A Flux-Conservative Finite Difference Scheme for the Numerical Solution of the Nonlinear Bioheat Equation

Abstract

We present a flux-conservative finite difference (FCFD) scheme for solving the nonlinear (bio)heat transfer in living tissue. The proposed scheme deals with steep gradients in the material properties for malignant and healthy tissues. The method applies directly on the raw medical image data without the need for sophisticated image analysis algorithms to define the interface between tumour and healthy tissues.We extend the classical finite difference (FD) method to cases with high discontinuities in the material properties. We apply meshless kernels, widely used in Smoothed Particle Hydrodynamics (SPH) method, to approximate properties in the off-grid points introduced by the flux conservative differential operators. The meshless kernels can accurately capture the steep gradients and provide accurate approximations. We solve the governing equations by using an explicit solver. The relatively small time-step applied is counterbalanced by the small computation effort required at each time step of the proposed scheme. The FCFD method can accurately compute the numerical solution of the bioheat equation even when noise from the image acquisition is present.Results highlight the applicability of the method and its ability to solve tumor ablation simulations directly on the raw image data, without the need to define the interface between malignant and healthy tissues (segmentation) or meshing.

Adaptive simulation of 3D thermometry maps for interventional MR-guided tumor ablation using Pennes’ bioheat equation and isotherms

“…Q m describes the metabolic heat generation rate, Q r the regional heat source and T represents the temperature at a given time point t. Regarding the density of blood itself ρ b there are different approaches. Some authors like Bourantas et al 21 treat the blood density individually in the simulation term. Other authors like Zhang et al 22 do not report the blood density in their optimization term because it is indirectly included in the blood perfusion rate w b .…”

Thermography Quantification of Human Perfusion Thermal Signature

“…The existence of temperature difference between the blood and tissue is considered as firm evidence of its function to remove or release heat [1,40]. Based on this theory, Pennes (1948) proposed his famous heat transfer model [41], which is called Pennes' bioheat equation [15,42]. Pennes suggested that the effect of blood flow in the tissue be modeled as a thermal source and physical sink term added to the conventional heat conduction equation [43].…”

“…The resultant peripheral vasoconstriction following changes in skin temperature from cooling and closing of local vasculature promote a larger central circulatory reserve that may be beneficial to reduce cardiac stress due to the increased venous return and stroke volume (SV) [41,47]. While the cardiovascular explanation is certainly plausible of the limited studies that have investigated post-exercise cooling [48], few studies report any differences in heart rates that are not explained by the passive nature of cold water immersion or differences in any cardiovascular measure during ensuing bouts of exercise [39,49].…”