In head-mounted eye tracking systems, the correct detection of pupil position is a key factor in estimating gaze direction. However, this is a challenging issue when the videos are recorded in real-world conditions, due to the many sources of noise and artifacts that exist in these scenarios, such as rapid changes in illumination, reflections, occlusions and an elliptical appearance of the pupil. Thus, it is an indispensable prerequisite that a pupil detection algorithm is robust in these challenging conditions. In this work, we present one pupil center detection method based on searching the maximum contribution point to the radial symmetry of the image. Additionally, two different center refinement steps were incorporated with the aim of adapting the algorithm to images with highly elliptical pupil appearances. The performance of the proposed algorithm is evaluated using a dataset consisting of 225,569 head-mounted annotated eye images from publicly available sources. The results are compared with the better algorithm found in the bibliography, with our algorithm being shown as superior.