Analysis of Diffusion-Controlled Dissolution from Polydisperse Collections of Drug Particles with an Assessed Mathematical Model

Normalized Surface Flux (Sh) and the “Diffusion Layer” (δ)

As discussed in W12, the rate at which pharmaceutical molecules leave the surface of a particle is described hydrodynamically by a nondimensional surface flux called the “Sherwood number” Sh:

(1)

(t) is the average flux of molecules (moles/time/area) from the particle surface into the bulk fluid, is the concentration of molecules (moles/volume) at the particle surface, is the average concentration of molecules in the “bulk” fluid surrounding the particle and confined by a “container” (moles in the bulk fluid divided by bulk fluid volume), is the diffusivity of the molecule in the medium and  is the particle radius at time t (if the particle is not spherical, R is “effective” particle radius). Although the term “Sherwood number” suggests a single value, Sh should be recognized as a normalized flux that varies from particle to particle and changes with time. Pharmaceutical molecules are released into the bulk in concert with reduction in particle radius, so and are both strong functions of time. The surface concentration, however, is driven by the surface kinetics. In the current model, we assume diffusion-driven dissolution and apply the common assumption that the fluid adjacent to the surface is saturated with surface concentration C_S given by the solubility of the drug molecule in the medium, a fixed value.

The right hand expression in Eq. 1 results from a specific definition of δ, a length scale that characterizes the thickness of the region of highest concentration adjacent to the particle surface (see W12 and Sugano 2010). When δ is defined as follows:

(2)

the normalized flux Sh is given equivalently by the ratio of the particle radius to diffusion layer thickness (). As illustrated in Figure 1, Eq. 2 defines δ by extrapolating the concentration  with a straight line from the particle surface with the true slope at the particle surface. The distance where this linear extrapolation crosses the bulk concentration is δ, a quantification of the thickness of the concentration layer adjacent to the particle surface. We shall return to Eq. 2 and Figure 1 later in our discussion of confinement effects.

Single Particle Dissolution

We show below that at the core of dissolution from a polydisperse collection of particles is dissolution from a single particle with consequent change in particle radius, . The rate of change in radius is determined by the rate at which molecules leave the particle surface and enter the bulk fluid surrounding the particle:

(3)

The concentration of molecules in the bulk fluid, , is obtained by integrating Eq. 3 to obtain the change in radius (and volume) for each particle in the collection from time step to and the loss of particle volume is converted to the addition of molecules to the bulk fluid () from which the new number of molecules in the bulk is obtained at the next time step and the bulk concentration is recalculated, a step that depends centrally on the confinement of the molecules in the bulk volume, . The updated value for is then incorporated into Eq. 3 at the next time instant and the calculation is continued forward in time. Thus, the accuracy of the predictions for the release of molecules into the bulk from the surfaces of collections of particles depends on the accuracy of the model for for each particle in the collection (see section A Polydisperse Hierarchical Model for Diffusion-Dominated Dissolution below). The time change in particle bulk concentration, , is strongly dependent on the confinement of the released molecules by the container into which the molecules are released. As discussed in the next section, this is one of two “confinement effects.”

The Core Single Particle QSM

Wang et al.1 showed that the QSM is both accurate and practical for predicting diffusion-driven dissolution from single spherical particles of volume confined by a spherical impermeable container of volume . This is a first-principles model based on the diffusion equation (Ficks second law) and applied to dissolution from a particle with the essential approximation that the nonsteady term in the equation is negligible. This is true when time is large compared with the diffusion time scale, —which, for dissolution from small pharmaceutical particles, tends to be very short compared with the time required for measurable change in bulk concentration. W12 showed that the QSM is an excellent approximation for predicting the time changes in bulk concentration because the error in neglecting the nonsteady term in the model for normalized surface flux when Eq. 3 is integrated in time occurs largely during the initial period of dissolution when the concentration field is first developing and C_b is very small compared with C_S, so the error in the prediction of surface flux occurs when there is little measureable effect on the bulk concentration.

The QSM therefore predicts accurately so long as the confinement of molecules added to the bulk fluid in is accurately incorporated into in Eq. 3 in as described in the paragraph above. The restriction of molecules to the region between the particle and container leads to a second “confinement effect” for which the QSM produces a very useful and accurate explicit mathematical expression (W12).

The QSM solution is developed by first solving for the concentration field surrounding a single spherical particle in an infinite medium with concentration at infinity and then adjusting to be consistent with the concentration field confined to a volume,. The steady form of the diffusion equation with the requirement that the concentration at the particle surface and infinity are C_S and is easy to integrate, to produce:

()

()

The QSM concept is to adjust in Eq. 4a to make the volume integral of over the container volume equal total number of molecules that have been released into the container to time t. Integrating over the volume between the particle and container surface and dividing by the bulk volume produces the required relationship between and (see W12):

(5)

where

(6)

The flux of molecules from the particle surface is given by (4b) with replaced by Eq 5. Equating this QSM prediction for flux with the general expression for flux given by Eq. 3 produces the QSM prediction for Sh, the normalized flux of molecules from a particle of volume in a container of volume :

(7)

Thus, in diffusion-driven dissolution, the normalized flux is dependent only on the relative volume of particle to container. When the container is sufficiently large compared with the particle, or the particle is sufficiently small compared with the container, that and Eq. 6 gives , Eq. 7 gives Sh ≈ 1. Thus, γ results from the influence of confinement of molecules in the “container” surrounding the particle on the rate of dissolution from a single particle. The second term in Eq. 7 is a modification to the Sherwood number from confinement that results from the solution of the diffusion equation. W12 have shown that this additional term is a very close approximation of the exact term calculated from an exact solution of diffusion-driven dissolution from a spherical particle in a spherical impermeable container. γ underlies a second confinement effect mentioned above and discussed in detail in the section Regimes that Define the Dissolution Process and Confinement below.

The Hierarchical Particle Dissolution Model

The accuracy of the model for nondimensional flux Sh (i.e., R/δ) in Eq. 3 is at the core of the accuracy of model predictions for dissolution. For pure diffusion from a particle in an infinite medium, Sh = 1, or : at each instant in time the diffusion layer thickness equals the particle radius. However, Eq. 7 shows that Sh is increased above one by the confinement of molecules within a finite volume . In addition, Sh can also be enhanced by flow, for example, by relative motion between the fluid and the particle surface (“slip”) enhancing mass transfer at the surface by convection. Thus, in our hierarchical modeling framework, we write, for each particle,

(8)

where

(9)

It can be shown from Eq. 6 that , so that . Thus, the “γ confinement effect” has the potential to be significant.

Equation 3 with Eq. 8 are at the core of our hierarchical model strategy. However, to represent realistic drug dissolution, the model must be extended to include polydisperse collections of small drug particles of different size. In the current study, we analyze diffusion-dominated dissolution, so (QSM) as per Eq. 7.

Evolution of the Particle Size Distribution

Consider the sudden mixing of a known distribution of spherical pharmaceutical particles in a liquid medium confined to an impermeable container of volume V_c. Diffusivity and solubility are assumed known at required levels of accuracy. As mentioned previously, we assume that the fluid adjacent to the particle surfaces is saturated with concentration equal to the measured solubility. Dissolution begins at time assuming no molecules initially in the bulk fluid surrounding the particles.

Let p(R,t) be the probability distribution function (PDF) of particle radii in the container at time t. Specifically, p(R,t) dR is the fraction of particles with radius between R and , so that . Let N(t) be the total number of particles at time t, so that is the number of particles between R and . Thus,

(12)

where is the number distribution function. The aim is to predict, from which the change in volume of all particles, the number of molecules introduced into the bulk and the bulk concentration are determined as a function of time.

By balancing the time rate of change in the number of particles from radius R to written in terms of with the time rate of change of particle radius between R and , the differential equation for the time evolution of the particle distribution function can be derived20:

(13)

The important observation in Eq. 13 is that the evolution of the number distribution function, and therefore the prediction of dissolution, contains at its core the time rate of change radii of the individual particles in the polydisperse collection. Thus, the accuracy of the prediction for depends centrally on the accuracy of the model for rate of change of particle radius for each particle j in the polydisperse collection, as given by Eqs. 10 and 11 above. From the solution for, is easily determined, as is the total volume of particles, :

(14)

Knowing the molar volume , it is now possible to calculate the number of molecules in the bulk fluid, and therefore the bulk concentration:

(15)

Equation 13 with Eqs. 10 and 11 must be solved on the computer to obtain the number distribution discretized in R and t from a specified initial distribution . Typically, must be obtained from measurements of particle volume fraction distribution using an instrument such as the Mastersizer (Malvern Instruments). Furthermore, to predict Eq. 9 a model is needed for the effective container volume surrounding each particle. These issues are discussed next.

Computation of the Particle Distribution Function Equation

Starting with, we solve Eq. 13 with Eqs. 10 and 11 in discretized form. is discretized as a function of discrete particle radii and over discretized time steps , where j and n are positive integers. There are at least two approaches to solve for discretized. The first is to discretize Eq. 13 using finite differencing in R and t and advance the discretized equation in time for directly, with replaced by the QSM Eqs. 10 and 11 at time t_n. The second approach is to solve Eq. 13 indirectly by discretizing into bins of fixed radius at and then integrating from the QSM Eqs. 10 and 11 to obtain at the next time step. The number of particles in each bin is fixed as the radii of each bin decrease with time and the particles in that bin dissolve and are removed from the distribution. Thus, as the change in bin sizes are calculated along with particle radii from Eq. 10, the change in discretized with time is determined. We found that the first approach was prone to numerical instability, so we used the second approach, which is similar conceptually to the algorithms described by Hintz and Johnson4 and Lindfors et al.25 that did not include Eq. 13 as the mathematical basis for prediction of time evolution of particle size distribution. Furthermore, to integrate the QSM Eqs. 10 and 11, an “effective particle volume” must be quantified. This is described next.

The “Homogeneously Mixed” Model Assumption

As pointed out above, we differ from previous approaches in the hierarchical approach taken and the use of the QSM at the core of our polydisperse dissolution model. Here, we consider confined dissolution by pure diffusion: the first two terms in Eq. 8 with Eq. 9 (or equivalently, Eq. 11). As discussed above, the second term in Eq. 8 (or 11) is a confinement effect for a single particle j in its effective confinement volume . The model used to determine is an issue worthy of discussion and refinement in future application of in the polydisperse dissolution model. In the current implementation, we apply the “homogeneously mixed” model assumption where the particles in each bin are assumed to be homogeneously distributed within the container volume V_c, and each effective confinement volume is assumed to have the same bulk concentration. That is, in our current model we assume that is the same surrounding all particles j at each time t, although the net bulk concentration varies with time.

This “homogeneously mixed” model assumption is effectively the same at that assumed by Hintz and Johnson4 and Lu et al.5 and is an element that should be refined in future models to take into account inhomogeneous concentrations of molecules in the bulk fluid, as is likely the case with in vivo dissolution in the gut.

The Initial Number Distribution Function

As described above, to initiate the calculation of the initial distribution must be estimated from measurements or must specified ad hoc. For example, in experiments developed by Weibull26 and reported in W12 and Lindfors et al.16 the Mastersizer instrument was used to measure the volume fraction of particle sizes within logarithmic bands of particle radii as illustrated in Figure 3. From the initial total volume of particles , the total concentration may be calculated.

To form, let be the volume fraction in bin j given as output from the Mastersizer and displayed in Figure 3. Thus, by construction, . For each bin from to , it is straightforward convert to bin widths on a linear scale (i.e., bins from to ). Note that

(16)

Then, in the limit and , so that Eq. 16 becomes:

(17)

where is the volumetric probability distribution function at the initial time and is the fraction of volume of the particles between R and R + dR at t = 0. Thus

(18)

where is the total volume occupied by the particles at the initial time, is the volume distribution function, and is the volume of particles between radii R and .

The volume distribution function is the particle distribution function multiplied by the volume of a particle with radius R, so that

(19)

where and N₀ are the PDF and number of particles at t = 0. Thus, the method to find from the Mastersizer outputs of logarithmically discretized volume fraction and total number of particles N₀ is as follows:

1. Form,
2. Using the center radius of each bin in Figure 3, interpolate (e.g., using cubic spline interpolation) to obtain at the desired particle radii R at whatever resolution is desired,
3. Use Eq. 19 to determine (and ) at the desired resolution.

In our computational experiments (below), we discretized R into 500 bins bounded by discretized radii values .

Generalized Distribution Functions and Average Particle Radii

Although Eqs. 17–19 were developed for the initial time , they actually apply at arbitrary time, t. Thus, as the time evolution of the particle number distribution function is obtained by integrating Eq. 13, so are the distribution functions, from Eq. 19, and . A similar process can be used to define the surface probability distribution function and surface distribution function , where is the total surface area of all particles at time t, and

(20)

Similarly to Eqs. 12 and 18,

(21)

Thus, from a prediction of, one can also construct, at each time t, the surface and volume probability and distribution functions, and , as well as the total number of particles , the total volume of particles , and the total surface area of particles, .

Average particle radius is commonly defined as the sum of radii over all particle divided by the number of particles, . However, this definition is not unique and one might also consider average radius based on the volume or surface area of the particles. Each of these definitions can be constructed from the prediction for by constructing the number probability distribution function p(R,t), volume probability distribution function, and surface probability distribution function and using the averaging property of the pdf:

(22)

(23)

(24)

Thus, from the prediction of the number distribution function most other quantifications of interest can be made. In our computational experiments (below), we show the differences in evolution of average particle radius using the three definitions above.

Specification of Initial Particle Size Distribution

To systematically vary the initial distribution of particle sizes, the initial size distribution should be specified in a form consistent with true distributions. As illustrated in Figure 3, realistic size distributions are typically well-represented as Gaussian functions of the log of the particle radii (log-normal). For our computational experiments, we therefore initiate our simulations with a log-normal volume fraction distribution similar to Figure 3, but much more finely resolved in (500 vs. 22 bins). For our results to be more generally applicable to a wide variety of particle sizes, we specify the distribution relative to nondimensional particle radius, where R is nondimensionalized by the radius at the peak in the distribution so that peaks at. The width of the distribution of particle sizes is correspondingly specified in terms of as illustrated in Figure 5a.

From Eq. 16, is proportional to the initial volumetric probability distribution function, , so that

(26)

where is the volume fraction of the total volume of the particles in the logarithmic band from to . The logarithmic band-widths are specified as uniform in ; both and peak at. To approximate measured particle distributions such as Figure 3, for example, is modeled as Gaussian in :

(27)

where the factor in front of the exponential ensures that the integral of over is unity as required by definition. (Note that varies from −∞ to +∞ as varies from 0 to ∞.) in Eq. 27 is symmetrical about the peak with its width controlled by, the variance of over . Inserting Eq. 27 into Eq. 19 produces , the initial condition which is used solve for and, from that solution, all other needed quantities during the dissolution process, as described above.

Figure 5 shows the two initial conditions for that were applied in the current study, a “narrow” distribution (N) with and a “wide” distribution (W) with . In both distributions, 500 logarithmic bins were used so that the bin widths were smaller with the narrow distribution than the wide. All particles in the same bin have the same radius and the bins farthest from the peak have volume fraction 0.001. These “farthest” bins are at for the narrow distribution and for the wide distribution. In Figure 5a, the volume fractions are given on a logarithmic scale. The corresponding initial pdf , plotted on a linear scale, is shown in Figure 5b. Because of the functional relationship between in Eq. 19, the PDF peaks at and the distribution exhibits a tail typical of manufactured formulations (e.g., Fig. 3). As the width of the distribution decreases, the dissolution process approaches that of a monodisperse distribution with single particle radius.

Effect of Initial Size Distribution on Drug Release

To evaluate the sensitivities of the dissolution process to the width of the distribution, we initiate simulations with the “narrow” (N) and “wide” (W) distribution of particles in Figure 5 and compare with a monodisperse collection of particles with initial radius , all at the same total concentration, . The evolution of bulk concentration varies with solubility C_S, diffusivity D_m, molar volume υ_m, and the initial radius at the center of the volume fraction distribution, . However, it can be shown (see section Generalized Dissolution with Appropriate Nondimensional Parameters below) that all variations can be represented as bulk concentration nondimensionalized by the solubility () against time nondimensionalized by the following time scale:

(28)

is the time it takes for a single particle of initial radius to completely dissolve in an unbounded medium (with no molecules initially in the bulk). This is obtained by integrating Eq. 3 from at to at with and .

Thus, in what follows a plot of dimensional bulk concentration versus time can be obtained from a plot of nondimensional against by forming from Eq. 28 and multiplying by C_S and by . Unless otherwise indicated, we use the properties of felodipine in the experiments of Lindfors et al.16 as given in section Model Validation above. However, as will be explained in more detail in the subsection below entitled Generalized Dissolution with Appropriate Nondimensional Parameters, when plotted and interpreted in proper nondimensional form, dissolution predictions are independent of drug properties.

In Figure 6, we plot the nondimensional changes in bulk concentration with respect to time initialized with wide (W), narrow (N), and monodisperse (M) distributions of drug particles at three total concentrations C_tot, one below the solubility and two above (C_S = 0.89 μM). Three manifestations of confinement are immediately apparent. The first is the occurrence of saturation when confined and total concentration exceeds solubility. The second is the sensitivity between the rate of change in bulk concentration and total concentration C_tot which, as shown by Eq. 25 and discussed in W12, quantifies the relative confinement of particles in the container at the initiation of dissolution. A more subtle manifestation is that complete dissolution () requires times longer than the unbounded dissolution time, . As C_tot decreases, dissolution time approaches , however at C_tot = 0.5 μM (C_tot/C_S = 0.56), the time to complete dissolution greatly exceeds the unbounded domain dissolution time. We shall return to this issue in the subsection below entitled The γ versus C_b Confinement Effects.

Figure 6 shows the effects of particle distribution on the dissolution process. Not surprisingly, as the distribution narrows, the dissolution process approaches that for a monodisperse distribution of particles. However, the existence of a distribution of particle sizes lengthens the time required for complete dissolution or saturation to occur. This is because particles larger than are the last to dissolve and do so in the presence of lower driving potential, (Eq. 10). Thus, although the narrow particle distribution closely approximates monodisperse dissolution at early times (Fig. 6b), the sensitivity to the existence of distributions of particle sizes is stronger in the final period of dissolution or saturation (Fig. 6a). The total time to dissolution or saturation is higher with a distribution of particles (at the same C_tot). We shall find that this is particularly true in the final periods of dissolution () as the final period is dictated by the dissolving of the largest particles.

Interestingly, although the rate of increase in bulk concentration is overall reduced by the existence of a range of particle sizes, the rate of increase in bulk concentration is initially increased because of the existence of a distribution of particle sizes. This is because the initial rate of increase in bulk concentration is dominated by dissolution of the smallest particles. It can be shown that the relative rates of addition of drug molecules to bulk concentration from groupings of drug particles with average radii and is proportional to , where and are the contributions to total concentration from the first and second groupings of particles. Thus, the smaller particles are highly favored to contribute to the bulk concentration at a faster rate so long as the contribution of each group of particles is not correspondingly out of balance. As indicated by Figure 5b, the log-normal distribution biases the number of particles to sizes smaller than the most probable. As a result, the smallest particles initially dominate the rate of increase in bulk concentration. Figure 6b shows that the cross over in the rate of increase in between polydisperse and monodisperse collections of particles occurs at later time with larger C_tot.

Similar characteristics were observed in Figure 4 when we compared data from an in vitro dissolution experiment with predictions using a monodisperse model versus predictions using the true initial polydisperse distribution of particle sizes. Clearly there exist potentially important details of the dissolution process that cannot be captured with a monodisperse model.

The time variations of total number of particles N(t), normalized by the initial number of particles, is shown in Figure 7 for different total concentrations and distribution widths. We observe a great sensitivity both to the total concentration and to the width of the size distribution of particles. Because the rate of change in particle radius is inversely proportional to particle radius (Eq. 10), the smallest particles reduce in radius at the most rapid rate. With both “wide” (W) and “narrow” (N) distributions, there is an initial period with no reduction in particles, as the smallest radius group dissolves. Because the wide distribution contains the smallest particles (relative to R*), the time period before particle numbers begin to reduce is shorter, and the relative change in number of particles is initially most rapid, with the wide distribution, at all C_tot. With complete dissolution (C_tot < C_S) the number of particle drop to zero; saturation (C_tot > C_S) implies that some particles never dissolve. With increasing confinement (increasing C_tot) the rate of reduction in particles is lower and, with saturation, the number of retained particles higher. The largest particles in the initial distribution are the last to dissolve or saturate.

Evolution of Particle Size Distribution

Over time, the smaller and larger particles decrease in radius at different rates (Eq. 10) and the shape of the particle size distribution changes as drug molecules are released into the bulk. The changes in particle size distribution for the six simulations in Figures 6 and 7 using the wide (W) and narrow initial distributions of Figure 5 are shown in Figure 8, where for each case the number distribution function is plotted at nondimensional time increments over the dissolution process for the three values of C_tot.

The distribution of particle sizes changes differently during dissolution depending on the level of confinement (υ_mC_tot) and the width of the initial distribution. When C_tot is below the solubility (C_S = 0.89 μM), all particles eventually dissolve and (Figs. 8a and 8b), whereas for C_tot > C_S, the dissolution process ends in saturation (Figs. 8c–8f). In all cases, the distribution of particle sizes is very different toward the end of the dissolution process in comparison with the initial distribution. In particular, because the rate at which the particle radius decreases is inversely proportional to the particle radius (Eq. 10 with Sh_j ≈ 1), the smallest particles in the distributions decrease in radius much more rapidly than do the largest particles, with the consequence that the distributions initially increase in width with time as the distribution rapidly extends towards zero radius. This initial increase in distribution width as the distribution extends in smaller radii is clear in all cases shown in Figure 8.

A consequence of the initial spreading of the distribution function toward smaller radius particles is that, over a period of time after the start of dissolution, the radius at the peak in the particle size distribution, R_peak, shifts to smaller values. The initial time period over which this shift in R_peak with time occurs depends on the initial width of the distribution and on C_tot and its value relative to C_S. The radius at the peak in the distribution either asymptotes to its smallest value or it shifts back toward larger values until complete dissolution of saturation occurs. In particular, when dissolution begins with an initially “narrow” distribution, the radius at the peak in the number distribution shifts continuously toward smaller R_peak until all particles have dissolved (Fig. 8a) or saturation has occurred (Figs. 8c and 8e) because the particle size is distributed in a sufficiently narrow region to approximate that from a monodisperse distribution—where all particle reduce together continuously to smaller values.

In contrast, the radius at the peak of an initially “wide” distributions decreases at first, but eventually shifts from decreasing to increasing R_peak at a point in time that depends on C_tot (Figs. 8b, 8d, and 8e). As with all distributions, the initial shift in R_peak to lower values results from the (Eq. 10). However, as the smallest particles move to zero radius, they begin to dissolve and disappear from the distribution. When this occurs, the distribution is anchored at R = 0, whereas the largest particle continue to reduce in size and the change in distribution with time is dictated by the rate of change in radius of the larger particles. If the distribution is sufficiently wide and the dissolution process is sufficiently long, particle numbers accumulate rapidly at the smallest scales as particle numbers decrease at the largest scales, with the consequence that the peak in the distribution shifts towards larger scales.

It should be noted that a plot roughly similar to Figure 8d is given in Johnson8 (their Fig. 2), although the conditions under which the simulation is performed are not given. Likely because the bin resolution is much cruder than in Figure 8 and the largest particles did not appear to reduce in diameter in their simulation, R_peak in their simulations moved rapidly to larger values. In contract, Figure 8 indicates that R_peak initially decreases as the smaller particles rapidly reduce in size and dissolve, before sometimes increasing as saturation is approached, depending on the initial width of the distribution. Still, the essential mechanism is similarly described.

Changes in Average Particle Radius

The changes in radii at the peaks in the particle size distribution to smaller or larger values result from the differential rates of change of larger versus smaller particles (Eq. 10) in relationship to the total concentration of molecules available for dissolution relative to the saturation concentration (C_tot/C_S) and the size of the container (υ_mC_tot). These changes are reflected in corresponding changes in average particle size during dissolution, as shown in Figure 9 where average particle radius is plotted against time using the three definitions for average radius given by Eqs. 22–24 and, in each case, comparing with the continuous reduction radius with an initially monodisperse collection of drug particles. We find that all definitions produce the same trends and that these trends follow the evolution of R_peak with time just discussed: the initially “narrow” distributions follow approximately the continually decreasing average radius of dissolution from the monodisperse collection, whereas the average particle radius with the initially “wide” distribution ultimately increases over time.

However, the initial reduction on average particle radius is observed only with the number-averaged radius. Neither the volume-averaged nor the surface-averaged definition for average particle radius is sensitive to the initial decrease in the peak in number distribution function shown in Figure 8. Furthermore, as the width of the initial particle size distribution increases, so does the difference between number-averaged radius, volume-averaged radius and surface-averaged radius. In fact, the volume-average radius can be a factor of two larger than the number-averaged particle radius with the “wide” distribution as volume-averaging weights the average toward the largest particles. Furthermore the proper representation of the polydisperse nature of the dissolution process has a large impact on predictions of time evolution of average particle size during dissolution. The improved accuracy in the prediction is particularly apparent when the particle distribution is relatively wide. In this case, average particle size increases over time, while the monodisperse model can only predict reductions in particle radius.

Interestingly, when C_tot < C_S and all particles ultimately dissolve, the final period of dissolution occurs with minimal change in bulk concentration (Figs. 9a and 9b). This results because as particles accumulate near zero radius in the distribution, the relative number of molecules in these near-zero radius particles becomes so small relative to the total number of molecules in the bulk that the bulk concentration changes very little at the end of the dissolution process. Consequently the average particle radii in Figures 9a and 9b drop rapidly to zero at the highest bulk concentration as the final particles completely dissolve.

Generalized Dissolution with Appropriate Nondimensional Parameters

The importance of the confinement effect is characterized by . relative to 1. However, when , approaches C_tot in the long time limit, so the confinement effect is characterized by at complete dissolution. The γ confinement effect is greatest at the initiation of dissolution and, as shown in Figure 10, increases with increasing . Thus, total concentration relative to solubility is an important relevant nondimensional ratio as it distinguishes total dissolution of drug particles () from saturation (). Furthermore, as described in W12 for single particles, and as discussed at length below for dissolution from distributions of particles, total concentration equaling solubility () produces a “saturation singularity” arising from “confinement effects.” Thus, is an important nondimensional parameter with clear physical interpretation. However, as discussed above, the γ confinement effect is itself determined by total concentration in the nondimensional form —that is, the product of and .

We conclude that the nondimensional parameters, and , are the key independent nondimensional parameters to characterize dissolution. This conclusion is validated by nondimensionalizing the diffusion equation as follows: (a) nondimensionalize the radial coordinate by , the radius at the peak in the initial particle distribution (Fig. 5); and (b) nondimensionalize time by (Eq. 28), the time required for a particle of initial radius to completely dissolve. It can be shown (and we shall demonstrate below) that when dissolution is diffusion-controlled, the evolution of collapses onto the same curve when is plotted against for the same nondimensional parameters and . Therefore, in the presentation of the dissolution process in figures below, and in Figures 6-11 above, we normalize concentrations by C_S and we plot appropriated nondimensionalized variables against or against as a function of . In this way, the results can be generalized to different drugs in any amount and in any container volume. (It should be noted that initial volume or number average particle radius could also be used to define τ_diss.)

Dissolution Regimes, Confinement, and the Role of Particle Size Distribution

In Figure 6, we plotted the increase in bulk concentration normalized by solubility against time nondimensionalized by , for felodipine, an extremely low solubility drug (C_S = 0.89 μM). Predictions were presented with complete dissolution ( = 0.56) and with saturation ( = 1.12 and 1.69), and comparing “narrow” (N), “wide” (W), and monodisperse distributions of felodipine particles at the initiation of dissolution. We found that the rate of increase in was a strong function of with higher corresponding to more rapid rates of increase in during the initial period of dissolution (relative to ). We further found some influence of the width of the particle size distribution.

Figure 6 suggests that the “dissolution time” for the dissolution process to end, either to complete dissolution ( < 1) or to saturation ( > 1), decreases with increasing total concentration. In Figure 12, we plot the nondimensional dissolution time t* as a function of for felodipine using the polydisperse model initialized with the same narrow (N) and wide (W) initial particle size distributions and compare with the monodisperse distribution (M). Note that a log–log scale is used. We immediately observe that the overall rate of dissolution relative to that for a single unbounded particle at the peak in the initial distribution, separates into three “regimes” that we label I, II, and III in Figure 12.

Regime I () identifies dissolution with small enough numbers of molecules in the bulk for “sink conditions” to apply to a reasonable level of approximation. Any total concentration levels that exceed approximately one tenth the saturation concentration C_S experience the bulk concentration confinement effect over the integrated dissolution process, a result independent of the width of the initial size distribution. In regime I, all particles eventually dissolve, with the nondimensional time required to complete dissolution strongly dependent on the width of the initial particle size distribution. Because the monodisperse collection includes only a single particle radius, the dissolution time in the low limit equals that for the same particle in an unbounded medium. Wider distributions require more time for the dissolution process to complete, by a factor of 3 for the initially narrow distribution and 11 for the wide distribution. These ratios, it turns out, are equal to , where is the maximum radius in the initial distribution. That is, in sink conditions, the time for complete dissolution is given by the time required for the largest particle in the distribution to dissolve: Eq. 28 with replaced by ,

(32)

Note that involves the nondimensional parameter but is independent of C_tot. This is because, in the limit , dissolution proceeds without influence from confinement.

That the max dissolution time given by Eq. 32 collapses the time to complete dissolution for sufficiently small is shown in Figure 13, where Figure 12 is replotted with replaced by . We conclude that when C_tot is below approximately 10% the solubility, sink conditions may be reasonably applied and the time to dissolution may be predicted by Eq. 32. When C_tot exceeds 0.1C_S, however, confinement effects are manifest and models of dissolution in which is neglected relative to C_S (Eq. 28) should be avoided. We also conclude from these two figures that the appropriate time scale to characterize the dissolution time is (Eq. 32) when (dissolution) and (Eq. 28) when (saturation).

The bulk concentration confinement effect appears in Figures 12 and 13 as begins to increase and approach what we refer to as “the saturation singularity,” a concept discussed in W12 for single particle dissolution (where it was shown that the QSM predicts dissolution times very accurately). Here, we find that saturation singularity is as strong for polydisperse collections of particles as it is for single particle dissolution. The singularity reflects the ever slower release of molecules from the particle surface as approaches C_S in the driving potential in Eq. 29 when the final state is given by . The consequence is large increases in time to saturation as . The saturation singularity may be approached from complete dissolution (regime II_d) or from saturation (regime II_S) in Figures 12 and 13. The influence of the saturation singularity that defines regime II covers a broad range of total concentration, from to . The increase in dissolution time when the dissolution process resides in regime II can be substantial, factors of 5–10 or more. The in vitro experimentalist may wish to avoid choosing total concentrations within regime II, surrounding the saturation singularity, in order to maintain dissolution times within reasonable practical limits.

In contrast with regime I, where particles dissolve over the dissolution time for a single particle in an unbounded fluid and the dependence on particle distribution is strong, Figure 12 shows that in region III the dissolution process saturates, the dependence on initial distribution width relatively weak when time is scaled on , and the distinction in dissolution between the initially narrow and monodisperse particle size distributions is much weaker. The time to saturation, , decreases rapidly with increasing C_tot. In fact, we discover that the saturation time follows a power-law relationship with :

(33)

where β = 4.8 for the monodisperse and narrow particle size distributions, and β = 3.6 for the wide distribution. These are shown in Figure 12 by the straight dotted lines that extend the M, N, and W lines to . The QSM model predicts the power law expression given by Eq. 33 in the limit of large enough to neglect the change in radius of the individual drug particles during the saturation process. It is interesting to note that the time to saturation given by Eq. 33 is given equivalently by

(34)

Comparing Eq. 34 with Eq. 32 we see that although in sink conditions (regime ), the time for complete dissolution is independent of C_tot and solely dependent on solubility through , in regime III, when the confinement effect is exceptionally strong, the time to saturation depends solely on C_tot through the nondimensional parameter and is unaffected by solubility C_S. Although regime I defines the limit of negligible confinement effect with no knowledge of C_tot, regime III defines the limit of dominant confinement effect with the impact of C_tot dominating over the influence of C_S. Regime II, therefore, may be regarded as a regime where the relative contributions of C_tot to C_S are apparent.

Equations 32 and 34 are useful in the design of in vitro dissolution experiments to estimate the length of time required for each experimental run. Clearly there is a strong dependence on initial particle size in addition to (regime III) or (regime I).

The γ versus Confinement Effects

The above discussion of regimes focused on the confinement effect. How does the influence of the γ confinement effect manifest in this integrating description of the dissolution process? Equation 31 suggests that the γ confinement effect, characterized by , should manifest at sufficiently large , depending on drug solubility times molar volume, . Figure 10 showed that because felodipine is a very low solubility drug, on average the γ confinement effect does not significantly manifest until . However, the γ confinement effect is expected to manifest at correspondingly smaller for higher solubility drugs as given by the nondimensional parameters .

To demonstrate the influence of the γ confinement effect, we plot in Figure 14 the time to complete dissolution or saturation as a function of for the monodisperse collections of particles with different values of from 0.1 to 1000 that for felodipine. As discussed above in the section Generalized Dissolution with Appropriate Nondimensional Parameters, the appropriate characteristic variable is the nondimensional parameter rather than the dimensional variables υ_m or C_S alone. To show that this is the case, we plot changes in by changing υ_m with C_S fixed and by changing C_S with υ_m fixed. Figure 14 shows that the same result is obtained if υ_m or C_S are changed by the same relative amount. Note, in particular, the inset in Figure 14 where the overlap between pairs of curves is obvious.

The γ confinement effect, however, is manifest by , the product of the nondimensional parameters and . For an exceptionally low solubility drug such as felodipine (the red curve), the γ confinement effect is observable in Figure 14 when and the curve deviates noticeably, albeit not greatly, from the = 0.1 curve. For higher solubility drugs, however, the effect manifests at much lower . Keeping in mind that the log scale reduces the apparent differences between curves, the γ confinement effect begins to appear significant (>10%–20%, say) when exceeds ∼10 when is ∼1000 greater than felodipine (). When is 100 time higher than felodipine (), the γ confinement effect becomes significant when exceeds roughly 100 (the inset indicates a 20% effect). For any drug, the γ confinement effect increases with increasing total concentration relative to solubility.

Figure 14 shows that dissolution time decreases with increasing total concentration when > 1 as a reflection of both the and γ confinement effects. When the γ confinement effect is negligible, the time to saturation can be predicted by Eqs. 33 or 34 for (regime III). The γ confinement effect reduces the saturation time even more. In Figure 15, we show that these reductions in time to saturation are a direct result of a more rapid rate of release of drug into the bulk fluid throughout the dissolution process. In this figure, we plot the increase in nondimensionalized bulk concentration over nondimensionalized time as separate functions of increasing and on a linear-log scale so that all variations can be observed on a single plot. We observe that the overall rate of increase in with nondimensional time is similar in all cases. The primary difference is that the rate of increase in bulk concentration systematically increases during the dissolution process, particularly in the initial period, both as increases and within groups as increases. We conclude that, during saturation, the entire dissolution process is strongly affected by confinement effects.