The reconstruction of charged particle trajectories in modern high energy experiments in particle physics is a complex and very time consuming task. At the same time the average available reconstruction time per particle is very limited due to the high data rate. However, using the massive parallel computing possibilities available on modern graphic processors, it should be feasible to reduce the needed reconstruction time significantly. This reduction could then be used to either allow for an improved reconstruction and hence improved precision or to reduce the costs in the worldwide distributed computing centers. We perform a feasibility study where the existing reconstruction software is implemented in a simplified test environment and modified to take advantage of the graphics processors. Using real and simulated data from the ATLAS experiment at the Large Hadron Collider (LHC) the possible speed improvements will be determined.