Background: REQUITE (validating pREdictive models and biomarkers of radiotherapy toxicity to reduce side effects and improve QUalITy of lifE in cancer survivors) is an international prospective cohort study. The purpose of this project was to analyse a cohort of patients recruited into REQUITE using a deep learning algorithm to identify patient-specific features associated with the development of toxicity, and test the approach by attempting to validate previously published genetic risk factors. Methods: The study involved REQUITE prostate cancer patients treated with external beam radiotherapy who had complete 2-year follow-up. We used five separate late toxicity endpoints: ≥grade 1 late rectal bleeding, ≥grade 2 urinary frequency, ≥grade 1 haematuria, ≥ grade 2 nocturia, ≥ grade 1 decreased urinary stream. Forty-three single nucleotide polymorphisms (SNPs) already reported in the literature to be associated with the toxicity endpoints were included in the analysis. No SNP had been studied before in the REQUITE cohort. Deep Sparse AutoEncoders (DSAE) were trained to recognize features (SNPs) identifying patients with no toxicity and tested on a different independent mixed population including patients without and with toxicity. Results: One thousand, four hundred and one patients were included, and toxicity rates were: rectal bleeding 11.7%, urinary frequency 4%, haematuria 5.5%, nocturia 7.8%, decreased urinary stream 17.1%. Twenty-four of the 43 SNPs that were associated with the toxicity endpoints were validated as identifying patients with toxicity. Twenty of the 24 SNPs were associated with the same toxicity endpoint as reported in the literature: 9 SNPs for urinary symptoms and 11 SNPs for overall toxicity. The other 4 SNPs were associated with a different endpoint. Conclusion: Deep learning algorithms can validate SNPs associated with toxicity after radiotherapy for prostate cancer. The method should be studied further to identify polygenic SNP risk signatures for radiotherapy toxicity. The signatures could then be included in integrated normal tissue complication probability models and tested for their ability to personalize radiotherapy treatment planning.

Aim: To identify the effect of single nucleotide polymorphism (SNP) interactions on the risk of toxicity following radiotherapy (RT) for prostate cancer (PCa) and propose a new method for polygenic risk score incorporating SNP-SNP interactions (PRSi). Materials and methods: Analysis included the REQUITE PCa cohort that received external beam RT and was followed for 2 years. Late toxicity endpoints were: rectal bleeding, urinary frequency, haematuria,

Within the framework of parameter dependent PDEs, we develop a constructive approach based on Deep Neural Networks for the efficient approximation of the parameter-to-solution map. The research is motivated by the limitations and drawbacks of state-of-the-art algorithms, such as the Reduced Basis method, when addressing problems that show a slow decay in the Kolmogorov n-width. Our work is based on the use of deep autoencoders, which we employ for encoding and decoding a high fidelity approximation of the solution manifold. In order to fully exploit the approximation capabilities of neural networks, we consider a nonlinear version of the Kolmogorov n-width over which we base the concept of a minimal latent dimension. We show that this minimal dimension is intimately related to the topological properties of the solution manifold, and we provide some theoretical results with particular emphasis on second order elliptic PDEs. Finally, we report numerical experiments where we compare the proposed approach with classical POD-Galerkin reduced order models. In particular, we consider parametrized advection-diffusion PDEs, and we test the methodology in the presence of strong transport fields, singular terms and stochastic coefficients.

Within the framework of precision medicine, the stratification of individual genetic susceptibility based on inherited DNA variation has paramount relevance. However, one of the most relevant pitfalls of traditional Polygenic Risk Scores (PRS) approaches is their inability to model complex high-order non-linear SNP-SNP interactions and their effect on the phenotype (e.g. epistasis). Indeed, they incur in a computational challenge as the number of possible interactions grows exponentially with the number of SNPs considered, affecting the statistical reliability of the model parameters as well. In this work, we address this issue by proposing a novel PRS approach, called High-order Interactions-aware Polygenic Risk Score (hiPRS), that incorporates high-order interactions in modeling polygenic risk. The latter combines an interaction search routine based on frequent itemsets mining and a novel interaction selection algorithm based on Mutual Information, to construct a simple and interpretable weighted model of user-specified dimensionality that can predict a given binary phenotype. Compared to traditional PRSs methods, hiPRS does not rely on GWAS summary statistics nor any external information. Moreover, hiPRS differs from Machine Learning-based approaches that can include complex interactions in that it provides a readable and interpretable model and it is able to control overfitting, even on small samples. In the present work we demonstrate through a comprehensive simulation study the superior performance of hiPRS w.r.t. state of the art methods, both in terms of scoring performance and interpretability of the resulting model. We also test hiPRS against small sample size, class imbalance and the presence of noise, showcasing its robustness to extreme experimental settings. Finally, we apply hiPRS to a case study on real data from DACHS cohort, defining an interaction-aware scoring model to predict mortality of stage II-III Colon-Rectal Cancer patients treated with oxaliplatin.

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Within the framework of parameter dependent Partial Differential Equations (PDEs), we develop a constructive approach based on Deep Neural Networks for the efficient approximation of the parameter-to-solution map. The research is motivated by the limitations and drawbacks of state-of-the-art algorithms, such as the Reduced Basis method, when addressing problems that show a slow decay in the Kolmogorov n n -width. Our work is based on the use of deep autoencoders, which we employ for encoding and decoding a high fidelity approximation of the solution manifold. To provide guidelines for the design of deep autoencoders, we consider a nonlinear version of the Kolmogorov n n -width over which we base the concept of a minimal latent dimension. We show that the latter is intimately related to the topological properties of the solution manifold, and we provide theoretical results with particular emphasis on second order elliptic PDEs, characterizing the minimal dimension and the approximation errors of the proposed approach. The theory presented is further supported by numerical experiments, where we compare the proposed approach with classical Principal Orthogonal Decomposition (POD)-Galerkin reduced order models. In particular, we consider parametrized advection-diffusion PDEs, and we test the methodology in the presence of strong transport fields, singular terms and stochastic coefficients.

Recently, deep Convolutional Neural Networks (CNNs) have proven to be successful when employed in areas such as reduced order modeling of parametrized PDEs. Despite their accuracy and efficiency, the approaches available in the literature still lack a rigorous justification on their mathematical foundations. Motivated by this fact, in this paper we derive rigorous error bounds for the approximation of nonlinear operators by means of CNN models. More precisely, we address the case in which an operator maps a finite dimensional input µ ∈ R p onto a functional output uµ : [0, 1] d → R, and a neural network model is used to approximate a discretized version of the input-to-output map. The resulting error estimates provide a clear interpretation of the hyperparameters defining the neural network architecture. All the proofs are constructive, and they ultimately reveal a deep connection between CNNs and the discrete Fourier transform. Finally, we complement the derived error bounds by numerical experiments that illustrate their application.

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