Variable thermal conductivity approach for bioheat transfer during thermal ablation

Figures & data

Table 1. Algorithm procedure.

Algorithm MATLAB programming procedure of FELWGM
Step 1:	Introduce time variable F0 and other expressions required in computation;
Step 2:	For $F_0\in [0,1)$;
Step 3:	Fixed the value of M and k;
Step 4:	Define Legendre Wavelet ψnm given in Equation (41)(41) $${\psi }_{nm}\left(F_0\right)=\{\begin{array}{cc}\sqrt{(m+1/2)}{2}^{k/2}{P}_m(2^k{F}_0-\widehat{n}),\hfill & \frac{\widehat{n}-1}{2^k}\le F_0\le \frac{\widehat{n}+1}{2^k}\hfill \\ 0,\hfill & \mathit{\text{otherwise}}\hfill \end{array},$$(41) and the matrices P, L, F given in Equations (43) (44)(44) $$L=\left(\begin{array}{ccccccc}1& \frac{1}{\sqrt{3}}& 0& 0& \dots & 0& 0\\ -\frac{1}{\sqrt{3}}& 0& \frac{1}{\sqrt{15}}& 0& \dots & 0& 0\\ 0& -\frac{1}{\sqrt{15}}& 0& \frac{1}{\sqrt{35}}& \ddots & 0& 0\\ 0& 0& -\frac{1}{\sqrt{35}}& 0& \ddots & 0& 0\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& 0& \dots & 0& \frac{\sqrt{2M-3}}{(2M-3)\sqrt{2M-1}}\\ 0& 0& 0& 0\dots & \frac{\sqrt{2M-1}}{(2M-1)\sqrt{2M-3}}& 0& \end{array}\right),$$(44) and (45)(45) $$F=\left(\begin{array}{cccc}2& 0& \dots & 0\\ 0& 0& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & 0\end{array}\right).$$(45) ;
Step 5:	C= ${\left[\begin{array}{c}{C}_{1,0},\dots ,C_{1,M-1},C_{2,0},\dots ,C_{2,M-1},C_{2^{k-1},0},\dots ,C_{2^{k-1},M-1}\end{array}\right]}^T,$ where C is unknown matrix of order $2^{k-1}M\times 2^{k-1}M$;
Step 6:	Solve Sylvester matrix Equation (47)(47) $$A{\mathrm{C}}^T{\mathrm{C}}_1-{\mathrm{C}}^T+D=0,$$(47) to obtain C^T, where $C^T=\mathit{\text{transpose}}(C)$ by using the command sylvester
Step 7:	$\Theta (F_0)=\mathit{\text{transpose}}(C)*P^2*\psi (F_0)+{\Theta }_0$, where ${\Theta }_0$ is initial temperature given in Equation (38)(38) $$\Theta \left(0\right)=\frac{K_1}{2},\text{and}\frac{d\Theta \left(0\right)}{d{F}_0}=0,$$(38) ;
Step 8:	end

Figure 1. Dimensionless tissue temperature response along dimensionless depth with different variable conductivity parameters K₁.

Figure 2. Dimensionless tissue temperature response along dimensionless depth with different values of time for $K_1=0.1$.

Figure 3. Dimensionless tissue temperature response along dimensionless depth for generalized coordinate systems.

Figure 4. Dimensionless tissue temperature profile for different value of relaxation time.

Figure 5. Dimensionless tissue temperature profile for different values of external heat source coefficient.

Figure 6. Plot of comparison in FELWGM solution and Exact solution.

Figure 7. Absolute error.

Table 2. The absolute error between the solutions obtained by FELWGM and analytical method.

Time	FELWGM	Exact solution	Absolute error
0	37.219842251500538	37.219842341320123	0.0090 $\times 1.0e-05$
100	37.319192808829554	37.319183170819544	0.9638 $\times 1.0e-05$
200	37.359167331208197	37.359166530128289	0.0801 $\times 1.0e-05$
300	37.369163646412476	37.369163921664174	0.0275 $\times 1.0e-05$
400	37.398959168730516	37.398959558330201	0.0390 $\times 1.0e-05$
500	37.409153152968211	37.409153429124091	0.0276 $\times 1.0e-05$