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Background: Amyotrophic Lateral Sclerosis (ALS) is a relentlessly progressive neurodegenerative condition with limited therapeutic options at present. Survival from symptom onset ranges from 3 to 5 years depending on genetic, demographic, and phenotypic factors. Despite tireless research efforts, the core etiology of the disease remains elusive and drug development efforts are confounded by the lack of accurate monitoring markers. Disease heterogeneity, late-stage recruitment into pharmaceutical trials, and inclusion of phenotypically admixed patient cohorts are some of the key barriers to successful clinical trials. Machine Learning (ML) models and large international data sets offer unprecedented opportunities to appraise candidate diagnostic, monitoring, and prognostic markers. Accurate patient stratification into well-defined prognostic categories is another aspiration of emerging classification and staging systems. Methods: The objective of this paper is the comprehensive, systematic, and critical review of ML initiatives in ALS to date and their potential in research, clinical, and pharmacological applications. The focus of this review is to provide a dual, clinical-mathematical perspective on recent advances and future directions of the field. Another objective of the paper is the frank discussion of the pitfalls and drawbacks of specific models, highlighting the shortcomings of existing studies and to provide methodological recommendations for future study designs. Results: Despite considerable sample size limitations, ML techniques have already been successfully applied to ALS data sets and a number of promising diagnosis models have been proposed. Prognostic models have been tested using core clinical variables, biological, and neuroimaging data. These models also offer patient stratification opportunities for future clinical trials. Despite the enormous potential of ML in ALS research, statistical assumptions are often violated, the choice of specific statistical models is seldom justified, and the constraints of ML models are rarely enunciated. Conclusions: From a mathematical perspective, the main barrier to the development of validated diagnostic, prognostic, and monitoring indicators stem from limited sample sizes. The combination of multiple clinical, biofluid, and imaging biomarkers is likely to increase the accuracy of mathematical modeling and contribute to optimized clinical trial designs.

Amyotrophic Lateral Sclerosis (ALS) is an inexorably progressive neurodegenerative condition with no effective disease modifying therapies. The development and validation of reliable prognostic models is a recognised research priority. We present a prognostic model for survival in ALS where result uncertainty is taken into account. Patient data were reduced and projected onto a 2D space using Uniform Manifold Approximation and Projection (UMAP), a novel non-linear dimension reduction technique. Information from 5,220 patients was included as development data originating from past clinical trials, and real-world population data as validation data. Predictors included age, gender, region of onset, symptom duration, weight at baseline, functional impairment, and estimated rate of functional loss. UMAP projection of patients shows an informative 2D data distribution. As limited data availability precluded complex model designs, the projection was divided into three zones with relevant survival rates. These rates were defined using confidence bounds: high, intermediate, and low 1-year survival rates at respectively ( ), ( ) and ( ). Predicted 1-year survival was estimated using zone membership. This approach requires a limited set of features, is easily updated, improves with additional patient data, and accounts for results uncertainty.

Structural model abstraction is a powerful technique for reducing the complexity of a state based enumeration analysis. We present in this paper new efficient Petri nets reductions. First, we define "behavioural" reductions (i.e. based on conditions related to the language of the net) which preserve a fundamental property of a net (i.e. liveness) and any formula of the (action-based) linear time logic that does not observe reduced transitions of the net. We show how to replace these conditions by structural or algebraical ones leading to reductions that can be efficiently checked and applied whereas enlarging the application spectrum of the previous reductions. At last, we illustrate our method on a significant and typical example of a synchronisation pattern of parallel programs.

Amyotrophic lateral sclerosis (ALS) is an inexorably progressive neurodegenerative condition with no effective diseasemodifying therapy at present. Given the striking clinical heterogeneity of the condition, the development and validation of reliable prognostic models is a recognised research priority. We present a prognostic model for functional decline in ALS where outcome uncertainty is taken into account. Patient data were reduced and projected onto a 2D space using Uniform Manifold Approximation and Projection (UMAP), a novel non-linear dimension reduction technique. Information from 3756 patients was included. Development data were sourced from past clinical trials. Real-world population data were used as validation data. Predictors included age, gender, region of onset, symptom duration, weight at baseline, functional impairment, and estimated rate of functional loss. UMAP projection of patients showed an informative 2D data distribution. As limited data availability precluded complex model designs, the projection was divided into three zones defined by a functional impairment range probability. Zone membership allowed individual patient prediction. Patients belonging to the first zone had a probability of 83% (± 3% ) to have an ALSFRS score over 20 at 1-year follow-up. Patients within the second zone had a probability of 89% (± 4% ) to have an ALSFRS score between 10 and 30 at 1 year follow-up. Finally, patients within the third zone had a probability of 88% (± 7% ) to have an ALSFRS score lower than 20 at 1 year follow-up. This approach requires a limited set of features, is easily updated, improves with additional patient data, and accounts for results uncertainty. This method could therefore be used in a clinical setting for patient stratification and outcome projection.

The limited amount of memory is the major bottleneck in model checking tools based on an explicit states enumeration. In this context, techniques allowing an efficient representation of the states are precious. We present in this paper a novel approach which enables to store the state space in a compact way. Though it belongs to the family of explicit storage methods, we qualify it as semi-explicit since all states are not explicitly represented in the state space. Our experiments report a memory reduction ratio up to 95% with only a tripling of the computing time in the worst case.

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