Heat conduction in live tissue during radiofrequency electrosurgery

Abstract

In this paper, we propose a method to model radiofrequency electrosurgery to capture the phenomena at higher temperatures and present the methods for parameter estimation. Experimental data taken from our surgical trials performed on in vivo porcine liver show that a non-Fourier Maxwell–Cattaneo-type model can be suitable for this application when used in combination with an Arrhenius-type model that approximates the energy dissipation in physical and chemical reactions. The resulting model structure has the advantage of higher accuracy than existing ones, while reducing the computation time required.

1. Introduction

Radiofrequency (RF) electrosurgery or ablation is a technology ubiquitous in medicine, used commonly in surgical procedures for tissue dissection, pathological delineation and often subsequent removal [1]. Ablation in medicine usually refers to the procedure of heating a tumour to deactivate the cancer cells, while in other engineering applications, ablation means burning off material such as heat shields on rockets under high heat. In the electrosurgery setting, ablation is closer to the second case, where tissue near the electrode is vaporized. The application of RF in surgery achieves the controlled heating of tissue with high power energy density by passing electrical currents through biological tissue. RF in surgery is used in cutting mode to dissect tissue and in coagulation mode to coagulate blood and related blood vessels to control bleeding. Coagulation mode may also be used for cutting purposes if the proper energy settings are used.

Since its inception approximately 100 years ago, electrosurgery has become a common tool for nearly all surgical procedures. With the rise in the use of the laparoscope (tubular tools to allow entry to human body cavities with smaller surgical incisions than those common in ‘pen’ surgery) and its related advance towards ‘robot-assisted surgery’, the electrosurgery tool has become the dominant surgical tool to dissect tissue, to treat or remove pathological specimens [2], accounting for half of surgical procedures [3]. Forthcoming advances in surgical technique must address the current problems inherent in the tools used in laparoscopy and robot-assisted surgery. Improvements in electrosurgery are vital as there is limited knowledge regarding its effect on tissue during dissection, the damage it causes along its cutting path, and the best means to apply electric current for a particular task, or incorporate engineering advances into solutions of the individual surgical problems [4]. As surgery adds aspects of ‘automation’ to the surgical armamentarium, understanding and modelling of the physics of electrosurgery assume primary importance for achieving accurate tissue heating with a minimum amount of unnecessary, dangerous, tissue damage.

There have been many attempts at the modelling of electrosurgery and RF ablation such as [1,5–7]. However, these modelling methods are either not accurate enough or too complicated and computationally intensive, precluding their use in the equipment available today. Some of the existing models could take 10–12 h to compute for a single procedure [8]. There are methods such as [9] that are aimed towards interactive surgical training. They can achieve real-time performance on coarser meshes at the expense of accuracy. There are also studies like [10] that incorporate tissue deformation. One of the biggest challenges in modelling is capturing the phase change and chemical reaction phenomena at temperatures above the boiling point of water. This work describes a model of the electrosurgery process in living tissue that is accurate and fast, predicting the tissue temperature during surgical procedures.

There are different approaches used to characterize the electrosurgery process. Ward et al. [11] approximated it with material removal under volumetric heating while completely ignoring heat transfer, while El-Kebir et al. [12] modelled the tissue denaturation process with pure heat transfer by approximating it as a Stefan problem. Madhukar et al. [7] proposed a heat conduction model using the telegraph equation for porcine muscle and blood. In the present work, we believe that the actual electrosurgery phenomena should be somewhere in between. While the model in [7,12] is suitable for unperfused tissue under stationary excitation at lower temperatures, here we will expand it to capture heat conduction in living soft tissue under RF electrosurgery due to a moving heat source. We present a modelling method that employs a non-Fourier type heat transfer model with some modifications to approximate the complicated physical and chemical reactions that occur during RF ablation at higher temperatures. Given the criticisms against non-Fourier type models questioning their usefulness [13–16], we will use experimental data on live porcine tissue to show the improvement in accuracy that could potentially be achieved using such a model. This work also covers the possible methods for fast and simple theoretically supported process dynamics parametrization that employs our model. Data-driven methods including machine learning are then discussed. This work constitutes the next step in the ongoing data-driven electrosurgical biophysics modelling effort by the present authors, with their previous results reported in [17,18]. Using the technique developed in [18], the work in [17] demonstrated the suitability of the hyperbolic Maxwell–Cattaneo model of heat propagation augmented by the volumetric parametrizable heat source to support a high-fidelity thermodynamic representation of the live tissue response to electrosurgical impact and introduced the parameter estimation framework for this model. El-Kebir et al. [17] also indicated that the phenomena associated with the pronounced appearance of Mach cones trailing the cutting probe tip, as seen in live tissue experiments, are associated with the electrosurgical action characterized by higher cutting speeds and larger applied power, and would be considered in future work. The latter task is addressed in the present paper through augmenting the Maxwell–Cattaneo model with the additional chemical dynamics and developing the corresponding model parametrization technique.

2. Modelling of the phenomena

RF heating of biological tissue is a coupled electrical–thermal problem where the electric field created by the electrodes generates heat that is then transferred through the tissue. We will first look at the heat transfer model.

Classical heat transfer is governed by Fourier’s Law of thermal conduction:

$$q=-\kappa \mathrm{\nabla }T,$$ 2.1

where q is the local heat flux, κ is the thermal conductivity and $\mathrm{\nabla }T$ is the temperature gradient.

Many studies have attempted to model RF ablation using this model (e.g. [5,6,19,20]). Effectively, equation (2.1) leads to a parabolic equation implying that all signals travel at infinite speeds. This modelling approach looks like a reasonable assumption because the times associated with electron–phonon and phonon–phonon interactions are extremely fast [21]. However, the simple Fourier model is not able to capture heat transfer in the electrosurgery process without significant modification or without a combination with other quite complicated multi-physics models. The reason for this inability is that biological tissues can be heated above 100°C during electrosurgery where evaporation and chemical reactions occur. To resolve this problem, some researchers have proposed models with variable parameters, such as thermal conductivity and specific heat [1,5] or modelling the tissue as a tri-phasic mixture [6]. The problem with all these models is that they are either not accurate enough or too complicated and time-consuming to compute. Most of the studies do not compare the simulations with experimental data on the entire temperature field. Instead, they only look at one-dimensional slices in space or time. Fits generated this way may seem promising but do not work for all situations. Therefore, we explore an alternative modelling method that offers speed and accuracy, and then evaluate the fit with the cross-correlation method [22] that accounts for the entire temperature field.

The model described in this work takes inspiration from the examination of infrared (IR) imagery taken at the time of electrosurgery illustrated in figure 1. With the electrosurgery tip cutting horizontally from left to right, the resulting temperature field has a cone-like outline, shown in green, which resembles the shock waves one finds in the study of supersonic flight. In simple terms, the image of the temperature field in this figure is similar to the waves found in hyperbolic non-Fourier type models that give a finite speed of propagation. The simplest of these is based on the Maxwell–Cattaneo heat conduction model

$$\tau \dot{q}+q=-\kappa \mathrm{\nabla }T,$$ 2.2

where τ is the relaxation time and the overdot denotes the time derivative [23,24]. With the energy balance

$$\rho c_p\dot{T}=-\mathrm{\nabla }\cdot q,$$ 2.3

where ρ and c_p are the density and specific heat at constant pressure, one arrives at the telegraph equation

$$\ddot{T}+\frac{1}{\tau }\dot{T}=c^2{\mathrm{\nabla }}^{2}T,$$ 2.4

where $c=\sqrt{\kappa /(\rho c_p\tau )}$ is commonly called the speed of second sound. Equation (2.4) is a hyperbolic equation that passes into a parabolic (diffusion) equation in the τ → 0 limit.

Figure 1.

Infrared image around the electrosurgery tip taken during electrosurgery on porcine liver at 15 W under pure cut mode with a ValleyLab Force FX generator.

In the Maxwell–Cattaneo model, the relaxation time τ represents the time lag for establishing equilibrium within an elemental volume [21]. We borrow this concept to cumulatively account for the effects of evaporation and chemical reactions during electrosurgery cuts at high temperatures, which absorb heat and could also be viewed as creating a time lag for establishing equilibrium.

There are criticisms against the Maxwell–Cattaneo model [13,14]. These authors claim that the infinite speed paradox is nonexistent and the Fourier heat equation is a good enough approximation. They also suggest that existing works, such as Kaminski [25] and Mitra [26], that attempt to validate the non-Fourier models have serious flaws and could be replaced with Fourier models if done correctly. In this paper, we do not try to demonstrate the infinite speed paradox. Rather, as a past president of the Royal Society tells us to feign no hypothesis [27], we will simply adapt the Maxwell–Cattaneo model to approximate a very complicated behaviour that involves many different phenomena such as phase change and chemical reactions that happen during electrosurgery and affect the heat transfer process. Or in other words, one could simply view this as a higher-order approximation by adding a second-order derivative in time to the classical heat transfer problem. Under this framework, the parameter τ is no longer a fixed material property and could be adapted in time as needed to provide a better fit as in [18].

It should be noted that the Maxwell–Cattaneo model alone is not sufficient to provide an accurate solution because the second-order term only changes the outline (shape) of the temperature field but does not account for the energy removed from the system that goes into the latent heat of vaporization and chemical reactions. As a result, one might get a solution with the correct shape but wrong magnitude. Also, there is heat removal through perfusion cooling in biological tissue and metabolic heat generation. Therefore, we need to introduce extra terms into (2.4).

To account for the energy absorbed by physical and chemical reactions, we can use the model of chemical kinetics [28]. The rate of change in concentration is

$$\dot{a}=k(1-a),$$ 2.5

where k is the reaction rate, usually modelled by the Arrhenius equation [29]:

$$k=A\hspace{0.17em}{\mathrm{e}}^{-E_a/RT}.$$ 2.6

In chemical kinetics, E_a would be the activation energy, A is a pre-exponential factor and R is the universal gas constant. However, in our case, these parameters have different meanings from what they were originally intended for because we are not dealing with a single chemical reaction but approximating the effect of the sum of several physical and chemical processes similar to [30]. Here, the parameters A and E_a/R are just tuning or fitting constants, and a can be viewed as an irreversible damage parameter, that measures how much the tissue has been denatured. As an analogy, one can view it as the ‘doneness’ of steak, with 0 being raw, 1 being charred, and medium-rare somewhere in between, such as commonly seen in food science studies [31].

Some researchers [1,32] use a similar model to assess tissue lesions during post-processing of simulation data. Instead of looking at isotherms of the maximum temperature, they account for the temperature history, as lower temperature for a longer duration and higher temperature for a shorter duration cause similar damage. To account for the energy removal, we add an extra term in equation (2.4) and combine it with the Pennes bio-heat equation [33] to obtain, after some rearrangement,

$$\tau \ddot{T}+\dot{T}=\alpha {\mathrm{\nabla }}^{2}T+Q-H\dot{a}-h_p+h_m.$$ 2.7

Here, α = κ/(ρc_p) is the diffusivity, Q is the heat source from the electrode, H measures the amount of heat absorbed per unit of damage, h_m is the metabolic heat generation which is insignificant and usually ignored [20] and h_p is cooling by perfusion:

$$h_p={\omega }_b(T-T_0)$$ 2.8

with ω_b being the perfusion rate and T₀ being the blood temperature. This effect could be important for surgery on live subjects near blood vessels.

Finally, since RF electrosurgery is a coupled electro-thermal problem, we also need to look at the electrical aspect. Since the frequency f used in RF ablation is usually between 300 kHz and 1 MHz, it gives wavelengths λ = c/f around hundreds of metres, which is much larger than the length scale of tissue under surgery, with c being the speed of electromagnetic waves in biological tissue. Thus, the volumetric heat source should always have a bell or Gaussian shape, with varying magnitude and width depending on power and waveform. Even though biological tissue can be capacitive and have waveform dependent impedance [34], in terms of the heat conduction calculations, we can assume that we will get an average power output from faster calculations on the power supply side. Therefore, a quasi-static approach can be employed [35]; the tissue can be approximated as purely resistive with the conservation of electrical charge:

$$\mathrm{\nabla }\cdot s\mathrm{\nabla }V=0,$$ 2.9

where s is the electrical conductivity and V is the electric potential. Then, the distributed source due to ionic heating (some works call it Joule heating, but we prefer the former term because heating is caused by ions in the tissue instead of free electrons) is

$$Q=J\cdot E,$$ 2.10

where J = sE is the current density and $E=\mathrm{\nabla }V$ is the electric field.

While this forms another boundary value problem that could be solved along with the heat equation (2.7), doing so would make the problem solution very time-consuming. Furthermore, the solution of the electrical boundary value problem should have a Gaussian shape, since it is simply a Laplace equation, unless there are multiple electrodes. As we are only considering a single electrosurgery probe in this paper, we can approximate the source by a three-dimensional (3D) Gaussian distribution as $Q(x)=Q_0\exp [-\sum _{n=1}^3{x}_i^2/(2{\sigma }^2)]$, where Q₀ is the magnitude of the source and σ is the radius of the source. Note that with this approximation, the source radius σ should be adjusted based on the power mode being selected at the electrosurgical generator because the electrical conductivity of biological tissue is waveform-dependent [36]. In the pure cut mode, the current should be concentrated extremely close to the tip of the probe; therefore we could approximate σ with the diameter of the electrosurgery tip. However, if the surgeon is using coagulation or spray mode to stop bleeding, this radius would be much larger and one could develop a lookup table for the approximation.

The resulting electrosurgery model using equation (2.7) coupled with equation (2.5) is much simpler than solving the coupled electro-thermal problem, especially compared to tracking the volume fraction of three phases and incorporating fluid dynamics. Moreover, the damage parameter that is calculated within the model can be very helpful in minimizing unnecessary tissue damage [18]. The details on the numerical method that can be used to solve equation (2.7) can be found in [37] and equation (2.5) just adds a simple integration which can be done with the Euler forward method and, since the damage parameter a ∈ [0, 1] by design, one does not need to worry about numerical stability. Using explicit methods like Runge–Kutta 4 and Euler forward, this model can be solved very quickly, whereas the classical Fourier heat equation and the electrical equation (2.9) would require implicit solvers.

While it is easy to see why our model is faster to compute compared to others, we would like to show that it is also well supported by the experimental results. Furthermore, we show the value of adding the second-order term to address some of the concerns about the usefulness of a non-Fourier type model [14]. Finally, in order to use this model to generate real-time simulations for helping with control decisions, a speedy parametrization of equation (2.7) is required. These are discussed in the following sections.

3. Demonstration of the value of non-Fourier models through experimental results

We have collected data from experimental trials conducted on in vivo porcine liver tissue performed at the University of Illinois at Chicago’s (UIC) Surgical Innovation Training Laboratory (SITL), in accordance with the NIH Vertebrate Animal Section standards. Details on the experimental design and setup can be found in [17]. In the experiment, cuts are made horizontally at 10 mm s⁻¹ (left to right) on the tissue at 5 mm depth in pure cut mode at 5 W power. Temperature data on the top surface of the tissue are collected with an IR thermographer and compared with simulation results obtained using the model presented in the previous section.

The fit of experimental data by the temperature field simulation result is quantified by the cross-correlation coefficient, ρ_rh [], of two panels such as in figure 2a,b with the details given below:

$${\rho }_{rh}:=\frac{1/I\sum _{i=1}^{I}r(i)h(i)-[1/I\sum _{i=1}^{I}r(i)][1/I\sum _{i=1}^{I}h(i)]}{\sqrt{\{1/I\sum _{i=1}^I{r}^2(i{)-[1/I\sum _{i=1}^{I}r(i)]}^2\}\{1/I\sum _{i=1}^I{h}^2(i)-{[1/I\sum _{i=1}^{I}h(i)]}^2\}}}.$$ 3.1

This formula allows a quantitative cross-comparison of two identically sized panels. Thus, both fields, r (experiment) and h (simulation), are obtained on rectangular domains (region of interest (ROI)) of the same lattice size, 30 × 20, so that I = 600, shown in figure 2. Since simulation data have much higher resolution, they are down-sampled. This method allows for a comparison of the temperature field over an area that contains many more data points than the number compatible with the common methods that only consider fixed locations or lines within the domain. To compensate for camera shake and body movement caused by the breathing of the subject, alignment is done by having the ROI fixed in the simulation result while shifting the ROI in the experimental data by several pixels vertically and horizontally in each frame until the best fit is found. Since the cut is about 5 cm long and we perform the comparison when the cutting electrode is outside our ROI, we place it on the starting end of the cut. Note that the experimental data at the tail end (r.h.s.) of the cut are not very clean due to bleeding, which could not be captured by our current model. We expect the temperature there to be higher but the experimental data show lower temperatures due to cooling caused by blood from the wound. With the assumption of axial symmetry, we place the ROI in the top left corner to avoid the contaminated region and redundant data points, and data fitting is only done for points inside the ROI. The cross-correlation coefficient calculation is performed at five time instances during t ∈ [3, 5] and then averaged to obtain the ‘fit’ for that cut.

Figure 2.

Comparison of (a) experimental data on in vivo porcine liver and (b) simulation result using our model at t = 5 s.

This process is repeated over arrays of different parameter combinations to minimize the difference between the experimentally measured and computed temperature data to find the optimum parameters. This optimization process is very time-consuming and therefore in the following section, we discuss possible methods for faster parameter estimation. Since the purpose of this section is to demonstrate the value of adding the second-order derivative in time (hyperbolic) compared to the classic heat equation (parabolic), we will show the sensitivity of the solution to changes in thermal conductivity κ and relaxation time τ while keeping all the other parameters constant.

Figure 3 illustrates the variation in computed cross-correlation coefficients when the parameters change. For example, in figure 3a, the fit gets better as the thermal conductivity increases to the optimum around 0.8 W (m K)⁻¹ and then decreases, while the relaxation time is kept at 0, meaning pure Fourier parabolic conduction. Then in figure 3b, the thermal conductivity is kept at the previously obtained optima of 0.8 W (m K)⁻¹ for various relaxation time values, and we observe a similar trend. Putting the two trends together in figure 3c, we observe that even at the optimum conductivity value, there is still room for improvement in the accuracy of the model, even though the sensitivity of the solution to τ is weaker than to κ, as expected.

Figure 3.

Sensitivity of the solution to changes in the parameters (a) thermal conductivity, (b) relaxation time, and (c) normalized increase in accuracy ${\hat{\rho }}_{rh}$ with both parameters on the same axis for comparison.

The computation was done in Matlab and the simulation for each cut takes about 10–30 s, depending on the parameter values. The computation time of existing models for electrosurgery/RF ablation can range from 1 h to 10 h, and some even longer [8,38,39]. These models use the parabolic heat equation which is stiff, therefore requiring more laborious methods such as implicit solvers or tiny time steps for numerical stability. Our model with the second-order term allows fast explicit solvers and larger time steps, while the incorporation of the Arrhenius equation provides a simper way to account for heat associated with reactions than tracking phase fractions. The current model working in series already achieves orders of magnitude reduction in computation time comparing to existing models mentioned earlier in this paragraph; further reduction can be accomplished through parallelization and precomputing of kernels of operators involved for robotic surgery in the future.

Here, we have demonstrated that adding a second-order derivative in time to the heat transfer model could potentially not only dramatically reduce the overall computation time of the multi-physics simulation but also significantly improve the accuracy. While one might argue that adding a higher-order term will increase the accuracy with a negligible improvement [13], we believe we have shown that this accuracy improvement is not as insignificant as one might expect. Additionally, we point out that we have kept the name of the parameter τ unchanged for convenience, although the physical meaning of this parameter in our case is not exactly the same as was originally intended. Since there is a debate about the validity of the Maxwell–Cattaneo model, as discussed in the previous section, one could simply view it as a fitting parameter. Next, we discuss potential ways of determining the value of τ without having to go through the tedious optimization or gradient descent process.

4. Parameter estimation methods

In order to use the model described in the previous section for simulation, one would need to determine the model parameter values. The physical properties such as density and specific heat should not vary too much from one individual to another, so they could be measured once and stored or be simply taken from the literature [40]. The variability in the constants used to approximate the latent heat of evaporation, blood perfusion rate, and size of heat source is limited so they could also come from a lookup table. There are many standard ways of measuring thermal conductivity [41]. The parameter τ, however, is not very easy to measure and it has a different physical meaning from what was originally intended by Maxwell–Cattaneo, or some might argue that it does not have any physical meaning at all. Therefore, these parameters should be estimated on a case-by-case basis according to geometry and excitation. One could imagine that the velocity of the ‘heat wave’ from a cut done at a lower speed might be different from that of a cut at a higher speed. During robot-assisted electrosurgery, it would be very useful if we were able to estimate this parameter and make adjustments as fast as possible. Therefore, in this section, we look at two types of methods that might help us with the estimation.

4.1. Analytical/theoretical method

First, we explore if it is possible to use an analytical method to measure and calculate the parameter τ. Since the equation we are using (2.7), or (2.4) in a simpler form, is hyperbolic, it should result in a wave-like motion. Therefore, the question becomes: Can we calculate the relaxation time from the motion of the ‘heat wave’? When we look at classic literature, Kamiński [25] proposed a method of estimating τ from the penetration time t_p. In his method, one would take temperature measurements at two locations at a distance x_p apart. Then the wave speed would be c = x_p/t_p; therefore the relaxation time could be calculated as

$$\tau =\frac{\alpha }{c^2}=\frac{\alpha t_p^2}{x_p^2}.$$ 4.1

This model has several flaws [15,16]. Experimentally, the measurement of penetration time is very difficult. Since there is noise in the signal, one would need to define a threshold to decide when the temperature started to increase; this threshold should be dependent on the strength of the excitation. In addition, the thermocouples have non-zero thermo-inertia and reaction time. Combined with the invasive nature of inserting the thermocouples, achieving consistent measurements is very challenging. In our opinion, using IR thermography might be able to offset some of these weaknesses, but there is still another problem with this method.

4.1.1. Three-dimensional harmonic wave solution

The simple wave speed calculation of distance over time used by Kamiński might be suitable for the pure wave equation, but it is not exactly correct for the telegraph equation. The solution to the one-dimensional telegraph equation has been derived mathematically in [42], along with the dispersion relation. Accordingly, propagation of a time-harmonic wave along an infinite waveguide should result in a ‘spatially attenuated and temporally periodic (SATP)’ wave solution [42], whose phase velocity is

$$V_p=\frac{\sqrt{2}c}{\sqrt{{(1+(1/{\omega }^2{\tau }^2))}^{1/2}+1}},$$ 4.2

where ω is the frequency of the wave. We expand the relation to three dimensions by exploring spherical symmetry and rewrite the 3D telegraph equation (2.4) in spherical coordinates:

$$\ddot{T}+\frac{1}{\tau }\dot{T}=c^2(T^{″}+\frac{2}{r}{T}^{\prime }),$$ 4.3

where a prime denotes the spatial derivative ∂/∂r. Substituting the SATP solution

$$T(r,t)=R(r)\exp (i\omega t)$$ 4.4

into (4.3) gives

$$R^{″}(r)+\frac{2}{r}{R}^{\prime }(r)+\frac{({\omega }^2\tau -i\omega )}{c^2\tau }R(r)=0.$$ 4.5

There is a wave solution when

$$-\frac{1}{r}\pm \sqrt{\frac{1}{r}-\frac{{\omega }^2\tau -i\omega }{c^2\tau }}$$ 4.6

has a non-zero imaginary part, and the general solution is

$$\begin{array}{r}R(r)=C_1\frac{1}{r}\hspace{0.17em}{\mathrm{e}}^{-r\sqrt{({\omega }^2\tau -i\omega )/c^2\tau }}+C_2\frac{1}{2r}\sqrt{\frac{c^2\tau }{({\omega }^2\tau -i\omega )}}\hspace{0.17em}{\mathrm{e}}^{r\sqrt{({\omega }^2\tau -i\omega )/c^2\tau }}.\end{array}$$ 4.7

Since the temperature far away from the electrode must be T₀, the body temperature, the solution for the temperature field should be bounded at r → ∞. C₂ must be 0 so the second term does not blow up. Thus, the solution can be rewritten as

$$R(r)=A\frac{1}{r}\hspace{0.17em}{\mathrm{e}}^{-\lambda r}=A\hspace{0.17em}{\mathrm{e}}^{(1/r\ln (1/r)-\lambda )r}=A\hspace{0.17em}{\mathrm{e}}^{-\hat{\lambda }r}.$$ 4.8

We realize that this is very similar to the one-dimensional solution from [42] and can therefore be written as [43]

$$\hat{\lambda }=\mu +\frac{i\omega }{V_p},$$ 4.9

where

$$\mu =\frac{\omega }{\sqrt{2}c}{[{(1+\frac{1}{{\omega }^2{\tau }^2})}^{1/2}-1]}^{-1/2}-\frac{1}{r}\ln \frac{1}{r}$$ 4.10a

and

$$V_p=\sqrt{2}c{[{(1+\frac{1}{{\omega }^2{\tau }^2})}^{1/2}+\hspace{0.17em}1]}^{-1/2}.$$ 4.10b

Here, μ is the attenuation and equation (4.10b) is the relation for the 3D phase velocity. Using the relations ω = V_pk and V_g = dω/dk, the group velocity can be determined as

$$V_g=4c^2\frac{k\tau }{\sqrt{4c^2{k}^2{\tau }^2+1}}-8c^4\frac{k^3{\tau }^3}{{(4c^2{k}^2{\tau }^2+1)}^{3/2}}.$$ 4.11

Note that this dispersion relation in 3D turns out to be the same as in 1D, albeit the attenuation in 3D has an extra term in it. We can also see this from equation (4.8). Compared with the 1D solution of X(x) = B e^−λx, the effect of changing from 1D to 3D is an extra 1/r term, which can be viewed as an additional attenuation that is stronger as one gets closer to the source. Also, note that this formulation of the problem in spherical coordinates has a singularity at r = 0. This makes prescribing a boundary condition at r = 0 impossible [44] because a single point value in 3D is not well defined. On the other hand, a heat source that is the size of a single point is also physically impossible. We resolve the problem by prescribing the boundary condition at a small distance away from r = 0, then solve the telegraph equation (4.3) numerically to verify the relation for phase velocity, and check whether the relaxation time τ can be calculated from it.

4.1.2. Numerical verification of the solution

We set the relaxation time, conductivity, density and specific heat all equal to unity. Then, the non-dimensionalized problem to solve is

$$\frac{\mathrm{\partial }T}{\mathrm{\partial }t}+\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{t}^2}=\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{r}^2}+\frac{2}{r}\frac{\mathrm{\partial }T}{\mathrm{\partial }r}$$ 4.12

on the domain [0, 8] discretized into 400 segments with Δr = 0.02. The initial and boundary conditions are

$$T(r,t=0)=0,$$ 4.13a

$$T(r=\mathrm{\Delta }r,t)=\sin (\omega t)$$ 4.13b

$$\mathrm{and}T(r=8,t)=0.$$ 4.13c

We then use a numerical method similar to the one described in [45] with second-order central differencing in space and the classic Runge–Kutta with Δt = 0.006 in time to perform the simulation.

Using a method similar to the one in [25], we measure the times t₁ and t₂ when the ‘heat wave’ reaches the locations r₁ and r₂, so the phase velocity can be simply calculated as V_p = (r₂ − r₁)/(t₂ − t₁). The resulting velocities V_p measured at different frequencies ω from the simulations are displayed in figure 4 and compared to the theoretical phase velocity:

$$V_p=\sqrt{\frac{2}{\sqrt{1+(1/{\omega }^2)}+1}}.$$ 4.14

Figure 4.

Numerical and theoretical calculation results of phase velocity V_p versus frequency ω for 3D telegraph equation (with κ = c = 1).

Figure 4 shows good agreement of the theoretical results with those obtained through simulation. It appears that this method of determination of relaxation time may be plausible. However, the result from the numerical experiment starts to deviate from the theoretical value at ω ≃ 0.5. The reason for this is that at lower frequencies (i.e. larger wavelengths), the magnitude of the waves, which are being used to calculate the phase velocities, is so low, or attenuated, that it is difficult to determine their locations accurately. In fact, for frequencies lower than ω ≃ 0.5, it is almost impossible to obtain a good measurement. The effect of going from 1D to 3D involves an extra attenuation term, as we have seen in equation (4.8). In the 3D case, the attenuation is so strong that even though the computer is able to distinguish the wavefronts, as shown in the previous section in figure 4, it would be nearly impossible to measure in a real-life experiment unless one can generate a setup that provides a 1D-like geometry to avoid the attenuation effect. Also, generating a high-frequency ‘heat wave’ in real life can be very challenging; one would need to have a source that is capable of both heating and cooling, which might require a complicated heating and refrigeration system.

4.2. Data-driven method

The previous subsection shows that estimating the relaxation time τ for robot-assisted surgery using classical analytical methods can be very challenging. To overcome this challenge, the present authors developed in [18] a data-driven machine-learning-type technique, referred to as attention-based noise-robust averaging (ANRA), for computing in real time the model parameters using the temperature data field directly acquired through an IR thermographer. Using ANRA, [17] demonstrated capability of estimating in real time the thermal diffusivity, the relaxation time, and the input power scaling in a Maxwell–Cattaneo model. However, due to image misalignment caused by breathing and uncertainties in the thermographer location, the thermal relaxation time estimate was obtained based on the stationary probe response, rather than a moving one. The noise-robust filtering of the IR thermographer data was used to reduce the image perturbations due to smoke, but the results were still sensitive to ablation-induced smoke perturbations. This technique works well for the relatively low cutting speeds and power values. A technique is therefore desirable that is capable of working at higher probe speeds and power values, and with an active heat source in the frame. In this section, we provide a simple demonstration of such a technique that uses a convolutional neural network (CNN) trained through numerical simulation to generate an estimation of the parameters quickly.

We train a neural network model that can identify these parameters when thermo-images are given. Since our experimental data contain strong noise and the properties of the tissues that were operated on are not clearly labelled, the parameters used in the equations are closer to fitting parameters than physical properties. Therefore, we decided that it is better to use numerical simulation results to train our neural network. We can simulate the solution of (2.7) with a range of α and τ pairs. Since these two parameters are different in magnitude by a factor of 100, we will first normalize them before training and convert them back to the actual value afterward.

We initially perform heat transfer simulations for 50 combinations of α and τ to generate dynamic thermal response data that cover the range of thermal properties of porcine tissues. Then, we build a convolutional neural network using the PyTorch package [46] in Python and train it with our simulation data. Figure 5 shows the layout of our CNN model. The purpose of the neural network is to estimate the coefficient for the second derivative in time in equation (2.7) using the time-series data. We take inspiration from the method in [17], which is similar to an inverse method, and group the thermal images into batches of three, since a second-order central differencing method would require three time steps. The temperature data are first passed through two convolutional and max-pool layers and then flattened. After that, the flattened array passes through three linear dense layers, with 21, 120 and 60 nodes, which turns the flattened data into two final outputs: prediction of α and τ (dense layers are not shown in the figure). The mean squared error is used as the loss function in optimization.

Figure 5.

Convolutional neural network diagram.

The histograms of α and τ parameters obtained from the CNN for electrosurgical cuts performed on the porcine liver are shown in figure 6. We approximate these to be Gaussian distributions and numerically fit the data to obtain the mean and standard deviation. The approximated normal distributions are shown in the insets of the corresponding plots (figure 6). Assuming tissue density of 800 kg m⁻³ and specific heat of 3 kJ (kg K)⁻¹, the mean thermal diffusivity estimated by the CNN corresponds to a thermal conductivity of κ = 0.82, which matches well with the result from the traditional fitting method in §3. We can see that even with the limited amount of experimental data, the neural network is able to provide estimations that could be used to help improve the accuracy of the simulations. With this brief demonstration of the CNN, we can see that this could be a promising method and research direction for the near future as more experimental data become available. Note that even though the data-driven methods are fast to compute, their accuracy can be very sensitive to training data and change in boundary conditions. A change to a curved cut, for example, instead of a straight one, can cause large errors. Therefore, more in-depth studies are needed in this area.

Figure 6.

Distributions of (a) α and (b) τ for porcine liver obtained using neural network.

5. Conclusion

RF electrosurgery is a technique that is widely employed in medical fields but the devices used remain inadequately modelled and less than well understood [47]. As a result, unnecessary thermal damage can occur. As robotic surgery becomes more popular and contemplates the eventual addition of limited device autonomy, understanding and modelling the physics of RF electrosurgery become more important. In this paper, we proposed a model of RF electrosurgery to capture the phenomena at higher temperatures and cutting speeds, and discussed the methods for parameter estimation. The model draws inspiration from Maxwell–Cattaneo non-Fourier type heat transfer and adapts it to electrosurgery on biological tissues, with the approximation of energy dissipation in physical and chemical reactions.

Experimental data taken from our surgical trials performed on in vivo porcine liver are used to support the use of a non-Fourier type model. It is shown that augmenting the Maxwell–Cattaneo heat transfer with an Arrhenius-type model that approximates the energy that goes into the desiccation and denaturation of the tissue, could improve the accuracy of the temperature field representation further, to expand the range of high-accuracy electrosurgical modelling beyond that covered in [17]. With the improvement in numerical stability, thanks to adding a second-order term in time and being able to avoid solving the coupled electrical–thermal problem or tracking volume fractions, the computation time required for the proposed model is much faster than that for the existing ones. We have demonstrated that, despite the debates on the validity of their physical meanings, the Maxwell–Cattaneo type models can be useful for approximating the complex multi-physics phenomena of heat conduction in living tissue under RF electrosurgery. Moreover, incorporating the thermal damage (reaction rate) computation into the heat transfer model reduces the amount of post-processing needed, allows for more accurate tissue property models, and opens the possibility for precise damage control during robot-assisted surgery in the future.

As a next step in our research, we aim to take advantage of modern data-driven methods and machine learning to further improve the accuracy and real-time computational capabilities of the model through parameter adaptation and assist decision-making in robotic surgery. Using the techniques proposed in the present work as well as in [17], we plan to build ionic (Joule) heating models for a range of electrosurgical generator modes.

Ethics

This project has been approved by the Animal Care Committee (ACC). The Committee functions are administrated through the Office of Animal Care and Institutional Biosafety (OACIB) within the Office of the Vice Chancellor for Research at the University of Illinois at Chicago. Protocol no. 22-097 has been carried out in strict adherence to all relevant guidance, regulations and ethical principles governing animal research.

Data accessibility

This article does not contain any additional data.

Declaration of AI use

We have not used AI-assisted technologies in creating this article.

Authors' contributions

J.R.: data curation, investigation, methodology, software, validation, visualization, writing—original draft, writing—review and editing; H.E.-K.: investigation, software; Y.L.: data curation, investigation; L.P.C.: investigation, resources; R.B.: conceptualization, investigation, methodology; G.M.A.C.: investigation, resources; E.B.: methodology, resources; P.C.G.: methodology, resources; R.B.: methodology, resources; J.B.: conceptualization, funding acquisition, methodology, project administration, resources, supervision, writing—review and editing; M.O.-S.: conceptualization, formal analysis, project administration, writing—review and editing.

All authors gave final approval for publication and agreed to be held accountable for the work performed therein.

Conflict of interest declaration

We declare we have no competing interests.

Funding

Experimental data reported in this publication were supported by the National Institute of Biomedical Imaging and Bioengineering of the National Institutes of Health under award no. R01EB029766, as part of the NSF/DHS/DOT/NIH/USDANIFA Cyber-Physical Systems Program. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.