Quantification and geometric analysis of coiling patterns in gastropod shells based on 3D and 2D image data

The morphology of gastropod shells has been a focus of analyses in ecology and evolution. It has recently emerged as an important issue in developmental biology, thanks to recent advancements in molecular biological techniques. The growing tube model is a theoretical morphological model for describing various coiling patterns of molluscan shells, and it is a useful theoretical tool to relate local tissue growth with global shell morphology. However, the growing tube model has rarely been adopted in empirical research owing to the difficulty in estimating the parameters of the model from morphological data. In this article, I solve this problem by developing methods of parameter estimation when (1) 3D Computed Tomography (CT) data are available and (2) only 2D image data (such as photographs) are available. When 3D CT data are available, the parameters can be estimated by fitting an analytical solution of the growing tube model to the data. When only 2D image data are available, we first fit Raup's model to the 2D image data and then convert the parameters of Raup's model to those of the growing tube model. To illustrate the use of these methods, I apply them to data generated by a computer simulation of the model. Both methods work well, except when shells grow without coiling. I also demonstrate the effectiveness of the methods by applying the model to actual 3D CT data and 2D image data of land snails. I conclude that the method proposed in this article can reconstruct the coiling pattern from observed data.