Estimating long-term multivariate progression from short-term data

Abstract

Motivation: Diseases that progress slowly are often studied by observing cohorts at different stages of disease for short periods of time. The Alzheimer's Disease Neuroimaging Initiative (ADNI) follows elders with various degrees of cognitive impairment, from normal to impaired. The study includes a rich panel of novel cognitive tests, biomarkers, and brain images collected every 6 months for as long as 6 years. The relative timing of the observations with respect to disease pathology is unknown. We propose a general semiparametric model and iterative estimation procedure to estimate simultaneously the pathological timing and long-term growth curves. The resulting estimates of long-term progression are fine-tuned using cognitive trajectories derived from the long-term "Personnes Agées Quid" study.

Results: We demonstrate with simulations that the method can recover long-term disease trends from short-term observations. The method also estimates temporal ordering of individuals with respect to disease pathology, providing subject-specific prognostic estimates of the time until onset of symptoms. When the method is applied to ADNI data, the estimated growth curves are in general agreement with prevailing theories of the Alzheimer's disease cascade. Other data sets with common outcome measures can be combined using the proposed algorithm.

Availability: Software to fit the model and reproduce results with the statistical software R is available as the grace package. ADNI data can be downloaded from the Laboratory of NeuroImaging.

Keywords: Growth curves; Multiple outcomes; Progression curves; Self-modeling regression; Semiparametric regression.

Figure 1

Dynamic biomarkers of the AD cascade hypothesized by Jack et al.[4].

Figure 2

Panel A. The three monotone functions depicted in bold are logistic, linear, and quadratic. Long-term trends are easily apparent because data is plotted with the unknown time shifts. The simulated data is not derived from real data and is intended for demonstration only. Panel B. Long-term trends are obscured because we observe the data in the short-term, without the unknown time shifts. Panel C. The algorithm described in Section 3 estimates long-term curves (red line) with good fidelity to the true target curves (dashed green line). Panel D. We repeated the experiment, generating new data and fitting the curves, 1000 times and plot the fitted curves in black.

Figure 3

The ADNI battery consists of a rich panel of biomarkers and assessments collected at six month intervals for up to 6 years. Subjects were entered into one of four diagnostic categories. The EMCI cohort was enrolled relatively recently. CSF measures are not collected from every ADNI volunteer. Some measures, such as florbetapir PET, have not been collected for as long. There is no obvious biological or clinical reference time point. The x-axis is time since first ADNI visit.

Figure 4

The top panels show each of the mean trajectories superimposed over the subject-level observations from N = 579 Amyloid+ and N = 570 ApoE e4 individuals colored by diagnosis. Colors in the top panel represent diagnosis at ADNI baseline, CN, EMCI, LMCI, and AD, and are generally sorted as expected. Shaded gray regions, where visible in the top panels, represent bootstrap 95% confidence bands. The middle panels show all of the trajectories at once. On the left, time has been shifted so that time zero represents the time at which mean CDRSB trajectory (not shown) meets the 80th percentile. On the right, time has been adjusted using long-term PAQUID MMSE trajectories so that time zero represents the estimated time to onset of dementia. The bottom panels show rates of change standardized by residual standard deviation.

Figure 5

Each panel repeats the same estimated trajectories in the Amyloid+ and ApoE ε4 carrier groups, as in Figure 4, and includes the Amyloid- and ApoE ε4 non-carrier comparison groups. The Amyloid-group consisted of n = 190 CN, 153 ECMI, 92 LMCI, and 13 AD; the ApoE ε4 allele non-carrier group consisted of n = 263 CN, 124 ECMI, 219 LMCI, and 75 AD. Shaded gray regions represent 95% confidence bands derived analytically, rather than by the bootstrap as in Figure 4.