Determination of mass transfer coefficients during freeze drying using modeling and parameter estimation techniques

Abstract

An approach using mathematical modeling and parameter estimation techniques was established to determine the best-fit mass transfer coefficients of the dried-layer resistance during primary drying of pharmaceuticals, in order to minimize experimental efforts. For modeling of the primary drying process, the equations associated with heat and mass transfer mechanisms established by Pikal were used as the model equations, whereas the parameter estimation was accomplished by Powell's nonlinear least-squares minimization algorithm. The advantages of this approach include that it can be applied to analyze various types of measurable quantities, such as the cumulative mass of sublimation M t, the temperature profiles at the bottom center of the frozen layer T b, and the pressure profiles of the vials. It can also be used for these quantities measured using the vials with different heat transfer coefficients. To test the performance of the proposed approach, hypothetical values of M t and T b with perturbed errors, were simulated using the modeling algorithm and a random number generator. These values were in turn used as the input data for parameter estimation. The results show that the best-fit mass transfer coefficients are successfully obtained using either the hypothetical M t or T b profiles, with appropriate initial guesses of the parameters. All computations in modeling and parameter estimation were developed in FORTRAN and compiled using Microsoft FORTRAN compiler, of which the source codes and documentation, with detailed mathematical equations and computation steps, are available upon request.