Prediction of surfactant-mediated dissolution of poorly soluble drugs from drug powder

Abstract

Beyond rough “what if” estimation, in vitro dissolution is infrequently predicted. The objective was to assess the predictability of a powder dissolution model with a single diffusion layer thickness model, where dissolution of various drugs was facilitated by several surfactant micelles. Powder dissolution of three poorly water soluble drugs (i.e., posaconazole, ritonavir, and griseofulvin) was measured into buffer, as well as four surfactant solutions [i.e., sodium lauryl sulfate (SLS), polysorbate 80 (PS80), polyoxyethylene (10) lauryl ether (POE10), and cetyltrimethylammonium bromide (CTAB)]. Drug solubility, micelle sizing, and powder sizing were also performed. Prediction of drug dissolution employed the film dissolution model, applied to spherical drug particle fractions of the percent weight particle size distribution, and with a surfactant-mediated dissolution component. There were two competing models for diffusion layer thickness: fixed thickness (i.e., h_fixed) and radius-dependent thickness (i.e., h_max) models. SLS, PS80, POE10, and CTAB increased the dissolution of posaconazole, ritonavir, and griseofulvin, compared to no-surfactant buffer. Results show that in vitro drug dissolution from various polydisperse powders into several surfactant solutions was successfully predicted using a surfactant-mediated dissolution model. The best diffusion layer thickness for the fixed thickness model and the radius-dependent model were separately found to be h_fixed = 12 μm and h_max = 12 μm, respectively, with h_fixed = 12 μm being the more preferred. Also, the powder dissolution model where powder was parameterized in terms of its entire particle size distribution was successful in predicting observed dissolution profiles using each h_fixed = 12 μm and h_max = 12 μm; model use of a mean particle size was also successful in prediction using h_fixed = 12 μm. Credibility assessment of the in vitro dissolution model was performed, including model verification and validation considerations in light of the question of interest, the context of use, and model risk.

1. Introduction

Predictions of dissolution profiles in this article build upon prior collective studies of Dressman and Fleisher, Johnson and colleagues, Polli and colleagues, and Okazaki et al. (Balakrishnan et al., 2004; Dressman and Fleisher, 1986; Hintz and Johnson, 1989; Okazaki et al., 2008; Oktay and Polli, 2024). Dressman and Fletcher devised a film model for spherical, mono-disperse particle dissolution into non-surfactant solution (Dressman and Fleisher, 1986). Johnson and colleagues extended this model to polydisperse drug particles, including the use of a radius-dependent diffusion layer thickness (Hintz and Johnson, 1989; Johnson, 2012).

Using mean particle size and entire particle size distribution (PSD), Okazaki et al. simulated and measured the dissolution of griseofulvin and danazol powders into various taurocholate solutions. The effective diffusivity values incorporated micelle diffusivity values of about 20–50 × 10⁻⁷ cm²/s (Okazaki et al., 2008). Without focusing on drug particles, Polli and colleagues devised and applied a surfactant-mediated dissolution model (Balakrishnan et al., 2004; Oktay and Polli, 2024). For example, predicted and observed dissolutions were similar for griseofulvin dissolution into several pharmaceutical surfactants; predicted dissolution employed both free drug diffusivity (e.g., 110 × 10⁻⁷ cm/s for griseofulvin) and micelle diffusivity [e.g., 12.9 × 10⁻⁷ cm/s for sodium lauryl sulfate (SLS) and 4.04 × 10⁻⁷ cm/s for Cremophor EL].

The film model for particle dissolution into either surfactant-free solution or surfactant solution has gained greater utilization, particularly in the applications of physiologically-based biopharmaceutical modeling (PBBM). This film model applied to particles is frequently referred to as the z-factor model (Hofsäss and Dressman, 2020; Takano et al., 2006). Particle size distribution considerations have been a common focus.

Historically, a lesser focus has been the impact of the surfactant type, including micelle diffusivity value, on dissolution (Balakrishnan et al., 2004). This lesser focus perhaps reflects the still developing understanding of surfactant micelles on drug dissolution (Jamil and Polli, 2022). We have previously described limitations of oral absorption PBBM and physiologically-based pharmacokinetic (PBPK) models, where such models to predict in vivo absorption of low water solubility compounds often do not employ realistic micelle diffusivity values (Jamil and Polli, 2022). However, a greater number of micelle diffusivity values are being reported (Oktay and Polli, 2022, 2024).

From solubility and micelle sizing studies, the present work utilizes the film model for particle dissolution and a film model for surfactant-mediated dissolution to predict profiles of three poorly water-soluble drugs. Previously, Takano et al. fitted dissolution profiles of various drugs from crushed tablets and capsule contents into Fasted Simulated Small Intestinal Fluid (FaSSIF) and reported z-factor parameter values (Takano et al., 2006). The film model for particle dissolution (i.e., z-factor model) can be described by $\frac{dM}{dt}=-z{M}_0^{\frac{1}{3}}{M}^{\frac{2}{3}}\left(C_s-\frac{M_0-M}{V}\right)$ where $z=\frac{3D}{h\rho r_0}$ is the particle dissolution z-factor, D is the drug diffusion coefficient in the media, $\mathrm{h}$ is the diffusion layer thickness, $\rho$ is the particle density, and $r_0$ is the initial particle radius. The film model used for dissolution prediction assumes that dissolution is controlled by a concentration gradient across an apparent diffusion layer thickness $\mathrm{h}$.

Two competing models to describe the thickness of the diffusion layer are the fixed diffusion layer thickness model and radius-dependent diffusion layer thickness model. The fixed diffusion layer thickness model employs the same thickness, regardless of particles size. The radius-dependent diffusion layer thickness model involves the thickness to be equal to the radius of the dissolving particle but to not exceed a maximum thickness (i.e., h_max). Johnson and colleagues as well as Okazaki et al. used a radius dependent diffusion layer thickness of 30 μm (i.e., h_max = 30 μm) to simulate dissolution (Hintz and Johnson, 1989; Okazaki et al., 2008). In the present study, as results show, a fixed h of 12 μm (i.e., h_fixed = 12 μm) provided best predictive accuracy across three drugs into several surfactant solutions.

The objective here was to assess the predictability of a particle dissolution model, where dissolution of various drugs was facilitated by surfactant micelles. Three different drugs were selected for particle dissolution with different degrees of solubility enhancement by surfactants (i.e., rank-order surfactant effect of posaconazole > ritonavir > griseofulvin). These drugs were studied in an effort to identify model parameters that may potentially be broadly applicable to other drug powders. Dissolution studies were conducted into five different media [i.e., no surfactant, 2 % SLS, 2 % polysorbate 80 (PS80), 2 % polyoxyethylene (10) lauryl ether (POE10), 2 % cetyltrimethylammonium bromide (CTAB)]. These surfactants were selected as they are used in compendial tests and exhibit differing solubilization and diffusivity properties (Jamil and Polli, 2022; Oktay and Polli, 2022, 2024). Two percent surfactant concentration was selected since 2% (and higher) is used in compendial dissolution tests for several products, such as ritonavir tablets, metolazone tablets, griseofulvin tablets, fenofibrate capsules, and acitretin capsules (USP, 2024).

Surfactant markedly increased posaconazole, ritonavir, and griseofulvin dissolution. Particle size distribution profiles [i.e., entire particle size distribution (PSD) and mean only] were used to predict dissolution for each drug.

In particular, the objective here focused on identifying a single thickness value (i.e., a single value for either h_fixed or h_max) that providing predictability across all dissolution scenarios. We are unaware of prior studies that examined the impact of diffusion layer thickness on predictive accuracy of dissolution into surfactant solutions.

Results show that the best thickness for the fixed model and the radius-dependent model were separately found to be h_fixed = 12 μm and h_max = 12 μm, respectively, with h_fixed = 12 μm being the more preferred and always providing acceptable results. Drug powder dissolution profiles at 30 and 60 min into surfactant were almost always accurately predicted using h_fixed = 12 μm. The only time point predictions using h_fixed = 12 μm that were not accurate was, using mean particle size, 30 min of griseofulvin into CTAB and 60 min of ritonavir dissolution into PS80, where dissolution was slightly over-predicted (i.e. too high). All profiles for posaconazole, griseofulvin and ritonavir at 30 and 60 min were accurately predicted using each entire PSD using h_fixed = 12 μm. Although less preferred, h_max = 12 μm provided acceptable predictions when entire PSD was used, although h_max = 12 μm using mean particle size only over-predicted observed profiles on several occasions. Findings have promise for improved predictions of dissolution of low solubility drugs into physiologically-relevant intestinal media and for improved PBBM.

2. Materials and methods

2.1. Overall approach and model context

Fig. 1 illustrates the overall workflow to predict dissolution for each drug and medium combination. Predictions involved estimating solubility, diffusivity, particle density, and initial particle size distribution, with dissolution modeled using film model with either a fixed diffusion layer thickness (i.e., with thickness h_fixed) or a radius-dependent thickness (i.e., with thickness h_max).

Fig. 1.

Overall workflow to predict drug particle dissolution. Predictions involved solubility, diffusivity, and particle density estimation, along with particle sizing measurements. Prediction employed either h_fixed or h_max models. Observed profiles were obtained across three drugs and four surfactant solutions, along with no surfactant. Accurate predictions were concluded when predictions were within 20 % dissolved of observed profile at 30 min and 60 min.

This study investigated the in vitro dissolution of three poorly water-soluble drugs (i.e., posaconazole, ritonavir, and griseofulvin) in no surfactant and surfactant media (i.e., 2 % SLS, 2 % PS80, 2 % POE10, and 2 % CTAB). Observed particle dissolution profiles were measured and compared to predicted profiles, each considering the entire particle size distribution (PSD) and mean particle size as represented by mean particle diameter (i.e., D₅₀, the diameter where 50 % of the drug powder has a smaller diameter). A prediction is denoted as being accurate when the predicted percent dissolved was within 20 % (i.e., +/−20 %) dissolved units of observed value, at the considered time (i.e., 30 min and 60 min). While a quality control specification for finished dosage forms of +/−10 % dissolved units is commonly used, this criterion was not relevant or reasonable here, given prediction of in vitro dissolution was the focus and given the current state of the art of prediction of in vitro dissolution. We are unaware of the prior use of +/−10 % to assess in vitro dissolution prediction accuracy.

In predicting drug particle dissolution for each drug/medium scenario, z-factor parameters were considered: initial PSD or initial D₅₀ (in terms of particle radius), particle density ($\rho$), diffusion layer thickness (h), and diffusivity [i.e., free drug diffusivity (D_D) if no surfactant, or effective diffusivity (D_eff) if surfactant present] for each medium. Particle dissolution was typically simulated using the Runge-Kutta RK45 numerical method. However, the Radau method was used when a more stiff method was required (i.e., for posaconazole and ritonavir dissolutions into no surfactant media where minimal dissolution occurred).

Additionally, drug solubility and micelle sizing studies were conducted to quantify drug solubility in various media and drug-loaded micelle diffusivity. Free drug diffusivity and effective diffusivity were estimated.

Three concepts in model credibility assessment are the question of interest, the context of use, and the model risk (Kuemmel et al., 2020). The question of interest here was: Can in vitro particle dissolution into surfactant solution be predicted assuming particles are spherical, assuming dissolution is limited by diffusion of free drug and drug-bound micelles, and using a single aqueous boundary layer (i.e., diffusion layer) that is either a fixed thickness or a radius-dependent thickness?

The planned context of use for this predictive dissolution model into surfactant solution was academic and without regulatory purpose. However, the planned model, if verified and validated, is expected to facilitate a better scientific understanding of in vitro drug power dissolution, including in drug development of low solubility compounds where surfactant systems are employed for quality control and prediction of in vivo product performance. None of the surfactant media were selected to predict the in vivo dissolution or in vivo performance of any particular finished dosage form or product, although each of these surfactants is used in compendial dissolution testing (USP, 2024).

Model risk is considered a function of model influence and decision consequence (FDA Guidance, 2023; Kuemmel et al., 2020). Both of these two considerations may be deemed to be low (1), medium (2), or high (3), and overall model risk score is defined as the sum minus 1. Here, the model would provide substantial impact on predicting in vitro dissolution; model influence was high (3). However, no adverse safety or efficacy outcomes would result from an incorrect model decision, therefore decision consequence was low (1). Hence, our overall model risk was medium (i.e., 4 – 1 or level 3).

2.2. Materials

Posaconazole (95 %) and ritonavir (>95 %) were purchased from Chemshuttle (Burlingame, CA). Griseofulvin (97 %) was purchased from Alfa Aesar (Ward Hill, MA). SLS, POE10, CTAB, and sodium phosphate monobasic dihydrate were purchased from Sigma Aldrich (St. Louis, MO). PS80 was purchased from Spectrum Chemicals (New Brunswick, NJ). LogP values for posaconazole, ritonavir, and griseofulvin are 6.72, 5.83, and 3.56, respectively (Coutinho et al., 2023).

2.3. Media for solubility, micelle diffusivity, and in vitro dissolution

Five different media at pH=6.8 were employed: 50 mM sodium phosphate monobasic dihydrate buffer without surfactant, and the same phosphate buffer with either 2 % SLS, 2 % PS80, 2 % POE10 or 2 % CTAB. Sodium phosphate monobasic dihydrate (7.8g ) was added to about 950 ml MilliQ water and mixed until dissolved. pH was adjusted to 6.8 using 0.1 N NaOH. Final volume was 1000ml. Media were prepared separately; a common stock solution was not used. Of note, some griseofulvin solubility and dynamic light scattering (DLS) were obtained from the literature (Patel et al., 2024).

2.4. Solubility studies

For each posaconazole, ritonavir, and griseofulvin, drug solubility was assessed in each of the five pH=6.8 media (i.e., no surfactant, 2 % SLS, 2 % PS80, 2 % POE10, and 2 % CTAB). Excess drug was dispersed in 10 ml of media and shaken for 72 h at 37 °C. pH was checked. Samples were filtered via a 0.45μm polyvinylidene difluoride (PVDF) 33 mm membrane filter (Millex filter unit) (Millipore; Burlington, MA). Dissolved drug was quantified by ultra-performance liquid chromatography (UPLC), as described below. Solubility studies were performed in triplicate.

2.5. Dynamic light scattering analysis for micelle size

Filtered samples from solubility studies were subjected to DLS, using a Zetasizer Nano ZSP (Malvern Instruments; Westborough, MA). Micelle radius was measured at 37 °C. A 12 mm square polystyrene disposable cuvette was used with 173° of backscattered light. The refractive index (1.330) and viscosity (0.890 cP) of distilled water were applied to instrument settings.

2.6. Calculation of diffusivity values

2.6.1. Free drug diffusivity

The diffusivity of free drug (${\mathrm{D}}_{\mathrm{D}}$) was calculated by using eqn (1), with griseofulvin as the reference (de Smidt et al., 1987):

$$D_D=D_{ref}{\left(\frac{M{W}_{ref}}{M{W}_D}\right)}^{\frac{1}{3}}$$ (1)

where ${\mathrm{D}}_{\text{ref}}=0.000008{\mathrm{c}\mathrm{m}}^2/\mathrm{s}\mathrm{e}\mathrm{c}$ and ${\mathrm{M}\mathrm{W}}_{\text{Ref}}=352.76\mathrm{g}/\mathrm{m}\mathrm{o}\mathrm{l}$. The molecular weight (MW) of posaconazole and ritonavir are 700.8 g/mol and 720.95 g/mol, respectively. ${\mathrm{D}}_{\mathrm{D}}$ was calculated to be 6.36 × 10⁻⁶ cm²/s, 6.30 × 10⁻⁶ cm²/s, and 8 × 10⁻⁶ cm²/s for posaconazole, ritonavir, and griseofulvin, respectively.

2.6.2. Drug-loaded micelle diffusivity

${\mathrm{D}}_{\mathrm{D}-\mathrm{M}}$ of drug-loaded micelles in surfactant media was calculated using the Stokes-Einstein equation:

$$D_{D-M}=\frac{k_{B}T}{6\pi \eta r}$$ (2)

where ${\mathrm{k}}_{\mathrm{B}}$ is the Boltzmann’s Constant (cm² •g/sec² •K), T is the absolute temperature (K), $\mathrm{\eta }$ is the viscosity of the media (g/cm•sec), and r is the hydrodynamic radius (cm) from DLS studies.

2.6.3. Effective drug diffusivity

For drug dissolution into surfactant solution, the effective drug diffusivity $D_{\text{eff}}$ was calculated from:

$$D_{eff}=f_f\cdot D_D+f_m\cdot D_{D-M}$$ (3)

where ${\mathrm{f}}_{\mathrm{f}}$ and ${\mathrm{f}}_{\mathrm{m}}$ are the fraction of free drug and fraction of micelle-bound drug, respectively (Oktay and Polli, 2024; Singh et al., 1968).

2.7. Drug particle size measurement

The particle size distribution of each drug powder was characterized (n = 3) using the Malvern Mastersizer 2000 laser diffraction particle size analyzer with a Scirocco 2000 dry dispersion unit (Malvern analytical; Malvern, UK). Dispersive air pressure was 1 bar.

2.8. In vitro drug particle dissolution

For each particle dissolution study (n = 3), 5 mg of either posaconazole, ritonavir, or griseofulvin was subjected to dissolution using USP apparatus type II (Hanson Virtual Instruments SR8 Plus dissolution tester; Chatsworth, CA) at 37 °C and 75 rpm. Five different media were employed (each using pH 6.8 phosphate buffer as main matrix): no surfactant, 2 % SLS, 2 % PS80, 2 % POE10, and 2 % CTAB. Dissolution volume was 100ml. Small volume vessels (150 ml) were used, with corresponding adapters and mini-paddles (Teledyne Hanson Research; Chatsworth, CA). All samples (1 ml) were filtered through a Millipore MillexHV hydrophilic PVDF 0.45 μm syringe filtered and was then analyzed by HPLC.

At time zero, powder was added to the pre-heated medium in the vessel, and stirring was initiated. However, for ritonavir, which exhibited the smallest particle size, particle aggregation (including drug floating and sticking to vessel and paddle shaft) was visually observed for PS80, POE10, and no surfactant. Hence, for ritonavir, studies for PS80, POE10, and no surfactant were conducted where power was pre-wetted, where this pre-wetting minimized this problem. Ritonavir powder was quickly dispersed into 1 ml of medium and transferred (with several washings) to the pre-heated medium in the vessel, and dissolution commenced; medium volume was adjusted so that total final volume was still 100ml.

2.9. Drug quantification of solubility and dissolution samples

Quantification of posaconazole employed an Acquity UPLC BEH 1.7 μm C-18, 50 mm × 2.1 mm column at 37 °C. The mobile phase comprised of 25 % 0.1 % triethylamine with pH adjusted to 3.0 with phosphoric acid, 75 % methanol. A flow rate of 0.4ml/min and an injection volume of 2 μl.

Quantification of ritonavir employed an Agilent Zorbax 5 μm C-18, 150 mm × 4.6 mm column at 37 °C. The mobile phase comprised of 53 % 0.3 % phosphoric acid, 47 % acetonitrile. A flow rate of 1 ml/min and an injection volume of 10 μl. Griseofulvin was quantified as previously described (Patel et al., 2024).

2.10. Prediction of drug particle dissolution

2.10.1. Surfactant-mediated dissolution film model

Overall, in vitro dissolution of 5 mg drug powder into surfactant and surfactant-free media was predicted using the z-factor method reported by Hintz and Johnson (Hintz and Johnson, 1989),

$$\frac{dM}{dt}=-z{M}_0^{\frac{1}{3}}{M}^{\frac{2}{3}}\left(C_s-\frac{M_0-M}{V}\right)$$ (4)

where $\mathrm{z}$ is the particle dissolution z-factor, $\mathrm{M}$ is the mass of solid drug at any time, $\mathrm{M}\mathrm{o}$ is the initial mass, ${\mathrm{C}}_{\mathrm{s}}$ is the solubility of drug, and $\mathrm{V}$ is the volume of the dissolution medium. $z=\frac{3D}{h\rho r_0}$, as noted above. Of note, $\mathrm{D}$ is the drug diffusion coefficient in the medium. In medium without surfactant, $\mathrm{D}$ is free drug diffusivity (i.e., ${\mathrm{D}}_{\mathrm{D}}$). In medium with surfactant, $\mathrm{D}$ is effective drug diffusivity (i.e., ${\mathrm{D}}_{\text{eff}}$) (Oktay and Polli, 2024; Singh et al., 1968). Correspondingly, in medium without surfactant, ${\mathrm{C}}_{\mathrm{s}}$ is free drug solubility. In medium with surfactant, ${\mathrm{C}}_{\mathrm{s}}$ is total drug solubility in the medium. Diffusion layer thickness h is discussed below. True density of posaconazole, ritonavir, and griseofulvin particles were calculated as previously described (Cao et al., 2008) and computed to be 1.26, 1.23, and 1.37 g/cm³.

2.10.2. Particle size parameterization

As described above, laser diffraction was employed to determine particle size distribution of each drug powder. Two different approaches were used to predict 5 mg drug particle dissolution: mean particle size only and entire PSD. Predictions based on mean only used drug D₅₀ to calculate a single initial drug ${\mathrm{r}}_0$ (i.e., treat powder as monodisperse where initial radius is half D₅₀). For each drug, D₅₀ was determined from the PSD as the diameter where 50 % of the powder mass is composed of smaller particles.

Predictions based on entire PSD were also performed. The initial drug mass (5 mg) was divided into discrete particle size fractions, based on observed mean PSD. For posaconazole, ritonavir, and griseofulvin, there were 27, 21, and 30 discrete particle size fractions (which are each taken to be monodisperse), respectively. Initial mass of each fraction was computed from its weight contribution to the overall PSD. Initial radius was from the observed PSD.

2.10.3. Model implementation in python

The film model dissolution Eq. (4) was encoded into python 3.11 programming language on Google Colaboratory (colab.google), a Jupyter Notebook service, using the Chrome browser. In python, the scipy ordinary differential equation function solve_ivp was used to perform numerical integration (Virtanen et al., 2020). Runga-Kutta RK45 numerical algorithm was used for integration. Of note, Radau method (and not Runga-Kutta RK45) was used to predict posaconazole and ritonavir dissolution into no surfactant media, where <3 % of drug dissolved and over 97 % remained undissolved; the Radau method gave more numerically stable results in these situations.

At time zero, the initial particle radius was assigned. Predictions based on mean particle size only involve a single differential equation. Predictions based on entire PSD employed a system of 27, 21, or 30 differential equations for posaconazole, ritonavir, and griseofulvin, respectively.

Particle $\mathrm{r}$ generally decreased with time and was needed at each integration step when employing the radius-dependent diffusion layer thickness model. In order to calculate $\mathrm{r}$ for each numerical integration step for any one particle fraction,

$$r=r_o{\left(\frac{M}{M_0}\right)}^{\frac{1}{3}}$$ (5)

2.10.4. Identification of best diffusion layer thickness

As results show, h_fixed = 12 μm and h_max = 12 μm were separately concluded as best diffusion layer parameters for the fixed diffusion layer thickness model and the radius-dependent diffusion layer thickness model, respectively. This finding was elucidated through two stages: a) a sensitivity analysis and b) inspection for an overall best single diffusion layer thickness across 15 (i.e., 3 drugs and 5 media) dissolution scenarios for each model. For each h_fixed and h_max models, “best” here considered results of sensitivity analysis, as well as accuracy assessment across all drug/media scenarios.

Firstly, in a sensitivity analysis for the fixed diffusion layer thickness model, particle dissolution was simulated for each drug and each media combination, using the entire PSD, over a range of h_fixed values (i.e., from 1 to 400 μm). Four hundred simulations were conducted for each drug/medium scenario, where h_fixed values differed by 1 μm. In comparing the simulated (i.e., predicted) profile to the observed profile, the root mean square error (RMSE) was calculated from

$$RMSE=\sqrt{\frac{\sum (\mathit{Observed}\mathrm{\%}\mathit{dissolved}-\mathit{Predicted}\mathrm{\%}\mathit{dissolved}{)}^2}{N}}$$

where $\mathrm{N}$ is total number of experimentally-observed data points (i.e., N = 12 for dissolution out to 300 min). Lower RMSE indicates closer agreement between simulated and observed profiles. With units of % dissolved, RMSE reflects an average difference between simulated and observed profiles. From this sensitivity analysis, the h_fixed that provided the lowest RMSE was identified and denoted the optimal h_fixed. Furthermore, the range of h_fixed values with RMSE within 10 % of the minimal RMSE was identified and denoted preferred range. This sensitivity analysis was also carried out for the r-dependent diffusion layer thickness model.

Secondly, for each diffusion layer thickness model, these preferred ranges of h_fixed (or h_max) values across the 15 dissolution scenarios were considered. For each model, 7, 12, and 16 μm were identified and further considered as the potential best single diffusion layer thickness.

3. Result and discussion

3.1. Overall dissolution predictions

Tables 1 and 2 summarize particle dissolution prediction results at 30 and 60 min using fixed diffusion layer thickness model with h_fixed = 12 μm for posaconazole, ritonavir, and griseofulvin into each of four surfactants, as well as no surfactant media. Powder was parameterized using entire PSD, as well as mean only. For 30 and 60 min, all profiles were accurately predicted using entire PSD (i.e., both time points in all 15 profiles). Using mean particle size, for each time point, 14 of the 15 profiles were predicted accurately, while griseofulvin dissolution into CTAB was over-predicted at 30 min and ritonavir dissolution into PS80 was over-predicted at 60 min. Overall, fixed diffusion layer thickness model with h_fixed = 12 μm was identified as the best and always provided acceptable results. Although less preferred, h_max = 12 μm provided acceptable predictions when entire PSD was used, although h_max = 12 μm using D₅₀ over-predicted observed profiles on several occasions.

Table 1.

Summary of particle dissolution prediction results for 30 min using h_fixed = 12μm. Dissolution of three drugs into four surfactant media, as well as no surfactant media, were studied. Accurate denotes the predicted percent dissolved at 30 min was within 20 % dissolved units of observed value. Observed particle dissolution employed USP II mini-paddle (75 rpm) using 150 ml vessel. Drug powder mass was 5 mg. Dissolution volume was 100 ml (pH 6.8). Predictions were performed separately using entire particle size distribution (PSD) and mean particle size. All profiles were accurately predicted using entire PSD for posaconazole, griseofulvin and ritonavir (i.e., 15 profiles). Using mean particle size, 14 of the 15 profiles were predicted accurately, while griseofulvin dissolution into CTAB was over-predicted.

Drug	Particle size parameterization	No surfactant	2 % SLS	2 % PS80	2 % POE10	2 % CTAB
Posaconazole		Accurate	Accurate	Accurate	Accurate	Accurate
Ritonavir	Entire PSD	Accurate	Accurate	Accurate	Accurate	Accurate
Griseofulvin		Accurate	Accurate	Accurate	Accurate	Accurate
Posaconazole		Accurate	Accurate	Accurate	Accurate	Accurate
Ritonavir	Mean Only	Accurate	Accurate	Accurate	Accurate	Accurate
Griseofulvin		Accurate	Accurate	Accurate	Accurate	Over-predicted

Table 2.

Summary of particle dissolution prediction results for 60 min h_fixed = 12μm. Accurate denotes the predicted percent dissolved at 60 min was within 20 % dissolved units of observed value. Methods were the same as in Table 1. All profiles were accurately predicted using entire PSD for posaconazole, griseofulvin and ritonavir (i.e., 15 profiles). Using mean particle size, 14 of the 15 profiles were predicted accurately, while ritonavir dissolution into PS80 was over-predicted.

Drug	Particle size parameterization	No surfactant	2 % SLS	2 % PS80	2 % POE10	2 % CTAB
Posaconazole		Accurate	Accurate	Accurate	Accurate	Accurate
Ritonavir	Entire PSD	Accurate	Accurate	Accurate	Accurate	Accurate
Griseofulvin		Accurate	Accurate	Accurate	Accurate	Accurate
Posaconazole		Accurate	Accurate	Accurate	Accurate	Accurate
Ritonavir	Mean Only	Accurate	Accurate	Over-predicted	Accurate	Accurate
Griseofulvin		Accurate	Accurate	Accurate	Accurate	Accurate

3.2. Effect of surfactant on drug solubility

Table 3 shows solubility of posaconazole, ritonavir, and griseofulvin with and without surfactants (i.e., SLS, PS80, POE10, and CTAB) at pH 6.8. Surfactants increase drug solubility by many fold. Rank-order impact was posaconazole > ritonavir > griseofulvin. Rank-order effect of surfactants was 2 %SLS > 2 %CTAB > 2 %POE10 > 2 %PS80. Posaconazole solubility increased by 14,820, 392, 550, and 8910-fold by SLS, PS80, POE10, and CTAB, respectively. Ritonavir solubility increased by 1730, 61.4, 71.6, and 631-fold. Griseofulvin solubility increases by 160, 11.3, 15.1, and 86.9-fold. Table S1 lists the fraction of free drug (${\mathrm{f}}_{\mathrm{f}}$) and fraction of drug that was micelle-incorporated (${\mathrm{f}}_{\mathrm{m}}$), under these solubility conditions. >90 % of posaconazole, ritonavir, and griseofulvin was micelle bound in each SLS, PS80, POE10, and CTAB.

Table 3.

Solubility of posaconazole, ritonavir, and griseofulvin with and without surfactant. Surfactant concentrations were 2 % SLS, 2 % PS80, 2 % POE10, and 2 % CTAB. Buffer was 50 mM phosphate buffer at pH 6.8. Mean±SEM from n = 3. Some griseofulvin data from literature (Patel et al., 2024).

Surfactant	Posaconazole Solubility (mg/ml)	Ritonavir Solubility (mg/ml)	Griseofulvin Solubility (mg/ml)
No surfactant	0.000228±0.000034	0.00148±0.00002	0.0103±0.0001
2 % SLS	3.38±0.45	2.56±0.04	1.65±0.02
2 % PS80	0.0893±0.0023	0.0909±0.0015	0.116±0.0005
2 % POE10	0.125±0.001	0.106±0.0005	0.155±0.001
2 % CTAB	2.031±0.033	0.919±0.014	0.895±0.009

3.3. DLS measurement and effective diffusivity

From DLS, diameters of posaconazole-loaded micelles were 3.72, 11.14, 8.26, and 6.60 nm for 2 % SLS, 2 % PS80, 2 % POE10, and 2 % CTAB, respectively (Table S2). Diameters of ritonavir-loaded micelles were 4.01, 11.44, 8.11, and 6.82 nm. Griseofulvin loaded micelles were 3.87, 12.06, 8.90, and 6.89 nm (Patel et al., 2024). Of note, these micelle sizes were obtained from solubility study samples, where solubility and particle dissolution studies were conducted under the same conditions (i.e., surfactant type and concentration, buffer and pH).

From micelle size, ${\mathrm{D}}_{\mathrm{D}-\mathrm{M}}$ and ${\mathrm{D}}_{\text{eff}}$ were computed (Eqs. (2) and 3) and tabulated in Table 4. In light of SLS providing the smallest micelle size, and PS80 being the largest, SLS > CTAB > POE10 > PS80 for ${\mathrm{D}}_{\mathrm{D}-\mathrm{M}}$. Because of the very high ${\mathrm{f}}_{\mathrm{m}}$ for posaconazole and ritonavir (i.e., >98 %), their ${\mathrm{D}}_{\text{eff}}$ were very similar to their ${\mathrm{D}}_{\mathrm{D}-\mathrm{M}}$. Since f_m was only about 90 % for griseofulvin in PS80 and POE10, their ${\mathrm{D}}_{\text{eff}}$ were somewhat larger than the corresponding ${\mathrm{D}}_{\mathrm{D}-\mathrm{M}}$. Micelle bound drug will have slower diffusivity compared to free drug diffusivity.

Table 4.

Drug diffusivity values in 2 % SLS, 2 % PS80, 2 % POE10, and 2 % CTAB. Shown are D_D–M values (i.e., drug-loaded micelle diffusivity) for each posaconazole, ritonavir, and griseofulvin. Diffusivities were derived from DLS measurement. Table S2 lists drug loaded micelle diameters. Also shown are their D_eff (i.e., effective diffusivity) with surfactant in 50 mM phosphate buffer at pH 6.8, per Eq. (3). Mean±SEM from n = 3.

Surfactant	Diffusivity parameter	Posaconazole diffusivity (×107) (cm2/s)	Ritonavir diffusivity (×107) (cm2/s)	Griseofulvin diffusivity (×107) (cm2/s)
2 % SLS	DD–M	15.3 ± 0.05	14.2 ± 0.03	15.0 ± 0.2
	Deff	15.3 ± 0.1	14.2 ± 0.03	15.4 ± 1.5
2 % PS80	DD–M	5.11±0.12	4.97±0.08	4.71±0.10
	Deff	5.22±0.11	5.94±0.07	11.4 ± 0.08
2 % POE10	DD–M	6.91±0.32	7.01±0.01	6.40±0.39
	Deff	6.99±0.33	7.81±0.02	11.3 ± 0.04
2 % CTAB	DD–M	8.63±0.26	8.34±0.09	8.25±0.16
	Deff	8.63±0.26	8.43±0.09	9.08±0.17

3.4. Drug particle size distribution and mean size

Fig. 2 plots the entire percent weight particle size distributions of posaconazole, ritonavir, and griseofulvin powders. Based on D₅₀, particle size rank-order was ritonavir (D₅₀=3 μm) < griseofulvin (D₅₀ = 9 μm) < posaconazole (D₅₀ = 10 μm), although each powder generally was composed of small particles. Also, posaconazole (D₅₀ = 10 μm and D90 = 50 μm) and griseofulvin (D50 = 9 μm and D90 = 50 μm) had PSDs with some large particles, compared to the mean particle size.

Fig. 2.

Particle size distribution of three drug powders. Panels A, B, and C are for posaconazole, ritonavir, and griseofulvin, respectively. Ritonavir powder exhibited the smallest particle size distribution particles of the three drugs.

3.5. Effect of surfactant on drug dissolution

Fig. 3 plots observed particle dissolution profiles of posaconazole, ritonavir, and griseofulvin in phosphate buffer (pH 6.8). Surfactants, especially 2 % SLS and 2 % CTAB, markedly enhanced the dissolution of posaconazole, ritonavir, and griseofulvin, reflecting SLS’s and CTAB’s solubilizing capacity and relatively small micelle size (i.e., relatively high micelle diffusivity) (Balakrishnan et al., 2004; Jamil and Polli, 2022; Oktay and Polli, 2022). PS80 and POE10 also markedly improved dissolution, although less so than SLS and CTAB, reflecting their lesser relative solubilizing capacities and relatively larger micelle sizes.

Fig. 3.

Observed particle dissolution profiles into phosphate buffer (pH 6.8) with and without surfactant. Panels A, B, and C are for posaconazole, ritonavir, and griseofulvin, respectively. Dissolution involved 5 mg of drug powder in 100ml. Five different media were no surfactant, 2 % SLS, 2 % PS80, 2 % POE10, and 2 % CTAB. Surfactant markedly increased posaconazole, ritonavir, and griseofulvin dissolution.

Of note, initial dissolution studies of ritonavir powder with PS80, POE10, and no surfactant showed visual powder aggregation, such that studies were conducted where power was pre-wetted. Fig. 3 shows results where ritonavir powder with PS80, POE10, and no surfactant were pre-wetted, and where aggregation was largely mitigated, and dissolution profiles were higher than with powder aggregation (data not shown).

In panel C of Fig. 3, only 20 % of griseofulvin dissolved (i.e., 1 of the 5 mg) in no surfactant media (100 ml), due to drug reaching its solubility limit of about 0.01 mg/ml. However, while the four surfactant media could completely dissolve all griseofulvin, dissolution over 300 min was incomplete, even though 70–80 % dissolved in the first 60 min. Dissolution slowed after about 70–80 % dissolved. This slower dissolution presumably reflects the slower dissolution of the particles whose initial size were relatively large. Griseofulvin (D₅₀ = 9 μm and D₉₀ = 50 μm) had a PSD with some large particles, compared to the mean particle size. That is, the larger griseofulvin particles appeared to result in a prolonged, flatter dissolution profile after smaller particles had dissolved.

3.6. Drug dissolution prediction: preliminary simulations

Preliminary predictions for each of the 15 dissolution scenarios (i.e., three drugs and five media), using fully developed code, were performed using h_fixed = 16 μm for the fixed diffusion layer thickness model and h_max = 16 μm for the r-dependent diffusion layer thickness model. h_max = 16 μm was used previously by Johnson to predict dissolution of cilostazol into water (Johnson, 2012).

The film model dissolution Eq. (4) was encoded into python on Google Colaboratory. Runga-Kutta RK45 numerical algorithm was used for numerical integration, except for the stiff situations involving posaconazole and ritonavir dissolution into no surfactant media, where <3 % of drug dissolved, such that the Radau method was used. z-factor parameter values for diffusivity D, thickness of diffusion layer h (h_fixed or h_max), and initial particle radius r_o were applied. In medium without surfactant, D was free drug diffusivity (i.e., ${\mathrm{D}}_{\text{D}}$), and C_s was the solubility of drug without surfactant (pH=6.8); in medium with surfactant, D was effective drug diffusivity (i.e., ${\mathrm{D}}_{\text{eff}}$), and C_s was the solubility of drug with surfactant (pH=6.8).

For each of the 15 drug/medium prediction scenarios, predictions were performed where powder was parameterized as the entire PSD, as well as mean only (i.e., using D₅₀). Accurate prediction denotes the predicted percent dissolved was within 20 % dissolved units of observed initial value, at the considered time (i.e., 30 min and 60 min).

These findings from using entire PSD and from using D₅₀ were generally favorable. These findings inspired a systematic sensitivity analysis where layer thickness was varied over a wide range.

3.7. Sensitivity analysis of diffusion layer thickness

Using entire PSD, sensitivity analysis was conducted on each the h_fixed model and h_max model. Figs. 4 and 5 show RMSE as a function of h_fixed value and h_max value, respectively. In each figure, profiles for posaconazole (panels A-E), ritonavir (panels F-J), and griseofulvin (panels K-O) are shown. Table S3 lists the optimal h_fixed and h_max values which provided the lowest RMSE for each of the 15 drug/media combinations.

Fig. 4.

Sensitivity analysis results for h_fixed. In comparing predicted and observed profiles, root mean square error (RMSE) is plotted as a function of fixed diffusion layer thickness. Media was no surfactant, 2 % SLS, 2 % PS80, 2 % POE10, and 2 % CTAB, respectively. Panels A-E are for posaconazole. Panels F-J are for ritonavir. Panels K-O are for griseofulvin.

Fig. 5.

Sensitivity analysis results for h_max. In comparing predicted and observed profiles, RMSE is plotted as a function of maximum diffusion layer thickness. Media was no surfactant, 2 % SLS, 2 % PS80, 2 % POE10, and 2 % CTAB, respectively. Panels A-E are for posaconazole. Panels F-J are for ritonavir. Panels K-O are for griseofulvin.

Fig. 4 was inspected in order to identify the overall best single h_fixed value across 15 drug/media combinations. Considered was the preferred range from each drug/media scenario (not shown), where RMSE was at most 10 % dissolved higher than the RMSE value from optimum h_fixed. For the h_fixed model, h_fixed = 12 μm was identified as the best thickness. In all 15 scenarios except posaconazole in 2 % SLS and 2 % PS80, h_fixed = 12 μm provided a RMSE that was within 10 % dissolved of the optimal thickness (i.e., within the preferred range). The preferred ranges were ≥ 14 μm for posaconazole in 2 % SLS and 3 μm - 10 μm for posaconazole in 2 % PS80, such that 12 μm was close.

For the h_max model, h_max = 12 μm was separately identified as the best single thickness. In all 15 scenarios, h_max = 12 μm provided a RMSE that was within 10 % dissolved of the optimal thickness (i.e., within the preferred range). Of note, in contrast to the fixed diffusion layer thickness model, a number of h_max values performed equally well. For example, hmax = 7 μm and hmax = 16 μm performed the same as hmax = 12 μm (Table S4).

3.8. Drug dissolution prediction for posaconazole: final simulations using h_fixed of 12 μm

Using h_fixed = 12 μm based on each entire PSD and mean only, Fig. 6 plots the 10 predicted (and five observed) posaconazole particle dissolution profiles. The 30 min and 60 min time points were used for accuracy assessment. In Fig 6, all five profiles were accurately predicted using entire PSD and mean particle size. Of note, with no surfactant, <1 % of drug dissolved (Fig 6 panel A).

Fig. 6.

Predicted particle dissolution profiles of posaconazole with and without surfactant using h_fixed = 12μm. Panels A, B, C, D, and E are for no surfactant, 2 % SLS, 2 % PS80, 2 % POE10, and 2 % CTAB, respectively. Dissolution involved 5 mg of API in 100 ml. One predicted profile considered the entire particle size distribution (denoted entire PSD) in Fig. 1. The other predicted profile assumed a monodisperse powder with a particle size equal to the mean size [i.e., D₅₀=10 μm (denoted mean only)]. Also shown is observed profile. All predicted profiles were within 20 % dissolved of observed profile at 30 min and 60 min.

From Fig. 2, posaconazole PSD exhibited some polydispersity (D₅₀ = 10 μm and D₉₀ = 50 μm), with some smaller and larger particles, compared to the mean particle size. In general, posaconazole predictions in Fig 6 using entire PSD where some particles were smaller than D₅₀ and others were larger than D₅₀ can be expected to be faster at early timepoint and slower at later timepoints, since smaller particles dissolve faster (than the mean) while larger particles dissolve slower (than mean). This general expectation is evident in panel C for posaconazole, where predicted initial profile from entire PSD was faster than predicted profile from mean only.

D₅₀ accurately predicted the observed profile, although D₅₀ insufficiently reflected the substantial population of markedly larger posaconazole particles, which dissolved slowly and incompletely in 300 min (about 78 % dissolved).

3.9. Drug dissolution prediction for ritonavir: final simulations using h_fixed of 12 μm

Using h_fixed = 12 μm based on each entire PSD and mean only, Fig. 7 plots the 10 predicted (and five observed) ritonavir particle dissolution profiles. All profiles were accurately predicted using entire PSD. The predicted profile for 60 min dissolution into 2 % PS80 using mean only was over-predicted, but accurate using entire PSD. Like for posaconazole, ritonavir dissolution into no surfactant medium was very incomplete due to low solubility.

Fig. 7.

Predicted particle dissolution profiles of ritonavir with and without surfactant using h_fixed = 12μm. Panels A, B, C, D, and E are for no surfactant, 2 % SLS, 2 % PS80, 2 % POE10, and 2 % CTAB, respectively. Dissolution involved 5 mg of API in 100ml. One predicted profile considered the entire particle size distribution (denoted entire PSD) in Fig. 1. The other predicted profile assumed a monodisperse powder with a particle size equal to the mean size [i.e., D₅₀ = 3 μm (denoted mean only)]. Also shown is observed profile. All predicted profiles were within 20 % dissolved of observed profile at 30 min and 60 min using entire PSD. Predicted profile was over-predicted for dissolution at 60 min into 2 % PS80 using mean only, but otherwise accurate using mean only.

In Fig. 7 panel C, predicted profile from entire PSD was faster than predicted profile from D₅₀ for the first 50 % dissolved (about first 10 min). More markedly, profile from mean only was much higher and slightly over-predicted than profile from entire PSD (and observed profile) for the last 80 % dissolved (i.e., after about 45 min). Predicted dissolution into surfactant reached 100 % dissolved using D₅₀ within 60 min. Predicted dissolution of ritonavir (D₅₀=3 μm and D₉₀=8 μm) into surfactant using entire PSD reached 90 % dissolved within 90 min; predicted profiles were not especially prolonged, reflecting all ritonavir particles were relatively small.

3.10. Drug dissolution prediction for griseofulvin: final simulations using h_fixed of 12 μm

Using h_fixed = 12 μm based on each entire PSD and mean only, Fig. 8 plots the 10 predicted (and five observed) griseofulvin particle dissolution profiles. Findings were similar to those of ritonavir. All profiles were accurately predicted using entire PSD. The predicted profile for 30 min dissolution into 2 % CTAB using mean only was over-predicted, but accurate using entire PSD.

Fig. 8.

Predicted particle dissolution profiles of griseofulvin with and without surfactant using h_fixed = 12μm. A, B, C, D, and E are for no surfactant, 2 % SLS, 2 % PS80, 2 % POE10, and 2 % CTAB, respectively. Dissolution involved 5 mg of API in 100 ml. One predicted profile considered the entire particle size distribution (denoted entire PSD) in Fig. 1. The other predicted profile assumed a monodisperse powder with a particle size equal to the mean size [i.e., D₅₀ = 9 μm (denoted mean only)]. Also shown is observed profile. All predicted profiles were within 20 % dissolved of observed profile at 30 min and 60 min using entire PSD. Predicted profile was over-predicted for dissolution at 30 min into 2 % CTAB using mean only, but otherwise accurate using mean only.

Like posaconazole, griseofulvin PSD (D₅₀=9 μm and D₉₀=50 μm) indicated large particles were present. In Fig 8 panel C, the predicted profile from mean only was higher than predicted profile from entire PSD (and observed profile) at about 80 % dissolved (i.e., at about 45 min). Like posaconazole, the profile using D₅₀ accurately predicted, although D₅₀ insufficiently reflected the substantial population of markedly larger griseofulvin particles, which dissolved slowly and incompletely in 300 min.

3.11. Drug dissolution prediction: final simulations using h_max of 12 μm

Using h_max = 12 μm based on each entire PSD and mean only, Figure S1–S3 plots the 10 predicted (and five observed) particle dissolution profiles for each posaconazole, ritonavir, and griseofulvin, respectively. Table S4 summarizes prediction accuracy at 30 and 60 min. Overall, h_max = 12 μm was less accurate than h_fixed = 12 μm (Tables 1 and 2), providing more over-predictions, particularly when D₅₀ was used. However, predictions based on entire PSD were adequate (i.e., 13 of 15 accurate for 30 min, and 14 of 15 accurate for 60 min).

3.12. Comparison of best values to literature

We are unaware of prior studies that examined the impact of diffusion layer thickness on predictive accuracy of dissolution into surfactant solutions. Prediction results at 30 min (Table 1) and 60 min (Table 2) using h_fixed = 12 μm for three drugs into four surfactants, as well as no surfactant media, were favorable, including whether entire PSD or only D₅₀ was used. Prediction results using h_max = 12 μm across all 15 scenarios (Table S4) was acceptable when powder was parameterized using entire PSD, although provided several over-predictions when D₅₀ was used. Overall, fixed diffusion layer thickness model with h_fixed = 12 μm was identified as the best single model.

Okazaki et al. used h_max = 30 μm to predict dissolution of griseofulvin and danazol into 3mM/0.75 mM taurocholate/egg lecithin medium (Okazaki et al., 2008). Also, using entire PSD, Jinno et al. simulated the dissolution of three cilostazol powders into FaSSIF and Fed Simulated Small Intestinal Fluid (FeSSIF) (Jinno et al., 2006). Perhaps since most particles were <30 μm, h = radius at all times in Jinno et al. A limitation of Jinno et al. was the use of a single diffusivity value of 67.3 × 10⁻⁷ cm²/s, which appears to have represented free drug diffusivity and not micelle diffusivity. Table S4 and S5 show prediction results using h_max = 30 μm and h_fixed = 30 μm, respectively. Neither preformed as well as h_fixed = 12 μm (Table 1). h_fixed = 30 μm performed worst than h_max = 30 μm, which matched prediction results h_max = 12 μm (Table S4).

These values of h_fixed = 12 μm and h_max = 12 μm in the present manuscript are within the range of values of prior findings. Johnson compared h_max values of 3, 15 and 16 μm in predicting the dissolution of small and large hydrocortisone powder into water, as well as small and large cilostazol powder into water (Johnson, 2012). Overall, h_max = 16 μm provided favorable predictions for all but small hydrocortisone powder, where h_max = 3 μm was the best.

Sheng et al. examined the effects of drug particle size (< 20–106 μm) and USP apparatus II paddle speed on particle diffusion layer thickness for fenofibrate dissolution into essentially water (Sheng et al., 2008). At 50 rpm, over the particle size range studied, h increased with particle radius (e.g., 21.0 μm for a particle with radius of 10 μm, and 8.0 μm for a particle with radius of 5 μm). At 100 rpm, h increased linearly with radius, but became constant when radius exceeded 23.7 μm. Studies in the present manuscript employed 75 rpm, which was selected as a representative paddle stirring speed, and stirring speed was not studied. It is possible that higher and lower speeds may result in slightly thinner and thicker layer thickness, potentially impacting prediction accuracy if h_fixed = 12 μm is used. Okazaki et al. and Jinno et al. each employed 50 rpm. Jinno et al. indicated that dissolution profiles from 50 to 200 rpm were equivalent.

4. Validation and verification of film dissolution model: assessment of model credibility

4.1. Model verification

The dissolution prediction model was encoded into python 3.11 programming language on Google Colaboratory, using the Chrome browser on a computer supported by the University of Maryland Baltimore. The scipy ordinary differential equation function solve_ivp was used to perform numerical integration using either the Runga-Kutta RK45 or the Radau method. Verification ensured that the model’s code accurately implemented the mathematical equations described by the film dissolution model. Predictions using D₅₀ only involve a single differential equation. Meanwhile, predictions using an entire PSD involves a system of differential equations (i.e., one for each initial particle size fraction, which were 27, 21, and 30 in number for posaconazole, ritonavir, and griseofulvin, respectively). Verification of the simpler model using D₅₀ only was performed before initiating coding for predictions using an entire PSD.

Several steps were taken to achieve and verify code correctness. The code was reviewed by a second individual to confirm that each component (e.g., differential equation) was implemented. By inspection of Eqs. (4) and (5), the underlying mathematical model did not require large lines of code. However, two scenarios where minimal drug dissolved due to low solubility (i.e., posaconazole and ritonavir dissolutions into no surfactant media) resulted in differential equations being prone to slight numerical integration instability, but remedied using the Radau method. Figure S4 plots predictions using Radau and Runga-Kutta (i.e., RK45) method for each posaconazole and ritonavir dissolution without surfactant. Runga-Kutta provided oscillations in percent dissolved at equilibrium (i.e., about 0.4 % or 3 % dissolved for posaconazole and ritonavir, respectively). Radau resulted in no oscillation.

Intermediate calculations were performed to confirm code accuracy. For example, in addition to computing the amount of drug dissolved with time, values of r and h with time were inspected, in light of r_o, and in light of h_max being set to be equal to r when r ≤ h_max but otherwise h = hmax when r > hmax.

4.2. Model validation

Following verification of the code and calculations, model validation was performed to assess its reliably to predict dissolution. Of note, in vitro dissolution is infrequently predicted, and criteria to assess dissolution prediction accuracy is not well developed. Predictions here build upon prior studies, as described above. All model elements have been previously described: the z-factor model for mono-disperse particle dissolution (Dressman and Fleisher, 1986; Okazaki et al., 2008), poly-disperse particle dissolution (Hintz and Johnson, 1989; Johnson, 2012), and a surfactant-mediated dissolution model (Balakrishnan et al., 2004; Oktay and Polli, 2024).

Studies here perhaps most closely follow Okazaki et al. where, using entire PSD and mean particle size, dissolution of griseofulvin and danazol powders into various taurocholate solutions were simulated and measured; the effective diffusivity values incorporated micelle diffusivity values of about 20–50 × 10⁻⁷ cm²/s (Okazaki et al., 2008).

The encoded dissolution model here was challenged to provide the same simulation results from Okazaki et al. In particular, the griseofulvin and danazol dissolution profiles into 3 mM/0.75 mM taurocholate/egg lecithin medium were predicted here, using the reported D₅₀, particle density, C_s, micelle ${\mathrm{D}}_{\text{eff}}$, and h_max values. Predictions here matched the reported simulations.

Also, the encoded dissolution model here was challenged to provide the same simulation results from Hintz and Johnson, where powder was polydisperse. In particular, the computer-estimated PSD of unmilled drug was used here to predict drug dissolution into water. Predictions here matched the reported simulations.

Predictions at each 30 and 60 min were considered accurate when the predicted percent dissolved was within 20 % dissolved units of observed value. We view this criterion as conservative and not as liberal, as in vitro dissolution is infrequently predicted. As a reference, models to predict pharmacokinetic profiles are often viewed as accurate when average fold error (AFE) or average absolute fold error (AAFE) are less than two to five (i.e., less than about a 2-fold to 5-fold error) (FDA Guidance, 2020; Parrott et al., 2022).

Parameter sensitivity analysis was also performed. Values for solubility (C_s), diffusivity (D), initial particle radius (r₀) were varied to confirm the effect of each parameter on simulated dissolution profiles (data not shown). This sensitivity analysis yielded the expected impact of each parameter on dissolution, confirming the model’s robustness under a range of parameter conditions.

Additionally, as detailed above, sensitivity analysis was also performed by varying fixed and radius-dependent diffusion layer thickness from 1 to 400 μm. Through these assessments, it was determined that setting h_fixed to 12 μm provided best predictive accuracy across all scenarios. Separately, for the radius-dependent diffusion thickness model, setting h_max to 12 μm provided adequate predictive accuracy across all scenarios, although was less accurate than h_fixed = 12 μm.

Additionally, further model refinement was performed, where each fixed diffusion layer thickness model and r-dependent diffusion layer thickness model were assessed for RMSE and accuracy using h_fixed = 7 μm, hfixed = 16 μm, hmax= 7 μm, and hmax= 16 μm, and compared to 12 μm. For h_fixed model, RMSE is listed in Table S6, and prediction accuracy is summarized in Table S7 and S8. Overall, for fixed diffusion layer thickness model, a slightly thinner layer tended to provide more over-predictions, while a slightly thicker layer tended to provide more under-predictions, as expected. Meanwhile, for the r-dependent diffusion layer thickness model in Table S4, accuracy did not change whether h_max was 7, 12, or 16 μm. The sameness in results reflects that the powder radii of the three drug was generally smaller than 16 μm. The D₅₀ of posaconazole, ritonavir, and griseofulvin was 10, 3, and 9 μm, with corresponding radii of 5, 1.5, and 4.5 μm, respectively. The D₉₀ of posaconazole, ritonavir, and griseofulvin was 50, 8, and 50 μm, with corresponding radii of 25, 4, and 25 μm, respectively.

Overall h_max yielded less accurate prediction than h_fixed, as several profiles were over-predicted.

4.3. Potential applications: mechanistic dissolution prediction into surfactant

A mechanistic model is a model based on an underpinning scientific law, theoretical or first principle, or phenomena that explains the system’s behavior. The model typically includes components that are relatable to the system and its dynamics. Meanwhile, empirical models are based on observation alone. Empirical models are common and useful, and often employ statistics (e.g., regression) to establish relationships.

Mechanistic models differ from empirical models in expected potential to accurately predict beyond the model’s training set. With the benefit of an underpinning model framework that reflects the system’s fundamental behavior, a mechanistic model can be expected to potentially perform extrapolations beyond the model’s training set. Meanwhile, an empirical model is generally expected to allow for interpolation but not extrapolation. Mechanistic models are generally preferable over empirical models in pharmaceutical research, since pharmaceutical research can present numerous scenarios that extend beyond situations that were available in model development.

PBPK modeling or PBBM implies the underpinning model is a mechanistic model (FDA Guidance, 2018). For example, oral PBBM models should provide a mechanistic framework of oral drug absorption (FDA Guidance, 2020). One consideration in a mechanistic approach to oral drug absorption modeling is small intestinal transit flow (Yu and Amidon, 1999).

A common application of PBPK modeling is enzyme-based drug-drug interactions (DDI) (Grimstein et al., 2019). Mechanistic DDI models have ability to anticipate DDI of drug combinations that were not part of the model development set.

A major application of oral PBBM is poorly water soluble drugs. Surfactant-mediated dissolution is a critical step in the absorption of poorly water soluble drugs. Applications of oral PBBM typically employ observed dissolution profiles, which are generally fitted to a model to parameterize the observed dissolution data (Mackie et al., 2024). However, aspects of the modeling may involve predictions not based on observed dissolution profiles, such as sensitivity analysis and specification identification. It is recognized that realistic parameterization of surfactant-mediated dissolution is needed in PBBM (Jamil and Polli, 2022; Pepin et al., 2024), and a greater number of micelle diffusivity values are being reported (Oktay and Polli, 2022, 2024). Here, a single value for the h_fixed diffusion layer thickness parameter was found to be acceptable for several poorly water soluble drugs and media conditions, suggesting a potentially wide applicability of the model.

5. Conclusion

Surfactant markedly enhanced drug dissolution. Results show that in vitro drug dissolution from polydisperse powders of various drugs into several surfactant solutions was successfully predicted using a surfactant-mediated dissolution model. The model employed the film model, applied to spherical drug particle fractions of the percent weight particle size distribution, and with a surfactant-mediated dissolution component. The best diffusion layer thickness for the fixed thickness model and the radius-dependent thickness model were separately found to be h_fixed = 12 μm and h_max = 12 μm, respectively, where both models were acceptable, but with h_fixed = 12 μm performing slightly better than h_max = 12 μm. Predictions using the entire PSD of the drug powder performed better than predictions based only on mean particle size, although prediction based on mean particle size were adequate for h_fixed = 12 μm but over-predicted for h_max = 12 μm. Credibility assessment of the in vitro dissolution model was performed, including model verification and validation considerations in light of the question of interest, the context of use, and model risk.

Supplementary Material

Supplementary materials

Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.ejps.2025.107052.

Abbreviations:

PBBM

physiologically-based biopharmaceutical modeling

PBPK

physiologically-based pharmacokinetic

PSD

particle size distribution

FaSSIF

Fasted Simulated Small Intestinal Fluid

FeSSIF

Fed Simulated Small Intestinal Fluid

POE10

polyoxyethylene (10) lauryl ether

PS80

polysorbate 80

SLS

sodium lauryl sulfate

CTAB

cetyltrimethylammonium bromide

PVDF

polyvinylidene difluoride

DLS

dynamic light scattering

Data availability

Data will be made available on request.