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Meshless method with operator splitting technique for transient nonlinear bioheat transfer in two-dimensional skin tissues.

Zhang ZW, Wang H, Qin QH - Int J Mol Sci (2015)

Bottom Line: In the analysis, the OSM is used first to separate the Laplacian operator and the nonlinear source term, and then the second-order time-stepping schemes are employed for approximating two splitting operators to convert the original governing equation into a linear nonhomogeneous Helmholtz-type governing equation (NHGE) at each time step.Subsequently, the RBF interpolation and the MFS involving the fundamental solution of the Laplace equation are respectively employed to obtain approximated particular and homogeneous solutions of the nonhomogeneous Helmholtz-type governing equation.Furthermore, the sensitivity of the coefficients in the cases of a linear and an exponential relationship of TDBPR is investigated to reveal their bioheat effect on the skin tissue.

View Article: PubMed Central - PubMed

Affiliation: Research School of Engineering, Australian National University, Acton, ACT 2601, Australia. zewei.zhang@anu.edu.au.

ABSTRACT
A meshless numerical scheme combining the operator splitting method (OSM), the radial basis function (RBF) interpolation, and the method of fundamental solutions (MFS) is developed for solving transient nonlinear bioheat problems in two-dimensional (2D) skin tissues. In the numerical scheme, the nonlinearity caused by linear and exponential relationships of temperature-dependent blood perfusion rate (TDBPR) is taken into consideration. In the analysis, the OSM is used first to separate the Laplacian operator and the nonlinear source term, and then the second-order time-stepping schemes are employed for approximating two splitting operators to convert the original governing equation into a linear nonhomogeneous Helmholtz-type governing equation (NHGE) at each time step. Subsequently, the RBF interpolation and the MFS involving the fundamental solution of the Laplace equation are respectively employed to obtain approximated particular and homogeneous solutions of the nonhomogeneous Helmholtz-type governing equation. Finally, the full fields consisting of the particular and homogeneous solutions are enforced to fit the NHGE at interpolation points and the boundary conditions at boundary collocations for determining unknowns at each time step. The proposed method is verified by comparison of other methods. Furthermore, the sensitivity of the coefficients in the cases of a linear and an exponential relationship of TDBPR is investigated to reveal their bioheat effect on the skin tissue.

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Results for the linear case of blood perfusion rate (The arrow is indicator of zoom in image of the three overlap points in temperature curves. So that the reader can view the curves in details clearly).
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ijms-16-02001-f003: Results for the linear case of blood perfusion rate (The arrow is indicator of zoom in image of the three overlap points in temperature curves. So that the reader can view the curves in details clearly).

Mentions: For the purpose of comparison, we consider that blood perfusion rate is a linear function of tissue temperature: , where and . 63 interpolation points and 32 boundary collocations (see Figure 2) are used to calculate the transient temperature distribution. Numerical results along the x-axis at three time instants ∆t = 50, 80, and 100 s are presented in Figure 3 to show the accuracy and stability of the second-order Adams-Bashforth and Adams-Moulton schemes. From Figure 3, it can be seen that the results from the proposed algorithm with fewer collocation points are in good agreement with the results from the ANSYS Transient thermal toolbox. The relative error of the results from the proposed method with respect to those from the Transient thermal toolbox of ANSYS is less than 0.5%.


Meshless method with operator splitting technique for transient nonlinear bioheat transfer in two-dimensional skin tissues.

Zhang ZW, Wang H, Qin QH - Int J Mol Sci (2015)

Results for the linear case of blood perfusion rate (The arrow is indicator of zoom in image of the three overlap points in temperature curves. So that the reader can view the curves in details clearly).
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4307345&req=5

ijms-16-02001-f003: Results for the linear case of blood perfusion rate (The arrow is indicator of zoom in image of the three overlap points in temperature curves. So that the reader can view the curves in details clearly).
Mentions: For the purpose of comparison, we consider that blood perfusion rate is a linear function of tissue temperature: , where and . 63 interpolation points and 32 boundary collocations (see Figure 2) are used to calculate the transient temperature distribution. Numerical results along the x-axis at three time instants ∆t = 50, 80, and 100 s are presented in Figure 3 to show the accuracy and stability of the second-order Adams-Bashforth and Adams-Moulton schemes. From Figure 3, it can be seen that the results from the proposed algorithm with fewer collocation points are in good agreement with the results from the ANSYS Transient thermal toolbox. The relative error of the results from the proposed method with respect to those from the Transient thermal toolbox of ANSYS is less than 0.5%.

Bottom Line: In the analysis, the OSM is used first to separate the Laplacian operator and the nonlinear source term, and then the second-order time-stepping schemes are employed for approximating two splitting operators to convert the original governing equation into a linear nonhomogeneous Helmholtz-type governing equation (NHGE) at each time step.Subsequently, the RBF interpolation and the MFS involving the fundamental solution of the Laplace equation are respectively employed to obtain approximated particular and homogeneous solutions of the nonhomogeneous Helmholtz-type governing equation.Furthermore, the sensitivity of the coefficients in the cases of a linear and an exponential relationship of TDBPR is investigated to reveal their bioheat effect on the skin tissue.

View Article: PubMed Central - PubMed

Affiliation: Research School of Engineering, Australian National University, Acton, ACT 2601, Australia. zewei.zhang@anu.edu.au.

ABSTRACT
A meshless numerical scheme combining the operator splitting method (OSM), the radial basis function (RBF) interpolation, and the method of fundamental solutions (MFS) is developed for solving transient nonlinear bioheat problems in two-dimensional (2D) skin tissues. In the numerical scheme, the nonlinearity caused by linear and exponential relationships of temperature-dependent blood perfusion rate (TDBPR) is taken into consideration. In the analysis, the OSM is used first to separate the Laplacian operator and the nonlinear source term, and then the second-order time-stepping schemes are employed for approximating two splitting operators to convert the original governing equation into a linear nonhomogeneous Helmholtz-type governing equation (NHGE) at each time step. Subsequently, the RBF interpolation and the MFS involving the fundamental solution of the Laplace equation are respectively employed to obtain approximated particular and homogeneous solutions of the nonhomogeneous Helmholtz-type governing equation. Finally, the full fields consisting of the particular and homogeneous solutions are enforced to fit the NHGE at interpolation points and the boundary conditions at boundary collocations for determining unknowns at each time step. The proposed method is verified by comparison of other methods. Furthermore, the sensitivity of the coefficients in the cases of a linear and an exponential relationship of TDBPR is investigated to reveal their bioheat effect on the skin tissue.

Show MeSH
Related in: MedlinePlus