Delineation of lung tumor from adjacent tissue from a series of magnetic resonance images (MRI) poses many difficulties due to the image similarities of the region of interest and surrounding area as well as the influence of respiration. However, accurate segmentation of the tumor region is essential in planning a radiation therapy to prevent healthy tissues from receiving excessive radiation. The manual delineation of the entire MRI sequence is tedious, time-consuming and costly. This study investigates how one can perform automatic tracking of tumor boundaries during radiation therapy using convolutional neural networks. We proposed to use a convolutional neural network architecture with modified Dice metric as the cost function. The proposed approach was evaluated over 600 images in comparison to expert manual contours. The proposed method yielded an average Dice score of $0.91 \pm 0.03$ and Hausdorff distance of $2.88 \pm 0.86$ mm. The proposed approach outperformed recent state-of-the-art methods in terms of accuracy in the delineation of the mobile tumors.