In the past few decades, to reduce the risk of X-ray in computed tomography (CT), low-dose CT image denoising has attracted extensive attention from researchers, which has become an important research issue in the field of medical images. In recent years, with the rapid development of deep learning technology, many algorithms have emerged to apply convolutional neural networks to this task, achieving promising results. However, there are still some problems such as low denoising efficiency, over-smoothed result, etc. In this paper, we propose the Edge enhancement based Densely connected Convolutional Neural Network (EDCNN). In our network, we design an edge enhancement module using the proposed novel trainable Sobel convolution. Based on this module, we construct a model with dense connections to fuse the extracted edge information and realize end-to-end image denoising. Besides, when training the model, we introduce a compound loss that combines MSE loss and multi-scales perceptual loss to solve the over-smoothed problem and attain a marked improvement in image quality after denoising. Compared with the existing low-dose CT image denoising algorithms, our proposed model has a better performance in preserving details and suppressing noise.