Methods for the Large Scale Simulation

1
Methods for the Large Scale Simulation of Blood
Cell Membranes Seth Green1 George Turkiyyah2
Duane Storti3 - University of Washington 1,3Depar
tment of Mechanical Engineering 2Deparment of
Civil Engineering 1sgreen_at_u.washington.edu
2george_at_ce.washington.edu 3storti_at_me.washington.ed
u
Simulation of Micropipette Suction Experiment

• Objective
• Simulate the mechanics of red blood cells for
  engineering design
• Criteria
• Computationally robust simulation
• Efficient solution strategy
• Scalable to large systems with many degrees of
  freedom
• Accurate representation of large cell membrane
  deformations

• Approach
• Subdivision surface-based finite element solver
• Constraints for simulating contact with other
  bodies
• Volume preservation to simulate incompressibility
  of internal fluid structure
• Verify results by simulating known experimental
  results

The images below show three simulation steps of a
spherical cell being sucked into a cylindrical
pipette. The cell is modeled as a thin membrane
(which resists both changes in area and changes
in curvature) surrounding a fluid center. A
volume preservation constraint is applied to
simulate the incompressible nature of the fluid
center. During simulation constraints are
applied around the rim of the pipette to ensure
that the cell surface slides tangent to the
pipette mouth, maintaining contact.
Efficiency
Subdivision surfaces provide a compact
representation for smooth surfaces of arbitrary
topology. The simple coarse base mesh shown
below contains only 42 nodal degrees of freedom,
yet is capable of representing each of the
complex deformed states shown beneath it. The
surface itself is always guaranteed to be
mathematically smooth under large deformations,
allowing for the accurate calculation of both
membrane and bending modes of deformation.
Subdivision representations provide a naturally
nested hierarchy of discretizations for a model
as illustrated above. Solutions may be obtained
quickly on a coarse discretization, but more
accurately (and slowly) on a fine one. By
considering each of these discretizations at once
we have implemented an efficient solver called
MGPCG (Multigrid-Preconditioned Conjugate
Gradient).
Increasing Pressure
Performance is illustrated in the plot on the
right. WIthout MGPCG the number of iterations
(computational effort) required to solve a system
of equations increases linearly with the number
of unknowns. Using our MGPCG method, this number
remains constant greatly speeding up the solution
process.
Accuracy
Bodies which are thin exhibit two characteristic
modes of deformation stretching which involves
changes in area, and bending which involves
changes in curvature. Often these two modes of
deformation will contribute vastly different
energies to the system imagine deforming a piece
of paper, it bends easily but it very difficult
to stretch. Subdivision models effectively
decouple stretching and bending deformations
allowing for greater numerical precision.
Subdivision models use only displacement degrees
of freedom, allowing for the simple application
of contact and volume-preservation constraints.
The plot at right shows that error decreases
quadratically with increasing refinement for a
benchmark problem in this case a cylinder
compressed by two opposing forces.
The plot below shows the relation between the
distance the cell tongue length (illustrated in
the diagram below left) and the applied load.
The character of the plot shows a large initial
slope followed by a leveling off of the curve,
indicating an increased resistance to
deformation. As the tongue length becomes larger
the total surface area of the cell must increase
to preserve volume. This deformation is resisted
by the cell membrane. The shape of this plot
correlates with experimental data.
Future Work Simulations would benefit from an
improved model of cell material behavior by
implementing the Evans-Skalak cell membrane
material model. More results will be helpful in
evaluating the accuracy of our approach compared
with experimental results.
Conclusions Subdivision surface-based finite
element techniques show great promise in
effectively simulating the mechanical behavior of
blood cells. Robust simulations of blood cell
behavior will aid engineers and physicians in
estimating blood damage due to cell rupture
caused by medical implants and artificial organs.
The subdivision surface paradigm provides a
compact, elegant and robust framework for
simulating the geometry and mechanics and
dynamics of blood cell membranes with efficient
solution strategies.