Modellansatz

This is one of two conversations which Gudrun Thäter recorded alongside the conference Women in PDEs which took place at our faculty in Karlsruhe on 27-28 April 2017. Marie Elisabeth Rognes was one of the seven invited speakers.

Marie is Chief Research Scientist at the Norwegian research laboratory Simula near Oslo. She is Head of department for Biomedical Computing there. Marie got her university education with a focus on Applied Mathematics, Mechanics and Numerical Physics as well as her PhD in Applied mathematics at the Centre for Mathematics for Applications in the Department of Mathematics at the University of Oslo.

Her work is devoted to providing robust methods to solve Partial Differential Equations (PDEs) for diverse applications. On the one hand this means that from the mathematical side she works on numerical analysis, optimal control, robust Finite Element software as well as Uncertainty quantification while on the other hand she is very much interested in the modeling with the help of PDEs and in particular Mathematical models of physiological processes. These models are useful to answer What if type-questions much more easily than with the help of laboratory experiments.

In our conversation we discussed one of the many applications - Cerebral fluid flow, i.e. fluid flow in the context of the human brain.

Medical doctors and biologists know that the soft matter cells of the human brain are filled with fluid. Also the space between the cells contains the water-like cerebrospinal fluid. It provides a bath for human brain. The brain expands and contracts with each heartbeat and appoximately 1 ml of fluid is interchanged between brain and spinal area. What the specialists do not know is: Is there a circulation of fluid? This is especially interesting since there is no traditional lymphatic system to transport away the biological waste of the brain (this process is at work everywhere else in our body). So how does the brain get rid of its litter? There are several hyotheses:

• Diffusion processes,
• Fast flow (and transport) along the space near blood vessel,
• Convection.

The aim of Marie's work is to numerically test these (and other) hypotheses. Basic testing starts on very idalised geometries. For the overall picture one useful simplified geometry is the annulus i.e. a region bounded by two concentric circles. For the microlevel-look a small cube can be the chosen geometry.

As material law the flow in a porous medium which is based on Darcy flow is the starting point - maybe taking into account the coupling with an elastic behaviour on the boundary.

The difficult non-mathematical questions which have to be answered are:

• How to use clinical data for estabilishing and testing models
• How to prescribe the forces

In the near future she hopes to better understand the multiscale character of the processes. Here especially for embedding 1d- into 3d-geometry there is almost no theory available.

For the project Marie has been awarded a FRIPRO Young Research Talents Grant of the Research Council of Norway (3 years - starting April 2016) and the very prestegious ERC Starting Grant (5 years starting - 2017).

• M.E. Rognes: Mathematics that cures us.TEDxOslo 3 May 2017
• Young academy of Norway
• ERC Starting Grant: Mathematical and computational foundations for modeling cerebral fluid flow 5 years
• P.E. Farrell e.a.: Dolfin adjoint (Open source software project)
• FEniCS computing platform for PDEs (Open source software project)
• Wikipedia on FEniCS
• Collection of relevant literature implemented in FEniCS