Research Article Analytical Model Based on a Cylindrical Geometry to Study RF Ablation with Needle-Like Internally Cooled Electrode

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1 Mathematical Problems in Engineering Volume, Article ID 83487, 6 pages doi:.55//83487 Research Article Analytical Model Based on a Cylindrical Geometry to Study RF Ablation with Needle-Like Internally Cooled Electrode Juan A. López Molina, María J. Riera, and Enrique Berjano Departamento de Matemática Aplicada, Instituto Uniersitario de Matemática Pura y Aplicada, Uniersitat Politècnica de València, 46 València, Spain Biomedical Synergy, Electronic Engineering Department, Uniersitat Politècnica de València, 46 València, Spain Correspondence should be addressed to Enrique Berjano, eberjano@eln.up.es Receied 4 July ; Accepted 9 September Academic Editor: Kwok W. Wong Copyright q Juan A. López Molina et al. This is an open access article distributed under the Creatie Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, proided the original work is properly cited. Radiofrequency RF ablation with internally cooled needle-like electrodes is widely used in medical techniques such as tumor ablation. The deice consists of a metallic electrode with an internal liquid cooling system that cools the electrode surface. Theoretical modeling is a rapid and inexpensie way of studying different aspects of the RF ablation process by the bioheat equation, and the analytical approach proides an exact solution to the thermal problem. Our aim was to sole 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 3mm from the electrode, which confirms preious experimental findings. We also obsered that the temperature distributions were similar for three coolant temperature alues 5 C, 5 C, and 5 C. The differences were only notable in temperature ery close to the probe. Finally, considering the 5 C line as a thermal lesion mark, we found that lesion diameter was around cm, which is exactly that obsered experimentally in perfused hepatic tissue and slightly smaller than that obsered in nonperfused ex io hepatic tissue.. Introduction Radiofrequency RF ablation with internally cooled needle-like electrodes is widely used for medical techniques such as tumor ablation, treatment of autonomously functioning thyroid nodules, and cardiac ablation to cure arrhythmias 3. The deice consists of an internally liquid cooled metallic electrode that cools the electrode surface see Figure a. Theoretical models proide a fast and inexpensie way of studying different aspects of the

2 Mathematical Problems in Engineering Cooled electrode Cooled electrode Tr,t)= T C r r a b Figure : 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 T C which corresponds with the coolant temperature inside the electrode. Accordingly, the theoretical model has one dimension D, that is, axis r. RF ablation process with the bioheat equation 4. Unlike numerical solutions such as those based on the finite element method, the analytical approach proides an exact solution to the thermal problem in a simplified scenario. In the case of a needle-like electrode, the model is based on an ideal conductor of infinite length totally immersed in homogeneous tissue see Figure b. As far as we know, only Haemmerich et al. 5 hae deeloped an analytical model of RF ablation with cooled-needle cylindrical electrodes, but this model had two important limitations: it only considered the steady-state solution and did not include the blood perfusion term, which is crucial in RF ablation of well-perfused organs such as the lier. A similar study including the blood perfusion term in the steady-state case was deeloped by Yue et al. in 6. The aim of our study was thus to sole analytically the transient time problem of RF ablation with a needle-like internally cooled cylindrical electrode in wellperfused tissue, that is, considering the blood perfusion term, which is essential in the bioheat equation. As far as we are aware, this is the first full analytical solution obtained for this problem.. Goerning Equations Temperature distribution in the tissue during RF ablation is mathematically obtained by soling the Pennes bioheat equation 4 : ηc T t k T S η bc b ω b T T b,.

3 Mathematical Problems in Engineering 3 where η, c, and k are the density, specific heat, and thermal conductiity of the tissue; respectiely, η b, c b,andω b are the density, specific heat, and perfusion coefficient of the blood, T b is the blood temperature, and S represents the heat sources. Assuming all quantities η, η b, c, c b, k, andω b to be constant and that the heat source is independent on the polar angle θ and working in cylindrical coordinates r, θ,. becomes ηc T T r, t k t r r, t ) T r, t S r, t η b c b ω b T r, t T b. r r. The heat source used in the study S was the electrical power density W/m 3 dissipated into the tissue throughout the temporal interal,. Due to the cylindrical geometry of the theoretical model, we employed the same formulation used by Haemmerich et al. 5, and hence we hae ηc T T r, t k t r r, t ) T r, t j r r r σr η bc b ω b T r, t T b,.3 where j is the current density at the conductor surface, σ is the electrical conductiity, and r is the electrode radius. The initial and boundary conditions in the case of an internally cooled electrode are T r, T b, r > r,.4 lim T r, t T b, t >,.5 r T r,t T C, t >,.6 where T C is the temperature fixed by the refrigeration of the electrode. We change to dimensionless ariables ρ r r, ξ αt, V ρ, ξ ) σk T r r j ρ, r ξ ) ) T b,.7 r α where α k/cη which leads us to the problem V ρ ) V V ρ ρ ξ βv ρ,.8

4 4 Mathematical Problems in Engineering where we hae defined β : η b c b ω b r /k with the following initial and boundary conditions: V ρ, ), ρ >,.9 lim ρ V ρ, ξ ), ξ >, ξ>,. V, ξ σk T C T b j r : B,. Dirichlet s boundary condition. 3. Preliminary Results about Bessel s Functions To sole the initial boundary alue problem.8,.9,.,., we shall need to use deep well-known properties of Bessel s and modified Bessel functions of complex argument, which we present now to facilitate reading the paper. First we recall the expression of the modified Bessel functions I z and K z of first and second class and order I z z ) K z I z log m m z m m m!γ m, / /m γ m m! z m, 3. and the expression of the Bessel function of second class and order Y z z ) J z log π ) m / /m γ z m, 3. m m! m γ is the Euler-Mascheroni constant which implies that z ) K z I z log Z z, Y z π 3.3 z ) ) log J z R z, 3.4 where Z z and R z are een holomorphic functions on C. Remark that Z zi R z. 3.5 Moreoer, we shall need the following relations: K z K z, 3.6 z C, I z J zi. 3.7

5 Mathematical Problems in Engineering 5 Finally we recall the asymptotic expansions for z of K z and the modified Bessel functions of first class and integer order I ν z see 7, 7..3, which will be necessary for delicate computations in the following sections: K z I ν z I ν z π z e z O z e z 4ν πz z e z ν / πi πz O e z 4ν πz z e z ν / πi πz O )), if Arg z 3π, 3.8 4ν ) 4ν 3 ) ) ) O 4! z z 3 z )), if π < Arg z < 3π, 3.9 4ν ) 4ν 3 ) ) ) O 4! z z 3 z )), if π < Arg z < 3π, if 3π < Arg z < π, 3. for eery ν N {}. 4. Resolution of the Initial-Boundary Value Problem Taking Laplace s transforms D ρ, s, β : L V ρ, ξ ρ, s, β with respect to ξ and using.9,., and 4., weobtain ρ d D dρ dd ρ dρ ) ρ D s, lim D ρ, s, β ), ρ D, s, β ) B s The homogeneous equation associated to 4. is a modified Bessel equation of order with general solution D ρ, s, β ) ) ) C s I ρ C s K ρ. 4.4 Since the Wronskian determinant of I z and K z is W I z K z /z formula 9 of 3.7 in 7, the method of ariation of parameters gies us C s s K ρ ) ρ, C s s I ρ ) ρ. 4.5

6 6 Mathematical Problems in Engineering So we take C s s C s s ρ ρ ) K d M s, I ) d M s, 4.6 where M s and M s are functions independent on ρ to be chosen in such a way that 4. and 4.3 hold. First, we remark the expected but nontriial fact that ) L : lim I ρ ρ ρ K ) d. 4.7 In fact, by L Hôpital s rule and formula 7 in Section 3.7 of 7, we hae L lim ρ ρ K ρ ) I ρ ) I ρ ). 4.8 Using the asymptotic expansions of I x, I x, andk x gien in Section 7.3 in 7, we obtain K x I lim x πxe x O /x O /x lim x I x x, πxe x e x O /x 4.9 and 4.7 follows easily. Analogously, ) ρ L : lim K ρ ρ I ) e x O /x O /x lim. x x e x O /x d lim ρ I ρ ) K ρ ) ρ K ρ ) 4. As a consequence of 4.7 and 4., we need to choose M s s K ) d. 4.

7 Mathematical Problems in Engineering 7 Then, to satisfy 4.3,weobtain ) B I M s ) ) sk sk K ) d, 4. obtaining finally D ρ, s, β ) ) s I ρ s K ρ ) ρ ρ K ) d I ) d B ) K ρ I s ) β ) sk K ) L [ ] ) f ρ, ξ, ρ L [ ] ) f ρ, ξ, d d ρ BL [ ] ) L [ ] ) f 4 ρ, ξ, f 3 ρ, ξ, d. K ρ ) ) sk d 4.3 To compute the inerse Laplace transform L D ρ, s, we proceed in seeral steps. All the inoled functions hae a branch point in s β, so we will use the Bromwich s contour of Figure where γ denotes a fixed positie real number such that all the used functions are holomorphic on the set Re s γ. We denote for subsequent use D β : C\, β. 4.. Computation of L f ρ, ξ, As ρ in the integral ρ L [ f ] ρ, ξ, ) d, 4.4 we begin computing, for eery ρ, ) F ρ, ξ, : L [ ] ) I f ρ, ξ, : L ρ )K s ) ρ, ξ, ). 4.5

8 8 Mathematical Problems in Engineering β γ Figure : Bromwich s integration contour. As ρ, it follows easily from the asymptotic expansions 3.8 and 3.9 that if s D β and is large enough in order that O /ρ /, O / /, and β/s / and, moreoer, Re s γ and Im s >, we hae the estimation I ρ )K ) 9 e ρ s β e ρ ) s β s 8 ρ s 9 e ρ s β cos Arg s β / e ρ ) s β cos Arg s β / 8 s ρ s β 9 ρ s, 3/ 4.6 because cos Arg / and β/s / /. A similar result holds if Im s < by 3.. Then we can to apply Bromwich s formula to find L f ρ, ξ,. The residue in the pole s is I R s lim se sξ s On the other hand, by 3. we obtain ρ )K ) s ) K β )I ρ β. 4.7 ) I ρ )K ) lim e sξ. s β, s D β s 4.8

9 Mathematical Problems in Engineering 9 Hence, β L [ ] ) f K β )I ρ β πi β e sξ πi s K s βi e sξ ) s K s βi )I ρ s βi ds ) )I ρ s βi ds. 4.9 Haing in mind 3.7 and 3.3, weobtain L [ ] ) f K β )I ρ β β πi β πi πi e yξ ) )) y J ρ y β )J y β iy y β dy πi e yξ ) )) y J ρ y β )J y β iy y β dy ) K β )I ρ β β e yξ ) y J ρ y β )J y β dy, 4. because the imaginary part of log z with z D β and Re z <βand Im z > has limit πi if z approach to the real axis. In definitie, after the change y x, F ρ, ξ, ) L [ f ] ) K β )I ρ β e xξ ) β x J ρ x β )J x β dx Computation of L f ρ, ξ, To find ρ L [ f ] ρ, ξ, ) d, 4. we hae ρ, and hence, as a consequence of the preious result, we obtain ) F ρ, ξ, : L [ ] ) I f ρ,t, : L )K ρ s ) ρ, ξ, ) 4.3 F, ξ, ρ ).

10 Mathematical Problems in Engineering 4.3. Computation of L f 3 ρ, ξ, Now we compute ) F 3 ρ, ξ : L [ ] ) K f 3 ρ, ξ : L ρ ) ) ρ, ξ ). sk 4.4 By 3.8, ifs D β, Re s γ,and is large enough in order that O /ρ / and O / /, we hae K ρ ) ) sk ρ s e ρ s β O /ρ ) O / ) e ρ s β cos Arg s β / 3, ρ s ρ s because cos Arg s β / and so Bromwich s inersion formula cannot be used directly to find F 3 ρ, ξ. To circument this complication we need to proceed in the way ) K F 3 ρ, ξ L ρ ) ) d K ρ ) sk dξ L ), 4.6 s K finding this last inerse with means of Bromwich s contour of Figure since K ρ ) 3 ) s K, ρ s 4.7 if is large enough. First, we remark that it follows easily from 3.3 that K ρ ) lim s β s ). K 4.8

11 Mathematical Problems in Engineering On the other hand, the function e sξ f 3 s /s has a pole of order in s. As the function e sξ K ρ ) ) K 4.9 has continuous partial deriaties of any order in some neighbourhood of s and eery fixed t, we can write d e dξ Res sξ K ρ ) ) s K s d d s e sξ K ρ ) lim dξ s ds ) s K lim s s ξ e sξ K ρ ) K ρ ) β ) ). K K β 4.3 Finally, since R z is an een function, by 3.3, 3.5, and 3.4, we hae lim r,ε πi ρ ) ) ds s K L esξ K ρ ) ) ds s K L esξ K β e sξ I ρ ) s βi log ρ ) ) s β/ π/ i Z ρ ) s βi πi s ) ) ) ds I s βi )log s β/ π/ i R s βi β e sξ I ρ ) s βi log ρ ) ) s β/ π/ i Z ρ ) s βi πi s ) ) ) ds I s βi )log s β/ π/ i R s βi β e sξ J ρ s β )i Y ρ ) s β πi s ) ) ds J s β i Y s β β e sξ J πi s π β ρ s β )i Y ρ ) s β J s β ) i Y s β ) ds e sξ J ρ ) s β )Y s β Y ρ ) s β )J s β s ) ) ds. J s β Y s β 4.3

12 Mathematical Problems in Engineering As a consequence, putting s x, weobtainfrom 4.6 that ) K F 3 ρ, ξ : L ρ ) ) ρ, ξ ) sk π e xξ β x K ρ ) β ) K β J ρ ) x β )Y x β Y J x β ) Y ρ ) x β )J x β ) dx. x β Computation of L f 4 ρ, ξ, To compute L [ f 4 ] ρ, ξ, ) d, 4.33 we hae and so, in order to find L [ ] ) K f 4 ρ, ξ, : L ρ )I )K ) sk ) ρ, ξ, ), 4.34 using the conolution theorem and our preious results, we can write ) F 4 ρ, ξ, : L [ ] ) f 4 ρ, ξ, K L ρ ) ) ρ, ξ, ) [ )] ρ, L ) I )K ξ, sk F 3 ρ, ξ ) d dξ L ) I K s ) ρ, ξ, ) ) d F 3 ρ, ξ dξ F, ξ, F ) ) 3 ρ, ξ e xξ J x β )J x β dx β ξ F ) ) ) 3 ρ, w e x ξ w J x β )J x β dx dw. β 4.35

13 Mathematical Problems in Engineering Complete Solution Collecting our preious results, by the second translation theorem, Fubini s theorem, and some natural computations, we obtain finally V ρ, ξ ) ) K β )I ρ β e xξ ) ) ρ β x J ρ x β )J x β dx d ρ ) K ρ β )I β e xξ ) ) β x J x β )J ρ x β dx d ) BF 3 ρ, ξ ξ F ) 3 ρ, w β ) ) ) e x ξ w J x β )J x β dx dw d, 4.36 that can be rewritten in the following way which is more suitable for numerical computations: V ρ, ξ ) ρ ) ρ K β )I ρ β d K K ) ) ρ β )I β d BF 3 ρ, ξ ) β )I ρ β d e xξ ) ) β x J ρ x β )J x β dx d ξ F ) 3 ρ, w β ) ) ) e x ξ w J x β )J x β dx dw d It is interesting to check that, as expected, actually we hae V, ξ B at eery time ξ>. In fact, as F 3, ξ ifξ>, in the computation of V, ξ, we can use Fubini s theorem and obtain explicitly the integration with respect to w, which is ξ e x ξ w dw x e xξ x Finally, after the ariable change z x β and the application of formula 5 in 7, Section 3.53, we obtain β ) J x β J ) x β dx x J z J z ) zdz I z β )K β, 4.39 β and this gies us V, ξ B.

14 4 Mathematical Problems in Engineering Figure 3: Temperatures at 6 s, 8 s, and 36 s and limit temperature thick line with a temperature of 5 C in the electrode surface and a current intensity 5 ma/mm Figure 4: Temperatures at 6 s, 8 s, and 36 s and limit temperature thick line with a temperature of 5 C in the electrode surface and a current intensity 5 ma/mm. 5. Results and Discussion Once the solution was achieed, we obsered that it was hard to make a direct plot of the temperatures with Mathematica 6. software Wolfram Research Inc., Champaign, IL, USA. The computer took around 4 hours for each plot at a fixed time t. Although we had continuously employed dimensionless ariables in the analytical solution of the problem, as here we considered the case of lier RF ablation, we used the following alues for the hepatic tissue characteristics: density η of 6 kg/m 3, specific heat c of 36 J/kg K, thermal conductiity k of.5 W/m K, and electrical conductiity σ of.33 S/m 8. Since this was a case of well-perfused tissue, we considered the following blood characteristics: density η b of kg/m 3, specific heat c b of 448 J/kg K, and a perfusion rate ω b of s 9. Blood temperature, and hence initial tissue temperature T b, was 37 C. For all the simulations, we used a current density j of alue 5 ma/mm. In order to assess the effect of different coolant temperatures on the temperature profile, we used different boundary temperatures on the electrode surface T C. Figure 3 shows the temperature profiles at different times for a coolant temperature of 5 C. Likewise Figures 4 and 5 show the same plots for coolant temperatures of 5 Cand5 C, respectiely.

15 Mathematical Problems in Engineering Figure 5: Temperatures at 6 s,8 s, and 36 s and limit temperature thick line with a temperature of 5 C in the electrode surface and a current intensity 5 ma/mm. The results showed that the maximal temperature in the tissue is reached 3mmfrom the electrode see Figures 3 5, which confirms preious experimental findings.wealso obsered that the temperature was rising until achieing a steady-state at infinite time thick lineinfigures3 5. We also obsered that the temperature distributions were similar for the three alues of coolant temperature 5 C, 5 C, and 5 C. The differences were only significant at temperatures ery close to the probe. This finding also agrees with preious experimental results in which little difference was obsered in lesion size when coolant temperature was aried 5. In this respect, if we considered the 5 C line as a thermal lesion mark in Figures 3 5, the lesion diameter would be around cm, which is exactly that obsered experimentally in perfused hepatic tissue and slightly smaller than that obsered in nonperfused ex io hepatic tissue 5. Our results were achieed with a current density of 5 ma/mm, and it seems obious that higher current density would cause bigger lesions. Howeer, once the tissue temperature reaches C, the charred tissue around the electrode creates a highly resistie electrical interface, which impedes further power deposition in the tissue 5. Since our analytical solution is not able to model the nonlinear processes inoled in these phenomena, we chose a alue of current density in our simulations to keep the tissue temperature always lower than C. This was also used in the preious analytical model by Haemmerich et al. 5. Temperature dependence of tissue electrical and thermal conductiity was not taken into account. Taking this into account would make the problem nonlinear and most likely impossible to sole analytically. Howeer, in real RF ablation experiments, this effect is present as well as phase transition, charring, and other hard to model effects. Therefore, although comparison of analytical solution to experimental results was faorable, future studies using numerical methods, such as finite element method, should be conducted. Howeer, it is necessary to point out that the solution in 4.37 is exact, while that the solution proided by FEM is an approximation. Since it is known that the density current pattern around a needle-like electrode is highly heterogeneous 8, 9, the alue of 5 ma/mm chosen for our simulations cannot be related with the alues of current usually employed in RF lier ablation - A. Inspiteof this, if we consider a 3 mm long and.75 mm radius electrode, the alue of 5 ma/mm proides a current total of.7 A, which is a bit smaller than the alues experimentally obsered.

16 6 Mathematical Problems in Engineering 6. Conclusion We hae soled 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 preious studies, which suggests the utility of the model and its analytical solution to study the thermal performance of this kind of electrode. Acknowledgments This work receied financial support from the Spanish Plan Nacional de I D i del Ministerio de Ciencia e Innoación Grant no. TEC8-369/TEC. The translation of this paper was funded by the Uniersitat Politècnica de València, Spain. References J. P. McGahan, S. Loh, F. J. Boschini et al., Maximizing parameters for tissue ablation by using an internally cooled electrode, Radiology, ol. 56, no., pp ,. J. H. Baek, W. J. Moon, Y. S. Kim, J. H. Lee, and D. Lee, Radiofrequency ablation for the treatment of autonomously functioning thyroid nodules, World Journal of Surgery, ol. 33, no. 9, pp , 9. 3 A. Thiagalingam, C. R. Campbell, A. C. Boyd, V. E. Eipper, D. L. Ross, and P. Kooor, Cooled intramural needle catheter ablation creates deeper lesions than irrigated tip catheter ablation, Pacing and Clinical Electrophysiology, ol. 7, no. 7, pp , 4. 4 E. J. Berjano, Theoretical modeling for radiofrequency ablation: state-of-the-art and challenges for the future, BioMedical Engineering Online, ol. 5, p. 4, 6. 5 D. Haemmerich, L. Chachati, A. S. Wright, D. M. Mahi, F. T. Lee Jr., and J. G. Webster, Hepatic radiofrequency ablation with internally cooled probes: effect of coolant temperature on lesion size, IEEE Transactions on Biomedical Engineering, ol. 5, no. 4, pp , 3. 6 K. Yue, X. Zhang, and F. Yu, Analytic solution of one-dimensional steady-state Pennes bioheat transfer equation in cylindrical coordinates, Journal of Thermal Science, ol. 3, no. 3, pp , 4. 7 G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library, Cambridge Uniersity Press, Cambridge, UK, I. Chang, Finite element analysis of hepatic radiofrequency ablation probes using temperaturedependent electrical conductiity, BioMedical Engineering Online, ol., p., 3. 9 I. A. Chang and U. D. Nguyen, Thermal modeling of lesion growth with radiofrequency ablation deices, BioMedical Engineering Online, ol. 3, no., p. 7, 4. F. Burdío, E. J. Berjano, A. Naarro et al., RF tumor ablation with internally cooled electrodes and saline infusion: what is the optimal location of the saline infusion? BioMedical Engineering Online, ol. 6, p. 3, 7. M. Ahmed, C. L. Brace, F. T. Lee Jr., and S. N. Goldberg, Principles of and adances in percutaneous ablation, Radiology, ol. 58, no., pp ,.

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