One way to deliver a timed dosage of a drug within the human body is to ingest a capsule and allow it to settle in the gastrointestinal system. Once inside the body, the capsule slowly releases the drug by a diffusion-limited process. A suitable drug carrier is a spherical bead of a nontoxic

Question

One way to deliver a timed dosage of a drug within the human body is to ingest a capsule and allow it to settle in the gastrointestinal system. Once inside the body, the capsule slowly releases the drug by a diffusion-limited process. A suitable drug carrier is a spherical bead of a nontoxic gelatinous material that can pass through the gastrointestinal system without disintegrating. A water-soluble drug (solute [latex]A[/latex] ) is uniformly dissolved within the gel and has an initial concentration [latex]C_{A_{o}}[/latex]. The drug loaded within the bead is the source for mass transfer, whereas the fluid surrounding the bead is the sink for mass transfer. This is an unsteady-state process, as the source for mass transfer is contained within the diffusion control volume itself.

Consider a limiting case where the resistance to film mass transfer of the drug through the liquid boundary layer surrounding the capsule surface to the bulk surrounding the fluid is negligible. Furthermore, assume that the drug is immediately consumed or swept away once it reaches the bulk solution so that in essence the surrounding fluid is an infinite sink. In this particular limiting case, [latex]c_{A s}[/latex] is equal to zero, so at a long time the entire amount of drug initially loaded into the bead will be depleted. If radial symmetry is assumed, then the concentration profile is only a function of the [latex]r[/latex] direction (Figure 27.4).

It is desired to design a spherical capsule for the timed release of the drug dimenhydrinate, commonly called Dramamine, which is used to treat motion sickness. A conservative total dosage for one capsule is [latex]10 \mathrm{mg}[/latex], where [latex]50 \%[/latex] of the drug must be released to the body within [latex]3 \mathrm{~h}[/latex]. Determine the size of the bead and the initial concentration of Dramamine in the bead necessary to achieve this dosage. The diffusion coefficient of Dramamine (species [latex]A[/latex] ) in the gel matrix (species [latex]B[/latex] ) is [latex]3 imes[/latex] [latex]10^{-7} \mathrm{~cm}^{2} / \mathrm{s}[/latex] at a body temperature of [latex]37^{\circ} \mathrm{C}[/latex]. The solubility limit of Dramamine in the gel is [latex]100 \mathrm{mg} / \mathrm{cm}^{3}[/latex], whereas the solubility of Dramamine in water is only [latex]3 \mathrm{mg} / \mathrm{cm}^{3}[/latex].

The model must predict the amount of drug released vs. time, bead diameter, initial concentration of the drug within the bead, and the diffusion coefficient of the drug within the gel matrix. The physical system possesses spherical geometry. The development of the differential material balance model and the assumptions associated with it follow the approach presented in Section 25.4.

Accepted Answer

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

[latex]\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} ight)[/latex] (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 [latex]\left(R_{A}=0 ight)[/latex]. The boundary conditions at the center [latex](r=0)[/latex] and the surface [latex](r=R)[/latex] of the bead are

[latex] \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} [/latex]

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

[latex]t=0, \quad c_{A}=c_{A o}, \quad 0 \leq r \leq R[/latex]

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

[latex]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} ight) e^{-D_{A B} n^{2} \pi^{2} t / R^{2}}, \quad r eq 0, n=1,2,3, \ldots [/latex] (27-19)

At the center of the spherical bead [latex](r=0)[/latex], the concentration is

[latex]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[/latex] (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, [latex]W_{A}[/latex], is the product of the flux at the surface of the bead [latex](r=R)[/latex] and the surface area of the spherical bead

[latex]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} ight)[/latex] (27-21)

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

[latex]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}}[/latex] (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 [latex]W_{A}[/latex] 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

[latex]m_{A o}=c_{A o} V=c_{A o} \frac{4}{3} \pi R^{3}[/latex]

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

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

After some effort, the result is

[latex]\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}}[/latex] (27-23)

The analytical solution is expressed as an infinite series summation that converges as " [latex]n[/latex] " 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 [latex]D_{A B} t / R^{2}[/latex] 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 [latex]D_{A B} t / R^{2}[/latex]. If the diffusion coefficient [latex]D_{A B}[/latex] is fixed for a given drug and gel matrix, then the critical engineering-design parameter we can manipulate is the bead radius [latex]R[/latex]. As [latex]R[/latex] increases, the rate of drug release decreases; if it is desired to release [latex]50 \%[/latex] of Dramamine from a gel bead within [latex]3 \mathrm{~h}[/latex], a bead radius of [latex]0.326 \mathrm{~cm}(3.26 \mathrm{~mm})[/latex] is required, as shown in Figure 27.5. Once the bead radius [latex]R[/latex] is specified, the initial concentration of Dramamine required in the bead can be backed out

[latex]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}}[/latex]

In summary, a 6.52-mm-diameter bead with an initial concentration of [latex]68.9 \mathrm{mg} / \mathrm{cm}^{3}[/latex] will dose out the required [latex]5 \mathrm{mg}[/latex] of Dramamine within [latex]3 \mathrm{~h}[/latex]. The concentration profile along the [latex]r[/latex] 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.

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

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