María J. Rivera

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.

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.

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).

Computer modelling of an impedance-controlled pulsing protocol for RF tumour ablation with a cooled electrode

Abstract Purpose: To develop computer models to mimic the impedance-controlled pulsing protocol implemented in radiofrequency (RF) generators used for clinical practice of radiofrequency ablation (RFA), and to assess the appropriateness of the models by comparing the computer results with those obtained in previous experimental studies. Methods: A 12-min RFA was modelled using a cooled electrode (17G, 3 cm tip) inserted in hepatic tissue. The short (transverse) diameter of the coagulation zone was assessed under in vivo (with blood perfusion (BP) and considering clamping) and ex vivo (at 21 °C) conditions. The computer results obtained by programming voltage pulses were compared with current pulses. Results: The differences between voltage and current pulses were noticeable: using current instead of voltage allows larger coagulation zones to be created, due to the higher energy applied by current pulses. If voltage pulses are employed the model can accurately predict the number of roll-offs, although the waveform of the applied power is clearly not realistic. If current voltages are employed, the applied power waveform matches well with those reported experimentally, but there are significantly fewer roll-offs. Our computer results were overall into the ranges of experimental ones. Conclusions: The proposed models reproduce reasonably well the electrical-thermal performance and coagulation zone size obtained during an impedance-controlled pulsing protocol.

Fourier, hyperbolic and relativistic heat transfer equations: a comparative analytical study

Parabolic heat equation based on Fouriers theory (FHE), and hyperbolic heat equation (HHE), has been used to mathematically model the temperature distributions of biological tissue during thermal ablation. However, both equations have certain theoretical limitations. The FHE assumes an infinite thermal energy propagation speed, whereas the HHE might possibly be in breach of the second law of thermodynamics. The relativistic heat equation (RHE) is a hyperbolic-like equation, whose theoretical model is based on the theory of relativity and which was designed to overcome these theoretical impediments. In this study, the three heat equations for modelling of thermal ablation of biological tissues (FHE, HHE and RHE) were solved analytically and the temperature distributions compared. We found that RHE temperature values were always lower than those of the FHE, while the HHE values were higher than the FHE, except for the early stages of heating and at points away from the electrode. Although both HHE and RHE are mathematically hyperbolic, peaks were only found in the HHE temperature profiles. The three solutions converged for infinite time or infinite distance from the electrode. The percentage differences between the FHE and the other equations were larger for higher values of thermal relaxation time in HHE.

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

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 ≈3 mm from the electrode, which confirms previous experimental findings. We also observed that the temperature distributions were similar for three coolant temperature values (5, 15 and 25). The differences were only notable in temperature very close to the probe. Finally, considering the 50 line as a thermal lesion mark, we found that lesion diameter was around 2 cm, which is exactly that observed experimentally in perfused hepatic tissue and slightly smaller than that observed in nonperfused (ex vivo) hepatic tissue.

Analytical validation of COMSOL Multiphysics for theoretical models of Radiofrequency ablation including the Hyperbolic Bioheat transfer equation

In this paper we outline our main findings about the differences between the use of the Bioheat Equation and the Hyperbolic Bioheat Equation in theoretical models for Radiofrequency (RF) ablation. At the moment, we have been working on the analytical approach to solve both equations, but more recently, we have considered numerical models based on the Finite Element Method (FEM). As a first step to use FEM, we conducted a comparative study between the temperature profiles obtained from the analytical solutions and those obtained from FEM. Regarding the differences between both methods, we obtain agreement in less than 5% of relative differences. Then FEM is a good alternative to model heating of biological tissues using BE and HBE in, for example, more complex and realistic geometries.

Operator ideals and tensor norms defined by a sequence space

We study the tensor norm defined by a sequence space λ and its minimal and maximal operator ideals associated in the sense of Defant and Floret. Our results extend the classical theory related to the tensor norms of Saphar [ 16 ]. They show the key role played by the finite dimensional structure of the ultrapowers of λ in this kind of problems.