{c}
{title2=$Q=k_H\sqrt t$}

The <Higuchi equation> models cumulative drug release proportional to the square root of time:
$$
Q(t)=k_H\sqrt t.
$$
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.<Mechanism of Sustained-Action Medication. Theoretical Analysis of Rate of Release of Solid Drugs Dispersed in Solid Matrices by T. Higuchi>[Publisher abstract, matrix mechanisms and 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> gives
$$
Q=(A-C_s/2)\ell,\qquad \frac{dQ}{dt}\simeq\frac{DC_s}{\ell}.
$$
Combining and integrating from zero released amount yields
$$
Q^2=D C_s(2A-C_s)t,\qquad k_H=\sqrt{D C_s(2A-C_s)}.
$$
This 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 $D=10^{-8}\,\mathrm{cm^2/s}$, $C_s=1\,\mathrm{mg/cm^3}$ and $A=10\,\mathrm{mg/cm^3}$, the model gives $Q(3600\,\mathrm{s})\approx0.0262\,\mathrm{mg/cm^2}$. At four hours it predicts twice that released amount, not four times. The depleted-layer depth at one hour is approximately $27.5\,\mathrm{\mu m}$. 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 $k_H$.


 Higuchi equation (source code)