Cole Electrical Impedance Model—A Critique and an Alternative

Cole Electrical Impedance Model—A Critique and an Alternative
Cole Electrical Impedance Model—A Critique and an
Sverre Grimnes and Ørjan G. Martinsen*
Abstract—The Cole single-dispersion impedance model is based upon a
constant phase element (CPE), a conductance parameter as a dependent
parameter and a characteristic time constant as an independent parameter.
Usually however, the time constant of tissue or cell suspensions is conductance dependent, and so the Cole model is incompatible with general relaxation theory and not a model of first choice. An alternative model with conductance as a free parameter influencing the characteristic time constant of
the biomaterial has been analyzed. With this free-conductance model it is
possible to separately follow CPE and conductive processes, and the nominal time constant no longer corresponds to the apex of the circular arc in
the complex plane.
Index Terms—Bioimpedance theory, biomaterial, cell suspension model,
Cole model, constant phase element (CPE), dielectric, electrical network
theory, electrical relaxation theory, tissue model.
Fig. 1. Single-dispersion equivalent circuits. (a) Debye circuit with ideal
components, (1). (b) Cole circuit with the ideal capacitor replaced by a CPE
with frequency-dependent components, (2) and (4).
] 1
Z = R1 + R0 0 R1 1 + (j!Z )
TheColeimpedancemodel was introduced in itsfinalform byKenneth
Cole in 1940 [1]. It is based upon the replacement of the ideal capacitor in
the Debye model [2] with a moregeneral constantphaseelement(CPE)as
shown in the equivalent circuit diagrams, Fig. 1. The idea of introducing
a CPE came after a number of findings, both in electrochemistry and with
tissue and cell suspensions, that measured impedance loci in the complex
plane were in the form of circular arcs with depressed circle centers, and
that such arcs were best modeled with a CPE.
The CPE together with a circular arc complex impedance analysis
were introduced by Cole as early as 1928 [3]. A Cole impedance model
consists however of three parts: an equation, an equivalent circuit, and
a complex impedance circular arc locus. The impedance model developed gradually in the 1930s until completed with the mathematical
equation [1]. In 1941 the Cole brothers introduced a similar model for
dielectric permittivity, the Cole-Cole model [4]. A Cole model is, therefore, an impedance model, a Cole-Cole model a permittivity model.
The popularity of the models is due to the condensed complex form
and the elegant and simple mathematical appearance, but as we shall see
this is unfortunately at the expense of general applicability. It is the purpose of this paper to present a critique of the Cole model, and to analyze
an alternative model with the time constant as a conductance dependent
parameter. Such a model is in general better suited for tissue and cell suspensions, and actually this model often implicitly has been used. Only
the basic Cole impedance model will be treated describing a single ideal
dispersion. Often a biomaterial shows multiple dispersions, but it is not
the purpose of this paper to analyze such systems, a more detailed description also of multiple-dispersion systems can be found in [5].
The Debye circuit [2] is shown on Fig. 1(a). The impedance of the
circuit is
Z = R1 +
Gvar + Gvar j!Z
= C=Gvar
Manuscript received June 6, 2003; revised May 9, 2004. Asterisk inidcates
corresponding author.
S. Grimnes is with the Department of Clinical and Biomedical Engineering,
N-0027 Oslo, Norway and also with the Department of Physics, University of
Oslo, N-0316 Oslo, Norway.
*Ø. G. Martinsen is with the Department of Physics, University of Oslo, POB
1048, Blindern, N-0316 Oslo, Norway (e-mail:
Digital Object Identifier 10.1109/TBME.2004.836499
where Z is complex impedance
; R is resistance at very high
frequencies, j is the imaginary unit, ! is the angular frequency [1/s],
Z is the characteristic relaxation time constant [s] of the circuit cor=Z ; C is the
responding to a characteristic angular frequency !Z
parallel capacitance [farad, F], and Gvar is an independent parameter
conductance [siemens, S].
The Cole empirical equation for the frequency dependence of tissue
or cell suspension complex impedance is [1]
where R0 is the resistance at very low frequencies, Z is the characteristic time constant of the system corresponding to a characteristic
angular frequency !Z
=Z , and is an exponent [dimensionless].
The product !Z is dimensionless, and j!Z represents a CPE
as long as is constant because
j = cos(=2) + j sin(=2)
where = 'CPE =90 .
The equivalent circuit of a CPE consists of a resistor and a capacitor,
both frequency dependent (nonideal) so that the phase 'CPE becomes
frequency independent. Equations (1) and (2) describe and define one
ideal dispersion. A more detailed description can be found in [5].
= G the Cole equation becomes
By introducing R0 0 R1
=1 1
Z = R1 +
1G + 1G(j!Z ) :
From (4), the capacitance CCPE of the CPE capacitor is found to be
CCPE = 1G (!Z ) sin(=2):
The dimension of G=! and, therefore, CCPE , is [S s] or [F]. An equivalent circuit is shown in Fig. 1(b).
As described in [5], (4) reveals that the Cole model presupposes a
CPE element in parallel with an ideal conductance G. However, the
CPE admittance G j!z is proportional to that same conductance
G. Thus, a parallel conductance is not an independent parameter in
the Cole model. Equation (5) shows that CCPE is proportional to G
with the consequence that z is independent of G, just as Z in (1) is
constant if C and Gvar vary by the same factor.
is related to the constant phase of the CPE according to (3), as well
as to the frequency exponent in the term ! . This double influence of reveals that the Cole model is based upon a Fricke compatible system.
According to Fricke’s law [5], [6], the phase angle ' and the frequency
exponent m of the impedance in many electrolytic systems are related
so that
1 (
' = m 1 90
Fricke's law:
In such cases it is common practice to set m .
An admittance version of the Cole equation [5] is shown in Fig. 2
0018-9294/04$20.00 © 2005EEE
0 G0
= G0 + 1 +G1(j!
Y )0
Fig. 2. Admittance Cole model. (a) Equivalent circuit according to (7) with
= R. (b) Complex plane Y -plot. ! is the characteristic
frequency, from which is found: =! .
( =1 )
A general equation without any constraints of Fricke compatibility
Gvar + Kj !m Z
where K is a real proportionality factor [S sm0 ] for the CPE admittance.
This equation is compatible neither with the Cole model, nor the
Fricke law, nor Kramers–Kronig transforms. It is not a very attractive
alternative, but shown here for the sake of completeness.
It is possible to stick to the condition that the system shall be Fricke
compatible and, therefore, also Kramers–Kronig compatible by introducing an independent ideal conductance Gvar in parallel with the CPE
(in the case of admittance: ideal resistance Rvar in series). These earlier proposed [5] equations are
Gvar + G1 (j!Z )
Y = G0 +
Rvar + R1 (j!Y )0
Fig. 3 shows the most adequate equivalent circuits.
To analyze (9) we use the Z -version as an example in the following.
The complex admittance of the CPE element is: CPE G1 j!Z .
When (9) is plotted in the Wessel plane (Fig. 4), the (angular) frequency
corresponding to the apex of an arc is the characteristic frequency.
From Fig. 4, we see that then the real impedance value is equal to
Z 0 R1
= R0 0 R1 . With that expression put into (9) the
measured characteristic frequency at the apex of the arc !Zm and the
measured relaxation time Zm is
+ 1 2(
= (
Zm = 1 = Z
The characteristic frequency corresponding to =Z is !Z . The Z parameter in (9) corresponds to the apex of the arc only when G1 Gvar :
=Z and that is the Cole case. However, with
then !Zm
G1 6 Gvar Z no longer corresponds to the apex of the arc, !Z has
become just a nominal characteristic frequency. The new time constant
Fig. 4. Free conductance model, a Z -plot example. Three parameters
R ;R ;'
define a circular arc locus. Two additional parameters
together with ! defines the frequency scale along the arc; for the Z-version of
(9) it is the product G .
Zm given in (10) shows that a change in Gvar now indeed influences
the time constant in accordance with relaxation theory: a higher value
of Gvar results in a shorter time constant Zm . The true time constant
=!Zm does not appear in (9) but is given by (10). With (10) it
is found that !Zm is the crossover frequency when the CPE admittance
YCPE G1 !Z becomes equal to Gvar .
= (
The Cole (2) has four parameters: R1 ; R0 ; Z , and ; and ! as the
variable. Equation (9) has five parameters, for the Z-version the parameters are: R1 ; Gvar ; G1 ; Z , and . The five parameters are independent parameters, Zm in (10) is a derived, dependent but measured
A. Forward Analysis
Z = R1 +
Fig. 3. General equivalent circuits for (9) with a free-conductance/resistance
parameter and a dependent parameter. (a) Z-version, (b) Y-version.
where Y is the complex admittance Y
Z [S], G is the conductance at very high frequencies, and G0 is the conductance at very
low frequencies.
Immittance values can be plotted in the complex (Argand or Wessel
[5]) plane, Fig. 2(b) shows an admittance and Fig. 4 an impedance example. The immittance loci for all the equations given in this paper
have the form of a circular arc.
The Cole equations are compatible with the Kramers–Kronig transforms [5]. A control of both Fricke [7] and Kramers–Kronig compatibility can be used for data quality evaluation.
Z = R1 +
Setting up a Cole model with the four parameters given is straightforward, so is setting up a model according to (9) with the five parameters
given. In the latter case the geometry of the Z-arc is determined by the
three parameters R1 ; Gvar , and . This is illustrated on Fig. 4 with
R0 =Gvar R1 and 'CPE 1 . Two additional parameters
together with ! define the frequency scale along a given arc: G1 and
Z . The true relaxation time Zm is found from (10).
= 90
B. Inverse Analysis
The inverse problem of determining the four parameters of the Cole
equation (2) from measurement results is simple; the parameters are
read directly from the Wessel plot. For the Z case: R0 ; R1 , and 'CPE
determine the arc geometry, and the characteristic time constant Z
=!Z is found from the measured frequency at the apex of the arc.
However, it may be impossible to interpret the results correctly if the
experiment continues and new sets of data are sampled. Let us take
the example that only !Z changes during an experiment. According
to (4) that cannot be due to a change in the ideal conductance G, a
conclusion which may be seriously wrong.
Determining all five parameters in (9) from measured data is not
possible. A circular arc is first fitted to the data in a Wessel plot. The
three geometrical parameters R ; Gvar , and are found from the arc
geometry. For the frequency scale on the arc, (10) is rearranged
G1 Z = Gvar :
Here the three parameters of the right hand side are found from the
Wessel plot: Gvar ; !Zm , and . Accordingly, the product G1 Z is
known, but not the individual values. As long as the product G1 Z and
all other parameters are constant, arcs with different G1 0 Z combinations are indistinguishable from each other. A fixed nominal value for
Z together with G1 as a free parameter is then a preferred alternative.
A fixed nominal value for G1 together with Z as a free parameter is
not preferred since Zm should be the dependent variable, not Z , (10).
G1 can then be selectively followed during an experiment with a fixed
nominal value for Z . From (10) we have
G1 = Gvar
(Z !Zm) :
The nominal time constant Z is not the true characteristic time constant Zm as found by measurement. In practice (10) is used in forward
analysis to find Zm with a given Z value. Equation (12) is used to
follow G1 values with a fixed nominal Z value.
In conclusion, the inverse analysis has shown that the four parameters of the Cole model can be determined directly from the measurement results, but the interpretation of the results may be seriously misleading. In the new model four parameters can also be determined; in
addition a fifth parameter G1 can be calculated and selectively followed
during an experiment.
C. An Example
Four skin impedance spectrographic complex data sets were obtained with three pregelled electrocardiogram-electrodes positioned
on the skin of a human underarm. The effective skin-wetted area was
3 cm2 and electrode center distances 4 cm. The first data set was taken
3 min after electrode onset to dry skin (Fig. 5 and Tables I and II), and
the subsequent three sets after 6, 9, and 16 min (Tables I and II). Each
spectrographic data set was measured in the frequency range 1 Hz–1
MHz with a Solartron model 1260/1294 setup and a measuring voltage
amplitude of 10 mV(rms). The four circular arcs were determined
with a ZView software package (Scribner Associates), Fig. 5 shows as
an example the arc found with the first data set (continuous line). With
the non-Cole model the G1 values were calculated with a nominal
With the non-Cole model (Table II) the samples showed a clear correlation with time between the and G1 (or G1 Z ) parameters. This is
in agreement with the commonly accepted electrical model of human
skin: the stratum corneum can be modeled by a CPE, and the sweat
duct filling by a dc conductance in parallel [9], [17]. The postulated
process is, therefore: First, the electrolyte penetrates the most superficial and electrically dominating layers of the stratum corneum resulting
in a rapid increase of the admittance G1 of the CPE. Then, gradually,
the sweat ducts are filled with contact electrolyte increasing the conductance path Gvar in parallel with the CPE. With the Cole model
(Table I) the four data sets showed no clear correlations between the
Z ; , and G parameters with time, making it impossible to postulate any particular process in the skin.
The Cole equation (2) or (4) is with the time constant as a conductance-independent parameter. In relaxation theory [7] the time constant
is the time necessary to discharge a resistance–capacitance network, so
that a large capacitance and resistance result in long time constants.
The Cole equation is, therefore, incompatible with general relaxation
theory. The alternative free-conductance model of (9) and (10) is in
Fig. 5. Skin impedance plotted in the complex plane (broken line), human
underarm. First spectrographic dataset as measured 3 min after electrode
positioning on dry skin. Best circular arc found by regression analysis
(continuous line) by the ZView program (Scribner Associates), calculated arc
data given in the right lower corner.
agreement because Gvar influences the discharge process of the CPE
capacitor so that higher conductance leads to higher characteristic frequency. This is true also if Z is regarded as a parameter describing
a distribution of relaxation time constants [7]. In biomaterials the capacitance values are usually much more stable than the resistance/conductance values. When that is the case, the Cole model with the time
constant independent of conductance is not a logical starting point for
model selection.
When data show that the time constant or characteristic frequency
indeed varies [8], this may be due to a variation in an independent
conductance according to the alternative model of (9), or a change in
the nominal time constant independently of conductance changes according to the Cole (2).
Some authors have explicitly used a model similar to (9). Yamamoto
et al. [9], for instance, called the independent parallel resistor R2 and
had it connected directly in parallel with the CPE as in Fig. 3(a). They
measured the skin impedance of about 60 persons on four different occasions, and found low correlation between R2 and the CPE capacitance, but a high correlation between R2 and , in agreement with
our argumentation. As long as many published varying time constant
results are due to changing conductance, and the authors do not discuss the use of Cole models based upon a time constant as an independent parameter, many of them implicitly actually may have used the
free-conductance model of (9).
An immittance equation should be in accordance with the theory
of physics and electrical networks [10]. The problem of dimensions
is already apparent in Frickes classical paper [6]. There, he presented
the formula for his frequency-dependent capacitance as Cp
!0m , which implies that either the constant cannot have the dimension of capacitance or that the formula is dimensionally heterogeneous.
Many authors have used such formulas, for instance [11]–[15]. However, network theory requires an angle in immittance equations, and the
problem of dimensions is solved if ! (angular frequency) is replaced
by ! (angle). may be seen as a simple frequency scaling factor, and
thereby looses some of its value as a biomaterial characteristic constant.
An alternative to Frickes formula is then Cp C1 ! 0m where C1
, compare
is a capacitance with the value corresponding to !
with (5). Several authors have also used the ! term alone but with hidden into a “constant” [9], [16]–[18], so that their equations actually
are homogeneous. This is in accordance with the frequency scale being
determined by the product G1 Z . Commercial software packages do
it the same way, e.g., in ZView (Scribner Associates) with the parameter CPE1-T being equal to G1 [S], and the value of Z chosen to be
Equations (1), (2), (4), and (7)–(9), in both impedance and admittance versions, have perfect circular arc loci in the immittance plane
and may, therefore, be equally preferable candidates for a fit to experimental data. Having the same descriptive power, a choice has to be
made on the basis of knowledge about the biomaterial examined and
the desired properties to be modeled. For instance, if the biomaterial
is the human skin in vivo, the impedance model of Fig. 3(a) and (9) is
a good starting point with the stratum corneum modeled by the CPE,
and the independent filling of the sweat ducts by Gvar , cf. the example
given in Section V. The Cole model of Fig. 1(b) is not a preferred model
because it does not have such an independent conductance. If the biomaterial is a cell suspension, the Y equivalent circuit of Fig. 3(b) and
(9) may be a good starting point. Then, Rvar models the cell interior,
and the CPE models the cell membranes. R as a dependent parameter according to the Cole Y-model (Fig. 2) is less preferable.
In this paper, we have limited our treatment to the CPE as an adequate model element for biomaterial dispersion analysis, without discussing its physical meaning. Cole himself was very well aware of the
poor explanatory power of his models, [1], [4]. Cole apparently with
time became less comfortable with his own model, and did not focus
on it in his summing-up book [19]. Fricke did not use the Cole equations.
= const
( )
The concept of dispersion is very fundamental to the field of
bioimpedance, and it is important to know the properties and limitations of the models describing the immittance spectra usually found.
The existence of a dependent conductance but an independent time
constant parameter in the Cole model makes it incompatible with
general relaxation theory and a more specialized model than usually
realized. The analysis of the alternative free-conductance model with
the relaxation time as a dependent parameter has shown that the model
is preferable both because it is in accordance with general relaxation
theory and actually measured data. Our results show that the time
constant in the non-Cole equations no longer corresponds to the apex
of the circular arc, and that the G1 and Z parameters cannot be
determined separately from measured data, only the product G1 Z .
However, it is possible to follow the relative changes in G1 selectively
during an experiment.
We find that dispersion analysis by circular arc plots in the complex
plane together with equivalent circuit modeling is still an important part
of the analysis of bioimmittance data.
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