Fundamentals of Skin Bioimpedances

The bioimpedances of tissues beyond the stratum corneum, which is the outermost layer of skin, contain crucial clinical information. Nevertheless, bioimpedance measurements of both the viable skin and the adipose tissue are not widely used, mainly because of the complex multilayered skin structure and the electrically insulating nature of the stratum corneum. Here, a theoretical framework is established for analyzing the impedances of multilayered tissues and, in particular, of skin. Then, strategies are determined for the system‐level design of electrodes and electronics, which minimize 4‐wire (or tetrapolar) measurement errors even in the presence of a top insulating tissue, thus enabling non‐invasive characterizations of tissues beyond the stratum corneum. As an example, non‐invasive measurements of bioimpedances of living tissues are demonstrated in the presence of parasitic impedances which are much (e.g., up to 350 times) higher than the bioimpedances of the living tissues beyond the stratum corneum, independently on extreme variations of the barrier (tape stripping) or of the skin–electrode contact impedances (sweat). The results can advance the development of bioimpedance systems for the characterization of viable skin and adipose tissues in several applications, including transdermal drug delivery and the assessment of skin cancer, obesity, dehydration, type 2 diabetes mellitus, cardiovascular risk, and multipotent adult stem cells.


Introduction
Skin is the outermost layer of the body and contributes to several key functions, including protection against external agents, complicate the electrical characterization of both viable skin and the underlying adipose tissue, thus limiting practical applications of skin bioimpedances. In fact, not only bioimpedance is much less used [9] than technologies (X-rays, computed tomography, positron emission tomography, …) which are much more expensive and potentially dangerous, but, as discussed later, it is generally easy to non-invasively measure the bioimpedance only for the stratum corneum (i.e., the outermost layer) and muscles, but not for viable skin and adipose tissues.
The multilayer structure of skin complicates the non-invasive and selective characterization of different layers, which is essential for extracting from measurements clinically relevant parameters which typically depend on individual layers (e.g., for the diagnosis of skin cancer, viable skin is the main layer of interest). As a second issue, the outermost layer (i.e., the only layer which is directly and non-invasively accessible) of the skin is the stratum corneum, which may be considered as constituted by dead cells and acts as an electrical barrier preventing the noninvasive electrical analysis of the deeper living tissues which perform crucial tasks and may also become sick. [11] In practice, the unknown and frequency-dependent ratio between the impedances of the stratum corneum and of the underlying living tissues may typically range between 100 and 1000. [11] The stratum corneum impedances are reduced and measurements are simplified by increasing the electrodes areas and the distances between the current carrying force electrodes. These strategies can result in significant currents flowing through deeper living tissues and enable electrical impedance tomography, [12,13] bioelectrical impedance analysis [14,15] and electrical impedance myography, [16] but are not suitable for the accurate evaluation of viable skin and adipose tissue, unless the stratum corneum is removed (e.g., tape stripping), penetrated by conductive needles or destroyed by certain medical conditions, so that the non-invasive electrical characterization of viable skin and adipose tissues is an open challenge.
There have been many attempts to solve this longstanding "stratum corneum dilemma". [11] Simulations with a simplified two-layer model show that at higher frequencies tissues below the stratum corneum can dominate the measurements, [17] but, first, low-frequency characteristics would still be hidden and, second, the relation between measurement depth and frequency would depend on unknown geometrical and electrical parameters of the tissues under analysis. The stratum corneum can also be invasively removed by aggressive gels or mechanical procedures such as tape stripping, at the cost of perturbing tissues, affecting measurements and introducing possibly intricate artifacts (e.g., osmotic transport of water [11] or slow penetration of gels through the skin-fat layer [18] ) or even health threats (e.g., in case of skin tumors). Conductive microneedles [19,20] can penetrate the stratum corneum and reach deeper living tissues. With this approach a 96.6% [21] sensitivity to melanoma has been demonstrated with the Nevisense system (SciBase AB, Stockholm, Sweden) which uses arrays of 150 μm high needles that are longer than typical stratum corneum thicknesses, but usually not sufficient for reaching blood vessels or nerve endings. Despite these promising results, some shortcomings may justify limited acceptance from dermatologists. [22] For instance, Nevisense has been approved by the FDA, but with limits (e.g., not suitable for curved or soft regions and only suitable for intact skin without inflammation, ulceration, scar, fibrosis, psoriasis, eczema or similar) and risks (e.g., bleeding or transferring malignant cells [23] ) associated to the microneedles and to the rigidity of the electrodes, which also requires soaking skin in saline solution prior to each measurement in order to improve the electrode-to-skin electrical contact.
Here, first, we systematically study impedances in multilayer structures and introduce a set of definitions (frequencydependent corner frequencies, current tubes, current tubes angles, relative tube currents, sensing tubes and sensing equipotential lines) which, together with equivalent circuits, symmetry considerations and the continuity of both the tangential electric field and the normal current density can facilitate the analysis and design of systems for measuring impedances in multilayer structures. Afterward, we introduce the normalized derivatives of the 4-wire (or tetrapolar) impedance (or admittance) with respect to the key electrical parameters and determine their fundamental properties, including their ability to easily identify the electrical parameters which mostly affect the measurements. Finally, we present strategies for overcoming the "stratum corneum dilemma" [11] and designing systems that can non-invasively measure bioimpedances of living tissues beyond the intact stratum corneum with conventional electrodes, without any pre-treatment and are suitable for measurements on soft and curved tissues and for integration in wearable systems. Theoretical analysis, SPICE simulations, test circuits and in vivo experiments demonstrate that bioimpedances beyond much (e.g., up to 350 times) higher parasitic impedances can be accurately measured. These results can greatly simplify the design of systems for enabling the accurate measurement of impedance in multilayer structures and can pave the way to accurate and non-invasive electrical characterizations of viable skin and adipose tissues for a wide variety of applications. Figure 1A schematically shows two electrodes in contact with a simplified 2D parallel four-layer scheme of the stratum corneum and of the underlying tissues, namely viable skin, adipose tissue and muscle. For a resistor, assuming, for simplicity, electrical homogeneity and isotropy, as illustrated in Figure 1B, the resistance of a parallelepiped structure (current parallel to the length L and orthogonal to the cross section S) is L/S = L/( S) where and are the electrical resistivity and conductivity, respectively. Analogously, for the same structure, in the Fourier domain, the ratio between the voltage and the current, namely the impedance Z, may be expressed as

Theory of Bioimpedance in Multilayer Structures
where Y, , , , and f are the admittance, the electrical conductivity, the dielectric constant, the impedivity or specific impedance (i.e., analogous to resistivity), the angular frequency and frequency, respectively. If the conductivity and the dielectric constant do not significantly depend on frequency, at frequencies much lower or much higher than the corner frequency /2 the impedance may be approximately considered as purely resistive or capacitive, respectively ( Figure 1B). However, conductivities and dielectric constants of biological tissues change with frequency. We can define a frequency-dependent corner frequency f C (f), equal to (f)/2 (f), so that, for a given tissue, at any frequency f x , if f C (f x ) is much higher or much lower than the frequency f x , (f x ) may be considered as purely real (resistive) or imaginary (capacitive), respectively. For each tissue, Figure 1C shows the magnitude of the impedivities corresponding to typical conductivities and dielectric constants [24][25][26] for the bandwidth (10 Hz, 1 MHz) and also illustrates that for some frequencies (including the most common 10 kHz to 100 kHz range) the stratum corneum tends to behave as a capacitor, whereas the adipose tissue and the muscle can be approximately considered as resistive ( Figure S1, Supporting Information). The magnitude of the stratum corneum impedivity decreases by orders of magnitude with increasing frequency (from 61.8 kΩ m at 10 Hz down to 37.6 Ω m at 1 MHz), but is always high. By contrast, both viable skin (4.36 to 1.63 Ω m) and muscles (4.94 to 1.95 Ω m) tend to have low impedivities, while the adipose tissue has high impedivities (77 to 39.8 Ω m) which, at high frequencies become comparable with the stratum corneum. Though bioimpedances depend on the geometries of both electrodes and tissues, the analysis of impedivities provides precious insight for design. Figure 1D,E shows typical FEM calculations found with typical impedivities ( Figure 1C) and thicknesses (20 μm, 1.2 mm, 1.2 mm and 10 mm for stratum corneum, viable skin, adipose tissue and muscle, respectively [24,27,28] ) for 2 mm electrodes parallel widths, with 1 cm distance between the force electrodes, d FE , and an input voltage with magnitude and frequency equal to 1 V and 100 kHz, respectively.
The idealized 2D parallel 4-layers system ( Figure 1A) can be divided into 4 zero-intersection current tubes ( Figure 1E), one for each layer. For a certain layer, the correspondent current tube is defined as the region which connects the current-carrying force electrodes and is delimited by the two current lines which, in correspondence of the device center line (i.e., the vertical straight line which is equidistant from the two force electrodes), exactly coincide with the two horizontal borders of the layer (Note S1, Supporting Information). For instance, the stratum corneum (not visible in Figure 1D,E because of its very small thickness) and the muscle current tubes are the shallowest and the deepest current tubes, respectively. The bottom border of the muscle current tube is determined by the domain boundary and is unimportant for the analysis provided that the domain is sufficiently large (for given electrodes). The total 2-wire impedance between the force electrodes is the parallel of all the current tubes impedances. For each layer, the relative tube current, defined as the ratio between the tube current and the total current quantifies the relative contribution of the current tube to the total current (Note S1, Supporting Information).
Due to the high stratum corneum impedivities, significant fractions of the total current can penetrate the stratum corneum and flow through deeper layers only if the stratum corneum thickness (typically 10 to 200 μm [29,30] ) is small in comparison with the distance between the electrodes or with the electrodes parallel width. [31] The penetration of significant current through the stratum corneum is a necessary, but not sufficient condition for measuring the electrical properties of deeper tissues. Unless large electrodes areas and high d FE are used, at typical frequencies (1 kHz to 1 MHz) the impedances of the current tubes associated with the deeper layers are easily dominated by the parts of the tube belonging to the high-impedivity stratum corneum, so that the 2-wire bioimpedance is almost independent on the (electrical and geometrical) properties of deeper tissues. As an additional issue, sweating results in extreme variability of the contact impedances between skin and dry electrodes. [32][33][34] These difficulties may be overcome by 4-wire (i.e., tetrapolar) impedance measurements [9,11,35] taken by adding two central (for simplicity, symmetric with respect to the center line) sense electrodes to the two outer force electrodes.
The sensing tube for a given layer can be defined as the restriction of the correspondent current tube included between the sensing equipotential lines (i.e., the equipotential lines passing through the sensing electrodes). The 4-wire impedance is the parallel connection of the sensing tubes impedances of all the layers.
At the interfaces between different layers, both the tangential component of the electric field, E t , and the normal component of the current density, J n (i.e., the product of and the normal component of the electric field, E n ), are continuous. [11] As a consequence, the large differences among the impedivities ( Figure 1C) translate into the typical behavior of the equipotential and current lines in Figure 1D, with equipotential lines becoming more horizontal when entering the (high , low E n ) adipose tissue and more vertical when entering the (low , high E n ) muscle.
Across all typical frequencies, the adipose tissue has much higher impedivities ( Figure 1C; Figure S1, Supporting Information) than viable skin. Therefore, unless the distance between the outer force electrodes, d FE , is very small (i.e., almost no current can penetrate below viable skin) or very large (see later), the adipose tissue current tube (blue lines in the inset in Figure 1E) carries a very small current, is very narrow when crossing the (lower impedivity) viable skin and only enters into the (higher impedivity) adipose tissue in proximity of the center with an almost isosceles triangular shape having identical adipose current tube angles AT , approximately equal to 45°( Figure 1E). In fact, the adipose layer can be approximately modeled by a (4N 2 + 3N) lumped impedances circuit which can be analyzed by means of a reduced 2N 2 lumped impedances half circuit ( Figure S4 , Supporting Information, represented with N = 3 only for illustrative purposes). For every N, assuming the antiparallel (i.e., flowing opposite to the current direction at the center of the device) components of the current are negligible, the homogeneity and the isotropy simplifying hypotheses for the adipose layer (identical impedances), the narrowness of the adipose tissue current tube in the viable skin (force electrodes represented by a single node), the existence of the central symmetry plane (vertical short-circuits and reduction to the equivalent half circuit), and the lower impedivity of the underlying muscle (horizontal short-circuits) result in an almost ideal 45°axis of symmetry ( Figure 1E; Figure S4 , Supporting Information). In practice, the adipose tissue current tube must have a flattened region at the center (instead of a vertex, Figure 1E) because the current lines at the center are necessarily horizontal (i.e., vertical equipotential lines), without consequences on the 45°angle (e.g., the inclusion of the flattened region would equally affect the vertical-horizontal and the horizontal paths). The isosceles triangle base is then somewhat larger than twice the adipose tissue thickness ( Figure S5 , Supporting Information). Remarkably, these considerations approximately determine the shapes of the adipose layer, the viable skin and the muscle current tubes (except the bottom border of the muscle current tube and the top border of the viable skin current tubes which, in practice, are irrelevant). FEM calculations confirm that AT is very close to 45°( Figure S6 , Supporting Information) independently on both frequency and the dimensions of the electrodes, except when d FE is so small that almost no current can penetrate below viable skin, thus resulting in AT much larger than 45°(e.g., see Figure S7 , Supporting Information, for d FE = 1 mm), or so large that the equipotential (current) lines become almost vertical (horizontal) for a very large portion of the device, thus resulting in AT much smaller than 45°(e.g., see Figure S8, Supporting Information, for d FE = 100 mm).

Analysis of Multilayer Structures with Normalized Derivatives
The significant series parasitic impedances due to the insulating nature of the stratum corneum would result in large errors for 2-wire characterizations of the tissues beyond the stratum corneum (Note S2, Supporting Information). In principle, this issue can be solved by 4-wire (or tetrapolar) bioimpedance measurements [36][37][38] which, only ideally, allow to measure an impedance of interest independently on the parasitic series impedances (Notes S3 and S4, Supporting Information). However, even with ideal 4-wire measurements, a key problem in multilayer structures is to understand which parameters significantly or negligibly contribute to the measured bioimpedance. Besides, from an applicative point of view, it may be crucial to design a system that is only sensitive to one parameter of interest. For instance, for the detection of cancer in viable skin, the system should ideally be able to measure, at a given frequency, the conductivity and/or the dielectric constant of viable skin. For these purposes, multilayer structures made of several stacked homogeneous layers ( Figure 1A) can be easily analyzed by means of the normalized derivatives of the 4-wire impedance Z 4W or, equivalently, of the 4-wire admittanceY 4W = 1/Z 4W with respect to the electrical parameters of the different layers (i.e., the dielectric constants k and the conductivities k of all the tissues). For instance, the normalized derivative of Y 4W with respect to x k (i.e., a given electrical parameter x k , namely the dielectric constant k or the conductivity k ) is Similarly, normalized derivatives can be useful in the more general case of a system made of several regions, each with an  arbitrary shape (i.e., not necessarily stacked rectangular regions), provided that each region is made of an electrically homogeneous and isotropic material (i.e., both the conductivity and the dielectric constant must be scalar and constant within the region). Such multiregion structure can be used for studying the effects of nonhomogeneities as in several cases a non-homogeneous layer may be approximated by different adjacent regions, with each region being internally homogeneous. A simple example for illustrating the practical relevance of this extension can be a multilayer structure ( Figure 1A) with a circular region of viable skin replaced by a different material (e.g., a cancer region) having its own electrical characteristics (different from the host viable skin). In the general case of multiregions structures, with each region made of an electrically homogeneous and isotropic material and, in particular, in multilayer structures made of several stacked homogeneous and isotropic layers ( Figure 1A) Y 4W is a homogeneous function of degree 1 of the conductivities and dielectric constants of all the layers. In fact, intuitively, if all the electrical parameters ( k , k ) of all the layers are multiplied by a scalar c, Y 4W is multiplied by the same scalar (e.g., if all the parameters For instance, if, for simplicity, we restrict to DC (or, equivalently, we only consider conductivities and neglect dielectric constants), whatever is the geometrical configuration of the multiregion structure, if we double the electrical conductivities of all the regions, Y 4W will also be doubled. More rigorously, if we consider a distributed equivalent circuit for the system, this result can be seen as the consequence of the linearity of the Kirchoff and Ohm laws. For homogeneous functions of degree 1, the Euler homogeneous function theorem states that which, dividing by Y 4W , results in the fundamental property or, equivalently, the sum of all the normalized derivatives is equal to 1. Clearly, dZ 4W /Z 4W is equal to (−dY 4W /Y 4W ) because of the equivalence Y 4W = 1/Z 4W , so that the normalized derivatives of Z 4W are the opposite inverse of the correspondent normalized derivatives of Y 4W and, therefore, hold similar properties. Moreover, since, for every k-th region, Y 4W depends on the impedivity of the region and, therefore, depends in the same way on k and on j k , we find We also verified these fundamental properties of the normalized derivatives by FEM simulations. As an example, Figure 2 shows the magnitudes of the normalized derivatives of the 4-wire impedance Z 4W (or, equivalently, of the 4-wire admittance Y 4W ) with respect to the electrical parameters of the stratum corneum, viable skin, adipose tissue and muscle computed at 100 kHz as a function of the electrodes parallel width. Clearly, for a certain electrodes parallel width, the sum of all the magnitudes can be larger than 1 (triangle inequality), while (as obvious from Equation (5), by considering that dZ 4W /Z 4W is equal to −dY 4W /Y 4W ) the sum of all the normalized derivatives is exactly equal to −1.
As evident, at very low values of d E , Z 4W mainly depends on the dielectric constant of the stratum corneum (which, at 100 kHz, is mainly capacitive, Figure 1C; Figure S1D, Supporting Information). For increasing values of d E , Z 4W becomes more dependent on deeper tissues and, with proper dimensions of the electrodes, Z 4W can provide information on both viable skin and the adipose tissue. The high impedivity of the stratum corneum results in a barrier that prevents the electrical characterization of the underlying layers. However, when d E increases, the stratum corneum becomes more and more negligible. In fact, as a rudimentary example, we can consider a simplified multilayer structure ( Figure 1A) made of only two constant-thickness layers, namely an ideal insulator (i.e., a perfect dielectric, with zero conductivity) on a good electrical conductor, with, on top of the insulator, two electrodes having parallel width equal to d E , constant orthogonal width and a distance equal to Nd E (with a constant N). The equivalent capacitance separating each electrode from the underlying conductor will approximately (more accurate values can be found by analyzing the distributed system with FEM analyses) be directly proportional to d E (as the area of the capacitance linearly increases with d E , whereas the insulator thickness is constant), thus resulting in an insulating impedance inversely proportional to d E (i.e., the insulation is less and less effective for increasing d E ). Moreover, for increasing d E , the distance between the electrodes (Nd E ) increases, so that the electrical length and, therefore, the resistance of the conductor increases. As a result, the barrier introduced by the insulator is less and less effective for increasing d E .
These FEM calculations also confirm that, as predicted by Equation (5), the normalized derivatives with respect to the conductivity k of the k-th layer is equal to the normalized derivatives with respect to the dielectric constant of the same layer, k , multiplied by k and divided by j k .
The contributions of the different tissues to the total 4-wire impedance can also be evaluated by using the selectivity, [11,35] which can be applied even for non-homogeneous layers. However, the normalized derivatives with respect to the electrical parameters of the different tissues have a much easier derivation and physical interpretation and a direct correspondence with electrical parameters. Besides, the Equation (4) also reveals the presence of an approximately parallel connection of the different layers, with a perfect correspondence (i.e., parallel connection) for large distances between the force electrodes. In fact, at sufficiently large d E the equipotential lines passing through the sense electrodes become almost vertical and, therefore, Y 4W can be approximately computed as the parallel of all the N parallelepiped admittances Y k of all the layers (in this case, the derivative of Y 4W with respect to the conductivity of a layer in Equation (4) would be equal to the layer thickness multiplied by the electrodes orthogonal width and divided by the distance between the sense electrodes, whereas the derivative of Y 4W with respect to the dielectric constant of a layer would be equal to the layer thickness multiplied by the electrodes orthogonal width, divided by the distance between the sense electrodes and multiplied by j ). In practice, the hypothesis on the homogeneity of all the layers, which is necessary for defining the normalized derivatives, may be partially removed as the inhomogeneities of electrical parameters which, if homogeneous, would have very small normalized derivatives, will obviously be irrelevant. Moreover, for design, simple and approximate models are often more useful than more accurate but more complex ones (whereas for analysis, more accurate calculations may be useful).

Strategies for System-Level Design
As shown in Figure 2, 4-wire measurements can, in principle, allow the characterization of the living tissues beyond the insulating stratum corneum. However, until now we have only considered an ideal 4-wire measurement, without taking into account the errors introduced by the unavoidable parasitics and the imperfect electronic interface. These non-idealities must also be carefully investigated because the high impedivity of the stratum corneum tends to increase their effects on the measurements (Notes S2-S4, Supporting Information).
Although, obviously, skin is a distributed system, as a first approximation, the measurement errors of 4-wire electronic interfaces and their dependence on the electrodes can be analyzed in the Fourier domain with the highly simplified lumped two-layers skin model and circuit shown in Figure 3A, where C F-GND (Z F-GND ) and C S -GND (Z S-GND ) are the parasitic capacitances (impedances) between ground and the force and sense electrodes, respectively, C SS and C FS are the parasitic capacitances between the sense electrodes and between adjacent force and sense electrodes, respectively, and Z SC (R SC¸CSC ) and Z B (R B , C B ) are the equivalent lumped impedances (resistance, capacitance) of the stratum corneum and of the living tissues beyond the stratum corneum, respectively. In striking contrast with conventional 4-wire measurements, small parasitics can introduce large errors because of the high stratum corneum impedivities (Notes S2-S4, Supporting Information). The first prerequisite for design is then to guarantee that parasitic impedances have much higher magnitude than Z SC (see Supporting Information for rigorous derivations). Under these circumstances, the orders of magnitude of the errors introduced by C F-GND , C S -GND , C SS and C FS can be estimated by separately considering each type of capacitances (Notes S5-S11, Figure S12, Supporting Information), neglecting the non-idealities of both the instrumentation amplifier and the op amp, and taking into account that |Z SC | ≫ |Z B | (high impedivity of the stratum corneum). A parasitic mechanism is negligible only if it does not significantly change the current through Z B (ideally close to V IN /2Z SC ) and does not introduce significant differences between the voltage of interest (across Z B ) and the voltage across the sensing electrodes (input voltage of the instrumentation amplifier). Under the assumption of ideal op amp, C F-GND does not introduce any error (Note S6, Supporting Information). The error due to C S -GND (Note S7, Supporting Information) is small because |Z S-GND | ≫ |Z SC |, so that, first, the error currents through C S,GND are much smaller than the current through Z B and, second, even considering imbalances among different Z SC , the voltage at each input terminal of the instrumentation amplifier is almost identical to the voltage at the corresponding terminal of Z B (the error may be further reduced if different Z SC and different C S,GND are similar so that the currents through the capacitances C S,GND would both be close to V IN /2(Z SC + Z S,GND ) and would introduce at the sensing nodes similar error voltages whose common mode may be Figure 3. System design. A) Simplified lumped two-layers skin model and 4-wire bioimpedance circuit with explicit illustration of the parasitic capacitive impedances between the sense electrodes (Z SS ), between the force electrodes (Z FF ) and between ground and the force (Z F-GND ) or sense (Z S-GND ) electrodes. B) Equivalent lumped model for calculating the error introduced by the parasitic impedances between adjacent sensing and force electrodes, Z FS (see Supporting Information for details and for the analysis of errors introduced by other parasitics). C) Semi-circuit illustrating how to actively shield the sense electrodes and to passively shield the force electrodes. D) False-color photo of 4-wire bioimpedance device with electrodes, actively (sensing) and passively (force) shielded connectors. E) Photo of an electronic system for wirelessly measuring bioimpedances beyond the stratum corneum, including a Bluetooth Low Energy (BLE) microcontroller, an AD5933 and the circuitry for the actively-shielded 4-wire bioimpedance measurement (C). F) 2-wire and G) 4-wire measurements on a test PCB which simulates the skin with discrete components according to the highly simplified lumped model shown in (A), with typical values for Z B (100 Ω) and Z SC (insets), with each line representing, for each frequency, the mean value of 10 subsequent measurements (relative standard deviation below 1%). rejected by the instrumentation amplifier CMRR). The error due to C SS (Note S8, Supporting Information) is small as, first, the series (2z SC + z SS ) is in parallel with the much smaller z B (error current through C SS much smaller than current through z B ) and, second, since |Z SS | ≫ |Z SC | the voltage across the sensing electrodes is almost identical to the voltage across z B . By contrast, with typical values, C FS is the most critical non-ideality (Notes S5-S11, Supporting Information). In fact, as evident from Figure 3B, since |Z SC | ≫ |Z B | the voltages at both the terminals of z B are close to V IN /2. As a result, the current (top to bottom) through both the impedances Z FS (and therefore through their seriesconnected Z SC impedances) is approximately equal to V IN /2Z FS , thus yielding a relative equivalent error in the estimation of Z B equal to Such a relative error, though |Z FS | ≫ |Z SC |, may be large as |Z SC | ≫ |Z B |, especially for compact electrodes (i.e., large Z SC ). The instrumentation amplifier has a common mode voltage (close to V IN /2) much higher than the differential input voltage (close to V IN Z B /2Z SC ) and thus introduces a relative equivalent error where A D and A CM are the differential and the common mode gain of the instrumentation amplifier. These relations show that the relative errors RE CFS and RE CMRR can be small, even if Z B is significantly lower than Z SC , only if the magnitudes of both Z FS and the instrumentation amplifier CMRR are sufficiently high. More accurate circuit analyses (see Supporting Information and Figure S12) and SPICE simulations ( Figure S13, Supporting Information) taking into account the non-idealities of both the op amp and the instrumentation amplifier confirm these results. As a consequence, first, proper design of electrodes, connectors and electronics must guarantee that parasitic impedances have much higher magnitudes than Z SC and that the relative error RE CFS is small enough (see Equation (7)), thus introducing fundamental limits to downsizing the electrodes areas. Second, as illustrated by the semi-circuit in Figure 3C and the false-colored photo of a practical device in Figure 3D, the effective value of C FS can be substantially reduced by actively shielding (bootstrap) the metals connecting the sense electrodes to the input terminals of the instrumentation amplifier with voltage buffers. Third, the force electrodes and their connectors can simply be passively shielded ( Figure 3C,D) in order to prevent the injection of current disturbances at the negative input terminal of the op amp ( Figure 3A) and to minimize the parasitic capacitances associated with the force electrodes (at the cost of increasing C F-GND which, with good op amps, may be made irrelevant). Fourth, the above discussions permit to select op amps (effective active and passive shielding as well as insensitivity to C F-GND across all the frequencies of interest) and the instrumentation amplifier (high enough CMRR across all the frequencies of interest, see Equation (8)). Fifth, as shown in Figure 3E, since the effectiveness of bootstrap is not ideal, especially at high frequencies, the connectors lengths must be minimized by microcontrollerbased, portable electronics close to the electrodes which can use Bluetooth Low Energy wireless communication ( Figure 3E). Each input terminal of the instrumentation amplifier must be biased by a DC voltage (within the input common mode voltage range of the instrumentation amplifier) by a sufficiently high resistor (e.g., 10 MΩ). Figure S14 (Supporting Information) shows a simplified scheme of the circuit. In order to prevent DC currents to flow toward the body, the positive input terminal of the op amp ( Figure 3A) must be biased at the DC component of the input voltage (instead of grounded) and blocking capacitors, in series to each electrode, may be included. Figure 3F shows measurements on a test PCB with reference components (typical values for bioimpedances with d E in the mm range) and confirms that our 4-wire system can enable the measurement of bioimpedances beyond much higher parasitic impedances (Note S12, Figure S15, Supporting Information).

Measurements of Bioimpedances of Living Tissues Beyond the Intact Stratum Corneum
Bioimpedances of living tissues beyond the stratum corneum should be measured with dry electrodes directly attached to the skin and without any intermediate gel. In fact, besides practical convenience, especially for wearable devices and longterm monitoring, [32,34,39] gels may result in osmotic transport of water [11] or even slowly penetrate the skin-fat layer [18] and therefore affect measurements. However, the high-impedivity air between dry electrodes and skin can significantly increase the parasitic contact impedances. Though the problem can be mitigated by epidermal devices [40][41][42][43][44] which can conform to skin, even 2wire impedances measured by soft and conformable electrodes ( Figure 4A) can still contain a large and time variant error [45] as sweat (i.e., conductive liquid) gradually improves the electrical skin-electrode contact ( Figure 4B), in analogy with wet electrodes. Clearly, if, despite the high stratum corneum impedivity, measurement errors can be kept low enough, 4-wire measurements can, in principle, be insensitive to the electrode-to-skin contact impedance. For instance, Figure 2 shows that with d E equal to 2 or 3 mm, if the errors introduced by the terminal parasitics are negligible (which, for terminal-parasitics of a certain order of magnitude, is more and more difficult for smaller www.advancedsciencenews.com www.advmat.de dimensions of the electrodes or, equivalently, for higher values of Z SC , Notes S5-S11, Supporting Information) the 4-wire impedance will be mainly dependent on the viable skin and on the adipose tissue and will, in particular, be almost completely independent on the stratum corneum. Consistently, Figure 4C,D shows the magnitudes of both the 2-wire, Z 2W , and the 4-wire, Z 4W , bioimpedances taken at different times with epidermal electrodes having d E = 2 mm ( Figure 4A). The magnitude and the weak variations with frequency confirm that Z 4W is sensitive to viable skin and adipose tissue ( Figure 1C,G; Figure S1, Supporting Information) and not to the stratum corneum. Sweating results in a large reduction of Z 2W but only slightly changes Z 4W , whose magnitude is in the order of 100 Ω as expected (values 100 to 1000 times smaller than the stratum corneum impedances [11] and also consistent with typical impedivities). Figure 4C,D confirms that bioimpedances (around 108 Ω at 15 kHz) beyond much (up to 38 kΩ, i.e., about 350 times) higher stratum corneum impedances can be measured. Similar results are found with d E = 3 mm ( Figure S16, Supporting Information). As schematically shown in Figure 4E, tape stripping consists in repeatedly attaching and abruptly removing, ex vivo or in vivo, a conventional adhesive tape to the skin, so that the outermost layers of the stratum corneum cells are gradually detached without significant pain. [24,27,[46][47][48] After 3 tape stripping procedures, we found a large reduction of the 2-wire bioimpedance followed by a slow recovery ( Figure 4F), with time scales consistent with the restoration of the stratum corneum. [49,50] In striking contrast, Figure 4G shows that 4-wire bioimpedances were unaffected by tape stripping (the small variations, after many hours, may be attributed to physiological changes, e.g. hydration), thus confirming that only tissues beyond the stratum corneum contribute to the measurement.

Conclusion
In this work, first, we have defined frequency-dependent corner frequencies, current tubes, current tubes angles, relative tube currents, sensing tubes and sensing equipotential lines. These definitions, equivalent circuits and symmetry considerations allow to approximately predict the shapes of the current tubes associated to living tissues beyond the stratum corneum and, in particular, show that the adipose current tube resembles an isosceles triangle with a base approximately twice as large as the layer thickness ( AT very close to 45°). Current tubes and the continuity of both E t and J n provide insight on equipotential and current lines. These concepts can greatly facilitate the analysis and design of systems for measuring impedances in multilayer structures.
Afterward, with reference to multilayer structures made of multiregions made of homogeneous and isotropic tissues, we have introduced the normalized derivatives of the 4-wire admittance Y 4W (and of the 4-wire impedance) with respect to the electrical parameters of all the tissues. By simply observing that Y 4W must be, at any given frequency, a homogeneous function of the conductivities and the dielectric constants of all the tissues, we have derived the fundamental properties of the normalized derivatives and have discussed their practical applications.
FEM calculations confirm that, if circuit errors can be made negligible, the bioimpedances of living tissues beyond the stratum corneum can be measured by 4-wire instrumentation ( Figure 2). Afterward, since the presence of terminal-parasitics (Note S4, Supporting Information) is expected to result in severe errors because of the insulating nature of the stratum corneum, we took advantage of a simplified lumped-element model for systematically investigating these errors. With this approach we identified design guidelines including conditions on the electrodes areas and parasitics, active shielding of the sensing electrodes and their connectors, passive (for simplicity) shielding of the force electrodes and their connectors, selection of components (op amps and instrumentation amplifier) and minimization of the connectors lengths by portable electronics. The effectiveness of the proposed design strategies has been initially validated by means of test PCBs with simplified lumped-element models (Figure 3). These results enabled the design of a portable system for measuring bioimpedances of living tissues beyond much higher stratum corneum impedances with conventional, noninvasive, dry and compact electrodes, without any pretreatment. In fact, based on FEM calculations (Figure 2), we designed electrodes with dimensions (d E equal to 2 or 3 mm) that, if the errors introduced by terminal-parasitics are made negligible (Figure 3), should enable the electrical characterization of the tissues beyond the stratum corneum by 4-wire bioimpedances. Consistently, in both sweating and tape stripping experiments we found initial values (before any perturbation) of the 4-wire bioimpedances (order of magnitude around 100 Ω, Figure 4D,G) much lower than 2-wire bioimpedances (order of magnitude around 35 and 10 kΩ at the lowest and highest frequencies of interest, respectively, Figure 4C before sweating and Figure 4F before tape stripping). Moreover, in both the sweating and tape stripping experiments, the 4-wire bioimpedances measurements do not show any detectable dependence on the stratum corneum as demonstrated by their almost perfect stability irrespective of the extreme variations of the barrier impedances associated to sweating ( Figure 4B-D) or to the removal and regeneration of the stratum corneum following repeated tape stripping procedures ( Figure 4E-G), thus confirming that the measurements only depend on the other tissues, namely the living tissues below the stratum corneum. Both the analysis of the 4-wire impedance during sweating or following tape-stripping have been proposed as criteria for experimentally assessing, in a very simple way, the ability of a certain system (electrode and electronic interface) to perform accurate 4-wire measurements which are insensitive to the stratum corneum.
In summary, we have provided a comprehensive set of definitions (frequency-dependent corner frequencies, current tubes, current tubes angles, relative tube currents, sensing tubes and sensing equipotential lines) and tools (normalized derivatives) for facilitating the design of 4-wire measurement systems for skin bioimpedances. Moreover, we have reported strategies for overcoming the longstanding "stratum corneum dilemma". This work can be an important step toward the widespread adoption of bioimpedance for the diagnosis and treatment of several diseases and dysfunctions of both skin and adipose tissues. Moreover, our results are not restricted to bioimpedances and can be applied to the measurement of impedances in multilayer structures or even in multiregion structures, with each region made of a homogeneous and isotropic material, and can therefore be of interest, more in general, for both material science and electronic devices.

Experimental Section
Materials: All the components of the 4-wire bioimpedance system were purchased from commercial sources. Printed circuit boards (PCBs) were designed with Autodesk Eagle and fabricated by an external company. FEM simulations were performed with Comsol Multiphysics. 0.5 and 1.1 mm poly(methyl methacrylate) (PMMA) foils were purchased from GoodFellow. 60 μm-thick and 12 mm-wide Kapton tape was purchased from TESA. Poly(dimethylsiloxane) (PDMS) (Sylgard 184) was purchased from Dow Corning. PMMA substrates and shadow masks were fabricated with a Minitech MiniMill/GX Minitech Machinery Corporation USA. PDMS thin films were spin-coated with a LAURELL 650M SPIN COATER. Metal pellets were used as received from Kurt J. Lesker. 3M TenflexTM 1500 tape was used for the tape stripping experiments.
Fabrication of Epidermal Devices: Kapton tape with adhesive on the back side was simply attached to the PMMA carrier substrate. A 50 μm PDMS film (PDMS, Sylgard 184, Dow Corning, mixed at 10:1 (w:w) prepolymer:cross-linker) was then spin-coated onto the substrate (1000 rpm, 1 min) and cured in oven at 65°C for 2 h. A bilayer of chromium and gold (10 and 200 nm, respectively) was thermally evaporated on the PDMS film through a 500 μm-thick PMMA shadow mask which was previously patterned with a micromilling machine. For passivation, Kapton tape was applied over the connecting traces and the metal shields. More details can be found in ref. [51].
Bioimpedance Measurements: Two versions of the circuit were used. The first version was used for preliminary tests on equivalent circuits and was connected to the PC and powered by a USB cable, while the second version, used for test on skin, was connected to the PC by a Bluetooth Low Energy (BLE) microcontroller and powered by two series-connected 3.6 V LiPo batteries. The circuit was connected to the PC by a Bluetooth Low Energy (BLE) microcontroller and powered by two series-connected 3.6 V LiPo batteries. The analog sections of both circuits were identical and were powered by the Arduino 5 V voltage regulator. The AD5933 (Analog Devices) was programmed to apply sinusoidal voltages with 0.6 V peakto-peak amplitude. The data were sent to the PC using the serial protocol RS-232, 9600 bps, 8N1 (USB connection, first version of the circuit for preliminary tests with equivalent circuits) or through a SoC (System on Chip) nRF52832 from Nordic Semiconductors that comprises a Bluetooth 5.0 transceiver (BLE) and an ARM Cortex M4 microcontroller (second version of the circuit for test on skin). The nRF52832 device gets the information from one UART port (RS-232 communication) and transmits to an Android cellphone using a UART-BLE service. The Android cellphone runs a custom app that implements the UART-BLE service, receives the data and exports the information as a standard .txt file.
For the experiments on the forearm, each curve representing the 2 wires or the 4 wires bioimpedances as function of frequency was obtained by averaging 10 consecutive frequency sweeps (available in the .txt file), each consisting of 81 bioimpedance measurements, with frequency changing from 15 to 95 kHz, with 1 kHz increment. The 10 frequency sweeps required for obtaining one (averaged) measurement of both the 2 wires and the 4 wires bioimpedances as functions of frequency, in total, require less than 1 minute. The average of 10 subsequent frequency sweeps was computed with MATLAB.
In the tape stripping experiments the bioimpedance measurements were taken immediately before and immediately after the procedure, consisting in three applications and abrupt removals of a 3M TenflexTM 1500 tape. Bioimpedance measurements were then taken after 2 h and after 8 h in order to monitor the stratum corneum restoration.
The experimental protocol was approved by the Ethics Committee of Policlinico Tor Vergata (protocol 218.18) and the participant gave informed consent.
Statistical Analysis: Bioimpedance measurements were taken by the AD5933 (Analog Devices), that performs the DFT (Discrete Fourier Transform) and returns the raw data which allow to determine, after calibration, the real and the imaginary parts of the impedance. For each measurement the AD5933 performs a frequency sweep from 15 to 95 kHz with a 1 kHz step (81 measurements in total). Each frequency sweep was repeated 10 times and then the mean value was computed, so that, for each frequency, the values shown in the graph were computed as the mean of ten measurements, one measurement for each frequency sweep. The relative standard deviation (i.e., the ratio between the standard deviation and the mean value) was always below 1%.
The software used for statistical analysis was MATLAB R2022b.
In the measurements, outliers were not found and all the data had been considered.
The device was designed to perform both 2-wire and 4-wire measurements. In fact, as shown in Figure S14 (Supporting Information), depending on the position of a configuration switch (blue switch), both the current flowing through the bioimpedance beyond the stratum corneum (blue switch connecting the output of the green op amp to the horizontal 18 kΩ resistor) or the voltage across the bioimpedances beyond the stratum corneum (blue switch connecting the voltage V S or, equivalently, the output of the red op amp to the horizontal 18 kΩ resistor) could be measured. As a result, the 2-wire impedance Z 2W could be determined as the ratio between the output stimulation voltage of the AD5933 (V OUT in Figure S14, Supporting Information) and the current flowing through the bioimpedance beyond the stratum corneum. Similarly, the 4wire impedance Z 4W could be determined as the ratio between the voltage across the bioimpedances beyond the stratum corneum and the current flowing through the bioimpedance beyond the stratum corneum. In order to perform a single measurement cycle, the blue switch must connect the output of the green op amp to the horizontal 18 kΩ resistor during the first sub-cycle (Z 2W measurement sub-cycle) and then the voltage V S (i.e., the output of the red op amp) to the horizontal 18 kΩ resistor during the second sub-cycle (Z 4W measurement sub-cycle).
In practice, in order to convert the raw data returned by AD5933 into impedance values, before the measurements, calibration must be performed, for each frequency, on known resistors according to the calibration procedure discussed in the AD5933 datasheet. For calibration, test PCBs were used comprising lumped electrical models of the skin, with all the impedances replaced by known resistors.
During calibration, the sensitivities of the op amp stages (circuits implemented by the red, green and purple op amps) must be fixed by choosing the positions of the red, green and purple switches in Figure S14 (Supporting Information) (e.g., for both the inverting voltage amplifiers implemented by the red and the green op amps in Figure S14 (Supporting Information), the voltage gain was −1 or −10 if the correspondent switch connected to the output of the correspondent op amp the 10 kΩ resistor or the 100 kΩ resistor, respectively; similarly, the sensitivity of the current-tovoltage converter implemented by the purple op amp in Figure S14 (Supporting Information) was 4.7 or 47 kΩ if the red switch connected to the output of the op amp the 4.7 kΩ resistor or the 47 kΩ resistor, respectively). The possibility to change the sensitivities allowed to optimize the signal-to-noise ratios (i.e., allowed to avoid excessively low levels of the signals entering the AD5933 and to avoid saturation of the op amp stages or of the AD5933).

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.