This paper focuses on contour tracking, an important problem in computer vision, and specifically on open contours that often directly represent a curvilinear object. Compelling applications are found in the field of bioimage analysis where blood vessels, dendrites, and various other biological structures are tracked over time. General open contour tracking, and biological images in particular, pose major challenges including scene clutter with similar structures (e.g., in the cell), and time varying contour length due to natural growth and shortening phenomena, which have not been adequately answered by earlier approaches based on closed and fixed end-point contours. We propose a model-based estimation algorithm to track open contours of time-varying length, which is robust to neighborhood clutter with similar structures. The method employs a deformable trellis in conjunction with a probabilistic (hidden Markov) model to estimate contour position, deformation, growth and shortening. It generates a maximum a posteriori estimate given observations in the current frame and prior contour information from previous frames. Experimental results on synthetic and real-world data demonstrate the effectiveness and performance gains of the proposed algorithm.