Last update: Sat Apr 14 02:01:36 MDT 2018
@Article{McQueen:2000:TDC,
author = "David M. McQueen and Charles S. Peskin",
title = "A three-dimensional computer model of the human heart
for studying cardiac fluid dynamics",
journal = j-COMP-GRAPHICS,
volume = "34",
number = "1",
pages = "56--60",
month = feb,
year = "2000",
CODEN = "CGRADI, CPGPBZ",
DOI = "https://doi.org/10.1145/604446.604453",
ISSN = "0097-8930",
bibdate = "Wed Oct 7 09:18:19 MDT 2009",
bibsource = "http://www.math.utah.edu/pub/tex/bib/siggraph2000.bib",
abstract = "In all areas of computational fluid dynamics (CFD),
proper treatment of the boundary conditions is
essential to computing fluid behavior correctly. In
many engineering problems, CFD is simplified by a
priori knowledge of the motion of the boundary. The
well-known parabolic velocity profile in
fully-developed flow of an incompressible Newtonian
fluid in a pipe of circular cross-section is easily
computed because the boundary (the pipe wall) is known
to be in a fixed location. Even in more complex
settings, such as flow around a ship's propeller, the
motion of the boundary (the propeller) can be specified
in advance.
By contrast, in most biological fluid dynamics problems
the boundaries are not rigid and their motions are the
result of forces imposed on them by the motion of the
surrounding fluid. The motion of the fluid, of course,
cannot be known without knowledge of the boundary
motion. The motion of the boundary and the motion of
the fluid form a coupled system; both motions must be
computed simultaneously, which makes biological CFD
difficult.\par
A particular problem of interest is the flow of blood
in the chambers of the human heart. The heart is an
organ whose muscular contractions pump blood around the
body. Simplifying somewhat, the heart consists of two
main pumping chambers that contract simultaneously. One
chamber, the left ventricle, accepts oxygen-enriched
blood from the lungs and pumps it to the body. The
other chamber, the right ventricle, accepts
oxygen-depleted blood from the body and pumps it to the
lungs. The inlet and outlet of each ventricle are
guarded by valves whose opening and closing guarantee
one-directional flow around the circulatory system.
There are a total of four valves. The valves generally
consist of two or three leaflets - membranes made of
very flexible but inextensible material. Familiar
examples of materials with this property would be paper
or fabric which can be easily bent or twisted but which
are not easily stretched. One edge of each valve
leaflet is securely attached to the wall of the heart,
but the other edge is free of attachment and can move
with the flow. Structures analogous to a valve leaflet
are a shirt pocket, with one edge (three sides of a
rectangular patch pocket) securely stitched to the
shirt and one edge free of attachment, or a flag, one
edge attached to the flag pole, the other edge free to
wave in the wind. When flow is passing through the
valve in the forward direction, the valve's leaflets
are positioned out of the way, permitting flow. When
flow attempts to pass in the reverse direction, the
leaflets come together, their free edges pressing
against the free edges of their neighbors to occlude
the flow passage.\par
The motion of the leaflets is not caused by muscles in
the valve. The outflow valves are entirely passive
structures with no muscular tissue whatsoever. Even in
the case of the inflow valves, whose free edges are
connected to the heart muscle by a sparse network of
tendons, the opening and closing motions result from an
interaction with the surrounding fluid. The forward
motion of the fluid pushes the leaflets aside out of
the main flow stream, but the inextensibilty of the
leaflet material prevents free motion of the fluid near
the leaflet, affecting the entire flow field. Reverse
motion of the fluid causes the leaflets to move back
into the flow passage where contact between neighboring
leaflets and the inextensibilty of the leaflet material
halts the flow. The highly interactive nature of the
fluid and leaflet motions makes this an especially
interesting and challenging CFD
problem.\par
Commercially available software packages intended for
engineering CFD are not equipped to handle this type of
dynamic interaction between boundary and fluid. We have
developed a numerical method (the 'Immersed Boundary
Method') which simultaneously computes the motion of a
fluid and the motion of an elastic boundary immersed
in, and interacting with, that fluid. In the Immersed
Boundary Method, the fluid is represented by Eulerian
velocities and pressures that are stored on a regular
three-dimensional computational lattice. The scale of
the heart chambers is such that blood can be treated as
a Newtonian fluid. Fluid dynamics is computed by
numerical solution of the Navier--Stokes equations,
including a body force. The boundary is represented by
elastic structures that are free to move continuously
in the space sampled by the computational lattice. The
essence of the method is to replace the elastic
boundary by the forces that result from its
deformations. These forces are applied to the lattice
in the neighborhood of the elastic boundary with the
aid of a numerical approximation to the Dirac delta
function. The fluid moves under the action of this body
force. The numerical delta function is then used again,
to interpolate the newly computed lattice velocities to
the locations of the boundary, and then the boundary is
moved at the interpolated velocity to a new location
(the no-slip condition). The process of computing
forces, then fluid motion and then new boundary
location is repeated cyclically in a time-stepping
procedure with a suitably chosen time step. The only
requirements for the method are the physical properties
of the fluid, the (possibly time-dependent) elastic
properties of the boundary, and the initial geometry of
the boundary. A complete description of the Immersed
Boundary Method can be found in [1, 2].",
acknowledgement = ack-nhfb,
fjournal = "Computer Graphics",
journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J166",
}