Drug Aggregation of Sparingly-Soluble Ionizable Drugs: Molecular Dynamics Simulations of Papaverine and Prostaglandin F2α.

Aggregation in aqueous solution can have important implications on both the in vivo exposure of a drug and its pharmaceutical manufacturability. However, the drug aggregates formed can be very small and, thus, difficult to interrogate experimentally. On the other hand, at higher supersaturations where larger aggregates are supported, the chemical system is inherently metastable and therefore likewise challenging to study from an experimental standpoint. Understanding aggregation behavior is further complicated in the case of ionizable drugs where, unlike neutral compounds, there can be uncertainty in the kinds of drug molecules (i.e., charged, neutral, or both) that become incorporated into various clusters, particularly at pH values near the pKa. In this paper, we apply physics-based all-atom molecular dynamics (MD) simulations to study aggregation in the weakly basic drug papaverine and in the weakly acidic drug prostaglandin F2α. We employ in silico tools to construct simulation workflows and comprehensive cluster analysis protocols to elucidate the size distributions and dynamics of the drug aggregates formed at both an experimentally relevant concentration and at high supersaturation. We build on a previously published treatment [Solubility of sparingly soluble ionizable drugs. Adv. Drug Deliv. Rev. 2007, 59, 568-590, DOI: 10.1016/j.addr.2007.05.008] to translate the predicted aggregate distributions of each ionized drug into corresponding pH-solubility curves that can be compared directly to experiment. Our findings show that the assumption of a single predominant (charged) aggregate can be misleading in interpreting experimental pH-solubility curves, as it does not adequately reflect the rich diversity revealed in our simulations. Beyond not accounting for the distribution of ionized drug-containing clusters actually observed in solution, for both drugs we find evidence that neutral drug molecules can also participate in the aggregation phenomena. Notably, we observe that many drug molecules remain as free monomers in solution even under simulated conditions designed to mimic those where there is significant deviation of the experimental pH-solubility curve from the Henderson-Hasselbalch (H-H) equation, often taken to be a clear signpost of drug aggregation.

Estimating the maximal solubility advantage of drug salts.

Salt formation can enable the development of poorly water-soluble drugs containing at least one ionizable moiety. Not only can salts offer a solubility enhancement that can sometimes far exceed that of other commonly used solubilization strategies applied across the pharmaceutical industry, they can simultaneously bestow additional benefits such as providing low-cost formulation options. The goal of this work is to put forth a simple methodology to enable one to accurately predict the maximal solubility advantage of acidic and basic drugs whose unionized conjugate (neutral parent molecule) is poorly soluble. While published equations leveraging the Henderson-Hasselbalch/H-H relationship reasonably estimate the thermodynamic solubility limit (in systems where there is no supersaturation), under physiologically relevant conditions the maximal/kinetic solubility can play an important role in determining oral bioavailability, as in the case of amorphous drugs. Under these circumstances, a higher solubility can be maintained for short durations through drug supersaturation provided that the precipitation is slow, thereby causing deviations from H-H predictions. It is possible also that, in some instances, supersaturation could coincide with behavior previously attributed to drug aggregation in solution. The proposed methodology utilizes speciation across the pH range to allow one to determine the maximal amount of ionized and unionized drug in solution at each pH. The calculation is easily extended to cases where the counterion serves as a competing weak acid, weak base, or as a common ion. Additionally, a more thorough assessment of the Gibbs free energy change associated with the solubilization of salts is also presented, as this energy describes the key driving force for the recrystallization of the neutral parent by triggering its nucleation. Lastly, to demonstrate applicability to real-world compounds containing multiple ionizable moieties, the complex pH-solubility profile of a drug maleate salt taken from the literature is simulated.

Predicting the Solubility Advantage of Amorphous Drugs: Effect of pH.

The solubility enhancement generated by an amorphous phase over its crystalline counterpart is unaffected by the pH of the solution (at fixed ionic strength and temperature), even for a drug containing ionizable moieties, provided that both the ionized and neutral species generated through dissolution of the solids remain dissolved.

Modelling sub-micron particle slip flow in liquid chromatography.

The profoundly fast flow rates, observed at unexpectedly low column back-pressures, of capillaries packed with colloidal silica-based stationary phases is explained using slip flow theory applied to the particles themselves rather than the capillary walls. An equation is proposed that leverages nanoscale thermodynamics, geometrical considerations and Hagen-Poiseuille flow without the traditional no-slip boundary condition. It is found that the proposed model successfully fits/predicts real-world data taken from the literature, presented in a plot of the observed flow (enhancement) as a function of the (diminishing) mean particle size, so long as the particles do not exceed ~1 µm. This finding lends support to the continued use of colloidal silica packings in liquid chromatographic applications to achieve efficient separations of molecules on the smallest scale.

Modeling Recrystallization Kinetics Following the Dissolution of Amorphous Drugs.

Amorphous phases are frequently employed to overcome the solubility limitation that is nowadays commonplace in developmental small-molecule drugs intended for oral administration. However, since the solubility enhancement has finite longevity (it is a "kinetic solubility" effect), characterizing its duration (i.e., the so-called "parachute" effect) can be important for optimizing a formulation with regard to its in vivo exposure. Two semiempirical models, based on dispersive kinetics theory, are evaluated for their ability to precisely describe experimental transients depicting a loss in supersaturation (initially generated by the dissolution of the amorphous phase) over time, as the solubilized drug recrystallizes. It is found that in cases where the drug solubility significantly exceeds that of the crystal at longer times, the mechanism has substantial "denucleation" (dissolution) character. On the other hand, "nucleation and growth" (recrystallization) kinetics best describe systems in which the recrystallization goes to completion within the experimental time frame. Kinetic solubility profiles taken from the recent literature are modeled for the following drugs: glibenclamide, indomethacin, loratadine, and terfenadine. In the last case, a combination of three different kinetic models, two classical ones plus the dispersive model, are used together in describing the entire dissolution-recrystallization transient of the drug, obtaining a fit of R2 = 0.993. By precisely characterizing the duration of the "parachute" in vitro (e.g., under biorelevant conditions), the proposed models can be useful in predicting trends and thereby guiding formulation development and optimization.

Can we trust kinetic methods of thermal analysis?

Solid-state reactions and phase transformations in which the activation energy changes with the extent of conversion, e.g. multi-step or dispersive kinetic processes, cannot be reliably treated using current thermal methods, particularly when it comes to handling non-isothermal data sets. New models that consider distributed reactivity and/or the potential temperature-dependence of the pre-exponential term in the Arrhenius equation are urgently needed and, based on them, the extension of current theory to appropriately treat non-isothermal data. Numerical methods could prove useful in overcoming the variable-separability problem that can hinder the analytic solution of more complex kinetic relationships.

Predicting the solubility enhancement of amorphous drugs and related phenomena using basic thermodynamic principles and semi-empirical kinetic models.

The accurate prediction of the solubility enhancement offered by neat amorphous drugs and amorphous solid dispersions, over their crystalline (API) counterparts, has been discussed in several landmark works dating back at least two decades. Against this backdrop, an assessment of the current state-of-the-art for rigorously, yet simply (circumventing computational methods), determining the amorphous:crystalline solubility ratio based on thermo-analytical quantities is presented herein. Included in this work is a brief survey of the literature together with a discussion of the advantages and shortcomings of some of the most popular approaches, to-date. While the focus is on neat amorphous drugs, both before and after moisture sorption, the methodology presented is readily extended to more complex (e.g. ternary) systems that form a single, homogeneous phase. Six key questions are addressed in the context of how to most accurately determine the amorphous:crystalline solubility ratio: (1) How is the lattice energy of the crystalline phase assessed? (2) What is the role of heat capacity? (3) How does the pKa impact the solubility ratio prediction (for ionizable drugs)? (4) How does one incorporate the effects of moisture sorption on the amorphous phase? (5) How might one characterize (predict) the rate of drug recrystallization under various conditions (since the duration of the solubility enhancement is a kinetic phenomenon)? (6) What is the best approach for linking the (loss in) solubility enhancement to the Tg-lowering of the amorphous drug (by water) and vice-versa? In addressing these questions, this work aims to put forth a standardized methodology for determining the amorphous solubility enhancement with improved accuracy.

On the Stability of Nano-formulations Prepared by Direct Synthesis: Simulated Ostwald Ripening of a Typical Nanocrystal Distribution Post-nucleation.

Compared to more traditional top-down processing, the less common route to preparing drug nanocrystals through direct synthesis, e.g., starting with the nucleation of dissolved drug molecules (bottom-up processing), can offer both speed and cost advantages that makes it worthy of investigation. The current, theoretical work puts forth a technical basis and simulated results that could provide additional impetus for conducting further experimental work in this area. Specifically, an asymmetrical particle size distribution generated through the nucleation of a typical small-molecule drug, mirrored after carbamazepine hydrate based on a recent work [Skrdla PJ. J Phys Chem C 116:214-25, 2012], is subject to growth over time by a particle-coarsening mechanism using simulations of Ostwald ripening. Compared to a symmetrical, Gaussian distribution under the same conditions (fixed, relatively low concentration of free drug molecules in solution), it is found that, at longer times, the asymmetrical distribution formed through nucleation broadens more slowly. This finding could represent an additional benefit of synthetic strategies for the preparation of nano-formulations, previously unreported.

Predicted amorphous solubility and dissolution rate advantages following moisture sorption: Case studies of indomethacin and felodipine.

Water is often readily absorbed by amorphous compounds, lowering their glass transition temperature (Tg) and facilitating their recrystallization (via nucleation-and-growth). At the same time, the increase in moisture content translates to a decrease in both the thermodynamic solubility and intrinsic dissolution rate, as compared to the corresponding dry (pure) amorphous phase, e.g. see [Murdande SB, Pikal MJ, Shanker RM, Bogner RH. 2010. Solubility advantage of amorphous pharmaceuticals: I. A thermodynamic analysis. J Pharm Sci 99:1254-1264.]. In the case of pure indomethacin and felodipine, the solubility advantage of each amorphous phase over its crystalline counterpart were previously determined to be 7.6 and 4.7, respectively, using a new methodology together with basic calorimetric data taken from the literature. Herein, we demonstrate that, theoretically, following the uptake of just ∼0.5% w/w water, the solubility ratios decrease to 6.9 and 4.5, in the same order. Moreover, as the predicted intrinsic dissolution rate (based on the Noyes-Whitney equation) is directly proportional to the solubility advantage of a given amorphous-crystalline pair, it decreases proportionately upon moisture uptake. Applying the methodology presented herein, one can directly predict the extent of Tg-lowering observed at any moisture content, for a given amorphous phase. Knowing that value, it is possible to estimate the relative decrease in the solubility and/or intrinsic dissolution rate of the plasticized phase compared to the pure glass, and vice-versa.

The amorphous state: first-principles derivation of the Gordon-Taylor equation for direct prediction of the glass transition temperature of mixtures; estimation of the crossover temperature of fragile glass formers; physical basis of the "Rule of 2/3".

Predicting the glass transition temperature (Tg) of mixtures has applications that span across industries and scientific disciplines. By plotting experimentally determined Tg values as a function of the glass composition, one can usually apply the Gordon-Taylor (G-T) equation to determine the slope, k, which subsequently can be used in Tg predictions. Traditionally viewed as a phenomenological/empirical model, this work proposes a physical basis for the G-T equation. The proposed equations allow for the calculation of k directly and, hence, they determine/predict the Tg values of mixtures algebraically. Two derivations for k are provided, one for strong glass-formers and the other for fragile mixtures, with the modeled trehalose-water and naproxen-indomethacin systems serving as examples of each. Separately, a new equation is described for the first time that allows for the direct determination of the crossover temperature, Tx, for fragile glass-formers. Lastly, the so-called "Rule of 2/3", which is commonly used to estimate the Tg of a pure amorphous phase based solely on the fusion/melting temperature, Tf, of the corresponding crystalline phase, is shown to be underpinned by the heat capacity ratio of the two phases referenced to a common temperature, as evidenced by the calculations put forth for indomethacin and felodipine.

Practical Estimation of Amorphous Solubility Enhancement Using Thermoanalytical Data: Determination of the Amorphous/Crystalline Solubility Ratio for Pure Indomethacin and Felodipine.

Use of amorphous phases can mitigate the low in vivo exposures of poorly soluble, crystalline active pharmaceutical ingredients. However, it remains challenging to accurately predict the solubility enhancement offered even by a pure amorphous phase relative to the crystalline form. In this work, a methodology is presented that allows estimation of the amorphous:crystalline solubility ratio, α, using only measured thermodynamic quantities for each of the pure phases. With this approach, α values of 7.6 and 4.7 were calculated for indomethacin and felodipine, respectively, correlating more closely than previous predictions with the experimentally measured values of 4.9 and 4.7 reported in the literature. There are 3 key benefits to this approach. First, it uses simple mathematical functions to more precisely relate the temperature variations in the heat capacity (Cp) to allow a more accurate estimation of the configurational energy difference between the 2 phases, whereas traditional models typically assume that Cp of both phases are constant(s). Second, the Hoffman equation is leveraged in translating the free energy of crystal lattice formation to the actual temperature of interest (selected to be 25°C/298K in this work), again, for better accuracy. Finally, as only 2 modulated differential scanning calorimetry scans are required (one for each phase), it is attractive from an experimental simplicity standpoint.

Disproportionation of a crystalline citrate salt of a developmental pharmaceutical compound: characterization of the kinetics using pH monitoring and online Raman spectroscopy plus quantitation of the crystalline free base form in binary physical mixtures using FT-Raman, XRPD and DSC.

The crystalline citrate salt (CS) of a developmental pharmaceutical compound, MK-Q, was investigated in this work from two different, but related, perspectives. In the first part of the paper, the apparent disproportionation kinetics were surveyed using two different slurry systems, one containing water and the other a pH 6.9 phosphate buffer, using time-dependent measurements of the solution pH or by acquiring online Raman spectra of the solids. While the CS is generally stable when stored as a solid under ambient conditions of temperature and humidity, its low pHmax (<3) facilitates rapid disproportionation in aqueous solution, particularly at higher pH values. The rate of disappearance of the CS was found to obey first-order (Noyes-Whitney/dissolution rate-limited) kinetics, however, the formation of the crystalline product form in the slurry system was observed to exhibit kinetics consistent with a heterogeneous nucleation-and-growth mechanism. In the second part of this paper, more sensitive offline measurements made using XRPD, DSC and FT-Raman spectroscopy were applied to the characterization of binary physical mixtures of the CS and free base (FB) crystalline forms of MK-Q to obtain a calibration curve for each technique. It was found that all calibration plots exhibited good linearity of response, with the limit of detection (LOD) for each technique estimated to be ≤7 wt% FB. While additional calibration curves would need to be constructed to allow for accurate quantitation in various slurry systems, the general feasibility of these techniques is demonstrated for detecting low levels of CS disproportionation.

Roles of nucleation, denucleation, coarsening, and aggregation kinetics in nanoparticle preparations and neurological disease.

Kinetic models for nucleation, denucleation, Ostwald ripening (OR), and nanoparticle (NP) aggregation are presented and discussed from a physicochemical standpoint, in terms of their role in current NP preparations. Each of the four solid-state mechanisms discussed predict a distinct time dependence for the evolution of the mean particle radius over time. Additionally, they each predict visually different particle size distributions (PSDs) under limiting steady-state (time-independent) conditions. While nucleation and denucleation represent phase transformation mechanisms, OR and NP aggregation do not. Thus, when modeling solid-state kinetics relevant to NP processing, either the time evolution of the mean particle radius or the fractional conversion data should be fit using appropriate models (discussed herein), without confusing/combining the two classes of models. Experimental data taken from the recent literature are used to demonstrate the usefulness of the models in real-world applications. Specifically, the following examples are discussed: the preparation of bismuth NPs, the synthesis of copper indium sulfide nanocrystals, and the aggregation of neurological proteins. Because the last process is found to obey reaction-limited colloid aggregation (RLCA) kinetics, potential connections between protein aggregation rates, the onset of neurological disease, and population lifespan dynamics are suggested by drawing a parallel between RLCA kinetics and Gompertz kinetics. The physical chemistry underpinning NP aggregation is investigated, and a detailed definition of the rate constant of aggregation, k(a), is put forth that provides insight into the origin of the activation energy barrier of aggregation. For the two nanocrystal preparations investigated, the initial kinetics are found to be well-described by the author's dispersive kinetic model for nucleation-and-growth, while the late-stage NP size evolution is dominated by OR. At intermediate times, it is thought that the two mechanisms both contribute to the NP growth, resulting in PSD focusing as discussed in a previous work [Skrdla, P. J. J. Phys. Chem. C2012, 116, 214-225]. On the basis of these two mechanisms, a synthetic procedure for obtaining monodisperse NP PSDs, of small and/or systematically targeted mean sizes, is proposed.

Activation energy distributions predicted by dispersive kinetic models for nucleation and denucleation: anomalous diffusion resulting from quantization.

The activation energy distributions underpinning the two complementary dispersive kinetic models described by the author in a recent work (Skrdla, P. J. J. Phys. Chem. A 2009, 113, 9329) are derived and investigated. In the case of nucleation rate-limited conversions, which exhibit "acceleratory" sigmoidal transients (a kind of S-shaped stretched exponential conversion profile), an activation energy distribution visually similar to the Maxwell-Boltzmann (M-B) distribution is recovered, consistent with the original derivation of that model. In the case of predominantly "deceleratory" conversions, the activation energy distribution is skewed from normal in the opposite direction. While the "M-B-like" activation energy distribution supports the empirical observation of a rate enhancement as a function of the conversion time in nucleation rate-limited processes, the complementary distribution, with its pronounced low-energy tail, reflects a slow-down in the specific rate as the conversion progresses, consistent with experimentally observed denucleation rate-limited conversions. Activation energy distributions were also plotted for real-world data (Qu, H.; Louhi-Kultanen, M.; Kallas, J. Cryst. Growth Des. 2007, 7, 724), depicting the impact of various additives on the nucleation rate-limited kinetics of the solvent-mediated phase transformation of the crystalline drug carbamazepine. Last, by coupling the author's dispersive kinetic description of the time-dependent activation energy for nucleation to the classical description of the critical nucleus energy provided by the Kelvin equation, an accelerated hopping mechanism for the diffusion of monomers to the growing embryo surface was observed. That hopping mechanism was rationalized by modifying the Einstein-Smoluchowski (E-S) equation to allow it to describe the "supra-brownian" molecular motion thought to lie at the heart of nucleation kinetics.

Semi-empirical description of the constant ß in the equation of state for interfacial tension.

A semi-empirical expression for the constant, ß, in the Equation of State (EOS) for interfacial tension is proposed that exhibits a dependence on the surface area, heat capacity and thermal expansion coefficient of each of the materials involved in the three phase equilibrium (at constant temperature and pressure). The "wetting function", which yields values between zero and unity as the contact angle, θ, varies from 0° to 180°, is also employed in the description of ß. Empirical data is used to support that equation as well as the initially proposed relationship leading to its derivation that is based entirely on Classical Nucleation theory (CNT) outcomes and a "difference of the squares" correction term for the liquid-vapor and solid-vapor interfacial tensions.

Observation of oscillatory behavior during the dissolution of a pharmaceutical compound and evidence for the existence of an inverse Ostwald rule.

Oscillatory kinetic behavior is reported for the first time in a solid-state phase transformation of an organic compound, specifically, in the dissolution of Form V crystals of etoricoxib in neat toluene. The dissolution involves kinetics of bistability due to the generation of a second polymorph, Form I, during the process. While an attempt is made to reconcile the periodic behavior with the oscillatory rate coefficient predicted by time-dependent Marcus theory, more successful simulations of the data are obtained using superimposed, elementary dispersive kinetic models for both dissolution and nucleation-and-growth. Contrastingly, the dissolution of Form V in toluene containing a suspected small quantity of methanol/base contaminant shows that the phase transformation of Form V to the less-stable Form I crystals is rate-limiting and that the process is well described by a single, elementary dispersive kinetic model (i.e., no oscillations are observed). The latter observation lends support to the bistability hypothesis and provides evidence for the existence of a 'corollary' to Ostwald's rule of stages.

Crystallizations, solid-state phase transformations and dissolution behavior explained by dispersive kinetic models based on a Maxwell-Boltzmann distribution of activation energies: theory, applications, and practical limitations.

The potential applications of dispersive kinetic models range from solid-state conversions to gas-phase chemical physics and to microbiology. Here, the derivation and application of two such models, for use in solid-state applications, is presented. The models are based on the concept of a Maxwell-Boltzmann distribution of activation energies. The ability of the models to fit/explain an assortment of asymmetric, sigmoidal conversion-versus-time transients presented in the recent literature, as well as to provide physicochemical interpretations of the kinetics via the two fit parameters, alpha and beta, makes them a powerful tool for understanding nucleation/denucleation rate-limited processes that are involved in many phase transformations, dissolutions and crystallizations.

Use of a quality-by-design approach to justify removal of the HPLC weight % assay from routine API stability testing protocols.

Due to the high method variability (typically > or = 0.5%, based on a literature survey and internal Merck experience) encountered in the HPLC weight percent (%) assays of various active pharmaceutical ingredients (APIs), it is proposed that the routine use of the test in stability studies should be discouraged on the basis that it is frequently not sufficiently precise to yield results that are stability-indicating. The high method variability of HPLC weight % methods is not consistent with the current ICH practice of reporting impurities/degradation products down to the 0.05% level, and it can lead to erroneous out-of-specification (OOS) results that are due to experimental error and are not attributable to API degradation. For the vast majority of cases, the HPLC impurity profile provides much better (earlier and more sensitive) detection of low-level degradation products. Based on these observations, a Quality-by-Design (QbD) approach is proposed to phase out the HPLC weight % assay from routine API stability testing protocols.

A high-temperature liquid chromatographic reactor approach for investigating the solvolytic stability of a pharmaceutical compound and an investigation of its retention behavior on a C18-modified zirconia stationary phase.

The solvolysis kinetics of a developmental active pharmaceutical ingredient (API) were investigated using a high temperature (HT)-HPLC reactor approach to determine whether it might be possible to use the technique to efficiently screen the relative stabilities of typical APIs (particularly those that are stable at the column temperatures achievable on most HPLC systems and over durations of less than 60 min-a reasonable upper limit for typical method run time). It was discovered that the on-column API degradation kinetics better obeyed a second-order model than a first-order one. Employing a newly developed mathematical treatment, the apparent activation energy for the process was determined to be 85.7+/-1.6 kJ/mol; the apparent frequency factor was found to be (3.9+/-0.4)x10(4) s(-1). The retention mechanism of the API on the C18-modified zirconia column (ZirChrom) Diamondbond-C18) was investigated using a van't Hoff analysis. It was discovered that the logarithm of the retention factor (following correction for the gradient elution of the assay method) exhibited a quadratic dependence on the reciprocal of the absolute temperature. While the retention was found to be predominantly enthalpically driven over the majority of temperatures investigated in this study (ranging from 40 to 200 degrees C), a regression fit of the curve predicted a maximum at approximately 20 degrees C, indicative of a transition from predominantly enthalpically controlled retention to a mainly entropically driven mechanism. A table summarizing the thermodynamic retention parameters at each experimental column temperature is provided. Finally, the preliminary application of the HT-HPLC reactor approach to the study of degradation kinetics of other APIs is discussed in terms of some unexpected findings obtained using the zirconia column.

Comparison of two types of dispersive kinetic approaches in relation to time-dependent marcus theory.

Two different approaches presented in recent literature for the treatment of dispersive kinetics for a first-order (F1) conversion mechanism are compared from a physicochemical perspective. The author's approach is found to be successful in describing activation energy trends as a function of time that can be predicted from a simple extension of Marcus theory. Thus, the approach is considered to have a sound fundamental basis.