Pennes Bioheat Equation Research Articles

Analytical solution of the bioheat equation for thermal response induced by any electrode array in anisotropic tissues with arbitrary shapes containing multiple-tumor nodules

The Pennes bioheat transfer equation is the most used model to calculate the temperature induced in a tumor when physical therapies like electrochemical treatment, electrochemotherapy and/or radiofrequency are applied. In this work, a modification of the Pennes bioheat equation to study the temperature distribution induced by any electrode array in an anisotropic tissue containing several nodules (primary or metastatic) with arbitrary shape is proposed. For this, the Green functions approach is generalized to include boundaries among two or more media. The analytical solution we obtain in a very compact way, under quite general suppositions, allows calculating the temperature distributions in the tumor volumes and their surfaces, in terms of heat sources, initial temperature and calorific sources at the boundary of tumors.

Non-Fourier transient thermal analysis of biological tissue phantoms subjected to high intensity focused ultrasound

The present study is concerned with the numerical analysis of the non-Fourier heat transfer phenomena in biological tissue samples subjected to high intensity focused ultrasound (HIFU). Due to the inherent non-homogeneous nature of biological samples, the well-known Pennes bioheat equation (PBHTE) is not strictly suitable for correctly predicting the temperature distribution in the tissue medium subjected to HIFU. In this respect, thermal wave model of bioheat transfer (TWMBT), which takes into account the finite propagation speed of thermal front, plays an important role in describing the observed wave-like behavior of heat transfer in biological tissue medium. The temperature profiles obtained from TWMBT show the presence of oscillations during the process of diffusion of thermal wave after reaching its maximum peak, depending upon the corresponding relaxation time. Results show that the thermal dosage with finite value of relaxation time gives lesser heat-affected area compared to the dosage calculated by considering the infinite propagation of thermal front as in the case of the conventional Fourier model. These observations make the present study important as otherwise, the target area is prone to remain under insufficiently heated conditions.

Towards Reinforced Brain Tumor Segmentation on MRI Images Based on Temperature Changes on Pathologic Area

Brain tumor segmentation is the process of separating the tumor from normal brain tissues; in clinical routine, it provides useful information for diagnosis and treatment planning. However, it is still a challenging task due to the irregular form and confusing boundaries of tumors. Tumor cells thermally represent a heat source; their temperature is high compared to normal brain cells. The main aim of the present paper is to demonstrate that thermal information of brain tumors can be used to reduce false positive and false negative results of segmentation performed in MRI images. Pennes bioheat equation was solved numerically using the finite difference method to simulate the temperature distribution in the brain; Gaussian noises of ±2% were added to the simulated temperatures. Canny edge detector was used to detect tumor contours from the calculated thermal map, as the calculated temperature showed a large gradient in tumor contours. The proposed method is compared to Chan–Vese based level set segmentation method applied to T1 contrast-enhanced and Flair MRI images of brains containing tumors with ground truth. The method is tested in four different phantom patients by considering different tumor volumes and locations and 50 synthetic patients taken from BRATS 2012 and BRATS 2013. The obtained results in all patients showed significant improvement using the proposed method compared to segmentation by level set method with an average of 0.8% of the tumor area and 2.48% of healthy tissue was differentiated using thermal images only. We conclude that tumor contours delineation based on tumor temperature changes can be exploited to reinforce and enhance segmentation algorithms in MRI diagnostic.

Theoretical investigation of thermal wave model of microwave ablation applied in prostate Cancer therapy

This paper presents an hyperbolic Pennes bioheat equation in cylindrical coordinate for modeling the Microwave Ablation (MWA) applied in prostate cancer. Due to recent reports, the number of patients of prostate cancer is growing by 15 million in the world each year. Since, it is shown that application of uniform microwave in prostate area of different patients may produce different temperature, the Pennes bioheat equation is considered to study the effect of perfusion term on produced temperature profiles. The solution method is Eigen value method which results in a closed form solution. The hyperbolic behavior of temperature profiles under high heat release against Fourier model is shown. Results show the importance of tissue perfusion term in estimation of temperature profiles and establish that the thermal tissue damage is expected to initiate from 1 to 3 mm above the catheter surface and to promote up to 7 mm. The solutions can be applied as a verification branch for other numerical works and can be very useful to reduce uncertainty about MWA treatments and improve the reliability of clinical protocols giving insight to the Surgeons.

Relationship between metabolic rate and blood perfusion under Fanger thermal comfort conditions

The one-dimensional steady Pennes (bioheat) equation was applied to analyze heat conduction inside a combined layer of human muscle and fat, under Fanger thermal comfort conditions. The bioheat equation was solved subject to two boundary conditions at the skin surface: a prescribed skin temperature satisfying the Fanger comfort criterion, and a prescribed heat flux obtained from the overall energy balance for the system. In addition to a fixed body core temperature, an adiabatic condition was imposed as an auxiliary condition at the core of the body, and a pair of equations were derived, relating the blood perfusion and the volumetric heat generation rate for a given activity level and environmental conditions. By solving the two equations, we determined the functional dependence of blood perfusion and metabolic heat generation on the human activity level. For convenience, we presented simple explicit expressions for the key relations, with the aid of asymptotic analyses. Additional results include the temperature distribution inside the muscle layer, and the effects of muscle and fat layer thickness on the heat transfer processes.

Feasibility and relevance of discrete vasculature modeling in routine hyperthermia treatment planning

Purpose: To investigate the effect of patient specific vessel cooling on head and neck hyperthermia treatment planning (HTP).Methods and materials: Twelve patients undergoing radiotherapy were scanned using computed tomography (CT), magnetic resonance imaging (MRI) and contrast enhanced MR angiography (CEMRA). 3D patient models were constructed using the CT and MRI data. The arterial vessel tree was constructed from the MRA images using the ‘graph-cut’ method, combining information from Frangi vesselness filtering and region growing, and the results were validated against manually placed markers in/outside the vessels. Patient specific HTP was performed and the change in thermal distribution prediction caused by arterial cooling was evaluated by adding discrete vasculature (DIVA) modeling to the Pennes bioheat equation (PBHE).Results: Inclusion of arterial cooling showed a relevant impact, i.e., DIVA modeling predicts a decreased treatment quality by on average 0.19 °C (T90), 0.32 °C (T50) and 0.35 °C (T20) that is robust against variations in the inflow blood rate (|ΔT| < 0.01 °C). In three cases, where the major vessels transverse target volume, notable drops (|ΔT| > 0.5 °C) were observed.Conclusion: Addition of patient-specific DIVA into the thermal modeling can significantly change predicted treatment quality. In cases where clinically detectable vessels pass the heated region, we advise to perform DIVA modeling.

Modeling Left Atrial Flow, Energy, Blood Heating Distribution in Response to Catheter Ablation Therapy.

Introduction: Atrial fibrillation (AF) is a widespread cardiac arrhythmia that commonly affects the left atrium (LA), causing it to quiver instead of contracting effectively. This behavior is triggered by abnormal electrical impulses at a specific site in the atrial wall. Catheter ablation (CA) treatment consists of isolating this driver site by burning the surrounding tissue to restore sinus rhythm (SR). However, evidence suggests that CA can concur to the formation of blood clots by promoting coagulation near the heat source and in regions with low flow velocity and blood stagnation.Methods: A patient-specific modeling workflow was created and applied to simulate thermal-fluid dynamics in two patients pre- and post-CA. Each model was personalized based on pre- and post-CA imaging datasets. The wall motion and anatomy were derived from SSFP Cine MRI data, while the trans-valvular flow was based on Doppler ultrasound data. The temperature distribution in the blood was modeled using a modified Pennes bioheat equation implemented in a finite-element based Navier-Stokes solver. Blood particles were also classified based on their residence time in the LA using a particle-tracking algorithm.Results: SR simulations showed multiple short-lived vortices with an average blood velocity of 0.2-0.22 m/s. In contrast, AF patients presented a slower vortex and stagnant flow in the LA appendage, with the average blood velocity reduced to 0.08–0.14 m/s. Restoration of SR also increased the blood kinetic energy and the viscous dissipation due to the presence of multiple vortices. Particle tracking showed a dramatic decrease in the percentage of blood remaining in the LA for longer than one cycle after CA (65.9 vs. 43.3% in patient A and 62.2 vs. 54.8% in patient B). Maximum temperatures of 76° and 58°C were observed when CA was performed near the appendage and in a pulmonary vein, respectively.Conclusion: This computational study presents novel models to elucidate relations between catheter temperature, patient-specific atrial anatomy and blood velocity, and predict how they change from SR to AF. The models can quantify blood flow in critical regions, including residence times and temperature distribution for different catheter positions, providing a basis for quantifying stroke risks.

An analytical solution for temperature distributions in hepatic radiofrequency ablation incorporating the heat-sink effect of large vessels

Fast prediction of the local thermal field induced by radiofrequency ablation (RFA) plays a critical role in hepatic RFA therapy. At present, it is still a challenging task to calculate and visualize the temperature distribution of RFA in real-time, especially when the heat-sink effect of adjacent large vessels is taken into account. To achieve this, the current investigation presented an analytical solution to calculate the temperature in RFA with an execution time of 0.05 s for three dimensional thermal field reconstruction. The presented temperature distribution is a combination of temperatures in homogeneous tissue and a quantification of the heat-sink effect of adjacent blood vessels. Temperatures in homogeneous tissue is calculated from a simplified Pennes bioheat equation, where several weighting parameters in the temperature expression are determined based on some reference point temperatures from the numerical simulation. The heat-sink effect is quantified based on a temperature factor, which measures the temperature difference between the vessel and the heated tissue, and a distance factor, which measures the distance to the vessel. The proposed method is validated to be able to gain similar temperature distributions to the numerical simulation but with its computational time being orders of magnitude smaller than that of numerical simulation, which improves the efficiency of interactive planning of RFA.

Validation of hybrid angular spectrum acoustic and thermal modelling in phantoms

In focused ultrasound (FUS) thermal ablation of diseased tissue, acoustic beam and thermal simulations enable treatment planning and optimization. In this study, a treatment-planning methodology that uses the hybrid angular spectrum (HAS) method and the Pennes’ bioheat equation (PBHE) is experimentally validated in homogeneous tissue-mimicking phantoms. Simulated three-dimensional temperature profiles are compared to volumetric MR thermometry imaging (MRTI) of FUS sonications in the phantoms, whose acoustic and thermal properties are independently measured. Additionally, Monte Carlo (MC) uncertainty analysis is performed to quantify the effect of tissue property uncertainties on simulation results. The mean error between simulated and experimental spatiotemporal peak temperature rise was +0.33°C (+6.9%). Despite this error, the experimental temperature rise fell within the expected uncertainty of the simulation, as determined by the MC analysis. The average errors of the simulated transverse and longitudinal full width half maximum (FWHM) of the profiles were –1.9% and 7.5%, respectively. A linear regression and local sensitivity analysis revealed that simulated temperature amplitude is more sensitive to uncertainties in simulation inputs than in the profile width and shape. Acoustic power, acoustic attenuation and thermal conductivity had the greatest impact on peak temperature rise uncertainty; thermal conductivity and volumetric heat capacity had the greatest impact on FWHM uncertainty. This study validates that using the HAS and PBHE method can adequately predict temperature profiles from single sonications in homogeneous media. Further, it informs the need to accurately measure or predict patient-specific properties for improved treatment planning of ablative FUS surgeries.

Influence of external head cooling on the head, core body and blood temperatures using 3D whole-body model

PurposeThe purpose of this study is to evaluate the impact of external head cooling on alleviating the heat stress in the human body by analyzing the temperatures of the core body (Tc), blood (Tblood) and head (Th) during exercise conditions using 3D whole body model.Design/methodology/approachComputational study is conducted to comprehend the influence of external head cooling on Tc, Tblood and Th. The Pennes bioheat and energy balance equations formulated for the whole-body model are solved concurrently to obtain Tc, Tblood and Th for external head cooling values from 33 to 233 W/m2. Increased external head cooling of 404 W/m2 is used to compare the numerical and experimental Th data.FindingsSignificant reductions of 0.21°C and 0.38°C are observed in Th with external head cooling of 233 and 404 W/m2, respectively. However, for external head cooling of 233 W/m2, lesser reductions of 0.03°C and 0.06°C are found in Tc and Tblood, respectively. Computational results for external head cooling of 404 W/m2 show a difference of 15 per cent in Th compared to experimental values from literature.Originality/valueThe development of stress because of heat generated within human body is major concern for athletes exercising at high intensities. This study provides an insight into the effectiveness of external head cooling in regulating the head and body temperatures during exercise conditions.

Generation and Thermal Simulation of a Digital Model of the Female Breast in Prone Position

Infrared (IR) breast thermography has been associated with the early detection of breast cancer (BC). However, findings in previous studies have been inconclusive. The upright position of subjects during imaging introduces errors in interpretation, because it blocks the optical access in the inframammary fold region and alters the temperature due to contact between breast and chest wall. These errors can be avoided by imaging breasts in prone position. Although the numerical simulations provide insight into thermal characteristics of the female breast with a tumor, most simulations in the past have used cubical and hemispherical breast models. We hypothesize that a breast model with the actual breast shape will provide true thermal characteristics that are useful in tumor detection. A digital breast model in prone position is developed to generate the surface temperature profiles for breasts with tumors. The digital breast model is generated from sequential magnetic resonance imaging (MRI) images and simulations are performed using finite volume method employing Pennes bioheat equation. We investigated the effect of varying the tumor metabolic activity on the surface temperature profile. We compared the surface temperature profile for various tumor metabolic activities with a case without tumor. The resulting surface temperature rise near the location of the tumor was between 0.665 and 1.023 °C, detectable using modern IR cameras. This is the first time that numerical simulations are conducted in a model with the actual breast shape in prone position to study the surface temperature changes induced by BC.

Evaluation of theranostic perspective of gold-silica nanoshell for cancer nano-medicine: a numerical parametric study.

Using gold-silica nanoshell as a reference nano-agent, this work has performed preliminary numerical parametric study to investigate the feasibility and if feasible the efficiency of using a single nano-agent to achieve theranostic goals. In total, seven generics of gold-silica nanoshells have been tested including the R[50,10] (radius of the silica core is 50 nm and thickness of the gold shell is 10 nm), R[40,15], R[55,25], R[40,40], R[75,40], R[104,23], and R[154,24] nanoshells. A planar tissue model has been constructed as the platform for parametric study. For mathematical modeling, radiant transport equation (RTE) has been applied to describe the interactions among laser lights, the hosting tissue, and the hosted nanoshells and Penne's bio-heat equation has been applied to describe the hyperthermia induced by such interactions. Effects of different nanoshell generics on the diffuse reflectance signal and hyperthermia temperature transition have been simulated, basing on which the potential of a certain nanoshell generic as theranostic nano-agent has been evaluated. It has been found that it is highly feasible for gold-silica nanoshells to be engineered for theranostic purpose and nanoshell generics that are preferentially scattering should be explored for good theranostic candidates. On the condition that nanoshell generic with the right optical properties has been located, a moderate nanoshell retention in the target tissue site is already sufficient to induce effective theranostic effects, which indicates that theranostic nano-medicine might not have a stringent requirement for the delivery technique. Among nanoshells that have been tested, the R[55,25] nanoshell seems to be a promising candidate as theranostic nano-agent. Further testing on it is highly recommended. Nanoshells that are preferentially absorbing such as the R[50,10] and R[40,15] nanoshells are efficient photothermal agent and could be used for therapeutic purpose only. However, it is not recommended that preferentially absorbing nanoshells being used for theranostic purpose due to possible negative effects such nanoshells might bring to the diffuse reflectance signal.

Influence of Blood Vessels on Temperature during High-Intensity Focused Ultrasound Hyperthermia Based on the Thermal Wave Model of Bioheat Transfer

The coupled effects of blood vessels and thermal relaxation time on temperature and thermal lesion region in biological tissue during high-intensity focused ultrasound (HIFU) hyperthermia are numerically investigated. Considering the non-Fourier behavior of heat conduction in biological tissue, the traditional Pennes bioheat equation was modified to thermal wave model of bioheat transfer (TWMBT). Consequently, a joint physical model, which combines TWMBT for tissue and energy transport equation for blood vessel, is presented to predict the evolution of temperature and the thermal lesion region. In this study, pulsatile blood flow is first introduced into numerical study of HIFU hyperthermia, and thermal relaxation time, ultrasonic focus location, blood vessel radius, and blood flow velocity are all taken into account. The results show that the thermal relaxation time plays a key role in the temperature and the thermal lesion region. Larger thermal relaxation time results in lower temperature and smaller thermal lesion region, which indicates that TWMBT leads to lower temperature and smaller thermal lesion region compared to Pennes bioheat transfer model. In addition, we found that the ultrasonic focus location and blood vessel radius significantly affected the temperature and thermal lesion region, while the heartbeat frequency and amplitude factor of pulsating blood flow as well as the average velocity of blood flow had only a slight effect.

Thermal management of flexible wearable electronic devices integrated with human skin considering clothing effect

The flexible wearable electronic devices can be directly integrated with human skin to monitor the vital signs of human body, which will be covered with clothes. An analytical model is developed in this paper to study the flexible wearable electronic devices integrated with human skin considering clothing effect. The coupling model of Pennes bio-heat transfer equation and Fourier heat conduction equation is adopted to consider the thermal behavior of the whole system. The clothes are modeled as a kind of composite material consisting of clothing fibers and air under the complicated boundary condition including convection, conduction and radiation between the environment. The predicted analytical temperature distribution of the system is validated by finite element analysis (FEA). And the influences of systematic parameters on characteristic temperatures, i.e., the temperature of electronic component and the maximum temperature at the device/skin interface are investigated comprehensively to discuss the design of heat dissipation and the clothing effect. The results presented are able to provide guidelines for the thermal management of flexible wearable electronic devices under real working conditions.

Independent assessment of MRI-induced temperature change and SAR distributions in phantoms

Background During an MR procedure, most of the transmitted RF power is transformed into heat referred to as the specific energy absorption rate(SAR). EU directive(IEC60602-3-33) mandates all scanners must measure SAR and provide safeguards for tissue-limits. Accurate estimation of SAR is critical in safeguarding vulnerable patients. Modern MRI systems can easily exceed safe SAR levels and our aim is to develop a protocol to verify patient specific SAR using a MR phantom heated solely by the RF fields. Methods Our phantom is a non-perfused material simplifying pennes-bioheat equation. The SAR at discreet points in the observation plane is determined by SAR ≈ C_agar Δ T / Δ t[W⧹kg] Where C_agar is 4200 J/kg.K, Δ T is the change in temperature and Δ t is the change in time. Δ T was determined via Proton Resonance Frequency Shift (PRF) thermometry. Δ T = ( ϕ - ϕ _ 0 ) /( α γ B_0 TE) Where.

Sweat effects on the thermal analysis of epidermal electronic devices integrated with human skin

Epidermal electronic devices (EEDs) are widely used in many applications especially directly integrated with human skin tissue to monitor the vital signs of human body. The thermal management of EEDs is very critical because the excessive temperature increase may cause discomfort or damage to human skin. Sweating can also significantly change the thermal environment of EED/skin system due to its function on the thermal transport process. A three-dimensional heat transfer analytical model is established to predict the thermal characteristics of EED/skin system considering the bio-heat transfer and effects of sensible sweat. In this model, the heat conductions in EEDs and in human skin are taken into account with Fourier heat transfer equation and Pennes bio-heat transfer equation, respectively, including the transport process of sweating. The results are validated by finite element analysis (FEA) and parameters studies such as velocity of sweat, thermal conductivity and thickness of substrate have been investigated. The results can help guide the thermal management of EED/skin system considering effects of sensible sweat.

Numerical investigation of convective cooling in minimizing skin burns during radiofrequency ablation of breast tumor

Skin burns is one of the most common complications reported during treatment of early-stage breast cancer with radiofrequency ablation (RFA) technique. In this regard, the present study aims to analyze the efficacy of forced convection surface cooling in minimizing the skin burns during RFA of breast tumor. The study considers a heterogeneous three-dimensional numerical model of breast that has been constructed based on the anatomical details available in the literature. A spherical tumor of 1.5 cm has been embedded in the numerical model of breast to represent in-situ tumor in its early stage. A programmable temperature-controlled RFA has been performed by incorporating the closed-loop feedback PID controller into the numerical model. The thermo-electric analysis has been performed using a finite element based commercial solver COMSOL Multiphysics® to obtain the temperature distribution by incorporating the coupled electric field distribution, Pennes bioheat equation and Arrhenius rate equation. The temperature dependent electrical and thermal conductivities of both the tissue and tumor along with damage dependent perfusion rate have been incorporated to achieve better correlation with the clinical RFA. The numerical simulation results revealed that, there is a significant reduction in the temperature at outer periphery of breast with increase in convective heat transfer coefficient. Further, the effects of variation in surface cooling on temperature distribution, input voltage requirement and treatment time required for complete tumor necrosis have been studied. The simultaneous application of forced convective surface cooling along with RFA could play a vital role in addressing the most common complication of skin burns during treatment of breast cancer.

Time-dependent perfusion coefficient estimation in a bioheat transfer problem

We consider the estimation of the time-dependent blood perfusion coefficient in the Pennes bioheat equation with Ionkin-type nonlocal boundary and integral energy over-determination conditions. In contrast to several methods that transform the original problem into an inverse source problem and then estimate the perfusion coefficient through numerical differentiation, we propose an alternative method in which the coefficient is estimated directly through a nonlinear minimization technique. In the method, the bioheat equation is solved by the method of lines based on a highly accurate pseudospectral approach, and perfusion coefficient values are estimated by the Levenberg–Marquardt method with the discrepancy principle as stopping rule. Numerical examples are presented to verify the accuracy and stability of the solution.

Modeling Skin Injury from Hot Rice Porridge Spills.

The present work analyzes skin burns from spills of hot rice and milk products. The traditional Norwegian rice porridge serves as an example. By testing spills on objects emulating an arm, it was concluded that spills were seldom thinner than 3 mm, and stayed in place due to the viscosity of the porridge for more than one minute. The Pennes bioheat equation was solved numerically for such spills, including heat conduction to the skin and convective heat losses from the porridge surface. Temperatures were analyzed in the porridge and skin layers, and the resulting skin injury was calculated based on the basal layer temperature. Parameters influencing burn severity, such as porridge layer thickness, porridge temperature, removal of the porridge and thermal effects of post scald tempered (15 °C) water cooling were analyzed. The spilled porridge resulted in a prolonged heat supply to the skin, and the skin injury developed significantly with time. The porridge temperature turned out to be the most important injury parameter. A 70 °C porridge temperature could develop superficial partial-thickness burns. Porridge temperatures at processing temperatures nearly instantly developed severe burns. It was demonstrated that prompt removal of the hot porridge significantly reduced the injury development. The general advice is to avoid serving porridge and similar products at temperatures above 65 °C and, if spilled on the skin, to remove it quickly. After such scald incidents, it is advised to cool the injured area by tempered water for a prolonged period to stimulate healing.

Computational Design of an RF Controlled Theranostic Model for Evaluation of Tissue Biothermal Response

Human disease management strategies are gradually shifting from the traditional approach towards personalized medicine. Thermal therapy comprising of clinical hyperthermia and therapeutic hypothermia has been subject of recent research. However, both have suffered in clinical trials because thermal monitoring is not impressive and delivering the appropriate amount of heat to the appropriate part of the patient’s tissues is challenging. Hence, this study is aimed at addressing these challenges by technically merging the principles of magnetic resonance relaxation to the Bioheat transfer phenomenon. The analytical solutions to the Pennes Bioheat equation are connected to radiofrequency (RF) power absorption in tissue that is dependent on the magnetic resonance relaxation parameters. A carefully selected RF field is used to control the model for hyperthermia and therapeutic hypothermia. The solution obtained is then applied to hyperthermia treatment of tumours and therapeutic hypothermia of the brain. The thermal profiles obtained do not only show excellent contrasts between different tissues but real time monitoring of tissue thermal response is very impressive. By producing individualized virtual tumours that may predict disease progression, the thermal profiles based on the controlled parameters developed in this study can contribute to the ongoing dialogue regarding the design of appropriate response criteria, provide a means to perform virtual clinical trials to assess the likely benefit of novel neurotherapeutics, and move neuro-oncology toward individualized treatment plans optimized for maximum benefit.