Multi-scale simulation is also important to the flow of blood within our bodies.  Recently, we have undertaken a collaboration with  Charles Taylor , a professor of  Biomedical Engineering at Stanford University.  Together, we are developing software to enable virtual planning of vascular surgery.  Taylor uses MRI scans to extract a solid model  of an arterial section which he later will use to discretize into finite elements using  SCOREC  mesh generators.  At the same time he can also use MRI scans to obtain the time-varying velocity field at a few arterial cross sections.  By using the experimentally obtained velocity field as a boundary condition, we then perform transient, finite element simulations of the flow within the arterial section.  The velocity fields at the other cross sections will be used to validate our simulations.

The process is being tested using pigs due to the similarity of their vascular system to that of humans.  First, a bypass is created around a section of the artery of a healthy pig.  Then, the original arterial section is restricted forcing a greater percentage of the flow to utilize the bypass (simulating a bypass of a diseased state).  Next, the aforementioned MRI data is collected.  Our simulation of this "real patient's" (pig's) arterial system is shown in this animation (15Mb).  As you can see the constriction, bypass, and pulsatile nature of the flow create rather complex flow patterns.  In the second animation we zoom (22Mb) in on the region close to the constriction and bypass reattachment.

While visualizing the flow patterns and the pressure field provides some insight into the quality of the bypass, more information can be gleaned from these simulations.  Recent research suggests that the shear stress on the various solids suspended within blood can affect the formation of clots and other arterial blockages.  Though it is currently impractical to model each solid particle in the blood suspension, we can develop transport equations that can keep track of the concentration of these suspensions. We can further develop transport equations that keep track of the shear stress "history" as a function of space and time.  Then, by incorporating models which relate this shear history to the physics of cell adhesion to the vessel walls (or to a pre-existing adhesion) a model for disease location and progression should be possible.  This model will most likely require information related to the average residence time as a function of space.  Our preliminary work in this area can be visualized in the following animation (24Mb).  Here, we consider five cardiac cycles from the flow shown in the first animation.  We initialize the entire flow field with a scalar given the value of 1.0.  Then, we bring in fluid at the inflow with a value of 0.0.  As the simulation progresses, the regions with slow or recirculating flow become evident since the value decays much more slowly there, indicating a higher residence time.

A second multi-scale aspect of this problem that we are researching involves the outflow boundary. In the physical system there are a series of branches which continue down to the capillary level.  Again, it is not practical to simulate down to this level so the domain must be truncated (as shown in the animations) and multi-scale models for the downstream vascular bed must be introduced.