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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

Lecture 11: Multiscale Bio-Modeling and Visualization

Organ Models I: Synapses and Transport Mechanisms

Chandrajit Bajaj

http://www.cs.utexas.edu/~bajaj


The Brain Organ System I


Axonal transport of membranous organelles


Action Potentials


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.


Neuro-Muscular_coupling (synapses)


Excitation-contraction coupling and relaxation in cardiac muscle

Transport in Myocytes


Transport Mechanisms






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


PDE based diffusion

Heat/Diffusion equation

the solution is where is a Gaussian of width

0div t

0)(t *

0)(t )(

02

0

t

Kt

(.)K


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)


Curvature Computations for Surfaces/Images/Volumes

F z = 1

• 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

• 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).

G = Structure Tensor. Rank of G is 1 and its Eigenvector (with nonzero Eigen-value ) is in the Normal directionkr Fk2

G’ is of Rank 2 and its Eigenvectors are in the tangent space of M with equal Eigenvalues kr Fk2


1. Mean Curvature Surface Diffusion

• The mean curvature flow is area shrinking.


2. Average Mean Curvature Surface Diffusion

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


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.


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.


Towards Anisotropic Surface/Image/Volume Diffusion

Early attempt: Perona-Malik model

where diffusivity becomes small for

large , i.e. at edges

or

0))div(g( t

22/1

1)(

g

)exp()(2

2

g

g


Anisotropic Diffusion I

Weickert’s anisotropic model:

Edges: Edge-normal vectors:

Diffusivity along edges Inhibit diffusivity across edges

||1

2

11

222/1

1


Anisotropic Volumetric Level Set Diffusion

Preuer and Rumpf’s level set method in 3D

0)(

Ddivt

A triad of vectors on the level set: two principal directions of curvature and the normal


Choice of Anisotropic Diffusion Tensor

Let be the principal curvature directions of

at point

v(1)(x);v(2)(x ); M(t)

x(t)

z = ëv(1)(x) + ì v(2)(x)Define tensor a, such that

az = g(k1)ëv(1)(x) + g(k2)ì v(2)(x)where

g(s) =2(1+õ2

s2)à1

1;ú s ô õ

s > õ;

is a given constant.õ > 0

N(x)If is the normal at then a vector

+ î N(x)

+ î N(x)

x(t)


Level set based Geometric Diffusion

Diffusion tensor

Diffusion along two principal directions of curvature on surface

No diffusion along normal direction

BBD

0

)(

)(,2

2,1

,12,1

G

GT


Anisotropic Volumetric Diffusion

• Three principal directions of curvature for volumes are used to construct the Diffusion tensor

],,[

)(00

0)(0

00)(

],,[ 321

3

2

1

321 vv

G

G

G

vvD T

• The principal directions of curvature are the unit eigenvectors of a matrix

• Principal curvatures are the corresponding eigenvalues

hA

},,{ 321

},,{ 321


A Finite Element approach to Anistropic Diffusion Filtering

a(x) is a symmetric, positive definite matrix (diffusion tensor)

@tx(t) à div(a(x)r M(t)x(t)) = 0

Variational (weak) form

))((

,0)),(()),(( )()()()(

tMC

txtx tTMtMtMtMt

(f ;g)M =R

M f gdx; (þ; )TM =R

M þT dx

where

• How to choose a(x) ?

• How to choose ?

Model


• 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.

Anisotropic Diffusion Filtering (contd)

• and are sparse.M nL n

M n

L n

M n + üL n

(Mn +üLn)C((n + 1)ü) = MnC(nü)

C(t) = [c1(t);ááá;cm(t)]M n = (þi;þj)M(nü)à ám

i;j=1

L n = (r M(nü)þi; r M(nü)þj)TM(nü)à ám

i;j=1


Spatial Discretization• Discretized Laplace-Beltrami Operator


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.

mpV

mpV

)(r

r4rrrrr mp

2

V

),;( mp Vr

is a Heaviside step-function equal to 0 on side I and 1 on side II, and

)(r

II sideon isr if)/)(exp()/4(

I sideon isr if)/)(exp()/4()(

)II(2

)I(22

a aaBcoreB

a aaBcoreB

qTkrUTk

qTkrUTkrk

is the coupling parameter varying between 0 and 1 to scale the protein charges. is the charge density of the solute.

)(rP


Poisson-Boltzmann voltage equation of the ion channel membrane system with asymmetrical solutions on sides I and II:

The step function is

r4rrrrr mp

2

bulk VH

bulkH

otherwise0

bulk in the isr if1)(rHbulk

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.


Poisson-Nernst-Plank equations:

r

rrrrJ eff

B

WTk

D

rr4rr c q

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.

)(rD )(r)(eff rW

)(rC

)(r

)(r

q


Additional Reading

1. C.Bajaj, G. Xu “Anisotropic Diffusion of Surfaces and Functions on Surfaces”, ACM Trans. On Graphics, 22, 4 – 32, 2003

2. G. Xu, Y. Pan, C. Bajaj “Discrete Surface Modelling Using PDE’s”, CAGD, 2005, in press

3. M. Meyer, M. Desbrun, P. Schroder, A. Barr, “Discrete Differential Geometry Operators for Triangulated 2-manifolds”, Proc. of Visual Math ’02, Germany

4. T. Weiss, “Cellular BioPhysics I: Transport ”, MIT Press, 1998


Finite Difference Solution of the PDE


Solution of the GPDEs (III)

• Time Direction Discretization – a semi-implicit Euler scheme.

We use a conjugate gradient iterative method with diagonal conditioning.