Dynamic and spatially multi-dimensional mathematical models of the primary and secondary drying stages of the freeze-drying of pharmaceutical crystalline and amorphous solutes in vials, are constructed and presented in this work. The models account for the removal of free and bound water and could also provide the geometric shape of the moving interface and its position. It is proved that the temperature of the moving interface can not be constant if the flux of heat flow to the sides of the vial is not zero. It is also proved that the slope of the free surface (moving interface) at the edge of the vial is always curved downward.