Mathematical Biology

Thermal therapies such as hyperthermia, laser ablation, and high-intensity focused ultrasound rely on delivering controlled heat to biological tissue. The temperature propagation in the tissue cannot be modelled accurately with the  Fourier's law, but should be modelled with the Maxwell–Cattaneo law instead, which gives rise to the hyperbolic bio-heat equation, that incorporates the assumption of the final speed propagation. In this talk I will discuss inverse source problems for such a hyperbolic model, where the goal is to recover the unknown space-dependent component of the heat source.

TBA

As cancer advances, cells often spread from the primary tumor to other parts of the body and form metastases. I'll present a conceptually elegant model of metastasis formation where each primary cell can initiate metastatic lesions which lesions then evolve as independent branching processes. We assume that the primary tumor is resected upon detection. Of fundamental importance is whether synchronous (detectable) or metachronous (undetectable) mets are present at this detection time, the distribution of their numbers and sizes. If there are only metachronous mets at detection, how long until these mets become detectable, leading to the relapse of the disease? We'll extend this model to cancers in which cells first need to evolve the ability to metastasize. Using sequence data from primary and corresponding met samples we'll propose that these intermediate cells are indeed present for certain primary-met pairs, why not there for others.

NOTES: unusual hour, etc (remove this line if not necessary).
The mitotic cell cycle governs DNA replication and cell division. The effectiveness of radiotherapy and chemotherapy depends on cell-cycle position, with increased resistance during DNA replication and mitosis. Thus, accurate mathematical models of the cell cycle are essential for understanding and predicting treatment response. However, mathematical modellers often face the problem of a lack of publicly available, sufficiently resolved, time-series datasets for parametrising models. In this work, we consider how the ability to collate population summary measurements across the literature, from different cell lines and/or experimental set ups, affects identifiability of parameters for a cell cycle model.

Initially synchronised cell populations gradually desynchronise over successive cycles, converging to balanced exponential growth (BEG) which is characterised by exponential population growth and steady, time-independent phase proportions. These proportions can be obtained from fluorescence-activated cell sorting (FACS) data. The increasing use of the Fluorescent Ubiquitination-based Cell Cycle Indicator (FUCCI) provides higher-resolution information on phase dynamics, such as minimum phase durations and variability.


We present an age-structured PDE model in which cell-cycle phase progression follows a delayed gamma distribution. We derive analytical expressions for BEG phase proportions and other FUCCI-observable quantities, and use them to assess how data availability influences parameter identifiability. When parameters are not uniquely identifiable, we determine identifiable parameter groupings, thereby determining the minimum amount of data that must be available for successfully fitting structured population models of the cell cycle.

NOTES: unusual hour, etc (remove this line if not necessary).
Effective application of mathematical models to interpret biological data and make accurate predictions often requires that model parameters are identifiable. Requisite to identifiability from a finite amount of noisy data is that model parameters are first structurally identifiable: a mathematical question that establishes whether multiple parameter values may give rise to indistinguishable model outputs. Approaches to assess structural identifiability of deterministic ordinary differential equation models are well-established, however tools for the assessment of the increasingly relevant stochastic and spatial models remain in their infancy.

I provide in this talk an introduction to structural identifiability, before presenting new frameworks for the assessment of stochastic and partial differential equations. Importantly, I discuss the relevance of our methodology to model selection, and more the practical and aptly named practical identifiability of parameters in the context of experimental data. Finally, I conclude with a brief discussion of future research directions and remaining open questions.

Every population consists of individuals that vary in many traits, and each trait may or may not be associated with fitness. Variation in fitness traits lends population studies prone to selective depletion biases. When an ageing cohort exhibits declining mortality, it could be individuals becoming healthier or selective depletion of the frail. In an epidemic, when growth in cumulative infections decelerates, it could be individuals cautiously changing behaviour or selective depletion of the most susceptible. In microbial populations, when an isogenic population is stressed by antimicrobial treatment and some cells survive, this could be due to individual cells switching between normal and persister phenotypes or antibiotic selectively killing cells that divide faster. In each case, the first explanation invokes individuals changing (1), while the second posits selection on pre-existing variation changing (2). While explanations of type (1) are intuitive and widely adopted, those of type (2) are more neutral and rarely considered due to cognitive biases and challenges in estimating all variation that matters. While both are plausibly operating in most real systems, neglect of (2) leads to over-attribution of results to (1), wrong predictions, bad policy decisions and poor reproducibility, negatively impacting science, economics and ethics.
To overcome this selective depletion bias, we propose a pragmatic approach to study design and analysis whereby we infer distributions of characteristics that respond to selection and reframe theories accordingly. The approach is based on remodelling selection (mathematically by introducing key parametric distributions into population dynamic models, and empirically by measuring quantities of interest along selection gradients) and statistical inference (by fitting mathematical models to data). The procedure is being tested in systems where trait distributions can be inferred from population trends as well as reconstructed directly from individual measurements. Results of this ongoing research will be presented, and the wider applicability discussed.

Antimicrobial resistance (AMR) is a global threat, and combating its spread is of paramount importance. AMR often results from cooperative behaviour involving shared drug protection [1]. Microbial communities typically evolve in volatile, spatially structured environments. Migration, fluctuations, and environmental variability therefore strongly influence AMR. While AMR is enhanced by cell migration under static conditions, this picture changes fundamentally in time-fluctuating, spatially structured environments [2].
In this seminar, we consider a two-dimensional metapopulation consisting of demes (subpopulations) in which drug-resistant and drug-sensitive cells evolve in a time-varying environment containing a toxin against which protection can be shared. Cells migrate between demes, thereby coupling them. When environmental changes and variations in deme composition occur on comparable timescales, strong population bottlenecks lead to fluctuation-driven extinction events, which are countered by migration. By assessing the combined influence of migration and environmental variability on AMR, we identify near-optimal conditions for fluctuation-driven resistance eradication and show that slow migration can accelerate and enhance AMR clearance.

The first part of this work was carried out with Lluís Hernández-Navarro, Matthew Asker, and Alastair Rucklidge; the second part was done with Lluís Hernández-Navarro, Kenneth Distefano and Uwe Täuber. This research was funded by the EPSRC and NSF. Project website: https://eedfp.com/

References:
[1] L. Hernández-Navarro, M. Asker, A. M. Rucklidge, and M. Mobilia, J. R. Soc. Interface 20, 20230393 (2023): https://royalsocietypublishing.org/doi/10.1098/rsif.2023.0393
[2] L. Hernández-Navarro, K. Distefano, U. C. Täuber, and M. Mobilia, bioRxiv (2024), 2024.12.30.630406 (for PLOS Computational Biology): https://doi.org/10.1101/2024.12.30.630406

In this talk we revisit the Wells-Riley model which has been widely used to estimate airborne infection risk in indoor settings. We consider a probabilistic framework which allows one to quantify infection risk as a probability distribution in the scenario that variation is observed in the model parameters (e.g. due to heterogeneity, or uncertainty in measurements). By using fitted Gamma distributions, we show how the classical Wells-Riley approach can lead to systematic inaccuracies. Population-related uncertainties (e.g. quanta emission rate, pulmonary rate, exposure time) can cause infection risk overestimation, whilst environmental uncertainties (e.g. ventilation rate) can lead to infection risk underestimation. Furthermore, we investigate the effects of simultaneously random parameters and the cases where one tends to dominate the stochasticity of the final distribution over the other. Considering entire risk distributions allows for more accurate risk assessments and can help prevent extreme infection events caused by significantly low ventilation or unusually high quanta emission.

Infectious diseases are often modelled via stochastic individual-level state-transition processes. As the transmission process is typically only partially and noisily observed, inference for these models generally follows a Bayesian data augmentation approach. However, standard data augmentation Markov chain Monte Carlo (MCMC) methods for individual-level epidemic models are often inefficient in terms of their mixing or challenging to implement. In this talk, I will introduce a novel data-augmentation MCMC method for discrete-time individual-level epidemic models, called the Rippler algorithm. I will explain how the Rippler algorithm works and how its performance compares to the standard and the state-of-the-art inference methods for individual-level models. I will also present results of application of the algorithm to data on AMR E. coli from Malawi.