 Diffusion-controlled drug release

Diffusion-controlled drug release occurs when molecular transport down a concentration gradient controls the relevant release time scale. It is a mechanism defined for a particular material, geometry and environment, rather than a name for every slowly changing release curve. A classic theoretical treatment by T. Higuchi distinguishes several matrix and geometrical models.^[ref]

A simple example is an initially uniform dissolved substance of concentration $c_0$ in a planar material, with constant diffusion coefficient $D$. At early times, approximate the material as the half-space $x>0$ and impose a perfect sink condition in drug release, $c(0,t)=0$. Solving the diffusion equation givesFick's first law then gives outward mass flux of magnitudeThis derivation is for dissolved material under the stated boundary condition; it does not identify every square-root law with the same solid-particle matrix model.

For a finite layer of thickness $L$ releasing through one face, the early-time fraction is approximately $F=2\sqrt{Dt/\pi }/L$. With illustrative $L=0.10\mathrm{mm}$ and $D=10^{-12}{\mathrm{m}}^2{\mathrm{s}}^{-1}$, the time scale $L^2/D$ is $10^4\mathrm{s}$, and at $100\mathrm{s}$ the approximation gives $F\approx 0.113$. Extending it to $10^4\mathrm{s}$ would give $F>1$, demonstrating its failure once the finite size matters. The initial mathematical singularity in $J(t)$ likewise reflects idealized initial and boundary conditions.

A useful experimental check changes thickness while keeping material and boundary conditions comparable: this diffusion model predicts a time scale proportional to $L^2$. Swelling, changing diffusivity, degradation or external transport can alter that scaling. A membrane used by the measurement apparatus can also limit transport, so observing a slow curve alone does not locate the controlling resistance inside the formulation.^[ref]