A comparison of approximate versus exact techniques for Bayesian parameter inference in nonlinear ordinary differential equation models

Quantifying in vitro B. anthracis growth and PA production and decay: a mathematical modelling approach

Protective antigen (PA) is a protein produced by Bacillus anthracis. It forms part of the anthrax toxin and is a key immunogen in US and UK anthrax vaccines. In this study, we have conducted experiments to quantify PA in the supernatants of cultures of B. anthracis Sterne strain, which is the strain used in the manufacture of the UK anthrax vaccine. Then, for the first time, we quantify PA production and degradation via mathematical modelling and Bayesian statistical techniques, making use of this new experimental data as well as two other independent published data sets. We propose a single mathematical model, in terms of delay differential equations (DDEs), which can explain the in vitro dynamics of all three data sets. Since we did not heat activate the B. anthracis spores prior to inoculation, germination occurred much slower in our experiments, allowing us to calibrate two additional parameters with respect to the other data sets. Our model is able to distinguish between natural PA decay and that triggered by bacteria via proteases. There is promising consistency between the different independent data sets for most of the parameter estimates. The quantitative characterisation of B. anthracis PA production and degradation obtained here will contribute towards the ambition to include a realistic description of toxin dynamics, the host immune response, and anti-toxin treatments in future mechanistic models of anthrax infection.

rAAV Manufacturing: The Challenges of Soft Sensing during Upstream Processing

Recombinant adeno-associated virus (rAAV) is the most effective viral vector technology for directly translating the genomic revolution into medicinal therapies. However, the manufacturing of rAAV viral vectors remains challenging in the upstream processing with low rAAV yield in large-scale production and high cost, limiting the generalization of rAAV-based treatments. This situation can be improved by real-time monitoring of critical process parameters (CPP) that affect critical quality attributes (CQA). To achieve this aim, soft sensing combined with predictive modeling is an important strategy that can be used for optimizing the upstream process of rAAV production by monitoring critical process variables in real time. However, the development of soft sensors for rAAV production as a fast and low-cost monitoring approach is not an easy task. This review article describes four challenges and critically discusses the possible solutions that can enable the application of soft sensors for rAAV production monitoring. The challenges from a data scientist's perspective are (i) a predictor variable (soft-sensor inputs) set without AAV viral titer, (ii) multi-step forecasting, (iii) multiple process phases, and (iv) soft-sensor development composed of the mechanistic model.

How quickly does a wound heal? Bayesian calibration of a mathematical model of venous leg ulcer healing

Chronic wounds, such as venous leg ulcers, are difficult to treat and can reduce the quality of life for patients. Clinical trials have been conducted to identify the most effective venous leg ulcer treatments and the clinical factors that may indicate whether a wound will successfully heal. More recently, mathematical modelling has been used to gain insight into biological factors that may affect treatment success but are difficult to measure clinically, such as the rate of oxygen flow into wounded tissue. In this work, we calibrate an existing mathematical model using a Bayesian approach with clinical data for individual patients to explore which clinical factors may impact the rate of wound healing for individuals. Although the model describes group-level behaviour well, it is not able to capture individual-level responses in all cases. From the individual-level analysis, we propose distributions for coefficients of clinical factors in a linear regression model, but ultimately find that it is difficult to draw conclusions about which factors lead to faster wound healing based on the existing model and data. This work highlights the challenges of using Bayesian methods to calibrate partial differential equation models to individual patient clinical data. However, the methods used in this work may be modified and extended to calibrate spatiotemporal mathematical models to multiple data sets, such as clinical trials with several patients, to extract additional information from the model and answer outstanding biological questions.

Cellular costs underpin micronutrient limitation in phytoplankton

Micronutrients control phytoplankton growth in the ocean, influencing carbon export and fisheries. It is currently unclear how micronutrient scarcity affects cellular processes and how interdependence across micronutrients arises. We show that proximate causes of micronutrient growth limitation and interdependence are governed by cumulative cellular costs of acquiring and using micronutrients. Using a mechanistic proteomic allocation model of a polar diatom focused on iron and manganese, we demonstrate how cellular processes fundamentally underpin micronutrient limitation, and how they interact and compensate for each other to shape cellular elemental stoichiometry and resource interdependence. We coupled our model with metaproteomic and environmental data, yielding an approach for estimating biogeochemical metrics, including taxon-specific growth rates. Our results show that cumulative cellular costs govern how environmental conditions modify phytoplankton growth.

Numerical method for parameter inference of systems of nonlinear ordinary differential equations with partial observations

Parameter inference of dynamical systems is a challenging task faced by many researchers and practitioners across various fields. In many applications, it is common that only limited variables are observable. In this paper, we propose a method for parameter inference of a system of nonlinear coupled ordinary differential equations with partial observations. Our method combines fast Gaussian process-based gradient matching and deterministic optimization algorithms. By using initial values obtained by Bayesian steps with low sampling numbers, our deterministic optimization algorithm is both accurate, robust and efficient with partial observations and large noise.

InfPolyn, a Nonparametric Bayesian Characterization for Composition-Dependent Interdiffusion Coefficients

Composition-dependent interdiffusion coefficients are key parameters in many physical processes. However, finding such coefficients for a system with few components is challenging due to the underdetermination of the governing diffusion equations, the lack of data in practice, and the unknown parametric form of the interdiffusion coefficients. In this work, we propose InfPolyn, Infinite Polynomial, a novel statistical framework to characterize the component-dependent interdiffusion coefficients. Our model is a generalization of the commonly used polynomial fitting method with extended model capacity and flexibility and it is combined with the numerical inversion-based Boltzmann-Matano method for the interdiffusion coefficient estimations. We assess InfPolyn on ternary and quaternary systems with predefined polynomial, exponential, and sinusoidal interdiffusion coefficients. The experiments show that InfPolyn outperforms the competitors, the SOTA numerical inversion-based Boltzmann-Matano methods, with a large margin in terms of relative error (10× more accurate). Its performance is also consistent and stable, whereas the number of samples required remains small.

Calibrating models of cancer invasion: parameter estimation using approximate Bayesian computation and gradient matching

We present two different methods to estimate parameters within a partial differential equation model of cancer invasion. The model describes the spatio-temporal evolution of three variables-tumour cell density, extracellular matrix density and matrix degrading enzyme concentration-in a one-dimensional tissue domain. The first method is a likelihood-free approach associated with approximate Bayesian computation; the second is a two-stage gradient matching method based on smoothing the data with a generalized additive model (GAM) and matching gradients from the GAM to those from the model. Both methods performed well on simulated data. To increase realism, additionally we tested the gradient matching scheme with simulated measurement error and found that the ability to estimate some model parameters deteriorated rapidly as measurement error increased.