Question 1
An active MRI (MRI stands for Magnetic Resonance Imaging) is a small device incorporated into metal surgical implants to make imaging via MRI more effective. However, MRI implants produce heat, and it is important to know whether the tissue temperature around the implant can rise too high. You are asked to do a simplified analysis to estimate the temperature rise in the tissue due to the MRI implant.
Assume that the MRI implant is spherical with a diameter of 40mm and is surrounded by the tissue. The implant has a rate of generation of 1 Watt that is conducted through the tissue at steady state.
Please note that the main interest is to know how the heat generated in the MRI implant is transferred through the tissue, and from that information, to find the highest temperature in the tissue. You can assume that the tissue temperature far away from the implant is maintained at 37°C, and at the boundary where the tissue touches the implant, there is a heat flux from the implant (i.e. 1 W/m^2) and that the generated heat flux is considered equal over the surface of the spherical implant. Assume that the thermal conductivity of the tissue is 0.5 W/m·K.
(a) Draw a schematic of the system.
(b) Write the governing equations in the proper coordinate system needed to solve the problem. Make suitable assumptions (and write them) to develop a model that could provide an analytical solution to the problem. In addition, write appropriate boundary conditions that will help you solve the problem. An important boundary condition is where the tissue touches the implant.
(c) Solve the resulting mathematical model obtained from (b), which is a differential equation + boundary conditions, to obtain the temperature profile in the tissue, i.e. temperature as a function of position (since the geometry is a sphere, position will be indicated by r).
(d) Find the maximum temperature in the tissue (under steady-state conditions) due to heat generated in the implant.
(e) How is that maximum temperature associated with the heat generation? For example, answer if the heat generation is doubled, does the maximum temperature double?
