Mathematical Modeling of the Human Brain

1. Front Matter

   Pages i-xvi

   PDF

2. Introduction

   • Kent-André Mardal, Marie E. Rognes, Travis B. Thompson, Lars Magnus Valnes

   Pages 1-6Open Access

3. Working with magnetic resonance images of the brain

   • Kent-André Mardal, Marie E. Rognes, Travis B. Thompson, Lars Magnus Valnes

   Pages 7-21Open Access

4. Getting started: from T1 images to simulation

   • Kent-André Mardal, Marie E. Rognes, Travis B. Thompson, Lars Magnus Valnes

   Pages 23-46Open Access

5. Introducing Heterogeneities

   • Kent-André Mardal, Marie E. Rognes, Travis B. Thompson, Lars Magnus Valnes

   Pages 47-80Open Access

6. Introducing Directionality with Diffusion Tensors

   • Kent-André Mardal, Marie E. Rognes, Travis B. Thompson, Lars Magnus Valnes

   Pages 81-96Open Access

7. Simulating Anisotropic Diffusion in Heterogeneous Brain Regions

   • Kent-André Mardal, Marie E. Rognes, Travis B. Thompson, Lars Magnus Valnes

   Pages 97-107Open Access

8. Concluding Remarks and Outlook

   • Kent-André Mardal, Marie E. Rognes, Travis B. Thompson, Lars Magnus Valnes

   Pages 109-110Open Access

9. Back Matter

   Pages 111-118

   PDF

This open access book bridges common tools in medical imaging and neuroscience with the numerical solution of brain modelling PDEs. The connection between these areas is established through the use of two existing tools, FreeSurfer and FEniCS, and one novel tool, the SVM-Tk, developed for this book. The reader will learn the basics of magnetic resonance imaging and quickly proceed to generating their first FEniCS brain meshes from T1-weighted images. The book's presentation concludes with the reader solving a simplified PDE model of gadobutrol diffusion in the brain that incorporates diffusion tensor images, of various resolution, and complex, multi-domain, variable-resolution FEniCS meshes with detailed markings of anatomical brain regions. After completing this book, the reader will have a solid foundation for performing patient-specific finite element simulations of biomechanical models of the human brain.

• Open Access
• magnetic resonance imaging
• Mesh generation
• mathematical modeling
• finite element methods
• scientific computing

“The book represents an excellent introduction and hands-on guide to this important and exciting field, for applied mathematicians and image processing practitioners … . It is, perhaps, most beneficial, to electrical and computer science majors who wish to rapidly immerse themselves in the field, in a manner that is mathematically correct and sound, yet also practical.” (Emil Saucan, zbMATH 1501.92001, 2023)

• Department of Mathematics, University of Oslo, Lysaker, Norway

  Kent-André Mardal

• Biocomp CBC 1-2118, Simula Research Laboratory, Fornebu, Norway

  Marie E. Rognes

• Mathematical Institute, Oxford University, Oxford, UK

  Travis B. Thompson

• Rikshospitalet, Oslo University Hospital, Oslo, Norway

  Lars Magnus Valnes