A comparison of four common mathematical models to assess baroreflex sensitivity

Different methods have been used to assess baroreflex gain in experiments where changes in the carotid sinus pressure or the arterial blood pressure using different techniques provoke a baroreflex response, usually a rapid variation of heart rate. Four mathematical models are most used in the literature: the linear regression, the piecewise regression, and two different four-parameter logistic equations: equation 1, Y = (A1-D1)/[1 + e^{B1(X - C1)} ] + D1; equation 2, Y = (A2-D2)/[1 + (X/C2)^B2 ] + D2. We compared the four models regarding the best fit to previously published data in all vertebrate classes. The linear regression had the worst fit in all cases. The piecewise regression generally exhibited a better fit than the linear regression, though it returned a similar fit when no breakpoints were found. The logistic equations showed the best fit among the tested models and were similar to each other. We demonstrate that equation 2 is asymmetric and the level of asymmetry is accentuated according to B2. This means that the baroreflex gain calculated when X = C2 is different from the actual maximum gain. Alternatively, the symmetric equation 1 returns the maximum gain when X = C1. Furthermore, the calculation of baroreflex gain using equation 2 disregards that baroreceptors may reset when individuals experience different mean arterial pressures. Finally, the asymmetry from equation 2 is a mathematical artifact inherently skewed to the left of C2, thus bearing no biological meaning. Therefore, we suggest that equation 1 should be used instead of equation 2.

Constructing longitudinal disease progression curves using sparse, short-term individual data with an application to Alzheimer's disease

In epidemiology, cohort studies utilised to monitor and assess disease status and progression often result in short-term and sparse follow-up data. Thus, gaining an understanding of the full-term disease pathogenesis can be difficult, requiring shorter-term data from many individuals to be collated. We investigate and evaluate methods to construct and quantify the underlying long-term longitudinal trajectories for disease markers using short-term follow-up data, specifically applied to Alzheimer's disease. We generate individuals' follow-up data to investigate approaches to this problem adopting a four-step modelling approach that (i) determines individual slopes and anchor points for their short-term trajectory, (ii) fits polynomials to these slopes and anchor points, (iii) integrates the reciprocated polynomials and (iv) inverts the resulting curve providing an estimate of the underlying longitudinal trajectory. To alleviate the potential problem of roots of polynomials falling into the region over which we integrate, we propose the use of non-negative polynomials in Step 2. We demonstrate that our approach can construct underlying sigmoidal trajectories from individuals' sparse, short-term follow-up data. Furthermore, to determine an optimal methodology, we consider variations to our modelling approach including contrasting linear mixed effects regression to linear regression in Step 1 and investigating different orders of polynomials in Step 2. Cubic order polynomials provided more accurate results, and there were negligible differences between regression methodologies. We use bootstrap confidence intervals to quantify the variability in our estimates of the underlying longitudinal trajectory and apply these methods to data from the Alzheimer's Disease Neuroimaging Initiative to demonstrate their practical use. Copyright © 2017 John Wiley & Sons, Ltd.