Question 27.2: One way to deliver a timed dosage of a drug within the human...

The general differential equation for mass transfer reduces to the following partial differential equation for the one-dimensional unsteady-state concentration profile c_{A}(r, t) :

\frac{\partial c_{A}}{\partial t}=D_{A B}\left(\frac{\partial^{2} c_{A}}{\partial r^{2}}+\frac{2}{r} \frac{\partial c_{A}}{\partial r}\right)      (27-18)

Key assumptions include radial symmetry, dilute solution of the drug dissolved in the gel matrix, and no degradation of the drug inside the bead \left(R_{A}=0\right). The boundary conditions at the center (r=0) and the surface (r=R) of the bead are

\begin{array}{lll} r=0, & \frac{\partial c_{A}}{\partial r}=0, & t \geq 0 \\ r=R, & c_{A}=c_{A s}=0, & t>0 \end{array}

At the center of the bead, we note the condition of symmetry where the flux N_{A}(0, t) is equal to zero. The initial condition is

t=0, \quad c_{A}=c_{A o}, \quad 0 \leq r \leq R

The analytical solution for the unsteady-state concentration profile c_{A}(r, t) is obtained by separationof-variables technique described earlier. The details of the analytical solution in spherical coordinates are provided by Crank. The result is

Y=\frac{c_{A}-c_{A o}}{c_{A s}-c_{A o}}=1+\frac{2 R}{\pi r} \sum_{n=1}^{\infty} \frac{(-1)^{n}}{n} \sin \left(\frac{n \pi r}{R}\right) e^{-D_{A B} n^{2} \pi^{2} t / R^{2}}, \quad r \neq 0, n=1,2,3, \ldots      (27-19)

At the center of the spherical bead (r=0), the concentration is

Y=\frac{c_{A}-c_{A o}}{c_{A s}-c_{A o}}=1+2 \sum_{n=1}^{\infty}(-1)^{n} e^{-D_{A B} n^{2} \pi^{2} t / R^{2}}, \quad r=0, n=1,2,3, \ldots         (27-20)

Once the analytical solution for the concentration profile is known, calculations of engineering interest can be performed, including the rate of drug release and the cumulative amount of drug release over time. The rate of drug release, W_{A}, is the product of the flux at the surface of the bead (r=R) and the surface area of the spherical bead

W_{A}(t)=4 \pi R^{2} N_{A r}=4 \pi R^{2}\left(-D_{A B} \frac{\partial c_{A}(R, t)}{\partial r}\right)     (27-21)

It is not so difficult to differentiate the concentration profile, c_{A}(r, t), with respect to radial coordinate r, set r=R, and then insert back into the above expression for W_{A}(t) to ultimately obtain

W_{A}(t)=8 \pi R c_{A o} D_{A B} \sum_{n=1}^{\infty} e^{-D_{A B} n^{2} \pi^{2} t / R^{2}}     (27-22)

The above equation shows that the rate of drug release will decrease as time increases until all of the drug initially loaded into the bead is depleted, at which point W_{A} will go to zero. Initially, the drug is uniformly loaded into the bead. The initial amount of drug loaded in the bead is the product of the initial concentration and the volume of the spherical bead

m_{A o}=c_{A o} V=c_{A o} \frac{4}{3} \pi R^{3}

The cumulative amount of drug release from the bead over time is the integral of the drug release rate over time

m_{A o}-m_{A}(t)=\int_{0}^{t} W_{A}(t) d t

After some effort, the result is

\frac{m_{A}(t)}{m_{A o}}=\frac{6}{\pi^{2}} \sum_{n=1}^{\infty} \frac{1}{n^{2}} e^{-D_{A B} n^{2} \pi^{2} t / R^{2}}      (27-23)

The analytical solution is expressed as an infinite series summation that converges as ” n ” goes to infinity. In practice, convergence to a single numerical value can be attained by carrying the series summation out to only a few terms, especially if the dimensionless parameter D_{A B} t / R^{2} is relatively large. It is a straightforward task to implement the infinite series summation on a spreadsheet program such as Excel (Microsoft Corporation).

A representative spreadsheet solution is provided in Table 27.1. Note that in Table 27.1 the terms within the series summation rapidly decay to zero after a few terms. The cumulative drug release vs. time profile is shown in Figure 27.5. The drug-release profile is affected by the dimensionless parameter D_{A B} t / R^{2}. If the diffusion coefficient D_{A B} is fixed for a given drug and gel matrix, then the critical engineering-design parameter we can manipulate is the bead radius R. As R increases, the rate of drug release decreases; if it is desired to release 50 \% of Dramamine from a gel bead within 3 \mathrm{~h}, a bead radius of 0.326 \mathrm{~cm}(3.26 \mathrm{~mm}) is required, as shown in Figure 27.5. Once the bead radius R is specified, the initial concentration of Dramamine required in the bead can be backed out

c_{A o}=\frac{m_{A o}}{V}=\frac{3 m_{A o}}{4 \pi R^{3}}=\frac{3(10 \mathrm{mg})}{4 \pi(0.326 \mathrm{~cm})^{3}}=\frac{68.9 \mathrm{mg}}{\mathrm{cm}^{3}}

In summary, a 6.52-mm-diameter bead with an initial concentration of 68.9  \mathrm{mg} / \mathrm{cm}^{3} will dose out the required 5  \mathrm{mg} of Dramamine within 3 \mathrm{~h}. The concentration profile along the r direction at different points in time is provided in Figure 27.6. The concentration profile was calculated by spreadsheet similar to the format given in Table 27.1. The concentration profile decreases as time increases and then flattens out to zero after the drug is completely released from the bead.