Towards uncertainty quantification of a model for cancer-on-chip experiments

This study is a first step towards using data-informed differential models to predict and control the dynamics of cancer-on-chip experiments. We consider a conceptualized one-dimensional device, containing a cancer and a population of white blood cells. The interaction between the cancer and the population of cells is modeled by a chemotaxis model inspired by Keller-Segel-type equations, which is solved by a Hybridized Discontinuous Galerkin method. Our goal is using (synthetic) data to tune the parameters of the governing equations and to assess the uncertainty on the predictions of the dynamics due to the residual uncertainty on the parameters remaining after the tuning procedure. To this end, we apply techniques from uncertainty quantification for parametric differential models. We first perform a global sensitivity analysis using both Sobol and Morris indices to assess how parameter uncertainty impacts model predictions, and fix the value of parameters with negligible impact. Subsequently, we conduct an inverse uncertainty quantification analysis by Bayesian techniques to compute a data-informed probability distribution of the remaining model parameters. Finally, we carry out a forward uncertainty quantification analysis to compute the impact of the updated (residual) parametric uncertainties on the quantities of interest of the model. The whole procedure is sped up by using surrogate models, based on sparse-grids, to approximate the mapping of the uncertain parameters to the quantities of interest.

💡 Research Summary

This paper presents a complete uncertainty quantification (UQ) workflow for a mathematical model of Cancer‑on‑Chip (CoC) experiments. Starting from a biologically motivated one‑dimensional Keller‑Segel‑type system, the authors describe the evolution of immune‑cell density u(x,t) and a chemoattractant φ(x,t). The equations include diffusion of both species (coefficients ν and µ), a chemotactic drift term χ ∂xφ for the immune cells, natural degradation of the chemoattractant (rate a), and a time‑decaying Gaussian source term representing dying cancer cells (amplitude kφ, decay ρ, spatial centre cφ, spread σφ). Initial conditions place immune cells as a Gaussian packet near x ≈ 750 µm and no chemoattractant initially. Homogeneous Neumann boundaries enforce a closed system.

Numerically, the system is solved with a Hybridizable Discontinuous Galerkin (HDG) method implemented in MATLAB. HDG guarantees element‑wise balance and exact conservation of the total immune‑cell mass, a crucial physical property. Because many forward solves are required for UQ, the authors construct surrogate models using sparse‑grid interpolation, training them on a modest set of HDG simulations and achieving sub‑percent prediction errors for the quantities of interest (QoIs).

The UQ analysis proceeds in three stages. First, a global sensitivity analysis (both Sobol variance‑based indices and Morris elementary effects) identifies seven parameters (χ, ν, µ, kφ, ρ, cφ, σφ) that dominate the variability of two QoIs: the centre‑of‑mass trajectory M(t) of the immune cells and the total chemoattractant amount I(t). Parameters with negligible impact are fixed to reduce dimensionality. Second, a Bayesian inverse problem is tackled using synthetic data generated by the HDG solver with added Gaussian noise. Uniform priors are assigned to the influential parameters, and a Metropolis‑Hastings MCMC sampler yields posterior distributions. The posterior is notably tighter for χ and ρ, indicating that the synthetic observations are most informative about chemotactic sensitivity and cancer‑cell decay rate. Third, forward propagation of the posterior uncertainty is performed via the sparse‑grid surrogates, producing time‑dependent confidence bands for M(t) and I(t). The results show large early‑time uncertainty in the immune‑cell centre of mass, which gradually diminishes, while the chemoattractant amount exhibits a smoother, yet still parameter‑dependent, decline.

The study demonstrates how a sophisticated numerical method (HDG), combined with modern UQ tools (global sensitivity, Bayesian inference, surrogate modeling), can turn a simplified CoC model into a predictive digital twin with quantified confidence. Limitations include reliance on synthetic rather than experimental data and the reduction of the true three‑dimensional microfluidic geometry to a one‑dimensional line. Future work will incorporate real video‑based measurements, extend the model to higher dimensions, and apply the framework to drug‑response prediction and experimental design.