Juliacon 2024

One of today's foremost challenges in analyzing clinical data is the integration of different data modalities. Patients -- especially in oncology -- undergo a wide range of diagnostics which yield diverse measurement types including both discrete and continuous quantities, often in a longitudinal setting, as well as time-to-event data.
In this project, we develop a computational approach for combining the different stochastic dynamics underlying such measurements into one mathematical model describing cancer patients' disease progression, and demonstrate the use of Bayesian inference for the resulting set of model parameters.
To illustrate this, we consider a simple example where one observes the size of a primary tumor, the number of different tissues affected by metastases, and the survival status of a patient. The first is growing continuously and modeled as an ordinary or stochastic differential equation, whereas the second and third are discrete events that are described as Poisson jump processes with intensities dependent on the other components. Additionally, we introduce measurement noise on the tumor growth observations. The resulting multidimensional process together with its discrete observations can be seen as a Hidden Markov Model whose parameters are to be estimated.
We examine various different models from this class, which mainly differ in the description of the continuous primary tumor growth, using simulated datasets. The first use deterministic growth laws through an ODE, the last uses a version of a geometric Brownian Motion to represent the tumor size. All models are further extendable via the parametrization of the transition rate parameters through different covariates.
For the simulation of the continuous processes modelling the tumor growth we use the functionalities of the SciML ecosystem in Julia. However, for the simulation of the two combined jump processes a self-implemented modified next reaction method algorithm showed the most accurate and reliable performance. Therefore, we couple the different simulation algorithms for robust creation of artificial data.
Model parameters are then learned using a likelihood-based Bayesian inference framework. In the simplest ODE case, it is possible to retrieve a closed form of the corresponding likelihood function, which can then be maximized. The posterior distribution of the parameters is then be obtained by the use of MCMC sampling methods. For the SDE case, we aim for the computation of a partly analytical and partly approximated likelihood function for the maximum-likelihood estimation and sampling. Furthermore, we will leverage advanced Particle filter algorithms, a class of Markov chain Monte Carlo methods targeting the posterior distribution by approximating the likelihood trough repeated sampling of a population of particles.
The performance of the model, including uncertainty quantification and robustness, is evaluated through an extensive simulation study. We showcase results obtained using an implementation solely in the Julia programming language. The SciML ecosystem provides us with a framework for expressing and solving continuous stochastic processes, whereas Julia's modularity makes it easy to combine these with our own simulation routines for jump processes. Additionally, it also allows a tidy integration of analytical and numerically approximated functions in the simulation-based inference scheme of particle filters. The latter benefits crucially from Julia' speed and parallelization.
In summary, this newly proposed approach of combining different dynamics into one stochastic model creates valuable groundwork for clinical trajectory analysis, as it allows for a more holistic description of patient outcomes. Furthermore, it sheds light on the interplay of different stochastic processes joined into one single process and starts the development of efficient simulation and estimation methods for such classes of combined stochastic models using the Julia programming language.
In future work, this can be further extended to study the effect of different therapy choices on patients' survival. Additionally, mixed effects can be included to better incorporate inter-individual variability.