A flux-conservative finite difference scheme for the numerical solution of the nonlinear bioheat equation

    Research output: Chapter in Book/Conference paperChapter

    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 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.

    LanguageEnglish
    Title of host publicationComputational Biomechanics for Medicine
    Subtitle of host publicationMeasurements, Models, and Predictions
    PublisherSpringer International Publishing AG
    Pages69-81
    Number of pages13
    ISBN (Electronic)9783319755892
    ISBN (Print)9783319755885
    DOIs
    Publication statusPublished - 14 May 2018

    Fingerprint

    Nonlinear equations
    nonlinear equations
    Tissue
    Fluxes
    Finite difference method
    Tumors
    Materials properties
    tumors
    gradients
    differential operators
    Image acquisition
    Ablation
    image analysis
    Image analysis
    ablation
    Hydrodynamics
    acquisition
    discontinuity
    heat transfer
    hydrodynamics

    Cite this

    Bourantas, G. C., Joldes, G. R., Wittek, A., & Miller, K. (2018). A flux-conservative finite difference scheme for the numerical solution of the nonlinear bioheat equation. In Computational Biomechanics for Medicine: Measurements, Models, and Predictions (pp. 69-81). Springer International Publishing AG. https://doi.org/10.1007/978-3-319-75589-2_7
    Bourantas, George C. ; Joldes, Grand R. ; Wittek, Adam ; Miller, Karol. / A flux-conservative finite difference scheme for the numerical solution of the nonlinear bioheat equation. Computational Biomechanics for Medicine: Measurements, Models, and Predictions. Springer International Publishing AG, 2018. pp. 69-81
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    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 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.",
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    Bourantas, GC, Joldes, GR, Wittek, A & Miller, K 2018, A flux-conservative finite difference scheme for the numerical solution of the nonlinear bioheat equation. in Computational Biomechanics for Medicine: Measurements, Models, and Predictions. Springer International Publishing AG, pp. 69-81. https://doi.org/10.1007/978-3-319-75589-2_7

    A flux-conservative finite difference scheme for the numerical solution of the nonlinear bioheat equation. / Bourantas, George C.; Joldes, Grand R.; Wittek, Adam; Miller, Karol.

    Computational Biomechanics for Medicine: Measurements, Models, and Predictions. Springer International Publishing AG, 2018. p. 69-81.

    Research output: Chapter in Book/Conference paperChapter

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    T1 - A flux-conservative finite difference scheme for the numerical solution of the nonlinear bioheat equation

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    N2 - 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.

    AB - 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.

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    Bourantas GC, Joldes GR, Wittek A, Miller K. A flux-conservative finite difference scheme for the numerical solution of the nonlinear bioheat equation. In Computational Biomechanics for Medicine: Measurements, Models, and Predictions. Springer International Publishing AG. 2018. p. 69-81 https://doi.org/10.1007/978-3-319-75589-2_7