Analysis and Interpretation of 24-h Blood Pressure Profiles: Appropriate Mathematical Models May Yield Deeper Understanding

The increasing use of ambulatory blood pressure monitoring (ABPM) in clinical practice and the availability of mathematical models focused on the quantitative assessment of different blood pressure (BP) patterns have offered new perspectives on the assessment of BP behavior over 24 h. In this context, further evidence has recently been offered by Head et al. who have described a new method for the analysis of 24-h BP profiles.1 Head's paper, in agreement with previous publications, emphasizes the value of applying statistical or mathematical models to derive more robust estimates of clinically relevant parameters (i.e., dipping status, morning BP surge) from ABPM data. The use of such modeling approaches may also allow for a more precise definition of BP control through treatment over 24 h and therefore improve the quantification of treatment effects on clinical outcomes employed in health economic analyses, commonly required by reimbursement agencies in Australia, United Kingdom, United States, Canada, and elsewhere.2 Nevertheless, the inappropriate application of mathematical modeling to ABPM data could be misleading. The dangers of a given modeling approach in this setting are closely linked to its underlying assumptions and to the potential risk of data under- or over-modeling, which may lead to conclusions that poorly reflect the clinically relevant information carried by raw data. Although most models may be useful in identifying some long-term variability patterns in the description of 24-h ambulatory BP (ABP) profiles, they all share the limitation of dealing with sparse BP data that may not reflect the actual BP fluctuations occurring in a given subject. This point calls for some caution in interpreting the results they provide, and the development of a robust physiological model remains the ultimate pre-requisite for their proper interpretation. Proper implementation of such a model, however, would require more extensive measurements that include the frequency content of beat-to-beat cycles that are not captured by 24-h ABPM. Different methods of varying complexity have been proposed over the years to analyze 24-h ABP profiles3, ranging from the simple assessment of differences between hourly BP changes in the time domain to more complex tools such as the square wave fit, the cosinor model, Fourier analysis with different number of harmonics, and the double logistic model as proposed by Head et al.1 A comparison of the results offered by these methods can be difficult, owing to the differences between the models in their mathematical approach and in the underlying assumptions applied. In fact, these approaches may largely differ in the degree of interpolation and the smoothing applied to the raw data, as well as in the different perspectives given to data analysis. Some approaches focus on the quantification of slow BP fluctuations at different times during the 24-h period, typically those between wakefulness and sleep, while others emphasize the assessment of average BP levels characterizing these periods. The method described by Head et al. focuses on day–night BP changes and, in particular, focuses on two components postulated to have particular clinical relevance, namely the nocturnal fall and the morning rise in BP. When comparing their method with other approaches, Head and colleagues highlight some possible disadvantages of other techniques based on data modeling such as the cosinor method, the square wave fit, and the partial Fourier approach, although they do not include any formal goodness of fit assessment when making these comparisons. The cosinor method, in its original version, has indeed been repeatedly criticized. It has been proposed as a method used to describe the entire 24-h profile by modeling it through the use of a single sinusoidal function. However, this assumption forces a symmetrical approximation of the circadian rhythm and is unable to account for the complexity of the BP variability associated with the occurrence of different behavioral patterns over 24 h. As a result of the evolution of computer performance, it is now possible to fit multiple cosine functions within the cosinor framework. Hence, modern use of both the cosinor and the Fourier approaches rely on equivalent mathematical methodology and deliver very similar estimates of circadian BP changes. Indeed, cosinor (or equivalently Fourier) approaches with multiple cosine functions have been used frequently to fit ABPM data.4 The number of harmonics needed to extract the relevant information from the raw data can be statistically evaluated, and when more than one harmonic is needed, the model provides a non-symmetrical approximation of the circadian rhythm. This type of approach has been shown to be reproducible and consistent, in spite of the inherent limitation represented by the sparse and uneven nature of BP sampling allowed by conventional noninvasive ABPM devices. It has been suggested that the semi-quantitative output of these methods might enable meaningful comparisons between changes in BP profiles induced by different interventions, thus offering a tool for comparative assessment of various antihypertensive treatments, both in terms of their efficacy in reducing average 24-h BP and in their ability to offer a smooth distribution of BP effects over 24 h. The double logistic model proposed by Head et al.5 focuses on the quantification of two average plateaus, consisting in daytime and night-time average BP values, and of the transitions between these plateaus. Therefore, this model is most adequate for describing the daytime and night-time BP data and thus for defining the dipping status, particularly when compared to the use of fixed-clock time intervals in this context. The parametric structure of the model is however very simple, and the main risk is that important features of BP fluctuations might be averaged out. This characteristic property might not allow such a method to properly account for the complexity of BP variations over time.3 Such complexity includes also the possible day-by-day variations in BP patterns due to behavioral and/or environmental factors; the latter being typically exemplified by seasonal influences on circadian BP fluctuations.6 All these factors are responsible for the reported limited reproducibility of 24-h BP changes,7,8 which represents an additional problem when considering mathematical approaches aimed at modeling 24-h BP profiles based on a single performance of 24-h ABPM. Besides dipping status, morning BP surge has also been shown to be associated with excess cardiovascular morbidity and mortality.9,10,11 The evidence provided on this issue may carry health as well as economic implications when assessing the benefits of antihypertensive treatment also in terms of its ability to reduce the steepness of BP increase in the morning. It therefore becomes important to have a 24-h ABP analysis model that is able to adequately describe the morning BP surge. Head and colleagues claim that one of the advantages of their model lies in the fact that it estimates the morning BP surge independently of the patterns or values obtained at other times of the day or night. However, the double logistic model imposes an arbitrary parametric shape to the morning rise and its junction with the daytime plateau. This assumption may negatively impact the assessment of other details of the BP profile. For example, the fact that BP could rise much higher in the early morning as compared with the daytime average is not taken into account. Moreover, the underlying assumption on the shape of the model that characterizes this approach could undermine its ability to reliably estimate the rate of BP increase during the morning surge. Thus, a model that does not impose any predefined shape to the circadian BP changes and that might fit the “local” morning BP rise using a non-parametric estimate is probably a better approach to describe this important but very short window in ABPM data. The tangent with the steepest slope to such a curve could then provide an appropriate estimate of the morning rise in BP (Figure 1). Model-based analyses of ABPM may offer the theoretical advantage of allowing identification of the statistically relevant portion of the signal contained in the overall ABPM data. Compared with empirical methods, modeling allows a distinction to be made between the main components of the BP signal and the “noisy” variability around them. This may significantly increase the precision in quantifying the clinically relevant parameters, while at the same time requiring less intensive signal sampling. However, if the BP sampling rate is low, an additional complication of models based on Fourier or cosinor analysis is represented by the possibility of an aliasing error, i.e., the possibility that high frequency components of the signal and noise appear as low frequency components in the sampled data (Shannon sampling theorem). This problem may be less important in case of estimates based on parametric models such as the double logistic model, although difficulties may be faced in case of sampling frequency equal or lower than one measure every 30 min, and when in addition some samples are rejected, as done also in the paper by Head et al.5 In fact, while the approach proposed by Head et al. may provide a robust model for estimating the average BP during the day and night-time, the rate of change between the plateaus cannot be robustly estimated due to sparse sampling and arbitrary removal of samples. Suboptimal control of 24-h BP and the consequent failure to achieve hypertension treatment goals is strongly associated with an increased incidence of target organ damage and adverse cardiovascular outcomes.12 This emphasizes the importance of a precise assessment of the 24-h BP coverage through treatment. Thus, appropriate mathematical and statistical modeling of 24-h BP profiles both before and during treatment could usefully be included not only in outcome studies and health economic analyses, but also in clinical programs, provided that their application to specific patient data is made after checking the patient's prior dosing history and compliance with the prescribed drug-dosing regimen. ABPM modeling approaches, such as those described by Head et al.1,5 may therefore represent a useful tool to quantify the 24-h BP effect, the duration, and the smoothness of a given antihypertensive regimen. These models may also help to assess whether components of the 24-h BP profile associated with adverse outcomes, such as a steeper morning BP surge or a blunted BP fall at night, may be better controlled by novel treatment strategies aimed at providing a smoother reduction of BP during the 24 h and beyond,11,12 even in case a dose is missed by the patient. However, with the background of all the above-mentioned limitations of a single performance of discontinuous 24-h ABPM, some caution is to be recommended both when choosing and applying a given mathematical model to describe the data. This is not only because of the possible technical problems related to the relatively low sampling frequency of conventional ABPM and the limited reproducibility of 24-h BP profiles in real-life conditions, but also because all models are “black box” models that often lack an underlying physiological explanation of the BP patterns to be analyzed. The situation is even more complex because of the acknowledged limitation that there is no single model that can adequately describe all components of a 24-h BP profile simultaneously. This highlights the need for future investigations aimed at comparing the differential abilities of various models to satisfactorily describe 24-h BP profiles, in terms of their pathophysiological and clinical relevance, in the frame of pharmacological and/or outcome studies.13 Indeed, from a clinician's perspective, the most useful model has to be the model that is able to provide the most accurate prediction. G. V. is employed by Novartis, manufacturer of pharmaceutical products.