Center for Computational Visualization

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Lecture 11 Multiscale Bio-Modeling and
VisualizationOrgan Models I Synapses and
Transport Mechanisms
Chandrajit Bajaj http//www.cs.utexas.edu/bajaj
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The Brain Organ System I
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Axonal transport of membranous organelles
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Action Potentials
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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.

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Neuro-Muscular_coupling (synapses)
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Transport in Myocytes
Excitation-contraction coupling and relaxation in
cardiac muscle
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Transport Mechanisms
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Transport Mechanisms
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Transport Mechanisms
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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

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PDE based diffusion

• Heat/Diffusion equation
• the solution is
• where is a Gaussian of width ?

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Generalized Geometric Surface Diffusion Models

• A curvature driven geometric evolution consists
  of finding a family M M(t) t gt 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)
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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).

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1. Mean Curvature Surface Diffusion

• The mean curvature flow is area shrinking.

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2. Average Mean Curvature Surface Diffusion

• The average mean curvature flow is volume
  preserving and area shrinking. The area shrinking
  stops if H h.

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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.

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4. Higher Order Diffusional Models

• The flow is volume preserving if K gt 2. The
  area/volume preserving/shrinking properties for
  the flows mentioned above are for closed surfaces.

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Towards Anisotropic Surface/Image/Volume Diffusion

• Early attempt Perona-Malik model
• where diffusivity becomes small for large
  , i.e. at edges
• or

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Anisotropic Diffusion I

• Weickerts anisotropic model
• Edges
• Edge-normal vectors
• Diffusivity along edges
• Inhibit diffusivity across edges

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Anisotropic Volumetric Level Set Diffusion

• Preu?er and Rumpfs level set method in 3D

A triad of vectors on the level set two
principal directions of curvature and the
normal
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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.
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Level set based Geometric Diffusion

• Diffusion tensor
• Diffusion along two principal directions of
  curvature on surface
• No diffusion along normal direction

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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

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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 ?

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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.
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Spatial Discretization

• Discretized Laplace-Beltrami Operator

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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.
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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.
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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.
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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 PDEs, 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

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Finite Difference Solution of the PDE
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Solution of the GPDEs (III)

• Time Direction Discretization a semi-implicit
  Euler scheme.
• We use a conjugate gradient iterative method
  with diagonal conditioning.