We are interested in modeling the thermal ablation of tumor tissue. This process is governed by the heat equation: where T(x, t) is the tissue temperature, k is the heat conductivity, and q(x) is a heat sink term that represents tissue cooling due to local blood flow qo if T(x) > To 0 otherwise When the tissue temperature exceeds 50 for more than 1 sec cells start to die, and above 60 they die instantly. We are interested in determining the size of the burnt region after 1 sec. Consider a 1D piece of tissue of length Lx. Suppose the tissue is initially at a uniform temperature To. At time t = 0, a region of size ? at the center of the tissue is heated instantaneously to a temperature T1. Assume the boundaries of the tissue are far away and at normal tissue temperature To Use the following values for the model parameters To = 370; T? = 200% k = 0.1 cm2/sec; Lx = 10.0 cm; ? = 0.2 cm; q,-0.01 o/sec; a) Write a code to solve the governing equation using the finite difference method. Use the appropriate stability condition to calculate the timestep size. Advance the solution for 1.0 second and plot the initial and final temperature distributions, along with a straight line at T 50o. b) By how much does your solution at t-1.0 sec change if you decrease the mesh size by half Is this change acceptable according to your engineering judgement? c) Compare results obtained with and without tissue cooling due to blood flow (qo 0). In particular, determine graphically how the extent of the burnt region (T> 50) changes with blood flow cooling. d) How can you increase the size of the burnt region at 1 sec? Run simulations to justify your answer We are interested in modeling the thermal ablation of tumor tissue. This process is governed by the heat equation: where T(x, t) is the tissue temperature, k is the heat conductivity, and q(x) is a heat sink term that represents tissue cooling due to local blood flow qo if T(x) > To 0 otherwise When the tissue temperature exceeds 50 for more than 1 sec cells start to die, and above 60 they die instantly. We are interested in determining the size of the burnt region after 1 sec. Consider a 1D piece of tissue of length Lx. Suppose the tissue is initially at a uniform temperature To. At time t = 0, a region of size ? at the center of the tissue is heated instantaneously to a temperature T1. Assume the boundaries of the tissue are far away and at normal tissue temperature To Use the following values for the model parameters To = 370; T? = 200% k = 0.1 cm2/sec; Lx = 10.0 cm; ? = 0.2 cm; q,-0.01 o/sec; a) Write a code to solve the governing equation using the finite difference method. Use the appropriate stability condition to calculate the timestep size. Advance the solution for 1.0 second and plot the initial and final temperature distributions, along with a straight line at T 50o. b) By how much does your solution at t-1.0 sec change if you decrease the mesh size by half Is this change acceptable according to your engineering judgement? c) Compare results obtained with and without tissue cooling due to blood flow (qo 0). In particular, determine graphically how the extent of the burnt region (T> 50) changes with blood flow cooling. d) How can you increase the size of the burnt region at 1 sec? Run simulations to justify your
