1. Numerical Modeling of Dynamics and Adhesion of Leukocytes
A. Golpayegan and N. Ashgriz
9/22/2018

2. Outline Introduction Leukocyte biomechanical model
Governing equations and the boundary conditions
Body forces
Numerical methodology
Interface tracking method
Solution procedure

3. Introduction Leukocytes are the main part of human defense system
They have a radius of about 8 micro meter
1% of the total volume of blood
They have larger volume and lower deformability than red blood cells
Leukocytes play an important role in the microcirculatory flows (they migrate to tissue in response to cuts or infections)

4. Ref: http://hsc.virginia.edu/medicine/basic-sci/biomed/ley/
Process
Rotation
Activation and merging
Deformation
Adhesion (Specific molecular binding)
Migration
Ref:

5. Micropipette Aspiration
A method to determine viscosity of leukocyte
A small suction pressure is applied
to deform leukocyte
Viscosity calculated based on amount of
deformation and pressure difference
Authors
operation condition
Cell
Apparent Viscosity (Poise)
Dong et al., 1988
small deformation
300
Evan and Yeung, 1989
large deformation
Dong et al., 1991
Hochmuth et al., 1993
600
Waugh and Tsai, 1994
Bathe et al. 2002 31

6. Apparent Viscosity Viscosity is a function of
Rate of deformation
Amount of deformation
Holding time in micropipette
Multi-layer model is capable of explaining behaviors
High deformation rate
Response of cytoplasm is rapid
Lower Viscosity
Nucleus deforms little

7. Mechanical Model Multi-component droplet
( Nucleus, cytoplasm and membrane)
Newtonian Fluid
Outer Layer (membrane)
Inner layer (Nucleus)
Middle layer (cytoplasm)
How can the changes in the observed viscosities be explained?

8. Apparent Viscosity(Continue)
Observation directing researchers conclude other than Newtonian behavior for leukocyte:
-Viscosity decreases when the cell recovers from smaller deformation (Hochmuth et al. 1993)
The cell is aspirated into a relatively large pipet and the nucleus does not have to deform very much.
Recovery is governed mainly by cytoplasmic layer producing lower viscosity
-Viscosity increases when the cell undergoes large deformation (Evan and Yeung, 1998)
- The cell is aspirated into a relatively small pipet and highly viscous nucleus must deform leading to a higher cell apparent.

9. Problem associated
Need to control the process (reinforce or prevent it)

10. Leukocyte Model
Multi-component droplet
( Nucleus, cytoplasm and membrane)
Newtonian Fluid
Interface membranes are represented by a surface tension model
Outer Layer (membrane)
Inner layer (Nucleus)
Middle layer (cytoplasm)

11. Outflow boundary condition
Boundary Conditions
No slip
Inflow and outflow
Surface tension acts as a boundary between each two fluids????
Computational Domain
Outflow boundary condition
Inflow
Outflow
Coarse grid
Fine grid
Wall effects fects

12. Governing Equations Momentum equation Continuity equation
Proper boundary condition and initial condition
Body forces:
Surface tension
Adhesion

13. Body force resulting from Surface tension
Continuum Surface model (CSF)
Surface tension is formulated as an equivalent body force :
Unit normal
Curvature

14. Body force resulting from Adhesion=
Bonds have a preferred bond length with minimal free energy. kf kr

15. Concentration of adhesive bonds
Diffuse
during
Adhesion
With the rate
“r”
Active Leukocytes
Passive Leukocytes
Concentration of adhesive bonds
Diffusion - reaction equation:

16. Reaction rate
r =
exp (
exp (

17. Non-specific interactions
Resistance of polymers to compression, osmotic tendency of water molecules
Electrostatic forces
exp

18. Numerical methodology
Fixed, staggered grids (Eulerian)
two-step projection method for pressure calculation
Boundary condition

19. Interface tracking using VOF
Initial interface geometry is used to compute volumetric fractions: B A C E D A B C D E

20. VOF Implementation Advection of “volumetric function” satisfies
Surface reconstruction
F- field used to determine normal vectors
Decide the type of “case” using normal
Position plane with known slope based upon volume fraction
Compute plane area and vertices
Fluid Advection
Compute flux across cell sides
Operator Split ( for x, y and z sweep)