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.
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.
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.
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.
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!
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.