Patient dataset
We used the patient dataset of a previously reported prospective cohort study of 191 adult patients, who were diagnosed with severe aortic valve disease in seven cardiology clinics in the wider Rotterdam area between 2006 and 2009, and who were followed for 2 years [
4]. Inclusion criteria were AVA ≤ 1 cm
2, peak transaortic jet velocity (Vmax) ≥ 4 m/s, or aortic valve/left ventricular outflow tract velocity time integral ratio ≥ 4. The patients were followed clinically, including BNP measurements, and echocardiographically at baseline and then after 6, 12 and 24 months. Baseline patient characteristics are displayed in Table
1. In total 561 BNP measurements were collected over a 2-year period (mean 0.9 years; range 0–2.5 years). During the follow-up period, 15 % of the patients (N = 28) died and 48 % (N = 91) received an aortic valve replacement of transcatheter aortic valve implantation.
Table 1
Baseline patient characteristics
Male gender (n, %) | 118, 62 % |
Age in years (mean, sd) | 72.6, 11.4 |
Symptomatic at study entry (n, %) | 132, 69 % |
Smoking (n, %) | 115, 60 % |
Hypertension (n, %) | 100, 52 % |
Diabetes (n, %) | 39, 20 % |
Dyslipidemia (n, %) | 93, 49 % |
AVA in cm2 (mean, sd) | 0.74, 0.27 |
LV ejection fraction in % (mean, sd) | 61, 6.7 |
Creatinine in micromol/L (mean, sd) | 89, 125 |
The study protocol was approved by the medical ethics committee of Erasmus University Medical Center (MEC 2006–066); all patients provided written informed consent.
Statistical methods
The development of a dynamic event prediction model that takes into account both baseline patient characteristics and longitudinal BNP measurement, requires that we first describe the evolution of BNP over time, correcting for baseline variables. Second, we use this information in a time-to-event model. Finally, using the combined model, we perform dynamic event predictions. In the next paragraphs we describe in detail the statistical methods that were employed in this 3-step process, and the rationale behind these methods.
First, we fitted a mixed-effects model to describe the evolution of BNP over time. Particularly, the model included time (years) and the baseline covariates: AVA (cm2), patient age (years), symptoms (yes/no), gender, transformed LV ejection fraction (%) and transformed creatinine (micromol/L). Transformation was done by dividing the values with the standard deviations of the specific covariates. Moreover, due to heterogeneity in the residuals plot the logarithmic scale of BNP was used. An advantage of the mixed-effects models is that they account for the positive correlation between the measurements that are observed within the same patient. For example, the values of BNP that are observed over time from the same patient are expected to be more correlated than between patients. Moreover, these models account for the biological variability in the longitudinal outcome. Specifically, if we measure BNP twice a day, we may not obtain the same result. By taking this into account using the mixed-effects model, more reliable results will be observed.
Second, to investigate the effect of the repeated BNP measurements on death and intervention probabilities, separate joint models of longitudinal and survival outcomes were constructed [
5,
6]. AVA, age, symptoms, gender, LV fraction and creatinine (all at baseline) were included as additional confounders. More details about the joint models are presented in the
Appendix.
Third, we considered the joint modeling framework and focused on the assessment of the predictive ability of our survival outcomes. Specifically, it was of interest to predict patient survival and aortic valve intervention-free for a new patient that has provided us with a set of BNP measurements and baseline characteristics, using the fitted joint model for all patients. Due to the fact that BNP is time-dependent and not constant between the visits and therefore providing longitudinal measurement up to a specific time
, assumes survival up to this time, it was more relevant to calculate the probability of surviving a future time point, given that the patient was alive until his last follow-up visit [
7,
8]. Using this approach, we applied the resulting joint modeling framework to two hypothetical patients: Mr. Jones and Mr. Smith and predicted their future survival and aortic valve intervention-free probabilities. Specifically, Mr. Jones is a 72 year old male, with creatinine value at baseline 92 micromol/L, AVA of 0.96 cm
2, LV ejection fraction 61 % and BNP values over time 64, 70, 72 and 78 pg/ml measured at 0.5, 0.9, 1.5 and 1.5 years. Moreover, he is asymptomatic at baseline. Additionally, Mr. Smith is a 79 year old male that has creatinine equal to 92 micromol/L, AVA equal to 0.61 cm
2, LV ejection fraction equal to 61 % and he is symptomatic at baseline. Finally, his BNP values are 381, 287, 1068 and 1070 pg/ml measured at 0, 0.9, 1.2 and 2 years.
Furthermore, we performed internal validation using a bootstrapping procedure (size of 1000). Specifically, we focused on discrimination, that is, how well can the model discriminate between patients who are about to experience the event within a time frame after the last measurement, from patients that are going to surpass this time frame. Since the patients were visiting their physician approximately every half year, we set this time frame. In particularly, we relied on the receiver operating characteristic (ROC) approach to assess the predictive ability of the longitudinal biomarker BNP [
7].
All analyses have been implemented in R-3.2.0, which can be downloaded as freeware at
http://www.r-project.org, using the JM package [
9].