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 (C) 2017 John Wiley & Sons, Ltd.