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

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

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)

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

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 1000-2000 Dong et al., 1991 200-2000 Hochmuth et al., 1993 600 Waugh and Tsai, 1994 1000-1500 Bathe et al. 2002 31

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

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?

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.

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

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)

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

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

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

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

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

Reaction rate r = exp ( exp (

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

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

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

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)