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 r_d of the pill in a patient is proportional to the product of the pill surface area and the concentration driving force:

r_d = kA (c_s – c_σq)

where c_σq is the concentration of the dissolved drug in the aqueous medium, C_s is the saturation value, A is the surface area of the pill, and k is the mass transfer coefficient. Because C_s >> c_σq, even if the pill dissolves completely, the rate of dissolution reduces to r_d = kAc_s.

(a) Derive a dynamic model that can be used to calculate pill mass 1W as a function of time. You can make the following simplifying assumptions:

(i) The rate of dissolution of the pill is given by r_d = kAc_s.

(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 = 2πrh.

(b) 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 r₀?

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p= 1.2g/ml C, = 500 g/L r0 = 0.4cm k = 0.016cm/min h = 1.8cm