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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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Sensitivity to constant a2 in the exponential case of blood perfusion rate.
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ijms-16-02001-f011: Sensitivity to constant a2 in the exponential case of blood perfusion rate.

Mentions: Again, we assume constant a1 to be 0.0005, while constant is set to be 0.03, 0.01, and 0.003. As we can see from Figure 11, when constant = 0.03, the temperature of the skin tissue becomes increasingly steep before the point (11.25 mm, 0), but the curve is flatter than temperature curves with smaller values of . Compared with the effect of the different values of in Figure 10, the increase in the value of causes a larger reduction of the peak value of the skin tissue temperature and the temperature becomes more stable from the location (11.25 mm, 0) to (26.25 mm, 0). In summary, an increase in the value of constant has a higher sensitivity to the temperature of skin tissue than an increase in the value of constant . Simultaneously, it is found that an increase in the blood perfusion rate causes the temperature of the skin tissue to reach its final steady state more quickly and reduces the peak value of the tissue temperature. That means that if the skin tissue absorbs a large quantity ofbiological heat from its environment, the blood perfusion effect causes the temperature to reach a certain value quickly and reduces the risk of burning of the skin tissue.


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)

Sensitivity to constant a2 in the exponential case of blood perfusion rate.
© Copyright Policy
Related In: Results  -  Collection

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

ijms-16-02001-f011: Sensitivity to constant a2 in the exponential case of blood perfusion rate.
Mentions: Again, we assume constant a1 to be 0.0005, while constant is set to be 0.03, 0.01, and 0.003. As we can see from Figure 11, when constant = 0.03, the temperature of the skin tissue becomes increasingly steep before the point (11.25 mm, 0), but the curve is flatter than temperature curves with smaller values of . Compared with the effect of the different values of in Figure 10, the increase in the value of causes a larger reduction of the peak value of the skin tissue temperature and the temperature becomes more stable from the location (11.25 mm, 0) to (26.25 mm, 0). In summary, an increase in the value of constant has a higher sensitivity to the temperature of skin tissue than an increase in the value of constant . Simultaneously, it is found that an increase in the blood perfusion rate causes the temperature of the skin tissue to reach its final steady state more quickly and reduces the peak value of the tissue temperature. That means that if the skin tissue absorbs a large quantity ofbiological heat from its environment, the blood perfusion effect causes the temperature to reach a certain value quickly and reduces the risk of burning of the skin tissue.

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