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

  • George C. BourantasEmail author
  • Grand R. Joldes
  • Adam Wittek
  • Karol Miller
Conference paper


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 tumor 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.


Flux-conservative finite difference Tumor ablation Bioheat equation Meshless method 



This research was supported partially by the Australian Government through the Australian Research Council’s Discovery Projects funding scheme (project DP160100714). The views expressed herein are those of the authors and are not necessarily those of the Australian Government or Australian Research Council. In addition, we wish to acknowledge the Raine Medical Research Foundation for funding G. R. Joldes through a Raine Priming Grant, and the Department of Health, Western Australia, for funding G. R. Joldes through a Merit Award.


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • George C. Bourantas
    • 1
    Email author
  • Grand R. Joldes
    • 1
    • 2
  • Adam Wittek
    • 1
  • Karol Miller
    • 1
    • 3
  1. 1.Intelligent Systems for Medicine Laboratory, School of Mechanical and Chemical EngineeringThe University of Western AustraliaCrawley/PerthAustralia
  2. 2.School of Engineering and Information TechnologyMurdoch UniversityMurdochAustralia
  3. 3.School of EngineeringCardiff UniversityCardiffUK

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