Center for Computational Visualization

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Lecture 5 Multiscale Bio-Modeling and
VisualizationCell Shapes, Sizes, Structures
Geometric Models
Chandrajit Bajaj http//www.cs.utexas.edu/bajaj
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Cells Their Form

• Evolutionary History of approx. 1.5 billion years
  ago
• Simple cells with their molecular machinery
  jumbled together in a single compartment -gt
  ancestors of modern bacteria
• Compartmented cells -gt yeast, plant, animal cells
  (tiny protozoa -gt mammals -gt tallest trees)
• Two basic types of cells
• Prokaryotes (before kernel)
• Eukaryotes (true kernel)

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The Tree of Life?
Eukaryotic cell
Archaebacteria cell
Prokaryotic cell
Ribosome
Viruses?
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Cells Structural/ Chemical Elements

• Fluid Sheets (membranes) enclose Cells
  Organelles
• Networks of Filaments maintain cell shape
  organize its contents
• Chemical composition...has an evolutionary
  resemblance (e.g. actin found in yeast to humans)

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Cells Vary in Sizes, Shape, Form and Function

• Operative Size is 1 µm (smallest is 0.3 µm and
  Largest gt 100 µm)
• Mycoplasms smallest plasma membrane
• Bacteria approx 1 µm in dia with more
  complicated layered membranes
• Plant Cells cell wall thickness is 0.1 to 10 µm
• Animal Cells 200 different cell types form the
  10n (n14) cells in the human body

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Neurononal Cells

• The neuron consists of a cell body (or soma) with
  branching dendrites (signal receivers) and a
  projection called an axon, which conduct the
  nerve signal. At the other end of the axon, the
  axon terminals transmit the electro-chemical
  signal across a synapse (the gap between the axon
  terminal and the receiving cell). A typical
  neuron has about 1,000 to 10,000 synapses.
• Types Sensory neurons or Bipolar neurons,
  Motorneurons or Multipolar neurons, Interneurons
  or Pseudopolare (Spelling) cells.
• Life span neurons cannot regrow after damage
  (except neurons from the hippocampus).
  Fortunately, there are about 100 billion neurons
  in the brain.

http//www.enchantedlearning.com/subjects/anatomy/
brain/Neuron.shtml
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Glial Cells

• Glial cells make up 90 percent of the brain's
  cells.
• Glial cells are nerve cells that don't carry
  nerve impulses. The various glial (meaning
  "glue") cells perform many important functions,
  including
• digestion of parts of dead neurons,
• manufacturing myelin for neurons,
• providing physical and nutritional support for
  neurons,
• and more.
• Types of glial cells include
• Schwann's Cells
• Satellite Cells
• Microglia
• Oligodendroglia
• Astroglia

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Functions performed by Cells

• Chemical e.g. manufacturing of proteins
• Information Processing e.g. cell recognition of
  friend or foe

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Neuromuscular Junction
http//fig.cox.miami.edu/cmallery/150/neuro/neuro
muscular-sml.jpg
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How do muscle cells function ?
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Human cardiac muscle cells
control of cardiac muscle contraction At this
level, we can see the interaction of molecules
(i.e. proteins, cell membrane molecules)to
understand how the nanoscale operations incur
microscale changes such as influx of sodium ions,
and Na/K ATPase pumping action.
http//www.bmb.leeds.ac.uk/illingworth/muscle/car
diac
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Cell Bio-Mechanics

• How does a cell maintain or change shape ?
• How do cells move ?
• How do cells transport materials internally ?
  What mechanisms and using what forces ?
• How do cells stick together ? Or avoid adhering ?
• What are stability limits of cells components ?

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More Reading

• Mechanics of the Cell, by D. Boal, Cambridge
  University Press, 2002
• Molecular Biology of the Cell, by B. Alberts, D.
  Bay, J. Lewis, M. Raff, K. Roberts, J. Watson,
  1994
• The Machinery of Life, D. Goodsell, Springer
  Verlag.
• Several Linear-NonLinear Finite Element Meshing
  Papers

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3D Geometric Modeling Techniques

• Segmentation from Imaging
• 2D segmentation lofting
• 3D segmentation into linear and non-linear finite
  elements
• Interactive Free-Form Design
• 2D splines lofting
• 2D splines revolution
• 3D curvilinear wireframe
• 3D linear and non-linear finite-elements

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2D Segmentation of Platelet Sub-structures
VolRover
Platelet Data courtesy Mike Marsh, Dr. Jose
Lopez, Dr. Wah Chiu, Baylor College of Medicine
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Lofting I Linear Boundary Elements

• To generate a boundary element triangular mesh
  from a set of cross-section polygonal slice data.
• Subproblems
• The correspondence problem
• The tiling problem
• The branching problem

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Boundary Segmentation from 3D EM

• Multi-seed Fast Marching Method
• Classify the critical points interior/exterior.
  
• Each seed initializes one contour, with its
  groups membership.
• Contours march simultaneously. Contours with same
  membership are merged, while contours with
  different membership stop each other.

bullfrog hair bundle tip link
C. Bajaj, Z. Yu, and M.Auer, J. Strutural
Biology, 2003. 144(1-2), pp. 132-143.
Data courtesy Dr. Manfred Auer
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Bull-Frog Inner Hair Cell Models
(Collaborators Manfred Auer, LBL
Sponsored by NSF-ITR, NIH
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Sub-problems

• Correspondence

• Tiling

• Branching

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Lofting II Tetrahedral Finite Elements

• To generate a 3D finite element tetrahedral mesh
  of the simplicial polyhedron obtained via the BEM
  construction of cross-section polygonal slice
  data.
• Subproblems
• The shelling of a polyhedron to prismatoids
• The tetrahedralization of prismatoids

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What is prismatoid?

• A prismatoid is a polyhedron having for bases
  two polygons in parallel planes, and for lateral
  faces triangles or quads with one side lying in
  one base, and the opposite vertex or side lying
  in the other base, of the polyhedron.

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Examples

• Knee joint (the lower femur, the pper tibia and
  fibula and the patella)
• Gouraud shaded
• The tetrahedralization

• Hip joint (the upper femur and the pelvic joint)
• Gouraud shaded
• The tetrahedralization

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Non-Linear Algebraic Curve and Surface Finite
Elements ?
a200
a110
a101
a002
a020
a011
The conic curve interpolant is the zero of the
bivariate quadratic polynomial interpolant over
the triangle
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Non-Linear Representations

• Explicit
• Curve y f(x)
• Surface z f(x,y)
• Volume w f(x,y,z)
• Implicit
• Curve f(x,y) 0 in 2D, ltf1(x,y,z)
  f2(x,y,z) 0gt in 3D
• Surface f(x,y,z) 0
• Interval Volume c1 lt f(x,y,z) lt c2
• Parametric
• Curve x f1(t), y f2(t)
• Surface x f1(s,t), y f2(s,t), z f3(s,t)

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Algebraic Curves Implicit Form
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A-spline segment over BB basis
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Regular A-spline Segments
If B0(s), B1(s), has one sign change, then the
curve is (a) D1 - regular curve. (b) D2 -
regular curve. (c) D3 - regular curve. (d) D4
- regular curve.
For a given discriminating family D(R, R1, R2),
let f(x, y) be a bivariate polynomial . If the
curve f(x, y) 0 intersects with each curve in
D(R, R1, R2) only once in the interior of R, we
say the curve f 0 is regular(or A-spline
segment) with respect to D(R, R1, R2).
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Examples of Discriminating Curve Families
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Constructing Scaffolds
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Input
G1 / D4 curves
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Lofting III Non-Linear Boundary Elements
Input contours
G2 / D4 curves
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Spline Surfaces of Revolution

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A-patch Surface (C1) Interpolant

• An implicit single-sheeted interpolant over a
  tetrahedron

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C1 Shell Elements
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C1 Quad Shell Surfaces can be built in a similar
way, by defining functions over a cube
C1 Shell Elements within a Cube
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Examples with Shell Finite Elements
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Extra Slides

• Details on Spline Interpolants

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Non-linear finite elements-3d

• Irregular prism
• Irregular prisms may be used to represent data.

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Non linear Transformation of mesh
s
x
v
u
XYZ space
UVS space
y
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Linear Interpolation on a line segment

• p0 p p1
• The Barycentric coordinates a (a0 a1) for any
  point p on line segment ltp0 p1gt, are given by

f
f1
fp
f0
which yields p a0 p0 a1 p1 and
fp a0 f0 a1 f1
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Linear interpolation over a triangle

• p0
• p1 p p2
• For a triangle p0,p1,p2, the Barycentric
  coordinates
• a (a0 a1 a2) for point p,

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Linear interpolant over a tetrahedron

• Linear Interpolation within a
• Tetrahedron (p0,p1,p2,p3)
• a ai are the barycentric coordinates of
  p
• p3
• p
• p0 p2
• p1

fp3
fp
fp2
fp0
fp1
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Other 3D Finite Elements (contd)

• Unit Prism (p1,p2,p3,p4,p5,p6)
• p1
• p2 p3
• p p4
• p5 p6

Note nonlinear
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Other 3D Finite elements

• Unit Pyramid (p0,p1,p2,p3,p4)
• p0
• p1 p2 p p3
• p4

Note nonlinear
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Other 3D Finite Elements

• Unit Cube (p1,p2,p3,p4,p5,p6,p7,p8)
• Tensor in all 3 dimensions
• p1 p2
• p3 p4
• p
• p5 p6
• p7 p8

Trilinear interpolant
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C1 Interpolant
Hermite interpolation f0
f1 f0
f1

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Incremental Basis Construction

• Define functions and gradients on the edges of a
  prism
• Define functions and gradients on the faces of a
  prism
• Define functions on a volume
• Blending

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Hermite Interpolant on Prism Edges
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Hermite Interpolation on Prism Faces
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Shell Elements (contd)

• The function F is C1 over ? and
  interpolates C1 (Hermite) data
• The interpolant has quadratic precision

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Side Vertex Interpolation
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C1 function construction (cont.)

• Blending
• where