A temporally relaxed theory of physically or chemically non-equilibrium solute transport in heterogeneous porous media.

Groundwater constitutes a critical component in providing fresh water for various human endeavors. Never-theless, its susceptibility to contamination by pollutants represents a significant challenge. A comprehensive understanding of the dynamics of solute transport in groundwater and soils is essential for predicting the spatial and temporal distribution of these contaminants.

Statistical modelling of goalkicking performance in the Australian Football League.

Objectives: Australian football goal kicking is vital to team success, but its study is limited. Develop and apply Bayesian models incorporating temporal, spatial and situational variables to predict shot outcomes. The models aim to (i) rank players on their goal kicking and (ii) create clusters of statistically similar players and rank these clusters to provide generalised recommendations about player types.

A stochastic mathematical model of 4D tumour spheroids with real-time fluorescent cell cycle labelling.

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 individual cells within the growing spheroid, giving rise to the notion of a four-dimensional (4D) tumour spheroid.

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

We compute profile likelihoods for a stochastic model of diffusive transport motivated by experimental observations of heat conduction in layered skin tissues. This process is modelled as a random walk in a layered one-dimensional material, where each layer has a distinct particle hopping rate. Particles are released at some location, and the duration of time taken for each particle to reach an absorbing boundary is recorded.

Diffusion in heterogeneous discs and spheres: New closed-form expressions for exit times and homogenization formulas.

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.

Finite transition times for multispecies diffusion in heterogeneous media coupled via first-order reaction networks.

Calculating how long a coupled multispecies reactive-diffusive transport process in a heterogeneous medium takes to effectively reach steady state is important in many applications. In this paper, we show how the time required for such processes to transition to within a small specified tolerance of steady state can be calculated accurately without having to solve the governing time-dependent model equations. Our approach is valid for general first-order reaction networks and an arbitrary number of species.

Advection improves homogenized models of continuum diffusion in one-dimensional heterogeneous media.

We propose an alternative homogenization method for one-dimensional continuum diffusion models with spatially variable (heterogeneous) diffusivity. Our method, which extends recent work on stochastic diffusion, assumes the constant-coefficient homogenized equation takes the form of an advection-diffusion equation with effective (diffusivity and velocity) coefficients. To calculate the effective coefficients, our approach involves solving two uncoupled boundary value problems over the heterogeneous medium and leads to coefficients depending on the spatially varying diffusivity (as usual) as well as the boundary conditions imposed on the heterogeneous model.

Drug delivery from microcapsules: How can we estimate the release time?

Predicting the release performance of a drug delivery device is an important challenge in pharmaceutics and biomedical science. In this paper, we consider a multi-layer diffusion model of drug release from a composite spherical microcapsule into an external surrounding medium. Based on this model, we present two approaches for estimating the release time, i.

New homogenization approaches for stochastic transport through heterogeneous media.

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.

Characteristic time scales for diffusion processes through layers and across interfaces.

This paper presents a simple tool for characterizing the time scale for continuum diffusion processes through layered heterogeneous media. This mathematical problem is motivated by several practical applications such as heat transport in composite materials, flow in layered aquifers, and drug diffusion through the layers of the skin. In such processes, the physical properties of the medium vary across layers and internal boundary conditions apply at the interfaces between adjacent layers.

Modelling mass diffusion for a multi-layer sphere immersed in a semi-infinite medium: application to drug delivery.

We present a general mechanistic model of mass diffusion for a composite sphere placed in a large ambient medium. The multi-layer problem is described by a system of diffusion equations coupled via interlayer boundary conditions such as those imposing a finite mass resistance at the external surface of the sphere. While the work is applicable to the generic problem of heat or mass transfer in a multi-layer sphere, the analysis and results are presented in the context of drug kinetics for desorbing and absorbing spherical microcapsules.

Calculating how long it takes for a diffusion process to effectively reach steady state without computing the transient solution.

Mathematically, it takes an infinite amount of time for the transient solution of a diffusion equation to transition from initial to steady state. Calculating a finite transition time, defined as the time required for the transient solution to transition to within a small prescribed tolerance of the steady-state solution, is much more useful in practice. In this paper, we study estimates of finite transition times that avoid explicit calculation of the transient solution by using the property that the transition to steady state defines a cumulative distribution function when time is treated as a random variable.

Quantifying the efficacy of first aid treatments for burn injuries using mathematical modelling and in vivo porcine experiments.

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.