^{1}

^{1}

^{2}

^{1}

^{2}

Radiofrequency (RF) ablation with internally cooled needle-like electrodes is widely used in medical techniques such as tumor ablation. The device consists of a metallic electrode with an internal liquid cooling system that cools the electrode surface. Theoretical modeling is a rapid and inexpensive way of studying different aspects of the RF ablation process by the bioheat equation, and the analytical approach provides an exact solution to the thermal problem.
Our aim was to solve analytically the RF ablation transient time problem with a needle-like internally cooled cylindrical electrode while considering the blood perfusion term. The results showed that the maximal tissue temperature is reached

Radiofrequency (RF) ablation with internally cooled needle-like electrodes is widely used for medical techniques such as tumor ablation [

(a) Cluster of three internally cooled needle-like electrodes used to ablate biological tissues by means of RF currents. (b) Analytical model representing a simplified scenario of an ideal conductor with infinite length totally immersed in homogeneous tissue. The internal cooling was modeled by means of a Dirichlet’s thermal boundary condition with a constant temperature

Temperature distribution in the tissue during RF ablation is mathematically obtained by solving the Pennes bioheat equation [

The heat source used in the study (^{3}) dissipated into the tissue throughout the temporal interval

The initial and boundary conditions in the case of an internally cooled electrode are

We change to dimensionless variables

To solve the initial boundary value problem (

Finally we recall the asymptotic expansions for

Taking Laplace’s transforms

So we take

First, we remark the expected but nontrivial fact that

As a consequence of (

To compute the inverse Laplace transform

Bromwich’s integration contour.

As

As

The residue in the pole

To find

Now we compute

First, we remark that it follows easily from (

Finally, since

As a consequence, putting

To compute

Collecting our previous results, by the second translation theorem, Fubini’s theorem, and some natural computations, we obtain finally

It is interesting to check that, as expected, actually we have

Once the solution was achieved, we observed that it was hard to make a direct plot of the temperatures with Mathematica 6.0 software (Wolfram Research Inc., Champaign, IL, USA). The computer took around 24 hours for each plot at a fixed time

Although we had continuously employed dimensionless variables in the analytical solution of the problem, as here we considered the case of liver RF ablation, we used the following values for the hepatic tissue characteristics: density ^{3}, specific heat ^{3}, specific heat ^{2}.

In order to assess the effect of different coolant temperatures on the temperature profile, we used different boundary temperatures on the electrode surface

Temperatures at 60 s, 180 s, and 360 s and limit temperature (thick line) with a temperature of 5°C in the electrode surface and a current intensity 5 mA/mm^{2}.

Temperatures at 60 s, 180 s, and 360 s and limit temperature (thick line) with a temperature of 15°C in the electrode surface and a current intensity 5 mA/mm^{2}.

Temperatures at 60 s,180 s, and 360 s and limit temperature (thick line) with a temperature of 25°C in the electrode surface and a current intensity 5 mA/mm^{2}.

The results showed that the maximal temperature in the tissue is reached

We also observed that the temperature distributions were similar for the three values of coolant temperature (5°C, 15°C, and 25°C). The differences were only significant at temperatures very close to the probe. This finding also agrees with previous experimental results in which little difference was observed in lesion size when coolant temperature was varied [

Our results were achieved with a current density of 5 mA/mm^{2}, and it seems obvious that higher current density would cause bigger lesions. However, once the tissue temperature reaches 100°C, the charred tissue around the electrode creates a highly resistive electrical interface, which impedes further power deposition in the tissue [

Since it is known that the density current pattern around a needle-like electrode is highly heterogeneous [^{2} chosen for our simulations cannot be related with the values of current usually employed in RF liver ablation (1-2 A). In spite of this, if we consider a 30 mm long and 0.75 mm radius electrode, the value of 5 mA/mm^{2} provides a current total of 0.7 A, which is a bit smaller than the values experimentally observed.

We have solved analytically the transient time problem of RF ablation with a needle-like internally cooled electrode by using the bioheat equation (i.e., considering the blood perfusion term). The temperature distributions computed from the theoretical model matched the experimental results obtained in previous studies, which suggests the utility of the model and its analytical solution to study the thermal performance of this kind of electrode.

This work received financial support from the Spanish “Plan Nacional de I + D + i del Ministerio de Ciencia e Innovación” Grant no. TEC2008-01369/TEC. The translation of this paper was funded by the Universitat Politècnica de València, Spain.