5.2 DISSOLUTION
The dissolution of a solid in a solvent is a rather complex process determined by a multiplicity of physicochemical properties of solute and solvent. Indeed, it can be considered as a consecutive process driven by energy changes [10] (Figure 5.1). The first step consists in the contact of the solvent with the solid surface (wetting), which leads to the production of a solid-liquid interface starting from solid-vapour one. The break of the molecular bonds of the solid (fusion) and molecules passage to the solid-liquid interface (solvation) are the second and third step, respectively. The final step implies the transfer of the solvated molecules from the interfacial region into the bulk solution (diffusion). Obviously, each step requires energy to be performed and the total energy required to solid dissolution is the sum of the energies relative to the four mentioned steps. Typically, the most important contribute to solid dissolution, in terms of energy required, is given by the fusion step. While solvation and diffusion depend on solid-solvent chemical nature and on dissolution environment conditions (temperature and mechanical agitation for example), wetting and fusion also depend on solid microstructure.
The classical interpretation of solid dissolution, the film theory [11], assuming a very large dissolution environment, suggests that, despite of agitation, the solid surface is surrounded by a liquid stagnant film whose constant thickness h decreases with increasing solid surface – bulk liquid relative velocity (usually referred to as bulk liquid hydrodynamic conditions). While the concentration of the solid molecules at the solid surface, Cs, is assumed to be equal to solid solubility in the dissolution medium, drug concentration Cb in the bulk solution is postulated homogeneous and smaller than Cs due to the mass transfer resistance exerted by the film. Indeed, it is assumed that inside the film, mass transfer occurs only due diffusion. Accordingly, the drug mass conservation law assuming Fick first law with constant diffusion coefficient for drug flux reads (see eq.4.139):
(5.1)
where t is time while Dd0 and C are, respectively, drug diffusion coefficient and concentration in the stagnant layer. Postulating steady state conditions () (this assumption is more than reasonable in the light of usual h and Dd0 values in drug dissolution) and assuming h very small in comparison with solid surface curvature radii (this means that solid surface can be considered a plane), eq.(5.1) becomes:
(5.2)
In the light of the set boundary conditions (drug concentration equal to Cs in X = 0 and Cb in X = h), eq.(5.2) solution provides the following linear drug concentration in the film thickness:
(5.3)
Accordingly, Cb (or mb, the amount of drug present in the dissolution medium of volume V) increase in the bulk dissolution medium becomes:
(5.4)
(5.4’)
where S is the solid surface area, J is drug flux while kd is the intrinsic drug dissolution constant expressed as the ratio of Dd0 and h. Accordingly, kd is strongly dependent on dissolution medium hydrodynamic conditions and this is the reason why some authors propose to define as “intrinsic dissolution constant” kd value for very high (theoretically, infinite) difference between bulk liquid velocity and solid surface according to an extrapolation procedure [12]. While eq.(5.4’) validity has been demonstrated on an experimental basis comparing it with a huge amount of data [13], its solution, assuming that initial drug bulk concentration Cb is zero, leads to [14]:
(5.5)
Interestingly, eq.(5.4’) makes clear that dissolution is the result of a kinetics (kd) and thermodynamic equilibrium (Cs) contributes and both of them are very important in determining dissolution rate. In addition, as this model holds after the development of a linear concentration profile in the limiting stagnant layer (steady state assumption), the wetting step, influencing the very beginning of the dissolution process, is not explicitly accounted for. The effect of this kinetics aspect implicitly reflects in lower kd values as discussed in paragraph 5.2.1. This brief introduction allows enucleating the fundamental aspects of dissolution and the topics that must be addressed in order to model it. Accordingly, in the light of eq.(5.4’) and of the energetic dissolution diagram shown in Figure 5.1, surface properties, hydrodynamic conditions, drug solubility and drug stability in the dissolution medium will be discussed in details in the following.
5.2.2.1 Rotating disk apparatus
This experimental set up, also known as intrinsic dissolution rate apparatus (IDR), implies a drug disk fixed on a rotating shaft (w is angular velocity) immersed in a very large dissolution medium (Figure 5.7). In order to greatly simplify the theoretical analysis, drug cylinder lateral surface is coated by a water impermeable membrane so that dissolution takes place only through the disk bottom plane surface. Shaft rotation provokes the dissolution medium movement driven by the perfect liquid adhesion to the rotating solid surface (see figure 5.7). Accordingly, at the liquid-solid interface, dissolution fluid moves exclusively in a rotational (same angular velocity w of the rotating disk) manner. As far as we detach from the interface, axial (vy), tangential (vf) and radial (vr) fluid velocity components are all non zero and that is why the fluid approaches the solid surface according to spiral trajectory. Obviously, the radial velocity component is due to centrifugal forces. As far as the distance from the disk rotating surface increases, the tangential and radial velocity components become smaller and smaller so that for a distance greater than d0 they vanish and the only non zero velocity component is the axial one. Practically, due to fluid continuity and perfect adhesion to the solid surface, the rotating disk recalls fluid from deeper dissolution environment and pushes it in the radial direction so that fluid stream lines describe a loop circulation (see figure 5.7). Assuming isothermal, stationary and Newtonian flow conditions and neglecting edge effects (disk radius much bigger than disk thickness), the description of the above mentioned physical frame requires the solution of the continuity (eq. 4.56) and momentum equation (eq. 4.70):
(4.56)
(4.70)
provided that the following boundary conditions are attained:
vr = 0; vf = wr; vy = 0 in y = 0 (disk surface) (5.51)
vr = 0; vf = 0; vy = -v0 in y ® ¥ (5.52)
While eq.(5.51) states that at the solid-liquid interface (y = 0) fluid moves only in the tangential direction (coherently with the solid surface), eq.(5.52) imposes that very far from the solid surface, fluid moves only in the axial direction at constant velocity –v0 (the minus remembers that fluid movement occurs upwards, i.e. in the negative y direction). Levich [13] demonstrates that the approximate analytical solution to this problem is:
(5.53)
(5.54)
(5.55)
where n is fluid kinematic viscosity and d0 may be regarded as the thickness of the hydrodynamic boundary layer on the disk surface. Within the boundary layer, the radial and tangential velocity components are not zero, while beyond that layer, only axial motion exists (see figure 5.7). In order to definitively get a solution to mass transport from the rotating disk apparatus, stationary mass balance equation must be considered:
(4.46’)
where D is the drug diffusion coefficient in the dissolution medium, C is drug concentration and v represents the convective field defined by eqs.(5.51) – (5.54). Assuming that C is function only of the distance y from disk surface and that it does not depend on either r or f, eq.(4.46’) simplifies in:
(5.56)
This equation must accomplish the following boundary conditions:
C(y = 0) = Cs (5.57)
C(y = ¥) = C0 (5.58)
where Cs and C0 are, respectively, drug solubility and drug concentration in the dissolution medium bulk. While eq.(5.57) states that drug concentration at the solid-liquid interface (no wettability problems occur) equates drug solubility, eq.(5.58) assumes that drug concentration in the dissolution medium bulk is homogeneous and equal to C0. To get eq.(5.56) solution, a double integration is needed:
1st integration (5.59)
2nd integration (5.60)
where a1 and a2 are constants to be determined on the basis of boundary conditions. While from eq.(5.57) it descends a2 = Cs, eq.(5.58) requires:
(5.61)
In the hypothesis of n/D >1 (this is the typical situation met in drug dissolution experiments) and recalling vy dependence on y (eqs.(5.53), (5.54)), Levich demonstrates that the right hand side term in eq.(5.61) is equal to J1 = 1.61166 D1/3 n1/6 w-1/2. Accordingly, we have a1 = (C0-Cs)/J1. As a consequence, the drug flux F leaving the disk surface is given by:
(5.62)
Finally, this equation allows the estimation of the diffusion boundary layer thickness d:
(5.63)
It is interesting noticing that, typically, d is much smaller than d0. Indeed, in the very common case of dissolution in an aqueous medium at 37°C, n = 7.02*10-7 m2/s while D can be assumed equal to 10-9 m2/s (usually, drug diffusion coefficient in water at 37°C ranges between 10-9 – 10-8 m2/s). Accordingly, d = 0.05 d0. Finally, one of the most important conclusions deriving from Levich analysis consists in the independence of the diffusion boundary layer thickness d on the distance from the rotation axis. In addition, this theory allows to express the intrinsic drug dissolution constant kd (appearing in eq.(5.5)) in function of drug (D), dissolution medium (n) and hydrodynamics (w) conditions:
(5.64)
Before discussing an application example of this theory, it is interesting focussing the attention on the hypotheses on which it relies. Apart from the already mentioned ones (isothermal, stationary and Newtonian flow conditions, negligible edge effects and perfect liquid adhesion to the rotating solid surface), others, equally important hypotheses, need to be discussed. First of all, laminar flow conditions must be attained this meaning that Reynolds number Re ((wRD)* RD /n), where RD is disk radius) has to be lower than 104. Practically, assuming RD = 0.01 m and n = 7.02*10-7 m2/s (water, 37°C), laminar flow conditions are attained up to w = 68.06 rad/s (650 rpm). On the other side, the independence of d on radial position holds only if local Re ((wr)*r/n), where r is local radial coordinate) exceeds 102. Below this value, d assumes smaller values than those predicted by eq.(5.63) so that a central zone exists where d is smaller than on the remaining part of disk surface and, consequently, drug flux is higher. Assuming n = 7.02*10-7 m2/s (water, 37°C) and different w and RD values, table 5.6 shows the ratio Ac/At between disk central surface where Re < 102 and, thus, eq.(5.63) no longer holds, and total disk surface. Assuming that Levich hypotheses are accomplished if Ac/At < 0.33, it follows that when RD = 0.005 m the minimum w should be 100 rpm while this limit lowers to 75 rpm in the case of RD = 0.0075 m while, for RD = 0.01, no minimum rotating velocity problems occur. Finally, Levich approach holds if the solid surface is perfectly smooth.
It is now useful to use Levich approach to estimate theophylline monohydrated (C7H8N4O2·H2O; Carlo Erba, Milano) diffusion coefficient DTHEO in water at 37°C [45]. A smooth cylindrical theophylline disk (5 mg, 1 cm diameter, compressed at 5 tons) is attached, by means of liquefied paraffin, to a rotating stainless steel disk of the same diameter. The whole apparatus is immersed in a 150 cm3 thermostatic water volume and disk motion is immediately started after immersion. Theophylline time concentration is continuously collected by means of a personal computer managing an UV detector (UV - VIS Spectrophotometer, Lambda 6, Perkin Elmer - USA) set at 271 nm for 300 s. Three different w are considered (11.52 rad/s; 15.71 rad/s; 18.32 rad/s). Dissolution data are fitted according to eq.(5.5) knowing that theophylline solubility Cs in water at 37°C is 12495 ± 104 mg/cm3. The only fitting parameter, kd, results (1.87 ± 0.001)*10-3 cm/s (w = 11.52 rad/s), (2.17 ± 0.0012)*10-3 cm/s (w = 15.71 rad/s) and (2.37 ± 0.02)*10-3 cm/s (w = 18.32 rad/s). These data are then fitted by means of eq.(5.64) to finally get DTHEO = (8.2 ± 0.6)*10-10 m2/s. The knowledge of DTHEO allows the determination of theophylline molecule hydrodynamic radius RH on the basis of the Stokes-Einstein equation:
(4.100)
where K is Boltzam constant, T is absolute temperature (37°C), h is water shear viscosity (6.97*10-4 Pa s). The result we get, RH = 4 Å, is perfectly in agreement with the estimation of theophylline van der Waals radius (3.7 Å) calculated according to molecular modelling techniques [46]. In addition, it is possible to further verify the result we got. Indeed, knowing that RH = 4 Å and h(25°C) = 8.94*10-4 Pa s, Stokes-Eistein equation allows to conclude that DTHEO(25°C) = (6.2 ± 0.4)*10-10 m2/s which is in perfect agreement with what found by De Smidt and co-workers [47].
References
13. Levich, V. G., Physicochemical Hydrodynamics, Prentice- Hall, 1962
45. Grassi, M., Colombo, I., and Lapasin, R., Experimental determination of the theophylline diffusioncoefficient in swollen sodium-alginate membranes. Journal of Controlled Release, 76 , 93, 2001.
46. Coviello, T. et al., Structural and rheological characterization of Scleroglucan/boraxhydrogel for drug delivery. International Journal of Biological Macromolecules, 32, 83, 2003.
47. de Smidt, J. H. et al., Dissolution of theophylline monohydrate and anhydrous theophylline in buffer solutions. J. Pharm. Sci., 75, 497, 1986.
Figure 5.7. Schematic representation of fluid streams lines induced by drug disk rotation. Close to the solid surface, the diffusion (d) and the hydrodynamic (d0) boundary layers form.
Table 5.6. Dependence of Ac/At (ratio between disk central surface (radius RC) where Re < 102 and, thus, eq.(5.63) no longer holds, and total disk surface) on disk radius RD and disk angular velocity w. Grey cells indicate unacceptable conditions for Levich theory to hold.
100*RD(m) => / 0.5 / 0.75 / 1.0w(rpm) / w(rad/s) / 100*RC(m) / At/Ac / At/Ac / At/Ac
50 / 5.23 / 0.37 / 0.55 / 0.24 / 0.14
75 / 7.85 / 0.30 / 0.36 / 0.16 / 0.09
100 / 10.47 / 0.26 / 0.27 / 0.12 / 0.07
150 / 15.71 / 0.21 / 0.17 / 0.08 / 0.04
200 / 20.94 / 0.19 / 0.14 / 0.06 / 0.04
250 / 26.18 / 0.17 / 0.11 / 0.05 / 0.03
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