Estimating electrical properties and the thickness of skin with electrical impedance spectroscopy: Mathematical analysis and measurements

A total of 120 young volunteers – non-smokers without any known skin diseases or allergies – participated with an equal distribution of men and women at 24 ± 3 years of age; on the day of the measurements they were asked to abstain from moisturizers.

Electrical impedance measurements were performed on the volar forearm of the volunteers in 2007 with a SciBase II impedance spectrometer and a non-invasive probe [15].

The non-invasive, circular probe comprises four electrodes as illustrated in Fig. 1: two voltage injection electrodes, one current detector and one guard electrode to decrease the impact of surface leakage currents. With this design, two-point measurements can be carried out. The dimensions of the electrode and the skin layers depicted in Fig. 1 can be found in Table 1.

Fig. 1

The SciBase II impedance spectrometer measures the impedance of the skin at 35 frequencies logarithmically distributed between 1.0 kHz and 2.5 MHz at five different depth settings; these are obtained by varying the applied voltage at the second injection electrode (boundary III in Fig. 1b) from 5 to 50 mV [16].

Before measurements, the skin of the subjects was inundated with a physiological saline solution of 0.9% sodium chloride mass concentration for 1 min. The relative air humidity was 36±7 %, and the ambient temperature was maintained at 21.7 ±0.3 C.

A mathematical model based on conservation of charge in both the time- and frequency-domain for an alternating, sinusoidal current in the rotation-symmetric gold-plated electrodes (EL) of the probe and the skin – stratum corneum (SC), viable skin (VS) and adipose tissue (AT) – in the vicinity of the probe was derived in our earlier work [8]. In short, the model can be expressed as follows in the frequency domain:

Here, J is the total current density, σ⁽ⁱ⁾ and εr ⁽ⁱ⁾ are the conductivity and relative permittivity of layer i (= SC,VS,AT,EL), φ is the potential, α is a coefficient for a given depth setting of the second injection electrode, V0 is the excitation voltage, j is the imaginary number, and ω is the angular frequency of the alternating current; the roman numerals denote the respective boundaries, which are highlighted in Fig. 1b.

The material properties are given by [17]:

where i denotes either viable skin, stratum corneum soaked with a saline solution with 0.9% NaCl for 1 min, and adipose tissue (relative permittivity); ρ is the resistivity; ν is the frequency; cj(i)$\mathfrak{c}_{j}^{\left( i \right)}$and dj(i)$\mathfrak{d}_{j}^{\left( i \right)}$are the material coefficients, given in table 2. These constitutive relations for the electrical material properties are valid between 1 kHz and 1 MHz. (N.B.: the dependent variable, φ, is a phasor; for more information, see our earlier work [8].)

In our search for closed-form analytical solutions that could allow for estimates of the stratum corneum thickness and its electrical properties, we start by noting the well-known fact that the impedance is dominated by the stratum corneum at low frequencies around 1 kHz [18, 19], which suggests that the potential drop in the viable skin and electrodes are negligible vis-à-vis the corresponding drop in the stratum corneum around this frequency. This is indeed the case, as can be shown with the following scaling analysis for the potential drops in the various skin layers and electrodes: the potential drops can be secured from an order-of-magnitude equivalent of the total current density, Eq. 1b, in each layer as

where [I] is a scale for the absolute value of the total current passing through the skin, which has been estimated from the experimentally measured impedance, Zmeas, as [I] ∼ V₀/|Z_meas|, [A_SC] is a scale for the area of the stratum corneum underneath the electrode, [A_VS] is a scale for the area that the current has to pass through in the radial direction in the viable skin, [A_EL] is a scale for the area of an electrode in contact with the skin; [Δφ_VS], [Δφ_EL] and [Δφ_SC] are the scales of the absolute values of the potential drop in the viable skin, electrodes and stratum corneum, respectively; and l_VS, l_EL and l_SC are length scales in the radial direction in the viable skin and in the z-direction in the electrodes and stratum corneum. Typical values for these at 1 kHz are as follows: l_VS ∼ 10⁻³ m, l_EL ∼ 10⁻⁴ m, l_SC ∼ 10⁻⁵ m, [A_VS] ∼ 2πl_VSh_VS ∼ 10⁻⁵ m², A_EL ∼ πw12$\pi w_{1}^{2}$∼ 10⁻⁶ m², A_SC ∼ A_EL, σ^(VS) ∼ 1 S m⁻¹, σ^(EL) ∼ 10⁷ S m⁻¹, σ^(SC) ∼ 10⁻⁶ S m⁻¹, ω = 2πν ∼ 10⁴ Hz, ε₀ ∼ 10⁻¹¹ F m⁻¹, εr(VS)$\varepsilon _{r}^{\left( \text{VS} \right)}$∼ 10⁶, εr(EL)$\varepsilon _{r}^{\left( \text{EL} \right)}$∼ 1, εr(SC)$\varepsilon _{r}^{\left( \text{SC} \right)}$∼ 10², and [I] ∼ 10⁻⁷ A for V₀ = 0.05 V. Here, we have approximated the scale for the areas of the electrodes with the area of the innermost electrode; the scale for the viable skin has been approximated as the area of the cylinder that the current has to pass through on its way through the viable skin from the injects to the sense electrode. In addition, we have used that the conductivity of layer i can be defined as follows: σ(i)=1/ρ(i)(Ω).${{\sigma }^{\left( i \right)}}={1}/{{{\rho }^{\left( i \right)}}}\;\left( \Omega \right).$

With these values, we find

Clearly, [Δφ_VS], [Δφ_EL] ≪ [Δφ_SC]. This, in turn, implies that we only need to consider the current passing through the stratum corneum underneath each electrode at frequencies around 1 kHz.

At this stage, it is also instructive to see how the potential drops distribute themselves at the other end of the frequency spectrum of the SciBase II probe: 1 MHz. Returning to the order-of-magnitude equivalents, Eqs. 2a-2c, and employing the typical magnitudes for the electrical parameters at 1 MHz, we obtain

We have here used that [I] ∼ 10−⁴ A from the numerical solution of the full model (see Numerics) since a significant portion of the total current will pass through the guard, whence Z_meas at this high frequency does not capture the scale of the total current; σ^(VS) ∼ 1 S m−¹, σ^(SC) ∼ 10−³ S m−¹, εr(VS)∼102$\varepsilon _{r}^{\left( \text{VS} \right)}\sim {{10}^{2}}$, εr(SC)∼102$\varepsilon _{r}^{\left( \text{SC} \right)}\sim {{10}^{2}}$(the other parameters remain the same). In this limit, the viable skin is thus contributing significantly to the measured impedance.

In view of the findings at the lower limit of 1 kHz, for which the impedance of the skin is governed at leading order by the stratum corneum, we should be able to reduce the considered domain comprising three skin layers, the electrodes and the alternating current passing through them to a simple case of currents orthogonal to the electrodes passing in and out through the stratum corneum. Those currents would in turn give rise to the leading-order potential drop. From the mathematical point of view, it should thus be possible to reduce the partial differential equation to either ordinary differential or algebraic equations. We aim for the latter and proceed with the following Ansatz:

Here, Δφ_k and I_k are the potential drop and current through the stratum corneum underneath electrode k respectively, as illustrated in Fig. 1c; and h_SC is the height of the stratum corneum. The current passing through each electrode, I_k, is positive when coming out (positive z-direction in Fig. 1) and going into (negative z-direction) the skin respectively.

The area for each electrode, A_k, is given by

In essence, Eq. 3a ensures that the total current is preserved, which can thus be seen as a representation of the conservation of charge, Eq. 1a;Eq. 3b is derived from the definition of the current density, J, in Eq. 1b, in which the gradient has been replaced with the potential drop and height of stratum corneum; and the potential drop for the three innermost electrodes – current detection (k = 1), guard (k = 2), and second injection electrode (k = 3) – in Eqs. 3c-3e over the stratum corneum underneath the electrodes originate from the boundary conditions, Eqs. 1c-1f.

We have thus reduced the complex-valued partial differential equation in the rotation-symmetric domain comprising stratum corneum, viable skin and adipose tissue together with constitutive relations and boundary conditions to a system of 8 linear algebraic equations with 8 unknowns: I_k and Δφ_k for k = 1, .., 4. Note that the Ansatz can be adjusted to other probe designs, such as rectangular probes, or probes with more or less electrodes.

After some algebra, we arrive at the following closed-form expression for the electrode currents:

where

with the constant, A, given by

which is the ratio of the areas of the three inner electrodes modified with the depth of the second injection electrode (k = 3) and the sum of the total electrode areas.

The impedance, Z, can then be evaluated from

where I₁ is the current passing through the sense electrode, resulting in

We will later analyze the findings from the impedance ₂in the form of the magnitude, 𝔐, and the phase, 𝔓, which can be expressed as

Finally, it is interesting to note that in the limit of 1 kHz the magnitude depends on the thickness of the stratum corneum, depth setting, frequency, material properties of stratum corneum, and the areas of the electrodes; whereas the phase is only a function of the frequency and material properties of the stratum corneum. Furthermore, both the magnitude and the phase are not functions of the applied voltage. This independence of the applied voltage has been confirmed by tests with the probe in the range from 50 mV to 250 mV, for which the measured impedance did not change at leading order (not shown here).

So far, we have derived closed-form expressions for the currents, impedance, magnitude and phase at frequencies around 1 kHz. Now, if we have access to an experimentally measured impedance, Z_meas, we can inverse the expressions to estimate electrical properties – conductivity and relative permittivity – of the stratum corneum provided that the latter’s thickness, h_SC, is known:

here, a = Re(Z_meas) and b = Im(Z_meas). (N.B.: b ≤ 0.)

If either the relative permittivity and/or the conductivity is known, then we are able to estimate the thickness of the stratum corneum, after some algebra, from either the imaginary part of the measured impedance

or the real part

or as an average of the two

For verification of the analytical solution in the limit of low frequencies (ν ∼ 1 kHz) and the region of validity, the mathematical model, Eqs. 1a-1i, was also implemented and solved in the commercial finite-element solver, COMSOL MULTIPHYSICS 3.5a [20]. In this context, verification refers to ensuring that we have not unwittingly discarded any leading-order terms (read: significant terms) when reducing the full set of equations to a simple algebraic counterpart.

In essence, the computational domain shown in Fig. 1 was resolved with a mesh of around 10⁵ triangular elements after a mesh-independence study to ensure mesh-independent results; the direct solver PARDISO was selected as the linear solver with a relative convergence tolerance of 10−⁶.

A typical run required around 2 seconds (wall-clock time) with 5 × 10⁴ degrees of freedom (quadratic Lagrange elements) on a 2.5 GHz workstation for a given frequency.

In order to verify the analytical solutions and establish their region of validity, we compare the predicted magnitude and phase with their counterparts from the full set of equations, Eqs. 1a-1i, through the frequency range 1-10³ kHz. As can be inferred from Fig. 2, the analytical solutions are able to predict the magnitude and phase of the EIS in the low to mid-frequency range up to around 10² kHz with an error less than 10%; after 10² kHz, the error increases rapidly up to around 25% and 40% for the magnitude and phase respectively.

The main causes for the increasing error with increasing frequency can be found in the underlying assumptions of the Ansatz: first, the assumption that the electrical impedance is dominated by the stratum corneum whilst the viable skin and subcutaneous fat are negligible becomes less accurate with increasing frequency, because it is well known that the impedance of the skin is governed by the stratum corneum only at lower frequencies whilst its influence decreases when the frequency increases [18, 19]; this was also demonstrated in the earlier scaling analysis. Second, the assumption that the currents pass straight through the stratum corneum is reasonable in the low- to mid-frequency range; at higher frequencies, however, the current density distributions underneath the electrodes becomes increasingly more non-uniform and thereby alter the active area of the electrodes (A₁−₄). Third, the assumption that the voltage drop will not occur solely over the stratum corneum becomes less accurate with increasing frequency, as was shown earlier with the scales for the potential drops in the scaling analysis around 10³ kHz.

Proceeding with verifying the individual currents for each electrode, as depicted in Fig. 3, we find that the real part agrees well throughout the frequency range, whereas the imaginary part starts to deviate above the earlier observed 10² kHz.

Fig. 2

Fig. 3

One of the reasons for the good agreement of the real part, Re(Ik)−AkσΔϕk/hSC,$\operatorname{Re}\left( {{I}_{k}} \right)-{{A}_{k}}\sigma \Delta {{{\phi }_{k}}}/{{{h}_{SC}}}\;,$as opposed to imaginary part of the current, Im(Ik)=−Akωε0εrΔϕk/hSC,$\left( {{I}_{k}} \right)=-{{A}_{k}}\omega {{\varepsilon }_{0}}{{\varepsilon }_{r}}\Delta {{{\phi }_{k}}}/{{{h}_{SC}}}\;,$could be that the latter has an explicit dependence on ν (ω = 2πν), which increases by three orders of magnitude from 1 to 10³ kHz; this, in turn, affects the error propagation as the frequency increases. Nonetheless, the region of validity of the analytical solutions extends by two orders of magnitude beyond 1 kHz to around 10² kHz compared to the numerical solution of the full set of equations, which is in line with our earlier intuition regarding the extent of the region of validity.

So far, we have verified and determined the region of validity of our Ansatz and resulting analytical closed-form expressions for the impedance measured during EIS of skin. We will now seek to further validate the mathematical model – solved numerically and analytically – by exploiting the different depth settings of the probe.

The measurements were carried out at five different depth settings, α_n = (0.1, 0.27, 0.58, 0.81, 1) as shown in Fig. 4

Fig. 4

Fig. 5

for the magnitude and the phase from the experimental measurements and predictions from the full model and analytical expressions. Here, several features are apparent: first and foremost is the good agreement between the predicted magnitude and phase for the depth setting, α = 0.1, which is due to the calibration of the conductivity and relative permittivity at this depth (see [17]); second is that the numerical solution of the full set of equations is not fully able to capture the shift in the phase as the frequency approaches 10³ kHz, although its magnitude adheres closely to the measured counterpart; third, and as expected, is that the closed-form expressions are limited to a region between 1 and 10² kHz in their validity; and fourth, is that the measured phase at different depth settings converges towards a single value for ν −→ 1 kHz, supporting the closed-form expression for the phase, Eq. 7b, which does not contain the depth, as opposed to the closed-form solution for the magnitude, Eq. 7a.

Returning to the analytical solution for the impedance, Eq. 6, and extending it for different depth settings as

we find that the depth settings and their impact reduce to a constant by taking the ratio of impedances at different depth settings:

which agrees well with the corresponding experimentally measured impedance ratios at 1 kHz and numerical and analytical counterpart, as can be inferred from Fig. 5. The maximum relative error between the measured and predicted ratios varies between 0 and 8 %. In addition, it is likely that the experimental measurements are imprecise around 1 kHz, since there is no physical justification for the ratios to suddenly change abruptly when ν → 1 kHz.

A closer look at the individual currents (not shown here) reveals that the current changes direction at the second injection electrode (k = 3) for α_4,5 since the applied voltage there is now 58% and 100% of the voltage at the primary injection electrode (k = 4). This demonstrates that the Ansatz is able to capture the dual role of the second injection electrode to act akin to a guard at low depths and as an inject at higher depth settings. In fact, a closer inspection of Eq. 4d reveals that Δ_φ3 changes sign when α = 1 −A.

From Eqs. 8a and 8b as well as by assuming the stratum corneum thickness to be 14 μm, we are able to estimate the resistivity and the relative permittivity for each individual subject separately. As can be inferred from Fig. 6, the closed-form analytical expressions follow the numerical counterparts up to around 10² kHz, after which the relative permittivity starts to deviate whilst the resistivity agrees throughout the frequency range.

Fig. 6

Fig. 7

We recall that the resistivity captures more information of the real part of the impedance, a, whilst the relative permittivity, as previously noted, has an explicit dependence on ω and includes more information of the imaginary part, b. Clearly, the same phenomenon observed for the real and imaginary part of the current in the analytical solution carries through for the estimates of the electrical properties. We therefore once more find that the region of validity extends beyond 1 kHz to around 10² kHz.

Finally, we estimate the thickness of the stratum corneum with Eqs. 9a-9c by assuming that the material properties are known. As illustrated in Fig. 7, the stratum corneum thickness estimate based on the real part of the impedance overestimates the skin thickness slightly, whilst the imaginary part underestimates the skin thickness somewhat; their combination, however, gives rise to an accurate distribution with 92% of the individuals between 6 and 20 μm, which corresponds well with literature values [8]. These findings suggest that one should thus employ both the real and imaginary parts of the measured impedance to ensure as accurate estimates as possible – this makes sense since we are tapping into more information from the EIS by including both contributions.

At this stage, it would have been useful to have access to other types of measurements of the stratum corneum thickness of the 120 subjects to further provide evidence of the fidelity of the predictions. This is especially the case since we have taken material properties that were calibrated for a median skin thickness of 14 μm.