A Simple One-Parameter Percent Dissolved Versus Time Dissolution Equation that Accommodates Sink and Non-sink Conditions via Drug Solubility and Dissolution Volume

Need for Applied Dissolution Equations that Can Accommodate Non-sink Conditions

IR in vitro dissolution media generally provide sink conditions, or at least 85% dissolution in 60 min (1). With the greater utilization of biorelevant media, there is a need for in vitro dissolution equations that accommodate and consider non-sink conditions, including incomplete dissolution due to insufficient solubility of all drug dose. Biorelevant media often provides a lower drug solubility than pharmaceutical surfactants.

For example, FaSSIF (pH = 6.5) increased carvedilol solubility only about 1.2-fold, compared to buffer, to 55.9 µg/ml (27). Meanwhile, 1% SLS (pH = 6.8) increased carvedilol solubility about 70-fold, compared to buffer, to about 1400 µg/ml (28). FaSSIF increased griseofulvin solubility only about 3%, compared to buffer, to about 12 µg/ml (29). Meanwhile, 1% SLS increased griseofulvin solubility about 85-fold to about 941 µg/ml (30). Posaconazole solubility at pH 6.5 is 0.27 µg/ml (31). FaSSIF and 0.3% SLS increased posaconazole solubility to 1.7 µg/ml and 31.2 µg/ml, respectively (32). Cinnarizine solubility at pH 6.5 is 0.16 µg/ml (33). FaSSIF-V2 and 2% SLS increased cinnarizine solubility to 2.82 µg/ml and 120 µg/ml, respectively (34).

Utility and Advantages of Polli Dissolution Equation

In general, one major application of equation fitting is to summarize data (i.e., data reduction) (35). Dissolution data is frequently subjected to equation fitting, with subsequent analysis such predicting absorption, physiologically based biopharmaceutics modeling (PBBM), in vitro-in vivo correlation, or comparison of dissolution profiles (36,37,38,39,40). For example, the FDA guidance on IR dissolution testing recognizes model-depended approaches compare profiles (1). The Hixson-Crowell equation has been used to compare profiles (5). In forecasting the in vivo impact of roller compaction scale up, dissolution data was fit to several dissolution equations, including a modified Higuchi equation (41, 42). Carducci has described the development of a real-time-release testing strategy that involves curve fitting to dissolution profiles, followed by regression of the curve fit parameters against the critical material attributes, critical processing parameters, and/or near-infrared (NIR) data (43). In similar cases, the Polli dissolution equation (i.e., Eq. 21, or Eq. 3) has potential application. Figure S5 shows the impact of kd value on potential “safe space” dissolution profiles into 900 ml when drug solubility cs = 0.1 mg/ml and dose = 10 mg.

Also, a shown here, the Polli dissolution equation has utility in fitting and summarizing dissolution data with a single parameter, when cs is known and regardless of sink condition prevailing or not prevailing. With only a fit single parameter, potential for model over-parameterization or poor model identifiability is low. Some other dissolution equations, such as Weibull equation, have at least two fitted parameters, which can be a disadvantage in data reduction, particularly if such data summarization is intended to be used in subsequent analysis (e.g., profile comparisons, a real-time-release testing strategy, parameter sensitivity assessment). The Polli dissolution equation is a relatively simple, non-differential equation. Excel software was used here to fit dissolution data. Included is an Excel file with instructions for the non-linear regression of the Polli dissolution equation to conventional dissolution data (i.e., percent dissolved versus time data). The Excel file requires the free and simple Excel Solver add-in. Since Eq. 21 is the solution to the differential form of the equation (i.e., Eq. 3), software that performs numerical integration is not needed.

Dissolution equations that are applied to percent dissolved versus time profiles often do not accommodate sink conditions, since sink conditions are commonplace (4,5,6,7,8,9,10,11). These dissolution equations have also been modified to allow for fitting of an extent of dissolution parameter when dissolution is incomplete (12, 13). Equation 21 intrinsically considers non-sink conditions. No extent of dissolution parameter was needed, although cs was needed.

Comparison of Polli Equation to the First-Order Dissolution Equation

A common equation for fitting percent dissolved versus time profiles is the familiar first-order Eq. (44, 45), whose differential and solution forms are, respectively:

$$^\!\left/ \!_\right.=-_M$$

(25)

$$\% \;dissolved=100\%\left(1-^_t}\right)$$

(26)

Equation 3, the differential form of the Polli question, has similarity to the above differential first-order dissolution equation (i.e., Eq. 25). Both are more applied equations than mechanistic equations, although both employ undissolved mass as the driving force for dissolution rate, which is often an acknowledgeable factor of dissolution rate. Under sink conditions, Eq. 3 yields Eq. 22 above, which is practically indistinguishable from the differential first-order equation. However, relative to Eq. 21, a limitation of the first-order equation is that its solution does not accommodate a solubility limit impact on percent dissolved. Of course, such a solubility limit impact can be added to the first-order equation if an impact of solubility on extent of dissolution had been observed or expected, yielding:

$$\% \;dissolved=100\%\left(\frac_}^_}\!\left/ \!_\right.}\right)\left(1-^_t}\right)$$

(27)

But, this decision-making is undesirable. Equation 21 has the advantage of not needing to conduct such decision-making, since potential non-sink condition effects are intrinsic to Eq. 21. Equation 21 can be employed in the presence and absence of sink conditions.

Of course, Eq. 21 will not be able to fit many dissolution profile shapes, such as sigmoidal profiles due to slow disintegration. kd is constant, although could be modified to be a function of time to accommodate other shapes. The Weibull equation is generally recognized as a function that has broad success in fitting a range of dissolution profile shapes. This advantage is in part due to it containing two fitted parameters (i.e., time factor τ and shape factor β), a potential disadvantage, particularly if such data reduction is intended to be used in subsequent analysis (e.g., dissolution profile comparisons, a real-time-release testing strategy, parameter sensitivity assessment). Additionally, like above for the first-order equation, the Weibull equation does not a priori accommodate non-sink condition effects.

Comparison of Polli Equation to the z-Factor Dissolution Rate Equation

The z-factor dissolution rate equation with sink conditions is (14, 15):

$$^\!\left/ \!_\right.=-z_^^\left(_-\frac_-M}\right)$$

(28)

where \(z=\frac_}\), where z is the particle dissolution z-factor, D is drug diffusion coefficient in the media, h is diffusion layer thickness, ρ is the particle density, and r0 is the initial particle radius. Although the units of kd and the particle dissolution z-factor can be considered the same (i.e., ml/mg per min), kd and z are not identical, by inspection of Eq. 3versus Eq. 28. Equations 3 and 28 differ from one another in that Eq. 3 includes \(_M\) term, while Eq. 28 includes \(z_^^\) term. In general, in fitting profiles, fitted kd values can be expected to be greater than fitted z values, since M < \(_^^\) when t > 0.

Although Eq. 28 was derived by considering spherical, uniform-sized particle dissolution, it is in practice an applied equation (14,16). For example, Hofsass and Dressman describe z-factor (i.e., z) as a hybrid parameter with potential pitfalls (16), particularly in application to solid oral dosage forms, where particles are typically neither spherical and uniform-sized, nor immediately available for dissolution.

Furthermore, particle size of drug in the formulation is often not available, such that applications of the z-factor dissolution rate equation often do not employ particle size, but rather utilize Eq. 28 for data reduction and simulation (46, 47). For example, Heimbach et al. determined the safe space for etoricoxib tablets by employing a PBBM model that used the z-factor dissolution rate equation for data reduction (46). Without regard to particle size, simulations at each pH 6.8 and 2.0 were conducted where z-factor was varied in a in stepwise fashion until the simulated Cmax was at least 20% reduced, to define the safe space. Similarly, the z-factor dissolution rate equation was used to fit in vitro dissolution data of lesinurad IR tablets, for subsequent pharmacokinetic modeling (47). However, particle distribution was not an input parameter set, but rather a fitted parameter set.

A limitation of Eq. 28 is that the equation is practically only available in the differential form, necessitating the need for software such as GastroPlus, SAAM II, or STELLA that can perform numerical integration (14,15,16,17). Compared to z-factor, the Polli equation has the advantage of not needing such software to fit dissolution profiles, as it is an analytical solution and hence does not need numerical integration methods but only regression. Of note, Pepin et al. have provided an Excel file with macros to perform numerical integration to fit the z-factor dissolution rate equation, in the context of an approach employing a certain input parameter set (47).

Table III compares characteristics of the Polli equation to the z-factor dissolution rate equation under non-sink conditions, and points towards the simplicity of the Polli equation as a strength. Other particle dissolution rate equations are the Johnson equation and the Wang-Flanagan equation (11,17). Like the z-factor dissolution rate equation, a well-appreciated strength of these equations is ability to simulate particle size effects on dissolution. However, they also require numerical integration to fit dissolution data (14,15,16,17). Most particle dissolution models assume spherical shape, which is typically not the case. Drug particle size distribution and shape are often not known in formulated tablets, attenuating the potential advantage of such particle dissolution models, and perhaps creating opportunity for model over-parameterization. Meanwhile, with only a single fitted parameter, Eq. 21 does not assume any particular particle distribution or shape, but only that drug mass is a driving force for dissolution, with drug solubility (i.e., non-sink conditions) potentially reducing dissolution. Likewise, while z in Eq. 28 can be considered an explicit function of drug diffusion coefficient, which may or may not be known (26, 29), Eq. 21 does not assume any particular drug transport phenomena beyond drug mass as a driving force for dissolution, with potential non-sink effects. Interesting, as noted above, FeSSGF here provided relatively small kd values, reflecting that FeSSGF colloids are large and slowly diffusing.

Table III Comparison of Characteristics of the Polli Dissolution Equation and the z-Factor Dissolution Rate Equation. Because It Is an Analytical, the Polli Equation Expresses Percent Dissolved as a Function of Time and Can Fit Dissolution Data via Non-linear Regression. Meanwhile, the z-Factor Dissolution Rate Equation Is a Differential Equation, Requiring Numerical Integration Methods to Fit the Equation to Dissolution Data

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