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In biomedical applications,understanding the effect of mechanical properties of the erythrocytes on oxygen delivery is important for identifying the biophysical origins of several diseases related to metabolic activities at the cellular level.In order to improve drug delivery efficiency,we also need to understand the effects of the physical and geometric properties of the liposome on the mass transfer rate.Our main objective in this paper is to extend the immersed boundary method for fluid flow problems with mass transfer across permeable moving interfaces.