  Amoeboid cell motility is essential for a wide range of biological processes
including wound healing, embryonic morphogenesis, and cancer metastasis. It
relies on complex dynamical patterns of cell shape changes that pose
long-standing challenges to mathematical modeling and raise a need for
automated and reproducible approaches to extract quantitative morphological
features from image sequences. Here, we introduce a theoretical framework and a
computational method for obtaining smooth representations of the spatiotemporal
contour dynamics from stacks of segmented microscopy images. Based on a
Gaussian process regression we propose a one-parameter family of regularized
contour flows that allows us to continuously track reference points (virtual
markers) between successive cell contours. We use this approach to define a
coordinate system on the moving cell boundary and to represent different local
geometric quantities in this frame of reference. In particular, we introduce
the local marker dispersion as a measure to identify localized membrane
expansions and provide a fully automated way to extract the properties of such
expansions, including their area and growth time. The methods are available as
an open-source software package called AmoePy, a Python-based toolbox for
analyzing amoeboid cell motility (based on time-lapse microscopy data),
including a graphical user interface and detailed documentation. Due to the
mathematical rigor of our framework, we envision it to be of use for the
development of novel cell motility models. We mainly use experimental data of
the social amoeba Dictyostelium discoideum to illustrate and validate our
approach.
