Mathematics of Planet Earth

The frontier of biology and medicine is defined by our ability to decipher the mechanisms that underlie basic phenomena. These phenomena may include cell motility and migration, cell division, cell reprogramming, and cell communication that may be manifested in a wide range of questions in development and disease. Thus, examples from stem cell, developmental, neural, and cancer biology have the potential to allow examination of basic biological processes within the context of real, in vivo phenomena. However, a major challenge has been the lack of a means to identify biologically tractable problems and link these problems to applications-oriented experts from imaging and mathematics.

The rate at which this frontier advances depends, at least in part, on how fast technology evolves and on how data is interpreted and translated into a better understanding of basic mechanisms. In the past 10 years, dramatic advances in imaging technology and mathematics have provided new tools and models for discovery that have enabled new observations and hypotheses to be tested. These tools, which are often designed for general applications, find their way into the hands of biologists who then see ways to use them. In some cases, specific mathematical models and applications drive innovations. The mathematical methods involved include PDEs, moving boundary value problems, dynamic geometric changes, optimal transport, stochastic modeling, and the analysis of large data sets. Advances in imaging technology that will be discussed include serial block-face scanning electron microscopy, superresolution microscopy, fluorescence resonance energy transfer (FRET)-based activity biosensors, detection of forces in cells and tissue, multispectral and multiphoton deep tissue imaging, and fluorescence light-sheet microscopy.

The goal of this workshop is to encourage biologists to describe tough questions and to jointly think about approaches that inspire new developments and interdisciplinary research collaborations. We plan to do this by combining input and discussion from experts in imaging technology and mathematics with cell, developmental and cancer biologists that share a passion for solving the riddles that underlie complex phenomena in dynamic living systems. We suggest that both groups of participants blend what is technically possible with what exists only in dream space, with the hope that together we will learn something new and be stimulated to explore new ways to visualize, model and better understand complex processes.

This workshop addresses the broad class of imaging problems in the life sciences that rely on shape or geometry to characterize biological processes and parameters. Of course, the strategy of observing shape and its relationships to biology is a classical undertaking, but in recent years, the availability of 3D imaging and better computational tools has opened up new possibilities for systematic, quantitative analyses of biological shape. This, in turn, has resulted in new demands for more fundamental approaches, based in mathematics, for quantifying and analyzing geometric objects. The problem of quantifying shapes arises in clinical science, where the shapes of neurological or musculoskeletal structures are thought to be related to growth, function, pathology, and degeneration. More recently, computational strategies for shape analysis have become widespread throughout the life sciences, with compelling applications in anthropology, cell and tissue biology, botany, etc.

The mathematical contributions to shape analysis have resulted in new tools for modeling or characterizing shapes and for analyzing both shape dynamics and the statistics of populations of shapes. However, the applications of these methods are typically limited by somewhat strong assumptions about the classes of shapes, such as smoothness, correspondence, and homogeneity or underlying simplifications in morphogenetic processes. This workshop focuses on the frontiers of this technology with an eye toward new applications, such as cell biology and biological morphogenesis, which have yet to benefit from robust, comprehensive approaches. Of particular interest are more general tools for handling nonmanifold shapes, such as networks or trees, as well as tools that can handle relatively heterogeneous collections of objects, such as those seen in cell or tissue biology. Also important is the analysis of dynamic shapes as in morphogenesis and regeneration, and the links to other data such as lineage, genomics, and proteomics. Participants will consist of life scientists with compelling scientific and clinical examples, engineers with computational tools for shape analysis, and mathematicians with insights into fundamental approaches for representing and quantifying shape.

Merging imaging modalities is increasingly important for biomedical questions related to time and space scales including function and anatomy. Integrating modalities from multiple scales can assist with understanding development and function, disease, diagnosis and treatment. This workshop will bring together researchers who are attempting to combine and integrate different imaging modalities to better understand anatomy, function and disease from the cellular to organ level.

Methodologies and challenges in combining imaging data from multiple sources, such as MRI, fMRI, DTI, PET, EEG, MEG, CT, ultrasound, NMR, x-ray diffraction, electron microscopy, proteomic and genomic data will be explored. Merging data from different modality time scales (functional time scales from nanoseconds to minutes; developmental time scales from embryonic to adult) and space scales (from microns to millimeters) present many mathematical questions. Interpretation, analysis and modeling of multi-modality data as it applies to development, disease models and therapies will also be explored. The heterogeneity of the data presents many difficult challenges that are suited for mathematical exploration.

The focus will include brain and cardiac imaging related to multiscale and bioscale data collection, merging data, modeling and analysis. This workshop will be of interest to mathematicians working in areas of statistical analysis, PDE modeling, inverse problems, differential geometry, computational visualization and multiscale problems. Biomedical researchers interested in merging imaging modalities to investigate questions related to genomics, gene expression and biomarkers and the role they play in macroscopic function would benefit from this workshop.

This workshop focuses on the challenges presented by the analysis and visualization of large data sets that are collected in biomedical imaging, genomics and proteomics. The sheer size of data (easily in the range of terabytes, and growing) requires computationally efficient techniques for the sampling, representation, organization, and filtering of data; ideas and techniques from signal processing, geometric and topological analysis, stochastic dynamical systems, machine learning and statistical modeling are needed to extract patterns and characterize features of interest. Visualization enables interaction with data, algorithms, and outputs.

Data sets from biomedical imaging, genomics and proteomics often have unique characteristics that differentiate them from other data sets, such as extremely high-dimensionality, high heterogeneity due to different data modalities (across different spatial and temporal scales, but also across different biological layers) that need to be fused, large stochastic components and noise, low sample size and possibly low reproducibility of per-patient data. These unique aspects, as well as the large size, pose challenges to many existing techniques aimed at solving the problems above.

The workshop will bring together biologists, computer scientists, engineers, mathematicians and statisticians working in a wide of areas of expertise, with the goal of pushing existing techniques, and developing novel ones, for tackling the unique challenges offered by large data sets in biomedical imaging.