Purpose: To develop mathematical models for primary and secondary drying steps of a lyophilization cycle in an array of vials on a lyophilizer shelf. To use the models to predict inhomogeneous lyophilization and providing a comparison to experimental data obtained under various process conditions. Methods: A combination of experimental and mathematical modeling was used to study and predict sublimation rate (for primary drying) and desorption rate (for secondary drying) at different vial locations on the shelf. One-dimensional mathematical models for primary drying and for secondary drying, applied to an individual vial, were combined with two-dimensional mathematical models for the shelf plate and interstices between vials. The resulting three-dimensional models of freeze-drying in a heterogeneous system consisting of shelf plate, vials and void space can simulate patterns of the inhomogeneous process during primary and secondary drying on the whole shelf. The models are represented by sets of coupled partial differential equations that were solved numerically. Experimentally determined temperature profiles at different vial locations, sublimation rate data obtained gravimetrically, cake moisture content obtained by the Karl-Fisher method, and the composition of gas in the lyophilizer measured with mass spectrometer and data loggers were used to confirm mathematical simulations. Results: The mathematical model for a single vial in the center of the shelf was employed for evaluation of overall heat transfer coefficient between the shelf and vial and for evaluation of mass transfer coefficient in the dried cake. The three-dimensional model was then used for evaluation of external heat transfer coefficient between the lyophilizer wall and array of vials as well as for evaluation of effective heat conductivity in the array of vials. Conclusions: Mathematical modeling successfully simulates and predicts patterns of inhomogeneous lyophilization on the whole shelf. The models are useful tools for the design, optimization and scale-up of freeze-drying.