Impact of Molecular Partitioning and Partial Equilibration on the

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The impact of molecular partitioning and partial equilibration on the estimation of diffusion coefficients from release experiments Reza Ghanbari, Salvatore Assenza, Patrick Züblin, and Raffaele Mezzenga Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.9b00510 • Publication Date (Web): 31 Mar 2019 Downloaded from http://pubs.acs.org on April 6, 2019

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The impact of molecular partitioning and partial equilibration on the estimation of diffusion coefficients from release experiments

Reza Ghanbari1†, Salvatore Assenza1†, Patrick Zueblin1, Raffaele Mezzenga1,2,* 1ETH

Zurich, Food and Soft Materials Science, Department of Health Science &

Technology, Institute of Food, Nutrition & Health, Schmelzbergstrasse 9, CH-8092 Zurich, Switzerland 2ETH

Zurich, Department of Materials, Wolfgang-Pauli-Strasse 10, 8093 Zurich, Switzerland

ACS Paragon Plus Environment

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ABSTRACT The present work addresses the effect of partial equilibration and molecular partitioning on the interpretation of release experiments. In this regard, it is shown how release profiles and the values of extracted transport parameters are affected by the time protocol chosen for sample collection by considering a series of experiments where the latter is systematically varied. Caffeine is investigated as a main model drug due to its similar affinity for water and lipids, while monolinolein-based lipid cubic phases are chosen as host matrices due to their wide employment in release studies. Our findings point to a progressive decline in diffusion rate upon increasing the time step, i.e. the gap in time between two consecutive pick-ups, which is a signature of increasing equilibration of caffeine concentration between the lipidic mesophase and the water phase. Furthermore, the amount of released molecules at the first pickup displays negligible changes for large time steps, indicating complete equilibration in such cases. A model is introduced based on Fick’s diffusion which goes beyond the assumption of perfect-sink conditions, a common feature of the typical theoretical approaches hitherto developed. The model is shown to account quantitatively for the experimental data, and is subsequently employed to clarify the interplay of the adopted release protocol with the various transport parameters in determining the final outcome of the release process. Particularly, two additional molecular drugs are considered, namely glucose and proflavine, which are respectively more hydrophilic and hydrophobic than caffeine, thus allowing elucidating the role of molecular partitioning.


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INTRODUCTION Lipid mesophases are a class of versatile lyotropic liquid crystals (LLCs) which have found growing use in numerous research and applied science areas, e.g. drug delivery, protein crystallization, membrane technology, material templating or complex food systems 1-12. The selforganization of the amphiphilic lipid molecules in presence of water drives the formation of a myriad of distinct structures as a function of the physico-chemical attributes of the environment and of the molecular structure of the lipids 8, 13-16. Being a mixture of lipids and water, the resulting mesophases can entrap molecules with different amphiphilicity. These characteristics have made them suitable candidates for drug delivery studies, where the release kinetics can be controlled by tuning the details of the system, particularly the geometrical details of the mesophase and the interactions of the drug with the LLC matrix

2, 6, 15, 17-25.

Among lipid mesophases, inverse

bicontinuous cubic phases (IBCPs) are especially suitable for drug-delivery purposes, as fast release is achieved thanks to their three-dimensional periodic porous network

25.

They are

characterized by a single lipid bilayer, which is arranged according to a minimal surface dividing the space into two congruent regions, where interwoven but non-communicating water channels are formed. The most frequent IBCPs are 𝐼𝑎3𝑑, 𝑃𝑛3𝑚 and 𝐼𝑚3𝑚, alternatively named as gyroid, diamond and primitive symmetries, which are distinguished from each other by their symmetrical features and corresponding connectivities of their labyrinthine water channels (3, 4 and 6, respectively) 6, 15, 20, 22, 26-29. The drug release from the mesophase is usually modelled assuming an underlying Fickian diffusion. One of the most common models is the Higuchi equation, which describes the release dynamics at its early stage

30-37.

A more refined approach based on the same physical picture

considers the whole time evolution of the release process

22, 31, 32.

Nonetheless, in several cases


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anomalous release profiles are observed, for which variations to the Higuchi equation have been employed for analysis 33-35, 37. For instance, the partitioning dynamics of the diffusing drug between the lipid and water phases in a LLC system has been evidenced to play a significant role in determining the release behavior 26, 38, 39. Strikingly, no attention has been dedicated to a similar partitioning phenomenon taking place in virtually all release experiments, which may partially be responsible for the observed anomalies. Indeed, the typical release protocol prescribes the periodic substitution of the content of the receiving chamber by fresh, drug-free solution. In this way, perfect-sink conditions are mimicked, which are also of direct relevance for applications in vivo. The various models implemented to interpret the release kinetics assume that the perfect-sink condition holds. Nevertheless, the finite amount of time between consecutive pick-ups gives space to the system for partial equilibration, achieved by molecular exchange between mesophase and water in both directions. The extent of such equilibration is expected to depend on the molecular details of the drug as well as its interaction with the host LLC matrix. To date, no study has systematically inspected neither the influence of the time window of pick-ups nor its interplay with the molecular features of the drug. This work fills this gap by examining the transport of caffeine as an amphiphilic model drug across a mesophase with 𝑃𝑛3𝑚 symmetry. Caffeine is often used as a model drug with stimulant effects on the central nervous system 39, 40. Its molecular structure makes it an interesting candidate due to active interactions with the components of lipid-based mesophases (Fig. 1a)

26, 37, 41.

The

dynamics of the release is studied as a function of the timing of pick-ups, highlighting the significant impact of this aspect onto the release kinetics. Particularly, a slower release rate is observed upon decreasing the rate of pick-ups. A model is introduced which is still based on Fickian diffusion, but drops the assumption of perfect-sink conditions, and is benchmarked against


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experimental data, showing excellent quantitative agreement despite a minimal tuning of parameters. The model is further employed to assess the impact of pick-up times on other model drug molecules, namely glucose and proflavine, characterized by high hydrophilicity and hydrophobicity, respectively. Our results indicate the relevance of the choice of pick-up times in determining the apparent release behaviour observed in experiments, and provide a robust methodology to extract meaningful information for the development of related applications.

MATERIALS AND EXPERIMENTAL METHODS Materials Dimodan U/J was used for preparing the mesophases as explained below. Dimodan is the commercial name of Monolinolein. It was received as a gift from Danisco (Copenhagen, Denmark) and contains more than 98 wt-% monoglycerides. Figure 1b illustrates the chemical structure of monolinolein. Caffeine, technically named 1,3,7-trimethylxanthine, is an organic molecule. Anhydrous caffeine was purchased from Sigma-Aldrich (Schnelldorf, Germany). Figure 1a displays the chemical structure of caffeine. Milli-Q grade water was used for all experimental setups.

Preparation of Caffeine solution and Mesophases Caffeine was dissolved in Milli-Q grade water at concentrattions of 0.5 and 1.0 𝑚𝑔 𝑚𝑙. In order to form a 𝑃𝑛3𝑚 cubic phase, a mixture containing 33 wt% caffeine solution and 67 wt% Dimodan was vortexed in Pyrex tubes. Homogeneization was achieved by centrifugation at a centrifugal force (RCF) of 1293 g for 45 min and 37 °C. Then, the mesophase was further equilibrated for 72 hours at 37 °C.


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Release Experiment For the release setup, the caffeine was embedded in the mesophase and the mixture was prepared as mentioned above. The mesophase was then transferred into a cylindrical tube and held in place by two long Teflon pistons. The tubes were then placed in the refrigerator overnight, enabling to remove the pistons without damaging the surface of the mesophase along with getting a disk of LLC with a thickness of 6.0 ± 0.1 mm. Both sides of the mesophase were then filled with caffeine solution and were let to equilibrate at 37 °C for three days. This equilibration step was confirmed by Small-angle X-ray Scattering measurements after 3 and 4 days to ensure a consistency in the symmetry of the mesophase. After equilibration, the two caffeine solutions were replaced with water. Following the time sequences prescribed by the chosen protocol (as detailed in the main text), both sides of the tubes were periodically replaced with MQ water (Fig. 2a). The tubes were maintained at 37 °C during the experiments.

Small-angle X-ray Scattering Small-angle x-ray scattering (SAXS) was used to identify the symmetry of the mesophases and determine the characteristic lattice parameter before and after the release experiment as well as to check for the consistency during the preparation of the experimental setup. The lattice parameter of each crystalline structure was estimated using the data collected from SAXS profiles. SAXS diffractograms were acquired using a Bruker microfocused X-ray source of wavelength λ=1.54 Å operating at 50 kV and 1 mA. The diffracted X-rays signal was collected on a gas-filled twodimensional detector. The scattering vector 𝑞 =

4𝜋 𝜆 sin 𝜃,

with 2θ being the scattering angle, was

calibrated using silver behenate. The q-range of interest included the values from 0.004 to 0.5 Å−1. Later, data were acquired and azimuthally averaged using the Saxs GUI software to yield onedimensional intensity versus scattering vector q. Samples were loaded in the Linkam hot-stage


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between two thin mica sheets and sealed by an O-ring, with a sample thickness of ca 1 mm. All the measurements were done at 37°C. Samples were equilibrated for 30 min prior to measurements, whereas scattered intensity was collected over 0.5 h. As an example, figure S1 illustrates the SAXS profile for a IBCP with the Pn3m symmetry before and after the release experiment of caffeine. It can be observed that the symmetry stays unaltered after the release experiment. The lattice parameter 𝑎 can be computed for using the information attained from the SAXS measurements using the following relationship: 𝑞∗

𝑎=

2𝜋 𝑞∗

2

(1)

corresponds to the first peak in the SAXS spectra. 𝑎 = 8.8 nm was obtained, in agreement with previous reports on the same systems 15, 22, 32.

Ultraviolet-visible Spectroscopy The amount of caffeine released from the mesophase was measured via Ultraviolet-visible (UVvis) Spectroscopy technique. An UV−vis spectrometer Varian Cary 300, version 10.00 with a temperature controller was used. The Caffeine solutions were excited detected at 276 nm. The measurements were conducted with a quartz cuvette at room temperature.

Numerical resolution of the multiphase diffusion equation The numerical resolution of the diffusion equation describing the model introduced in the present work was obtained as follows. Let us label the left chamber, mesophase and right chamber as corresponding to 𝛼 = 1,2,3. Knowing the concentration profiles at a given time, 𝑐𝛼(𝑥,𝑡), 𝛼 = 1,2,3, enables computing the updated profiles at a subsequent time 𝑡 + 𝑑𝑡 (with no pick-up taking place between 𝑡 and 𝑡 + 𝑑𝑡), where 𝑑𝑡 is the integration step, by discretizing the set of partial differential equations. In this regard, a discretization space interval 𝑑𝑥 = 0.01 min (𝐿𝑐,𝐿𝑚) was considered,


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where 𝐿𝑐 and 𝐿𝑚 are the size of each chamber and the mesophase, respectively. As for the time step, 𝑑𝑡 = min (𝑑𝑡𝐴,𝑑𝑡𝐵) was chosen, where 𝑑𝑡𝐴 = 0.01 𝑑𝑥2/max (𝐷𝑐,𝐷𝑚) and 𝑑𝑡𝐵 = 0.01 Δ𝑡, with Δ𝑡 being the pick-up time (see main text), while 𝐷𝑐 and 𝐷𝑚 are the diffusion coefficients within the chamber and the mesophase, respectively. For any internal point 𝑥 within each chamber and the mesophase, the updating rule was 42 𝑐𝛼(𝑥 + 𝑑𝑥,𝑡) ― 2𝑐𝛼(𝑥,𝑡) + 𝑐𝛼(𝑥 ― 𝑑𝑥,𝑡) 𝑐𝛼(𝑥,𝑡 + 𝑑𝑡) = 𝑐𝛼(𝑥,𝑡) + 𝐷𝛼 𝑑𝑡 𝑑𝑥2

(2)

where 𝐷1 = 𝐷3 = 𝐷𝑐 and 𝐷2 = 𝐷𝑚. The initial conditions where enforced by setting 𝑐1(𝑥,0) = 𝑐3 (𝑥,0) = 0 and 𝑐2(𝑥,0) = 𝐶𝑖. The reflecting walls at the end of the receiving chambers were implemented by setting (after integration of the internal points was performed) 𝑐1 𝐿𝑚

(― (

𝐿𝑚 2

2

)

(

― 𝐿𝑐,𝑡 + 𝑑𝑡 = 𝑐1 ―

𝐿𝑚 2

)

(

and

― 𝐿𝑐 + 𝑑𝑥,𝑡 + 𝑑𝑡

𝑐3

𝐿𝑚 2

)

+ 𝐿𝑐,𝑡 + 𝑑𝑡 = 𝑐3

)

+ 𝐿𝑐 ― 𝑑𝑥,𝑡 + 𝑑𝑡 . The conditions at the interface were obtained by solving the set of algebraic

equations corresponding to their discretization. For instance, discretization of the conditions of local equilibrium and continuity of flux at the interface between the first chamber and the mesophase (Eq. 19 and Eq.21) leads to the following system:

{

(

𝑐1 ―

(

𝑐1 ― 𝐷1

𝐿𝑚 2

) (

,𝑡 + 𝑑𝑡 ― 𝑐1 ― 𝑑𝑥

𝐿𝑚 2

𝐿𝑚 2

) )

(

,𝑡 + 𝑑𝑡 = 𝑘 𝑐2 ―

(

― 𝑑𝑥,𝑡 + 𝑑𝑡

𝑐2

𝐿𝑚

)

,𝑡 + 𝑑𝑡 (3) 2 𝐿𝑚 𝐿𝑚 ― + 𝑑𝑥,𝑡 + 𝑑𝑡 ― 𝑐2 ― ,𝑡 + 𝑑𝑡 2 2

= 𝐷2

) (

) (4)

𝑑𝑥

where 𝑘 is the partition coefficient and denotes the ratio of the drug concentration in the mesophase and the receiving chamber. The algebraic resolution of this system leads to


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{

(

𝐿𝑚

𝑐1 ―

(

𝑐2 ―

𝐿𝑚 2

2

)

(

𝐷1 𝑐1 ―

)

𝐿𝑚

(

,𝑡 + 𝑑𝑡 = 𝑘 𝑐2 ― 𝐿𝑚 2

2

)

,𝑡 + 𝑑𝑡

)

(

(5)

― 𝑑𝑥,𝑡 + 𝑑𝑡 + 𝐷2 𝑐2 ―

,𝑡 + 𝑑𝑡 =

𝐿𝑚 2

)

+ 𝑑𝑥,𝑡 + 𝑑𝑡

(6)

𝑘 𝐷1 + 𝐷2

Analogously, the conditions for the interface between the mesophase and the second chamber were implemented as

{

𝐿𝑚

(

𝑐3

(

𝐿𝑚

𝑐2

2

)

2

)

𝐿𝑚

(

𝐷3 𝑐1

2

𝐿𝑚

(

,𝑡 + 𝑑𝑡 = 𝑘 𝑐2

2

) (

,𝑡 + 𝑑𝑡

)

+ 𝑑𝑥,𝑡 + 𝑑𝑡 + 𝐷2 𝑐2

,𝑡 + 𝑑𝑡 =

(7)

𝐿𝑚 2

)

― 𝑑𝑥,𝑡 + 𝑑𝑡

(8)

𝑘 𝐷3 + 𝐷2

Finally, if a pick-up was prescribed at a given time 𝑡, it was implemented before integration by imposing 𝑐1(𝑥,𝑡) = 𝑐3(𝑥,𝑡) = 0. Integration of the concentration in the chambers was beforehand performed to compute the amount of released molecules.

RESULTS AND DISCUSSION The cubic phases considered in our experiments are obtained as a mixture of monolinolein and caffeine solutions. The Small-angle X-ray Scattering profiles reveal that the symmetry of the mesophases is the four-folded Pn3m phase, where the positioning of wave vectors is represented in figure S1 (the reflections are spaced as

2,

3,

4,

6,

8 and

9). The choice of this

mesophase is due to its popularity as a model drug delivery system for release studies in view of its thermodynamic stability in excess water conditions

3, 6, 20, 23, 24, 43, 44,

thus enabling a critical

assessment of published data by means of the present results. As illustrated in figure 2a, the release setup consists in a cylindrical tube where a mesophase loaded with caffeine is put in contact with two receiving chambers. This geometry enables treating the


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release kinetics as a one-dimensional diffusion process. A standard approach in the literature to estimate the diffusion coefficient is the Higuchi model, which focuses on the early stage of the release. The Higuchi equation reads 31, 36: 0.5

( )

𝑄(𝑡) = 4

𝐷 𝑚𝑡

𝜋𝐿2𝑚

(9)

where 𝐷𝑚 and 𝐿𝑚 are the effective diffusion coefficient within and the thickness of the mesophase. Here, we adopt a more complete treatment including the whole time range of the release kinetics, based on Fick’s second law of diffusion 31. More specifically, considering a reference frame with

(

origin at the center of the tube, the mesophase is identified by the domain ―

𝐿𝑚

2,+

𝐿𝑚 2

) (Fig. 2b).

A typical approach assumes homogenous initial distribution of the drug within the LLC and perfect sink conditions at the interfaces between the mesophase and the water chambers 31, 32. The diffusion problem thus reads ∂𝐶(𝑥,𝑡) ∂2𝐶(𝑥,𝑡) = 𝐷𝑚 ∂𝑡 ∂𝑥2 𝐶(𝑥,0) = 𝐶𝑖

(

𝐶 ±

𝐿𝑚 2

)

,𝑡 = 0

[

𝑥∈ ―

𝐿𝑚 𝐿𝑚 , 2 2

]

(10)

(11) (12)

where 𝐶(𝑥,𝑡) stands for the concentration profile at the position 𝑥 and at time 𝑡, while 𝐶𝑖 is the initial drug concentration in the mesophase. This problem can be solved straightforwardly by separation of variables 30. Then, the relative amount of released molecules 𝑄(𝑡) can be computed as 𝐿𝑚

∫2 ―

𝑄(𝑡) = 1 ―

𝐿𝑚

𝐶(𝑥,𝑡) 𝑑𝑥

2

𝐶𝑖 𝐿𝑚

(13)


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which finally yields the following formula 30: ∞

𝑄(𝑡) = 1 ―

∑ 𝑛

8

(

𝑒𝑥𝑝 ― 𝐷𝑚 2 2 (2𝑛 + 1) 𝜋 =0

)

(2𝑛 + 1)2𝜋2𝑡 𝐿𝑚2

(14)

The perfect-sink condition is an approximation which is well-suited for applications, as in vivo the size of the delivering LLC is always negligible when compared to the host environment. Nevertheless, for studies in vitro such as in the present case, the size of the receiving chambers is often comparable to that of the mesophase, which may have significant consequences on the estimated values of the diffusion coefficient of the drug. Indeed, the typical protocol implemented to mimic perfect-sink conditions consists in periodically emptying the receiving chambers and refilling them with drug-free solution. Measuring the concentration of the drug in the removed solution enables monitoring the release kinetics. For this process to actually result into perfectsink conditions, the time Δ𝑡 between two consecutive pick-ups needs to be sufficiently low to prevent partial equilibration between the mesophase and the receiving chambers, i.e. to avoid the return of released molecules into the LLC. As we show below, this detail can have significant consequences on the outcome of a release experiment.

Release at different time sequences The impact of partial equilibration was assessed by considering release experiments where the parameter ∆t was systematically varied. Results are displayed in figure 3a. Upon increasing the time step, the release profiles show progressive delay in reaching the plateau, evidencing a systematic reduction of the release rate. The extent of partial equilibration can be appreciated by considering the amount of molecules measured at the first pickup. As shown in figure 3d, this quantity is an increasing function of Δ𝑡, as it is expected from a release kinetics. Nevertheless, for


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the largest time steps a plateau is reached, which is in contrast to such expectations, and is a clear signal of the complete equilibration of the system in the corresponding time frame. Following the perfect-sink approximation, each release profile was fitted with the aid of Eq. (14), by considering the first 100 terms and tuning the effective diffusion coefficient 𝐷𝑚. As expected from a qualitative comparison of the release profiles, the optimized value of 𝐷𝑚 decreases upon increasing the time step (see table 1 and Fig. 3c). When compared to the case Δ𝑡 = 0.5 h, the estimated value of 𝐷𝑚 shows a decrease of 20%-30% for ∆𝑡 = 1 ― 4 h. Even more direct consequences are observed for larger values of the time step, reaching an underestimation of 80% for the extreme case ∆𝑡 = 48 h. Eq. (9) was also applied to compute 𝐷𝑚 for ∆𝑡 = 0.5, 1 ― 4 h and the results show similar values as those obtained by means of Eq. (14). However, the coefficient of determination was not satisfactory, particularly in the case of ∆𝑡 = 0.5 (see table 1). For the following analysis, the more complete approach provided by Eq. (14) was thus preferred. 𝐷𝑚 is an intrinsic feature of the caffeine/mesophase system, thus its value should not depend on the chosen time protocol. These results thus highlight the relevance of the latter for a correct estimation of 𝐷𝑚, which can be severely underestimated due to partial equilibration between consecutive pickups. In the present case, re-entrance of released molecules into the mesophase is promoted by a similar affinity of caffeine for LLC and water phases, which is the result of a significant exchange of molecules between lipid bilayer and water channels within the mesophase 𝑒𝑞 (Fig. 4a). This can be quantified by the partition coefficient 𝑘 = 𝑐𝑒𝑞 𝑤 /𝑐𝑚 , defined as the ratio of 𝑒𝑞 concentrations in water and mesophase in contact and at equilibrium, here denoted as 𝑐𝑒𝑞 𝑚 and 𝑐𝑤 𝑐𝑒𝑞 𝑤

respectively 32. In the case of caffeine, 𝑘 can be computed starting from the known value 𝑘 = 𝑐𝑒𝑞 𝑙

≃ 1.23 of its partition coefficient for pure monolinolein and water

26,

where 𝑐𝑒𝑞 𝑙 indicates the


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equilibrium concentration in the pure lipid phase. Indeed, if 𝜑𝑙𝑖𝑝 ≃ 0.62 is the volume fraction of lipids within the LLC (computed starting from the weight composition, i.e. 63%w/w, and 𝑒𝑞 𝑒𝑞 considering 1.05 g/cm3 for the lipid density 15), the condition 𝑐𝑒𝑞 𝑚 = (1 ― 𝜑𝑙𝑖𝑝)𝑐𝑤 + 𝜑𝑙𝑖𝑝 𝑐𝑙 holds. 𝑒𝑞 Making use of the definition of 𝑘, one can thus write 𝑐𝑒𝑞 𝑚 = (1 ― 𝜑𝑙𝑖𝑝) 𝑐𝑤 + 𝑘 𝑘(1 ― 𝜑𝑙𝑖𝑝) + 𝜑𝑙𝑖𝑝

𝜑𝑙𝑖𝑝 𝑐𝑒𝑞 𝑤

𝑐𝑒𝑞 𝑤

𝑘

𝑚

⇒𝑘 = 𝑐𝑒𝑞 =

≃ 1.13. This value of the partition coefficient is close to one, indicating a similar

affinity of caffeine for water and LLCs dictated by the favourable environment provided by the lipids. For comparison, glucose is a hydrophilic molecule of similar size to caffeine and yields 𝑘 = 4.14 in the same mesophase 32. Further indications on the relevance of the affinity of caffeine for lipids comes from a comparison between the estimated 𝐷𝑚 and theoretical predictions. Particularly, if one unphysically assumes that caffeine only diffuses through the water channels, simulation results predict

45

𝐷𝑚 = 1.80 ⋅

10 ―6 cm2/s (following the analysis performed in Ref.[43], a layer of bound water of thickness ∼ 0.6 nm was assumed). This value is larger than the best estimation offered by the case Δ𝑡 = 0.5 h (see table 1), which indicates that caffeine is spending a significant amount of time within the lipid bilayer, where the diffusion coefficient is expected to be ∼ 10 ―7 cm2/s or lower

23,

thus

resulting into a smaller overall value of the effective diffusion coefficient 𝐷𝑚.

Numerical modeling of the release experiments To reconcile the change in kinetics observed in experiments with modelling based on the diffusion equation, a more thorough description of the system is needed. In this regard, one can adapt an approach successfully employed in a similar setup

32,

where the finite extension of the water

chambers is taken into account. Specifically, the diffusion problem now becomes (see Fig.2a)


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∂𝐶(𝑥,𝑡) ∂𝑡

=

{

𝐷𝑚 𝐷𝑐

∂2𝐶(𝑥,𝑡)

∂𝑥 ∂2𝐶(𝑥,𝑡)

(15)

2

[

(16) 𝑖𝑓 𝑥 ∈ ―

∂𝑥2

Page 14 of 30

𝑖𝑓 𝑥 ∈ [ ―

𝐿𝑚

― 𝐿𝑐, ―

2

𝐿𝑚 2

] 𝑜𝑟 𝑥 ∈ [

𝐿𝑚 𝐿𝑚

2,2] 𝐿𝑚 𝐿𝑚 2,2

+ 𝐿𝑐]

where 𝐷𝑐 and 𝐿𝑐 are the diffusion coefficient of caffeine in water and the size of each chamber, respectively. This set of equations is complemented by the initial condition

{

𝐶(𝑥,0) =

[

𝐶𝑖

𝑖𝑓 𝑥 ∈ ―

0

𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

𝐿𝑚 𝐿𝑚 , 2 2

]

(17)

Moreover, reflecting walls are imposed at the end of the receiving chambers: 𝐿𝑚 ∂𝐶(𝑥,𝑡) ± ± 𝐿𝑐,𝑡 = 0 ∂𝑥 2

(

)

(18)

Finally, assuming local equilibrium and continuity of flux (that is, mass conservation) at the water/mesophase interfaces, the following boundary conditions need to be imposed:

(

𝐶 𝑥

―

―

𝐿𝑚

)

(

2 ,𝑡 = 𝑘 𝐶 𝑥

+

𝐿𝑚

―

( (

𝐶 𝑥

+ 𝐿𝑚

∂𝐶 𝑥

2 ―

𝐷𝑐

(

where

―

and

+

―

)

(

,𝑡 = 𝑘 𝐶 𝑥 𝐿𝑚 2

∂𝑥

)

2

)

,𝑡 (20)

(

∂𝐶 𝑥

+

= 𝐷𝑚

+ 𝐿𝑚

2

― 𝐿𝑚

,𝑡

∂𝑥 ∂𝐶 𝑥

𝐷𝑐

) (19)

2 ,𝑡

)

∂𝐶 𝑥 = 𝐷𝑚

𝐿𝑚 2

)

,𝑡

(21)

∂𝑥

(

,𝑡

―

― 𝐿𝑚

2

)

,𝑡

∂𝑥

(22)

denote approaching the indicated quantity from below or above, respectively. This

diffusion problem is more complex than the original formulation based on a perfect sink. Although possible, the analytical resolution of this set of equations is cumbersome, and is further complicated by the pick-up protocol

32.

For the present purposes, it is sufficient to consider a

numerical approach (see Methods). In practice, the diffusion problem was solved between two


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consecutive pick-ups. Then, the replacement of the solution in the receiving chamber was implemented by computing the total amount of molecules present therein and subsequently setting the concentration to zero in the corresponding domains (Fig. 5). To compare numerical and experimental data, the values of the parameters of the model need to be fixed. The geometric parameters were set according to the details of the experimental setup as 𝐿𝑐 = 3 cm and 𝐿𝑚 = 0.6 cm. The partition coefficient was set to 𝑘 = 1.13 as indicated by the calculations above. As for the diffusion coefficient of caffeine in water, values from literature 46 measured at 25°C were employed to compute 𝐷𝑐 = 8.5 ⋅ 10 ―6 cm2/s in the present conditions (37°C) following the prescription from Ref.[30]. Finally, 𝐷𝑚 = 1.492 ⋅ 10 ―6 cm2/s was estimated by considering the dataset relative to the shortest timestep Δ𝑡 = 0.5 h (see table 1), as this protocol is the one most closely mimicking perfect-sink conditions. The predicted release curves for the various values of Δ𝑡 are reported as black continuous lines in figure 6, together with the experimental points (red diamonds). The excellent agreement obtained without further fitting conclusively shows that, once the details of the experimental setup are thoroughly taken into account, Fickian diffusion quantitatively captures the systematic delay observed in our experiments, and appoint the present model as an informative tool to assess the impact of the release protocol on the estimated value of the diffusion coefficient. From a physical perspective, the delay is ultimately caused by the partial equilibration between mesophase and receiving chamber that takes place between consecutive pick-ups. It is expected that this effect is more prominent for molecules characterized by a large diffusion coefficient or strong affinity for the mesophase, as both these factors facilitate re-entrance of released molecules into the LLC. To further assess this point, the model was employed to simulate the release kinetics of other two model drug molecules, namely glucose and proflavine, which are characterized by


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strong hydrophilicity and hydrophobicity, respectively. In the case of glucose, the parameters were set to the values determined in the literature 32, namely 𝐷𝑐 = 10.3 ⋅ 10 ―6 cm2/s , 𝐷𝑚 = 2.2 ⋅ 10 ―6 cm2/s and 𝑘 = 4.14. For proflavine, 𝐷𝑐 = 11.5 ⋅ 10 ―6 cm2/s was computed starting from the measured value

47

at 20°C; 𝐷𝑚 = 0.178 ⋅ 10 ―6 cm2/s was taken from Ref.[35]; 𝑘 = 0.13 was

computed as above starting from the measured value 𝑘 = 0.093 for proflavine partitioning between pure monolinolein and water 26. It is worth noting the extremely low value of the partition coefficient, indicating the strong hydrophobicity of proflavine, which is further supported by the small value of 𝐷𝑚, incompatible with a diffusion process taking place only in the water channels, which is predicted to give 𝐷𝑚 = 2.71 ⋅ 10 ―6 cm2/s, i.e. more than ten times the measured value 45. For both molecules, the outcome of a release experiment was simulated by means of our model for different values of the time step Δ𝑡. Then, each of the obtained release profiles was fitted by means of Eq. (14) to extract an apparent value of the diffusion coefficient within the mesophase, 𝐷𝑎𝑝𝑝 𝑚 . As shown in figure 7, both for glucose (cyan squares) and proflavine (red diamonds) a systematic decrease of the apparent diffusion coefficient is observed as time steps of larger duration are considered, although in the former case the hydrophicility of the molecule results into a weaker effect of protocol, in line with our expectations. For instance, in the case of glucose a reduction of up to 20% is observed for Δ𝑡 = 1 ― 5 h, while a decrease of 40% is obtained at the largest inspected value Δ𝑡 = 16 h. Noteworthy, also for the fastest pace (Δ𝑡 = 0.5 h) the estimated diffusion coefficient is lower than its actual value (𝐷𝑡𝑟𝑢𝑒 𝑚 , black horizontal line), thus indicating that small underestimations of the diffusion coefficient are inherent to the release setup ( ∼ 7% for glucose). Due to its hydrophobicity, proflavin is more affected by partial equilibration. A reduction of up to 35% is observed for Δ𝑡 = 1 ― 5 h, which goes to more than 50% for Δ𝑡 = 16 h. To further assess the effects of hydrophobicity and diffusion


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speed on the partial equilibration of the system, a third simulation was implemented for an ideal chimeric molecule with partition coefficient equal to that of proflavine and diffusion properties inherited from glucose (yellow circles). This combines the two factors that yield fast partial equilibration, and as a result a strong dampening effect is observed for all time steps: even for Δ 𝑡 = 0.5 h, a reduction of 65% of the apparent diffusion coefficient is observed. The effect of the diffusion coefficient on the partial equilibration can be appreciated by comparing the results of the chimera with proflavine; analogously, the impact of the partition coefficient alone can be inspected by considering chimera and glucose data. Both quantities affect dramatically the apparent value of the diffusion coefficient, thus highlighting the relevance of partial equilibration in the analysis of release data.

CONCLUSIONS This work has explored the effect of partial equilibration on the release kinetics of drugs from lipidic mesophases. The role of molecular interactions between the diffusing particle and the host matrix has been analyzed by considering caffeine as a model drug, due to its known affinity for lipids. Varying the time protocol for pick-ups in a typical release experiment showed the stark effect of partial equilibration on the estimated values of the effective diffusion coefficient 𝐷𝑚, which were found to decrease for slower pick-up rates. These results show that the widelyemployed perfect-sink condition may lead to a strong underestimation of the transport properties of the host mesophase as extracted by experimental data, an issue which needs to be thoroughly addressed for the development of related applications with controlled properties. A model based on Fick’s diffusion and which goes beyond the perfect-sink approximation was introduced and demonstrated to account quantitatively for the set of release data obtained from


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Page 18 of 30

experiments. Furthermore, it was employed to assess the role of diffusivity and mesophase affinity on the extent of the systematic underestimation of parameters extracted from fits. Although focused on lipid mesophases, our results concern the whole field of release studies, as the phenomena here described are ultimately due to the finite size of the receiving chamber and the mutual solubilty of the target molecule in the host matrix and in the receiving solution, which are common features of all release experiments. The model here developed is thus of general value, and can be used to obtain more accurate estimation of transport parameters of the hosting environment, or at least minimize the impact of the experimental protocol, thus paving the way to a closer control of related applications.

ASSOCIATED CONTENT Supporting information The supplmentary information can be found online: AUTHOR INFORMATION †These

authors contributed equally

Corresponding Author *[email protected] Funding Sources This work was supposted by the Swiss National Science Foundation under Grant No. 200021_162355.


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REFERENCES 1. Vallooran, J. J.; Handschin, S.; Bolisetty, S.; Mezzenga, R., Twofold Light and Magnetic Responsive Behavior in Nanoparticle-Lyotropic Liquid Crystal Systems. Langmuir 2012, 28 (13), 5589-5595. 2. Boyd, B. J.; Whittaker, D. V.; Khoo, S. M.; Davey, G., Lyotropic liquid crystalline phases formed from glycerate surfactants as sustained release drug delivery systems. Int J Pharm 2006, 309 (1-2), 218-26. 3. Guo, C.; Wang, J.; Cao, F.; Lee, R. J.; Zhai, G., Lyotropic liquid crystal systems in drug delivery. Drug Discov Today 2010, 15 (23-24), 1032-40. 4. Zhou, M. J.; Nemade, P. R.; Lu, X. Y.; Zeng, X. H.; Hatakeyama, E. S.; Noble, R. D.; Gin, D. L., New type of membrane material for water desalination based on a cross-linked bicontinuous cubic lyotropic liquid crystal assembly. Journal of the American Chemical Society 2007, 129 (31), 9574-9575. 5. Seddon, A. M.; Lotze, G.; Plivelic, T. S.; Squires, A. M., A Highly Oriented Cubic Phase Formed by Lipids under Shear. Journal of the American Chemical Society 2011, 133 (35), 1386013863. 6. Nazaruk, E.; Miszta, P.; Filipek, S.; Gorecka, E.; Landau, E. M.; Bilewicz, R., Lyotropic Cubic Phases for Drug Delivery: Diffusion and Sustained Release from the Mesophase Evaluated by Electrochemical Methods. Langmuir 2015, 31 (46), 12753-12761. 7. Mezzenga, R.; Meyer, C.; Servais, C.; Romoscanu, A. I.; Sagalowicz, L.; Hayward, R. C., Shear rheology of lyotropic liquid crystals: A case study. Langmuir 2005, 21 (8), 3322-3333. 8. Angelov, B.; Angelova, A.; Ollivon, M.; Bourgaux, C.; Campitelli, A., Diamond-type lipid cubic phase with large water channels. Journal of the American Chemical Society 2003, 125 (24), 7188-7189. 9. Clogston, J.; Craciun, G.; Hart, D. J.; Caffrey, M., Controlling release from the lipidic cubic phase by selective alkylation. J Control Release 2005, 102 (2), 441-461. 10. Werber, J. R.; Osuji, C. O.; Elimelech, M., Materials for next-generation desalination and water purification membranes. Nature Reviews Materials 2016, 1 (5), 16018. 11. Salvati Manni, L.; Zabara, A.; Osornio, Y. M.; Schoppe, J.; Batyuk, A.; Pluckthun, A.; Siegel, J. S.; Mezzenga, R.; Landau, E. M., Phase behavior of a designed cyclopropyl analogue of monoolein: implications for low-temperature membrane protein crystallization. Angew Chem Int Ed Engl 2015, 54 (3), 1027-31.


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12. N.Garti, P. S., and R. Mezzenga, Self-Assembled Supramolecular Architectures: Lyotropic Liquid Crystals. John Wiley & Sons: 2012; Vol. 3. 13. Kulkarni, C. V.; Wachter, W.; Iglesias-Salto, G.; Engelskirchen, S.; Ahualli, S., Monoolein: a magic lipid? Phys. Chem. Chem. Phys. 2011, 13 (8), 3004-3021. 14. Qiu, H.; Caffrey, M., Lyotropic and thermotropic phase behavior of hydrated monoacylglycerols: Structure characterization of monovaccenin. Biophysical Journal 1998, 102 (24), 4819-4829. 15. Negrini, R.; Mezzenga, R., Diffusion, Molecular Separation, and Drug Delivery from Lipid Mesophases with Tunable Water Channels. Langmuir 2012, 28 (47), 16455-16462. 16. Speziale, C.; Ghanbari, R.; Mezzenga, R., Rheology of Ultraswollen Bicontinuous Lipidic Cubic Phases. Langmuir 2018, 34 (17), 5052-5059. 17. Bisset, N. B.; Boyd, B. J.; Dong, Y. D., Tailoring liquid crystalline lipid nanomaterials for controlled release of macromolecules. Int J Pharm 2015, 495 (1), 241-8. 18. Boyd, B. J., Characterisation of drug release from cubosomes using the pressure ultrafiltration method. International Journal of Pharmaceutics 2003, 260 (2), 239-247. 19. Burrows, R.; Collett, J. H.; Attwood, D., The Release of Drugs from Monoglyceride-Water Liquid-Crystalline Phases. Int J Pharmaceut 1994, 111 (3), 283-293. 20. Clogston, J.; Caffrey, M., Controlling release from the lipidic cubic phase. Amino acids, peptides, proteins and nucleic acids. J Control Release 2005, 107 (1), 97-111. 21. Fong, C.; Le, T.; Drummond, C. J., Lyotropic liquid crystal engineering-ordered nanostructured small molecule amphiphile self-assembly materials by design. Chemical Society Reviews 2012, 41 (3), 1297-1322. 22. Ghanbari, R.; Assenza, S.; Saha, A.; Mezzenga, R., Diffusion of Polymers through Periodic Networks of Lipid-Based Nanochannels. Langmuir 2017, 33 (14), 3491-3498. 23. Meikle, T. G.; Yao, S.; Zabara, A.; Conn, C. E.; Drummond, C. J.; Separovic, F., Predicting the release profile of small molecules from within the ordered nanostructured lipidic bicontinuous cubic phase using translational diffusion coefficients determined by PFG-NMR. Nanoscale 2017, 9 (7), 24712478. 24. Mulet, X.; Boyd, B. J.; Drummond, C. J., Advances in drug delivery and medical imaging using colloidal lyotropic liquid crystalline dispersions. Journal of Colloid and Interface Science 2013, 393, 1-20. 25. Negrini, R.; Fong, W. K.; Boyd, B. J.; Mezzenga, R., pHresponsive lyotropic liquid crystals and their potential


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therapeutic role in cancer treatment. Chem Commun (Camb) 2015, 51 (30), 6671-4. 26. Martiel, I.; Baumann, N.; Vallooran, J. J.; Bergfreund, J.; Sagalowicz, L.; Mezzenga, R., Oil and drug control the release rate from lyotropic liquid crystals. J Control Release 2015, 204, 78-84. 27. Lee, K. W. Y.; Nguyen, T. H.; Hanley, T.; Boyd, B. J., Nanostructure of liquid crystalline matrix determines in vitro sustained release and in vivo oral absorption kinetics for hydrophilic model drugs. Int J Pharmaceut 2009, 365 (1-2), 190199. 28. van 't Hag, L.; Li, X.; Meikle, T. G.; Hoffmann, S. V.; Jones, N. C.; Pedersen, J. S.; Hawley, A. M.; Gras, S. L.; Conn, C. E.; Drummond, C. J., How Peptide Molecular Structure and Charge Influence the Nanostructure of Lipid Bicontinuous Cubic Mesophases: Model Synthetic WALP Peptides Provide Insights. Langmuir 2016, 32 (27), 6882-94. 29. Ghanbari, R.; Assenza, S.; Mezzenga, R., The interplay of channel geometry and molecular features determines diffusion in lipidic cubic phases. J Chem Phys 2019, 150 (9), 094901. 30. Crank, J., The Mathematics of Diffusion. Oxford University Press: Oxford, (1979). 31. Siepmann, J.; Peppas, N. A., Higuchi equation: Derivation, applications, use and misuse. Int J Pharmaceut 2011, 418 (1), 612. 32. Antognini, L. M.; Assenza, S.; Speziale, C.; Mezzenga, R., Quantifying the transport properties of lipid mesophases by theoretical modelling of diffusion experiments. Journal of Chemical Physics 2016, 145 (8), 084903. 33. Paul, D. R., Elaborations on the Higuchi model for drug delivery. Int J Pharmaceut 2011, 418 (1), 13-17. 34. Paul, D. R., Solute Release from Membrane Matrix Composites. J Membrane Sci 1984, 21 (2), 203-207. 35. Paul, D. R., Modeling of Solute Release from Laminated Matrices. J Membrane Sci 1985, 23 (2), 221-235. 36. Higuchi, T., Rate of Release of Medicaments from Ointment Bases Containing Drugs in Suspension. J Pharm Sci 1961, 50 (10), 874-875. 37. Negrini, R.; Sanchez-Ferrer, A.; Mezzenga, R., Influence of electrostatic interactions on the release of charged molecules from lipid cubic phases. Langmuir 2014, 30 (15), 42808. 38. Kramer, S. D.; Lombardi, D.; Primorac, A.; Thomae, A. V.; Wunderli-Allenspach, H., Lipid-bilayer permeation of druglike compounds. Chem Biodivers 2009, 6 (11), 1900-16.


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39. Paloncyova, M.; Berka, K.; Otyepka, M., Molecular insight into affinities of drugs and their metabolites to lipid bilayers. J Phys Chem B 2013, 117 (8), 2403-10. 40. Preedy, V. R., Caffeine: Chemistry, Analysis, Function and Effects, . The Royal Society of Chemistry: Cambridge, UK, (2012). 41. Vallooran, J. J.; Negrini, R.; Mezzenga, R., Controlling anisotropic drug diffusion in lipid-Fe3O4 nanoparticle hybrid mesophases by magnetic alignment. Langmuir 2013, 29 (4), 9991004. 42. Kreyszig, E., Advanced engineering mathematics. 10th ed.; John Wiley: USA, (2011). 43. Nguyen, T. H.; Hanley, T.; Porter, C. J. H.; Boyd, B. J., Nanostructured liquid crystalline particles provide long duration sustained-release effect for a poorly water soluble drug after oral administration. J Control Release 2011, 153 (2), 180-186. 44. Turner, D. C.; Wang, Z. G.; Gruner, S. M.; Mannock, D. A.; Mcelhaney, R. N., Structural Study of the Inverted Cubic Phases of Di-Dodecyl Alkyl-Beta-D- Glucopyranosyl-Rac-Glycerol. Journal De Physique Ii 1992, 2 (11), 2039-2063. 45. Assenza, S.; Mezzenga, R., Curvature and bottlenecks control molecular transport in inverse bicontinuous cubic phases. J Chem Phys 2018, 148 (5), 054902. 46. Green, D. W.; Perry, R. H., Perry's Chemical Engineers' Handbook, Eighth Edition. McGraw-Hill (2008). 47. Malkia, A.; Liljeroth, P.; Kontturi, K., Membrane activity of ionisable drugs - a task for liquid-liquid electrochemistry? Electrochem Commun 2003, 5 (6), 473-479.


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Page 24 of 30

Figures

Figure 1. Molecular structure of a: Caffeine and b: Monolinolein

Figure 2. Experimental release setup; a: The simulation of perfect sink conditions for release experiments b: the schematic of the lipid-based mesophase c: the ideal sink conditions at infinitely small time gradients


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Figure 3. a: Release profiles of caffeine at different time steps and corresponding fits using equation (14) b: Release profiles for ∆𝑡 = 0.5,1,2,3,4 hours using equation (9) c: Effective diffusion coefficient extracted from equation (14) (blue circle) and equation (9) (red diamond) d: The amount of released caffeine % at first pickup vs time step

Figure 4. a: Partitioning of the caffeine between lipid-based mesophase and the water reservoir at a macroscopic level b: the partitioning of the caffeine between the lipidic and the water domains.


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Page 26 of 30

Figure 5. Representative concentration profile obtained from our numerical model immediately before (black continuous line) and after (dashed cyan line) a pickup. The interfaces between the mesophase and the receiving chambers are depicted as vertical dotted lines. A discontinuous jump in concentration is there observed, corresponding to a partition coefficient 𝑘 ≠ 1. For the particular choice considered to the figure, the jump goes from about 0.4 to 0.1, when going from the solution to the mesophase. Integration of the concentration profile in the receiving chamber immediately before the pickup (yellow areas) gives the amount of released molecules ideally measured in this simulated experiment. As the pick-up does not involve the mesophase, the concentration profiles overlap exactly in the corresponding region.


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Figure 6. Mosaic showing the comparison between experimental data (red diamonds, compare Fig.3a) and theoretical prediction according to the numerical model (black continuous lines).


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Page 28 of 30

Figure 7. Effect of time step Δ𝑡 on the apparent diffusion coefficient 𝐷𝑎𝑝𝑝 𝑚 extracted from fitting simulated release 𝑎𝑝𝑝 experiments by means of Eq.(14) normalized with the actual value of diffusion coefficient 𝐷𝑡𝑟𝑢𝑒 (𝐷𝑚 𝐷𝑡𝑟𝑢𝑒 𝑚 ) for 𝑚 glucose (cyan squares), proflavine (red diamonds) and the chimera molecule with combined properties (yellow circles).


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Tables

Table 1. Effective Diffusion Coefficient for release profiles using Eq. (14), left column and Eq. (9) right column.

𝛥𝑡

2

𝐷𝑚( × 10 ―6 𝑐𝑚 /𝑠)

𝑅2

Eq. (14)

2

𝐷𝑚( × 10 ―6 𝑐𝑚 /𝑠)

𝑅2

Eq. (9)

0.5

1.492 ± 0.174

0.96

1.494 ± 0.205

0.68

1

1.196 ± 0.095

0.99

1.267 ± 0.157

0.91

2

1.043 ± 0.111

0.99

1.087 ± 0.245

0.88

3

1.003 ± 0.03

0.99

1.023 ± 0.144

0.99a

4

0.994 ± 0.048

0.99

0.976 ± 0.21

0.96

5

1.070 ± 0.013

0.99

12

0.976 ± 0.037

0.99

16

0.685 ± 0.0282

0.99

24

0.547 ± 0.04

0.99

48

0.301 ± 0.025

0.99


29

Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 30

Table of Contents/Abstract Graphic


30

Impact of Molecular Partitioning and Partial Equilibration on the

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