Publications & Writing

Journal articles

  • A. Eisenträger, D. Vella, and I. M. Griffiths. Particle capture efficiency in a multi-wire model for high gradient magnetic separation. Applied Physics Letters, 105(3): 033508, 2014. doi:10.1063/1.4890965.
    High gradient magnetic separation (HGMS) is an efficient way to remove magnetic and paramagnetic particles, such as heavy metals, from waste water. As the suspension flows through a magnetized filter mesh, high magnetic gradients around the wires attract and capture the particles removing them from the fluid. We model such a system by considering the motion of a paramagnetic tracer particle through a periodic array of magnetized cylinders. We show that there is a critical Mason number (ratio of viscous to magnetic forces) below which the particle is captured irrespective of its initial position in the array. Above this threshold, particle capture is only partially successful and depends on the particle’s entry position. We determine the relationship between the critical Mason number and the system geometry using numerical and asymptotic calculations. If a capture efficiency below 100% is sufficient, our results demonstrate how operating the HGMS system above the critical Mason number but with multiple separation cycles may increase efficiency.
  • B. Wirth, I. Sobey, and A. Eisenträger. Conditions for choking in a poroelastic flow model. IMA Journal of Applied Mathematics, 79(2): 254-273, 2014. doi:10.1093/imamat/hxs062. First published online: September 4, 2012.
    We consider a steady Darcy flow of fluid through a section of poroelastic material under an applied pressure difference. In porous media flow, the superficial fluid velocity and applied pressure gradient are related by a permeability such that, when the porous material is rigid, specifying either the flow rate or the applied pressure difference allows the determination of the other quantity and flow will occur for all non-zero applied pressure gradients. When the underlying solid matrix is a deformable elastic material and the fluid permeability can vary with applied strain on the solid matrix, then the permeability can reduce to zero and the solid matrix become impermeable, so that flow is not possible and no steady solution exists. We examine existing steady poroelastic equations for a number of different well-known elastic models and permeability–strain relations and consider solutions for the resulting steady strain and fluid flow. We solve the governing equations numerically for some example situations and propose a sufficient condition on the functional form of the permeability–strain relation for solutions to exist for any specified applied pressure.
  • A. Eisenträger, I. Sobey, and M. Czosnyka. Parameter estimations for the cerebrospinal fluid infusion test. Mathematical Medicine and Biology, 30(2): 157-174, 2013. doi: 10.1093/imammb/dqs001. First published online: February 16, 2012.
    We consider parameter estimation for a single compartment model of a cerebrospinal fluid (CSF) infusion test using an inverse power law cerebral compliance depending on intracranial pressure (ICP). A least squares optimization is used to solve the inverse problem of estimating model parameter values from ICP observed during an infusion test. The optimization is applied to synthetic test data and to clinical ICP data from a number of infusion tests. We show that in tests that successfully reach a new plateau pressure, the resistance to CSF outflow and elastance can be reliably estimated using an automated least squares process. We find that a single infusion test does not provide enough resolution to distinguish between compliance models.
  • I. Sobey, A. Eisenträger, B. Wirth, and M. Czosnyka. Simulation of cerebral infusion tests using a poroelastic model. International Journal of Numerical Analysis and Modeling, Series B, 3(1):52–64, 2012.
    In an infusion test the apparent rate of cerebrospinal fluid (CSF) production is temporarily increased through injection of fluid directly into the CSF system with the result that CSF pressure rises, in theory to a new plateau average, and the change in pressure level gives a measure of resistance to CSF outflow and the rate of approach to the plateau gives information about cerebral compliance. In the first part of this paper we give details of a two-fluid (blood and CSF) spherically symmetric poroelastic model that can simulate an infusion test which includes oscillations in blood pressure. This model has been applied to clinical data where the infusion rate and arterial blood pressure are input and an oscillatory CSF pressure is computed along with spatial parenchyma displacement, strain and local changes in CSF content. In the later part of this paper, the poroelastic model is simplified by spatial integration resulting in a one-compartment model that includes blood pressure oscillations but which, when they are ignored, reduces to a well known one-compartment model. When the arterial pressure pulsations are included, their interaction with a non-linear compliance results in solutions that have to be interpreted very carefully to predict parameter values.

Conference proceedings

  • A. Eisenträger and I. Sobey. Multi-Fluid Poroelastic Modelling of CSF Flow Through the Brain. In C. Hellmich, B. Pichler and D. Adam, editors, Poromechanics V: Proceedings of the Fifth Biot Conference on Poromechanics, pages 2148-2157, 2013. doi: 10.1061/9780784412992.253.
    Cerebrospinal fluid (CSF) fills a system of cavities at the center of the brain, surrounds the brain and the spinal cord, and is in free communication with the interstitial fluid of the brain. Disturbances in CSF dynamics can lead to diseases causing severe brain damage. Biomechanical modelling of CSF flow through the brain tissue is crucial to improve our understanding of this type of disease. While poroelastic theory has repeatedly been used to model different aspects of CSF-brain interactions, recently the indirect interaction between blood and CSF has brought multi-fluid modelling into focus of research. We review two types of multi-fluid poroelastic models of the brain and show how they can be combined to form a generalized model. In addition, we relate a modification of one of these models to single-compartment models used in clinical practice. This yields further information about the spatial fluid distribution and long term behavior.
  • I. Sobey, A. Eisenträger, B. Wirth, and M. Czosnyka. Multi-fluid poro-elastic modelling of the CSF infusion test. In C.T. Lim and J.C.H. Goh, editors, 6th World Congress of Biomechanics (WCB 2010), volume 31 of IFMBE Proceedings, pages 362–365. Springer, 2010. doi:10.1007/978-3-642-14515-5_93.
    We have developed a multi-fluid model for flow of cerebro-spinal fluid (CSF) in the brain which incorporates both CSF and blood pressure variations and has general application for study of hydrocephalus, and particularly here, for simulation of a CSF infusion test. Existing poro-elastic models for CSF flow treat the parenchyma and CSF as a continuum where very slow fluid flow interacts with and can affect the deformation of the underlying elastic substructure. A spherical model with CSF production at the centre of the brain, which allows flow through the aqueduct and very slow flow through the parenchyma before absorption at the outer periphery of the brain, has provided plausible results for hydrocephalus. However, while that model provides new predictions about the interior state of the parenchyma, it is only applicable for long time scales that neglect blood pulsations on the cardiac cycle time scale. Here we describe a model which, by adding a blood compartment to the usual poro-elastic framework, allows blood pulsations to be included and so to attempt a time accurate simulation of CSF pressure fluctuations during an infusion test.


  • A. Eisenträger, I. Sobey, and M. Czosnyka. Parameter estimations for the
    cerebrospinal fluid infusion test
    . Technical report, Mathematical Institute,
    University of Oxford, Dec. 2011. This is a previous version of the above article in Mathematical Medicine and Biology.
    We consider a single compartment model for a cerebrospinal fluid infusion test using first, an inverse power law cerebral compliance with constant elastance and second, a more recently developed compliance model with elastance depending on intracranial pressure (ICP). A least squares optimisation is used to solve the inverse problem of estimating parameter values from ICP observed during an infusion test. The optimisation is applied to synthetic data and to clinical ICP data from eleven infusion tests. From consideration of synthetic data we show that it is not, in general, possible to distinguish between compliance models and attempting to extract parameter estimates for an extended compliance model is not well posed with different parameter sets having ICP time dependence that cannot be distinguished using current ICP measurements. This conclusion is also confirmed in examination of clinical data.
  • B. Wirth, I. Sobey, and A. Eisenträger. A note on the solution of a poroelastic problem. Technical report, Mathematical Institute, University of Oxford, Feb. 2010. This is a previous version of the above article “Conditions for choking in a poroelastic flow model”.
    We examine the forced flow through a poroelastic matrix. This apparently simple problem turns out to exhibit some interesting difficulties, and in some circumstances the problem may not be well-posed. It can be regarded as the inverse of that problem where a given pressure difference is applied across a section of poroelastic material. There are two difficulties that arise. In one situation, for certain choices of the function relating permeability to the elastic deformation, compression of the matrix pores results in choking as the underlying matrix permeability deceases. In the second situation, the underlying elastic model fails because the Jacobian of the deformation becomes negative. We examine a sufficient condition on the functional form of the permeability-strain relation for solutions to exist for all flow rates.


  • A. Eisenträger. Finite Element Simulation of a Poroelastic Model of the CSF System in the Human Brain during an Infusion Test. DPhil thesis, University of Oxford, 2012. PDF available on request.
    Cerebrospinal fluid (CSF) fills a system of cavities at the centre of the brain, known as ventricles, and the subarachnoid space surrounding the brain and the spinal cord. In addition, CSF is in free communication with the interstitial fluid of the brain tissue. Disturbances in CSF dynamics can lead to diseases that cause severe brain damage or even death. So-called infusion tests are frequently performed in the diagnosis of such diseases. In this type of test, changes in average CSF pressure are related to changes in CSF volume through infusion of known volumes of additional fluid.

    Traditionally, infusion tests are analysed with single compartment models, which treat all CSF as part of one compartment and balance fluid inflow, outflow and storage through a single ordinary differential equation. Poroelastic models of the brain, on the other hand, have been used to simulate spatial changes with disease, particularly of the ventricle size, on larger time scales of days, weeks or months. Wirth and Sobey (2008) developed a two-fluid poroelastic model of the brain in which CSF pressure pulsations are linked to arterial blood pressure pulsations. In this thesis, this model is developed further and simulation results are compared to clinical data.

    At first, the functional form of the compliance, which governs the storage of CSF in single compartment models, is examined by comparison of two different compliance models with clinical data. The derivations of a single-fluid and a two-fluid poroelastic model of the brain in spherical symmetry are laid out in detail and some of the parameters are related to the compliance functions considered earlier. The finite element implementation of the two-fluid model is described and finally simulation results of the average CSF pressure response and the pressure pulsations are compared to clinical data.

  • A. Eisenträger. FE simulations for the plate equation on large deformations. Diplomarbeit, Technische Universität Chemnitz, Fakultät für Mathematik, 2008.
    The aim of this thesis is a model for the plate deformations, under consideration of large strains, and the implementation of a suitable FE simulation. Starting from nonlinear static 3D elasticity theory and introducing the Kirchhoff assumptions, the total energy of a deformed plate is described solely by the displacements of its midsurface. Minimizing this energy leads to a nonlinear variational problem, which can be solved numerically, using Newton’s method and the finite element method. For this purpose, the formulae of the energy functional and its necessary two linearizations are provided. With the further assumption that the normal of the midsurface does not change, an FE implementation is derived, with bilinear element functions in the in-plane-direction and bicubic element functions, from the BognerFox-Schmidt-element, in the out-of-plane-direction.

Presentation slides

Other writing

  • M. Dalwadi, E. Dubrovina, A. Eisenträger, A. Lee, J. Maestri, B. Matejczyk, D. O’Kiely, M. Stamper, S. Thomson. Toxic Chemicals and their Neutralising Agents in Porous Media, Sep. 2014. Report for the European Study Group with Industry ESGI100 in Oxford.
    Executive Summary
    The UK Government Decontamination Service advises central Government on the national capability for the decontamination of buildings, infrastructure, transport and open environment, and be a source of expertise in the event of a chemical, biological, radiological and nuclear (CBRN) incident or major release of HazMat materials. The study group constructed mathematical models to describe the depth to which a toxic chemical may seep into an initially dry porous substrate, and of the neutralisation process between a decontaminant and the imbibed chemical.
    The group recognised that capillary suction was the dominant process by which the contaminant spreads in the porous substrate. Therefore, in the first instance the absorption of the contaminant was modelled using Darcy’s law. At the next level of complication a diffuse interface model based on Richards’ equation was employed. The results of the two models were found to agree at early times, while at later times we found that the diffuse interface model predicted the more realistic scenario in which the contaminant has seeped deeper into the substrate even in the absence of further contaminant being supplied at the surface.
    The decontamination process was modelled in two cases; first, where the product of the decontamination reaction was water soluble, and the second where the reaction product formed soluble in the contaminant phase and of similar density. These simple models helped explain some of the key physics involved in the process, and how the decontamination process might be optimised. We found that decontamination was most effective in the first of these two cases.
    The group then sought to incorporate hydrodynamic effects into the reaction model. In the long wavelength limit, the governing equations reduced to a one-dimensional Stefan model similar to the one considered earlier. More detailed approximations and numerical simulations of this model were beyond the scope of this study group, but provide an entry point for future research in this area.
  • A. Eisenträger. Modelling Flow of Cerebrospinal Fluid, Oct. 2009. Report for the transfer of status from Probationary Research Student to DPhil Student, University of Oxford. PDF available on request.
  • A. Eisenträger. Simulating porous flow with chebfun, Oct. 2009. Special topic for the lecture course Approximation Theory, University of Oxford.
  • A. Eisenträger. Geometrisch exakte Modellierung einer elastischen dünnen Platte mittels Energieminimierung, Jan. 2008. Praktikumsbericht im Rahmen des Modellierungsseminars 2007/2008, Technische Universität Chemnitz.
  • A. Eisenträger. Nonmonotone Line Search and Trust Region Methods for Optimization, Apr. 2007. Special topic for lecture course Continuous Optimisation, University of Oxford.
  • R. Borsdorf and A. Eisenträger. Sequence Spaces, SS 2005, Technische Universität Chemnitz. Translation of “Folgenräume” by A. Eisenträger.
  • R. Borsdorf and A. Eisenträger. Folgenräume, WS 2004/2005. Vortrag im Seminar “Vektorwertige Funktionen”, Technische Universität Chemnitz.