The concept of cellular automata with the dynamic structure of a cellular space (DCA) is proposed. The DCA extends the capabilities of classical cellular automata (CA), and allows using the cellular automata approach to the problems of simulating the biological tissues growth. A DCA differs from a classical CA in that its cells are not located on a regular lattice, and intercellular connections are explicitly described by the neighborhood matrix. In addition, insertion and partition operators are introduced for the DCA. These operators allow one to dynamically change the cell space-structure. Based on this extension, the DCA-model of the apical meristem escape of Arabidopsis Thaliana growth is constructed, being a parallel composition of the two DCA: the asynchronous two-dimensional DCA simulating self-regulation in biological cells, and the synchronous one-dimensional DCA simulating growth and division of biological cells. The results of computer simulations have shown that the behavior of the proposed DCA-model matches the behavior of the existing model based on the composition of differential equations and the method of L-system (Lindenmayer system). Furthermore, the proposed DCA-model allows one to simulate growth of individual biological cells and to visualize the substances dynamics in these cells (decay, synthesis and diffusion).