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

• Published in: International journal of numerical modelling Vol. 31; no. 5
• Main Authors: Milan, Hugo F.M., Gebremedhin, Kifle G.
• Format: Journal Article
• Language: English
• Published: Bognor Regis Wiley Subscription Services, Inc 01.09.2018
• Subjects: bio‐heat transfer; complex geometries; Formulations; Heat transfer; Nodes; Numerical methods; Parallelepipeds; Partial differential equations; Pyramids; Rectangles; Skin; Skin temperature; Tetrahedra; Transmission-line matrix; transmission‐line modeling; Triangles
• ISSN: 0894-3370; 1099-1204
• DOI: 10.1002/jnm.2455

Transmission‐line modeling (TLM) is a numerical method used to solve coupled partial differential equations in time domain. Using TLM, the equations of the problem and a small control volume are represented by an electric network composed of nodes. In this paper, a general TLM node theory for bio‐heat transfer that simplifies to existing node formulations is presented. This formulation allows for the use of irregular node geometries, such as irregular quadrangles and hexahedrons, which are necessary to accurately model complex geometries of biological systems. Using irregular hexahedron nodes, solutions of skin‐surface temperatures of a breast with tumor are demonstrated. This theory is based on 2 assumptions. The first is that the unit vector perpendicular to the border of the node is similar to the unit vector from the center of the node to the midpoint of its border, which is exact for regular geometries (eg, rectangles in 2‐D, and parallelepipeds in 3‐D) and approximated for most irregular geometries used in practice (eg, triangles and quadrangles in 2‐D, and tetrahedrons and hexahedrons in 3‐D). This first assumption does not, however, hold for all geometries such as 3‐D pyramids. The second assumption, which is controlled by an error parameter, is that the dispersive effect intrinsically modeled by the transmission‐line inductor is negligible compared with the diffusive effect modeled by the resistor of the node. This general TLM node formulation facilitates algorithm development and increases flexibility, increasing the applicability of TLM to solve complex geometric problems typical of most biological systems.