Chandrajit Bajaj cs.utexas/~bajaj

1. Lecture 11: Multiscale Bio-Modeling and VisualizationOrgan Models I: Synapses and Transport Mechanisms Chandrajit Bajaj http://www.cs.utexas.edu/~bajaj Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

2. The Brain Organ System I Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

3. Axonal transport of membranous organelles Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

4. Action Potentials Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

5. Neuronal Synapses Neurons must be triggered by a stimulus to produce nerve impulses, which are waves of electrical charge moving along the nerve fibers. When the neuron receives a stimulus, the electrical charge on the inside of the cell membrane changes from negative to positive. A nerve impulse travels down the fiber to a synaptic knob at its end, triggering the release of chemicals (neurotransmitters) that cross the gap between the neuron and the target cell, stimulating a response in the target. Synapse The communication point between neurons (the synapse, enlarged at right) comprises the synaptic knob, the synaptic cleft, and the target site. Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

6. Neuro-Muscular_coupling (synapses) Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

7. Transport in Myocytes Excitation-contraction coupling and relaxation in cardiac muscle Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

8. Transport Mechanisms Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

9. Transport Mechanisms Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

10. Transport Mechanisms Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

11. Diffusion based Transport Mechanisms • Diffusion: the random walk of an ensemble of particles from regions of high concentration to regions of lower concentration • Conduction: heat migrates from regions of high heat to regions of low heat Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

12. PDE based diffusion • Heat/Diffusion equation the solution is where is a Gaussian of width Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

13. Generalized Geometric Surface Diffusion Models • A curvature driven geometric evolution consists of finding a family M = {M(t): t >= 0} of smooth closed immersed orientable surface in IR3 which evolve according to the flow equation Where x(t) – a surface point on M(t) Vn(k1, k2, x) – the evolution speed of M(t) N(x) – the unit normal of the surface at x(t) Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

14. Curvature Computations for Surfaces/Images/Volumes • If surface M in 3D is the level set F(x,y,z) = 0 of the 3D Map Principal curvatures/directions are the Eigen-values/vectors of • G = Structure Tensor. Rank of G is 1 and its Eigenvector (with nonzero Eigen-value ) is in the Normal direction • G’ is of Rank 2 and its Eigenvectors are in the tangent space of M with equal Eigenvalues • If Surface M in 3D is the graph of an Image F(x,y) in 2D. Principal Curvatures are Eigenvalues of H Principal Curvature directions are Eigenvectors of C with Similar for 3D Images or Maps F(x,y,z) (Volumes). Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

15. 1. Mean Curvature Surface Diffusion • The mean curvature flow is area shrinking. Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

16. 2. Average Mean Curvature Surface Diffusion • The average mean curvature flow is volume preserving and area shrinking. The area shrinking stops if H = h. Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

17. 3. Isotropic L-B Surface Diffusion • The surface flow is area shrinking, but volume preserving. The area stops shrinking when the gradient of H is zero. That is, H is a surface with constant mean curvature. Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

18. 4. Higher Order Diffusional Models • The flow is volume preserving if K >= 2. The area/volume preserving/shrinking properties for the flows mentioned above are for closed surfaces. Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

19. Towards Anisotropic Surface/Image/Volume Diffusion • Early attempt: Perona-Malik model where diffusivity becomes small for large , i.e. at edges or Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

20. Anisotropic Diffusion I Weickert’s anisotropic model: Edges: • Edge-normal vectors: • Diffusivity along edges • Inhibit diffusivity across edges Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

21. Anisotropic Volumetric Level Set Diffusion Preuer and Rumpf’s level set method in 3D • A triad of vectors on the level set: • two principal directions of curvature • and the normal Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

22. ( 1 ) ( 2 ) v ( x ) ; v (x ú 1 ; g ( s ) = 2 1 s à 2 ( 1 + ) ; 2 õ Choice of Anisotropic Diffusion Tensor Let be the principal curvature directions of at point ) ; If is the normal at then a vector Define tensor a, such that where is a given constant. Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

23. Level set based Geometric Diffusion • Diffusion tensor • Diffusion along two principal directions of curvature on surface • No diffusion along normal direction Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

24. Anisotropic Volumetric Diffusion • Three principal directions of curvature for volumes are used to construct the Diffusion tensor • The principal directions of curvature are the unit eigenvectors of a matrix • Principal curvatures are the corresponding eigenvalues Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

25. ) ( ) ( ¶ q + Ñ Ñ q = x ( t ), x ( t ), 0 , t M ( t ) M ( t ) M ( t ) TM ( t ) ¥ " q Î C ( M ( t )) A Finite Element approach to Anistropic Diffusion Filtering Model a(x) is a symmetric, positive definite matrix (diffusion tensor) Variational (weak) form where • How to choose a(x) ? • How to choose q ? Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

26. n n n ( M + ü L ) C ( ( n + 1 ) ü ) = M C ( n ü ) Anisotropic Diffusion Filtering (contd) • and are sparse. • is symmetric and positive definite. • is symmetric and nonnegative definite. • is symmetric and positive definite. The linear system is solved by a conjugate gradient method. Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

27. Spatial Discretization • Discretized Laplace-Beltrami Operator Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

28. The linearized Poisson-Boltzmann equation for the total average electrostatic potential in the presence of a membrane potential where is the position-dependent dielectric constant at point r, is the total average electrostatic potential at point r, with potential charges scaled by , and imposed membrane potential , which governs the movement of charged species across the cell membrane. is a Heaviside step-function equal to 0 on side I and 1 on side II, and is the coupling parameter varying between 0 and 1 to scale the protein charges. is the charge density of the solute. Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

29. Poisson-Boltzmann voltage equation of the ion channel membrane system with asymmetrical solutions on sides I and II: The step function is The “pore” region is the region from which all ions other than the permeating species are excluded and the “bulk” region contains the electrolytic solutions. Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

30. Poisson-Nernst-Plank equations: where is the diffusion coefficient, is the density, is an effective potential acting on the ions, is the charge density of the channel, is the position-dependent dielectric constant at point r, is the average electrostatic potential arising from all the interactions in the system, is the charge of the ions. Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

31. Additional Reading • C.Bajaj, G. Xu “Anisotropic Diffusion of Surfaces and Functions on Surfaces”, ACM Trans. On Graphics, 22, 4 – 32, 2003 • G. Xu, Y. Pan, C. Bajaj “Discrete Surface Modelling Using PDE’s”, CAGD, 2005, in press • M. Meyer, M. Desbrun, P. Schroder, A. Barr, “Discrete Differential Geometry Operators for Triangulated 2-manifolds”, Proc. of Visual Math ’02, Germany • T. Weiss, “Cellular BioPhysics I: Transport ”, MIT Press, 1998 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

32. Finite Difference Solution of the PDE Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

33. Solution of the GPDEs (III) • Time Direction Discretization – a semi-implicit Euler scheme. We use a conjugate gradient iterative method with diagonal conditioning. Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin