Exact solutions to the multi-region time-dependent bioheat equation with transient heat sources and boundary conditions

Numerical simulation in magnetic resonance imaging radiofrequency dosimetry

Magnetic Resonance Imaging (MRI) employs a radiofrequency electromagnetic field to create pictures on a computer. The prospective biological consequences of exposure to radiofrequency electromagnetic fields (RF EMFs) have not yet been demonstrated, and there is not enough evidence on biological hazards to offer a definite response concerning possible RF health dangers. Therefore, it is crucial to research the health concerns in reaction to RF EMFs, considering the entire exposure in terms of patients receiving MRI. Monitoring increases in temperaturein-vivothroughout MRI scan is extremely invasive and has resulted in a rise in the utilization of computational methods to estimate distributions of temperatures. The purpose of this study is to estimate the absorbed power of the brain exposed to RF in patients undergoing brain MRI scan. A three-dimensional Penne's bio-heat equation was modified to computationally analyze the temperature distributions and potential thermal effects within the brain during MRI scans in the 0.3 T to 1.5 T range (12.77 MHz to 63.87 MHz). The instantaneous temperature distributions of thein-vivotissue in the brain temperatures measured at a time, t = 20.62 s is 0.2 °C and t = 30.92 s is 0.4 °C, while the highest temperatures recorded at 1.03 min and 2.06 min were 0.4 °C and 0.6 °C accordingly. From the temperature distributions of thein-vivotissue in the brain temperatures measured, there is heat build-up in patients who are exposed to electromagnetic frequency ranges, and, consequently, temperature increases within patients are difficult to prevent. The study has, however, indicated that lengthier imaging duration appears to be related to increasing body temperature.

Special function form solutions of multi-parameter generalized Mittag-Leffler kernel based bio-heat fractional order model subject to thermal memory shocks

The primary objective of this research is to develop a mathematical model, analyze the dynamic occurrence of thermal shock and exploration of how thermal memory with moving line impact of heat transfer within biological tissues. An extended version of the Pennes equation as its foundational framework, a new fractional modelling approach called the Prabhakar fractional operator to investigate and a novel time-fractional interpretation of Fourier's law that incorporates its historical behaviour. This fractional operator has multi parameter generalized Mittag-Leffler kernel. The fractional formulation of heat flow, achieved through a generalized fractional operator with a non-singular type kernel, enables the representation of the finite propagation speed of heat waves. Furthermore, the dynamics of thermal source continually generates a linear thermal shock at predefined locations within the tissue. Introduced the appropriate set of variables to transform the governing equations into dimensionless form. Laplace transform (LT) is operated on the fractional system of equations and results are presented in series form and also expressed the solution in the form of special functions. The article derives analytical solutions for the heat transfer phenomena of both the generalized model, in the Laplace domain, and the ordinary model in the real domain, employing Laplace inverse transformation. The pertinent parameter's influence, such as α, β, γ, a0, b0, to gain insights into the impact of the thermal memory parameter on heat transfer, is brought under consideration to reveal the interesting results with graphical representations of the findings.

Temperature-controlled power modulation compensates for heterogeneous nanoparticle distributions: a computational optimization analysis for magnetic hyperthermia

Purpose:

To study, with computational models, the utility of power modulation to reduce tissue temperature heterogeneity for variable nanoparticle distributions in magnetic nanoparticle hyperthermia.

Methods:

Tumour and surrounding tissue were modeled by elliptical two- and three-dimensional computational phantoms having six different nanoparticle distributions. Nanoparticles were modeled as point heat sources having amplitude-dependent loss power. The total number of nanoparticles was fixed, and their spatial distribution and heat output were varied. Heat transfer was computed by solving the Pennes’ bioheat equation using finite element methods (FEM) with temperature-dependent blood perfusion. Local temperature was regulated using a proportional-integral-derivative (PID) controller. Tissue temperature, thermal dose and tissue damage were calculated. The required minimum thermal dose delivered to the tumor was kept constant, and heating power was adjusted for comparison of both the heating methods.

Results:

Modulated power heating produced lower and more homogeneous temperature distributions than did constant power heating for all studied nanoparticle distributions. For a concentrated nanoparticle distribution, located off-center within the tumor, the maximum temperatures inside the tumor were 16% lower for modulated power heating when compared to constant power heating. This resulted in less damage to surrounding normal tissue. Modulated power heating reached target thermal doses up to nine-fold more rapidly when compared to constant power heating.

Conclusions:

Controlling the temperature at the tumor-healthy tissue boundary by modulating the heating power of magnetic nanoparticles demonstrably compensates for a variable nanoparticle distribution to deliver effective treatment.

Bioheat transfer problem for one-dimensional spherical biological tissues

Based on the Pennes bioheat transfer equation with constant blood perfusion, we set up a simplified one-dimensional bioheat transfer model of the spherical living biological tissues for application in bioheat transfer problems. Using the method of separation of variables, we present in a simple way the analytical solution of the problem. The obtained exact solution is used to investigate the effects of tissue properties, the cooling medium temperature, and the point-heating on the temperature distribution in living bodies. The obtained analytical solution can be useful for investigating thermal behavior research of biological system, thermal parameter measurements, temperature field reconstruction and clinical treatment.

Thermal Distribution of Ultrasound Waves in Prostate Tumor: Comparison of Computational Modeling with In Vivo Experiments

Ultrasound irradiation to a certain site of the body affects the efficacy of drug delivery through changes in the permeability of cell membrane. Temperature increase in irradiated area may be affected by frequency, intensity, period of ultrasound, and blood perfusion. The aim of present study is to use computer simulation and offer an appropriate model for thermal distribution profile in prostate tumor. Moreover, computer model was validated by in vivo experiments. Method. Computer simulation was performed with COMSOL software. Experiments were carried out on prostate tumor induced in nude mice (DU145 cell line originated from human prostate cancer) at frequency of 3 MHz and intensities of 0.3, 0.5, and 1 w/cm2 for 300 seconds. Results. Computer simulations showed a temperature rise of the tumor for the applied intensities of 0.3, 0.5 and 1 w/cm2 of 0.8, 0.9, and 1.1°C, respectively. The experimental data carried out at the same frequency demonstrated that temperature increase was 0.5, 0.9, and 1.4°C for the above intensities. It was noticed that temperature rise was very sharp for the first few seconds of ultrasound irradiation and then increased moderately. Conclusion. Obtained data holds great promise to develop a model which is able to predict temperature distribution profile in vivo condition.

Study of the one dimensional and transient bioheat transfer equation: multi-layer solution development and applications

In this work we derive an analytical solution given by Bessel series to the transient and one-dimensional (1D) bioheat transfer equation in a multi-layer region with spatially dependent heat sources. Each region represents an independent biological tissue characterized by temperature-invariant physiological parameters and a linearly temperature dependent metabolic heat generation. Moreover, 1D Cartesian, cylindrical or spherical coordinates are used to define the geometry and temperature boundary conditions of first, second and third kinds are assumed at the inner and outer surfaces. We present two examples of clinical applications for the developed solution. In the first one, we investigate two different heat source terms to simulate the heating in a tumor and its surrounding tissue, induced during a magnetic fluid hyperthermia technique used for cancer treatment. To obtain an accurate analytical solution, we determine the error associated with the truncated Bessel series that defines the transient solution. In the second application, we explore the potential of this model to study the effect of different environmental conditions in a multi-layered human head model (brain, bone and scalp). The convective heat transfer effect of a large blood vessel located inside the brain is also investigated. The results are further compared with a numerical solution obtained by the Finite Element Method and computed with COMSOL Multiphysics v4.1©.

Solution to the bioheat equation for hyperthermia with La(1-x)Ag(y)MnO(3-delta) nanoparticles: the effect of temperature autostabilization

This work aimed to analyze the possibility and performance of the temperature controlled hyperthermia based on AC heating of magnetic nanoparticles with low Curie temperature. Temperature dependence of dynamic magnetic susceptibility has been studied experimentally on fine powders of La(0.8)Ag(0.15)MnO(2.95) in the frequency range of 0.5-2.0 MHz. Critical drop of the AC magnetic losses was found in the vicinity of the Curie point, T(C) = 42 degrees C. The obtained data was used in the numerical analysis of the bioheat equations under typical conditions of the hyperthermia treatment. The mathematical model includes a spherical tumor containing magnetic particles and surrounded by concentric healthy tissue, with account made for the blood perfusion. The calculations performed for various AC power, tumor sizes and doping geometries predict effective autostabilization of the temperature at T congruent with T(C) inside the tumor and steep temperature profile at the interface with the healthy tissue.

Transient solution to the bioheat equation and optimization for magnetic fluid hyperthermia treatment

Two finite concentric spherical regions were considered as the tissue model for magnetic fluid hyperthermia treatment. The inner sphere represents the diseased tissue containing magnetic particles that generate heat when an alternating magnetic field is applied. The outer sphere represents the healthy tissue. Blood perfusion effects are included in both the regions. Analytical and numerical solutions of the one-dimensional bioheat transfer equation were obtained with constant and spatially varying heat generation in the inner sphere. The numerical solution was found to be in good agreement with the analytical solution. In an ideal hyperthermia treatment, all the diseased tissues should be selectively heated without affecting any healthy tissue. The present work optimized the magnetic particle concentration in an attempt to achieve the ideal hyperthermia conditions. It was found that, for a fixed amount of magnetic particles, optimizing the magnetic particle distribution in the diseased tissue can significantly enhance the therapeutic temperature levels in the diseased tissue while maintaining the same level of heating in the healthy tissue.

Phenomenological model of diffuse global and regional atrophy using finite-element methods

The main goal of this work is the generation of ground-truth data for the validation of atrophy measurement techniques, commonly used in the study of neurodegenerative diseases such as dementia. Several techniques have been used to measure atrophy in cross-sectional and longitudinal studies, but it is extremely difficult to compare their performance since they have been applied to different patient populations. Furthermore, assessment of performance based on phantom measurements or simple scaled images overestimates these techniques' ability to capture the complexity of neurodegeneration of the human brain. We propose a method for atrophy simulation in structural magnetic resonance (MR) images based on finite-element methods. The method produces cohorts of brain images with known change that is physically and clinically plausible, providing data for objective evaluation of atrophy measurement techniques. Atrophy is simulated in different tissue compartments or in different neuroanatomical structures with a phenomenological model. This model of diffuse global and regional atrophy is based on volumetric measurements such as the brain or the hippocampus, from patients with known disease and guided by clinical knowledge of the relative pathological involvement of regions and tissues. The consequent biomechanical readjustment of structures is modelled using conventional physics-based techniques based on biomechanical tissue properties and simulating plausible tissue deformations with finite-element methods. A thermoelastic model of tissue deformation is employed, controlling the rate of progression of atrophy by means of a set of thermal coefficients, each one corresponding to a different type of tissue. Tissue characterization is performed by means of the meshing of a labelled brain atlas, creating a reference volumetric mesh that will be introduced to a finite-element solver to create the simulated deformations. Preliminary work on the simulation of acquisition artefacts is also presented. Cross-sectional and longitudinal sets of simulated data are shown and a visual classification protocol has been used by experts to rate real and simulated scans according to their degree of atrophy. Results confirm the potential of the proposed methodology.

Three-dimensional model on thermal response of skin subject to laser heating

A three-dimensional (3D) multilayer model based on the skin physical structure is developed to investigate the transient thermal response of human skin subject to laser heating. The temperature distribution of the skin is modeled by the bioheat transfer equation, and the influence of laser heating is expressed as a source term where the strength of the source is a product of a Gaussian shaped incident irradiance, an exponentially shaped axial attenuation, and a time function. The water evaporation and diffusion is included in the model by adding two terms regarding the heat loss due to the evaporation and diffusion, where the rate of water evaporation is determined based on the theory of laminar boundary layer. Cryogen spray cooling (CSC) in laser therapy is studied, as well as its effect on the skin thermal response. The time-dependent equation is discretized using the finite difference method with the Crank-Nicholson scheme and the stability of the numerical method is analyzed. The large sparse linear system resulted from discretizing the governing partial differential equation is solved by a GMRES solver and the expected simulation results are obtained.

An inverse problem approach to the estimation of volume change

We present a new technique for determining structure-by-structure volume changes, using an inverse problem approach. Given a pre-labelled brain and a series of images at different time-points, we generate finite element meshes from the image data, with volume change modelled by means of an unknown coefficient of expansion on a per-structure basis. We can then determine the volume change in each structure of interest using inverse problem optimization techniques. The proposed method has been tested with simulated and clinical data. Results suggest that the presented technique can be seen as an alternative for volume change estimation.

A fully coupled binary biochemical reactive-diffusion model with analytic solution

Coupled multicomponent biochemical reactive diffusion underlies a variety of biological signalling processes and pharmacokinetic applications, such as paracrine signalling involving "cocktails" comprised of growth promoter/inhibitor factors and proteases associated with tumor angiogenesis, invasion and metastasis, extravascular drug delivery, and polymeric controlled-release drug codelivery design. Here, we present a model and develop a new analytic solution to illustrate the spatiotemporal behavior associated with fully coupled binary biochemical reactive diffusion. The complete coupling renders the solution appreciably more complex in structure and behavior than solutions for unicomponent or partially coupled models. Concentration behavior is illustrated by the computational simulation of binary-species tumor angiogenesis factor reactive-diffusion in the extravascular tissue matrix. The computational results indicate that (a) steady-state concentration profiles are achieved within 1 h of a change in factor production; (b) in the steady state, the spatial profiles of the two components tend to be similar; (c) exceedingly steep concentration gradients, involving several orders-of-magnitude differences in concentration over a few tenths of a millimeter, can occur in the vicinity of boundary sources due to inter-species reaction; (d) the concentration profiles of the two species differ from unicomponent predictions due to the simultaneous mass interchange between the two species. The analytic solution predictions are also used to provide a first-ever validation of a time-dependent, binary-component Crank-Nicholson numerical solution. The ability to quantitatively model interacting and often strongly varying concentration levels as a function of time and position can serve as a powerful complementary tool to experimental analyses for assessing disease state and interventional pharmacological efficacy, especially when the spatial scales on which in vivo behavior occurs taxes the limits of imaging capabilities.

Analytical study on bioheat transfer problems with spatial or transient heating on skin surface or inside biological bodies

Several closed form analytical solutions to the bioheat transfer problems with space or transient heating on skin surface or inside biological bodies were obtained using Green's function method. The solutions were applied to study several selected typical bioheat transfer processes, which are often encountered in cancer hyperthermia, laser surgery, thermal comfort analysis, and tissue thermal parameter estimation. Thus a straightforward way to quantitatively interpret the temperature behavior of living tissues subject to constant, sinusoidal, step, point or stochastic heatings etc. both in volume and on boundary were established. Further solution to the three-dimensional bioheat transfer problems was also given to illustrate the versatility of the present method. Implementations of this study to the practical problems were addressed.