Problem 3: Many drugs designed to treat disorders in the brain fail because they do not adequately pass the blood-brain barrier. Consider a drug that, when injected into the blood (volume Vb=5L ), rapidly exchanges with many tissues of the body (total volume Vt=20L ) such that the blood and these tissues can be considered to be in equilibrium. The drug exchanges by diffusion much more slowly with the brain (volume Vh=1L(h ' for head, since we used b for blood already)) with rate constant k=0.001L/min. A. (5) Develop a model for the distribution of drug as a function of time following a bolus injection of mass, minj,=100mg into the blood at t=0. a. State all assumptions and explain and justify any approximations you make. b. Define any symbols you use (and give their units) that weren't given in the problem description. c. Develop the model equations using symbols for the various parameters rather than numerical values. B. (5 pts) Using the numerical data given, plot the concentration of drug in the brain during the first 4 hours after injection. You may solve this analytically and then just plot the solution, or you can perform a simulation. C. (5 pts) Focused ultrasound stimulation can be used to increase the permeability of the blood brain barrier for about 4 hours and can do it selectively in a region of interest. Assume that using this approach, a region of the brain with volume 0.1L is targeted, and the permeability is increased 10 fold such that k remains 0.001L/min (permeability is increased because of the ultrasound, but the contact area between blood and brain is reduced because the volume of the tissue is reduced). Under these conditions, find the concentration of the drug in the target tissue region. Plot the concentration in the target tissue for the first 4 hours and compare with the response for the case without ultrasound stimulation. Comment on the effectiveness of the focused ultrasound