Lecture 11: Multiscale Bio-Modeling and Visualization Organ 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

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

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

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

Neuronal Synapses Synapse The communication point between neurons (the synapse, enlarged at right) comprises the synaptic knob, the synaptic cleft, and the target site. 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. Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

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

Excitation-contraction coupling and relaxation in cardiac muscle 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

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

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

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

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

where is a Gaussian of width  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

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

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

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

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

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

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

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

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

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

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

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

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

A Finite Element approach to Anistropic Diffusion Filtering Model a(x) is a symmetric, positive definite matrix (diffusion tensor) Variational (weak) form )) ( , ) ), t M C x TM ¥ Î " = Ñ + ¶ q 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

Anisotropic Diffusion Filtering (contd) M n + ü L ) C 1 = 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

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

is the charge density of the solute. 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

the “bulk” region contains the electrolytic solutions. 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

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

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

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

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