Higuchi equation ($Q=k_H\sqrt{t}$)

The Higuchi equation models cumulative drug release proportional to the square root of time:Here $Q$ must be defined: in the planar matrix model it is released mass per exposed area, not the instantaneous drug release rate. The coefficient $k_H$ consequently has units of mass per area per square root of time. Takeru Higuchi’s 1963 primary paper distinguishes drug dispersed in a homogeneous matrix from drug leached from a granular matrix and considers planar and spherical geometries.^[ref]

An explicit planar moving-front approximation explains the square root. Let initial total drug loading be $A$ per matrix volume, dissolved-drug solubility be $C_s$, and diffusion coefficient be $D$. Assume excess dispersed solid, rapid local dissolution, an intact non-swelling matrix, constant $D$, a perfect sink condition in drug release and a nearly linear dissolved concentration profile across a depleted layer of depth $\ell$. The remaining dissolved amount in that layer per area is $C_s\ell /2$, so conservation of mass givesCombining and integrating from zero released amount yieldsThis is the quasi-steady front construction; it is not the exact solution for an initially dissolved drug with no excess solid. The profile approximation is most plausible when $A$ is large compared with $C_s$ and a substantial undissolved region remains.

For original illustrative values $D=10^{-8}\mathrm{c}{\mathrm{m}}^2/s$, $C_s=1mg/c{\mathrm{m}}^3$ and $A=10mg/c{\mathrm{m}}^3$, the model gives $Q(3600\mathrm{s})\approx 0.0262mg/c{\mathrm{m}}^2$. At four hours it predicts twice that released amount, not four times. The depleted-layer depth at one hour is approximately $27.5\mu \mathrm{m}$. A finite matrix cannot sustain the law indefinitely: once the front approaches the far boundary, the assumed geometry and drug inventory must be reconsidered.

A useful experiment plots corrected released mass per area against $\sqrt{t}$ and checks whether the apparent slope changes with medium flow or formulation thickness. A straight segment alone does not identify the mechanism. External transport resistance, swelling or changing diffusivity can invalidate the model even if a short data segment looks nearly linear. Sampling correction in drug release testing and a specified exposed area are therefore part of interpreting a fitted $k_H$.