introduction:

Introduction:• Gravitational forces resulting from microgravity, take off and landing of spacecraft are experienced by individual cells in the living organism.

• Such stresses alter cell shape, cytoskeleton organization and internal pre-stresses in the cell tissue matrix.

• Spaceflight is associated with a significant increase in the number of circulating blood cells including leukocyte, B cells and T-helper cells and their motion through capillaries.

• Prior studies have shown that the stresses due to the spaceflight lead to a sympathetic nervous system-mediated redistribution of circulating leukocytes.

• In addition, study of the cell migration is relevant to several other biological processes such as embryogenesis, and cell division.

• Obtaining the properties of human blood cell is necessary to have a better understanding of the deformability of human cells, in particular the leukocytes, under various stress conditions such as those in a spaceflight and microgravity.

• Properties of a drop, surface tension and viscosity can be determined based on the dynamical behavior and shape deformation during motion through a nozzle.

Numerical technique:• Full Navier-stokes and continuity equations for an incompressible and Newtonian fluid are solved numerically.

• To solve the flow equations within the drop, the numerical model needs to track the location of the liquid interface.

.0

ft

fV

Interface Tracking model (Volume-Of-Fluid):

• For each cell a volumetric function f defined, representing the amount of the fluid present in that cell.

. fluidoutside0

, fluidinside1f

• The surface cells are defined as the cell with 0<f<1.

• Properties used in the Navier-stokes equation for the surface cell are calculated based on the value of f.

• A teach time step the unit normal vectors are calculated and the function f is advected:

f

fn

Surface Tension model:

• The surface tension was modeled using Continuum Surface Force (CSF).

• Surface tension was reformulated as a body force in the Navier-Stokes equation.

• For this purpose for each computational cell, curvature k was calculated.

rrxrrx dnS

ST F

Internal obstacle modeling:

• Internal obstacles are modeled as a special case of two phase flow.

• The fluid volume fraction is defined as θ, and the obstacle volume fraction is defined as 1-θ.

• The internal obstacle is characterized as a fluid with infinite density and zero velocity.

• θ is independent of time.

• θ = 1, not an obstacle, open to the flow..

• θ =0, is an obstacle, close to the flow.

• The Navier- stokes equations are modified and solved based on considering the obstacle:

0.( V)

bFVV.VV

2.p

t

Results:• The following figures represent a drop with radius of 1.15mm simulating the cell moving toward a passage.

• The nozzle has a conic angle of 35.5°. The outer diameter of the nozzle is equal to 0.86 mm.

• The drop properties are: surface tension 0.073 N/m, and kinematics viscosity of 8.95×10-5 m/s2.

Amirreza Golpaygan, Ali Jafari & Nasser Ashgriz

Department of Mechanical and Industrial Engineering

University of Toronto

σ= 0.146 N/m ( 2 ×Water surface tension)

L/D= 2.128

t= 6.8ms

σ= 0.073 N/m (Water surface tension)

L/D= 2.93

t= 10.2ms

σ= 0.0365 N/m (1/2 ×Water surface tension)

L/D= 3.75

t= 13.4ms

Proposed Model:• In order to study the cell cytoskeleton deformation during the cell migration, cell is modeled as viscous liquid drop with interfacial tension moving through a controlled surface environment.

• The viscous liquid drop represents the cell which has been forced to migrate through a nozzle representing capillaries in the tissue of human body.

• The morphological changes in the drop shape represent changes in the cytoskeleton of the cell.

• The viscosity of liquid drop is representative of the resistance of the cytoskeleton to the shape deformation.

• A drop with the diameter D and initial velocity of V moving toward a nozzle with the conic angle of 2α and the diameter d at its outlet.

• Inertia, surface tension, viscosity, and wall effects are the parameters which determine the dynamics of the drop and its shape.

V

D d

H

2α

Cell shape is the most critical determinant of cell function.

Numerical Simulation of Deformation and Shape Recovery of Drops Passing Through a Capillary

Multiphase Flow and Spray System Laboratory http://www.mie.utoronto.ca/labs/mfl

t=0ms

t=18ms

t=8ms t=13.5ms

t=4ms

t=30ms

Initial velocity is equal to 1m/sec. The inertia of the drop is not enough to overcome the opposite forces from the wall, and effects of the surface tension.

• The outcome is determined based on the balance of the forces.

• The inertia of the drop forces it against the resistance from the wall resisting its forward motion, and the resistance from the surface tension against deformation.

• The viscosity of the drop acts as the internal friction which is another barrier against the inertia.

By increasing inertia of the drop, changing the velocity to 1.5 m/sec from 1 m/sec in previous case, drop moves through the nozzle. Drop starts to oscillate freely to get its initial shape as a spherical drop.

t=0ms

t=8.5ms

t=2.5ms

t=10.2ms

t=0ms

t=1ms

• Characteristic length of the drop is defined as the elongated length of the drop (L) after deformation over its in initial diameter (D).

σ(N/m) t (ms) L/D

0.146 6.8 2.128

0.073 10.2 2.93

0.0365 13.4 3.75

Conclusion:• A 3-dimensional computational model for a cell migrating through a channel with the shape of nozzle is presented. The cell is modeled as a viscous drop. For the liquid viscous drop, full Navier-stokes equations considering surface tension and internal obstacle are solved.

• The results of simulation for the shape deformation and recovery are presented.

• The work is in progess to obtain a correlation for the changes in the cell viscosity with the changes in the cell’s cytoskeletal structure in order to gain a qualitative description of the cytoskeletal deformation process of the cell.

The velocity vectors for the drop with initial velocity of 1.5 m/sec. After the nozzle, the drop continues oscillation to gain its initial shape.

The velocity vectors for the drop with initial velocity of 1 m/sec. The viscous effect and wall effects damp the inertia, therefore the drop oscillates inside the nozzle.

t = 0ms t = 1.5mst = 0.5ms

t = 3.5ms t = 9mst = 7.5ms

t = 10.5ms t = 15.5mst =13ms

t = 17ms t = 25mst = 18.5ms

t = 0ms t = 1.5mst = 0.5ms

t = 3.5ms t = 9mst = 7.5ms

t = 10.5ms t = 15.5mst =13ms

t = 17ms t = 25mst = 18.5ms

• F represent present body force, surface tension.