Dissolution kinetics: models, fitting, interpretation and the Turkish school

Introduction

Dissolution kinetics studies ask two questions together: which equation describes the curve best and what do the coefficients of that equation say about the release mechanism? Focusing on the first question alone and picking the model with the highest R² is a common but inadequate habit (Costa & Sousa Lobo, 2001; Askarizadeh et al., 2023). This guide covers the most frequently used models, how they are fitted, and how the results should be compared and interpreted. Its central thesis, in line with the literature, is this: linearised regression is instructive and quick, but for parameter estimation non-linear least squares is more reliable on most real data sets (Costa et al., 2003; Dash et al., 2010; Zhang et al., 2010).

The tool's approach in one sentence: every model is first seeded from its linearised form, then refined non-linearly on the raw release percentage; all goodness-of-fit criteria are computed in the same space; models are ranked by AIC, which penalises the number of parameters; and non-physical results are flagged as warnings. The definitions used are the same as those of DDSolver, the field's widely used software (Zhang et al., 2010); the comparison of the results with DDSolver outputs is given in the relevant section.

A short history

The first quantitative description of dissolution rate is the 1897 work of Noyes and Whitney: the rate is proportional to the difference between the saturation concentration and the instantaneous concentration (Noyes & Whitney, 1897). Hixson and Crowell (1931) derived the cube-root law by accounting for the shrinking surface of the dissolving solid. Wagner (1969) explained why most tablet and capsule dissolution profiles appear to follow first-order kinetics: the surface area shrinks exponentially and the curve approaches an asymptote. Higuchi (1961, 1963) showed that release from matrix systems by Fickian diffusion is linear in the square root of time; under the name "square-root kinetics" this became the most widely used equation in pharmaceutical technology.

Langenbucher (1972) proposed the Weibull distribution for linearising dissolution curves; in the Turkish literature this equation was long referred to as RRSBW (Rosin–Rammler–Sperling–Bennet–Weibull). Baker and Lonsdale (1974) derived expressions for spherical matrices and Hopfenberg (1976) for surface-eroding geometries. Korsmeyer and co-workers (1983) and Ritger and Peppas (1987) introduced the power law and the mechanistic interpretation of the exponent n; Peppas and Sahlin (1989) introduced the two-term equation that separates the diffusional and relaxational contributions. The 2001 review by Costa and Sousa Lobo gathered this body of work under one roof and remains the most cited reference text today. DDSolver by Zhang and co-workers (2010) is an Excel add-in that applies the same model set with non-linear fitting and AIC and has become the de facto standard of the field.

The tradition of kinetic evaluation in Turkey took shape within this timeline: the (Bt)^a kinetics derived by Ağabeyoğlu in 1978 is an independent and earlier application of the power law; the Ege University school (Ertan, Karasulu, Özyazıcı) used the model set systematically in in vitro–in vivo correlation and extended the Hopfenberg exponent to new geometries; the Ankara school (Yüksel, Kanık, Baykara) tested profile-comparison methods statistically. Details are given in the section The Turkish school.

Data preparation

The tool expects cumulative release percentage (0–100) against time. The time unit is free (minutes, hours) but must be the same in every row; the units of the coefficients follow from the chosen time unit. The point t = 0, F = 0 is not entered; it is assumed. If replicates (up to six vessels) are entered, the mean and standard deviation are computed and the analysis is run on the mean; if only the mean is available, it can be typed directly into the "Mean" column. Copy-and-paste from a spreadsheet works; the decimal separator may be a comma or a point.

• Fraction or percentage? If values between 0 and 1 are entered, a warning is shown. Because the saturating models assume a 100 % ceiling, an analysis run on fractions gives meaningless results.
• Decreasing profile. If the values fall with time, the data are taken to be "drug remaining", converted to release as 100 − F, and this is reported as a warning.
• Values above 100 % are not discarded; they are used as they are and a warning is given. The cause may be analytical deviation or a label-claim problem; in the saturating models it affects the Fmax variant.
• Number of points. At least 5–6 time points are recommended; models with 2–3 parameters "memorise" fewer points and the degrees of freedom fall. No analysis is run with fewer than three points.
• Coverage. The points should include both the rising part of the release and the plateau; plateau points alone cannot determine the rate constant, and early points alone cannot determine Fmax.
• Nature of the replicates. Replicates at the same time point should come from independent vessels; re-reading the same vessel is not a replicate.
• F ≤ 60 % for Korsmeyer–Peppas. The derivation of the power law holds for the early phase of release; n should be determined from this region only (Ritger & Peppas, 1987; Costa & Sousa Lobo, 2001). The tool applies this by default; the "all points" option is for special cases such as comparison with DDSolver.

Model catalogue

The tool fits 16 base models. Some have a physical derivation; others are empirical expressions proposed to capture the shape of the curve. Since dissolution, wetting, swelling, erosion and diffusion operate simultaneously in tablet and matrix systems, a single equation should not be expected to represent the whole process (Siepmann & Siepmann, 2008; Ojsteršek et al., 2024). In the table F is the release percentage and t is time; the "Variant" column shows which extensions (Tlag, F0, Fmax) can be applied to the model.

Deliberately absent from the catalogue: Quadratic (a polynomial without a mechanism that can decrease; present in DDSolver, absent in Costa & Sousa Lobo; removed because it came last in the ranking on every trial set) and duplicates of the older linearisation-based models (RRSBW, Langenbucher, modified Langenbucher, (Bt)^a; see the equivalence table).

Variants: Tlag, F0, Fmax

Three extensions model three deviations frequently seen in real data without changing the base equation. Which model may take which extension is restricted by physical meaning; combinations that are not permitted are not offered in the interface.

• Tlag (lag time). t → (t − Tlag); F = 0 for t < Tlag. It represents a waiting phase such as coating dissolution, wetting or capsule opening, and can be added to all models. A negative value is allowed and often means an "intercept": some release had already occurred before the first measurement point. In the Turkish literature Weibull + Tlag is known as "modified Langenbucher" (Ertan et al., 2000).
• F0 (initial burst). F → F0 + base. Meaningful only in the models without a ceiling (Zero-order, Higuchi, Korsmeyer–Peppas); adding F0 to a saturating model pushes the asymptote above 100 %, which is why Fmax is used there instead. In Zero-order and Higuchi a small negative F0 is a legitimate fit that mimics a lag. In Korsmeyer–Peppas, however, if F0 is left unbounded the solver finds a logarithmic curve with the triple F0 → −∞, k_KP → +∞, n → 0: the sum of squares falls but none of the coefficients is meaningful. For this reason F0 ≥ 0 is enforced in KP.
• Fmax (incomplete-release plateau). The 100 in the equation → Fmax. Selected only in the saturating models (First-order, Weibull); Logistic (Fmax) and Gompertz (Fmax) already carry Fmax in their equation. Fmax > 100 % is not physical and is flagged. If the data have not reached a plateau, Fmax and the rate constant compensate each other (small k, large Fmax ≈ a straight line); if Fmax exceeds the highest observed release by more than 25 %, a "plateau not observed" warning is given and the T25–T90 derived from that fit should be regarded as extrapolation.

Fitting procedure: seed, Levenberg–Marquardt, Nelder–Mead

The traditional route is to bring the equation into a straight-line form by an algebraic transformation and read the coefficients from the slope and intercept: for first order, the logarithm of the amount remaining is plotted against time; for Higuchi, release against √t; for Korsmeyer–Peppas, log F against log t. The method can be applied with a spreadsheet and is valuable in teaching; most of the Turkish literature reported its coefficients this way. But the transformation changes the error structure: while the measurement error is roughly constant on the release-percentage scale, small values are magnified on the logarithmic scale and least squares gives weight to the wrong points (Costa et al., 2003). Moreover, R² values in linearised spaces cannot be compared between models, because each model has a different dependent variable.

The tool therefore uses linearisation only for the initial estimate (seed). The coefficients are then sought iteratively on the raw release percentage so as to minimise the sum of squares between observed and computed F. The solver is the Levenberg–Marquardt algorithm (Marquardt, 1963): it switches between a Gauss–Newton step and a gradient-descent step via a damping parameter, behaving safely far from the solution and quickly near it. If it fails or cannot improve on the seed, the derivative-free Nelder–Mead simplex (Nelder & Mead, 1965) takes over; it is more robust for piecewise-defined models (F = 0 before Tlag) and for the numerical root solution of Baker–Lonsdale. Neither solver ever returns a result worse than the seed. For models linear in the coefficients (Zero-order, Higuchi, Peppas–Sahlin-2) the seed is already the global optimum; the solver confirms it.

Parameter bounds follow from physical meaning: rate constants and exponents cannot be negative, Fmax ≥ 0, F0 ≥ 0 in KP; Tlag and the Peppas–Sahlin coefficients are free. Weighting (1, 1/F, 1/F²) exists in the engine but is not exposed in the interface; where the variance differs markedly between time points, weighted least squares has been shown to improve the estimates (Costa et al., 2003).

Goodness-of-fit criteria and ranking

All criteria are computed in F(%) space over the same N observations, so that the models can be compared with one another. The definitions are the same as in DDSolver (Zhang et al., 2010); the results were compared with DDSolver outputs (below).

Two rules keep the ranking honest. First, because Korsmeyer–Peppas is fitted to a different data set (F ≤ 60 %), its AIC cannot be compared with the others; it is presented separately in the "Mechanism analysis" table and receives no Akaike weight. Second, ranking the same curve under two names distorts the weights (the denominator inflates and the other models shrink unfairly); for this reason Hopfenberg is not entered in the ranking for the slab (= Zero-order) and sphere (= Hixson–Crowell) geometries.

Secondary parameters and profile metrics

For the selected model, T25, T50, T75, T80, T90 (times to reach the corresponding percentage) are computed from the inverse of the fitted equation; where no closed-form inverse exists (Baker–Lonsdale, Makoid–Banakar) they are solved numerically. If the target cannot be reached by the curve, Non Calc is written: Fmax < target, or the Makoid–Banakar peak staying below the target, for instance. In the Tlag variant the times shift by Tlag; in the F0 variant the target is lowered by F0. Times beyond the last measurement are an extension of the curve and make a claim about an unmeasured region. For Weibull, Td (time to 63.2 % release) is also reported.

The profile summary does not depend on any model and serves to compare two formulations with a single number (Costa & Sousa Lobo, 2001): AUC, the area under the curve (trapezoidal rule, from t = 0); DE, dissolution efficiency, the ratio of the area under the curve to the rectangle representing 100 % release over the same period (Khan, 1975); MDT, mean dissolution time, the average of the interval midpoints weighted by the amount released. For the similarity of two profiles the right tool is not these either but the model-independent f₁/f₂ factors (Moore & Flanner, 1996); the F1/F2 tool on this site is for that.

Glossary of warning flags

The warnings shown next to a model in the results panel say that the fit is unreliable despite a low sum of squares. Their meanings and what to do:

• Fmax > 100 %. The plateau exceeds the physical limit; the parameters may be compensating each other (e.g. Weibull b < 1 with Fmax = 125). Prefer the form without Fmax or a different model.
• Plateau not observed. Fmax is far above the highest observed release; the data have not reached saturation. Fmax and T25–T90 are extrapolations; a longer measurement period is needed.
• F0 out of range. F0 < −20 or > 100: the model has degenerated on these data and the coefficients cannot be interpreted.
• n ≈ 0 (Korsmeyer–Peppas) / m ≈ 0 (Peppas–Sahlin). The power law is degenerate: the curve is almost constant or logarithmic; the mechanistic interpretation is invalid.
• k ≈ 0 (Makoid–Banakar). The damping term is ineffective; the model has reduced to Korsmeyer–Peppas and the extra parameter only worsens the AIC.
• Degrees of freedom ≤ 1. The number of parameters is very close to the number of points; the coefficients "memorise" the data. Select fewer variants or enter more points.
• Fitted with all points (KP). The F ≤ 60 % filter has been switched off; the interpretation of n may be unreliable.
• Geometry notes. Hopfenberg slab = Zero-order, sphere = Hixson–Crowell; hemisphere and triangle are the proposal of Karasulu et al. (2000) and are not in the classical set of Costa & Sousa Lobo.
• Convergence failed. No solver produced a valid candidate; the result is the seed. Review the data coverage and the suitability of the model.

Model selection and interpretation

There is no single best model or single best criterion for every data set (Dash et al., 2010). A sound choice looks at complementary indicators together: AIC and Akaike weight, R²_adj, random scatter of the residuals along the time axis, physical plausibility of the coefficients and visual fit of the curve (Yuksel et al., 2000; Ojsteršek et al., 2024). The role of AIC is this: the raw sum of squares always favours the model with more parameters; AIC corrects this bias and answers the question "is the more complex model really better, or merely more flexible?"

The best-fitting model is not proof of the mechanism. Empirical models (Weibull, Makoid–Banakar, Peppas–Sahlin) fit almost any curve; a low AIC says "describes the data well", not "release happens this way". For a mechanistic claim, diagnostic indicators such as the Korsmeyer–Peppas n, the Peppas–Sahlin k₂/k₁ ratio and the Weibull shape parameter should be read together with knowledge of the formulation (Papadopoulou et al., 2006; Siepmann & Siepmann, 2008). Which model comes out on top depends on the formulation: among different systems examined by the same method, sigmoid models surpassed Higuchi in one study while Higuchi was the most suitable in another (Yuksel et al., 2000; Baishya, 2017). Rather than imposing a fixed model on a product family, the choice should be made afresh for every profile.

To sum up: linearisation is a good starting point for understanding a model, not for reporting coefficients. Coefficients should be estimated by non-linear fitting to the raw data, models should be compared by AIC, warning flags should be taken seriously, and the result should be interpreted together with the residuals and knowledge of the formulation.

Worked example: a 12-point profile

The data below are one of the example profiles distributed with the DDSolver add-in (Zhang et al., 2010): a 20-hour extended-release profile with 12 measurement points. If you paste the same data into the Kinetic Analysis tool and press "Calculate" with the default options (no variants, no weighting, F ≤ 60 % filter on for Korsmeyer–Peppas), you obtain the results below. The purpose of this section is to show how to read the numbers, not to prove a new result.

What do the weights say? The first three models share 90 % of the weight; none is dominant on its own. This means "the data cannot distinguish these three curves": Probit and Logistic capture a slightly sigmoid start, First-order and Weibull (β ≈ 1) an exponential approach. Among indistinguishable candidates, the one with the fewest parameters is the reasonable choice as the reporting model: First-order, k₁ = 0.139 h⁻¹. If the sigmoid start has a meaning for the formulation (coating, wetting delay), Probit or Weibull is preferred and the justification is written down. On the same data DDSolver gives k₁ = 0.1385 h⁻¹ and AIC 56.28 for First-order; this tool finds 0.139 and 56.26. The two results are the same in practice; the small difference comes from the solvers' stopping criteria (below).

What do the residuals say? Although the R²_adj values are close to one another at around 0.99, the residual sequences tell different stories. The residuals of First-order are negative at hour 1, positive at hours 3–4, negative at hours 8–12 and positive at the end: the classic "S" pattern, the real curve being slightly more sigmoid than the model. Probit, conversely, holds the middle well and the tail poorly (+3.2 and +4.6 at hours 18–20). No model leaves the residuals random; this shows that expecting a single "true model" is not realistic for these data. The residual plot in the results panel should always be inspected.

Korsmeyer–Peppas and the F ≤ 60 % rule. With the filter on, KP is fitted only to the first five points (F ≤ 58) and gives n = 0.83: anomalous transport. If the filter is switched off and all twelve points are used, n falls to 0.52 and the interpretation becomes "close to Fickian diffusion". The difference is large enough to reverse the mechanistic interpretation; that is the rationale of the rule (Ritger & Peppas, 1987; Costa & Sousa Lobo, 2001). This is why KP does not enter Table 4 and is presented separately in the "Mechanism analysis" table (k_KP = 13.8, n = 0.834, R²_adj = 0.935).

Secondary parameters. For the first four models T50 falls in the range 4.5–5.0 h and T80 in 11.6–11.8 h; these values are almost independent of the model and can be reported with confidence. T90, on the other hand, varies between 16.6 (First-order) and 20.5 h (Logistic): it depends on the shape of the tail of the curve and should therefore be given together with the model name. Model-independent profile summary: AUC 1315 %·h, DE 65.8 %, MDT 6.16 h.

Sanity check. Zero-order comes last in the list with R²_adj = 0.65 and AIC 98; its residuals run systematically from +25 to −22. This shows that the ranking does indeed exclude the clearly wrong models. Higuchi (R²_adj = 0.963) staying in the middle is also expected: the square-root law loses its validity in the region F > 60 %, whereas here seven of the twelve points lie in that region.

Common mistakes and decision flow

The following are the problems most frequently encountered in publications and theses; most are prevented in the tool by a flag or a design decision, but the final decision rests with the user.

• Selecting a model by linearised R². R² values computed in ln F, √t or log–log space cannot be compared with one another; each transformation weights the residuals differently. The comparison should be made in F(%) space, after non-linear fitting (Zhang et al., 2010).
• Comparing models with different numbers of parameters by R². A three-parameter Weibull never gives a lower R² than a two-parameter First-order. The comparison criterion should be AIC (or AICc, MSC); R²_adj is only a partial correction.
• Fitting Korsmeyer–Peppas to all points and interpreting n. The power law is derived for F ≤ 60 %; in the example above n shifts from 0.83 to 0.52. If the filter is off, the tool warns.
• Adding the point (0, 0) as data. The origin is not a measurement; adding it inflates N and misleads the AIC and the degrees of freedom. Most models already pass through F(0) = 0; those that do not (the F0 variant) exist precisely to estimate this point.
• Including plateau points without limit. Points taken after release has reached 100 % carry no kinetic information but affect the residual sum and the rate constant. For Higuchi and KP the validity limit is 60 %; in the other models, points after the first one reaching the plateau are usually discarded and this decision is reported.
• Fitting the mean profile and ignoring variability. The mean of six units gives a single curve; the differences between units are lost. Either each unit is fitted separately and the mean and standard deviation of the parameters are given, or, when the mean profile is used, the standard deviation of the points is reported alongside (Costa & Sousa Lobo, 2001).
• Too many parameters for too few points. Squeezing a three-parameter model plus Tlag and Fmax onto a five-point profile reduces the degrees of freedom to 0–1; the coefficients memorise the data. The tool raises the "DF ≤ 1" flag. As a rule, N − p ≥ 3 should be required.
• Switching variants on without justification. Tlag, F0 and Fmax are each a physical claim: delayed wetting, rapid initial release, incomplete release. They should be selected because they have a counterpart in the formulation, not because they lower the AIC.
• Mixing up the time unit. k₁ = 0.139 h⁻¹ and 0.0023 min⁻¹ are the same rate. Coefficients should always be reported with the unit; when comparing two studies the units must be matched.
• Presenting extrapolation as measurement. If the last measurement is at hour 12, T90 = 19 h is an extension of the curve. The tool marks this; it should be marked in the report too.
• Taking the best model as proof of the mechanism. A low AIC describes the curve well; it does not say why release happens that way (above).

Suggested decision flow

1. Check the data: time order, duplicated points, unit, post-plateau points, values above 100 %.
2. Run all models without variants. Do not interpret flagged models (Fmax > 100, F0 out of range, degenerate exponent, not converged), however high they stand in the ranking.
3. Look at the Akaike weight. If a single model has w ≥ 0.5, that is the model. Otherwise note the models sharing the weight as "equivalent" and make the one with the fewest parameters the reporting model.
4. Examine the residual plot. If there is a systematic pattern (S shape, one-sided tail) and it has a counterpart in the formulation, try the relevant variant; confirm that the AIC falls and the pattern disperses.
5. For the mechanism question read the Korsmeyer–Peppas n (F ≤ 60 %) and, if needed, the Peppas–Sahlin k₂/k₁ ratio; select the geometry thresholds (slab, cylinder, sphere).
6. Report T50/T80/T90 with the model name and the unit; mark extrapolated values. Add AUC, DE, MDT for a model-independent summary.
7. If the question is the similarity of two formulations, use f₂ instead of comparing kinetic parameters (below).

f₂ and kinetic modelling: where they diverge

The question whether two profiles are similar and the question how a profile releases call for different tools. For similarity, the regulatory standard is the model-independent difference factor f₁ and similarity factor f₂ of Moore and Flanner (1996):

f₁ = Σ|R_t − T_t| / Σ R_t × 100     f₂ = 50 · log₁₀{ [1 + (1/n) Σ (R_t − T_t)²]^−1/2 × 100 }

R and T are the mean releases of the reference and test profiles at the same n time points. According to the FDA guidance, f₂ between 50 and 100 (mean difference ≤ 10 %) means similarity; at least three points, at most one point above 85 % release, and a coefficient of variation not exceeding 20 % at the early points and 10 % at the later ones are required (FDA, 1997). Shah and co-workers (1998) examined the statistical properties of f₂ and proposed a bootstrap confidence interval when variability is high; Polli and co-workers (1997) and Anderson and co-workers (1998) compared model-independent, model-dependent and ANOVA-based approaches.

The two approaches are not mutually exclusive, but their places should not be confused. The inference "the k₁ values of the two formulations differ, therefore the profiles differ" is invalid: in the example above, three models give three different rate constants for the same profile. Conversely, two profiles with f₂ ≥ 50 may release by different mechanisms; f₂ is shape-blind. When Yuksel and co-workers (2000) compared the three approaches on the same data, they found the model-dependent method more sensitive in showing differences between formulations and f₂ more stable for decision making. The practical rule: the equivalence decision with f₂, characterisation with a kinetic model; reporting both together is the most informative route. The F1/F2 tool on this site computes f₁, f₂ and the model-independent summaries from the same data.

The Turkish school and name equivalents

In Turkey, the kinetic evaluation of dissolution data took shape at three centres, and all three used the same model set: zero order, first order, Hixson–Crowell, RRSBW, Higuchi (Q√t), Hopfenberg, Langenbucher, modified Langenbucher and (Bt)^a. This set was also the list of this site's desktop engine dating from 1995. For an international reader: RRSBW is the Weibull function under the name Rosin–Rammler–Sperling–Bennet–Weibull, "Langenbucher" is the same function in its linearised form, "modified Langenbucher" is Weibull with a lag time, and (Bt)^a is the power law written with the rate inside the bracket.

The Ankara school. The line that began with İzgü and Ağabeyoğlu's 1974 mathematical study of diffusion rate from ointment bases turned into a direct contribution to kinetics in Ağabeyoğlu's 1978 associate-professorship (doçentlik) thesis: when sulfamethizole data obtained with a continuous-flow cell did not fit the Langenbucher linearisation, the exponent of time was set free and the (Bt)^a kinetics was derived (Ağabeyoğlu, 1978). This is an application of the power law that precedes Korsmeyer and co-workers (1983) and is independent of them; with the transformation k_KP = B^a, n = a it is the same curve as Korsmeyer–Peppas. Ağabeyoğlu's chapter "Biyofarmasötik" (Biopharmaceutics) in Modern Farmasötik Teknoloji (2007) is the most comprehensive Turkish-language treatment of the subject, from Noyes–Whitney to Wagner's theory and from the Kitazawa and Higuchi equations to the f₂ calculation. Yüksel, Kanık and Baykara (2000) systematically tested ANOVA-based, model-dependent and model-independent profile-comparison methods and showed the limits of the discriminating power of f₂. Studies on marketed products (Kaynar Özdemir et al., 1989; Baloğlu & Hızarcıoğlu, 2001) use the same model set; the best fit was most often found with modified Hixson–Crowell and RRSBW.

The Ege school. The nitrofurantoin and theophylline studies of Ertan, Karasulu and Güneri established the systematic use of the model set in in vitro–in vivo correlation: the "inverse kinetics" method with modified Langenbucher (Ertan et al., 2000), the comparison of nine models by coefficient of determination and the improvement of correlation accuracy in ultra-extended-release theophylline (Karasulu et al., 2003), and accelerated dissolution experiments (Ertan et al., 2011). Karasulu, Ertan and Köse (2000) adapted the equation of Katzhendler and co-workers (1997) to HPMC tablets of different geometries and proposed the values 1.5 for the hemisphere and 4 for the triangle for the Hopfenberg exponent; these values appear in the geometry list of the tool. Özyazıcı, Gökçe and Ertan (2006) applied the power law and the Fickian/relaxational distinction to lipid matrices.

The Istanbul school. The formulation studies of Araman, Özsoy, Cevher and Güngör "evaluate release data kinetically" (Güngör et al., 2003; Orlu et al., 2006); they are examples of application rather than sources of methodology. At Hacettepe, RRSBW kinetics was used together with factorial design, and the Farmakokinetik textbook of Hıncal and Kaş (1986) provided the theoretical ground.

In the Turkish literature, model selection was made almost exclusively by the coefficient of determination. That this guide puts AIC first does not diminish the value of those studies; on the same data r² and AIC usually select the same model. The difference appears when models with unequal numbers of parameters are compared (e.g. Weibull against first order); there r² always favours the one with more parameters.

Comparison with DDSolver

DDSolver (Zhang et al., 2010) is the most widely used and most cited software of the field; it applies the same models with non-linear fitting and AIC/MSC. This tool does not claim to replace DDSolver; the aim was to check, against an independent reference, that the definitions and the solver work correctly. For this purpose 32 result sheets produced with DDSolver (37 formulations; Zero-order, First-order, Higuchi, Korsmeyer–Peppas and the F0/Tlag/Fmax variants) were taken into the repository as permanent test data and are compared again in every release. Observations:

• The definitions are the same. When the parameters found by DDSolver are put into this tool's equations, SS, R, R², R²_adj, MSE, AIC and MSC come out identical in all 37 sets; that is, the equations, the point rules and the goodness-of-fit criteria are shared.
• Solver difference. DDSolver uses Nelder–Mead; this tool refines with Levenberg–Marquardt after linear seeding and, if necessary, with Nelder–Mead. In the 9 sets linear in the coefficients the results are identical; in the remaining 28 sets the sums of squares differ by small amounts, and these differences come from the solvers' stopping criteria. The difference between k₁ = 0.1385 and 0.139 h⁻¹ in the worked example is typical.
• Korsmeyer–Peppas. DDSolver fits KP to all points; this tool applies the F ≤ 60 % rule by default. The comparison is made with the "all points" option.
• MSC detail. In the MSC calculation DDSolver drops the points with F = 0 from the total sum of squares, but does not drop them in the R² calculation; this tool uses all points in both criteria. The difference is seen only in data where the row t = 0, F = 0 has been entered.

The practical conclusion: the two programs use the same definitions; if you see a small coefficient difference between DDSolver and this tool, its source is the solver difference. Cite the software you used in publications; if you work with DDSolver, this tool can be used to cross-check results quickly.