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Mathematical and computational foundations for modeling cerebral fluid flow.

Periodic Reporting for period 3 - Waterscales (Mathematical and computational foundations for modeling cerebral fluid flow.)

Reporting period: 2020-04-01 to 2021-09-30

Your brain has its own waterscape: whether you are reading or sleeping, fluid flows through or along the brain and clears waste in the process. These physiological processes are crucial for the well-being of the brain. In spite of their importance we understand them but little. Mathematics and numerics could play a crucial role in gaining new insight. Indeed, medical doctors express an urgent need for multiscale modeling of water transport through the brain, to overcome limitations in traditional techniques. Surprisingly little attention has been paid to the numerics of the brain’s waterscape however, and fundamental knowledge is missing.

In response, the Waterscales ambition is to establish the mathematical and computational foundations for predictively modeling fluid flow and solute transport through the brain across scales – from the cellular to the organ level. The project aims to bridge multiscale fluid mechanics and cellular electrophysiology to pioneer new families of mathematical models that couple macroscale, mesoscale and microscale flow with glial cell dynamics. For these models, we will design numerical discretizations that preserve key properties and that allow for whole organ simulations. To evaluate predictability, we will develop a new computational platform for model adaptivity and calibration. The project is multidisciplinary combining mathematics, mechanics, scientific computing, and physiology.

If successful, this project enables the first in silico studies of the brain’s waterscape across scales. The new models would open up a new research field within computational neuroscience with ample opportunities for further mathematical and more applied study. The processes at hand are associated with neurodegenerative diseases e.g. dementia and with brain swelling caused by e.g. stroke. The Waterscales project will provide the field with a sorely needed, new avenue of investigation to understand these conditions, with tremendous long-term impact.
Since its beginning in April 2017, the Waterscales project has lead to the development of new mathematical models, methods and software allowing for in-silico investigations of brain mechanics and associated neurological disorders such as dementia, stroke and epilepsy.

Traditional mathematical models for excitable cells such as heart or brain cells are commonly based on the assumption that ionic concentrations are spatially constant. However, understanding variations in ionic concentrations are important in connection with neurological conditions such as e.g. migraine or epilepsy. To allow for studying concentration variations in brain tissue in-silico, the Waterscales team have designed new mathematical models and numerical methods for describing the distribution and evolution of ion concentrations in brain tissue at the microscale.

At a macroscale, we envision that fluid flow in porous and elastic media such as biological tissue in general and the brain in particular can be modelled mathematically via generalized poroelasticity equations. These equations are complicated time-dependent partial differential equations typically requiring hours on supercomputers to solve. To speed up the solution process, the Waterscales team have designed more efficient solution methods. These technological developments allow larger simulations to be run faster, and can be applied both to in-silico studies of physiology and also in e.g. geoscience.

Mixed dimensional partial differential equations are systems of differential equations coupling solution fields defined over domains of different topological dimensions. For instance, in physiology, such equations can model blood flow in a three-dimensional vessels interacting with an essentially two-dimensional vessel wall. Similarly, in geology, fluid flow through faults and fractures in rocks can be modelled via such equations. The Waterscales team have introduced computer algorithms and new technology for the simulation of such equations. The technology has been implemented in the FEniCS finite element software and is openly available.

Influx and clearance of substances in the brain occur by a combination of diffusion and convection, but the relative importance of these mechanisms is unclear. Accurate modeling of tracer distributions in the brain relies on parameters that are partially unknown and with literature values varying by several orders of magnitude. In response, the Waterscales team have quantified the variability of tracer distribution in the brain resulting from uncertainty in diffusion and convection model parameters. Even when uncertainties are taken into account, our findings show that diffusion alone is not sufficient to explain transport of tracer deep into the white matter as seen in experimental data. A glymphatic-type velocity field may increase transport if a large-scale directional structure is included in the circulation.

Finally, by organization of minisymposia, conferences and seminars, the Waterscales project has created meeting grounds for scientific discussion and knowledge exchange on the topic of cerebral fluid flow and transport across scientific disciplines and across the world.
In the upcoming period, the Waterscales team will continue creating new mathematical models and methods allowing for better understanding of brain mechanics and physiology. We will focus on models and methods at the cellular level aiming to understand the interactions between electrophysiology and mechanics in glial cells, neurons and extracellular space. We will design new models and efficient solution methods for investigating flow along paravascular spaces. Finally, we will evaluate the current state-of-the-art in modelling the brain’s waterscape across scales and compare with experimental and clinical data.
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