Biothermal Heating on Human Skin by Millimeter and Sub-Terahertz Waves in Outdoor Environment—A Theoretical Study

Abstract

The frequency band in the millimeter-wave (MMW) and sub-terahertz (sub-THz) range has shown great potential in mobile communication technology due to the advantages of ultra-large bandwidth and ultra-high data rates. Based on the increasing research activities on MMW/sub-THz waves, biological safety at relevant frequencies must be explored, especially when high-power illumination occurs. Here, its non-ionizing nature plays a vital role, which makes it safe for humans at low illumination powers. However, under high power, the biothermal heating on the skin surface is still a main concern, and lots of research has been conducted in a laboratory. In this article, we analyze the thermal heating effect of human skin in outdoor environments, where atmospheric conditions can significantly impact the propagation of MMW/sub-THz waves. Our analysis is based on rat skin, which has a similar structure to human skin. A theoretical model combining Pennes’ bioheat transfer equation (BHTE), the ITU model, and the Mie scattering theory is developed. Good agreement between calculation results and measured data confirms the efficiency of this model. The influence of rainfall rate, humidity, operating frequency, illumination time, power density, and propagation distance is presented and discussed.

1. Introduction

As the outer tissue of the human body, the skin acts as an interface with the environment and thus is the first to be exposed to electromagnetic radiation, especially at high illumination powers. When the irradiation power density is higher than 10 mW/cm², the thermal effect is mainly studied, which is directly manifested by increasing skin temperature and damage to biological tissues [1]. There have been several safety guidelines/standards set by the International Commission on Non-Ionizing Radiation Protection (ICNIRP) and the IEEE International Committee on Electromagnetic Safety (ICES) Technical Committee (TC) 95 [2,3,4] to protect humans from excessive exposure. Reference levels of 20 mW/cm² at 6 GHz to 10 mW/cm² at 300 GHz have been derived for local body exposure based on laboratory measurements or simulations [5,6,7]. However, further research on skin response in the atmospheric environment is required for outdoor applications of MMW/sub-THz techniques, such as remote security inspection, high-speed data coverage, and active denial system.

MMW/sub-THz waves suffer from high sensitivity to outdoor weather conditions [8,9,10,11,12,13,14,15]. The path availability and channel performance are reduced even in clear weather, as atmosphere gases and air turbulence can lead to serious channel degradation [13]. Outdoor measurements just on channel deterioration have been conducted [11,16], but no confirmative conclusions could be obtained. That’s because of the challenges in measuring the impact of weather conditions over a long observation time and the difficulties in obtaining the re-occurrence of similar weather conditions to verify the results of independent measurements due to the time-varying rainfall rates and raindrop size distributions, as an example. Several groups have tried to build weather-emulating chambers in a controlled laboratory environment to generate reproducible weather for experimental measurements [13,17]. However, there are some distinctions between emulated and actual weather [8]. Therefore, a combined approach relying on both theoretical models and case studies is still valuable and necessary.

In this article, we present a comprehensive theoretical analysis that explores the biothermal heating effect of MMW/sub-THz waves on human skin under outdoor weather conditions. To the best of our knowledge, similar research has not yet been conducted, highlighting the importance of our investigation for future applications of this frequency band in outdoor settings. The methodology we propose can be adapted and applied to various weather conditions by measuring or determining the relevant parameters. Additionally, we emphasize that studying the thermal responses of rat skin provides a feasible approach to understanding the potential effects on human skin, considering their structural similarities. The article is organized as follows: Section 2 presents the dielectric properties of skin and water, and proposes theoretical models for the propagation of MMW/sub-THz waves in clear weather and rain. Section 3 shows theoretical models to predict the thermal heating effect and Section 4 presents theoretical models for thermal heating in outdoor weather with investigations on the influence of humidity and rainfall rate. Section 5 shows the conclusion and implications of this work.

2. Dielectric Properties and Channel Modeling

A description of skin response to MMW/sub-THz waves occurring outdoors requires detailed information about the skin and weather conditions (rain condition is considered here), such as the dielectric properties of skin and rainwater, temperature, humidity, and rainfall rate.

2.1. Dielectric Properties of Skin and Rainwater

The skin is the largest organ of mammals and its structure is composed of three interconnected layers of tissue: the epidermis as the outermost layer, the dermis as the middle layer, and the hypodermis as the innermost layer [18]. In this work, we consider the epidermis and dermis (with the thickness in the range of 0.06–0.1 mm and 1.2–2.8 mm, respectively [19]) in the theoretical model due to their ability to limit the penetration depth of MMW/sub-THz waves as discussed later. The dielectric properties of skin are characterized by measurements of its relative complex dielectric constant $\mathsf{\epsilon }={\mathsf{\epsilon }}^{\prime }-j{\mathsf{\epsilon }}^{″}$ and have been proposed to be described by the single Debye model [20], the four Cole-Cole dispersion model [21] and other combined methods [22,23,24]. The four Cole-Cole model is considered the most widely used method based on open online databases [25]. It takes the skin to be a uniformly isotropic passive conducting medium [21] and can be expressed as

$${\epsilon }_{\mathit{skin}}{=\epsilon }_{\infty }+{{\displaystyle \sum }}_{n=1}^4\frac{\Delta {\epsilon }_n}{{1+(j\omega \tau }_n{)}^{{1-\alpha }_n}}+\frac{\sigma }{{j\omega \epsilon }_0}$$

(1)

Here, the parameter ω refers to the angular frequency and τ refers to the relaxation time. Parameter ε_∞ represents the dielectric constant in the high-frequency limit; α₁, α₂, α₃, and α₄ indicate the broadening due to dispersion; σ is the ionic conductivity of the skin; and ε₀ represents the dielectric property of free space. For dry skin, the parameters are set as τ₁ = 7.23 ps, τ₂ = 32.48 ns, ε_∞ = 4.0, α₁ = 0.0, α₂ = 0.2, Δε₁ = 32.0, Δε₂ = 1100, and σ = 0.0002. For wet skin, parameters are τ₁ = 7.96 ps, τ₂ = 79.58 ns, τ₃ = 1.59 μs, τ₄ = 1.592 ms, ε_∞ = 4.0, α₁ = 0.1, α₂ = 0.0, α₃ = 0.16, α₄ = 0.2, Δε₁ = 39.0, Δε₂ = 280, Δε₃ = 3.0 × 10⁴, Δε₄ = 3.0 × 10⁴, and σ = 0.0004. The sweat duct inside the skin, with its radius ranging between 45 μm and 50 μm [26], is neglected due to its smaller size compared with the wavelength of MMW/sub-THz waves. We estimated the penetration depth by using the general expression p = (67.52/f[MHz]){[(ε′)² + (ε″)²]^1/2 − ε′}^−1/2 based on Equation (1) and show its dependence on operating frequency in Figure 1a. The penetration depth decreases exponentially, which agrees well with the measured values set in the ICNIRP guidelines [2].

The dielectric properties of rainwater are mainly determined by the vibration mode of hydrogen bonds between water molecules. The bonds resonate and thus exert strong absorption on MMW/sub-THz waves. The absorbed energy is converted into kinetic energy for the irregular motion of water molecules and then transformed into thermal energy with an increase in mutual collision frequency [27]. The dielectric properties of water can be described by the Double-Debye dielectric model (D3M), which was first developed for saline water based on experimental measurements [28,29,30,31] and can also be reduced to pure water [32] in the frequency range up to 1000 GHz [33]. It can be expressed [34] as

$${\epsilon }_{\mathit{water}}{=\epsilon }_{w0}-f·{{\displaystyle \sum }}_{n=1}^2\frac{{\epsilon }_{w(n-1)}{-\epsilon }_{\mathit{wn}}}{{f+\mathit{j}\gamma }_n}$$

(2)

with ${\epsilon }_{w0}\left(T\right)=77.66-103.3·T_{\mathit{in}}$ as the static dielectric constant of water and $T_{\mathit{in}}=1-300/(273.15+T)$ as the modified relative inverse temperature variable. Parameters ${\epsilon }_{w1}=0.0671{\epsilon }_{w0}$ and ${\epsilon }_{w2}=3.52+7.52·T_{\mathit{in}}$ are the first and second high-frequency constants (f→∞), respectively_. Parameters ${\gamma }_1=20.20+146.4·T_{\mathit{in}}+316·T_{\mathit{in}}{}^2$ and ${\gamma }_2=39.8{\gamma }_1$ denote the primary and secondary relaxation frequencies. T_w is the water temperature in °C, and f is the operating frequency in GHz.

As shown in Figure 1a, the penetration depth in skin is always higher than that in water because of the smaller dielectric constant of skin [23]. The penetration depth is so sensitive to the incident frequency that higher frequencies cannot arrive at the skin surface if the skin has a thick water layer covering it due to falling rain droplets. A calculation of the transmissivity through the air-dry skin and water-wet skin interfaces is conducted and shown in Figure 1a by using Fresnel’s law with a horizontal monopolarization (hh) configuration. A higher transmission through the water-skin interface is observed, due to the smaller dielectric difference between air (free space) and skin on both sides as the water content of skin tissue can be up to 70% [35]. In the following sub-section, we conduct further calculations by assuming a layer of in vivo skin tissue instead of an in vitro one [36] and neglect the variation of its water content due to its self-adjusting capability.

2.2. Channel Models

In our analysis of rainy weather conditions, we consider the presence of rainwater adhering to bare skin, forming a uniform and planar water layer. When the power is focused onto a small area of the skin, both the skin and water layers can be treated as uniform and planar. The water layer introduces power losses, including reflection and absorption losses, on the MMW/sub-THz waves. We estimate the total power loss resulting from these effects by the equation

$$A_{water} \left(\mathrm{dB}\right)=10\log T_{tr}$$

(3)

with T_tr being the total transmissivity through the water layer, which can be derived by using a multi-layer model [36]. As shown in Figure 1b, a three-layer structure (air-water-skin) is presented and can be estimated by the Fresnel principle simply, which has been confirmed in reference [36], when each layer is considered to be uniform. An expression for the total transmissivity from air into the skin can be expressed by the equation [37]

$$T_{tr}=\frac{{(1+\rho }_{12}{\left)\right(1+\rho }_{23}{)e}^{{-jk}_2{\delta \cos \theta }_2}}{{1+\rho }_{12}{\rho }_{23}{e}^{{-j2k}_2{\delta \cos \theta }_2}},$$

(4)

where ρ₁₂ and ρ₂₃ are the reflection coefficients of the air-water interface and the water-skin interface, respectively. Parameter k₂ is the wave number in the water layer, δ is the thickness of the water layer, and θ₂ is the refraction angle in the water layer corresponding to the incident angle θ₁, as shown in Figure 1b. The exponential term represents the propagation factor in the water layer between its top boundary (air-water interface) and bottom boundary (water-skin interface) along the angle θ₂. Here, we neglect the scattering loss due to random wrinkles of the skin surface with a maximum depth of 0.3 mm [36], which is much smaller than the wavelength of the MMW/sub-THz waves. Thus, the total transmissivity relates to the reflection by the air-water interface and the water-skin interface, and also to the absorption by the water layer. Figure 1c shows a calculation of total transmissivity through the water layer with different incident angles. It increases first and then starts to decrease for frequencies above 100 GHz. By comparing it with the evolution curve of transmissivity in Figure 1a, we can attribute the difference between both trends to the absorption loss by the water layer. To further confirm this, we calculate the total transmissivity through a water layer with different thicknesses and plot it in Figure 1d. The total transmissivity increases with respect to frequency when the water layer is thinner than 0.2 mm, and follows an opposite trend when the thickness (δ) is larger than 0.2 mm, due to the higher absorption loss by the increased water layer thickness.

In clear weather, the gaseous composition of the atmosphere, mostly water vapor, strongly affects the propagation of MMW/sub-THz waves. The gaseous attenuation is commonly described as the summation of spectral absorption lines of water vapor and oxygen, and a water vapor continuum component. A line-by-line calculation is provided by the ITU recommendation sector (ITU-R) [38] based on the physical model MPM93 [39]. This method is proven to be valid at frequencies up to 500 GHz [40]. The results shown in Figure 2 indicate the ability to reduce the influence of atmospheric attenuation by setting the incident frequencies away from the water vapor absorption resonances. It should also be noted that higher terahertz frequencies are not preferred due to the large gaseous attenuation, especially when the relative humidity (RH) is higher than 60% in rainy weather.

The attenuation due to falling rain can generally be predicted by using the Mie scattering theory. This is reasonable because the ordinary size of raindrop particles is in the range from mm to cm [41], which is comparable to the MMW/sub-THz wavelength. The scattering characteristics can be described by the absorption and scattering coefficients. Two important underlying assumptions are usually considered: firstly, each scatter is independent of the other, and secondly, multiple scattering can be neglected. Under those assumptions, the attenuation suffered in rain [42] can be determined using the following equation

$${\kappa }_{rain}=4{.343\times 10}^3{{\displaystyle \int }}_0^{\infty }{\sigma }_{ext}{(n}_r,r,\lambda \left)N\right(r)dr (\mathrm{dB}/\mathrm{km})$$

(5)

where n_r is the refractive index of the rain particles, r is the particle radius, λ is the carrier wavelength, and N(r) is the residual size distribution of raindrops, which is one of the main roles influencing the accuracy of theoretical predictions [8]. The parameter ${\sigma }_{\mathrm{ext}}$ refers to the total extinction cross-section in the expression ${\sigma }_{\mathit{ext}}{=\sigma }_{\mathit{abs}}{+\sigma }_{\mathit{sca}}$, with σ_abs and σ_sca as cross-sections of absorption and scattering, respectively. All three terms depend on the incident wavelength (λ), refractive index (n_r), and radius (r) of water droplets. The power attenuation due to falling rain is calculated and shown in Figure 2. Here, we use the Marshall-Palmer [M-P] raindrop size distribution, which was also found to be reasonable for snow [43]. The Joss distribution was first obtained by measuring the size distribution of raindrops in Switzerland using a disdrometer [44]. The specific model proposed by ITU-R is based on measured data and empirical conduction [45]. A moderate falling rain (rain rate ranging from 2.6 to 7.5 mm/h) exerts a higher attenuation than the atmospheric gases when channels operate below 200 GHz. This indicates that the influence of falling rain on the biothermal heating effect cannot be neglected from the viewpoint of channel propagation. The Third Generation Partnership Project (3GPP) TS 38.104 has defined 24.25–52.6 GHz as frequency range 2 (FR2) for 5G NR [46]. In consideration of the significant potential demonstrated by certain terahertz frequency band windows, such as 140 GHz, 220 GHz, and 340 GHz, in various short-range wireless communication applications (e.g., V2X, high-speed transportation communication, and backhaul/fronthaul), we have included calculations for the 140 GHz frequency in addition to the 33.5 GHz frequency used in [47]. This selection is based on several factors, including lower transmission losses and higher penetration depth in water and skin layers, as illustrated in Figure 1 and Figure 2.

3. Thermal Heat Modeling

The biothermal heating effect by MMW/sub-THz waves on the skin is a complex process. The relevant parameters include frequency, power density, irradiation time, heat conduction, blood flow velocity in skin tissue, convection between skin surface and air, sweat evaporation and metabolic rate, etc. [48]. When electromagnetic powers are focused onto the skin surface, they will be absorbed and converted into heat, which is equivalent to establishing a heat source inside the skin tissue. The illumination spot inside the skin tissue acts as a heat source and the energy it generates can be expressed by the equation $Q_r{\left(x\right)=(I}_0\Lambda /L)·e^{-x/L}$ [48], with x as the depth of the source, I₀ as the illumination power density, and L as the penetration depth in the skin. The parameter ${\Lambda =\left|\tau \right|}^2$ refers to the power transfer coefficient and can be calculated by Fresnel’s principle with the transmission coefficient as τ = 2η₂cosθ₁/(η₂cosθ₁ + η₁cosθ₂). Here parameters η₁ and η₂ are the wave impedance of electromagnetic waves in air and skin, respectively. Parameters θ₁ refers to the angle of incidence and θ₂ to the angle of reflection.

The classical Bioheat Transfer Equation (BHTE) models, such as Pennes’ BHTE, Weinbaum-Jiji (W-J) BHTE, and the thermal wave model of bioheat transfer (TWMBT), can be adapted to express the heat transfer mechanism inside the tissue. The Pennes’ BHTE model lays a solid and favorable foundation for establishing and studying the distribution of tissue temperature, due to its superiorities in simple structure, strong practicality, and wide range of applications. The W-J BHTE model is based on the fine anatomy of skin tissue [49] and takes into account the changes in number density, size, blood flow velocity, and direction of blood vessels in the tissue’s microstructure along with the penetration direction of electromagnetic waves. Its solution is more applicable to micro-vascular tissue due to its complexity but is not as efficient for a single skin layer. The TWMBT model considers the finite speed of heat transmission and can reflect the fluctuation characteristics of heat flow [50], which is not necessary or required here. In this work, we employ Pennes’ BHTE model, which is considered to be more reasonable in describing the microstructure and temperature distribution of subcutaneous areas caused by external heat sources [51]. It is also regarded as a common method to study the dependence of the biothermal heating effect on specific parameters, such as temperature, frequency, illumination time, and power density.

There are several assumptions for establishing the model: (1) the convective heat transfer process of blood occurs in the dermis with capillaries, (2) the tissue and blood in capillaries have the same temperature and are equal to the tissue temperature, and (3) the blood perfusion rate in local capillaries is isotropic. We also consider the skin tissue as a semi-infinite, homogeneous, and isotropic medium, with the heat exchange as a one-dimensional process, which is available to describe the heat transfer due to the thermal heating effect [52]. Thus, Pennes’ BHTE model can be simplified to be a one-dimensional form as

$$\rho c\frac{\partial T}{\partial t}=k\frac{{\partial }^{2}T}{\partial x^2}{+W}_b{c}_b{(T}_b{-T)+Q}_m{+Q}_r$$

(6)

where parameters ρ, c, and k are the density, specific heat, and thermal conductivity of the skin, respectively. Parameter W_b refers to the blood perfusion rate per unit volume with c_b as the blood-specific heat and T_b as the arterial blood temperature. Parameter T is the skin temperature, Q_m is metabolic heat production, and Q_r is the electromagnetic wave heat source term. Under the assumption that the heat generated by the skin metabolism is negligible compared to the effect of blood perfusion on the skin temperature distribution [52], the term Q_m in Equation (6) can be ignored. The corresponding initial and boundary conditions for Equation (6) are set as ${\left.T(x,t)\right|}_{t=0}{=T}_0$, ${\left.T(x,t)\right|}_{{x=L}_s}{=T}_c$, and ${\left.-k\partial T/\partial x\right|}_{x=0}{=h(T-T}_e)$ with T₀, T_c, T_e, L_s, and h representing the initial skin temperature, the core skin temperature, the ambient temperature, the skin thickness, and the total heat transfer coefficient between the skin tissue and the environment, respectively. The core skin temperature can also be called deep tissue temperature, which refers to the subcutaneous tissue temperature. Our work referred to the skin temperature range in [53,54], so 28.85 °C was selected as the initial skin temperature. Some of the values of these parameters used in the following calculations can be found in

Table 1. Values of Physical Parameters for Theoretical Calculation [25,51,53,54,55,56].

Parameter	Value	Parameter	Value
ρ (kg/m3)	1200	T0 (°C)	28.85
c (J/kg/°C)	3580	Tc (°C)	36.5
k (W/m/°C)	0.4	Te (°C)	12.5
Wb (kg/s/m3)	0.59	Ls (cm)	1.0
cb (J/kg/°C)	4200	h (W/m2·K)	13
Tb (°C)	36.5

.

In order to check the efficiency of the model by Equation (6), we estimate the temporal evolution of temperature due to the illumination of a 33.5 GHz channel onto a layer of skin, with the power density changing from 284 mW/cm² to 664 mW/cm². In Figure 3a, the variation of skin temperature versus illumination time is plotted. The temperature increases significantly for longer illumination time and higher power density and then comes to be a constant (equilibrium). The calculation agrees well with the measured data obtained by Wang [47], who employed a layer of in vitro rat skin in the laboratory under the illumination of a continuous wave at 33.5 GHz and launched out from a lens antenna. The agreement between our calculations and the measured data indicates the efficacy of Pennes’ BHTE model in analyzing the temperature evolution of rat skin under electromagnetic radiation, provided that the total heat transfer coefficient (h) is determined as shown in Figure 3b. These results support the transferability of our findings to discussions on the thermal heating effect of human skin, despite its thickness being 5–6 times greater than that of rat skin. It is important to consider that as body temperature rises, mammals regulate their temperature through physiological activities, including increased heat dissipation, which results in an augmented heat transfer between the skin and the surrounding environment. Consequently, the heat transfer coefficient increases in response. Based on the curves shown in Figure 3b, which are calculated by fitting the measured data in Figure 3a, we can see that the coefficient h evolves with ∆T along the same trend before stabilizing (h_max). This means there should be a correlation between the heat transfer coefficient and the temperature difference ∆T (between the skin and ambient temperature), which has been demonstrated theoretically in reference [57].

To see the influence of a rainwater layer on human skin on a rainy day, we predicted the variation of thermal heating with/without a planar water layer by employing Equation (6). In the case of water layer coverage, we adopted the dielectric properties of wet skin. The heat transfer coefficient of rats in Figure 3b was used here because it has been proven to be close to that of human skin [58]. Calculations were completed using the parameters in Table 1 and setting the thickness of the planar water layer to 0.2 mm due to the relatively stable transmissivity over the whole frequency range as indicated in Figure 1d. The results are shown in Figure 3c. It is obvious that the temperature increases faster when there is a water layer. We attribute this to the insulation effect of the water layer, which has a much higher specific heat capacity than air [59]. It should also be noted that the temperature increases with respect to the operating frequency. This is due to the larger absorption coefficient by the uniform and planar skin layer at higher frequencies, as indicated in Figure 6(b) of reference [36]. The same phenomenon can also be observed in Figure 3d, which shows a temporal variation of human skin temperature due to the illumination of channels operating at 33.5 GHz and 140 GHz. However, this does not mean a higher frequency is preferred for thermal heating devices because it suffers limitations of low penetration depth, as shown in Figure 1a; high propagation loss, as shown in Figure 2; and low output power [60].

4. Outdoor Thermal Heating Performance

To evaluate the thermal heating performance of MMW/sub-THz waves in rainy weather, we combined the theoretical models for thermal heating by Equation (6), channel propagation by Equations (3)–(5), and free space path loss (FSPL). Calculations were conducted by setting a rain rate of 5 mm/hr and a planar and uniform water layer of 0.2 mm covering the surface of human skin surface due to falling rain, a normal incidence configuration for the largest transmissivity, as shown in Figure 1c, as well as a relative humidity (RH) of 30% in clear weather and 60% in rain, output power of 60 dBm at the transmitter side launched out from an antenna with a gain of 50 dBi, and an illumination time of 240 s. Results are plotted in Figure 4.

A negative correlation between skin temperature and channel propagation distance is shown in Figure 4a, due to the distance dependence of channel propagation losses, as shown in Figure 2. The temperature increase caused by the 140 GHz channel is smaller than that by the 33.5 GHz channel due to the higher propagation losses and lower transmissivity into the skin tissue, as shown in Figure 1a. This means the thermal heating effect by sub-THz frequencies would not be an urgent concern for human skin if the propagation distance or output power density is limited. It should also be noted that the equilibrium temperature in rain is much lower than that in clear weather, which is opposite to our predictions in Figure 3d, due to the power losses caused by the water layer covering the skin surface. This indicates that the influence of falling rain and water vapor humidity is not so obvious when compared with power loss caused by the water layer.

To further investigate the dependence of the equilibrium temperature by the 33.5 GHz signal on the rainfall rate and relative humidity, we calculated evolutional curves due to the variation of rainfall rate and humidity and show the results in Figure 4b,c respectively. The horizontal lines imply the negligible influence of rain conditions, which exert negligible propagation degradation compared to the FSPL and the power losses due to the presence of a water layer on the skin surface. This means the major deterioration of the thermal-heating effect in rain is from the rainwater falling on the skin surface and the output power density at the transmitter side.

5. Conclusions

The evaluation of the biothermal heating effect on human skin is crucial for the application of MMW/sub-THz waves in future wireless communications and biological weapon scenarios in outdoor environments. In this study, we extensively investigate the efficiency of MMW/sub-THz waves for biothermal heating under outdoor weather conditions, considering the challenges posed by water vapor and falling rain. To accomplish this, we propose a comprehensive theoretical model that incorporates Pennes’ bioheat transfer equation (BHTE), the Mie scattering theory, and the ITU-R model. The proposed model is validated using measured data and is found to accurately evaluate the heating effect. We systematically explore the impact of various factors, including water vapor humidity, rainfall rate, propagation distance, operating frequency, illumination time, and power density, on the biothermal heating effect. We observed that the heat transfer coefficient is highly sensitive to the temperature differences between the skin and the surrounding environment. In the presence of falling rain, the thermal heating effect of MMW/sub-THz waves is diminished due to the formation of a water layer on the skin surface. However, this water layer does not cause significant channel propagation losses when the distance between the power source and the target (skin) is limited.

It is important to note that our work has been validated by measurements conducted on rat skin, and the propagation models employed in our analysis have been widely justified. Although there are certain assumptions in our study with limited justifications, the persuasive nature of this work lies in the validation of the thermal heating model using rat skin measurements and the extensive justification of the propagation models. Consequently, we are confident in the reliability and validity of our findings.

In conclusion, this work confirms the current and near-future applications of sub-THz frequencies in biothermal heating. The developed theoretical model, validated by rat skin measurements, is also deemed suitable for evaluating in vitro-in vivo terahertz communication channels.