Bioheat Transfer Problems Research Articles

Deep operator networks for bioheat transfer problems with parameterized laser source functions

To numerically characterize optothermal interactions at the laser-tissue interface during tissue imaging or surgery, the bioheat transfer partial differential equation (PDE) needs to be solved with a laser excitation source function. Considering various possible experimental scenarios, the PDEs become parametric and need to be solved for several optical and laser parameters. Classical numerical solvers that are used for performing such simulations are computationally expensive and time consuming. In this work, we use the recent deep operator network (DeepONet) framework for learning the mapping between infinite dimensional function spaces of the parameterized laser source functions and the solutions of the parameterized bioheat transfer PDE. We propose a new neural network architecture, Res-DeepONet, for training a purely physics-informed DeepONet with laser heating functions comprising of Gaussian beam profiles. We also showcase the versatility and robustness of our network architecture within purely data-driven and hybrid training frameworks for applications where labeled ground truth data may be accessible. The proposed neural network architecture exhibits excellent generalization performance on unseen Gaussian source functions with 10× higher accuracy when compared to traditional feedforward network architectures. Furthermore, zero-shot performance studies highlight the exceptional extrapolation capabilities of our network with 50× higher accuracy compared to traditional DeepONet architectures at inference times that are two orders of magnitude faster than classical solvers.

Magnetic resonance imaging of blood perfusion rate based on Helmholtz decomposition of heat flux.

Objective.Thermal property (TP) maps of human tissues are useful for tumor treatment and diagnosis. In particular, the blood perfusion rate is significantly different for tumors and healthy tissues. Noninvasive techniques that reconstruct TPs from the temperature measured via magnetic resonance imaging (MRI) by solving an inverse bioheat transfer problem have been developed. A few conventional methods can reconstruct spatially varying TP distributions, but they have several limitations. First, most methods require the numerical Laplacian computation of the temperature, and hence they are sensitive to noise. In addition, some methods require the division of a region of interest (ROI) into sub-regions with homogeneous TPs using prior anatomical information, and they assume an unmeasurable initial temperature distribution. We propose a novel robust reconstruction method without the division of an ROI or the assumption of an initial temperature distribution.Approach.The proposed method estimates blood perfusion rate maps from relative temperature changes. This method avoids the computation of the Laplacian by using integral representations of the Helmholtz decomposition of the heat flux.Main Result.We compare the reconstruction results of the conventional and proposed methods using numerical simulations. The results indicate the robustness of the proposed method.Significance.This study suggests the feasibility of thermal property mapping with MRI using the robust proposed method.

Detection of multiple tumors through temperature response of a human brain using inverse bioheat transfer based on swarm optimization

In this article, an innovative inverse regularized framework has been proposed for a bio-medical problem to detect tumors in the human brain via temperature. Most of the thermal data obtained through infrared thermography is sensitive towards errors, thus a regularized approach is sought from the literature. The best inversion algorithm, WOA, together with a hybrid GWOCS are utilized. Pennes model is used for the formulation of heat transfer within the human brain. Using inverse analysis, the unknown blood perfusion rate ωi is retrieved in the regularized environment. The current research reported that at positions where tumor cells are present, the temperature rises. Moreover, with an increase in time, the temperature of the cancer cells increases, whereas no change in the temperature of tissue without tumor is seen. This observation marks the presence of tumor. Elastic net regularization (λ=0.1, α=10−2) for WOA and (λ=0.9, α=10−4) for GWOCS shows the least relative error. After regularization, the perfusion rate is retrieved. A clear distinction of two tumors from the brain tissue is observed. The obtained perfusion rate is used to reconstruct the temperature profile. An excellent matching of the reconstructed temperature field and the exact field is obtained, even when the forward data contain measurement errors. The obtained set of perfusion rate is appropriate for tumor detection, up to 7% error in the measured data. Thus, the proposed comparative procedure is found suitable for the bio-heat transfer problem.

A closed‐form solution of DPL bioheat transfer problem with time‐periodic boundary conditions

AbstractA closed‐form (long‐time) solution of one‐dimensional dual‐phase lag bioheat transfer problem with consistent time‐periodic boundary conditions (BCs) is presented in this paper for planar, cylindrical, and spherical skin tissue for a newly developed solution methodology. The steady‐periodic solution is composed of a steady‐state part and an oscillating part; corresponding to the constant and oscillating parts of BCs, respectively. Using the superposition principle, these two parts are split into two problems, which are solved separately. The steady‐state part is fairly straightforward to obtain, while for the oscillating part, an alternate Laplace transform (LT) approach is proposed in this work. It is demonstrated that for sinusoidal BCs, a closed‐form solution in the time domain can be obtained by evaluating an approximate convolution integral, which emulates the effect of the inverse LT. The obtained closed‐form solution is free of any series summation or numerical inversion, thereby, making it computationally very efficient compared with conventional LT and eigenfunctions‐based approaches. The current methodology is verified with the established eigenfunctions expansion‐based methodology. It can be seen that the long‐time solutions obtained by these two approaches are almost identical. The verified methodology is further extended for the time‐periodic nonsinusoidal BCs. The ease of implementation and simplicity of the new methodology for both sinusoidal and nonsinusoidal BCs is demonstrated using a few test cases. It is evident from the results that the developed methodology leads to an efficient and accurate solution.

Analysis under Uncertainty with the Monte Carlo Method Applied to a Bioheat Transfer Problem with Coupled Deep Brain Stimulation Lead

This article deals with an analysis of uncertainties applied to a bioheat transfer problem containing a deep brain stimulation lead. The classic two-dimensional bioheat transfer equation in cylindrical coordinates was considered in the mathematical formulation. The electric potential was solved with a Laplace equation to incorporate the DBS lead effects. Thus, the solution for the electric potential was coupled to the temperature problem, considering an external heat transfer rate. The analysis under uncertainties was performed by the Monte Carlo method considering different types of uncertainties for all parameters of the mathematical model. The uncertainties were chosen according to the information available in the literature in order to analyze the problem more realistically. The solutions showed a significant variation in the temperature profile over time when considering the random variations in the parameters.

The cooling effect of blood flow during hyperthermia treatment

The amount and duration of the applied heat in hyperthermia treatment are critical for cancer survivors. The challenge is to use a mechanism dealing with the tumor cells only while keeping healthy tissues unharmed. The aim of this paper is to predict the blood temperature distribution in main dimensions during hyperthermia process by deriving a new analytical solution of unsteady flow that adequately covers the cooling factor. We adopted a separation of variable method to solve the bio-heat transfer problem of unsteady blood flow. The solution is similar to Pennes’ equation, except that it is for blood rather than tissue. We also performed computational simulations with varied flow conditions and thermal energy transports. The blood cooling effects were calculated with vessel’s diameter, tumor’s zone length, pulsating period and flow velocity. The cooling rate rises by around 133% if the tumor zone’s length is extended four times the diameter of 0.5 (mm), but it is seemingly fixed with this distance if the diameter is equal or larger than 4 (mm). Likewise, the temporal variations of temperature disappear if the blood vessel has a diameter of 4 (mm) or more. Pre-heating or post-cooling techniques perform effectively given the theoretical solution; under particular conditions, the reduction percentages of the cooling effect are between 130% and 200%, respectively.

Kalman filter temperature estimation with a photoacoustic observation model during the hyperthermia treatment of cancer

Hyperthermia with laser heating can be a noninvasive treatment for tumors near the body surface. Thermal damage of tissues depends on temperature levels and time of exposure to high temperatures, thus requiring accurate techniques to measure the temperature in the target region of interest. In this computational work, we couple the laser hyperthermia treatment of cancer to the photoacoustic temperature measurement of internal tissues, by solving a state estimation problem with the Kalman filter. The region of interest is considered to be continuously heated by a laser for the hyperthermia treatment, while another laser is sequentially pulsed over the same heated surface to generate acoustic pressure waves. The observation model is derived from the photoacoustic problem, which is linear with respect to the local internal temperatures of the tissues. The evolution model given by the bioheat transfer problem is also supposed linear and uncertainties are modeled as additive and Gaussian. Therefore, the Kalman filter provides the optimal solution of the state estimation problem considered in this work. Simulated nonintrusive acoustic pressure measurements are used in inverse analyses for two test cases, with different dimensions and locations of a tumor in a two-dimensional region. The simulated tumors are supposed loaded with nanoparticles, in order to enhance the local absorption of the laser energy and to reduce thermal damage to healthy cells. The Kalman filter estimated accurate and stable temperature distributions, even in the tumor region that involved large gradients. Moreover, uncertainties in the Kalman filter estimates were smaller than those resulting from the direct simulation of the photoacoustic thermometry problem.

An inverse analysis of the brain cooling process in neonates using the particle filter method

Purpose This study deals with the computational simulation and inverse analysis of the cooling treatment of the hypoxic-ischemic encephalopathy in neonates. A reduced-order model is implemented for real-time monitoring of the internal body temperatures. The purpose of this study is to sequentially estimate the transient temperatures of the brain and other body regions with reduced uncertainties. Design/methodology/approach Pennes’ model was applied in each body element, and Fiala’s blood pool concept was used for the solution of the forward bioheat transfer problem. A state estimation problem was solved with the Sampling Importance Resampling (SIR) algorithm of the particle filter method. Findings The particle filter method was stable and accurate for the estimation of the internal body temperatures, even in situations involving large modeling and measurement uncertainties. Research limitations/implications The proposed reduced-order model was verified with the results of a high-fidelity model available in the literature. Validation of the proposed model and of the solution of the state estimation problem shall be pursued in the future. Practical implications The solution of the state estimation problem with the reduced-order model presented in this paper has great potential to perform as an observer of the brain temperature of neonates, for the analysis and control of the systemic cooling treatment of neonatal hypoxic-ischemic encephalopathy. Social implications The main treatment for hypoxic-ischemic encephalopathy in neonates is the cooling of affected regions. Accurate and fast models might allow the development of individualized protocols, as well as control strategies for the cooling treatment. Originality/value This paper presents the application of the SIR algorithm for the solution of a state problem during the systemic cooling of a neonate for the treatment of the hypoxic-ischemic encephalopathy.

Solving bio-heat transfer multi-layer equation using Green’s Functions method

An analytical method using Green’s Functions for obtaining solutions in bio-heat transfer problems, modeled by Pennes’ Equation, is presented. Mathematical background on how treating Pennes’ equation and its μ2T term is shown, and two contributions to the classical numbering system in heat conduction are proposed: inclusion of terms to specify the presence of the fin term, μ2T, and identify the biological heat transfer problem. The presentation of the solution is made for a general multi-layer domain, deriving and showing general approaches and Green’s Functions for such n number of layers. Numerical examples are presented to simplify human skin as a two-layer domain: dermis and epidermis, accounting metabolism as a heat source, and blood perfusion only at the dermis. Time-independent summations in the series-solution are written in closed forms, leading to better convergence along the boundaries. Details on obtaining the two-layer solution and its eigenvalues are presented for boundary conditions of prescribed temperature inside the body and convection at the surface, such as its intrinsic verification.

Thermal Effects of 5G Frequency EM Waves on Ocular Tissue

This research article aims to investigate the thermal effects on human eye due to exposure of EM (electromagnetic) field at 5G frequencies. A secondary purpose of the article is to show that SPICE, one of the most commonly used tools in circuit analysis, can be used to solve complex EM and bioheat transfer problems. Both electric field and temperature distributions in human eye are simulated with transmission line modelling (TLM) method. Also, finite element method (FEM) calculations are carried out for comparison. Temperature distributions are calculated for 30 minutes of EM radiation exposure. As a subject of the simulation, a human is chosen that the power density of incoming field at the human eye is assumed as the maximum of the ICNIRP general public exposure limits. A good consistency between TLM and FEM based simulations is found. A new transmission line model based on Debye parametric model is generated for simulation of EM field distribution in the tissues. With this generated transmission line, the results for the frequencies of interest can be obtained via single simulation. Also, a new transmission line model is developed for the temperature distribution simulation in which the effects of noncompressible fluid flow such as non-unidirectional blood flow in tissues and flow in aqueous humor are taken into consideration.

A new boundary element algorithm for a general solution of nonlinear space-time fractional dual-phase-lag bio-heat transfer problems during electromagnetic radiation

The main aim of this paper is to propose a new boundary element method (BEM) formulation for solving the nonlinear space-time fractional dual-phase-lag bio-heat transfer problems during electromagnetic radiation. Due to the advantages of BEM, such as not requiring a discretization of the interior of the treated region and providing a low RAM and CPU time. BEM is therefore a flexible and efficient tool for modeling bio-heat transfer problems. The efficiency of our proposed methodology has been improved by applying the communication-avoiding versions of the Arnoldi (CA-Arnoldi) preconditioner for solving the resulting linear systems arising from the BEM to reduce the iterations number and CPU time. Numerical results are depicted graphically to show the effects of time-fractional derivative order and space-fractional derivative order on the nonlinear temperature distributions. The numerical results also show the significant differences between the nonlinear temperature distributions of the classical Fourier, single-phase-lag, and dual-phase-lag bio-heat conduction models. To demonstrate the validity and accuracy of the proposed BEM methodology, numerical solutions for two-dimensional (2D) special case of the nonlinear space-time fractional dual phase lag bio-heat transfer problems are obtained and compared to experimental, Legendre wavelet collocation method (LWCM) and Fractional order Legendre functions and Galerkin method (FOLFs-GM).

Evaluation of a radial basis function node refinement algorithm applied to bioheat transfer modeling

Many bioheat transfer problems involve linear/non-linear equations with non-linear or time-dependent boundary conditions. For heat transfer problems, the presence of time and space-dependent functions under Neumann and Mixed type boundary conditions characterize trivial applications in bioengineering, such as thermotherapies, laser surgeries, and burn studies. This greatly increases the complexity of the numerical solution in several problems, requiring fast and accurate numerical solutions. This paper has a main objective evaluate an adaptive mesh refinement radial basis function method strategy for the classical Penne's bioheat transfer modeling. Our numerical results had errors of ~0.1% compared to analytical solutions. Thus, the proposed methodology is accurate and has a low computational cost. For step function heating, two RBF shape parameters were applied, again achieving excellent results. The distributions of the nodes in the solution domain show that the primary source of error in the numerical solutions came from the boundary conditions. This finding should arouse the interest of engineers and scientists in the development of new strategies for problems involving boundary conditions with periodic functions.

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.

Time-dependent lowest term estimation in a 2D bioheat transfer problem with nonlocal and convective boundary conditions

ABSTRACT A solution method for an inverse problem of determining the time-dependent lowest order coefficient of the 2D bioheat Pennes equation with nonlocal boundary conditions and total energy integral overdetermination condition recently appeared in literature is analysed and improved. Improvements include convective boundary condition into the model, the development of an accurate forward solver, accurate determination of total energy, and the proposal of a method for the numerical treatment of the inverse problem from noisy data. In the method, the 2D bioheat Pennes equation is solved by the method of lines based on a highly accurate pseudospectral approach, and sought coefficient values are estimated by the Levenberg–Marquardt method with the discrepancy principle as stopping rule. Numerical experiments are reported to illustrate effectiveness of the proposed method.

Analytical analysis of the dual-phase-lag model of bio-heat transfer with pulse boundary heat flux on skin tissue

ABSTRACT The evolution of thermal response in the biological tissue is very important to the effective medical treatment processes. The purpose of this work is to seek an analytical method to solve the transient bio-heat transfer problem. The governing equations with the pulse boundary heat flux on skin tissue are first proposed in the context of the dual-phase-lag model (DPL model). An analytical procedure, based on Laplace transform technique and its limit theorem, is then introduced to solve these equations analytically. The exact solution of the temperature distribution is finally obtained and is used to reveal the temporal and spatial effects on the transient heat transfer through skin tissues. The thermal damage is further studied and some evaluation is also conducted by comparing with the results from the classical Pennes model of bio-heat transfer (PMBHT).

Real-time temperature estimation with enhanced spatial resolution during MR-guided hyperthermia therapy

In this article, the temperature of biological tissues is estimated during hyperthermia therapy, while accounting for uncertainties in the bioheat transfer problem and in the available measurements. A state estimation problem is solved with the Steady-State Kalman Filter. The Pennes bioheat transfer model and the PRF-Shift Magnetic Resonance Thermometry are used as evolution and observation models, respectively. Instead of using the direct inversion of the measured data as with the PRF-Shift Magnetic Resonance Thermometry, the state-estimation framework allows for enhancing the spatial resolution of the estimated temperature variation and reducing the related uncertainties. Since the time consuming steps of the Steady-State Kalman Filter can be performed offline, the recursive solution of the state estimation problem is performed with computational times smaller than the simulated physical times. Synthetic measurements are used for the state estimation problem in a region of the human forearm, for radiofrequency and laser-diode heat sources of the hyperthermia therapy.

Approximate Bayesian computation applied to the identification of thermal damage of biological tissues due to laser irradiation

This paper deals with the solution of an inverse bioheat transfer problem, by using Approximate Bayesian Computation (ABC). A Sequential Monte Carlo (SMC) method is applied for simultaneous model selection and model calibration (estimation of the model parameters) by using synthetic measurements. Two competing models are considered in the analysis of the thermal damage of biological tissues. The results show that the ABC-SMC algorithm provides accurate results for the model selection and estimation of the thermal damage model parameters.

A unique technique for analytical solution of 2-D dual phase lag bio-heat transfer problem with generalized time-dependent boundary conditions

The eigenfunction-based solutions for various heat conduction models (Fourier and non-Fourier) exhibit mismatch at the boundaries due to the discrepancy between the boundary conditions (BCs) and the corresponding eigenfunction profiles. This mismatch leads to spurious oscillations and erroneous solutions near the boundaries. Recently developed finite integral transform (FIT) based analytical solution of the dual phase lag (DPL) bio-heat transfer problem with generalized time-dependent BCs in a 2-D planar skin tissue phantom also encounters the similar mismatch, especially for Dirichlet BCs. The present study aims at removing such mismatch from eigenfunction-based solutions, by the homogenization of generalized time-dependent BCs which is essentially carried out by subtracting an auxiliary function from the temperature of the domain of interest. This novel treatment yields a modified problem in terms of the modified temperature with homogeneous BCs and contains a modified source. For this, the above-mentioned auxiliary functions need to satisfy a set of conditions at the boundaries. However, the functions satisfying such conditions may not be unique. Therefore, an additional condition (pseudo-steady state condition) is imposed with a view to obtaining relatively simple, unique auxiliary functions. The current study also explores the homogenization process for 2-D bio-heat transfer problem for the tissue phantom in conjunction with the use of orthogonal eigenfunction expansion method (OEEM). The present developed solution approach is demonstrated using two test problems comprising of (a) constant surface temperature and (b) sinusoidal heat flux on the surface, respectively. In case of constant surface temperature, the present approach gets rid of large spurious oscillations near the boundary. In case of surface heat flux, temperature prediction by the present approach shows good agreement in the domain compare to that predicted by the FIT approach. Besides that, the flux distribution predicted on the boundary by present approach is a realistic non-zero value, whereas, the FIT approach produces zero heat flux for the same applied non-zero heat flux at the boundary. Moreover, for both the test problems, the developed novel approach facilitates significant reduction in number of terms in the series summation as compared to the corresponding FIT solutions, for the same level of accuracy.

Trefftz method in solving the pennes’ and single-phase-lag heat conduction problems with perfusion in the skin

Purpose The purpose of this paper is to derive the Trefftz functions (T-functions) for the Pennes’ equation and for the single-phase-lag (SPL) model (hyperbolic equation) with perfusion and then comparing field of temperature in a flat slab made of skin in the case when perfusion is taken into account, with the situation when a Fourier model is considered. When considering the process of heat conduction in the skin, one needs to take into account the average values of its thermal properties. When in biological bodies relaxation time is of the order of 20 s, the thermal wave propagation appears. The initial-boundary problems for Pennes’ model and SPL with perfusion model are considered to investigate the effect of the finite velocity of heat in the skin, perfusion and thickness of the slab on the rate of the thermal wave attenuation. As a reference model, the solution of the classic Fourier heat transfer equation for the considered problems is calculated. A heat flux has direction perpendicular to the surface of skin, considered as a flat slab. Therefore, the equations depend only on time and one spatial variable. Design/methodology/approach First of all the T-functions for the Pennes’ equation and for the SPL model with perfusion are derived. Then, an approximate solutions of the problems are expressed in the form of a linear combination of the T-functions. The T-functions satisfy the equation modeling the problem under consideration. Therefore, approximating a solution of a problem with a linear combination of n T-functions one obtains a function that satisfies the equation. The unknown coefficients of the linear combination are obtained as a result of minimization of the functional that describes an inaccuracy of satisfying the initial and boundary conditions in a mean-square sense. Findings The sets of T-functions for the Pennes’ equation and for the SPL model with perfusion are derived. An infinite set of these functions is a complete set of functions and stands for a base functions layout for the space of solutions for the equation used to generate them. Then, an approximate solutions of the initial-boundary problem have been found and compared to find out the effect of finite velocity of heat in the skin, perfusion and thickness of the slab on the rate of the thermal wave attenuation. Research limitations/implications The methods used in the literature to find an approximate solution of any bioheat transfer problems are more complicated than the one used in the presented paper. However, it should be pointed out that there is some limitation concerning the T-function method, namely, the greater number of T-function is used, the greater condition number becomes. This limitation usually can be overcome using symbolic calculations or conducting calculations with a large number of significant digits. Originality/value The T-functions for the Pennes’ equation and for the SPL equation with perfusion have been reported in this paper for the first time. In the literature, the T-functions are known for other linear partial differential equations (e.g. harmonic functions for Laplace equation), but for the first time they have been derived for the two aforementioned equations. The results are discussed with respect to practical applications.

Numerical Solution of Bioheat Transfer Problems with Transient Blood Temperature

In the heat treatment process, blood perfusion starts up a negative feedback mechanism. The blood temperature undergoes a transient process before onset of equilibrium and then changes the situation of temperature distribution. In substance, the blood temperature undergoes a transient process for heat exchange between blood and tissue. For more fully exploring the heat transfer behavior of biological tissue, this paper analyzes the bioheat transfer problems with the nonconstant blood temperature based on the Pennes bioheat equation. A numerical scheme based on the Laplace transform is proposed for solving the present problems.