Bayesian Updating of constitutive parameters under hybrid uncertainties with a novel surrogate model applied to biofilms

TL;DR

The paper tackles the challenge of calibrating constitutive parameters for multispecies biofilm growth under hybrid (epistemic and aleatory) uncertainty. It develops a Bayesian updating framework that uses a Time-Separated Stochastic Mechanics reduced-order model (TSM-ROM) to propagate aleatory uncertainty with a single-loop inference, aided by Transitional MCMC for efficient sampling. The approach is demonstrated on two- and four-species biofilm models, including a hierarchical updating scheme that reduces dimensionality in the high-parameter case and a validation with time-dependent antibiotics. Results show accurate recovery of parameters, meaningful inter-parameter correlations, and substantial computational savings, enabling rapid, uncertainty-aware calibration with predictive power for varying environmental conditions.

Abstract

Accurate modeling of bacterial biofilm growth is essential for understanding their complex dynamics in biomedical, environmental, and industrial settings. These dynamics are shaped by a variety of environmental influences, including the presence of antibiotics, nutrient availability, and inter-species interactions, all of which affect species-specific growth rates. However, capturing this behavior in computational models is challenging due to the presence of hybrid uncertainties, a combination of epistemic uncertainty (stemming from incomplete knowledge about model parameters) and aleatory uncertainty (reflecting inherent biological variability and stochastic environmental conditions). In this work, we present a Bayesian model updating (BMU) framework to calibrate a recently introduced multi-species biofilm growth model. To enable efficient inference in the presence of hybrid uncertainties, we construct a reduced-order model (ROM) derived using the Time-Separated Stochastic Mechanics (TSM) approach. TSM allows for an efficient propagation of aleatory uncertainty, which enables single-loop Bayesian inference, thereby avoiding the computationally expensive nested (double-loop) schemes typically required in hybrid uncertainty quantification. The BMU framework employs a likelihood function constructed from the mean and variance of stochastic model outputs, enabling robust parameter calibration even under sparse and noisy data. We validate our approach through two case studies: a two-species and a four-species biofilm model. Both demonstrate that our method not only accurately recovers the underlying model parameters but also provides predictive responses consistent with the synthetic data.

Figures (18)

• Figure 1: Parameters with different combinations of aleatory and epistemic uncertainties (modified from dannert2023).
• Figure 2: Model realizations corresponding to the input given by the prior samples of case I from \ref{['tab:priors-case-I']}.
• Figure 3: Dataset with $N_{\mathrm{data}} = 20$ volume fractions of living bacteria for two species, $\overline{\phi}_1(t)$ and $\overline{\phi}_2(t)$. Individual realizations are shown as yellow and blue lines, each ending at different steps, indicated by the dots. The data is generated from 1000 samples of the underlying true distribution of parameters with mean values $\bm{\theta}^{*} = [1,\ 0.1,\ 1,\ 1,\ 2]$ along with $\mathrm{CoV}^a = 0.5 \%$.
• Figure 4: Comparison of posterior samples of the mean values of the five material parameters $\bm{\theta} = [a_{11}, \ a_{12}, \ a_{22}, \ b_{1}, \ b_{2}]$ of case I calibrated with a model with $\mathrm{CoV}^a = 0.5 \%$ (purple) and $\mathrm{CoV}^b = 2 \%$ (blue).
• Figure 5: Comparison of the posterior output obtained with $\mathrm{CoV}^a$ (\ref{['fig:post-output-I']}) and $\mathrm{CoV}^b$ (\ref{['fig:post-output-I-higher']}).