(Top Row, left to right) Models of the HIV-Envelope Spike Multi-Protein; Electro-static potential (red = negative, blue = positive) via Poisson Boltzmann of the Machupo-Virus; Quality meshed models of a spiny Hippocampal dendrite (yellow) and axon (green);Visualization from a different view of the dendrite and axon, together with a slice of the ssTEM; (Bottom Row): Model of the inner workings of a cell with a collection of ribosomes and phage-29 at different stages of formation.
Tue 1:00 - 4:00p or by appt. via email@example.com
The course will teach you the basic mathematics, algorithms, techniques and tools of imaging, geometric and physiological modeling and visualization with applications in the biomedical sciences and engineering. Bio-medical modeling (or Biomodeling) and visualization has roots in medical illustration and communication for the health sciences, with branches of application to mathematical modeling and computer simulation of artificial life. In this course we shall emphasize computational image processing, harmonic analysis, computational topology, computational geometry (algebraic and differential), group theory, polynomial spline approximations, computer graphics, data analysis, together with aesthetic choices involved in producing effective scientific animations. The emphasis shall be on spatial realism, and the programmatic use of multi-scale modeling, analysis and visualization to quantitatively depict "how things work" at the molecular, and cellular scales.
Exercises on image processing, geometric and physiological modeling analysis and visualization at multiple scales, shall be drawn from virology (viral envelopes, capsids, proteins, nucleic acids), and neurology (brain, hippocampus, neuropil, axons, dendrites, glial cells, ion-channels, neurotransmitters), and their interactions (molecular energetics and force fields, molecular flexibility, synaptic transmission, synaptic spillover).
Mathematical Preliminaries: Linear algebra, Barycentric coordinates, Mean-value coordinates, Algebraic (polynomial) splines, Parametrization, Singularities, Differential forms, Discrete exterior calculus, Motion groups, Radon and Fourier transforms
Models: Surface and Volumetric representations, Point-based, Clouds, Weighted Delaunay triangulations, Voronoi diagrams, Octrees, Complementary Space (Pockets, Tunnels, Voids)
Maps: X-ray diffraction imaging, Electron microscopy, CT/MRI, Voxel and Continuum Representations
Images & Maps: Forward and Inverse Problems, Contrast Transfer Corrections, Symmetry and
Anisotropy considerations, Compression
Maps2Models: Filtering, Contrast enhancement, Alignment, Classification, Symmetry detection, Static & active Contouring, Segmentation, Medial axis, Skeletonization, Clustering, Matching
Models2Analytics I: Point cloud and Cross-sectional Contour Reconstruction, Surface and Volumetric finite element meshing, Spline representations, Feature identification, Symmetry detection, Shape segmentation, Matching & Complementary Docking, Flexibility, Multi-component Assemblies & Reassembly,
Models2Analytics II: Bonded and non-bonded Molecular Energetics, Forces, Poisson-Boltzmann Electrostatics, Poisson-Nernst Planck ElectroDiffusion, Electric Cable Models, Numerical Quadrature, Cubature, Fast Multipole Methods, fast Fourier techniques, Discrete differential operators, de-Rham Diagrams, Integral equations
Analytics2Informatics/Visualization : Differential/integral/Topological/Combinatorial Properties, Active sites, Hydrogen bond Networks, Branching structures, Packings & Tilings, Contour trees, Comparative Structural analysis, Multi-dimensional Transfer Functions, Visible Surface and Volume rendering, Functions on Surface, Quantifying Uncertainity
Case Studies: Molecular recognition, Electrical signaling amongst neurons.
You will be graded on periodic written homework assignments (60%), a research and/or programming project with a written report and final presentation (40%).
Syllabus & Lectures
Please see Canvas for the syllabus and lecture notes.
Assignments & Exercises
Please see Canvas for the assignments and exercises.
Primary Texts & References
Background Mathematical, Modeling References
Suggested Modeling, Analysis and Visualization Projects
I Molecular Forces and Recognition (Computational Drug Design and Discovery)
II Neuronal Structure and Plasticity (Electrical Circuits in the Hippocampus: Form and Function)
Modeling, Analysis and Visualization Software
Graphical User Tools
Software Libraries and Command-Line Utilities
Server Based Codes
Additional Suggested Reading