the parameters f(Vff) and g(Vff) were of critical importance to the calculations in this study. Their introduction makes it possible to model the kinetics of multi-dimensional freeze-drying, which are applicable to most situations where foods and agricultural materials are freeze-dried. In this study, it was assumed that f(Vff) and g(Vff) are simply a function of Vff, regardless of the kinetics of drying. This assumption must be validated in future work. In practice, the deformation of the drying product (e.g., due to shrinkage) is likely to have an influence on the drying kinetics. Apple was selected for this work because the degree of its deformation during drying is rather limited. For this reason, we set the value for Aext to a constant. However, occasional shrinkage of specimens was observed. Shrinkage lead to a decrease in Aext, thus causing an overestimation of the mean thickness of the dried cake (i.e., View the MathML source, and resulting in an overestimation of the drying time. The model could be improved by allowing Aext to vary as a function of the degree of shrinkage. However, shrinkage depends largely on the type of food material, so the acquisition of As(Vff),Abtm(Vff) and Aext(Vff) from various materials may be a useful future study.