Robust parametric modeling of Alzheimer's disease progression

Abstract

Quantitative characterization of disease progression using longitudinal data can provide long-term predictions for the pathological stages of individuals. This work studies the robust modeling of Alzheimer's disease progression using parametric methods. The proposed method linearly maps the individual's age to a disease progression score (DPS) and jointly fits constrained generalized logistic functions to the longitudinal dynamics of biomarkers as functions of the DPS using M-estimation. Robustness of the estimates is quantified using bootstrapping via Monte Carlo resampling, and the estimated inflection points of the fitted functions are used to temporally order the modeled biomarkers in the disease course. Kernel density estimation is applied to the obtained DPSs for clinical status classification using a Bayesian classifier. Different M-estimators and logistic functions, including a novel type proposed in this study, called modified Stannard, are evaluated on the data from the Alzheimer's Disease Neuroimaging Initiative (ADNI) for robust modeling of volumetric magnetic resonance imaging (MRI) and positron emission tomography (PET) biomarkers, cerebrospinal fluid (CSF) measurements, as well as cognitive tests. The results show that the modified Stannard function fitted using the logistic loss achieves the best modeling performance with an average normalized mean absolute error (NMAE) of 0.991 across all biomarkers and bootstraps. Applied to the ADNI test set, this model achieves a multiclass area under the ROC curve (AUC) of 0.934 in clinical status classification. The obtained results for the proposed model outperform almost all state-of-the-art results in predicting biomarker values and classifying clinical status. Finally, the experiments show that the proposed model, trained using abundant ADNI data, generalizes well to data from the National Alzheimer's Coordinating Center (NACC) with an average NMAE of 1.182 and a multiclass AUC of 0.929.

Keywords: Alzheimer’s disease; Bayesian classifier; Cerebrospinal fluid; Disease progression modeling; Generalized logistic function; Kernel density estimation; M-estimation; Magnetic resonance imaging; Positron emission tomography.

Figure 1:

An illustration of the AD progression modeling method proposed by Jedynak et al. (2012, 2015). Left: A Sigmoid function is fitted to the biomarker measurements of each subject. Middle: The biomarker trajectories are aligned by linearly transforming subject age to DPS. Right: The aligned biomarker fit is obtained for all subjects.

Figure 2:

An illustration of how the proposed method (red curves) tackles the existing biomarker curve-fitting problems using simulated data generated based on logistic functions and additive white Gaussian noise. Left: A flexible function is used to fit the asymmetric shape of the simulated data points. Middle: A constrained function is utilized to estimate the exact dynamic range of the biomarker. Right: A robust estimator is applied to fit a curve to the simulated data contaminated with outliers.

Figure 3:

An illustration of data partitioning and evaluation approaches. The values in the tables indicate the within-class ratio of the sampled data.

Figure 4:

Estimated curves per bootstrap (in gray) for the ADNI biomarkers using the modified Stannard function and the logistic loss. The average of the bootstrapped curves per biomarker is shown as the black curve.

Figure 5:

The average of the normalized curves of the ADNI biomarkers across 100 bootstraps.

Figure 6:

Temporal ordering of the ADNI biomarkers in the disease course obtained using inflection points and quantified through 100 bootstraps. The values in the matrix represent the frequency of occurrences (probabilities) and the units in the x-axis indicate the relative ordering of the biomarkers.

Figure 7:

Estimated class-conditional likelihoods using the DPSs obtained from 100 ADNI-trained bootstraps. The box plots indicate the 25th to 75th percentiles of the estimated inflection points per biomarker, centrally marked with the median, and they are extended to the most extreme non-outlier inflection points using dashed lines.

Figure 8:

Estimated curves per bootstrap (in gray) for the NACC biomarkers using the modified Stannard function and the logistic loss. The average of the bootstrapped curves per biomarker is shown as the black curve. The last subfigure shows the average of the normalized curves of the NACC biomarkers across 100 bootstraps.