Problem #2 (50 points) In medical applications the chief objectives for drug delivery are (i) to deliver the drug to the correct location in the patient's body, and (ii) to obtain a specified drug concentration profile in the body through a controlled release of the drug over time. Drugs are often administered as pills. In order to derive a simple dynamic model of pill dissolution, assume that the rate of dissolution rd of the pill in a patient is proportional to the product of the pill surface area and the concentration driving force: r₁=KA(C.-C.) where Ca is the concentration of the dissolved drug in the aqueous medium, C. is the saturation value, A is the surface area of the pill, and k is the mass transfer coefficient. Because C. >> C., even if the pill dissolves completely, the rate of dissolution reduces to r.=KAC. 1. Derive a dynamic model that can be used to calculate pill mass M as a function of time. (20 points) You can make the following simplifying assumptions: i. The rate of dissolution of the pill is given by r.=KAC. ii. The pill can be approximated as a cylinder with radius r and height h. It can be assumed that h/r » 1. Thus, the pill surface area can be approximated as A = 2xrh. 2. Solve the model you have derived (10 points) 3. For the conditions given below, how much time is required for the pill radius r to be reduced by 90% from its initial value of ro (10 points)? p = 1.2 g/ml To = 0.4 cm h = 1.8 cm C. = 500 g/L k = 0.016 cm/min 4. Does the shape have an effect on the dissolution rate? If so what shape selection would be best? (10 points)