Solve bioheat transfer equation

Hello,

I want to write a code in matlab which solves bioheat transfer equation for a given heat source and other parameters. In which the source would be laser source;

ρC_ρ ∂T/∂t-κ∇^2 T=Q-Q_ρ

[Image Analyst]

No question was asked, and you didn't specify which variable/symbol was the "given heat source" and which of the variables/symbols were the "other parameters" for which you know the values of.

If you have a question, then ask it, and attach your data and code to read it in with the paperclip icon after you read this:

TUTORIAL: How to ask a question (on Answers) and get a fast answer 🙂

[soso]

thank you, Q is heat source, The temperature of tissue is T, the temperature of blood is Tb, the specific heat capacity is Cb, density of blood is Pb P is the blood perfusion, k is thermal conductivity, C is heat capacity, p is density, and Q_ρ=Pρ_b C_b (T-T_b) is for blood, and temperature changes with time is the output of equation

[Image Analyst]

That's just an equation. It's like saying the Pythagorean theorem is "a^2 + b^2 = c^c. Solve it."

What is there to actually "solve" for? Which is the unknown? Do you have numerical values for any of the variables you gave, like

Q_p = 10
C_b = 14

or whatever?

[soso]

This equation is to solve temperature change with time, which is T. T is the unknown.

I have:

rho_blood=1000 % kg/m³
Cp_blood=3000 % J/(kg·K)
omega_blood= 0.7 % 1/s
T_blood= 310.15 % K
eps_diel=30
eps_cat=32
f= 10e14
P_in= 12 % mw
Cp=3400 % J/(kg·K)
rho=1000 % kg/m³
k=0.4 % W/(m·K)
A= 5.5e39 % (1/s) frequency factor
dE = 2.3e5 % J/mol activation energy
sigma= 0.05 % S/m

[Torsten]

Use "pdepe" to solve. Examples how to use the code can be found here:

https://de.mathworks.com/help/matlab/ref/pdepe.html

You will have to specify the temporal and spatial range of integration plus the initial and boundary conditions for T.

[Sam Chak]

Hi @soso,

Well, now you have given more than the number of 'parameters' .

[soso]

Hi@Sam Chak, yes because some of them for other equation, you can use whatever needed in this equation

[Raj Kumar]

what is your external heat source? Is it from laser beam or photon fluence?

Answer 1

[Raj Kumar]

% Simulation Timing

t_final = 30.0; % Total laser exposure heating time (seconds)

dt = 0.01; % Time step size (seconds)

n_steps = ceil(t_final / dt);

% Re-verify FDM Stability Criteria using the corrected size constraints

alpha = k_cond / (rho * c_p);

stability_check = alpha * dt * (1/dx_m^2 + 1/dy_m^2 + 1/dz_m^2);

if stability_check >= 0.5

error('FDM instability detected! Decrease time-step variable dt.');

end

% Initialize Temperature Fields using the exact safe matrix sizes

T = ones(nx, ny, nz) * T_b;

T_next = T;

% Run Explicit Time-Stepping Finite Difference Loop

fprintf('Computing Transient Temperature Rise (%d steps)...\n', n_steps);

for step = 1:n_steps

% Enforce Internal Voxel 3D Conduction Laplacians

% Ignoring outer edge ghost voxels to implement simpler insulation boundary

for i = 2:nx-1

for j = 2:ny-1

for k = 2:nz-1

d2T_dx2 = (T(i+1,j,k) - 2*T(i,j,k) + T(i-1,j,k)) / dx_m^2;

d2T_dy2 = (T(i,j+1,k) - 2*T(i,j,k) + T(i,j-1,k)) / dy_m^2;

d2T_dz2 = (T(i,j,k+1) - 2*T(i,j,k) + T(i,j,k-1)) / dz_m^2;

% Pennes Equation: dT/dt = (k*Laplacian + Q_perfusion + Q_metabolic + Q_laser)/(rho*cp)

Q_perf = w_b * rho_b * c_b * (T_b - T(i,j,k));

dT_dt = (k_cond * (d2T_dx2 + d2T_dy2 + d2T_dz2) + Q_perf + Q_m + Q_laser(i,j,k)) / (rho * c_p);

T_next(i,j,k) = T(i,j,k) + dt * dT_dt;

end

end

end