Properties of a general quaternion-valued gradient operator and its applications to signal processing

The gradients of a quaternion-valued function are often required for quaternionic signal processing algorithms. The HR gradient operator provides a viable framework and has found a number of applications. However, the applications so far have been limited to mainly real-valued quaternion functions and linear quaternionvalued functions. To generalize the operator to nonlinear quaternion functions, we define a restricted version of the HR operator, which comes in two versions, the left and the right ones. We then present a detailed analysis of the properties of the operators, including several different product rules and chain rules. Using the new rules, we derive explicit expressions for the derivatives of a class of regular nonlinear quaternion-valued functions, and prove that the restricted HR gradients are consistent with the gradients in the real domain. As an application, the derivation of the least mean square algorithm and a nonlinear adaptive algorithm is provided. Simulation results based on vector sensor arrays are presented as an example to demonstrate the effectiveness of the quaternion-valued signal model and the derived signal processing algorithm

Filtering and Tracking with Trinion-Valued Adaptive Algorithms

A new model for three-dimensional processes based on the trinion algebra is introduced for the first time. Compared with the pure quaternion model, the trinion model is more compact and computationally more efficient, while having similar or comparable performance in terms of adaptive linear filtering. Moreover, the trinion model can effectively represent the general relationship of state evolution in Kalman filtering, where the pure quaternion model fails. Simulations on real-world wind recordings and synthetic data sets are provided to demonstrate the potentials of this new modeling method

Dapagliflozin and quality of life measured using the EuroQol 5-Dimension questionnaire in patients with HFrEF and HFmrEF/HFpEF

Aims: Although much is known about the usefulness of heart failure (HF)-specific instruments for assessing patient well-being, less is known about the value of generic instruments for the measurement of health-related quality of life (HRQL) in HF. The aim of this study was to assess the relationship between the EuroQol 5-dimension 5-level (EQ-5D-5L) visual analogue scale (VAS) and index scores, clinical characteristics, and outcomes in patients with HF and the effect of dapagliflozin on these scores. Methods and results: We performed a patient-level pooled analysis of the DAPA-HF and DELIVER trials, which investigated the effectiveness and safety of dapagliflozin in patients with HF and reduced ejection fraction (HFrEF) and mildly reduced/preserved ejection fraction (HFmrEF/HFpEF), respectively. Patients reporting higher (better) EQ-5D-5L VAS and index scores had a lower prevalence of comorbidities, including atrial fibrillation and hypertension, than patients with a worse score. They were also more likely to have better investigator-reported (New York Heart Association class) and patient-self-reported (Kansas City Cardiomyopathy Questionnaire) health status and lower median N-terminal pro-B-type natriuretic peptide levels. Compared to patients with the lowest scores (Q1), those with higher EQ-5D-5L VAS scores had better outcomes: the hazard ratio for the composite of cardiovascular death or worsening HF was 0.81 (95% confidence interval 0.72–0.91) in Q2, 0.74 (0.65–0.84) in Q3, and 0.62 (0.54–0.72) in Q4. The risk of each component of the composite outcome, and all-cause death, was also lower in patients with better scores. Similar findings were observed for the index score. Treatment with dapagliflozin improved both EQ-5D-5L VAS and index scores across the range of ejection fraction. Conclusions: Both higher (better) EQ-5D-5L VAS and index scores were associated with better outcomes. Dapagliflozin treatment improved EQ-5D-5L VAS and index scores, irrespective of ejection fraction

Prognostic models for mortality and morbidity in heart failure with preserved ejection fraction

Importance Accurate risk prediction of morbidity and mortality in patients with heart failure with preserved ejection fraction (HFpEF) may help clinicians risk stratify and inform care decisions. Objective To develop and validate a novel prediction model for clinical outcomes in patients with HFpEF using routinely collected variables and to compare it with a biomarker-driven approach. Design, Setting, and Participants Data were used from the Dapagliflozin Evaluation to Improve the Lives of Patients With Preserved Ejection Fraction Heart Failure (DELIVER) trial to derive the prediction model, and data from the Angiotensin Receptor Neprilysin Inhibition in Heart Failure With Preserved Ejection Fraction (PARAGON-HF) and the Irbesartan in Heart Failure With Preserved Ejection Fraction Study (I-PRESERVE) trials were used to validate it. The outcomes were the composite of HF hospitalization (HFH) or cardiovascular death, cardiovascular death, and all-cause death. A total of 30 baseline candidate variables were selected in a stepwise fashion using multivariable analyses to create the models. Data were analyzed from January 2023 to June 2023. Exposures Models to estimate the 1-year and 2-year risk of cardiovascular death or hospitalization for heart failure, cardiovascular death, and all-cause death. Results Data from 6263 individuals in the DELIVER trial were used to derive the prediction model and data from 4796 individuals in the PARAGON-HF trial and 4128 individuals in the I-PRESERVE trial were used to validate it. The final prediction model for the composite outcome included 11 variables: N-terminal pro–brain natriuretic peptide (NT-proBNP) level, HFH within the past 6 months, creatinine level, diabetes, geographic region, HF duration, treatment with a sodium-glucose cotransporter 2 inhibitor, chronic obstructive pulmonary disease, transient ischemic attack/stroke, any previous HFH, and heart rate. This model showed good discrimination (C statistic at 1 year, 0.73; 95% CI, 0.71-0.75) in both validation cohorts (C statistic at 1 year, 0.71; 95% CI, 0.69-0.74 in PARAGON-HF and 0.75; 95% CI, 0.73-0.78 in I-PRESERVE) and calibration. The model showed similar discrimination to a biomarker-driven model including high-sensitivity cardiac troponin T and significantly better discrimination than the Meta-Analysis Global Group in Chronic (MAGGIC) risk score (C statistic at 1 year, 0.60; 95% CI, 0.58-0.63; delta C statistic, 0.13; 95% CI, 0.10-0.15; P < .001) and NT-proBNP level alone (C statistic at 1 year, 0.66; 95% CI, 0.64-0.68; delta C statistic, 0.07; 95% CI, 0.05-0.08; P < .001). Models derived for the prediction of all-cause and cardiovascular death also performed well. An online calculator was created to allow calculation of an individual’s risk. Conclusions and Relevance In this prognostic study, a robust prediction model for clinical outcomes in HFpEF was developed and validated using routinely collected variables. The model performed better than NT-proBNP level alone. The model may help clinicians to identify high-risk patients and guide treatment decisions in HFpEF

Dapagliflozin and total heart failure events in patients with mildly reduced or preserved ejection fraction heart failure in the DELIVER trial

Importance: In the Dapagliflozin Evaluation to Improve the Lives of Patients With Preserved Ejection Fraction Heart Failure (DELIVER) trial, dapagliflozin reduced the risk of time to first worsening heart failure (HF) event or cardiovascular death in patients with HF with mildly reduced or preserved ejection fraction (EF). Objective: To evaluate the effect of dapagliflozin on total (ie, first and recurrent) HF events and cardiovascular death in this population. Design, Setting, and Participants: In this prespecified analysis of the DELIVER trial, the proportional rates approach of Lin, Wei, Yang, and Ying (LWYY) and a joint frailty model were used to examine the effect of dapagliflozin on total HF events and cardiovascular death. Several subgroups were examined to test for heterogeneity in the effect of dapagliflozin, including left ventricular EF. Participants were enrolled from August 2018 to December 2020, and data were analyzed from August to October 2022. Interventions: Dapagliflozin, 10 mg, once daily or matching placebo. Main Outcomes and Measures: The outcome was total episodes of worsening HF (hospitalization for HF or urgent HF visit requiring intravenous HF therapies) and cardiovascular death. Results: Of 6263 included patients, 2747 (43.9%) were women, and the mean (SD) age was 71.7 (9.6) years. There were 1057 HF events and cardiovascular deaths in the placebo group compared with 815 in the dapagliflozin group. Patients with more HF events had features of more severe HF, such as higher N-terminal pro-B-type natriuretic peptide level, worse kidney function, more prior HF hospitalizations, and longer duration of HF, although EF was similar to those with no HF events. In the LWYY model, the rate ratio for total HF events and cardiovascular death for dapagliflozin compared with placebo was 0.77 (95% CI, 0.67-0.89; P &lt;.001) compared with a hazard ratio of 0.82 (95% CI, 0.73-0.92; P &lt;.001) in a traditional time to first event analysis. In the joint frailty model, the rate ratio was 0.72 (95% CI, 0.65-0.81; P &lt;.001) for total HF events and 0.87 (95% CI, 0.72-1.05; P =.14) for cardiovascular death. The results were similar for total HF hospitalizations (without urgent HF visits) and cardiovascular death and in all subgroups, including those defined by EF. Conclusions and Relevance: In the DELIVER trial, dapagliflozin reduced the rate of total HF events (first and subsequent HF hospitalizations and urgent HF visits) and cardiovascular death regardless of patient characteristics, including EF. Trial Registration: ClinicalTrials.gov Identifier: NCT03619213.</p