Dissolution Kinetics Model Fitting (Higuchi, Weibull and More)

About the Dissolution Kinetics Model Fitting (Higuchi, Weibull and More)

Fits dissolution or release data to the models of the dissolution literature — zero order, first order, Higuchi, Hixson–Crowell, Korsmeyer–Peppas, Weibull, Baker–Lonsdale, Hopfenberg and Peppas–Sahlin — by non-linear least squares, with an optional lag time and plateau, and ranks them by adjusted R², AIC or MSC. It reads the release exponent n of the Korsmeyer–Peppas fit to the first 60 % of release for your geometry (thin film, cylinder or sphere) and the shape of the Weibull curve, and adds the model-independent mean dissolution time (MDT), dissolution efficiency (DE), t50 and t80.

Paste the times and the % released — one column per batch or vessel — and get the ranking, the parameters with standard errors and the chart, for a formulation or QC report. Without a pass the result is a free preview: the best model, the ranking and the mechanism reading, with your data on the chart; the parameters, the goodness of fit, the fitted curves, the PDF and Excel report and printing unlock with a Premium pass.

Common uses

• Comparing the release mechanism of matrix tablet batches or formulations in development.
• Reporting MDT, DE, t50 and t80 of dissolution profiles for a development or stability report.
• Checking a spreadsheet’s or another program’s fit with a second, independent calculation.

The models

• Zero order F = k0·t · First order F = Fmax·(1 − e^(−k1·t)) · Higuchi F = kH·√t · Hixson–Crowell F = 100·[1 − (1 − kHC·t)³]
• Korsmeyer–Peppas F = kKP·tⁿ · Weibull F = Fmax·[1 − exp(−t^β ÷ α)], with Td = α^(1/β) the time at which 63.2 % of Fmax is released
• Baker–Lonsdale (3/2)·[1 − (1 − F/100)^(2/3)] − F/100 = kBL·t · Hopfenberg F = 100·[1 − (1 − kHB·t)ⁿ], n = 1, 2 or 3 for a slab, cylinder or sphere · Peppas–Sahlin F = k1·t^m + k2·t^(2m), m the Fickian exponent of the geometry

F is the % released and t the time; Fmax is 100 % unless you fit a plateau, and a lag time replaces t with t − Tlag. The point at time 0 is not fitted: every model starts there.

How the fits are ranked

Each model is fitted by non-linear least squares (Levenberg–Marquardt, equal weights), from several starting values derived from the linearised forms. Adjusted R² = 1 − (1 − R²)·(n − 1) ÷ (n − p) allows for the number of parameters p, so a model is not favoured only for having more; AIC = n·ln(SSR) + 2p, lower is better; MSC = ln(Σ(y − ȳ)² ÷ SSR) − 2p ÷ n, higher is better. A model with as many parameters as points is not ranked. The goodness of fit is not the whole answer: a mechanistic model must also make sense for the dosage form.

Reading the release exponent and the Weibull shape

Korsmeyer–Peppas is fitted to the points up to 60 % released (the cut-off can be changed), and n is read with the limits Ritger and Peppas derived: for a thin film 0.5 is Fickian diffusion and 1.0 Case II transport; for a cylinder 0.45 and 0.89; for a sphere 0.43 and 0.85 — in between, anomalous (non-Fickian) transport; above the Case II value, super Case II. The Weibull β describes the curve: 1 is exponential, above 1 a sigmoid curve with a turning point, below 1 a parabolic curve that is steeper at first.

MDT, DE, t50 and t80

• MDT = Σ t̄·ΔM ÷ Σ ΔM, with t̄ the middle of each sampling interval and ΔM the % released in it (from time 0).
• DE = the area under the profile up to a time ÷ (the reference × that time) × 100, by trapezoids: the reference is 100 % of the label claim or, as some software defines it, the highest % dissolved.
• t50 and t80 by linear interpolation between the points, and from the best model.

Sources

• Zhang et al.: DDSolver, an add-in program for modeling and comparison of drug dissolution profiles — the model forms, adjusted R², AIC, MSC, MDT and DE
• Costa and Sousa Lobo: modeling and comparison of dissolution profiles00095-1) — the models and model-independent parameters
• Ritger and Peppas: a simple equation for description of solute release, I90034-4) and II90035-6) — the release exponent for slabs, cylinders and spheres
• Peppas and Sahlin: coupling of diffusion and relaxation90306-2) — the Peppas–Sahlin model
• Dash et al.: kinetic modeling on drug release from controlled drug delivery systems — the 60 % rule and the reading of n

Limitations

• A fit describes the data; it does not prove a mechanism, and models with more parameters can fit noise.
• Equal weights only; weighted fits and profile comparisons (f1, f2, bootstrap) are not part of this tool.
• Korsmeyer–Peppas needs at least 3 points up to the cut-off for the release exponent; the verdict says when there are fewer.
• MySmartCoPilot is not affiliated with or endorsed by DDSolver or its authors; their paper is cited for the published model forms and criteria.
• The cut-off is read on the % released as you enter it. The power law describes the first 60 % of the amount finally released (Mt/M∞ ≤ 0.6), so for a profile that levels off well below 100 %, enter the cut-off for its plateau: 48 for a plateau at 80 %.

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