REPORT ON THE LONI/IPAM WORKSHOP

ON

MATHEMATICS AND MODELING IN BRAIN MAPPING

http://www.ipam.ucla.edu/programs/gbm2001/May21-24.html

by

Paul Thompson, Ph.D.

Assistant Professor of Neurology, UCLA Laboratory of Neuroimaging (LONI)

Overview

On May 24, 2001, an exciting Workshop was held at UCLA entitled: Mathematics and Modeling in Brain Mapping. Nine nationally renowned experts in the brain mapping field came to speak at the Workshop; they gave detailed and well-illustrated tutorial presentations on recent progress and current challenges in the field. Over 70 participants attended the meeting, many of them researchers in mathematics, biomedical engineering, computer science, radiology, neuroscience, and medicine. This one-day Workshop was part of a week-long meeting on Imaging and Medicine in the Neurosciences, and many of the speakers from the other symposia during the week participated in some extremely interesting discussion and question sessions. All talks were recorded, and were telecast via a video link to an adjacent room. This accommodated additional attendees for keynote presentations. The quality of the presentations and audience feedback was outstanding. The Workshop helped to create a first-class multidisciplinary environment for students and experts in fields to interact, and learn about the latest developments in the mathematics of brain mapping.

Format and Goals

The goal of the one-day brain mapping Workshop was to introduce students and researchers interested in imaging, mathematics and medicine, to the latest developments in the field. Speakers were selected by the Organizing Committee based on their knowledge of the field and their track record of research at the interface of neuroimaging and mathematics. The meeting was chaired by Paul Thompson and Arthur Toga of the UCLA Laboratory of Neuroimaging (LONI), and was funded by a National Center for Research Resources (NCRR) Grant. The Workshop was held at the Institute for Pure and Applied Mathematics (IPAM) on the UCLA campus in Westwood. The Organizing Committee consisted of Simon Cherry (UCLA School of Medicine), Guillermo Sapiro (University of Minnesota), Paul Thompson (UCLA School of Medicine, LONIR), Arthur W. Toga (UCLA School of Medicine, LONIR), and Eitan Tadmor (IPAM & UCLA Mathematics).

Invited Lectures: Morning Session

After coffee and registration for the meeting, participants were welcomed to IPAM by its Co-Director, Eitan Tadmor, professor of mathematics at UCLA. Paul Thompson chaired the morning session and introduced the keynote speaker, Dr. Arthur Toga, who gave an excellent talk on 'The Informatics and Computational Anatomy of Brain Mapping'.

Arthur Toga's talk was a superb introduction to the day. He introduced many key themes that would be taken up by later speakers. First, he underscored the need for 3D coordinate systems to store information on the structure and function of the brain. He traced the development of these systems from their beginnings - as a guide for neurosurgeons - into an international standard of reference that now supports the comparison, communication and dissemination of brain data across laboratories worldwide. Pointing out the challenges of indexing brain imaging data from individuals whose anatomy varies widely, Dr. Toga introduced the idea of a probabilistic atlas, which stores statistics on the variability of brain structure and function in entire populations. He showed tensor visualizations that encode the parameters of anatomic variability in normal populations, and use this information to detect anomalies in new subjects. He also described the motivation for image warping, or nonlinear registration, algorithms. These mathematically sophisticated algorithms elastically reconfigure images from different subjects so that they can be compared with one another, or with a standard anatomical atlas. Dr. Toga also illustrated the enormous diversity of brain maps and imaging methods that has spurred the development of population-based digital brain atlases. Visualization and animations were shown indicating the richness of data that modern imaging provides as a basis for mathematical and computational analysis.

The keynote talk began an interesting discussion of mathematical challenges in the handling of brain mapping data. Dr. David Mumford (Brown University, Division of Applied Mathematics) asked whether, and to what degree, the anatomy of the brain presents a guide to its function. Arthur Toga described the development of probabilistic reference systems, or atlases, that attempt to model anatomic variability first, and then model functional variability relative to the underlying anatomy. Maps were described that reveal the function of the brain in terms of activation imaging, optical reflectance, emission tomography, and unique perspectives provided by cryosection imaging. Dr. Toga emphasized that the first steps in understanding functional variation were to develop algorithms to model, and control for, anatomic variations across subjects. He outlined several projects, now underway, to elucidate the statistical dependency between functional variation and the complex pattern of anatomical structure in healthy and diseased populations.

Additional discussion points were raised by Dr. Gary Christensen (University of Iowa) related to how topological differences in anatomy among subjects, such as interrupted sulci, could be accommodated by algorithms modeling structural variability using a 'deformable template' paradigm. Other questions related to the basis of measures of brain function using PET and functional activation imaging.

Dr. Ron Kikinis (Surgical Planning Laboratory, Harvard University and Brigham Women's Hospital) gave a remarkable talk on intraoperative brain mapping, entitled: 'Biomechanical Modeling of the Brain'. He outlined developments at the SPL in collaboration with Joachim Dengler, Simon Warfield and others, for high dimensional warping, or elastic registration of imaging data acquired intraoperatively. This deformable mapping problem is motivated by the need to overlay pre-operative brain maps with imaging data acquired intraoperatively in tumor patients. This overlay is often complicated, as considerable brain shift and deformation occurs. He emphasized, as Dr. Christos Davatzikos would underscore later in the day, that the accurate tracking of the brain intraoperatively is difficult, as it requires assumptions, or empirical data, on the mechanics of tissue deformation. Dr. Kikinis showed data from a finite element algorithm that was able to track, and predict, brain deformations in real-time, using an elastic deformation principle. He showed validation data and compelling animations tracking brain change, with selective overlay of vascular, functional, diffusion imaging, and tissue segmented data. Visualizations of the local strain tensor of the deformation, based on work with Dr. Simon Warfield, were also presented. Extending the tensor theme, Dr. Kikinis also showed some remarkable tensor visualizations based on diffusion imaging, a comparatively new MRI technique capable of representing the local diffusion of water in the brain as a tensor map. He showed how the principal eigenvector of these tensor fields generally indicates the principal directions of local fiber trajectories in the brain. Dr. Kikinis also showed a new method for flattening the surface topography of the cortex, developed in collaboration with Steve Haker (SPL/University of Minnesota), Allen Tannenbaum (Georgia Tech) and colleagues. This technique is able to produce conformal parameterizations of the cortical surface, generating area-preserving mappings by using principles of Monge-Kantorovich optimal transport, for surface warping and image registration.

After a short coffee break, the group reconvened for additional talks on new techniques for brain mapping, image analysis and visualization.

Dr. Anders Dale (Harvard Medical School, Massachusetts General Hospital) is a pioneer in the development of techniques for event-related fMRI and cortical surface mapping. He gave an excellent lecture on the reconstruction of MEG/EEG brain data. He described collaborative work with Dr. Bruce Fischl and colleagues at MGH on the construction of accurate models of the human cortex for fMRI visualization and analysis, based on the generation of triangulated surface meshes. The flattening of these anatomical meshes was then used to develop a common spherical reference space for the plotting of functional neuroimaging data, which could then be aligned across subjects for comparison and integration. Registration of data across subjects was described using flows in the cortical parameter space, driven by the overlay of surface curvature maps. Dr. Dale also described pioneering work combining fMRI/EEG data with simultaneous EEG/MEG data. This produced spatio-temporal maps that combined the temporal information available in EEG and MEG with the spatial resolution available with fMRI. He compared the use of equivalent current dipole (ECD) and distributed source models for solving the inverse problem, and illustrated these multi-modality reconstructions using experimental data mapping the distinct regions of the human visual cortex.

Dr. Jim Brinkley (University of Washington) gave an exciting talk on the use of novel visualization tools for intraoperative mapping. He described a registration approach that his group had developed and explored for cortical stimulation mapping (CSM) of language areas in the brain. He showed a variety of new tools that could be used for databasing and understanding of language data acquired intraoperatively. He also described a flexible system he and his colleagues developed for analyzing brain images, based on a range of algorithms for automated and manually-assisted segmentations. First, he described a constraint-based technique for skull stripping (editing the scalp and non-brain tissues from neuroimaging data) that combines learned shape knowledge of the cortical envelope with local interactive control. Dr. Brinkley described the integration of this technique in a working system that has been used to map over 40 patients to-date. Finally he described the development of a web-based experiment management system for organizing and visualizing multi-modality language mapping data.

Invited Lectures: Afternoon Session

After lunch, Dr. Paul Thompson gave a talk entitled: 'Mathematical Challenges in Population-Based Brain Mapping'. He described his recent work with his colleagues at the UCLA Laboratory of Neuroimaging, developing mathematical algorithms (1) to uncover disease-specific patterns of brain structure and function in human populations, and (2) to reveal the effects of medication, and even genetics, in altering these patterns. Dr. Thompson introduced methods for averaging anatomical data across individuals, and measuring anatomic variability at the cortex. This produced average maps of brain structure revealing disease-specific patterns that were often not apparent in individual subjects. He also described techniques that have allowed the construction of probabilistic atlases. These atlases store detailed information on how the brain varies across age and gender, across time, in health and disease, and in large human populations. Dr. Thompson introduced a mathematical framework based on covariant partial differential equations (PDEs), harmonic flows, and high-dimensional random fields to encode variations in cortical patterning, asymmetry and tissue distribution in a population-based brain image database (N=200 scans). Illustrative visualizations showed how this reference information could be used to detect disease-specific abnormalities in Alzheimer's disease and schizophrenia, including dynamic changes and response to medication over time. Patterns of cortical organization, asymmetry, and disease-specific trends were also resolved. Finally, Dr. Thompson introduced four-dimensional (4D) maps that store probabilistic information on the dynamics of brain change in development and disease. He emphasized that digital brain atlases show considerable promise in identifying general patterns of structural and functional variation in diseased populations. He stressed that the algorithms were beginning to reveal how brain structure and function depend on demographic, genetic, clinical and therapeutic parameters, and were identifying new aspects of development and disease processes.

An extremely impressive lecture was given by Mr. Faisal Beg, an advanced graduate student in the laboratory of Dr. Michael Miller, at the Johns Hopkins Center for Imaging Science. Mr. Beg described very ingenious work in which fluid transformations are used to measure differences in brain anatomy across subjects. He explained how a highly automated fluid segmentation technique can be used to deform an anatomic template onto brain scans from new subjects. This allows automated labeling of brain structures and shape analysis. He explained how the requirement that mappings be diffeomorphic leads to a formulation of image matching problem as a fluid flow problem, in which a functional expressing the intensity similarity between two images and a mapping energy are jointly minimized. He explained how the regularity of mappings can be constrained by using a mapping energy which results from applying a self-adjoint second-order partial differential operator, such as the Laplacian, to the velocity vector field. The time-integral of the resulting velocity field describes the spatial flow of a deformable anatomical template. He explained how these energy functionals could also be used to develop a measure of shape similarity for Macaque brain data, and for mitochondrial data, and showed validation data and interesting examples of each.

Mr. Beg's talk gave rise to some excited discussions from the partial differential equations group at UCLA on whether total-variation (TV) models could also be used to regularize image flows for similar applications. Dr. Tony Chan (UCLA Mathematics, IPAM Director) asked questions on the domain of definition of the similarity energy and their boundary conditions. Dr. Stan Osher (UCLA Mathematics) noted the relevance of recent work on flows from the modeling of pollution data, and suggested alternate forms for the differential operator regularizing the flow, and for the image similarity metric. Mr. Beg also gave interesting descriptions of the parallel hardware architecture, and efficient data representations, which he and his colleagues have developed to greatly accelerate the fluid matching algorithms.

After a short coffee break, Dr. Jim Gee (University of Pennsylvania) gave an ingenious talk on his recent work on the registration of tensor-valued brain imaging data. The talk was fascinating due to the interaction of two mathematical ideas. One was the use of 3D deformation maps that are often employed to register one imaging dataset with another; the second was the use of tensor-valued data to drive these deformation mappings. The tensor matching problem was motivated by the challenge of registering diffusion tensor magnetic resonance images. These images contain orientation information that must be handled appropriately when images are transformed spatially. Dr. Gee presented solutions for global transformations of 3D images up to 12-parameter affine complexity and then indicated how the methods can be extended to higher order transformations. He also discussed relevant comparative measures of similarity between diffusion tensors and a new formulation for their implementation. The entire presentation was an excellent introduction to a very challenging problem and the mathematics required to solve it.

Dr. Christos Davatzikos (Johns Hopkins School of Medicine) gave a very impressive and stimulating talk on "Deformable Shape Models for Computational Anatomy". He began by indicating how deformable models had found widespread applicability in diverse fields, for the past 15 years. He reviewed two challenging applications, developing deformable models for (1) computational anatomy, and (2) modeling soft tissue deformability in surgical applications. On the former topic, Dr. Davatzikos presented a shape-based approach he has pioneered for deformable registration of medical images. He showed examples applying the approach to computational morphometry of the brain, as well as applications to the spine and the prostate. He developed a second interesting theme for combining statistical shape models and biomechanics, with the goal of predicting diverse anatomical deformations. He showed how the shape of a deformable structure in different configurations could be inferred by building covariance matrices to store information on shape change. Dr. Davatzikos showed intriguing applications of these techniques in predicting and tracking deformations in image-guided surgical environments.

Finally, Dr. Gary Christensen (University of Iowa) gave a very exciting talk on 'Synthesizing Average Brain Shape and Validation'. He introduced a pioneering approach he had developed with colleagues at the University of Iowa and Washington University, using continuum-mechanical models to capture shape differences among different brains. This approach has generated great interest and acclaim due to its ability to deform anatomical atlases onto new imaging data with precision and extremely high automation. Dr. Christensen described his recent work on 'consistent' image registration. In this approach, both the forward transformation of one brain to another and its inverse are reconciled to produce robust, symmetric mappings. He described how this formulation leads to a powerful technique for synthesizing a 3D MRI image of the 'average brain' from a set of MRI images collected from a subject population. He described a series of interesting experiments in which he examined the error associated with the choice of template selected from the population used to synthesize the average population shape. Dr. Christensen showed compelling data visualizing average image templates synthesized for a population of adult brains using a consistent linear-elastic image registration algorithm. Each data set from the population was used as the template to synthesize a population average, and the displacement variance examined. The resulting algorithms were shown to be able to create crisp, well-defined anatomical templates with the average morphology of a population of subjects. Exciting data was also shown in a cranio-facial application. High-resolved cranio-facial atlases were synthesized with the average geometry for a group of individuals.

Feedback and Evaluation Questionnaires

An evaluation sheet was distributed to the participants at the beginning of the day, and was returned by those who had attended the entire session. The evaluation was designed to obtain feedback on the quality of the talks, selection of speakers, and to find out what participants liked best, and what could be improved. 20 questionnaires were returned and indicated extremely high enthusiasm for the event. Those returning the questionnaire were uniformly enthusiastic. They rated the talks as outstanding, and the level of difficulty of the material about right. Nobody rated any category less favorably than 'Excellent'. The quality of the talks and choice of speakers was praised, specifically with regard to the level of interest and educational value of the information presented. Some were highly pleased with the venue, the mix of talks, the graphics, and the overall format. Several respondents hoped that we would organize such a workshop again.

We also requested feedback on what aspects of the Workshop could be improved, or whether we should have done anything differently. Some suggested the coffee could be improved (!). On a more serious note, a couple of respondents suggested that they would like to have lecture materials, covering the material presented in the talks. In response to these suggestions (and following IPAM's excellent policy), lecture materials have now been made available by the speakers via the Web (see below). We also hope that this will allow a wider audience to continually benefit from the Workshop, with online access to a range of educational materials.

Availability of Lecture Materials

Lecture materials were made available by the speakers, and included video files, animations, lecture notes, slides and links to related publications. These lecture materials may be downloaded via the Internet here: http://www.ipam.ucla.edu/programs/gbm2001/May21-24.html

Funding and Acknowledgments

The Workshop was made possible by a P41 Resource grant to the UCLA Laboratory of Neuroimaging (LONI) from the National Center for Research Resources (NCRR). Special thanks go to the speakers and participants for their time, to IPAM staff for their provision of the venue, excellent organization and audio-visual support, and to members of the Organizing Committee, IPAM and LONI who assisted with the scheduling and logistics of the Workshop.