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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: 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.

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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Temperature variation vs. time along x-axis for the exponent-form blood perfusion rate.
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ijms-16-02001-f007: Temperature variation vs. time along x-axis for the exponent-form blood perfusion rate.

Mentions: More numerical results are now presented to illustrate temperature distribution in the solution domain caused by different temperature-dependent blood perfusion rates. In Figure 6 and Figure 7, the temperature distribution in the skin tissue along the x-axis at different times is presented. It is found that the steady state in Figure 6 is reached much earlier (the linear case, at about 1600 s) than that in Figure 7 (the exponential case, at about 8000 s). It is also noted that the slope of the steady-state temperature curve along the x-axis increases and then decreases from the left side to the right side for both linear and exponential cases. However, the slope for the linear case appears greater than that for the exponential case in the region close to the left surface, which has a lower environmental temperature, whereas the slope for the linear case becomes less than that for the exponential case in the region close to the right surface, which has a higher body core temperature. Moreover, the exponential-form blood perfusion rate produces a higher interior temperature in the region close to x = 18.75 mm than that for the linear-form rate. The main reason is that the exponential-form blood perfusion rate generally has a lower value of the blood perfusion rate than the linear-form with the coefficients given above. In the region close to the left surface, where the skin tissue temperature is evidently lower than the blood temperature, the greater blood perfusion rate means that more heat flows from blood to skin tissue, causing a rapid increase of the tissue temperature. Thus there is greater temperature gradient in this region for the linear case than the exponential case. When the tissue temperature exceeds the blood temperature, a greater blood perfusion rate causes more heat to flow from tissue to blood and causes the tissue temperature to decrease.


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)

Temperature variation vs. time along x-axis for the exponent-form blood perfusion rate.
© Copyright Policy
Related In: Results  -  Collection

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

ijms-16-02001-f007: Temperature variation vs. time along x-axis for the exponent-form blood perfusion rate.
Mentions: More numerical results are now presented to illustrate temperature distribution in the solution domain caused by different temperature-dependent blood perfusion rates. In Figure 6 and Figure 7, the temperature distribution in the skin tissue along the x-axis at different times is presented. It is found that the steady state in Figure 6 is reached much earlier (the linear case, at about 1600 s) than that in Figure 7 (the exponential case, at about 8000 s). It is also noted that the slope of the steady-state temperature curve along the x-axis increases and then decreases from the left side to the right side for both linear and exponential cases. However, the slope for the linear case appears greater than that for the exponential case in the region close to the left surface, which has a lower environmental temperature, whereas the slope for the linear case becomes less than that for the exponential case in the region close to the right surface, which has a higher body core temperature. Moreover, the exponential-form blood perfusion rate produces a higher interior temperature in the region close to x = 18.75 mm than that for the linear-form rate. The main reason is that the exponential-form blood perfusion rate generally has a lower value of the blood perfusion rate than the linear-form with the coefficients given above. In the region close to the left surface, where the skin tissue temperature is evidently lower than the blood temperature, the greater blood perfusion rate means that more heat flows from blood to skin tissue, causing a rapid increase of the tissue temperature. Thus there is greater temperature gradient in this region for the linear case than the exponential case. When the tissue temperature exceeds the blood temperature, a greater blood perfusion rate causes more heat to flow from tissue to blood and causes the tissue temperature to decrease.

Bottom Line: 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.

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