Tetrahedral node for Transmission-Line Modeling (TLM) applied to Bio-heat Transfer

Abstract

Transmission-Line Modeling (TLM) is a numerical method used to solve complex and time-domain bio-heat transfer problems. In TLM, parallelepipeds are used to discretize three-dimensional problems. The drawback in using parallelepiped shapes is that instead of refining only the domain of interest, a large additional domain would also have to be refined, which results in increased computational time and memory space. In this paper, we developed a tetrahedral node for TLM applied to bio-heat transfer that does not have the drawback associated with the parallelepiped node. The model includes heat source, blood perfusion, boundary conditions and initial conditions. The boundary conditions could be adiabatic, temperature, heat flux, or convection. The predicted temperature and heat flux were compared against results from an analytical solution and the results agreed within 2% for a mesh size of 69,941 nodes and a time step of 5ms. The method was further validated against published results of maximum skin-surface temperature difference in a breast with and without tumor and the results agreed within 6%. The published results were obtained from a model that used parallelepiped TLM node. An open source software, TLMBHT, was written using the theory developed herein and is available for download free-of-charge.

General node for transmission‐line modeling (TLM) method applied to bio‐heat transfer

“…This paper presents a general TLM formulation for bio‐heat transfer that can be simplified to previously published formulations . This formulation assumes that the unit vector normal to the border of the node is similar to the unit vector pointing from the center of the node to the midpoint of the border (Equation ), which becomes exact for regular geometries such as rectangles and parallelepipeds.…”

“…Because the formulation described in this study reduces to published formulations, it can be said that the following node geometries are already validated: lines and graded lines in 1‐D; squares, rectangles, and triangles in 2‐D; cubes, parallelepipeds, and tetrahedrons in 3‐D. The validation performed in this paper is for non‐previously described node elements, which are irregular quadrangle in 2‐D (Figure A), irregular hexahedron in 3‐D (Figure B), and pyramids in 3‐D (Figure C).…”

“…It is important to note that, with minor modifications, Equations to can be simplified to the previously published formulations …”

“…To validate irregular hexahedrons and pyramids, temperature and heat flux computed using TLM (Tables and ) were compared with values obtained from an analytical solution of the problem shown in Figure B (for the analytical solution, refer to Milan and Gebremedhin) with the following parameters: L = 1 mm, H = 0.75 mm, T z = 0.375 mm, ρ = 1200 kg/m 3 , c = 3200 J/(kg °C), k = 0.3 W/(m °C), ω b = 10 −4 s −1 , ρ b = 1052 kg/m 3 , c b = 3600 J/(kg °C), T b = 37°C, q ′ ′ ′ = 500 W/m 3 , T c = 37°C, T ∞ = 100°C, = 10 000 W/m 2 , and = 100 000 W/m 2 .…”

“…The general TLM node was used to solve the effect of the presence of a breast tumor on skin‐surface temperature of a tissue. The problem was solved using regular and irregular hexahedrons, and the results were compared with values obtained previously using tetrahedrons and cubes . The problem geometry is shown in Figure (which is the same geometry given in Figure 5 of Milan and Gebremedhin and Figure 11 of Amri et al) and has the following dimensions: subcutaneous fat thickness, T fat = 5 mm; gland thickness, H = 45 mm; side lengths, L = 20 mm; tumor center depth = 20 mm; tumor diameter, D , varying from 10 mm to 30 mm in steps of 2.5 mm.…”