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(β)P(t|y), along with p = αy(β) (α and β were constant); the solution was p(t|y) = αy(β) exp(−αy(β)t). 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(0.24788)t). The cardiovascular event-free rate was computed as P(t) = 1/60Σ(i=1,…,60) exp(−0.000485y(i)(0.24788)t), 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.