Electrical Signals in a Heart
By Prof. Simonetta Filippi and Dr. Christian Cherubini from Università Campus Biomedico di Roma, Italy.
Making sense of the contractions and dilations in the heart requires far more than simply considering how mechanical deformations force blood through hollows in cardiac tissue. The transfer of charged ions in heart tissue produces electrical pulses that trigger the mechanical contractions. In a healthy heart the electrical pulses attenuate or dampen naturally-this yields a rhythmic heart beat. When the electrical signals reenter and amplify, it disturbs the normal steady pulse. We typically refer to certain of the more severe and acute disturbances by the name arrhythmia.
Dr. Christian Cherubini
Dr. Simonetta Filippi
Prof. Simonetta Filippi and Dr. Christian Cherubini from Università Campus Biomedico di Roma (Italy) use COMSOL Multiphysics to model how electrical signals propagate in cardiac tissue. They base their work on mathematical models that describe complex phenomena of excitable media. With excitable media, elementary segments or cells can have a well-defined rest state, a threshold for excitation, and a diffusive transfer between nearest neighbors. The excitable media modeling concept applies to many different complex interacting processes, including nerve pulses, the spreading of forest fires, and certain types of chemical reactions. With these diverse phenomena, signals below a certain threshold die out, while signals above the threshold propagate without damping. The particular excitable media theory Filippi and Cherubinie employ, the FitzHugh-Nagumo equations and the Complex Ginzburg-Landau equations, allow them to explore how interesting patterns like spiral waves that produce effects similar to those of cardiac arrhythmia.
Complex system approach for robust process modeling
The FitzHugh-Nagumo equations characterize many important characteristics of electrical signal propagation in cardiac tissue. In the heart, ionic current triggers the rhythmic muscle contractions that pump blood in and out. The ions move (diffusive transfer) through small pores or gates in the cellular membrane, which either are open (excited state) or closed (rest state). While the state of the membrane gates is random on a microscopic scale, the probability of a given state is a continuous function of the voltage, which gives an averaged macroscopic continuum description of the current flow. Using the FitzHugh-Nagumo equations requires information about an activator and an inhibitor. In these heart models the activator variable corresponds to the electric potential, and the inhibitor describes the voltage-dependent probability of the pores in the membrane being open and ready to transmit ionic current.
The complex Landau-Ginzburg equations simplify modeling the particular point of transition from periodic behavior to a chaotic state with oscillations that increase in amplitude and frequency. First used to describe superconductivity, the complex Landau-Ginzburg equations also characterize the formation of vortices and oscillating chemical reactions. With an activator-inhibitor form that is similar to the Fitzhugh-Nagumo model, the complex Landau-Ginzburg equations simulate how rhythmic pulsing transitions to dynamic spiral waves in cardiac tissue.
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COMSOL Multiphysics solution to complex-systems equations applied to electrical signals in cardiac tissue. The Fitzhugh-Nagumo model gives an initial pulse that leads to a reentrant wave that travels without damping around the cardiac tissue in a characteristic spiral pattern (Fitzhugh-Nagumo equations). The Complex Landau-Ginzburg equations utilize equation parameters and initial conditions that give spiral patterns of a complexity that grows over time. Left figure shows a side view of the model results. In the right image the geometry is tilted to demonstrate the spiral pattern. View animation
Built on a Real Heart Geometry
Even though the equations for this complex system are so unusual that they are not offered standard on any packaged software, Filippi and Cherubini created their COMSOL models quickly by typing intuitive symbol language for the math into the PDE-based application modes of COMSOL Multiphysics. Despite the strongly nonlinear nature of these two equation systems, their models achieve reliable results. The Filippi and Cherubini worked out the modeling of these complex systems concepts on a simple geometry that is topologically similar to a real heart----a sphere with two semispherical chambers. They built the model shown here on a real heart geometry introduced into COMSOL Multiphysics with the COMSOL CAD Import Module and solved in on a 64-bit platform. Filippi and Cherubini achieved a high accuracy for the time intervals of interest by refining meshes and high order elements they implemented through simple point-and-click GUI choices. Future models will include temperature effects.
Similar heart models from Filippi and Cherubini come standard with COMSOL Multiphysics 3.2. The 3D time-dependent Fitzhugh-Nagumo and complex Landau-Ginzberg examples are set up on a geometry that is topologicially similar to a real heart. They solve within 20 minutes on a conventional PC. Refining that model is easy. With easy-to-implement options you increase the order of the elements, refine the mesh, or lower the tolerances and solve on one of the COMSOL Multiphysics 64-bit platforms.
About the researchers
Dr. Simonetta Filippi and Dr. Christian Cherubini are members of the Engineering faculty at the University Campus Biomedico UCBM and the International Centre for Relativistic Astrophysics ICRA, both of Rome. Their combined work in numerical modeling spans a wide range of phenomena and includes unraveling astrophysics phenomena. No newcomer to our software, Dr. Cherubini has used COMSOL Multiphysics to tackle general relativity problems important for analyzing black holes and gravitation.