Our paper on Automated adjoints of coupled PDE-ODE systems is now published online in the SIAM Journal on Scientific Computing (SISC). 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 discuss an extension to the FEniCS finite element software for expressing and efficiently solving such coupled systems. Given an ODE described using an augmentation of the Unified Form Language (UFL) and a discretization 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. We demonstrate the capabilities of the approach on examples from cardiac electrophysiology and mitochondrial swelling.
Our paper on Towards personalized computer simulation of breast cancer treatment: a multi-scale pharmacokinetic and pharmacodynamic model informed by multi-type patient data has been published online in Cancer Research. (The bioRxiv preprint is also available.) For a popular science account, read about the study in the research magazine Apollon.
Abstract: The usefulness of mechanistic models to disentangle complex multi-scale cancer processes such as treatment response has been widely acknowledged. However, a major barrier for multi-scale models to predict treatment outcomes in individual patients lies in their initialization and parametrization which need to reflect individual cancer characteristics accurately. In this study we use multi-type measurements acquired routinely on a single breast tumor, including histopathology, magnetic resonance imaging, and molecular profiling, to personalize parts of a complex multi-scale model of breast cancer treated with chemotherapeutic and anti-angiogenic agents. The model accounts for drug pharmacokinetics and pharmacodynamics. We developed an open-source computer program that simulates cross-sections of tumors under 12-week therapy regimens and use it to individually reproduce and elucidate treatment outcomes of four patients. Two of the tumors did not respond to therapy, and model simulations were used to suggest alternative regimens with improved outcomes dependent on the tumor’s individual characteristics. It was determined that more frequent and lower doses of chemotherapy reduce tumor burden in a low proliferative tumor while lower doses of anti-angiogenic agents improve drug penetration in a poorly perfused tumor. Furthermore, using this model we were able to predict correctly the outcome in another patient after 12 weeks of treatment. In summary, our model bridges multi-type clinical data to shed light on individual treatment outcomes.
Our new preprint on Uncertainty quantification of parenchymal tracer distribution using random diffusion and convective velocity fields is now out on bioRxiv! My first time submitting to bioRxiv rather than good old arXiv.
Over the last decade, there has been a significant renewed interest in the waterscape of the brain; that is, the physiological mechanisms governing cerebrospinal fluid (CSF) and interstitial fluid (ISF) flow in (and around) the brain. A number of new theories have emerged, but a great deal of uncertainty relating to the roles of diffusion, convection and clearance within the brain remains. With this study, we aimed to rigorously quantify how the aforementioned uncertainties in the physiological parameters and in ISF flow affect the spread of a tracer into the brain. We assumed movement of tracer in the brain to occur by diffusion and/or convection. To account for uncertainty and variability, we circumvented the lack of precise parameter values by modelling velocity and diffusivity as Matérn stochastic fields. We then set up a PDE model with these stochastic (random) fields as coefficients and quantify the uncertainty in the model prediction via the Monte Carlo (MC) method.
Our paper on mixed finite element methods for nearly incompressible multiple-network poroelasticity has now appeared online in SISC.
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 which fenicsproject # Ready to go! fenicsproject run
Our paper on Fluid dynamics in syringomyelia cavities: Effects of heart rate, CSF velocity, CSF velocity waveform and craniovertebral decompression was published earlier this fall in The Neuroradiology Journal.
How fluid moves during the cardiac cycle within a syrinx (a fluid-filled cyst in the spinal cord) may affect its development. We measured syrinx fluid velocities before and after craniovertebral decompression in a patient and simulated syrinx fluid velocities for different heart rates, syrinx sizes and cerebrospinal fluid (CSF) flow velocities in a model of syringomyelia. With phase-contrast magnetic resonance we measured CSF and syrinx fluid velocities in a Chiari patient before and after craniovertebral decompression. With an idealized two-dimensional model of the subarachnoid space (SAS), cord and syrinx, we simulated fluid movement in the SAS and syrinx with the Navier-Stokes equations for different heart rates, inlet velocities and syrinx diameters. In the patient, fluid oscillated in the syrinx at 200 to 210 cycles per minute before and after craniovertebral decompression. Velocities peaked at 3.6 and 2.0 cm per second respectively in the SAS and the syrinx before surgery and at 2.7 and 1.5 cm per second after surgery. In the model, syrinx velocity varied between 0.91 and 12.70 cm per second. Increasing CSF inlet velocities from 1.56 to 4.69 cm per second increased peak syrinx fluid velocities in the syrinx by 151% to 299% for the three cycle rates. Increasing cycle rates from 60 to 120 cpm increased peak syrinx velocities by 160% to 312% for the three inlet velocities. Peak velocities changed inconsistently with syrinx size. In conclusion, CSF velocity, heart rate and syrinx diameter affect syrinx fluid velocities, but not the frequency of syrinx fluid oscillation. Craniovertebral decompression decreases both CSF and syrinx fluid velocities.
Our new paper on A mixed finite element method for nearly incompressible multiple-network poroelasticity is now available on arXiv!
In this paper, we present and analyze a new mixed finite element formulation of a general family of quasi-static multiple-network poroelasticity (MPET) equations. The MPET equations describe flow and deformation in an elastic porous medium that is permeated by multiple fluid networks of differing characteristics. As such, the MPET equations represent a generalization of Biot’s equations, and numerical discretizations of the MPET equations face similar challenges. Here, we focus on the nearly incompressible case for which standard mixed finite element discretizations of the MPET equations perform poorly. Instead, we propose a new mixed finite element formulation based on introducing an additional total pressure variable. By presenting energy estimates for the continuous solutions and a priori error estimates for a family of compatible semi-discretizations, we show that this formulation is robust in the limits of incompressibility, vanishing storage coefficients, and vanishing transfer between networks. These theoretical results are corroborated by numerical experiments. Our primary interest in the MPET equations stems from the use of these equations in modelling interactions between biological fluids and tissues in physiological settings. So, we additionally present physiologically realistic numerical results for blood and tissue fluid flow interactions in the human brain.
Congratulations to Cécile Daversin-Catty for winning the Best postdoctoral presentation award at the FEniCS’18 conference at the University of Oxford from March 21-23 2018! Cécile, postdoctoral fellow with the Waterscales project, presented her exciting and versatile work on mixed-dimensional coupled finite elements in FEniCS.
Overall, the Waterscales and Waterscape projects were well represented in the FEniCS’18 program with additional talks from Waterscape post doc Travis Thompson (on Stokes-Biot stable H(div)-based mixed finite element methods for generalized poroelasticity), from SUURPh PhD candidate Eleonora Piersanti (on Parameter-robust discretization and preconditioning of the multiple-network poroelasticity equations), and a poster presentation from Waterscales PhD candidate Ada Ellingsrud (on Numerical modelling of the role of glial cells in cerebral interstitial fluid movement). Well done Waterscapers!
Our new paper on Efficient white noise sampling and coupling for multilevel Monte Carlo with non-nested meshes is now available on arXiv!
When solving stochastic partial differential equations (SPDEs) driven by additive spatial white noise, the efficient sampling of white noise realizations can be challenging. In this paper, we present a new sampling technique that can be used to efficiently compute white noise samples in a finite element method and multilevel Monte Carlo (MLMC) setting. The key idea is to exploit the finite element matrix assembly procedure and factorize each local mass matrix independently, hence avoiding the factorization of a large matrix. Moreover, in a MLMC framework, the white noise samples must be coupled between subsequent levels. We show how our technique can be used to enforce this coupling even in the case of non-nested mesh hierarchies. We demonstrate the efficacy of our method with numerical experiments. We observe optimal convergence rates for the finite element solution of the elliptic SPDEs of interest in 2D and 3D and we show convergence of the sampled field covariances. In a MLMC setting, a good coupling is enforced and the telescoping sum is respected.
This week I’ve been in Odense for a PhD defense (Congratulations to Dr. Christian Valdemar Hansen!), while today I am giving at a talk at Nansensenteret i Bergen for a meeting organized by Norges Tekniske Vitenskapsakademi and Tekna. I will talk about the brain’s waterscape/Hjernens vannveier, of course. This presentation targets a semi-academic, semi-technical audience and the slides are publicly available.