- #1
eprparadox
- 138
- 2
Hello!
My goal here is to plot the solution to the bioheat equation for a tumor as a function of time. I'm plotting this for a fixed radius at r = 0 (the very center of the tumor).
The equation to solve is this:
[tex]
\rho_1c_1 \frac{\partial T}{\partial t} = 3(\frac{\partial }{\partial r}(\frac{\partial T}{\partial r}) + \frac{w_{b1}c_b}{\lambda_1}((T_b - T) + \frac{P\lambda_1}{w_{b1}c_b})
[/tex]
The numerical solution to this equation is given in a paper I'm reading. Using the central finite difference method, they get:
[tex]
T_0^{n+1} = (1 - 6 \frac{\lambda_1}{\rho_1c_1}(\frac{\Delta t}{\Delta r^2}))T_0^n + 6(\frac{\lambda_1}{\rho_1c_1}\frac{\Delta t}{\Delta r^2}) + (\frac{\Delta t}{\rho_1c_1})w_{b1}c_b((T_b - T_0^n) + \frac{P\lambda_1}{w_{b1}c_b})
[/tex]Note: the lambda, rho, c constants are thermal conductivity, density and specific heat of the tumor and anything with a subscript b refers to blood.
The graph presented in the paper for the above discrete solution (for 0 < t < 600 seconds) has the Temperature linear for about the first 100 seconds and then it begins to taper off after that (see attached photo).
But when I plot this solution in mathematica, it's just linearly grows without bound. And I don't know why.
I was thinking that maybe I needed to solve the PDE for some sort of steady state solution and then plot the sum of this steady state + the transient solution above. But the paper doesn't include a steady state solution for the PDE so that got me confused.
Any insight into this would be great.
Thanks so much.
My goal here is to plot the solution to the bioheat equation for a tumor as a function of time. I'm plotting this for a fixed radius at r = 0 (the very center of the tumor).
The equation to solve is this:
[tex]
\rho_1c_1 \frac{\partial T}{\partial t} = 3(\frac{\partial }{\partial r}(\frac{\partial T}{\partial r}) + \frac{w_{b1}c_b}{\lambda_1}((T_b - T) + \frac{P\lambda_1}{w_{b1}c_b})
[/tex]
The numerical solution to this equation is given in a paper I'm reading. Using the central finite difference method, they get:
[tex]
T_0^{n+1} = (1 - 6 \frac{\lambda_1}{\rho_1c_1}(\frac{\Delta t}{\Delta r^2}))T_0^n + 6(\frac{\lambda_1}{\rho_1c_1}\frac{\Delta t}{\Delta r^2}) + (\frac{\Delta t}{\rho_1c_1})w_{b1}c_b((T_b - T_0^n) + \frac{P\lambda_1}{w_{b1}c_b})
[/tex]Note: the lambda, rho, c constants are thermal conductivity, density and specific heat of the tumor and anything with a subscript b refers to blood.
The graph presented in the paper for the above discrete solution (for 0 < t < 600 seconds) has the Temperature linear for about the first 100 seconds and then it begins to taper off after that (see attached photo).
But when I plot this solution in mathematica, it's just linearly grows without bound. And I don't know why.
I was thinking that maybe I needed to solve the PDE for some sort of steady state solution and then plot the sum of this steady state + the transient solution above. But the paper doesn't include a steady state solution for the PDE so that got me confused.
Any insight into this would be great.
Thanks so much.