lecture 6: multiscale bio-modeling and visualization cell modeling ii: deformable models

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

Lecture 6: Multiscale Bio-Modeling and Visualization

Cell Modeling II: Deformable Models

Chandrajit Bajaj

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


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 ?


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)


Biological Motivation

endocytosis/exocytosis mitosis – cytokinesis

cell-cell interaction

Many biological processes at the cellular level involve the interaction of deformable interfaces. The deformation and interaction of cells is fundamental in many biological functions and processes. Any process involving cell motility involves deformation of the cell membrane.

http://faculty.uca.edu/~johnc/MEMBRANE.htm

• http://www.hopkinsmedicine.org/cellbio/robinson/

Goal– Accurate modeling, simulation, and visualization of such

phenomena


Baker’s Yeast: Saccharomyces Cerevisiae

• a & b: cytoplasm• c: nuclear membrane• d: nuclear matrix• e: nuclear pore• f & g:

mitochondrion• h: endoplasmic

reticulum• i: golgi apparatus• j: coated vesicle• k: vacuole


Cytoplasm: Cytoskeleton

CytoskeletonYeast cytoplasm is

crisscrossed with structural filaments, together forming the cytoskeleton.

microtubule

actin filament

Intermediate filament


Cytoplasm: Protein Synthesis

Protein synthesis

Underway in the cytoplasm. The machinery of

translation, consisting of ribosomes, tRNA and tRNA synthetases, builds proteins from mRNA in the cytoplasm.

Ribosomes line tRNA molecules along the mRNA strand and links up amino acids.

actin filament Intermediate filament


Mitochondria: power generators

Mitochondrial Membranes

• two concentric membranes

cytoplasm

Outer membrane Inner membrane

Pore-forming protein

Energy-producing protein


Mitochondrial Interior

Mitochondria are filled to their limit with proteins, ribosomes, and nucleic acids.

Inner membrane

Pyruvate dehydrogenase complex: enzyme that links glycolysis with the citric acid cycle by converting pyruvate to acetyl-CoA.


Nucleus: library

• Functions of nucleus– Storing the delicate

strands of DNA and protecting them from the rigors of the cytoplasm

– Where transcription of DNA into mRNA performed

• Specialized in archiving, reading, and regulating the use of info held in DNA

A double nuclear membrane Cytoplasm

Nuclear interior

DNA


Nuclear Interior

DNA

RNA polymerase molecules


Nuclear Pore

Nuclear pore

A complex of mRNA and protein is seen emerging through the elaborate gate of proteins. Because of the large size of the pore complex, only four of the eight symmetric gate elements fit into the image.


Endoplasmic Reticulum

Ribosomes bind to special transport proteins in the endoplasmic reticulum membrane. The newly built proteins are guided inside the endoplasmic reticulum. The proteins inside have begun their journey to the vacuole or cell surface.


Golgi Apparatus

Protein journey path

When proteins pass through the stacked disks of Golgi apparatus, they are successively modified and sorted. Long polysaccharides are added to membrane proteins and small digestive enzymes are gathered by membrane-bound sorting proteins.


Coated Vesicle

After passing through Golgi apparatus, proteins are transported to their ultimate destination in small vesicles. Triskelions associate into a geodesic structure on the surface of the Golgi apparatus membrane and pull off vesicles.

triskelion


Vacuole

The vesicle dumps its cargo of digestive enzymes and membrane proteins after reaching the vacuole.

Interior of the vacuole


Deformable Bodies

Two Droplets in Shear Flow

INITIAL BUBBLE RADIUS: 1.5 cmCapillary Number = 3.0Viscosity ratio = 1.0Surface tension = 1.5 dynes/cm^2Flow Rate = 1.0 /s


Interface Dynamics I

• A simplistic model for these phenomena utilizes Stokes equations for two-phase flow

0u 02 gup

cos

p fluid pressure

u fluid velocity

fluid vis ity

fluid density

g body forceterm acting on fluid

Stokes equation describes slow, viscous, steady flows characterized by low Reynolds number.

Can also be used to model quasi-steady flow in which flow changes due to changing boundary interfaces

Parameters


Interface Dynamics II

• Consider closed interfaces (droplets) that are immersed in a viscous fluid

(1) (2)u u

Boundary Conditions• velocity is continuous

(1) (2)u u

i

j

j

iijij x

u

x

up

• stress is discontinuous

( )f x is a model dependent term: could include effects from surface

tension, gravity or other body forces, or membrane elasticity

nxfn ˆ)(ˆ)2()1(

Stress defined as


Interface Dynamics III

• For interfacial dynamics in Stokes flow an integral representation is used to represent the velocity at the boundary

0 0 0

0

2 1ˆ( ) ( ) ( , ) ( ) ( )

1 8

1ˆ( ) ( , ) ( )

8

k k kj js

i ikj j

u x u x G x x f x n x d

u x T x x n x d

.

.

/ cos

( ) 2 ( )

int

1ˆ

2

droplet susp fluid

susp fluid dropletm

m

vis ity ratio

u asymptotic flow

f x g

erface tension

mean curvature n

3

00

0

1),(

r

xxxx

rxxG ji

ijij

5

000

0 6,r

xxxxxxxxT kji

ijk

Green’s Functions

EQN(*)

[5] C. Pozrikidis. 2000

Boundary Integral Formulation

0xxr


Algorithm Sketch

1. Specify an initial configuration and implicit representation of interfaces. Initialize fluid parameters and surrounding flow

2. Extract a “quality” adaptive mesh of the interface

3. Use a BEM to calculate interfacial velocities and analyze associated errors.

4. Check Oracle for topology change and modify implicit function if topology change should occur.

5. Evolve interface based on interfacial velocities

6. Check for geometric interference and adjust interface evolution if necessary.

7. Iterate steps 2 to 6 till stopping criterion is satisfied


Algorithm Sketch

1. velocity computation

2. quality meshing & refinement

3. topology control

• Details

4. interface update

5. interference check


0. Initial Configuration

• Setup initial interfaces as linear boundary element contour mesh M of an implicit function

• M provides the following:

– Vertex positions

– Face Connectivity

– Vertex Normals

– Mean Curvatures at vertices

– Minimum vertex list

• Fluid parameters:

cos , cos / , ,s sfluid vis ity vis ity ratio surfacetension gravity g


1. Velocity Computation IBoundary Element Method• Surface mesh representing the boundary is created• We attach a discretized EQN(*) at each vertex of the mesh• We are left with a linear system which we solve for the interfacial velocities at the mesh vertices

vertices of the mesh serve as boundary element nodes where velocities are computed

1

1

2 1ˆ( ) ( ) ( , ) ( ) ( )

1 8

1ˆ( ) ( , ) ( )

8

1.. .

.

e

e

N

j L k L jk L k ees

N

i ijk L k ee

u x u x G x x f x n x d

u x T x x n x d

L num vertices

N num faces

1

2 1 1 1ˆ( ) ( ) ( , ) ( ) ( )

1 8 4 1e

N

j L j L jk L k ees

F x u x G x x f x n x d

1

ˆ( ) ( ) ( ( ) ( )) ( , ) ( ) 4 ( )N

j L j L i i L ijk L k e j Le

u x F x K u x u x T x x n x d u x


1. Velocity Computation II

• Calculate the velocities at each vertex by iterating

• Start vectors for the iteration can be derived from previously computed boundary element velocities.

• Previously computed velocities are used in the integral on the right hand side for each iteration. Iteration is done until a convergence criteria is met.

1

1ˆ( ) ( ) ( ( ) ( )) ( , ) ( )

1 4 1 4e

N

j L j L i i L ijk L k ee

Ku x F x u x u x T x x n x d

K K


• Flow is incompressible so we can check that velocity is divergence free

• This also means that volume is preserved so we can calculate it, and check that its change is within acceptable tolerance

• A rough measure of local error can be made if we calculate the following quantity at each face,

• is the velocity calculated from the boundary element method linearly interpolated to the quadrature point and is taken to as the face average of the velocities at the vertices.

• The quantity gives us a measure of the variation of the velocity over each face. If this does not meet a specified tolerance then that face can be refined.

1. Velocity Computation II: Error Estimation

0u n d

1

3V r d

220( )

m

bem mE m u u d

ubem

u0


2. Quality Quad Meshing• Extending the dual contouring method to adaptive and

quality quad/hex meshing– The selection the starting octree level to guarantee the

correct topology– Crack-free and adaptive quad/hex meshing without hanging

nodes– Using geometric flows to improve mesh quality

• Mesh Adaptivity1. Feature sensitive error function2. Various areas users are interested in 3. Finite element solutions (bubble)4. User defined

References:• Y. Zhang, C. Bajaj. Adaptive and Quality Quadrilateral/Hexahedral Meshing from Volumetric Data. Computer Methods in Applied Mechanics and Engineering (CMAME), in press, 2005. •Y. Zhang, C. Bajaj. Adaptive and Quality Quadrilateral/Hexahedral Meshing from Volumetric Data. 13th International Meshing Roundtable, pp365-376, 2004.

Template :


3. Topology Control I : Function and Mesh Refinement

• Octree refinement & coarsening Contouring– Coarsening : remove high-frequency and change

topology in close areas

< Up-Sampling > < Down-Sampling >


3. Topology Control II: OracleTopology Decision • We invent an “Oracle” to determine whether topology change occurs during simulation.

• If two interfaces or two non-local regions of one surface are close enough within a distance bound, we assume the close regions are in a contact status.

• An octree spatial decomposition is used to identify contact regions.

• The Oracle’s implementation and how it handles this contact status can be changed depending on the specific problem being modeled.

The Oracle basically provides three outputs:

– No Change

– Coalescence

– Breakup

<coalescence>

<no topology change>


4. Interface Update I:TimeStep Restriction

• Once velocities are computed at the vertices of the mesh and the Oracle is satisfied we advect using

1.. .ii

dxv i num vertices

dt

For stability the timestep is taken as

t K x

x is the minimum size of a face on the mesh and K is a O(1) constant taken to be 0.5

[7] Rallison `81


4. Interface Update II: Semi-Lagrangian Level Set Method

• The interface is passively advected with the interface velocities by updating the level set data with the ‘level set equation’

• The level set data is stored in an octree data structure• To find the new value of the level set at the grid point located at we find the point that maps into by evolving

backward with the velocity

• Trace the point backward one timestep to find the location that maps to it.

• Interpolate from neighboring cells to find the value at this point.

0iiv

t

),( xtv n

txtvxyxtvt

yx nn

),(),(

The old value at y is the new value at x

( )t t

x

y


4. Interface Update II: Velocity Extension & Redistancing

• Lift velocity to nearest grid points by finding the closest point on the mesh and then interpolating velocity from surrounding vertices.

• Once grid points of cells intersecting the interface have velocities, march outwards from the interface solving for F in the following equation.

• F is a speed normal to the interface.

21

Reinitialization of level set data to a signed distance function is also done as we march outwards by solving

F 0


5. Interference Checking: Topology and Remeshing

• Interfacial Remeshing during a Droplet Coalescence


Flow Scenarios• To observe interesting effects in bubble dynamics such as drop break up, drop

coalescence, and drop slippage, initially spherical drops are embedded within various Stokes flows. Some common flows are the following,

Straining Flows• Straining flows act to pull apart the droplet.

Shearing Flows

Convergent Flows• Convergent flows are the opposite of straining flows in that they tend to squeeze droplets

together

),2

1(),( zrauu zr

a Determines strength of the flow

),0,0(),,( yauuu zyx

),2

1(),( zrauu zr

(r,z) are cylindrical coordinates


Coalescence

• Coalescence can occur if two droplets are placed in a convergent flow and allowed enough time for the thin fluid film between the drops to be drained.

• Non-hydrodynamic effects such as Van der Waals forces are only relevant at the moment of interface rupture.

• The dynamics of coalescence up to this point can be understood solely from the hydrodynamics.

• The strength of the asymptotic Stokes flow determines the rate at which the thin fluid film is drained


Breakup

• Drop breakup can occur in shearing and straining flows as well as buoyancy driven motion.

• Breakup is governed by three dimensionless physical parameters: capillary number, Bond number, viscosity ratio

R = undeformed drop radius. Other quantities are as they have been defined previously

• The capillary number, Ca, governs drop breakup in shearing flow. Breakup will occur after long enough time when Ca = O(1).

• In buoyancy driven motion, Bo is the parameter which governs drop breakup. When Bo = O(1) then gravity-induced drop breakup can occur.

aR

Cas

2gR

B

s


Example Scenarios

Straining Flow Converging Flow

INITIAL BUBBLE RADIUS: .95 cmCapillary Number = 3.8Viscosity ratio = 1.0 Surface tension = 0.5 dynes/cm^2Flow Rate = 1.0 /s

INITIAL BUBBLE RADIUS: 0.43 cmCapillary Number = .258Viscosity Ratio = 1.0Surface Tension = 1.0 dynes/cm^2Flow Rate = 1.0 /s


Simulation ResultsStraining Flow Converging Flow


BEM & Meshing ReferencesBoundary Element Fluid Simulation

[1] Z.A. Zinchenko,M.A. Rother, and R.H. Davis. A novelboundary integral algorithm for viscous interaction ofdeformable drops. Phys. Fluids, 9:1493–1511, 1996.[2] V. Cristini, J. Blawzdziewicz, and M. Loewenberg. Anadaptive mesh algorithm for evolving surfaces : simulationsof drop breakup and coalescence. Journal ofComputational Physics, 168:445–463, 2001.[3] J.-M. Hong and Kim C.-H. Animation of bubbles inliquid. In In Proceedings of Eurographics 2003, pages253–262, 2003.[4] M.Loewenberg and E.J. Hinch. Numerical simulationof a concentrated emulsion in shear flow. J. FluidMech., 321:395, 1996.[5] C. Pozrikidis. Boundary integral and singularity methodsfor linearized viscous flow. Cambridge UniversityPress, 1992.[6] C. Pozrikidis. Interfacial dynamics for stokes flow. J.Comp. Phys, 169:250–301, 2000. [7] J.M. Rallison. A numerical study of the deofrmationand burst of a viscous drop in general shear flows. J.Fluid Mech., 109:465–482, 1981.[8] E. Kita and N. Kamiya. A new adaptive boundary elementrefinement based on simple algorithm. Mech ResCommun., 18(4).[9] J.Tausch. Sparse BEM for potential theory and Stokes flowusing variable second order wavelets. Comp. Mech, 1, pp 312--318 2003.[10] J.Tausch. Rapid solution of Stokes flow using multiscale galerkin BEM. PAMM, Proc. Appl. Math. Mech. 1, pp 8-112002.

Meshing

[1] J. C. Carr, R. K. Beatson, J. B. Cherrie, T. J. Mitchell,W. R. Fright, B. C.McCallum, and T. R. Evans. Reconstructionand representation of 3d objects with radialbasis functions. In ACM SIGGRAPH, pages 67–76, 2001. [2] T. Ju, F. Losasso, S. Schaefer, and J.Warren. Dual contouringof hermite data. In Proceedings of SIGGRAPH,pages 339–346, 2002.[3] Leif Kobbelt, Thilo Bareuther, and Hans-Peter Seidel.Multiresolution shape deformations for meshes withdynamic vertex connectivity. volume 19, pages 249–260, 2000.[4] Jens Vorsatz, Christian Rössl, and Hans-Peter Seidel.Dynamic remeshing and applications. In ACM Symposiumon Solid Modeling and Applications, pages 167–175, 2003.[5] G. Yngve and G. Turk. Robust creation of implicitsurfaces from polygonal meshes. IEEE Transactionson Visualization and Computer Graphics, (4):346–359,2002.[6] Yongjie Zhang and Chandrajit Bajaj. Adaptive andquality quadrilateral/hexahedral meshing from volumetricdata. To Appear in Computer Methods in AppliedMechanics and Engineering (CMAME), 2005.[7] Yongjie Zhang, Chandrajit Bajaj, and Bong-Soo Sohn.3D finite element meshing from imaging data. To Appearin the special issue of Computer Methods in AppliedMechanics and Engineering (CMAME) on UnstructuredMesh generation, 2005.


Level Set References

• [1] D. Adalsteinsson and J. Sethian. A fast level set method for propagating• interfaces. J. Comp. Phys., 118:269-277, 1995.• [2] D. Enright, R. Fedkiw, J. Ferziger, and I. Mitchell. A hybrid particle• level set method for improved interface capturing. J. Comp. Phys.,• 183:83-116, 2002.• [3] D. Enright, S. Marschner, and R. Fedkiw. Animation and rendering of• complex water surfaces. ACM Trans. on Graphics (SIGGRAPH 2002• Proceedings), 21:736-744, 2002.• [4] S. Osher and R. Fedkiw. Level Set Methods and Dynamic Implicit• Surfaces. Springer-Verlag, New York, 2002.• [5] S. Osher and J. Sethian. Fronts propagating with curvature dependent• speed: Algorithms based on hamiliton-jacobi formulations. J. Comp.• Phys., 79:12-49, 1988.• [6] D. Peng, B. Merriman, S. Osher, H.-K. Zhao, and M. Kang. A pde-• based fast local level set method. J. Comp. Phys., 155:410-438, 1999.• [7] W. Rider and D. Kothe. Reconstructing volume tracking. J. Comp.• Phys., 141:112-152, 1998.• [8] J. Sethian. A fast marching level set method for monotonically advanc-• ing fronts. Proc. Natl. Acad. Sci., 93:1591{1595, 1996.• [9] J. Sethian. Fast marching methods. SIAM Rev., 41:199-235, 1999.• [10] C. Shu and S. Osher. Effcient implementation of essentially non-• oscillatory shock capturing schemes. J. Comp. Phys., 77:439-471, 1988.• [11] J. Strain. Semi-lagrangian methods for level set equations. J. Comp.• Phys., 151:498-533, 1999.• [12] J. Strain. A fast modular semi-lagrangian method for moving interfaces.• J. Comp. Phys., 161:512-536, 2000.• [13] M. Sussman and E. Fatemi. An effcient, interface-preserving level set• redistancing algorithm and its application to interfacial incompressible• fuid fow. SIAM J. Sci. Comput., 20:1165-1191, 1999.• [14] D. Enright, F. Lossasso, R. Fedkiw. 2005. A fast and• accurate semi-lagrangian particle level set method. Computers• and Structures, 83, 479–490.• [15] R. Fedkiw, J. Stam, H.W. Jensen. Visualization of smoke. In Proc. Of ACM SIGGRAPH 2001, 15-22.


Method Overview for MultiphaseCharged, Viscous Fluid Flows

Some preliminary Thoughts


Assumptions of the Model

• Consider two electrically charged fluids of different viscosities and different dielectrics

• Model the hydrodynamics with the Stokes equation which governs viscous flow

• Model the electrostatic interaction with Poisson-Boltzmann equation


Governing Equation for Stokes Flow

02 gup

• p is the pressure• u is the fluid velocity• The last term represents a long range body force. For this model this force will be electrostatic.


Poisson-Boltzmann Equation

kTre

rrrff /)(4

))(sinh()()()( 2

0)()()( frr

where,

B.Honig and A.Nicholls, Science 268, 1144 (1995)


Boundary Element Method

• With an explicit representation for the boundary we choose the boundary element method as our numerical solver

• We first solve the electrostatic boundary value problem

• Use the result to solve the hydrodynamic problem


Overview of solver

• Discretize the boundary interface• Assign a charge density to each boundary element on the interface

• Solve using a point collocation scheme

• Calculate the electrostatic potential at all collocation nodes.


Boundary Traction

• The difference in traction at the boundary is given by,

))((ˆ2 xgnf psimi

gThe terms are the electrostatic forces per

unit volume on the particle and suspending fluid

C.Pozrikidis, J. Comp. Phy. 169, 250-301 (2001)


Coupling Electrostatics with Interfacial Dynamics

• Having solved for the electrostatic potential via BEM we calculate the electrostatic force per unit volume at each boundary collocation node

• This is added to the boundary traction difference

• The velocities at points on the boundary are solved via BEM.

• The interface is updated


Additional Notes

• We may have multiple resolutions of the boundary element mesh: one for the electrostatic BEM solver and another for the hydrodynamic BEM solver.

• We could also use finite element or finite difference methods to calculate the electrostatic potential and then couple to the BEM fluid solver.

lecture 6: multiscale bio-modeling and visualization cell modeling ii: deformable models

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