We present an analytical solution to the electrothermal mathematical model of radiofrequency ablation of biological tissue using a cooled cylindrical electrode. The solution presented here makes use of the method of separation of variables to solve the problem. Green’s functions are used for the handling of nonhomogeneous terms, such as effect of electrical currents circulation and the nonhomogeneous boundary condition due to cooling at the electrode surface. The transcendental equation for determination of eigenvalues of this problem is solved using Newton’s method, and the integrals that appear in the solution of the problem are obtained by Simpson’s rule. The solution obtained here has the possibility of handling different functional dependencies of the source term and nonhomogeneous boundary condition. The solution provides a tool to understand the physics of the problem, as it shows how the solution depends on different parameters, to provide mathematical tools for the design of surgical procedures and to validate other modeling techniques, such as the numerical methods that are frequently used to solve the problem.
Radiofrequency (RF) ablation is a surgical technique considered as marginally invasive, used, for instance, to destroy malignant tissue in bland organs, such as liver, kidney, and lungs. The objective of this treatment is to heat the tumor to a temperature above a threshold where the tissue is damaged. Initially, this procedure was limited to small volumes of tissue damage. Rossi et al. [
Much of the studies that have been performed in relation to RF current ablation have relied on experimental techniques and numerical methods. Very few analytical solutions have been reported in relation to this phenomenon. Haemmerich et al. [
Figure
(a) Physical situation modeled: an RF applicator is inserted in the tissue and the active electrode is placed in the center of a spherical tumor. (b) Analytical model has one dimension (
Consider then the case of a fragment of tissue in the shape of a concentric annulus with inner radius
Properties of liver tumor tissue used in this study [
Property  Nomenclature  Magnitude  Units 

Tissue density 

1000  kg 
Tissue thermal conductivity 

0.512  W 
Tissue specific heat 

4200  J 
Blood density 

1000  Kg 
Blood perfusion 

0.0015 

Blood specific heat 

3600  J 
Tissue electrical conductivity 

0.188  S 
Metabolic heat generation 

700  W 
The governing equation for this problem is Pennes’ Bioheat equation [
The calculation of
The heat generation per unit volume from RF currents circulation,
Considering that in this problem the voltage is only a function of the radius and
With these equation and set of boundary conditions, the solution can be stated as
The electrical potential
And the heat generation per unit volume is determined from
Considering a temperature of tissue
Using a change of variable
This problem is nonhomogeneous in the equation itself, because of term
To complete the solution, suitable Green’s function must be obtained. The solution of an associated homogeneous equation and boundary conditions provides a way to identify Green’s function for this problem. We take the auxilliary equation
The method of separation of variables can be used to solve (
By defining
Equation (
The equation for
Writing again
Applying the initial condition expressed in (
The solution to the problem in terms of Green’s function can be stated as
A comparison of terms, after replacing
Once Green’s function for the problem is obtained, the solution to (
Since, in this case
The final form of the solution is
Equation (
First 20 eigenvalues of transcendental equation (
Eigenvalue number 


1  27.7412491590629 
2  59.7215157304870 
3  91.6538083942138 
4  123.5335630443243 
5  155.3754428957278 
6  187.1895966699011 
7  218.9827041355604 
8  250.7593126035128 
9  282.5226419505083 
10  314.2750529359010 
11  346.0183280786956 
12  377.7538469185017 
13  409.4826995869618 
14  441.2057629309977 
15  472.9237530699340 
16  504.6372626347768 
17  536.3467877607103 
18  568.0527480482156 
19  599.7555015887882 
20  631.4553564558279 
The integrals that appear in the solution are evaluated using Simpson’s rule, using a fine mesh of 397 divisions of the region
In order to validate the analytical solution presented in the previous section, a comparison of that solution as
Comparison of solution obtained from (
Temperature distribution of tissue subject to different conditions of blood perfusion during RF ablation for (a)
The solution was also validated by comparing it to the solution obtained by López Molina et al. [
Figure
Figure
Temperature distribution of tissue subject to different conditions of blood perfusion during RF ablation for
The solution provided by (
The first derivative of (
The previous equation defines the maximum temperature within the tissue. It can be solved to determine the critical radius,
Considering the case
Figure
Voltage needed to produce rolloff and volume of tissue damage at that time for different blood perfusion rates.
Figure
Figure
Voltage needed to produce rolloff and volume of tissue damage at that time for different electrode temperatures.
One advantage of the method of solution presented here is the generality of the method in terms of the type of source term
Consider the case where the tissue electrical conductivity is not uniform but is layered in two regions:
Expression (
Figure
Temperature distribution of tissue with two different layers subject to RF ablation. (a)
In a similar way, the solution expressed by (
The present model is limited to constant properties of tissue and for that reason does not consider temperature dependent properties and does not work for the region where the tissue vaporizes. Our previous experience in computer modeling in relation to the temperature dependence of thermal conductivity is that its effect is really negligible before tissue vaporization [
The 1D model used here does not model the particularities of the electrode tip and connection with the plastic catheter; tissue at those zones absorbs higher RF power (edge effect) and hence is desiccated during the first seconds. In contrast, in the 1D model, the RF power is equally distributed along all the tissue next to the electrode surface. For this reason the thermal dynamics of a 1D model is different to that of 2D/3D models. The utility of the 1D model proposed here is to have a fast and easytouse tool to study the effect of the design parameters involved in RF ablation with internally cooled needlelike electrodes (see Section
Results from Figure
However, computer results from the 1D model are quantitatively in agreement with experimental findings. For instance, Figure
The model obtained here is 1D, while most of the experimental and numerical results available are for 2D/3D cases. The plane at the midheight of the active part of the electrode represents the only part of the electrode that behaves quasi one dimensionally, because of symmetry. It has been observed [
Similarly, Figure
Figure
This study presents an analytical solution for the RF ablation of tissue from a cooled cylindrical electrode. The solution matches well with other available analytical solutions and with approximate numerical solutions to the problem. Unlike other available time dependent analytical solutions, this is for a finite spatial domain and is more general, because of the possibility of handling different functional dependencies of the source term and nonhomogeneous boundary condition. Application of boundary condition at the far field,
The authors declare that they have no competing interests.
This work was supported by the Universidad Autónoma de San Luis Potosí (México), which granted Ricardo RomeroMéndez a sabbatical leave to do research in the field of biomedical engineering, and the Government of Spain through the “Plan Estatal de Investigación, Desarrollo e Innovación Orientada a los Retos de la Sociedad” (Grants TEC 201452383C3R and TEC 201452383C31R).