Exploring diurnal variation using piecewise linear splines: an example using blood pressure

In this large population based study we present an alternative method of modelling 24 h BP that can easily be applied to any physiological process that follows a circadian cycle. Our novel but simple approach utilising a piecewise linear random-effects model, with an adjustment to ensure that the average level is the same at the beginning and end of each 24-h period, offers a practical alternative to other methodological modelling techniques for researchers exploring circadian patterns. The flexible model has the ability to capture overall average, group and individual trajectories (in addition to being capable of examining slopes at different periods of the day).

Despite the large amounts of literature relating to BP, those specifically modelling 24 h ABPM remain sparse. Our method offers new measures of short-term BP variability as we can quantify the variation about an individual’s trajectory but it also allows examination of the variation in slopes between individuals (random-effects). Our results indicated that after adjustment for age, sex and BMI the sharpest fall in BP occurred just after an individual went to sleep and the steepest rise occurred just after waking. Although there was a significant difference on average between those with and without microalbuminuria we found there was no overall improvement in fit after including interaction effects with the spline terms. However interestingly we found that the variation after awaking, representing what is known as the morning surge was considerably larger than the other periods of the day.

It has been acknowledged there is not a generally accepted “standard” method of analysing 24-h ABPM [21]. Cosinor analysis has been highlighted as the most common approach [14‐17] while fourier analysis [18], has also been implemented which are both based on the idea that any time series can be described by a series of cosine (and sine) waves of various frequencies [38]. It has been suggested that these methods impose too many restrictions on the shape of the profile and have been shown to fit real profiles poorly [39]. Wang et al. [40] suggest problems with fitting a sinusoidal function to a circadian pattern include (1) that the pattern over time may not be symmetric; that is, the peak and nadir may not be separated by 12 h and/or the amplitude and width of the peak may differ from those of the nadir, (2) sometimes there are local minimum and maximum points. Additionally Wang et al. [41] suggests that the sinusoidal function is too restrictive and “rhythms with a shape closely approximating a cosine curve are uncommon” [42]. Alternative methods have examined restricted cubic splines and more recently orthonormal polynomials [13, 21]. As we highlighted previously these approaches may model the data quite well and their curvature nature may look graphically appealing but it is difficult to understand and compare their resulting coefficients.

Piecewise regression which allows separate slopes to be fitted to observations before and after a certain period or event (knot points) has been cited as a useful tool that should be implemented more often in the context of epidemiological studies [43] but has not, to the best of our knowledge been used with ABPM data or other physiological processes that are circadian. The benefit of this method as opposed to polynomials is that the regression coefficients represent something meaningful directly without the need for further manipulation of the results—in our context, the rate of increase or decrease of BP for a certain time of day. The position of the knot points can easily be altered depending on the requirements of a specific study. For example if we were examining the effect of dialysis on BP in haemodialysis patients we could fix knot points at the time their dialysis began and at period(s) a number of hours later.

The morning is recognised as the most important period in relation to cardiovascular diseases [44] and cardiovascular events occur more frequently in this period [44‐46]. In our study we found that the steepest rise (slope) occurred during the period just after waking which is in line with the literature, thus verifying that our method is capturing known features of the data. It is suggested that the abrupt steep rise in BP may explain the link between cardiovascular events and the morning period [29]. In a review of morning surge with cardiovascular risk, three different definitions of morning surge were identified, all of which simply use BP differences where they subtracted some average night value minus an average of morning BP readings [45]. We argue that our method offers a more accurate estimate, as by definition of a slope we can specifically quantify the rate of “surge”. In fact, Parati et al. argue that a method that would be capable of capturing a slope similar to one purposed by our method would provide an accurate method of estimating the morning surge [47]. Considering that morning surge has been cited as a predictor of stroke and advanced target organ damage independent of ambulatory BP and nocturnal BP [44, 46], accurately quantifying it remains an important issue, particularly when we are assessing the benefits of antihypertensive medication in their ability to reduce this steep rise. This may not only have health implications but also financial benefits. A similar argument could be put forward for the dipping effect at night which is usually quantified just as a ratio of the mean BP between night-day periods. The slope at night obtained by our approach may represent a more accurate measure but further work would be needed to explore this.

Kario argues that the perfect 24 h BP control is not limited to reducing mean BP but includes restoring disrupted circadian BP rhythms and reducing exaggerated BP variability [44]. As highlighted previously most studies examining BPV have concentrated on summary measures of variability such as SD over 24 h or separated into day and night values [5‐8, 11, 12, 24]. With the use of our mixed-effects model we were are able to obtain superior measures of BPV that take into account the temporal nature of the data. We were able to quantify the variation about an individual’s trajectory but also the variation in slopes between individuals. Our work highlighted that the largest variation between individuals occurred during the morning surge period. Adjusting for age, sex and BMI did not help explain this variation. Similarly the presence of microalbuminuria had little impact on the variation. Ideally we would have preferred to explore if the variation could in part predict CV events but as data is currently only available for wave one, we have been restricted to explore a surrogate marker in microalbuminuria and have acknowledged this as a limitation. Further work is warranted to include CVD endpoints but perhaps an underlining physiological phenomenon of BP is that it is most variable in the morning possibly because this period of the day has an abrupt rise. Although some of the knots are subject-specific, others are at common fixed locations which may not represent the best position for a specific individual and this assumption is recognised as a limitation. In addition to the average plot, we have attempted to incorporate our knowledge of the underlying pattern of BP to help inform our knot positions as suggested by Howe et al. [48].

As debate remains in relation to how to correctly quantify short-term BPV [24, 49] our approach offers new alternatives that utilise the full power of ABPM that is often lost when using summary measures such as SD because it only reflects the dispersion of measurements around a single value (mean) not accounting for the order in which BP measurements were obtained [12, 50]. As argued by Bamberger et al. there are a large and increasing variety of mathematical equations available which allow us to test specific hypotheses about, and estimate parameters of, growth or change (BP) with varying purposes such as summarizing the shape of change across a sample or determining patterns in variability [51]. Despite this however, Diehl et al. suggests that summary measures and multilevel models described here are two of the most common techniques used specifically for exploring BP changes [52]. They also suggest the use of dynamic developmental models with multiple time scales as another approach to address questions related to intraindividual variability. This is quite similar to the multilevel model framework but is slightly more flexible and can include more complex scenarios. These models are characterized by time-dependent parametric changes occurring at different time scales, where each time scale defines a distinct level within a hierarchy of time scales [53]. In the context of our work this approach may be useful if we had additional data that could incorporate another time scale, such as BP over years e.g. the lifespan. The benefit from this would be that we could compare and relate BP fluctuations over 24 h to that over the lifespan. However, in this study we are only interested in 24 h variation where we only have recordings taken every 30 min. With only one time scale, we believe that the use of the traditional mixed-effects model is appropriate for our analysis. We argue that our approach, which has the ability to determine variation over specific periods of the day offers a novel measure of variability in the analysis of 24 h BP which may have benefits when attempting to determine the optimal timing of antihypertensive medication administration in future studies. Finally, as was briefly alluded to, the approach and discussion outlined is not restricted to the use of BP and can easily be implemented on any physiological process that demonstrates a circadian cycle. BP is not the only biological process where disruptions to circadian rhythms are clinical relevant. Wang et al. found that those with Cushing syndrome exhibited no circadian rhythm of cortisol, while those with depression showed a dampened rhythm compared to the normal group [41]. Liu et al. found that larger short-term fluctuations in intraocular pressure are more common in glaucoma [54]. Similar to the morning BP surge, it was found that intraocular pressure was higher in the morning and more prevalent in those with glaucoma. This suggests that our approach may be beneficial to the exploration of other biological rhythms that have similar features to that of BP.