2.2. Dielectric Properties of Skin

Biological tissue is a very heterogeneous material, and various components influence the dielectric properties, such as concentration of free water, bound water, and molecular compositions. The dielectric dispersion characteristics (or Debye-type dispersion) of biological tissues are commonly approximated by a Cole–Cole model [7,21,22,26]. The human skin is employed as a multilayer model due to separable dispersive dielectric behavior. In this model, the focus was on three layers; the outermost layer, the stratum corneum, the viable skin, which combines living epidermis and dermis, and the hypodermis as subcutaneous fat. Each skin layer is represented by a complex relative permittivity εr=ε′−jε″ characterized by the Cole–Cole model:

where j is the imaginary unit, ε∞ is the optical permittivity, n is the number of dispersions, Δεn is the nth permittivity increment, τn is the nth relaxation time, αn is the nth broadening parameter, ω is the angular frequency, σDC is the static conductivity, and ε0 is the dielectric permittivity of the vacuum.

The frequency-dependent relationship between the conductivity and relative permittivity is

where the conductivity σ and the relative permittivity εr are complex quantities [26].

In [21], it was found that bound water has a negligible influence on the dielectric properties of the SC, and recommends simplifying the model to

where the free parameters are caused by the free water in SC and the slow parameters are assumed to be caused by protein polarization.The optical permittivity directly depends on the permittivity increment of free water

The value 3.3 in (13) represents the relative permittivity of dry SC, which is a typical value for the dry protein [27], and the value 73.2 is the relative permittivity of free water [21].

As stated in [21], the dielectric properties of the SC vary with the hydration of the skin. Further, a linear dependency between the free water content of the SC and the permittivity increment of the free water Δεfree was found. With the assumption that all dielectric parameters from (12) undergo the same linear dependency, the hydration relation can be fitted with linear regression for each parameter. To relate the dielectric parameter directly to the water activity, the water concentration Cw is formulated in dependency of the water activity aw using the Equations (7) and (8).

We obtain

where Cw is the resulting water concentration in the SC by the given water activity aw. Applying the linear regression to the dielectric parameters, the equation becomes:

with y→ as the sample vector representing the dielectric parameters, Cw→ is the vector of water concentrations at measurement of the dielectric parameters, and β0 and β1 are the parameters to be estimated. The linear regression fit resulted in a scaling of Cw (with β0 set to 0) because the provided measurements only represent the hydrated state of the SC, and no samples for dry tissue are given. The samples and fit parameters are listed in Table 1. For a complete derivation of the fit, see Section S2 in the Supplementary Materials.

Cole–Cole parameters from [21] at three hydration levels were used to approximate the SC hydration dependency. β lists the calculated scaling value resulting from the linear regression. ε∞=3.3+273.2Δεfree can be calculated from Δεfree.

The interpolation equations take the form

where each dielectric parameter has its own scaling parameter β. The quantities Δε^free, τ^free, Δε^slow, and τ^slow stand for the interpolated dielectric parameters.

The final estimated relative permittivity for the SC is formulated as