In silico screening for solid dispersions: the trouble with solubility

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In silico screening for solid dispersions: the trouble with solubility parameters and #FH Eleanor Turpin, Vincenzo Taresco, Wathiq Al-Hachami, Jonathan Booth, Kevin Treacher, Simone Tomasi, Cameron Alexander, Jonathan C Burley, Charles A. Laughton, and Martin C. Garnett Mol. Pharmaceutics, Just Accepted Manuscript • DOI: 10.1021/acs.molpharmaceut.8b00637 • Publication Date (Web): 24 Aug 2018 Downloaded from http://pubs.acs.org on August 25, 2018

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Molecular Pharmaceutics

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, UK

++

AstraZeneca, Macclesfield UK, SK10 RNA, UK

KEYWORDS: Hildebrand solubility parameter, Hansen solubility parameter, Flory Huggins parameter, Group Contribution methods, Molecular dynamics methods, Solid Dispersions.

ABSTRACT

The problem of predicting small molecule-polymer 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

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approach, nor 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. 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 crystals1. Miscibility is also important in a variety of pharmaceutical dosage forms;2 and more recently its importance has been recognised 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.


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These problems can be overcome by converting a crystalline drug into an amorphous drug that does not have to 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 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


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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 meta-stable phase where a drug-rich and polymer-rich phase co-exist 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


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cohesive energy density (CED), the energy per unit volume required to vapourise molecules in a condensed phase (Equation 1). (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), hydrogen bonding component (δhbond) and a dispersive component (δdisp) (Equation 2). (2) = + +

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 estimated from all-atom molecular dynamics (MD), a simulation method that propagates the evolution of a molecular system through time using classical mechanics.


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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 are 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 ∆δdrug-polymer less than 7.0 MPa1/2 is expected to be miscible and ∆δdrug-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 (Equation 2), however Charles Hansen12 advocates that the three parameters should be combined into a so-called radius Ra, using Equation 3, where a factor of four gives additional weighting to the dispersive term. (3) = 4

− "# $ +

+

− "# $

− "# $

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


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experimental data, the magnitude of Ra should still be indicative of compatibility and produce a ranking of most miscible drug-polymer combinations.

The Flory-Huggins parameter The Flory-Huggins theory is a statistical thermodynamics model that describes the behaviour of polymer-solvent solutions. The major result of Flory-Huggins theory is that the free energy of mixing, ∆Gmix, can be calculated using the following relation26 (4) ∆& ' ) "# ln ) "# ) ln ) = + ( , "# ,

+ -) "# )

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.


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A widely used relation for the Flory-Huggins parameter is given in Equation 5:27 (5) -=

. Δ0 ' (

where Vref is the volume of a lattice site, taken to be the volume of a monomer unit, and ∆Hmix is the enthalpy of mixing. One method extensively used for evaluating χ in silico for a drugpolymer system27 is to substitute ∆Hmix by ∆δ2 (Equation 6), the square of the difference in solubility parameters. (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 Equation 7 (7) Δ0 ' = 1

Δ Δ 2 − ) 1 2 '

Δ − ) "# 1 2 "#


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where 3 65 7 is cohesive energy density and φ is the volume fraction of the relevant material in 4

the mixed system. Unlike some other theoretical tools, such as equation of state models like 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 drugexcipient pairs into three categories by the size of ∆δ, stating that ∆δ < 2.0 MPa1/2 are fully


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miscible. However, this was not supported by the DSC results published alongside, which show that lacidipine is immiscible with PVP-30 despite ∆δ = 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 ∆δ2 as an estimate of the heat of mixing in Equation 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 loadings. This was followed up by two further studies using the MD method to calculate χ for mixtures of one of the same co-block 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


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and geometry optimisations of drug molecules with polymer dimers or trimers to identify and analyse 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 drugpolymer combinations), their proposed method is only suitable for analyzing materials already identified for use in solid dispersions and not for predicting drug-polymer choices.

Lead author and

#

Method(

# drugs reference

Calculated parameter(s) polymers

s)

Greenhalgh25

1

4

GC

Solubility parameters

Forster 24

2

6

GC


Dwan'Isa 38

8

3

GC

χ from solubility parameters

Patel 40

2

1

MD

χ from MD

Mahmud 39

1

3

GC


Patel 41

2

1

MD

χ from MD

Kasimova 43

4

1

MD

χ from MD

Thakral 44

3

1

GC


Xiang 19

1

1

MD

Solubility parameters, χ from MD Solubility parameters, χ from Eslami 45

1

2

MD MD

Maniruzzaman22

6

4

GC, QM



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Bansal46

2

1

GC


Erlebach 47

3

1

MD

χ from MD Solubility parameters, χ from

Xiang

21

1

1

MD MD

Chakravarty48

1

1

GC, MD


Gupta49

1

1

GC,MD


Paudel50

1

1

GC


Table 1. Summary of literature on predicting drug-polymer compatibility in silico.

Table 1 summarises 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 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, 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


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(indomethacin-PVP and felodipine-HPMC) so cannot be used to draw conclusions about the wider predictive power of this method. He and Ho 51 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 delivery26. 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 paper to address that need.

In this current work therefore, we calculate solubility parameters for nine drugs (aspirin, caffeine, carbamazepine, finasteride, flufenamic acid, flutamide, mefenamic acid, salicylamide and theophylline) and six polymers (PGA, PGA-C4, PGA-C8, PGA-Phe, PVP, PVP/VA) using group contribution methods and molecular dynamics 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 is compared with experimentally derived miscibility limits for all 54 pairings. This represents a far larger chemical space than studied previously, facilitated by the


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use of high throughput 2D printing that has rapidly provided us with a dataset to correlate with computational predictions.

MATERIALS AND METHODS Drugs and Polymers Drugs Caffeine (CAF) (CAS 58-08-2), 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 53078-9) were purchased from SIGMA-Aldrich (UK). Mefenamic acid (MEF) (CAS 61-68-7) and finasteride (FIN) (CAS 98319-26-7) were obtained from Alfa Aesar (US, Ward Hill, MA). Carbamazepine (CBZ) (CAS 298-46-4) was purchased from MP Biomaterials, LLC (France). Table 2 shows the structure and properties of the drug molecules used in this study. In 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 materials crystallize upon initial cooling and are considered non-glass formers; Class II materials form glasses upon melt-quenching, 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. The drug molecules chosen for this study are 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 towards crystallisation.


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Mass DRUGS

(Da)

Tm (°C)

Structure

Aspirin 180.20

135

194.19

237

236.27

192

372.55

259

281.23

135

Caffeine

Carbamazepine

Finasteride

Flufenamic acid


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Flutamide 276.21

110

241.29

230

137.10

139

180.20

272

Mefenamic acid

Salicylamide

Theophylline

Table 2. Properties of drug molecules used in this study. Mass and melting temperature (Tm) are taken from the Chemical Abstracts Service Registry, retrieved via SciFinder (scifinder.cas.org). Polymers Polyvinylpyrrolidone-vinyl acetate (PVP/VA) copolymer (Kollidon VA64) was received from BASF, whilst Polyvinylpyrrolidone was purchased from Sigma-Aldrich and both the materials were used as received. PVP and PVPVA 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 polymer with a free -OH on the repeating unit, that


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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 non-functionalized PGA.

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-C880% a = 0.2, b = 0.8; for PGA-Phe a = b = 0.5. Drug-polymer mixtures


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A miniaturized, high-throughput assay method was used to assess the miscibility limit of the drug-polymer mixtures. Using the method outlined in our previous paper,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 (Sigma-Aldrich 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 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 Polarising Optical Microscopy (POM) after 7 days (Advanced Polarizing HS1 Microscope, 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 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 crystallisation (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


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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 neighbouring 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 miscibility limit has previously been used by Parekh et al using a film casting technique.59

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 crystalisation (AAAC); slightly crystalline (AACC); (second line) mainly crystalline (ACCC); and completely crystalline (CCCC). Computational Methods Group Contribution Methods


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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/). 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 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 co-polymers such as modified PGA.

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. Hoy,61 van Krevelen.62 The recommended method for current calculations is the Yamamoto-Molecule Breaking algorithm (Y-MB). New molecules are split into sub-groups 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


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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 co-polymers. 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

Supplementary Material 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 chance 0.5 following a probability test; for the co-polymers, 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 PGA-C4-40% 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 1ns 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


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dynamics at 750 K with a time step of 1 fs. Periodic boundary conditions were applied to the unit cell, electrostatics were treated using 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 K 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, 300K) 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 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 Equation 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. (8) 8 = =

∑ ?@ 〈 − ;=

In silico screening for solid dispersions: the trouble with solubility

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