The perimeter of uniform and geometric words: a probabilistic analysis

Let a word be a sequence of $n$ i.i.d. integer random variables. The perimeter $P$ of the word is the number of edges of the word, seen as a polyomino. In this paper, we present a probabilistic approach to the computation of the moments of $P$. This is applied to uniform and geometric random variables. We also show that, asymptotically, the distribution of $P$ is Gaussian and, seen as a stochastic process, the perimeter converges in distribution to a Brownian motionComment: 13 pages, 7 figure

A Liouville theorem for some Bessel generalized operators

In this paper we establish a Liouville theorem in $\mathcal{H'}_{\mu}$ for a wider class of operators in $(0,\infty)^{n}$ that generalizes the $n$-dimensional Bessel operator. We will present two different proofs, based in two representation theorems for certain distributions "supported in zero"

Relation of a Maximal Exercise Test to Change in Exercise Tolerance During Cardiac Rehabilitation

The purpose of this study was to test the hypothesis that an individualized exercise training target heart rate (HR) based on a maximal graded exercise test (GXT) is associated with greater improvements in exercise tolerance during cardiac rehabilitation (CR) compared with no GXT. In this retrospective study, we identified patients who completed 9 to 36 visits of CR between 2001 and 2016, with a length of stay ≤18 weeks and a visit frequency of 1 to 3 days per week. Patients were grouped based on whether their exercise was guided by a target HR determined from a GXT. To assess the relation between GXT and change in exercise training metabolic equivalents of task (METs), we used generalized linear models adjusted for age, gender, race, referral reason, CR visits, CR frequency, METs at start, CR location, and year of participation. Out of 4,455 patients (37% female, 48% White, median age = 62 years), 53% were prescribed a target HR based on a GXT. Compared with no GXT, a GXT was associated with a significantly greater increase in covariate-adjusted METs during CR and percentage change from start (+0.44 METs [95% confidence interval [CI] 0.38 to 0.51] and +17% [95% CI 14% to 19%], respectively). In a sensitivity analysis limited to patients with 24 to 36 visits at ≥2 days per week (n = 1,319), a GXT was associated with a significantly greater increase in covariate-adjusted exercise training METs (+0.51 [95% CI 0.36 to 0.66]; +19% [95% CI 13% to 24%]). In conclusion, to maximize the potential increase in exercise capacity during CR, patients should undergo a GXT to determine an individualized exercise training target HR

Predicting diabetes mellitus using SMOTE and ensemble machine learning approach: The Henry Ford ExercIse Testing (FIT) project

Machine learning is becoming a popular and important approach in the field of medical research. In this study, we investigate the relative performance of various machine learning methods such as Decision Tree, Naïve Bayes, Logistic Regression, Logistic Model Tree and Random Forests for predicting incident diabetes using medical records of cardiorespiratory fitness. In addition, we apply different techniques to uncover potential predictors of diabetes. This FIT project study used data of 32,555 patients who are free of any known coronary artery disease or heart failure who underwent clinician-referred exercise treadmill stress testing at Henry Ford Health Systems between 1991 and 2009 and had a complete 5-year follow-up. At the completion of the fifth year, 5,099 of those patients have developed diabetes. The dataset contained 62 attributes classified into four categories: demographic characteristics, disease history, medication use history, and stress test vital signs. We developed an Ensembling-based predictive model using 13 attributes that were selected based on their clinical importance, Multiple Linear Regression, and Information Gain Ranking methods. The negative effect of the imbalance class of the constructed model was handled by Synthetic Minority Oversampling Technique (SMOTE). The overall performance of the predictive model classifier was improved by the Ensemble machine learning approach using the Vote method with three Decision Trees (Naïve Bayes Tree, Random Forest, and Logistic Model Tree) and achieved high accuracy of prediction (AUC = 0.92). The study shows the potential of ensembling and SMOTE approaches for predicting incident diabetes using cardiorespiratory fitness data

Chronotropic incompetence and long-term risk of heart failure: The henry ford exercise testing project

Background: Chronotropic incompetence (CI) has been associated with cardiovascular mortality. However, its relationship with long-term risk of heart failure (HF) is not well studied. Methods: We included 43,098 participants (mean age 51.7±12.3 years, 47.5% females, 66.6% white) of the FIT project who completed a clinically indicated exercise test between 1991 and 2009. Patients with coronary artery disease, prior HF or on heart rate reducing medications were excluded. Incident HF was defined by having a documented diagnosis in 3 separate clinical encounters. CI was defined by inability to achieve 85% of maximal age predicted heart rate (calculated by 220 - age) with exercise. Multivariable adjusted Cox models were used to assess the independent association of CI with incident HF. Results: At baseline, 5,249 (12.2%) had CI. After a mean follow-up duration of 10.9 ± 4.6 years, 1329 (3.2%) experienced new onset HF. The cumulative incidence of HF was 397 (7.6%) among CI patients compared with 992 (2.6%) among chronotropic competent patients. Figure 1 shows the cumulative incidence of heart failure by CI status. In multivariable Cox regression models, CI was associated with increased risk of incident heart failure (HR 1.73; 95% CI 1.48 – 2.03, p \u3c0.001) after adjusting for confounders. There were no interactions by age, sex, race, body mass index. Conclusions: Our study shows that CI is an independent risk factor for HF. Further research is needed to determine whether CI could be a therapeutic target for HF

Tracking Cardiac Rehabilitation Utilization in Medicare Beneficiaries: 2017 UPDATE

PURPOSE: This study updates cardiac rehabilitation (CR) utilization data in a cohort of Medicare beneficiaries hospitalized for CR-eligible events in 2017, including stratification by select patient demographics and state of residence. METHODS: We identified Medicare fee-for-service beneficiaries who experienced a CR-eligible event and assessed their CR participation (≥1 CR sessions in 365 d), engagement, and completion (≥36 sessions) rates through September 7, 2019. Measures were assessed overall, by beneficiary characteristics and state of residence, and by primary (myocardial infarction; coronary artery bypass surgery; heart valve repair/replacement; percutaneous coronary intervention; or heart/heart-lung transplant) and secondary (angina; heart failure) qualifying event type. RESULTS: In 2017, 412 080 Medicare beneficiaries had a primary CR-eligible event and 28.6% completed ≥1 session of CR within 365 d after discharge from a qualifying event. Among beneficiaries who completed ≥1 CR session, the mean total number of sessions was 25 ± 12 and 27.6% completed ≥36 sessions. Nebraska had the highest enrollment rate (56.1%), with four other states also achieving an enrollment rate \u3e50% and 23 states falling below the overall rate for the United States. CONCLUSIONS: The absolute enrollment, engagement, and program completion rates remain low among Medicare beneficiaries, indicating that many patients did not benefit or fully benefit from a class I guideline-recommended therapy. Additional research and continued widespread adoption of successful enrollment and engagement initiatives are needed, especially among identified populations

The association of clinical indication for exercise stress testing with all-cause mortality: the FIT Project

INTRODUCTION: We hypothesized that the indication for stress testing provided by the referring physician would be an independent predictor of all-cause mortality. MATERIAL AND METHODS: We studied 48,914 patients from The Henry Ford Exercise Testing Project (The FIT Project) without known congestive heart failure who were referred for a clinical treadmill stress test and followed for 11 ±4.7 years. The reason for stress test referral was abstracted from the clinical test order, and should be considered the primary concerning symptom or indication as stated by the ordering clinician. Hierarchical multivariable Cox proportional hazards regression was performed, after controlling for potential confounders including demographics, risk factors, and medication use as well as additional adjustment for exercise capacity in the final model. RESULTS: A total of 67% of the patients were referred for chest pain, 12% for shortness of breath (SOB), 4% for palpitations, 3% for pre-operative evaluation, 6% for abnormal prior testing, and 7% for risk factors only. There were 6,211 total deaths during follow-up. Compared to chest pain, those referred for palpitations (HR = 0.72, 95% CI: 0.60-0.86) and risk factors only (HR = 0.72, 95% CI: 0.63-0.82) had a lower risk of all-cause mortality, whereas those referred for SOB (HR = 1.15, 95% CI: 1.07-1.23) and pre-operative evaluation (HR = 2.11, 95% CI: 1.94-2.30) had an increased risk. In subgroup analysis, referral for palpitations was protective only in those without coronary artery disease (CAD) (HR = 0.75, 95% CI: 0.62-0.90), while SOB increased mortality risk only in those with established CAD (HR = 1.25, 95% CI: 1.10-1.44). CONCLUSIONS: The indication for stress testing is an independent predictor of mortality, showing an interaction with CAD status. Importantly, SOB may be associated with higher mortality risk than chest pain, particularly in patients with CAD

Capillary Density of Skeletal Muscle: A Contributing Mechanism for Exercise Intolerance in Class II–III Chronic Heart Failure Independent of Other Peripheral Alterations

AbstractOBJECTIVESThe study was conducted to determine if the capillary density of skeletal muscle is a potential contributor to exercise intolerance in class II–III chronic heart failure (CHF).BACKGROUNDPrevious studies suggest that abnormalities in skeletal muscle histology, contractile protein content and enzymology contribute to exercise intolerance in CHF.METHODSThe present study examined skeletal muscle biopsies from 22 male patients with CHF compared with 10 age-matched normal male control patients. Aerobic capacities, myosin heavy chain (MHC) isoforms, enzymes, and capillary density were measured.RESULTSThe patients with CHF demonstrated a reduced peak oxygen consumption when compared to controls (15.0 ± 2.5 vs. 19.8 ± 5.0 ml·kg−1·min−1, p <0.05). Using cell-specific antibodies to directly assess vascular density, there was a reduction in capillary density in CHF measured as the number of endothelial cells/fiber (1.42 ± 0.28 vs. 1.74 ± 0.35, p = 0.02). In CHF, capillary density was inversely related to maximal oxygen consumption (r = 0.479, p = 0.02). The MHC IIx isoform was found to be higher in patients with CHF versus normal subjects (28.5 ± 13.6 vs. 19.5 ± 9.4, p <0.05).CONCLUSIONSThere was a significant reduction in microvascular density in patients with CHF compared with the control group, without major differences in other usual histologic and biochemical aerobic markers. The inverse relationship with peak oxygen consumption seen in the CHF group suggests that a reduction in microvascular density of skeletal muscle may precede other skeletal muscle alterations and play a critical role in the exercise intolerance characteristic of patients with CHF

Cardiorespiratory Fitness and Incident Stroke Types: The FIT (Henry Ford ExercIse Testing) Project

OBJECTIVE: To study the association between cardiorespiratory fitness (CRF) and incident stroke types. PATIENTS AND METHODS: We studied a retrospective cohort of patients referred for treadmill stress testing in the Henry Ford Health System (Henry Ford ExercIse Testing Project) without history of stroke. CRF was expressed by metabolic equivalents of task (METs). Using appropriate International Classification of Diseases, Ninth Revision codes, incident stroke was ascertained through linkage with administrative claims files and classified as ischemic, hemorrhagic, and subarachnoid hemorrhage (SAH). Multivariable-adjusted Cox proportional hazards models examined the association between CRF and incident stroke. RESULTS: Among 67,550 patients, mean ± SD age was 54±13 years, 46% (n=31,089) were women, and 64% (n=43,274) were white. After a median follow-up of 5.4 (interquartile range 2.7-8.5) years, a total of 7512 incident strokes occurred (6320 ischemic, 2481 hemorrhagic, and 275 SAH). Overall, there was a graded lower incidence of stroke with higher MET categories. Patients with METs of 12 or more had lower risk of overall stroke [0.42 (95% CI, 0.36-0.49)], ischemic stroke [0.69 (95% CI, 0.58-0.82)], and hemorrhagic stroke [0.71 (95% CI, 0.52-0.95)]. CONCLUSION: In a large ethnically diverse cohort of patients referred for treadmill stress testing, CRF is inversely associated with risk for ischemic and hemorrhagic stroke