Chandrajit Bajaj cs.utexas/~bajaj
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Transcript of Chandrajit Bajaj cs.utexas/~bajaj
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
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005
The Brain Organ System I
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005
Axonal transport of membranous organelles
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Action Potentials
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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 VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005
Neuro-Muscular_coupling (synapses)
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005
Excitation-contraction coupling and relaxation in cardiac muscle
Transport in Myocytes
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005
Transport Mechanisms
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005
Transport Mechanisms
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Transport Mechanisms
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005
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 VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005
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
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005
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 VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005
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
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005
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.
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005
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 VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005
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 VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005
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
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005
Anisotropic Diffusion I
Weickert’s anisotropic model:
Edges: Edge-normal vectors:
Diffusivity along edges Inhibit diffusivity across edges
||1
2
11
222/1
1
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005
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
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005
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)
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005
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
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005
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
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005
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
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005
• 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
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005
Spatial Discretization• Discretized Laplace-Beltrami Operator
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005
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
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005
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.
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005
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
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005
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
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005
Finite Difference Solution of the PDE
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005
Solution of the GPDEs (III)
• Time Direction Discretization – a semi-implicit Euler scheme.
We use a conjugate gradient iterative method with diagonal conditioning.