Our paper on A Cell-Based Framework for Numerical Modeling of Electrical Conduction in Cardiac Tissue is now (open access) published! In this paper, we study a mathematical model of cardiac tissue based on explicit representation of individual cells. In this EMI model, the extracellular (E) space, the cell membrane (M), and the intracellular (I) space are represented as separate geometrical domains. This representation introduces modeling flexibility needed for detailed representation of the properties of cardiac cells including their membrane. In particular, we will show that the model allows ion channels to be non-uniformly distributed along the membrane of the cell. Such features are difficult to include in classical homogenized models like the monodomain and bidomain models frequently used in computational analyses of cardiac electrophysiology.
Our paper on High-resolution data assimilation of cardiac mechanics applied to a dyssynchronous ventricle is now published! Computational models of cardiac mechanics, personalized to a patient, offer access to mechanical information above and beyond direct medical imaging. Additionally, such models can be used to optimize and plan therapies in-silico, thereby reducing risks and improving patient outcome. Model personalization has traditionally been achieved by data assimilation, which is the tuning or optimization of model parameters to match patient observations. Current data assimilation procedures for cardiac mechanics are limited in their ability to efficiently handle high-dimensional parameters. This restricts parameter spatial resolution, and thereby the ability of a personalized model to account for heterogeneities that are often present in a diseased or injured heart. In this paper, we address this limitation by proposing an adjoint gradient–based data assimilation method that can efficiently handle high-dimensional parameters. We test this procedure on a synthetic data set and provide a clinical example with a dyssynchronous left ventricle with highly irregular motion. Our results show that the method efficiently handles a high-dimensional optimization parameter and produces an excellent agreement for personalized models to both synthetic and clinical data.
The European Conference on Numerical Mathematics and Advanced Applications (ENUMATH) was organized by the University of Bergen at Voss this week, Sept 25-29 2017. The Waterscapes research teams were well represented at the conference with the plenary open lecture on “The operator preconditioning framework with various applications to interstitial fluid flow and the aging human brain” given by Waterscape collaborator Kent-Andre Mardal, and I presenting a plenary lecture on “Compatible discretizations in our hearts and mind”. Slides available on SlideShare:
The Cutting Edge festival is “Norway’s largest interdisciplinary business jam session for scientists, technologists, entrepreneurs, investors and policy makers”. The 2017 edition had a session on “Brain and Aging” where I presented the core vision of the Waterscapes projects: to improve our fundamental understanding of fluid flow and solute transport in brain tissue using mathematical modelling. Here are the Cutting Edge 2017 slides (all simulations were produced using the Waterscapes solver collection), and the talk:
Every ballet ensemble or opera has a primadonna. Most football teams have a primadonna. Hey, probably every work place has at least one primadonna. You know, the one that gets and takes all of the attention. In the brain, the neuron is the primadonna, the one that everyone cares about. But, I think we all know that in a ensemble or a football team or a work place, it is all the different pieces playing together that really matters. So, in the brain, we should also care about the other types of brain cells those called glial cells, which very rarely anyone has heard of, we should care about the water-filled spaces between cells, we should care about the brain’s blood vessels. And this is precisely I call the brain’s waterscape: the fluid-filled pathways, the canals and the water channels within the brain.
We should care about these because they are all crucial for the well-being of the brain, for the well-being of the primadonna neurons. And we know that a number of brain disorders, like Alzheimer’s, like Parkinson’s, like oedema, like Stroke are associated with abnormalities in the brain’s waterscape. But at the same time, we know surprisingly little about how the fundamental mechanisms underlying the waterscape, about the fundamental mechanisms for water balance and water-enhanced clearance of metabolic waste in the brain. One reason for this is that experimental investigations into the mechanics of the human brain are extremely challenging: both from a practical and an ethical viewpoint.
So in view of this challenge, I and my research team, we are spear-heading a different approach into investigating the brain’s waterscape, a mathematical approach. We are developing and using mathematical and computational models to describe fluid flow and transport within brain tissue. Our mathematical models are physics-based, so what we do is to combine basic laws of physics for how fluids move and how fluid can transport solutes and how fluids interact with structures like biological cells, and we express these as mathematical equations. These equations can then predict how the modelled system will evolve over time.
This is the simplest mathematical model that I know of. It is called the diffusion equation, and it describes how a solute could spread in brain tissue. The solute could be like injected dye or injected drugs or natural by-products like protein fragments from the brain’s metabolism. These models consist of three main components: an unknown, something known and the link between the known and the unknown. The unknown, or the output, is what we want to predict or compute, in this case the concentration of the solute over time and space. The known, or the input parameters, could be the geometry of the brain, the initial amount of the solute, and some hopefully quantifyiable properties of the brain tissue in question. And the physical laws that we choose to include provides the link between the two.
Now, this looks like a very innocent equation, but it actually impossible to solve using pen and paper only in general. So, we actively develop and use high performance numerical simulation technology to compute the output of this model. These illustrations on your far right show how this model predicts that dye injected inside a brain would spread over time. Now, we can add complexity to this model … we can account for additional solute transport … we can account for more complex physics, but the core idea remains the same.
Now, the remarkable thing about computational models is that we can experiment and make predictions in a way that would never be possible in physical experiments. The first aspect we aim at is a better understanding of the fundamental mechanisms for how metabolic waste and other solutes can be cleared from the brain, which is essential to understand how normal clearance and, to understand what fails and leads to accumulation of waste in Alzheimer’s patients for instance, and also to understand how treatment drugs can be delivered into the brain. Using these computational models we can test clinical hypotheses, that may be impossible to test in a lab or in a patient, to see if existing hypotheses make physical sense. In this way, we envision using mathematical models to drive basic medical science forward.
But we can go much further than that. We can make predictions under changing scenarios. For instance, we know that arteries stiffen and that brains shrink as we age. And we can represent this behaviour in the model by representing it as a change in the input parameters of our equations. We can thus make predictions for aging brains, and link these predictions with features of the disease. And if we can understand the underlying cause of a disease, that will be a crucial step on the way to cure or prevent diseases, like Alzheimer’s, where we have no cure today.
And going even further, another possibility is to create personalized models. By including person-specific information in the input parameters of our models, like person-specific geometries, we can create simulations and make predictions specific to an individual. This means that we can evaluate therapies and even design optimal therapies tailored to highly stratified patient groups or even tailored to each and every one of us.
So, in the next years, I envision that mathematical models of the brain’s waterscape can provide a more practical, more ethical and less expensive avenue to understand the brain’s mechanisms, how these vary from individual to individual, and ultimately towards designing, improving and tailoring medical treatments.
And as such find a way to keep the brain’s primadonnas, the neurons, happy and healthy. Thank you.
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 Ms. Ada Ellingsrud joined the Waterscales project as a PhD candidate at Simula Research Laboratory on Aug 15 2017. Ada received a Masters degree in Applied Mathematics from the University of Oslo in 2015. Her thesis investigated preconditioning of unified mixed discretizations of the coupled Darcy-Stokes problem. Within Waterscales, Ada will be developing and studying multiscale models and methods for fluid flow and transport in brain tissue bridinging glial cell dynamics, osmotic pressures and tissue level flow.
Our paper on Inverse estimation of cardiac activation times via gradient-based optimisation is now published online in the International Journal for Numerical Methods in Biomedical Engineering! In this study, we use a PDE-constrained optimal control approach to numerically investigate the identifiability of an initial activation sequence from synthetic observations of the extracellular potential using the bidomain approximation and 2D representations of heart tissue. Our results demonstrate that activation times and duration of several stimuli can be recovered even with high levels of noise, that it is sufficient to sample the observations at the ECG-relevant sampling frequency of 1 kHz, and that spatial resolutions that are coarser than the standard in electrophysiological simulations can be used.