Analytical Solution to the Transient 1D Bioheat Equation in a Multilayer Region with Spatial Dependent Heat Sources

2011 ◽  
Author(s):  
Dário B. Rodrigues ◽  
Pedro J. d.S. Pereira ◽  
Paulo M. Limão-Vieira ◽  
Paolo F. Maccarini
Keyword(s):  
Analytical Solution ◽  
Heat Sources ◽  
Bioheat Equation

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

2019 ◽  
Vol 65 (3) ◽  
pp. 284
Author(s):  
E. J. Roca Oria ◽  
L. E. Bergues Cabrales ◽  
And J. Bory Reyes
Keyword(s):  
Analytical Solution ◽  
Initial Temperature ◽  
Electrode Array ◽  
Thermal Response ◽  
Electrochemical Treatment ◽  
Bioheat Transfer ◽  
Heat Sources ◽  
Bioheat Equation ◽  
Multiple Tumor ◽  
Temperature Distributions

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.

scholarly journals Analytical solution of one-dimensional Pennes’ bioheat equation

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 1084-1092
Author(s):  
Hongyun Wang ◽  
Wesley A. Burgei ◽  
Hong Zhou
Keyword(s):  
Analytical Solution ◽  
Radiofrequency Energy ◽  
Bioheat Equation ◽  
Surface Cooling ◽  
One Dimensional ◽  
The Fourier Transform ◽  
The One ◽  
Parameter Values ◽  
Mathematical Derivation ◽  
Effect Of Surface

Abstract Pennes’ bioheat equation is the most widely used thermal model for studying heat transfer in biological systems exposed to radiofrequency energy. In their article, “Effect of Surface Cooling and Blood Flow on the Microwave Heating of Tissue,” Foster et al. published an analytical solution to the one-dimensional (1-D) problem, obtained using the Fourier transform. However, their article did not offer any details of the derivation. In this work, we revisit the 1-D problem and provide a comprehensive mathematical derivation of an analytical solution. Our result corrects an error in Foster’s solution which might be a typo in their article. Unlike Foster et al., we integrate the partial differential equation directly. The expression of solution has several apparent singularities for certain parameter values where the physical problem is not expected to be singular. We show that all these singularities are removable, and we derive alternative non-singular formulas. Finally, we extend our analysis to write out an analytical solution of the 1-D bioheat equation for the case of multiple electromagnetic heating pulses.

scholarly journals Exact Analytical Solution for 3D Time-Dependent Heat Conduction in a Multilayer Sphere with Heat Sources Using Eigenfunction Expansion Method

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Nemat Dalir
Keyword(s):  
Heat Conduction ◽  
Analytical Solution ◽  
Eigenfunction Expansion ◽  
Radial Direction ◽  
Exact Analytical Solution ◽  
Internal Heat ◽  
Expansion Method ◽  
Time Dependent ◽  
Heat Sources ◽  
Eigenfunction Expansion Method

An exact analytical solution is obtained for the problem of three-dimensional transient heat conduction in the multilayered sphere. The sphere has multiple layers in the radial direction and, in each layer, time-dependent and spatially nonuniform volumetric internal heat sources are considered. To obtain the temperature distribution, the eigenfunction expansion method is used. An arbitrary combination of homogenous boundary condition of the first or second kind can be applied in the angular and azimuthal directions. Nevertheless, solution is valid for nonhomogeneous boundary conditions of the third kind (convection) in the radial direction. A case study problem for the three-layer quarter-spherical region is solved and the results are discussed.

Analytical solution for laser short-pulse heating of two-dimensional solids: volumetric and surface heat source considerations

2013 ◽  
Vol 91 (7) ◽  
pp. 522-529
Author(s):  
B.S. Yilbas ◽  
A.Y. Al-Dweik
Keyword(s):  
Analytical Solution ◽  
Heat Source ◽  
Energy Transport ◽  
Pulse Heating ◽  
Short Pulse ◽  
Thermal Response ◽  
Solid Substrate ◽  
Surface Heat ◽  
Heat Sources ◽  
Power Intensity

An analytical solution for lattice temperature distribution in a metallic solid subjected to laser short-pulse heating is presented. The method of similarity solution is adopted for the solution of the diffusive–ballistic energy equation. Volumetric and surface heat sources are each incorporated separately in the analysis. The material thermal response due to both heat sources during the short heating period is analyzed. It is found that a volumetric heat source resulted in smaller temperature increase in the irradiated material than a surface heat source, despite the same laser power intensity being used in both cases. This is attributed to energy transport mechanisms taking place in the solid substrate due to volumetric and surface heat sources.

Analytical Solution for the Prediction of Temperature Distributions During Radio-Frequency Ablation of Cardiac Tissue

2003 ◽  
Author(s):  
Ryan T. Roper ◽  
Matthew R. Jones
Keyword(s):  
Analytical Solution ◽  
Radio Frequency ◽  
Numerical Solutions ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Tissue Temperature ◽  
Bench Mark ◽  
Bioheat Equation ◽  
Surface Integral ◽  
Temperature Distributions

One of the most effective methods of treatment for cardiac arrhythmias is radio-frequency (RF) ablation. Many studies have shown that the tissue temperature distribution is the key factor influencing lesion shape and size, and that accurate prediction of this distribution is essential to the further improvement of the procedure. Temperature distributions can be obtained by solving the bioheat equation, which has been done in several studies using numerical techniques. This paper describes the development of an analytical solution that can be used as a bench mark for subsequent numerical solutions. Using integral transforms, the bioheat equation is reduced to an ordinary differential equation with time as the independent variable. The solution has the form of a surface integral within another surface integral. An integration routine that extends the trapezoidal method of integration in two dimensions to an analogous method in three dimensions has been developed in order to evaluate the analytical solution. A C program was written to implement this method, and the program was validated using a surface integral with a known analytical solution. The program was then used to generate temperature profiles at various time values and for different convection coefficients.

scholarly journals A computational algorithm and the method of determining the temperature field along the length of the rod of variable cross section

2018 ◽  
Vol 8 (1) ◽  
pp. 170-175 ◽  
Author(s):  
Anarbay Kudaykulov ◽  
A. A. Tashev ◽  
A. Askarova
Keyword(s):  
Temperature Field ◽  
Analytical Solution ◽  
Cross Section ◽  
Computational Algorithm ◽  
Approximate Analytical Solution ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Heat Sources ◽  
Reliable Operation ◽  
Limited Length

Abstract Bearing elements of a number of strategic equipment are limited length rods with a variable cross-section. Most of them are exposed to certain types of heat sources. To ensure reliable operation of these equipment it is necessary to know the temperature field along the length of the rod with a variable cross section. This paper proposes a computational algorithm and method to determine the temperature field along the length of the rod with limited length and variable cross-section. They are based on fundamental laws of energy conservation. Also obtained is an approximate analytical solution of the problem.

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