Heat conduction in porcine muscle and blood: experiments and time-fractional telegraph equation model

Abstract

This paper presents experimental evidence for the damped-hyperbolic nature of transient heat conduction in porcine muscle tissue and blood. An examination of integer order and Maxwell–Cattaneo heat conduction models indicates that the latter, in effect resulting in a time-fractional telegraph (TFT) equation, provides the best fit to transient heat phenomena in such materials. The numerical method is verified on Dirichlet and Neumann initial boundary value problems using existing analytical results. Overall, the TFT equation captures the wave-like nature of heat conduction and temperature profiles obtained in experiments, while reducing the need for further tunable parameters.

1. Introduction

Over the last few decades, electrosurgery has emerged as the main tool for carrying out most of the surgical procedures. Electrosurgical impact on the tissue is accomplished by applying high voltage at high frequency to the interface between the surgical probe and the tissue boundary. Depending on the voltage, frequency and duty cycle, interaction of an electrosurgical probe with the tissue can take many forms—ranging from heavy charring of the tissue surface to coagulation of blood at the probe/tissue interface. The main desired clinical effect is produced through explosive boiling of the fluid in the tissue cells, induced by the high-temperature arc of ionized air between the tissue and the probe [1]. This boiling results in the rupture of cells and creation of an incision at the probe/tissue interface. Despite the many advantages offered by electrosurgery, major complications can occur due to its incorrect application [2–4]. One of these is prolonged and/or excessively strong exposure to current, which results in damage due to excessive heat transfer to surrounding tissues. To guard against such complications, a comprehensive understanding and predictive tools for heat propagation in biological tissues are needed. In this paper, we provide the first step towards this goal by first, simplifying the heat transfer to pure conduction, and then through the use of experimental data, capturing the resulting process by means of the time-fractional telegraph (TFT) equation.

It has been understood for some time that heat conduction in biological materials is of a non-Fourier type [5]. This implies that thermal signals have a finite speed of propagation and that heat equilibrates only after a delay (or lag) once a thermal gradient has been imposed across a material volume element. The best known model of non-Fourier heat conduction is the Maxwell–Cattaneo [6–8] flux law in three dimensions given by

$$(1+{\tau }_0\frac{\mathrm{\partial }}{\mathrm{\partial }t})\mathit{q}=-k\mathbf{\nabla }T,$$ 1.1

where k, q and T are the thermal conductivity, heat flux vector and absolute temperature, respectively. Here, τ₀ is the thermal relaxation time, representing the time lag required to establish equilibrium conditions once a thermal gradient has been applied to a material volume element. Combining equation (1.1) with the balance of energy under a rigid body assumption:

$$\rho c_p\frac{\mathrm{\partial }T}{\mathrm{\partial }t}+\mathbf{\nabla }\cdot \mathit{q}=0,$$ 1.2

where ρ and c_p are the mass density and specific heat at constant pressure, respectively, results in a hyperbolic partial differential equation governing the heat propagation:

$$\frac{\mathrm{\partial }T}{\mathrm{\partial }t}+{\tau }_0\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{t}^2}=\kappa {\mathbf{\nabla }}^{2}T.$$ 1.3

Here, κ = k/ρc_p is the so-called thermal diffusivity. This is the telegraph equation which can also be thought of as a damped wave equation, e.g. [9].

Note that the heat conduction according to (1.3) occurs in a wave-like manner rather than by a diffusion process. The limiting case of τ₀ → 0 gives back the well-known Fourier law of parabolic heat conduction. The phenomenon of the second sound was first observed in experiments conducted at very low temperatures for small time intervals, especially for liquid helium [10]. In such problems of short times and very high heat fluxes, the inclusion of a non-zero relaxation time significantly improves predictions [11,12]. Since then, research has found the range of this relaxation time to be of the order of 10⁻¹⁰ s for gases and 10⁻¹⁴ s for metals, with liquids and insulators having values within this range [13–15]. Recent studies [5,16] have shown that biological tissues have a much larger relaxation time: of the order of tens of seconds. In light of this, the Fourier model of heat conduction is invalid for transient heat phenomena and the more general Maxwell–Cattaneo (MC) model appears appropriate. For example, studies by Dai et al. [17] and Zhang [18] have demonstrated the applicability of this approach to the ‘bioheat transfer model’ to understand skin burn injury. Heat transfer according to (1.3) is known as the second sound, the first sound being the mechanical wave such as in hyperbolic thermoelasticity, e.g. [19].

Fractional order calculus has received increased interest recently due to its generality over traditional integer order PDEs. As suggested by Magin [20], fractional order models provide a simpler way to describe the complex microstructures of biomaterials. Indeed, research has shown that fractional order models work especially well to describe dielectric, magnetic and viscoelastic materials in both time and frequency domains. For the case of viscoelasticity of biomaterials, fractional order models can capture complex phenomena with fewer constants than other traditional models [21]. The focus of the present work is to use the TFT equation which has been studied extensively in the literature as a generalization of the telegraph equation for describing damped wave propagation. This equation has been applied in various fields through the use of both analytical [22] and numerical techniques [23–28].

In this paper, we present the first application of the TFT equation to heat conduction observed in living biological materials. We investigate the applicability of both the integer and the fractional order models to experimental data conducted on in vitro porcine muscle tissue and blood.

2. Experimental set-up

We present a brief overview of the experimental methods used in the work. The experiments were carried out on freshly collected porcine muscle tissue (§2.1) and blood samples (§2.2) procured from the Meat Sciences Lab at UIUC. The apparatus used in these experiments was verified by performing experiments with glycerin and water, followed by a comparison of experimental and numerical results.

2.1. Tissue experiments

The specimen was a cylindrical sample of porcine muscle tissue of 66 mm in diameter and 40 mm in height (figure 1). Temperature sensing resistance temperature detectors (RTDs) were placed at 10 mm and 20 mm along a line from the centre of the sample. The sample was placed in a cylindrical measurement chamber which is 3D printed using ABS plastic. A resistor of $400\hspace{0.17em}\mathrm{\Omega }$ was placed along the central line of the meat sample which provided a constant heat flux. A direct current of 25 V is provided through the resistor. For an axisymmetric model—along any ray—the initial boundary conditions can be approximated as:

$$\text{ICs}:\hspace{0.17em}T(r,0)=T_0,T_t(r,0)=0$$ 2.1a

$$\text{and}\text{BCs}:\hspace{0.17em}{T}_r(0,t)=-Q_{0}H(t),T_r(l,t)=0.$$ 2.1b

Here T₀ is the initial, constant temperature of the meat sample and Q₀ is the constant heat flux supplied by the resistor. H(t) is the Heaviside step function. We assume that the radial symmetry of the problem can be reduced to the one-dimensional problem studied later. We perform verification experiments using glycerin, presented in figure 2, and find that the integer order model captures the experimental response accurately.

Figure 1.

Schematic of cylindrical apparatus used in the measurement of porcine muscle tissue. Heat flux provided by a resistor (R) placed along the central line of the sample; temperature is monitored using the two RTDs (T₁ and T₂) embedded at fixed distances away from the central line. All external surfaces of the sample are insulated.

Figure 2.

Comparison of experimental and integer order model for glycerin used as verification. Temperature is monitored at the T₁ location, cf. figure 1. (Online version in colour.)

2.2. Blood experiments

Additional experiments included porcine blood placed into an insulated channel of square cross-section, having dimensions 3 mm × 3 mm × 40 mm, where heat flux was provided by a resistor at one end (figure 3). The temperature was recorded with two thermocouples placed at a distance of 5 mm and 15 mm from the resistor along the channel. Constant heat flux was provided with a $10\hspace{0.17em}\text{k}\mathrm{\Omega }$ resistor placed at one end of the channel. A direct current of 25 V was supplied through the resistor. The boundary conditions are presented in equation (2.2). However, in this problem, we maintain a limit on the maximum temperature attainable by limiting the heat flux.

$$\text{ICs}:\hspace{0.17em}T(x,0)=T_0,T_t(x,0)=0$$ 2.2a

$$\text{and}\text{BCs}:\hspace{0.17em}{T}_x(0,t)=-Q_{0}H(t),T_x(l,t)=0.$$ 2.2b

Figure 3.

Schematic of the fluid channel used in the measurement of porcine blood. Heat flux provided by a resistor (R) placed at one end of the channel; temperature is monitored using the two RTDs (T₁ and T₂) embedded at fixed distances away from the resistor. All external surfaces of the sample are insulated.

3. Problem formulation and numerical methods

In this section, we elaborate on the definition of the problem and outline the methodology for the numerical approach. We first start with an integer order telegraph equation in §3.1 (based on the formulation from [29]) and extend it to the TFT equation in §3.2 (based on the formulation from [27,28]).

3.1. Integer order telegraph equation

We consider an axisymmetric model assuming isotropic material properties to simplify the formulation. In this manner, we are able to consider the solution along any ray r ∈ [0, l] such as that running through both RTDs, as depicted in figure 1. The absolute temperature along the ray is denoted by T = T(r, t). We apply boundary conditions at r = 0 and r = l. For simplicity, we assume the convective heat transfer to be negligible.

3.1.1. Scheme construction

The problem definition, introduced in equation (1.3), can be cast into the following axisymmetric initial boundary value problem (IBVP). The heat equation in polar coordinates becomes:

$$\frac{\mathrm{\partial }T}{\mathrm{\partial }t}+{\tau }_0\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{t}^2}=\kappa (\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{r}^2}+\frac{1}{r}\frac{\mathrm{\partial }T}{\mathrm{\partial }r}),(r,t)\in [0,l]\times [0,\mathrm{\infty });$$ 3.1a

$$T(r,0)=f(r),T_t(r,0)=g(r)r\in [0,l]$$ 3.1b

$$\text{and}T(0,t)=p(t),T(l,t)=q(t)t>0.$$ 3.1c

We use the finite difference method to solve this IBVP and employ a five-point, or centred-difference stencil. First, we discretize the space–time domain with a grid of M and K points in space and time, with spacing Δr and Δt. Thus, the continuous quantities above are replaced by their discrete counterparts where r → r_m = m(Δr), t → t_k = k(Δt), $T(r_m,t_k)\approx T_m^k$.

Next, we use the centred-difference finite difference stencil which gives us a second-order equation, $\mathcal{O}[{(\mathrm{\Delta }r)}^2+{(\mathrm{\Delta }t)}^2]$. Thus, equation (3.1a) becomes

$$\frac{T_m^{k+1}-T_m^{k-1}}{2\mathrm{\Delta }t}+{\tau }_0\left[\frac{T_m^{k+1}-2T_m^k+T_m^{k-1}}{{(\mathrm{\Delta }t)}^2}\right]=\kappa \frac{T_{m+1}^k-2T_m^k+T_{m-1}^k}{{(\mathrm{\Delta }r)}^2}+\frac{\kappa }{r_m}\frac{T_{m+1}^k-T_{m-1}^k}{2\mathrm{\Delta }r}.$$ 3.2

Using the relation r_m = m(Δr) and rearranging the above to solve for the next time-step implies:

$$T_m^{k+1}=\frac{\kappa (\mathrm{\Delta }t)R[T_{m+1}^k(1+S)-2T_m^k+T_{m-1}^k(1-S)]+{\tau }_0[2T_m^k-T_m^{k-1}]+(\mathrm{\Delta }t/2)T_m^{k-1}}{{\tau }_0+\mathrm{\Delta }t/2},$$ 3.3

where R = Δt/(Δr)² and S = 1/2m.

Finally, we need a way to approximate the initial conditions in equation (3.1b) to start the numerical scheme. For this, we expand $T_m^1$ using a Maclaurin expansion and consider the first two terms to get an approximation with error $\mathcal{O}{(\mathrm{\Delta }t)}^3$, similar to Burden & Faires [30]. We find that

$$T_m^1\approx T_m^0+(\mathrm{\Delta }t)g(r_m)+\frac{1}{{\tau }_0}\frac{{(\mathrm{\Delta }t)}^2}{2}[f^{″}(r_m)-g(r_m)].$$ 3.4

3.1.2. Verification

To perform verification, we use two different sets of initial conditions for which the exact analytic solution is known. The first problem corresponds to the Dirichlet boundary conditions:

$$T(r,0)=f(r)=T_0\sin (\pi r),T_t(r,0)=g(r)=0$$ 3.5a

$$\text{and}T(0,t)=p(t)=0,T(1,t)=q(t)=0,$$ 3.5b

where T₀ is a prescribed positive constant.

The exact analytic solution for this IBVP is given in Mickens & Jordan [29] using the method of separation of variables. We compare this with the implemented numerical method in figure 4a.

Figure 4.

Verification for integer order telegraph equation comparing exact analytic solution (solid lines) with computed solution (circles) for t = 0.01 and t = 0.2. (a) Dirichlet boundary value problem and (b) Neumann boundary value problem.

The second problem involves a Neumann boundary value problem with spatial gradients of temperature as the boundary conditions (i.e. replacing equation (3.1c)):

$$T(r,0)=f(r)=T_0,T_t(r,0)=g(r)=0$$ 3.6a

$$\text{and}{T}_r(0,t)=p(t)=-Q_{0}H(t),T_r(1,t)=q(t)=0.$$ 3.6b

where T₀ and Q₀ are positive constants and H(t) is the Heaviside step function.

The exact analytic solution is determined using Green’s function as given in Polyanin & Nazaikinskii [31]. This is compared to the numerical method in figure 4b.

3.2. Fractional order telegraph equation

In this section, we consider the fractional order telegraph equation which is a generalized version of the integer order equation developed in the preceding section. First, we present a derivation of the TFT equation from the balance of energy equation and the fractional MC equation. We note that the final form of the equation thus derived is consistent with the underlying governing equations, something which would not be achieved if another form of this equation were used. Next, we present the numerical method used as well as the verification.

We consider a thin rod of constant cross-section with homogeneous material properties along its span x ∈ [0, l] (figure 5). The absolute temperature of the rod is denoted by T = T(x, t). The rod is subjected to boundary conditions at its ends, whereas the lateral surface is fully insulated. We assume that the heat conduction within the rod is governed by the MC heat flux law. Again, we assume the convective heat transfer to be negligible.

Figure 5.

One-dimensional rod model with insulated lateral surfaces.

3.2.1. Derivation of time-fractional telegraph equation from governing equations

To model the channel problem of figure 3, we introduce an equation of Maxwell–Cattaneo type with a fractional order time derivative for the heat flux, which reads

$$(1+{\tau }_0\frac{{\mathrm{\partial }}^{\alpha }}{\mathrm{\partial }{t}^{\alpha }})q=-k\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{x}^2},$$ 3.7

where we generally employ the Caputo fractional derivative

$$\frac{{\mathrm{\partial }}^{\alpha }T}{\mathrm{\partial }{t}^{\alpha }}=\frac{1}{\mathit{\Gamma }(n-\alpha )}{\int }_0^t\frac{{\mathrm{\partial }}^{n}T(x,s)}{\mathrm{\partial }{s}^n}\hspace{0.17em}\frac{\text{d}s}{{(t-s)}^{\alpha -n+1}},n-1\le \alpha <n;$$ 3.8

with Γ( · ) being the gamma function and n = 1.

To derive a governing (or field) equation, we differentiate (3.7) with respect to x to find:

$$q^{\prime }+{\tau }_0\frac{{\mathrm{\partial }}^{\alpha }{q}^{\prime }}{\mathrm{\partial }{t}^{\alpha }}=-k\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{x}^2}.$$ 3.9

Next, a time differentiation of order α applied to (1.2) yields:

$$\rho c_{\hspace{0.17em}p}\frac{{\mathrm{\partial }}^{\alpha +1}T}{\mathrm{\partial }{t}^{\alpha +1}}+\frac{{\mathrm{\partial }}^{\alpha }{q}^{\prime }}{\mathrm{\partial }{t}^{\alpha }}=0$$ 3.10

which, together with (3.9), leads to

$$\rho c_{\hspace{0.17em}p}{\tau }_0\frac{{\mathrm{\partial }}^{\alpha +1}T}{\mathrm{\partial }{t}^{\alpha +1}}-q^{\prime }=k\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{x}^2}.$$ 3.11

A repeated use of (1.2) gives the TFT equation:

$$\frac{\mathrm{\partial }T}{\mathrm{\partial }t}+{\tau }_0\frac{{\mathrm{\partial }}^{\alpha +1}T}{\mathrm{\partial }{t}^{\alpha +1}}=\kappa \frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{x}^2},$$ 3.12

where κ = k/ρc_p is the thermal diffusivity as before and 0 ≤ α < 1. This equation is known as the generalized Cattaneo equation of type 4 as defined by Povstenko [32,33].

3.2.2. Scheme construction

The general fractional order IBVP, with 0 ≤ α < 1, is

$$\frac{\mathrm{\partial }T}{\mathrm{\partial }t}+{\tau }_0\frac{{\mathrm{\partial }}^{\alpha +1}T}{\mathrm{\partial }{t}^{\alpha +1}}=\kappa \frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{x}^2}+\mathit{\Phi }(x,t),(x,t)\in [0,l]\times [0,\mathrm{\infty });$$ 3.13a

$$T(x,0)=f(x),T_t(x,0)=g(x)x\in [0,l]$$ 3.13b

$$\text{and}T(0,t)=p(t),T(l,t)=q(t)t>0,$$ 3.13c

where Φ(x, t) represents the source terms.

Note that setting α = 1, returns equation (3.1), i.e. the integer order telegraph equation.

Following the steps outlined in Sun & Wu [27] and Li & Cao [28], it is possible to discretize the fractional derivatives using a grid similar to that proposed in §3.1. We choose a grid of K points in time and M points in space with grid spacing Δt and Δx, respectively. Thus, we have that 0 < t₀ < · · · < t_k < · · · < t_K, 0 < x₀ < · · · < x_m < · · · < x_M = l and $T(x_m,t_k)\approx T_m^k$.

For convenience, we introduce the following notations:

$$T_m^{k-1/2}=\frac{1}{2}(T_m^k-T_m^{k-1}),{\delta }_t{T}_m^{k-1/2}=\frac{1}{\mathrm{\Delta }t}(T_m^k-T_m^{k-1})$$ 3.14a

$$\begin{array}{rl}\text{and}& {\delta }_x{T}_{m-1/2}^k=\frac{1}{\mathrm{\Delta }x}(T_m^k-T_{m-1}^k),\\ \hspace{0.17em}& {\delta }_x^2{T}_m^k=\frac{1}{\mathrm{\Delta }x}({\delta }_x{T}_{m+1/2}^k-{\delta }_x{T}_{m-1/2}^k),\end{array}$$ 3.14b

where $T_m^{k-1/2}$ is the time average of T at the two points (x_m, t_k) and (x_m, t_k−1) and ${\delta }_t{T}_m^{k-1/2}$ is the difference quotient of T based on these two points, very similar to a first-order finite difference approximation. Similarly, ${\delta }_x{T}_{m-1/2}^k$ is the first-order difference operator between the two points (x_m, t_k) and (x_m−1, t_k), and ${\delta }_x^2{T}_m^k$ is the second-order difference operator between the points (x_m−1, t_k), (x_m, t_k) and (x_m+1, t_k).

Now, we follow the method proposed by Sun & Wu [27] and adapted by Li & Cao [28] for the TFT equation to formulate a finite difference scheme for equations (3.13a–c). We introduce two new variables to transform the original equation into a low-order system of equations. The difference scheme is as follows:

$$\begin{array}{rl}\hspace{0.17em}& \frac{{\tau }_0}{\mathit{\Gamma }(1-\alpha )}\frac{1}{\mathrm{\Delta }t}\{[a_0+\frac{\mathrm{\Delta }t}{{\tau }_0}\mathit{\Gamma }(1-\alpha )]{\delta }_t{T}_m^{k-1/2}.\\ \hspace{0.17em}& -\sum _{\hspace{0.17em}j=1}^{k-1}(a_{k-j-1}-a_{k-j}){\delta }_t{T}_m^{\hspace{0.17em}j-1/2}-a_{k-1}{g}_m\}=\kappa {\delta }_x^2{T}_m^{k-1/2}+{\mathit{\Phi }}_m^{k-1/2},\\ \hspace{0.17em}& 1\le m\le M-1,k\ge 1;\\ \hspace{0.17em}& T_m^0=f_m,0\le m\le M,\end{array}$$ 3.15

where

$$\begin{array}{rl}{a}_{\hspace{0.17em}p}& :={\int }_{t_{\hspace{0.17em}p}}^{t_{\hspace{0.17em}p+1}}{t}^{-\alpha }\hspace{0.17em}\text{d}t=\frac{1}{1-\alpha }[t_{\hspace{0.17em}p+1}^{1-\alpha }-t_{\hspace{0.17em}p}^{1-\alpha }]\\ \hspace{0.17em}& =\frac{\mathrm{\Delta }{t}^{1-\alpha }}{1-\alpha }[{(p+1)}^{1-\alpha }-p^{1-\alpha }]\end{array}$$ 3.16

and

$$f_m:=f(x_m);g_m:=g(x_m);{\mathit{\Phi }}_m^{k-1/2}:=\mathit{\Phi }(x_m,\frac{t_k+t_{k-1}}{2}).$$

At each time step, t_k, we can show that equation (3.15) is a tridiagonal system of linear algebraic equations of the form:

$$\begin{array}{r}\left[\begin{array}{cccccc}{B}_1& C_1& \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}& 0\\ A_2& B_2& C_2& \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}\\ \hspace{0.17em}& A_3& B_3& \ddots & \hspace{0.17em}& \hspace{0.17em}\\ \hspace{0.17em}& \hspace{0.17em}& \ddots & \ddots & \hspace{0.17em}& C_{M-2}\\ 0& \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}& A_{M-1}& B_{M-1}\end{array}\right]\left[\begin{array}{c}{T}_1^k\\ T_2^k\\ T_3^k\\ ⋮\\ T_{M-1}^k\end{array}\right]=\left[\begin{array}{c}{D}_1\\ D_2\\ D_3\\ ⋮\\ D_{M-1}\end{array}\right]\end{array}$$ 3.17

where

$$A_m=-\frac{\kappa }{2\mathrm{\Delta }{x}^2},$$ 3.18a

$$B_m=\frac{{\tau }_0}{\mathit{\Gamma }(1-\alpha )}\frac{a_0}{\mathrm{\Delta }{t}^2}+\frac{1}{\mathrm{\Delta }t}+\frac{\kappa }{\mathrm{\Delta }{x}^2},$$ 3.18b

$$C_m=-\frac{\kappa }{2\mathrm{\Delta }{x}^2}$$ 3.18c

$$\begin{array}{rl}\text{and}{D}_m& =\frac{{\tau }_0}{\mathit{\Gamma }(1-\alpha )}\frac{1}{\mathrm{\Delta }t}\{[\frac{a_0}{\mathrm{\Delta }t}+\frac{1}{{\tau }_0}\mathit{\Gamma }(1-\alpha )]T_m^{k-1}\\ \hspace{0.17em}& +\sum _{\hspace{0.17em}j=1}^{k-1}(a_{k-j-1}-a_{k-j})(T_m^{\hspace{0.17em}j}-T_m^{\hspace{0.17em}j-1})+a_{k-1}{g}_m\}\\ \hspace{0.17em}& +\frac{\kappa }{2\mathrm{\Delta }{x}^2}(T_{m+1}^{k-1}-2T_m^{k-1}+T_{m-1}^{k-1})+{\mathit{\Phi }}_m^{k-1/2}.\end{array}$$ 3.18d

This tridiagonal system of equations can readily be solved for the unknown temperature field at time t_k on the interior of the domain. The temperature values at the endpoints of the domain are prescribed from the boundary conditions imposed.

3.2.3. Verification

We use the example from Li & Cao [28] to verify the numerical method implemented. The IBVP is defined as

$$\begin{array}{rl}\hspace{0.17em}& \frac{\mathrm{\partial }T}{\mathrm{\partial }t}+\frac{{\mathrm{\partial }}^{\alpha +1}T}{\mathrm{\partial }{t}^{\alpha +1}}=\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{x}^2}+\mathit{\Phi }(x,t),\\ & (x,t)\in [0,1]\times [0,1];\end{array}$$ 3.19a

$$T(x,0)=0,T_t(x,0)=0x\in [0,1]$$ 3.19b

$$\text{and}T(0,t)=0,T(l,t)=0t>0.$$ 3.19c

The source term is

$$\mathit{\Phi }(x,t)=2(x^2-x)t[\frac{t^{-\alpha }}{\mathit{\Gamma }(2-\alpha )}+1]-2t^2.$$ 3.20

This example has an exact solution given by

$$T(x,t)=x(x-1)t^2.$$

The comparison between the exact solution and computational results is given in figure 6 for two different values of α. The plots show that the computed solution compares favourably with the exact solution.

Figure 6.

Verification of the fractional order telegraph equation by comparing exact analytic solution (solid lines) with computed solution (circles) for t = 0.5 and t = 1. (a) α = 0.5, Δx = 1/64, Δt = 1/1000; (b) α = 0.8, Δx = 1/32, Δt = 1/1000.

4. Results and discussion

In this section, we apply the numerical models described in §3 to the experiments performed on porcine tissue and blood. The results obtained from the models and confirmed by the experiments offer compelling evidence that heat conduction in biological materials is governed by a wave-like process.

4.1. Heat conduction in muscle

In the numerical model, we choose the thermal diffusivity for porcine muscle tissue as κ = 1.4 × 10⁻⁷ m² s⁻¹ [5]. The value of the relaxation time τ₀ has been determined experimentally by the method of least-squares fit for different values of τ₀. For this experiment, we found τ₀ = 6.9 s which yields the speed of second sound as c = 2 mm s⁻¹.

The results from the integer order telegraph equation are compared to the experimental results in figure 7. The solid lines represent the experimental data measured by the two RTDs (T₁ and T₂) located at 10 mm and 20 mm away from the centre of the sample. We find the integer order telegraph equation (3.1) able to accurately determine the experimentally observed temperature changes. The effect of the thermal relaxation time τ₀ is clearly evidenced in these results. For the first RTD T₁, there is a noticeable time lag before the heat propagates from the heat source and reaches the RTD. This is the so-called heat wave that travels at a finite speed. As the heat wave propagates through the material, it arrives at the RTD T₂ at a later time. The speed of propagation is found to be inversely proportional to τ₀.

Figure 7.

Experimental data (solid lines) measured by the two RTDs (T₁ and T₂) compared to the results obtained from the integer order telegraph equation (dashed and dotted lines) for porcine tissue. The Fourier model, corresponding to τ₀ = 0 (dashed-dotted line), is also presented. (Online version in colour.)

As a comparison, we plot the results obtained from the Fourier model, i.e. τ₀ = 0 (dash-dotted line in figure 7), at the position of the first RTD. The heat wave propagates almost instantaneously and the temperature rises before it is physically possible. This result is consistent with the literature, thus confirming the wave-like nature of heat conduction in biological tissue.

One discrepancy present in our model is the assumption of perfect insulation. However, as can be seen in the experimental results measured at RTD T₁, the rate of increase of temperature decreases as time increases. However, our model predicts a constant rate of temperature increase. This difference may be attributed to some heat loss through the walls of the test chamber. So far, we have modelled the heat loss as an exponential decay of the input heat source. However, we can also include a loss term in (3.1a), as follows:

$$\frac{\mathrm{\partial }T}{\mathrm{\partial }t}+{\tau }_0\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{t}^2}=\kappa \frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{x}^2}-\gamma T,$$ 4.1

where the constant γ would need to be tuned to match the experimentally observed results. We plan to investigate this change in the future.

4.2. Heat conduction in blood

The thermal properties of blood are taken as: k = 0.505 W⁻¹ m⁻¹ K⁻¹, c_p = 3617 J⁻¹ kg °C⁻¹ and ρ = 1050 kg m⁻³ [34,35]. The thermal relaxation time τ₀ = 20s^α was determined from experiments.

The results from the experiments are presented in figure 8. We find that the integer order equation (α = 1) fails to produce realistic results for this case. We therefore also plot the results of the fractional order telegraph equation. The value of α = 0.5 has been found to produce the best results. The generalization of the fractional order derivatives gives us a further tunable parameter to match the experimental results. Without this capability, we would be forced to introduce additional terms to the integer order telegraph equation, such as third- and fourth-order time and space derivatives.

Figure 8.

Experimental data (solid lines) measured by the two thermocouples (T₁ and T₂) compared to the results obtained from both the integer and fractional order telegraph equation (dashed and dotted lines) for porcine blood. (Online version in colour.)

Additionally, the heat wave is still observed in the fractional order telegraph equation. We see that the thermocouple T₁ registers the heat source before T₂ and both have τ₀ ≠ 0.

5. Conclusion

In this paper, we have studied the heat conduction in biological materials, specifically porcine muscle tissue and blood. As noted in the literature, Fourier’s Law of heat conduction has the drawback of infinite speed of propagation leading to non-physical solutions. For this reason, with respect to the porcine muscle tissue, the use of the Maxwell–Cattaneo flux law is proposed which introduces a thermal relaxation time, τ₀ resulting in the telegraph equation. The latter equation is a hyperbolic PDE which predicts that the heat conduction occurs in a wave-like manner rather than only through a diffusion process. Further, with respect to porcine blood, we generalize the integer order telegraph equation by introducing fractional derivatives yielding the TFT equation where the fractional order of the derivatives allows us to capture heat propagation phenomena with fewer constants than traditional models.

Experiments performed on porcine muscle tissue and blood clearly demonstrate the second sound effect in biological materials. The heat wave has a finite speed of propagation and the thermal response manifests itself only after some time delay. We note that a fractional derivative of order α = 0.5 offers the most appropriate model for heat conduction in blood while an integer model is sufficient to describe heat conduction in muscle. We do not investigate the biological reason for the choice of these parameters in this work, instead we are leaving this for future publications. Nevertheless, without the use of a fractional order model, we would have been forced to introduce more terms with integer order derivatives to match the experimental results.

As a next step in this research, we aim to enrich our model to capture the explosive boiling and charring that occurs in tissues during electrosurgery. We also hope to account for the boundary layer evolution between burnt and unburnt tissues.

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Authors' contributions

A.M. performed the numerical modelling and prepared the manuscript; Y.P., W.K. and H.J.S. performed experimental studies; R.B., L.P.C., J.B. and M.O.S. provided guidance, designed experiments and prepared the manuscript.

Competing interests

We declare we have no competing interest.

Funding

Research reported in this publication was supported in part by the National Institute of Biomedical Imaging and Bioengineering of the National Institutes of Health under award number R01EB029766. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.