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A coupled finite-volume immersed boundary method for the simulation of bioheat transfer in 3D complex tumor

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Abstract

Numerical schemes based on the immersed boundary method (IBM) offer the advantages of avoiding the body-conformal structured or unstructured grid generation process in the complex tumor morphology. This study uses the finite-volume immersed boundary method (FV-IBM) to solve bioheat physics in actual liver tumor tissue. IBM is employed in this methodology to enforce the boundary effect on the non-body conformal Cartesian grid. The finite-volume method (FVM) is used as a numerical technique to discretize the governing equations. The validation and verification of the FV-IB method have shown that the scheme is second-order accurate. Furthermore, the numerical results in the spherical tumor model are in good agreement with previously reported results for steady and transient cases. Results for MNPs-based hyperthermia investigation with two heat source (Gaussian and uniform) distribution patterns in the liver tumor are in good agreement with the numerical solution of COMSOL Multiphysics. Thus, a simple and robust FV-IBM-based numerical scheme is proposed to solve the bioheat models in arbitrary tissue shapes.

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All data generated or analysed during this study are included in this published article.

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Acknowledgements

The research is performed in partial fulfillment of the requirements for PhD degree by Amritpal Singh from Thapar Institute of Engineering and Technology (TIET) Patiala.

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This research received no specific grant from the public, commercial, or not-for-profit funding agencies.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Thapar Institute of Engineering and Technology, Patiala, 147004, India

    Amritpal Singh & Neeraj Kumar

  2. Virginia Tech-TIET Center of Excellence in Emerging Materials, Thapar Institute of Engineering and Technology, Patiala, 147004, India

    Neeraj Kumar

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Correspondence to Neeraj Kumar.

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Appendix

Appendix

\(\left({A}_{p}-{\chi }_{e}{\alpha }_{e} {A}_{e}- {\chi }_{w}{\alpha }_{w} {A}_{w} -{\chi }_{n}{\alpha }_{n} {A}_{n}- {\chi }_{s}{\alpha }_{s} {A}_{s}\right){T}_{p}^{t+1}= (\left(1-{\chi }_{e}\right){A}_{e}+ {\chi }_{w}{\beta }_{w} {A}_{w}){T}_{e}^{t+1}+(\left(1-{\chi }_{w}\right){A}_{w}{+{\chi }_{e}{\beta }_{e} {A}_{e}) T}_{w}^{t+1}+ (\left(1-{\chi }_{n}\right){A}_{e}+ {\chi }_{s}{\beta }_{s} {A}_{s}){T}_{n}^{t+1}+((1-{\chi }_{s}){A}_{s}{+{\chi }_{n}{\beta }_{n} {A}_{n}) T}_{s}^{t+1}+ {B}_{p}+{\chi }_{e}{\gamma }_{e}{A}_{e}+ {\chi }_{w}{\gamma }_{w}{A}_{w}+{\chi }_{n}{\gamma }_{n}{A}_{n}+ {\chi }_{s}{\gamma }_{s}{A}_{s}\) A.1

For \({p}_{1}\) IB node (as shown in Fig. 1).

For east and north nodes: \({\chi }_{e}=1\) and \({\chi }_{n}\) \(=\) 1.

For west and south nodes: \({\chi }_{w}=0\) and \({\chi }_{s}\) \(=\) 0.

\(\left({A}_{p}-{\alpha }_{e} {A}_{e}- {\alpha }_{n} {A}_{n}\right){T}_{p}^{t+1}= ({A}_{w}{+{\beta }_{e} {A}_{e}) T}_{w}^{t+1}+({A}_{s}{+{\beta }_{n} {A}_{n}) T}_{s}^{t+1}+{B}_{p}+{\gamma }_{e}{A}_{e}+{\gamma }_{n}{A}_{n}\) A.2

For regular node, the \(\chi\) value in all direction is zero (‘p’ node, shown in Fig. 1).

\({A}_{p}{T}_{p}^{t+1}= {A}_{e}{T}_{e}^{t+1}+{A}_{w}{ T}_{w}^{t+1}+ {A}_{n}{T}_{n}^{t+1}+{A}_{s}{ T}_{s}^{t+1}+ {B}_{p}\) A.3

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Singh, A., Kumar, N. A coupled finite-volume immersed boundary method for the simulation of bioheat transfer in 3D complex tumor. Engineering with Computers 39, 3743–3758 (2023). https://doi.org/10.1007/s00366-023-01797-9

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