ORCID Profile
0000-0001-6254-313X
Current Organisation
Queensland University of Technology
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Research Topics
In Research Link Australia (RLA), "Research Topics" refer to ANZSRC FOR and SEO codes. These topics are either sourced from ANZSRC FOR and SEO codes listed in researchers' related grants or generated by a large language model (LLM) based on their publications.
Top 5 Research Topics
ANZSRC Field of Research (FoR)
Applied Mathematics | Biological Mathematics | Biological mathematics | Applied mathematics | Stochastic Analysis And Modelling | Systems biology | Differential, Difference And Integral Equations | Biological Mathematics | Microbiology not elsewhere classified | Biological physics
ANZSRC Socio-Economic Objective (SEO)
Cancer and Related Disorders | Skin and Related Disorders | Expanding Knowledge in the Mathematical Sciences | Child health | Biological sciences | Mathematical sciences |
Related Links
Publications
Simplified approach for calculating moments of action for linear reaction-diffusion equations
Publisher: American Physical Society (APS)
Date: 15-11-2013
A model for mesoscale patterns in motile populations
Publisher: Elsevier BV
Date: 04-2010
Simulating invasion with cellular automata: Connecting cell-scale and population-scale properties
Publisher: American Physical Society (APS)
Date: 17-08-2007
Tactically-driven nonmonotone travelling waves
Publisher: Elsevier BV
Date: 05-2008
New mathematical modelling tools for co-culture experiments: when do we need to explicitly account for signalling molecules?
Publisher: Cold Spring Harbor Laboratory
Date: 14-01-2020
DOI: 10.1101/2020.01.13.905414
Abstract: Mathematical models are often applied to describe cell migration regulated by diffusible signalling molecules. A typical feature of these models is that the spatial and temporal distribution of the signalling molecule density is reported by solving a reaction–diffusion equation. However, the spatial and temporal distributions of such signalling molecules are not often reported or observed experimentally. This leads to a mismatch between the amount of experimental data available and the complexity of the mathematical model used to simulate the experiment. To address this mismatch, we develop a discrete model of cell migration that can be used to describe a new suite of co–culture cell migration assays involving two interacting subpopulations of cells. In this model, the migration of cells from one subpopulation is regulated by the presence of signalling molecules that are secreted by the other subpopulation of cells. The spatial and temporal distribution of the signalling molecules is governed by a discrete conservation statement that is related to a reaction–diffusion equation. We simplify the model by invoking a steady state assumption for the diffusible molecules, leading to a reduced discrete model allowing us to describe how one subpopulation of cells stimulates the migration of the other subpopulation of cells without explicitly dealing with the diffusible molecules. We provide additional mathematical insight into these two stochastic models by deriving continuum limit partial differential equation descriptions of both models. To understand the conditions under which the reduced model is a good approximation of the full model, we apply both models to mimic a set of novel co–culture assays and we systematically explore how well the reduced model approximates the full model as a function of the model parameters.
A Bayesian Sequential Learning Framework to Parameterise Continuum Models of Melanoma Invasion into Human Skin
Publisher: Springer Science and Business Media LLC
Date: 15-11-2018
DOI: 10.1007/S11538-018-0532-1
Abstract: We present a novel framework to parameterise a mathematical model of cell invasion that describes how a population of melanoma cells invades into human skin tissue. Using simple experimental data extracted from complex experimental images, we estimate three model parameters: (i) the melanoma cell proliferation rate, [Formula: see text] (ii) the melanoma cell diffusivity, D and (iii) [Formula: see text], a constant that determines the rate that melanoma cells degrade the skin tissue. The Bayesian sequential learning framework involves a sequence of increasingly sophisticated experimental data from: (i) a spatially uniform cell proliferation assay (ii) a two-dimensional circular barrier assay and (iii) a three-dimensional invasion assay. The Bayesian sequential learning approach leads to well-defined parameter estimates. In contrast, taking a naive approach that attempts to estimate all parameters from a single set of images from the same experiment fails to produce meaningful results. Overall, our approach to inference is simple-to-implement, computationally efficient, and well suited for many cell biology phenomena that can be described by low-dimensional continuum models using ordinary differential equations and partial differential equations. We anticipate that this Bayesian sequential learning framework will be relevant in other biological contexts where it is challenging to extract detailed, quantitative biological measurements from experimental images and so we must rely on using relatively simple measurements from complex images.
Mechanical Cell Competition in Heterogeneous Epithelial Tissues
Publisher: Springer Science and Business Media LLC
Date: 26-09-2020
The role of mechanical interactions in EMT
Publisher: IOP Publishing
Date: 12-05-2021
Abstract: The detachment of cells from the boundary of an epithelial tissue and the subsequent invasion of these cells into surrounding tissues is important for cancer development and wound healing, and is strongly associated with the epithelial–mesenchymal transition (EMT). Chemical signals, such as TGF- β , produced by surrounding tissue can be uptaken by cells and induce EMT. In this work, we present a novel cell-based discrete mathematical model of mechanical cellular relaxation, cell proliferation, and cell detachment driven by chemically-dependent EMT in an epithelial tissue. A continuum description of the model is then derived in the form of a novel nonlinear free boundary problem. Using the discrete and continuum models we explore how the coupling of chemical transport and mechanical interactions influences EMT, and postulate how this could be used to help control EMT in pathological situations.
Rapid calculation of maximum particle lifetime for diffusion in complex geometries
Publisher: AIP Publishing
Date: 07-03-2018
DOI: 10.1063/1.5019180
Abstract: Diffusion of molecules within biological cells and tissues is strongly influenced by crowding. A key quantity to characterize diffusion is the particle lifetime, which is the time taken for a diffusing particle to exit by hitting an absorbing boundary. Calculating the particle lifetime provides valuable information, for ex le, by allowing us to compare the timescale of diffusion and the timescale of the reaction, thereby helping us to develop appropriate mathematical models. Previous methods to quantify particle lifetimes focus on the mean particle lifetime. Here, we take a different approach and present a simple method for calculating the maximum particle lifetime. This is the time after which only a small specified proportion of particles in an ensemble remain in the system. Our approach produces accurate estimates of the maximum particle lifetime, whereas the mean particle lifetime always underestimates this value compared with data from stochastic simulations. Furthermore, we find that differences between the mean and maximum particle lifetimes become increasingly important when considering diffusion hindered by obstacles.
Identifying cell-to-cell variability in internalisation using flow cytometry
Publisher: Cold Spring Harbor Laboratory
Date: 25-11-2021
DOI: 10.1101/2021.11.24.469957
Abstract: Biological heterogeneity is a primary contributor to the variation observed in experiments that probe dynamical processes, such as internalisation. Given that internalisation is a critical process by which many therapeutics and viruses reach their intracellular site of action, quantifying cell-to-cell variability in internalisation is of high biological interest. Yet, it is common for studies of internalisation to neglect cell-to-cell variability. We develop a simple mathematical model of internalisation that captures the dynamical behaviour, cell-to-cell variation, and extrinsic noise introduced by flow cytometry. We calibrate our model through a novel distribution-matching approximate Bayesian computation algorithm to flow cytometry data of internalisation of anti-transferrin receptor antibody in a human B-cell lymphoblastoid cell line. Our model reproduces experimental observations, identifies cell-to-cell variability in the internalisation and recycling rates, and, importantly, provides information relating to inferential uncertainty. Given that our approach is agnostic to s le size and signal-to-noise ratio, our modelling framework is broadly applicable to identify biological variability in single-cell data from internalisation assays and similar experiments that probe cellular dynamical processes.
Critical timescales and time intervals for coupled linear processes
Publisher: Cambridge University Press (CUP)
Date: 2013
DOI: 10.1017/S1446181113000059
Abstract: In 1991, McNabb introduced the concept of mean action time (MAT) as a finite measure of the time required for a diffusive process to effectively reach steady state. Although this concept was initially adopted by others within the Australian and New Zealand applied mathematics community, it appears to have had little use outside this region until very recently, when in 2010 Berezhkovskii and co-workers [A. M. Berezhkovskii, C. S le and S. Y. Shvartsman, “How long does it take to establish a morphogen gradient?” Biophys. J. 99 (2010) L59–L61] rediscovered the concept of MAT in their study of morphogen gradient formation. All previous work in this area has been limited to studying single-species differential equations, such as the linear advection–diffusion–reaction equation. Here we generalize the concept of MAT by showing how the theory can be applied to coupled linear processes. We begin by studying coupled ordinary differential equations and extend our approach to coupled partial differential equations. Our new results have broad applications, for ex le the analysis of models describing coupled chemical decay and cell differentiation processes.
Mechanistic and experimental models of cell migration reveal the importance of cell-to-cell pushing in cell invasion
Publisher: IOP Publishing
Date: 27-05-2019
Abstract: Moving fronts of cells are essential for development, repair and disease progression. Therefore, understanding and quantifying the details of the mechanisms that drive the movement of cell fronts is of wide interest. Quantitatively identifying the role of intercellular interactions, and in particular the role of cell pushing, remains an open question. In this work, we report a combined experimental-modelling approach showing that intercellular interactions contribute significantly to the spatial spreading of a population of cells. We use a novel experimental data set with PC-3 prostate cancer cells that have been pretreated with Mitomycin-C to suppress proliferation. This allows us to experimentally separate the effects of cell migration from cell proliferation, thereby enabling us to focus on the migration process in detail as the population of cells recolonizes an initially-vacant region in a series of two-dimensional experiments. We quantitatively model the experiments using a stochastic modelling framework, based on Langevin dynamics, which explicitly incorporates random motility and various intercellular forces including: (i) long range attraction (adhesion) and (ii) finite size effects that drive short range repulsion (pushing). Quantitatively comparing the ability of this model to describe the experimentally observed population-level behaviour provides us with quantitative insight into the roles of random motility and intercellular interactions. To quantify the mechanisms at play, we calibrate the stochastic model to match experimental cell density profiles to obtain estimates of cell diffusivity, D , and the litude of intercellular forces, f 0 . Our analysis shows that taking a standard modelling approach which ignores intercellular forces provides a poor match to the experimental data whereas incorporating intercellular forces, including short-range pushing and longer range attraction, leads to a faithful representation of the experimental observations. These results demonstrate a significant role of cell pushing during cell front movement and invasion.
Identifying density-dependent interactions in collective cell behaviour
Publisher: Cold Spring Harbor Laboratory
Date: 07-2021
DOI: 10.1101/811257
Abstract: Scratch assays are routinely used to study collective cell behaviour in vitro . Typical experimental protocols do not vary the initial density of cells, and typical mathematical modelling approaches describe cell motility and proliferation based on assumptions of linear diffusion and logistic growth. Jin et al. (2016) find that the behaviour of cells in scratch assays is density-dependent, and show that standard modelling approaches cannot simultaneously describe data initiated across a range of initial densities. To address this limitation, we calibrate an in idual based model to scratch assay data across a large range of initial densities. Our model allows proliferation, motility, and a direction bias to depend on interactions between neighbouring cells. By considering a hierarchy of models where we systematically and sequentially remove interactions, we perform model selection analysis to identify the minimum interactions required for the model to simultaneously describe data across all initial densities. The calibrated model is able to match the experimental data across all densities using a single parameter distribution, and captures details about the spatial structure of cells. Our results provide strong evidence to suggest that motility is density-dependent in these experiments. On the other hand, we do not see the effect of crowding on proliferation in these experiments. These results are significant as they are precisely the opposite of the assumptions in standard continuum models, such as the Fisher-Kolmogorov equation and its generalisations.
The role of mechanical interactions in EMT
Publisher: Cold Spring Harbor Laboratory
Date: 10-12-2020
DOI: 10.1101/2020.12.09.418434
Abstract: The detachment of cells from the boundary of an epithelial tissue and the subsequent invasion of these cells into surrounding tissues is important for cancer development and wound healing, and is strongly associated with the epithelial-mesenchymal transition (EMT). Chemical signals, such as TGF- β , produced by surrounding tissue can be up-taken by cells and induce EMT. In this work, we present a novel cell-based discrete mathematical model of mechanical cellular relaxation, cell proliferation, and cell detachment driven by chemically-dependent EMT in an epithelial tissue. A continuum description of the model is then derived in the form of a novel nonlinear free boundary problem. Using the discrete and continuum models we explore how the coupling of chemical transport and mechanical interactions influences EMT, and postulate how this could be used to help control EMT in pathological situations.
Travelling waves for a velocity-jump model of cell migration and proliferation
Publisher: Elsevier BV
Date: 08-2013
DOI: 10.1016/J.MBS.2013.04.010
Abstract: Cell invasion, characterised by moving fronts of cells, is an essential aspect of development, repair and disease. Typically, mathematical models of cell invasion are based on the Fisher-Kolmogorov equation. These traditional parabolic models cannot be used to represent experimental measurements of in idual cell velocities within the invading population since they imply that information propagates with infinite speed. To overcome this limitation we study combined cell motility and proliferation based on a velocity-jump process where information propagates with finite speed. The model treats the total population of cells as two interacting subpopulations: a subpopulation of left-moving cells, L(x,t), and a subpopulation of right-moving cells, R(x,t). This leads to a system of hyperbolic partial differential equations that includes a turning rate, Λ⩾0, describing the rate at which in iduals in the population change direction of movement. We present exact travelling wave solutions of the system of partial differential equations for the special case where Λ=0 and in the limit that Λ→∞. For intermediate turning rates, 0<Λ<∞, we analyse the travelling waves using the phase plane and we demonstrate a transition from smooth monotone travelling waves to smooth nonmonotone travelling waves as Λ decreases through a critical value Λcrit. We conclude by providing a qualitative comparison between the travelling wave solutions of our model and experimental observations of cell invasion. This comparison indicates that the small Λ limit produces results that are consistent with experimental observations.
Exact series solutions of reactive transport models with general initial conditions
Publisher: Elsevier BV
Date: 05-2014
A free boundary mechanobiological model of epithelial tissues
Publisher: Cold Spring Harbor Laboratory
Date: 04-07-2020
DOI: 10.1101/2020.07.02.185686
Abstract: In this study, we couple intracellular signalling and cell–based mechanical properties to develop a novel free boundary mechanobiological model of epithelial tissue dynamics. Mechanobiological coupling is introduced at the cell level in a discrete modelling framework, and new reaction–diffusion equations are derived to describe tissue–level outcomes. The free boundary evolves as a result of the underlying biological mechanisms included in the discrete model. To demonstrate the accuracy of the continuum model, we compare numerical solutions of the discrete and continuum models for two different signalling pathways. First, we study the Rac–Rho pathway where cell– and tissue–level mechanics are directly related to intracellular signalling. Second, we study an activator–inhibitor system which gives rise to spatial and temporal patterning related to Turing patterns. In all cases, the continuum model and free boundary condition accurately reflect the cell–level processes included in the discrete model.
Depth-averaging errors in reactive transport modeling
Publisher: American Geophysical Union (AGU)
Date: 02-2009
DOI: 10.1029/2008WR007356
Using Experimental Data and Information Criteria to Guide Model Selection for Reaction–Diffusion Problems in Mathematical Biology
Publisher: Springer Science and Business Media LLC
Date: 27-02-2019
DOI: 10.1007/S11538-019-00589-X
Abstract: Reaction-diffusion models describing the movement, reproduction and death of in iduals within a population are key mathematical modelling tools with widespread applications in mathematical biology. A erse range of such continuum models have been applied in various biological contexts by choosing different flux and source terms in the reaction-diffusion framework. For ex le, to describe the collective spreading of cell populations, the flux term may be chosen to reflect various movement mechanisms, such as random motion (diffusion), adhesion, haptotaxis, chemokinesis and chemotaxis. The choice of flux terms in specific applications, such as wound healing, is usually made heuristically, and rarely it is tested quantitatively against detailed cell density data. More generally, in mathematical biology, the questions of model validation and model selection have not received the same attention as the questions of model development and model analysis. Many studies do not consider model validation or model selection, and those that do often base the selection of the model on residual error criteria after model calibration is performed using nonlinear regression techniques. In this work, we present a model selection case study, in the context of cell invasion, with a very detailed experimental data set. Using Bayesian analysis and information criteria, we demonstrate that model selection and model validation should account for both residual errors and model complexity. These considerations are often overlooked in the mathematical biology literature. The results we present here provide a straightforward methodology that can be used to guide model selection across a range of applications. Furthermore, the case study we present provides a clear ex le where neglecting the role of model complexity can give rise to misleading outcomes.
Accurate and efficient discretizations for stochastic models providing near agent-based spatial resolution at low computational cost
Publisher: The Royal Society
Date: 23-10-2019
Abstract: Understanding how cells proliferate, migrate and die in various environments is essential in determining how organisms develop and repair themselves. Continuum mathematical models, such as the logistic equation and the Fisher–Kolmogorov equation, can describe the global characteristics observed in commonly used cell biology assays, such as proliferation and scratch assays. However, these continuum models do not account for single-cell-level mechanics observed in high-throughput experiments. Mathematical modelling frameworks that represent in idual cells, often called agent-based models, can successfully describe key single-cell-level features of these assays but are computationally infeasible when dealing with large populations. In this work, we propose an agent-based model with crowding effects that is computationally efficient and matches the logistic and Fisher–Kolmogorov equations in parameter regimes relevant to proliferation and scratch assays, respectively. This stochastic agent-based model allows multiple agents to be contained within compartments on an underlying lattice, thereby reducing the computational storage compared to existing agent-based models that allow one agent per site only. We propose a systematic method to determine a suitable compartment size. Implementing this compartment-based model with this compartment size provides a balance between computational storage, local resolution of agent behaviour and agreement with classical continuum descriptions.
Identifying cell-to-cell variability in internalization using flow cytometry
Publisher: The Royal Society
Date: 05-2022
Abstract: Biological heterogeneity is a primary contributor to the variation observed in experiments that probe dynamical processes, such as the internalization of material by cells. Given that internalization is a critical process by which many therapeutics and viruses reach their intracellular site of action, quantifying cell-to-cell variability in internalization is of high biological interest. Yet, it is common for studies of internalization to neglect cell-to-cell variability. We develop a simple mathematical model of internalization that captures the dynamical behaviour, cell-to-cell variation, and extrinsic noise introduced by flow cytometry. We calibrate our model through a novel distribution-matching approximate Bayesian computation algorithm to flow cytometry data of internalization of anti-transferrin receptor antibody in a human B-cell lymphoblastoid cell line. This approach provides information relating to the region of the parameter space, and consequentially the nature of cell-to-cell variability, that produces model realizations consistent with the experimental data. Given that our approach is agnostic to s le size and signal-to-noise ratio, our modelling framework is broadly applicable to identify biological variability in single-cell data from internalization assays and similar experiments that probe cellular dynamical processes.
Distinguishing between mean-field, moment dynamics and stochastic descriptions of birth-death-movement processes
Publisher: Elsevier BV
Date: 02-2014
Examining Go-or-Grow Using Fluorescent Cell-Cycle Indicators and Cell-Cycle-Inhibiting Drugs
Publisher: Elsevier BV
Date: 03-2020
Quantitative analysis of tumour spheroid structure
Publisher: eLife Sciences Publications, Ltd
Date: 29-11-2021
DOI: 10.7554/ELIFE.73020
Abstract: Tumour spheroids are common in vitro experimental models of avascular tumour growth. Compared with traditional two-dimensional culture, tumour spheroids more closely mimic the avascular tumour microenvironment where spatial differences in nutrient availability strongly influence growth. We show that spheroids initiated using significantly different numbers of cells grow to similar limiting sizes, suggesting that avascular tumours have a limiting structure in agreement with untested predictions of classical mathematical models of tumour spheroids. We develop a novel mathematical and statistical framework to study the structure of tumour spheroids seeded from cells transduced with fluorescent cell cycle indicators, enabling us to discriminate between arrested and cycling cells and identify an arrested region. Our analysis shows that transient spheroid structure is independent of initial spheroid size, and the limiting structure can be independent of seeding density. Standard experimental protocols compare spheroid size as a function of time however, our analysis suggests that comparing spheroid structure as a function of overall size produces results that are relatively insensitive to variability in spheroid size. Our experimental observations are made using two melanoma cell lines, but our modelling framework applies across a wide range of spheroid culture conditions and cell lines.
Multifidelity multilevel Monte Carlo to accelerate approximate Bayesian parameter inference for partially observed stochastic processes
Publisher: Elsevier BV
Date: 11-2022
Mathematical models incorporating a multi-stage cell cycle replicate normally-hidden inherent synchronization in cell proliferation
Publisher: The Royal Society
Date: 08-2019
Abstract: We present a suite of experimental data showing that cell proliferation assays, prepared using standard methods thought to produce asynchronous cell populations, persistently exhibit inherent synchronization. Our experiments use fluorescent cell cycle indicators to reveal the normally hidden cell synchronization, by highlighting oscillatory subpopulations within the total cell population. These oscillatory subpopulations would never be observed without these cell cycle indicators. On the other hand, our experimental data show that the total cell population appears to grow exponentially, as in an asynchronous population. We reconcile these seemingly inconsistent observations by employing a multi-stage mathematical model of cell proliferation that can replicate the oscillatory subpopulations. Our study has important implications for understanding and improving experimental reproducibility. In particular, inherent synchronization may affect the experimental reproducibility of studies aiming to investigate cell cycle-dependent mechanisms, including changes in migration and drug response.
Parameter estimation and uncertainty quantification using information geometry
Publisher: The Royal Society
Date: 04-2022
Abstract: In this work, we: (i) review likelihood-based inference for parameter estimation and the construction of confidence regions and (ii) 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 .
Experimental and Modelling Investigation of Monolayer Development with Clustering
Publisher: Springer Science and Business Media LLC
Date: 13-04-2013
DOI: 10.1007/S11538-013-9839-0
Abstract: Standard differential equation-based models of collective cell behaviour, such as the logistic growth model, invoke a mean-field assumption which is equivalent to assuming that in iduals within the population interact with each other in proportion to the average population density. Implementing such assumptions implies that the dynamics of the system are unaffected by spatial structure, such as the formation of patches or clusters within the population. Recent theoretical developments have introduced a class of models, known as moment dynamics models, which aim to account for the dynamics of in iduals, pairs of in iduals, triplets of in iduals, and so on. Such models enable us to describe the dynamics of populations with clustering, however, little progress has been made with regard to applying moment dynamics models to experimental data. Here, we report new experimental results describing the formation of a monolayer of cells using two different cell types: 3T3 fibroblast cells and MDA MB 231 breast cancer cells. Our analysis indicates that the 3T3 fibroblast cells are relatively motile and we observe that the 3T3 fibroblast monolayer forms without clustering. Alternatively, the MDA MB 231 cells are less motile and we observe that the MDA MB 231 monolayer formation is associated with significant clustering. We calibrate a moment dynamics model and a standard mean-field model to both data sets. Our results indicate that the mean-field and moment dynamics models provide similar descriptions of the 3T3 fibroblast monolayer formation whereas these two models give very different predictions for the MDA MD 231 monolayer formation. These outcomes indicate that standard mean-field models of collective cell behaviour are not always appropriate and that care ought to be exercised when implementing such a model.
Author response: Quantitative analysis of tumour spheroid structure
Publisher: eLife Sciences Publications, Ltd
Date: 19-11-2021
Identifiability analysis for stochastic differential equation models in systems biology
Publisher: Cold Spring Harbor Laboratory
Date: 11-08-2020
DOI: 10.1101/2020.08.10.245233
Abstract: Mathematical models are routinely calibrated to experimental data, with goals ranging from building predictive models to quantifying parameters that cannot be measured. Whether or not reliable parameter estimates are obtainable from the available data can easily be overlooked. Such issues of parameter identifiability have important ramifications for both the predictive power of a model, and the mechanistic insight that can be obtained. Identifiability analysis is well-established for deterministic, ordinary differential equation (ODE) models, but there are no commonly-adopted methods for analysing identifiability in stochastic models. We provide an accessible introduction to identifiability analysis and demonstrate how existing ideas for analysis of ODE models can be applied to stochastic differential equation (SDE) models through four practical case studies. To assess structural identifiability , we study ODEs that describe the statistical moments of the stochastic process using open-source software tools. Using practically-motivated synthetic data and Markov-chain Monte Carlo (MCMC) methods, we assess parameter identifiability in the context of available data. Our analysis shows that SDE models can often extract more information about parameters than deterministic descriptions. All code used to perform the analysis is available on Github.
A novel mathematical model of heterogeneous cell proliferation
Publisher: Springer Science and Business Media LLC
Date: 12-03-2021
Reproducibility of scratch assays is affected by the initial degree of confluence: Experiments, modelling and model selection
Publisher: Elsevier BV
Date: 02-2016
DOI: 10.1016/J.JTBI.2015.10.040
Abstract: Scratch assays are difficult to reproduce. Here we identify a previously overlooked source of variability which could partially explain this difficulty. We analyse a suite of scratch assays in which we vary the initial degree of confluence (initial cell density). Our results indicate that the rate of re-colonisation is very sensitive to the initial density. To quantify the relative roles of cell migration and proliferation, we calibrate the solution of the Fisher-Kolmogorov model to cell density profiles to provide estimates of the cell diffusivity, D, and the cell proliferation rate, λ. This procedure indicates that the estimates of D and λ are very sensitive to the initial density. This dependence suggests that the Fisher-Kolmogorov model does not accurately represent the details of the collective cell spreading process, since this model assumes that D and λ are constants that ought to be independent of the initial density. Since higher initial cell density leads to enhanced spreading, we also calibrate the solution of the Porous-Fisher model to the data as this model assumes that the cell flux is an increasing function of the cell density. Estimates of D and λ associated with the Porous-Fisher model are less sensitive to the initial density, suggesting that the Porous-Fisher model provides a better description of the experiments.
Exact sharp-fronted solutions for nonlinear diffusion on evolving domains
Publisher: IOP Publishing
Date: 24-10-2023
Chemotactic and diffusive migration on a nonuniformly growing domain: numerical algorithm development and applications
Publisher: Elsevier BV
Date: 08-2006
Profile likelihood analysis for a stochastic model of diffusion in heterogeneous media
Publisher: The Royal Society
Date: 06-2021
Abstract: 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 these 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 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.
Identifiability analysis for stochastic differential equation models in systems biology
Publisher: The Royal Society
Date: 12-2020
Abstract: Mathematical models are routinely calibrated to experimental data, with goals ranging from building predictive models to quantifying parameters that cannot be measured. Whether or not reliable parameter estimates are obtainable from the available data can easily be overlooked. Such issues of parameter identifiability have important ramifications for both the predictive power of a model, and the mechanistic insight that can be obtained. Identifiability analysis is well-established for deterministic, ordinary differential equation (ODE) models, but there are no commonly adopted methods for analysing identifiability in stochastic models. We provide an accessible introduction to identifiability analysis and demonstrate how existing ideas for analysis of ODE models can be applied to stochastic differential equation (SDE) models through four practical case studies. To assess structural identifiability , we study ODEs that describe the statistical moments of the stochastic process using open-source software tools. Using practically motivated synthetic data and Markov chain Monte Carlo methods, we assess parameter identifiability in the context of available data. Our analysis shows that SDE models can often extract more information about parameters than deterministic descriptions. All code used to perform the analysis is available on Github .
New travelling wave solutions of the Porous–Fisher model with a moving boundary
Publisher: IOP Publishing
Date: 07-02-2020
Curvature dependences of wave propagation in reaction–diffusion models
Publisher: The Royal Society
Date: 12-2022
Abstract: Reaction–diffusion waves in multiple spatial dimensions advance at a rate that strongly depends on the curvature of the wavefronts. These waves have important applications in many physical, ecological and biological systems. In this work, we analyse curvature dependences of travelling fronts in a single reaction–diffusion equation with general reaction term. We derive an exact, non-perturbative curvature dependence of the speed of travelling fronts that arises from transverse diffusion occurring parallel to the wavefront. Inward-propagating waves are characterized by three phases: an establishment phase dominated by initial and boundary conditions, a travelling-wave-like phase in which normal velocity matches standard results from singular perturbation theory and a dip-filling phase where the collision and interaction of fronts create additional curvature dependences to their progression rate. We analyse these behaviours and additional curvature dependences using a combination of asymptotic analyses and numerical simulations.
Biologically-informed neural networks guide mechanistic modeling from sparse experimental data
Publisher: Public Library of Science (PLoS)
Date: 12-2020
DOI: 10.1371/JOURNAL.PCBI.1008462
Abstract: Biologically-informed neural networks (BINNs), an extension of physics-informed neural networks [1], are introduced and used to discover the underlying dynamics of biological systems from sparse experimental data. In the present work, BINNs are trained in a supervised learning framework to approximate in vitro cell biology assay experiments while respecting a generalized form of the governing reaction-diffusion partial differential equation (PDE). By allowing the diffusion and reaction terms to be multilayer perceptrons (MLPs), the nonlinear forms of these terms can be learned while simultaneously converging to the solution of the governing PDE. Further, the trained MLPs are used to guide the selection of biologically interpretable mechanistic forms of the PDE terms which provides new insights into the biological and physical mechanisms that govern the dynamics of the observed system. The method is evaluated on sparse real-world data from wound healing assays with varying initial cell densities [2].
Mathematical Model of Tumour Spheroid Experiments with Real-Time Cell Cycle Imaging
Publisher: Springer Science and Business Media LLC
Date: 20-03-2021
Discrete and Continuum Approximations for Collective Cell Migration in a Scratch Assay with Cell Size Dynamics
Publisher: Springer Science and Business Media LLC
Date: 25-01-2018
DOI: 10.1007/S11538-018-0398-2
Abstract: Scratch assays are routinely used to study the collective spreading of cell populations. In general, the rate at which a population of cells spreads is driven by the combined effects of cell migration and proliferation. To examine the effects of cell migration separately from the effects of cell proliferation, scratch assays are often performed after treating the cells with a drug that inhibits proliferation. Mitomycin-C is a drug that is commonly used to suppress cell proliferation in this context. However, in addition to suppressing cell proliferation, mitomycin-C also causes cells to change size during the experiment, as each cell in the population approximately doubles in size as a result of treatment. Therefore, to describe a scratch assay that incorporates the effects of cell-to-cell crowding, cell-to-cell adhesion, and dynamic changes in cell size, we present a new stochastic model that incorporates these mechanisms. Our agent-based stochastic model takes the form of a system of Langevin equations that is the system of stochastic differential equations governing the evolution of the population of agents. We incorporate a time-dependent interaction force that is used to mimic the dynamic increase in size of the agents. To provide a mathematical description of the average behaviour of the stochastic model we present continuum limit descriptions using both a standard mean-field approximation and a more sophisticated moment dynamics approximation that accounts for the density of agents and density of pairs of agents in the stochastic model. Comparing the accuracy of the two continuum descriptions for a typical scratch assay geometry shows that the incorporation of agent growth in the system is associated with a decrease in accuracy of the standard mean-field description. In contrast, the moment dynamics description provides a more accurate prediction of the evolution of the scratch assay when the increase in size of in idual agents is included in the model.
An analytical solution for diffusion and nonlinear uptake of oxygen in a spherical cell
Publisher: Elsevier BV
Date: 07-2012
Travelling wave analysis of cellular invasion into surrounding tissues
Publisher: Elsevier BV
Date: 12-2200
Simplified method for including spatial correlations in mean-field approximations
Publisher: American Physical Society (APS)
Date: 10-06-2013
Quantitative analysis of tumour spheroid structure
Publisher: Cold Spring Harbor Laboratory
Date: 06-08-2021
DOI: 10.1101/2021.08.05.455334
Abstract: Tumour spheroids are common in vitro experimental models of avascular tumour growth. Compared with traditional two-dimensional culture, tumour spheroids more closely mimic the avascular tumour microenvironment where spatial differences in nutrient availability strongly influence growth. We show that spheroids initiated using significantly different numbers of cells grow to similar limiting sizes, suggesting that avascular tumours have a limiting structure in agreement with untested predictions of classical mathematical models of tumour spheroids. We develop a novel mathematical and statistical framework to study the structure of tumour spheroids seeded from cells transduced with fluorescent cell cycle indicators, enabling us to discriminate between arrested and cycling cells and identify an arrested region. Our analysis shows that transient spheroid structure is independent of initial spheroid size, and the limiting structure can be independent of seeding density. Standard experimental protocols compare spheroid size as a function of time however, our analysis suggests that comparing spheroid structure as a function of overall size produces results that are relatively insensitive to variability in spheroid size. Our experimental observations are made using two melanoma cell lines, but our modelling framework applies across a wide range of spheroid culture conditions and cell lines.
Small-scale spatial structure influences large-scale invasion rates
Publisher: Cold Spring Harbor Laboratory
Date: 22-10-2019
DOI: 10.1101/814582
Abstract: Local interactions among in idual members of a population can generate intricate small-scale spatial structure, which can strongly influence population dynamics. The two-way interplay between local interactions and population dynamics is well understood in the relatively simple case where the population occupies a fixed domain with a uniform average density. However, the situation where the average population density is spatially varying is less well understood. This situation includes ecologically important scenarios such as species invasions, range shifts, and moving population fronts. Here, we investigate the dynamics of the spatial stochastic logistic model in a scenario where an initially confined population subsequently invades new, previously unoccupied territory. This simple model combines density-independent proliferation with dispersal, and density-dependent mortality via competition with other members of the population. We show that, depending on the spatial scales of dispersal and competition, either a clustered or a regular spatial structure develops over time within the invading population. In the short-range dispersal case, the invasion speed is significantly lower than standard predictions of the mean-field model. We conclude that mean-field models, even when they account for non-local processes such as dispersal and competition, can give misleading predictions for the speed of a moving invasion front.
Theoretical analysis and physical interpretation of temporal truncation errors in operator split algorithms
Publisher: Elsevier BV
Date: 02-2008
Quantifying the roles of random motility and directed motility using advection-diffusion theory for a 3T3 fibroblast cell migration assay stimulated with an electric field
Publisher: Springer Science and Business Media LLC
Date: 17-03-2017
Laboratory and Numerical Investigation of Flow and Transport Near a Seepage-Face Boundary
Publisher: Wiley
Date: 09-2003
DOI: 10.1111/J.1745-6584.2003.TB02407.X
Abstract: Laboratory and numerical modeling investigations were completed to study the unconfined ground water flow and transport processes near a seepage-face boundary. The laboratory observations were made in a radial sand tank and included measurements of the height of the seepage face, flow velocity near the seepage face, travel time distribution of multiple tracer slugs, and streamlines. All the observations were reliably reproduced with a three-dimensional, axi-symmetric, variably saturated ground water flow model. Physical data presented in this work demonstrate and quantify the importance of three-dimensional transport patterns within a seepage-face zone. The results imply that vertically averaged flow models that employ Dupuit approximations might introduce error in the analysis of localized solute transport near a seepage-face boundary. The experimental dataset reported in this work will also be of interest for those who are attempting to validate a numerical algorithm for solving ground water and contaminant discharge patterns near a surface-water boundary.
Cell proliferation and migration explain pore bridging dynamics in 3D printed scaffolds of different pore size
Publisher: Cold Spring Harbor Laboratory
Date: 13-03-2020
DOI: 10.1101/2020.03.12.989053
Abstract: Tissue growth in bioscaffolds is influenced significantly by pore geometry, but how this geometric dependence emerges from dynamic cellular processes such as cell proliferation and cell migration remains poorly understood. Here we investigate the influence of pore size on the time required to bridge pores in thin 3D-printed scaffolds. Experimentally, new tissue infills the pores continually from their perimeter under strong curvature control, which leads the tissue front to round off with time. Despite the varied shapes assumed by the tissue during this evolution, we find that time to bridge a pore simply increases linearly with the overall pore size. To disentangle the biological influence of cell behaviour and the mechanistic influence of geometry in this experimental observation, we propose a simple reaction–diffusion model of tissue growth based on Porous-Fisher invasion of cells into the pores. First, this model provides a good qualitative representation of the evolution of the tissue new tissue in the model grows at an effective rate that depends on the local curvature of the tissue substrate. Second, the model suggests that a linear dependence of bridging time with pore size arises due to geometric reasons alone, not to differences in cell behaviours across pores of different sizes. Our analysis suggests that tissue growth dynamics in these experimental constructs is dominated by mechanistic crowding effects that influence collective cell proliferation and migration processes, and that can be predicted by simple reaction–diffusion models of cells that have robust, consistent behaviours.
How much information can be obtained from tracking the position of the leading edge in a scratch assay?
Publisher: The Royal Society
Date: 06-08-2014
Abstract: Moving cell fronts are an essential feature of wound healing, development and disease. The rate at which a cell front moves is driven, in part, by the cell motility, quantified in terms of the cell diffusivity D , and the cell proliferation rate λ . Scratch assays are a commonly reported procedure used to investigate the motion of cell fronts where an initial cell monolayer is scratched, and the motion of the front is monitored over a short period of time, often less than 24 h. The simplest way of quantifying a scratch assay is to monitor the progression of the leading edge. Use of leading edge data is very convenient because, unlike other methods, it is non-destructive and does not require labelling, tracking or counting in idual cells among the population. In this work, we study short-time leading edge data in a scratch assay using a discrete mathematical model and automated image analysis with the aim of investigating whether such data allow us to reliably identify D and λ . Using a naive calibration approach where we simply scan the relevant region of the ( D , λ ) parameter space, we show that there are many choices of D and λ for which our model produces indistinguishable short-time leading edge data. Therefore, without due care, it is impossible to estimate D and λ from this kind of data. To address this, we present a modified approach accounting for the fact that cell motility occurs over a much shorter time scale than proliferation. Using this information, we ide the duration of the experiment into two periods, and we estimate D using data from the first period, whereas we estimate λ using data from the second period. We confirm the accuracy of our approach using in silico data and a new set of in vitro data, which shows that our method recovers estimates of D and λ that are consistent with previously reported values except that that our approach is fast, inexpensive, non-destructive and avoids the need for cell labelling and cell counting.
A one-dimensional individual-based mechanical model of cell movement in heterogeneous tissues and its coarse-grained approximation
Publisher: The Royal Society
Date: 07-2019
Abstract: Mechanical heterogeneity in biological tissues, in particular stiffness, can be used to distinguish between healthy and diseased states. However, it is often difficult to explore relationships between cellular-level properties and tissue-level outcomes when biological experiments are performed at a single scale only. To overcome this difficulty, we develop a multi-scale mathematical model which provides a clear framework to explore these connections across biological scales. Starting with an in idual-based mechanical model of cell movement, we subsequently derive a novel coarse-grained system of partial differential equations governing the evolution of the cell density due to heterogeneous cellular properties. We demonstrate that solutions of the in idual-based model converge to numerical solutions of the coarse-grained model, for both slowly-varying-in-space and rapidly-varying-in-space cellular properties. We discuss applications of the model, such as determining relative cellular-level properties and an interpretation of data from a breast cancer detection experiment.
Cell proliferation drives neural crest cell invasion of the intestine
Publisher: Elsevier BV
Date: 02-2007
DOI: 10.1016/J.YDBIO.2006.10.017
Abstract: A general mathematical model of cell invasion is developed and validated with an experimental system. The model incorporates two basic cell functions: non-directed (diffusive) motility and proliferation to a carrying capacity limit. The model is used here to investigate cell proliferation and motility differences along the axis of an invasion wave. Mathematical simulations yield surprising and counterintuitive predictions. In this general scenario, cells at the invasive front are proliferative and migrate into previously unoccupied tissues while those behind the front are essentially nonproliferative and do not directly migrate into unoccupied tissues. These differences are not innate to the cells, but are a function of proximity to uninvaded tissue. Therefore, proliferation at the invading front is the critical mechanism driving apparently directed invasion. An appropriate system to experimentally validate these predictions is the directional invasion and colonization of the gut by vagal neural crest cells that establish the enteric nervous system. An assay using gut organ culture with chick-quail grafting is used for this purpose. The experimental results are entirely concordant with the mathematical predictions. We conclude that proliferation at the wavefront is a key mechanism driving the invasive process. This has important implications not just for the neural crest, but for other invasion systems such as epidermal wound healing, carcinoma invasion and other developmental cell migrations.
Unpacking the Allee effect: determining individual-level mechanisms that drive global population dynamics
Publisher: Cold Spring Harbor Laboratory
Date: 18-09-2019
DOI: 10.1101/774000
Abstract: We present a solid theoretical foundation for interpreting the origin of Allee effects by providing the missing link in understanding how local in idual-based mechanisms translate to global population dynamics. Allee effects were originally proposed to describe population dynamics that cannot be explained by exponential and logistic growth models. However, standard methods often calibrate Allee effect models to match observed global population dynamics without providing any mechanistic insight. By introducing a stochastic in idual-based model, with proliferation, death, and motility rates that depend on local density, we present a modelling framework that translates particular global Allee effects to specific in idual-based mechanisms. Using data from ecology and cell biology, we unpack in idual-level mechanisms implicit in an Allee effect model and provide simulation tools for others to repeat this analysis.
Comment on Local accumulation times for source, diffusion, and degradation models in two and three dimensions [J. Chem. Phys. 138, 104121 (2013)]
Publisher: AIP Publishing
Date: 03-07-2013
DOI: 10.1063/1.4811832
Mean-field descriptions of collective migration with strong adhesion
Publisher: American Physical Society (APS)
Date: 29-05-2012
Unpacking the Allee effect: determining individual-level mechanisms that drive global population dynamics
Publisher: The Royal Society
Date: 09-2020
Abstract: We present a solid theoretical foundation for interpreting the origin of Allee effects by providing the missing link in understanding how local in idual-based mechanisms translate to global population dynamics. Allee effects were originally proposed to describe population dynamics that cannot be explained by exponential and logistic growth models. However, standard methods often calibrate Allee effect models to match observed global population dynamics without providing any mechanistic insight. By introducing a stochastic in idual-based model, with proliferation, death and motility rates that depend on local density, we present a modelling framework that translates particular global Allee effects to specific in idual-based mechanisms. Using data from ecology and cell biology, we unpack in idual-level mechanisms implicit in an Allee effect model and provide simulation tools for others to repeat this analysis.
Design and interpretation of cell trajectory assays
Publisher: The Royal Society
Date: 06-11-2013
Abstract: Cell trajectory data are often reported in the experimental cell biology literature to distinguish between different types of cell migration. Unfortunately, there is no accepted protocol for designing or interpreting such experiments and this makes it difficult to quantitatively compare different published datasets and to understand how changes in experimental design influence our ability to interpret different experiments. Here, we use an in idual-based mathematical model to simulate the key features of a cell trajectory experiment. This shows that our ability to correctly interpret trajectory data is extremely sensitive to the geometry and timing of the experiment, the degree of motility bias and the number of experimental replicates. We show that cell trajectory experiments produce data that are most reliable when the experiment is performed in a quasi-one-dimensional geometry with a large number of identically prepared experiments conducted over a relatively short time-interval rather than a few trajectories recorded over particularly long time-intervals.
Modelling the movement of interacting cell populations: A moment dynamics approach
Publisher: Elsevier BV
Date: 04-2015
DOI: 10.1016/J.JTBI.2015.01.025
Abstract: Mathematical models describing the movement of multiple interacting subpopulations are relevant to many biological and ecological processes. Standard mean-field partial differential equation descriptions of these processes suffer from the limitation that they implicitly neglect to incorporate the impact of spatial correlations and clustering. To overcome this, we derive a moment dynamics description of a discrete stochastic process which describes the spreading of distinct interacting subpopulations. In particular, we motivate our model by mimicking the geometry of two typical cell biology experiments. Comparing the performance of the moment dynamics model with a traditional mean-field model confirms that the moment dynamics approach always outperforms the traditional mean-field approach. To provide more general insight we summarise the performance of the moment dynamics model and the traditional mean-field model over a wide range of parameter regimes. These results help distinguish between those situations where spatial correlation effects are sufficiently strong, such that a moment dynamics model is required, from other situations where spatial correlation effects are sufficiently weak, such that a traditional mean-field model is adequate.
Mechanical cell competition in heterogeneous epithelial tissues
Publisher: Cold Spring Harbor Laboratory
Date: 09-12-2019
DOI: 10.1101/869495
Abstract: Mechanical cell competition is important during tissue development, cancer invasion, and tissue ageing. Heterogeneity plays a key role in practical applications since cancer cells can have different cell stiffness and different proliferation rates than normal cells. To study this phenomenon, we propose a one-dimensional mechanical model of heterogeneous epithelial tissue dynamics that includes cell-length-dependent proliferation and death mechanisms. Proliferation and death are incorporated into the discrete model stochastically and arise as source/sink terms in the corresponding continuum model that we derive. Using the new discrete model and continuum description, we explore several applications including the evolution of homogeneous tissues experiencing proliferation and death, and competition in a heterogeneous setting with a cancerous tissue competing for space with an adjacent normal tissue. This framework allows us to postulate new mechanisms that explain the ability of cancer cells to outcompete healthy cells through mechanical differences rather than by having some intrinsic proliferative advantage. We advise when the continuum model is beneficial and demonstrate why naively adding source/sink terms to a continuum model without considering the underlying discrete model may lead to incorrect results.
Model-based data analysis of tissue growth in thin 3D printed scaffolds
Publisher: Cold Spring Harbor Laboratory
Date: 25-03-2021
DOI: 10.1101/2021.03.25.436898
Abstract: Tissue growth in three-dimensional (3D) printed scaffolds enables exploration and control of cell behaviour in biologically realistic geometries. Cell proliferation and migration in these experiments have yet to be explicitly characterised, limiting the ability of experimentalists to determine the effects of various experimental conditions, such as scaffold geometry, on cell behaviour. We consider tissue growth by osteoblastic cells in melt electro-written scaffolds that comprise thin square pores with sizes that we deliberately vary. We collect highly detailed temporal measurements of the average cell density, tissue coverage, and tissue geometry. To quantify tissue growth in terms of the underlying cell proliferation and migration processes, we introduce and calibrate a mechanistic mathematical model based on the Porous-Fisher reaction-diffusion equation. Parameter estimates and uncertainty quantification through profile likelihood analysis reveal consistency in the rate of cell proliferation and steady-state cell density between pore sizes. This analysis also serves as an important model verification tool: while the use of reaction-diffusion models in biology is widespread, the appropriateness of these models to describe tissue growth in 3D scaffolds has yet to be explored. We find that the Porous-Fisher model is able to capture features relating to the cell density and tissue coverage, but is not able to capture geometric features relating to the circularity of the tissue interface. Our analysis identifies two distinct stages of tissue growth, suggests several areas for model refinement, and provides guidance for future experimental work that explores tissue growth in 3D printed scaffolds. Advances in 3D printing technology have led to cell culture experiments that realistically capture natural biological environments. Despite the necessity of quantifying cell behaviour with parameters that can be compared between experiments, many existing mathematical models of tissue growth in these experiments neglect information relating to population size. We consider tissue growth by cells on 3D printed scaffolds that comprise square pores of various sizes in this work. We apply a relatively simple mathematical model based on the Porous-Fisher reaction-diffusion equation to interpret highly detailed measurements relating to both the cell density and the quantity of tissue deposited. We analyse the efficacy of such a model in capturing cell behaviour seen in the experiments and quantify cell behaviour in terms of parameters that carry a biologically meaningful interpretation. Our analysis identifies important areas for model refinement and provides guidance for future data-collection and experimentation that explores tissue growth in 3D printed scaffolds.
Distinguishing between mechanisms of cell aggregation using pair-correlation functions
Publisher: Elsevier BV
Date: 07-2014
DOI: 10.1016/J.JTBI.2014.02.033
Abstract: Many cell types form clumps or aggregates when cultured in vitro through a variety of mechanisms including rapid cell proliferation, chemotaxis, or direct cell-to-cell contact. In this paper we develop an agent-based model to explore the formation of aggregates in cultures where cells are initially distributed uniformly, at random, on a two-dimensional substrate. Our model includes unbiased random cell motion, together with two mechanisms which can produce cell aggregates: (i) rapid cell proliferation and (ii) a biased cell motility mechanism where cells can sense other cells within a finite range, and will tend to move towards areas with higher numbers of cells. We then introduce a pair-correlation function which allows us to quantify aspects of the spatial patterns produced by our agent-based model. In particular, these pair-correlation functions are able to detect differences between domains populated uniformly at random (i.e. at the exclusion complete spatial randomness (ECSR) state) and those where the proliferation and biased motion rules have been employed - even when such differences are not obvious to the naked eye. The pair-correlation function can also detect the emergence of a characteristic inter-aggregate distance which occurs when the biased motion mechanism is dominant, and is not observed when cell proliferation is the main mechanism of aggregate formation. This suggests that applying the pair-correlation function to experimental images of cell aggregates may provide information about the mechanism associated with observed aggregates. As a proof of concept, we perform such analysis for images of cancer cell aggregates, which are known to be associated with rapid proliferation. The results of our analysis are consistent with the predictions of the proliferation-based simulations, which supports the potential usefulness of pair correlation functions for providing insight into the mechanisms of aggregate formation.
Designing and interpreting 4D tumour spheroid experiments
Publisher: Springer Science and Business Media LLC
Date: 24-01-2022
DOI: 10.1038/S42003-022-03018-3
Abstract: Tumour spheroid experiments are routinely used to study cancer progression and treatment. Various and inconsistent experimental designs are used, leading to challenges in interpretation and reproducibility. Using multiple experimental designs, live-dead cell staining, and real-time cell cycle imaging, we measure necrotic and proliferation-inhibited regions in over 1000 4D tumour spheroids (3D space plus cell cycle status). By intentionally varying the initial spheroid size and temporal s ling frequencies across multiple cell lines, we collect an abundance of measurements of internal spheroid structure. These data are difficult to compare and interpret. However, using an objective mathematical modelling framework and statistical identifiability analysis we quantitatively compare experimental designs and identify design choices that produce reliable biological insight. Measurements of internal spheroid structure provide the most insight, whereas varying initial spheroid size and temporal measurement frequency is less important. Our general framework applies to spheroids grown in different conditions and with different cell types.
Velocity-jump processes with proliferation
Publisher: IOP Publishing
Date: 05-12-2013
Exact sharp-fronted travelling wave solutions of the Fisher–KPP equation
Publisher: Elsevier BV
Date: 04-2021
Simulation and inference algorithms for stochastic biochemical reaction networks: from basic concepts to state-of-the-art
Publisher: The Royal Society
Date: 02-2019
Abstract: Stochasticity is a key characteristic of intracellular processes such as gene regulation and chemical signalling. Therefore, characterizing stochastic effects in biochemical systems is essential to understand the complex dynamics of living things. Mathematical idealizations of biochemically reacting systems must be able to capture stochastic phenomena. While robust theory exists to describe such stochastic models, the computational challenges in exploring these models can be a significant burden in practice since realistic models are analytically intractable. Determining the expected behaviour and variability of a stochastic biochemical reaction network requires many probabilistic simulations of its evolution. Using a biochemical reaction network model to assist in the interpretation of time-course data from a biological experiment is an even greater challenge due to the intractability of the likelihood function for determining observation probabilities. These computational challenges have been subjects of active research for over four decades. In this review, we present an accessible discussion of the major historical developments and state-of-the-art computational techniques relevant to simulation and inference problems for stochastic biochemical reaction network models. Detailed algorithms for particularly important methods are described and complemented with Matlab ® implementations. As a result, this review provides a practical and accessible introduction to computational methods for stochastic models within the life sciences community.
Models of collective cell motion for cell populations with different aspect ratio: Diffusion, proliferation and travelling waves
Publisher: Elsevier BV
Date: 07-2012
Learning differential equation models from stochastic agent-based model simulations
Publisher: The Royal Society
Date: 03-2021
Abstract: Agent-based models provide a flexible framework that is frequently used for modelling many biological systems, including cell migration, molecular dynamics, ecology and epidemiology. Analysis of the model dynamics can be challenging due to their inherent stochasticity and heavy computational requirements. Common approaches to the analysis of agent-based models include extensive Monte Carlo simulation of the model or the derivation of coarse-grained differential equation models to predict the expected or averaged output from the agent-based model. Both of these approaches have limitations, however, as extensive computation of complex agent-based models may be infeasible, and coarse-grained differential equation models can fail to accurately describe model dynamics in certain parameter regimes. We propose that methods from the equation learning field provide a promising, novel and unifying approach for agent-based model analysis. Equation learning is a recent field of research from data science that aims to infer differential equation models directly from data. We use this tutorial to review how methods from equation learning can be used to learn differential equation models from agent-based model simulations. We demonstrate that this framework is easy to use, requires few model simulations, and accurately predicts model dynamics in parameter regions where coarse-grained differential equation models fail to do so. We highlight these advantages through several case studies involving two agent-based models that are broadly applicable to biological phenomena: a birth–death–migration model commonly used to explore cell biology experiments and a susceptible–infected–recovered model of infectious disease spread.
How osteons form: A quantitative hypothesis-testing analysis of cortical pore filling and wall asymmetry
Publisher: Cold Spring Harbor Laboratory
Date: 27-10-2023
Comparison of finite difference and finite element solutions to the variably saturated flow equation
Publisher: Elsevier BV
Date: 2003
Designing combination therapies using multiple optimal controls
Publisher: Elsevier BV
Date: 07-2020
Interpreting how nonlinear diffusion affects the fate of bistable populations using a discrete modelling framework
Publisher: The Royal Society
Date: 06-2022
Abstract: Understanding whether a population will survive or become extinct is a central question in population biology. One way of exploring this question is to study population dynamics using reaction–diffusion equations, where migration is usually represented as a linear diffusion term, and birth–death is represented with a nonlinear source term. While linear diffusion is most commonly employed to study migration, there are several limitations of this approach, such as the inability of linear diffusion-based models to predict a well-defined population front. One way to overcome this is to generalize the constant diffusivity, D , to a nonlinear diffusivity function D ( C ) , where C 0 is the population density. While the choice of D ( C ) affects long-term survival or extinction of a bistable population, working solely in a continuum framework makes it difficult to understand how the choice of D ( C ) affects survival or extinction. We address this question by working with a discrete simulation model that is easy to interpret. This approach provides clear insight into how the choice of D ( C ) either encourages or suppresses population extinction relative to the classical linear diffusion model.
Sensitivity of Edge Detection Methods for Quantifying Cell Migration Assays
Publisher: Public Library of Science (PLoS)
Date: 24-06-2013
Assessing the role of spatial correlations during collective cell spreading
Publisher: Springer Science and Business Media LLC
Date: 16-07-2014
DOI: 10.1038/SREP05713
Abstract: Spreading cell fronts are essential features of development, repair and disease processes. Many mathematical models used to describe the motion of cell fronts, such as Fisher's equation, invoke a mean–field assumption which implies that there is no spatial structure, such as cell clustering, present. Here, we examine the presence of spatial structure using a combination of in vitro circular barrier assays, discrete random walk simulations and pair correlation functions. In particular, we analyse discrete simulation data using pair correlation functions to show that spatial structure can form in a spreading population of cells either through sufficiently strong cell–to–cell adhesion or sufficiently rapid cell proliferation. We analyse images from a circular barrier assay describing the spreading of a population of MM127 melanoma cells using the same pair correlation functions. Our results indicate that the spreading melanoma cell populations remain very close to spatially uniform, suggesting that the strength of cell–to–cell adhesion and the rate of cell proliferation are both sufficiently small so as not to induce any spatial patterning in the spreading populations.
Velocity-jump models with crowding effects
Publisher: American Physical Society (APS)
Date: 28-12-2011
Multiple types of data are required to identify the mechanisms influencing the spatial expansion of melanoma cell colonies
Publisher: Springer Science and Business Media LLC
Date: 12-2013
The impact of experimental design choices on parameter inference for models of growing cell colonies
Publisher: The Royal Society
Date: 08-2018
DOI: 10.1098/RSOS.180384
Abstract: To better understand development, repair and disease progression, it is useful to quantify the behaviour of proliferative and motile cell populations as they grow and expand to fill their local environment. Inferring parameters associated with mechanistic models of cell colony growth using quantitative data collected from carefully designed experiments provides a natural means to elucidate the relative contributions of various processes to the growth of the colony. In this work, we explore how experimental design impacts our ability to infer parameters for simple models of the growth of proliferative and motile cell populations. We adopt a Bayesian approach, which allows us to characterize the uncertainty associated with estimates of the model parameters. Our results suggest that experimental designs that incorporate initial spatial heterogeneities in cell positions facilitate parameter inference without the requirement of cell tracking, while designs that involve uniform initial placement of cells require cell tracking for accurate parameter inference. As cell tracking is an experimental bottleneck in many studies of this type, our recommendations for experimental design provide for significant potential time and cost savings in the analysis of cell colony growth.
New homogenization approaches for stochastic transport through heterogeneous media
Publisher: AIP Publishing
Date: 23-01-2019
DOI: 10.1063/1.5067290
Abstract: The diffusion of molecules in complex intracellular environments can be strongly influenced by spatial heterogeneity and stochasticity. A key challenge when modelling such processes using stochastic random walk frameworks is that negative jump coefficients can arise when transport operators are discretized on heterogeneous domains. Often this is dealt with through homogenization approximations by replacing the heterogeneous medium with an effective homogeneous medium. In this work, we present a new class of homogenization approximations by considering a stochastic diffusive transport model on a one-dimensional domain containing an arbitrary number of layers with different jump rates. We derive closed form solutions for the kth moment of particle lifetime, carefully explaining how to deal with the internal interfaces between layers. These general tools allow us to derive simple formulae for the effective transport coefficients, leading to significant generalisations of previous homogenization approaches. Here, we find that different jump rates in the layers give rise to a net bias, leading to a non-zero advection, for the entire homogenized system. Ex le calculations show that our generalized approach can lead to very different outcomes than traditional approaches, thereby having the potential to significantly affect simulation studies that use homogenization approximations.
Population dynamics with spatial structure and an Allee effect
Publisher: The Royal Society
Date: 10-2020
Abstract: Population dynamics including a strong Allee effect describe the situation where long-term population survival or extinction depends on the initial population density. A simple mathematical model of an Allee effect is one where initial densities below the threshold lead to extinction, whereas initial densities above the threshold lead to survival. Mean-field models of population dynamics neglect spatial structure that can arise through short-range interactions, such as competition and dispersal. The influence of non-mean-field effects has not been studied in the presence of an Allee effect. To address this, we develop an in idual-based model that incorporates both short-range interactions and an Allee effect. To explore the role of spatial structure we derive a mathematically tractable continuum approximation of the IBM in terms of the dynamics of spatial moments. In the limit of long-range interactions where the mean-field approximation holds, our modelling framework recovers the mean-field Allee threshold. We show that the Allee threshold is sensitive to spatial structure neglected by mean-field models. For ex le, there are cases where the mean-field model predicts extinction but the population actually survives. Through simulations we show that our new spatial moment dynamics model accurately captures the modified Allee threshold in the presence of spatial structure.
Coalescence of interacting cell populations
Publisher: Elsevier BV
Date: 08-2007
DOI: 10.1016/J.JTBI.2007.02.020
Abstract: We analyse the coalescence of invasive cell populations by studying both the temporal and steady behaviour of a system of coupled reaction-diffusion equations. This problem is relevant to recent experimental observations of the dynamics of opposingly directed invasion waves of cells. Two cell types, u and v, are considered with the cell motility governed by linear or nonlinear diffusion. The cells proliferate logistically so that the long-term total cell density, u+v approaches a carrying capacity. The steady-state solutions for u and v are denoted u(s) and v(s). The steady solutions are spatially invariant and satisfy u(s)+v(s)=1. However, this expression is underdetermined so the relative proportion of each cell type u(s) and v(s) cannot be determined a priori. Various properties of this model are studied, such as how the relative proportion of u(s) and v(s) depends on the relative motility and relative proliferation rates. The model is analysed using a combination of numerical simulations and a comparison principle. This investigation unearths some novel outcomes regarding the role of overcrowding and cell death in this type of cell migration assay. These observations have relevance to experimental design and interpretation regarding the identification and parameterisation of mechanisms involved in cell invasion.
A new and accurate continuum description of moving fronts
Publisher: IOP Publishing
Date: 06-03-2017
Quantifying the roles of cell motility and cell proliferation in a circular barrier assay
Publisher: The Royal Society
Date: 06-05-2013
Abstract: Moving fronts of cells are essential features of embryonic development, wound repair and cancer metastasis. This paper describes a set of experiments to investigate the roles of random motility and proliferation in driving the spread of an initially confined cell population. The experiments include an analysis of cell spreading when proliferation was inhibited. Our data have been analysed using two mathematical models: a lattice-based discrete model and a related continuum partial differential equation model. We obtain independent estimates of the random motility parameter, D , and the intrinsic proliferation rate, λ , and we confirm that these estimates lead to accurate modelling predictions of the position of the leading edge of the moving front as well as the evolution of the cell density profiles. Previous work suggests that systems with a high λ / D ratio will be characterized by steep fronts, whereas systems with a low λ / D ratio will lead to shallow diffuse fronts and this is confirmed in the present study. Our results provide evidence that continuum models, based on the Fisher–Kolmogorov equation, are a reliable platform upon which we can interpret and predict such experimental observations.
Cell invasion with proliferation mechanisms motivated by time-lapse data
Publisher: Elsevier BV
Date: 09-2010
Population dynamics with threshold effects give rise to a diverse family of Allee effects
Publisher: Cold Spring Harbor Laboratory
Date: 02-04-2020
DOI: 10.1101/2020.04.02.021741
Abstract: The Allee effect describes populations that deviate from logistic growth models and arises in applications including ecology and cell biology. A common justification for incorporating Allee effects into population models is that the population in question has altered growth mechanisms at some critical density, often referred to as a threshold effect . Despite the ubiquitous nature of threshold effects arising in various biological applications, the explicit link between local threshold effects and global Allee effects has not been considered. In this work, we examine a continuum population model that incorporates threshold effects in the local growth mechanisms. We show that this model gives rise to a erse family of Allee effects and we provide a comprehensive analysis of which choices of local growth mechanisms give rise to specific Allee effects. Calibrating this model to a recent set of experimental data describing the growth of a population of cancer cells provides an interpretation of the threshold population density and growth mechanisms associated with the population.
Optimal control of acute myeloid leukaemia
Publisher: Elsevier BV
Date: 06-2019
DOI: 10.1016/J.JTBI.2019.03.006
Abstract: Acute myeloid leukaemia (AML) is a blood cancer affecting haematopoietic stem cells. AML is routinely treated with chemotherapy, and so it is of great interest to develop optimal chemotherapy treatment strategies. In this work, we incorporate an immune response into a stem cell model of AML, since we find that previous models lacking an immune response are inappropriate for deriving optimal control strategies. Using optimal control theory, we produce continuous controls and bang-bang controls, corresponding to a range of objectives and parameter choices. Through ex le calculations, we provide a practical approach to applying optimal control using Pontryagin's Maximum Principle. In particular, we describe and explore factors that have a profound influence on numerical convergence. We find that the convergence behaviour is sensitive to the method of control updating, the nature of the control, and to the relative weighting of terms in the objective function. All codes we use to implement optimal control are made available.
Incorporating spatial correlations into multispecies mean-field models
Publisher: American Physical Society (APS)
Date: 20-11-2013
Analytical Model for Computing Residence Times Near a Pumping Well
Publisher: Wiley
Date: 05-2003
DOI: 10.1111/J.1745-6584.2003.TB02604.X
Abstract: An analytical solution for calculating the residence time of fluid flowing toward a pumping well in an unconfined aquifer has been developed. The analytical solution was derived based on a radial, steady-state, Dupuit-Forchheimer flow model. The resulting integral expression involved computing the imaginary error function, for which a simple series expansion is proposed. The validity of the analytical expression is demonstrated by testing its results against numerical results for an ex le problem. The analytical solution compared favorably with the numerical approximation.
How long does it take for aquifer recharge or aquifer discharge processes to reach steady state?
Publisher: Elsevier BV
Date: 09-2013
Comparing methods for modelling spreading cell fronts
Publisher: Elsevier BV
Date: 07-2014
DOI: 10.1016/J.JTBI.2014.02.023
Abstract: Spreading cell fronts play an essential role in many physiological processes. Classically, models of this process are based on the Fisher-Kolmogorov equation however, such continuum representations are not always suitable as they do not explicitly represent behaviour at the level of in idual cells. Additionally, many models examine only the large time asymptotic behaviour, where a travelling wave front with a constant speed has been established. Many experiments, such as a scratch assay, never display this asymptotic behaviour, and in these cases the transient behaviour must be taken into account. We examine the transient and the asymptotic behaviour of moving cell fronts using techniques that go beyond the continuum approximation via a volume-excluding birth-migration process on a regular one-dimensional lattice. We approximate the averaged discrete results using three methods: (i) mean-field, (ii) pair-wise, and (iii) one-hole approximations. We discuss the performance of these methods, in comparison to the averaged discrete results, for a range of parameter space, examining both the transient and asymptotic behaviours. The one-hole approximation, based on techniques from statistical physics, is not capable of predicting transient behaviour but provides excellent agreement with the asymptotic behaviour of the averaged discrete results, provided that cells are proliferating fast enough relative to their rate of migration. The mean-field and pair-wise approximations give indistinguishable asymptotic results, which agree with the averaged discrete results when cells are migrating much more rapidly than they are proliferating. The pair-wise approximation performs better in the transient region than does the mean-field, despite having the same asymptotic behaviour. Our results show that each approximation only works in specific situations, thus we must be careful to use a suitable approximation for a given system, otherwise inaccurate predictions could be made.
Implementation and acceleration of optimal control for systems biology
Publisher: Cold Spring Harbor Laboratory
Date: 17-03-2021
DOI: 10.1101/2021.03.17.435721
Abstract: Optimal control theory provides insight into complex resource allocation decisions. The forward-backward sweep method (FBSM) is an iterative technique commonly implemented to solve two-point boundary value problems (TPBVPs) arising from the application of Pontryagin’s Maximum Principle (PMP) in optimal control. In this review we discuss the PMP approach to optimal control and the implementation of the FBSM. By conceptualising the FBSM as a fixed point iteration process, we leverage and adapt existing acceleration techniques to improve its rate of convergence. We show that convergence improvement is attainable without prohibitively costly tuning of the acceleration techniques. Further, we demonstrate that these methods can induce convergence where the underlying FBSM fails to converge. All code used in this work to implement the FBSM and acceleration techniques is available on GitHub at github.com/Jesse-Sharp/Sharp2021 .
A continuum mathematical model of substrate-mediated tissue growth
Publisher: Springer Science and Business Media LLC
Date: 02-03-2022
DOI: 10.1007/S11538-022-01005-7
Abstract: We consider a continuum mathematical model of biological tissue formation inspired by recent experiments describing thin tissue growth in 3D-printed bioscaffolds. The continuum model, which we call the substrate model , involves a partial differential equation describing the density of tissue, $${\\hat{u}}(\\hat{{\\mathbf {x}}},{\\hat{t}})$$ u ^ ( x ^ , t ^ ) that is coupled to the concentration of an immobile extracellular substrate, $${\\hat{s}}(\\hat{{\\mathbf {x}}},{\\hat{t}})$$ s ^ ( x ^ , t ^ ) . Cell migration is modelled with a nonlinear diffusion term, where the diffusive flux is proportional to $${\\hat{s}}$$ s ^ , while a logistic growth term models cell proliferation. The extracellular substrate $${\\hat{s}}$$ s ^ is produced by cells and undergoes linear decay. Preliminary numerical simulations show that this mathematical model is able to recapitulate key features of recent tissue growth experiments, including the formation of sharp fronts. To provide a deeper understanding of the model we analyse travelling wave solutions of the substrate model, showing that the model supports both sharp-fronted travelling wave solutions that move with a minimum wave speed, $$c = c_{\\mathrm{min}}$$ c = c min , as well as smooth-fronted travelling wave solutions that move with a faster travelling wave speed, $$c c_{\\mathrm{min}}$$ c c min . We provide a geometric interpretation that explains the difference between smooth and sharp-fronted travelling wave solutions that is based on a slow manifold reduction of the desingularised three-dimensional phase space. In addition, we also develop and test a series of useful approximations that describe the shape of the travelling wave solutions in various limits. These approximations apply to both the sharp-fronted and smooth-fronted travelling wave solutions. Software to implement all calculations is available at GitHub .
A stochastic mathematical model of 4D tumour spheroids with real-time fluorescent cell cycle labelling
Publisher: Cold Spring Harbor Laboratory
Date: 29-11-2021
DOI: 10.1101/2021.11.28.470300
Abstract: In vitro tumour spheroid experiments have been used to study avascular tumour growth and drug design for the last 50 years. Unlike simpler two-dimensional cell cultures, tumour spheroids exhibit heterogeneity within the growing population of cells that is thought to be related to spatial and temporal differences in nutrient availability. The recent development of real-time fluorescent cell cycle imaging allows us to identify the position and cell cycle status of in idual cells within the growing population, giving rise to the notion of a four-dimensional (4D) tumour spheroid. In this work we develop the first stochastic in idual-based model (IBM) of a 4D tumour spheroid and show that IBM simulation data qualitatively and quantitatively compare very well with experimental data from a suite of 4D tumour spheroid experiments performed with a primary human melanoma cell line. The IBM provides quantitative information about nutrient availability within the spheroid, which is important because it is very difficult to measure these data in standard tumour spheroid experiments. Software required to implement the IBM is available on GitHub, github.com/ProfMJSimpson/4DFUCCI .
Modelling count data with partial differential equation models in biology
Publisher: Cold Spring Harbor Laboratory
Date: 12-09-2023
Correcting mean-field approximations for birth-death-movement processes
Publisher: American Physical Society (APS)
Date: 05-10-2010
Crowded transport within networked representations of complex geometries
Publisher: Springer Science and Business Media LLC
Date: 25-10-2021
DOI: 10.1038/S42005-021-00732-Y
Abstract: Transport in crowded, complex environments occurs across many spatial scales. Geometric restrictions can hinder the motion of in iduals and, combined with crowding, can have drastic effects on global transport phenomena. However, in general, the interplay between crowding and geometry in complex real-life environments is poorly understood. Existing analytical methodologies are not always readily extendable to heterogeneous environments and, in these situations, predictions of crowded transport behaviour rely on computationally intensive mesh-based approaches. Here, we take a different approach based on networked representations of complex environments in order to provide an efficient framework to explore the interactions between environments’ geometry and crowding. We demonstrate how this framework can be used to extract detailed information both at the level of the in idual as well as of the whole population, identify the environments’ topological features that enable accurate prediction of transport phenomena, and provide insights into the design of optimal environments.
Invading and receding sharp-fronted travelling waves
Publisher: Springer Science and Business Media LLC
Date: 21-02-2021
Migration of breast cancer cells: Understanding the roles of volume exclusion and cell-to-cell adhesion
Publisher: American Physical Society (APS)
Date: 10-2010
Exits in order: How crowding affects particle lifetimes
Publisher: AIP Publishing
Date: 27-06-2016
DOI: 10.1063/1.4953913
Abstract: Diffusive processes are often represented using stochastic random walk frameworks. The amount of time taken for an in idual in a random walk to intersect with an absorbing boundary is a fundamental property that is often referred to as the particle lifetime, or the first passage time. The mean lifetime of particles in a random walk model of diffusion is related to the amount of time required for the diffusive process to reach a steady state. Mathematical analysis describing the mean lifetime of particles in a standard model of diffusion without crowding is well known. However, the lifetime of agents in a random walk with crowding has received much less attention. Since many applications of diffusion in biology and biophysics include crowding effects, here we study a discrete model of diffusion that incorporates crowding. Using simulations, we show that crowding has a dramatic effect on agent lifetimes, and we derive an approximate expression for the mean agent lifetime that includes crowding effects. Our expression matches simulation results very well, and highlights the importance of crowding effects that are sometimes overlooked.
Models of collective cell behaviour with crowding effects: Comparing lattice-based and lattice-free approaches
Publisher: The Royal Society
Date: 13-06-2012
Abstract: In idual-based models describing the migration and proliferation of a population of cells frequently restrict the cells to a predefined lattice. An implicit assumption of this type of lattice-based model is that a proliferative population will always eventually fill the lattice. Here, we develop a new lattice-free in idual-based model that incorporates cell-to-cell crowding effects. We also derive approximate mean-field descriptions for the lattice-free model in two special cases motivated by commonly used experimental set-ups. Lattice-free simulation results are compared with these mean-field descriptions and with a corresponding lattice-based model. Data from a proliferation experiment are used to estimate the parameters for the new model, including the cell proliferation rate, showing that the model fits the data well. An important aspect of the lattice-free model is that the confluent cell density is not predefined, as with lattice-based models, but an emergent model property. As a consequence of the more realistic, irregular configuration of cells in the lattice-free model, the population growth rate is much slower at high cell densities and the population cannot reach the same confluent density as an equivalent lattice-based model.
Reply to comment on computing residence times for flow towards a pumping well: Nomograph solution and validity of the small draw-down approximation, Technical Note published in Hydrogeology Journal (2005) 13:889-894
Publisher: Springer Science and Business Media LLC
Date: 28-11-2007
Extended logistic growth model for heterogeneous populations
Publisher: Elsevier BV
Date: 05-2018
DOI: 10.1016/J.JTBI.2018.02.027
Abstract: Cell proliferation is the most important cellular-level mechanism responsible for regulating cell population dynamics in living tissues. Modern experimental procedures show that the proliferation rates of in idual cells can vary significantly within the same cell line. However, in the mathematical biology literature, cell proliferation is typically modelled using a classical logistic equation which neglects variations in the proliferation rate. In this work, we consider a discrete mathematical model of cell migration and cell proliferation, modulated by volume exclusion (crowding) effects, with variable rates of proliferation across the total population. We refer to this variability as heterogeneity. Constructing the continuum limit of the discrete model leads to a generalisation of the classical logistic growth model. Comparing numerical solutions of the model to averaged data from discrete simulations shows that the new model captures the key features of the discrete process. Applying the extended logistic model to simulate a proliferation assay using rates from recent experimental literature shows that neglecting the role of heterogeneity can, at times, lead to misleading results.
Exact solutions of linear reaction-diffusion processes on a uniformly growing domain: Criteria for successful colonization
Publisher: Public Library of Science (PLoS)
Date: 18-02-2015
Exact calculations of survival probability for diffusion on growing lines, disks, and spheres: The role of dimension
Publisher: AIP Publishing
Date: 04-09-2015
DOI: 10.1063/1.4929993
Abstract: Unlike standard applications of transport theory, the transport of molecules and cells during embryonic development often takes place within growing multidimensional tissues. In this work, we consider a model of diffusion on uniformly growing lines, disks, and spheres. An exact solution of the partial differential equation governing the diffusion of a population of in iduals on the growing domain is derived. Using this solution, we study the survival probability, S(t). For the standard non-growing case with an absorbing boundary, we observe that S(t) decays to zero in the long time limit. In contrast, when the domain grows linearly or exponentially with time, we show that S(t) decays to a constant, positive value, indicating that a proportion of the diffusing substance remains on the growing domain indefinitely. Comparing S(t) for diffusion on lines, disks, and spheres indicates that there are minimal differences in S(t) in the limit of zero growth and minimal differences in S(t) in the limit of fast growth. In contrast, for intermediate growth rates, we observe modest differences in S(t) between different geometries. These differences can be quantified by evaluating the exact expressions derived and presented here.
Editorial: Infectious disease modelling
Publisher: Cambridge University Press (CUP)
Date: 10-2012
An analytical framework for quantifying aquifer response time scales associated with transient boundary conditions
Publisher: Elsevier BV
Date: 11-2014
Multilevel rejection sampling for approximate Bayesian computation
Publisher: Elsevier BV
Date: 08-2018
Designing and interpreting 4D tumour spheroid experiments
Publisher: Cold Spring Harbor Laboratory
Date: 19-08-2021
DOI: 10.1101/2021.08.18.456910
Abstract: Tumour spheroid experiments are routinely used to study cancer progression and treatment. Various and inconsistent experimental designs are used, leading to challenges in interpretation and reproducibility. Using multiple experimental designs, live-dead cell staining, and real-time cell cycle imaging, we measure necrotic and proliferation-inhibited regions in over 1000 4D tumour spheroids (3D space plus cell cycle status). By intentionally varying the initial spheroid size and temporal s ling frequencies across multiple cell lines, we collect an abundance of measurements of internal spheroid structure. These data are difficult to compare and interpret. However, using an objective mathematical modelling framework and statistical identifiability analysis we quantitatively compare experimental designs and identify design choices that produce reliable biological insight. Measurements of internal spheroid structure provide the most insight, whereas varying initial spheroid size and temporal measurement frequency is less important. Our general framework applies to spheroids grown in different conditions and with different cell types.
Computationally efficient framework for diagnosing, understanding, and predicting biphasic population growth
Publisher: Cold Spring Harbor Laboratory
Date: 29-07-2022
DOI: 10.1101/2022.07.27.501797
Abstract: Throughout the life sciences, biological populations undergo multiple phases of growth, often referred to as biphasic growth for the commonly-encountered situation involving two phases. Biphasic population growth occurs over a massive range of spatial and temporal scales, ranging from microscopic growth of tumours over several days, to decades-long re-growth of corals in coral reefs that can extend for hundreds of kilometres. Different mathematical models and statistical methods are used to diagnose, understand, and predict biphasic growth. Common approaches can lead to inaccurate predictions of future growth that may result in inappropriate management and intervention strategies being implemented. Here we develop a very general computationally efficient framework, based on profile likelihood analysis, for diagnosing, understanding, and predicting biphasic population growth. The two key components of the framework are: (i) an efficient method to form approximate confidence intervals for the change point of the growth dynamics and model parameters and, (ii) parameter-wise profile predictions that systematically reveal the influence of in idual model parameters on predictions. To illustrate our framework we explore real-world case studies across the life sciences.
Profile likelihood-based parameter and predictive interval analysis guides model choice for ecological population dynamics
Publisher: Cold Spring Harbor Laboratory
Date: 06-09-2022
DOI: 10.1101/2022.09.05.506701
Abstract: Calibrating mathematical models to describe ecological data provides important insight via parameter estimation that is not possible from analysing data alone. When we undertake a mathematical modelling study of ecological or biological data, we must deal with the trade-off between data availability and model complexity. Dealing with the nexus between data availability and model complexity is an ongoing challenge in mathematical modelling, particularly in mathematical biology and mathematical ecology where data collection is often not standardised, and more broad questions about model selection remain relatively open. Therefore, choosing an appropriate model almost always requires case-by-case consideration. In this work we present a straightforward approach to quantitatively explore this trade-off using a case study exploring mathematical models of coral reef regrowth after some ecological disturbance, such as damage caused by a tropical cyclone. In particular, we compare a simple single species ordinary differential equation (ODE) model approach with a more complicated two-species coupled ODE model. Univariate profile likelihood analysis suggests that the both models are practically identifiable. To provide additional insight we construct and compare approximate prediction intervals using a new parameter-wise prediction approximation, confirming both the simple and complex models perform similarly with regard to making predictions. Our approximate parameter-wise prediction interval analysis provides explicit information about how each parameter affects the predictions of each model. Comparing our approximate prediction intervals with a more rigorous and computationally expensive evaluation of the full likelihood shows that the new approximations are reasonable in this case. All algorithms and software to support this work are freely available as jupyter notebooks on GitHub so that they can be adapted to deal with any other ODE-based models.
Small-scale spatial structure affects predator-prey dynamics and coexistence
Publisher: Springer Science and Business Media LLC
Date: 29-06-2020
Pathlines in exclusion processes
Publisher: American Physical Society (APS)
Date: 27-03-2009
Computationally efficient mechanism discovery for cell invasion with uncertainty quantification
Publisher: Public Library of Science (PLoS)
Date: 16-11-2022
DOI: 10.1371/JOURNAL.PCBI.1010599
Abstract: Parameter estimation for mathematical models of biological processes is often difficult and depends significantly on the quality and quantity of available data. We introduce an efficient framework using Gaussian processes to discover mechanisms underlying delay, migration, and proliferation in a cell invasion experiment. Gaussian processes are leveraged with bootstrapping to provide uncertainty quantification for the mechanisms that drive the invasion process. Our framework is efficient, parallelisable, and can be applied to other biological problems. We illustrate our methods using a canonical scratch assay experiment, demonstrating how simply we can explore different functional forms and develop and test hypotheses about underlying mechanisms, such as whether delay is present. All code and data to reproduce this work are available at github.com/DanielVandH/EquationLearning.jl .
A free boundary model of epithelial dynamics
Publisher: Elsevier BV
Date: 11-2019
Bayesian uncertainty quantification for data-driven equation learning
Publisher: The Royal Society
Date: 10-2021
Abstract: Equation learning aims to infer differential equation models from data. While a number of studies have shown that differential equation models can be successfully identified when the data are sufficiently detailed and corrupted with relatively small amounts of noise, the relationship between observation noise and uncertainty in the learned differential equation models remains unexplored. We demonstrate that for noisy datasets there exists great variation in both the structure of the learned differential equation models and their parameter values. We explore how to exploit multiple datasets to quantify uncertainty in the learned models, and at the same time draw mechanistic conclusions about the target differential equations. We showcase our results using simulation data from a relatively straightforward agent-based model (ABM) which has a well-characterized partial differential equation description that provides highly accurate predictions of averaged ABM behaviours in relevant regions of parameter space. Our approach combines equation learning methods with Bayesian inference approaches so that a quantification of uncertainty can be given by the posterior parameter distribution of the learned model.
Interpreting scratch assays using pair density dynamics and approximate Bayesian computation
Publisher: The Royal Society
Date: 09-2014
DOI: 10.1098/RSOB.140097
Abstract: Quantifying the impact of biochemical compounds on collective cell spreading is an essential element of drug design, with various applications including developing treatments for chronic wounds and cancer. Scratch assays are a technically simple and inexpensive method used to study collective cell spreading however, most previous interpretations of scratch assays are qualitative and do not provide estimates of the cell diffusivity, D , or the cell proliferation rate, λ . Estimating D and λ is important for investigating the efficacy of a potential treatment and provides insight into the mechanism through which the potential treatment acts. While a few methods for estimating D and λ have been proposed, these previous methods lead to point estimates of D and λ , and provide no insight into the uncertainty in these estimates. Here, we compare various types of information that can be extracted from images of a scratch assay, and quantify D and λ using discrete computational simulations and approximate Bayesian computation. We show that it is possible to robustly recover estimates of D and λ from synthetic data, as well as a new set of experimental data. For the first time, our approach also provides a method to estimate the uncertainty in our estimates of D and λ . We anticipate that our approach can be generalized to deal with more realistic experimental scenarios in which we are interested in estimating D and λ , as well as additional relevant parameters such as the strength of cell-to-cell adhesion or the strength of cell-to-substrate adhesion.
Population Dynamics with Threshold Effects Give Rise to a Diverse Family of Allee Effects
Publisher: Springer Science and Business Media LLC
Date: 06-2020
DOI: 10.1007/S11538-020-00756-5
Abstract: The Allee effect describes populations that deviate from logistic growth models and arises in applications including ecology and cell biology. A common justification for incorporating Allee effects into population models is that the population in question has altered growth mechanisms at some critical density, often referred to as a threshold effect . Despite the ubiquitous nature of threshold effects arising in various biological applications, the explicit link between local threshold effects and global Allee effects has not been considered. In this work, we examine a continuum population model that incorporates threshold effects in the local growth mechanisms. We show that this model gives rise to a erse family of Allee effects, and we provide a comprehensive analysis of which choices of local growth mechanisms give rise to specific Allee effects. Calibrating this model to a recent set of experimental data describing the growth of a population of cancer cells provides an interpretation of the threshold population density and growth mechanisms associated with the population.
Are in vitro estimates of cell diffusivity and cell proliferation rate sensitive to assay geometry?
Publisher: Elsevier BV
Date: 09-2014
DOI: 10.1016/J.JTBI.2014.04.026
Abstract: Cells respond to various biochemical and physical cues during wound-healing and tumour progression. in vitro assays used to study these processes are typically conducted in one particular geometry and it is unclear how the assay geometry affects the capacity of cell populations to spread, or whether the relevant mechanisms, such as cell motility and cell proliferation, are somehow sensitive to the geometry of the assay. In this work we use a circular barrier assay to characterise the spreading of cell populations in two different geometries. Assay 1 describes a tumour-like geometry where a cell population spreads outwards into an open space. Assay 2 describes a wound-like geometry where a cell population spreads inwards to close a void. We use a combination of discrete and continuum mathematical models and automated image processing methods to obtain independent estimates of the effective cell diffusivity, D, and the effective cell proliferation rate, λ. Using our parameterised mathematical model we confirm that our estimates of D and λ accurately predict the time-evolution of the location of the leading edge and the cell density profiles for both assay 1 and assay 2. Our work suggests that the effective cell diffusivity is up to 50% lower for assay 2 compared to assay 1, whereas the effective cell proliferation rate is up to 30% lower for assay 2 compared to assay 1.
A practical guide to pseudo-marginal methods for computational inference in systems biology
Publisher: Elsevier BV
Date: 07-2020
Quantifying the role of different surface coatings in experimental models of wound healing
Publisher: Elsevier BV
Date: 07-2020
Survival, extinction, and interface stability in a two-phase moving boundary model of biological invasion
Publisher: Elsevier BV
Date: 12-2023
A free boundary mechanobiological model of epithelial tissues
Publisher: The Royal Society
Date: 11-2020
Abstract: In this study, we couple intracellular signalling and cell-based mechanical properties to develop a novel free boundary mechanobiological model of epithelial tissue dynamics. Mechanobiological coupling is introduced at the cell level in a discrete modelling framework, and new reaction–diffusion equations are derived to describe tissue-level outcomes. The free boundary evolves as a result of the underlying biological mechanisms included in the discrete model. To demonstrate the accuracy of the continuum model, we compare numerical solutions of the discrete and continuum models for two different signalling pathways. First, we study the Rac–Rho pathway where cell- and tissue-level mechanics are directly related to intracellular signalling. Second, we study an activator–inhibitor system which gives rise to spatial and temporal patterning related to Turing patterns. In all cases, the continuum model and free boundary condition accurately reflect the cell-level processes included in the discrete model.
Three-dimensional experiments and individual based simulations show that cell proliferation drives melanoma nest formation in human skin tissue
Publisher: Cold Spring Harbor Laboratory
Date: 08-01-2018
DOI: 10.1101/244582
Abstract: Melanoma can be diagnosed by identifying nests of cells on the skin surface. Understanding the processes that drive nest formation is important as these processes could be potential targets for new cancer drugs. Cell proliferation and cell migration are two potential mechanisms that could conceivably drive melanoma nest formation. However, it is unclear which one of these two putative mechanisms plays a dominant role in driving nest formation. We use a suite of three-dimensional (3D) experiments in human skin tissue and a parallel series of 3D in idual-based simulations to explore whether cell migration or cell proliferation plays a dominant role in nest formation. In the experiments we measure nest formation in populations of irradiated (non-proliferative) and non-irradiated (proliferative) melanoma cells, cultured together with primary keratinocyte and fibroblast cells on a 3D experimental human skin model. Results show that nest size depends on initial cell number and is driven primarily by cell proliferation rather than cell migration. We find that nest size depends on initial cell number, and is driven primarily by cell proliferation rather than cell migration. All experimental results are consistent with simulation data from a 3D in idual based model (IBM) of cell migration and cell proliferation.
Reversible signal transmission in an active mechanical metamaterial
Publisher: The Royal Society
Date: 07-2019
Abstract: 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 time scale of the reaction mechanism on the signal transmission speed, which is essential for controlling the material.
A Bayesian computational approach to explore the optimal duration of a cell proliferation assay
Publisher: Springer Science and Business Media LLC
Date: 28-06-2017
DOI: 10.1007/S11538-017-0311-4
Abstract: Cell proliferation assays are routinely used to explore how a low-density monolayer of cells grows with time. For a typical cell line with a doubling time of 12 h (or longer), a standard cell proliferation assay conducted over 24 h provides excellent information about the low-density exponential growth rate, but limited information about crowding effects that occur at higher densities. To explore how we can best detect and quantify crowding effects, we present a suite of in silico proliferation assays where cells proliferate according to a generalised logistic growth model. Using approximate Bayesian computation we show that data from a standard cell proliferation assay cannot reliably distinguish between classical logistic growth and more general non-logistic growth models. We then explore, and quantify, the trade-off between increasing the duration of the experiment and the associated decrease in uncertainty in the crowding mechanism.
Models of collective cell spreading with variable cell aspect ratio: A motivation for degenerate diffusion models
Publisher: American Physical Society (APS)
Date: 02-02-2011
Continuum descriptions of spatial spreading for heterogeneous cell populations: Theory and experiment
Publisher: Elsevier BV
Date: 12-2019
DOI: 10.1016/J.JTBI.2019.109997
Abstract: Variability in cell populations is frequently observed in both in vitro and in vivo settings. Intrinsic differences within populations of cells, such as differences in cell sizes or differences in rates of cell motility, can be present even within a population of cells from the same cell line. We refer to this variability as cell heterogeneity. Mathematical models of cell migration, for ex le, in the context of tumour growth and metastatic invasion, often account for both undirected (random) migration and directed migration that is mediated by cell-to-cell contacts and cell-to-cell adhesion. A key feature of standard models is that they often assume that the population is composed of identical cells with constant properties. This leads to relatively simple single-species homogeneous models that neglect the role of heterogeneity. In this work, we use a continuum modelling approach to explore the role of heterogeneity in spatial spreading of cell populations. We employ a three-species heterogeneous model of cell motility that explicitly incorporates different types of experimentally-motivated heterogeneity in cell sizes: (i) monotonically decreasing (ii) uniform (iii) non-monotonic and (iv) monotonically increasing distributions of cell size. Comparing the density profiles generated by the three-species heterogeneous model with density profiles predicted by a more standard single-species homogeneous model reveals that when we are dealing with monotonically decreasing and uniform distributions a simple and computationally efficient single-species homogeneous model can be remarkably accurate in describing the evolution of a heterogeneous cell population. In contrast, we find that the simpler single-species homogeneous model performs relatively poorly when applied to non-monotonic and monotonically increasing distributions of cell sizes. Additional results for heterogeneity in parameters describing both undirected and directed cell migration are also considered, and we find that similar results apply.
A computational modelling framework to quantify the effects of passaging cell lines
Publisher: Public Library of Science (PLoS)
Date: 27-07-2017
Communication: Distinguishing between short-time non-Fickian diffusion and long-time Fickian diffusion for a random walk on a crowded lattice
Publisher: AIP Publishing
Date: 05-05-2016
DOI: 10.1063/1.4948782
Abstract: The motion of cells and molecules through biological environments is often hindered by the presence of other cells and molecules. A common approach to modeling this kind of hindered transport is to examine the mean squared displacement (MSD) of a motile tracer particle in a lattice-based stochastic random walk in which some lattice sites are occupied by obstacles. Unfortunately, stochastic models can be computationally expensive to analyze because we must average over a large ensemble of identically prepared realizations to obtain meaningful results. To overcome this limitation we describe an exact method for analyzing a lattice-based model of the motion of an agent moving through a crowded environment. Using our approach we calculate the exact MSD of the motile agent. Our analysis confirms the existence of a transition period where, at first, the MSD does not follow a power law with time. However, after a sufficiently long period of time, the MSD increases in proportion to time. This latter phase corresponds to Fickian diffusion with a reduced diffusivity owing to the presence of the obstacles. Our main result is to provide a mathematically motivated, reproducible, and objective estimate of the amount of time required for the transport to become Fickian. Our new method to calculate this crossover time does not rely on stochastic simulations.
A stochastic mathematical model of 4D tumour spheroids with real-time fluorescent cell cycle labelling
Publisher: The Royal Society
Date: 04-2022
Abstract: In vitro tumour spheroids have been used to study avascular tumour growth and drug design for over 50 years. Tumour spheroids exhibit heterogeneity within the growing population that is thought to be related to spatial and temporal differences in nutrient availability. The recent development of real-time fluorescent cell cycle imaging allows us to identify the position and cell cycle status of in idual cells within the growing spheroid, giving rise to the notion of a four-dimensional (4D) tumour spheroid. We develop the first stochastic in idual-based model (IBM) of a 4D tumour spheroid and show that IBM simulation data compares well with experimental data using a primary human melanoma cell line. The IBM provides quantitative information about nutrient availability within the spheroid, which is important because it is difficult to measure these data experimentally.
Characterizing and minimizing the operator split error for Fisher's equation
Publisher: Elsevier BV
Date: 07-2006
Characterizing transport through a crowded environment with different obstacle sizes
Publisher: AIP Publishing
Date: 04-02-2014
DOI: 10.1063/1.4864000
Abstract: Transport through crowded environments is often classified as anomalous, rather than classical, Fickian diffusion. Several studies have sought to describe such transport processes using either a continuous time random walk or fractional order differential equation. For both these models the transport is characterized by a parameter α, where α = 1 is associated with Fickian diffusion and α & 1 is associated with anomalous subdiffusion. Here, we simulate a single agent migrating through a crowded environment populated by impenetrable, immobile obstacles and estimate α from mean squared displacement data. We also simulate the transport of a population of such agents through a similar crowded environment and match averaged agent density profiles to the solution of a related fractional order differential equation to obtain an alternative estimate of α. We examine the relationship between our estimate of α and the properties of the obstacle field for both a single agent and a population of agents we show that in both cases, α decreases as the obstacle density increases, and that the rate of decrease is greater for smaller obstacles. Our work suggests that it may be inappropriate to model transport through a crowded environment using widely reported approaches including power laws to describe the mean squared displacement and fractional order differential equations to represent the averaged agent density profiles.
Computationally efficient mechanism discovery for cell invasion with uncertainty quantification
Publisher: Cold Spring Harbor Laboratory
Date: 13-05-2022
DOI: 10.1101/2022.05.12.491596
Abstract: Parameter estimation for mathematical models of biological processes is often difficult and depends significantly on the quality and quantity of available data. We introduce an efficient frame-work using Gaussian processes to discover mechanisms underlying delay, migration, and proliferation in a cell invasion experiment. Gaussian processes are leveraged with bootstrapping to provide uncertainty quantification for the mechanisms that drive the invasion process. Our frame-work is efficient, parallelisable, and can be applied to other biological problems. We illustrate our methods using a canonical scratch assay experiment, demonstrating how simply we can explore different functional forms and develop and test hypotheses about underlying mechanisms, such as whether delay is present. All code and data to reproduce this work are available at github.com/DanielVandH/EquationLearning.jl . 1 In this work we introduce uncertainty quantification into equation learning methods, such as physics-informed and biologically-informed neural networks. Our framework is computationally efficient and applicable to problems with unknown nonlinear mechanisms that we wish to learn from experiments where only sparse noisy data are available. We demonstrate our methods on a canonical scratch assay experiment from cell biology and show the underlying mechanisms can be learned, providing confidence intervals for functional forms and for solutions to partial differential equation models believed to describe the experiment.
Exact time-dependent solutions of a Fisher–KPP-like equation obtained with nonclassical symmetry analysis
Publisher: Elsevier BV
Date: 10-2022
Modelling of the SDF-1/CXCR4 regulated in vivo homing of therapeutic mesenchymal stem/stromal cells in mice
Publisher: PeerJ
Date: 27-08-2018
DOI: 10.7287/PEERJ.PREPRINTS.27144V1
Abstract: Background. Mesenchymal stem/stromal cells (MSCs) are a promising tool for cell-based therapies in the treatment of tissue injury. The stromal cell-derived factor-1 (SDF-1)/CXC chemokine receptor 4 (CXCR4) axis plays a significant role in directing MSC homing to sites of injury. However in vivo MSC distribution following intravenous transplantation remains poorly understood, potentially h ering the precise prediction and evaluation of therapeutic efficacy. Methods. A murine model of partial ischemia/reperfusion (I/R) is used to induce liver injury, increase the hepatic levels of SDF-1, and study in vivo MSC distribution. Hypoxia-preconditioning increases the expression of CXCR4 in human bone marrow-derived MSCs. Quantitative assays for human DNA allow us to examine the in vivo kinetics of intravenously infused human MSCs in mouse blood and liver. A mathematical model-based system is developed to characterize in vivo homing of human MSCs in mouse models with SDF-1 levels in liver and CXCR4 expression on the transfused MSCs. The model is calibrated to experimental data to provide novel estimates of relevant parameter values. Results. Images of immunohistochemistry for SDF-1 in the mouse liver with I/R injury show a significantly higher SDF-1level in the I/R injured liver than that in the control. Correspondingly, the ELISA results illustrate a higher MSC dose in the I/R injured liver than the normal liver. CXCR4 is overexpressed in hypoxia-preconditioned MSCs. An increased number of hypoxia-preconditioned MSCs in the I/R injured liver is observed from the ELISA results. The model simulations align with the experimental data of control and hypoxia-preconditioned human MSC distribution in normal and injured mouse livers, and accurately predict the experimental outcomes with different MSC doses. Discussion. The modelling results suggest that SDF-1 in organs is an effective in vivo attractant for MSCs through the SDF-1/CXCR4 axis and reveals the significance of the SDF-1/CXCR4 chemotaxis on in vivo homing of MSCs, especially under hypoxic preconditioning. The impact of the liver and MSC conditions on passive homing is small. This in vivo modelling approach allows qualitative characterization and prediction of the MSC homing to normal and injured organs on the basis of clinically accessible variables, such as the MSC dose and SDF-1 concentration in blood. This model could also be adapted to abnormal conditions and/or other types of circulating cells to predict in vivo homing patterns.
Moments of action provide insight into critical times for advection-diffusion-reaction processes
Publisher: American Physical Society (APS)
Date: 25-09-2012
Modeling proliferative tissue growth: A general approach and an avian case study
Publisher: American Physical Society (APS)
Date: 16-09-2008
Predicting population extinction in lattice-based birth–death–movement models
Publisher: The Royal Society
Date: 06-2020
Abstract: The question of whether a population will persist or go extinct is of key interest throughout ecology and biology. Various mathematical techniques allow us to generate knowledge regarding in idual behaviour, which can be analysed to obtain predictions about the ultimate survival or extinction of the population. A common model employed to describe population dynamics is the lattice-based random walk model with crowding (exclusion). This model can incorporate behaviour such as birth, death and movement, while including natural phenomena such as finite size effects. Performing sufficiently many realizations of the random walk model to extract representative population behaviour is computationally intensive. Therefore, continuum approximations of random walk models are routinely employed. However, standard continuum approximations are notoriously incapable of making accurate predictions about population extinction. Here, we develop a new continuum approximation, the state-space diffusion approximation , which explicitly accounts for population extinction. Predictions from our approximation faithfully capture the behaviour in the random walk model, and provides additional information compared to standard approximations. We examine the influence of the number of lattice sites and initial number of in iduals on the long-term population behaviour, and demonstrate the reduction in computation time between the random walk model and our approximation.
A Bayesian sequential learning framework to parameterise continuum models of melanoma invasion into human skin
Publisher: Cold Spring Harbor Laboratory
Date: 19-03-2018
DOI: 10.1101/284612
Abstract: We present a novel framework to parameterise a mathematical model of cell invasion that describes how a population of melanoma cells invades into human skin tissue. Using simple experimental data extracted from complex experimental images, we estimate three model parameters:(i) the melanoma cell proliferation rate, λ (ii) the melanoma cell diffusivity, D and (iii) δ , a constant that determines the rate that melanoma cells degrade the skin tissue. The Bayesian sequential learning frame-work involves a sequence of increasingly-sophisticated experimental data from:(i) a spatially uniform cell proliferation assay (ii) a two-dimensional circular barrier assay and, (iii) a three-dimensional invasion assay. The Bayesian sequential learning approach leads to well-defined parameter estimates. In contrast, taking a naive approach that attempts to estimate all parameters from a single set of images from the same experiment fails to produce meaningful results. Overall our approach to inference is simple-to-implement, computationally efficient, and well-suited for many cell biology phenomena that can be described by low dimensional continuum models using ordinary differential equations and partial differential equations. We anticipate that this Bayesian sequential learning framework will be relevant in other biological contexts where it is challenging to extract detailed, quantitative biological measurements from experimental images and so we must rely on using relatively simple measurements from complex images.
Efficient inference and identifiability analysis for differential equation models with random parameters
Publisher: Public Library of Science (PLoS)
Date: 28-11-2022
DOI: 10.1371/JOURNAL.PCBI.1010734
Abstract: Heterogeneity is a dominant factor in the behaviour of many biological processes. Despite this, it is common for mathematical and statistical analyses to ignore biological heterogeneity as a source of variability in experimental data. Therefore, methods for exploring the identifiability of models that explicitly incorporate heterogeneity through variability in model parameters are relatively underdeveloped. We develop a new likelihood-based framework, based on moment matching, for inference and identifiability analysis of differential equation models that capture biological heterogeneity through parameters that vary according to probability distributions. As our novel method is based on an approximate likelihood function, it is highly flexible we demonstrate identifiability analysis using both a frequentist approach based on profile likelihood, and a Bayesian approach based on Markov-chain Monte Carlo. Through three case studies, we demonstrate our method by providing a didactic guide to inference and identifiability analysis of hyperparameters that relate to the statistical moments of model parameters from independent observed data. Our approach has a computational cost comparable to analysis of models that neglect heterogeneity, a significant improvement over many existing alternatives. We demonstrate how analysis of random parameter models can aid better understanding of the sources of heterogeneity from biological data.
Quantifying spatial structure in experimental observations and agent-based simulations using pair-correlation functions
Publisher: American Physical Society (APS)
Date: 12-08-2013
Mean exit time for diffusion on irregular domains
Publisher: IOP Publishing
Date: 09-1923
Abstract: Many problems in physics, biology, and economics depend upon the duration of time required for a diffusing particle to cross a boundary. As such, calculations of the distribution of first passage time, and in particular the mean first passage time, is an active area of research relevant to many disciplines. Exact results for the mean first passage time for diffusion on simple geometries, such as lines, discs and spheres, are well-known. In contrast, computational methods are often used to study the first passage time for diffusion on more realistic geometries where closed-form solutions of the governing elliptic boundary value problem are not available. Here, we develop a perturbation solution to calculate the mean first passage time on irregular domains formed by perturbing the boundary of a disc or an ellipse. Classical perturbation expansion solutions are then constructed using the exact solutions available on a disc and an ellipse. We apply the perturbation solutions to compute the mean first exit time on two naturally-occurring irregular domains: a map of Tasmania, an island state of Australia, and a map of Taiwan. Comparing the perturbation solutions with numerical solutions of the elliptic boundary value problem on these irregular domains confirms that we obtain a very accurate solution with a few terms in the series only. MATLAB software to implement all calculations is available at github.com/ProfMJSimpson/Exit_time .
Quantifying the efficacy of first aid treatments for burn injuries using mathematical modelling and in vivo porcine experiments
Publisher: Springer Science and Business Media LLC
Date: 07-09-2017
DOI: 10.1038/S41598-017-11390-Y
Abstract: First aid treatment of burns reduces scarring and improves healing. We quantify the efficacy of first aid treatments using a mathematical model to describe data from a series of in vivo porcine experiments. We study burn injuries that are subject to various first aid treatments. The treatments vary in the temperature and duration. Calibrating the mathematical model to the experimental data provides estimates of the thermal diffusivity, the rate at which thermal energy is lost to the blood, and the heat transfer coefficient controlling the loss of thermal energy at the interface of the fat and muscle. A limitation of working with in vivo experiments is the difficulty of measuring variations in temperature across the tissue layers. This limitation motivates us to use a simple, single layer mathematical model. Using the solution of the calibrated mathematical model we visualise the temperature distribution across the thickness of the tissue. With this information we propose a novel measure of the potential for tissue damage. This measure quantifies two important factors: (i) the volume of tissue that rises above the threshold temperature associated with the accumulation of tissue damage and (ii) the duration of time that the tissue remains above this threshold temperature.
Distinguishing between directed and undirected cell motility within an invading cell population
Publisher: Springer Science and Business Media LLC
Date: 07-02-2009
DOI: 10.1007/S11538-008-9381-7
Abstract: Cell invasion is the basis of several fundamental biological systems including developmental morphogenesis and disease progression. Invasion processes involve combined cell motility and proliferation. Standard experimental approaches to characterize invasion systems focus on measuring population-level wavespeed data. However, continuum models which incorporate either directed or undirected motility both give rise to traveling wave solutions with a well-defined wavespeed in terms of the motility parameters. Therefore, such population-level models and experimental data cannot be used to determine whether the motility is directed or undirected. This is a major impediment limiting our ability to interpret experimental observations of cell invasion. We demonstrate how to overcome this difficulty using in idual-level data and discrete models. This approach can be used to interpret and design time-lapse imaging data to determine whether the cell motility is directed or undirected. Making a distinction between directed and undirected motility has profound implications regarding our ability to design strategies to manage development and disease associated with cell invasion.
Theoretical analysis of the worthiness of Henry and Elder problems as benchmarks of density-dependent groundwater flow models
Publisher: Elsevier BV
Date: 2003
Practical parameter identifiability for spatio-temporal models of cell invasion
Publisher: The Royal Society
Date: 03-2020
Abstract: We examine the practical identifiability of parameters in a spatio-temporal reaction–diffusion model of a scratch assay. Experimental data involve fluorescent cell cycle labels, providing spatial information about cell position and temporal information about the cell cycle phase. Cell cycle labelling is incorporated into the reaction–diffusion model by treating the total population as two interacting subpopulations. Practical identifiability is examined using a Bayesian Markov chain Monte Carlo (MCMC) framework, confirming that the parameters are identifiable when we assume the diffusivities of the subpopulations are identical, but that the parameters are practically non-identifiable when we allow the diffusivities to be distinct. We also assess practical identifiability using a profile likelihood approach, providing similar results to MCMC with the advantage of being an order of magnitude faster to compute. Therefore, we suggest that the profile likelihood ought to be adopted as a screening tool to assess practical identifiability before MCMC computations are performed.
Modeling transport through an environment crowded by a mixture of obstacles of different shapes and sizes
Publisher: Elsevier BV
Date: 05-2016
Extinction of Bistable Populations is Affected by the Shape of their Initial Spatial Distribution
Publisher: Springer Science and Business Media LLC
Date: 20-12-2021
DOI: 10.1007/S11538-021-00974-5
Abstract: The question of whether biological populations survive or are eventually driven to extinction has long been examined using mathematical models. In this work, we study population survival or extinction using a stochastic, discrete lattice-based random walk model where in iduals undergo movement, birth and death events. The discrete model is defined on a two-dimensional hexagonal lattice with periodic boundary conditions. A key feature of the discrete model is that crowding effects are introduced by specifying two different crowding functions that govern how local agent density influences movement events and birth/death events. The continuum limit description of the discrete model is a nonlinear reaction-diffusion equation, and we focus on crowding functions that lead to linear diffusion and a bistable source term that is often associated with the strong Allee effect. Using both the discrete and continuum modelling tools, we explore the complicated relationship between the long-term survival or extinction of the population and the initial spatial arrangement of the population. In particular, we study different spatial arrangements of initial distributions: (i) a well-mixed initial distribution where the initial density is independent of position in the domain (ii) a vertical strip initial distribution where the initial density is independent of vertical position in the domain and, (iii) several forms of two-dimensional initial distributions where the initial population is distributed in regions with different shapes. Our results indicate that the shape of the initial spatial distribution of the population affects extinction of bistable populations. All software required to solve the discrete and continuum models used in this work are available on GitHub .
Lattice-free descriptions of collective motion with crowding and adhesion
Publisher: American Physical Society (APS)
Date: 23-12-2013
Mathematical model of tumour spheroid experiments with real-time cell cycle imaging
Publisher: Cold Spring Harbor Laboratory
Date: 07-12-2020
DOI: 10.1101/2020.12.06.413856
Abstract: Three-dimensional (3D) in vitro tumour spheroid experiments are an important tool for studying cancer progression and potential drug therapies. Standard experiments involve growing and imaging spheroids to explore how different experimental conditions lead to different rates of spheroid growth. These kinds of experiments, however, do not reveal any information about the spatial distribution of the cell cycle within the expanding spheroid. Since 2008, a new experimental technology called fluorescent ubiquitination-based cell cycle indicator (FUCCI), has enabled real time in situ visualisation of the cell cycle progression. FUCCI labelling involves cells in G1 phase of the cell cycle fluorescing red, and cells in the S/G2/M phase of the cell cycle fluorescing green. Experimental observations of 3D tumour spheroids with FUCCI labelling reveal significant intratumoural structure, as the cell cycle status can vary with location. Although many mathematical models of tumour spheroid growth have been developed, none of the existing mathematical models are designed to interpret experimental observations with FUCCI labelling. In this work we extend the mathematical framework originally proposed by Ward and King (1997) to develop a new mathematical model of FUCCI-labelled tumour spheroid growth. The mathematical model treats the spheroid as being composed of three subpopulations: (i) living cells in G1 phase that fluoresce red (ii) living cells in S/G2/M phase that fluoresce green and, (iii) dead cells that do not fluoresce. We assume that the rates at which cells pass through different phases of the cell cycle, and the rate of cell death, depend upon the local oxygen concentration in the spheroid. Parameterising the new mathematical model using experimental measurements of cell cycle transition times, we show that the model can capture important experimental observations that cannot be addressed using previous mathematical models. Further, we show that the mathematical model can be used to quantitatively mimic the action of anti-mitotic drugs applied to the spheroid. All software required to solve the nonlinear moving boundary problem associated with the new mathematical model are available on GitHub .
Modelling cell guidance and curvature control in evolving biological tissues
Publisher: Cold Spring Harbor Laboratory
Date: 12-07-2020
DOI: 10.1101/2020.07.10.197020
Abstract: Tissue geometry is an important influence on the evolution of many biological tissues. The local curvature of an evolving tissue induces tissue crowding or spreading, which leads to differential tissue growth rates, and to changes in cellular tension, which can influence cell behaviour. Here, we investigate how directed cell motion interacts with curvature control in evolving biological tissues. Directed cell motion is involved in the generation of angled tissue growth and anisotropic tissue material properties, such as tissue fibre orientation. We develop a new cell-based mathematical model of tissue growth that includes both curvature control and cell guidance mechanisms to investigate their interplay. The model is based on conservation principles applied to the density of tissue synthesising cells at or near the tissue’s moving boundary. The resulting mathematical model is a partial differential equation for cell density on a moving boundary, which is solved numerically using a hybrid front-tracking method called the cell-based particle method. The inclusion of directed cell motion allows us to model new types of biological growth, where tangential cell motion is important for the evolution of the interface, or for the generation of anisotropic tissue properties. We illustrate such situations by applying the model to simulate both the resorption and infilling components of the bone remodelling process, and to simulate root hair growth. We also provide user-friendly MATLAB code to implement the algorithms.
Computationally efficient framework for diagnosing, understanding and predicting biphasic population growth
Publisher: The Royal Society
Date: 12-2022
Abstract: Throughout the life sciences, biological populations undergo multiple phases of growth, often referred to as biphasic growth for the commonly encountered situation involving two phases. Biphasic population growth occurs over a massive range of spatial and temporal scales, ranging from microscopic growth of tumours over several days, to decades-long regrowth of corals in coral reefs that can extend for hundreds of kilometres. Different mathematical models and statistical methods are used to diagnose, understand and predict biphasic growth. Common approaches can lead to inaccurate predictions of future growth that may result in inappropriate management and intervention strategies being implemented. Here, we develop a very general computationally efficient framework, based on profile likelihood analysis, for diagnosing, understanding and predicting biphasic population growth. The two key components of the framework are as follows: (i) an efficient method to form approximate confidence intervals for the change point of the growth dynamics and model parameters and (ii) parameter-wise profile predictions that systematically reveal the influence of in idual model parameters on predictions. To illustrate our framework we explore real-world case studies across the life sciences.
Small-scale spatial structure affects predator-prey dynamics and coexistence
Publisher: Cold Spring Harbor Laboratory
Date: 18-12-2019
DOI: 10.1101/2019.12.17.880104
Abstract: Small-scale spatial variability can affect community dynamics in many ecological and biological processes, such as predator-prey dynamics and immune responses. Spatial variability includes short-range neighbour-dependent interactions and small-scale spatial structure, such as clustering where in iduals aggregate together, and segregation where in iduals are spaced apart from one another. Yet, a large class of mathematical models aimed at representing these processes ignores these factors by making a classical mean-field approximation, where interactions between in iduals are assumed to occur in proportion to their average density. Such mean-field approximations amount to ignoring spatial structure. In this work, we consider an in idual based model of a two-species community that is composed of consumers and resources . The model describes migration, predation, competition and dispersal of offspring, and explicitly gives rise to varying degrees of spatial structure. We compare simulation results from the in idual based model with the solution of a classical mean-field approximation, and this comparison provides insight into how spatial structure can drive the system away from mean-field dynamics. Our analysis reveals that mechanisms leading to intraspecific clustering and interspecific segregation, such as short-range predation and short-range dispersal, tend to increase the size of the resource species relative to the mean-field prediction. We show that under certain parameter regimes these mechanisms lead to the extinction of consumers whereas the classical mean-field model predicts the coexistence of both species.
Do pioneer cells exist?
Publisher: Public Library of Science (PLoS)
Date: 21-01-2014
Revisiting the Fisher–Kolmogorov–Petrovsky–Piskunov equation to interpret the spreading–extinction dichotomy
Publisher: The Royal Society
Date: 09-2019
Abstract: The Fisher–Kolmogorov–Petrovsky–Piskunov model, also known as the Fisher–KPP model, supports travelling wave solutions that are successfully used to model numerous invasive phenomena with applications in biology, ecology and combustion theory. However, there are certain phenomena that the Fisher–KPP model cannot replicate, such as the extinction of invasive populations. The Fisher–Stefan model is an adaptation of the Fisher–KPP model to include a moving boundary whose evolution is governed by a Stefan condition. The Fisher–Stefan model also supports travelling wave solutions however, a key additional feature of the Fisher–Stefan model is that it is able to simulate population extinction, giving rise to a spreading–extinction dichotomy . In this work, we revisit travelling wave solutions of the Fisher–KPP model and show that these results provide new insight into travelling wave solutions of the Fisher–Stefan model and the spreading–extinction dichotomy. Using a combination of phase plane analysis, perturbation analysis and linearization, we establish a concrete relationship between travelling wave solutions of the Fisher–Stefan model and often-neglected travelling wave solutions of the Fisher–KPP model. Furthermore, we give closed-form approximate expressions for the shape of the travelling wave solutions of the Fisher–Stefan model in the limit of slow travelling wave speeds, c ≪1.
Growth and adaptation mechanisms of tumour spheroids with time-dependent oxygen availability
Publisher: Cold Spring Harbor Laboratory
Date: 24-04-2022
DOI: 10.1101/2022.04.24.489294
Abstract: Tumours are subject to external environmental variability. However, in vitro tumour spheroid experiments, used to understand cancer progression and develop cancer therapies, have been routinely performed for the past fifty years in constant external environments. Furthermore, spheroids are typically grown in ambient atmospheric oxygen (normoxia), whereas most in vivo tumours exist in hypoxic environments. Therefore, there are clear discrepancies between in vitro and in vivo conditions. We explore these discrepancies by combining tools from experimental biology, mathematical modelling, and statistical uncertainty quantification. Focusing on oxygen variability to develop our framework, we reveal key biological mechanisms governing tumour spheroid growth. Growing spheroids in time-dependent conditions, we identify and quantify novel biological adaptation mechanisms, including unexpected necrotic core removal, and transient reversal of the tumour spheroid growth phases.
Lattice-Free Models of Cell Invasion: Discrete Simulations and Travelling Waves
Publisher: Springer Science and Business Media LLC
Date: 17-08-2013
DOI: 10.1007/S11538-013-9885-7
Abstract: Invasion waves of cells play an important role in development, disease, and repair. Standard discrete models of such processes typically involve simulating cell motility, cell proliferation, and cell-to-cell crowding effects in a lattice-based framework. The continuum-limit description is often given by a reaction-diffusion equation that is related to the Fisher-Kolmogorov equation. One of the limitations of a standard lattice-based approach is that real cells move and proliferate in continuous space and are not restricted to a predefined lattice structure. We present a lattice-free model of cell motility and proliferation, with cell-to-cell crowding effects, and we use the model to replicate invasion wave-type behaviour. The continuum-limit description of the discrete model is a reaction-diffusion equation with a proliferation term that is different from lattice-based models. Comparing lattice-based and lattice-free simulations indicates that both models lead to invasion fronts that are similar at the leading edge, where the cell density is low. Conversely, the two models make different predictions in the high-density region of the domain, well behind the leading edge. We analyse the continuum-limit description of the lattice-based and lattice-free models to show that both give rise to invasion wave type solutions that move with the same speed but have very different shapes. We explore the significance of these differences by calibrating the parameters in the standard Fisher-Kolmogorov equation using data from the lattice-free model. We conclude that estimating parameters using this kind of standard procedure can produce misleading results.
Mean exit time in irregularly-shaped annular and composite disc domains
Publisher: IOP Publishing
Date: 02-03-2022
Abstract: Calculating the mean exit time (MET) for models of diffusion is a classical problem in statistical physics, with various applications in biophysics, economics and heat and mass transfer. While many exact results for MET are known for diffusion in simple geometries involving homogeneous materials, calculating MET for diffusion in realistic geometries involving heterogeneous materials is typically limited to repeated stochastic simulations or numerical solutions of the associated boundary value problem (BVP). In this work we derive exact solutions for the MET in irregular annular domains, including some applications where diffusion occurs in heterogenous media. These solutions are obtained by taking the exact results for MET in an annulus, and then constructing various perturbation solutions to account for the irregular geometries involved. These solutions, with a range of boundary conditions, are implemented symbolically and compare very well with averaged data from repeated stochastic simulations and with numerical solutions of the associated BVP. Software to implement the exact solutions is available on GitHub.
Looking inside an invasion wave of cells using continuum models: Proliferation is the key
Publisher: Elsevier BV
Date: 12-2006
DOI: 10.1016/J.JTBI.2006.06.021
Abstract: Recently, a suite of cell migration assays were conducted to investigate the migration of neural crest (NC) cells along the gut during the development of the enteric nervous system (ENS). The NC cells colonise the gastro-intestinal tract as a rostro-caudal wave. Local behaviour was shown to be controlled by position relative to the leading edge of the wavefront. The assays involved chick-quail grafting techniques allowing the total invading population to be considered as a two-species system. A two-species continuum model with logistic proliferation and a migration mechanism is developed here to simulate the chick-quail graft experiments and provide a means of looking at the processes occurring within the invasion wave. Five migration mechanisms are considered--linear diffusion, two cases of nonlinear diffusion, chemokinesis and chemotaxis. The model results agree with the experimental observations, regardless of the specific type of migration mechanism. The results show that NC cell invasion is driven by proliferation and cell motility at the leading edge of the wave. Furthermore, logistic proliferation exerts the dominant control on the system. This observation is confirmed by analysing some simplified invasion models. Once the basic experiments were mathematically replicated, the mathematical models were used in turn to make some predictions that were yet to be experimentally tested. This involved conducting a sensitivity analysis of the system by interrupting the proliferation and/or migration ability of the leading edge. Numerical results show that the system is stable against these changes. Of the three experiments suggested, one was carried out and the experimental results were concordant with the theoretical predictions. The outcome of two other suggested experiments are predicted and left for future experimental validation.
Analysis of split operator methods applied to reactive transport with Monod kinetics
Publisher: Elsevier BV
Date: 09-2007
Cell proliferation and migration explain pore bridging dynamics in 3D printed scaffolds of different pore size
Publisher: Elsevier BV
Date: 09-2020
Mathematical and experimental insights into the development of the enteric nervous system and Hirschsprung's Disease
Publisher: Wiley
Date: 05-2007
Estimating parameters of a stochastic cell invasion model with fluorescent cell cycle labelling using Approximate Bayesian Computation
Publisher: Cold Spring Harbor Laboratory
Date: 21-04-2021
DOI: 10.1101/2021.04.20.440712
Abstract: We develop a parameter estimation method based on approximate Bayesian computation (ABC) for a stochastic cell invasion model using fluorescent cell cycle labeling with proliferation, migration, and crowding effects. Previously, inference has been performed on a deterministic version of the model fitted to cell density data, and not all the parameters were identifiable. Considering the stochastic model allows us to harness more features of experimental data, including cell trajectories and cell count data, which we show overcomes the parameter identifiability problem. We demonstrate that, whilst difficult to collect, cell trajectory data can provide more information about the parameters of the cell invasion model. To handle the intractability of the likelihood function of the stochastic model, we use an efficient ABC algorithm based on sequential Monte Carlo. Rcpp and MATLAB implementations of the simulation model and ABC algorithm used in this study are available at ichaelcarr-stats/FUCCI .
Diffusion in heterogeneous discs and spheres: New closed-form expressions for exit times and homogenization formulas
Publisher: AIP Publishing
Date: 21-08-2020
DOI: 10.1063/5.0010810
Abstract: Mathematical models of diffusive transport underpin our understanding of chemical, biochemical, and biological transport phenomena. Analysis of such models often focuses on relatively simple geometries and deals with diffusion through highly idealized homogeneous media. In contrast, practical applications of diffusive transport theory inevitably involve dealing with more complicated geometries as well as dealing with heterogeneous media. One of the most fundamental properties of diffusive transport is the concept of mean particle lifetime or mean exit time, which are particular applications of the concept of first passage time and provide the mean time required for a diffusing particle to reach an absorbing boundary. Most formal analysis of mean particle lifetime applies to relatively simple geometries, often with homogeneous (spatially invariant) material properties. In this work, we present a general framework that provides exact mathematical insight into the mean particle lifetime, and higher moments of particle lifetime, for point particles diffusing in heterogeneous discs and spheres with radial symmetry. Our analysis applies to geometries with an arbitrary number and arrangement of distinct layers, where transport in each layer is characterized by a distinct diffusivity. We obtain exact closed-form expressions for the mean particle lifetime for a diffusing particle released at an arbitrary location, and we generalize these results to give exact, closed-form expressions for any higher-order moment of particle lifetime for a range of different boundary conditions. Finally, using these results, we construct new homogenization formulas that provide an accurate simplified description of diffusion through heterogeneous discs and spheres.
Computing residence times for flow towards a pumping well: Nomograph solution and validity of the small draw-down approximation
Publisher: Springer Science and Business Media LLC
Date: 09-09-2005
Optimal control of acute myeloid leukaemia
Publisher: Cold Spring Harbor Laboratory
Date: 28-09-2018
DOI: 10.1101/429811
Abstract: Acute myeloid leukaemia (AML) is a blood cancer affecting haematopoietic stem cells. AML is routinely treated with chemotherapy, and so it is of great interest to develop optimal chemotherapy treatment strategies. In this work, we incorporate an immune response into a stem cell model of AML, since we find that previous models lacking an immune response are inappropriate for deriving optimal control strategies. Using optimal control theory, we produce continuous controls and bang-bang controls, corresponding to a range of objectives and parameter choices. Through ex le calculations, we provide a practical approach to applying optimal control using Pontryagin’s Maximum Principle. In particular, we describe and explore factors that have a profound influence on numerical convergence. We find that the convergence behaviour is sensitive to the method of control updating, the nature of the control, and to the relative weighting of terms in the objective function. All codes we use to implement optimal control are made available.
Corrected mean-field models for spatially dependent advection-diffusion- reaction phenomena
Publisher: American Physical Society (APS)
Date: 26-05-2011
Identification of two‐phase recovery for interpretation of coral reef monitoring data
Publisher: Wiley
Date: 27-10-2022
Abstract: The world's coral reefs are under threat as climate change causes increases in frequency and severity of acute thermal stress. This is compounded by chronic pressures including rises in sea surface temperature, overfishing and decline in water quality. Monitoring to understand the recovery dynamics of corals is paramount to enable effective management of coral reefs. While detailed mechanistic models provide insight into reef recovery patterns, colony scale monitoring is not viable for reefs over a large geographical extent, such as the Great Barrier Reef (GBR). Consequently, aggregated coral cover data are utilised in practice and phenomenological analysis directly applicable to these monitoring programmes is essential for reef health reporting. These analyses are especially challenging for assessment of recovery potential of reefs reduced to very low coral cover ( %) after disturbance since standard modelling assumptions may not hold. Through the application of an innovative diagnostic approach modified from methods used in cancer cell biology, we found that almost 50% of reefs recovering from low cover exhibited a previously undocumented initial phase of slower growth per unit cover before sigmoid coral cover recovery trajectories were observed. Without properly accounting for these reduced growth periods, the expected performance of reefs may be overestimated immediately after disturbance events. The presence of two‐phase recovery patterns has a profound negative impact on the continued provision of ecological services from these reefs as major disturbance frequencies increase. Projections show that if the time between major disturbances is years, then reefs with two‐phase growth are never likely to reach 15% cover. Synthesis and applications . This work provides a method to detect two‐phase recovery, the tendency for observed reef recovery to be slower than expected after a major disturbance. This phenomenon is observed across the Great Barrier Reef with serious implications for recovery potential as major disturbances occur more frequently due to climate change. Identification of reefs prone to two‐phase recovery can assist the triage of reefs for intervention actions in response to disturbances. Understanding mechanisms will inform interventions and management actions targeted towards the unique marine ecosystems across the world's reefs.
A multiscale approach to estimate the cellular diffusivity during food drying
Publisher: Elsevier BV
Date: 12-2021
Understanding time scales of diffusive fluxes and the implication for steady state and steady shape conditions
Publisher: American Geophysical Union (AGU)
Date: 05-01-2017
DOI: 10.1002/2016GL071914
Formation and growth of co-culture tumour spheroids: new compartment-based mathematical models and experiments
Publisher: Cold Spring Harbor Laboratory
Date: 22-12-2022
DOI: 10.1101/2022.12.21.521515
Abstract: Co-culture tumour spheroid experiments are routinely performed to investigate cancer progression and test anti-cancer therapies. Therefore, methods to characterise and interpret coculture spheroid growth are of great interest. However, co-culture spheroid growth is complex. Multiple biological processes occur on overlapping timescales and different cell types within the spheroid may have different characteristics, for ex le proliferation rate or response to nutrient availability. There is no standard, widely-accepted mathematical model of such complex spatio-temporal growth processes. Typical approaches to analyse these experiments focus on the late-time temporal evolution of spheroid size and overlook early-time spheroid formation, spheroid structure and geometry. Here we make theoretical and practical contributions. We develop a general framework, based on mathematical and statistical modelling, to analyse a series of co-culture experiments. Using a range of different mathematical models we provide new biological insights about spheroid formation, growth, and structure. In addition, we extend a general class of compartment-based monoculture spheroid mathematical models to describe multiple populations and provide mechanistic biological insight. The framework is well-suited to analyse spheroids grown with multiple different cell types and the new class of mathematical models provide opportunities for further mathematical and biological insights.
Nonlinear diffusion and exclusion processes with contact interactions
Publisher: American Physical Society (APS)
Date: 11-01-2010
Identifying density-dependent interactions in collective cell behaviour
Publisher: The Royal Society
Date: 04-2020
Abstract: Scratch assays are routinely used to study collective cell behaviour in vitro . Typical experimental protocols do not vary the initial density of cells, and typical mathematical modelling approaches describe cell motility and proliferation based on assumptions of linear diffusion and logistic growth. Jin et al. (Jin et al . 2016 J. Theor. Biol. 390 , 136–145 ( doi:10.1016/j.jtbi.2015.10.040 )) find that the behaviour of cells in scratch assays is density-dependent, and show that standard modelling approaches cannot simultaneously describe data initiated across a range of initial densities. To address this limitation, we calibrate an in idual-based model to scratch assay data across a large range of initial densities. Our model allows proliferation, motility, and a direction bias to depend on interactions between neighbouring cells. By considering a hierarchy of models where we systematically and sequentially remove interactions, we perform model selection analysis to identify the minimum interactions required for the model to simultaneously describe data across all initial densities. The calibrated model is able to match the experimental data across all densities using a single parameter distribution, and captures details about the spatial structure of cells. Our results provide strong evidence to suggest that motility is density-dependent in these experiments. On the other hand, we do not see the effect of crowding on proliferation in these experiments. These results are significant as they are precisely the opposite of the assumptions in standard continuum models, such as the Fisher–Kolmogorov equation and its generalizations.
A hybrid model for studying spatial aspects of infectious diseases
Publisher: Cambridge University Press (CUP)
Date: 10-2012
DOI: 10.1017/S1446181112000296
Abstract: We consider a hybrid model, created by coupling a continuum and an agent-based model of infectious disease. The framework of the hybrid model provides a mechanism to study the spread of infection at both the in idual and population levels. This approach captures the stochastic spatial heterogeneity at the in idual level, which is directly related to deterministic population level properties. This facilitates the study of spatial aspects of the epidemic process. A spatial analysis, involving counting the number of infectious agents in equally sized bins, reveals when the spatial domain is nonhomogeneous.
Nonmonotone chemotactic invasion: High-resolution simulations, phase plane analysis and new benchmark problems
Publisher: Elsevier BV
Date: 07-2007
Critical time scales for advection-diffusion-reaction processes
Publisher: American Physical Society (APS)
Date: 23-04-2012
Three-dimensional experiments and individual based simulations show that cell proliferation drives melanoma nest formation in human skin tissue
Publisher: Springer Science and Business Media LLC
Date: 27-03-2018
Practical parameter identifiability for spatiotemporal models of cell invasion
Publisher: Cold Spring Harbor Laboratory
Date: 12-11-2019
DOI: 10.1101/839282
Abstract: We examine the practical identifiability of parameters in a spatiotemporal reaction-diffusion model of a scratch assay. Experimental data involves fluorescent cell cycle labels, providing spatial information about cell position and temporal information about the cell cycle phase. Cell cycle labelling is incorporated into the reaction–diffusion model by treating the total population as two interacting subpopulations. Practical identifiability is examined using a Bayesian Markov chain Monte Carlo (MCMC) framework, confirming that the parameters are identifiable when we assume the diffusivities of the subpopulations are identical, but that the parameters are practically non-identifiable when we allow the diffusivities to be distinct. We also assess practical identifiability using a profile likelihood approach, providing similar results to MCMC with the advantage of being an order of magnitude faster to compute. Therefore, we suggest that the profile likelihood ought to be adopted as a screening tool to assess practical identifiability before MCMC computations are performed.
An analytical method to solve a general class of nonlinear reactive transport models
Publisher: Elsevier BV
Date: 05-2011
Population dynamics with spatial structure and an Allee effect
Publisher: Cold Spring Harbor Laboratory
Date: 28-06-2020
DOI: 10.1101/2020.06.26.173153
Abstract: Allee effects describe populations in which long-term survival is only possible if the population density is above some threshold level. A simple mathematical model of an Allee effect is one where initial densities below the threshold lead to population extinction, whereas initial densities above the threshold eventually asymptote to some positive carrying capacity density. Mean field models of population dynamics neglect spatial structure that can arise through short-range interactions, such as short-range competition and dispersal. The influence of such non mean-field effects has not been studied in the presence of an Allee effect. To address this we develop an in idual-based model (IBM) that incorporates both short-range interactions and an Allee effect. To explore the role of spatial structure we derive a mathematically tractable continuum approximation of the IBM in terms of the dynamics of spatial moments. In the limit of long-range interactions where the mean-field approximation holds, our modelling framework accurately recovers the mean-field Allee threshold. We show that the Allee threshold is sensitive to spatial structure that mean-field models neglect. For ex le, we show that there are cases where the mean-field model predicts extinction but the population actually survives and vice versa. Through simulations we show that our new spatial moment dynamics model accurately captures the modified Allee threshold in the presence of spatial structure.
Persistence as an optimal hedging strategy
Publisher: Cold Spring Harbor Laboratory
Date: 20-12-2019
DOI: 10.1101/2019.12.19.883645
Abstract: Bacteria invest in a slow-growing subpopulation, called persisters, to ensure survival in the face of uncertainty. This hedging strategy is remarkably similar to financial hedging, where ersifying an investment portfolio protects against economic uncertainty. We provide a new theoretical foundation for understanding cellular hedging by unifying the study of biological population dynamics and the mathematics of financial risk management through optimal control theory. Motivated by the widely accepted role of volatility in the emergence of persistence, we consider several novel models of environmental volatility described by continuous-time stochastic processes. This allows us to study an emergent cellular hedging strategy that maximizes the expected per-capita growth rate of the population. Analytical and simulation results probe the optimal persister strategy, revealing results that are consistent with experimental observations and suggest at new opportunities for experimental investigation and design. Overall, we provide a new way of conceptualising and modelling cellular decision-making in volatile environments by explicitly unifying theory from mathematical biology and finance.
Multi-species simple exclusion processes
Publisher: Elsevier BV
Date: 02-2009
Exact solutions for diffusive transport on heterogeneous growing domains
Publisher: The Royal Society
Date: 08-2023
Abstract: From the smallest biological systems to the largest cosmological structures, spatial domains undergo expansion and contraction. Within these growing domains, diffusive transport is a common phenomenon. Mathematical models have been widely employed to investigate diffusive processes on growing domains. However, a standard assumption is that the domain growth is spatially uniform. There are many relevant ex les where this is not the case, such as the colonization of growing gut tissue by neural crest cells. As such, it is not straightforward to disentangle the in idual roles of heterogeneous growth and diffusive transport. Here we present exact solutions to models of diffusive transport on domains undergoing spatially non-uniform growth. The exact solutions are obtained via a combination of transformation, convolution and superposition techniques. We verify the accuracy of these solutions via comparison with simulations of a corresponding lattice-based random walk. We explore various domain evolution functions, including linear growth, exponential growth and contraction, and oscillatory evolution. Provided the domain size remains positive, we find that the derived solutions are valid. The exact solutions reveal the relationship between model parameters, such as the diffusivity and the type and rate of domain evolution, and key statistics, such as the survival and splitting probabilities.
Implementation and acceleration of optimal control for systems biology
Publisher: The Royal Society
Date: 08-2021
Abstract: Optimal control theory provides insight into complex resource allocation decisions. The forward–backward sweep method (FBSM) is an iterative technique commonly implemented to solve two-point boundary value problems arising from the application of Pontryagin’s maximum principle (PMP) in optimal control. The FBSM is popular in systems biology as it scales well with system size and is straightforward to implement. In this review, we discuss the PMP approach to optimal control and the implementation of the FBSM. By conceptualizing the FBSM as a fixed point iteration process, we leverage and adapt existing acceleration techniques to improve its rate of convergence. We show that convergence improvement is attainable without prohibitively costly tuning of the acceleration techniques. Furthermore, we demonstrate that these methods can induce convergence where the underlying FBSM fails to converge. All code used in this work to implement the FBSM and acceleration techniques is available on GitHub at github.com/Jesse-Sharp/Sharp2021 .
Spatial Moment Description of Birth–Death–Movement Processes Incorporating the Effects of Crowding and Obstacles
Publisher: Springer Science and Business Media LLC
Date: 10-08-2018
DOI: 10.1007/S11538-018-0488-1
Abstract: Birth-death-movement processes, modulated by interactions between in iduals, are fundamental to many cell biology processes. A key feature of the movement of cells within in vivo environments is the interactions between motile cells and stationary obstacles. Here we propose a multi-species model of in idual-level motility, proliferation and death. This model is a spatial birth-death-movement stochastic process, a class of in idual-based model (IBM) that is amenable to mathematical analysis. We present the IBM in a general multi-species framework and then focus on the case of a population of motile, proliferative agents in an environment populated by stationary, non-proliferative obstacles. To analyse the IBM, we derive a system of spatial moment equations governing the evolution of the density of agents and the density of pairs of agents. This approach avoids making the usual mean-field assumption so that our models can be used to study the formation of spatial structure, such as clustering and aggregation, and to understand how spatial structure influences population-level outcomes. Overall the spatial moment model provides a reasonably accurate prediction of the system dynamics, including important effects such as how varying the properties of the obstacles leads to different spatial patterns in the population of agents.
Estimating parameters of a stochastic cell invasion model with fluorescent cell cycle labelling using approximate Bayesian computation
Publisher: The Royal Society
Date: 09-2021
Abstract: We develop a parameter estimation method based on approximate Bayesian computation (ABC) for a stochastic cell invasion model using fluorescent cell cycle labelling with proliferation, migration and crowding effects. Previously, inference has been performed on a deterministic version of the model fitted to cell density data, and not all parameters were identifiable. Considering the stochastic model allows us to harness more features of experimental data, including cell trajectories and cell count data, which we show overcomes the parameter identifiability problem. We demonstrate that, while difficult to collect, cell trajectory data can provide more information about the parameters of the cell invasion model. To handle the intractability of the likelihood function of the stochastic model, we use an efficient ABC algorithm based on sequential Monte Carlo. Rcpp and MATLAB implementations of the simulation model and ABC algorithm used in this study are available at ichaelcarr-stats/FUCCI .
Assessment of a non-traditional operator split algorithm for simulation of reactive transport
Publisher: Elsevier BV
Date: 09-2005
Growth and adaptation mechanisms of tumour spheroids with time-dependent oxygen availability
Publisher: Public Library of Science (PLoS)
Date: 12-01-2023
DOI: 10.1371/JOURNAL.PCBI.1010833
Abstract: Tumours are subject to external environmental variability. However, in vitro tumour spheroid experiments, used to understand cancer progression and develop cancer therapies, have been routinely performed for the past fifty years in constant external environments. Furthermore, spheroids are typically grown in ambient atmospheric oxygen (normoxia), whereas most in vivo tumours exist in hypoxic environments. Therefore, there are clear discrepancies between in vitro and in vivo conditions. We explore these discrepancies by combining tools from experimental biology, mathematical modelling, and statistical uncertainty quantification. Focusing on oxygen variability to develop our framework, we reveal key biological mechanisms governing tumour spheroid growth. Growing spheroids in time-dependent conditions, we identify and quantify novel biological adaptation mechanisms, including unexpected necrotic core removal, and transient reversal of the tumour spheroid growth phases.
Diffusive and chemotactic cellular migration: Smooth and discontinuous traveling wave solutions
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2005
DOI: 10.1137/040604066
Designing combination therapies using multiple optimal controls
Publisher: Cold Spring Harbor Laboratory
Date: 21-11-2019
DOI: 10.1101/850693
Abstract: Strategic management of populations of interacting biological species routinely requires interventions combining multiple treatments or therapies. This is important in key research areas such as ecology, epidemiology, wound healing and oncology. Despite the well developed theory and techniques for determining single optimal controls, there is limited practical guidance supporting implementation of combination therapies. In this work we use optimal control theory to calculate optimal strategies for applying combination therapies to a model of acute myeloid leukaemia. We consider various combinations of continuous and bang-bang (discrete) controls, and we investigate how the control dynamics interact and respond to changes in the weighting and form of the pay-off characterising optimality. We demonstrate that the optimal controls respond non-linearly to treatment strength and control parameters, due to the interactions between species. We discuss challenges in appropriately characterising optimality in a multiple control setting and provide practical guidance for applying multiple optimal controls. Code used in this work to implement multiple optimal controls is available on GitHub.
Does sea-level rise have an impact on saltwater intrusion?
Publisher: Elsevier BV
Date: 10-2011
The impact of experimental design choices on parameter inference for models of growing cell colonies
Publisher: Cold Spring Harbor Laboratory
Date: 02-08-2017
DOI: 10.1101/171710
Abstract: To better understand development, repair and disease progression it is useful to quantify the behaviour of proliferative and motile cell populations as they grow and expand to fill their local environment. Inferring parameters associated with mechanistic models of cell colony growth using quantitative data collected from carefully designed experiments provides a natural means to elucidate the relative contributions of various processes to the growth of the colony. In this work we explore how experimental design impacts our ability to infer parameters for simple models of the growth of proliferative and motile cell populations. We adopt a Bayesian approach, which allows us to characterise the uncertainty associated with estimates of the model parameters. Our results suggest that experimental designs that incorporate initial spatial heterogeneities in cell positions facilitate parameter inference without the requirement of cell tracking, whilst designs that involve uniform initial placement of cells require cell tracking for accurate parameter inference. As cell tracking is an experimental bottleneck in many studies of this type, our recommendations for experimental design provide for significant potential time and cost savings in the analysis of cell colony growth.
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