The goal of this study is a better understanding of the interaction between cells and a solidification front during a cryopreservation process. This technique of freezing is commonly used to conserve biological material for long periods at low temperatures. However the biophysical mechanisms of cell injuries during freezing are difficult to understand because a cell is a very sophisticated microstructure interacting with its environment. We have developed a finite element model to simulate the response of cells to an advancing solidification front. A special front-tracking technique is used to compute the motion of the cell membrane and the ice front during freezing. The model solves the conductive heat transfer equation and the diffusion equation of a solute on a domain containing three phases: one or more cells, the extra-cellular solution and the growing ice. This solid phase growing from a binary salt solution rejects the solute in the liquid phase and increases the solute gradient around the cell. This induces the shrinkage of the cell. The model is used to simulate the engulfment of one cell modelling a red blood cell by an advancing solidification front initially planar or not is computed. We compare the incorporation of a cell with that of a solid particle.