Time-dependent behavior and modeling of drug particle depots in nonionic surfactant systems

Time-dependent behavior and modeling of drug particle depots in nonionic surfactant systems Dan Trunov a,* , Kateˇ rina Neubergerov´ a a , Jan Heyda b , Gabriela ˇ Zelonkov´ a b , Josef Ber´ anek c , Ondˇ rej Dammer c , Miroslav ˇ So´ oˇ s a,* a Department of Chemical Engineering, University of Chemistry and Technology, Technick´ a 3, 166 28 Prague 6 – Dejvice, Czech Republic b Department of Physical Chemistry, University of Chemistry and Technology in Prague, Technick´ a 3, 166 28, Prague, Czech Republic c Zentiva, k.s., U Kabelovny 130, 102 00, Prague 10, Czech Republic ARTICLE INFO Keywords: Nonionic surfactants Flux of Valsartan Solubilization kinetics Predictive modeling Drug delivery ABSTRACT The solubilization and transport of poorly water-soluble drugs present significant engineering challenges due to the interplay of physicochemical, colloidal, and interfacial processes in biological environments. In particular, the gastrointestinal tract represents a dynamic system where dissolution, bile salt–mediated micellization of poorly soluble drugs, particle precipitation and aggregation, and membrane permeation occur concurrently and govern overall drug bioavailability. In this study, we investigate the behavior of Valsartan, a poorly soluble model drug, in the presence of four nonionic surfactants to understand its colloidal states and solubilization mechanisms. Valsartan was found to exist in multiple colloidal states: as freely dissolved molecules, micellesolubilized drug, and precipitated particles. A combination of molecular dynamics (MD) simulations and experimental techniques was used to investigate Valsartan–surfactant interactions. Quantitative analysis revealed that micelle encapsulation capacities ranged from 2 to 7 molecules per micelle, leading to solubilization enhancement factors of up to 57-fold compared with the aqueous solubility of Valsartan, with encapsulation efficiency correlating strongly with the surfactant’s critical micelle concentration (CMC). While measurements of Valsartan flux across a model membrane confirmed that higher surfactant concentrations reduce free drug transport, this observation serves as contextual support for the focus of the study. Building on these results, we developed and validated a chemical engineering model that integrates dissolution, particle formation, micellization, and membrane permeation kinetics, accurately predicting free drug flux within ±5 % of experimental values. This integrated approach provides both molecular-level insight and a practical predictive tool for the engineering design and optimization of drug delivery systems. 1. Introduction Newly synthesized active pharmaceutical ingredients (APIs) often exhibit poor water solubility, leading to low oral bioavailability and presenting major challenges for formulation engineering. Among the many strategies developed to address this issue, the use of solubilization enhancers, particularly surfactants, has proven especially effective [1, 2]. Surfactants improve drug wetting [3], dispersion [4], and stabilization, and facilitate the solubilization of lipophilic compounds [5,6]. They also play a critical role in suppressing recrystallization during supersaturation [7]. The solubilization capacity of surfactants is influenced by their molecular structure, the properties of the drug, and environmental factors such as temperature, pH, and ionic strength [8]. In systems utilizing highly soluble amorphous drug forms, supersaturated solutions can develop, making the presence of surfactants essential to control precipitation and manage particle size distribution [9]. The resulting drug nanoparticles can enhance dissolution and improve absorption when dissolution is the rate-limiting step, especially in oral delivery applications [10,11]. Building upon our previous work [9], we demonstrated that nonionic surfactants can control the size of Valsartan particles formed from supersaturated solutions, yielding particles ranging from nanometer to micrometer scale. These include amorphous drug-rich particles and drug-loaded micelles. At elevated surfactant concentrations, micelle formation enables complete drug solubilization and a marked increase in apparent solubility. These drug-loaded micelles may act as depots, enabling sustained or controlled release [12]. * Corresponding authors. E-mail addresses: [email protected] (D. Trunov), [email protected] (M. ˇ So´ oˇ s). Contents lists available at ScienceDirect Results in Engineering journal homepage: www.sciencedirect.com/journal/results-in-engineering https://doi.org/10.1016/j.rineng.2025.107957 Received 11 August 2025; Received in revised form 21 October 2025; Accepted 28 October 2025 Results in Engineering 28 (2025) 107957 Available online 28 October 2025 2590-1230/© 2025 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ). Studies have shown that while micellar systems are effective at stabilizing poorly soluble compounds, they may hinder semi permeability diffusion, as only the unencapsulated drug fraction contributes to permeation [13]. Therefore, excessive surfactant concentrations can reduce drug permeability across biological membranes, potentially limiting bioavailability [14–16]. This solubility-permeability interplay is a major consideration in the engineering of colloid based drug delivery systems, where careful surfactant selection and concentration optimization are crucial to avoid compromising drug bioavailability. As demonstrated, the interplay between micelle size [17], aggregation number [18], and drug-surfactant interactions strongly impacts drug release kinetics and flux across membranes [19,20]. For example, in an ex vivo intestinal permeability study, increasing concentrations of Tween 80 improved solubility but corresponded to decreased permeation rates for poorly soluble drugs, highlighting the encapsulation–flux tradeoff [21]. Understanding these relationships is essential for rational engineering of drug delivery systems. We selected four widely used nonionic surfactants, Tween 20, Tween 40, Tween 80, and Kolliphor HS 15, due to their distinct hydrophilic–lipophilic balance (HLB) values and polyethylene glycol (PEG) chain lengths enable systematic evaluation of how surfactant structure affects drug solubilization and permeability. Tween surfactants (polysorbates) are established pharmaceutical excipients with regulatory approval and defined intake limits, while Kolliphor HS 15 (polyoxyl 15 hydroxystearate) is commonly applied in oral and parenteral formulations for poorly soluble drugs [22]. The European Food Safety Authority (EFSA) set a group acceptable daily intake (ADI) of 25 mg/kg body weight/day for polysorbates, with pharmaceutical exposures typically far below this level [23]. Similarly, the FDA permits their use in food and drug products, with maximum daily intakes of 300–475 mg [24]. Valsartan was selected as the model of poor water-soluble drug due to its Biopharmaceutics Classification System (BCS) class II designation, low oral bioavailability, and therapeutic role as an angiotensin II receptor blocker in hypertension and heart failure [25]. Its limited aqueous solubility makes it well suited to investigate the interplay between solubility, micellization, and permeability, while ensuring that the findings have practical implications for oral drug delivery and formulation design. In this study, we systematically evaluated the influence of four nonionic surfactants on Valsartan solubility, particle formation, and membrane permeation. Using a controlled cellulose membrane system, we quantified drug flux under defined temperature, pH, and ionic strength conditions. At low nonionic surfactant concentrations, micrometer-sized particles dominate; at higher concentrations, nanoparticle formation and micellar encapsulation were observed. Importantly, increasing surfactant concentration reduced free drug flux, indicating competitive partitioning between freely dissolved, particleassociated, and micelle-bound states. To elucidate these mechanisms, we combine molecular dynamics (MD) simulations with experimental data to estimate aggregation numbers, binding free energies, and micelle loading capacities. We then introduce a chemical engineering model integrating dissolution, particle formation, micellization, and membrane permeation kinetics. This integrated framework allows prediction of free drug concentration profiles and provides a tool for optimizing nonionic surfactant selection and dosage. To our knowledge, this is the first comprehensive investigation that simultaneously addresses solubility, particle dynamics, micellar encapsulation, and membrane permeation for multiple nonionic surfactants within a unified predictive model. 2. Materials and methods 2.1. Materials Valsartan free acid was kindly provided by Zentiva, k.s. (Prague, Czech Republic). Hydrochloric acid (35 wt. %), high purity sodium hydroxide and ethanol for UV (99 wt. %) were purchased from PENTA, Czech Republic. Polyethylene glycol sorbitan monolaurate (Tween 20 and Tween 40 as C 12 E 20 and C 16 E 20 ) and Polyethylene glycol sorbitan monooleate (Tween 80 as C 18 E 20 ) were obtained from Sigma-Aldrich. Polyethylene glycol (15)-hydroxystearate (Kolliphor HS 15 as C 18 E 15 ) was purchased from BASF, Germany. Detailed information about the used nonionic surfactants and their abbreviations is presented in Table 1. The chemical structures of all nonionic surfactants and valsartan are illustrated in Fig. 1. High purity deionized water (specific conductivity 1.8 μ S/cm, Aqual, Czech Republic) was used for the preparation of all aqueous solutions. 2.2. Preparation of valsartan particle depots in the donor compartment (anti-solvent precipitation) A stock solution of Valsartan in ethanol-water mixture (8:2) with concentration 24.2 g/L was rapidly mixed with water solution in the donor compartment at a fixed temperature of 37 ◦C, in a mass ratio of water to stock solution 10:1 using magnetically stirred bar with diameter 10 mm operated at 700 rpm. The water solution has a pH of 4.8 with ethanol fraction of 8 % to fully protonate molecules of Valsartan (LH − and L 2forms), as the dissociation constants of Valsartan are pKa 3.9 and pKa 4.6 [26]. Additionally, selected pH is used to simulate gastric acid with a peptone meal [27]. To study the impact of the nonionic surfactant concentration on the behavior of the supersaturated Valsartan solution in the donor compartment, the used concentration of nonionic surfactants was ranging from 0 to 300 times the nonionic surfactant’s CMC. Such concentration range of nonionic surfactants was selected based on common application in drug delivery [28,29]. The maximum final Valsartan concentration was equal to 2.2 g/L. The selected preparation method resulted in the formation of particles with an amorphous character, as demonstrated previously [9]. All stock solutions were filtered through a 200 nm PTFE syringe filter prior to mixing to remove impurities and reduce potential nucleation sites. 2.3. Characterization techniques and methods 2.3.1. Critical micellar concentration (CMC) To enable comparison of the obtained results, the concentration of each nonionic surfactant was normalized to its CMC. The CMC was determined in aqueous solution at pH 4.8 containing 8 % ethanol, under constant temperature conditions (37 ◦C), consistent with Section 2.2. Measurements were performed using a 3D LS spectrometer from LS Instruments, Switzerland (dynamic light scattering) according to Sadeghi [30]. Micelle size was measured in a 1 cm round glass cuvette at five different detector angles (30◦, 60◦, 90◦, 120◦, and 150◦) with 60 s data collection intervals. Multi-angle detection is essential for accurately characterizing micelles across different size ranges, with lower angles providing sensitivity to larger aggregates and higher angles improving resolution of smaller populations. During the experiments, the count rate was maintained at approximately 200 kHz. A reduction in count rate indicated insufficient micelle population, since scattering intensity is proportional to the number of micelles formed. Both micelle size and scattering count rate were evaluated as a function of surfactant concentration (see Supplementary Information, Figures SI 1(a–h)). From these data, the CMC values were extracted. All measurements were carried out in triplicate to ensure reproducibility, and the obtained values are summarized in Table 1. 2.3.2. Determination of particle size distribution in the donor compartment The evolution of particle sizes was determined by dynamic light scattering (DLS) using 2D pseudo cross correlation, modulated 3D cross correlation [31], or a sample goniometer setup [32]. As demonstrated previously [9], these techniques enable accurate measurement of highly turbid samples without dilution, which could otherwise alter particle size, while simultaneously suppressing multiple light scattering. In D. Trunov et al. Results in Engineering 28 (2025) 107957 2 modulated 3D cross-correlation, two light beams are employed to isolate single-scattering contributions, whereas the goniometer setup allows measurements of turbid samples in short-path square cells, thereby minimizing multiple scattering effects. Particle size polydispersity was evaluated by varying the scattering angle between 30◦and 150◦The laser intensity was adjusted to maintain a scattering count rate of approximately 200 kHz, using a 660 nm Cobolt modulated DPSS laser at 37 ◦C. Viscosity and refractive index corrections were applied to account for the ethanol content. Measurements were performed in a 1 cm round cuvette with 30 s data acquisition intervals, while highly turbid samples were analyzed in a rotating square cuvette to further reduce the optical path length. Mean particle size and polydispersity index were derived from the autocorrelation functions using the cumulants method [33]. In case of large particle size (typically above one micron), the measurement of the particle size distribution was done with an in-house built video optical microscope (OM) using high-speed camera pixelink PL-D725MU-T (Copenhagen, DK). Images of particles were collected using optical lens with 20×and 60×magnification. The non-diluted solution was flowing through the glass capillary with an optical path of 200 µm. The particle size distribution was evaluated from the obtained images with an in-house developed image analysis program [9]. 2.3.3. UV–Vis spectroscopy analysis Valsartan absorbance in its water solution was measured by Agilent Cary 60 UV–Vis (California, USA). The liquid measurement was performed with an optical stainless-steel probe with removable tips with 40 and 10 mm path length to cover the broad range of Valsartan concentrations. Ultraviolet light was used for detection range of wavelengths from 200 to 400 nm using a scan speed of 400 nm/min. The wavelength 242 nm was used to measure the Valsartan absorbance and then its concentration was calculated from the freshly prepared calibration curve. The wavelength of 400 nm was used for the baseline correction to remove any potential interference. This wavelength was selected since there was no interference from the nonionic surfactants, particles or ethanol present in the samples. 2.3.4. Determination of valsartan concentration in the donor compartment The concentration of the dissolved Valsartan in nonionic surfactant containing solutions at pH =4.8 and its amount present in the nonionic surfactant micelles was measured at the fixed temperature 37 ◦C. Vial content was mixed at least two days to ensure equilibrium of Valsartan excess with the liquid phase. The undissolved material was removed by a 200 nm PTFE syringe filter (typically used during USP-based dissolution studies) and the supernatant was immediately dilute 100×with water at pH =4.8. Similar approach was also used for determination of dissolved Valsartan in supersaturated solutions where samples withdrawn at selected time intervals were filtered through 200 nm PTFE syringe filters followed by 100×dilutions with water at pH =4.8. All prepared and collected solutions were analyzed by UV–Vis spectroscopy (see Section 2.3.3). To account for micelle-encapsulated valsartan, which can affect the concentration of freely dissolved drug, the water-soluble fraction of valsartan was separated using the ultrafiltration procedure described by Schott [34]. Therefore, the solution was centrifuged through the cellulose membrane Amicon Ultra – 0.5 mL, Ultracel 3 kDa (Merck Millipore, IRL). In particular, supersaturated solution of Valsartan was taken from the donor compartment at predefined time intervals, followed by centrifugation for 2 min at a speed of 8 000 RPM using the Sorvall Legend X1R (ThermoFisher, USA). Consequently, the supernatant was immediately diluted 100×with water at a pH of 4.8 followed by the Valsartan concentration determination via UV–Vis spectroscopy (see Section 2.3.3). 2.3.5. Diffusion cell measurements The Valsartan diffusion rate through the porous membrane was evaluated using an in-house built diffusion cell (see Fig. 2). The donor and acceptor compartments were separated by a passive dialysis nonionic cellulose membrane with a Molecular Weight Cut Off (MWCO) [35] value of 6 – 8 kDa and connected with an orifice 3 cm in diameter (area of 7.069 cm 2 ). The membrane with relatively larger pores was selected to allow free diffusion of Valsartan (~0.4 kDa) without introducing an additional rate-limiting barrier. This classical membrane was selected to minimize nonspecific interactions between the membrane and the drug, micelles, or solvents. The volume of each compartment was equal to 77 cm 3 . Supersaturated Valsartan solution at pH 4.8 was prepared in the donor compartment while the acceptor compartment was filled only with water solution at pH 4.8. A circulating water bath was used to control the temperature of both compartments at 37 ◦C. Both chambers were stirred using a cross-shaped magnetic stir bar operated at 300 RPM to reduce the membrane dominant resistance to Table 1 Properties of the nonionic surfactants. Nonionic surfactant Number a (alkyl group) Number b (ethoxy group) Summary structure (C a E b )M w (g/mol) HLB (-) CMC ( μ g/cm 3 ) Micelles size (nm) Tween 20 12 20 C 12 E 20 1228 16.7 318 6.8 ±0.4 Tween 40 16 20 C 16 E 20 1283 15.6 147 8.5 ±0.8 Tween 80 18 20 C 18 E 20 1310 15 29 11.7 ±1.0 Kolliphor HS 15 18 15 C 18 E 15 345 15.2 17 12.4 ±0.8 Fig. 1. Chemical structures of nonionic surfactants and Valsartan. D. Trunov et al. Results in Engineering 28 (2025) 107957 3 mass transfer. Valsartan flux across the membrane was determined from the time evolution of Valsartan concentration measured in the acceptor compartment by UV–Vis probe (see Section 2.3.3). 2.4. Flux calculation based on the diffusion cell measurements The Fick’s first law of diffusion [36] may be used to describe the mass transport of Valsartan across the membrane. Considering only the initial stage of transport and small amount of drug transported, we can describe the flux of API through the membrane J in a side-by-side diffusion cell as [37]: J=Deff H× (CD−CA) = Peff × (CD−CA)(1) where the flux depends on the effective permeability coefficient Peff and the concentration difference between the donor (CD)and the acceptor (CA)compartment [38]. The effective permeability coefficient is calculated as the ratio of effective diffusion coefficient Deff and the total thickness of the membrane and aqueous diffusion layer resistance in acceptor and donor compartments H. Experimentally, the diffusive flux can be evaluated from the change of the drug concentration over specific period of time dC/dt across the MWCO membrane with an area A as [39]: J=VA A×dC dt (2) where VA is the volume in acceptor compartment and A is the surface of the membrane. When combining Eq. (1) and 2, it is possible to calculate the drug accumulation in the acceptor compartment when knowing VA, A and Peff according to: dCA dt =A VA Peff (CD−CA) = keff (CD−CA)(3) where keff is the effective mass transport coefficient through the porous medium. The effective permeability contains the contribution of boundary layer at donor compartment, in the membrane, and in the acceptor boundary layer and is given by the equation: Peff =Deff H=1 PD+HM DM+PA (4) where PD and PA are permeability coefficient of the aqueous diffusion layer resistance on the donor and acceptor side of the membrane and HM is the membrane thickness. The total amount of the dissolved drug molecules (STOT) can be evaluated based on the drug aqueous solubility in the absence of nonionic surfactant (Sw) and its amount present in the micelles (Sm): STOT(C) = Sw+Sm=Sw+kCS(5) where Sm is linearly proportional to the concentration of nonionic surfactant in the solution (CS)via experimentally determined constant k [40]. 2.5. Molecular dynamic (MD) simulation and analysis MD simulations were performed in GROMACS (version 2022) simulation package [41] to provide microscopic insight into micelle-Valsartan interactions of experimentally investigated systems. First, we parameterized C 12 E 20 and Valsartan (in an uncharged state) molecules for simulations in aqueous environment, i.e., to be compatible with common three-site water models. In our case model SPC/E was used as this model provides good description of solution structure and thermodynamics, is frequently use for investigation of specific interactions, and performed well in combination with poly-ethylene glycol [42–44]. In the next step, we used (non)-bonded interaction parameters from GAFF force-field in combination with partial charges calculated by RESP method on locally minimized geometries employing GAUSSIAN software [45–48]. Quality of the molecular models was tested on poor solubility of Valsartan and on spontaneous micelle formation propensity of C 12 E 20 in water, which were confirmed (see Figures SI 2 in Supplementary Information). Considering the experimentally measured micelle sizes and known composition, a sufficiently large cubic simulation box with a 12 nm side was used to accommodate ≈55 000 of SPC/E water molecules [49] and varying number (1–50) of C 12 E 20 molecules to investigate micelle formation and stability (homogenous solution, and micelle-like initial conditions were prepared by PACKMOL [50]). In the simulation this resulted in ≈170 000 atoms in all investigated simulations. After construction, prior to any dynamic simulations, the system was minimized (with steepest descend algorithm) to remove potential close contacts between particles. All MD simulations employed 2 fs time step and were performed in isothermal-isobaric ensemble at experimental conditions (T=310 K, p=1 bar), which were controlled by velocity-rescale for canonical sampling thermostat [44] and Parrinello-Rahman barostat [51] with coupling times 0.1 ps and 2 ps, respectively. LINCS algorithm was used to constrain all bonds involving hydrogen atoms [52]. While 1 nm cut-off was used for short range Lennard-Jones and electrostatic interactions, particle mesh Ewald (PME) method on a 0.16 nm grid accounted for a long range electrostatics [53]. Direct (unbiased) simulations of a 100–500 ns length were used to determine micelle size and structure, or to equilibrate systems for subsequent free energy calculations (see details below). Due to slow conformational changes of large molecules (especially in crowded environment of the micelle) sampling frequency of 10 ps was used, resulting in 10 000 – 50 000 of samples for subsequent analysis. The interaction energies in the micelle interior are too high and molecular diffusion is slow, prohibiting determination of binding free energies between Valsartan and micelle by direct sampling in MD simulations. To overcome this issue, we employed non-equilibrium pulling simulation technique (steered MD) [54], with very slow pulling speed, so that the free energy profiles are approaching the equilibrium limit. In selected cases, the computationally demanding, equilibrium, umbrella sampling method was applied as a reference or for refining purposes. This is a working protocol in binding studies of biophysics and soft matter systems [55–57]. On a practical side, at first, we thoroughly tested different definitions of pulling groups, pulling speeds and direction (reversibility), and harmonic force constants, Fig. 2. Schematic of the in-vitro diffusion cell in the presence of the micelles and particles. The donor compartment is located on the left side and acceptor compartment is located on the right side. D. Trunov et al. Results in Engineering 28 (2025) 107957 4 resulting in choice of optimal setup. A reference group was set to the center of mass of the micelle (formed of C 12 E 20 and if applicable also remaining Valsartan molecules), while the pulling group was set as the center of mass of the pulled Valsartan molecule. Force constant and the pulling speed of the harmonic potential were set to kpull =1000 kJ mol −1 nm −2 , and vpull =0.05 nm ns −1 , respectively. Consequently, within the ca 100 ns pulling simulation the VAL molecule slowly moved from micelle interior r~1.0 nm to a micelle exterior with r~6.0 nm. For ensemble averaging, multiple replications (5–10) of the pulling simulation were used. These differed by initial conditions, namely by pulling different Valsartan molecules, or by selecting uncorrelated (Δt>10 ns) geometries from the equilibrium simulation. Obtained force-distance data were analyzed by in-house script [57] being in quantitative agreement with an application of weighted histogram analysis method (WHAM) as implemented in GROMACS [58]. Since the pulling is performed along a radial distance, we had to account for the r – dependence of the translational entropy, volume element (4 π r2dr) by subtracting 2kBTln(r), to obtain the free energy profiles [57]. To refine the binding free energy of Valsartan to the micelle, umbrella sampling simulations were performed along the radial coordinate defined by the center of mass of the micelle distance between the micelle (C 12 E 20 ) and the Valsartan molecule. Configurations from the pulling trajectory were extracted every 0.1 nm from ~1.0 to ~5.0 nm, resulting in an ensemble of well overlapping equidistantly spaced umbrella windows. Each window was sampled for at least 10 ns under isothermalisobaric conditions (T=310 K, p=1 bar). Valsartan molecule was harmonically restrained at a given distance from the center of mass of a micelle with a constant force of 1000 kJ mol −1 nm −2 , allowing it to sample ca 0.15 nm region (window) around the reference position (full width at half maximum of ~0.2 nm). WHAM analysis was applied to obtain the unbiased potential of mean force (PMF), which was corrected for the radial geometry by subtracting the volume entropy by the factor of 2kBTln(r) Standard GROMACS routines (e.g., gmx rdf, polystat, wham, mindist) together with in-house scripts were used for analysis of simulation data, determining radial distribution functions, hydration numbers, free energy profiles, or radius of gyration. 3. Results and discussion 3.1. Distribution of valsartan into micelles, particles and solution To study behavior of Valsartan under supersaturated conditions in the presence of nonionic surfactants, we prepared series of supersaturated solutions containing various amount of nonionic surfactants. In Fig. 3-A is presented example of the particle size evolution during period of 60 min after formation of supersaturated Valsartan solution. As can be seen, there is a strong impact of C 12 E 20 amount on the size of formed particles. In particular, for low amount of nonionic surfactant, corresponding to 1×CMC, Valsartan forms micron-sized particles which grow slowly during the period of 60 min. When nonionic surfactant amount increases to 10×CMC, there is reduction of the particles size to approximately 100 nm measured at 5 min followed by slow growth to nearly 300 nm over the period of 60 min. When nonionic surfactant amount is increased even further to 100×CMC, formed particles have size around 10 nm corresponding to the size of C 12 E 20 micelles (see Table 1). Sizes of all other nonionic surfactants are presented in the Supplementary Information (Figures SI 3 a–d). Since the goal of this work is to characterize the impact of various nonionic surfactants on drug solubility and investigate possibility to use precipitated particles as drug depot, in Fig. 3–Bare presented results of the apparent drug solubility in the used media, i.e., water at pH 4.8. As can be seen, Valsartan concentration dissolved in the acidified water is constant for nonionic surfactant concentration below the CMC, while there is a strong increase of the apparent drug solubility when nonionic surfactant concentration exceeds CMC. In addition, Valsartan apparent solubility depends on the type of nonionic surfactant (increase of carbon atoms within the hydrophobic tail). Since it was previously observed [14–16] that presence of nonionic surfactant might reduce the amount of drug diffusing through the membranes, we performed flux measurement using setup shown in Fig. 2. Obtained results for C 12 E 20 are summarized in Fig. 4-A over the period of 48 h. As can be seen, the amount of Valsartan in the acceptor compartment is approximately twice as high for the solution with nonionic surfactant concentration equal to 1×CMC compared to the solution with nonionic surfactant concentration well above its CMC (100×). Please note, that under all conditions, the measured concentration of Valsartan in the acceptor compartment was below the thermodynamic Valsartan solubility as indicated by the blue dashed line in Fig. 4-A, confirming sink conditions. Despite this, approximately after 12 h, the concentration of Valsartan starts to deviate from linear growth, which is related to the establishment of osmotic pressure equilibrium in both chambers. Therefore, based on these results only the measurement of the flux in region I was considered in this work. Summary of the obtained data measured for all studied nonionic surfactants as presented in Fig. 4–B. As can be seen, in all cases we see a significant decrease of the flux with increased nonionic surfactant concentration independent on the type of used nonionic surfactant. This observation is in line with previously reported trends [16] however, over much broader range of nonionic surfactant concentrations and nonionic surfactant types. Since such decay in flux is connected with drug distribution between particles, micelles and dissolved drug, in what follows to reduce the system complexity we would focus on the case of very high nonionic surfactant concentration (≥100×CMC). Under such circumstances the drug will be present in the dissolved form or absorbed within the Fig. 3. (A) Evolution particle sizes of Valsartan in the presence of nonionic surfactant C 12 E 20 . The concentration of Valsartan was fixed at concentration of 2.2 g/L. (B) Apparent solubility of the drug in the presence of four nonionic surfactants. D. Trunov et al. Results in Engineering 28 (2025) 107957 5 micelles. In fact, presence of micelles was confirmed for all tested nonionic surfactants at concentrations ≥100×CMC by DLS measurement (see Supporting Information Figure SI 3). To determine the free drug in the solution, an ultracentrifugation filtration method was applied (see Section 2.3.4) to remove any drug present inside the micelles. Fig. 5 shows the actual Valsartan concentration in the liquid without Valsartan present in the micelles versus the initial Valsartan concentration used in the experiment. Please note that sum of Valsartan amount measured in the solution and that remaining on the filter corresponding to Valsartan within the micelles confirm no losses of the drug. As can be seen in Fig. 5, if the initial concentration of drug in the final mixture is approximately below 280 μ g cm −3 (vertical black dashed line), there is a negligible amount of free drug in the liquid of the donor compartment and the total flux through the porous membrane is negligible (see Figures SI 3 in Supplementary Information). On the other hand, when the initial concentration of Valsartan in the final mixture is greater than 280 μ g cm −3 , the concentration of the drug in the free liquid starts to increase, however being well below Valsartan thermodynamic solubility of in the absence of nonionic surfactant as indicated by the dashed blue line in Fig. 5. These results confirm that Valsartan molecules accumulate in the micelles and the amount of the drug is dependent on the drug-surfactant ratio. Furthermore, variation of the flux depending on the amount of nonionic surfactant shown in Fig. 4–B, suggesting that once the drug is accumulated inside the micelles, the diffusion of the API through the micellar membrane becomes a transport-controlling factor. As illustrated, both the free drug concentration in the donor compartment and the flux decrease are a function of nonionic surfactant concentration in the solution. According to Eq. (1), this relationship is governed primarily by the membrane permeation coefficient, which depends on the physicochemical properties of the permeating molecule (Valsartan) and the characteristics of the membrane. This observation suggests that permeation experiments, in combination with Eq. (1), can be employed to estimate the free concentration of Valsartan. It is important to note that this approximation is valid only within a short initial time interval, during which the drug concentration in the donor compartment remains significantly higher than that in the acceptor compartment. Under these conditions, plotting the measured flux as a function of the actual free Valsartan concentration, determined via ultracentrifugation, reveals a linear correlation for the nonionic surfactant C 12 E 20 (see Fig. 6). This linearity is associated with a relative error of approximately 8 %, as indicated by the blue dashed lines in the figure. This approach is particularly useful for quantifying the free concentration of Valsartan in complex systems where the drug exists in multiple forms, such as free molecules, micelles, and particulate aggregates. Accordingly, Eq. (1) can be applied in combination with flux measurements to determine the free Valsartan concentration in the donor compartment under such conditions. The results for additional nonionic surfactants are summarized in Fig. 6 and represented by red crosses. 3.2. Capacity of the micelle at steady state It can be considered that the free Valsartan concentration is the fluxcontrolling factor. While this is true, the more critical question is not how much drug remains outside the micelle, but rather the micelle’s capacity for drug encapsulation. To explore this phenomenon, we Fig. 4. (A) Measured Valsartan concentration over time in the acceptor compartment for C 12 E 20 nonionic surfactant. (B) Valsartan flux measured for various combinations of nonionic surfactant types and concentrations. Fig. 5. Initial and actual concentration of the free solute drug in solution in the presence of 100×CMC nonionic surfactant C 12 E 20 . These results correspond to steady state concentration in donor compartment, i.e. in our case corresponds to 60 min. Fig. 6. Measured flux of Valsartan versus the actual free solute drug concentration in the donor compartment for various nonionic surfactant concentrations. The slope corresponds to the effective permeability of the drug based on the Eq. (1). Blue lines correspond to the maximum error bars equivalent amount of 8 percent. D. Trunov et al. Results in Engineering 28 (2025) 107957 6 employed advanced computational methods using MD simulations to investigate drug–surfactant chain interactions. Specifically, the binding free energy, and, consequently, the localization of drug molecules within the micelle. As a result, we were able to predict the number of drug molecules encapsulated per micelle using two simulation approaches: non-equilibrium slow pulling and equilibrium umbrella sampling. Quantification of the initial micelle state, such as radius of gyration, aggregation number, and structural characteristics of C 12 E 20 nonionic surfactants is presented in detail in Supporting Information, Section 2 Section 2. For the initial state prior to the pulling simulation, it was assumed (and supported by hundreds of ns long equilibration MD) that the micelle was already formed and the Valsartan molecule was localized at the interface of hydrophobic core of the micelle. These positions, which corresponds to a local free energy minimum, are shown at representative configuration in Fig. 7. 3.2.1. Non-equilibrium approach of valsartan packaging inside a micelle Radially resolved free energy profile of Valsartan molecule in the micelle was obtained by a non-equilibrium pulling simulation approach and is illustrated in Fig. 8. As mentioned previously, the minimum free energy corresponds to the most thermodynamically stable location of Valsartan, which is at the surface of the hydrophobic core inside the micelle, being partially exposed to the hydrated PEG corona of C 12 E 20 . This location satisfies interactions of dominantly hydrophobic (biphenyl, alkyl), but partially also polar (tetrazole, carboxyl, carbonyl) nature of a Valsartan molecule. As the Valsartan molecule moves out of the micelle through the hydrated PEG corona region, we observed an increase in free energy until the point when the Valsartan molecule reaches the corona/bulk water interface. At this point, located approximately at a distance of 3.5 nm from the micelle center of mass, the net interaction between the nonionic surfactant chains and Valsartan vanishes and the free energy reaches its bulk value (set to zero for clarity). For better visualization please see Figure SI 7 in Supporting information indicating radial distribution function of various components of the studied system. The strong binding of Valsartan to the hydrophobic core is associated with major dehydration of the vast nonpolar Valsartan molecule (see Figures SI 7 in Supplementary Information). The simulation presented in Fig. 8 illustrates the situation when only one Valsartan molecule was incorporated into the micelle. To study the impact of the number of Valsartan molecules we repeated the above discussed protocol and performed simulation with larger number of embedded Valsartan molecules, i.e., using 5 and 10 molecules per micelle, and collected free energy minima of individual Valsartan molecules. These results are schematically summarized in Fig. 9. When pulling the Valsartan molecules across the micelle with occupancy of Fig. 7. Representative structure of the micelle C 12 E 20 with presence of 5 Valsartan molecules. The structure is shown in licorice representation with oxygen atoms in red, carbon in light blue and hydrogen in white. The Valsartan molecules (represented in dark blue) are localized in the hydrophilic PEO corona (thin line representation) of the micelle and are stabilized with additional hydrophobic interactions with the nonpolar core (black spheres representation) of the micelle. Fig. 8. Illustration of a free energy profile (PMF(r)), obtained by slow pulling simulation (vpull =0.05 nm/ns), of one Valsartan molecule from micelle interior, where it is attached to the hydrophobic core (at r≈1.2 nm), to the aqueous solution (r> ∼4 nm). The inserts represent simulation snapshots at selected distances (r∼1.2,2.6,3.0,4.0 nm), corresponding to the Valsartan location and orientation with respect to the micelle. Micelle hydrophobic core is shown in space-filling (gray), hydrophilic moieties in lines (carbon cyan, oxygen red, hydrogens not shown for clarity) and Valsartan molecules in spacefilling (nitrogen blue, hydrogen white). As a coordinate, the distance between the center of mass of the micelle and center of mass of the pulled Valsartan molecule is used. The accuracy of the PMF(r) form a single slow pulling simulation is <1kT. However, PMF(r) profile is more sensitive to a state of slow modes (C 12 E 20 hydrophilic and hydrophobic chains dynamics), which impact micelle shape and thus indirectly also a CM distance and location of PMF minimum. Averaged PMF(r), with highlighted error bars, was constructed from 6 independent pulling simulations. Fig. 9. Summary of potential binding free energies of Valsartan molecules in micelles containing 25 C 12 E 20 molecules and 1, 5 and 10 Valsartan molecules, respectively. Binding free energies were extracted as the minima of free energy profiles with dashed lines indicating results variation for different pulling experiment. Error bars represents realization of pulling experiment with different Valsartan molecule, or initial conditions. The open blue triangle represents the maximum amount of Valsartan per micelle extracted from calculation. D. Trunov et al. Results in Engineering 28 (2025) 107957 7 one or five molecules, independent on the pulled Valsartan molecule, the minimum free energy is equal to -45 ±8 kT. The drug molecules were predominantly located at the interface of the hydrophobic core and the hydrophilic part (palisade layer), thus forming the first thermodynamically stable layer. Once this layer is fully occupied, subsequent drug molecules reside further from the core, and are less strongly bound and become most likely to be released from the micelle, i.e., the micelle is being oversaturated. In fact, such scenario is indirectly observed when pulling experiment was realized on micelles filled with 10 molecules of Valsartan. In particular, we found that some Valsartan molecules when pulled out from the micelle requires more energy (72 ±7 kT) due to interactions with one or more drug molecules in the upper layer. This deeper free energy minima were observed for three Valsartan molecules, which is interpreted as three molecules reside in the upper (less bound) layer. Therefore, the free energy is divided into two stable regions as shown in Fig. 9 (as dashed lines). The energy of the upper level is comparable with that observed in pulling of 1 and 5 Valsartan molecules, while the appearance of deeper (second) minima for 10 Valsartan molecules indicates a different regime, i.e., an overfilling of a micelle. Since this behavior was observed for 3 out of 10 Valsartan molecules, we can estimate the maximum drug capacity per micelle, i.e., full occupancy of the first thermodynamic drug layer, in the range between five to seven molecules (indicated by blue arrows in Fig. 9). Further support for this interpretation and conclusion is provided in the following section. 3.2.2. Enhanced umbrella sampling simulation to characterize valsartanmicelle interactions Due to the irregular and often non-spherical shape of the micelle, especially when multiple Valsartan molecules are encapsulated, distance from the micelle center of mass is not a reliable metric for evaluating drug localization. Instead of relying on spatial criteria alone, we focused on the number of hydrophobic contacts between Valsartan, the C 12 E 20 nonionic surfactants, and water, as a more meaningful measure of molecular stability. Therefore, umbrella sampling method was employed to construct equilibrium free energy landscapes of Valsartan within the micelle. To establish a reference free energy state, we first calculated the binding free energy of a single Valsartan molecule in its aggregated state, which was found to be approximately −15 kT (Fig. 10A). In comparison, detachment from the micellar hydrophobic core required 27 kT (Fig. 10–B, blue line), resulting in a net difference of ~12 kT. This value represents the energy threshold beyond which Valsartan is more likely to leave the micelle and rejoin the bulk aggregate (or being dissolved in a monomeric free form at respectively low concentration). Monitoring the loss of hydrophobic contacts was found sufficient to track this transition, and a critical distance of 2.2 nm from the micelle center of mass was identified as the point at which molecules begin to escape (Fig. 10–B, red line). This behavior suggests that when any Valsartan molecules intermittently lose contact with the micelle core (overcomes 2.2 nm distance), its free energy increases so that it becomes probable (following Boltzmann probability, P∼eΔG/RT) to be released from the micelle into the aqueous solution or to aggregate in the bulk. This metrics is further used as an indicator of micelle encapsulation capacity and oversaturation. For further evaluation, the number of nonpolar contacts was selected as the primary parameter to determine the micelle’s capacity. As shown in Fig. 11, the hydrophobic contacts (interactions) between the Valsartan molecule and the micelle were tracked over 100 ns and then translated into number of hydrophobic contacts distribution. From this analysis, it is evident that the Valsartan molecule resides in contact with the hydrophobic core, due to the ~40 of hydrophobic contacts (Fig. 10, red line). The same procedure was applied for micelles solubilizing 5 and 10 Valsartan molecules, and the probability distributions are presented in Fig. 11. When 5 Valsartan molecules are incorporated into the micelle, the probability of hydrophobic contacts slightly increases as the drug molecules loose translational entropy and are localized closer to the micelle core. This region is thermodynamically more stable, reducing the probability of drug molecules leaving the micelle. However, when the micelle becomes oversaturated as it is for 10 drug molecules, the probability distribution of hydrophobic contacts splits into two distinct regions. This observation aligns with our results of nearequilibrium pulling simulations, where two layers of Valsartan molecules were identified based on two free energy minima (levels) observed for oversaturated micelles. The first broad region (N=70 ±20) represents a more stable drug configuration, where molecules shift closer to the hydrophobic core, enhancing drug stability in the micelle. The second region, of lower number of hydrophobic interactions (N=20 ±10), indicates Valsartan occurrence at the micelle periphery, from which a release of a drug molecule to the bulk solution becomes probable. Notably, this behavior was observed for four out of ten Valsartan molecules, suggesting that the maximum micelle capacity is approximately six drug molecules, as determined from enhanced sampling simulations. 3.2.3. Experimental validation of micelles capacity Combining the experimentally determined amount of Valsartan in solution (from flux measurements) and number of nonionic surfactant molecules forming a micelle obtained from MD simulation data (see Figure SI 5, 25 for C 12 E 20 , 44 for C 18 E 20 , and 38 for C 18 E 15 ), we estimated the average number of Valsartan molecules per micelle (see Supporting Information, Section 3 Section 3 for detailed calculations). Our analysis revealed that the maximum number of Valsartan molecules per C 12 E 20 micelle at steady state (at 60 min after micelle formation) is Fig. 10. (A) Illustration of a free energy profile (PMF(r)), obtained by umbrella sampling simulation, of pulling one molecule Valsartan from Valsartan aggregate (8 molecules) interior to the aqueous solution (r> ∼2 nm). The inserts represent simulation snapshots at selected distances (r∼0.5,1.2,1.8,2.6 nm), corresponding to the Valsartan location and orientation with respect to the Valsartan aggregate. (B) Potential of mean force (in blue) and number of contacts (within 0.4 nm, in red) vs. distance between Valsartan and micelle (mass center is used). The red line represents the number of contacts between hydrophobic functional groups of Valsartan and C 12 E 20 . All curves and uncertainties are evaluated from umbrella sampling simulation. D. Trunov et al. Results in Engineering 28 (2025) 107957 8 approximately 5.3 ±0.4. Interestingly, during the initial phase of micelle formation (2 min), the encapsulation efficiency was slightly higher, with an estimate of 6.2 ±0.3 Valsartan molecules per micelle. Notably, both values are in excellent agreement with MD simulations, which predict that the maximum number of Valsartan molecules effectively accommodated per micelle falls within the range of 5 to 7 molecules. The same calculation approach was applied to determine the solubilization capacity for Valsartan within the micelles of other nonionic surfactants (either determining aggregation number from our MD simulations for nonionic surfactant C 18 E 20 and C 18 E 15 or from the literature data [59] for nonionic surfactant C 16 E 20 ). The results are summarized in Fig. 12 and Table 2. As can be seen from Fig. 12, the solubilization capacity of the micelles increases with an increase of CMC point. Since the CMC is inversely proportional to the number of carbon atoms of the hydrophobic chain of the nonionic surfactant, there would be decrease of the solubilization capacity with the increase number of carbon chains in the nonionic surfactant molecule (see Table 1). Simultaneously, the number of PEG groups influences the micelle’s drug loading capacity, decreasing proportionally as the number of hydrophilic groups is reduced. 3.3. Kinetics of solubilization 3.3.1. Static mechanism of micelle solubilization At steady state, drug molecules are predominantly located at the interface between PEG chains and the hydrophobic core of the formed micelle. However, during the initial phase of surfactant-micelle formation, the drug primarily exists as aggregate clusters, which, in real-world scenarios, correspond to nanoor microparticles dispersed in solution. This assumption was supported by representative snapshots of the system taken over a timescale of several seconds, as presented in Figure SI 9 of the Supporting Information. It was found that, when the solution of ethanol containing dissolved Valsartan to water-micelle solution was added, there is a formation of a clearly visible cloud of particles, which disappears shortly after 3 s of mixing (Figure SI 10-A). Please note, when no Valsartan inside of ethanol solution was present (see Figure SI 10-B), there was no particle cloud formation, confirming that the nonionic surfactant alone is not responsible for any particle’s formation, which might originate from phase separation of alcohol rich phase present in the solvent solution. To elucidate the mechanism during very early stage of micelle solubilization, we conducted MD simulations where a pre-formed Valsartan aggregate was placed to a simulation box with a randomly distributed nonionic surfactant molecules (at concentration above CMC) (as illustrated in Fig. 13). During the simulation process, which held up to 1.5 µs, we identified two distinct stages of system behavior. At an early stage, the Valsartan cluster remains intact, while nonionic surfactant molecules quickly self-associate and also interact with Valsartan cluster, until a moment, when a hydrophobic micelle core begins forming at the drugwater interface. At the late stage, the nonionic surfactant molecules diffuse into the drug cluster, gradually disrupting and separating individual drug molecules, driven by the higher binding affinity between the drug and micelle (previously discussed). As shown in Fig. 14, two distinct clusters were formed after 0.6 µs of simulation, and four clusters at 1.5 µs of simulation. During the initial solubilization stage, Valsartan molecules are homogeneously distributed within the micelle’s hydrophobic core, resulting in a higher solubilization capacity (Table 2). Once the micelle becomes saturated, additional drug molecules can no longer be incorporated and instead form stable Valsartan aggregates outside the micelle, characterized by reduced nonpolar interactions. This final equilibrium state, referred to as steady state, is difficult to capture within the limited time frame of conventional MD simulations. Therefore, to further investigate the dynamics of micelle solubilization, we used engineering model to capture such dynamics. 3.3.2. Dynamic process of micelle solubilization As was illustrated in Fig. 3-A, depending on the amount of nonionic surfactant, there is an observed formation of micelles, particles or their combination during the initial time period of the studied system. Corresponding evolution of the Valsartan concentration in the donor compartment is presented in Fig. 15-A. As can be seen, for low nonionic surfactant concentration equal to 1×CMC we observed a decrease in the Valsartan amount with time until the steady state was reached. This Fig. 11. (Left image) time evolution of a number of hydrophobic contacts between one Valsartan molecule (1x Val) and a hydrophobic core of a C 12 E 20 micelle, which is used to evaluate a distribution of a number of hydrophobic contacts. (Right image) distribution of number of hydrophobic contacts per Valsartan molecule for varying number of Valsartan molecules in the micelle (1x, 5x, and 10x Val), where 20 hydrophobic contacts represent the minimum threshold for Valsartan molecule to be released from the micelle, and more than 50 hydrophobic contacts represent tight (stable) association with the core of the micelle. Fig. 12. Amount of Valsartan molecules per micelle as a function of CMC value of the relevant nonionic surfactant. These results correspond to steady state concentration in donor compartment, i.e., in our case corresponds to 60 min. D. Trunov et al. Results in Engineering 28 (2025) 107957 9