We investigate the polymerization kinetics of rod-like polymer filaments interacting with a distribution of monomer in one spatial dimension (e.g. along a narrow tube). We consider a variety of possible cases, including competition by the filament tips for the available monomer, and behaviour analogous to "treadmilling" in which the polymer adds subunits to one end and loses them at the other end so as to maintain a constant length. Applications to biological polymers such as actin filaments and microtubules are discussed.