= 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.<Mechanism of Sustained-Action Medication. Theoretical Analysis of Rate of Release of Solid Drugs Dispersed in Solid Matrices by T. Higuchi>[Original abstract and publication, pp. 1145–1149]{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> gives
$$
c(x,t)=c_0\operatorname{erf}\!\left(\frac{x}{2\sqrt{Dt}}\right).
$$
<Fick's first law> then gives outward <mass flux> of magnitude
$$
J(t)=c_0\sqrt{\frac{D}{\pi t}},\qquad
\frac{M_{\mathrm{rel}}(t)}{A}=\int_0^tJ(s)\,ds
=2c_0\sqrt{\frac{Dt}{\pi}}.
$$
This 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 $D=10^{-12}\,\mathrm{m^2\,s^{-1}}$, the time scale $L^2/D$ is $10^4\,\mathrm s$, and at $100\,\mathrm s$ the approximation gives $F\approx0.113$. 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.<Adaptive Perfusion: A Novel In Vitro Drug Release Testing Method for Complex Drug Products by the FDA>[Technology Summary, limitations of passive membrane transport]{ref}
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