high impedance the response is a constant value resulting from the voltage divider formed by R∞ , R1 , R2, and RL, (2) an increasing magnitude region resulting from the decrease in impedance of the CPEs, and (3) at high frequencies, when the CPEs behave as short circuits with very low impedance the response is a constant value resulting from the voltage divider formed by R∞ and RL.
Fig. 2 Circuit for the collection of the magnitude response of the double-dispersion Cole impedance
2 Methodology 2.1 Double‑dispersion magnitude response The typical method to extract the Cole impedance parameters requires direct measurement of the impedance of a connected material. Requiring the hardware to apply a sinusoidal voltage or current at various frequencies to excite the tissue and then measure the magnitude and phase of the corresponding response for further data processing. However, in [9] and [20], a single-dispersion Cole impedance was used as a component in a filter and integrator circuit, respectively, from which the impedance parameters were estimated directly from the collected magnitude response when a sinusoidal voltage was applied to a connected tissue. This method reduces the hardware to extract the impedance parameters by removing the need to measure the phase response. The accuracy of these methods were improved using a NLSF in [13] over the direct calculations. The application of a NLSF method is extended here to extract the parameters of a double-dispersion Cole impedance when used as a component in the circuit shown in Fig. 2. The transfer function of this circuit is given by
T (s) =
sα1 +α2 + a1 sα1 + a2 sα2 + a3 RL · α +α R∞ + RL s 1 2 + b1 sα1 + b2 sα2 + b3
(4)
2.2 Nonlinear least squares fitting To extract the impedance parameters from a collected magnitude response dataset, a NLSF numerical method can be applied that attempts to solve the problem
min f0 (x) = x
n
|T (x, ωj )| − yj
j=1
2
(5)
s.t. x > 0 where x is the set of impedance parameters to minimize fo (x), |T (x, ωj )| is the magnitude response given by (4) at frequency ωj , yj is the collected magnitude response at frequency ωj, and n is the total number of data points in the collected response. This routine aims to find the impedance parameters that would ideally reduce the least squares error to zero, but more realistically reduces it to the level of noise within the dataset. The constraint is added to the problem because negative resistances and capacitances are
(3)
Table 1 Magnitude response coefficients of the circuit in Fig. 2 given by (4) Coefficient
Values
X0
ωα1 +α2 cos
X1
ωα1 +α2 sin
ωα1 +α2 cos
ωα1 +α2 sin
X2 X3
(α1 +α2 )π 2
+
ωα1 C2 R2
cos
(α1 +α2 )π 2
+
ωα1 C2 R2
sin
(α1 +α2 )π 2
(α1 +α2 )π 2
α1 π 2
α1 π 2
ωα2 C1 R1
+
+
ωα2 C1 R1
cos sin
2 +RL cos +ωα1 CR2 R0 +R 2 (R0 +RL )
α1 π
2 +RL sin + ωα1 CR2 R0 +R 2 (R0 +RL )
α1 π
2
2
α2 π 2
+
1 C1 C2 R1 R2
α2 π 2
1 +RL +ωα2 CR1 R0 +R cos 1 (R0 +RL )
α2 π
1 +RL + ωα2 CR1 R0 +R sin 1 (R0 +RL )
2
1 +R2 +RL + C1RC0 2+R R1 R2 (R0 +RL )
α2 π 2
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Med Biol Eng Comput
not physically possible, and a negative value of α1,2 would indicate fractional-order inductive characteristics and not fractional-order capacitive characteristics. To begin, the NLSF method requires an initial condition, x0, from which to start iteratively solving toward a solution x ∗. This NLSF when implemented in MATLAB using the default settings will quit when any of the following conditions are satisfied during each iteration:
R1 + R2 ≥
RL − |T (ω1 )|(R∞ + RL ) |T (ω1 )|
(7)
Finally, the range of values reported in the literature is used to create the bounds to generate the initial conditions of the CPEs such that 0.5 ≤ α1,2 ≤ 1 and 0.1 nF≤ C1,2 ≤ 1 mF. 3 Results
• The number of function evaluations exceeds 100 · m, where m is the number of impedance parameters in x; • The number of iterations of the algorithm exceeds 400; • The change in the value of x is less than 1 × 10−6; • The change in the function value is less than 1 × 10−6. • After quitting due to meeting any of the ending criteria, the solver returns a possible solution. However, there may be many local minima that satisfy the ending criteria without being the true global solution. Therefore, the solver has the potential to return solutions that are not the true impedance parameters. One method to overcome this problem is to run the solver multiple times, each time using a new randomly generated x0. The intent is to generate a initial starting condition for the solver that returns the global solution when the ending criteria are met and avoids returning a local minima. Therefore, by solving (5) from each x0 and selecting the solution set that yields the lowest least squares error increases the likelihood of finding the global solution. The number of the applied initial conditions should be increased when the impedance parameters span a very large range of values because there is a greater chance of local minimums being returned as possible solutions using the least squares fitting. 2.3 Initial conditions To increase the likelihood of generating initial conditions that are close to the solution, characteristics of the magnitude response are used to create boundaries in which to generate them. We note from (4) that at ω = 0 and ω = ∞ the magnitude will be |T (0)| = RL /(R∞ + R1 + R2 + RL ) and |T (∞)| = RL /(R∞ + RL ). Since the collected magnitude response will not extend from ω = 0 to ω = ∞ but is instead finite in the range 0 < ω1 < ωn < ∞, where ω1 and ωn are the lowest and highest frequencies of the collected magnitude response, the values collected will only be an approximation of their ideal counterparts. Therefore, they can be used as lower and upper boundaries when generating R∞ , R1, and R2 for the initial conditions such that
R∞ ≤
RL (1 − |T (ωn )|) |T (ωn )|
13
(6)
3.1 MATLAB simulations The accuracy of extracting double-dispersion impedance parameters from the magnitude response data is assessed by extracting the parameters from 100 MATLAB simulations of (4) with increasing levels of Gaussian white noise with signal-to-noise (SNR) ratios from 30 dB to 80 dB in 10 dB steps. For all simulations, the datasets were generated from 10 Hz to 25 MHz with 50 logarithmically spaced datapoints using impedance parameters R∞ = 42.9 �, R1 = 71.6 �, R2 = 16.5 �, α1 = 0.507, α2 = 0.766, C1 = 3.086 µF, C2 = 89.29 µF, which were previously extracted from sheep intestinal tissue in [24]. The algorithm described in Sect. 2.2 and implemented for these simulations is given in Fig. 3. The algorithm begins by setting a global error value, Eg = ∞, which is used after each iteration to evaluate whether the returned impedance parameters provide the least error compared to the magnitude dataset of all iterations. Eg is initially set to infinity to ensure the solution of the NLSF first iteration, which will return a finite value, is saved. Next, an initial condition (x0) is randomly generated within the described boundaries and used as the starting condition for the NLSF. After executing, a solution (x ∗) is returned and used to determine the current least squares error (Ec). If the current least squares error is less than the global error (Ec < Eg), the current solution is saved as global solution (xg = x ∗). This process continues if the current iteration is less than the defined maximum iterations (n = 20 for these simulations). When the current iteration exceeds the defined maximum, the algorithm ends returning the current value of xg as the possible solution. The NLSF was implemented using the MATLAB function lsqcurvefit, which implements the trustregion-reflective algorithm [7]. The details of the trustregion-reflective algorithm are extensively described in [7] and the MATLAB online documentation for those readers interested in a thorough description of the solver and its process. The evolution of the mean error for each parameter while the NLSF applies 20 initial conditions to the magnitude data is shown in Fig. 4a. The impact of the number of initial conditions is clear showing a significant decrease in the relative error of
Med Biol Eng Comput Start Set global error
Generate random initial condition
Apply NLSF
Calculate Error
Update impedance parameters
Compare least squares error
Yes
No Check iteration No Yes End
Fig. 3 Algorithm implemented in MATLAB to extract the impedance parameters using NLSF from simulated data with 20 applied initial conditions
each parameter between the first and twentieth applied initial conditions. After solving using the first initial condition, the extracted parameters show relative errors of [57, 43, 5518, 150, 227, 1.05 × 107 , 6.66 × 105 ] % for [R∞ , R1 , R2 , α1 , α2 , C1 , C2 ], respectively, with reductions to [0.165, 0.135, 0.215, 0.133, 0.213, 0.872, 1.18] % after the application of 13 initial conditions with no significant decrease in the error after this point. The use of multiple initial conditions shows reductions in relative errors from 3 to 7 orders of magnitude over the single initial condition. The impact of increasing levels of noise on the relative error of the extracted parameters is given in Table 2. When the SNR of the noise decreases to 40 dB, there is a significant increase in the relative error of the extracted parameters with all showing larger than 1 % relative error compared to the ideal values.
An example of a simulated dataset with 60 dB SNR (solid line) compared to simulations using the extracted parameters (circles) is given in Fig. 4b. From this dataset, the datapoints collected at 10 Hz and 25 MHz are |T (10 Hz)| = −7.263 dB = 0.6322 and |T (25 MHz)| = −3.9827 dB = 0.4334, respectively. Therefore, from (6) and (7), the lower and upper boundaries for the generation of the initial conditions are R∞ ≤ 58.17 and R1 + R2 ≥ 72.58 . Applying the NLSF with 20 initial conditions yields relative errors of [0.025, 0.029, 0.204, 0.008, 0.273, 0.095, 1.73] % for [R∞ , R1 , R2 , α1 , α2 , C1 , C2 ], respectively. The simulated dataset using the extracted parameters shows less than 0.05 % relative error with the simulated noisy dataset over the entire frequency range. The impedance parameters previously extracted from simulations were within a small range of resistance values (R < 100 �); though it is still possible to extract the impedance parameters from datasets generated with resistances from a larger range. However, to maintain the same level of accuracy, it is necessary to increase the number of applied initial conditions. The evolution of the mean error for each impedance parameter while the NLSF applies 60 initial conditions to 100 magnitude datasets randomly generated such that R∞,1,2 ≤ 1 k�, 0.5 ≤ α1,2 ≤ 1, 0.1 nF ≤ C1,2 ≤ 1 mF with 60 dB SNR Gaussian white noise is shown in Fig. 4c. The impact of the number of initial conditions is clear showing a significant decrease in the relative error of each parameter between the first and sixtieth applied initial conditions. After solving using 20 initial conditions, the extracted parameters show relative errors of [0.227, 0.176, 0.487, 0.253, 0.224, 4.45, 1.33] % for [R∞ , R1 , R2 , α1 , α2 , C1 , C2 ], respectively, with reductions to [0.123, 0.121, 0.176, 0.126, 0.104, 1.60, 0.559] % after 60 initial conditions. An example of a simulated dataset with parameters [R∞ , R1 , R2 , α1 , α2 , C1 , C2 ] = [885 , 899 , 626 , 0.5414, 0.7653, 19 nF, 5.1 µF], RL = 100, and 60 dB SNR Guassian white noise (solid line) compared to simulations using the extracted parameters (circles) is given in Fig. 4d. Applying the NLSF with 60 initial conditions yields relative errors of [5.44, 5.37, 0.02, 0.36, 0.06, 7.21, 0.36] % for [R∞ , R1 , R2 , α1 , α2 , C1 , C2 ], respectively. This same process can be scaled up to extract the parameters from even larger ranges of impedance values by further increasing the number of applied initial conditions to the NLSF. 3.2 PSPICE simulations While there are currently no CPEs available for simulation in standard circuit simulator software packages,
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(a)
(c)
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Error Error Error
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Error Error
Error Error
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Average Relative Error (%)
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0
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Error Error
Error Error Error
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(d)
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40
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10 6
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10 7
Simulated Fitted
−26.5
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35
−26
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30
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−27
10 2
25
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Magnitude (dB)
−4
15
Initial Conditions
Initial Conditions
10 2
10 3
10 4
10 5
10 6
10 7
Frequency (Hz)
Frequency (Hz)
Fig. 4 Relative error evolution of the extracted impedance parameters and simulated noisy response (solid) compared to the response using the extracted parameters (circles) for different double-dispersion impedance parameter ranges
Table 2 Average relative error and standard deviations of impedance parameters extracted from 100 MATLAB simulations of (4) with 30– 80 dB SNR Gaussian white noise SNR (dB)
Relative error (%) ± SD R∞
R1
R2
α1
α2
C1
C2
30 40 50 60 70
4.92 ± 3.87 1.29 ± 1.12 0.453 ± 0.398 0.165 ± 0.125 0.049 ± 0.037
3.84 ± 3.02 1.09 ± 0.912 0.366 ± 0.322 0.135 ± 0.098 0.039 ± 0.028
6.25 ± 5.15 2.06 ± 1.70 0.641 ± 0.426 0.215 ± 0.0148 0.063 ± 0.045
3.54 ± 2.59 1.04 ± 0.924 0.358 ± 0.283 0.133 ± 0.095 0.036 ± 0.026
5.88 ± 4.41 1.88 ± 1.43 0.598 ± 0.416 0.213 ± 0.140 0.060 ± 0.042
23.6 ± 20.7 7.02 ± 6.36 2.44 ± 1.83 0.872 ± 0.615 0.240 ± 0.164
34.9 ± 33.2 10.1 ± 8.21 3.23 ± 2.18 1.18 ± 0.816 0.330 ± 0.236
80
0.013 ± 0.01
0.010 ± 0.008
0.017 ± 0.013
0.009 ± 0.007
0.017 ± 0.015
0.062 ± 0.047
0.098 ± 0.084
there are many methods to create an approximation of sα that include continued fraction expansions (CFEs) as well as rational approximation methods [23], which can be implemented for simulation. These methods present a large array of approximations with the accuracy and approximated frequency band increasing as the order of the approximation increases. Here, a CFE method [18] was selected to model the CPEs for PSPICE simulations. Collecting twenty terms of the CFE yields a tenth-order approximation of the fractional capacitor that can be
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physically realized using the RC ladder network in Fig. 5a when n = 10. The component values required to realize tenth-order approximated fractional capacitors with values C = 3.086 µ F, α = 0.507 and C = 89.29 µF, α = 0.766 centered at f = 60 kHz are given in Table 3. The absolute error of the approximations magnitude compared to the ideal responses is given in Fig. 5b. Both approximations show deviations of less than 0.1 dB over 3 decades but show larger deviations in the frequency
Med Biol Eng Comput
(a)
Absolute Magnitude Error (dB)
(b)
1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 10 2
10 3
10 4
10 5
10 6
10 7
Frequency (Hz)
Fig. 5 a RC ladder structure to realize a nth order integer approximation of a fractional-order capacitor and b the error of tenth-order approximated fractional capacitors with values C = 3.086 µF, α = 0.507 (solid) and C = 89.29 µF, α = 0.766 (dashed) centered at f = 60 kHz
bands from 100 to 1,410 Hz and 2.4 to 25 MHz, reaching maximums of 1.182 dB at 379 Hz and 9.52 MHz for C = 3.086 µF, α = 0.507 and 1.246 dB at 265 Hz and 13.4 MHz for C = 89.29 µF, α = 0.766. The circuit in Fig. 2 with a double-dispersion Cole impedance (R∞ = 42.9 , R1 = 71.6 , R2 = 16.5 , α1 = 0.507, α2 = 0.766, C1 = 3.086 µF, C2 = 89.29 µF), RL = 100 and CPEs approximated using the component values in Table 3 was simulated using an OPA655 (400 MHz unity gain bandwidth) operational amplifier. For this simulation, a sinusoidal voltage, with peak-to-peak amplitude of 1 V and zero DC offset was applied at 55 logarithmically spaced frequencies from 100 Hz to 25 MHz. The
collected PSPICE simulated magnitude response is given in Fig. 6a as a dashed line compared against the ideal response using (4), given as a solid line. From the PSPICE simulated response, parameters of [R∞ , R1 , R2 , α1 , α2 , C1 , C2] were extracted as [50.3 , 62.3 , 17.5 , 0.588, 0.769, 11.12 µF, 63.53 µF] using 20 initial conditions with the NLSF. Compared to the ideal parameters, the extracted parameters show relative errors of [17.1, 13.1, 6.09, 16.0, 0.33, 64.0, 28.9] %, respectively. The much larger relative errors of the PSPICE simulation compared to the MATLAB simulations are attributed to the deviations introduced using the approximations of the fractional capacitors. These approximations combined with the nonidealities introduced by the OPA655 result in deviations greater than 0.75 % above 4 MHz reaching a maximum of 3.14 % at 25.1 MHz. Note though that the response using the extracted parameters shows a very good fit with the PSPICE response showing less than 0.5 % error until 1 MHz and less than 2 % error over the entire response. The relative errors of the PSPICE response compared to the ideal and the response using the extracted parameters compared to the PSPICE response are given in Fig. 6b as squares and circles, respectively. 3.3 Experimental results To validate the extraction method, the magnitude response of the circuit in Fig. 2 was collected when apples were used as the double-dispersion Cole impedances. This magnitude response was collected using an Agilent 54622D mixed signal oscilloscope when a 1 V peak-to-peak sinusoidal waveform with frequencies from 100 mHz to 1 MHz, in 64 logarithmic steps, was applied by a Agilent 33250A. In the experimental set-up, a OPA275 op amp with 9 MHz gain bandwidth product was used to buffer the output voltage, vo, when RL = 497 . Note that RL was selected such that the lowest frequency measurement (at 0.1 Hz) would be
Table 3 Values to realize tenth-order approximated fractional capacitors with values C = 3.086 µF, α = 0.507 and C = 89.29 µF, α = 0.766 centered at f = 60 kHz CPE
R0
R1
R2
R3
R4
R5
R6
R7
R8
R9
R10
C = 3.086 µF α = 0.507
21.9
45.8
49.5
56.0
67.0
85.4
118.7
185.5
349.7
951.8
8,681
C = 89.29 µF α = 0.766
3.88 m
15.5 m
23.6 m
33.5 m
48.2 m
72.7 m
120 m
227 m
546 m
2.21
89.2
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
C = 3.086 µF α = 0.507
1.33 n
5.12 n
11.0 n
18.5 n
26.8 n
35.2 n
43.0 n
49.4 n
53.7 n
55.3 n
C = 89.29 µF
4.89 µ
12.2 µ
20.2 µ
27.8 µ
34.0 µ
37.8 µ
38.6 µ
35.7 µ
28.4 µ
12.5 µ
CPE
α = 0.766
All resistor and capacitor values given in [ ] and [F], respectively
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Med Biol Eng Comput −4 −4.5
(a) −10
Ideal PSPICE Fitted
−15
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−5 −5.5 −6 −6.5
−25 −30 −35 −40 −45 −50
−7
−55 −1
−7.5
10 2
10
10 3
10 4
10 5
10 6
−15
Magnitude (dB)
Relative Error (%)
10
1
−1
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−2
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10
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−20 −25 −30 −35 −40
−3
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−45
−4
10 3
10 4
10 5
10 6
−50 −1
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10 7
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0
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1
greater than 1 mV (the minimum value measurable by the Agilent 54622D). From the experimental magnitude responses, parameters [R∞ , R1 , R2 , C1 , C2 , α1 , α2] were extracted as [371 , 5.28 k , 58.6 k , 112 nF, 24.2 µF, 0.768, 0.762] and [327 , 5.56 k , 16.8k , 95.6 nF, 49.2 µF, 0.778, 0.638] for apple 1 and 2 , respectively, using the NLSF with 30 applied initial conditions. The MATLAB simulated responses using these parameters are given in Fig. 7a, b as circles and squares, respectively. These simulated responses show excellent agreement with the experimental results, with the relative errors between the experimental and simulated datapoints given in Fig. 7c as solid and dashed lines for apple 1 and apple 2, respectively. The simulated results show less than 3 % relative error over the entire collected datasets.
4 Discussion This work extends applying a NLSF to collected magnitude response datasets of the single-dispersion Cole
(c)
101
Relative Error (%)
Fig. 6 a Simulated ideal (solid), PSPICE (dashed) magnitude responses when (R∞ = 42.9 �, R1 = 71.6 �, R2 = 16.5 �, α1 = 0.507, α2 = 0.766, C1 = 3.086 µF, C2 = 89.29 µF) and RL = 100 compared to the simulated response using the extracted parameters (circles) and b relative errors of the PSPICE simulation compared to the ideal (squares) and simulated response using the extracted parameters compared to the PSPICE response (circles)
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Fig. 7 Experimentally collected step responses of a apple 1 (solid) and b apple 2 (dashed) compared to MATLAB simulated responses using extracted impedance parameters (circles and squares) and c relative errors of apple 1 and 2 simulated responses compared to the experimental
impedance model to include the double-dispersion model. Including the double-dispersion model increases the complexity and frequency range of materials that can be characterized and investigated with this indirect measurement method. Conventional techniques typically require the direct measurement of either the magnitude and phase or voltage and current of a connected tissue excited by a sinusoidal input. Though, extracting the impedance parameters entirely from magnitude data eliminates the need to measure the phase response entirely. This can potentially reduce the hardware and cost while increasing the portability of instrumentation to characterize
Med Biol Eng Comput
biological tissues, which will reduce the barriers to conduct research in this field. The simulations and experimental results confirm that extracting the impedance parameters of the double-dispersion Cole impedance model using a NLSF applied to magnitude response datasets is both possible and accurate within certain bounds. These bounds require a high SNR of the magnitude data for accurate extraction of the impedance parameters. From the average relative errors given in Table 2 when the SNR decreases below 40 dB, there is a significant increase in the relative error of the extracted parameters, which all show larger than 1 % error compared to the ideal values reaching a maximum of 23.6 and 34.9 % for C1 and C2, respectively, when the SNR is 30 dB. Therefore, an instrument to implement this extraction method must be designed to maximize the SNR to maintain a high extraction accuracy. The evolution error demonstrated in Fig. 4a and c illustrates the need for multiple initial conditions to be applied to the NLSF, with larger numbers required for wider ranges of values of the impedance parameters. Further design considerations when implementing this method include: 1. the optimum number of collected datapoints in the magnitude response with larger datasets increasing the collection and processing time; 2. the frequency range of the collected response with a wider range will yield closer initial estimates of R∞ and R1 + R2 which could reduce processing time but require higher precision circuitry; 3. the range of values required for RL such that the response can be adjusted to lie within the most sensitive input range of the measurement circuitry; 4. the optimum number of initial conditions to apply with higher numbers increasing the accuracy but also the processing time; and 5. the impact of the physical circuitry (operational amplifier bandwidth and nonlinearities, electrode impedance, analog-to-digital conversions, etc) on the extraction accuracy. The effort to simulate a biological tissue using CPEs to collect the magnitude response in PSPICE also highlights the need to expand circuit simulation tools to include fractional-order elements to simplify working with these models. Current methods require significant effort from measurement of fractional impedances to implementation of circuit models for simulation. Expanding these tools will remove the expertize required to implement approximated fractional-order circuit models and encourage researchers to conduct simulations in the field of bioimpedance.
In conclusion, a NLSF applied to magnitude response datasets to extract the impedance parameters that characterize a double-dispersion Cole impedance model has been outlined and verified in simulation and with experimentally collected results. The potential reduction of the required circuitry by eliminating phase measurements has significant benefits to furthering the goal of low-cost instrumentation of bioimpedance research and warrants continued investigation. Acknowledgments Todd Freeborn would like to acknowledge Canada’s National Sciences and Engineering Research Council (NSERC), Alberta Innovates—Technology Futures, and Alberta Advanced Education & Technology for their financial support of this research through their graduate student scholarships.
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