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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 $k_H$.

Primary paper: pubmed.ncbi.nlm.nih.gov/14088963/ . T. Higuchi, Journal of Pharmaceutical Sciences 52, 1145–1149 (December 1963); DOI doi.org/10.1002/jps.2600521210 . Original scan hosted in USPTO proceedings: ptacts.uspto.gov/ptacts/public-informations/petitions/1519735/download-documents?artifactId=Oa66ferdR3w2b5pS9-xX0Mcec5DHvuJE3_Ze0mytFlVRg2kA16YpxPc . The original abstract distinguishes diffusion in a homogeneous matrix from solvent leaching of a granular matrix and considers planar and spherical geometry. Initial-only byline retained.

Verified author identity: the University of Kansas Kenneth Spencer Research Library lists this exact 1963 paper in Personal papers of Takeru Higuchi, PP 359, Box 10 Folder 13, archives.lib.ku.edu/repositories/3/archival_objects/124437 . Publisher page onlinelibrary.wiley.com/doi/abs/10.1002/jps.2600521210 retains the literal byline T. Higuchi and University of Wisconsin pharmacy affiliation. Person and empty Work canonicalized to Takeru Higuchi; old person and Work IDs retained as adjacent synonyms. Preserve the published initial-only source title/byline.

Publisher abstract: onlinelibrary.wiley.com/doi/abs/10.1002/jps.2600521210 . It explicitly relates the one-dimensional release rates to the earlier ointment-base equation and distinguishes those results from spherical-pellet release, which it says is not first-order. This is the verified abstract locator for the historical context of the Higuchi equation.

Takeru Higuchi, credited as T. Higuchi in his 1963 paper, developed mathematical descriptions of diffusion-controlled drug release from solid matrices. The University of Kansas archive holds that exact paper among his personal papers, establishing the connection between the initial-only byline and the full name.^[ref]

The paper distinguished a uniform matrix serving as the diffusional medium from a granular matrix penetrated by a solvent. It considered planar and spherical geometries, rather than treating all sustained release as a single first-order process.^[ref] The resulting models connect formulation geometry and material properties with a measurable drug release rate.

A simplified planar derivation illustrates the Higuchi equation. Assume a stationary matrix with constant diffusion coefficient $D$, excess undissolved drug, a perfect sink condition in drug release at its exposed face, and a slowly moving depletion boundary at depth $h(t)$. Approximate the dissolved-drug concentration as linear from zero at the exposed face to its solubility $C_s$ at that boundary. If initial total loading per unit volume is $C_0$, the amount released per exposed area isbecause the depleted layer retains a triangular dissolved-drug profile. The outward mass flux is approximately $D{C}_s/h$, so mass conservation givesThus the cumulative released amount is proportional to $\sqrt{t}$ in this model, whereas its derivative decreases as $t^{-1/2}$. The high-loading approximation $C_0\gg C_s$ makes the slowly moving boundary assumption more plausible; the result is not a general solution for an arbitrary dissolving or swelling material.

For an original illustration, let $Q=k_H\sqrt{t}$ with $k_H=2\mathrm{mg}\mathrm{c}{\mathrm{m}}^{-2}{\mathrm{h}}^{-1/2}$. The cumulative release is $2\mathrm{mg}\mathrm{c}{\mathrm{m}}^{-2}$ at one hour and $4\mathrm{mg}\mathrm{c}{\mathrm{m}}^{-2}$ at four hours, but the instantaneous rates per area are respectively $1$ and $0.5\mathrm{mg}\mathrm{c}{\mathrm{m}}^{-2}{\mathrm{h}}^{-1}$. These are chosen model values. They show why sustained release need not be constant-rate release. Finite thickness, depletion of the solid phase, changing solubility or external transport resistance can end this regime. Measuring release tests the model's transport assumptions; a square-root fit alone does not uniquely identify them.