A three-dimensional computer model of the human heart for studying cardiac fluid dynamics

• Authors: David M. McQueen;Charles S. Peskin
• Venue: ACM SIGGRAPH Computer Graphics
• Year: 2000

In all areas of computational fluid dynamics (CFD), propertreatment of the boundary conditions is essential to computingfluid behavior correctly. In many engineering problems, CFD issimplified by a priori knowledge of the motion of the boundary. Thewell-known parabolic velocity profile in fully-developed flow of anincompressible Newtonian fluid in a pipe of circular cross-sectionis easily computed because the boundary (the pipe wall) is known tobe in a fixed location. Even in more complex settings, such as flowaround a ship's propeller, the motion of the boundary (thepropeller) can be specified in advance.By contrast, in most biological fluid dynamics problems theboundaries are not rigid and their motions are the result of forcesimposed on them by the motion of the surrounding fluid. The motionof the fluid, of course, cannot be known without knowledge of theboundary motion. The motion of the boundary and the motion of thefluid form a coupled system; both motions must be computedsimultaneously, which makes biological CFD difficult.A particular problem of interest is the flow of blood in thechambers of the human heart. The heart is an organ whose muscularcontractions pump blood around the body. Simplifying somewhat, theheart consists of two main pumping chambers that contractsimultaneously. One chamber, the left ventricle, acceptsoxygen-enriched blood from the lungs and pumps it to the body. Theother chamber, the right ventricle, accepts oxygen-depleted bloodfrom the body and pumps it to the lungs. The inlet and outlet ofeach ventricle are guarded by valves whose opening and closingguarantee one-directional flow around the circulatory system. Thereare a total of four valves. The valves generally consist of two orthree leaflets - membranes made of very flexible but inextensiblematerial. Familiar examples of materials with this property wouldbe paper or fabric which can be easily bent or twisted but whichare not easily stretched. One edge of each valve leaflet issecurely attached to the wall of the heart, but the other edge isfree of attachment and can move with the flow. Structures analogousto a valve leaflet are a shirt pocket, with one edge (three sidesof a rectangular patch pocket) securely stitched to the shirt andone edge free of attachment, or a flag, one edge attached to theflag pole, the other edge free to wave in the wind. When flow ispassing through the valve in the forward direction, the valve'sleaflets are positioned out of the way, permitting flow. When flowattempts to pass in the reverse direction, the leaflets cometogether, their free edges pressing against the free edges of theirneighbors to occlude the flow passage.The motion of the leaflets is not caused by muscles in thevalve. The outflow valves are entirely passive structures with nomuscular tissue whatsoever. Even in the case of the inflow valves,whose free edges are connected to the heart muscle by a sparsenetwork of tendons, the opening and closing motions result from aninteraction with the surrounding fluid. The forward motion of thefluid pushes the leaflets aside out of the main flow stream, butthe inextensibilty of the leaflet material prevents free motion ofthe fluid near the leaflet, affecting the entire flow field.Reverse motion of the fluid causes the leaflets to move back intothe flow passage where contact between neighboring leaflets and theinextensibilty of the leaflet material halts the flow. The highlyinteractive nature of the fluid and leaflet motions makes this anespecially interesting and challenging CFD problem.Commercially available software packages intended forengineering CFD are not equipped to handle this type of dynamicinteraction between boundary and fluid. We have developed anumerical method (the "Immersed Boundary Method") whichsimultaneously computes the motion of a fluid and the motion of anelastic boundary immersed in, and interacting with, that fluid. Inthe Immersed Boundary Method, the fluid is represented by Eulerianvelocities and pressures that are stored on a regularthree-dimensional computational lattice. The scale of the heartchambers is such that blood can be treated as a Newtonian fluid.Fluid dynamics is computed by numerical solution of theNavier-Stokes equations, including a body force. The boundary isrepresented by elastic structures that are free to movecontinuously in the space sampled by the computational lattice. Theessence of the method is to replace the elastic boundary by theforces that result from its deformations. These forces are appliedto the lattice in the neighborhood of the elastic boundary with theaid of a numerical approximation to the Dirac delta function. Thefluid moves under the action of this body force. The numericaldelta function is then used again, to interpolate the newlycomputed lattice velocities to the locations of the boundary, andthen the boundary is moved at the interpolated velocity to a newlocation (the no-slip condition). The process of computing forces,then fluid motion and then new boundary location is repeatedcyclically in a time-stepping procedure with a suitably chosen timestep. The only requirements for the method are the physicalproperties of the fluid, the (possibly time-dependent) elasticproperties of the boundary, and the initial geometry of theboundary. A complete description of the Immersed Boundary Methodcan be found in [1, 2].