Applications of mathematics to medical questions have grown substantially both in number and in sophistication in the last several years. These have developed with, and sometimes because of, a combination of advances in imaging technology, new computational resources, and advanced software.

The slim volume under review here looks specifically at modeling the human brain using magnetic resonance imaging (MRI) and applying finite element techniques to simulate brain processes. While several software libraries are available for solving partial differential equations (PDEs), doing that over brain domains with complex geometries is a considerable practical barrier. The folds of the brain’s cortex are intricate structures. Generating a mesh for finite element modeling of those structures that is physiologically useful is very difficult. This book addresses some of the issues.

The goal of the book is to provide connections between the tools of medical imaging, neuroscience and the numerical solution of PDEs arising in brain modeling. The authors begin by describing a model problem that is designed to illustrate how their techniques apply to modeling the brain; this provides a focus for their development. They want to study how a solute concentration diffuses through a region Ω of the brain. This solute could be a metabolic waste protein such as amyloid. The mathematical model of this process is a time-dependent PDE with concentration \( u = u(x, t) \), where \( u \) satisfies a diffusion equation with diffusion tensor \( D: u_{t} − \mbox{div} D \nabla u = f\) in \( (0, T ] \times \Omega \) with boundary conditions \( u = u_{d} \) on \( (0, T ] \times \partial \Omega \), and \( u(0, ·) = u_{0} \).

With their model problem set out, the authors begin to fill in the background. Brain physiology and imaging are introduced first, with enough description of brain anatomy to make the material that follows comprehensible. The basics of magnetic resonance imaging (MRI) are discussed first, and then three variations are described. The last of these (diffusion tensor imaging mRI) is an imaging method that can detect water molecule movement patterns. From this the diffusion tensor coefficients can be determined, so this is the mode of MRI operation that the authors need to solve the model problem. Even with this advanced MRI it is necessary to modify the surface model file that results by re-meshing, smoothing and avoiding surface intersections and missing facets. In addition, to solve their model problem, the authors need to mesh different regions of the brain to differentiate between gray and white brain matter. This is a very complicated process.

The model problem that the authors describe is in an area of current research, one that aims to address how the presence and movement of some fluid in the brain might contribute to a neurodegenerative disease. Other questions with more immediate clinical applications are also relevant to their research. One of them is an important question in the treatment of epilepsy. It is a kind of inverse problem to electroencephalography: how to determine the source in the brain of an epileptic seizure. Most likely this requires less sophistication in imaging and meshing.

This would not be the first place for a newcomer to learn some neuroscience and mathematical modeling techniques for the brain. One place to start might be *Models of the Mind* by Lindsay; this offers a more basic introduction to the questions of neuroscience for those new to the field. The current book provides the minimum needed to go forward, but many readers would want more.

This book is part of the Simula SpringerBriefs on Computing Series, and its contents reflect that. It has a relatively extensive discussion of computer software directed toward a fairly narrow field. Its benefit to mathematical readers is the integration of a medical question, the corresponding mathematical development and a computer implementation. The treatment of how finite element meshes are devised for complicated surfaces is of particular value.

Bill Satzer (

bsatzer@gmail.com), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications. He did his PhD work in dynamical systems and celestial mechanics.