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Thermal field and tissue damage analysis of moving laser in cancer thermal therapy

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Abstract

In this paper, a closed-form analytical solution of hyperbolic Pennes bioheat equation is obtained for spatial evolution of temperature distributions during moving laser thermotherapy of the skin and kidney tissues. The three-dimensional cubic homogeneous perfused biological tissue is adopted as a media and the Gaussian distributed function in surface and exponentially distributed in depth is used for modeling of laser moving heat source. The solution procedure is Eigen value method which leads to a closed form solution. The effect of moving velocity, perfusion rate, laser intensity, absorption and scattering coefficients, and thermal relaxation time on temperature profiles and tissue thermal damage are investigated. Results are illustrated that the moving velocity and the perfusion rate of the tissues are the main important parameters in produced temperatures under moving heat source. The higher perfusion rate of kidney compared with skin may lead to lower induced temperature amplitude in moving path of laser due to the convective role of the perfusion term. Furthermore, the analytical solution can be a powerful tool for analysis and optimization of practical treatment in the clinical setting and laser procedure therapeutic applications and can be used for verification of other numerical heating models.

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Abbreviations

n :

Fourier counter

q :

Heat flux vector

t :

Time (s)

x :

Cartesian coordinate (m)

y :

Cartesian coordinate (m)

z :

Cartesian coordinate (m)

v :

Constant velocity (ms−1)

l :

Dimension of biological tissue (m)

Q :

Heat source function

P :

Constant power

P r :

Reference power

T :

Temperature (°C)

T b :

Blood temperature (°C)

C t :

Tissue specific heat (JKg−1 ° C−1)

C b :

Blood specific heat (JKg−1 ° C−1)

T m :

Reference temperatures (°C)

h :

Dimensionless convection coefficient

X :

Dimensionless coordinate

Y :

Dimensionless coordinate

Z :

Dimensionless coordinate

w :

Propagation speed (ms−1)

α :

Thermal diffusivity (m2s−1)

μ :

Absorption depth coefficient(m−1)

ρ t :

Tissue density (Kgm−3)

ρ b :

Blood density (Kgm−3)

ϖ b :

Blood perfusion rate (s−1)

κ :

Thermal conductivity (Wm−1 ° C−1)

τ q :

Thermal relaxation time(s)

σ :

Spot radius(m)

τ :

Dimensionless time

θ :

Dimensionless temperature

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Funding

This research is done in the Iran University of Science and Technology. There is not any funding source applicable.

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  1. School of Railway Engineering, Iran University of Science and Technology, Tehran, Iran

    Ali Kabiri & Mohammad Reza Talaee

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  1. Ali Kabiri
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Appendix

Appendix

The simplified relation of Rmnk(τ) is as follows:

$$ {R}_{mnk}\left(\tau \right)=\left[\frac{\psi_0\pi \sigma w\left(\beta {e}^{\left[\beta L\right]}-\beta Cos\left[L{\mu}_k\right]+{\mu}_k Sin\left[L{\mu}_k\right]\right)}{2{\alpha}^2{L}^4\left({\beta}^2+{\mu_k}^2\right)}\right]\times {e}^{-\left[\beta L+\frac{im\pi \left(L-\tau v\right)}{L}+\frac{2{\alpha}^2\left({L}^2-2 L\tau v+2{\tau}^2{v}^2\right)}{\sigma^2{w}^2}+\frac{\left({m}^2+{n}^2\right){\pi}^2{\sigma}^2{w}^2}{8{\alpha}^2{L}^2}\right]} $$
$$ \left[2\alpha L v\sqrt{\frac{2}{\pi }}{e}^{\left[-\frac{im\pi \tau v}{L}+\frac{2{\alpha}^2\tau v\left(-2L+\tau v\right)}{\sigma^2{w}^2}+\frac{m^2{\pi}^2{\sigma}^2{w}^2}{8{\alpha}^2{L}^2}\right]}\left(2{e}^{\left[ im\pi +\frac{2{\alpha}^2{L}^2}{\sigma^2{w}^2}\right]}-{e}^{\left[\frac{4{\alpha}^2 L\tau v}{\sigma^2{w}^2}\right]}\left(1+{e}^{\left[2 im\pi \right]}\right)\right)+\sigma w{e}^{\left[\frac{im\pi \left(L-2\tau v\right)}{L}+\frac{2{\alpha}^2\left({L}^2-2 L\tau v+2{\tau}^2{v}^2\right)}{\sigma^2{w}^2}\right]}\left({e}^{\left[\frac{2 im\pi \tau v}{L}\right]}\left[2L+ im\pi v\right]\left(\mathit{\operatorname{erf}}\left[\frac{\sqrt{2}\alpha \left(L-\tau v\right)}{\sigma w}-\frac{im\pi \sigma w}{2\sqrt{2}\alpha L}\right]+\mathit{\operatorname{erf}}\left[\frac{\sqrt{2}\alpha \tau v}{\sigma w}+\frac{im\pi \sigma w}{2\sqrt{2}\alpha L}\right]\right)+\left[2L- im\pi v\right]\left(\mathit{\operatorname{erf}}\left[\frac{\sqrt{2}\alpha \tau v}{\sigma w}-\frac{im\pi \sigma w}{2\sqrt{2}\alpha L}\right]+\mathit{\operatorname{erf}}\left[\frac{\sqrt{2}\alpha \left(L-\tau v\right)}{\sigma w}+\frac{im\pi \sigma w}{2\sqrt{2}\alpha L}\right]\right)\right)\right]\left[\mathit{\operatorname{erf}}\left[\frac{\sqrt{2}\alpha L}{\sigma w}-\frac{in\pi \sigma w}{2\sqrt{2}\alpha L}\right]+\mathit{\operatorname{erf}}\left[\frac{\sqrt{2}\alpha L}{\sigma w}+\frac{in\pi \sigma w}{2\sqrt{2}\alpha L}\right]\right] $$

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Kabiri, A., Talaee, M.R. Thermal field and tissue damage analysis of moving laser in cancer thermal therapy. Lasers Med Sci 36, 583–597 (2021). https://doi.org/10.1007/s10103-020-03070-7

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