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We compute profile likelihoods for a stochastic model of diffusive transport motivated by experimental observations of heat conduction in layered skin tissues. This process is modelled as a random walk in a layered one-dimensional material, where each layer has a distinct particle hopping rate. Particles are released at some location, and the duration of time taken for each particle to reach an absorbing boundary is recorded. To explore whether this data can be used to identify the hopping rates in each layer, we compute various profile likelihoods using two methods: first, an exact likelihood is evaluated using a relatively expensive Markov chain approach; and, second we form an approximate likelihood by assuming the distribution of exit times is given by a Gamma distribution whose first two moments match the expected moments from the continuum limit description of the stochastic model. Using the exact and approximate likelihoods we construct various profile likelihoods for a range of problems. In cases where parameter values are not identifiable, we make progress by re-interpreting those data with a reduced model with a smaller number of layers., 41 pages, 11 figures

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 on GitHub., 31pages, 12 figures

Calculating how long a coupled multi-species 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. Three numerical examples are presented to confirm the analysis and investigate the efficacy of the approach. A key finding is that for sequential reactions our approach works better provided the two smallest reaction rates are well separated. MATLAB code implementing the methodology and reproducing the results in the paper is made available., 8 pages, 3 figures, accepted version of paper published in Physical Review E

In this work we consider a recent experimental data set describing heat conduction in living porcine tissues. Understanding this novel data set is important because porcine skin is similar to human skin. Improving our understanding of heat conduction in living skin is relevant to understanding burn injuries, which are common, painful and can require prolonged and expensive treatment. A key feature of skin is that it is layered, with different thermal properties in different layers. Since the experimental data set involves heat conduction in thin living tissues of anesthetised animals, an important experimental constraint is that the temperature within the living tissue is measured at one spatial location within the layered structure. Our aim is to determine whether this data is sufficient to reliably infer the heat conduction parameters in layered skin, and we use a simplified two-layer mathematical model of heat conduction to mimic the generation of experimental data. Using synthetic data generated at one location in the two-layer mathematical model, we explore whether it is possible to infer values of the thermal diffusivity in both layers. After this initial exploration, we then examine how our ability to infer the thermal diffusivities changes when we vary the location at which the experimental data is recorded, as well as considering the situation where we are able to monitor the temperature at two locations within the layered structure. Overall, we find that our ability to parameterise a model of heterogeneous heat conduction with limited experimental data is very sensitive to the location where data is collected. Our modelling results provide guidance about optimal experimental design that could be used to guide future experimental studies.NomenclatureA brief description of all variables used in the document are given in Table 1.Table 1:Variable nomenclature and description.

International audience; Numerous problems involving gradient-driven transport processes—e.g., Fourier's and Darcy's law—in heterogeneous materials concern a physical domain that is much larger than the scale at which the coefficients vary spatially. To overcome the prohibitive computational cost associated with such problems, the well-established Distributed Microstructure Model (DMM) provides a two-scale description of the transport process that produces a computationally cheap approximation to the fine-scale solution. This is achieved via the introduction of sparsely distributed micro-cells that together resolve small patches of the fine-scale structure: a macroscopic equation with an effective coefficient describes the global transport and a microscopic equation governs the local transport within each micro-cell. In this paper, we propose a new formulation, the Extended Distributed Microstructure Model (EDMM), where the macroscopic flux is instead defined as the average of the microscopic fluxes within the micro-cells. This avoids the need for any effective parameters and more accurately accounts for a non-equilibrium field in the micro-cells. Another important contribution of the work is the presentation of a new and improved numerical scheme for performing the two-scale computations using control volume, Krylov subspace and parallel computing techniques. Numerical tests are carried out on two challenging test problems: heat conduction in a composite medium and unsaturated water flow in heterogeneous soils. The results indicate that while DMM is more efficient, EDMM is more accurate and is able to capture additional fine-scale features in the solution.

Diffusion in a composite slab consisting of a large number of layers provides an ideal prototype problem for developing and analysing two-scale modelling approaches for heterogeneous media. Numerous analytical techniques have been proposed for solving the transient diffusion equation in a one-dimensional composite slab consisting of an arbitrary number of layers. Most of these approaches, however, require the solution of a complex transcendental equation arising from a matrix determinant for the eigenvalues that is difficult to solve numerically for a large number of layers. To overcome this issue, in this paper, we present a semi-analytical method based on the Laplace transform and an orthogonal eigenfunction expansion. The proposed approach uses eigenvalues local to each layer that can be obtained either explicitly, or by solving simple transcendental equations. The semi-analytical solution is applicable to both perfect and imperfect contact at the interfaces between adjacent layers and either Dirichlet, Neumann or Robin boundary conditions at the ends of the slab. The solution approach is verified for several test cases and is shown to work well for a large number of layers. The work is concluded with an application to macroscopic modelling where the solution of a fine-scale multilayered medium consisting of two hundred layers is compared against an “up-scaled” variant of the same problem involving only ten layers.

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 that provide useful indicators of the release time, i.e. the time required for the drug-filled capsule to be depleted. Both approaches make use of temporal moments of the drug concentration versus time curve at the centre of the capsule, which provide useful insight into the timescale of the process and can be computed exactly without explicit calculation of the full transient solution of the multi-layer diffusion model. The first approach, which uses the zeroth and first temporal moments only, provides simple algebraic expressions involving the various parameters in the model (e.g. layer diffusivities, mass transfer coefficients, partition coefficients) to characterize the release time while the second approach yields an asymptotic estimate of the release time that depends on consecutive higher moments. Through several test cases, we show that both approaches provide a computationally-cheap and useful measure to compare \textit{a priori} the release time of different composite microcapsule configurations., 15 pages, 4 figures, submitted

We propose an alternative 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. Computational experiments comparing our advection-diffusion homogenized model to the standard homogenized model demonstrate that including an advection term in the homogenized equation leads to improved approximations of the solution of the original heterogeneous model., Comment: 7 pages, 2 figures, accepted version of paper published in Physical Review E

This paper presents a simple tool for characterising the timescale 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. To characterise the timescale, we use the concept of mean action time, which provides the mean timescale at each position in the medium by utilising the fact that the transition of the transient solution of the underlying partial differential equation model, from initial state to steady state, can be represented as a cumulative distribution function of time. Using this concept, we define the characteristic timescale for a multilayer diffusion process as the maximum value of the mean action time across the layered medium. For given initial conditions and internal and external boundary conditions, this approach leads to simple algebraic expressions for characterising the timescale that depend on the physical and geometrical properties of the medium, such as the diffusivities and lengths of the layers. Numerical examples demonstrate that these expressions provide useful insight into explaining how the parameters in the model affect the time it takes for a multilayer diffusion process to reach steady state., Comment: 15 pages, 2 figures, accepted version of paper published in Physical Review E

Highlights - We study the boundary conditions (BCs) in a macroscale model of layered diffusion. - A set of corrected macroscale BCs are derived using the method of volume averaging. - For example: a Dirichlet BC at the microscale leads to a Robin BC at the macroscale. - Resulting macroscale field more accurately captures averaged microscale field. - Resulting reconstructed field is in excellent agreement with true microscale field. Abstract This paper investigates the form of the boundary conditions (BCs) used in macroscale models of PDEs with coefficients that vary over a small length-scale (microscale). Specifically, we focus on the one-dimensional multilayer diffusion problem, a simple prototype problem where an analytical solution is available. For a given microscale BC (e.g., Dirichlet, Neumann, Robin, etc.) we derive a corrected macroscale BC using the method of volume averaging. For example, our analysis confirms that a Robin BC should be applied on the macroscale if a Dirichlet BC is specified on the microscale. The macroscale field computed using the corrected BCs more accurately captures the averaged microscale field and leads to a reconstructed microscale field that is in excellent agreement with the true microscale field. While the analysis and results are presented for one-dimensional multilayer diffusion only, the methodology can be extended to and has implications on a broader class of problems.

We develop a new semi-analytical method for solving multilayer diffusion problems with time-varying external boundary conditions and general internal boundary conditions at the interfaces between adjacent layers. The convergence rate of the semi-analytical method, relative to the number of eigenvalues, is investigated and the effect of varying the interface conditions on the solution behaviour is explored. Numerical experiments demonstrate that solutions can be computed using the new semi-analytical method that are more accurate and more efficient than the unified transform method of Sheils [Appl. Math. Model., 46:450-464, 2017]. Furthermore, unlike classical analytical solutions and the unified transform method, only the new semi-analytical method is able to correctly treat problems with both time-varying external boundary conditions and a large number of layers. The paper is concluded by replicating solutions to several important industrial, environmental and biological applications previously reported in the literature, demonstrating the wide applicability of the work., Comment: 24 pages, 8 figures, accepted version of paper published in Applied Mathematics and Computation

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 \textit{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. In total, three approaches are studied: (i) mean action time (ii) mean plus one standard deviation of action time and (iii) a new approach derived by approximating the large time asymptotic behaviour of the cumulative distribution function. The new approach leads to a simple formula for calculating the finite transition time that depends on the prescribed tolerance $\delta$ and the $(k-1)$th and $k$th moments ($k \geq 1$) of the distribution. Results comparing exact and approximate finite transition times lead to two key findings. Firstly, while the first two approaches are useful at characterising the time scale of the transition, they do not provide accurate estimates for diffusion processes. Secondly, the new approach allows one to calculate finite transition times accurate to effectively any number of significant digits, using only the moments, with the accuracy increasing as the index $k$ is increased., Comment: 17 pages, 2 figures, accepted version of paper published in Physical Review E