Derivation of a mathematical expression for predicting the time to cardiac events in patients with heart failure: a retrospective clinical study

Elucidation of the Strongest Predictors of Cardiovascular Events in Patients with Heart Failure

BACKGROUND

In previous retrospective studies, we identified the 50 most influential clinical predictors of cardiovascular outcomes in patients with heart failure (HF). The present study aimed to use the novel limitless-arity multiple-testing procedure to filter these 50 clinical factors and thus yield combinations of no more than four factors that could potentially predict the onset of cardiovascular events. A Kaplan-Meier analysis was used to investigate the importance of the combinations.

METHODS

In a multi-centre observational trial, we prospectively enrolled 213 patients with HF who were hospitalized because of exacerbation, discharged according to HF treatment guidelines and observed to monitor cardiovascular events. After the observation period, we stratified patients according to whether they experienced cardiovascular events (rehospitalisation or cardiovascular death).

FINDINGS

Among 77,562 combinations of fewer than five clinical parameters, we identified 151 combinations that could potentially explain the occurrence of cardiovascular events. Of these, 145 combinations included the use of inotropic agents, whereas the remaining 6 included the use of diuretics without bradycardia or tachycardia, suggesting that the high probability of cardiovascular events is exclusively determined by these two clinical factors. Importantly, Kaplan-Meier curves demonstrated that the use of inotropes or of diuretics without bradycardia or tachycardia were independent predictors of a markedly worse cardiovascular prognosis.

INTERPRETATION

Patients treated with either inotropic agents or diuretics without bradycardia or tachycardia were at a higher risk of cardiovascular events. The uses of these drugs, regardless of heart rate, are the strongest clinical predictors of cardiovascular events in patients with HF.

The impact of creating mathematical formula to predict cardiovascular events in patients with heart failure

Since our retrospective study has formed a mathematical formula, α = f(x₁, …, x₂₅₂), where α is the probability of cardiovascular events in patients with heart failure (HF) and x₁ is each clinical parameter, we prospectively tested the predictive capability and feasibility of the mathematical formula of cardiovascular events in HF patients. First of all, to create such a mathematical formula using limited number of the parameters to predict the cardiovascular events in HF patients, we retrospectively determined f(x) that formulates the relationship between the most influential 50 clinical parameters (x) among 252 parameters using 167 patients hospitalized due to acute HF; the nonlinear optimization could provide the formula of α = f(x₁, …, x₅₀) which fitted the probability of the actual cardiovascular events per day. Secondly, we prospectively examined the predictability of f(x) in other 213 patients using 50 clinical parameters in 3 hospitals, and we found that the Kaplan–Meier curves using actual and estimated occurrence probabilities of cardiovascular events were closely correlated. We conclude that we created a mathematical formula f(x) that precisely predicted the occurrence probability of future cardiovascular outcomes of HF patients per day. Mathematical modelling may predict the occurrence probability of cardiovascular events in HF patients.

Non-linear Equation using Plasma Brain Natriuretic Peptide Levels to Predict Cardiovascular Outcomes in Patients with Heart Failure

Brain natriuretic peptide (BNP) is the most effective predictor of outcomes in chronic heart failure (CHF). This study sought to determine the qualitative relationship between the BNP levels at discharge and on the day of cardiovascular events in CHF patients. We devised a mathematical probabilistic model between the BNP levels at discharge (y) and on the day (t) of cardiovascular events after discharge for 113 CHF patients (Protocol I). We then prospectively evaluated this model on another set of 60 CHF patients who were readmitted (Protocol II). P(t|y) was the probability of cardiovascular events occurring after >t, the probability on t was given as p(t|y) = −dP(t|y)/dt, and p(t|y) = pP(t|y) = αy^βP(t|y), along with p = αy^β (α and β were constant); the solution was p(t|y) = αy^β exp(−αy^βt). We fitted this equation to the data set of Protocol I using the maximum likelihood principle, and we obtained the model p(t|y) = 0.000485y^0.24788 exp(−0.000485y^0.24788t). The cardiovascular event-free rate was computed as P(t) = 1/60Σ_i=1,…,60 exp(−0.000485y_i^0.24788t), based on this model and the BNP levels y_i in a data set of Protocol II. We confirmed no difference between this model-based result and the actual event-free rate. In conclusion, the BNP levels showed a non-linear relationship with the day of occurrence of cardiovascular events in CHF patients.