Problem $3$

In medical applications, a common objective is to deliver the active pharmaceutical ingredient $($API$)$ to

the target area and keep the concentration in the so$-$called "therapeutic window" where it is most

effective ${?}^1.$ One of the most rudimentary and common ways of administering drugs to the body are in the

form of oral pills. To achieve a simple model of pill dissolution, we may assume that the rate of dissolution

$(r_d)$ is directly proportional to the product of the pill's surface area $(A),$ the aqueous concentration of the

API in the stomach $(c_{aq}),$ and the concentration of the API on the surface of the pill itself $(c_s)$ :

$r_d=kA(c_s-c_{aq})$

We may further assume that the concentration in the stomach is negligible $(c_s{c}_{aq}),$ which makes the

rate of dissolution expression boil down to $r_d=kA{c}_s.$

By performing an unsteady$-$state mass balance, derive a dynamic model that describes the size of

the pill, and hence its mass $m,$ as a function of time. You may assume: $[5]$

The rate of dissolution is given by the expression derived above: $r_d=kA{c}_s.$

The pill can be approximated as a sphere with radius $r$ and has constant density $.$

The volume and surface area of a sphere are given as $V=\frac{4}{3}{r}^3$ and $A=4r^2.$

The specific parameters for a certain type of pill are provided below. For this pill, solve for the

mass of the pill as a function of time $m(t)$ and plot the resulting mass profile as a function of time.

Do this task two ways and compare your results $($mind the units$)$:

a$.$ Analytically using separation of variables $($this system is not nasty, I swear$).[5]$

b$.$ Setting the ODE up in MATLAB and solving for $m(t)$ numerically using ode $45.[5]$

$=1\mathrm{.}21\frac{g}{mL},r_0=0\mathrm{.}77cm,c_s=450\frac{g}{L}$

$k=0\mathrm{.}0175\frac{cm}{\text{m}\text{i}\text{n}}$

The so$-$called "therapeutic window" of an API is the time where it is at an optimal concentration

in the bloodstream $($or affected area$)$ that is neither too high nor too low. For this drug, based on

the rate of dissolution $r_d$ and known uptake kinetics, it is known that the therapeutic window is

when the drug is between $10\%$ and $90\%$ of its initial mass. Once a patient ingests the pill,

determine how long it takes for the API to enter the therapeutic window, and the length of the

window itself. $[5]$