In Silico Screening for Solid Dispersions: The Trouble with Solubility

Article Cite This: Mol. Pharmaceutics XXXX, XXX, XXX−XXX

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In Silico Screening for Solid Dispersions: The Trouble with Solubility Parameters and χFH Eleanor R. Turpin,† Vincenzo Taresco,† Wathiq A. Al-Hachami,† Jonathan Booth,‡ Kevin Treacher,‡ Simone Tomasi,‡ Cameron Alexander,† Jonathan Burley,† Charles A. Laughton,† and Martin C. Garnett*,† †

School of Pharmacy, University of Nottingham, University Park, Nottingham NG7 2RD, U.K. AstraZeneca, Macclesfield SK10 RNA, U.K.

Mol. Pharmaceutics Downloaded from pubs.acs.org by UNIV OF SUNDERLAND on 09/30/18. For personal use only.

‡

S Supporting Information *

ABSTRACT: The problem of predicting small moleculepolymer compatibility is relevant to many areas of chemistry and pharmaceutical science but particularly drug delivery. Computational methods based on Hildebrand and Hansen solubility parameters, and the estimation of the Flory− Huggins parameter, χ, have proliferated across the literature. Focusing on the need to develop amorphous solid dispersions to improve the bioavailability of poorly soluble drug candidates, an innovative, high-throughput 2D printing method has been employed to rapidly assess the compatibility of 54 drug-polymer pairings (nine drug compounds in six polymers). In this study, the first systematic assessment of the in silico methods for this application, neither the solubility parameter approach nor the calculated χ, correctly predicted drug−polymer compatibility. The theoretical limitations of the solubility parameter approach are discussed and used to explain why this approach is fundamentally unsuitable for predicting polymer−drug interactions. Examination of the original sources describing the method for calculating χ shows that only the enthalpic contributions to the term have been included, and the corrective entropic term is absent. The development and application of new in silico techniques, that consider all parts of the free energy of mixing, are needed in order to usefully predict small molecule−polymer compatibility and to realize the ambition of a drug− polymer screening method. KEYWORDS: Hildebrand solubility parameter, Hansen solubility parameter, Flory−Huggins parameter, group contribution methods, molecular dynamics methods, solid dispersions

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INTRODUCTION Predicting the miscibility of polymers with small molecules is important in producing polymeric composites and, in particular, for pharmaceutical applications in a very wide range of drug delivery systems. For example, polymer substrates are used for dispersions of liquid crystals.1 Miscibility is also important in a variety of pharmaceutical dosage forms;2 and more recently its importance has been recognized in preparing drug loaded polymer micelle nanoparticles.3 In this present work, we are focused on a major area where an understanding of the interactions of polymers with small molecules is urgently needed. This is the development of solid dispersions containing active pharmaceutical ingredients (API) to aid the development of new and effective medicines. The modern drug discovery and development process has resulted in an increasing number of drugs that have poor pharmaceutical properties. Specifically, up to 75% of these candidate drugs are poorly soluble in water4 or have a low physical stability, resulting in difficulties with dissolution in the gastrointestinal tract, leading to poor bioavailability for oral formulations. These problems can be overcome by converting a crystalline drug into an amorphous drug that does not have to © XXXX American Chemical Society

overcome the lattice energy before dissolution, or alternatively, by using a smaller drug particle size. Both of these objectives can be achieved by an amorphous or molecular dispersion of the drug within a polymer.5 For pharmaceutical applications, this amorphous material should preferably remain in a thermodynamically stable state for the duration of the shelf life of the medicine. If the drug is incompatible with the excipient, the material could continue to undergo structural relaxation after glass formation, leading to the formation of separate polymer and drug domains, which may then allow for the recrystallization of the drug phase.6 This would result in a change in the properties of the medicine, which would be unsuitable from both a patient safety and regulatory point of view. If the carrier is a polymer with a high glass transition temperature (Tg), this may kinetically trap the drug and so reduce the rate of phase separation and recrystallization.7 For development of these formulations, much effort is spent on finding the optimum Received: Revised: Accepted: Published: A

June 18, 2018 August 1, 2018 August 24, 2018 August 24, 2018 DOI: 10.1021/acs.molpharmaceut.8b00637 Mol. Pharmaceutics XXXX, XXX, XXX−XXX

Article

Molecular Pharmaceutics

estimated from all-atom molecular dynamics (MD), a simulation method that propagates the evolution of a molecular system through time using classical mechanics. Most modern force fields used in MD simulations, such as GROMOS,13 OPLS,14,15 CHARMM,16,17 AMBER,18 and COMPASS,19 do not have a functional form that includes an explicit term for hydrogen bonding, but instead hydrogen bonds are treated implicitly within the polar term; therefore, the Hansen parameters from MD only include δpolar and δdisp. The difference in the magnitude of the solubility parameters of materials is used to make predictions about their miscibility. One method is to use the absolute difference between δdrug and δpolymer to predict compatibility, and a commonly used rule of thumb is a Δδdrug−polymer less than 7.0 MPa1/2 is expected to be miscible, and Δδdrug−polymer greater than 10 MPa1/2 is expected to be immiscible.12,20−25 The individual Hansen parameters can be combined to make a single value equivalent to the Hildebrand parameter (eq 2); however, Charles Hansen12 advocates that the three parameters should be combined into a so-called radius, Ra, using eq 3, where a factor of 4 gives additional weighting to the dispersive term.

combination of drugs and excipients, but all aspects of this formulation, the selection of compatible polymers, determination of ratio of drug to polymer, and determination of stability of the formulation, are not trivial. Consequently, at present, there are only 24 medicines on the market that make use of such a polymer excipient to deliver the amorphous form of the active pharmaceutical ingredient in the formulation.8−10 Drug−polymer solid dispersions can be produced via a number of manufacturing techniques, such as hot melt extrusion, milling, vapor deposition, and 2D printing.11 Important factors for identifying the stability of drug−polymer mixtures produced by these manufacturing techniques are the miscibility limit and the solubility limit, which are related by their dependence on compatibility between the two molecules. The miscibility limit is the ratio below which a mixture exists in a single, stable phase and above which the different components become phase separated; the solubility limit is the ratio above which the drug becomes crystalline. Between the miscibility limit and the solubility limit, there may exist, depending on the properties of the drug and polymer materials, a metastable phase where a drug-rich and polymer-rich phase coexist at equilibrium, i.e., phase separation occurs. Depending on the properties of the drug and polymer, crystallization may occur either before phase separation, or the drug may remain amorphous or become crystalline after phase separation. Phase separation, domain formation, and crystallization in a solid sample can be identified via a number of experimental techniques, including polarized optical microscopy (POM), Raman spectroscopy, TOF-SIMs, atomic force microscopy (AFM), and differential scanning calorimetry (DSC). However, all of these methods are time and cost intensive, and it would be highly desirable to be able to screen for the extent of drug−polymer compatibility in silico prior to committing to experimental work. In Silico Methods Used for Predicting Small Molecule−Polymer Compatibility. Hildebrand and Hansen Solubility Parameters. Two theories that have been widely applied to the problem of predicting polymer−small molecule compatibility in silico are the solubility parameter approach and the estimation of the Flory−Huggins interaction parameter. The first, Hildebrand and Hansen solubility parameters, are primarily used to indicate if a solute is miscible with a solvent and have been extended to polymer-based systems. The Hildebrand solubility parameter, δ, is the square root of the cohesive energy density (CED), the energy per unit volume required to vaporize molecules in a condensed phase (eq 1). δ=

Evaporization V

=

CED

Ra = 4(δdisp − polymer − δdisp − drug)2 + (δpolar − polymer − δpolar − drug)2 + (δ hbond − polymer − δ hbond − drug)2

In the Hansen computer program (HSPiP12), Ra is compared to the radius of an experimentally derived sphere that excludes immiscible solvents; however, in the absence of the necessary experimental data, the magnitude of Ra should still be indicative of compatibility and produce a ranking of the most miscible drug−polymer combinations. Flory−Huggins Pparameter. The Flory−Huggins theory is a statistical thermodynamics model that describes the behavior of polymer−solvent solutions. The major result of Flory−Huggins theory is that the free energy of mixing, ΔGmix, can be calculated using the following relation26 ϕdrug ln ϕdrug ϕpolymer ln ϕpolymer ΔGmix = + + χϕdrug ϕpolymer RT mdrug mpolymer (4)

where ϕ is drug or polymer volume fraction, and m is the ratio of the volume of the drug or polymer to a lattice site. The Flory− Huggins parameter, χ, is a dimensionless quantity that characterizes the strength and favorability of polymer−solvent interactions compared with polymer−polymer and solvent− solvent interactions. Compounds are predicted to undergo phase separation if χ > 0.5 and to be miscible if χ < 0.5.26 The Flory−Huggins theory has been used to describe drug−polymer dispersions by treating the drug as a solvent for the polymer. A widely used relation for the Flory−Huggins parameter is given in eq 5:27

(1)

It was proposed that when the value of δ for two materials in amorphous phases is similar, they should be miscible. The Hildebrand approach was found to perform poorly for solutions containing polar species and was extended by Hansen12 by decomposing δ into a polar component (δpolar), hydrogenbonding component (δhbond), and a dispersive component (δdisp) (eq 2). δ=

2 2 2 δpolar + δ hbond + δdisp

(3)

χ= (2)

Vref ΔHmix RT

(5)

where Vref is the volume of a lattice site, taken to be the volume of a monomer unit, and ΔHmix is the enthalpy of mixing. One method extensively used for evaluating χ in silico for a drug− polymer system27 is to substitute ΔHmix by Δδ2 (eq 6), the square of the difference in the solubility parameters.

Solubility parameters can be calculated using group contribution (GC) methods, where the value of a parameter is given by the sum of the contributions from the different moieties that make up the molecule of interest. A value for cohesive energy density, and therefore solubility parameters, can also be B

DOI: 10.1021/acs.molpharmaceut.8b00637 Mol. Pharmaceutics XXXX, XXX, XXX−XXX

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Molecular Pharmaceutics Δδ 2 = (δpolymer − δdrug)2

Table 1. Summary of Literature on Predicting Drug− Polymer Compatibility in Silico

(6)

It has been proposed that the Flory−Huggins parameter can also be evaluated from MD by simulating pure drug, pure polymer, and an example of the mixed system.22,27 The enthalpy of mixing is estimated using the relation in eq 7 ΔHmix

where

lead author and reference

i ΔE y i ΔE y = jjj v zzz − ϕpolymer jjj v zzz − ϕdrug V k {mix k V { polymer ij ΔEv yz jj zz k V {drug

(7)

Ev V

( ) is the cohesive energy density, and ϕ is the volume

fraction of the relevant material in the mixed system. Unlike some other theoretical tools, such as equation of state models like the statistical associating fluid theory (SAFT)28 or the locally correlated lattice theory (LCL),29 the solubility parameter approach and estimation of the Flory−Huggins parameter can be considered as predictive from first-principles as they require no input beyond the molecular structure. Applications of Predictions of Solubility Parameters and χ. Recent examples (since the start of 2017) of applications of the methods described in the Introduction above include predicting the mixing of impurities with crude oil;30,31 developments in the theoretical understanding of new classes of material;32,33 the effect of water and sensitizers on polymer components of solar cell components;34,35 and the selection of polymers for filtering pollutants from water,36,37 illustrating that these methods are in wide use across many fields. In the remainder of this Introduction section, an overview is given of studies that have been conducted on predicting the miscibility of drug−polymer mixtures using the in silico methods, and an evaluation made on the quality and usefulness of the predictions. In this mini-review, only work that compares prediction with experimental data is included; studies that calculate predictive parameters in isolation have been disregarded. Forster et al.24 calculated solubility parameters from GC methods for two drugs (indomethacin and lacidipine), 6 polymers, and 5 other small molecule excipients. They grouped their drug−excipient pairs into three categories by the size of Δδ, stating that Δδ < 2.0 MPa1/2 are fully miscible. However, this was not supported by the DSC results published alongside, which show that lacidipine is immiscible with PVP-30 despite Δδ = 1.6 (Tables 1 and 324). The largest GC study with corresponding experimental data is from Dwan’Isa et al.,38 who calculated Hansen solubility parameters for polyethylene glycol (PEG), random copolyesters of e-caprolactone (CL) and trimethylene carbonate (TMC), and a selection of 19 drugs. They then used Δδ2 as an estimate of the heat of mixing in eq 4, and estimated the Flory−Huggins parameter for a subset of eight poorly soluble drugs with the polymers; this was compared with experimentally determined solubility of the drugs within polymer micelles. Although the prediction is described as useful, as it clusters the drugs into three groups of decreasing solubility, it does not correctly rank the drugs from most soluble to least. Mahmud et al.39 used solubility parameters from GC methods to calculate χ for mixtures of cucurbitacin I with three polymers (PCL, PChCL, and PBCL) that form the core of block copolymer micelles with MePEO; however, they found the resulting in silico prediction to be inadequate as it did not correctly rank the polymers compared with experimental drug

# drugs

# polymers

Greenhalgh25

1

4

GC

Forster24

2

6

GC

Dwan’Isa38

8

3

GC

Patel40 Mahmud39

2 1

1 3

MD GC

Patel41 Kasimova43 Thakral44

2 4 3

1 1 1

MD MD GC

Xiang19

1

1

MD

Eslami45

1

2

MD

Maniruzzaman22

6

4

GC, QM

Bansal46

2

1

GC

Erlebach47 Xiang21

3 1

1 1

MD MD

Chakravarty48

1

1

GC, MD

Gupta49

1

1

GC, MD

Paudel50

1

1

GC

method(s)

calculated parameter(s) solubility parameters solubility parameters χ from solubility parameters χ from MD χ from solubility parameters χ from MD χ from MD χ from solubility parameters solubility parameters, χ from MD solubility parameters, χ from MD χ from solubility parameters χ from solubility parameters χ from MD solubility parameters, χ from MD χ from solubility parameters solubility parameters χ from solubility parameters

loadings. This was followed up by two further studies using the MD method to calculate χ for mixtures of one of the same coblock polymers (PEO-b-PCL) with fenofibrate and nimodipine,40 and cucurbitacin I and B.41 Although an effort was made at method evaluation in the first of these studies, by comparing the values of χ found via GC to those from MD, the study was limited by the size of the system considered (two drugs and one polymer). Maniruzzaman et al.23,42 have also made an attempt at method evaluation after finding that calculations of χ from solubility parameters for six drugs and four polymers did not correctly predict the miscibilities of paracetamol and ibuprofen with PVP, or of paracetamol with Eudragit E PO (cationic acrylic copolymer) (EPO). They proposed using quantum mechanical (QM) level energy calculations and geometry optimizations of drug molecules with polymer dimers or trimers to identify and analyze hydrogen-bonding patterns and strengths. However, no efforts were made to take account of the wider polymeric environment experienced by a drug molecule. Noting this deficiency and in the absence of a systematic evaluation (results reported for only 9 of the 24 possible drug−polymer combinations), their proposed method is only suitable for analyzing materials already identified for use in solid dispersions and not for predicting drug−polymer choices. Table 1 summarizes the available literature that focuses on using in silico methods to predict drug−polymer compatibility. The largest of these studies involves 8 drugs with 3 polymers for GC methods37 and 4 drugs with 1 polymer for MD.42 The table illustrates how historically only a limited chemical space has C


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Table 2. Properties of Drug Molecules Used in This Studya

been considered, and consequently none of the available studies are able to draw definitive conclusions about the usefulness of in silico techniques. For example, in a study by Kasimova et al.,42 χ is calculated using the MD method, and a strong correlation is identified between χ and the experimental drug loading of a micelle. However, the study covers only a single polymer (MPEG-hexPLA) with four drugs (cyclosporine A, griseofulvin, ketoconazole, and quercetin dehydrate), and like many studies in this area, contains the following caveat in the concluding remarks “...the presented procedure could be used [to estimate solubility]...once fitting with a few test cases has been verified and proven...”. Similarly, studies on MD simulations of drug− polymer mixtures by Xiang and Anderson19,21 calculate χ and use it to illustrate drug−polymer miscibility, but each study only contains single drug−polymer mixtures (indomethacin-PVP and felodipine-HPMC), so they cannot be used to draw conclusions about the wider predictive power of this method. He and Ho51 conducted a review on the use and challenges of amorphous solid dispersions in drug delivery, including the use of in silico screening of polymers using solubility parameters and Flory−Huggins. They identify that in silico predictions are currently inadequate, and that the optimization of amorphous drug dispersions should rely on experimental screening. In 2012, Huynh et al. reviewed the use of computational approaches for rational design of nanoparticles for drug delivery.26 Their review begins with the use of Hildebrand, Hansen, and Flory−Huggins parameters, calculated from GC methods and MD, for predicting drug−polymer compatibility for material selection; it then goes on to an assessment of all-atom and coarse grain MD for interpreting nanoparticle structure and dynamics. They identify that there is a strong need for systematic studies to compare and evaluate in silico methods, and to test their predictive power, which is currently lacking in the literature. It is the aim of this article to address that need. Therefore, in this current work, we calculate solubility parameters for nine drugs (aspirin, caffeine, carbamazepine, finasteride, flufenamic acid, flutamide, mefenamic acid, salicylamide, and theophylline) and six polymers (PGA, PGAC4, PGA-C8, PGA-Phe, PVP, and PVP/VA) using GC methods and MD simulations of single component systems. We also calculated the Flory−Huggins interaction parameter for each drug with unmodified PGA from additional MD simulations of the mixed systems. These are used to make predictions about compatibility, which are compared with experimentally derived miscibility limits for all 54 pairings. This represents a chemical space far larger than that studied previously, facilitated by the use of high throughput 2D printing that has rapidly provided us with a data set to correlate with computational predictions.

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MATERIALS AND METHODS Drugs and Polymers. Drugs. Caffeine (CAF) (CAS 58-082), flutamide (FLU) (CAS 13311-84-7), aspirin (CAS 50-78-2), salicylamide (CAS 65-45-2), teophylline (CAS 58-55-9), and flufenamic acid (FLA) (CAS 530-78-9) were purchased from SIGMA-Aldrich (U.K.). Mefenamic acid (MEF) (CAS 61-68-7) and finasteride (FIN) (CAS 98319-26-7) were obtained from Alfa Aesar (Ward Hill, MA). Carbamazepine (CBZ) (CAS 29846-4) was purchased from MP Biomaterials, LLC (France). Table 2 shows the structure and properties of the drug molecules used in this study. In the work by Baird et al.,52 and extended by Alhalaweh et al.,53 drug molecules were assigned a category to describe their propensity for glass forming following a heating/cooling/heating cycle monitored by DSC. Class I

a Mass and melting temperature (Tm) are taken from the Chemical Abstracts Service Registry, retrieved via SciFinder (scifinder.cas.org).

materials crystallize upon initial cooling and are considered nonglass formers; Class II materials form glasses upon meltquenching, but recrystallize when heated above the glass transition temperature (Tg); and Class III materials have a strong propensity for glass formation and remain in an amorphous phase following heating of melt-quenched material. D


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Figure 1. Polymer structures. (Top) Kollidon 64 (PVP/VA), where n = 0.6, m = 0.4; for PVP, n = 1, m = 0. (Bottom) PGA and the substitutions of R: unmodified PGA, PGA-C4, PGA-C8, and PGA-Phe. For PGA-C4−40% and PGA-C8−40%, a = 0.6, b = 0.4; for PGA-C4−80% and PGA-C8−80%, a = 0.2, b = 0.8; and for PGA-Phe, a = b = 0.5.

Figure 2. Examples of different drug recrystallization profiles are presented. All of the pictures are reported both without (top part) and with (bottom part) a polarized filter: (l−r, first line) completely amorphous (AAAA); some minor crystallization (AAAC); slightly crystalline (AACC); (second line) mainly crystalline (ACCC); and completely crystalline (CCCC).

polymer with a free −OH on the repeating unit, that can be easily functionalized to tune physical properties of the polymer and alter drug−polymer interactions. PGA and its variants were synthesized by following the same synthetic pathways previously reported by our group.54−56 The structures of the polymers are shown in Figure 1. In order to maintain the possible highest physical−chemical reproducibility in terms of polymeric platform, PGA modifications were obtained from a common batch of nonfunctionalized PGA. Drug−Polymer Mixtures. A miniaturized, high-throughput assay method was used to assess the miscibility limit of the

The drug molecules chosen for this study were all Class I in order to provide a meaningful test of the ability of the polymers to stabilize an amorphous state of a material that is known to have a strong tendency toward crystallization. Polymers. Polyvinylpyrrolidone-vinyl acetate (PVP/VA) copolymer (Kollidon VA64) was received from BASF, while polyvinylpyrrolidone (PVP) was purchased from Sigma-Aldrich, and both of the materials were used as received. PVP and PVP/ VA were chosen for this study as they are commercially available and widely used in previous formulation studies for polymer dispersions. Poly(glycerol-adipate) (PGA) is an amphiphilic E


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The second GC method used to calculate Hansen solubility parameters is from HSPiP version 5 (www.hansen-solubility. com). This has been produced by a collaboration led by Charles Hansen, who developed the original theory, and is a suite of tools to aid solvent choice for formulation using Hansen parameters. Within the DIY module of the software, a number of different algorithms are available for HSP prediction, most for backward compatibility with early ideas for GC methods to calculate HSPs, e.g., Hoy61 and van Krevelen.62 The recommended method for current calculations is the Yamamoto-molecule breaking algorithm (Y-MB). New molecules are split into subgroups by an automated procedure, which then form the input for a neural network that fits the Hansen parameters for the complete molecule. The input accepts many common molecular formats, such as SMILES, sdd, mol2, and pdb, and again results are instantaneous. HSPiP calculates Hansen solubility parameters for polymers using a SMILES string of the monomer as input with an “X” dummy atom symbol representing the cut bonds between repeating units. As with the ProPred implementation, we have used a simple weighted sum to represent copolymers. Molecular Dynamics Methods. All simulations and analysis of the drug, polymer, and drug−polymer systems were performed in GROMACS using parameters from the CHARMM-36 force field.63,64 Please see the Supporting Information for discussion of force field parameter assignment. Each polymer is initially in a fully extended conformation, with all backbone dihedrals trans, and an end-to-end distance of ∼780 Å for a PGA-based 60mer. Each monomer unit has its side chain chirality set to R from S, with a chance 0.5 following a probability test; for the copolymers, the side chain modification (or monomer choice for PGA/VA) was added with a probability test based on the weighting, e.g., 0.4 chance of modification for PGAC4−40% and 0.8 chance for PGA-C4−80%. In order to generate more compact conformations that can be packed into a simulation unit-cell, each chain undergoes 1 ns of dynamics in vacuum conditions at 700 K. Unit cells containing 20 different compacted polymer molecules were constructed using PACKMOL.65 Following the method used by Belmares et al.,66 the size of the initial unit cell was chosen to be 100% of a target density (1.21 g cc−1 for unmodified PGA at room temperature). The amorphous materials undergo a cycle where the size of the cell is increased by increments of 1.0 Å to reach 75% of the target density, then decreased by increments of 0.5 Å to reach 125% of the target density, and finally returned to 100% of the target density in steps of 1.0 Å. At each phase of the cycle, the system undergoes 500 steps of steepest descent minimization followed by 200 ps of NVT dynamics at 750 K with a time step of 1 fs. Periodic boundary conditions were applied to the unit cell, electrostatics were treated using the particle Ewald method, and temperature was controlled using a Nose-Hover thermostat with τ set to 0.2 ps. The unit cell then underwent a further 5000 steps of energy minimization using the conjugate gradient algorithm with a tolerance value of 4 kJ mol−1 nm−1 prior to an NPT annealing stage that reduced the temperature from 750 to 300 K at a rate of 0.1 K ps−1. A further equilibration phase of 1000 steps of steepest descent followed by 10 ns of NPT dynamics at 300 K, to allow the box size to fully relax, preceded the 10 ns production phase (NPT ensemble, 300 K) used for analysis. This procedure was repeated to simulate systems of 125 drug molecules; systems of 50% w/w drug/polymer ratio of drugs and 10 PGA 60mers; and 100mers of PVP and PVP/VA. Target densities were the density of the amorphous drug where known (if unknown the

drug−polymer mixtures. Using the method outlined in our previous article,57 arrays were produced by 2D inkjet printing of microdots containing drug−polymer solution in w/w drug/ polymer ratio ranging from 5 to 90%, giving 9 different drug− polymer ratios for each drug−polymer combination. Briefly, DMSO solutions of both polymers and drugs (10 mg/mL) were printed onto gold-coated microscope glass slides (SigmaAldrich 643203-5EA; layer thickness 100 Å; 99.999% (Au)), reaching a fixed final spot size of 625−700 ng for each formulation by employing a piezo electric inkjet printer (Sciflexarray S5, Scienion). Drug and polymer solutions were printed in a stepwise fashion, by first printing a set volume of drug solution, and then printing the polymer solution on top of the drug wet spot. Polymer solution volumes were tuned to reach the final drug/polymer ratio. To dispense all of the relevant formulations, only 2−3 μg of materials were required per drug−polymer combination. Total DMSO evaporation from the arrays was allowed by leaving the slides overnight inside the printer cage (55% relative humidity (RH) and 25 °C). After DMSO evaporation, drug crystallization was monitored by using POM after 7 days (Advanced Polarizing HS1Microscope, Prior LuxPOLTM with 12 V, and a 30 W halogen lamp with variable brightness control). The arrays were stored in a desiccator employing silica as the drying agent, in order to avoid the effects of humidity on drug recrystallization. Crystallization of the spots, as evaluated by POM, were categorized using the method first described by Van Eerdenburgh and Taylor.58 As shown in Figure 2, five classes of drug crystallization can be observed: (i) completely amorphous (AAAA); (ii) some minor crystallization (AAAC); (iii) slightly crystalline (AACC); (iv) mainly crystalline (ACCC); and (v) completely crystalline (CCCC). Because of the small size, the entire spot can be viewed and assessed in one snapshot. The miscibility limit is judged to be the drug−polymer ratio that corresponds to AAAC; i.e., the mixture with the lowest drug concentration at which crystallization is first observed. If the classification is not continuous, and the designation between neighboring spots moves from, for example, AAAA to AACC, then the miscibility limit is taken to the ratio halfway between the spots. A similar definition for the miscibility limit has previously been used by Parekh et al. using a film casting technique.59 Computational Methods. Group Contribution Methods. Hansen parameters have been calculated via group contribution methods using two different computational packages. The first is the ProPred tool in the ICAS suite of software (www.capec.kt. dtu.dk/Software/ICAS-and-its-Tools/). The Hansen solubility parameter calculations in ProPred implement the GC methodology of Marrero and Gani.60 In this implementation, a molecular structure is considered as three orders of grouping: the first level is small molecular fragments, and the second and third levels then improve the groupings by taking account of effects, such as isomerization, and interactions between groups. The solubility parameter is then determined by summing the contributions assigned to the three group levels from a parameter database. The molecular input can either be created in a molecular builder or from a SMILES string, and the calculation is essentially instantaneous. ProPred also has a utility for where the user can indicate where the cut bonds are on the repeating monomer unit. However, as this can only evaluate a single, infinitely repeating unit, a simple weighted sum is used to evaluate parameters for copolymers, such as modified PGA. F


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Molecular Pharmaceutics Table 3. Drug Percentage Corresponding to AAAC under POM, Indicating Miscibility Limit of Drug within Polymer polymer drug

PGA

PGA-C4−40%

PGA-C4−80%

PGA-C8−40%

PGA-C8−80%

PGA-Phe

PVP

PVP/VA

aspirin caffeine carbamazepine finasteride flufenamic acid flutamide mefenamic acid salicylamide theophylline

60 8 90 15 20 83 5 33 5

55 3 13 8 33 83 5 68 3

55 5 15 5 40 83 15 60 5

45 3 15 10 25 83 20 40 3

55 3 15 10 33 83 20 40 3

55 3 90 25 33 83 8 40 3

68 5 60 5 75 83 50 40 15

50 5 60 10 60 90 33 40 10

crystal formation by the drugs in this present study and the low Tg of the polymers, we are assuming that the difference between the solubility limit and the miscibility limit is negligible, as there is unlikely to be a metastable amorphous, phase-separated mixture. The experimental miscibility was determined using a recently published and validated high-throughput method.54 In this method, the evaporation of solvent took 2 h so that there was ample time for interaction between drug and polymer to occur, and we also demonstrated that, when free drug was used in this method, full crystallization of the drugs in this study occurred within 24 h, so that we were measuring the system at equilibrium. We are therefore confident that the experimental miscibility has been accurately determined by this method. Table 3 shows the experimentally determined miscibility limit of each drug with the six polymers. Caffeine and theophylline, which have a very similar structure, have a low miscibility limit with all of the polymers tested here, indicating that none of the polymers stabilize either compound in an amorphous state. Conversely, flutamide has a high miscibility limit with all of the polymers, and this drug only recrystallizes at very high loadings (>83%), despite being identified as a nonglass former in its pure form.49,50 Carbamazepine shows the strongest response to the composition of the polymer and has a high miscibility limit in unmodified PGA and PGA-Phe, but a low limit in the case of PGA-C4 and PGA-C8. The degree of modification of the alkyl chain of PGA has a limited effect on the drug miscibility limit, and there are only very small differences between the PGA-C4− 40% and PGA-C4−80% columns and the PGA-C8−40% and PGA-C8−80% columns. The overall spread of miscibility limits from immiscible (3%) to highly miscible (90%) and range of responses to polymer type and composition means that this is a suitable data set for testing predictive power, as the test cases span a range of possible interactions that a successful screening method should be able to differentiate. Solubility Parameter Predictions. Figure 3 shows Ra calculated from MD as an example of solubility parameter predictions against experimental miscibility limits; the other examples (Ra and Δδ from ProPred, HSPiP, and MD) are in Figures S1−S4. A negative correlation is expected for these plots, as a small difference in the solubility parameter is supposedly indicative of high miscibility and a large difference in the solubility parameters of immiscibility. However, none of these plots show this expected correlation. The raw Hansen and Hildebrand parameters are shown in Figures S2−S4; Figure S2 contains the data from ProPred, which was unable to output any values for the PGA-Phe monomer or theophylline, and is missing δpolar for the VA monomer, despite them containing common and simple chemical moieties. Because of this, it is not possible to calculate Ra from the ProPred parameters. These

value for a drug with a similar molecular weight was used); the density of PGA; and the density of PVP (1.20 g cc−1), respectively. The relation in eq 862 is used to calculate solubility parameters from MD, where n is the number of molecules in the simulation unit cell, angle brackets indicate time average, Ei is the energy of an individual molecule in the gas phase (intramolecular terms), Ec is the total energy of all the molecules in the cell (intra- plus intermolecular terms), and Vc is the volume of the simulation cell. δ=

Ev = V

n

∑i = 1 ⟨Ei −

Ec

n⟩

⟨Vc⟩

(8)

Following precedence from the literature (see Table 1), the Flory−Huggins parameter was calculated from MD using eqs 5 and 9, a simplification of eq 7, that assumes that there is no change in volume upon mixing ij 1 yz z(E − E ΔHmix = jjj drug − Epolymer ) j V zzz mix k mix {

(9)

From this relation, the value of ΔHmix can then be evaluated using the average potential energy of the drug, polymer, and mixed system for E and the volume of the unit cell in the mixed simulations.19 The Flory−Huggins parameter χMD has been calculated from MD simulations of 50% w/w drug−polymer ratio mixtures of the nine drugs and unmodified PGA, using eqs 5 and 9 at T = 298 K. In terms of the free energy of mixing, the miscibility limit corresponds to the value of ϕdrug when ΔGmix changes from negative, favoring mixing, to positive, favoring phase separation, i.e., when ΔGmix = 0. By setting eq 4 equal to 0 and using the volume of a PGA repeating unit (0.283 nm3) for the volume of a lattice site and using the experimental miscibility limits for ϕdrug and (1 − ϕdrug) for ϕpolymer, we can model the value of χ at room temperature to obtain a value for χexp.

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RESULTS Determination of Experimental Miscibility. The use of nonglass forming Class I drugs allows us to use this estimation of the miscibility limit with confidence, as it is unlikely that a printed microarray spot, that is uniformly amorphous under POM, would contain separate regions of amorphous drug material and polymer if the drug is not able to form a glass in isolation. All of the PGA polymers have a low Tg and are liquid at room temperature. This ensures that there is minimal kinetic trapping of the API within the polymers. Given the strong propensity for G


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Figure 4. ΔGmix/RT against the drug fraction calculated using χMD and eq 4 for unmodified PGA. Figure 3. Ra calculated from MD data against the experimental miscibility limit.

missing values illustrate one of the problems of GC methods: generally, they rely on “black box” software where the user is dependent on a predetermined database which cannot easily be extended for new molecular types or chemical families. Flory−Huggins Parameters. The Flory−Huggins parameter has been calculated from MD simulations of 50% w/w drug−polymer ratio mixtures of the nine drugs and unmodified PGA, using eqs 5 and 9 at T = 298 K (Table 4). In terms of the Table 4. χ Calculated from MD (χMD) and Experimental Miscibility Limit Data (χexpt) for Drugs and Unmodified PGAa drug

χMD

χexpt

ΔχMD‑expt

aspirin caffeine CBZ finasteride flufenamic acid flutamide mefenamic acid salicylamide theophylline

−0.01 −0.73 −1.97 0.47 -9.50 0.94 -0.89 2.91 -0.76

1.47 3.11 0.44 1.09 0.77 0.48 2.46 2.52 3.87

1.48 3.84 2.41 0.62 10.27 0.46 3.35 0.39 4.63

Figure 5. ΔGmix/RT against the drug fraction calculated using χexpt and eq 4 for unmodified PGA.

ΔχMD‑expt is the absolute difference between the two values of χ.

a

free energy of mixing, the miscibility limit corresponds to the value of ϕdrug when ΔGmix changes from negative, favoring mixing, to positive, favoring phase separation, i.e., when ΔGmix = 0. By setting eq 4 equal to 0, using the volume of a PGA repeating unit (0.283 nm3) for the volume of a lattice site and using the experimental miscibility limits from Table 3 for ϕdrug and (1 − ϕdrug) for ϕpolymer, we can model the value of χ at room temperature (Table 4). The energy of the mixing diagrams shown in Figures 4 and 5 do not correspond, and therefore the predicted values of χMD do not correlate to χexpt (Figure 6). Figure 4 shows ΔGmix/RT as a function of ϕdrug using χ calculated from MD, and Figure 5 shows the same value calculated from experimental data. Flutamide, for example, is predicted to have a positive free energy of mixing and be incompatible with PGA, but experimentally it has a high miscibility limit of 83% drug loading; similarly, theophylline, mefenamic acid, and caffeine poorly mix with PGA and begin to recrystallize at 5−8% drug loading, but they are predicted here to have a comparable

Figure 6. Comparison of χMD and χexpt for the drugs with unmodified PGA.

response to the highly miscible aspirin and carbamazepine. Six of the drugs have negative values for χMD, indicating complete miscibility across all values of ϕdrug, which is not seen experimentally for any drug−polymer pairing.

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DISCUSSION Many areas of drug delivery involving polymers are dependent on utilizing or understanding polymer−drug compatibility. This is particularly important in solid dispersions where a stable and homogeneous formulation is required. Within this field, the H


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and 0.10 for MD vs HSPiP, HSPiP vs ProPred, and MD vs ProPred, respectively. From the review of literature in the Introduction, a problem of overfitting data and overinterpretation of theory in this area has been identified. For example, Greenhalgh11 is a widely cited paper (212 citations on Web of Science, July 2017) that is the source of the rule that Δδ less than 7.0 MPa1/2 is expected to be miscible and Δδ greater than 10 MPa1/2 is expected to be immiscible. However, this is a study of dispersions of ibuprofen and xylitol in six other sugars (sucrose, xylose, maltose, mannose, sorbitol, and dextran) and three PEG-based polymeric surfactants (lutrol F68, polysorbate, and polyoxyethylene 40 stearate). While it is correct that in this particular study there was a trend where Δδ < 7.0 MPa1/2 corresponded to miscibility and Δδ > 10.0 MPa1/2 to immiscibility, this work was limited to a very small chemical space of a single drug mixed with a single molecular family, and there is no justification for extrapolating it onto all drugs and potential excipients. Theoretical Limitations Mean that Solubility Parameters Are Unsuitable for Estimating Drug−Polymer Compatibility. By considering the fact that solubility parameters are the square root of the cohesive energy density, i.e., the intermolecular energy per unit volume, it can be understood why they are not able to correctly describe drug− polymer systems. Solubility parameters are based upon the idea that materials with similar δ values should be miscible. A material with a high δ requires more energy for dispersal than is gained by mixing with a material with a low δ, leading to immiscibility, and materials with similar values of δ gain sufficient energy on mutual dispersion to allow mixing.11,72 However, the solubility parameter approach fails to account for unique factors that might be important to mixing, such as temperature and concentration dependence, viscosity, ionic interactions, and electrostatics.26,73 Although the decomposition by the Hansen parameters tries to address this, in part, by considering the contribution to cohesive energy density by polar, dissipative and hydrogen-bonding forces, it cannot take account of compensatory interspecies interactions, such as the presence of hydrogen bond donors in one material and acceptors in the other. A more fundamental problem with the approach is that cohesive energy density is not a defined concept for molecules that cannot be vaporized, such as polymers, and there is no agreed way to split δ into the three Hansen components. State of the art methods to experimentally determine cohesive energy density, and thus solubility parameters, are inverse gas chromatography,74 where data can easily be misinterpreted due to the tendency of probe molecules to depart from ideal gas behavior,75 or indirect techniques that estimate a Hansen sphere by measuring phase separation in a range of solvents and then inferring values for δ from these reference solutions.76 Both of these methods require extensive reference data and parametrization, and they can only be considered as internally consistent; there is little justification for extrapolation into new chemical space, such as the field of drug−polymer interactions, as the way that cohesive energy density is defined leads to different solubility parameter values. The theoretical limitation of solubility parameters, i.e., that they are based upon the cohesive energy density of pure compounds, the energy of moving a molecule from the bulk to vacuum, and therefore can neither include any interspecies interactions nor have an agreed meaning for polymers, explains

polymers are often chosen to kinetically trap the drugs using a high Tg polymer to prevent phase separation and maintain amorphicity. Additionally in these studies, drugs are often chosen which tend to remain amorphous, leading to difficulties in establishing whether the presence of a polymer or the preparation method is responsible for stabilizing the drug in an amorphous phase.67−69 However, it would be more effective to use polymers which were miscible with the drug to achieve this outcome. The drugs in the present study were also selected to be ones that crystallize readily, so providing a further challenge to amorphicity. Different workers within this field have used both solubility parameters and the Flory−Huggins χ parameter to screen or predict outcomes of miscibility of different polymer−drug mixtures with varying degrees of confidence in their results. To understand the state of the art, we have conducted an evaluation of in silico methods for predicting drug−polymer compatibility. For this work, we have applied GC methods and MD simulation to a varied data set. Two of the polymers chosen are commonly used in formulating solid dispersions (PVP and PVP/VA), while the others are based on a new class of experimental polymers which can be easily modified to yield a range of different physicochemical properties (PGA variants).51 The drugs were chosen from a study which set out to derive a set of drugs from a varied chemical space for studying solid dispersions.70 As discussed in the Introduction to this article, repeatedly across the literature, the calculation of a limited number of solubility parameters or a single Flory−Huggins parameter is extrapolated to conclude that these techniques are useful for designing new drug delivery systems, despite no systematic evidence of predictive power.37,44−46,71 From the data presented in this current article, and given the inconsistencies highlighted in the review of literature in the Introduction, it is our conclusion that these methods cannot be trusted to predict drug−polymer interactions for the purposes of screening or excipient selection. There are two potential explanations for the failure of these methods to correctly predict drug−polymer miscibility. The first is that the values of the Hildebrand and Hansen parameters and χ estimated in silico are incorrect; the second is that, even if correct values are calculated, they are not a valid description of the systems under study. Accuracy of the Estimation of Solubility Parameters and Reproducibility of Data. Tables S2−S4 give the raw Hansen and Hildebrand parameters calculated via the two GC methods and by MD. Choosing mefenamic acid as an example, as fenamates are a well-known and widely studied class of drug, there is little consistency in the values for the Hildebrand parameter (ProPred, 14.2 MPa1/2; HSPiP, 25.5 MPa1/2; and MD, 13.6 MPa1/2), despite this value representing a real, physical quantity (square root of the cohesive energy density). There is also no agreement on how the cohesive energy density can be decomposed into the constituent parts described by the Hansen parameters, with the ratio of δdisp:δpolar:δhbond as 2.5:1.0:1.6 for HSPiP, 4.8:1.0:1.8 for ProPred, and 1.8:1.0 for MD (MD only includes δdisp and δpolar terms as hydrogen bonding is treated implicitly as part of the electrostatics in modern force fields). Figures S5−S7 compare Δδ for all drug− polymer combinations calculated via the three methods; as these methods should be different ways of calculating the same parameter, a positive linear correlation is expected with a slope of one and an intercept through the origin. However, the consistency of the predictions is poor, and no correlation is observed; the values for r2 from a linear regression are 0.05, 0.20, I


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Molecular Pharmaceutics Table 5. Values of α and β from Journal Articles that Use the Melting Point Depression Method of Lin and Huang6a lead author and reference Lin6 Tian82 Donnelly83 Baghel84

Li85 Bansal45

drug

polymer

value of β

value of α

percentage β at 300 K* (%)

felodopine felodopine felodopine felodopine felodopine dipyridamole cinnarizine cinnarizine felodopine aceclofenac

PAA HPMCAS soluplus PVP/VA PVP PAA PVP PAA Eudragit EPO soluplus

−18.84 −18.77 −14.42 −13.89 −33.55 −10.68 −8.53 −19.11 −126.56 −3.39

8105.0 7830.4 5744.7 4984.4 13699.7 3247.3 3161.2 7520.1 53038.6 1709.0

41 42 43 45 42 49 45 43 42 37

(*) Percentage β contributes to the sum of the absolute values of β and α at 300 K.

a

many variables, including temperature and the nature of specific interactions. Estimates of χ from MD Simulations Are Not Reliable. In the present study, the values of χ calculated from MD did not correlate with the values of χ calculated from the experimentally derived drug−polymer miscibility limits. A possible reason is that the force field parameters for the drug molecules were not verified and used directly from the ParamChem server without further optimization, and therefore the simulations may not adequately describe the molecular systems. However, flutamide, for example, has the best parameter and charge scores (Table 1, Supporting Information), meaning it is likely to be a reasonable parametrization, yet χflutamide‑PGA from MD is 0.94, predicting immiscibility, in contrast to an experimentally high miscibility of 83%. A more significant problem with using MD to calculate χ is that the method used here, and across the literature,20,21,39,81 is still based upon eq 6, and only considers simple enthalpic contributions and not the corrective entropy term. The first two terms of eq 4 describe the contribution of combinatorial entropy to the free energy of mixing; however, these terms do not account for entropy, or other energetic effects, arising from the specific chemical nature of the polymer, which are included as part of χ. For example, if a polymer is involved in a hydrogen bonding network with drug molecules that restrict the backbone into certain constrained conformations, this will have a configurational entropic penalty due to the reduced number of chain configurations available. Therefore, while evaluating ΔHmix from eq 7 does represent an improvement over using Δδ2, as it explicitly accounts for the energetic contributions from drug−polymer interactions, there is still no assessment of the corrective entropic term in eq 10a. Although it is difficult to estimate the size of the entropic component of χ, the melting point depression method of Lin and Huang6 may be illustrative. Using DSC to measure how the melting temperature of the drug in the drug−polymer mixtures (Tm) decreases with increased drug loading, eq 11 can be used to estimate χ(T) for a series of drug−polymer ratios

why this approach is fundamentally unsuitable for predicting the interactions of drug−polymer mixtures. Solubility Parameters Cannot Be Used To Reliably Estimate ΔHmix or χ. Across the literature, eq 5 has been used to estimate χ from solubility parameters 11,22,37,38,45 by substituting Δδ2 for ΔHmix. There are two issues with this method: the first is that it propagates the inaccuracies discussed in the previous subsection by using solubility parameters to evaluate the enthalpy of mixing. The second is that, even if solubility parameters are available that correctly evaluate the cohesive energy density of a material and ΔHmix, careful examination of the source citations shows that this is a misinterpretation of the Flory−Huggins theory. Four sources are given for this expression: Hansen;11 Hildebrand and Scott;77 Case and Honeycutt,78 who themselves cite Hildebrand and Scott; and Lindvig et al.,79 who do not cite anyone as a source for the relation. But the expressions relating solubility parameters to χ, reported by both Hansen and Hildebrand and Scott, include an entropic term that has been lost from later works that use them as a reference. Hildebrand and Scott provide eqs 10a−10c. Eq 10a, where χH and χS are the enthalpic and entropic contributions to χ respectively, is generally true, and allows for the fact that χ, χH, and χS can be functions. χS is described as the entropic term, but it may better considered as a general corrective term that includes an entropic component. Eq 10b is true over a limited temperature range, where it is reasonable to assume that χs is a constant independent of temperature; in this form, χH = αT and χS = β. χ = χH + χS

(10a)

χ=

α +β T

(10b)

χ=

VΔδ 2 + χS RT

(10c)

1 1 R ij 1 jjlnϕ + ijjj1 − yzzz(1 − ϕ) − 0 =− j Tm H m Δ Tm k { k y + χ (1 − ϕ)2 zzzz {

In the original text from 1950, Hildebrand and Scott themselves warn that eq 10c, where δ2 is used for ΔHmix, is “certainly incorrect in detail” but say that it had still been useful in interpreting quantitative data on polymer swelling. Hansen also reports eq 10c, and that χS (or β) has the “accepted value” of 0.34. The original source for this value, however, is a paper from 196480 that is concerned with the methodology for calculating χ and produces this value to fit the specific data set studied (a selection of nine polymers within common solvents); there is no suggestion that all of this should be considered a universal value for all polymer−solvent systems, as this term is a function of

(11)

where T0m is the melting temperature of the pure drug in the crystal, ΔH is the heat of fusion of the drug, ϕ is the volume fraction of drug, and m is the ratio of the volume of the polymer to the lattice site. These values of χ(T) are then fitted to eq 10b to find the values of α, the enthalpic term, and β, the entropic J


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which avoid these complications and offer the possibilities of greater accuracy. Some of these possibilities are discussed by Anderson, who makes a case for the use of PC-SAFT,2 the use of other equation of state theories, such as LCL,28 or the use of a quasi chemical partition function.87 In addition to changing the theoretical approach to polymer drug interactions, there is a widely recognized need to develop an in silico method for screening polymer excipients to identify those that might stabilize a drug molecule in an amorphous solid dispersion analogous with virtual screens to identify hit compounds that bind with a target protein. An ideal screening method would include the following attributes: (1) Completely ab initio: uses the chemical structure as the only input, no need for experimental information. (2) Able to discriminate between miscible and immiscible better than a random assignment. (3) A scoring function for ranking miscible “hits”. (4) Fast to run calculations. (5) Possible to automate and script to run calculations with minimal setup. We consider the first two criteria essential and the remaining three desirable. Currently there are no methods available that fulfill the first two criteria. The methods tested in this study can be described as truly ab initio, but they were unable to correctly predict, let alone score, drug−polymer compatibility. Currently, the state of the art methods for drug−polymer screenings are high-throughput solvent casting48 or 2D printing54 of a range of drug−polymer combinations, such as the method used in this article for assessing drug−polymer miscibility. New computational ideas and methods are needed, which take better account of both local entropic effects and intraspecies interactions, if the aim of drug−polymer screening in silico is to be achieved.

term, which also includes any other corrections needed for the free energy. (n.b. in the literature describing the melt depression method, α is usually referred to as B and β as A). Table 5 shows the values of α and β from a selection of journal articles that use the melting point depression method; the final column shows the relative proportions of the terms at 300 K, indicating that, while the temperature dependent enthalpic term is the dominant energetic component in χ, the corrective/entropic term also makes a large contribution and should not be neglected. Examining the final column of Table 4 shows that there is no consistency in the absolute difference of the values of χMD, which we would assume provides an estimate of χH, and χexpt, the latter including all components of χ, for each drug−unmodified PGA pairing. This suggests that the current method for estimating χ in silico is lacking a significant energetic component of the drug and polymer interaction. These analyses above, which are largely a response to the data generated in the present work, should not be viewed in isolation. Previous articles in the literature have also highlighted failings of both the solubility parameters and Flory−Huggins models in describing polymer−drug interactions accurately, e.g., Anderson.86 Many of the criticisms stem from the inability of either the solubility parameter or χ parameter approaches to take into account hydrogen bonding, which is a prominent feature in many polymer−drug combinations used in amorphous solid dispersions. A key criticism is the way these models constrain the value of χAB to positive values because of the geometric mean assumption in the solubility parameter theory. A further criticism relates to the assumption in the derivation of the Flory−Huggins χ parameter that all contact sites are available. However, the difficulty of pairing H-bonds becomes more difficult, especially at high concentrations, thus leading to nonrandom mixing and violating the mean field assumption.

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CONCLUSIONS The leading methods for predicting polymer−small molecule compatibility ab initio, that is methods that do not need any experimentally measured values as input, are not able to describe correctly the miscibility of a selection of nine nonglass forming drugs with six polymers. Despite numerous small-scale studies (see Table 1) that suggest successful application of these methods in the area of drug−polymer formulation for one or two pairings, this is the first systematic evaluation of solubility parameters and χ calculated in silico for a large, varied data set. Solubility parameter approaches are based upon estimating the cohesive energy density, the energy of vaporization per unit volume, and assume that the cost of removing a molecule from the pure bulk will be compensated for when placed in a material with a similar cohesive energy density. This cannot tell us about the free energy of the mixing of polymers with drugs as it neglects specific intraspecies interactions, such as hydrogen bonding, and furthermore, cohesive energy density is an undefined quantity for polymers. The Flory−Huggins theory of the free energy of mixing does take account of the entropic contribution of polymers and intraspecies interactions, but the computational methods widely in use for estimating χ have lost the corrective entropy term originally included by Hildebrand and Scott,73 seemingly through a series of citations that have not reviewed the original sources. These findings and analyses also complement other criticisms of these theories by other studies, and make the case that neither solubility parameters nor the F− H χ parameters have any place in the estimation of polymer− drug compatibility. We should therefore turn to other models

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.molpharmaceut.8b00637. Discussion of parametrization, ParamChem penalties for drug molecules and PVP/VA monomers, raw Hansen parameters calculated using the HSPiP program, raw Hansen and Hildebrand parameters calculated using ProPred, raw Hansen and Hildebrand parameters calculated using molecular dynamics, Δδ calculated from ProPred data against the experimental miscibility limit, Ra calculated from HSPiP data against the experimental miscibility limit, Δδ calculated from HSPiP data against the experimental miscibility limit, Δδ calculated from MD data against the experimental miscibility limit, comparison of Hildebrand parameters calculated from HSPiP vs ProPred, comparison of Hildebrand parameters calculated from MD vs HSPiP, and comparison of Hildebrand parameters calculated from MD vs ProPred (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Vincenzo Taresco: 0000-0003-4476-8233 Cameron Alexander: 0000-0001-8337-1875 K


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All-Atom Empirical Potential for Molecular Modeling and Dynamics Studies of Proteins. J. Phys. Chem. B 1998, 102, 3586−3616. (18) Case, D. A.; Cheatham, T. E.; Darden, T.; Gohlke, H.; Luo, R.; Merz, K. M.; Onufriev, A.; Simmerling, C.; Wang, B.; Woods, R. J. The Amber Biomolecular Simulation Programs. J. Comput. Chem. 2005, 26, 1668−1688. (19) Sun, H. COMPASS: An Ab Initio Force-Field Optimized for Condensed-Phase ApplicationsOverview with Details on Alkane and Benzene Compounds. J. Phys. Chem. B 1998, 102, 7338−7364. (20) Huynh, L.; Grant, J.; Leroux, J. C.; Delmas, P.; Allen, C. Predicting the Solubility of the Anti-Cancer Agent Docetaxel in Small Molecule Excipients Using Computational Methods. Pharm. Res. 2008, 25, 147−157. (21) Xiang, T. X.; Anderson, B. D. Molecular Dynamics Simulation of Amorphous Hydroxypropylmethylcellulose and Its Mixtures With Felodipine and Water. J. Pharm. Sci. 2017, 106, 803−816. (22) Xiang, T. X.; Anderson, B. D. Molecular Dynamics Simulation of Amorphous Indomethacin-Poly(Vinylpyrrolidone) Glasses: Solubility and Hydrogen Bonding Interactions. J. Pharm. Sci. 2013, 102, 876− 891. (23) Maniruzzaman, M.; Pang, J.; Morgan, D. J.; Douroumis, D. Molecular Modeling as a Predictive Tool for the Development of Solid Dispersions. Mol. Pharmaceutics 2015, 12, 1040−1049. (24) Forster, A.; Hempenstall, J.; Tucker, I.; Rades, T. Selection of Excipients for Melt Extrusion with Two Poorly Water-Soluble Drugs by Solubility Parameter Calculation and Thermal Analysis. Int. J. Pharm. 2001, 226, 147−161. (25) Greenhalgh, D. J.; Williams, A. C.; Timmins, P.; York, P. Solubility Parameters as Predictors of Miscibility in Solid Dispersions. J. Pharm. Sci. 1999, 88, 1182−1190. (26) Leroy, F. Molecular Driving Forces. Statistical Thermodynamics in Biology, Chemistry, Physics, and Nanoscience. Soft Mater. 2013, 11, 231−231. (27) Huynh, L.; Neale, C.; Pomès, R.; Allen, C. Computational Approaches to the Rational Design of Nanoemulsions, Polymeric Micelles, and Dendrimers for Drug Delivery. Nanomedicine 2012, 8, 20−36. (28) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. SAFT: Equation-of-State Solution Model for Associating Fluids. Fluid Phase Equilib. 1989, 52 (C), 31−38. (29) Lipson, J. E. G.; White, R. P. Connecting Theory and Experiment to Understand Miscibility in Polymer and Small Molecule Mixtures. J. Chem. Eng. Data 2014, 59, 3289−3300. (30) Santos, D. C.; Filipakis, S. D.; Rolemberg, M. P.; Lima, E. R. A.; Paredes, M. L. L. Asphaltene Flocculation Parameter in Brazilian Crude Oils and Synthetic Polar and Nonpolar Mixtures: Experimental and Modeling. Fuel 2017, 199, 606−615. (31) Alimohammadi, S.; Sayyad Amin, J.; Nikooee, E. Estimation of Asphaltene Precipitation in Light, Medium and Heavy Oils: Experimental Study and Neural Network Modeling. Neural Comput. Appl. 2017, 28, 679−694. (32) Xiao, T.; Zhang, X.; Wu, J.; Yang, J.; Yang, Y. Aliphatic-AlcoholInduced Opaque-to-Transparent Transformation and Application of Solubility Theory in a Bis-Dipeptide-Based Supramolecular Gel. ChemPlusChem 2017, 82, 879−887. (33) Radhakrishna, M.; Basu, K.; Liu, Y.; Shamsi, R.; Perry, S. L.; Sing, C. E. Molecular Connectivity and Correlation Effects on Polymer Coacervation. Macromolecules 2017, 50, 3030−3037. (34) Du, X.; Jiao, X.; Rechberger, S.; Perea, J. D.; Meyer, M.; Kazerouni, N.; Spiecker, E.; Ade, H.; Brabec, C. J.; Fink, R. H.; et al. Crystallization of Sensitizers Controls Morphology and Performance in Si-/C-PCPDTBT-Sensitized P3HT:ICBA Ternary Blends. Macromolecules 2017, 50, 2415−2423. (35) Wang, D.; Cornelius, C. J. Modeling Ionomer Swelling Dynamics of a Sulfonated Polyphenylene, Pentablock Copolymers, and Nafion. J. Polym. Sci., Part B: Polym. Phys. 2017, 55, 435−443. (36) Asgarpour Khansary, M.; Shirazian, S.; Asadollahzadeh, M. Polymer-Water Partition Coefficients in Polymeric Passive Samplers. Environ. Sci. Pollut. Res. 2017, 24, 2627−2631.

Charles A. Laughton: 0000-0003-4090-3960 Martin C. Garnett: 0000-0002-4365-4499 Notes

All raw data created during this research are openly available from the corresponding author ([email protected]. uk) and at the University of Nottingham Research Data Management Repository (https://rdmc.nottingham.ac.uk), and all analyzed data supporting this study are provided as Supporting Information accompanying this article. The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

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REFERENCES

This work was funded by EPSRC grant number EP/L013835/1. C.A. thanks EPSRC for a Leadership Fellowship grant number EP/H005625/1X. We thank Christine Grainger-Boultby, Tom Booth, and Paul Cooling for technical support. Also, thanks to Dipak Gordhan for help with printing some of the 2D arrays.

(1) Patnaik, S. S.; Pachter, R. A Molecular Simulations Study of the Miscibility in Binary Mixtures of Polymers and Low Molecular Weight Molecules. Polymer 2002, 43, 415−424. (2) Hancock, B. C.; York, P.; Rowe, R. C. The Use of Solubility Parameters in Pharmaceutical Dosage Form Design. Int. J. Pharm. 1997, 148, 1−21. (3) Liu, J.; Xiao, Y.; Allen, C. Polymer-Drug Compatibility: A Guide to the Development of Delivery Systems for the Anticancer Agent, Ellipticine. J. Pharm. Sci. 2004, 93, 132−143. (4) Di, L.; Kerns, E. H.; Carter, G. T. Drug-like Property Concepts in Pharmaceutical Design. Curr. Pharm. Des. 2009, 15, 2184−2194. (5) Leuner, C.; Dressman, J. Improving Drug Solubility for Oral Delivery Using Solid Dispersions. Eur. J. Pharm. Biopharm. 2000, 50, 47−60. (6) Lin, D.; Huang, Y. A Thermal Analysis Method to Predict the Complete Phase Diagram of Drug-Polymer Solid Dispersions. Int. J. Pharm. 2010, 399 (1−2), 109−115. (7) Hancock, B. C.; Shamblin, S. L.; Zografi, G. Molecular Mobility of Amorphous Pharmaceutical Solids Below Their Glass Transition Temperatures. Pharm. Res. 1995, 12 (6), 799−806. (8) Vo, C. L. N.; Park, C.; Lee, B. J. Current Trends and Future Perspectives of Solid Dispersions Containing Poorly Water-Soluble Drugs. Eur. J. Pharm. Biopharm. 2013, 85, 799−813. (9) Janssens, S.; Van den Mooter, G. Review: Physical Chemistry of Solid Dispersions. J. Pharm. Pharmacol. 2009, 61, 1571−1586. (10) ICH. European Medicines Agency; 2006; pp 1−15. (11) Zhang, X.; Xing, H.; Zhao, Y.; Ma, Z. Pharmaceutical Dispersion Techniques for Dissolution and Bioavailability Enhancement of Poorly Water-Soluble Drugs. Pharmaceutics 2018, 10, 74. (12) Hansen, C. Hansen Solubility Parameters; CRC Press, 2007. (13) Christen, M.; Hünenberger, P. H.; Bakowies, D.; Baron, R.; Bürgi, R.; Geerke, D. P.; Heinz, T. N.; Kastenholz, M. A.; Kräutler, V.; Oostenbrink, C.; et al. The GROMOS Software for Biomolecular Simulation: GROMOS05. J. Comput. Chem. 2005, 26, 1719−1751. (14) Jorgensen, W. L.; Maxwell, D. S.; Tirado-Rives, J. Development and Testing of the OPLS All-Atom Force Field on Conformational Energetics and Properties of Organic Liquids. J. Am. Chem. Soc. 1996, 118, 11225−11236. (15) Jorgensen, W. L.; Tirado-Rives, J. The OPLS Potential Functions for Proteins. Energy Minimizations for Crystals of Cyclic Peptides and Crambin. J. Am. Chem. Soc. 1988, 110, 1657−1666. (16) Brooks, B. R.; Bruccoleri, R. E.; Olafson, B. D.; States, D. J.; Swaminathan, S.; Karplus, M. CHARMM: A Program for Macromolecular Energy, Minimization, and Dynamics Calculations. J. Comput. Chem. 1983, 4, 187−217. (17) MacKerell, A. D.; Bashford, D.; Bellott, M.; Dunbrack, R. L.; Evanseck, J. D.; Field, M. J.; Fischer, S.; Gao, J.; Guo, H.; Ha, S.; et al. L


Article


Crystallization Tendency of Drug Molecules. Mol. Pharmaceutics 2014, 11, 3123−3132. (54) Taresco, V.; Creasey, R. G.; Kennon, J.; Mantovani, G.; Alexander, C.; Burley, J. C.; Garnett, M. C. Variation in Structure and Properties of Poly(Glycerol Adipate) via Control of Chain Branching during Enzymatic Synthesis. Polymer 2016, 89, 41−49. (55) Taresco, V.; Suksiriworapong, J.; Creasey, R.; Burley, J. C.; Mantovani, G.; Alexander, C.; Treacher, K.; Booth, J.; Garnett, M. C. Properties of Acyl Modified Poly(Glycerol-Adipate) Comb-like Polymers and Their Self-Assembly into Nanoparticles. J. Polym. Sci., Part A: Polym. Chem. 2016, 54, 3267−3278. (56) Taresco, V.; Suksiriworapong, J.; Styliari, I. D.; Argent, R. H.; Swainson, S. M. E.; Booth, J.; Turpin, E.; Laughton, C. A.; Burley, J. C.; Alexander, C.; et al. New N-Acyl Amino Acid-Functionalized Biodegradable Polyesters for Pharmaceutical and Biomedical Applications. RSC Adv. 2016, 6, 109401−109405. (57) Taresco, V.; Louzao, I.; Scurr, D.; Booth, J.; Treacher, K.; McCabe, J.; Turpin, E.; Laughton, C. A.; Alexander, C.; Burley, J. C.; et al. Rapid Nanogram Scale Screening Method of Microarrays to Evaluate Drug-Polymer Blends Using High-Throughput Printing Technology. Mol. Pharmaceutics 2017, 14, 2079−2087. (58) Van Eerdenbrugh, B.; Taylor, L. S. An Ab Initio Polymer Selection Methodology to Prevent Crystallization in Amorphous Solid Dispersions by Application of Crystal Engineering Principles. CrystEngComm 2011, 13, 6171−6178. (59) Parikh, T.; Gupta, S. S.; Meena, A. K.; Vitez, I.; Mahajan, N.; Serajuddin, A. T. M. Application of Film-Casting Technique to Investigate Drug-Polymer Miscibility in Solid Dispersion and Hot-Melt Extrudate. J. Pharm. Sci. 2015, 104, 2142−2152. (60) Marrero, J.; Gani, R. Group-Contribution Based Estimation of Pure Component Properties. Fluid Phase Equilib. 2001, 183−184, 183−208. (61) Hoy, K. L. New Values of the Solubility Parameters from Vapor Pressure Data. J. Paint Technol. 1970, 42, 76−118. (62) Van Krevelen, D. W.; Te Nijenhuis, K. Cohesive Properties and Solubility. Properties of Polymers 2009, 189−227. (63) Brooks, B. R.; Brooks, C. L.; Mackerell, A. D.; Nilsson, L.; Petrella, R. J.; Roux, B.; Won, Y.; Archontis, G.; Bartels, C.; Boresch, S.; et al. CHARMM: The Biomolecular Simulation Program. J. Comput. Chem. 2009, 30, 1545−1614. (64) Vanommeslaeghe, K.; Hatcher, E.; Acharya, C.; Kundu, S.; Zhong, S.; Shim, J.; Darian, E.; Guvench, O.; Lopes, P.; Vorobyov, I.; Mackerell, A. D. CHARMM General Force Field: A Force Field for Drug-like Molecules Compatible with the CHARMM All-Atom Additive Biological Force Fields. J. Comput. Chem. 2009, 31, 671−690. (65) Martinez, L.; Andrade, R.; Birgin, E. G.; Martínez, J. M. PACKMOL: A Package for Building Initial Configurations for Molecular Dynamics Simulations. J. Comput. Chem. 2009, 30, 2157− 2164. (66) Belmares, M.; Blanco, M.; Goddard, W. A.; Ross, R. B.; Caldwell, G.; Chou, S. H.; Pham, J.; Olofson, P. M.; Thomas, C. Hildebrand and Hansen Solubility Parameters from Molecular Dynamics with Applications to Electronic Nose Polymer Sensors. J. Comput. Chem. 2004, 25, 1814−1826. (67) Chokshi, R. J.; Shah, N. H.; Sandhu, H. K.; Malick, A. W.; Zia, H. Stabilization of Low Glass Transition Temperature Indomethacin Formulations: Impact of Polymer-Type and Its Concentration. J. Pharm. Sci. 2008, 97, 2286−2298. (68) Patel, R.; Patel, M. Preparation, Characterization, and Dissolution Behavior of a Solid Dispersion of Simvastatin with Polyethylene Glycol 4000 and Polyvinylpyrrolidone K30. J. Dispersion Sci. Technol. 2008, 29, 193−204. (69) Higashi, K.; Seo, A.; Egami, K.; Otsuka, N.; Limwikrant, W.; Yamamoto, K.; Moribe, K. Mechanistic Insight into the Dramatic Improvement of Probucol Dissolution in Neutral Solutions by Solid Dispersion in Eudragit e PO with Saccharin. J. Pharm. Pharmacol. 2016, 68, 655−664. (70) Nurzyńska, K.; Booth, J.; Roberts, C. J.; McCabe, J.; Dryden, I.; Fischer, P. M. Long-Term Amorphous Drug Stability Predictions Using

(37) Khansary, M. A.; Mellat, M.; Saadat, S. H.; Fasihi-Ramandi, M.; Kamali, M.; Taheri, R. A. An Enquiry on Appropriate Selection of Polymers for Preparation of Polymeric Nanosorbents and Nanofiltration/Ultrafiltration Membranes for Hormone Micropollutants Removal from Water Effluents. Chemosphere 2017, 168, 91−99. (38) Latere Dwan’Isa, J. P.; Rouxhet, L.; Préat, V.; Brewster, M. E.; Ariën, A. Prediction of Drug Solubility in Amphiphilic Di-Block Copolymer Micelles: The Role of Polymer-Drug Compatibility. Pharmazie 2007, 62, 499−504. (39) Mahmud, A.; Patel, S.; Molavi, O.; Choi, P.; Samuel, J.; Lavasanifar, A. Self-Associating Poly(Ethylene Oxide)-b-Poly(Cholesteryl Carboxylate-Caprolactone) Block Copolymer for the Solubilization of Stat-3 Inhibitor Cucurbitacin I. Biomacromolecules 2009, 10, 471−478. (40) Patel, S.; Lavasanifar, A.; Choi, P. Application of Molecular Dynamics Simulation to Predict the Compatability between WaterInsoluble Drugs and Self-Associating Poly(Ethylene Oxide)-b-Poly(εCaprolactone) Block Copolymers. Biomacromolecules 2008, 9, 3014− 3023. (41) Patel, S. K.; Lavasanifar, A.; Choi, P. Roles of Nonpolar and Polar Intermolecular Interactions in the Improvement of the Drug Loading Capacity of PEO-b-PCL with Increasing PCL Content for Two Hydrophobic Cucurbitacin Drugs. Biomacromolecules 2009, 10, 2584− 2591. (42) Maniruzzaman, M.; Morgan, D. J.; Mendham, A. P.; Pang, J.; Snowden, M. J.; Douroumis, D. Drug-Polymer Intermolecular Interactions in Hot-Melt Extruded Solid Dispersions. Int. J. Pharm. 2013, 443, 199−208. (43) Kasimova, A. O.; Pavan, G. M.; Danani, A.; Mondon, K.; Cristiani, A.; Scapozza, L.; Gurny, R.; Möller, M. Validation of a Novel Molecular Dynamics Simulation Approach for Lipophilic Drug Incorporation into Polymer Micelles. J. Phys. Chem. B 2012, 116, 4338−4345. (44) Thakral, S.; Thakral, N. K. Prediction of Drug-Polymer Miscibility through the Use of Solubility Parameter Based FloryHuggins Interaction Parameter and the Experimental Validation: PEG as Model Polymer. J. Pharm. Sci. 2013, 102, 2254−2263. (45) Eslami, M.; Nikkhah, S. J.; Hashemianzadeh, S. M.; Sajadi, S. A. S. The Compatibility of Tacrine Molecule with Poly(n-Butylcyanoacrylate) and Chitosan as Efficient Carriers for Drug Delivery: A Molecular Dynamics Study. Eur. J. Pharm. Sci. 2016, 82, 79−85. (46) Bansal, K.; Baghel, U. S.; Thakral, S. Construction and Validation of Binary Phase Diagram for Amorphous Solid Dispersion Using Flory−Huggins Theory. AAPS PharmSciTech 2016, 17, 318−327. (47) Erlebach, A.; Ott, T.; Otzen, C.; Schubert, S.; Czaplewska, J.; Schubert, U. S.; Sierka, M. Thermodynamic Compatibility of Actives Encapsulated into PEG-PLA Nanoparticles: In Silico Predictions and Experimental Verification. J. Comput. Chem. 2016, 37, 2220−2227. (48) Chakravarty, P.; Lubach, J. W.; Hau, J.; Nagapudi, K. A Rational Approach towards Development of Amorphous Solid Dispersions: Experimental and Computational Techniques. Int. J. Pharm. 2017, 519, 44−57. (49) Gupta, J.; Nunes, C.; Vyas, S.; Jonnalagadda, S. Prediction of Solubility Parameters and Miscibility of Pharmaceutical Compounds by Molecular Dynamics Simulations. J. Phys. Chem. B 2011, 115, 2014− 2023. (50) Paudel, A.; Van Humbeeck, J.; Van Den Mooter, G. Theoretical and Experimental Investigation on the Solid Solubility and Miscibility of Naproxen in Poly(Vinylpyrrolidone). Mol. Pharmaceutics 2010, 7, 1133−1148. (51) He, Y.; Ho, C. Amorphous Solid Dispersions: Utilization and Challenges in Drug Discovery and Development. J. Pharm. Sci. 2015, 104, 3237−3258. (52) Baird, J. A.; Van Eerdenbrugh, B.; Taylor, L. S. A Classification System to Assess the Crystallization Tendency of Organic Molecules from Undercooled Melts. J. Pharm. Sci. 2010, 99, 3787−3806. (53) Alhalaweh, A.; Alzghoul, A.; Kaialy, W.; Mahlin, D.; Bergström, C. A. S. Computational Predictions of Glass-Forming Ability and M


Article

Molecular Pharmaceutics Easily Calculated, Predicted, and Measured Parameters. Mol. Pharmaceutics 2015, 12, 3389−3398. (71) Ahmad, S.; Johnston, B. F.; Mackay, S. P.; Schatzlein, A. G.; Gellert, P.; Sengupta, D.; Uchegbu, I. F. In Silico Modelling of DrugPolymer Interactions for Pharmaceutical Formulations. J. R. Soc., Interface 2010, 7, S423−S433. (72) Hildebrand, J. H. Solubility. J. Am. Chem. Soc. 1916, 38, 1452− 1473. (73) DeBoyace, K.; Wildfong, P. L. D. The Application of Modeling and Prediction to the Formation and Stability of Amorphous Solid Dispersions. J. Pharm. Sci. 2018, 107, 57−74. (74) DiPaola-Baranyi, G.; Guillet, J. E. Estimation of Polymer Solubility Parameters by Gas Chromatography. Macromolecules 1978, 11, 228−235. (75) Huang, J.-C. Methods to Determine Solubility Parameters of Polymers at High Temperature Using Inverse Gas Chromatography. J. Appl. Polym. Sci. 2004, 94, 1547−1555. (76) Milliman, H. W.; Boris, D.; Schiraldi, D. A. Experimental Determination of Hansen Solubility Parameters for Select POSS and Polymer Compounds as a Guide to POSS-Polymer Interaction Potentials. Macromolecules 2012, 45, 1931−1936. (77) Porter, J. D. The Solubility of Nonelectrolytes, 3rd ed.; Hildebrand, J. H., Scott, R. L., Eds.; Reinhold: New York, 1951; Vol. 113, pp 450− 451. (78) Case, J. D.; Honeycutt, F. Will My Polymers Mix?: Methods for Studying Polymer Miscibility. Trends Polym. Sci. 1994, 2, 259. (79) Lindvig, T.; Michelsen, M. L.; Kontogeorgis, G. M. A FloryHuggins Model Based on the Hansen Solubility Parameters. Fluid Phase Equilib. 2002, 203, 247−260. (80) Blanks, R. F.; Prausnitz, J. M. Thermodynamics of Polymer Solubility in Polar and Nonpolar Systems. Ind. Eng. Chem. Fundam. 1964, 3, 1−8. (81) Xiang, T. X.; Anderson, B. D. Molecular Dynamics Simulation of Amorphous Hydroxypropyl-Methylcellulose Acetate Succinate (HPMCAS): Polymer Model Development, Water Distribution, and Plasticization. Mol. Pharmaceutics 2014, 11, 2400−2411. (82) Tian, Y.; Booth, J.; Meehan, E.; Jones, D. S.; Li, S.; Andrews, G. P. Construction of Drug-Polymer Thermodynamic Phase Diagrams Using Flory-Huggins Interaction Theory: Identifying the Relevance of Temperature and Drug Weight Fraction to Phase Separation within Solid Dispersions. Mol. Pharmaceutics 2013, 10, 236−248. (83) Donnelly, C.; Tian, Y.; Potter, C.; Jones, D. S.; Andrews, G. P. Probing the Effects of Experimental Conditions on the Character of Drug-Polymer Phase Diagrams Constructed Using Flory-Huggins Theory. Pharm. Res. 2015, 32, 167−179. (84) Baghel, S.; Cathcart, H.; O’Reilly, N. J. Theoretical and Experimental Investigation of Drug-Polymer Interaction and Miscibility and Its Impact on Drug Supersaturation in Aqueous Medium. Eur. J. Pharm. Biopharm. 2016, 107, 16−31. (85) Li, S.; Tian, Y.; Jones, D. S.; Andrews, G. P. Optimising Drug Solubilisation in Amorphous Polymer Dispersions: Rational Selection of Hot-Melt Extrusion Processing Parameters. AAPS PharmSciTech 2016, 17, 200−213. (86) Anderson, B. D. Predicting Solubility/Miscibility in Amorphous Dispersions: It Is Time to Move Beyond Regular Solution Theories. J. Pharm. Sci. 2018, 107, 24−33. (87) Prange, M. M.; Hooper, H. H.; Prausnitz, J. M. Thermodynamics of Aqueous Systems Containing Hydrophilic Polymers or Gels. AIChE J. 1989, 35, 803−813.

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