Pranav Bhagirath

153 Computing volume potentials for noninvasive imaging of cardiac excitation Computing BVP Given the source potential distribution on the epicardial surface, the resulting volume potential distribution is governed by the equations below. The current density J as a function of conductivity s and field strength E is given by ohms law: J = s E = s grad(P), where P is the potential [1] Apart from the heart there are no current sources in the thorax so: div(J) == 0 [2] From [1] and [2] follows Laplace’s equation: div(s grad(P)) == 0 [3] Multiplying by a test function T leads to the following variational form: ∫ div(s grad(P)) T dV == ∫ 0 * T dV == 0 [4] Partial integration yields (n is the unit surface normal): ∫ div (s grad(P) T dV == ∫ s grad(P).grad(T)dV - ∫ s grad(P).n T dS [5] From [4] and [5] follows: ∫ s grad(P).grad(T) dV == ∫ s grad(P).n T dS [6] With J.n = 0 and [1] at the skin this becomes: ∫ s grad(P).grad(T) dV = 0 [7] To yield a non-trivial solution, the source potentials at the heart surface are applied as boundary conditions. There are many general-purpose FEM tools available to solve these equations. FEniCS [15], freely available for research purposes, was selected. This software package allowed the aforementioned equation to be specified in a very natural form. All work is done by the following lines of code: RHS=sigma*inner(grad(trialFunction),grad (testFunction)) * dx [1] LHS = Constant (0)* testFunction * dx [2] A,b = assemble_system(LHS,RHS,boundaryCond,keep_diagonal = True) [3] solve(A,potential.vector(),b,’gmres’,’default’) [4]