Mathematical modelling of ionic electrodiffusion and water movement is emerging as a powerful avenue of investigation to provide new physiological insight into brain homeostasis. However, in order to provide solid answers and resolve controversies, the accuracy of the predictions is essential. Ionic electrodiffusion models typically comprise non-trivial systems of non-linear and highly coupled partial and ordinary differential equations that govern phenomena on disparate time scales. Here, we study numerical challenges related to approximating these systems. We consider a homogenized model for electrodiffusion and osmosis in brain tissue and present and evaluate different associated finite element-based splitting schemes in terms of their numerical properties, including accuracy, convergence, and computational efficiency for both idealized scenarios and for the physiologically relevant setting of cortical spreading depression (CSD). We find that the schemes display optimal convergence rates in space for problems with smooth manufactured solutions. However, the physiological CSD setting is challenging: we find that the accurate computation of CSD wave characteristics (wave speed and wave width) requires a very fine spatial and fine temporal resolution.
In these studies, we consider systems of equations describing flow in multiple, interacting fluid networks with different permeabilities. The resulting discrete systems of equations can easily become very large, and efficient solution techniques are key. In cases where the inter-network interactions dominate the intra-network permeabilities, we observed that the systems of equations became increasingly hard to solve. In particular, the number of solver iterations would increase with increasing discrete resolution and interaction parameters. However, by creating an algorithm for defining tailored changes of variables, we were able to define robust preconditioners with uniformly bounded iteration counts.
Personal favorite: The lovely trick for how to define parameter-tailored changes of variables via classical matrix analysis including simultaneous diagonalization by congruence.
Infusion testing is a common procedure to determine whether shunting will be beneficial in patients with normal pressure hydrocephalus. The method has a well-developed theoretical foundation and corresponding mathematical models that describe the CSF circulation from the choroid plexus to the arachnoid granulations. Here, we investigate to what extent the proposed glymphatic or paravascular pathway (or similar pathways) modifies the results of the traditional mathematical models.
We used a two-compartment model consisting of the subarachnoid space and the paravascular spaces. For the arachnoid granulations, the cribriform plate, capillaries and paravascular spaces, resistances were calculated and used to estimate flow before and during an infusion test. Next, pressure in the subarachnoid space and paravascular spaces were computed. Finally, different variations to the model were tested to evaluate the sensitivity of selected parameters.
At baseline, we found a very small paravascular flow directed into the subarachnoid space, while 60% of the fluid left through the arachnoid granulations and 40% left through the cribriform plate. However, during the infusion, paravascular flow reversed and 25% of the fluid left through these spaces, while 60% went through the arachnoid granulations and only 15% through the cribriform plate.
The relative distribution of CSF flow to different clearance pathways depends on intracranial pressure (ICP), with the arachnoid granulations as the main contributor to outflow. As such, ICP increase is an important factor that should be addressed when determining the pathways of injected substances in the subarachnoid space.
I have finally updated my laptop from the arcane Ubuntu 16.04 to 18.04. First thing to configure: fluxbox :heart: of course. Second, xterm and emacs font, size and color. And third, custom FEniCS installation! Very easy this time around, but recording it here for future reference:
# Install curl
sudo apt install curl
# Download FEniCS project script
curl -s https://get.fenicsproject.org | bash
# fenicsproject script is installed as /foo/.local/bin/fenicsproject
# Add to e.g. .bashrc:
# export PATH=/foo/.local/bin/fenicsproject:$PATH
# Check that you are using the expected version of fenicsproject, by
# examining output of
# Ready to go!
Our preprint on Automated adjoints of coupled ODE-PDE systems is now available on arXiv! Mathematical models that couple partial differential equations (PDEs) and spatially distributed ordinary differential equations (ODEs) arise in biology, medicine, chemistry and many other fields. In this paper we present extension to the FEniCS and dolfin-adjoint softwares for expressing and efficiently solving such coupled systems. Given an ODE described using an augmentation of the Unified Form Language (UFL) and a discretisation described by an arbitrary Butcher tableau, efficient code is automatically generated for the parallel solution of the ODE. The high-level description of the solution algorithm also facilitates the automatic derivation of the adjoint and tangent linearization of coupled PDE-ODE solvers. The supplementary code for the paper is also openly available.
I am happy to announce that Dr. Travis Thompson joined the Waterscape project as a postdoctoral fellow at Simula Research Laboratory on June 8 2017. Travis received a PhD in Mathematics from Texas A&M university in 2013; his research areas are numerical analysis, and scientific computing. Recently published work is focused on the construction of a-priori error estimates, and solver development, for finite element methods applied to problems in computational fluid dynamics, and biomechanics. Within Waterscape, Travis will be studying a-priori and a-posteriori error analysis of mixed finite element methods for generalized poroelasticity aiming at accurate, robust and property-preserving methods.
Late last year, I received an email from a member of the TEDxOslo organization committee, asking if I would be interested in giving a TEDx talk at TEDxOslo 2017 taking place in the National Theatre on May 3 2017. Enjoy the end result here:
I am truly grateful and honoured that the European Research Council has awarded me with a 5-year Starting Grant within Mathematics (PE1) to fund the Waterscales project, a project dedicated to the mathematical and computational foundations for modeling cerebral fluid flow.
The Waterscales vision
Over the next decades, mathematics and numerics could play a crucial role in gaining new insight into the mechanisms driving water transport through the brain. Indeed, medical doctors express an urgent need for multiscale modeling and simulation – to overcome fundamental limitations in traditional techniques. Surprisingly little attention has been paid to the numerics of the brain’s waterscape however, in stark contrast to the role of simulation in other fields of neuroscience, and key mathematical models and methods are missing. To address this important challenge, the overall ambition of the Waterscales project is to establish the mathematical, numerical and computational foundations for predictively modeling fluid flow and solute transport through the brain across spatiotemporal scales – from the cellular to the organ level.