Cosinor model for systolic blood pressure


In the below panels, we show:

A primary risk factor for adverse cardiovascular events is elevated arterial blood pressure. This is the main reason why regulatory authorities require 24-hour ambulatory blood pressure measurements (ABPM) to be included in any New Drug Application submission.
Studying blood pressure (BP) is not an easy task as both circadian and inter-subject variability have to be accounted for, when analyzing this type of data, to extract a clinical meaningful ‘signal’ from the noise.
The concepts of circadian rhythm and inter-subject variability are introduced in the first two panels before moving to the core topic of BP statistical modeling. The third panel presents the data and models, while the fourth panel gives you the possibility to experiment the effect of changing dose on the SBP response. In the last panel we discuss the value of radial vs. Cartesian display with circadian data.

*Click on a section header* to open the panel. The below charts are best rendered on modern browsers (Chrome, Safari, Firefox).

I. Circadian rhythm and normal range

*In this panel, a typical time profile of systolic blood pressure (SBP) is displayed.*

Blood pressure, like other human biological markers, is characterized by predictable changes during the 24 hours, for the most part in synchrony with the rest-activity cycle [1]. This circadian variation represents the influence of:

  • intrinsic factors: gender, autonomic nervous system tone, vasoactive hormones, and hematologic and renal variables,
  • extrinsic factors: sleep/wake routine, physical activity, ambient temperature/humidity, emotional state, alcohol or caffeine consumption, and of course, medical treatment - in treated patients.
Over a 24-hour period, a sinusoidal rhythm is observed that varies about the mean (MESOR) and generally peaks during the waking hours and deeps during sleep. Medicinal treatments affect the MESOR, and sometimes the time to peak (acrophase).

Some specific features of the 24h BP pattern are linked to the triggering of cardiac and cerebrovascular events. The day/night variation of blood pressure (BP) has been used to classify patients into nocturnal ‘dippers’ (≥10% drop in BP overnight – 23:00 to 07:00) and ‘non-dippers’. It has been shown that non-dippers, as well as patients with increased morning surge, are at increased risk for serious cardiovascular adverse events [2].

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II. Inter-individual variability

*Click on the button to get new SBP profiles.*

The purpose of this animation is to give you a sense of the differences in BP time profiles between patients.
Despite the circadian rhythm of BP being typically described as sinusoidal, a large inter-subject variability is usually observed in clinical setting. Each patient is following a different time profile, with her/his own mean SBP level (MESOR), amplitude, and peaks or nadirs. To identify the sources of variability in the data and quantify them, we use statistical models with random effects.

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III. SBP data and ‘cosinor’ model

*Click on a check-box to add or remove data and model corresponding to each dose group.*

In the below figure, we display the SBP data observed at end-of-study in patients who have received a medical treatment, either at dose 0.4 mg, 2 mg, 10 mg, or placebo once daily for 12 weeks. The observed data are plotted as dots, and the best-fitting model to describe these data is plotted as a plain curve.

As you can see, as the dose increases, the MESOR decreases; these patients receive an anti-hypertensive treatment :). The SBP of each participant was measured every 15 minutes over 24h at the occasion of each visit to the physician during the trial.

Data :

Model :


0.4 mg qd

2 mg qd

10 mg qd

The model we use to describe the data consists in a series of harmonic terms [3]:

\(\displaystyle{ Y_{ij}=MESOR_i \times \left( 1+\sum_{k=1}^K Amp_{ik} \times cos \left( \frac{2\pi}{24} \times k \times \left( t_j-\phi_{ik} \right) \right) \right) + \varepsilon_{ij} \\ MESOR_i=MESOR \times exp\left( \eta_{1i} \right) \\ Amp_{ik}=Amp_k \times exp\left( \eta_{2ki} \right) \\ \phi_{ik}=\phi_k \times exp\left( \eta_{(2k+1)i} \right) }\)
where $Y_{ij}$ is the SBP observed at time $t_j$ in patient i, $MESOR_i$ represents the mean level over 24h, $Amp_{ik}$ are amplitudes and $\phi_{ik}$ are phase shifts of the cosine terms. The terms $\eta_{1i}$, $\eta_{2ki}$, and $\eta_{(2k+1)i}$ reflect the differences in BP profiles between individuals. These terms are drawn from a multivariate normal distribution centered on zero. An additional term, $\varepsilon$, is capturing the residual unexplained variance in BP and is assumed to be independent identically distributed normal: $\varepsilon_{ij}$~ iid N(0, $\sigma^2$).
The number of harmonic, K, is usually equal to 2 or 3 depending on the clinical setting, amount and quality of data.

This type of model can be easily implemented in any statistical programming environment e.g. R/nlme, NONMEM, Phoenix/nlme, SAS/NLMIXED.

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IV. Predicting dose effect on SBP dynamics

*Experiment by yourself:*

  • mouse over the dose-MESOR (figure on the right)
  • change the model parameter values (in the control panel)

In early development, a major objective of the analysis BP data is to define the dose-concentration-effect relationship for the drug, in order to guide dosage form development (e.g. extended-release vs immediate-release) and to define the dosing regimen for the pivotal phase III trials.

The effect of drugs on phase shifts (i.e. changes in peak and nadir locations on the x-axis) has rarely been modeled, but to study chronotherapy [1]. More frequent are the cases where the drug effect – i.e. dose-response – is factored in the BP model as a covariate influencing the MESOR.

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V. Cartesian vs. radial plot

*Use your mouse to get your head around the radial plot (on the right)*

In this last panel, two graphical representations of the same phenomenon are presented. One the left-hand side, the ‘conventional’ plot displays the SBP data using Cartesian coordinates. On the right-hand side, the ‘radial’ plot (a.k.a. polar plot) displays the same data using polar coordinates. Many criticisms have been formulated towards polar plots [6]. However, there are conditions under which this format may be acceptable; depiction of circadian data is one as it mirrors the cognitive image we have of a clock. In addition, a (24h) loop can be displayed in a radial plot; something a Cartesian plot cannot do.

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