Graded meshes in bio-thermal problems with transmission-line modeling method

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.…”

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

“…It is important to note that, with minor modifications, Equations 3 to 15 can be simplified to the previously published formulations. [11][12][13][14][15][16][17][18][19] 2.2 | Scattering, connection, boundary conditions, initial conditions, temperature calculation, and heat flux calculation Scattering is computed as the resultant voltage in the impedances Z n of the Thévenin's equivalent circuit ( Figure 3) of the general TLM node circuit ( Figure 2):…”

“…[7][8][9] Because of these advantages, TLM has become an increasingly popular method to solve problems in heat transfer in biological tissues. [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] The limitation of the TLM method is that, for every control volume geometry, a new circuit node must be developed. The first developed TLM node formulation was for regular geometries (squares and cubes) 15,16 and was later expanded to include graded elements (rectangles and parallelepipeds) 11 and orthogonal curvilinear elements (such as cylinders).…”

“…[10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] The limitation of the TLM method is that, for every control volume geometry, a new circuit node must be developed. The first developed TLM node formulation was for regular geometries (squares and cubes) 15,16 and was later expanded to include graded elements (rectangles and parallelepipeds) 11 and orthogonal curvilinear elements (such as cylinders). 29,30 However, bio-heat-transfer problems have inherently complex geometries that are difficult to accurately model using the node geometries previously described.…”