gms | German Medical Science

Background: When considering longitudinal data of individuals, personalised modelling means that each individual receives a customised dynamic model. Yet, biomedical time-series data from individuals are often characterised by a sparse, highly irregular time grid of measurements and individual-specific development patterns, which complicates the corresponding modelling task.

Our work is motivated by a scenario from the NAKO epidemiological cohort. Specifically, an extensive characterisation with measurements of many variables at a baseline time point is available for each individual, but only a smaller subset of these variables is measured again at an individually differing second time point, resulting in a very sparse (only two time points) and irregular time grid.

Methods: Inspired by recent advances on combining black-box deep learning with explicit mechanistic modelling by differential equations, we propose a generative deep learning model that captures individual dynamics in a low-dimensional latent representation as solutions of ordinary differential equations (ODEs) in such a challenging setting.

We employ a variational autoencoder (VAE) to obtain a low-dimensional representation of the central factors of variation governing the development patterns in the data in a non-linear, unsupervised way. We constrain the representation to model smooth trajectories by imposing an ODE system on the latent space. To infer an individual-specific set of ODE parameters and thus capture the individual development patterns, we use the additional variables measured only at baseline. In an extension of the model, we enrich every individual's information by assigning to it a batch of individuals with similar underlying development patterns. The combination of all second time point measurements in the batch then serves as proxy information on the common dynamics at multiple time points. Thus, we exploit the irregularity of second measurement time points to address the problem of strong time sparsity.

Results: Using simulated data, we show that our model can correctly recover individual trajectories from two-dimensional ODE systems with two or four unknown parameters in linear as well as non-linear systems and accurately infers groups of individuals with similar trajectories.

Conclusion: In conclusion, our model provides an individual-level understanding of the underlying dynamics governing each simulated individual's development rather than estimating average effects. This has the potential to assess interventions based on the knowledge of full individual-specific dynamical systems.

The authors declare that they have no competing interests.

The authors declare that an ethics committee vote is not required.