Drug Nanocrystals : Theoretical Background of Solubility Increase and Dissolution Rate Enhancement

Regardless of the administration route, the key factor for the success and reliability of any formulation is drug bioavailability, defined as the rate and extent at which the active drug is absorbed from a pharmaceutical form and becomes available at the site of drug action.1 Although metabolism and physiological factors highly affect drug absorption by living tissues, bioavailability is strongly dependent on drug permeability through cell membranes and on drug dissolution in physiological fluids. In particular, dissolution becomes the most important factor in terms of bioavailability for poorly water-soluble – highly permeable drugs (class II drugs according to the Amidon classification2). This aspect is of paramount importance if we know that, at present, about 40 % of the drugs being in the development pipelines are poorly soluble, up to 60 % of synthetized compounds are poorly soluble,3 and 70 % of the potential drug candidates are discarded due to low bioavailability related with poor solubility in water.4 Examples of commonly marketed drugs that are poorly soluble in water (less than 100 mg cm–3) include analgesics, cardiovasculars, hormones, antivirals, immune suppressants and antibiotics.5 Since it is well known that solubility increases with drug nanocrystal size decrease,5 this paper focuses on the theoretical relation existing between nanocrystal size, solubility and dissolution rate. To this end, a thermodynamic model relating to nanocrystal dimensions (radius), melting temperature/ enthalpy and solubility will be presented and discussed.


Introduction
Regardless of the administration route, the key factor for the success and reliability of any formulation is drug bioavailability, defined as the rate and extent at which the active drug is absorbed from a pharmaceutical form and becomes available at the site of drug action. 1Although metabolism and physiological factors highly affect drug absorption by living tissues, bioavailability is strongly dependent on drug permeability through cell membranes and on drug dissolution in physiological fluids.In particular, dissolution becomes the most important factor in terms of bioavailability for poorly water-soluble -highly permeable drugs (class II drugs according to the Amidon classification 2 ).This aspect is of paramount importance if we know that, at present, about 40 % of the drugs being in the development pipelines are poorly soluble, up to 60 % of synthetized compounds are poorly soluble, 3 and 70 % of the potential drug candidates are discarded due to low bioavailability related with poor solubility in water. 4Examples of commonly marketed drugs that are poorly soluble in water (less than 100 mg cm -3 ) include analgesics, cardiovasculars, hormones, antivirals, immune suppressants and antibiotics. 5nce it is well known that solubility increases with drug nanocrystal size decrease, 5 this paper focuses on the theoretical relation existing between nanocrystal size, solubility and dissolution rate.To this end, a thermodynamic model relating to nanocrystal dimensions (radius), melting temperature/ enthalpy and solubility will be presented and discussed.

Materials and methods
Vinpocetine (C 22 H 26 N 2 O 2 , molecular weight M w = 350.45(-)), a kind gift from Linnea SA ( Riazzino-Locarno, CH, Switzerland), is a semisynthetic derivative of the Vinca minor L. alkaloid vincamine 6 used for the treatment of cognitive disorders and related symptoms such as cerebral infarction, cerebral hemorrhage residual and cerebral arteries cirrhosis. 7This base-type drug (pKa = 7.1) 8 was chosen as a model drug because it is practically insoluble in water (1.6 mg mL -1 in pH = 7.4 at 37 °C). 9Vinpocetine melting temperature (T m∞ = 149.6 °C), melting enthalpy (DH m∞ = 94600 J kg -1 ) and DC p (difference in specific heat capacity at constant pressure between the liquid ( l p C ) and the solid ( s p C ), DC p = 374 J kg -1 °C-1 ) were measured by a differential scanning calorimeter (Mod.TA 4000, equipped with a measuring cell DSC 20 Mettler).
The calibration of the instrument was performed with indium.A sample, containing about 2 mg of vinpocetine was placed in pierced aluminum pans and heated at a scanning rate of 10 °C per minute under air atmosphere.Solid vinpocetine density (r s = 1268 kg m -3 ) was measured by a helium picnometer (Quantachrome, Boynton Beach, Florida) while liquid vinpocetine density (r l = 1217 kg m -3 ) was determined as the ratio between the molecular weight M w and the liquid vinpocetine molar volume (M v ).M v (= 287.5• 10 -6 m 3 mol -1 ) was evaluated according to the group contribution approach 10,11 implemented by the Breitkreutz software. 12etermination of the solid vinpocetine -vapor surface tension ( sv  g ) was performed according to the following equation of state: 13 where q i is the contact angle of liquid "i" on solid vinpocetine, b is a model fitting parameter and lv i g is the liquid i -vapor surface tension.Four liquids of decreasing polarity (1-water, lv 1 g = 72.8mJ m -2 ; 2-ethylene glycol, lv 2 g = 48 mJ m -2 ; 3-dimethyl sulfoxide, lv 3 g = 44 mJ m -2 ; 4-diiodomethane, lv 4 g = 50.8mJ m -2 ) 5 were considered.Eq. ( 1) fitting to experimental cos(q i ) yielded to sv  g = 38.4mJ m -2 and b = 1.5• 10 -4 (m 4 mJ -2 ).Contact angles (water, q 1 = 82°; ethylene glycol, q 2 = 61°; dimethyl sulfoxide, q 3 = 37°; diiodomethane, q 4 = 33°) were measured by the tensiometer G10, Kruss, GmbH, Hamburg, D. Determination of the liquid vinpocetine -vapor surface tension ( lv  g ) was performed according to the Parachor method: 14 The solid-liquid vinpocetine ( sl  g ) surface tension was evaluated according to the Young equation for pure substances (q = 0): 15 sl sv lv 2 7.2 (mJ m ) The solid vinpocetine -water ( sl 1 g ) surface tension was evaluated according to the Young equation 15 (q 1 = 82°): The Tolman length d 16 for vinpocetine (d is equal to 1/3 of vinpocetine molecule diameter 17 ) was evaluated resorting to the knowledge of liquid vinpocetine molar volume M v and assuming a spherical shape for the vinpocetine molecule: where N A is the Avogadro number.Finally, the Crystallographic Information File (CIF) of vinpocetine was extracted from the International Union of Crystallography database. 18Vinpocetine shows a monoclinic crystal system with the following unit cell parameters: a = 0.8907 nm, a a = 90°, b = 0.953 nm, b b = 106.57°,c = 0.11286 nm, g c = 90°, unit cell volume 0.9182 nm 3 .Vinpocetine unit cell corresponds to a sphere of radius approximately equal to 0.5 nm.

Drug dissolution
The dissolution of a solid in a solvent is a rather complex process determined by a multiplicity of physicochemical properties of the solute and solvent. 5Typically, for a fixed solid-solvent couple, solid solubility, liquid environment fluido-dynamic conditions, dissolution surface and solid wettability are the key factors affecting the dissolution kinetics. 5,19In addition, for what concerns solid solubility, temperature and pH play predominant roles whilst pressure is generally less important.However, pH becomes the key parameter for the crowded class of small organic drugs behaving as weak acids or bases. 20Indeed, as the solubility of the dissociated form can be much higher than that of the un-dissociated form, the relation between environmental pH and drug pKa being fundamental.For example, nimesulide (pKa = 6.5), 21a weak acid nonsteroidal anti-inflammatory drug, increases its solubility by about one order of magnitude, from pH = 1.2 (37 °C: solubility ~ 10 mg mL -1 ; nimesulide is un-dissociated) to pH = 7.5 (37 °C: ~ 100 mg mL -1 ; nimesulide is dissociated). 22When temperature and pH are fixed, it is well known that solubility also depends on the solid melting temperature and enthalpy 23 that, in turn, depend on solid crystals dimensions, as discussed in the next sections 4 and 5.In particular, the smaller the crystals, the lower their melting temperature/enthalpy and, consequently, the higher their solubility.Accordingly, in order to discern the effect of equilibrium (solid solubility) and kinetics (fluido-dynamic conditions, wettability, and dissolution surface) factors, let us focus on the dissolution process of a solid phase subdivided in many spherical particles, having the same diameter.In addition, we assume that particles do not aggregate during dissolution and that the initial amount of the solid phase (M 0 ) is so high that at equilibrium (i.e. when solid concentration reaches solubility in the liquid phase) the variation of the solid particles radius is negligible.Accordingly, for the generic solid particle, the physical situation we are referring to is depicted in Fig. 1.Dissolution can be considered as a consecutive process driven by four energy changes DE i : 1) Contact of the solvent with the solid surface (wetting), which implies the production of a solid-liquid interface starting from solid-vapor one (DE 1 = DE w ) 2) Breakdown of molecular bonds of the solid (fusion; (DE 2 = DE f )) 3) Solid molecules transfer to the solid-liquid interface (solvation; (DE 3 = DE s )) 4) Movement of the solvated molecules from the interfacial region into the bulk solution (diffusion; (DE 4 = DE d )).The sum of the energies relative to the four mentioned steps represents the total energy required for solid dissolution.Obviously, the higher the dissolution energy required, the lower the dissolution kinetics.In order to connect these four energy contributes to the equilibrium and the kinetics parameters ruling solid dissolution, we need a mathematical model linking the time evolution of the solid concentration profile (C(r)) in the stagnant layer with the solid molecules concentration in the bulk liquid (C b ).Obviously, C(r) is given by the solution of the so-called second Fick's law: where t is time, r is the radial position and D is the solid molecules diffusion coefficient in the stagnant layer (D can be considered constant in the stagnant layer thickness 5 ).Assuming that a pseudo-stationary condition can be rapidly achieved in the stagnant layer (this hypothesis is supported by the numerical solution of eq. ( 6) assuming usual values for D (~ 10 -10 m 2 s -1 ) 5 and stagnant layer thickness (h SL ) lower than 20 mm), eq. ( 6) becomes: The solution of eq. ( 7), subjected to the following initial boundary conditions: initial: boundary: leads to the solid molecule concentration profile C(r) in the stagnant layer surrounding the solid particles (see details in Appendix): where R c is the particle radius (supposed time independent; see Fig. 1), R s is the stagnant layer radius (supposed time independent; see Fig. 1), 1/k m is the solid-liquid interface mass transfer resistance while is the so-called dissolution constant. 5While eq. ( 8) simply states that the stagnant layer is initially drug empty, eq. ( 9) affirms that, in R s , the solid molecules concentration equals the bulk one.Finally, eq. ( 10) affirms that solid molecules flux at the solid-liquid interface (R c ) depends on both k m and the difference between drug solubility (C s ) and actual local drug concentration (C(R c )).An inspection of eq. ( 11) reveals that C(R c ) develops from a fraction of C s (for t = 0, C b = 0 and to C s for a very long time.This is due to the mass transfer resistance (1/k m ) that accounts also for solid surface wettability problems.Indeed, when this resistance is zero, In order to complete the dissolution model, it is necessary to consider the following equation:

F i g . 1 -Solid drug dissolution in a solvent liquid phase implies overcoming four energy barriers represented by solid wetting (DE w ), breakdown (fusion) of solid molecular bonds (DE f ), drug molecules solvation (DE s ) and drug molecules diffusion through the solvent stagnant layer to reach the bulk solvent (DE d ). These energies affect, in different manners, the mass transfer coefficient (k m ) at the solid/liquid interface, the dissolution constant k d and the drug solubility C s (DE f ). It is important to stress that, due to possible solid surface wetting problems, the solid molecule concentration at the solid/liquid interface (C(R c )) can be lower than C s , as shown by eq. (11).
Eq. ( 12) imposes that the increase rate of the solid molecules concentration in the bulk liquid (volume V) depends on the mass flux in R s and on the release surface ( 2 p s 4N R p ), being N p the number of particles considered and r s their density.Eq. ( 12) solution (see details in Appendix) is: Assuming, for the sake of simplicity, that the ratio R c /R s does not sensibly depend on R c , eq. ( 13) indicates that a reduction of particles radius (R c ), being constant V and M 0 , implies the improvement of the dissolution kinetics as the dissolution surface increases proportionally to M 0 /(R c r s ).However, it will be demonstrated in the following sections 4 and 5 that R c reduction also implies a dissolution kinetics increase due to C s increase.It is worth noting that this last effect becomes important only for very small particles (R c < 50 nm).On the basis of eq. ( 13) and Fig. 1, it is possible to argue that k m and, thus, 1/k m , depend on DE w , DE f and DE s (k m → ∞ when crystal breakdown is very fast, no wettability problems occur (solid/liquid contact angle less than ≈ 30°) and solvation is very rapid).At the same time, k d depends on DE d (the lower this energy, the higher the solid molecules mobility (D) in the stagnant layer), while C s depends only on DE f (at fixed temperature and pH) as discussed in sections 4 and 5.

Nanocrystal solubility dependence on melting enthalpy/temperature
The peculiar behavior of very small crystals is explained by the different characteristics of the surface and bulk phases.Indeed, due to their less confined arrangement (fewer inter-atomic bonds), 24 surface atoms/molecules are characterized by a higher energy content with respect to bulk atoms/molecules.Consequently, lattice breakdown on crystal surface would require less energy and would be favored over bulk lattice breakdown.This theoretical interpretation, supported by experimental data and molecular dynamics simulations, 25 clearly reveals the importance of surfaces as stand-alone phases.Obviously, the effect of surfaces becomes relevant (i.e., appreciable at the macroscopic level) only when the number of surface atoms/molecules is not negligible compared to that of the bulk atoms/molecules, which happens when the crystal surface/volume ratio is very high, i.e. when crystal size falls in the nanometer range. 26The immediate consequence of the different behavior of surface atoms/molecules is that nanocrystals are characterized by a decreasing melting temperature/enthalpy as the nanocrystal radius (R c ) decreases.The thermodynamic equilibrium condition between a liquid phase (solvent) and a solid phase (drug) yields to the C s dependence on solid crystals radius R c , provided that the solid surface properties are properly accounted for. 26,27Assuming that only the drug partitions between the two phases (this means that the solvent does not go inside the drug), the equilibrium condition requires that the solid drug fugacity ( s d f ) and liquid phase drug fugacity ( l d f ) are equal: where g d and X d are, respectively, the drug activity coefficient and solubility (molar fraction) in the solvent, l d f is the drug fugacity in the reference state, while s d f is the drug fugacity in the solid state.Thus, X d is given by: If we assume that l d f is the fugacity of the pure drug in the state of undercooled liquid at system temperature (T) and pressure (P), the ratio s l d d f f can be evaluated as the variation of the molar Gibbs free energy (Dg da ) between the state of undercooled liquid drug (state d) and solid (nanocrystalline) drug (state a) (see Fig. 2):

 
where R is the universal gas constant, while the enthalpy (Dh ba , Dh cb , Dh dc ) and entropy (Ds ba , Ds cb , Ds dc ) variations can be evaluated as: where T m is the nanocrystal melting temperature, s p c and l p c are, respectively, the solid (drug) and liquid (drug) molar specific heat at constant pressure while Dg m , Dh m and Ds m are, respectively, the molar nanocrystals melting Gibbs free energy, enthalpy and entropy.Upon rearrangement of eqs.( 16)- (19), drug solubility can be expressed as a function of determinable parameters: 26,27 where Dc p = l p c -s p c .
Finally, molar solubility X d can be expressed as the mass/volume solubility C s : where M d and M s are drug and solvent molecular weight, respectively, while r sol is solvent density.In order to understand the effect of nanocrystal radius R c decrease on X d or C s , the dependence of Dh m and T m on R c has to be necessarily expressed.

Melting enthalpy/temperature dependence on nanocrystal radius
The physics of nanocrystals melting can be described by the three different mechanisms 28 reported in Fig. 3.According to the Homogeneous Melting approach (HM), the solid (spherical) crystal is in equilibrium with a liquid (spherical) phase having the same mass and both are embedded in the vapor phase.The Liquid Skin Melting theory (LSM) assumes the formation of a thin liquid layer over the solid core.The thickness of the liquid layer remains constant until the solid core completely melts.On the contrary, in the Liquid Nucleation and Growth approach (LNG), the liquid layer thickness increases as it approaches the melting temperature.Accordingly, the solid core melting takes place when the liquid layer thickness is no longer negligible in comparison to the solid core radius.
In a previously published work, 26 we theoretically demonstrated the connection existing among melting temperature/enthalpy and nanocrystals radius (R c ).In particular, efforts were made to properly express the infinitesimal, reversible, variation of the internal energy E (dE) referred to the closed system composed of solid drug, liquid drug and vapor drug.In that demonstration, the connection among dE and the LSM, LNG, HM frames (i.e. the relations existing among the solid, liquid and vapor phases; see Fig. 3) was introduced only in a late stage of the demonstration.In this work, on the contrary, we would like to follow a more physically-oriented strategy aimed to immediately consider the relations existing among the three phases according to the LSM, LNG, HM mechanisms.We believe that the coincidence of the final results, coming from the two approaches, strengthens the whole theoretical approach.

LSM and LNG mechanisms
The starting point is the definition of dE for the closed system (no matter or energy exchanges with Homogeneous Melting Liquid Skin Melting Liquid Nucleation and Growth the surroundings, i.e. system volume, entropy and moles number are constant) composed of one component and 3 phases (s solid, l liquid, v vapor) 15 whose spatial organization is that of the LNG and LSM melting mechanisms (see Fig. 3): where E s , E l , E v , E sl , E lv represent the solid, liquid, vapor, solid/liquid and liquid/vapor phase internal energy, respectively.Assuming that a) the contribution of the first and the second curvatures to system internal energy, E, is negligible (this is strictly true for planes and spheres), and that b) thermal and chemical equilibrium is attained among the bulk and surface phases (same temperature and chemical potential in all bulk and surface phases), eq. ( 22) becomes: where P is pressure, V is volume, A is interfacial area, g is interfacial tension while superscripts sl and lv are solid-liquid and liquid-vapor, respectively.In the light of the closed system hypothesis (dV l = -dV v -dV s ), we have: The system equilibrium condition (dE = 0) implies:   These relations can be inserted in the Gibbs-Duhem equation referring to the solid, liquid and vapor phases: where s s , s l and s v indicate the molar entropy of the solid, liquid and vapor phases, respectively, m 1 is the chemical potential (equal for all three phases in hypothesis (b)), v s , v l and v v are, respectively, the molar volume of the solid, liquid and vapor phases, and T is temperature.Subtracting eq. ( 28) from eq. ( 27) and eq.( 29) from eq. ( 28) leads to: Inserting in eqs.( 30)-( 31) the expressions of dP v and dP s coming from eqs. ( 25) and ( 26) leads to:   Equating eqs.( 32) to (33), we have: where Ds m is the molar melting entropy (s l -s s ).In order to make eq.( 35) operative, it is necessary to evaluate the two differential terms involving areas, volumes and surface tensions.In particular, for spherical crystals, we have: where R sl and R lv represent the curvature radius proper to the solid-liquid and liquid-vapor interface, respectively (see Fig. 3).In addition, the surface tension g dependence on the interface curvature radius, as suggested by Tolman, 16 is as follows: where g ∞ and g are, respectively, the surface tension competing to a flat surface (infinite curvature radius) and a surface of curvature radius R c , while d is the Tolman length whose order of magnitude should correspond to the effective molecular diameter m d and it is usually assumed 17 to be m d /3.On the basis of eqs.( 36)-(39), eq. ( 35) becomes: where Dh m is the molar melting enthalpy.The integration of eq. ( 40) between the melting temperature of the infinitely large crystal (T m∞ , R lv →∞, R sl → ∞) and the melting temperature (T m ) of the crystal of radius R sl , leads to: or, expressing the enthalpy per unit mass instead of per mole: where r l and r s are, respectively the density of the liquid and solid phases.Establishing a relation between R lv and R sl is now essential.To this end, a generalized version of the strategy adopted by Coceani 27 can be undertaken.This approach relies upon the fact that, very often, drug crystals' melting occurs in the presence of a drug amorphous solid that becomes liquid before the melting temperature.Indeed, as the glass transition temperature of the amorphous drug is lower than the crystals' melting temperature (whatever R sl ), the amorphous drug will be liquid before the nanocrystals melting.Thus, at melting point, the nanocrystals mass fraction, X nc , can be evaluated as: Consequently, we have: It is easy to verify that the LSM condition occurs for X nc ≈ 1 (the amorphous, liquid phase is virtually absent; R lv ≈ R c ), while LNG condition takes place for X nc ≈ 0 (nanocrystal melting occurs in a virtually infinite liquid phase, R lv ≈ ∞).For 0 < X nc < 1, we can account for what usually happens during the melting of organic drug crystals embedded in a polymeric stabilizing carrier. 27When X nc is unknown, its determination implies an iterative procedure based on the knowledge of the differential scanning calorimetry trace referring to the melting of the amorphous/crystal drug. 26,27t is interesting to note that the results of the approach presented in this paper lead to what was previously demonstrated following a different path. 26

HM mechanism
In the case of the homogeneous melting, the expression of the infinitesimal, reversible, variation of the internal energy E reads: It is worth noting that eq. ( 45) differs from eq. ( 23) in that the term g sl dA sl is substituted by the term g sv dA sv .Indeed, in the HM case, the solid/liquid interface no longer exists (see Fig. 3), while it is replaced by the solid/vapor interface.Starting from eq. ( 45) and following a procedure analogue to that shown for the LSM and LNG approaches, we obtain: Remembering that in the HM case the solid and liquid phases are characterized by the same mass, the relation between R lv and R sl (= R c ) is:

Results
In order to integrate eq. ( 42) and ( 46), it is necessary to know the DH m dependence on R c and T m .Therefore, the thermodynamic relation used by Zhang and co-workers, 29 holding regardless of nanocrystal nature (organic or inorganic) and characterized by easily determinable parameters, seems appropriate: where DH m∞ is the specific melting enthalpy (J Kg -1 ) referred to the infinite radius drug crystal, DC p (J kg -1 K -1 ) is the difference between the liquid and solid drug specific heat capacity at constant pressure.As DC p is almost temperature invariant, the integral in eq. ( 48) can be approximated by DC p (T m∞ -T m ).Figs. 4A and 4B indicate, respectively, the reduction of vinpocetine melting temperature (T m ) and melting enthalpy (DH m ) with crystal radius (R c ) according to eq. ( 42) in the case of liquid skin melting (X nc = 1), liquid nucleation and growth (X nc ≈ 0) mechanisms.It can be seen that, for both mechanisms, an appreciable effect of R c reduction takes place for R c < 50 nm, while T m and DH m undergo a considerable reduction for R c < 10 nm.On the contrary, the effect of X nc seems limited, and this is mainly due to the fact that vinpocetine density does not significantly change from the solid to the liquid state (see eq. ( 42)).Interestingly, we found similar results using other poorly water soluble small organic drugs, i.e., nifedipine, griseofulvin 26 and nimesulide. 27In the case of griseofulvin and nifedipine, model predictions were substantially confirmed by means of the X rays evaluation of nanocrystal dimensions. 26ig. 5A shows, according to eq. ( 20), the increase of vinpocetine nanocrystal solubility (C s ) with R c reduction for two different X nc values (1, LNG; ≈ 0, LNG).Also in this case, an appreciable effect takes place for R c < 50 nm and the effect exerted by X nc is not so relevant.For the smallest nanocrystals, C s is about 3 times that of the infinitely large vinpocetine crystals (C s∞ ).
1][32][33] Some useful considerations can be drawn by making a comparison between our model and the well-known Ostwald-Freundlich model: 34 sl s 1 vs   20) and the T m and DH m trend shown in Fig. 4A, B. C s is the solubility of the vinpocetine nanocrystal of radius R c , while C s∞ is the solubility of an infinitely large vinpocetine crystal.X nc is the nanocrystal mass fraction (simulations referred to 37 °C).B) Increase of vinpocetine solubility according to our approach (eq.( 20); solid lines) and the Ostwald-Freundlich approach (eq.( 49); dashed line) (simulations referred to 37 °C).
Assuming T = 37 °C, M vs = (= 276 • 10 -6 m 3 mol -1 ) and the sl 1 g value evaluated in section 2 (Materials and Methods), eq. ( 49) prediction is shown in Fig. 5B (dashed line) jointly with the predictions of our approach (solid thin and thick lines).It can be noticed that for small crystals eq. ( 49) leads to an unreasonable solubility increase keeping in mind that, in general, the solubility increase of amorphous drugs (not too dissimilar from vinpocetine) have been theoretically estimated in the range (7 -100). 35On the contrary, the prediction of our model seems more reasonable.Nevertheless, we recognize that the difficulty connected to the experimental measurement of nanocrystals solubility leaves this question still open.

Conclusions
In this paper, we demonstrated the theoretical correctness of our previous approach aimed at the evaluation of melting temperature/enthalpy depression with decreasing nanocrystals radius.Indeed, by means of a more physically oriented approach, we confirmed the theoretical results of our previous work.In addition, considering these results in the frame of drug dissolution, we proved that nanocrystals can play a very important role in the pharmaceutical field as they can enhance the bioavailability of poorly soluble -highly permeable drugs.Indeed, the concomitant decrease in melting temperature and enthalpy has a double beneficial effect on drug dissolution and, ultimately, on drug bioavailability.On the one hand, melting enthalpy reduction (DH m = DE f at constant temperature and pressure, see Fig. 1) improves the dissolution process by reducing one of the most important energy-related steps involved in drug dissolution (i.e.breakdown of the crystalline network (step 2 in Fig. 1), connected to the mass transfer resistance 1/k m ).On the other hand, it increases drug solubility (see eq. ( 20) and Fig. 5A).As a result, the dissolution rate can be increased considerably.
Interestingly, despite the mathematical (not physical) complexity connected to the theoretical study of the relation existing between nanocrystals size and solubility, the preparation of drug nanocrystals is no longer problematic since many reliable technologies are available nowadays. 36,37This is of particular interest if we know that, until now, about 70 % of the potential drug candidates are discarded due to low bioavailability related with poor solubility in water. 4Obviously, the problem of bioavailability enhancement of poorly soluble drugs also requires the estimation of drug solubility in the drug amorphous state that can be considered as a nanocrystal of vanishing dimensions.In this respect, the work of Bogner and co-workers 35 is a very interesting theoretical contribution.

APPENDIX
The solution of eq. ( 7) implies a double integration in space: first: second: The integration constants A 1 and A 2 can be evaluated by means of the boundary conditions expressed by eqs.( 9) and (10): From eqs. ( 9) and ( 10), we have: The solution of the linear system represented by eqs.(A3) and (A4) leads to the determination of the integration constants A 1 and A 2 : Inserting these expressions for A 1 and A 2 into eq.(A2), leads to eq. (11).
Starting from eq. ( 11), it is possible to solve eq. ( 12) since we now have an analytical expression for the spatial concentration derivative: Substitution of eq.(A6) into eq.( 12) leads to: Remembering that the number of particles is given by N p = M 0 /((4/3)prR c 3 ) and that k d = D/h SL , eq. (A7) can be rearranged to give: This first order differential equation can be solved by means of standard techniques to give eq.( 13).-drug diffusion coefficient in the liquid stagnant layer, m 2 s -1 E -system internal energy, J E s -solid phase internal energy, J E l -liquid phase internal energy, J E v -vapor phase internal energy, J E sl -solid/liquid phase internal energy, J E lv -liquid/vapor phase internal energy, J M vs -solid vinpocetine molar volume, m 3 mol -1 M s -solvent molecular weight, -M w -drug molecular weight, -M 0 -initial solid drug amount considered in dissolution, kg N A -Avogadro number, -N p -numbers of solid spherical crystals involved in the dissolution process, -P -pressure, Pa P l -liquid phase pressure, Pa P s -solid phase pressure, Pa P v -vapor phase pressure, Pa P s -Parachor, J 0.25 m 2.5 /mol -1 r -radial position, m R -universal gas constant, J mol -molar liquid drug entropy, J mol -1 K -1 s s -molar solid drug entropy, J mol -1 K -1 s v -molar vapor drug entropy, J mol -1 K -1 t -time, s T -temperature, K T m∞ -melting temperature of an infinitely wide drug crystal, K V -bulk liquid volume, m 3 V l -liquid phase volume, m 3 V s -solid phase volume, m 3 V v -vapor phase volume, m 3 X d -drug molar fraction solubility, -X nc -nanocrystal mass fraction, -G r e e k l e t t e r s a -eq.( 12) parameter, s -1 a a -angle between directions a and b, degrees b -model fitting parameter (eq.( 1)) b b -angle between directions a and c, degrees d -Tolman length, m DC p -difference between the liquid and the solid drug specific heat capacity at constant pressure, J kg -1 °C-1 DE 1 = DE w -energy variation related to solid drug wetting, J DE 2 = DE f -energy variation related to the breakdown of molecular bonds of the solid, J DE 3 = DE s -energy variation related solid molecules solvation, J DE 4 = DE d -energy variation related to the movement of the solvated molecules from the interfacial region (solid drug-liquid) into the bulk liquid solution, J Dg da -molar Gibbs energy variation from state "d" (under-cooled liquid drug at T and P) and state "a" (drug nanocrystals at T and P), J mol -1 Dg m -melting molar Gibb energy, J mol -1 Dh da -molar enthalpy variation between state "d" (under-cooled liquid drug at T and P) and state "a" (drug nanocrystals at T and P), J mol -1 Dh ba -molar enthalpy variation between state "b" (drug nanocrystals at T m and P) and state "a" (drug nanocrystals at T and P), J mol -1 Dh cb -molar enthalpy variation between state "c" (liquid drug at T m and P) and state "b" (drug nanocrystals at T m and P), J mol -1 Dh dc -molar enthalpy variation from state "d" (under-cooled liquid drug at T and P) to state "c" (liquid drug at T m and P), J mol -1 Dh m -molar melting enthalpy of nanocrystals, J mol -1 DH m∞ -specific melting enthalpy of infinitely wide crystals, J kg -1 DH m -specific melting enthalpy of nanocrystals, J kg -1 Ds m -nanocrystals melting molar entropy, J mol -1 Ds da -molar entropy variation between state "d" (under-cooled liquid drug at T and P) and state "a" (drug nanocrystals at T and P), J mol -1 Ds ba -molar entropy variation between state "b" (drug nanocrystals at T m and P) and state "a" (drug nanocrystals at T and P), J mol -1 Ds cb -molar entropy variation between state "c" (liquid drug at T m and P) and state "b" (drug nanocrystals at T m and P), J mol -1 Ds dc -molar entropy variation from state "d" (under-cooled liquid drug at T and P) to state "c" (liquid drug at T m and P), J mol -1 g c -angle between directions b and c, degrees g d -drug activity coefficient, - g -solid drug -liquid drug surface tension (flat surface), J m -2 m 1 -chemical potential of species "1", J mol -1 n l -liquid drug molar volume, m 3 mol -1 n s -solid drug molar volume, m 3 mol -1 n v -vapor drug molar volume, m 3 mol -1 r l -liquid drug density, kg m -3 r s -solid drug density, kg m -3 r sol -solvent density, kg m -3 q i -contact angle of liquid "i" on solid drug, degrees R e f e r e n c e s

F i g . 2 -
Thermodynamic cycle used to estimate the ratio s l d d f f (solid drug fugacity in the reference state "a" / drug fugacity in the state of undercooled liquid (state "d")).This ratio is linked to the Gibbs free energy variation between states "a" and "d" (Dg da ), by the relation: s and h indicate, respectively, the molar Gibbs energy, entropy and enthalpy.T m is the nanocrystals melting enthalpy.Adapted from ref. 26.

F i g . 3 -
Melting mechanisms: Homogeneous Melting, Liquid Skin Melting, Liquid Nucleation and Growth.R v indicates the radius of the vapor phase while R sv , R lv and R sl are, respectively, the solid-vapor, liquid-vapor and solid-liquid interface radii.Adapted from ref.26.

F i g . 4 -
A) Reduction of melting temperature T m with crystal radius R c in the case of Vinpocetine.B) Reduction of the melting enthalpy DH m with crystal radius R c in the case of Vinpocetine.X nc is the nanocrystal mass fraction.Prediction according to eq. (42).

F i g . 5 -
A) Increase of vinpocetine solubility with nanocrystals radius R c decrease according to eq. (

L
i s t o f s y m b o l s a -vinpocetine unit cell side, m A lv -liquid drug -vapor drug interface area, m 2 A sl -solid drug -liquid drug interface area, m 2 A sv -solid drug -vapor drug interface area, m 2 b -vinpocetine unit cell side, m c -vinpocetine unit cell side, m C(r) -drug concentration profile in the stagnant layer, kg m -3 C b -drug concentration in the bulk liquid, kg m -3 C s -nanocrystals drug solubility in the bulk liquid, kg m -3 C s∞ -infinitely wide crystals drug solubility in the bulk liquid, kg m -3 l p C -liquid drug specific heat capacity at constant pressure, J kg -1 °C-1 l p c -molar liquid drug heat capacity at constant pressure, J mol -1 °C-1 s p C -solid drug specific heat capacity at constant pressure, J kg -1 °C-1 s p c -molar solid drug heat capacity at constant pressure, J mol -1 °C-1 D fugacity in the solid state, Pa s d f -pure solid drug fugacity, Pa l d f -drug fugacity in the liquid state, Pa l d f -drug fugacity in the reference state, Pa k d -dissolution constant, m s -1 h SL -stagnant layer thickness, m k m -solid drug-liquid mass transfer coefficient, m s -1 M d -drug molecular weight,m d -drug molecular diameter, m M v -liquid vinpocetine molar volume, m 3 mol -1 lv i g -liquid "i" -vapor surface tension (flat surface), J m -2 lv g -liquid drug -vapor drug surface tension, J m -2 sl i g -solid drug -liquid "i" surface tension (flat surface), J m -2 sl g -solid drug -liquid drug surface tension, J m -2 lv  g -liquid vinpocetine -vapor surface tension (flat surface), J m -2 sv g -solid drug -vapor drug surface tension, J m -2 sv  g -solid drug -vapor surface tension (flat surface), J m -2 sl 