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This article describes a non-parametric clustering algorithm with an outlier removal step. Our method is based on tools from topological data analysis: we define a new filtration on metric spaces which is a variant of the Vietoris–Rips filtration that adds information about the points' nearest neighbor to the persistence diagram. We prove a stability theorem for this filtration, and evaluate our method on point cloud and graph datasets, showing that it can compete with state-of-the-art methods while being non-parametric.

IntroductionThe aim of this study was to realize a systematic review of the different ways, both clinical and instrumental, used to evaluate the effects of the surgical correction of an equinovarus foot (EVF) deformity in post-stroke patients.MethodsA systematic search of full-length articles published from 1965 to June 2021 was performed in PubMed, Embase, CINAHL, Cochrane, and CIRRIE. The identified studies were analyzed to determine and to evaluate the outcomes, the clinical criteria, and the ways used to analyze the impact of surgery on gait pattern, instrumental, or not.ResultsA total of 33 studies were included. The lack of methodological quality of the studies and their heterogeneity did not allow for a valid meta-analysis. In all, 17 of the 33 studies involved exclusively stroke patients. Ten of the 33 studies (30%) evaluated only neurotomies, one study (3%) evaluated only tendon lengthening procedures, 19 studies (58%) evaluated tendon transfer procedures, and only two studies (6%) evaluated the combination of tendon and neurological procedures. Instrumental gait analysis was performed in only 11 studies (33%), and only six studies (18%) combined it with clinical and functional analyses. Clinical results show that surgical procedures are safe and effective. A wide variety of different scales have been used, most of which have already been validated in other indications.DiscussionNeuro-orthopedic surgery for post-stroke EVF is becoming better defined. However, the method of outcome assessment is not yet well established. The complexity in the evaluation of the gait of patients with EVF, and therefore the analysis of the effectiveness of the surgical management performed, requires the integration of a patient-centered functional dimension, and a reliable and reproducible quantified gait analysis, which is routinely usable clinically if possible.

In the past few years, light, affordable wearable inertial measurement units have been providing to clinicians and researchers the possibility to quantitatively study motor degeneracy by comparing gait trials from patients and/or healthy subjects. To do so, standard gait features can be used but they fail to detect subtle changes in several pathologies including multiple sclerosis. Multiple sclerosis is a demyelinating disease of the central nervous system whose symptoms include lower limb impairment, which is why gait trials are commonly used by clinicians for their patients' follow-up. This article describes a method to compare pairs of gait signals, visualize the results and interpret them, based on topological data analysis techniques. Our method is non-parametric and requires no data other than gait signals acquired with inertial measurement units. We introduce tools from topological data analysis (sublevel sets, persistence barcodes) in a practical way to make it as accessible as possible in order to encourage its use by clinicians. We apply our method to study a cohort of patients suffering from progressive multiple sclerosis and healthy subjects. We show that it can help estimate the severity of the disease and also be used for longitudinal follow-up to detect an evolution of the disease or other phenomena such as asymmetry or outliers.