A fast and simple algorithm for the calculation of convective heat transfer by large vessels in three-dimensional inhomogeneous tissues

Abstract
A fast and simple algorithm has been presented for the calculation of time-dependent temperature distributions in inhomogeneous vascularized tissue. Three-dimensional anatomical data of tissues and vessel structures are decomposed into elementary cubic nodes by a special digitizing routine with vessels represented by connected strings of vessel nodes. Vessel cross sections may be irregular shaped and/or tapered. Conductive and convective heat transfer was calculated through use of the heat balance technique on each cubic node resulting in an explicit finite difference computational scheme. Employing a three time level scheme, the Fourier stability criterion is circumvented allowing arbitrary time steps to be defined in the algorithm. Time steps as large as 100 times the Fourier restricted one still result in stable and convergent calculations of the stationary temperature distribution. Vessels with different flows and diameters are incorporated by performing a vessel specific second discretization step in time. Using the new algorithm as a mathematical tool the thermal equilibration length of vessel segments have been established under a broad range of geometrical and flow conditions. Validation followed from comparing transient and stationary temperature distributions derived by the proposed algorithm to those from an accurate cylindrical numerical model. Predicted values for the thermal equilibration lengths are compared to an analytical expression and phantom experiments. The algorithm is incorporated in a thermal model being the main part of our hyperthermia treatment planning system.

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