The Local Effect Model (LEM) is a track-structure model that was developed to predict the biological response of a cell to irradiation with any ion. Because it needs to be studied both experimentally and theoretically, a mathematical formalization of the LEM based on three main postulates and three secondary approximations is proposed for a more detailed analysis. The general relationship that links cell survival to the mean number of lethal events is deduced. A Monte Carlo simulation is also proposed to calculate the local dose. It is shown that the local dose is highly heterogeneous even for uniform X irradiations. This observation raises questions about the estimation of the density of ion-induced lethal events from the expression of cell survival after exposure to X rays. Finally, it is shown that a strict theory of local effects based solely on local dose cannot reproduce nonlinear structures in cell survival curves, such as the shoulders observed after low-LET irradiation.