Effect of variable heat transfer coefficient on tissue temperature next to a large vessel during radiofrequency tumor ablation
 Icaro dos Santos^{1}Email author,
 Dieter Haemmerich^{2, 3},
 Cleber da Silva Pinheiro^{1} and
 Adson Ferreira da Rocha^{1}
https://doi.org/10.1186/1475925X721
© dos Santos et al; licensee BioMed Central Ltd. 2008
Received: 26 February 2008
Accepted: 11 July 2008
Published: 11 July 2008
Abstract
Background
One of the current shortcomings of radiofrequency (RF) tumor ablation is its limited performance in regions close to large blood vessels, resulting in high recurrence rates at these locations. Computer models have been used to determine tissue temperatures during tumor ablation procedures. To simulate large vessels, either constant wall temperature or constant convective heat transfer coefficient (h) have been assumed at the vessel surface to simulate convection. However, the actual distribution of the temperature on the vessel wall is nonuniform and timevarying, and this feature makes the convective coefficient variable.
Methods
This paper presents a realistic timevarying model in which h is a function of the temperature distribution at the vessel wall. The finiteelement method (FEM) was employed in order to model RF hepatic ablation. Two geometrical configurations were investigated. The RF electrode was placed at distances of 1 and 5 mm from a large vessel (10 mm diameter).
Results
When the ablation procedure takes longer than 1–2 min, the attained coagulation zone obtained with both timevarying h and constant h does not differ significantly. However, for short duration ablation (5–10 s) and when the electrode is 1 mm away from the vessel, the use of constant h can lead to errors as high as 20% in the estimation of the coagulation zone.
Conclusion
For tumor ablation procedures typically lasting at least 5 min, this study shows that modeling the heat sink effect of large vessels by applying constant h as a boundary condition will yield precise results while reducing computational complexity. However, for other thermal therapies with shorter treatment using a timevarying h may be necessary.
Keywords
Background
Radiofrequency (RF) tumor ablation is a treatment modality that uses radiofrequency electric current in an attempt to destroy cancer cells by localized heating of tumors. RF ablation is used for liver cancer, with increasing use in other organs such as kidney, lung, bone, and adrenal gland [1].
The heat due to RF heating causes tissue necrosis at predictable temperatures in relatively predictable volumes. During an RF ablation procedure, an active electrode is inserted percutaneously (i.e. through a small incision in the skin), during laparoscopy, or during open surgery with imageguidance into the tumors. Grounding pads are positioned at the patient's thighs or back muscles. RF energy is applied and current flows from the active electrode to the grounding pads. Thus, the patient becomes an element of the electrical circuit.
High RF current densities around the electrode results in resistive heating in surrounding tissue. While it takes several hours to induce cell necrosis at 43°C [2], at 50°C cell death occurs within 2–3 min [3]. The tissue adjacent to the electrode is rapidly heated, and the remainder of the tissue is heated by thermal conduction, which is a slow process. Typically, a single ablation takes 12 – 35 min depending on device type [4]. If the tissue temperature near the electrode is excessive, the tissue desiccates, and this process results in an electrically insulating layer preventing further energy deposition. As a result, the phenomenon limits the volume of the tissue that can be treated. A difference between tumor ablation and hyperthermia treatments (where lower temperatures of 43 – 45°C are used) is that during ablation most of the tissue volume that undergoes necrosis is at temperatures between 50 and 100°C. Therefore, the prevalent effect of cell death is coagulation necrosis conversely to hyperthermia treatments where a number of biologically complex mechanisms are in effect [5].
A shortcoming of current RF ablation devices is the limited performance adjacent to large blood vessels (diameter > 3 mm). When the active electrode is inserted near large vessels, the blood flow drags thermal energy away from the target tissue [6]. This is a heat sink effect that can change both the shape and maximum volume that can be treated. In fact, the distance of the blood vessels from the tumor determines the location of the maximal tissue temperature. As a result, tumors in the vicinities of large vessels are associated with high recurrence rates [7]. A considerable number of mathematical models have been suggested to describe heat transfer between tissue and vasculature [8–20] most of them are not directly applicable to liver tissue due to the specific blood supply of the liver. Most of the blood perfusing the liver (roughly 70%) is venous blood [21]; in addition, there are no countercurrent vessels (i.e. venous and arterial vessels adjacent with opposing blood flow) in the liver as present in other tissues, and countercurrent vessels are an integral assumption of most heat transfer models. As a result, most computational models of hightemperature tumor ablation simulate microvascular perfusion using the commonly used Pennes formulation [8], and model large vessels separately since the Pennes model does not describe large vessel perfusion accurately, similar to a previous study [22].
In order to estimate the heat sink effect of large vessels, many simulations performed hitherto assigned either constant temperature at the surface of the vessel [6, 23] or constant convective heat transfer coefficient, h, throughout the RF ablation procedure [24]. In order to estimate the value of h, the simulations performed implied that the flow in the vessel is laminar, the thermal boundary layer is fullydeveloped and h remains constant throughout the procedure. Other works support that hypothesis that the flow is laminar in large vessels [25, 26]. However, since the heated region during the procedure varies in time and the vessel heated length is small, the thermal boundary layer is not fullydeveloped. Thus, both the vessel wall temperature and h varies during the ablation procedure and cannot be considered constant [27]. Since the magnitude of h is a parameter that may significantly impact on the shape and size of the coagulation zone obtained during hepatic RF ablation, it is important to accurately model the timevarying behavior of h during the RF ablation of liver tumors in order to correctly determine the coagulation zone size. Thus, the overall objective of this work is to theoretically evaluate the impact of the timevarying h on the accuracy of the simulations of RF tumor ablation procedures.
Methods and models
where ρ, c and k are, respectively, the density (kg*m^{3}), the specific heat (J*kg^{1}*K^{1}) and the thermal conductivity (W*m^{1} K^{1}) of the liver tissue. $\overrightarrow{J}$(A*m^{2}) and $\overrightarrow{E}$(V*m^{1}) are current density and the electric field intensity and can be calculated with Laplace's equation, using a quasistatic approximation. T is the temperature of the tissue, T _{ bl }is the temperature of the blood, ρ _{ bl }is the blood density, c _{ bl }is specific heat of the blood and w _{ bl }is blood perfusion (s^{1}). Q _{ m }(W*m^{3}) is the energy generated by metabolic processes and was neglected since it is small if compared to the other terms [28]. Microvascular perfusion was included in the model, considering the Pennes model [8]. Although the perfusion in normal and tumorous tissue can vary greatly, this paper focus on the temperature field close to large vessels and on the behavior of the timevarying h during RF ablation rather than simulating a clinical situation. The parameter w _{ bl }used in this model was 6.4.10^{3} s^{1} [29], which is in the range of the perfusion for cirrhotic human liver tissue [30]. Equation 1 states that RF current flowing through the tissue is converted into thermal energy, which in turn cause tissue injury.
Thermal and electric properties of the materials.
Finite element region  Material  ρ(kg*m^{3})  c(J*kg^{1}*K^{1})  k(W*m^{1}*K^{1})  σ(S*m^{1}) 

Electrode  NickelTitanium  6450  840  18  1.10^{8} 
Trocar  Stainless steel  21500  132  71  4.10^{6} 
Insulated trocar  Polyurethane  70  1045  0.026  1.10^{5} 
Tissue  Blood  1000  4180  0.543  0.667 
Tissue  Liver  1060  3600  0.512  0.333 
(4)
where T _{ t }is the electrode tip temperature, T _{ s }is the set temperature measured at the tip of the electrode where maximum temperatures occur, and K _{ p }and K _{ i }are control parameters of the PI controller. The control parameters determine the behavior of the algorithm, e.g., response time, overshoot, swinging. The parameters we used for the PI controller were K _{ p }= 0:4 and K _{ i }= 0:04. These values were obtained based on tests performed with the dynamic system. In order to implement the time integral of the temperature difference between the electrode tip and the set temperature (equations 3 and 4), we added an ordinary differential equation in Femlab.
In this study, the maximum hepatic tissue temperature was kept at 90°C and the maximum power applied during the procedure was 28 W. We used the 50°Cisotherm to determine the coagulation zone boundary as has been done in previous models of RF tumor ablation [6, 24, 28, 37]. We used the temperature at the tip of the electrode for convergence test. We refined the mesh and run the analysis again. We searched the optimal mesh size in order to keep the simulation fast while keeping the accuracy better than 0.01°C at the tip of the electrode. We tested 2 mesh sizes: 65,759 and 13,714. We concluded that the 13,714element model satisfied our requirement. The time step size was chosen at 0.05 s, so that the maximum temperature change was smaller than 0.01°C during each step. The simulations were performed on a PC with a 2.4 GHz PENTIUM Celeron CPU, with 1 GB of RAM and 30 GB of hard disk space. For post processing, we employed the builtin module in FEMLAB and MATLAB. The computation time required for 10 min simulation was about 3 hours.
We placed the electrode in two different configurations: (a) 1 mm away from the vessel (shown in Figure 1) and (b) 5 mm away from the vessel. For each of these two configurations, we performed three studies.

The behavior of the convective heat coefficient on the vessel;

The evolution of the coagulation zone volume when the timevarying behavior of h is taken into account and when h is constant and equal to the value of the timevarying h at the end of the 10 minutes ablation;

The behavior of the maximum temperature at the vessel wall with both constant and timevarying h.
Results
The behavior of the convective heat coefficient at the vessel wall
The evolution of the coagulation zone volume for varying h and for constant h cases
The behavior of the maximum temperature at the vessel wall for varying h and for constant h
Discussion
The aim of this study was to evaluate the impact of the timevarying behavior of the convective heat transfer coefficient h for the determination of the coagulation zone during RF tumor ablation procedures. We have simulated two different behaviors of the convective heat transfer coefficient: constant and variable. We also studied two configurations with the electrode either 1 mm, or 5 mm distant from a large vessel 10 mm in diameter [38]. When the electrode is 1 mm away from the vessel, h initially rapidly increases up to 6000 W*m^{2} *K^{1} followed by a sharp decrease to approximately 554 W*m^{2} *K^{1} within 60 s (Figure 3). When the electrode is 5 mm away from the vessel, the convective coefficient rapidly increases to approximately 7000 W*m^{2} *K^{1} and then rapidly decreases to 336 W*m^{2} *K^{1} (Figure 4). This behavior occurs because the thermal energy reaches the vessel rapidly due to close proximity to the electrode. Thus, h initially increases due to the increase in temperature on the vessel wall, and after a while the temperature increase extends along the vessel and the convective heat transfer coefficient decreases. Hence, the maximum value of h for the first case occurs around 0.2 s, and for the second case it occurs around 0.5 s after start of the ablation. This behavior becomes more obvious by observing Equation 2. Initially, the vessel wall is not heated yet, and h is 0 W*m^{2} *K^{1} because there is no heat transfer to the vessel. Then the temperature at the vessel starts to increase and also the heated length grows larger typically up to 11 cm. Initially the heated length is very small compared to the temperature difference T _{1}  T _{0}, thus h sharply increases. After a few seconds, the length starts to increase faster than the difference (T _{1}  T _{0}) and h starts to decrease until both (T _{1}  T _{0}) and h show little change because equilibrium between heat loss and heating is reached. Recall that the thermal boundary layer is very small at the entrance region. Thus, h is very high [39]. It is noteworthy that if one wants to measure the dynamic behavior of the heat convection coefficient, the instrumentation must have a fast dynamic response and a very high static range as one can see from figures 3 and 4. We also analyzed the coagulation zone for the two cases of timevarying h and constant h. In the first few seconds and when the electrode is close to the vessel, the timevarying h has considerably higher values than the constant h and the relative error in coagulation zone volume due to the assumption of a constant h is very high (22%). For RF liver ablation this may not be important since initially the ablation volume is small (with a small absolute error), and the volume near the beginning of the treatment is not of high clinical value. As the time increases, the timevarying h sharply decreases, approaching its final value after 80 s for the 1 mm case and after 120 s for 5 mm case. Thus, the final coagulation zone volume is nearly identical in both cases. We also investigated the maximum temperature at the wall of the vessel, where tumor recurrence is possible due to insufficient temperatures. The maximum error in temperature is around 6% for the first 50–100 s, but then again is close to zero at the end of the ablation. Consequently, for tumor ablation procedures typically lasting at least 5 min, a very important result is the fact that modeling the heat sink effect of large vessels by applying constant h as a boundary condition will yield accurate results for RF ablation while reducing computational complexity. For other thermal therapies with shorter treatment times (e.g. 45 – 60 s long cardiac ablation close to coronary vessels) using a timevarying h might be required but needs to be further investigated.
Conclusion
Previous studies considered a constant heat transfer coefficient throughout the ablation procedure. In this work, simulations were performed using a more realistic, timevarying analytical expression of the convective heat transfer coefficient, which depends on the blood velocity and on the temperature distribution on the vessel wall. The simulations showed that the assumption of a constant convective coefficient leads to precise results when it is used for typical ablation procedures. Only during the first 1–2 min, a timevarying coefficient produces noticeable different results. However, this has no clinical impact for RF liver ablation procedure, which typically takes over 5 min.
Declarations
Acknowledgements
We would like to thank the University of Brasilia, CNPq, CAPES and FAPDF for providing support for this research. Part of this work was conducted in a facility constructed with support from the National Institutes of Health, Grant Number C06 RR018823 from the Extramural Research Facilities Program of the National Center for Research Resources.
Authors’ Affiliations
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