It is common for studies on the concentration of procalcitonin to report data by day without including the time of sampling [4, 7, 23]. In the dataset used for our study, within-day clock times of birth and procalcitonin sampling were not recorded and could not be obtained retrospectively, so we used an imputation approach to improve the resolution of the sampling time. A model based approach was used to investigate imputation scenarios on the description of the time course of procalcitonin.

Multiple imputation is commonly used when covariate data is missing [24]. Imputation is repeated multiple times and each dataset is analysed separately. It is superior to a single imputation as uncertainty can be included so that the imputed value is not considered the true value [24]. In this study random error was introduced in the imputation replicates assuming a uniform distribution of birth and sample times. Imputation methods typically impute missing covariates, while we imputed a missing part of the independent variable time after birth.

In our study, imputation spreads the observation times across the day, which is a more accurate description of clinical sample collection times than reporting observations by study day. Overall, compared with not using imputation (Scenario 0), Scenario 3 had the best performance when evaluated by both ∆OFV and VPC.

The peak procalcitonin concentration was much lower in Scenario 0, which has no samples between 24 and 48 h after birth. This likely reflects a misinterpretation of the observed concentrations because the lack of samples in this period means information is missing on the time and magnitude of peak concentration. Also, all concentrations from the first postnatal day are grouped into the 24 h time point, which will influence the parameter estimates leading to a later and lower peak concentration (Fig. 5).

Scenario 3 showed an improvement when the distribution of OFVs were compared pairwise with the other scenarios (Fig. 3) supporting the box-whisker plot statistics in Fig. 2. Scenario 3 had the earliest peak, around 22 h after birth, similar to other reports (21–24 h) with timed observations during the day [15, 25]. This shows that knowledge of typical clinical practice is a good way to inform imputation scenarios. The scenarios were imputed based on clinical sampling procedures. If a biomarker is known to have diurnal variation then this could be considered as part of an imputation scenario but, to our knowledge, diurnal variation has not been described for procalcitonin.

Some maturation half times (TM50PCT) from the bootstrap of Scenario 0 were lower than postmenstrual ages in the dataset, and the average Hill exponent was large (10.3), implying an abrupt square wave change in maturation from 0 to adult values which does not seem biologically plausible. This highlights that without imputation, it is difficult to accept the parameter estimates associated with the time course. Based on Scenario 0 with a short average half life of the birth effect (TelB) 4.26 h compared with Scenario 3 (5.67 h) and long average lag time (18.7 h) compared with Scenario 3 (12.0 h) (Table 2) reveals the loss of information based on just using study day rather than clock time.

The random effects in Scenario 0 come from re-sampling based on observations (procalcitonin) while in the other scenarios the random effects come from resampling from the independent variable (clock time). The sources of random effects are different, so no attempt is made to apply tests such as the likelihood ratio test. The OFV distributions shown in Fig. 2a are provided to give a graphical indication of the goodness of fit of Scenarios 1 to 3 compared with Scenario 0. The purpose is to illustrate that the scenarios with imputed clock times and a lag time in the model appear to give a better description of the procalcitonin concentrations.

Within each scenario the 95% confidence interval for the lag time estimate did not include zero (Fig. 2b), showing that the lag time is greater than zero. The presence of a lag time between the time of the birth event and a detectable change in CPCT was expected. Clinical studies have shown that it takes 3–6 h for procalcitonin concentrations to increase after bacterial stimuli (measured at 3, 6 and 24 h) [26], endotoxin administration (measured at 2 h intervals for 8 h) [14] or surgical trauma (measured at 6 h post-surgery) [27]. Other studies have reported that procalcitonin concentrations are low at birth and take time to rise, but the delay was not quantified [15, 28]. Scenario 3 has the shortest average lag time (12.0 h) compared to the other imputation scenarios (averages: 15.3–18.7 h; Table 2), however, this is still much longer than the delay seen following endotoxin administration or infection (3–4 h) [11, 29]. The lag time parameter estimate is design dependent and appropriate sampling times are needed to inform the onset of the rise. The use of imputed clocktimes explains why the imputation scenarios 1 to 3 have shorter lag times than Scenario 0 (no imputation). It should be noted that the lag time does not directly reflect a physiological process but is an approximation of a delay that might be better described by a transit chain process [30].

It is reassuring that Scenario 3 was detected as the most appropriate imputation scenario because it aligns most closely with what occurs in clinical practice. In an attempt to further align with clinical practice, Scenario 3 was extended so that the first measurement time each day was between 8:00 and 10:00, however this did not significantly change parameter estimates such as lag time. This study demonstrates that the use of clock time, informed by clinical practice, and study day improves imputation over other approaches when the exact sampling time is unavailable. As Scenario 3 was based on clinical practice at the study hospital, and not on properties of procalcitonin or a sampling procedure unique to this biomarker, this scenario could be used for other biomarkers collected within the same blood sample.

In the original study, the samples were collected as part of routine patient care and the data was collected for a purpose other than pharmacometric modelling, so sampling times were likely unimportant for the original purpose. However, when biomarker concentrations are expected to change rapidly, the direction of change and the duration over which it occurs is likely important, so recording sample clock times and the study day is a good policy for clinical investigators working in this area.

A limitation of this approach is that the results of the imputation scenarios are based on the assumptions made during the imputation process. Therefore, assumptions have been centred around what is clinically feasible for sample collection. A potential limitation is that the imputation approach may not completely address the lack of clock times because actual clock times are unknown.

An alternative approach we could have taken is to complete an entirely simulation based imputation study in which the clock times were determined, then removed in order to investigate the appropriateness of the different imputation scenarios. However, any results for this simulation would still be speculative as the dataset true clock times are not known.

We have applied the imputation approach to studies of procalcitonin [31, 32] and of c-reactive protein following birth and/or surgery where clock times were missing [31]. In all cases, the imputed datasets based on sampling times informed by clinical practice have been shown to improve the interpretation of the time course of biomarker concentrations following these events. Our analysis presents a framework for imputing clock times that could be used in other similar studies where times are missing.