Macarena Trujillo

Modeling the heating of biological tissue based on the hyperbolic heat transfer equation

In modern surgery, a multitude of minimally intrusive operational techniques are used which are based on the point heating of target zones of human tissue via laser or radiofrequency currents. Traditionally, these processes are modeled by the bioheat equation introduced by Pennes, who considers Fouriers theory of heat conduction. We present an alternative and more realistic model established using the hyperbolic equation of heat transfer. To demonstrate some features and advantages of our proposed method, we apply the results obtained to different types of tissue heating with high energy fluxes, in particular radiofrequency heating and pulsed laser treatment of the cornea to correct refractive errors. We hope that the results from our approach will help with refining surgical interventions in this novel field of medical treatment.

Review of the mathematical functions used to model the temperature dependence of electrical and thermal conductivities of biological tissue in radiofrequency ablation

Abstract Purpose: Although theoretical modelling is widely used to study different aspects of radiofrequency ablation (RFA), its utility is directly related to its realism. An important factor in this realism is the use of mathematical functions to model the temperature dependence of thermal (k) and electrical (σ) conductivities of tissue. Our aim was to review the piecewise mathematical functions most commonly used for modelling the temperature dependence of k and σ in RFA computational modelling. Materials and methods: We built a hepatic RFA theoretical model of a cooled electrode and compared lesion dimensions and impedance evolution with combinations of mathematical functions proposed in previous studies. We employed the thermal damage contour D63 to compute the lesion dimension contour, which corresponds to Ω = 1, Ω being local thermal damage assessed by the Arrhenius damage model. Results: The results were very similar in all cases in terms of impedance evolution and lesion size after 6 min of ablation. Although the relative differences between cases in terms of time to first roll-off (abrupt increase in impedance) were as much as 12%, the maximum relative differences in terms of the short lesion (transverse) diameter were below 3.5%. Conclusions: The findings suggest that the different methods of modelling temperature dependence of k and σ reported in the literature do not significantly affect the computed lesion diameter.

Effect of the thermal wave in radiofrequency ablation modeling: an analytical study

To date, all radiofrequency heating (RFH) theoretical models have employed Fouriers heat transfer equation (FHTE), which assumes infinite thermal energy propagation speed. Although this equation is probably suitable for modeling most RFH techniques, it may not be so for surgical procedures in which very short heating times are employed. In such cases, a non-Fourier model should be considered by using the hyperbolic heat transfer equation (HHTE). Our aim was to compare the temperature profiles obtained from the FHTE and HHTE for RFH modeling. We built a one-dimensional theoretical model based on a spherical electrode totally embedded and in close contact with biological tissue of infinite dimensions. We solved the electrical-thermal coupled problem analytically by including the power source in both equations. A comparison of the analytical solutions from the HHTE and FHTE showed that (1) for short times and locations close to the electrode surface, the HHTE produced temperatures higher than the FHTE, however, this trend became negligible for longer times, when both equations produced similar temperature profiles (HHTE always being higher than FHTE); (2) for points distant from the electrode surface and for very short times, the HHTE temperature was lower than the FHTE, however, after a delay time, this tendency inverted and the HHTE temperature increased to the maximum; (3) from a mathematical point of view, the HHTE solution showed cuspidal-type singularities, which were materialized as a temperature peak traveling through the medium at a finite speed. This peak rose at the electrode surface, and clearly reflected the wave nature of the thermal problem; (4) the differences between the FHTE and HHTE temperature profiles were smaller for the lower values of thermal relaxation time and locations further from the electrode surface.

Relationship between roll-off occurrence and spatial distribution of dehydrated tissue during RF ablation with cooled electrodes.

Purpose: To study the relationship between roll-off (sudden increase in impedance) and spatial distribution of dehydrated tissue during RF ablation using a cooled electrode (temperatures around 100°C). Methods: We used a double approach: (1) theoretical modelling based on the finite element method, and (2) 20 ablations using an experimental study on ex vivo excised bovine liver in which we measured impedance progress and temperature at three points close to the electrode surface: 0.5 (T1), 1.5 (T2) and 2.5 (T3) mm from the tip. T2 was located exactly at the centre of the 30 mm long electrode. Results: Temperatures at T1 and T3 quickly rose to 100°C (at ≈20 and 40 s, respectively), while at the rise at T2 was somewhat slower, stabilized around 50 s and reached a maximum value of 99°C at about 60 s. Impedance reached a minimum of 65 Ω (plateau), began increasing at 50 s and continued rising throughout the procedure, reaching a value equal to the initial value at 70 s. Likewise, computed impedance dropped to ≈73 Ω (plateau), began increasing at 50 s and reached an impedance value equal to the initial value at ≈78 s, which approximately coincided with the time when the entire zone surrounding the electrode was within the 100°C isotherm. Conclusion: There is a close relationship between the moment at which roll-off occurs and the time when the entire electrode is completely encircled by the dehydrated tissue. The mid-electrode zone is the last in which tissue desiccation occurs.

Assessment of Hyperbolic Heat Transfer Equation in Theoretical Modeling for Radiofrequency Heating Techniques

Theoretical modeling is a technique widely used to study the electrical-thermal performance of different surgical procedures based on tissue heating by use of radiofrequency (RF) currents. Most models employ a parabolic heat transfer equation (PHTE) based on Fourier’s theory, which assumes an infinite propagation speed of thermal energy. We recently proposed a one-dimensional model in which the electrical-thermal coupled problem was analytically solved by using a hyperbolic heat transfer equation (HHTE), i.e. by considering a non zero thermal relaxation time. In this study, we particularized this solution to three typical examples of RF heating of biological tissues: heating of the cornea for refractive surgery, cardiac ablation for eliminating arrhythmias, and hepatic ablation for destroying tumors. A comparison was made of the PHTE and HHTE solutions. The differences between their temperature profiles were found to be higher for lower times and shorter distances from the electrode surface. Our results therefore suggest that HHTE should be considered for RF heating of the cornea (which requires very small electrodes and a heating time of 0.6 s), and for rapid ablations in cardiac tissue (less than 30 s).

Thermal modeling for pulsed radiofrequency ablation: Analytical study based on hyperbolic heat conduction

The objectives of this study were to model the temperature progress of a pulsed radiofrequency (RF) power during RF heating of biological tissue, and to employ the hyperbolic heat transfer equation (HHTE), which takes the thermal wave behavior into account, and compare the results to those obtained using the heat transfer equation based on Fourier theory (FHTE). A theoretical model was built based on an active spherical electrode completely embedded in the biological tissue, after which HHTE and FHTE were analytically solved. We found three typical waveforms for the temperature progress depending on the relations between the dimensionless duration of the RF pulse delta(a) and the expression square root of lambda(rho-1), with lambda as the dimensionless thermal relaxation time of the tissue and rho as the dimensionless position. In the case of a unique RF pulse, the temperature at any location was the result of the overlapping of two different heat sources delayed for a duration delta(a) (each heat source being produced by a RF pulse of limitless duration). The most remarkable feature in the HHTE analytical solution was the presence of temperature peaks traveling through the medium at a finite speed. These peaks not only occurred during the RF power switch-on period but also during switch off. Finally, a physical explanation for these temperature peaks is proposed based on the interaction of forward and reverse thermal waves. All-purpose analytical solutions for FHTE and HHTE were obtained during pulsed RF heating of biological tissues, which could be used for any value of pulsing frequency and duty cycle.

Analytical thermal–optic model for laser heating of biological tissue using the hyperbolic heat transfer equation

In this paper, we solve in an analytical way the thermal-optic coupled problem associated with a 1D model of non-perfused homogeneous biological tissue irradiated by a laser beam. We consider a laser pulse duration of 200 micros and study the temperatures of areas very close to the point of laser beam application. We consider that these values of the temporal and spatial variables mean that the problem has to be solved by means of the hyperbolic heat conduction model instead of the classic or parabolic model. We therefore obtain the solution using both models and apply the temperature profiles obtained to a specific biological tissue for comparison. Finally, we theoretically study the effect of the thermal relaxation time on the temperature profiles in the tissue for both heating and cooling phases (i.e. during and after laser application).

Radiofrequency cardiac ablation with catheters placed on opposing sides of the ventricular wall: computer modelling comparing bipolar and unipolar modes.

Abstract Purpose: The aim of this study was to compare the efficacy of bipolar (BM) vs. unipolar (UM) mode of radiofrequency ablation (RFA) in terms of creating transmural lesions across the interventricular septum (IVS) and ventricular free wall (VFW). Materials and methods: We built computational models to study the temperature distributions and lesion dimensions created by BM and UM on IVS and VFW during RFA. Two different UM types were considered: sequential (SeUM) and simultaneous (SiUM). The effect of ventricular wall thickness, catheter misalignment, epicardial fat, and presence of air in the epicardial space were also studied. Results: Regarding IVS ablation, BM created transmural and symmetrical lesions for wall thicknesses up to 15 mm. SeUM and SiUM were not able to create transmural lesions with IVS thicknesses ≥12.5 and 15 mm, respectively. Lesions were asymmetrical only with SeUM. For VFW ablation, BM also created transmural lesions for wall thicknesses up to 15 mm. However, with SeUM and SiUM transmurality was obtained for VFW thicknesses ≤7.5 and 12.5 mm, respectively. With the three modes, VFW lesions were always asymmetrical. In the scenario with air or a fat tissue layer on the epicardial side, only SiUM was capable of creating transmural lesions. Overall, BM was superior to UM in IVS and VFW ablation when the catheters were not aligned. Conclusions: Our findings suggest that BM is more effective than UM in achieving transmurality across both ventricular sites, except in the situation of the epicardial catheter tip surrounded by air or placed over a fat tissue layer.

Feasibility study of an internally cooled bipolar applicator for RF coagulation of hepatic tissue: Experimental and computational study

Purpose: To study the capacity of an internally cooled radiofrequency (RF) bipolar applicator to create sufficiently deep thermal lesions in hepatic tissue. Materials and methods: Three complementary methodologies were employed to check the electrical and thermal behaviour of the applicator under test. The experimental studies were based on excised bovine (ex vivo study) and porcine liver (in vivo study) and the theoretical models were solved by means of the finite element method (FEM). Results: Experimental and computational results showed good agreement in terms of impedance progress and lesion depth (4 and 4.5 mm respectively for ex vivo conditions, and ≈7 and 9 mm respectively for in vivo conditions), although the lesion widths were overestimated by the computer simulations. This could have been due to the method used to assess the thermal lesions; the experimental lesions were assessed by the white coagulation zone, whereas the tissue damage function was used to assess the computational lesions. Conclusions: The experimental results suggest that this applicator could create in vivo lesions to a depth of around 7 mm. It was also observed that the thermal lesion is mainly confined to the area between both electrodes, which would allow lesion width to be controlled by selecting a specific applicator design. The comparison between the experimental and computational results suggests that the theoretical model could be usefully applied in further studies of the performance of this device.

Irreversible electroporation of the liver: is there a safe limit to the ablation volume?

Irreversible electroporation is a fast-growing liver ablation technique. Although safety has been well documented in small ablations, our aim is to assess its safety and feasibility when a large portion of liver is ablated. Eighty-seven mice were subjected to high voltage pulses directly delivered across parallel plate electrodes comprising around 40% of mouse liver. One group consisted in 55 athymic-nude, in which a tumor from the KM12C cell line was grown and the other thirty-two C57-Bl6 non-tumoral mice. Both groups were subsequently divided into subsets according to the delivered field strength (1000 V/cm, 2000 V/cm) and whether or not they received anti-hyperkalemia therapy. Early mortality (less than 24 hours post-IRE) in the 2000 V/cm group was observed and revealed considerably higher mean potassium levels. In contrast, the animals subjected to a 2000 V/cm field treated with the anti-hyperkalemia therapy had higher survival rates (OR = 0.1, 95%CI = 0.02–0.32, p < 0.001). Early mortality also depended on the electric field magnitude of the IRE protocol, as mice given 1000 V/cm survived longer than those given 2000 V/cm (OR = 4.7, 95%CI = 1.8–11.8, p = 0.001). Our findings suggest that ionic disturbances, mainly due to potassium alterations, should be warned and envisioned when large volume ablations are performed by IRE.