Pennes Bioheat Equation Research Articles

Analysis of hyperbolic Pennes bioheat equation in perfused homogeneous biological tissue subject to the instantaneous moving heat source

The one-dimensional hyperbolic Pennes bioheat equation under instantaneous moving heat source is solved analytically based on the Eigenvalue method. Comparison with results of in vivo experiments performed earlier by other authors shows the excellent prediction of the presented closed-form solution. We present three examples for calculating the Arrhenius equation to predict the tissue thermal damage analysis with our solution, i.e., characteristics of skin, liver, and kidney are modeled by using their thermophysical properties. Furthermore, the effects of moving velocity and perfusion rate on temperature profiles and thermal tissue damage are investigated. Results illustrate that the perfusion rate plays the cooling role in the heating source moving path. Also, increasing the moving velocity leads to a decrease in absorbed heat and temperature profiles. The closed-form analytical solution could be applied to verify the numerical heating model and optimize surgery planning parameters.

Fast calculation of 3D radiofrequency ablation zone based on a closed-form solution of heat conduction equation fitted by ex vivo measurements

Fast calculation or simulation of the ablation zone induced by radiofrequency ablation (RFA) has a critical role in hepatic RFA planning and therapy. However, it remains challenging to approximate the ablation zone in real time, especially when more than one probe is involved in one ablation session. This paper presents a novel computational technique to calculate the 3D ablation zone of one probe RFA and two-probe switching RFA. The main idea is to get an approximate solution of the temperature distribution from a simplified Pennes bioheat equation, and further fit the solution to the coagulation measurements on ex vivo porcine liver. With a closed-form solution of temperature distribution, the calculation of the ablation zone is as simple as the commonly used ellipsoidal model, but it allows a more realistic prediction of combined ablation zones with different inter-probe spacing. The new approximation technique could potentially replace the original ellipsoidal model in the intervention planning step.

Pulsed Ultrasound Assisted Thermo-Therapy for Subsurface Tumor Ablation: A Numerical Investigation

AbstractHigh-intensity focused ultrasound (HIFU) is a promising therapy for thermal ablation and hyperthermia, characterized by its non-invasiveness and high penetration depth. Effective HIFU thermo-therapy requires the ability to accurately predict temperature elevation and corresponding thermal dose distribution in target tissues. We report a parametric numerical study of the thermal response and corresponding of thermal dose in a soft tissue in response to ultrasound. We compared the predictions of tissue models with two, three, and seven layers, to ultrasound-induced heating at duty cycles ranging from 0.6 and 0.9. Further, two tumor sizes and transducer powers (10 W and 15 W) were considered. The inhomogeneous Helmholtz equation was coupled with the Pennes bio-heat equation to predict heating in response to pulsed ultrasound. Necrotic lesion size was calculated using the cumulative equivalent minute (CEM) thermal dose function. In-vitro experiments were performed with agar-based tissue phantoms as a preliminary validation of the numerical results. The simulations conducted with the seven-layered model predicted up to 33.5% lower peak pressure amplitude than the three-layered model. As the ultrasound pulse width decreased with the equivalent sonication time fixed, the corresponding magnitude of the peak temperature and the rate of temperature rise decreased. Pulsed ultrasound resulted in the increased volume of necrotic lesions for an equivalent time of sonication. The findings of this study highlight the dependence of HIFU-induced heating on target geometry and acoustic properties and could help guide the choice of suitable ultrasound exposure parameters for further studies.

Influence of temperature-dependent acoustic and thermal parameters and nonlinear harmonics on the prediction of thermal lesion under HIFU ablation.

According to the traditional method of high intensity focused ultrasound (HIFU) treatment, the acoustic and thermal characteristic parameters of constant temperature (room temperature or body temperature) are used to predict thermal lesion. Based on the nonlinear spherical beam equation (SBE) and Pennes bio-heat transfer equation, and a new acoustic-thermal coupled model is proposed. The constant and temperature-dependent acoustic and thermal characteristic parameters are used to predict thermal lesion, and the predicted lesion area are compared with each other. Moreover, the relationship between harmonic amplitude ratio (P2/P1) and thermal lesion is studied. Combined with the known experimental data of acoustic and thermal characteristic parameters of biological tissue and data fitting method, the relationship between acoustic and thermal characteristic parameters and temperature is obtained; and the thermal lesion simulation calculation is carried out by using the acoustic and thermal characteristic parameters under constant temperature and temperature- dependent acoustic and thermal characteristic parameters, respectively. The simulation results show that under the same irradiation condition, the thermal lesion predicted by temperature-dependent acoustic and thermal characteristic parameters is larger than that predicted by traditional method, and the thermal lesion increases with the decrease of harmonic amplitude ratio.

Mathematical modeling of temperature variation in breast tissue with and without tumor/cyst during menstrual cycle

The thermoregulation of human female body is influenced by hormonal and physiological changes in the body during the menstrual cycle. The fluctuation of estrogen and progesterone hormones, release in the follicular phase and the luteal phase of menstrual cycle, respectively play an important role in the growth of breast ducts and lobules (milk glands). The imbalance of these hormones causes breast tumors/cysts. The body core temperature, blood perfusion and metabolism rate are higher in the luteal phase than the follicular phase of menstrual cycle. In the present work, a tumor/cyst is assumed to be in the glandular layer. A two-dimensional Pennes bioheat equation is solved to find the temperature variation in breast tissue with and without tumor/cyst during the menstrual cycle by using the finite element method. The results show that the temperature of each layer of breast tissue in the luteal phase is higher than the follicular phase in the case of normal breast, tumorous breast and breast with cyst.

Analysis of the effect of external heating in the human tissue: A finite element approach

Abstract Thermal therapy which involves either raising or lowering tissue temperature to treat malignant cells needs precise acknowledgment of thermal history inside the biological system to ensure effective treatment. For this purpose, this study presents a two-dimensional unsteady finite element model (FEM) of the bioheat transfer problem based on Pennes bio-heat equation to analyze the thermal response of tissue subject to external heating. Crank-Nikolson scheme was used for the unsteady solution. A finite element code was developed using C language to calculate results. The obtained numerical result was compared with the analytical and other numerical results available in the literature. A good agreement was found from the comparison. Temperature distribution inside the human body due to constant and sinusoidal spatial and surface heating were analyzed. Response to point heating was also investigated. Moreover, a sensitivity analysis was carried out to know the effect of various parameters, i.e. blood temperature, thermal conductivity, and blood perfusion rate on tissue temperature. The outcome of this study will be helpful for the researchers and physicians involved in the thermal treatment of human tissue.

The radially symmetric Green’s function for Dirichlet problem of Pennes equation outside of a circular domain

We find a general method to obtain the radially symmetric solutions of Dirichlet problem for Pennes bioheat equation in the exterior domain of a circle through the computation of Green’s function of a naturally related operator. We apply this technique to solve a problem in radio-frequency ablation.

Thermal field and tissue damage analysis of moving laser in cancer thermal therapy.

In this paper, a closed-form analytical solution of hyperbolic Pennes bioheat equation is obtained for spatial evolution of temperature distributions during moving laser thermotherapy of the skin and kidney tissues. The three-dimensional cubic homogeneous perfused biological tissue is adopted as a media and the Gaussian distributed function in surface and exponentially distributed in depth is used for modeling of laser moving heat source. The solution procedure is Eigen value method which leads to a closed form solution. The effect of moving velocity, perfusion rate, laser intensity, absorption and scattering coefficients, and thermal relaxation time on temperature profiles and tissue thermal damage are investigated. Results are illustrated that the moving velocity and the perfusion rate of the tissues are the main important parameters in produced temperatures under moving heat source. The higher perfusion rate of kidney compared with skin may lead to lower induced temperature amplitude in moving path of laser due to the convective role of the perfusion term. Furthermore, the analytical solution can be a powerful tool for analysis and optimization of practical treatment in the clinical setting and laser procedure therapeutic applications and can be used for verification of other numerical heating models.

Buoyancy Effects on Human Skin Tissue Thermoregulation due to Environmental Influence

This paper theoretically examines the impact of thermal buoyancy on human skin tissue’s blood flow, heat exchange and their interaction with the surrounding environment using a two phase mathematical model that relies on continuity, momentum and energy conservation equations in continuum mechanics. The tissue blood flows and heat transfer characteristics are determined numerically based on Darcy’s Brinkman model for a saturated porous medium coupled with modified Pennes bioheat equation while analytical approach is employed to tackle the model of interacting surrounding environmental buoyancy driven air flow with heat sink. The influence of embedded biophysical parameters on the skin tissue’s blood flow rate and temperature distribution together with friction coefficient at skin tissue surface and Nusselt number are display graphically and discussed quantitatively. It is found that a boost in thermal buoyancy enhances skin tissue heat transfer and blood flow rates.

Towards efficient design optimization of a miniaturized thermoelectric generator for electrically active implants via model order reduction and submodeling technique.

Thermoelectric generators (TEG) convert the thermal energy into electrical energy and are under investigation as a power supply for medical implants. To improve the performance of TEG, the design optimization process through finite element model simulation is preferred by biomedical engineers. This paper aims to provide an efficient method of speeding up the design optimization process of TEG. A three-dimensional realistic human torso model incorporating the TEG is investigated, where the internal heat transfer in human tissue is characterized by Pennes bioheat equation. In addition, convection, radiation, and evaporation effects at the skin surface are applied to identify the heat transfer effects between the human body and the environment. To speed up finite element simulation of the large-scale human torso model, projection-based model order reduction (MOR) is applied for generation of a compact but highly accurate model. Parametric MOR (pMOR) further enables generating a parameter-independent compact model. For an efficient design optimization of TEG, this compact human torso model is applied within a thermal submodeling approach. Its temperature distribution results are back-projected and used as boundary conditions for the TEG submodel. The achieved speed-up in simulation time, demonstrated in this work, clearly indicates that the design optimization process of TEG is more efficient with the combination of MOR and submodeling techniques.

Simulation the distribution of thermodynamic temperatures and microwave radiation of the human head

Background and ObjectiveThe influence of biophysical parameters on the formation of microwave radiation of the human head is poorly studied. Existing approaches to modeling microwave radiation of the human head have limitations associated with simplifying the geometry of human anatomy. The article proposes methodological solutions for numerical modeling of microwave radiation of the brain biological tissues using the geometry obtained from MRI data. MethodsThe geometrical characteristics of biological tissues in model are determined using an MRI image of the head. The methodology proposed in the article allows simulation of a human body voxel models performed the Pennes bio-heat transfer equation using the Fenix software package. ResultsModeling evaluations have shown that anatomical tissues heterogeneities on the surface of the head form temperature gradient of up to 2.0 K, and changes of the microwave radiation up to 0.3 K. ConclusionsVerification data made by IR thermograph practically coincide with the results of numerical modeling. The fluctuations of the brain microwave radiation are not only the result of thermal processes in its tissues, but are determined by the dynamics of its thermoregulation processes and are an indicator of changes in the physiological processes occurring in it.

Influence of dynamic tissue properties on temperature elevation and lesions during HIFU scanning therapy: Numerical simulation**Project partially supported by the National Natural Science Foundation of China (Grant Nos. 11774088 and 11474090), the Natural Science Foundation of Hunan Province, China (Grant Nos. 2016JJ3090 and 2018JJ3557), and the Scientific Research Fund of Hunan Provincial Education Department, China (Grant Nos. 16B155 and 17B025).

When large tumors are treated, ablation of the entire volume of tumors requires multiple treatment spots formed by high intensity-focused ultrasound (HIFU) scanning therapy. The heating effect of HIFU on biological tissue is mainly reflected in temperature elevation and tissue lesions. Tissue property parameters vary with temperature and, in turn, the distribution of temperature as well as the heating effects change accordingly. In this study, an HIFU scanning therapy model considering dynamic tissue properties is provided. The acoustic fields and temperature fields are solved combining the Helmholtz wave equation with Pennes bio-heat transfer equation based on the finite element method (FEM) to investigate the effects of various tissue properties (i.e., the attenuation coefficient, acoustic velocity, thermal conductivity, specific heat capacity, density, and blood perfusion rate) on heating performance. Comparisons of the temperature distribution and thermal lesions under static and dynamic properties are made based on the data of tissue property parameters varying with temperature. The results show that the dynamic changes of thermal conductivity, specific heat capacity, and acoustic velocity may account for the decrease of temperature elevation in HIFU treatment, while the dynamic changes of attenuation coefficient, density, and blood perfusion rate aggravate the increase of temperature on treatment spots. Compared with other properties, the dynamic change of attenuation coefficient has a greater impact on tissue temperature elevation. During HIFU scanning therapy, the temperature elevation and tissue lesions of the first treatment spot are smaller than those of the subsequent treatment spots, but the temperature on the last treatment spot drops faster during the cooling period. The ellipsoidal tissue lesion is not symmetrical; specifically, the part facing toward the previous treatment spot tends to be larger. Under the condition of the same doses, the temperature elevation and the size of tissue lesions under dynamic properties present significant growth in comparison to static properties. Besides, the tissue lesion begins to form earlier with a more unsymmetrical shape and is connected to the tissue lesion around the previous treatment spot. As a result, lesions around all the treatment spots are connected with each other to form a closed lesion region. The findings in this study reveal the influence of dynamic tissue properties on temperature elevation and lesions during HIFU scanning therapy, providing useful support for the optimization of treatment programs to guarantee higher efficacy and safety.

Numerical Study of Transient Bio-Heat Transfer Model With Heat Transfer Coefficient and Conduction Effect in Cylindrical Living Tissue

The human thermal comfort is affected by the body’s heat exchange mechanism conduction, convection, radiation, and evaporation. The mode of heat transfer between the body and environment depends upon the human internal physiological phenomena, together with the boundary conditions. The present paper provides the comprehensive overview of the thermoregulatory system of human body and studies the numerical solution of unsteady-state one dimensional Pennes bio-heat equation with appropriate boundary conditions. The solution is used to observe the temperature profiles at different thermal conductivities, and different heat transfer coefficients in the living tissue at the various time steps. Various physical and physiological factors across the cylindrical living tissue have been incorporated in the model.

BIOHEAT EQUATION WITH FOURIER AND NON-FOURIER HEAT TRANSPORT LAWS: APPLICABILITY TO HEAT TRANSFER IN HUMAN TISSUES

The paper is dedicated to mathematical problem formulations for the heat propagation in biological tissues based on the Fourier and non-Fourier laws at different boundary conditions. The heating of the tissues is provided by external heat sources like low intensity lasers or light-emitting diodes which are widely used in contemporary medical care. Numerical computations on the standard Pennes bioheat equation with Fourier heat conduction give the temperature curves for both heating and thermal relaxation processes that do not correspond to the in vivo measurement data on human skin tissue. It is shown the modified bioheat equation based on the Guyer-Krumhansl heat conduction with correct formulation of the boundary conditions produces realistic temperature curves when the distributed heat sources and sinks in the tissue are accounted for. The former corresponds to the metabolic heat and temperature dependent chemical reactions, while the latter is provided by the heat convection with blood microcirculation system. The proposed model gives realistic two time temperature curves. The perspective applications of the novel mathematical formulation are discussed.

Numerical two-dimensional thermal analysis of the human skin using the multigrid method

Heat can be used as an adjuvant treatment of many diseases and also as a powerful tool to help diagnose cancers, with the advantage to be a noninvasive exam. Some tumors may be best diagnosed by evaluating body temperature distribution, for instance, it is observed that local temperatures of the skin over a tumor are higher than the average skin temperature. Certainly, it is expected from medical diagnostics to be, early, fast and very precise. Especially if the health problem is a tumor, it is necessary to know the shape and the size of the cancer. Thermal images can provide further information about the tumor, generally, the thermal diagnostic is made comparing images of the region with a bioheat model. In this context, the present study shows interesting results about the multigrid method applied to solve the Pennes bioheat equation in two dimensions, using a non-stationary and steady state cases for the skin heath and with melanoma. The multigrid method presented itself as an extremely efficient and fast tool to solve the bioheat equation with refined grids that provide good spatial precision.

MRI-based numerical modeling strategy for simulation and treatment planning of nanoparticle-assisted photothermal therapy

Nanoparticle-assisted photothermal therapy (NPTT) has recently emerged as a promising alternative to traditional thermal therapy methods. Computational modeling for simulation and treatment planning of NPTT seems to be essential for clinical translation of this modality. Non-invasive identification of nanoparticle distribution within the tissue is a key perquisite for accurate prediction of NPTT in real conditions. In the present study, we have developed a magnetic resonance imaging (MRI)-based numerical modeling strategy for simulation and treatment planning of NPTT. To this end, we have utilized the core-shell γ-Fe2O3@Au nanoparticle comprising a gold layer with plasmonic properties and a magnetic core that enables to track the location of this structure via MRI. The map of nanoparticle distribution in the tumor derived from T2-weighted MR image was imported into a finite element simulation software, and Pennes bioheat equation and Arrhenius damage model were applied to simulate the temperature and damage distributions, respectively. The validation of the model developed herein was assessed by monitoring the superficial and the central temperature variations of the tumor in experiment. Both the numerical modeling and experimental study proved that a localized heating and then a focused damage could be achieved due to nanoparticle inclusion. There is quite satisfactory agreement between the numerical and experimental results. The model developed in this study has a good capability to be used as a promising planning method for NPTT of cancer.

Neural network methodology for real-time modelling of bio-heat transfer during thermo-therapeutic applications.

Real-time simulation of bio-heat transfer can improve surgical feedback in thermo-therapeutic treatment, leading to technical innovations to surgical process and improvements to patient outcomes; however, it is challenging to achieve real-time computational performance by conventional methods. This paper presents a cellular neural network (CNN) methodology for fast and real-time modelling of bio-heat transfer with medical applications in thermo-therapeutic treatment. It formulates nonlinear dynamics of the bio-heat transfer process and spatially discretised bio-heat transfer equation as the nonlinear neural dynamics and local neural connectivity of CNN, respectively. The proposed CNN methodology considers three-dimensional (3-D) volumetric bio-heat transfer behaviour in tissue and applies the concept of control volumes for discretisation of the Pennes bio-heat transfer equation on 3-D irregular grids, leading to novel neural network models embedded with bio-heat transfer mechanism for computation of tissue temperature and associated thermal dose. Simulations and comparative analyses demonstrate that the proposed CNN models can achieve good agreement with the commercial finite element analysis package, ABAQUS/CAE, in numerical accuracy and reduce computation time by 304 and 772.86 times compared to those of with and without ABAQUS parallel execution, far exceeding the computational performance of the commercial finite element codes. The medical application is demonstrated using a high-intensity focused ultrasound (HIFU)-based thermal ablation of hepatic cancer for prediction of tissue temperature and estimation of thermal dose.

Mathematical simulation of temperature distribution in tumor tissue and surrounding healthy tissue treated by laser combined with indocyanine green

BackgroundPhotothermal therapy is a local treatment method for cancer and the heat energy generated from it could destroy the tumor cells. This study is aimed to investigate the temperature distribution in tumor tissue and surrounding health tissue of tumor bearing mice applying mathematical simulation model. Tumor bearing mice treated by laser combined with or without indocyanine green. Monte Carlo method and the Pennes bio-heat equation were used to calculate the light distribution and heat energy. COMSOL Multiphysic was adopted to construct three dimensional temperature distribution model.ResultsThis study revealed that the data calculated by simulation model is in good agreement with the surface temperature monitored by infrared thermometer. Effected by the optical parameters and boundary conditions of tissue, the highest temperature of tissue treated by laser combined with indocyanine green was about 65 °C which located in tumor tissue and the highest temperature of tissue treated by laser was about 43 °C which located under the tumor tissue. The temperature difference was about 20 °C. Temperature distribution in tissue was not uniform. The temperature difference in different parts of tumor tissue raised up to 15 °C. The temperature of tumor tissue treated by laser combined with indocyanine green was about 20 °C higher than that of the surrounding healthy tissue.ConclusionsReasonably good matching between the calculated temperature and the measured temperature was achieved, thus demonstrated great utility of our modeling method and approaches for deepening understand in the temperature distribution in tumor tissue and surrounding healthy tissue during the laser combined with photosensitizer. The simulation model could provide guidance and reference function for the effect of photothermal therapy.

Numerical modelling of skin tumour tissue with temperature-dependent properties for dynamic thermography

Dynamic thermography has been clinically proven to be a valuable diagnostic technique for skin tumour detection as well as for other medical applications, and shows many advantages over static thermography. Numerical modelling of heat transfer phenomena in biological tissue during dynamic thermography can aid the technique by improving process parameters or by estimating unknown tissue parameters based on measurement data. This paper presents a new non-linear numerical model of multilayer skin tissue containing a skin tumour together with thermoregulation response of the tissue during the cooling-rewarming process of dynamic thermography. The thermoregulation response is modelled by temperature-dependent blood perfusion rate and metabolic heat generation. The aim is to describe bioheat transfer more realistically. The model is based on the Pennes bioheat equation and solved numerically using a subdomain BEM approach treating the problem as axisymmetrical. The paper includes computational tests for Clark II and Clark IV tumours, comparing the models using constant and temperature-dependent properties which showed noticeable differences and highlighted the importance of using a local thermoregulation model. Results also show the advantage of using dynamic thermography for skin tumour screening and detection at an early stage. One of the contributions of this paper is a complete sensitivity analysis of 56 model parameters based on the gradient of the surface temperature difference between tumour and healthy skin. The analysis shows that size of the tumour, blood perfusion rate, thermoregulation coefficient of the tumour, body core temperature and density and specific heat of the skin layers in which the tumour is embedded are important for modelling the problem, and so have to be determined more accurately to reflect realistic skin response of the investigated tissue, while metabolic heat generation and its thermoregulation are not.

Numerical method to evaluate the survival rate of malignant cells considering the distribution of treatment temperature field for magnetic hyperthermia

Abstract Magnetic hyperthermia ablates malignant cells by heating a tumor region when magnetic nanoparticles (MNPs) are subjected to a high-frequency magnetic field. The treatment effect for magnetic hyperthermia can be evaluated by the mortality of malignant cells, which is mainly determined by the temperature profile of therapeutic target, treatment duration, Arrhenius kinetic coefficients, and also the material of MNPs. Both of these interests should be designed properly for a clinical consideration. This paper evaluates the survival rate of malignant cells by using an improved Arrhenius model, in which both treatment temperature profile and therapeutic duration are all taken into consideration. The treatment temperature profile for a proposed geometric model is predicted by solving Pennes bio-heat transfer equation using finite element method, in which a heat source in form of a Gaussian distribution is used as the power dissipation of MNPs. In addition, a kind of MNPs with low Curie temperature is considered to further improve the therapeutic effect for magnetic hyperthermia in this paper. Simulation results demonstrate that the temperature profile considering the MNPs with low Curie temperature can be more homogeneous than the case using Fe 3 O 4 particles if their power dissipation are increased properly with respect to a critical value for the Fe 3 O 4 particles. This better distribution in treatment temperature will then result in an obviously positive effects in malignant ablation for magnetic hyperthermia. Meanwhile, results also show that the survival rate of tumor cells is not only relevant to the therapeutic temperature and duration, but also depends more on the kinetic parameters due to the individual characteristic of malignant cells.