Pennes Bioheat Model Research Articles

Fractional diffusion equations interpolate between damping and waves

Abstract The behaviour of the solutions of the time-fractional diffusion equation,
based on the Caputo derivative, is studied and its dependence on the fractional
exponent is analysed. The time-fractional convection-diffusion equation is also solved
and an application to Pennes bioheat model is presented. Generically, a wave-like
transport at short times passes over to a diffusion-like behaviour at later times.

Effect of high blood flow on heat distribution and ablation zone during microwave ablation-numerical approach.

Microwave ablation has become a viable alternative for cancer treatment for patients who cannot undergo surgery. During this procedure, a single-slot coaxial antenna is employed to effectively deliver microwave energy to the targeted tissue. The success of the treatment was measured by the amount of ablation zone created during the ablation procedure. The significantly large blood vessel placed near the antenna causes heat dissipation by convection around the blood vessel. The heat sink effect could result in insufficient ablation, raising the risk of local tumor recurrence. In this study, we investigated the heat loss due to large blood vessels and the relationship between blood velocity and temperature distribution. The hepatic artery, with a diameter of 4 mm and a height of 50 mm and two branches, is considered in the computational domain. The temperature profile, localized tissue contraction, and ablation zones were simulated for initial blood velocities 0.05, 0.1, and 0.16 m/s using the 3D Pennes bio-heat equation, temperature-time dependent model, and cell death model, respectively. Temperature-dependent blood velocity is modeled using the Navier-Stokes equation, and the fluid-solid interaction boundary is treated as a convective boundary. For discretization, we utilized elements for the wave propagation model, elements for the Pennes bio-heat model, and elements for the Navier-Stokes equation, where represents the computational domain. The simulated results show that blood vessels and blood velocity have a significant impact on temperature distribution, tissue contraction, and the volume of the ablation zone.

Incorporating vascular-stasis based blood perfusion to evaluate the thermal signatures of cell-death using modified Arrhenius equation with regeneration of living tissues during nanoparticle-assisted thermal therapy

Cellular and biological tissue heating may result in reversible (or repairable injury) and irreversible (or lethal) thermal cell-death in living biological tissues. Continuous regeneration of living human tissues due to the continuous supply of oxygen through arterial blood must be taken into account to counter balance the thermal degradation at quasi-static thermal conditions. This study incorporates vascular-stasis based non-linear blood perfusion for magnetic nanoparticle assisted thermal therapy to model the thermal by-stander effect - a hyperthermia-induced deep infiltration of nanoparticles in the targeted tissue domain. Pennes bioheat model based on Fourier heat conduction theory is four-way coupled with Arrhenius and non-Arrhenius kinetic models of cell-death with healthy cells regeneration. In determination of treatment endpoint, the kinetic model must be coupled with quantitative and qualitative pathological biomarkers of thermal damage. Nanoparticle distribution volume increases by 39.62% after possible rupturing of cell membrane during heating. The release of intracellular solution by dead cells during heating promotes nanoparticle migration from the region of higher concentration to the regions of lower concentration thereby 80% enhancement in interstitial space and five-fold increase in diffusion coefficient. For such redistribution phenomenon, the heating time is sufficient to reduce the oxygen in erythrocytes (red blood cells) and maximize the necrosis zone inside tumour. However, at the interface, the regeneration of healthy cells triggers an immune response of biological tissue towards continued heating to suppress, prevent and restrict further accumulation of thermal damage within damage bounds of Ω ≤ 1. While modelling the kinetics of thermal damage of tumour, one must include and should not ignore the partial self-regeneration of connecting normal human tissues at the tumour periphery due to continuous matching of oxygen demands in the healthy tissue by the arterial blood.

RF heating measurement using MR thermometry and field monitoring: Methodological considerations and first in vivo results.

A MR thermometry (MRT) method with field monitoring is proposed to improve the measurement of small temperature variations induced in brain MRI exams. MR thermometry experiments were performed at 7 Tesla with concurrent field monitoring and RF heating. Images were reconstructed with nominal k-space trajectories and with first-order spherical harmonics correction. Experiments were performed in vitro with deliberate field disturbances and on an anesthetized macaque in 2 different specific absorption rate regimes, that is, at 50% and 100% of the maximal specific absorption rate level allowed in the International Electrotechnical Commission normal mode of operation. Repeatability was assessed by running a second separate session on the same animal. Inclusion of magnetic field fluctuations in the reconstruction improved temperature measurement accuracy in vitro down to 0.02°C. Measurement precision in vivo was on the order of 0.15°C in areas little affected by motion. In the same region, temperature increase reached 0.5 to 0.8°C after 20 min of heating at 100% specific absorption rates and followed a rough factor of 2 with the 50% specific absorption rate scans. A horizontal temperature plateau, as predicted by Pennes bioheat model with thermal constants from the literature and constant blood temperature assumption, was not observed. Inclusion of field fluctuations in image reconstruction was beneficial for the measurement of small temperature rises encountered in standard brain exams. More work is needed to correct for motion-induced field disturbances to extract reliable temperature maps.

Computational study on constant and sinusoidal heating of skin tissue using radial basis functions

This paper concerns the study of heat distribution in skin tissue with constant and sinusoidal heat flux at the skin surface. Three types of heat transfer models, i.e., Pennes bioheat model, single-phase lag model, and dual-phase lag model based on Fourier and non-Fourier heat conduction, are taken into consideration. Also, Fourier, as well as non-Fourier boundary conditions, are considered at both the end of skin tissue for the single and dual-phase lag model of heat conduction. The resulting models are solved using finite difference and radial basis function (RBF) approximations for the temporal and spatial variables, respectively. The effects of these three models and other parameters involved in models on heat transfer in skin tissue are studied. The effects of Fourier and non-Fourier boundary conditions on heat transfer for single and dual-phase lag models are also studied.

Thermomechanical analysis of a triple layered skin structure in presence of nanoparticle embedding multi-level blood vessels

Concerning a target-specific malignant cell damage sparing healthy tissue, the advancements in laser and nanotechnology are one of the recent challenges in non-invasive cancer treatment processes. In addition, proper assessment of nociceptive pain induced in the ablation process by simulating the tissue thermomechanical response in a more realistic situation is a useful step prior to real clinical practice. In the present study a nanoparticle infused lesion in a 3-D triple-layered healthy skin structure embedding thermally significant multi-level small arteries and veins is chosen as the computational domain. Dual-phase-lag (DPL) and Pennes bioheat model coupled with equilibrium equations are solved using COMSOL Multiphysics (Bangalore, India) to compute the thermal and mechanical fields in the tissue. An in-vitro temperature measurement of laser irradiated agar-based tissue phantom is also performed to validate the predicted temperature. The aim is to develop a comparative analysis between Intravenous (IV) and Intratumoral (IT) infusion schemes of nanoparticles in terms of precise and uniform necrosis achieved along with stress, strain, and deformation induced in tissue during laser therapy. Numerical results show that the IT scheme offers a useful outcome in terms of precise tumor ablation with minimum thermally induced skin deformation in comparison to the IV scheme. Additionally, the DPL model predicts a more realistic temperature profile along with the induced tissue mechanical response in contrast to the Pennes bioheat model. Most importantly, the small veins have comparatively lower cooling effects than the small arteries in the composite skin structure. Nevertheless, the present study can improve the pre-operational computational modeling of the tissue damage and the nociceptive pain involved during laser therapy prior to clinical application.

Exact Analytical Formulation of Three-Dimensional Pennes Bioheat Model in Regional Hyperthermia with Modified Initial Condition

The present research work is intended to develop an exact analytical solution of three-dimensional Pennes bioheat model subjected to regional hyperthermia. Existing literature survey indicates that a large number of research papers have been reported in one-dimensional and two-dimensional approaches, but researchers have rarely focused on three-dimensional modelling apart from a few. In practical sense, energy always propagates in multi-dimensional manner; hence, three-dimensional analysis would be considered as better approach in comparison with existing one and two-dimensional investigation particularly for living tissues. Living tissues are highly susceptible to temperature variation. As suggested by the treatment protocol of thermal therapy, regional hyperthermia occurs in large-sized organs (prostrate, breast, lung, etc.) which are deep seated inside the human body and in such cases three-dimensional analysis is essential. To solve three-dimensional Pennes bioheat model, a hybrid analytical scheme comprising of ‘shift of variables’ and ‘finite Fourier transform’ has been employed in the present research paper. The prime novelty of this research work is implementation of spatially dependent initial condition as highly non-homogeneous and non-uniform anatomical structures are observed in living tissues, whereas existing research work has been carried out on the constant temperature initial condition. The boundary conditions are imposed in the present modelling in relation with the realistic approach subjected to thermal therapies. The research output has been validated and justified with the published numerical research work, and it indicates the maximum temperature deviation of 0.215%.

Abstract 691: Simulating TTFields-induced temperature changes within a patient’s brain - a proof of concept study

Abstract Introduction: TTFields is an antimitotic cancer treatment that utilizes low intensity (1-3 V/cm) alternating electric fields in the intermediate frequency (100-300 kHz) that are delivered in two orthogonal directions using 2 pairs of transducer arrays. TTFields are currently approved for Glioblastoma Multiforme (GBM). Preclinical studies show that the effect of TTFields is intensity-dependent with a therapeutic threshold of 1 V/cm, and that as the intensity rises so does the efficacy. However, delivery of high field intensities could result in temperature increases within the body that lead to thermal damage to tissue. In practice, during treatment with TTFields, tissue heating is limited by a closed loop that adjusts the delivery of TTFields to ensure that skin temperature remains within the safety limits. Since heating limits the intensity at which TTFields is delivered, accurate models of tissue heating during TTFields therapy could provide insights into new approaches for the delivery of TTFields. Since measuring the temperature spatial distribution within a patient’s brain is extremely challenging, simulating the heat flow within the head during treatment is a more practical approach. Here, we present preliminary results from a study aimed at developing a valid computational model of heat transfer in the head during TTFields therapy. Method: A realistic computational head model of a healthy male was used in this study. Virtual transducer arrays were placed on the head. To simulate the delivery of TTFields to the brain, we used ZMT's Sim4Life v4.0 electro-quasi-static solver. The results of the electric field simulation were then used as heat source for the heat simulation. The heat simulation solves the Pennes bio-heat model, taking into account tissue heat capacity and conductivity, and also accounting for heat convection through blood perfusion. Sensitivity analysis was performed to establish the sensitivity of the temperature distribution within the brain to the thermal properties of the various tissue types. Results: Average temperatures within the brain tissue were well below 38° C for all simulations. Temperature distributions within the brain were highly sensitive to the perfusion coefficient assigned to the cerebrospinal fluid as well as other tissues.Conclusions: We have established a computational framework for investigating heat transfer in the brain during TTFields therapy. Further work is needed to correctly establish the thermal properties of the various tissue types. This may require experimental measurements in which the heat transfer model is fine-tuned and validated. Citation Format: Ariel Naveh, Ze'ev Bomzon. Simulating TTFields-induced temperature changes within a patient’s brain - a proof of concept study [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2019; 2019 Mar 29-Apr 3; Atlanta, GA. Philadelphia (PA): AACR; Cancer Res 2019;79(13 Suppl):Abstract nr 691.

Numerical analysis of an enhanced cooling rate cryopreservation process in a biological tissue

Cryopreservation is the method of preservation of biological tissues for future references without causing significant damages to their physical and functional properties. This can be done by exposing them to very low cryogenic temperature that involves a greater heat removal rate. A two dimensional numerical model is developed to study the temperature distribution, cooling rate attained and movement of the freezing front during the cryopreservation process. The Pennes Bio-heat model is used for current study. The Finite Volume Method is employed for discretization of the governing differential equations while the Tri-Diagonal Matrix Algorithm is used to solve the discretized algebraic equations to find temperature distribution inside the domain. The Enthalpy-Porosity method is used to track the solid-liquid interfaces during the freezing process. The current model is first validated with the result of the existing literature. In the present work, freezing of tissue is done from one and two sides in two separate cases and the resulting temperature distribution inside the tissue and cooling rate in the two cases are compared. It is found that the freezing rate of tissue is enhanced about two times when it is freezed from two sides as compared to freezing from single side. Further, it is observed that a lower value of blood perfusion rate causes a lower value of the final temperature of the tissue after freezing. Thus, it can be concluded that the tissue with high blood perfusion rate is to be freezed in a lower cooling medium temperature. In the present condition, metabolic heat generation plays no significant role in the temperature distribution inside the healthy tissue.

A computational theoretical model for radiofrequency ablation of tumor with complex vascularization

Radiofrequency ablation (RFA) for liver tumors is a minimally invasive procedure that uses electrical energy and heat to destroy cancer cells. One of the critical factors that impedes its successful outcome is the thermal heat sink effects from complex vascular systems that give rise to incomplete destruction of the target tumor tissue, resulting in therapy failure. To better understand the thermal influence of the complex vascular system during RFA, this work proposes the employment of two 3D fractal tree-like branched networks to investigate which key factors of the tree-like vascular system impact heating process. A three-dimensional finite difference analysis is employed to simulate the RFA treatment. Based on the data acquired from the measured experiments, the simulated results derived from combining the Pennes bioheat model and the boundary condition-enforced immersed boundary method (IBM) have demonstrated close agreement with experimental data with a maximum discrepancy of ±8.3%. We employed the orthogonal design approach to analyze 3 factors, namely, the blood vessel's volume, the average distance between probe center and the blood vessel system and the number of the selected part's branches at three different levels. Results have revealed that the distance between RFA probe and blood vessel plays a major role during the heating process compared with the other two factors. In addition, both the ablating rates and the volume of damaged tissue are slightly reduced with increasing number of blood vessel branches.

Dual phase lag bio-heat transfer during cryosurgery of lung cancer: Comparison of three heat transfer models

The Pennes bio-heat model is based on Fourier's law of heat conduction, which assumed that a thermal signal propagate with infinite speed. This gives contradiction in physical situation. Also, the hyperbolic bio-heat model considers the micro scale response in time, but it does not explain the micro scale response in space. Therefore, to consider the thermal behaviour which is not captured by the Fourier's law and to take into account the microstructural effect in space, a dual phase lag (DPL) bio-heat conduction model would be advantageous. In this paper, a two dimensional DPL model is proposed to study the phase change heat transfer process during cryosurgery of lung cancer. The governing equations are solved numerically by enthalpy based finite difference method. The non-ideal behaviour of tissue and heat source terms, metabolism and blood perfusion are also considered. This study is made to examine the effects of phase lags in heat flux and temperature gradient on interface positions and temperature distribution during freezing process. A comparative study of DPL, parabolic and hyperbolic conduction models is thoroughly investigated. It is found that the phase lags of temperature gradient and heat flux have significant effect on interface positions and temperature distribution.

Incorporating an immersed boundary method to study thermal effects of vascular systems during tissue cryo-freezing

In this paper, the three-dimensional thermal effects of a clinically-extracted vascular tissue undergoing cryo-freezing are numerically investigated. Based on the measured experimental temperature field, the numerical results of the Pennes bioheat model combined with the boundary condition-enforced immersed boundary method (IBM) agreed well with experimental data with a maximum temperature discrepancy of 2.9°C. For simulating the temperature profile of a tumor sited in a dominantly vascularized tissue, our model is able to capture with ease the thermal effects at specified junctions of the blood vessels. The vascular complexity and the ice-ball shape irregularity which cannot be easily quantified via clinical experiments are also analyzed and compared for both two-dimensional and three-dimensional settings with different vessel configurations and developments. For the three-dimensional numerical simulations, a n-furcated liver vessels model from a three-dimensional segmented volume using hole-making and subdivision methods is applied. A specific study revealed that the structure and complexity of the vascular network can markedly affect the tissue's freezing configuration with increasing ice-ball irregularity for greater blood vessel complexity.

Characterization of fiber-optic light delivery and light-induced temperature changes in a rodent brain for precise optogenetic neuromodulation.

Understanding light intensity and temperature increase is of considerable importance in designing or performing in vivo optogenetic experiments. Our study describes the optimal light power at target depth in the rodent brain that would maximize activation of light-gated ion channels while minimizing temperature increase. Monte Carlo (MC) simulations of light delivery were used to provide a guideline for suitable light power at a target depth. In addition, MC simulations with the Pennes bio-heat model using data obtained from measurements with a temperature-measuring cannula having 12.3 mV/°C of thermoelectric sensitivity enabled us to predict tissue heating of 0.116 °C/mW on average at target depth of 563 μm and specifically, a maximum mean plateau temperature increase of 0.25 °C/mW at 100 μm depth for 473 nm light. Our study will help to improve the design and performance of optogenetic experiments while avoiding potential over- and under-illumination.

Numerical modeling and optimization of the cryosurgery operations

Numerical computation of the three dimensional problem of the freezing interface propagation during the cryosurgical impact coupled with the multi-objective optimization mechanism is used in order to improve the efficiency and safety of the cryosurgery operations performing. The heat transfer in soft tissue during the thermal exposure to low temperature is described by the Pennes bioheat model. The finite volume method combined with the control volume approximation of the heat fluxes is applied for the cryosurgery numerical modeling on the tumor tissue of a quite arbitrary shape. The flux relaxation approach is used for the stability improvement of the explicit finite difference schemes. The generalized method of effective traversals is proposed for the searching of the Pareto front segments as the multi-objective optimization problem solutions.

Thermal Analysis of the Increasing Subcutaneous Fat Thickness Within the Human Skin—A Numerical Study

This article reports a numerical study on the thermal response of skin with increasing fat thickness. The study considers the Pennes bioheat model for skin to simulate the thermal recovery phase after the removal of undercooled condition. Based on the surface thermal maps of a three-dimensional skin, the change in fat thickness within the skin is characterized. Sensitivity study reveals that the major variation in the thermal pattern at the skin surface is mainly due to the change in fat thickness. Possible random noise associated with background disturbances is also considered to determine the associated signal-to-noise ratio.

Real-time tumor ablation simulation based on the dynamic mode decomposition method.

The dynamic mode decomposition (DMD) method is used to provide a reliable forecasting of tumor ablation treatment simulation in real time, which is quite needed in medical practice. To achieve this, an extended Pennes bioheat model must be employed, taking into account both the water evaporation phenomenon and the tissue damage during tumor ablation. A meshless point collocation solver is used for the numerical solution of the governing equations. The results obtained are used by the DMD method for forecasting the numerical solution faster than the meshless solver. The procedure is first validated against analytical and numerical predictions for simple problems. The DMD method is then applied to three-dimensional simulations that involve modeling of tumor ablation and account for metabolic heat generation, blood perfusion, and heat ablation using realistic values for the various parameters. The present method offers very fast numerical solution to bioheat transfer, which is of clinical significance in medical practice. It also sidesteps the mathematical treatment of boundaries between tumor and healthy tissue, which is usually a tedious procedure with some inevitable degree of approximation. The DMD method provides excellent predictions of the temperature profile in tumors and in the healthy parts of the tissue, for linear and nonlinear thermal properties of the tissue. The low computational cost renders the use of DMD suitable for in situ real time tumor ablation simulations without sacrificing accuracy. In such a way, the tumor ablation treatment planning is feasible using just a personal computer thanks to the simplicity of the numerical procedure used. The geometrical data can be provided directly by medical image modalities used in everyday practice.

Generalised polynomial chaos-based uncertainty quantification for planning MRgLITT procedures

Purpose: A generalised polynomial chaos (gPC) method is used to incorporate constitutive parameter uncertainties within the Pennes representation of bioheat transfer phenomena. The stochastic temperature predictions of the mathematical model are critically evaluated against MR thermometry data for planning MR-guided laser-induced thermal therapies (MRgLITT). Methods: The Pennes bioheat transfer model coupled with a diffusion theory approximation of laser tissue interaction was implemented as the underlying deterministic kernel. A probabilistic sensitivity study was used to identify parameters that provide the most variance in temperature output. Confidence intervals of the temperature predictions are compared to MR temperature imaging (MRTI) obtained during phantom and in vivo canine (n = 4) MRgLITT experiments. The gPC predictions were quantitatively compared to MRTI data using probabilistic linear and temporal profiles as well as 2-D 60 °C isotherms. Results: Optical parameters provided the highest variance in the model output (peak standard deviation: anisotropy 3.51 °C, absorption 2.94 °C, scattering 1.84 °C, conductivity 1.43 °C, and perfusion 0.94 °C). Further, within the statistical sense considered, a non-linear model of the temperature and damage-dependent perfusion, absorption, and scattering is captured within the confidence intervals of the linear gPC method. Multivariate stochastic model predictions using parameters with the dominant sensitivities show good agreement with experimental MRTI data. Conclusions: Given parameter uncertainties and mathematical modelling approximations of the Pennes bioheat model, the statistical framework demonstrates conservative estimates of the therapeutic heating and has potential for use as a computational prediction tool for thermal therapy planning.

Kalman Filtered MR Temperature Imaging for Laser Induced Thermal Therapies

The feasibility of using a stochastic form of Pennes bioheat model within a 3-D finite element based Kalman filter (KF) algorithm is critically evaluated for the ability to provide temperature field estimates in the event of magnetic resonance temperature imaging (MRTI) data loss during laser induced thermal therapy (LITT). The ability to recover missing MRTI data was analyzed by systematically removing spatiotemporal information from a clinical MR-guided LITT procedure in human brain and comparing predictions in these regions to the original measurements. Performance was quantitatively evaluated in terms of a dimensionless L(2) (RMS) norm of the temperature error weighted by acquisition uncertainty. During periods of no data corruption, observed error histories demonstrate that the Kalman algorithm does not alter the high quality temperature measurement provided by MR thermal imaging. The KF-MRTI implementation considered is seen to predict the bioheat transfer with RMS error < 4 for a short period of time, ∆t < 10 s, until the data corruption subsides. In its present form, the KF-MRTI method currently fails to compensate for consecutive for consecutive time periods of data loss ∆t > 10 sec.

Calculation of SAR and temperature rise in a high-resolution vascularized model of the human eye and orbit when exposed to a dipole antenna at 900, 1500 and 1800 MHz

The eye is considered to be a critical organ when determining safety standards for radiofrequency radiation. With a detailed anatomy of the human eye and orbit inserted in a whole-head model, the specific absorption rates (SARs) and thermal effects were determined under exposure to a dipole antenna representing a mobile phone operating at 900, 1500 and 1800 MHz with an output power of 1 W. The temperature rise was calculated by taking the blood flow into account either by the Pennes bioheat model or by including the discrete vasculature (DIVA). In addition, a simple spherical model using constant heat transfer coefficients was used. Peak SARs in the humour are 4.5, 7.7 and 8.4 W kg−1 for 900, 1500 and 1800 MHz respectively. Averaged over the whole eyeball, the SARs are 1.7, 2.5 and 2.2 W kg−1. The maximum temperature rises in the eye due to the exposure are 0.22, 0.27 and 0.25 °C for exposure of 900, 1500 and 1800 MHz, respectively, calculated with DIVA. For the Pennes bioheat model, the temperature rises are slightly lower: 0.19, 0.24, 0.22 °C respectively. For the simple spherical model, the maximum temperature rises are 0.15, 0.22 and 0.20 °C. The peak temperature is located in the anterior part of the lens for 900 MHz and deeper in the eye for higher frequencies, and in the posterior part of the lens for 1500 MHz and close to the centre of the eyeball for 1800 MHz. For these RF safety applications, both DIVA and the Pennes bioheat model could be used to relate the SAR distributions to the resulting temperature distributions. Even though, for these artificial exposure conditions, the SAR values are not in compliance with safety guidelines, the maximum temperature rises in the eye are too small to give harmful effects. The temperature in the eye also remains below body core temperature.

Modelling the impact of blood flow on the temperature distribution in the human eye and the orbit: fixed heat transfer coefficients versus the Pennes bioheat model versus discrete blood vessels

Prediction of the temperature distribution in the eye depends on how the impact of the blood flow is taken into account. Three methods will be compared: a simplified eye anatomy that applies a single heat transfer coefficient to describe all heat transport mechanisms between the sclera and the body core, a detailed eye anatomy in which the blood flow is accounted for either by the bioheat approach, or by including the discrete vasculature in the eye and the orbit. The comparison is done both for rabbit and human anatomies, normo-thermally and when exposed to homogeneous power densities. The first simplified model predicts much higher temperatures than the latter two. It was shown that the eye is very hard to heat when taking physiological perfusion correctly into account. It was concluded that the heat transfer coefficient describing the heat transport from the sclera to the body core reported in the literature for the first simplified model is too low. The bioheat approach is appropriate for a first-order approximation of the temperature distribution in the eye when exposed to a homogeneous power density, but the discrete vasculature down to 0.2 mm in diameter needs to be taken into account when the heterogeneity of the temperature distribution at a mm scale is of interest.