DATA MANIPULATION

2. DATA MANIPULATION

2.1. Calculation of Cumulative Release For an analytical system in which volume is constant, as is

the case for in situ methods and recirculatory continuous flow systems with in-line concentration measurement, cumulative release may be determined as follows:

C n V R n ¼ 100

D where R n is the percentage cumulative release at time-point

n, C n the concentration at time-point n, and D is the drug con- tent of the sample; for a regulatory test, it is normal practice that the test comprise, multiple determinations of a single unit dose, and for D to represent the labeled dose.

Where the release test involves the withdrawal of a sample of dissolution medium in which the formulation is homogeneously dispersed, and the test is continued with diminished volume, the formulation:dissolution medium ratio is unaffected by the sampling operation and hence the concen- tration of dissolved drug at subsequent time-points is the

140 Clark et al.

same as would be the case in the constant volume procedure described above. Cumulative release is given by

C n V 0 R n ¼ 100

D where R n ,C n , and D are as previously described and V 0 is the

initial volume of dissolution medium. Where the method involves in-situ filtration such that supernatant medium containing dissolved drug is withdrawn and not replaced, a correction factor must be applied as follows:

n ¼ 100

where R n ,C n , and D are as previously described, and V s is the volume of supernatant medium withdrawn at each time-point.

2.2. Mathematical Description of Release Profile In general, drug dissolution from solids can be described

using the Noyes–Whitney equation as modified by Nernst and Brunner:

dM DS ðC s

¼ dt

h where M is the amount of drug dissolved in time t, D is the

diffusion coefficient of the solute in the dissolution medium, S is the surface area of the expressed drug, h is the thickness of the diffusion layer, C s is the solubility of the solute and C t is the concentration of the solute in the medium at time t; the equation may be simplified by assuming that, for dissolution testing under sink conditions, C t is zero.

This model assumes that a layer of saturated solution forms instantly around a solid particle, and that the dissolu- tion rate-controlling step is transport across this so-called diffusion layer. Ficks law describes the diffusion process:

m DADC ¼ t

In Vitro=In Vivo Release 141

where m =t is the mass flow rate (mass m diffusing in time t), D is the diffusion constant, DC is the concentration difference, A is the cross-sectional area, and L is the diffusion path length.

The cube root law developed by Hixson and Crowell takes into account Fick’s law and may be considered to describe the dissolution of a single spherical particle under sink conditions:

3 rh where w is particle weight at time t, w 0 is the initial particle

weight, k 1 =3 is the composite rate constant, r is the density of the particle, and D, C s , and h are as previously defined. The expressions above may be used as the basis for mathematical models of drug release (28).

2.3. Comparison of Release Profiles

A comparison of dissolution profiles may be necessary to support changes in formulation, site, scale or method of man- ufacture. The comparison should be based on at least 12 units of reference (prechange) and test (postchange) product.

A common procedure is the model-independent approach (29,30), which involves calculation of a difference factor (f 1 ) and a difference factor (f 2 ) to compare profiles. This approach is suitable where the dissolution profile is based on three or more time-points, only one of which occurs after 85% dissolu- tion; the time-points for the reference and test batches must

be the same and no modification to the release test is permis- sible. The reference profile may be based on the mean dissolu- tion values for the last prechange batch, or the last two or more consecutively manufactured prechange batches; for mean data to be meaningful, the RSD should be < 20% for the initial time-point and <10% for subsequent time-points.

The difference factor (f 1 ) calculates the percentage difference between the two curves at each time-point and is

a measurement of the relative error between the two curves: P n

jR t ¼ 100

f 1 t ¼1

P n t ¼1 R t

142 Clark et al.

where n is the number of time-points, R t is the dissolution value of the reference (prechange) batch at time t, and T t is the dissolution value of the test (postchange) batch at time

t. For curves to be considered similar, f 1 should be close to zero and within the range 0–15. The similarity factor (f 2 ) is a logarithmic reciprocal square root transformation of the sum of squared error and is a measurement of the similarity between the two curves:

f 2 ¼ 50 log q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > :

½1 þ ð1=nÞ 2

t ¼1 ðR t

where n, t, R t , and T t are as defined for the calculation of difference factor.

For curves to be considered similar, the similarity factor (f 2 ) should be close to 100 and within the range 50–100. Where batch-to-batch variation within the reference and test batches is greater than 15% RSD, misleading results may arise and an alternative approach is preferable. A model- dependent method, involving the derivation of a mathemati- cal function to describe the dissolution profile followed by determination of the statistical distance between the refer- ence and test batches (31), may be used to compare the test and reference profiles taking into account variance and covar- iance of the data sets and allowing the use of different sampling schemes for the reference and test lots.

A comparison of anova-based, model-dependent, and model-independent methodologies for immediate release tablets (32) concluded that the anova-based and model-depen- dent methods have narrower limits and are more discrimina- tory than the similarity =difference factor methods.