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A heuristic model to evaluate the dielectric properties of human tissues at microwave band based on water and solid content


At the molecular level, the body is composed, on average, of water for 62%, fat for 15%, 17% of protein, and 6% of minerals. In this work, we propose a heuristic methodology using hydration models as a base to realize an automatic and noninvasive procedure to estimate an ad hoc map of the complex dielectric permittivity of a generic human tissue in the frequency range of microwaves based on their solid and water content.


In silico models of the human body are increasingly used in dosimetry studies as well as in medical applications of electromagnetic fields. However, up to now, human body models are mostly derived from magnetic resonance imaging (MRI) through the segmentation of the different anatomical elements by expert teams of biologists and physicians [12, 61]. Moreover, the dielectric properties of tissues, to be used in combination with the geometrical models, are mostly obtained through ex vivo measurements [23, 27,28,29, 73]. Accordingly, this procedure limits the availability of human models, particularly useful in medical applications where the exact knowledge of the patient’s anatomy and related electromagnetic properties would be of great benefit for the success of the treatment. As an example, in hyperthermia treatment planning (HTP) [47], simulations are carried out on the patients’ anatomy to obtain the optimum hyperthermia protocol. However, the same should be performed in microwave thermal ablation, or microwave-based diagnostic applications, e.g., microwave tomography.

The relationship between water content and dielectric permittivity of human tissues and their electromagnetic interactions is well known [2, 4, 71, 79, 83]; however, even if a dielectric model of the human body was derived based on this assumption [60], an automatic tool was never devised. Several works were performed in the past to link the dielectric properties of the human body tissues with their water content [79, 83]. In these works, the dielectric properties of biological tissues were measured from audio frequencies to microwaves, and some mathematical models were proposed to predict their values as a function of the frequency to understand the dielectric relaxation phenomena in tissues. Also, it is assumed that tissue’s total water content belongs to two pools: 70% can be considered ‘free water,’ while 30% is the hydration water considered to be bound to the tissue’s solid content and supposed to have the same dielectric properties of the dry protein [79, 83].

Proteins in aqueous solutions

Proteins are essential for life. These are the building blocks of cells and, virtually, part of every biological process. In organisms, proteins with biological functions usually exist in solution and many of their physical and functional properties are strongly influenced by the solvent. Therefore, it is vital to examine proteins within their common environment [89]. Since a large fraction of proteins exists in the aqueous intra-cellular or extra-cellular environment, the quantitative characterization of protein dynamics in aqueous solutions is essential for the understanding of living systems at a molecular level and many studies regard protein in aqueous solution in controlled conditions. Understanding the equilibrium existing between water and proteins within human tissues, thereby deducing a map of the tissues dielectric permittivity, is central to this study.

Water in hydrated proteins is often classified into three main groups: internal (or included) water, hydration water, and bulk (or free) water. The molecules of the water encounter highly heterogeneous protein surface sites, both for structure and electrostatics and characterization of the dynamics of the water molecules in the hydration layer. These aspects influence the dielectric properties of the hydration water. Both local curvature and chemical heterogeneity of the solute interface, e.g., clefts and pockets and hydrophobic or hydrophilic sites, influence the structure and dynamics of water. Internal water molecules occupy cavities within the protein and are present in most globular proteins [85, 88]. For most purposes, internal water molecules are best regarded as an integral part of the protein, even though they exchange with external water molecules, typically on a time scale of 0.1–10 μs [36, 37]. Forming up to four hydrogen bonds with neighbor molecules, reorientation of a water molecule is possible only when a sufficient number of hydrogen bonds are broken. The dielectric relaxation, in this way, is controlled by the lifetime τHB of H-bonds [43]. As a consequence of the presence of many factors influencing its dynamics, the hydration water within the hydration shell has a pronounced nonexponential character that reveals a broad distribution of reorientation times [85].

Many works have investigated the range of the hydration water and the geometry of the water compartments and their distribution around the hydrated macromolecules [38, 46, 48]. The water molecules next to the biomolecular interface, i.e., typically within 2.8–3.5 Å generally define the first layer of water molecules in the hydration shell. A single water molecule occupies 2.8 Å, and the molecules from 2.8 to 5.6 Å are referred to as the second layer and so on [48]. Laage and co-workers have shown that the length-scale over which water adapts its bulk-water characteristics can vary between 2 and 10 hydration layers from the biomolecular surface depending on the parameters that have been under observation [48]. We have considered reasonable Persson and Halle’s thesis [69] which uses a 5 Å water-carbon cutoff, and a 4 Å water-water cutoff has demonstrated that the best assignation is for the first shell 4-5 Å thick, and the higher shells 2.8–4 Å tall. The different properties of water that are under the influence of the surface recover bulk-like characteristics at a different distance from the source of the perturbation. As a consequence, the experimental results need proper interpretation depending on the specificity of the property under observation [38].

Many works concluded that not only the hydration shell dynamics for all globular proteins can be rationalized by the same local topological and chemical factors but also, for many globular proteins, the hydration shell as a whole can have similar underlying distributions of reorientation times and, hence, similar overall dynamics. Fogarty and Laage performed molecular dynamics simulations of dilute aqueous solutions of four globular proteins, which cover a wide range of functions and molecular weights (from 9 to 59 kDa). They showed that all four proteins have very similar hydration shell dynamics, despite their wide range of sizes and functions, and differing secondary structures. They demonstrated that this arises from the similar local surface topology and surface chemical composition of the four proteins and that such local factors alone are sufficient to rationalize the hydration shell dynamics [20]. Within this framework, Cameron and colleagues have shown that in many hydrated proteins and cells and, in particular, in hydrated collagen the hydration compartments are the same, in number, size, and physical characteristics. These water compartments can be defined using their hydration fraction limits h = grams of water for 1 g of dry mass [24,25,26]. Cameron and Fullerton concluded that the dynamics of water molecules in solution are affected even up to 10–11 Å from the protein surface and that the hydration water layer can be divided into 2–3 shells 3.5 Å thick. In addition, the results seem to give proof that within 6–7 Å reside water molecules in interaction with polar sites of the protein's surface, while between 7 Å and 9 Å, there are water molecules influenced by hydrophobic sites. Based on these results, they presented the molecular stoichiometric hydration model (SHM) which interprets the broadening of the hydration water dynamics in terms of a hydration monolayer up to h = 1.6 g/g, divided into four water compartments with a well-fixed relative size. The dynamics of these molecules are slower than the bulk water that resides in the “compartment” defined by h > 1.6 g/g. The SHM predicts and explains the commonly cited and measured “bound” water fraction of 0.2–0.4 g of water/g of dry mass on proteins and, in particular, in tendon/collagen type I [25, 26] to which an h = 1.6 g/g, i.e., 1.6 g of water for 1 g of collagen correspond to a hydration layer thickness of 5.66 Å from the protein surface that is about two water layers. Cameron and colleagues demonstrate that the results of their experiments on model bovine collagen samples using different techniques converge with one another and with the results of different researchers on other proteins and cells and confirm the assignments of their hydration model [6]. The SHM model considers a monolayer with hm = 1.6 g/g divided 50% into primary hydration with hpr = 0.8 g/g on polar hydrophilic sites, and 50% into secondary hydration over hydrophobic surfaces hse = 0.8 g/g. In particular, the primary hydration water molecules hydrogen-bonded to collagen polar side chains have hpsc = 0.54 g/g with greater free energy and hpmc = 0.26 g/g relative to the protein main chain hydration with the greater free energy binding which includes the Ramachandran water-bridge, with capacity hRa = 0.0656 g/g. As proof of the validity of their model, they conducted measurements on proteins with variable hydration, isotherm rehydration from the vapor phase, NMR (nuclear magnetic resonance) water titration, NMR freezing point depression, high G-force dehydration, and hydration force (osmotic compression) and, in particular, confirmed the SHM model for the type I collagen [26]. In addition, they showed that many other techniques confirmed their results, such as X-ray scattering, neutron scattering, dielectric spectroscopy, and NMR [7].

Collagen is the most abundant protein in mammals, making up about 25% to 35% of the whole-body protein content, whose 90% is type I. For these characteristics, collagen is central in our study. There are nearly 28 types of collagens, but collagen type I is the most common in skin, cornea, artery walls, bone, teeth, tendon, ligaments, vascular ligature, and organs [30, 49, 77, 80, 84].

Dielectric spectroscopy experiments

Dielectric spectroscopy can be used to obtain information on the dynamics of proteins. Among the many techniques, dielectric spectroscopy has the advantage to investigate the arrangement of water in confined systems, more generally in interfacial or restricted environments, over a wide time scale, providing information on the orientational dynamics of molecular dipoles and covering all kinds of polarization fluctuations in the milli- to picosecond time scales [9, 23]. Usually, the dielectric spectrum in protein solutions displays three main features, denoted as β, γ, and δ dispersion, which represent dielectric relaxation processes at well-separated timescales. In the past decades, many researchers have focused their efforts on this field. These studies provided useful information on the physicochemical properties of hydrated biomolecules and showed their dielectric spectrum from 1 MHz to tens of GHz. The early dielectric spectroscopy experiments conducted by Harvey and Hoekstra [39] on hydrated powders of lysozyme revealed two distinct dispersions with a relaxation time of near 1 ns (100–200 MHz) and near 0.02 ns (7–8 GHz), respectively [39]. This dispersion is related to the hydration water that Grant named δ-relaxation [32]. Many computer simulations and experiments with different techniques have been conducted so far to explain the rationale of the δ-dispersion [5, 9, 10, 18, 21, 33,34,35, 38, 46, 62, 64,65,66, 68, 72, 85]. The molecular dynamics (MD) simulations performed by Oleinikova, Cametti, Wolf, and, Steinhauser’s group were particularly interesting [19, 68, 89]. However, so far, no consensus on the physical origins of the δ-dispersions has yet been found.

We have focused on the works produced by Grant, Oleinikova, Cametti, and Wolf [9, 33, 34, 68, 89] that report on dielectric spectroscopy measurements and molecular dynamics simulations of protein in an aqueous solution at different concentrations and controlled conditions to characterize the δ-dispersion main features. Grant studied globular proteins in water solutions and, in particular, with Takashima measured the dielectric dispersion of Bovine Serum Albumin solutions (BSA molecular weight is 66.46 kDa at pH = 5.07 and T = 293.15 K), from MHz to GHz and concentrations from 724 to 1083 mg/mL (10.9–16.3 mM or h = 0.64–0.18), and collected results at different temperature and pH. They, using the mixture theory, have quantified the bound water fraction related to the δ-dispersion and verified that the bound water static permittivity εSB ranges from 100 to 300 depending on the concentration [33].

Oleinikova research group has used dielectric spectroscopy to study the dynamics of ribonuclease A in aqueous solutions (RNase A molecular weight 13.7 kDa at pH 5.5 and T = 298.15 K) and concentration from 0.36 to 4.48 mM (or from 0 to 59.96 mg/mL or h = 0.005–0.06) [68]. They have decoded the complexity of the dielectric spectrum, comparing it to other experiments such as NMR techniques and molecular dynamics simulation of the autocorrelation function Φ(t) of the total collective dipole moment of the sample.

In particular, they decomposed Φ(t) in three self-components ΦPP(t), ΦHH(t), and ΦBB(t) and three cross-correlation components ΦPH(t), ΦPB(t), and ΦHB(t). The ΦPP(t) was assigned to the β-dispersion due to the protein tumbling, with τrelax =20 ns and frelax = 1 MHz, the ΦBB(t) to the γ-dispersion due to the bulk water reorientation. They confirmed the existence of three different states of hydration water. The ΦHH(t) is related to the δ3 dispersion, with τrelax =35 ps and frelax = 4.55 GHz, due to hydration water orientation polarization. Specifically, ΦPP(t) is related to the δ2 dispersion, with τrelax =500 ps and frelax = 318.5 MHz, concerning polar side chain fluctuation and, finally, the δ1 dispersion, with τrelax =2 ns and frelax = 79.6 MHz, to protein water cross-correlation ΦPH(t), which occurs near the β-protein relaxation. The δ1 assignment was supported by MD simulation. In this picture, the cross-correlation between water and protein plays a major role confirmed later by Steinhauser’s group [5]. Cametti obtained the dielectric spectra of Lysozyme in aqueous solutions (lysozyme molecular weight is 14.3 kDa and partial specific volume is 0.73 mL/mg at pH = 5.5 and T = 293.15 K and) and concentration from 1 to 125 mg/mL (0.1–8.7 mM that corresponds to a hydration h > 7.26) in the range of 1 MHz-50 GHz [9]. Wolf and his team studied the dynamics of aqueous lysozyme solutions in the frequency range from 1 MHz to 40 GHz in relation to concentration and temperatures ranging from 275 to 330 K, providing data on the temperature dependences of the β, δ, and γ-relaxations [89].


Given the negligible size of the cells making up the tissue, compared to the frequency range taken into consideration the electromagnetic homogenization theory can be applied [22, 50, 51, 53, 82]. We have used the Maxwell Garnett mixing formulae to perform the simulations and evaluate the tissue’s permittivity and conductivity [49, 54, 90] and compare our different models. The effective permittivity of a material made by different constituents, such as protein solution or biological tissue, can be derived as a function of the constituent’s fractional volumes and their permittivity at the frequency of interest [58, 59, 82]. For the binary system, we have used the Maxwell Garnett Eq. (1):

$${\epsilon}_{eff}^{\ast }(f)={\epsilon}_{host}^{\ast }(f)+3\phi {\epsilon}_{host}^{\ast}\frac{\epsilon_i^{\ast }(f)-{\epsilon}_{host}^{\ast }(f)}{\epsilon_i(f)+2{\epsilon}_{host}^{\ast }(f)-\phi \left[{\epsilon}_i^{\ast }(f)-{\epsilon}_{host}^{\ast }(f)\right]}$$

ϵi(f) and \({\epsilon}_{host}^{\ast}(f)\) are the complex permittivity of the inclusions and the host material at the same frequency f, while ϕ is the fractional volume of the inclusion. For a multiphase system made of N types of inclusions, we use Eq. (2) [58, 59, 82]:

$${\epsilon}_{eff}^{\ast }(f)={\epsilon}_{host}^{\ast }(f)+3{\epsilon}_{host}^{\ast }(f)\frac{\sum_{n=1}^N{\phi}_n\frac{\epsilon_{i.n}^{\ast }(f)-{\epsilon}_{host}^{\ast }(f)}{\epsilon_{i.n}^{\ast }(f)+2{\epsilon}_{host}^{\ast }(f)}}{1-\sum_{n=1}^N{\phi}_n\frac{\epsilon_{i.n}^{\ast }(f)-{\epsilon}_{host}^{\ast }(f)}{\epsilon_{i.n}^{\ast }(f)+2{\epsilon}_{host}^{\ast }(f)}}$$

To use the Maxwell Garnett formulas, we need to know the volume fraction and the Debye parameters of all the elements of our tissue models. We have computed the permittivity and conductivity of a generic inclusion—hydration water compartment protein and lipid contents—using the Debye formula Eq. (3). The Debye formula Eq. (3) models the complex relative permittivity of fluids and other materials ε as a function of the frequency f, [13]. Equations (4) and (5) show the real and the imaginary parts of relative permittivity:

$${\varepsilon}^{\ast }(f)={\epsilon}^{\prime }(f)-j{\varepsilon}^{"}(f)={\varepsilon}_{\infty }+\frac{\varepsilon_s-{\varepsilon}_{\infty }}{1+ j\omega \tau}$$
$${\varepsilon}^{\prime }(f)={\varepsilon}_{\infty }+\frac{\varepsilon_s-{\varepsilon}_{\infty }}{1+{\omega}^2{\tau}^2}={\varepsilon}_{\infty }+\frac{\varepsilon_s-{\varepsilon}_{\infty }}{1+{\left({}^{f}\!\left/ \!{}_{{f}_r}\right.\right)}^2}$$
$${\varepsilon}^{"}(f)=\frac{\left({\varepsilon}_s-{\varepsilon}_{\infty}\right)\omega \tau}{1+{\omega}^2{\tau}^2}=\frac{\upsigma_d(f)+{\upsigma}_s}{\varepsilon_02\pi \textrm{f}}$$

In Eqs. (3)–(5), ω = 2πf is the pulsation of the external field, τ is the relaxation time of the generic dipole of the system, εs is the static permittivity of the material and ε is its permittivity at field frequencies for ωτ 1, σd(f) is the frequency-dependent conductivity arising from dielectric polarization, while σs is the steady-state conductivity. The quantity εs − ε is the change in the permittivity from very low frequencies to very high frequencies, compared to the relaxation frequency \({f}_r=\frac{1}{2\pi \tau}\).

The tissue models

In our procedure, the water in the hydration shell is divided into the four compartments (see Table 1) defined by the hydration SHM model in Fig. 1, proposed by Cameron and Fullerton [6,7,8]. In Table 1, there are the protein hydration limits and the four hydration compartment intervals hi defined by the SHM used to compute the size of any compartments D hi in the protein hydration layer declared in Fig. 1.

Table 1 Protein hydration limits and the four hydration compartment intervals hi defined by the SHM used to compute the size of any compartments in the protein hydration layer declared in Fig. 1
Fig. 1
figure 1

Scheme of the Cameron and Fullerton SHM hydration model

Native tendon hydration has monolayer coverage on collagen hm = 1.6 g/g which divides into primary hydration on polar surfaces hpp = 0.8 g/g and secondary hydration hs = 0.8 g/g bridging over hydrophobic surfaces.

We introduced four hydration models based on homogenization theory to represent a generic human tissue as a mixture of hydrated proteins and adipocytes in a host of free water. The four hydration models, shown in Fig. 2, were then compared to one another. The hydration compartments in the protein hydration layer have been distributed into one to four shells. A generic protein with water in the hydration shell is constituted of four compartments defined by the hydration SHM model proposed by Cameron and Fullerton [6,7,8]. We have arbitrarily named these water species as super bound water (SBW), bound water slow (BWS), bound water fast (BWF), and structured water (STRW). The size of any compartment of the “system” was then computed with the hydration intervals computed using the hydration limits of the SHM model, assuming for water, protein, and lipids a mass density equal to 0.997 g/ml, 1.39 g/ml, and 0.905 g/ml, respectively [56, 76]. Moreover, 1.10 g/ml was assumed for water over strongly hydrophilic solute and 0.85–0.90 g/ml for water over strongly hydrophobic solute [16, 19]. For a generic adipocyte fat cell, the model introduced in Said’s work was adopted [76]. The fat globules are water-coated and have an outer radius of a = 100 μm and an inner radius of b = 80 μm. We have imposed a hydration fraction h = 0.278 g of which 70% is structured water and 30% free water.

Fig. 2
figure 2

ad Depiction of the models 1–4 to homogenize the tissues. These are not in scale: hydrated protein size is 1–300 nm. We model an adipocyte, with a size around 20–300 μm, composed of a water coated fat globule with an outer radius a = 100 μm and an inner radius b = 80 μm

All proposed models are characterized by mixture 1 in common. Mixture 1 is a core-shell system. Dry lipid and hydrated lipid are present in the core and shell layer, respectively. In model 1 (see Fig. 2a), the protein system is a homogenized core-shell system. The core is a two-phase system: dry protein-SBW and a three-phase shell. The following three constituents: BW slow, BW fast, and STRW (mixture 2) are homogeneously distributed. In mixture 3, the Maxwell Garnett (MG) formula for a multi-phase system was used. Finally, a two-phase MG model was used between mixture 1 and mixture 2 to obtain the permittivity of model 1. In model 2 (see Fig. 2b), the protein system has three independent layers. Mixture 4 is composed of a two-phase system core with dry protein-SBW, a two-phase intermediate layer with BW slow and BW fast, and then an STRW layer. Each layer was modeled with the MG equation, including a top layer, obtaining mixtures 2, 3, and 4 each time. Finally, as in the previous case, the tissue was homogenized using the multi-phase MG equation. In model 3 (see Fig. 2c), as in the first model, the protein system has two independent layers (core-shell). The core consists of dry protein. The external layer is composed of BW slow, BW fast, SBW, and STRW (mixture 2). Mixture 2 is homogeneously distributed around the core. Mixture 3 was obtained using the MG model. Finally, the tissue’s permittivity and conductivity were obtained with the homogenization of mixture 1, the hydrated lipid with mixture 3, and the hydrated protein. Model 4 presents the hydrated protein as a 4-layers system (see Fig. 2d). The core consists of a two-phase: SBW-dry protein. All the outer layers were considered monophasic and independent, in particular, from the innermost layer to the outermost one; there are BW slow, BW fast, and STRW, obtaining mixtures 1, 2, 3, and 4, respectively. As in all previous cases, the tissue homogenization has been computed using the MG model on mixture 1 and mixture 4. We modeled an adipocyte around 20–300 μm composed of a water-coated fat globule with an outer radius a = 100 μm and an inner radius b = 80 μm.

The SBW compartment is related to the internal water molecules present in the protein cavities. For most purposes, internal water molecules are considered an integral part of the protein, even though they undergo the exchange mechanism with external water molecules, typically on a time scale of 0.1–10 μs [36, 37]. These molecules are added to our models to the protein compartment with the same Debye parameters. The BW water molecules are related to the δ-dispersion revealed by Haggis and Buchanan 10 years after Oncley’s work showed the presence of β- and γ-relaxations in the late 1930s and 1940s. This dispersion is due to the dielectric relaxation of water near the protein surface. Detailed MD simulations interpret the broadening of this dispersion due to a faster contribution related to the protein-protein tern of the total electric dipole moment autocorrelation interaction and a slower one related to the component due to the protein-water interaction. Steinhauser’s group, analyzing the collective nature of the dielectric experiment through the MD simulation, found three additional δ-terms related to the water-water self-correlation and a δ-term related to protein self-correlation in addition to the β- and γ-relaxations and the main δ-term due to the water protein cross-correlation [5]. Beyond the β- and γ-dispersions, we consider the two δ-dispersions related to protein-protein and water-proteins and the one near the γ-dispersion related to the structured water influenced by hydrophobic interactions. We have named bound water slow the compartment related to δ1 and bound water fast the compartment related to δ2; these are computed as average of δ1 and δ2 [68].

The structured water compartment is indeed related to δ3; we have computed about the γ-dispersion. With structured water, we intend the water molecules near hydrophobic sites. In the SHM hydration model, this compartment is related to the hydration interval h = (0.8–1.6) g of water per g of solid. The water interacting with hydrophobic groups forms a “clathrate” structure. Computer simulations indicate that such modifications in water structure can extend at least 10 Å into the bulk liquid from the hydrophobic surface [71]. This compartment has a little slower dynamic than the bulk water [41, 67, 74]. Structured water molecules face hydrophobic sites and are a little slower than the free water molecules [11, 20, 31, 57, 63, 70]. We imposed the relaxation time of these water molecules to τstrw= (1.4 – 2) × τfree water [75].

Ultimately, the water molecules that are not affected by the presence of a solute are considered free. In our study, the Debye parameters for pure water at different temperatures were related to the data reviewed by Kaatze [42]. These data are shown in Table 2.

Table 2 Debye parameters for pure water at different temperatures. These data were reviewed by Kaatze [42]

Debye parameters assignment

The SHM hydration model to reproduce the results of valuable studies on protein aqueous solution was first analyzed. We, then, considered these heterogeneous systems as a collection of spherical particles (the proteins) of complex dielectric constant εp*(f) covered by a hydration shell composed of four water compartments with complex dielectric constant εhi*(f), uniformly distributed in a continuous medium of free water with its own complex dielectric constant εm*(f). In particular, the results of Oleinikova and co-workers were used [68], which studied the spectrum of RNase A in water at different concentrations and the spectrum of Lysozyme in aqueous solution reported by Cametti and colleagues both in the frequency range from 1 MHz to tens of GHz [9].

Then, we optimized the Debye parameters to reproduce the dielectric response of tendon/collagen in the frequency range 0.01–10 GHz. Finally, the complex permittivity of a generic human tissue assessing our results in comparison to the golden standard was estimated [29].

The size of the four compartments of our four mixture models, based on the hydration limits of the SHM model proposed by Cameron and Fullerton, was computed [6]. A generic hydrate system, protein aqueous solution, or human tissue, as a mixture of solid and water components, was modeled. The host is composed of free water; the inclusions are the hydrated proteins and lipids and the hydration water compartments. The mixture’s complex permittivity is computed using the MG mixture formulas. The volume fraction and the complex permittivity of any components of the mixture are the input of the model. The complex permittivity was computed with the Debye formula for each component ranging from 0.01 to 10 GHz assigning the Debye parameters: εs, ε, σi, and, τrelax.

The aim was to reproduce the Oleinikova results about the dielectric spectrum of RNase A in an aqueous solution at concentrations of 4.34 mM, 298.15 K, and pH 5.5. We have imposed the relaxation times of any compartments using the Oleinikova results related to the dispersions β, γ, and the three δ1, δ2, and δ3. Then, the static permittivity εs and ε of any compartment were observed and optimized to reproduce the Oleinikova results.

Using the same parameters, we have, then, tested our procedure with the four protein hydration models shown in Fig. 2, reproducing the results of Cametti on lysozyme aqueous solutions. In these cases, we have scaled the εs considering for RNase A μ0 = 280 D and MW = 13690 Da, while for lysozyme μ0 = 210 D and MW = 14300 Da. For this purpose, Onsager-Oncley’s model relation, which connects the proportionality between the ∆β increment and the effective dipole moment volume density of the protein solution, was used [72, 78]:

$$\Delta {\varepsilon}^{\prime }={N}_Ac{g}_k\ {\mu_0}^2/\left(2{\varepsilon}_0 MWkT\right)$$

where NA is Avogadro’s number, k is the Boltzmann constant, c is the concentration in kg/m3 of the polar molecule in the solvent, and gk is a parameter introduced by Kirkwood [44, 45] to account for molecular associations and correlation effects between the motions of solute and solvent molecules. The μ0 is the dipole moment of an isolated molecule, gk μ02= μ2eff is the effective squared dipole moment per molecule in the ensemble, C (mM) = c(mg/ml)/MW(g/mole) is the concentration expressed in millimoles per liter, while NAC (mM) is the protein/dipole volume density.

Next, using again the SHM hydration model, we have optimized the free parameters of our models to best attain the dielectric spectrum of the tendon/collagen. The tendon in human tissue is composed on average of 61.54% water, 37.46% proteins and other solid not lipids, and 1% lipids. This average composition corresponds to a “concentration” of 420 mg/mL. We used Onsange-Oncley’s model to correct the ∆β computed to reproduce the RNase A dispersion and the scale factor to compute the εs for the protein considered in the simulation. In the case of collagen type I with μ0 = 15000 D and molecular weight (MW) = 324000 Da, considering for RNase A μ0 = 280 D and molecular weight (MW) = 13700 Da, the following was obtained:

$${\upmu}_0^2/\textrm{MW}\ \left[\textrm{collagen},\textrm{type}\ \textrm{I}\right]/{\upmu}_0^2/\textrm{MW}\ \left[\textrm{RNaseA}\right]=722.614/5.515=131.$$

For more details on the methods, refer to the “Supporting information” document.

Debye parameters assignation for RNase A and human tissues

In Tables 3 and 4, there are the Debye parameters optimized to reproduce the dielectric spectrum of 4.34 mM RNase A aqueous solution and tendon tissue in which 100 g are, on average, divided as follows: 61.54 g of water, 1 g of lipids, and 37.46 g of collagen type I that correspond to a concentration of 245.20 mg/mL or 1.47 mM. In particular, for tendon/collagen type I, RNase A, and lysozyme, we have found εSB ≈100–180 in line with the values found by Grant [33, 34].

Table 3 Debye parameters chosen to reproduce the RNase A aqueous solution at 4.38 mM concentration
Table 4 Debye parameters optimized for tendon/collagen type I about the average tissue composition reviewed by Duck [17] and regarding the average dielectric tendon characteristics collected by Gabriel and co-workers [29], our golden standard. We propose to use these parameters to reproduce the complex permittivity (ε′ and σ) of a generic human tissue based on its water and solid content (see Supporting information, Eq. (S2))


The current procedure and models were assessed through simulations using a homemade MatLab script. As previously mentioned, all the models are based on MG formulas. In the cases where more than two spheres were considered, the Transmission-Matrix technique and Morse–Feshbach formulas were implemented numerically [3, 14, 15, 50,51,52, 55].

The input of our simulations is the protein concentration for protein solutions obtained from the Oleinikova and Cametti works on RNase A and lysozyme aqueous solutions, while the tissue’s content in terms of water, lipid, and solid fractions are obtained from the Duck review [17]. To correct the Debye parameters for using the protein solutions at 20 °C, the values were rescaled at 37 °C for human tissues [89] (for more information, refer to Supporting information materials). In Fig. 3, the simulations for the protein aqueous solutions made with model 3 are shown. In Fig. 4, the simulations for tendon, on the left, and adipose tissues, on the right, made with model 3 are also shown. We use these simulations to validate our procedure and modelization of protein hydrated systems. We compare the deviation from the experimental traces published independently by Oleinikova for RNase A aqueous solutions, Cametti for lysozyme aqueous solutions, and the traces published by Gabriel for human tissues (their results are the golden standard we consider in these simulations). We have computed the root mean square relative error percentage (RMSRE%) in order to assess the results of our simulations performed using a homemade MatLab script. In our simulation, the deviation for the permittivity is less than 10%, while for the conductivity, the errors % of our results compared to the experimental are around 20–30% in the frequency range 0.01–10 GHz.

Fig. 3
figure 3

RNase A 4.38 mM aqueous solution (on top) and 7.69 mM lysozyme aqueous solution (on bottom) results of simulation made with model 3. The golden standards are from Oleinikova and Cametti data [9, 68]

Fig. 4
figure 4

From the top simulation results for the human tendon (a and b), human muscle (c and d), and adipose tissue (e and f). In the left column, the imaginary part of the complex dielectric permittivities is represented, and on the right column, the conductivity is represented

The assessment of our models 1–4 were conducted with simulations using MG formulation and computing the root mean squared relative error % (RMSRE%) in Eq. (6) compared to the golden standard.

$$RMSRE\%=100\ \sqrt{\frac{1}{N}\sum_{i=1}^N\frac{{\left[{x}^m-{x}^{GS}\right]}^2}{x^{GS}}}$$

Tables 5 and 6 show the RMSRE% for ε′ and σ evaluated with the simulations, using the 4 homogenization models, in two different frequency ranges 0.01–10 GHz and 1–10 GHz for a 4.34 mM RNase A and a 7.79 mM lysozyme aqueous solution. From these tables, it can be seen that for both protein solutions, model 3 is the one that provides a lower RMSRE%, i.e., the model with all water species (SBW, BWS, BWF, and STRW) homogenized in a single layer around the dry protein

Table 5 RNase A (4.38 mM aqueous solution) model 1-4 simulations. Average RMSRE% for ε’ and σ of on the ranges 0.01-10 GHz and 1-10 GHz (in green is highlighted the best average RMSRE%)
Table 6 Lysozyme (7.68 mM aqueous solution) model 1–4 simulations. Average RMSRE% for ε′ and σ of on the ranges 0.01–10 GHz and 1–10 GHz. εs (protein) rescaled for [μ2/MW (Lysozyme)/μ2/MW (RNaseA)]

In Table 7 is model 1–4 assessment for human tissue tendon, muscle, and fat, while the results for seven selected human tissues are shown in Table 8. The golden standards are data from Oleinikova’s and Cametti’s experimental results on the protein solutions [9, 68] and Gabriel’s experimental data on human tissues [29]. Simulation results are shown; model 3 gives the best results for both protein solutions and most human tissues simulations.

Table 7 Selected human tissues (tendon, muscle, and fat) simulation results. RMSRE% for both ε′ and σ of model 3 for εs (protein) rescaled for [μ2/MW (collagen type I)/μ2/MW (RNaseA)] on the range 0.01–10 GHz and 1–10 GHz
Table 8 Seven selected human tissues model 1–4 simulations. Average RMSRE% for ε′ and σ of on the ranges 0.01–10 GHz and 1–10 GHz. εs (protein) rescaled for [μ2/MW (Collagen)/μ2/MW (RNaseA)]

In this work, we are interested to the dielectric relaxation processes at lower microwaves occurring from 1 GHz to 10 GHz, where the dipolar relaxation of tissue water occurs and from 0.1 to 4.5 GHz from dielectric relaxation of the “bound” water near the protein surface, and the protein polar sidechains. We have used the Onsager-Oncley’s equation to improve our model connecting the lower limit of our band of interest (0.1-10 GHz) to the β-dispersion (1 kHz to several MHz).

In this way, we have considered the Δε’ increment associated with the polarization of proteins and other organic macromolecules. In reality the β-dispersion is the result of more mechanisms, for example the ionic conductivity of cell membranes that our model does not control. This is the reason of the high deviation for ε’ of muscle from the Gold Standard at low frequencies.


The aim was to prove the feasibility of realizing an automatic non-invasive methodology to estimate the dielectric properties in vivo of a generic human tissue and to create both a permittivity and a conductivity map in the microwave frequency range from 1 to 10 GHz. We propose a heuristic method as a base to realize an automatic and non-invasive procedure to obtain an ad hoc dielectric mapping of human tissues in vivo, with a standard magnetic resonance imaging (MRI) scanner for medical imaging to quantify the water (free and bound), the lipid content, and the bone content. The homogenization theory was adopted in the study because, in the considered frequency range, the protein sizes were always much shorter than λmin ~ 5 × 10−3 m.

This method can be, particularly, useful in medical applications where the exact knowledge of the patient’s anatomy and related electromagnetic properties would be of great benefit for the success of the treatment. Collagen type I is considered the prototype of the human tissue proteins in our models. First, the evaluation of the complex permittivity of a 4.34 mM RNase A aqueous solution at room temperature was performed to fix the Debye parameters. Our methodology was validated by evaluating the dielectric spectrum of a 7.69 mM lysozyme aqueous solution and the permittivity and conductivity of tendon/collagen human tissue in the frequency range from 0.01 to 10 GHz.

Then, using the collagen/tendon-optimized Debye parameters and MG mixture formulas, we evaluate the complex permittivity of human tissues. Comparing our results to a golden standard, it was found that the estimated permittivity and conductivity showed a 13.2% RMSRE% on average.

Since we have used data relative to the average human tissue composition and average human tissue permittivity and conductivity from two different databases, we believe that these promising preliminary results lead to the next phase of this research.

The next step of this research will consist of involving animal ex vivo samples. The aim will be, first of all, to quantify the real sample composition in terms of water and the solid fraction, and this will be achieved using specific quantitative magnetic resonant (qMRI) sequences [40, 81]. In addition, with these MRI sequences, we can measure the free and bound water fractions in the sample [40, 81]. Finally, we will try to implement an MRI procedure to estimate the pH of the sample [86, 87, 91]. In this way, we are confident to be able to better fix the static conductivity of the sample, sI, that we have arbitrarily fixed at σi = 0.250 S/m in this first investigation, taking the value reported in the literature for living tendon tissue at T = 37 °C [29].

Availability of data and materials

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.



Bovine serum albumin


Bound water fast


Bound water slow


Hyperthermia treatment planning


Molecular dynamics


Maxwell Garnett


Magnetic resonance imaging


Molecular weight


Nuclear magnetic resonance


Quantitative magnetic resonant imaging


Root mean square relative error percentage


Ribonuclease A


Super bound water


Stoichiometric hydration model


Structured water


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Tannino, M., Mangini, F., Dinia, L. et al. A heuristic model to evaluate the dielectric properties of human tissues at microwave band based on water and solid content. J. Eng. Appl. Sci. 70, 43 (2023).

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