Question 2 of 6: Transdermal Controlled-Release Design for an Antihypertensive Patch

Question 2: Transdermal Controlled-Release Design for an Antihypertensive Patch (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(a) Both proposed designs are one-dimensional diffusion problems through the same layered geometry — an outer, impermeable backing; a drug-bearing layer; and an inner boundary at the skin, which acts as a perfect sink because the underlying tissue is very well vascularized (given concentration ≈ 0). The two methods differ only in how the drug-bearing layer is loaded, and that difference changes the governing equation completely: Method 1's drug is dissolved (loaded at exactly its solubility limit, with no excess), so the reservoir is finite and depletes as it releases — a transient diffusion problem. Method 2's drug is dispersed as excess solid, far above what the boundary membrane can dissolve (1000 mg/mL loading against a 0.4 mg/mL membrane solubility), so the membrane's inner face is held at a constant (saturated) concentration for as long as any undissolved solid remains — a steady-state, zero-order diffusion problem.

Given.

Find. The total release rate delivered by each design at $t=28$ h, and which design (with what film thickness, for Method 1) satisfies the 200 µg/day requirement.

Approach. Compute the patch area; solve Method 1 as one-dimensional transient (Fickian) diffusion from a finite slab with a reflecting outer face and a sink inner face (exact Fourier-series solution, cross-checked against the short-time semi-infinite approximation); solve Method 2 as steady-state Fick's-first-law diffusion through the rate-controlling membrane; compare both to the 200 µg/day floor and to each other's time-dependence.

1. Patch area. $$A = \frac{\pi d^2}{4} = \frac{\pi (3\text{ cm})^2}{4} = 7.069\text{ cm}^2$$
2. Method 2 — steady-state flux through the rate-controlling membrane. Because the matrix loading (1000 mg/mL) vastly exceeds the membrane's solubility limit (0.4 mg/mL), undissolved drug persists in the matrix and holds the membrane's upstream face at the constant saturated concentration $C_s$ for as long as any solid remains; the downstream (skin) face is held at zero. This is ordinary Fick's first law across the membrane at steady state: $$J_2 = \frac{D_m C_s}{L_m} = \frac{(2\times10^{-8}\text{ cm}^2/\text{s})(0.4\text{ mg/mL})}{0.02\text{ cm}} = 4.00\times10^{-7}\ \text{mg/(cm}^2\text{s)}$$ Converting to daily units and multiplying by the patch area, $$\dot m_2 = J_2 A = \left(4.00\times10^{-7}\ \tfrac{\text{mg}}{\text{cm}^2\text{s}}\right)(86400\ \tfrac{\text{s}}{\text{day}})(1000\ \tfrac{\mu\text{g}}{\text{mg}})(7.069\ \text{cm}^2) = \boxed{244\ \mu\text{g/day}}$$ This value is constant once the membrane reaches its own quasi-steady concentration profile, after a short lag time $\tau = L_m^2/(6D_m) \approx 0.93$ h — negligible next to the 24–28 h dosing interval — and it stays essentially constant afterward because the excess solid in the matrix keeps replenishing the membrane's upstream face. Method 2 therefore delivers $244\ \mu\text{g/day}$ at $t=28$ h (indeed, at any $t$ beyond about 1 h), comfortably above the $200\ \mu\text{g/day}$ floor.
3. Method 1 — transient release from the finite, depleting gelatin reservoir. Here the drug is dissolved, not dispersed, so the gelatin film holds a fixed total amount ($C_0$ throughout a film of thickness $L$) with no replenishment; as diffusion proceeds toward the sink at the skin-side face, the local concentration falls and the release rate declines with time. With the outer face reflecting ($\partial C/\partial x = 0$ at $x=0$) and the inner face a perfect sink ($C=0$ at $x=L$), the boundary-value problem is the classic finite-slab diffusion problem (Crank), and the flux delivered at the sink face is $$J_1(t) = \frac{2DC_0}{L}\sum_{n=0}^{\infty}\exp\!\left[-\frac{D(2n+1)^2\pi^2 t}{4L^2}\right]$$ While the diffusion penetration depth $\sqrt{Dt}$ is still much smaller than $L$ (i.e., the reservoir "does not yet know" the outer face is sealed), this reduces to the familiar semi-infinite short-time form $$J_1(t) \approx C_0\sqrt{\dfrac{D}{\pi t}}$$ At $t=28\text{ h}=100{,}800\text{ s}$, the penetration depth is $\sqrt{D_1 t}=\sqrt{(3\times10^{-7})(100{,}800)}=0.174\text{ cm}$ — i.e., roughly the middle of the allowed $0.1$–$1$ cm thickness range, so the film thickness genuinely matters here (unlike Method 2, where the outer 0.1–1 cm gelatin thickness is irrelevant because it is only a reservoir, not the diffusion path).
4. Choosing the film thickness. Evaluating the exact finite-slab series at $t=28$ h for several candidate thicknesses (within the given $0.1$–$1$ cm range):

   Method 1 — release rate at t = 28 h vs. film thickness L

   L (cm)	Release rate at 28 h (µg/day)	Fraction of loaded drug already released
   0.10 (thinnest allowed)	2.5	≈100% (essentially exhausted)
   0.17 (threshold)	200 (exactly meets the floor)	≈94%
   0.20	340	87%
   0.50 (selected design)	713	39%
   1.00 (thickest allowed)	713	20%

   A film at the thin end of the allowed range (0.1 cm) is already almost completely depleted by 28 h and delivers only about 2.5 µg/day — a severe, silent under-dose. The minimum thickness that still clears the 200 µg/day floor at 28 h is $L\approx0.17$ cm; choosing $L=0.5$ cm (still well inside the allowed $0.1$–$1$ cm range) gives a comfortable margin, $\boxed{713\ \mu\text{g/day at }t=28\text{ h}}$, and is close enough to the semi-infinite plateau that the design is not sensitive to small manufacturing variation in thickness.

Both methods can be made to satisfy the numerical 200 µg/day spec at 28 h, but they are not engineering equivalents. Method 1's release rate follows a $1/\sqrt{t}$ decay from a genuinely depleting reservoir: it would deliver a very large initial burst (well over ten times the required rate in the first hour) and then decline continuously for as long as the patch is worn, so any dosing beyond the intended interval, or any manufacturing variation that makes the film a little thinner than designed, erodes the margin above the 200 µg/day floor or defeats it altogether (as the $L=0.1$ cm case shows outright). For a drug whose therapeutic action is to lower blood pressure, an uncontrolled early burst is a genuine safety concern (risk of acute hypotension), not merely an inconvenience. Method 2, by contrast, is a rate-controlling-membrane design fed by an excess (dispersed) reservoir: once the short lag transient has passed it delivers an essentially constant, zero-order rate that is insensitive to how much drug remains in the matrix (only to the membrane thickness and the two diffusion/solubility constants, which are manufacturing-controlled to tight tolerance). Method 2 is therefore the preferred design — it meets the release-rate specification with a stable, predictable, non-bursting profile appropriate for a chronic antihypertensive.

(b) Alternative method. An osmotic-pump patch (a miniature Rose–Nelson/Alzet/DUROS-type reservoir) would be a stronger alternative to both proposed designs. A rigid, semipermeable outer membrane admits water from the skin/tissue at a rate fixed by the osmotic-pressure difference across it (set by an internal osmagent, not by the drug's own solubility); the resulting steady influx of water displaces a fixed volume of saturated drug solution out through a small calibrated orifice at a truly constant, zero-order rate for as long as any solid drug (plus osmagent) remains undissolved inside — the delivery rate is decoupled from the drug's own diffusivity and solubility entirely, unlike Method 2, whose rate is fixed by $D_m$ and $C_s$ and therefore cannot be tuned without changing materials. This removes Method 1's burst/depletion problem, removes Method 2's dependence on the drug's own solubility in the membrane (which fixes the achievable rate for a given membrane thickness), and gives the designer an independent rate-setting parameter (orifice diameter) that can be adjusted without reformulating the drug reservoir — a genuine engineering advantage when, as in this scenario, the same drug substance also needs different release profiles for its other (glaucoma, GI) indications.