  Three-dimensional cultures of cells are gaining popularity as an in vitro
improvement over 2D Petri dishes. In many such experiments, cells have been
found to organize in aggregates. We present new results of three-dimensional in
vitro cultures of breast cancer cells exhibiting patterns. Understanding their
formation is of particular interest in the context of cancer since metastases
have been shown to be created by cells moving in clusters. In this paper, we
propose that the main mechanism which leads to the emergence of patterns is
chemotaxis, i.e., oriented movement of cells towards high concentration zones
of a signal emitted by the cells themselves. Studying a Keller-Segel PDE system
to model chemotactical auto-organization of cells, we prove that it is subject
to Turing instability under a time-dependent condition. This result is
illustrated by two-dimensional simulations of the model showing spheroidal
patterns. They are qualitatively compared to the biological results and their
variability is discussed both theoretically and numerically.
