The article presents a comprehensive study on the asymptotic solution of electromagnetic heating of skin tissue, with a particular focus on the lateral heat conduction. It addresses the challenge of modeling the temperature evolution in skin tissue exposed to microwave energy, highlighting the importance of understanding the biological effects of electromagnetic exposures. The authors employ asymptotic analysis to construct approximate formulas for predicting temperature distribution, leveraging the separation of scales between the depth and lateral length scales. The solution is derived using a two-term asymptotic approximation, which simplifies the problem into a set of one-dimensional initial boundary value problems (IBVPs). The article emphasizes the practicality of the approach, providing closed-form solutions in terms of parameter-free single-variable functions that can be computed conveniently. Numerical verification is conducted to validate the accuracy of the asymptotic solutions, demonstrating their effectiveness in capturing the true solution quantitatively. The study concludes by highlighting the theoretical and computational advantages of the asymptotic solution, making it a valuable tool for predicting temperature distribution in skin tissue exposed to electromagnetic radiation.
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Abstract
We study the temperature evolution in the three-dimensional skin tissue exposed to an electromagnetic beam of millimeter wavelength. The skin absorption coefficient of the beam frequency determines how deep the electromagnetic energy penetrates into the skin tissue, which gives a sub-millimeter penetration depth for a 94 GHz wave. In contrast, in the lateral directions perpendicular to the depth, the beam size is usually much larger than the penetration depth. Based on this separation of length scales, we establish an asymptotic formulation in which each term has separable dependences on the depth coordinate and on the lateral coordinates. We solve it analytically to obtain a two-term asymptotic solution of the temperature distribution in the three-dimensional skin tissue. This closed-form analytical solution provides a practical and accurate way of predicting the temperature. When the beam size is moderately larger than the penetration depth (a ratio of 20), the effect of lateral heat conduction is well captured in the asymptotic solution with maximum error less than 0.0017 in the normalized temperature of magnitude well above 1.
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1 Introduction
Heat transfer induced by electromagnetic radiation through living organisms or tissues is a subject of great interest due to its extensive applications in industry and modern medicine [1‐4]. Safety concern prompts a better understanding of the biological effects of electromagnetic exposures. Modeling studies of heating skin tissue by microwave energy in the frequency range of 30–300 GHz were reviewed by Foster et al. in [5, 6]
We consider the thermal effect of an electromagnetic beam on the skin tissue. The temperature evolution is governed by an initial boundary value problem (IBVP) of the three-dimensional heat equation with a heat source distributed inside the skin tissue corresponding to the absorption of electromagnetic energy. Mathematically, the solution of the IBVP is given by a four-dimensional convolution (3 spatial dimensions + time) of the heat source and the fundamental solution. So far there is no closed-form expression of this 4D convolution in terms of parameter-free single-variable functions that can be computed conveniently, such as, \(\text {erfc}(\; )\), \(\exp (\; )\). We are interested in developing practical and accurate approximate formulas for predicting the temperature at any given spatial location, and at any given time, using only parameter-free single-variable functions.
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In this study, we use asymptotic analysis to construct approximate formulas for the temperature distribution. The asymptotic formulation is based on the separation of scales. Specifically, we consider the case where the depth scale of electromagnetic heating is much smaller than the lateral length scale. The skin absorption coefficient for the beam frequency determines how deep the electromagnetic energy penetrates into the skin tissue, which gives a sub-millimeter scale in the depth direction. In most applications, in the lateral directions perpendicular to the depth, the size of the beam cross section is much larger. In addition, the heat source (the absorbed electromagnetic energy per volume) as a function of the 3D coordinates has separable dependences on the depth coordinate and on the lateral coordinates. Together these two factors allow us to formulate a two-term asymptotic approximation, in which each term has separable dependences on the depth coordinate and on the lateral coordinates. For each term, the dependence on the lateral coordinates is calculated from the given beam power density over its cross section. For each term, the dependence on the depth coordinate is governed by a one-dimensional IBVP of the heat equation. We use the fundamental solution of the heat equation and the tool of convolution to derive closed-form solutions of these one-dimensional IBVPs in terms of parameter-free single-variable functions. It is important to clarify that due to the lateral heat conduction, the separable dependences of the heat source do not carry to the exact temperature solution. Correspondingly, the sum of the two asymptotic terms does not have separable dependences even though each term in the approximation does.
This paper is organized into five additional sections. Section 2 describes the IBVP of the 3D heat equation governing the skin temperature evolution in electromagnetic heating. We carry out non-dimensionalization in Sect. 3. The dimensionless governing equation contains a small parameter and is otherwise parameter free. In Sect. 4, we exploit the small parameter to establish a two-term asymptotic formulation and derive the 1D IBVP governing each term. We then solve each IBVP to obtain a closed-form analytical solution in terms of parameter-free single-variable functions. The asymptotic analysis is followed by numerical verification in Sect. 5. In Sect. 6, we summarize main results.
2 Governing equation of skin temperature
We first establish the coordinate system. We consider the case of a flat skin surface. The normal direction into the skin tissue is selected as the positive z-direction with \(z=0\) at the skin surface. The z-coordinate represents the depth into the skin. In a plane perpendicular to the z-axis (i.e., with \(z = z_0\)), the 2D coordinates are denoted by \(\textbf{r}=(x, y)\). The skin surface is such a plane with \(z=0\). We consider the situation where the incident electromagnetic beam is perpendicular to the skin surface (i.e., along the z-direction). Figure 1 shows a schematic diagram of the skin tissue, the coordinate system, and the incident beam. We set \(\textbf{r} = (0, 0)\), the origin of the xy-plane, to the incident beam center on the skin surface. In the skin tissue, the 3D coordinates of a point are described by \((\textbf{r}, z) = (x, y, z), \; z \ge 0\).
Fig. 1
A schematic diagram of the skin tissue, the coordinate system, and the incident beam. The skin tissue is below the surface labeled \(z=0\)
We introduce mathematical quantities and notations as follows.
\((\textbf{r}, z) = (x, y, z), \text{ with } \; z \ge 0\) is the 3D coordinates of a point in the skin tissue.
\(T(\textbf{r}, z, t)\) is the temperature at position \((\textbf{r}, z)\) at time t.
\(P_d(\textbf{r})\) is the beam power density passing through the skin surface at position \(\textbf{r}=(x, y)\). \(P_d(\textbf{r})\) is the power density absorbed into the skin tissue (which becomes heat). It has the expression \(P_d(\textbf{r}) = \alpha P_d^\text {(i)}(\textbf{r})\) where \(P_d^\text {(i)}(\textbf{r})\) is the incident beam power density and \(\alpha \) is the fraction of beam power absorbed into the skin.
Let \(P_d(\textbf{r}, z)\) denote the power density penetrating beyond depth z into the skin tissue. The boundary condition of \(P_d(\textbf{r}, z)\) at \(z=0\) is
\(\mu \) is the skin absorption coefficient for the electromagnetic beam. In physics, \(\mu \) is the fraction of beam power absorbed per depth and it has the dimension of \(1/[\text {length}]\). Mathematically, \(P_d(\textbf{r}, z)\) is governed by
The rate of change in heat per volume of skin is \(\rho _m C_p\frac{d T}{d t}\) where \(\rho _m\) is the mass density and \(C_p\) the specific heat capacity. Conservation of energy gives
$$\begin{aligned} \rho _m C_p\frac{d T}{d t} = \left( \begin{array}{c} \text {rate of net heat flow} \\ \text {per volume} \end{array} \right) + \underbrace{\quad P_d(\textbf{r}) \mu e^{-\mu z} \quad }_{ \text {heating power per volume}}. \end{aligned}$$
(5)
The rate of net heat flow per volume is
$$\begin{aligned} \left( \begin{array}{c} \text {rate of net heat flow} \\ \text {per volume} \end{array} \right) = k \nabla ^2 T = k \Big (\frac{\partial ^2T}{\partial x^2} +\frac{\partial ^2T}{\partial y^2}+\frac{\partial ^2T}{\partial z^2} \Big ) \end{aligned}$$
(6)
where k is the heat conductivity of the skin.
Combining (5) and (6), we obtain the governing equation for T(x, y, z, t).
For a 94 GHz beam, the penetration depth of electromagnetic wave into the skin tissue is \(z_s = 1/\mu = 0.16\, \text {mm}\) [7]. As given in (3), the electromagnetic power density passing through depth z decays exponentially in \(z/z_s\). The electromagnetic power is practically all absorbed within a few multiple of \(z_s \) in depth. The electromagnetic heating is virtually limited to a few multiple of \(z_s \) in depth. Since the thickness of skin tissue is much larger than a few multiple of \(z_s\), we approximately treat the skin tissue as a semi-infinite body extending to \(z=+\infty \). We need to point out that this approximation is valid only when the heating duration is relatively short such that the conduction does not spread the heat significantly beyond a few multiple of \(z_s\) in depth into the skin. For a low-intensity electromagnetic heating over a long time duration, this assumption is no longer valid.
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To solve the heat equation (7) in a semi-infinite body, we need to impose an initial condition at \(t=0\) and a boundary condition at the skin surface (\(z=0\)). At \(t=0\), we assume that the skin temperature is uniform both in lateral coordinates (x, y) and in depth z, which is called the baseline temperature and is denoted by \(T_\text {base}\). At the skin surface (\(z=0\)), we neglect the heat loss due to black body radiation and/or convection cooling by the surrounding air. The assumption of zero heat flux at \(z=0\) is justified when the skin surface heat loss flux is relatively much smaller than the electromagnetic energy flux (flow per time per area), which is given by the absorbed beam power density \(P_d(\textbf{r})\). Thus, the assumption of zero heat flux at \(z=0\) is valid when the beam power density is high. For safety reason, a high-intensity electromagnetic heating dictates a short exposure duration to limit the amount of total energy density absorbed into the skin. To summarize, the electromagnetic heating is governed by the initial boundary value problem (IBVP):
As we discussed in Sect. 2, the depth scale is \(z_s = 1/\mu = 0.16\, \text {mm}\). In the depth direction, the electromagnetic heating is practically limited to a few multiple of \(z_s\). We compare the depth scale \(z_s\) to the lateral length scale \(r_s\) of the beam power density \(P_d({\textbf {r}})\), which is characterized by the spot size of beam cross section. For a Gaussian beam, the lateral length scale \(r_s\) is given by the effective mode radius of the beam, \(w_\text {eff}\), defined as [8]
In most electromagnetic heating situations, the lateral length scale is in the range of centimeters [10], much larger than the depth scale: \(r_s \gg z_s \). In this study, we exploit this separation of scales and use asymptotic analysis to solve IBVP (8). The asymptotic solution is based on the analytical solution we obtained in a previous study [11] in the case where the heat conduction is neglected in the lateral directions.
To facilitate the analysis, we carry out non-dimensionalization. We first identify the physical dimensions of various quantities in (8), in terms of the four basic physical dimensions: [mass], [length], [time] and [temperature].
Using combination of these quantities, we construct scales for z, \(\textbf{r}\), t, \(P_d\), and T in (8), and use the scales to carry out non-dimensionalization. Recall that the lateral length scale \(r_s\) comes from the beam power density, which is different from the depth scale \(z_s\).
Non-dimensional depth and lateral length variables:
$$\begin{aligned}&\Delta T = T_\text {target}-T_\text {base}, \quad T_{nd}(x_{nd}, y_{nd}, z_{nd}, t_{nd}) = \frac{T(x, y, z, t)-T_\text {base}}{\Delta T} \\&P_s= k \mu \Delta T, \quad P_{d,nd}({\textbf {r}}_{nd}) =\frac{P_d({\textbf {r}})}{P_s}, \end{aligned}$$
where \(T_\text {target} \) is a reference target temperature in the heating problem, for example, the activation temperature of the skin thermal nociceptors.
The non-dimensional governing equation for \(T_{nd}\) is
Remarkably, (11) has only one parameter: \(\frac{z_s}{r_s}\). We consider the case where the lateral length scale is much larger than the depth scale: \(r_s \gg z_s\). We introduce
For conciseness, we drop the subscript \(_{nd}\) in all non-dimensional variables and functions. For clarity, we shall use \(X_{phy}\) to denote the physical quantity. For example, the non-dimensional temperature is \(T = \frac{T_{phy}-T_{\text {base},phy}}{\Delta T_{phy}} \). The non-dimensional version of IBVP (8) is
In this section, we seek an asymptotic solution of IBVP (13). Our approach consists of three parts. In part 1 of the analysis (Sect. 4.1), we write out the form of an asymptotic solution and derive the governing equations for the terms in the asymptotic formulation.
Notice that in IBVP (13), the small parameter appears in the form of \(\varepsilon ^2\). Based on that, we attempt to determine an asymptotic solution of the form:
$$\begin{aligned} T(x, y, z, t) = T^{(0)}(x, y, z, t) + \varepsilon ^2 T^{(1)}(x, y, z, t) + \cdots . \end{aligned}$$
(14)
Substituting into (13) and balancing O(1) terms give the IBVP governing \(T^{(0)}\).
$$\begin{aligned} {\left\{ \begin{array}{ll} \bigg (\frac{\partial }{\partial t } - \frac{\partial ^2 }{\partial z^2} \bigg ) T^{(0)} = \underbrace{\;\; P_{d}(\textbf{r}) e^{-z} \;\; }_{\text {source term for } T^{(0)}} \\ \\ \frac{\partial T^{(0)}(x, y, z, t)}{\partial z} \bigg |_{z=0}=0, \qquad \lim _{|(x, y, z)| \rightarrow \infty } T^{(0)}(x, y, z, t) = 0 \\ \\ T^{(0)}(x, y, z, 0) = 0 \end{array}\right. }. \end{aligned}$$
(15)
Balancing \(O(\varepsilon ^2)\) terms in (13) yields the IBVP governing \(T^{(1)}\).
$$\begin{aligned} {\left\{ \begin{array}{ll} \bigg (\frac{\partial }{\partial t } - \frac{\partial ^2 }{\partial z^2} \bigg ) T^{(1)} = \underbrace{\Big (\frac{\partial ^2 T^{(0)}}{\partial x^2} +\frac{\partial ^2 T^{(0)}}{\partial y^2}\Big )}_{\text {source term for } T^{(1)}} \\ \\ \frac{\partial T^{(1)}(x, y, z, t)}{\partial z} \bigg |_{z=0}=0 , \qquad \lim _{|(x, y, z)| \rightarrow \infty } T^{(1)}(x, y, z, t) = 0 \\ \\ T^{(1)}(x, y, z, 0) = 0 \end{array}\right. }. \end{aligned}$$
(16)
In (15), the source term \(P_d(\textbf{r}) e^{-z}\) has separable dependences on \(\textbf{r}\) and on z. IBVP (15) is linear with homogeneous initial and boundary conditions, and \(\textbf{r}\) is not a differential variable in (15). Thus, the separable dependences of the source term carry to solution \(T^{(0)}\).
$$\begin{aligned} T^{(0)}(x, y, z, t) = P_d(\textbf{r}) U^{(0)}(z, t), \end{aligned}$$
(17)
where \(U^{(0)}(z, t)\) is a function of (z, t) only and satisfies the IBVP
In [11], we solved an extended version of (18) that includes the effects of blood perfusion and cooling at skin surface via black body radiation and air convection. Here, we rephrase the solution process in a more elegant approach. We utilize the fundamental solution of the heat equation and the notation of convolution. This new approach will be directly relevant for solving IBVP (21).
When solving (18) and (21), we first disregard the condition at infinity. At the end, we verify that the solution found indeed satisfies this condition. It is straightforward to find a particular solution satisfying the nonhomogeneous differential equation with the homogeneous boundary condition at skin surface.
The zero derivative at \(z=0\) allows us to extend u(z, t) to an even function of z for \(z \in (-\infty , +\infty )\). After the even extension, u(z, t) satisfies the IVP
To evaluate the convolution in (26), we derive several properties of the fundamental solution \(\Phi (z, t)\), the general convolution, and the convolution with \(\Phi (z, t)\).
Property 1: The fundamental solution \(\Phi (z, t)\) satisfies the scaling property
Property 3: Applying Property 1 on \(\Phi (\bullet , t)\) and then applying Property 2 on the convolution with \(f(\beta \bullet )\), we obtain another scaling property
In integral (26), \(u_0(z) \) consists of 4 terms. We calculate the convolution of each term. For the first term \(\big ((e^{-\bullet } H(\bullet ))*\Phi (\bullet , t)\big )(z) \), we calculate it directly.
It is straightforward to verify that function \(U^{(0)}(z, t)\) obtained above indeed satisfies the condition at infinity: \(\displaystyle \lim _{z \rightarrow +\infty } U^{(0)}(z, t) = 0\).
In part 3 of the analysis (next subsection), we solve for \(U^{(1)}(z, t)\) from IBVP (21).
It is interesting to note that (21) has the same form as (18); only the source term is much more complicated. We follow the same general procedure to solve IBVP (21) for \(U^{(1)}(z, t)\). The first step is to make the differential equation in (21) homogeneous. We need to find a particular solution of the differential equation with source term \(U^{(0)}(z, t)\). We use the superposition of several particular solutions to match \(U^{(0)}(z, t)\). For these particular solutions, we enforce only the zero derivative condition at skin surface.
In IBVP (18), \(U^{(0)}\) satisfies \((\partial /\partial t -\partial ^2 /\partial z^2)U^{(0)} = e^{-z}\). To produce a source term containing \(U^{(0)}(z, t)\), we try \((t-1)U^{(0)}\), which satisfies
We need to get rid of \((t-1) e^{-z}\). To produce a source term containing \(\big (-(t-1) e^{-z} \big ) \), we recall the particular solution given in (22): \((\partial /\partial t -\partial ^2 /\partial z^2)(e^{-z}+z) = -e^{-z}\). We modify it and try \(t (e^{-z}+z)\), which satisfies
We extend v(z, t) to an even function of z for \(z \in (-\infty , +\infty )\). After the even extension, v(z, t) satisfies the heat equation with initial condition \(v(z, 0) = \big (z^3\,H(z)+(-z)^3\,H(-z) \big )\). The solution of IVP is given by a convolution
The detailed derivation of these integrals is included in Appendix. Applying Property 3 with \(\beta = -1\) on result (45), solves the second part in (44).
Here again, it is straightforward to verify that function \(U^{(1)}(z, t)\) obtained above indeed satisfies the condition at infinity: \(\displaystyle \lim _{z \rightarrow +\infty } U^{(1)}(z, t) = 0\).
4.4 A two-term asymptotic solution
In electromagnetic heating, the nondimensional temperature, representing the normalized temperature increase, is always nonnegative. We combine the results of previous subsections to write out a two-term asymptotic solution of IBVP (13) that guarantees the positivity.
$$\begin{aligned}&T(x, y, z, t) = T^{(0)}(x, y, z, t) + \varepsilon ^2 T^{(1)}(x, y, z, t) + \cdots \nonumber \\&\qquad = T^{(0)}(x, y, z, t) \exp \Big (\varepsilon ^2 \frac{T^{(1)}(x, y, z, t)}{T^{(0)}(x, y, z, t)} \Big ) + \cdots \end{aligned}$$
We examine the accuracy of expression (50) as an approximate solution to IBVP (13). Since we do not have a closed-form expression for the solution of (13) in terms of parameter-free functions that can be efficiently computed, we use a finite difference method on a high-resolution numerical grid to solve IBVP (13). We use the accurate numerical solution as the surrogate “exact solution” when evaluating the performance of asymptotic solutions.
In the test problem, we use the Gaussian beam given in (55) with \(P_d^{(0)} = 1\) and select the ratio of length scales \(\varepsilon = 0.1\). We examine the accuracy of, respectively, the leading term asymptotic \(T^{(0)}\) and the two-term asymptotic \(T^{(0)} \exp \big (\varepsilon ^2 T^{(1)}/T^{(0)} \big )\) in (50).
Fig. 2
Comparison of the “exact solution” and the two asymptotic approximations of the non-dimensional temperature as functions of time t
Fig. 3
Comparison of the “exact solution” and the two asymptotic approximations of the non-dimensional temperature as functions of spatial variables at time \(t_f = 4\)
Fig. 4
Comparison of errors in the leading term and the two-term asymptotics for \(\varepsilon = 0.1\) (top row) and \(\varepsilon = 0.05\) (bottom row)
Figure 2 compares the “exact solution” and the two asymptotic approximations for T vs t at \((x, y, z)=(0,0,0)\) (left panel) and at \((x, y, z)=(0,0,1)\) (right panel). Figure 3 plots the “exact solution” and the two asymptotic approximations at time \(t_f = 4\) as functions of spatial variables: T vs z at \((x, y) = (0,0)\) (left panel) and T vs x at \((y, z) = (0, 0)\) (right panel). It is clear that while the leading term asymptotic captures the main features of the true solution qualitatively, the two-term asymptotic approximation is much more accurate, capable of capturing the true solution quantitatively. In Figs. 2 and 3, the ratio of length scales is \(\varepsilon = 0.1\). That is, the lateral length scale (beam radius) is 10 times the depth scale (penetration depth). For a 94GHz beam, the penetration depth is 0.16mm and \(\varepsilon = 0.1\) corresponds to a beam radius of 1.6mm. The leading term asymptotic is based on neglecting the effect of lateral heat conduction. The discrepancy between the leading term asymptotic and the true solution is attributed to the lateral heat conduction. Figures 2 and 3 demonstrate that in this parameter regime, the lateral heat conduction has a significant effect on the temperature evolution. This effect is well captured in the two-term asymptotic (compare the red and blue lines in Figs. 2 and 3).
In Fig. 4, we examine the errors of the two asymptotic approximations for \(T(0, 0, z, t_f)\) vs z (left column) and for \(T(x, 0, 0, t_f)\) vs x (right column) at \(t_f = 4\). The top row shows the results of \(\varepsilon = 0.1\), the bottom row the results of \(\varepsilon = 0.05\). At a given beam frequency, the penetration depth is fixed, solely determined by the frequency. In that situation, a smaller value of ratio \(\varepsilon \) corresponds to a larger beam spot size. When the ratio \(\varepsilon \) is reduced, the error of the two-term asymptotic decreases much more rapidly than that of the leading term. At \(\varepsilon = 0.05\), the maximum error of the leading term asymptotic is about 0.05 while the maximum error of the two-term asymptotic is less than 0.0017.
6 Concluding remarks
We studied the temperature evolution in skin tissue exposed to a millimeter wave radiation. The three-dimensional temperature distribution is governed by (i) heat source from the absorbed electromagnetic energy, (ii) heat conduction in the depth direction, and (iii) heat conduction in lateral directions perpendicular to the depth.
The absorbed electromagnetic power per volume decays exponentially in the depth direction with a characteristic length scale, called the penetration depth. For a 94 GHz electromagnetic wave, the skin absorption coefficient gives a sub-millimeter penetration depth. Due to this small depth scale, the heat conduction in the depth direction is intrinsically tangled with the absorbed power in the heat transfer process. In most applications, the characteristic length scale in lateral directions is much larger than the penetration depth. This separation of scales provides an opportunity of constructing asymptotic solutions. This opportunity is further enhanced by that the absorbed power per unit volume has separable dependences on the depth coordinate (z) and on the lateral coordinates (\(\textbf{r}\)).
The ratio of the penetration depth to the lateral length scale gives a small parameter (\(\varepsilon \)). We carried out non-dimensionalization on the governing initial boundary value problem of the 3D heat equation. The non-dimensional system contains the small parameter \(\varepsilon \) and a normalized beam power density distribution \(P_d(\textbf{r})\), but it is otherwise parameter free. Even though the heat source term has separable dependences on z and on \(\textbf{r}\), due to the effect of lateral heat conduction, the separable dependences are lost in the temperature solution of the 3D heat equation. That is why so far there is no closed-form exact solution of the 3D problem in terms of parameter-free single-variable functions. We exploited the presence of small parameter \(\varepsilon \), we formulated a two-term asymptotic approximation, and we derived the IBVP governing each term in the asymptotic. It turns out that each term in the asymptotic has separable dependences on z and on \(\textbf{r}\), which reduces the governing equation for each term to a 1D IBVP. We used the fundamental solution of the heat equation and the tool of convolution to solve these 1D problems. In this way, we arrived at a closed-form analytical expression for the two-term asymptotic in terms of parameter-free single-variable functions that can be computed conveniently.
To verify the accuracy of asymptotic solutions, we need an “exact solution” to compare to. Since there is no close-form exact solution for the 3D initial boundary value problem, we computed a very accurate numerical solution of the 3D problem with a finite difference method on a fine 3D grid. We treated this accurate numerical solution as the surrogate exact solution when evaluating the performance of asymptotic approximations. The leading term asymptotic neglects the lateral heat conduction. For a moderate \(\varepsilon = 0.1\), the leading term asymptotic deviates significantly from the true solution; the two-term asymptotic, on the other hand, accurately captures the effect of lateral heat conduction.
Finally, we point out the theoretical and computational values of the asymptotic solution obtained in this paper. The asymptotic solution has a close-form expression in terms of parameter-free single-variable functions. With this analytical setting, we can readily obtain derivatives of the 3D temperature analytically without losing numerical accuracy; we can conveniently study the variation trends of the 3D temperature with respect to spatial position and time. The effect of lateral heat conduction is clearly identified by the second term in the two-term asymptotic. In the analytical asymptotic, it is easy to see the effects of the beam frequency (via the characteristic penetration depth), the size of beam spot (via the characteristic lateral length scale), and the beam power density distribution as a function of normalized skin surface variables. The exact solution of the 3D temperature does have a close-form expression in terms of a 4D convolution with the Green’s function (the fundamental solution). But this theoretical convolution formula is impractical in computing the 3D temperature. Discretizing and evaluating the 4D convolution are computationally even more expensive than solving the initial boundary value problem directly using a finite difference discretization. Numerically, the finite difference method is fairly efficient if we only need to solve the 3D initial boundary value problem a few times. When we need to solve the 3D problem repeatedly for different beam setups, the finite different method is slow. In contrast, the two-term asymptotic provides a practical and accurate way of predicting the 3D temperature distribution in skin tissue when it is exposed to a millimeter wave radiation. This asymptotic approximation is especially relevant in the regime where the beam spot size is moderately larger than the penetration depth.
Acknowledgements
The authors acknowledge the Joint Intermediate Force Capabilities Office of U.S. Department of Defense and the Naval Postgraduate School for supporting this work. The views expressed in this document are those of the authors and do not reflect the official policy or position of the Department of Defense or the U.S. Government.
Declarations
Conflict of interest
The authors declare no competing interests.
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