Nonlinear modeling of oral glucose tolerance test response to evaluate associations with aging outcomes

Abstract

As people age, their ability to maintain homeostasis in response to stressors diminishes. Physical frailty, a syndrome characterized by loss of resilience to stressors, is thought to emerge due to dysregulation of and breakdowns in communication among key physiological systems. Dynamical systems modeling of these physiological systems aims to model the underlying processes that govern response to stressors. We hypothesize that dynamical systems model summaries are predictive of age-related declines in health and function. In this study, we analyze data obtained during 75-gram oral-glucose tolerance tests (OGTT) on 1,120 adults older than 50 years of age from the Baltimore Longitudinal Study on Aging. We adopt a two-stage modeling approach. First, we fit OGTT curves with the Ackerman model-a nonlinear, parametric model of the glucose-insulin system-and with functional principal components analysis. We then fit linear and Cox proportional hazards models to evaluate whether usual gait speed and survival are associated with the stage-one model summaries. We also develop recommendations for identifying inadequately-fitting nonlinear model fits in a cohort setting with numerous heterogeneous response curves. These recommendations include: (1) defining a constrained parameter space that ensures biologically plausible model fits, (2) evaluating the relative discrepancy between predicted and observed responses of biological interest, and (3) identifying model fits that have notably poor model fit summary measures, such as [Formula: see text], relative to other fits in the cohort. The Ackerman model was unable to adequately fit 36% of the OGTT curves. The stage-two regression analyses found no associations between Ackerman model summaries and usual gait speed, nor with survival. The second functional principal component score was associated with faster gait speed (p<0.01) and improved survival (p<0.01).

Fig 1. Adequate Ackerman model fits.

Estimated effective periods for the model fits by panel: A—1.9 hours, B—3.3 hours, C—4.9 hours, D—8.4 hours.

Fig 2. Inadequate Ackerman model fits.

Reasons for inadequate fit classification by panel: A—$\widehat{A}$ on boundary of parameter space, B—$\widehat{k}$ on boundary of parameter space, C—$\widehat{\omega }$ on boundary of parameter space, D—$\widehat{A}$ and $\widehat{k}$ on boundary of parameter space and small $R_{\text{pseudo}}^2$, E—large underestimation in max glucose concentration, F—small $R_{\text{pseudo}}^2$.

Fig 3. Predicted and observed maximum glucose concentrations.

Black line corresponds to perfect prediction. Grey lines correspond to a 10% relative difference. Blue triangles outside of the grey lines indicate fits for which the predicted maximum glucose concentration occurred later than the end of the OGTT.

Fig 4. Functional PCA plots.

Panel A—Proportion variance explained by principal component. Panels B-D—Eigenfunctions’ deviations about the mean curve. The dashed red line shows a predicted OGTT curve for a 1 standard deviation increase in the respective fPC score with all other scores held at 0. The dotted blue line shows the same but for a 1 standard deviation decrease in the respective fPC score.

Fig 5. Ackerman parameters and fPC scores scatterplot matrix.

Scatterplots and correlations between estimated Ackerman model parameters and fPC scores for OGTT curves which the Ackerman model adequately fit.

Fig 6. Predicted OGTT curves from Ackerman and fPCA fits against observed data.

Observed data are shown as black points, the Ackerman model fit as a solid red line, and the fPCA fit as a blue dashed line. Panel A—Adequate Ackerman model fit. Predicted Ackerman and fPCA fit largely agree. Panel B—Abnormal OGTT curve which neither the Ackerman nor fPCA fit model closely. Panel C—Boundary Ackerman model fit ($\widehat{k}=0$). Ackerman and fPCA fit model the observed data closely. Panel D—Extrapolated max Ackerman model fit. The Ackerman fit models the observed data more closely, but the fPCA fit appears more reasonable.