1. bookVolume 9 (2018): Issue 1 (January 2018)
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1891-5469
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Mechanistic multilayer model for non-invasive bioimpedance of intact skin

Published Online: 18 Aug 2018
Volume & Issue: Volume 9 (2018) - Issue 1 (January 2018)
Page range: 31 - 38
Received: 19 Mar 2018
Journal Details
License
Format
Journal
eISSN
1891-5469
First Published
01 Jan 2010
Publication timeframe
1 time per year
Languages
English
Abstract

An approximate semi-analytical solution based on a Hankel transform of a mechanistic model for electrical impedance spectroscopy (EIS) is derived for a non-invasive axisymmetric concentric probe with m electrodes measuring the response of n layers of human skin. We validate the semi-analytical solution for the case when the skin is treated as a three-layer entity - (i) stratum corneum, (ii) viable skin comprising living epidermis and dermis and (iii) adipose tissue – on the volar forearm in the frequency range 1 kHz to 1 MHz with experimental EIS measurements of 120 young subjects. Overall, we find good agreement for both the mean magnitude and phase of the impedance as well as the natural variability between subjects. Finally, the semi-analytical solution is verified with the full set of equations solved numerically: Good agreement is found for the point-wise potential distribution in the three skin layers.

Keywords

Introduction

The development of a detailed, continuous mathematical model of an electrical-impedance-spectroscopy (EIS) measurement of human skin requires knowledge of both the EIS probe design as well as the properties of the adjacent skin layers. While the former is straightforward, the latter is more demanding due to the inherent complexity of human skin. This complexity arises from the form and functionality of human skin.

In essence, human skin comprises multiple layers of cells and tissues; these can be grouped into three main layers with their own thicknesses and electrical properties, as illustrated in Fig. 1: (i) a stratified, cellular epidermis; (ii) an underlying dermis made up of connective tissue that is separated from the epidermis through the dermo-epidermal junction; and (iii) the hypodermis, also known as the fatty subcutaneous layer or adipose tissue. Each of these three layers, in turn, consist of one or more sublayers; e.g., epidermis of thin skin comprises stratum corneum, stratum granulosum, stratum spinosum and stratum basale. Besides natural variations between individuals that result in differing thicknesses and electrical properties, other factors come into play such as aging, skin moisture and variation of blood flow due to ambient conditions.

Fig. 1

Schematic of skin layers.

In our earlier work, we derived a mathematical model based on conservation of charge for EIS measurements of human skin [1,2,3] between 1 kHz and 1 MHz by grouping the skin layers into three layers: viz., stratum corneum (SC), viable skin (VS) and adipose tissue (AT). The model, which was calibrated and validated with experimental EIS measurements, was solved numerically [1,2,3] and semianalytically [3] in the entire frequency domain and analytically [2,3] at around 1 kHz. Similar continuous models have also been solved numerically [4,5,6,7,8].

Here, we extend our previous work by generalizing the mathematical model and its solution to account for n layers and m electrodes, as shown in Fig. 2. We then demonstrate the model for three layers with a four-electrode probe, one sense with one or multiple injection electrodes, by verifying with the full set of equations solved numerically and validating with experimentally measured impedance data from an earlier study [9].

Fig. 2

Schematic of skin comprising n layers and a circular EIS probe with m electrodes.

Materials and methods
Mathematical formulation

The potential distribution, current density and measured impedance in cylinder coordinates [3] in the skin can be written as

1rrrΦr+2Φz2=0,$$\begin{array}{} \displaystyle \frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial\Phi}{\partial r}\right)+\frac{\partial^{2}\Phi}{\partial_{z^{2}}}=0, \end{array}$$

Jr=σi,effΦr,$$\begin{array}{} \displaystyle J_{r}=-\displaystyle \sigma_{i,eff}\frac{\partial\Phi}{\partial r}, \end{array}$$

Jz=σi,effΦz,$$\begin{array}{} \displaystyle J_{z}=-\displaystyle \sigma_{i,eff}\frac{\partial\Phi}{\partial z}, \end{array}$$

Z=V02π0R1Jz(0rR1,Hn)rdr1,$$\begin{array}{} \displaystyle Z=V_{0}\left[2\displaystyle \pi\int\limits_{0}^{R_{1}}J_{z}(0\leq r\leq R_{1},\mathcal{H}_{n})rdr\right]^{-1}, \end{array}$$

subject to the following external boundary conditions for j = 1, …, m electrodes:

Φ(R2j2rR2j1,Hn)=Vj,$$\begin{array}{} \displaystyle \Phi(R_{2j-2}\leq r\leq R_{2j-1},\, \mathcal{H}_{n})=V_{j}, \end{array}$$

Φz(R2j1rR2j,Hn)=0,(jm)$$\begin{array}{} \displaystyle \frac{\partial\Phi}{\partial z}(R_{2j-1}\leq r\leq R_{2j},\mathcal{H}_{n})=0,\, (j\neq m) \end{array}$$

Φz(R2m1r,Hn)=0,$$\begin{array}{} \displaystyle \frac{\partial\Phi}{\partial z}(R_{2m-1}\leq r,\, \mathcal{H}_{n})=0, \end{array}$$

limrΦ(r,z)=0,$$\begin{array}{} \displaystyle \lim_{r\rightarrow\infty}\Phi(r,z)=0, \end{array}$$

Φz(r,0)=0,$$\begin{array}{} \displaystyle \frac{\partial\Phi}{\partial z}(r,0)=0, \end{array}$$

Φr(0,z)=0;$$\begin{array}{} \displaystyle \frac{\partial\Phi}{\partial r}(0,z)=0; \end{array}$$

and constitutive relations for the individual skin layers, i = 1, …, n:

σi,eff=σ(i)+iωε0εr(i),$$\begin{array}{} \displaystyle \sigma_{i,eff}=\sigma^{(i)}+i\omega\varepsilon_{0}\varepsilon_{r}^{(i)}, \end{array}$$

The radii, Rj, for the electrodes are defined for j = 1, …, m:

R0=0,$$\begin{array}{} \displaystyle R_{0}=0, \end{array}$$

R1=w1,$$\begin{array}{} \displaystyle R_{1}=w_{1}, \end{array}$$

Rj=Rj1+wj,(j=2,,m).$$\begin{array}{} \displaystyle R_{j}=R_{j-1}+w_{j},~~ (j=2,\ldots,m) . \end{array}$$

For convenience, we also define the cumulative height, 𝓗i, of skin layer i = 1, …, n:

Hi=k=1ihk.$$\begin{array}{} \displaystyle \mathcal{H}_{i}=\sum_{k=1}^{i}h_{k}. \end{array}$$

In the above equations, r and z are the cylindrical coordinates (see Fig. 2), Φ is the complex-valued potential, Jr and Jz are the current densities in the r and z direction respectively, m and n are the number of electrodes in the EIS probe and layers of skin respectively; σi,eff, εr(i)$\begin{array}{} \displaystyle \varepsilon_{r}^{(i)} \end{array}$ and

σ(i) are the effective, complex-valued electrical conductivity, the relative permittivity and conductivity of layer i, respectively; ε0 is the relative permittivity of vacuum, i is the imaginary unit, ω = 2πν is the angular frequency, ν is the frequency, Z is the impedance, Vj is the prescribed potential at a given electrode, j; wj is the width of or between an electrode as illustrated in Fig. 2 and hk is the height of skin layer k.

Note that we do not solve for the potential distribution in the electrodes, because the potential drop there is negligible [3].

Analysis

Let’s start by exploiting that the current density distributions underneath the m electrodes are near-to constant because of the high resistance to current of the stratum corneum in the considered frequency range [3]. This allows us to rewrite the boundary conditions, Eqs. 5-7, between the probe and the uppermost skin layer n, stratum corneum, as (we drop the subindex `eff’ for notational convenience in the analysis)

σnΦ(r,Hn)z=j=1mIjAj[U(R2j1r)U(R2j2r)],$$\begin{array}{} \displaystyle -\sigma_{n}\frac{\partial\Phi(r,\mathcal{H}_{n})}{\partial z}=\sum_{j=1}^{m}\frac{I_{j}}{A_{j}}\bigg[U(R_{2j-1}-r)\\ \displaystyle\qquad\qquad\quad\,-U(R_{2j-2}-r)\bigg], \end{array}$$

where U() denotes the Heaviside function and Ij is the total current at the jth electrode; the area of the jth electrode is

Aj=π(R2j12R2j22).$$\begin{array}{} \displaystyle A_{j}=\pi(R_{2_{j-1}}^{2}-R_{2_{j}-2}^{2}). \end{array}$$

Strictly speaking, we are no longer solving exactly the same set of equations with the rewritten boundary condition, Eq. 15, whence we will refer to our generalized solution as approximate in nature.

From the conservation of current for the entire system, we know that

j=1mIj=0.$$\begin{array}{} \displaystyle \sum_{j=1}^{m}I_{j}=0. \end{array}$$

We can now proceed with a Hankel transform of zeroth order, H0 {}, for the potential,

Ψ(ξ,z)=H0{Φ(r,z)}=0rΦ(r,z)J0(ξr)dr,$$\begin{array}{} \displaystyle \Psi(\xi,z)=H_{0}\big\{\Phi(r,z)\big\}=\int_{0}^{\infty}{r\Phi}(r,z)J_{0}(\xi r)dr, \end{array}$$

which is ideally suited for the axisymmetric Laplace equation, Eq. 1, on our unbounded domain:

H0{ΔΦ(r,z)}=ξ2Ψ(ξ,z)+2z2Ψ(ξ,z)=0,$$\begin{array}{} \displaystyle H_{0} \big\{\Delta\Phi(r,z)\big\}=-\xi^{2}\Psi(\xi,z)+\frac{\partial^{2}}{\partial z^{2}}\Psi(\xi,z)=0, \end{array}$$

with the solution for each skin layer, i,

Ψi(ξ,z)=Ai(ξ)eξz+Bi(ξ)eξz,$$\begin{array}{} \displaystyle \Psi_{i}(\xi,\, z)=A_{i}(\xi)e^{-\xi z}+B_{i}(\xi)e^{\xi z}, \end{array}$$

and its z-derivative,

Ψi(ξ,z)=Ai(ξ)ξeξz+Bi(ξ)ξeξz;$$\begin{array}{} \displaystyle \Psi'_{i}(\xi,z)=-A_{i}(\xi)\xi e^{-\xi z}+B_{i}(\xi)\xi e^{\xi z}; \end{array}$$

here, Ai (ξ) and Bi (ξ) are two functions that we will soon determine, Δ is the Laplace operator, and J0 is the 0th - order Bessel function of the first kind. Between layers, we know that the flux and potential are continuous; i.e.

σiΨi(ξ,Hi)z=σi+1Ψi+1(ξ,Hi)z,$$\begin{array}{} \displaystyle \sigma_{i}\frac{\partial\Psi_{i}(\xi,\mathcal{H}_{i})}{\partial z}=\sigma_{i+1}\frac{\partial\Psi_{i+1}(\xi,\mathcal{H}_{i})}{\partial z}, \end{array}$$

Ψi(ξ,Hi)=Ψi+1(ξ,Hi),$$\begin{array}{} \displaystyle \Psi_{i}(\xi,\mathcal{H}_{i})=\Psi_{i+1}(\xi,\mathcal{H}_{i}), \end{array}$$

for 1 ≤ in − 1. In addition, we need to satisfy the two boundary conditions at z = 0,

Ψ1(ξ,0)z=0,$$\begin{array}{} \displaystyle \frac{\partial\Psi_{1}(\xi,0)}{\partial z}=0, \end{array}$$

and at z =𝓗n,

σnΨn(ξ,Hn)z=k=1mIkκk(ξ),$$\begin{array}{} \displaystyle -\sigma_{n}\frac{\partial\Psi_{n}(\xi,\mathcal H_{n})}{\partial z} =\sum_{k=1}^{m}I_{k}\kappa_{k}(\xi) , \end{array}$$

with

κj(ξ)=R2j1J1(R2j1ξ)R2j2J1(R2j2ξ)π(R2j12R2j22)ξ,$$\begin{array}{} \displaystyle \kappa_{j}(\xi)=\frac{R_{2{j-1}}J_{1}(R_{2j-1}\xi)-R_{2j-2}J_{1}(R_{2j-2}\xi)}{\pi(R_{2j-1}^{2}-R_{2j-2}^{2})\xi}, \end{array}$$

here we have used that the Hankel transform of zeroth order of a Heaviside function is defined as

H0{U(ar)}=aξJ1(aξ),$$\begin{array}{} \displaystyle H_{0}\displaystyle \big\{U(a-r)\big\}=\frac{a}{\xi}J_{1}(a\xi), \end{array}$$

with J1 and a (> 0) denoting the 1st-order Bessel function of the first kind and a constant respectively [10].

Note that we have also switched to the subscript, k, in the summation over the electrodes in Eq. 25 to avoid confusion with the matrix notation we will introduce later. Let us solve the potential of the first layer and then work our way up to the uppermost n-th layer. Starting with the boundary condition, Eq. 24 at z = 0, we have

Ψ1(ξ,0)=A1(ξ)ξ+B1ξ(ξ)=0,$$\begin{array}{} \displaystyle \Psi_{1}'(\xi,0)=-A_{1}(\xi)\xi+B_{1}\xi(\xi)=0, \end{array}$$

which results in

A1(ξ)=B1(ξ)e2ξH0Λ1,$$\begin{array}{} \displaystyle A_{1}(\xi)=B_{1}(\xi)e^{2\xi \mathcal{H}_{0}}\Lambda_{1}, \end{array}$$

with

Λ1=1$$\begin{array}{} \displaystyle \Lambda_{1}=1 \end{array}$$

H0=0;$$\begin{array}{} \displaystyle \mathcal{H}_{0}=0; \end{array}$$

the role of e2ξ𝓗0 Λ1 (=1) will become apparent as we proceed with our analysis. Proceeding to the other boundary condition for the first layer, Eq. 23, we have for z = 𝓗1 that

Ψ1(ξ,H1)=Ψ2(ξ,H1),$$\begin{array}{} \displaystyle \Psi_{1}(\xi,\mathcal{H}_{1})=\Psi_{2}(\xi,\mathcal{H}_{1}), \end{array}$$

i.e.,

B1(ξ)(Λ1eξH1+eξH1)=A2(ξ)eξH1+B2(ξ)eξH1,$$\begin{array}{} \displaystyle B_{1}(\xi)(\Lambda_{1}e^{-\xi \mathcal{H}_{1}}+e^{\xi \mathcal H_{1}})=A_{2}(\xi)e^{-\xi \mathcal{H}_{1}}+B_{2}(\xi)e^{\xi \mathcal{H}_{1}}, \end{array}$$

from which we find (N.B.: 𝓗1 = h1)

B1(ξ)=A2(ξ)e2ξH1+B2(ξ)Λ1e2ξh1+1.$$\begin{array}{} \displaystyle B_{1}(\displaystyle \xi)=\frac{A_{2}(\xi)e^{-2\xi \mathcal{H}_{1}}+B_{2}(\xi)}{\Lambda_{1}e^{-2\xi h_{1}}+1}. \end{array}$$

Moving on to the second layer, we find after satisfying the continuity of flux at z = 𝓗1,

σ1Ψ1(ξ,H1)z=σ2Ψ2(ξ,H1)z,$$\begin{array}{} \displaystyle \sigma_{1}\frac{\partial\Psi_{1}(\xi,\mathcal H_{1})}{\partial z}=\sigma_{2}\frac{\partial\Psi_{2}(\xi,\mathcal{H}_{1})}{\partial z}, \end{array}$$

that

A2(ξ)=B2(ξ)e2ξH1Λ2(ξ),$$\begin{array}{} \displaystyle A_{2}(\xi)=B_{2}(\xi)e^{2\xi\mathcal H_{1}}\Lambda_{2}(\xi), \end{array}$$

with

Λ2(ξ)=σ1(1Λ1e2ξh1)σ2(1+Λ1e2ξh1)σ1(Λ1e2ξh11)σ2(1+Λ1e2ξh1).$$\begin{array}{} \displaystyle \Lambda_{2}(\xi)=\frac{\sigma_{1}(1-\Lambda_{1}e^{-2\xi h_{1}})-\sigma_{2}(1+\Lambda_{1}e^{-2\xi h_{1}})}{\sigma_{1}(\Lambda_{1}e^{-2\xi h_{1}}-1)-\sigma_{2}(1+\Lambda_{1}e^{-2\xi h_{1}})}. \end{array}$$

Between the second and third layer, z = 𝓗2, we require continuity of the potential,

Ψ2(ξ,H2)=Ψ3(ξ,H2),$$\begin{array}{} \displaystyle \Psi_{2}(\xi,\mathcal{H}_{2})=\Psi_{3}(\xi,\, \mathcal{H}_{2}), \end{array}$$

which reveals

B2(ξ)=A3(ξ)e2ξH2+B3(ξ)Λ2(ξ)e2ξh2+1.$$\begin{array}{} \displaystyle B_{2}(\displaystyle \xi)=\frac{A_{3}(\xi)e^{-2\xi \mathcal{H}_{2}}+B_{3}(\xi)}{\Lambda_{2}(\xi)e^{-2\xi h_{2}}+1}. \end{array}$$

Let us explicitly continue with the third layer and then generalize the solution for layer i, where 1 ≤ in − 1. In the third layer, we find

A3(ξ)=B3(ξ)e2ξH2Λ3(ξ),$$\begin{array}{} \displaystyle A_{3}(\xi)=B_{3}(\xi)e^{2\xi \mathcal{H}_{2}}\Lambda_{3}(\xi), \end{array}$$

and

B3(ξ)=A4(ξ)e2ξH3+B4(ξ)Λ3(ξ)e2ξh3+1,$$\begin{array}{} \displaystyle B_{3}(\displaystyle \xi)=\frac{A_{4}(\xi)e^{-2\xi \mathcal{H}_{3}}+B_{4}(\xi)}{\Lambda_{3}(\xi)e^{-2\xi h_{3}}+1}, \end{array}$$

with

Λ3(ξ)=σ2(1Λ2e2ξh2)σ3(1+Λ2e2ξh2)σ2(Λ2e2ξh21)σ3(1+Λ2e2ξh2),$$\begin{array}{} \displaystyle \Lambda_{3}(\xi)=\frac{\sigma_{2}(1-\Lambda_{2}e^{-2\xi h_{2}})-\sigma_{3}(1+\Lambda_{2}e^{-2\xi h_{2}})}{\sigma_{2}(\Lambda_{2}e^{-2\xi h_{2}}-1)-\sigma_{3}(1+\Lambda_{2}e^{-2\xi h_{2}})}, \end{array}$$

after satisfying the boundary conditions with the second and fourth skin layer.

We can now generalize the solution as

Ai(ξ)=Bi(ξ)e2ξHi1Λi(ξ),$$\begin{array}{} \displaystyle A_{i}(\xi)=B_{i}(\xi)e^{2\xi\mathcal H_{i-1}}\Lambda_{i}(\xi), \end{array}$$

and

Bi(ξ)=Ai+1(ξ)e2ξHi+Bi+1(ξ)Λi(ξ)e2ξhi+1,$$\begin{array}{} \displaystyle B_{i}( \xi)=\frac{A_{i+1}(\xi)e^{-2\xi \mathcal{H}_{i}}+B_{i+1}(\xi)}{\Lambda_{i}(\xi)e^{-2\xi h_{i}}+1}, \end{array}$$

for 1 ≤ in − 1, with

Λ1=1,$$\begin{array}{} \displaystyle \Lambda_{1}=1, \end{array}$$

for i = 1, and the recursive formula for Λi,

Λi(ξ)=σi1(1Λi1e2ξhi1)σi(1+Λi1e2ξhi1)σi1(Λi1e2ξhi11)σi(1+Λi1e2ξhi1),$$\begin{array}{} \displaystyle \Lambda_{i}(\xi)=\frac{\sigma_{i-1}(1-\Lambda_{i-1}e^{-2\xi h_{i-1}})-\sigma_{i}(1+\Lambda_{i-1}e^{-2\xi h_{i-1}})}{\sigma_{i-1}(\Lambda_{i-1}e^{-2\xi h_{i-1}}-1)-\sigma_{i}(1+\Lambda_{i-1}e^{-2\xi h_{i-1}})}, \end{array}$$

for 2 ≤ in − 1.

In the n-th layer, we find after satisfying the continuity of potential with the n − 1 layer and the boundary condition, Eq. 25, between the electrodes and the uppermost skin layer,

An(ξ)=Bn(ξ)e2ξHn1Λn(ξ),$$\begin{array}{} \displaystyle A_{n}(\xi)=B_{n}(\xi)\mathrm{e}^{2\xi\mathcal H_{n-1}}\Lambda_{n}(\xi), \end{array}$$

by setting

σn1Ψn1(ξ,Hn1)z=σnΨn(ξ,Hn1)z;$$\begin{array}{} \displaystyle \sigma_{n-1}\frac{\partial\Psi_{n-1}(\xi,\mathcal{H}_{n-1})}{\partial z}=\sigma_{n}\frac{\partial\Psi_{n}(\xi,\mathcal{H}_{n-1})}{\partial z}; \end{array}$$

and

Bn(ξ)=eξHnk=1mIkκk(ξ)σnξΛn(ξ)eξhn1,$$\begin{array}{} \displaystyle B_{n}(\xi)=\frac{e^{-\xi\mathcal H_{n}}\sum\limits_{k=1}^{m}I_{k}\kappa_{k}(\xi)}{\sigma_{n}\xi\left[ \Lambda_{n}(\xi)e^{-\xi h_{n}}-1\right]}, \end{array}$$

from the uppermost boundary condition at z = 𝓗n for layer n:

σnΨn(ξ,Hn)z=k=1mIkkk(ξ).$$\begin{array}{} -\displaystyle \sigma_{n}\frac{\partial\Psi_{n}(\xi,\mathcal{H}_{n})}{\partial z}=\sum_{k=1}^{m}I_{k}k_{k}(\xi). \end{array}$$

At this stage, we can write the local, pointwise transformed potential in layer i as

Ψi(ξ,z)=Bi(ξ)Λi(ξ)eξ(2Hi1z)+eξz.$$\begin{array}{} \displaystyle \Psi_{i}(\xi,z)=B_{i}(\xi)\left[\Lambda_{i}(\xi)e^{\xi(2H_{i-1}-z)}+e^{\xi z}\right]. \end{array}$$

In this expression, the coordinate, z, is bounded for each layer as

Hi1z(in layeri)Hi.$$\begin{array}{} \displaystyle \mathcal{H}_{i-1}\leq z (\text{in layer}~ i) \leq \mathcal{H}_{i}. \end{array}$$

Now, taking the inverse Hankel transform,

Υi(r,z)=H01{Ψi(ξ,z)}=0ξΨi(ξ,z)J0(ξr)dξ,$$\begin{array}{} \displaystyle \Upsilon_{i}(r,z)=H_{0}^{-1}\{\Psi_{i}(\xi,z)\}=\int_{0}^{\infty}\xi\Psi_{i}(\xi,z)J_{0}(\xi r)d\xi, \end{array}$$

we find the local potential distribution throughout the viable skin as

Υi(r,z)=0ξBi(ξ)Λi(ξ)eξ(2Hi1z)+eξZJo(ξr)dξ$$\begin{array}{} \displaystyle \Upsilon_{i}(r,z)=\int_{0}^{\infty}\xi B_{i}(\xi)\left[\Lambda_{i}(\xi)e^{\xi(2\mathcal{H}_{i-1}-z)}+e^{\xi_{Z}}\right]J_{o}(\xi r)d\xi \end{array}$$

and

Φi(r,z)=Υi(r,z)+c.$$\begin{array}{} \displaystyle \Phi_{i}(r,z)=\Upsilon_{i}(r,z)+c. \end{array}$$

We have m + 1 unknows: I1 to Im and c; the latter constant provides a reference point for the potential since we have in the approximate solution only solved for Neuman boundary conditions. In order to determine the unknowns, we return to our pointwise Dirichlet boundary conditions for the potentials underneath the electrodes, Eq. 5, and rewrite Eqs. 54 and 55 in terms of the average potential underneath the jth electrode in the uppermost skin layer, n:

Φn,javg=2πAjR2j2R2j1rΥn(r,Hη)dr+c=Vj,$$\begin{array}{} \displaystyle \Phi_{n,j}^{avg}=\frac{2\pi}{A_{j}}\int_{R_{2j-2}}^{R_{2j-1}}r\Upsilon_{n}(r,\mathcal{H}_{\eta})dr+c=V_{j}, \end{array}$$

or, after integration once, as

Φn,javg=2πσn0k=1mIkκk(ξ)κj(ξ)Ω(ξ)dξ+c=Vj,$$\begin{array}{} \displaystyle \Phi_{n,j}^{avg}=\frac{2\pi}{\sigma_{n}}\int_{0}^{\infty}\left[\sum_{k=1}^{m}I_{k}\kappa_{k}(\xi)\right]\kappa_{j}(\xi)\Omega(\xi)d\xi+c=V_{j}, \end{array}$$

where

Ω(ξ)=Λn(ξ)e2ξhn+1Λn(ξ)e2ξhn1.$$\begin{array}{} \displaystyle \Omega(\xi)=\frac{\Lambda_{n}(\xi)e^{-2\xi h_{n}}+1}{\Lambda_{n}(\xi)e^{-2\xi h_{n}}-1}. \end{array}$$

During the integration, we used that

J0(ar)rdr=raJ1(ar)+constant;$$\begin{array}{} \displaystyle \int J_{0}(ar)rdr=\frac{r}{a}J_{1}(ar)+ \text{constant}; \end{array}$$

J1(0)=0.$$\begin{array}{} \displaystyle J_{1}(0)=0. \end{array}$$

The overall conservation of current, Eq. 17, allows us to reduce the number of unknowns by one, as we can write

Im=j=1m1Ij.$$\begin{array}{} \displaystyle I_{m}=-\displaystyle \sum_{j=1}^{m-1}I_{j}. \end{array}$$

Introducing the expression for the current of the outermost electrode, Im, into Eq. 57, yields the following system of m linear equations:

Ax=b,$$\begin{array}{} \displaystyle \mathbf{Ax}=\mathbf{b}, \end{array}$$

with

A=A1,1A1,2A1,m11A2,1A2,2A2,m11Am1,1Am1,2Am1,m11Am,1Am,2Am,m11,$$\begin{array}{} \displaystyle \mathbf{A}=\left[\begin{array}{lllll} A_{1,1} & A_{1,2} & \cdots & A_{1,m-1} & 1\\ A_{2,1} & A_{2,2} & \cdots & A_{2,m-1} & 1\\ ~~~~\vdots & ~~~~\vdots & \ddots & ~~~~~\vdots & \vdots\\ A_{m-1,1} & A_{m-1,2} & \cdots & A_{m-1,m-1} & 1\\ A_{m,1} & A_{m,2} & \cdots & A_{m,m-1} & 1 \end{array}\right], \end{array}$$

x=IlI2Imlc,$$\begin{array}{} \displaystyle \mathbf{x}=\left[\begin{array}{l} ~~I_{\mathrm{l}}\\ ~~I_{2}\\ ~~~\vdots\\ I_{m-\mathrm{l}}\\ ~~~c \end{array}\right], \end{array}$$

b=V1V2VmlVm,$$\begin{array}{} \displaystyle \mathbf{b}=\left[\begin{array}{l} ~~V_{1}\\ ~~V_{2}\\ ~~~~\vdots\\ V_{m-\mathrm{l}}\\ ~~V_{m} \end{array}\right], \end{array}$$

and

Ai,j=2πσn0κj(ξ)κm(ξ)κi(ξ)Ω(ξ)dξ.$$\begin{array}{} \displaystyle A_{i,j}=\displaystyle \frac{2\pi}{\sigma_{n}}\int_{0}^{\infty}\left[\kappa_{j}(\xi)-\kappa_{m}(\xi)\right]\kappa_{i}(\xi)\Omega(\xi)d\xi. \end{array}$$

Once the linear system of equations together with the improper integrals given by Eq. 66 have been solved, the impedance can be found from

ZH=V0I1,$$\begin{array}{} \displaystyle Z_{H}=\displaystyle \frac{V_{0}}{I_{1}}, \end{array}$$

assuming that V0 is set as reference potential for the measurements and that the first electrode is used as the sense. We’ll give one example of this in the next subsection. If the electrode system is different, one needs to couple the right current with the given potential to determine the impedance.

Solution for n=3, m=4

This solution corresponds to EIS measurements in our earlier experimental study [9] and the model thereof with skin encompassing three layers (n=3): stratum corneum, viable skin and adipose tissue, as illustrated in Fig. 3.

Fig. 3

Schematic of a 4-electrode probe and skin modeled as a three-layer entity.

Here, the first electrode is the sense, the second serves as the guard, the third is the second inject with a potential set as αV0 and the fourth is the primary inject with a potential of V0; α is referred to as a depth setting on the probe; i.e. m=4. The boundary conditions for the complexvalued Laplace equation reduce to

Φ(0rR1,H3)=0,$$\begin{array}{} \displaystyle \Phi(0\leq r\leq R_{1},\ \mathcal{H}_{3})=0, \end{array}$$

Φ(R2rR3,H3)=0,$$\begin{array}{} \displaystyle \Phi(R_{2}\leq r\leq R_{3},\, \mathcal{H}_{3})=0, \end{array}$$

Φ(R4rR5,H3)=αV0,$$\begin{array}{} \displaystyle \Phi(R_{4}\leq r\leq R_{5},\, \mathcal{H}_{3})=\alpha V_{0}, \end{array}$$

Φ(R6rR7,H3)=V0,$$\begin{array}{} \displaystyle \Phi(R_{6}\leq r\leq R_{7},\,\mathcal{H}_{3})=V_{0}, \end{array}$$

Φz(R1rR2,H3)=0,$$\begin{array}{} \displaystyle \frac{\partial\Phi}{\partial z}(R_{1}\leq r\leq R_{2},\, \mathcal{H}_{3})=0, \end{array}$$

Φz(R3rR4,H3)=0,$$\begin{array}{} \displaystyle \frac{\partial\Phi}{\partial z}(R_{3}\leq r\leq R_{4},\,\mathcal{H}_{3})=0, \end{array}$$

Φz(R5rR6,H3)=0,$$\begin{array}{} \displaystyle \frac{\partial\Phi}{\partial z}(R_{5}\leq r\leq R_{6},\,\mathcal{H}_{3})=0, \end{array}$$

Φz(R7r,H3)=0,$$\begin{array}{} \displaystyle \frac{\partial\Phi}{\partial z}(R_{7}\leq r,\,\mathcal{H}_{3})=0, \end{array}$$

and

limrΦ(r,z)=0,$$\begin{array}{} \displaystyle \lim_{r\rightarrow\infty}\Phi(r,z)=0, \end{array}$$

Φz(r,0)=0,$$\begin{array}{} \displaystyle \frac{\partial\Phi}{\partial z}(r,0)=0, \end{array}$$

Φr(0,z)=0.$$\begin{array}{} \displaystyle \frac{\partial\Phi}{\partial r}(0,z)=0. \end{array}$$

Our generalized solution becomes

Λ1=1,$$\begin{array}{} \displaystyle \Lambda_{1}=1, \end{array}$$

Λ2(ξ)=σ11Λ1e2ξh1σ21+Λ1e2ξh1σ1Λ1e2ξh11σ21+Λ1e2ξh1$$\begin{array}{} \displaystyle \Lambda_{2}(\xi)=\frac{\sigma_{1}\left(1-\Lambda_{1}e^{-2\xi h_{1}}\right)-\sigma_{2}\left(1+\Lambda_{1}e^{-2\xi h_{1}}\right)}{\displaystyle\sigma_{1}\left(\Lambda_{1}e^{-2\xi h_{1}}-1\right)-\sigma_{2}\left(1+\Lambda_{1}e^{-2\xi h_{1}}\right)} \end{array}$$

Λ3(ξ)=σ21Λ2e2ξh2σ31+Λ2e2ξh2σ2Λ2e2ξh21σ31+Λ2e2ξh2,$$\begin{array}{} \displaystyle \Lambda_{3}(\xi)=\frac{\sigma_{2}\left(1-\Lambda_{2}e^{-2\xi h_{2}}\right)-\sigma_{3}\left(1+\Lambda_{2}e^{-2\xi h_{2}}\right)}{\displaystyle\sigma_{2}\left(\Lambda_{2}e^{-2\xi h_{2}}-1\right)-\sigma_{3}\left(1+\Lambda_{2}e^{-2\xi h_{2}}\right)}, \end{array}$$

where

Ω(ξ)=Λ3(ξ)e2ξh3+1Λ3(ξ)e2ξh31.$$\begin{array}{} \displaystyle \Omega(\xi)=\frac{\Lambda_{3}(\xi)e^{-2\xi h_{3}}+1}{\Lambda_{3}(\xi)e^{-2\xi h_{3}}-1}. \end{array}$$

The individual average currents for the four electrodes are as follows (we have dropped the comma notation for the subscripts of the elements in A for notational convenience; b = [0, 0, αV0, V0]T):

I1=V0D(A12A23+A33A12A32A13A23αA42+A13αA42A13αA22A12A43α+A12A23α+A43αA22+A32A23A33A22+A13A22),$$\begin{array}{} I_{1}=\displaystyle \frac{V_{0}}{D}(-A_{12}A_{23}+A_{33}A_{12}-A_{32}A_{13}-A_{23}\alpha A_{42}\\ \quad+A_{13}\alpha A_{42}-A_{13}\alpha A_{22}-A_{12}A_{43}\alpha+A_{12}A_{23}\alpha\\ \quad~~+A_{43}\alpha A_{22}+A_{32}A_{23}-A_{33}A_{22}+A_{13}A_{22})\ , \end{array}$$

I2=V0D(A11A33A11A43αA11A23+A11A23αA13A21αA31A13+A41A13α+A43A21αA23αA41+A31A23+A13A21A33A21),$$\begin{array}{} I_{2}=-\displaystyle \frac{V_{0}}{D}(A_{11}A_{33}-A_{11}A_{43}\alpha-A_{11}A_{23}+A_{11}A_{23}\alpha\\ \quad~-A_{13}A_{21}\alpha-A_{31}A_{13}+A_{41}A_{13}\alpha+A_{43}A_{21}\alpha\\ \qquad-A_{23}\alpha A_{41}+A_{31}A_{23}+A_{13}A_{21}-A_{33}A_{21}) , \end{array}$$

I3=V0D(αA41A12αA41A22A11αA42+A32A11A31A12+αA42A21αA12A21A32A21+A12A21+A31A22A11A22+A11αA22),$$\begin{array}{} I_{3}=\displaystyle \frac{V_{0}}{D}(\alpha A_{41}A_{12}-\alpha A_{41}A_{22}-A_{11}\alpha A_{42}+A_{32}A_{11}\\ \qquad-A_{31}A_{12}+\alpha A_{42}A_{21}-\alpha A_{12}A_{21}-A_{32}A_{21}\\ \qquad+A_{12}A_{21}+A_{31}A_{22}-A_{11}A_{22}+A_{11}\alpha A_{22}), \end{array}$$

and

I4=I1I2I3,$$\begin{array}{} I_{4}=-I_{1}-I_{2}-I_{3}, \end{array}$$

with the constants

c=V0D(A13A21αA42+A23A11αA42A22A11A43αA33A12A21A31A13A22A11A32A23+A11A33A22+A31A12A23+A32A13A21+A41A13αA22+A43αA12A21A23αA41A12),$$\begin{array}{} c=\displaystyle\frac{V_{0}}{D}(-A_{13}A_{21}\alpha A_{42}+A_{23}A_{11}\alpha A_{42}-A_{22}A_{11}A_{43}\alpha\\ \qquad\quad-A_{33}A_{12}A_{21}-A_{31}A_{13}A_{22}-A_{11}A_{32}A_{23}\\ \qquad\quad+A_{11}A_{33}A_{22}+A_{31}A_{12}A_{23}+A_{32}A_{13}A_{21}\\ \quad~+A_{41}A_{13}\alpha A_{22}+A_{43}\alpha A_{12}A_{21}-A_{23}\alpha A_{41}A_{12}), \end{array}$$

and

D=A33A41A12A33A41A22A13A21A42+A31A13A42A31A13A22A23A41A12A31A23A42A33A12A21+A32A41A23A32A41A13+A12A21A43+A31A12A23A31A12A43+A31A22A43+A22A41A13+A33A42A21+A11A33A22+A11A32A43A11A33A42+A32A13A21A32A43A21A11A22A43+A11A23A42A11A32A23).$$\begin{array}{} D=A_{33}A_{41}A_{12}-A_{33}A_{41}A_{22}-A_{13}A_{21}A_{42} \\ ~~+A_{31}A_{13}A_{42}-A_{31}A_{13}A_{22}-A_{23}A_{41}A_{12} \\ ~~-A_{31}A_{23}A_{42}-A_{33}A_{12}A_{21}+A_{32}A_{41}A_{23} \\ ~~-A_{32}A_{41}A_{13}+A_{12}A_{21}A_{43}+A_{31}A_{12}A_{23} \\ ~~-A_{31}A_{12}A_{43}+A_{31}A_{22}A_{43}+A_{22}A_{41}A_{13} \\ ~~+A_{33}A_{42}A_{21}+A_{11}A_{33}A_{22}+A_{11}A_{32}A_{43} \\ ~~-A_{11}A_{33}A_{42}+A_{32}A_{13}A_{21}-A_{32}A_{43}A_{21} \\ ~-A_{11}A_{22}A_{43}+A_{11}A_{23}A_{42}-A_{11}A_{32}A_{23}). \end{array}$$

Note that the entries, Aij, in the above expressions entail solving the improper integral, Eq. 66, twelve times.

The parameters are given in Table 1.

Base-case dimensions and material parameters for the 4-electrode probe and 3 skin layers (n=3, m=4). (The outer radius of the skin is only used in the numerical solution for verification purposes.)

Current detection width, w11 mm[2]
Ceramic width, w2, w4, w60.15, 0.15, 1.9 mm[2]
Guard width, w30.3 mm[2]
Secondary inject width, w50.5 mm[2]
Primary inject width, w70.5 mm[2]
Outer radius of the probe, R4.5 mm[2]
Outer radius of the skin, Rskin20 mm
Stratum corneum thickness, h314 μm[11]
Viable skin thickness, h21.2 mm[11]
Adipose tissue thickness, h11.2 mm[11]
Inject voltage, V00.05 V[2]
Depth setting, α0.1[2]
Electrical permittivity in vacuum, ε08.85 ×107 Fm-1[12]

For the skin, we can write the constitutive relations for the relative permittivity and the conductivity (S m-1) as follows [9] based on the dimensionless logarithmic frequency, N = log10(ν/1 Hz).

In the stratum corneum, i=3, we have

log10(σ(3))=0.011516×N5+0.25509×N42.2537×N3+9.9955×N221.404×N+11.803,$$\begin{array}{} \quad~\log_{10}(\sigma^{(3)})=-0.011516\times N^{5}+0.25509\times N^{4}\\ -2.2537\times N^{3}+9.9955\times N^{2}-21.404\times N+11.803, \end{array}$$

log10(εr(3))=0.011319×N5+0.23953×N42.0255×N3+8.5278×N217.961×N+17.57,$$\begin{array}{} \quad\,\log_{10}(\varepsilon_{r}^{(3)})=-0.011319\times N^{5}+0.23953\times N^{4}\\ -2.0255\times N^{3}+8.5278\times N^{2}-17.961\times N+17.57, \end{array}$$

In the viable skin, i=2, we have

log10(σ(2))=0.015178×N50.34167×N4+3.0088×N312.952×N2+27471×N23.688,$$\begin{array}{} \quad~~\log_{10}(\sigma^{(2)})=0.015178\times N^{5}-0.34167\times N^{4}\\ +3.0088\times N^{3}-12.952\times N^{2}+27_{-}471\times N-23.688, \end{array}$$

log10(εr(2))=0.044465×N5+0.92429×N47.7649×N3+32.847×N270.466×N+67.61,$$\begin{array}{} \quad~\log_{10}(\varepsilon_{r}^{(2)})=-0.044465\times N^{5}+0.92429\times N^{4}\\ -7.7649\times N^{3}+32.847\times N^{2}-70.466\times N+67.61, \end{array}$$

And finally in the adipose tissue, i=1, we have

σ(1)=1/54.9236,$$\begin{array}{} \sigma^{(1)}=1/54.9236, \end{array}$$

log10(εr(1))=0.00052423×N50.00040926×N4+0.11797×N30.77214×N2+0.083142×N+7.584$$\begin{array}{} \quad\,\log_{10}(\varepsilon_{r}^{(1)})=-0.00052423\times N^{5}-0.00040926\times N^{4}\\ +0.11797\times N^{3}-0.77214\times N^{2}+0.083142\times N+7.584{\small\angle} \end{array}$$

Numerics

We solve the model, Eqs. 1-13, in the finite-element solver Comsol Multiphysics 5.2a [13] and the approximate semianalytical model, Eqs. 79-88, and its pointwise solution in the skin layers, Eq. 55, in the general-purpose numerical software Matlab R2015 [14].

Results and discussion

We have thus far derived an approximate semianalytical solution of the mechanistic model for EIS measurements of human skin. Let us proceed by validating it with a previous study comprising 120 young subjects (24 ± 3 years old), who had no known skin diseases or allergies; and were asked to abstain from applying moisturizers on the day of the measurements. Before the measurements were taken on the volar forearm of the subjects, their skin was soaked in physiological saline solution (at a concentration of 0.9% sodium chloride in mass) for 1 min [9].

From the full set of 35 impedance measurements distributed logarithmically from 1 kHz to 2.5 MHz, we select a subset within the frequency range of 1 kHz to 1 MHz and a depth setting, α, of 0.1 for this validation. Overall, as can be inferred from Fig. 4, we see a good agreement between the semi-analytical solution (lines) and the measured means (symbols) for both the magnitude and the phase throughout the frequency spectrum. Furthermore, the semi-analytical solutions for ± 1 standard deviation for the thickness of stratum corneum (dashed lines) follow the ± 1 standard deviations from the experiments (error bars) well for the magnitude and manages to capture the increasing variations, larger standard deviations, in the measured phase as we move towards 1 MHz.

Fig. 4

The mean magnitude (▲) and phase (•) from the experimental EIS measurements [9] with the error bars indicating one standard deviation. The lines are the predicted counterparts from the approximate semi-analytical solution, where (–) represent a thickness of 14 μm SC and (- -) denotes a thickness of either 11 or 17 μm. These values for the thickness of SC correspond to the mean and ± one standard deviation.

As we proceed to verify the individual currents at each electrode, we observe that the real part has good agreement throughout the frequency range, while the imaginary part has while the imaginary part has good agreement for most of the range and has a slight deviation of around 9% towards 1 MHz, as depicted in Fig. 5. By employing Hankel transform, we obtain a lower error with our analytical solution, as compared to the previously used analytical solution [2].

Fig. 5

The individual currents, Ik, in terms of their (a) real and (b) imaginary parts for the numerical solution of the complete model (symbols) and the corresponding analytical counterpart (lines), for I4 (★), I2 (▲), I1 (■) and I3 (▼).

One advantage of the analysis is that we can estimate not only the overall measured impedance in the form of the magnitude and phase, but also quantify the local potential distribution throughout the various skin layers of interest. It turns out that the approximation of the uppermost boundary conditions between the stratum corneum and the electrodes is valid throughout the frequency spectrum we consider here, as illustrated in Fig. 6 at 1 MHz. In particular, we are able to capture the one-dimensional current in the stratum corneum in the z-direction, which has also been shown through a scaling analysis [3].

Fig. 6

The absolute value of the potential distribution at 1 MHz for n=3 and m=4: The numerical solution of the full set of equations for SC (a), VS (c), and AT (e); and approximate semianalytical solution for SC (b), VS (d) and AT (f).

Conclusions

We have secured approximate semi-analytical solutions of a generalized continuous mathematical model that considers EIS with a probe comprising m electrodes and the adjacent skin divided into n layers.

Good agreement for the frequency range of 1 kHz to 1 MHz was found between the semi-analytical solution and the numerical solution of the full set of equations assuming a three-layer skin entity as well as an experimental series of 120 subjects that were measured with an EIS probe.

This particular probe has four electrodes – one sense, one guard and two injects – but we note that one can easily modify the boundary conditions such that one can model different types of measurements, such as a pure two-point measurement.

These solutions lend themselves well to studies that seek to calibrate the material properties of more skin layers than the three we consider here. In particular, it would be of interest to split up our viable skin into living epidermis and dermis since they should have significantly different material properties.

Finally, with regards the semi-analytical nature of the solutions, we note that if we manage to find closed-form expressions – even approximate ones – of the improper integrals, Aij, that arose when we pulled the solution back to the original cylindrical domain, we would be able to achieve an analytical solution that does not require any numerical integration at all.

Fig. 1

Schematic of skin layers.
Schematic of skin layers.

Fig. 2

Schematic of skin comprising n layers and a circular EIS probe with m electrodes.
Schematic of skin comprising n layers and a circular EIS probe with m electrodes.

Fig. 3

Schematic of a 4-electrode probe and skin modeled as a three-layer entity.
Schematic of a 4-electrode probe and skin modeled as a three-layer entity.

Fig. 4

The mean magnitude (▲) and phase (•) from the experimental EIS measurements [9] with the error bars indicating one standard deviation. The lines are the predicted counterparts from the approximate semi-analytical solution, where (–) represent a thickness of 14 μm SC and (- -) denotes a thickness of either 11 or 17 μm. These values for the thickness of SC correspond to the mean and ± one standard deviation.
The mean magnitude (▲) and phase (•) from the experimental EIS measurements [9] with the error bars indicating one standard deviation. The lines are the predicted counterparts from the approximate semi-analytical solution, where (–) represent a thickness of 14 μm SC and (- -) denotes a thickness of either 11 or 17 μm. These values for the thickness of SC correspond to the mean and ± one standard deviation.

Fig. 5

The individual currents, Ik, in terms of their (a) real and (b) imaginary parts for the numerical solution of the complete model (symbols) and the corresponding analytical counterpart (lines), for I4 (★), I2 (▲), I1 (■) and I3 (▼).
The individual currents, Ik, in terms of their (a) real and (b) imaginary parts for the numerical solution of the complete model (symbols) and the corresponding analytical counterpart (lines), for I4 (★), I2 (▲), I1 (■) and I3 (▼).

Fig. 6

The absolute value of the potential distribution at 1 MHz for n=3 and m=4: The numerical solution of the full set of equations for SC (a), VS (c), and AT (e); and approximate semianalytical solution for SC (b), VS (d) and AT (f).
The absolute value of the potential distribution at 1 MHz for n=3 and m=4: The numerical solution of the full set of equations for SC (a), VS (c), and AT (e); and approximate semianalytical solution for SC (b), VS (d) and AT (f).

Base-case dimensions and material parameters for the 4-electrode probe and 3 skin layers (n=3, m=4). (The outer radius of the skin is only used in the numerical solution for verification purposes.)

Current detection width, w11 mm[2]
Ceramic width, w2, w4, w60.15, 0.15, 1.9 mm[2]
Guard width, w30.3 mm[2]
Secondary inject width, w50.5 mm[2]
Primary inject width, w70.5 mm[2]
Outer radius of the probe, R4.5 mm[2]
Outer radius of the skin, Rskin20 mm
Stratum corneum thickness, h314 μm[11]
Viable skin thickness, h21.2 mm[11]
Adipose tissue thickness, h11.2 mm[11]
Inject voltage, V00.05 V[2]
Depth setting, α0.1[2]
Electrical permittivity in vacuum, ε08.85 ×107 Fm-1[12]

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