Mathematical functions and their properties as relevant to the biomechanical modeling of cell and tissue damage

The extrapolation of biological damage from a biomechanical model requires that a closed-form mathematical damage threshold function (DTF) be included in the model. A DTF typically includes a generic load variable, being the critical load (e.g., pressure, strain, temperature) causing irreversible tissue or cell damage, and a generic time variable, which represents the exposure to the load (e.g., duration, strain rate). Despite the central role that DTFs play in biomechanical studies, there is no coherent literature on how to formulate a DTF, excluding the field of heat-induced damage studies. This technical note describes six mathematical function types (Richards, Boltzmann, Morgan-Mercer-Flodin, Gompertz, Weibull, Bertalanffy) that are suitable for formulating a wide range of DTFs. These functions were adapted from the theory of restricted growth, and were fitted herein to describe biomechanical damage phenomena. Relevant properties of each adapted function type were extracted to allow efficient fitting of its parameters to empirical biomechanical data, and some practical examples are provided.