Mass transport analysis of the particle drifting effect in a biphasic diffusion apparatus

Abstract

Formulations containing colloidal particles, such as micelles, nanocrystals, and amorphous drug nanoparticles, are widely used to enhance the oral absorption of poor soluble drugs. The underlying mechanism was proposed to be the particle drifting effect, where the particles effectively reduce the diffusional resistance of the aqueous boundary layer by releasing the free drug near the surface of the absorption site, such as a membrane, the intestinal mucosa, or an interface. However, it remains challenging to appropriately interpret experimental data or to accurately predict enhanced permeation rate provided by particle drifting effect. In this study, we developed an integrated dissolution-permeation model to quantitatively analyze the particle drifting effect. Using a biphasic experimental setup, amorphous drug nanoparticles of several poorly soluble model drugs were evaluated. The particle drifting effect was modeled by coupling the Wang-Flanagan particle dissolution model with a stagnant-film permeation model. Results suggested that drug nanoparticles at low concentrations did not alter the diffusional profile or the fitted permeability coefficient. At high particle concentrations, a flux plateau was observed, signifying non-sink dissolution of the particles. The fitted interfacial permeability coefficient increased with increasing particle concentration, confirming reduced diffusional resistance of the aqueous boundary layer by the presence of drug particles. High number of particles also altered the fitted partition coefficient of the drug due to saturation of the free drug in the aqueous boundary layer adjacent to the liquid interface. The mass transport model was able to predict the particle drifting effect for systems with high particle concentrations or extremely poorly soluble drugs where experimental evaluations become challenging. Combined with a differential equation-based pharmacokinetic model, in vivo drug absorption of a model drug enzalutamide was predicted at different doses with satisfaction. This work provides mechanistic understanding of the diffusional profiles obtained through the biphasic setup, and may contribute to more accurate oral bioavailability prediction for formulations that contain amorphous drug nanoparticles.

Graphical Abstract

INTRODUCTION

There is a growing trend of newly approved small molecule oral drugs that go beyond Lipinski’s Rule of 5, particularly in terms of lipophilicity and molecular weight.¹ These new drugs often suffer from low aqueous solubility, leading to slow dissolution rates and subsequently, low oral bioavailability. As a result, various formulation strategies have been developed to address this issue.² One of these approaches involves the use of colloidal particles in formulations, such as nanocrystal formulations^3,4, surfactant micelles², as well as amorphous drug nanoparticles⁵. Amorphous nanoparticles can be generated from a variety of formulations, such as through the dissolution of amorphous solid dispersions⁵, the gastric transit of weakly basic drugs⁶, or the digestion of lipid-based formulations⁷. These strategies have been proven to promote drug absorption and enable the drug’s successful development.^8,9

In addition to the drug’s solubility, another critical factor determining oral drug absorption is permeability¹⁰. A drug’s permeation rate is determined by its intrinsic permeability through the intestinal membrane, as well as the diffusional resistance from the aqueous boundary layer, also known as the unstirred water layer (UWL), adjacent to the intestinal membrane^11–13. The slower one of these two interdependent steps determines the overall permeability of the drug. For unionized poorly soluble drugs, UWL permeability is usually the rate-limiting step. In recent years, the importance of UWL has been gaining increasing attention in explaining the improved bioavailability provided by formulations containing colloidal particles.^4,14,15 Although colloidal species have been shown to improve a drug’s bioavailability^4,13–19, such bioavailability enhancement cannot be sufficiently explained by increased dissolution rate due to increased surface area. Alternatively, colloidal particles can serve as drug reservoirs, dissolve and replenish the free drug concentration in the UWL to reduce its diffusional resistance, effectively increasing the drug’s membrane permeation rate and absorption.²⁰ The reservoir effect in the diffusion boundary layer provided by bile micelles was reported in the 1970s¹³, and is now more commonly referred as the particle drifting effect for micro- and nano-sized drug particles²⁰.

While many studies have reported enhanced permeation rates provided by these colloidal formulations and linked them to the particle drifting effect^16,21,22, few have attempted to predict the extent of the particle drifting effect,²³ with most of these efforts being empirical. Most recently, a mass transport model was developed to describe the membrane transport of ibuprofen crystals.²³ However, particle dissolution was treated as a bulk solution process outside the UWL; also, experimental validation was limited to a single drug. Therefore, a mechanistic mass transport model more accurately describing the role of UWL in the particle drifting effect, together with experimental evaluations using a variety of drugs and particles, is needed.

In this study, we integrated the two kinetic processes involved, dissolution in the aqueous boundary layer and free drug permeation across the liquid interface, in the mass transport model, and performed mass transport analysis of the particle drifting effect. Amorphous drug nanoparticle was used as a model system due to its simplicity in preparation and particle size control. Experiments were performed using a biphasic diffusion setup. Anacetrapib, atazanavir, clotrimazole, enzalutamide, and felodipine were used as model drugs to evaluate the impact of particle concentration and size on mass transport. Their molecular structures are shown in Figure 1.

Figure 1.

Molecular structure of model drugs

THEORY

Diffusion Boundary Layer (DBL)

Drug dissolution and absorption are often controlled by diffusion and can be mathematically modeled using the concept of the diffusion boundary layer (DBL). Introduced in 1904 by Nernst and Brunner²⁴, the concept of DBL has been extensively used to model mass transport phenomena in various processes, including electrochemical reaction, corrosion, environmental pollution, drug dissolution, and membrane permeation among others.²⁵ The DBL, also often referred to as the UWL, the aqueous boundary layer (ABL), or concentration boundary layer (CBL), is a hypothetical layer of stagnant water adjacent to a solid surface where mass transport occurs through diffusion only rather than convection. This concept is used to simplify the mathematical modeling of diffusion-controlled dissolution. Although mechanistically far from reality, this approach helps describe complex dissolution and permeation mechanisms using simple mathematical expressions, leading to its wide acceptance in the scientific community.²⁶

In contrast, the hydrodynamic boundary layer (HBL), also present adjacent to the solid surface, is a real layer formed by viscous forces, creating velocity and concentration gradients across the dissolving solid’s surface.

In this work, we used the simplified diffusion boundary layer for mass transport modeling purposes.

Dissolution Model

The mathematical modeling of dissolution dates back to 1897, when Noyes and Whitney proposed their equation (Eq-1) based on constant surface area of dissolving solids ($\frac{dC}{dt}$ is dissolution rate, $K_{NW}$ is a constant, $C_S$ is the saturation solubility of the dissolving solid, and $C$ is the bulk concentration).²⁷ Later in 1900, Brunner and Tolloczko modified this equation to account for the impact of surface area ($S_A$) on the overall dissolution rate (Eq-2, $K_B$ is a constant)²⁴. Bruner continued his work on dissolution with the Nobel Laureate Walther Nernst, and published the Nernst-Brunner equation (Eq-3) together in 1904, where the proportionality constant $K_B$ in the previous equation was replaced with $K_B=\frac{V}{Dh}$, ($D$ is the diffusion coefficient of the solute, $V$ is the volume of the solvent, and $h$ is the thickness of the diffusion boundary layer).²⁴ This equation introduced the concept of DBL in dissolution modeling.

$$\frac{dC}{dt}=K_{NW}\times \left(C_s-C\right)$$ 1

$$\frac{dC}{dt}=K_B\times S_A\times \left(C_s-C\right)$$ 2

$$\frac{dC}{dt}=\frac{D}{Vh}\times S_A\times \left(C_s-C\right)$$ 3

In 1931, Hixson and Crowell observed that particle shape and surface area change during dissolution, affecting particle dissolution rate. They linked such surface area decrease to particle weight and proposed the Hixson-Crowell cube-root law.²⁸ In 1963, Niebergall et al. noted that some dissolving particles did not conform to the cube-root law and suggested a square-root law²⁹. In the same year, Higuchi and Hiestand proposed a two-thirds-root law for the dissolution of finely divided powders³⁰. Finally, in 1999, Wang and Flanagan unified these three laws by proposing a general solution for diffusion-controlled dissolution of spherical particles under both sink and non-sink conditions, and explained that each of the three laws are special cases of the general solution under sink conditions.²⁶

While dissolution is often modeled as a diffusion-controlled process, significant convection can cause model failure, especially at non-sink conditions, where suspended particles experience enhanced convection due to slip, shear, and confinement.³¹ Convective factors like agitation strength, slip/shear and fluid viscosity can be incorporated into dissolution modeling, but often these models are more complex and require dimensionless parameters such as Sherwood, Reynolds, and Schmidt numbers³². However, if the particle size is significantly smaller than the dimension of the DBL, as in the case of colloidal drug particles investigated in this study, convective effects are negligible and dissolution is diffusion-controlled, making these systems suitable for diffusion-controlled modeling.^23,26,31

Dissolution-Permeation Model

Mass transport across liquid-liquid interfaces can also be modeled using a diffusion-controlled stagnant film model. In this model, two immiscible fluids are in contact through a stationary interface, across which mass transport occurs solely via diffusion. Two stagnant films (DBL), one on each side of the interface, along with the mass transport resistance provided by the interface, as shown in Figure 2, constitute the overall resistance of a solute moving across the liquid interface.

Figure 2.

Schematic showing the structure of the dissolution-permeation model

In our previously published work³³, we found that the particle drifting effect shows the kinetics of a sequential reaction: particle dissolution followed by interfacial permeation of the free drug (Figure 2). In the present work, the Wang-Flanagan model²⁶ was used to describe the particle dissolution step, and the stagnant-film model³⁴ was used to model subsequent solute transport across the liquid interface.

Due to the high solubility of poorly soluble drugs in the organic phase, a discontinuity in the concentration gradient with a sharp rise in drug concentration in the organic side of the interface is expected²⁵. Such discontinuity in solute concentration was corrected using the octanol-water partition coefficient of the drug, based on the assumptions that the partition kinetics is instantaneous, and the partition coefficient remains constant across different drug concentrations.

Additionally, for poorly soluble drugs, the diffusional resistance provided by the organic DBL was assumed to be negligible³⁴.

Nomenclature

$A$ Area of the aqueous-organic interface

$c_{\mathit{\text{DBLa}}}$ Average free drug concentration in the aqueous DBL

$c_{\mathit{\text{DBLp}}}$ Particle concentration in the aqueous DBL

$c_f$ Amorphous solubility of the drug making up the drug particle

$c_0$ Initial drug concentration in the donor (aqueous) solution

$c_o$ Bulk concentration of the dissolved drug in the organic phase

$c_p$ Concentration of undissolved drug particles in the aqueous phase

$D_f$ Diffusion coefficient of free drug in the aqueous phase

$h$ Thickness of the diffusion boundary layer surrounding the particles

$h_{\mathit{\text{DBLa}}}$ Thickness of the aqueous DBL

$K$ Partition coefficient of the drug in the aqueous and organic media at the liquid interface

$M_p$ Mass of each individual drug particle

$N$ Total number of undissolved particles

$P_I$ Overall permeability of the drug across the interface (aqueous and organic DBLs and the interface)

$P{e}_a$ Effective permeability of the drug across the aqueous DBL

$P{e}_i$ Effective permeability of the drug across the aqueous-organic interface

$P{e}_o$ Effective permeability of the drug across the organic DBL

$r$ Radius of the drug particle

$S_A$ Surface area of the solid-liquid interface

$V_a$ Volume of the aqueous phase

$V_{\mathit{\text{DBLa}}}$ Volume of the aqueous DBL

$V_o$ Volume of the organic phase

$V_p$ Volume of each individual drug particle

$\rho$ Density of the drug particle

Model structure

Amorphous drug nanoparticles were generated by spontaneous precipitation of the dissolved drug above its miscibility gap with water, where the amorphous particles were in equilibrium with the saturated free drug in solution. Therefore, at the beginning of biphasic diffusion experiment (time 0), the suspended nanoparticles in the aqueous phase were not dissolving. With the introduction of the organic phase, the free drug diffuses from the aqueous to the organic phase, leading to decreased free drug concentration in the aqueous diffusion boundary layer (DBL), thereby initiating particle dissolution in the DBL.

In this setup, the overall interfacial permeability ($P_I$) is given by:

$$\frac{1}{P_I}=\frac{1}{P{e}_a}+\frac{1}{P{e}_i}+\frac{1}{P{e}_o}$$ 4

For poorly soluble drugs, Eq. 4 reduces to:³⁴

$$\frac{1}{P_I}\approx \frac{1}{P{e}_a}$$ 5

Where $P{e}_a$ is defined as:

$$P{e}_a=\frac{D_f}{h_{\mathit{\text{DBLa}}}}$$ 6

Using the Wang-Flanagan model, the average dissolution rate of each individual particle can be described using equation 7, assuming that particle distribution and dissolution in the DBL were homogeneous:

$$\frac{dM}{dt}=-4\pi r^2{D}_f\left(\frac{1}{r}+\frac{1}{h}\right)\left(c_f-c_{\mathit{\text{DBLa}}}\right)$$ 7

Where $\frac{dM}{dt}$ is the rate of dissolution.

For nanoparticles with $\text{r}\ll \text{h}$, particle dissolution rate reduces to equation 8:

$$\frac{dM}{dt}\approx -4\pi r{D}_f\left(c_f-c_{\mathit{\text{DBLa}}}\right)$$ 8

For $N$ number of particles, the total particle dissolution rate is:

$$\frac{dM\left(\mathit{\text{total}}\right)}{dt}=-4\pi r{D}_f\left(c_f-c_{\mathit{\text{DBLa}}}\right)\times N$$ 9

Where $N$ is defined as:

$$N=\frac{c_p{V}_{\mathit{\text{DBLa}}}}{M_p}=\frac{3c_p{V}_{\mathit{\text{DBLa}}}}{4\pi r^3\rho }$$ 10

The particle dissolution rate can then be converted to changes in particle concentration ($c_{\mathit{\text{DBLp}}}$) in the aqueous diffusion boundary layer. Assuming homogeneous mixing, $c_{\mathit{\text{DBLp}}}$ also equals the particle concentration in the bulk solution.

$$\frac{d{C}_p}{dt}=\frac{1}{V_{\mathit{\text{DBLa}}}}\cdot \frac{dM\left(\mathit{\text{total}}\right)}{dt}=-\frac{3D_f}{r^2\rho }\cdot c_p\left(c_f-c_{\mathit{\text{DBLa}}}\right)$$ 11

If we write $\gamma$ as a constant defined by the properties of the drug particle:

$$\gamma =\frac{3D_f}{\rho }$$ 12

Eq. 11 can be simplified to:

$$\frac{d{C}_p}{dt}=-\gamma \cdot \frac{1}{r^2}\cdot c_p\left(c_f-c_{\mathit{\text{DBLa}}}\right)$$ 13

As the particles dissolve, their radius shrinks. For each dissolving particle, the dissolution rate can be written as a function of particle radius:

$$\frac{dM}{dt}=\frac{d\left(\rho \cdot V_p\right)}{dt}=4\pi \rho r^2\frac{dr}{dt}$$ 14

Combining equation 8 and 14, we obtain the radius of each dissolving particle as a function of time:

$$\frac{dr}{dt}=-\frac{D_f}{\rho }\cdot \frac{1}{r}\left(c_f-c_{\mathit{\text{DBLa}}}\right)$$ 15

Since the organic DBL resistance is negligible, the diffusional flux of the drug across the liquid interface can be written as:

$$J=P_I\left(c_{\mathit{\text{DBLa}}}-\frac{C_0}{K}\right)$$ 16

The drug concentration in the aqueous diffusion boundary layer as a function of time can be described as:

$$\frac{d{c}_{\mathit{\text{DBLa}}}}{dt}=\gamma \cdot \frac{1}{r^2}\cdot c_p\left(c_f-c_{\mathit{\text{DBLa}}}\right)-\frac{A}{V_o}{P}_I\left(c_{\mathit{\text{DBLa}}}-\frac{C_o}{K}\right)$$ 17

The drug concentration in the bulk organic solution as a function of time can be described as:

$$\frac{d{C}_o}{dt}=\frac{A}{V_o}{P}_I\left(c_{\mathit{\text{DBLa}}}-\frac{C_o}{K}\right)$$ 18

The initial conditions for systems with the free drug only and without amorphous drug nanoparticles are:

$$\begin{gathered} c_p\left(0\right)=0 \\ {c}_{\mathit{\text{DBLa}}}\left(0\right)=c_0 \\ {c}_o\left(0\right)=0 \end{gathered}$$

The initial conditions for systems with amorphous drug nanoparticles are:

$$\begin{gathered} c_p\left(0\right)=c_0-c_f \\ {c}_{\mathit{\text{DBLa}}}\left(0\right)=c_f \\ {c}_o\left(0\right)=0 \end{gathered}$$

By solving Eqs. 13, 15, 17 and 18 simultaneously, using the initial conditions listed above, $c_p$, $c_{\mathit{\text{DBLa}}}$, and $c_o$ can be resolved as a function of time.

MATERIALS

Model drugs anacetrapib (ACP), atazanavir (ATZ), enzalutamide (ENZ), and felodipine (FLP) were purchased from ChemShuttle (Wuxi, China). Clotrimazole (CTZ) was purchased from Sigma Aldrich (St. Louis, MO).

High-performance liquid chromatography (HPLC) grade dimethyl sulfoxide (DMSO) and 1-decanol (98+%), were purchased from Alfa Aesar (Ward Hills, MA). HPLC grade acetonitrile (ACN) was purchased from Fisher Chemical (Fair Lawn, NJ). Sodium phosphate monobasic monohydrate and sodium phosphate dibasic dihydrate were obtained from Sigma-Aldrich (St. Louis, MO). Hydroxypropyl methylcellulose acetate succinate (HPMCAS) HF grade was a kind gift from Shin-Etsu Chemical Co. Ltd. (Totowa, NJ).

In 1 L of water, 4.434 g of sodium phosphate monobasic monohydrate and 3.186 g of sodium phosphate dibasic dihydrate were dissolved to prepare the 50 mM pH 6.5 phosphate buffer needed for solubility and biphasic diffusion tests. Using a 3 M HCl solution, the solution’s final pH was brought to pH 6.5. A Micropure UV water purification system from Thermo Fisher Scientific (Waltham, USA) was used to produce reverse osmosis water ($18.5\text{M}\Omega$).

METHODS

Biphasic Diffusion Experiments

Experimental data from our previously published work showing unsaturated mass transport were used to train and validate our dissolution-permeation model³³.

Additionally, high particle concentration experiments showing saturated mass transport were carried out in this study to test the model. Amorphous nanoparticles of 300 nm, stabilized using $100\mu \text{g/mL}$ HPMCAS, were prepared using atazanavir, felodipine, and clotrimazole as model drugs at different concentrations to evaluate the impact of particle concentration on interfacial mass transport. The experimental setup for the biphasic setup is depicted in Figure 2. Briefly, a 100 mL water-jacketed beaker was used with 45 mL of 1-decanol and 45 mL of pH 6.5 buffer. An overhead stirrer with two impellers, each having three blades, was used to simultaneously stir both liquid layers at 90 rpm or 45 rpm. The distance from the liquid interface to both impeller blades were kept the same. For all the experiments, the stirrer blade position remained constant. A fiber optics dip probe connected to a 400 UV/Vis spectrometer (SI Photonics, Tucson, AZ) was used to measure the drug concentration in the organic phase. To capture the complete diffusional profile, all experiments were recorded until most of the drug partitioned in the decanol phase, or an asymptote was observed (near equilibrium). The drug concentration at each time point was calculated using a calibration curve that covered the detected concentration range.

Model Fitting

The family of differential equations defining our model (Eqs 13, 15, 17 and 18) were solved in MATLAB^® using the ode45 function that combines a fourth and fifth order Runge-Kutta (RK) method based on Dormand-Prince pair to solve differential equations numerically.³⁵ Experimentally obtained drug concentration in the decanol at each time point ($c_o$) was fitted to the numerical solution to obtain the overall interfacial permeability ($P_I$) and partition coefficient ($K$). An empirical equation reported previously³⁶ was used to calculate the drug’s aqueous diffusion coefficient: $\text{logD}=-4.131–0.453\text{log}{\text{M}}_{\text{w}}$. All parameters used in model fitting is shown in Table 1.

Table 1.

Parameters used for model fitting

Drug	Molecular Weight	Amorphous Solubility ($\mu \text{g/ml}$)	Density ($\mu \text{g/ml}$)	LogP	Diffusion Coefficient (cm2/sec)	$\gamma =\frac{3D_f}{\rho }\left({\text{cm}}^5/\text{h}/\mu \text{g}\right)$
Anacetrapib	637.51	0.133	116000037	9.238	3.9654E-06	3.69192E-08
Atazanavir	704.869	39.533	116000037	4.539	3.78898E-06	3.52767E-08
Clotrimazole	344.84	9.933	116000037	6.140	5.23848E-06	4.87721E-08
Enzalutamide	464.44	40.5433	116000037	2.9833	4.57735E-06	4.26167E-08
Felodipine	384.25	11.233	116000037	4.541	4.98783E-06	4.64384E-08

The particle drifting effect (PDE) can significantly increase the overall mass transport. This was reflected in the model as an increase in fitted permeability coefficient ($P_I$). The dimensionless Saturation Index of a drug is defined as the ratio of total aqueous drug concentration (colloid + molecularly dispersed drug) to the amorphous solubility of the drug ($c_f$).

For experiments with a saturation index below 3, where particle dissolution in the DBL occurred at near-sink conditions, literature logP values were used in the model, and only $P_I$ was fitted. This was the preferred method as it only leaves one unknown parameter ($P_I$) to be fitted. However, at high saturation indices, non-sink dissolution was observed, which led to the flux plateau and incomplete mass transfer observed. Drug particles dissolve in the DBL, leading to elevated free drug concentration in the aqueous solution near the aqueous-organic liquid interface. To account for this, a reduction in apparent partition coefficient K was needed. This was achieved through simultaneous fitting. Hence, for experiments with Saturation Indices higher than 3, both $P_I$ and $K$ were simultaneously fitted to the experimental data to account for the dynamic changes in $P_I$ and $K$ with increasing particle concentration.

Apparent Flux Calculation

To better visualize the particle drifting effect, apparent flux ($J_{app}$) values at each experimental condition were calculated using Fick’s First law of diffusion:

$$J_{app}=\frac{D_f}{h_{\mathit{\text{DBLa}}}}{c}_f=P_I{c}_f$$ 19

Model fitted ${\text{P}}_{\text{I}}$ values at each particle size and concentration were used to calculate apparent flux.

Predicting drug absorption in vivo

A direct differential equation based in vitro-in vivo correlation (IVIVC) method⁴² was used to predict drug absorption in vivo. It is a one-stage IVIVC method in which differential equations that make up an underlying pharmacokinetic model (such as a one-compartment or a two-compartment model) can be paired with an appropriate dissolution model and are directly solved using numerical methods.

Biphasic diffusion profiles and mouse plasma drug concentration data were used for model development and fitting. The biphasic diffusional profile at different doses was calculated using the mass transport model developed in this study, and pharmacokinetics data was summarized in a recent publication.⁴³ The permeability coefficient ${\text{P}}_{\text{I}}$ and partition coefficient K were extrapolated based on biphasic experimental data obtained using atazanavir given its similar amorphous solubility to enzalutamide. Mice intravenous (IV) data was fitted into a two-compartment pharmacokinetic model to establish pharmacokinetic parameters ($K_e,K_{12}$, and $K_{21}$), and drug absorption was predicted using the family of differential equations defined by Eqs 20, 21, 22, and 23.

$$\frac{d{C}_{p1}}{dt}=r\left(t\right)-K_e{C}_{p1}-K_{12}{C}_{p1}+K_{21}{C}_{p2}$$ 20

$$\frac{d{C}_{p2}}{dt}=K_{12}{C}_{p1}-K_{21}{C}_{p2}$$ 21

$$r_p\left(t\right)=\frac{d{C}_o}{dt}$$ 22

$$r\left(t\right)=S_r{S}_t{r}_p\left(S_t.t\right)$$ 23

Where $C_{p1}$ is the plasma drug concentration in mice, $C_{p2}$ is the drug concentration in the second compartment in vivo, $r\left(t\right)$ is the rate of drug absorbed into systemic circulation, $K_e$ is the elimination rate constant in vivo, $K_{12}$ is the rate constant for drug transport from first to second compartment in vivo, $K_{21}$ is the rate constant for drug transport from second to first compartment in vivo, ${\text{r}}_{\text{p}}\left(t\right)$ is the overall rate of drug permeation into decanol phase, $S_r$ is the amplitude scaling factor, and $S_t$ is the time scaling factor. Here we assume there is no delay in drug absorption in vivo.

Pharmacokinetic experimental data from another work from our group⁴³ was utilized. All animal experiments were approved by the Institutional Animal Care and Use Committee (IACUC) at the University of Connecticut under protocol numbers A22-046 and A23-031. Plasma drug concentration profile obtained after intravenous injection of enzalutamide in mice was subjected to two compartment analysis using WinNonlin (Certara, St. Louis, MI) to determine the elimination rate constant $K_e$ and compartmental rate transfer coefficients $K_{12}$ and $K_{21}$.

The coefficients $S_r$ and $S_t$ in equation 23 are the amplitude and time scaling factor that allows us to appropriately scale the drug permeation rate into decanol to the rate of drug absorption in vivo.

The simulated output from our biphasic model ($\frac{d{C}_o}{dt}$) was appropriately scaled to $\text{time}=S_t.t$ using shape-preserving piecewise cubic interpolation function in MATLAB^®. $S_r$ and $S_t$ were determined by globally fitting the plasma drug concentration obtained after oral administration of enzalutamide formulation containing only the free drug (absence of any particles) to the family of differential equations consisting of equation 13, 15, 17, 18, 20, 21, 22 and 23, at both 90 rpm and 45 rpm. The amplitude scaling factor $S_r$ scales the overall amplitude of drug absorption between the in vitro (biphasic) and in vivo data. Thus, it was kept constant across different stirring rates. The time scaling factor $S_t$ scales the time-dependent rate of mass transport between the in vitro (biphasic) and in vivo data, which is inversely proportional to the thickness of the unstirred water layer formed in the in vitro experiments. The ratio of the unstirred water layer thickness at 45 rpm versus 90 rpm was determined to be 1.8779. Hence, the following constraints were applied during global fitting: $S_{r90}=S_{r45}$ and $S_{t45}=S_{t90}\times 1.8779$. The $S_r$ and $S_t$ values obtained through global fitting of all six sets of free drug data (0.5, 1, and 2 mg/kg doses for each rpm) was then used as model input to the same equations to predict the drug absorption of formulation containing nanoparticles (higher doses).

The in vivo amorphous solubility of enzalutamide measured⁴³ was greater than its intrinsic aqueous solubility. Therefore, in vitro and in vivo particle concentrations were calculated using total enzalutamide concentration subtracting its corresponding amorphous solubility at each condition. The total flux was then calculated by adding the free drug and nanoparticle contributions together, considering the amorphous solubility of enzalutamide in vivo.

RESULTS

Free drug diffusion

The free drug diffusion data for each drug, except for anacetrapib due to experimental difficulty to detect its extremely low solubility, was fitted into the model to determine the interfacial permeability coefficient $P_I$ of the free drug.

Although the solubility of drugs tested ranges from $9\text{to}40\mu \text{g/ml}$, the permeability coefficients were similar (Figure 3). Such results confirm that the DBL is the rate-limiting barrier to mass transport for free drug diffusion for these poorly soluble drugs. These permeability values correspond to a DBL thickness between $44–55\mu \text{m}$ at 90 rpm and $83–91\mu \text{m}$ at 45 rpm.

Figure 3.

Fitted permeability ($P_I$) values for the free drug

The Particle Drifting Effect

Impact of particle concentration

Amorphous nanoparticles of 300 nm were prepared at different concentrations to evaluate the impact of particle concentration on the particle drifting effect. Atazanavir, clotrimazole, and felodipine were used as model drugs. As shown in Figure 4, excellent agreements were observed between model fitting and experimental data at conditions with both high and low Saturation Indices. The enhancement of mass transport in the presence of particles, driven by the particle drifting effect at different particle concentration, is also evident in the increasing apparent flux (${\text{J}}_{\text{app}}$) observed as particle concentration rises (Figure 5A).

Figure 4.

Impact of particle concentration on PDE: representative diffusional profiles showing good agreements between model fitting and experimental data (90 rpm).

Figure 5:

Impact of total drug concentration on apparent flux, permeability and fitted partition coefficient (90 rpm).

At low particle concentrations, all three drugs demonstrated an overall permeability coefficient (${\text{P}}_{\text{I}}$) close to the free drug permeability coefficient, indicating minimal impact of the particle drifting effect on overall mass transport.

However, at high particle concentrations, all three drugs demonstrated significant increases in ${\text{P}}_{\text{I}}$ compared to the free drug. Moreover, drugs with higher amorphous solubility (${\text{S}}_{\text{a}}$) exhibited higher ${\text{P}}_{\text{I}}$ values. For instance, at a Saturation Indices of approximately 20, ${\text{P}}_{\text{I}}$ values for atazanavir (${\text{S}}_{\text{a}}:39.5\mu \text{g}/\text{mL}$), felodipine (${\text{S}}_{\text{a}}:11.2\mu \text{g}/\text{mL}$), and clotrimazole (${\text{S}}_{\text{a}}:9.9\mu \text{g}/\text{mL}$) were 17.6 cm/hr, 10.58 cm/hr, and 8.51 cm/hr, respectively, while that for the free drug was approximately 3 cm/hr. Additionally, at high particle concentrations, drug transport to the organic phase slowed down eventually, reaching a plateau with concentrations much lower than the total drug concentration (Figure 4). At the end of the experiment, drug nanoparticles were still visibly present in the aqueous phase. As shown in the model calculations, particle concentrations gradually declined to an asymptote. The asymptotic value increased with increasing particle concentration. The calculated free drug concentration also remained constant as a function of time, indicating saturation of the bulk solution in the presence of amorphous nanoparticles.

At conditions where no or low numbers of particles were present, the partition coefficient K used was calculated from the logP of the model drug and remained unchanged. However, at high particle concentrations, due to saturated mass transport, the apparent partitioning behavior of the drug was altered. The fitted partition coefficient K was plotted as a function of particle concentration as shown in Figure 5C. Compared to the drug’s theoretical partition coefficient P, the K value decreased remarkably. This was due to the saturation of the drug in the aqueous DBL adjacent to the aqueous-organic interface due to the particle drifting effect. In this case, the interface and organic DBL resistance became the rate limiting step. As particle concentration continue to increase, K value showed an increasing trend (Figure 5C).

Impact of particle size

For model drugs atazanavir, enzalutamide, and anacetrapib, amorphous drug nanoparticles with varying sizes were used to evaluate the impact of particle size on the particle drifting effect. For all particle size experiments, model fitting showed good agreement with experimental data (Figure 6). For atazanavir and enzalutamide, the effect of particle size on apparent flux was insignificant. However, for anacetrapib, increasing particle size negatively impacted apparent flux (Figure 7A).

Figure 6.

Impact of particle size on PDE: representative diffusional profiles showing good agreements between model fitting and experimental data (90 rpm).

Figure 7 :

Impact of particle size on apparent flux, permeability and fitted partition coefficient (90 rpm).

The total drug concentration of atazanavir used in the particle size experiments was $100\mu \text{g/ml}$, corresponding to a Saturation Index of approximately 2.53. At this concentration, all particle size experiments demonstrated complete drug diffusion into the decanol phase. Unlike higher Saturation Index experiments, a flux plateau was not observed in this case. The fitted permeability coefficient (${\text{P}}_{\text{I}}~3.5$) was very close to that of the free drug (3.09 ± 0.07), indicating minor PDE (Figure 6). Particle size did not have significant impact on $P_I$ values. Model calculations also confirmed complete particle dissolution and a subsequent decline in bulk free drug concentration.

For enzalutamide, the total drug concentration used was $140\mu \text{g/ml}$. This concentration corresponds to a Saturation Index of 3.45. Under this condition, the flux plateau was not observed. However, the fitted permeability coefficient (${\text{P}}_{\text{I}}~5.8$) was slightly higher than that of the free drug (3.25 ± 0.08). Similar to atazanavir, the permeability coefficient was unaltered by particle size variations. Complete particle dissolution and a subsequent decline in bulk free drug concentration was confirmed by the model.

The total concentration of anacetrapib used was $100\mu \text{g/ml}$, corresponding to a Saturation Index greater than 100. At this condition, amorphous nanoparticles of all sizes demonstrated incomplete dissolution. The fitted permeability coefficients were significantly higher than that of the free drug. Increasing particle size led to decreased ${\text{P}}_{\text{I}}$, whereas the fitted partition coefficient K remained unaffected by particle size (Figure 7C). The model predicted bulk free drug concentration remained constant, and particle concentration graduate decreased without complete dissolution even with extended time. As particle size increased, the rate and extent of such decline decreased (Figure 6).

DISCUSSION

Improbable possibility of direct particle transfer

Two possible mechanisms could facilitate solute transfer from drug particles in the aqueous solution into the decanol phase: (1) particle dissolution followed by free drug diffusion across the liquid interface, where mass transfer occurs only through the dissolved free drug; and (2) direct particle migration into the decanol phase. The authors acknowledge that mechanism II could occur under specific conditions, such as excessively vigorous stirring or inadequate particle wetting. However, for the system tested in this study, mechanism I appears to be the predominant, if not exclusive, mass transport mechanism.

In our previous study where identical experimental conditions were employed³³, we reported photos and video clips showing clear liquid interfaces and the absence of particle transfer using a pink dye and 300 nm silica nanoparticles, along with computational fluid dynamics simulation results. The observed solubility-dependent particle drifting effect also conformed with mechanism I, dissolution-driven solute transfer³³. In the present study, we further evaluated the impact of particle concentration on the particle drifting effect. The saturation in mass transport and flux plateau observed at high particle concentrations also conform with dissolution-driven solute transfer. If direct particle migrations were indeed the major mass transport mechanism, solute transfer should consistently follow first-order kinetics without any plateau (zero-order kinetics). Additionally, since the developed mass transport model is based on particle dissolution, direct particle migration across the interface would lead to model failure. The excellent in vitro-in vivo relationships obtained between biphasic and mice data summarized in another study from our group⁴³, along with discussions provided in the flux plateau sections below, also confirms dissolution-driven solute transfer in vivo. Therefore, for amorphous drug nanoparticles, the biphasic diffusion experimental setup appears to be an appropriate and useful tool to evaluate drug absorption under proper stirring conditions.

Flux plateau and non-sink dissolution

Analysis of the experimental data revealed two distinct types of drug transport from the aqueous phase to the organic phase. The first type was observed at low particle concentrations, where nearly all the drug in the aqueous phase, including the molecularly dissolved drug and drug nanoparticles, was transported to the decanol phase. The diffusional profiles resembled that of a single-phase process. The second type was observed at higher particle concentrations, where the drug transport process remained incomplete even after extended runtimes. Under these conditions, nanoparticles remained in the aqueous buffer at the end of the experiment. In these cases, the diffusional profiles appeared to be “biphasic”: Initial mass transfer from the aqueous phase to the organic phase was rapid. However, a second phase was observed in the diffusional profiles, where the mass transfer rate declined significantly. Solute transport continued at a very slow rate, appearing to be a “plateau” approaching an asymptote.

Such two-stage diffusion profile was also reported elsewhere. However, it was modeled using a simplified first order rate constant⁴⁴ assuming that the slower reaction was also attributed to partitioning. Through our previous investigations³³, we revealed the particle drifting effect to be a sequential reaction consisting of particle dissolution and free drug permeation. Initially, particle dissolution is rapid, and the particle drifting effect is limited by free drug permeation (diffusion) across the aqueous DBL adjacent to the liquid interface. Such rapid solute transfer constitutes the initial phase in the “biphasic” diffusional profiles. As more particles dissolve in the aqueous DBL, the free drug concentration in the DBL quickly increases. As shown in Eq.9, an increase in $c_{\mathit{\text{DBLa}}}$ would reduce particle dissolution rate. As $c_{\mathit{\text{DBLa}}}$ continues to increase, particle dissolution rate continues to decrease, leading to reduced solute transport with time, appearing as the “flux plateau”. In this case, particle dissolution at non-sink conditions becomes the slow reaction, dominating the particle drifting effect.

Violation of Model Assumptions

At high particle concentrations, we observed change in the drug’s partition coefficient K and interfacial permeability coefficient ${\text{P}}_{\text{I}}$ in model fitting. Of course, the drug’s physicochemical properties should remain the same. Therefore, these changes were attributed to violations of several model assumptions with details discussed below.

As shown in Figure 2, four steps are involved in the transfer of drug from amorphous nanoparticles to the bulk decanol phase. The first step is the mass transport from the particle surface to the aqueous boundary layer surrounding each particle. This is dependent on particle radius (or surface area) and the concentration gradient across the particle’s diffusion boundary layer, as defined in equation 13. After crossing the particle’s diffusion boundary layer, the dissolved drug must then diffuse across the aqueous DBL (described by $P{e}_a$) near the decanol-water interface, the aqueous-organic interface (described by $P{e}_i$), and the organic DBL (described by $P{e}_o$). The model developed in this work assumes that the partition of the drug between water and decanol is instantaneous and independent of the total drug concentration in the aqueous solution. Additionally, for poorly soluble drugs, the diffusional resistance across the organic DBL is also assumed to be negligible.³⁴ In this case, diffusion across the aqueous DBL $\left(P{e}_a\right)$ becomes the rate determining step of the overall reaction.

The fitted permeability coefficient at each particle concentration was plotted as a function of Saturation Index in the donor (aqueous) solution (Figure 8). For experiments at low particle concentrations (Saturation Index < 3 in our experiments), PDE only led to slight increase in overall drug permeation rate across the aqueous DBL. This was observed in the fitted permeability coefficients (${\text{P}}_{\text{I}}$) for atazanavir and enzalutamide, both of which were slightly higher than the free drugs’ ${\text{P}}_{\text{I}}$. Under these conditions, interfacial permeation ($P{e}_i$) and mass transport across the organic DBL ($P{e}_o$) are much faster to solute transport across the aqueous DBL ($P{e}_a$). In this case, particle dissolution was not the rate limiting step, and no flux plateau was observed.

Figure 8.

Permeability coefficient as a function of Saturation Index. A) 90 rpm B) 45 rpm

At high particle concentrations (Saturation Index ⩾ 3 in our investigations), PDE led to significant increases in $P{e}_a$ as demonstrated by the increase in fitted ${\text{P}}_{\text{I}}$ values (Figure 8). Experimentally, this was reflected as a flux plateau in the diffusional profile, indicating the rate limiting step being particle dissolution. In the model, this was also reflected as a decrease in the partition coefficient K of the drug as shown in Figure 5C. In this case, PDE increased drug permeation across the aqueous DBL to such an extent that it was no longer the rate-determining step ($P{e}_a\gg P{e}_i$ or $P{e}_o$). Therefore, interfacial resistance and/or the organic DBL resistance became significant in overall mass transport. In many biological and pharmaceutical systems, interfacial drug transport was found to require high free energy input and can be the rate-limiting step⁴⁵. Also, as the aqueous DBL resistance becomes negligible, the organic DBL resistance may become the major diffusion barrier. In both cases, Equation 5 cannot be approximated from Equation 4, as $P{e}_i$ and $P{e}_o$ can significantly impact overall flux. Therefore, at high particle concentrations, these assumptions are violated. In the model, this was reflected as a reduction in the partitioning coefficient K, where the drug concentration in the aqueous DBL was increased due to particle dissolution, but the drug concentration in the decanol phase was not immediately increased to the same proportion.

Nonlinear nature of PDE

For the free drug, the overall mass transport rate across the liquid interface is dictated by the thickness of the aqueous DBL. In the presence of particles, the overall interfacial permeability coefficient ${\text{P}}_{\text{I}}$ of the drug increases due to PDE.

Limited by particle dissolution in the saturable aqueous boundary layer, PDE is also saturable. The extent to which drug particles saturate PDE depends on both particle concentration and the solubility of the drug that makes up the particle. Therefore, compared to drugs with low solubility such as clotrimazole, higher solubility drugs such as atazanavir require higher particle concentration to saturate the flux provided by PDE. This effect is normalized by converting the total drug concentration to Saturation Index. However, particle dissolution rate is still dependent on the solubility of the drug. Hence, at a particular Saturation Index, drug particles with higher solubility have faster dissolution rates, leading to higher ${\text{P}}_{\text{I}}$ values observed (Figure 8).

At conditions with low Saturation Indices (<3), particle concentration in the aqueous DBL is not high enough to cause a significant increase in flux due to PDE. Hence, we observed negligible increase in the permeability coefficient ${\text{P}}_{\text{I}}$ for all three drugs. However, as the Saturation Index increases, particle dissolution in the aqueous DBL increases significantly, thereby increasing the overall flux due to PDE. This increase is reflected as increase in the permeability coefficient of all three drugs as shown in Figure 5b.

At high particle concentrations, particle dissolution in the aqueous DBL slows down due to non-sink dissolution. Under these conditions, the observed overall mass transport is slow, suggesting that particle dissolution rate is saturated. Due to this limit, further increase in particle concentration would simply increase the number of dissolving particles but not increase the overall mass transport rate significantly. Hence, the rate of change in the overall particle dissolution rate with increasing particle concentration becomes inversely proportional to the number of particles in the aqueous DBL. Since particle dissolution provides the source of permeability (${\text{P}}_{\text{I}}$) increase due to PDE, and the number of particles is linearly proportional to the Saturation Index of the system, mathematically, such an inversely proportional relationship can be described as:

$$\frac{dy}{dx}\propto \frac{1}{x}$$ 24

$$y\propto \mathit{\text{ln}}\left(x\right)+c$$ 25

where y is the rate of mass transport provided by particle dissolution, and x is the Saturation Index.

By integrating equation Eq. 24, we obtain Eq. 25, which suggests a log-linear dependence of permeability (${\text{P}}_{\text{I}}$) on the Saturation Index. Indeed, such a log-linear relationship was observed in Figure 8 at Saturation Indices higher than 3, where particle dissolution becomes the rate limiting step in PDE. For enzalutamide, biphasic experiments were conducted with varying particle sizes at a single concentration. Therefore, only one enzalutamide data point is shown in Figure 8A. Additionally, we performed experiments at a different stirring rate (45 rpm), and similar log-linear relationships were obtained (Figure 8B).

Bioavailability prediction

Experimentally, it may be difficult to determine flux at these conditions due to saturated UV signal at high solute concentrations, or slow and incomplete solute transport in the decanol phase due to extremely low drug solubility. Therefore, the log-linear relationship between permeability coefficient ${\text{P}}_{\text{I}}$ and saturation index can be used to predict the particle drifting effect at conditions where PDE is limited by particle dissolution. Combined with pharmacokinetic models, the mass transport model developed in this work can then be used to predict drug absorption in vivo. An example is provided below.

Amorphous nanoparticles-releasing amorphous solid dispersions of enzalutamide were dosed to mice, and the pharmacokinetics data was summarized in another publication.⁴³ Given the low bile salt concentration in the mouse GI, and that the amorphous solubility of enzalutamide remained nearly unchanged at this condition, this system was treated as a binary system containing only the free drug and amorphous drug nanoparticles. Since enzalutamide has similar amorphous solubility as atazanavir, the model developed using atazanavir data from this work was used for predictions. The validity of this approach was also confirmed by the agreement of between enzalutamide and atazanavir data shown in Figure 8. Also, since the data in Figure 7 suggested no particle size impact on PDE for atazanavir, the size of amorphous particles generated from the enzalutamide formulation in vivo was not monitored.

Here in this example, the mass transport model was used to calculate the diffusional profile of enzalutamide at different doses. This was then used as model input for the pharmacokinetic model. The free drug data (0.5, 1, and 2 mg/kg), as shown in Figure 10A, B, and C, were fitted to obtain the amplitude and time scaling factors ${\text{S}}_{\text{r}}$ and ${\text{S}}_{\text{t}}$. They were then used in the model to predict drug absorption at all other doses where amorphous drug nanoparticles were formed. Due to the limited plasma sampling capacity of mice, four mice were used to construct each plasma drug concentration profile. Consequently, the in vivo data exhibited significant variability. However, visual observation reveals excellent agreements between the predicted (both 45 rpm and 90 rpm) and observed plasma drug concentrations across all doses (Figure 10D–F, J–L), confirming the potential of the proposed model to establish IVIVC for nanoparticle-containing formulations that exhibit PDE.

Figure 10.

Predicted vs experimental plasma drug concentration after oral administration of enzalutamide amorphous solid dispersions in mice using in vitro data obtained at (A-F) 90 rpm, and (G-L) 45 rpm

Impact of stirring rate and flux plateau

Since the particle drifting effect occurs in the aqueous DBL, the magnitude of flux enhancement should be sensitive to the thickness of the DBL. A thicker DBL would result in slower solute diffusion and lower flux. In this case, higher flux enhancement would be expected from the particle drifting effect as it circumvents the diffusional resistance of the aqueous DBL. The thickness of the DBL, or unstirred water layer, in humans and conscious animals, was reported to be in the range of approximately $30–100\mu \text{m}$^46–49. Therefore, we performed in vitro experiments at different stirring rates. The unstirred water layer thickness at 90 rpm and 45 rpm in the biphasic setup we used was calculated to be $44–55\mu \text{m}$ and $83–91\mu \text{m}$, respectively, conforming with the unstirred water layer thickness in vivo. From a drug absorption rate perspective, 90 rpm appeared to be the more appropriate condition as the time scaling factor obtained is close to 1.

Within the physiologically relevant DBL thickness range, satisfactory predictions were obtained at both rpms, confirming the robustness of the biphasic assay. At 45 rpm, we observed a higher extent of particle drifting effect for enzalutamide, but no significant changes for felodipine (Figure 9). We attributed this to the different mass transport mechanisms. The particle drifting effect is considered a sequential reaction of particle dissolution followed by free drug permeation. Therefore, the slower reaction dictates the rate of the overall reaction. For enzalutamide, free drug permeation across the aqueous DBL dominated the overall reaction due to the drug’s moderately high amorphous solubility³³, whereas felodipine demonstrated dissolution-limited particle drifting effect. By changing the stirring rate from 90 to 45 rpm in this case, the particle drifting effect remained permeation-limited and dissolution-limited for enzalutamide and felodipine, respectively. Therefore, changing the unstirred water layer thickness would only impact the particle drifting effect for enzalutamide but not felodipine. If the mass transport mechanism shifts from permeation-limited to dissolution-limited, altered particle drifting effect is also expected, and further investigations are warranted to demonstrate this experimentally.

Figure 9:

Normalized apparent flux at different stirring rates. A) atazanavir shows increased PDE at lower rpm, and B) felodipine does not show any change in PDE at different stirring rates

The DBL thickness also impacted the observed flux plateau. An example is shown in Figure 11. Within the free drug concentration ranges, flux follows Fick’s first law of diffusion and is inversely proportional to DBL thickness. At the stirring rate decreases, diffusion slows, but the shape of the diffusional profile remains unchanged. Therefore, when the time axis is appropriately scaled, diffusional profiles obtained at different stirring rates superimpose (Figure 11B). However, at high drug concentrations, where a flux plateau occurs, changes in stirring rate affect both the diffusion rate and the shape of the diffusional profile. Despite different stirring rates, the time required to achieve this plateau was similar (Figure 11C). This similarity arises because the particle concentration within the DBL remains constant at a given drug concentration, resulting in a consistent time duration needed to saturate the aqueous DBL. However, the plateau level was lower at slower stirring rates, possibly due to a thickened DBL in the decanol phase. In this case, applying the same time-scaling does not give overlapping diffusional profiles (Figure 11D). Despite such differences, bioavailability predictions using diffusional profiles obtained at both stirring conditions resulted in satisfactory results (Figure 10).

Figure 11:

The impact of stirring rate on flux plateau: A and B) actual vs time-scale adjusted atazanavir free drug permeation profiles; C and D) actual vs time-scale adjusted atazanavir diffusional profiles in the presence of amorphous drug nanoparticles

We also performed model simulations without considering the flux plateau. As shown in Figure S1, without the flux plateau, the model over predicted drug absorption at high doses, particularly at 10 and 15 mg/kg. These results suggested saturated mass transport and the occurrence of flux plateau in vivo, conforming with the dissolution-driven mass transport mechanism. Thus, it is critically important to account for incomplete mass transport and the resulting flux plateau in bioavailability predictions involving the particle drifting effect, particularly at doses with high particle concentrations.

Model limitations

While the presented model offers mathematical equations to explain and predict the particle drifting effect at different drug concentration, its applicability to predict bioavailability enhancement through PDE is subject to several limitations.

A primary limitation is the model’s dependence on the drug’s in vivo solubility. In the case of enzalutamide, more than two-fold increase in its amorphous solubility was observed in vivo. However, it is not always feasible to perform dedicated animal studies or human clinical trials to estimate in vivo amorphous solubility. For drugs with moderately high amorphous solubility such as enzalutamide, inaccurate estimation of this phase boundary would result in significant errors in model prediction. For drugs with extremely low solubility where the free drug contribution is negligible, this may be less problematic.

As stated in the previous section, the thickness of the unstirred water layer could impact on the extent of particle drifting effect observed. While we demonstrated the robustness of the biphasic assay for drugs with the same mass transport mechanisms, both dissolution-limited (felodipine) and permeation-limited (enzalutamide), drugs showing shifted mass transport mechanisms may exhibit altered particle drifting effect.

Currently, this model remains semi-empirical as both permeability coefficient $P_I$ and partition coefficient $K$ were fitted using experimental data for drugs with different solubility. Further investigations are needed to survey more model drugs to understand the dynamic changes in $P_I$ and $K$ and achieve apriori prediction.

CONCLUSION

This study presents the theory and derivation of a mechanistic dissolution-permeation model for solute transport provided by amorphous drug nanoparticles in a biphasic setup. Experimental data and mass transport analysis revealed saturated mass transport at high particle concentrations, due to non-sink dissolution of the particles in the aqueous diffusion boundary layer. Fitted drug partition coefficient also showed a large decline at high particle concentrations, confirming saturation of the drug in the aqueous boundary layer. Flux enhancement provided by the particle drifting effect increases nonlinearly with increasing drug concentration. The model revealed a log-linear relationship between permeability and saturation index, which can be utilized for flux predictions. Combined with pharmacokinetic models, it was demonstrated that the biphasic experimental setup combined with the mass transport model may be appropriate tools to predict oral drug absorption in vivo.

DATA SHARING

All raw data used in this work are available upon request.