Analyzing spatio-temporal dependencies between different types of events is highly relevant to numerous biological phenomena (e.g. signalling and trafficking) especially as advances in probes and microscopy have facilitated imaging of dynamic processes in living cells. For many types of events, the segmented areas can overlap spatially and temporally forming random clumps. In this paper, we model binary image sequences of two different event types as a realization of a bivariate temporal random set and propose a non-parametric approach to quantify spatial and spatio-temporal interrelations using the pair-correlation, cross-covariance and the Ripley IK functions. Based on these summary statistics we propose a randomization procedure to test independence between event types by applying random toroidal shifts and Monte Carlo tests. A simulation study assessed the performance of the proposed estimators and showed that these statistics capture the spatio-temporal dependencies accurately. The estimation of the spatio-temporal interval of interactions was also obtained. The method was successfully applied to analyze the interdependencies of several endocytic proteins using image sequences of living cells and validated the procedure as a new way to automatically quantify dependencies between proteins in a formal and robust manner.

Total Internal Reflection Fluorescence Microscopy (TIRFM) greatly facilitates to imaging the first steps of endocytosis, a process whereby cells traffic cargo from the cell surface to endosomes. Using TIRFM, fluorescent-tagged endocytic proteins are observed as overlapped areas forming random clumps of different sizes, shapes and durations. A common procedure to segment these objects consists of thresholding the original gray-level images to produce binary sequences in which a pixel is covered or not by a given fluorescent-tagged protein. This binary logic is not appropriate because it leaves a free tuning parameter to be set by the user which can influence on the conclusions of the statistical analysis. Instead, we have adopted a more realistic approach, in which segmented binary images are modelled as a fuzzy temporal random set. Here, we propose some measures of spatio-temporal interactions based on the fuzzy counterparts of the pair-correlation function and the Ripley K-function. We used a randomization procedure to test for independence. Our results show that this procedure will permit biologists to examine and quantify the interactions between endocytic proteins robustly.

Counting spatially and temporally overlapping events in image sequences and estimating their shape-size and duration features are important issues in some applications. We propose a stochastic model, a particular case of the nonisotropic 3D Boolean model, for performing this analysis: the temporal Boolean model. Some probabilistic properties are derived and a methodology for parameter estimation from time-lapse image sequences is proposed using an explicit treatment of the temporal dimension. We estimate the mean number of germs per unit area and time, the mean grain size and the duration distribution. A wide simulation study in order to assess the proposed estimators showed promising results. The model was applied on biological image sequences of in-vivo cells in order to estimate new parameters such as the mean number and duration distribution of endocytic events. Our results show that the proposed temporal Boolean model is effective for obtaining information about dynamic processes which exhibit short-lived, but spatially and temporally overlapping events.