arXiv subjects

Alexander P Browning

Publications and source records attributed to Alexander P Browning.

15 recordsLinked to original sources

arXiv2026

Adaptive therapy under parametric, structural, and measurement uncertainty

Adaptive therapy has emerged as a promising treatment strategy that exploits within-tumour competition to delay disease progression. Implementation, however, typically relies on indirect measurements of tumour burden and must account for potentially substantial patient heterogeneity. In this work, we capture patient-to-patient variability, parameter uncertainty, and imperfect biomarker measurements with a mathematical and statistical model that we calibrate to clinical prostate cancer data using a Bayesian inference framework. We use the resulting virtual cohort to demonstrate that, within the simple but now well-established Lotka-Volterra-based model, adaptive therapy robustly improves time-to-progression for the subset of patients that are predicted to eventually progress by the model. To account for other risk factors associated with larger tumour volumes, we introduce a new metric based on the risk of metastasis that demonstrates how adaptive therapy may be disadvantageous when sustained tumour burden is also considered. Given the ubiquity of uncertainty in oncology, we then describe several future modelling directions that also capture uncertainty in the temporal evolution of the underlying tumour or biomarker dynamics. Finally, we demonstrate how model misspecification and non-identifiability can lead to unreliable predictions, especially if uncertainty is inadequately captured.

q-bio.TO↗

arXiv2026

Structural functional identifiability and model discovery in differential equation models

Differential equation models are widely used to describe, interpret, and predict dynamical phenomena across science and engineering. In practice, however, the governing dynamics are rarely fully known and must be inferred from observational data. Traditionally, inverse problems in differential equation modelling have focused on estimating unknown parameter values. In this setting, structural identifiability determines whether parameter values can, in principle, be uniquely recovered from ideal observations and is, therefore, a prerequisite for meaningful inference. More recently, the integration of machine learning with mechanistic modelling has enabled the discovery of unknown equations, functions, and constitutive relationships, substantially expanding the space of admissible models. This raises a fundamental question: under what conditions can unknown functional components be uniquely recovered from data? In this paper, we generalise the classical notion of structural parameter identifiability to functional identifiability. We first identify broad classes of models for which unique functional recovery is impossible. We then show how functional identifiability can be assessed for differential equation models using differential algebra-based techniques which are well-established as a means of assessing structural identifiability for ordinary differential equation-based models. Our framework reveals new phenomena that arise in the transition from parametric to functional inference and have no analogue in the classical setting. Finally, we characterise functional identifiability in several common model classes. Taken together, our results demonstrate that functional identifiability provides a theoretical foundation for modern inverse problems in differential equation modelling, particularly those that use machine learning representations of unknown system components.

math.ST↗

arXiv2026

Framing structural identifiability in terms of parameter symmetries

A key step in mechanistic modelling of dynamical systems is to conduct a structural identifiability analysis. This entails deducing which parameter combinations can be estimated from a given set of observed outputs. The standard differential algebra approach answers this question by re-writing the model as a higher-order system of ordinary differential equations that depends solely on the observed outputs. Over the last decades, alternative approaches for analysing structural identifiability based on Lie symmetries acting on independent and dependent variables as well as parameters, have been proposed. However, the link between the standard differential algebra approach and that using full symmetries remains elusive. In this work, we establish this link by introducing the notion of parameter symmetries, which are a special type of full symmetry that alter parameters while preserving the observed outputs. Our main result states that a parameter combination is locally structurally identifiable if and only if it is a differential invariant of all parameter symmetries of a given model. We show that the standard differential algebra approach is consistent with the concept of structural identifiability in terms of parameter symmetries. We present an alternative symmetry-based approach for analysing structural identifiability using parameter symmetries. Lastly, we demonstrate our approach on two well-known models in mathematical biology.

math.OC↗

arXiv2025

Exact identifiability analysis for a class of partially observed near-linear stochastic differential equation models

Stochasticity plays a key role in many biological systems, necessitating the calibration of stochastic mathematical models to interpret associated data. For model parameters to be estimated reliably, it is typically the case that they must be structurally identifiable. Yet, while theory underlying structural identifiability analysis for deterministic differential equation models is highly developed, there are currently no tools for the general assessment of stochastic models. In this work, we present a differential algebra-based framework for the structural identifiability analysis of linear and a class of near-linear partially observed stochastic differential equation (SDE) models. Our framework is based on a deterministic recurrence relation that describes the dynamics of the statistical moments of the system of SDEs. From this relation, we iteratively form a series of necessarily satisfied equations involving only the observed moments, from which we are able to establish structurally identifiable parameter combinations. We demonstrate our framework for a suite of linear (two- and $n$-dimensional) and non-linear (two-dimensional) models. Most importantly, we define the notion of structural identifiability for SDE models and establish the effect of the initial condition on identifiability. We conclude with a discussion on the applicability and limitations of our approach, and potential future research directions in this understudied area.

stat.ME↗

arXiv2025

A cautionary tale of model misspecification and identifiability

Mathematical models are routinely applied to interpret biological data, with common goals that include both prediction and parameter estimation. A challenge in mathematical biology, in particular, is that models are often complex and non-identifiable, while data are limited. Rectifying identifiability through simplification can seemingly yield more precise parameter estimates, albeit, as we explore in this perspective, at the potentially catastrophic cost of introducing model misspecification and poor accuracy. We demonstrate how uncertainty in model structure can be propagated through to uncertainty in parameter estimates using a semi-parametric Gaussian process approach that delineates parameters of interest from uncertainty in model terms. Specifically, we study generalised logistic growth with an unknown crowding function, and a spatially resolved process described by a partial differential equation with a time-dependent diffusivity parameter. Allowing for structural model uncertainty yields more robust and accurate parameter estimates, and a better quantification of remaining uncertainty. We conclude our perspective by discussing the connections between identifiability and model misspecification, and alternative approaches to dealing with model misspecification in mathematical biology.

stat.ME↗

arXiv2025

Phenotypic heterogeneity in temporally fluctuating environments

Many biological systems regulate phenotypic heterogeneity as a fitness-maximising strategy in uncertain and dynamic environments. Analysis of such strategies is typically confined both to a discrete set of environmental conditions, and to a discrete (often binary) set of phenotypes specialised to each condition. In this work, we extend theory on both fronts to encapsulate a potentially continuous spectrum of phenotypes arising in response to environmental fluctuations that drive changes in the phenotype-dependent growth rate. We consider two broad classes of stochastic environment: those that are temporally uncorrelated (modelled by white-noise processes), and those that are correlated (modelled by Poisson and Ornstein-Uhlenbeck processes). For tractability, we restrict analysis to an exponential growth model, and consider biologically relevant simplifications that pertain to the timescale of phenotype switching relative to fluctuations in the environment. These assumptions yield a series of analytical and semi-analytical expressions that reveal environments in which phenotypic heterogeneity is evolutionarily advantageous.

q-bio.PE↗

arXiv2025

Reducing phenotype-structured PDE models of cancer evolution to systems of ODEs: a generalised moment dynamics approach

Intratumour phenotypic heterogeneity is nowadays understood to play a critical role in disease progression and treatment failure. Accordingly, there has been increasing interest in the development of mathematical models capable of capturing its role in cancer cell adaptation. This can be systematically achieved by means of models comprising phenotype-structured nonlocal partial differential equations, tracking the evolution of the phenotypic density distribution of the cell population, which may be compared to gene and protein expression distributions obtained experimentally. Nevertheless, given the high analytical and computational cost of solving these models, much is to be gained from reducing them to systems of ordinary differential equations for the moments of the distribution. We propose a generalised method of model-reduction, relying on the use of a moment generating function, Taylor series expansion and truncation closure, to reduce a nonlocal reaction-advection-diffusion equation, with general phenotypic drift and proliferation rate functions, to a system of moment equations up to arbitrary order. Our method extends previous results in the literature, which we address via two examples, by removing any \textit{a priori} assumption on the shape of the distribution, and provides a flexible framework for mathematical modellers to account for the role of phenotypic heterogeneity in cancer adaptive dynamics, in a simpler mathematical framework.

q-bio.PE↗

arXiv2025

Structural identifiability of linear-in-parameter parabolic PDEs through auxiliary elliptic operators

Parameter identifiability is often requisite to the effective application of mathematical models in the interpretation of biological data, however theory applicable to the study of partial differential equations remains limited. We present a new approach to structural identifiability analysis of fully observed parabolic equations that are linear in their parameters. Our approach frames identifiability as an existence and uniqueness problem in a closely related elliptic equation and draws, for homogeneous equations, on the well-known Fredholm alternative to establish unconditional identifiability, and cases where specific choices of initial and boundary conditions lead to non-identifiability. While in some sense pathological, we demonstrate that this loss of structural identifiability has ramifications for practical identifiability; important particularly for spatial problems, where the initial condition is often limited by experimental constraints. For cases with nonlinear reaction terms, uniqueness of solutions to the auxiliary elliptic equation corresponds to identifiability, often leading to unconditional global identifiability under mild assumptions. We present analysis for a suite of simple scalar models with various boundary conditions that include linear (exponential) and nonlinear (logistic) source terms, and a special case of a two-species cell motility model. We conclude by discussing how this new perspective enables well-developed analysis tools to advance the developing theory underlying structural identifiability of partial differential equations.

math.AP↗

arXiv2024

Smoothing in linear multicompartment biological processes subject to stochastic input

Many physical and biological systems rely on the progression of material through multiple independent stages. In viral replication, for example, virions enter a cell to undergo a complex process comprising several disparate stages before the eventual accumulation and release of replicated virions. While such systems may have some control over the internal dynamics that make up this progression, a challenge for many is to regulate behaviour under what are often highly variable external environments acting as system inputs. In this work, we study a simple analogue of this problem through a linear multicompartment model subject to a stochastic input in the form of a mean-reverting Ornstein-Uhlenbeck process, a type of Gaussian process. By expressing the system as a multidimensional Gaussian process, we derive several closed-form analytical results relating to the covariances and autocorrelations of the system, quantifying the smoothing effect discrete compartments afford multicompartment systems. Semi-analytical results demonstrate that feedback and feedforward loops can enhance system robustness, and simulation results probe the intractable problem of the first passage time distribution, which has specific relevance to eventual cell lysis in the viral replication cycle. Finally, we demonstrate that the smoothing seen in the process is a consequence of the discreteness of the system, and does not manifest in system with continuous transport. While we make progress through analysis of a simple linear problem, many of our insights are applicable more generally, and our work enables future analysis into multicompartment processes subject to stochastic inputs.

q-bio.MN↗

arXiv2024

Structural identifiability analysis of linear reaction-advection-diffusion processes in mathematical biology

Effective application of mathematical models to interpret biological data and make accurate predictions often requires that model parameters are identifiable. Approaches to assess the so-called structural identifiability of models are well-established for ordinary differential equation models, yet there are no commonly adopted approaches that can be applied to assess the structural identifiability of the partial differential equation (PDE) models that are requisite to capture spatial features inherent to many phenomena. The differential algebra approach to structural identifiability has recently been demonstrated to be applicable to several specific PDE models. In this brief article, we present general methodology for performing structural identifiability analysis on partially observed reaction-advection-diffusion (RAD) PDE models that are linear in the unobserved quantities. We show that the differential algebra approach can always, in theory, be applied to such models. Moreover, despite the perceived complexity introduced by the addition of advection and diffusion terms, identifiability of spatial analogues of non-spatial models cannot decrease in structural identifiability. We conclude by discussing future possibilities and the computational cost of performing structural identifiability analysis on more general PDE models.

q-bio.QM↗

arXiv2022

Geometric analysis enables biological insight from complex non-identifiable models using simple surrogates

An enduring challenge in computational biology is to balance data quality and quantity with model complexity. Tools such as identifiability analysis and information criterion have been developed to harmonise this juxtaposition, yet cannot always resolve the mismatch between available data and the granularity required in mathematical models to answer important biological questions. Often, it is only simple phenomenological models, such as the logistic and Gompertz growth models, that are identifiable from standard experimental measurements. To draw insights from the complex, non-identifiable models that incorporate key biological mechanisms of interest, we study the geometry of a map in parameter space from the complex model to a simple, identifiable, surrogate model. By studying how non-identifiable parameters in the complex model quantitatively relate to identifiable parameters in surrogate, we introduce and exploit a layer of interpretation between the set of non-identifiable parameters and the goodness-of-fit metric or likelihood studied in typical identifiability analysis. We demonstrate our approach by analysing a hierarchy of mathematical models for multicellular tumour spheroid growth. Typical data from tumour spheroid experiments are limited and noisy, and corresponding mathematical models are very often made arbitrarily complex. Our geometric approach is able to predict non-identifiabilities, subset non-identifiable parameter spaces into identifiable parameter combinations that relate to individual data features, and overall provide additional biological insight from complex non-identifiable models.

q-bio.QM↗

arXiv2022

Parameter estimation and uncertainty quantification using information geometry

In this work we: (1) review likelihood-based inference for parameter estimation and the construction of confidence regions; and, (2) explore the use of techniques from information geometry, including geodesic curves and Riemann scalar curvature, to supplement typical techniques for uncertainty quantification such as Bayesian methods, profile likelihood, asymptotic analysis and bootstrapping. These techniques from information geometry provide data-independent insights into uncertainty and identifiability, and can be used to inform data collection decisions. All code used in this work to implement the inference and information geometry techniques is available on GitHub.

stat.ME↗

arXiv2022

Population Calibration using Likelihood-Free Bayesian Inference

In this paper we develop a likelihood-free approach for population calibration, which involves finding distributions of model parameters when fed through the model produces a set of outputs that matches available population data. Unlike most other approaches to population calibration, our method produces uncertainty quantification on the estimated distribution. Furthermore, the method can be applied to any population calibration problem, regardless of whether the model of interest is deterministic or stochastic, or whether the population data is observed with or without measurement error. We demonstrate the method on several examples, including one with real data. We also discuss the computational limitations of the approach. Immediate applications for the methodology developed here exist in many areas of medical research including cancer, COVID-19, drug development and cardiology.

stat.ME↗

arXiv2021

Profile likelihood analysis for a stochastic model of diffusion in heterogeneous media

We compute profile likelihoods for a stochastic model of diffusive transport motivated by experimental observations of heat conduction in layered skin tissues. This process is modelled as a random walk in a layered one-dimensional material, where each layer has a distinct particle hopping rate. Particles are released at some location, and the duration of time taken for each particle to reach an absorbing boundary is recorded. To explore whether this data can be used to identify the hopping rates in each layer, we compute various profile likelihoods using two methods: first, an exact likelihood is evaluated using a relatively expensive Markov chain approach; and, second we form an approximate likelihood by assuming the distribution of exit times is given by a Gamma distribution whose first two moments match the expected moments from the continuum limit description of the stochastic model. Using the exact and approximate likelihoods we construct various profile likelihoods for a range of problems. In cases where parameter values are not identifiable, we make progress by re-interpreting those data with a reduced model with a smaller number of layers.

physics.bio-ph↗

arXiv2019

Reversible signal transmission in an active mechanical metamaterial

Mechanical metamaterials are designed to enable unique functionalities, but are typically limited by an initial energy state and require an independent energy input to function repeatedly. Our study introduces a theoretical active mechanical metamaterial that incorporates a biological reaction mechanism to overcome this key limitation of passive metamaterials. Our material allows for reversible mechanical signal transmission, where energy is reintroduced by the biologically motivated reaction mechanism. By analysing a coarse grained continuous analogue of the discrete model, we find that signals can be propagated through the material by a travelling wave. Analysis of the continuum model provides the region of the parameter space that allows signal transmission, and reveals similarities with the well-known FitzHugh-Nagumo system. We also find explicit formulae that approximate the effect of the timescale of the reaction mechanism on the signal transmission speed, which is essential for controlling the material.

cond-mat.soft↗