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Disease Progression Modeling and Prediction through Random Effect Gaussian Processes and Time Transformation

Marco Lorenzi, Maurizio Filippone, Daniel C. Alexander, Sebastien Ourselin

arXiv:1701.01668v1stat.AP

TL;DR

Disease progression models need to jointly represent heterogeneous biomarker changes and support probabilistic staging despite missing observations and uncertain individual timing. This paper uses a Bayesian Gaussian-process model with random effects and time reparameterization, reporting plausible Alzheimer’s progression and predictive staging performance on unseen, incomplete clinical data.

  • Problem

    Jointly modeling heterogeneous biomarker trajectories and predicting disease stage remain challenging because individual onset varies, age and visit date are biased time references, and probabilistic prediction with missing observations needs further investigation.

    Source: §1 · pp. 2–3

  • Method

    A Bayesian Gaussian-process regression framework jointly models imaging and clinical biomarker time series with individual random effects and time reparameterization.

    Source: §1 · p. 2

  • Results

    The model gave a biologically plausible description of Alzheimer’s evolution across the disease span and statistically significant staging prediction with high classification accuracy on unseen, incomplete data.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The approach showed remarkable modeling and predictive performance on a large clinical cohort and is presented as a promising instrument for clinical-trial data analysis.

    Source: §5 · p. 11

    The study focuses on disease staging as a time translation, although the framework can represent more complex time transformations.

    Source: §5 · p. 11

Abstract

from arXiv · show

The development of statistical approaches for the joint modelling of the temporal changes of imaging, biochemical, and clinical biomarkers is of paramount importance for improving the understanding of neurodegenerative disorders, and for providing a reference for the prediction and quantification of the pathology in unseen individuals. Nonetheless, the use of disease progression models for probabilistic predictions still requires investigation, for example for accounting for missing observations in clinical data, and for accurate uncertainty quantification. We tackle this problem by proposing a novel Gaussian process-based method for the joint modeling of imaging and clinical biomarker progressions from time series of individual observations. The model is formulated to account for individual random effects and time reparameterization, allowing non-parametric estimates of the biomarker evolution, as well as high flexibility in specifying correlation structure, and time transformation models. Thanks to the Bayesian formulation, the model naturally accounts for missing data, and allows for uncertainty quantification in the estimate of evolutions, as well as for probabilistic prediction of disease staging in unseen patients. The experimental results show that the proposed model provides a biologically plausible description of the evolution of Alzheimer's pathology across the whole disease time-span as well as remarkable predictive performance when tested on a large clinical cohort with missing observations.

1 Introduction

Joint modeling of biomarker trajectories matters for understanding pathology and staging patients, but individual disease onset makes age and visit date biased time references. The paper proposes a Bayesian Gaussian-process framework for progression modeling and probabilistic prediction, reporting biologically plausible Alzheimer’s progression and staging predictions on unseen, incomplete data.

1 Introduction

  • Age and visit date are biased time references because pathology onset varies across individuals, complicating longitudinal staging and biomarker integration.Progression modeling aims both to quantify biomarker dynamics and temporal relationships and to stage patients from individual observations.Source: §1 · p. 2
  • Earlier approaches left probabilistic prediction less investigated; event-based staging discretized biomarker changes and omitted longitudinal observations, while another random-effect method required known disease onset.These constraints limit generalization to general clinical populations.Source: §1 · p. 2
  • The proposed Bayesian Gaussian-process regression jointly models imaging and clinical biomarker progressions from individual time series, with random effects and time reparameterization.It supports non-parametric evolution estimates and flexible random-effect and time-transformation specifications.Source: §1 · p. 2
  • The framework naturally handles missing data and quantifies uncertainty in biomarker evolutions and parameters, enabling probabilistic staging prediction for unseen patients.The time transformation is constrained through a prior on temporal derivatives, and the covariance structure supports block-wise computation.Source: §1 · p. 3
  • On a cohort of 517 amyloid-positive ADNI individuals, the model described Alzheimer’s progression plausibly across the disease span and achieved statistically significant staging prediction and high classification accuracy on unseen, incomplete data.The results are reported for clinical and multivariate imaging measurements.Source: §1 · p. 3

2 Gaussian process-based random effect modeling of longitudinal progressions

The model represents multivariate longitudinal biomarker observations with group-level Gaussian-process evolution, individual random effects, and observational noise. It accommodates subject-specific timing and covariance choices, while imposing monotonicity on group trajectories but allowing individual departures.

2 Gaussian process-based random effect modeling of longitudinal progressions

  • Each individual’s multivariate observations are modeled as a group-wise biomarker evolution plus correlated Gaussian random effects and independent, heteroskedastic Gaussian observational noise.The group-wise evolution is a zero-mean Gaussian process; the individual random effects are Gaussian-distributed correlated signals.Source: §2 · p. 3
  • The group-level covariance is block diagonal across biomarkers, with each block modeling within-biomarker temporal covariance using a negative squared exponential function.Its parameters represent marginal variance and temporal length scale.Source: §2 · pp. 3–4
  • The random-effect covariance can flexibly model subject-specific biomarker progression; the stated choices include a linear form and an i.i.d. Gaussian form.For the linear form, the individual’s average observational time is used when more than four measurements are available.Source: §2 · p. 4
  • With two or three measurements, the covariance is assigned an i.i.d. Gaussian form, and otherwise set to zero to keep model complexity compatible with limited follow-up data.The stated choices are motivated by stability concerns.Source: §2 · p. 4
  • Individual observations are mapped from absolute time τ to group-wise time through a warping function; the paper focuses on a linear shift whose value encodes relative disease stage.The linear shift is motivated by the generally short follow-up available per individual.Source: §2 · p. 4
  • A prior on group-level temporal derivatives enforces quasi-monotonic biomarker trajectories, while individual trajectories may remain non-monotonic to reflect specific clinical conditions.The derivative constraint is motivated by steady progression from normal to pathological values; enforcing it at the individual level would add model complexity.Source: §2 · p. 5

3 Joint Model: marginal likelihood and inference

This section develops posterior inference for the joint Gaussian-process model and uses posterior predictions to estimate biomarker distributions and individual disease stage. Expectation propagation handles the non-Gaussian derivative likelihood, while block covariance structure supports efficient optimization.

3.1 Approximated inference with Expectation Propagation

  • Because the derivative likelihood is non-Gaussian, direct posterior inference is analytically intractable; expectation propagation substitutes local unnormalized Gaussian approximations.The approximation parameters are updated by iterative moment matching.Source: §3.1 · p. 6, Estimating the EP parameters · p. 6

3.2 Marginal Likelihood and hyper-parameter estimation

  • Hyperparameters and individual time-shifts are alternately optimized by conjugate-gradient descent; block-structured covariance gradients reduce matrix size and improve stability and efficiency.Derivative-point positions are updated as the GP domain changes, and Gaussian priors regularize parameters.Source: §3.2 · p. 7

3.3 Prediction of observations and individual staging

  • At each time point, the model yields a Gaussian posterior distribution for biomarker values, enabling probabilistic predictions.The posterior parameters are computed from the observed data and the GP covariance structure.Source: §3.3 · p. 7
  • Bayes’ formula yields a probabilistic individual-stage estimate from biomarker observations, and the covariance specification accommodates incomplete biomarker data.The prediction uses a uniform prior over individual time and a Gaussian observation model.Source: §3.3 · p. 7

4 Experiments

Synthetic tests show that estimated disease stages track their ground truth across varied noise, sample-size, and biomarker settings. In the ADNI application, the model produces plausible biomarker trajectories and distinguishes clinical groups using predicted disease time.

4.1 Synthetic multivariate progressions

  • Across synthetic settings, staging correlations were generally high and increased with lower noise, more biomarkers, and larger samples, except for a slight decrease at σ = 0.1 with N = 100.The experiments varied noise, biomarker count, and sample size, repeating each configuration 10 times.Source: §4.1 · p. 8

4.2 Longitudinal modelling of Alzheimer’s disease progression

  • Clinical validation selected amyloid-positive ADNI individuals with baseline CSF Aβ below 192 pg/ml; testing included subjects with missing biomarkers and stable MCI.Training required at least one measurement for each of the included biomarkers.Source: §4.2 · pp. 8–9
  • The estimated progression spans roughly 20 years, with group-wise time-shifts separating NL, MCI, and AD; converter-versus-stable MCI classification accuracy was 0.84.Figure 1 shows mean ± SD biomarker trajectories, and Figure 2 shows posterior time-shift predictions for training and testing data.Source: §4.2 · p. 10, Figure 1 · p. 10, Figure 2 · p. 11
  • For AD vs NL, MCI vs NL, and AD vs MCI, effect sizes were (1.96, 1.36, 0.57) with this model versus (1.74, 1.18, 0.47) with grace.Individual time-shifts from the two methods were strongly correlated (R2 = 0.89, p <1e-16).Source: §4.2 · p. 11

5 Conclusions

The authors present a unified Gaussian-process approach for disease progression modeling and probabilistic prediction, including staging unseen patients. They identify limits of the current time-transformation focus and outline extensions for richer biomarker and imaging models.

5 Conclusions

  • The GP-based approach supports probabilistic staging of unseen patients, accounts for missing data, and quantifies uncertainty in biomarker evolutions.The authors report remarkable modeling and predictive performance on a large clinical cohort and describe the approach as promising for clinical-trial data analysis.Source: §5 · p. 11
  • The current study models disease staging as a time translation, while allowing more complex transformations when each individual has enough time points.The authors describe the time-translation focus as shared with.Source: §5 · p. 11
  • Future extensions include quantifying biomarker-specific predictive effects, modeling differential subpathology progressions, and adding spatio-temporal covariance for images and scalar biomarkers.The authors propose automatic relevance determination for feature selection, Gaussian-process mixtures for subpathologies, and a Kronecker-form covariance model for joint data.Source: §5 · p. 12
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