Cryst Growth Des.2017 Aug;17(8):4072-4083

**Exploring the shape influence on melting temperature, enthalpy and solubility of organic drug nano-crystals by a thermodynamic model**

*Gianluca Chiarappa ^{1}, Andrea Piccolo^{1}, Italo Colombo^{1}, Dritan Hasa^{2}, Dario Voinovich^{3}, Mariarosa Moneghini^{3}, Gabriele Grassi^{4}, Rossella Farra^{1}, Michela Abrami^{4}, Paola Posocco^{1}, Sabrina Pricl^{1}, Mario Grassi^{1*}.*

^{1} Dept. of Engineering and Architecture, Trieste University, Piazzale Europa 1, Trieste I-34127, Italy.

^{2} Dept. of Chemistry, Lensfield Road, Cambridge, CB2 1EW, United Kingdom.

^{3} Dept. of Chemical and Pharmaceutical Sciences, Trieste University, Piazzale Europa 1, Trieste I-34127, Italy.

^{4 }Dept. of Life Sciences, Cattinara Hospital, Trieste University, Piazzale Europa 1, Trieste I-34127, Italy.

**KEYWORDS** Melting, Drug, Nano-crystal, Mathematical model, Molecular dynamics.

**INTRODUCTION**

Drug bioavailability is defined as the rate and extent to which an active substance or an active moiety is absorbed from a pharmaceutical form and becomes available at the site of action. Its enhancement represents one of the most important challenges for what concerns orally administered drugs as 70% of potential drug candidates are discarded due to low bioavailability related with poor solubility in water. Typically, poorly soluble drugs (water solubility less than 100 mg cm^{-3}) include analgesics, cardiovascular drugs, hormones, antivirals, immune suppressants and antibiotics.

Several techniques are commonly employed to improve the bioavailability of poorly water-soluble drugs. In this frame, size reduction, drug encapsulation in the lipid matrix of nano/microspheres, drug complexation with cyclodextrins in solution or in the presence of the molten drug, drug dissolution in the dispersed lipophilic phase of oil/water emulsions/ micro-emulsions or drug loading into a polymeric matrix represent the most common. In particular, among other approaches such as solvent swelling, supercritical carbon dioxide and cryo-milling, drug grinding in presence of a stabilizing agent (typically a polymer) represents a profitable approach to increase the solubility of poorly soluble drugs. The advantage of this approach (co-grinding) consists in avoiding the use of harmful solvents whose elimination from the final formulation is very time and money consuming. Basically, co-grinding implies the reduction of drug crystals into nanocrystals that are embedded inside the stabilizing polymer matrix (nanocrystals are not stable and tend to come back to the more stable macro-crystal condition). The reason why nanocrystals are more soluble than macro-crystals relies on the different arrangement of surface and bulk phases. In fact, surface atoms/molecules have fewer bonds than the bulk ones. Consequently, surface lattice destruction necessitates less energy and is favored in comparison to the bulk one. Of course, the effect of surface atoms/molecules becomes significant and experimentally determinable only when the ratio between crystal surface/volume is not negligible, this occurring in the nanometer range. For the same reason, nanocrystals melting temperature (*T*_{m}) and enthalpy (Δ*H*_{m}) are lower than those of macro-crystals.

While the experimental determination of melting properties in connection with crystal size has been performed by many authors, the evaluation of solubility dependence on crystal dimension is still controversial due to experimental measurement difficulties. Indeed, manufacturing processes can introduce lattice defects, little impurities are able to affect solubility and poly-dispersed crystals experience Ostwald ripening (i.e., the growth of larger crystals at the expense of the smaller ones), this reflecting in an asymptotic solubility diminution. Thus, the theoretical determination of drug solubility vs. nano-crystals size has become mandatory.

The aim of our research was to build up a thermodynamic model able to account for melting temperature/enthalpy/solubility dependence on crystal size. In particular, the attention was focused on the fact that not only crystal dimension is important but also crystal geometry (sphere, cylinder, parallelepiped) can play an important role in determining nanocrystals melting temperature/enthalpy/solubility.

**THERMODYNAMIC MODEL**

**Melting Properties**

This model relies on the evaluation of the infinitesimal, reversible, variation of the internal energy, *E*, for closed systems composed of *k* components and 3 phases:

where *E*^{s}, *E*^{l} and* E*^{v} represent the internal energies of the solid, liquid and vapor phases, respectively, while *E*^{sv}, *E*^{sl} and* E*^{lv} are the internal energies of the solid/vapor, solid/liquid and liquid/vapor interfaces, respectively. The elaboration of Eq. (1) leads to the following equation relating the melting properties:

where Δ*H*_{m} is the specific melting enthalpy (J/kg), ρ_{s} and ρ_{l} are the density of the solid and liquid drug phases, respectively, γ^{lv}, γ^{sl}, *A*^{lv} and *A*^{sl} are the surface energy and the areas of the liquid/vapor and solid/liquid interfaces, respectively, while *V*^{v} and *V*^{s} are the vapor and solid volumes, respectively. Once crystal geometry is known, the two derivatives, d*A*^{lv}/d*V*^{v} and d*A*^{sl}/d*V*^{s}, can be evaluated and the model final form can be deduced. In order to simultaneously get the *T*_{m} and Δ*H*_{m} dependence on crystal dimension, it is necessary to evaluate Δ*H*_{m} dependence on *T*_{m}. At this purpose, the classic thermodynamic approach employed by Zhang and co-workers (Zhang, M. et al., Phys. Rev. B 2000, 62, 10548–10557), may be considered.

**Solubility**

Once *T*_{m} and Δ*H*_{m} dependence on crystal shape and dimension is known, the classical thermodynamic relation linking *T*_{m}, Δ*H*_{m} and solubility can be used:

where *X*_{d }is the drug molar solubility, γ_{d} is the drug activity coefficient, Δ*h*_{m} and Δ*c _{p}* are, respectively, the drug molar melting enthalpy and the difference between the solid (drug) and the liquid (drug) molar specific heat at constant pressure,

*M*

_{d}and

*M*

_{s}are, respectively, the drug and the solvent molecular weight, ρ

_{sol}is the solvent density,

*R*is the universal gas constant, while

*C*

_{s}is the mass/volume nano-crystal solubility. Assuming, for the sake of simplicity, that the activity coefficient, γ

_{d}, varies only slightly with concentration, Eq. (3) allows to evaluate the ratio

*C*

_{s}/

*C*

_{s∞}, where

*C*

_{s∞}is the mass/volume solubility of the infinitely large crystal.

**RESULTS AND DISCUSSION**

The proposed model was tested on poorly water soluble drugs such as nimesulide (non-steroidal anti-inflammatory drug with pain and fever reducing properties), nifedipine (a drug used to manage angina, high blood pressure, Raynaud’s phenomenon, and premature labor) and griseofulvin (antifungal drug used to treat a number of types of dermatophytoses). In all drugs, a clear dependence of *T*_{m} and Δ*H*_{m} on crystal shape and volume was observed. These outcomes were validated by means of molecular dynamics tests. Interestingly, we observed that, being constant the nanocrystal volume, dish-like nanocrystals show a more pronounced reduction of *T*_{m} and Δ*H*_{m} than rod-like nanocrystals while cubic nanocrystals are characterized by lowest reduction of *T*_{m} and Δ*H*_{m}. All these evidences can be simply explained looking at the nanocrystal surface/volume ratio. Indeed, this ratio is minimum for a cubic crystal while it rapidly increases for thinner and thinner dish-like nanocrystals, and it moderately increases for longer and longer rod-like nanocrystals. In other words, the fraction of surface atoms/molecules of a cubic crystal is minimum.

The findings of this paper reflect in two crucial and practical aspects characterizing the nano-crystals based delivery systems, i.e. the nano-crystals size distribution inside the polymeric carrier and nano-crystals water solubility.

Figure 1. Size distribution assuming nimesulide nanocrystals to be spherical (thick line) or cubic (thin line) in the case of a system realized by co-grinding nimesulide and polyvinylpyrrolidone (ratio 1:3 wt/wt) for half an hour in a planetary mill (Coceani, N. et al., Chem. Eng. Sci. 2012, 71, 345–355).R_{sphere}is the radius of the equivalent sphere sharing the same volume of the considered crystal.

Figure 1 clearly shows that nanocrystal shape is very important for the determination of nanocrystal size distribution inside the stabilizing polymer. When cubic nanocrystals are considered, the size distribution moves towards bigger dimension. Remembering that nimesulide nanocrystal looks like a cube rather than a sphere, the shifting towards bigger dimension seems absolutely reasonable as nimesulide unit cell half side is around 0.77 nm (= *R*_{sphere} in Figure 1). Thus, the nanocrystals size distribution referring to spherical nanocrystals seems too close to the nimesulide unit cell half side.

Figure 2. Ideal (i.e. constant activity coefficient γ_{d}) relative increase of griseofulvine (cubic shape) solubility (C_{s}) as a function of nanocrystal size represented by the radius of the equivalent sphere (R_{sphere}) having the same volume of the nanocrystal.C_{s∞}is the solubility of an infinitely large nanocrystal (i.e. a macrocrystal).

Figure 2 makes clear that when nanocrystal dimension (*R*_{sphere}) decreases, griseofulvine solubility can increase up to almost ten times the solubility of griseofulvine macrocrystals.

**CONCLUSIONS**

Despite the complexity of some theoretical/mathematical aspects, the developed model is very easy to use as it translates in a small executable file that can be run on every personal computer. The input file requires to know the geometrical crystal characteristics (the two ratios among the three parallelepiped dimension, or the height/radius ratio for a cylinder), the specific melting enthalpy (Δ*H*_{m∞}(J/kg)) an temperature (*T*_{m∞ }(K)) of the infinitely large crystal, the surface energy of the plane solid/vapor and liquid/vapor interfaces ( γ_{∞}^{SV }and γ_{∞}^{lv } respectively), the solid and liquid drug specific heat capacities at constant pressure (J/kg K) ( and respectively), whose difference is almost constant and temperature independent, and the density of the solid and liquid drug phases (ρ_{s} and ρ_{l}, respectively). In the case of the determination of the nanocrystals size distribution and nanocrystals mass fraction (*X*_{nc}) inside the particles of a polymeric stabilizing agent, it is also necessary having the Differential Scanning Calorimeter (DSC) trace reporting the relation between temperature (°C) and the power (mW) supplied by DSC. Bye means of these information, our model is able to provide in few seconds, the crystals size distribution, *X*_{nc}, the melting temperature/enthalpy and the solubility dependence on crystals size. All these information are very important for the optimization and the designing of nanocrystals based delivery systems.