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Advances in microscope hardware in the last couple of decades have made it possible to acquire large data sets with image sequences of living cells grown in cell culture. This has led to a demand for automated ways of analyzing the acquired images. This article presents a new algorithm for tracking cells and constructing cell lineages in such image sequences. The algorithm uses information from the entire sequence to make local decisions about cell tracks and can therefore make more robust decisions than algorithms that process the data sequentially. It also incorporates image-based likelihoods of cell division and cell death into the tracking, without having to resort to separate detection algorithms or post processing of tracks. The algorithm consists of a scoring function to rank tracks and an iterative algorithm that searches for the highest scoring tracks, in a computationally efficient way, using the Viterbi algorithm.