## 1 Introduction  
Computed tomography (CT) provides the photon attenuation information that is required in positron emission tomography (PET). Therefore, CT has been used for PET attenuation correction (AC) and for external beam radiation therapy (EBRT) planning since the appearance of the first PET/CT in 1997 [1]. In recent years, the number of combined PET/MR scanners has increased among medical centres [2]. Thus, the interest in replacing the CT scanners with magnetic resonance (MR) imaging has raised since MR provides greater tissue contrast as well as other complementary information such as perfusion or diffusion. Additionally, MR decreases the use of ionizing radiation, specially in the AC for PET imaging. MR Early developments made use of fat and water separation or other specific MR sequences (i.e., UTE, ZTE) to estimate AC maps [3, 4]. However, the AC maps estimated using MR imaging have wide discrepancies with the AC maps calculated using the CT [5, 6]. The last decade has seen a renewed interest in MR-only workflows for PET imaging and radiotherapy. This way, several works have proposed the synthesis of CT volumes from MR images (pseudo-CT) using computer vision techniques. The first approaches used traditional image processing and machine learning strategies, such as segmentation-based methods [7, 8, 9, 10, 11], atlas-based methods [12, 13, 14, 15, 16, 17] or learning-based methods [18, 19, 20]. These approaches present several disadvantages, such as the need of an accurate spatial normalization to a template space, the assumption of mostly normal anatomy, or the problem of accommodating a large amount of training data. These problems have been solved in the last few years with the advent of new techniques based on Deep Convolutional Neural Networks (DCNN). The use of DCNNs has also improved the quality of the results while reducing the time for pseudo-CT synthesis. According to our knowledge, the first approach that adopted deep learning to generate a pseudo-CT from an MR scan was presented by [21]. This work proposed a DCNN that used a U-net architecture [22] performing 2D convolutions. The network received axial slices from a T1-weighted volume of the head as input, and tried to generate the corresponding slices from a registered CT scan. Their architecture incorporated unpooling layers in the up-sampling steps of the U-net and Mean Absolute Error (MAE) as loss function. Their results were compared to an atlas based method [23], obtaining favorable results in accuracy and computational time (close to real-time). In contrast, the work of [24] explored a similar architecture to segment air, bone, and soft tissue instead of generating the continuous pseudo-CT. They used a set of 3D T1-weighted head volumes as well. In this case, they incorporated a nearest-neighborhood interpolation in the up-sampling steps of the U-net. Additionally, as they were classifying instead of regressing, they employed a multi-class cross-entropy loss metric, which is usually easier to optimize than the L1 or L2 error. The deep learning approach performed better than a Dixon-based approach and took less than 0.5 min to generate the pseudo-CT segmentation. Several works made use of more sophisticated network architectures and training pipelines. The work by [25] proposed a 3D neural network with dilated convolutions to avoid the use of pooling operations. They also explored the advantages of residual connections [26] and auto-context refinement [27]. For training, they employed an adversarial network that tried to differentiate between real CTs and pseudo-CTs [28]. They used 3D patches from T1 volumes from head and pelvis as input, and compared their results against traditional methods such as atlas registration, sparse representation and Random Forest with auto-context. Their method outperformed all of these traditional methods. A similar approach was proposed by [29], who trained a 2D DCNN that incorporated the residual blocks from Resnet [26] and an adversarial strategy [28] for training.