###### Abstract
PDw, by quantifying the amount of water protons contributing to each voxel [9], provides a good distinction between fat and fluid [21]. ADC yields a quantitative map of the water diffusion characteristics of the prostatic tissue: PCa typically has packed and dense regions with intra- and inter-cellular membranes that influence water motion [4; 8]. Lastly, DCE sequences depict the patient's vascular system in detail, since tumors exhibit a highly vascularized micro-environment, by exploiting a Gadolinium-based contrast medium [22]. In Deep Learning applications to medical image analysis, some challenges are still present [23]; namely, (i) the lack of large training data sets, (ii) the absence of reliable ground truth data, and (_iii_) the difficulty in training large models [24]. Nonetheless, some factors are consistently present in successful models [25]: expert knowledge, novel data preprocessing or augmentation techniques, and the application of task-specific architectures. Despite the growing interest in developing novel models for the task of PCa, little effort has been devoted to the addition of new types of layers. Recently, the Semantic Learning Machine (SLM) [26; 27; 28] neuroevolution algorithm was successfully employed to replace the backpropagation algorithm commonly used in the Fully-Connected (FC) layers of Convolutional Neural Networks (CNNs) [29; 30]. When compared with backpropagation, SLM achieved higher classification accuracy in PCa detection as well as a training speed-up of one order of magnitude. A CNN architecture developed for the classification task is typically composed of regular layers, which perform convolutions, dot products, batch normalization, or pooling operations. This work integrates a Conditional Random Field (CRF) model as a Recurrent Neural Network (RNN) [31], generally exploited in segmentation, to the PCa classification problem. In accord with the latest clinical trends aiming at decreasing contrast medium usage [4], we analyzed only the non-contrast-enhanced mpMRI sequences to assess also the feasibility of our methodology from a patient safety and health economics perspective [32]. Research Questions. We specifically address two questions:  
* Can the CRF-CNN be integrated into a state-of-the-art CNN as an end-to-end approach? * Can the smoothing effect of CRFs increase the classification performance of CNNs in PCa detection? Contributions. Our main contributions are the following:  
* A hybrid end-to-end trainable network that combines CRF-RNN [31] and a state-of-the-art CNN, namely XmasNet [33], without requiring a two-phase training procedure. * The proposed CRF-XmasNet architecture generally outperforms the baseline architecture XmasNet [33] in terms of PCa classification on mpMRI. * The proposed end-to-end integration of CRF-RNN provides better generalization ability when compared to a two-phase implementation, using a CRF as a postprocessing step. In particular, the proposed approach aimed at outperforming XmasNet, a network specifically created for dealing with prostate cancer MRI data. This network is the most state-of-the-art for this kind of application. Thus, outperforming it would be a valuable contribution in the medical field, as it shows how the integration of CRFs in XmasNet could improve the performance of the commonly used XmasNet architecture. The manuscript is organized as follows. Section 2 introduces the theoretical foundations of CRFs and the Deep Neural Network (DNN) architectures underlying the devised method. Section 3 presents the characteristics of the analyzed prostate mpMRI dataset, as well as the proposed method. Section 4 shows and critically analyzes the achieved experimental results. Finally, Section 5 concludes the paper and suggests future research avenues.