Cardiac Mechanics and Electrophysiology in a Unified Mathematical and Computational Framework
Charles Peskin, CIMS
(joint work with Boyce Griffith and David McQueen)

      The heart is a mechanical and fluid-mechanical system that is coordinated and controlled by electrical activity intrinsic to the
heart itself.  The immersed boundary (IB) method was introduced to study the fluid-structure interaction of the heart valves, but has
since been elaborated into a modeling framework for the heart as a whole, including the blood flow in the cardiac chambers, the passive
elasticity of the flexible heart valve leaflets, and the active elasticity of the muscular heart walls.  The IB method employs a fixed
Cartesian grid for the storage of the Eulerian velocity and pressure field of the entire system, and a moving collection of fibers that cut
through the Cartesian grid and serve to model the collagen fibers of the valve leaflets and the muscle fibers of the heart walls.
Interaction between the Lagrangian fiber variables and the Eulerian fluid-like variables stored on the fixed Cartesian grid is modeled
with the help of a smoothed representation of the Dirac delta function, which is used in the application of Lagrangian fiber force
to update the Eulerian velocity field, and also in the interpolation of the Eulerian velocity field that is needed to update the Lagrangian
positions of the fibers points.
      The bidomain equations of cardiac electrophysiology separately track the spatially distributed extracellular and intracellular
voltages and currents, which are coupled by capacitive and ionic transmembrane currents.  Although intracellular and extracellular
conductivity are both influenced by the fiber direction, this influence is very strong intracellularly and much weaker in the
extracellular domain.  The IB formulation of the bidomain equations mirrors that of the mechanical IB method described above.  In
particular, the electrical IB method employs a fixed Cartesian grid for the extracellular electrical variables, and a moving system of
fibers to keep track of intracelluar voltages and currents as well as membrane associated variables in a Lagrangian manner.  As in the
mechanical IB method, the regularized Dirac delta function is used to describe interaction between Lagrangian and Eulerian variables.  In
particular, it is used in the application of Lagrangian transmembrane current to the extracellular space, and in the evaluation of Eulerian
extracellular voltage at a fiber point for purposes of computing the transmembrane current.  These operations are exactly analogous to the
application of force and the interpolation of velocity, respectively. A noteworthy feature of the IB formulation of the bidomain equations
is that the extracelluar space extends naturally beyond the myocardium itself and includes both the blood within in the cardiac chambers and
also the tissue external to the heart, both of which are electrically conducting media.  For this reason, an IB bidomain computation
automatically produces an electrocardiogram.