A convolutional neural network (ConvNet) is usually trained and then tested using images drawn from the same distribution. To generalize a ConvNet to various tasks often requires a complete training dataset that consists of images drawn from different tasks. In most scenarios, it is nearly impossible to collect every possible representative dataset as a priori. The new data may only become available after the ConvNet is deployed in clinical practice. ConvNet, however, may generate artifacts on out-of-distribution testing samples. In this study, we present Targeted Gradient Descent (TGD), a novel fine-tuning method that can extend a pre-trained network to a new task without revisiting data from the previous task while preserving the knowledge acquired from previous training. To a further extent, the proposed method also enables online learning of patient-specific data. The method is built on the idea of reusing a pre-trained ConvNet's redundant kernels to learn new knowledge. We compare the performance of TGD to several commonly used training approaches on the task of Positron emission tomography (PET) image denoising. Results from clinical images show that TGD generated results on par with training-from-scratch while significantly reducing data preparation and network training time. More importantly, it enables online learning on the testing study to enhance the network's generalization capability in real-world applications.