# Fitting Controlled Release and Dissolution Data

Introduction

The parameters derived from fitting models of controlled release mechanisms to experimental data may be used to characterize the release properties (1). Some of the same equations may be used to fit dissolution profiles and pairwise comparisons made using confidence intervals (2). A custom curve fit library, Release.jfl, was created in SigmaPlot to fit five published nonlinear models for the controlled release of pharmaceuticals (1).

The different models represent different encapsulation geometries and different release mechanisms. The “right click, Fit Curve” feature of SigmaPlot easily allows rapid fitting of all five models to data for comparison. Download the self-extracting file Release.zip and double click on it to obtain the custom curve fit library and graph example. An example of the fit of this model to published data (3) is shown in the graph above.

The Custom Fit Library

The equations implemented in Release.jfl are shown in the table below. F is the percentage of drug released at time t.

 Name Equation Parameters Baker and Lonsdale (4) k Peppas (5) k, n Hixon and Crowell (6) k Higuchi Square Root of Time (7) k First Order (8) k

The model developed by Baker and Lonsdale is an implicit function. Two ways to approach an implicit function fit are described. First, in an approximate but nevertheless accurate approach, TableCurve 2D was used to “invert” this function to provide an explicit function approximation accurate to 0.03%. Automatic initial parameter estimation functions are used for each of these equations.

A particularly clever approach, using the ape() function for the Baker & Lonsdale equation is described here. Second, SigmaPlot can now fit implicit functions using the new implicit() transform command. So the problem can now be solved directly without approximation. The fit commands for fitting the implicit Baker & Lonsdale equation are described here.

Fitting Equations

It is easy to fit the five models to data and compare the results. You simply right click on a graph data point for the data to be fitted, select the equation and click Finish. Then repeat this for each equation. The first time you fit a curve you will need to click the Back button and Browse for the Release.jfl fit library. This produces the Controlled Release equation library with the five models shown in the following dialog.

The fit of the five models to Shakula and Price’s theophylline microsphere data is shown in the following figure. The numbers in the legend are the sums of squares of the residuals. The Baker and Lonsdale equation provides the best fit by far. Even the Peppas equation with its extra degree of freedom (two parameters) fits less well.

Replicate Measurements

Data with replicate measurements can be graphed and fit using the replicate data format X, Y Replicate. Data with time (X) in column 1 and 12 replicate % release data values (Y Replicate) in columns 2 through 13 is graphed using symbol values computed from Row Means and Standard Deviation for the error calculation. The replicates are entered rowwise for each time value.

The resulting mean-plus-error-bar graph is shown below. This data can then be fit using the same X, Y Replicate format in the Regression Wizard. To do this right click on one of the symbols in the graph and select Fit Curve. Select the Baker and Lonsdale equation and other options in the Regression Wizard. Since you right clicked on a graph that has been formatted using X, Y Replicate, this format will be automatically selected for the curve fit. Be sure to select Extend Fit to Axes to start the fit line at the origin.

Summary

The Regression Wizard design of SigmaPlot allows the creation of custom curve fit libraries. The procedure for creating a controlled release library for analyzing data from drug dissolution tests was shown to be a straightforward modification of equations in the SigmaPlot Fit Library. The regression library Release.jfl was created using the procedure described. Once the library has been created the different functions can be rapidly fitted to data by right clicking on a data point and selecting Fit Curve.

References

• Lu, D.R., Abu-Izza, K., and Mao, F., Nonlinear Data Fitting for Controlled Release Devices: An Integrated Computer Program, Internat. J. Pharmaceutics 129 (1996) 243-251.
• Polli, J.E. and Rekhi, G.S., Methods to compare dissolution profiles, Drug Information Journal, 30, (1996), 1113-1120.
• Shukla, A.J. and Price, J.C., Effect of drug loading and molecular weight of cellulose acetate propionate on the release characteristics of theophylline microspheres. Pharm. Res., 8 (1991) 1369-1400.
• Baker, R.W., and Lonsdale, H.S., Controlled released: mechanisms and rates. In Tanquary, A.C. and Lacey, R.E. (Ed.), Controlled Release of Biologicallyl Active Agents, Plenum Press, New York, 1974, pp. 15-71.
• Peppas, N.A., Analysis of Fickian and non-Fickian drug release from polymers. Pharm. Acta Helv., 60 (1985) 110-111.
• Hixon, A.W. and Crowell, J.H., Dependence of reaction velocity upon surface and agitation: I-Theoretical condideration. Ind. Eng. Chem., 23 (1931) 923-931.
• Higuchi, T., Rate of release of medicaments from ointment bases containing drugs in suspension. J. Pharm. Sci., 50 (1961) 874-875.
• Shah, M.V., De Gennaro, M.D. and Suryakasuma, H., An evaluation of albumin microcapsules prepared using a multiple emulsion technique. J. Microencapsul., 4 (1987) 223-238.