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

Lecture 5: Multiscale Bio-Modeling and Visualization

Cell Shapes, Sizes, Structures: Geometric Models

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 September 2005

Cells : Their Form

• Evolutionary History of approx. 1.5 billion years ago– Simple cells with their molecular machinery jumbled together in a single compartment -> ancestors of modern bacteria

– Compartmented cells -> yeast, plant, animal cells (tiny protozoa -> mammals -> tallest trees)

• Two basic types of cells– Prokaryotes (before kernel)– Eukaryotes (true kernel)

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

The Tree of Life?

Prokaryotic cell

Eukaryotic cell

Archaebacteria cell

RibosomeViruses?

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

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

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

Cell’s 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)

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

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

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

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

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

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

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

Cell’s Vary in Sizes, Shape, Form and Function

Operative Size is 1 μm (smallest is 0.3 μm and Largest > 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 (n=14) cells in the human body

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

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

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

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

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

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

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

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

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

Functions performed by Cells

• Chemical – e.g. manufacturing of proteins

• Information Processing – e.g. cell recognition of friend or foe

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

Neuromuscular Junction

http://fig.cox.miami.edu/~cmallery/150/neuro/neuromuscular-sml.jpg

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

How do muscle cells function ?

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

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/#cardiac

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

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 cell’s components ?

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

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

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

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

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

2D Segmentation of Platelet Sub-structures

Platelet Data courtesy: Mike Marsh, Dr. Jose Lopez, Dr. Wah

Chiu, Baylor College of Medicine

VolRover

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

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

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

Boundary Segmentation from 3D EM• Multi-seed Fast Marching Method

– Classify the critical points interior/exterior. – Each seed initializes one contour, with its group’s membership. – Contours march simultaneously. Contours with same membership are

merged, while contours with different membership stop each other.

C. Bajaj, Z. Yu, and M.Auer, J. Strutural Biology, 2003. 144(1-2), pp. 132-143.

bullfrog hair bundle tip link

Data courtesy: Dr. Manfred Auer

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

Bull-Frog Inner Hair Cell Models

(Collaborators: Manfred Auer, LBL

**Sponsored by NSF-ITR, NIH

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

Sub-problems

• Correspondence

• Tiling

• Branching

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

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

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

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.

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

Knee joint (the lower femur, the pper tibia and fibula and the patella)

(a) Gouraud shaded

(b) The tetrahedralization

Hip joint (the upper femur and the pelvic joint)

(a) Gouraud shaded

(b) The tetrahedralization

Examples

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

Non-Linear Algebraic Curve and Surface Finite Elements ?

a200

a020 a002

a101a110

a011

The conic curve interpolant is the zero of the bivariate quadratic polynomial interpolant over the triangle

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

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, <f1(x,y,z) = f2(x,y,z) = 0> in 3D

•Surface: f(x,y,z) = 0•Interval Volume: c1 < f(x,y,z) < c2

– Parametric•Curve: x = f1(t), y = f2(t)•Surface: x = f1(s,t), y = f2(s,t), z = f3(s,t)

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

Algebraic Curves: Implicit Form

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

A-spline segment over BB basis

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

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).

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

Examples of Discriminating Curve Families

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

Constructing Scaffolds

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

Input:

G1 / D4 curves:

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

Lofting III : Non-Linear Boundary Elements

Input contours G2 / D4 curves

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

Spline Surfaces of Revolution

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

A-patch Surface (C^1) Interpolant

• An implicit single-sheeted interpolant over a tetrahedron

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

C^1 Shell Elements

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

C^1 Quad Shell Surfaces can be built in a similar way, by defining functions over a cube

C^1 Shell Elements within a Cube

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

Examples with Shell Finite Elements

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

Extra Slides

Details on Spline Interpolants

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

Non-linear finite elements-3d

sNon linearTransformationof mesh

UVS space XYZ spacev u

x

y

z

))()1(()1)()1(( svuvusvuvuzyx

543210 pppppp

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

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

Linear Interpolation on a line segment

p0 p p1

The Barycentric coordinates α = (α0 α1) for any point p on line segment <p0 p1>, are given by

)),(),(,

),(),((

10

0

10

1

ppdistppdist

ppdistppdist

which yields p = α0 p0 + α1 p1

and fp = α0 f0 + α1 f1

ff1f0fp

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

Linear interpolation over a triangle

p0

p1 p p2

For a triangle p0,p1,p2, the Barycentric coordinates

α = (α0 α1 α2) for point p, )),,(),,(,

),,(),,(,

),,(),,((

210

10

210

20

210

21

pppareappparea

pppareappparea

pppareappparea

2

0ii pp

2

0ii fpfp

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

Linear interpolant over a tetrahedron

Linear Interpolation within a • Tetrahedron (p0,p1,p2,p3)

α = αi are the barycentric coordinates of pp3

pp0 p2

p1

3

0ii pp

fp0

fp1

fp3

fp23

0ii fpfp

fp

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

Other 3D Finite Elements (contd)

• Unit Prism (p1,p2,p3,p4,p5,p6)

p1

p2 p3

p p4

p5 p6

))(1()(6

43

3

1 iiii ptptp

Note: nonlinear

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

Other 3D Finite elements

• Unit Pyramid (p0,p1,p2,p3,p4)

p0

p1 p2 p p3

p4

)))1()(1())1(()(1( 43210 pssptpssptuupp

Note: nonlinear

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

Other 3D Finite Elements

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

p1 p2

p3 p4

pp5 p6

p7 p8

)))1()(1())1(()(1()))1()(1())1(((

8765

4321

pssptpssptupssptpssptup

Trilinear

interpolant

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

Hermite interpolation f0’ f1’ f0 f1

C1 Interpolant

f (t) = f 0H30(t) + f 0

0H31(t) + f 1H3

2(t) + f 01H3

3(t)

0 1

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

• 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

Incremental Basis Construction

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

Hermite Interpolant on Prism Edges

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

Hermite Interpolation on Prism Faces

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

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

Shell Elements (contd)

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

Side Vertex Interpolation

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

• Blending

where

Wi(b1;b2;b3) = (b2b3)ì +(b1b3)ì +(b1b2)ì(bjbk)ì

; ì > 1

P WiD i(b1;b2;b3;õ) + (b1b2b3)2E(b1;b2;b3;õ)

C1 function construction (cont.)