Martin Burger Institut für Numerische und Angewandte Mathematik European Institute for Molecular Imaging CeNoS Computing Transmembrane Potentials from BSPM

Martin Burger Computing Transmembrane Potentials 2 Maastricht, Based on Joint work with Bjorn Fredrik Nielsen (SIMULA Research Labs, Oslo) Computational results for ischemia have been obtained at SIMULA using results from Rikshospitalet Oslo Funding of collaboration by DAAD

Martin Burger Computing Transmembrane Potentials 3 Maastricht, ECGI – State of the Art ECG-Imaging consists in computing the epicardial potential from body surface potential measurements (cf. Rudy et al) Needs solution of a Cauchy-Problem for the Poisson- equation Severely ill-posed, hence natural restrictions on the possibly quality to achieve Is the epicardial potential the most interesting quantity for cardiologists ?

Martin Burger Computing Transmembrane Potentials 4 Maastricht, Transmembrane Potentials & Bidomain Model In the 1970s the bidomain model has been developed by Geselowitz, Miller, Schmitt & Tung It has become a standard model for simulating electrical signal propagation on the heart and torso Still active model development, in particular related to channel activation

Martin Burger Computing Transmembrane Potentials 5 Maastricht, Transmembrane Potentials & Bidomain Model Poisson equation in torso Fully coupled system of PDEs and ODES in the myocardium

Martin Burger Computing Transmembrane Potentials 6 Maastricht, Transmembrane Potentials & Bidomain Model Transmembrane Potential s... state vector of cells and gating variables F … specific channel / gating model conductivity tensors

Martin Burger Computing Transmembrane Potentials 7 Maastricht, Bidomain Model & ECG Inversion Fundamental Questions: - Can we partly overcome the severe ill-posedness ? - Can we use the results of bidomain modelling for inversion ? - Can we look into the heart ?

Martin Burger Computing Transmembrane Potentials 8 Maastricht, Bidomain Model & ECG Inversion Basic approaches to do so: - Use results of bidomain modelling and simulations in special situations to obtain prior knowledge that can be used in the inversion. In particular the resulting structure of transmembrane potentials can be a suitable prior. Still via linear inversion - Use (parametrized) bidomain model directly for inversion. Distributed parameters appearing in the bidomain model are the unknowns to be found (potentials fully determined by potential). Yields nonlinear inversion

Martin Burger Computing Transmembrane Potentials 9 Maastricht, Paradigm I: Ischemia Example (due to SIMULA) Possible prior knowledge from bidomain modelling (or from medical insight): Transmembrane potential takes two different and well-defined values in healthy and ischemic regions during the resting phase v equals approximately -95mV in healthy and -75mV in ischemic regions cf. Simulation result (right)

Martin Burger Computing Transmembrane Potentials 10 Maastricht, Paradigm I: Ischemia Example (due to SIMULA) Use the prior to invert for transmembrane potential with additional assumption: The transmembrane potential is restricted to those taking only the two predefined values Second (linear) equation in bidomain model is used to define u e from v

Martin Burger Computing Transmembrane Potentials 11 Maastricht, Paradigm I: Ischemia Example (due to SIMULA) Linear inversion from Cauchy data has huge nullspace ! Prior for transmembrane potential (only two different values) allows to (hopefully) eliminate nullspace

Martin Burger Computing Transmembrane Potentials 12 Maastricht, Paradigm I: Ischemia Example (due to SIMULA) Linear inversion from Cauchy data has huge nullspace ! Prior for transmembrane potential (only two different values) allows to (hopefully) eliminate nullspace

Martin Burger Computing Transmembrane Potentials 13 Maastricht, Paradigm I: Ischemia Example (due to SIMULA) Computational results from B.Nielsen‘s group with real data from Rikshospitalet Oslo Geometry model from MR image Reconstructed ischemic regions for two patients, coherent with results of szintigraphy #1 #13

Martin Burger Computing Transmembrane Potentials 14 Maastricht, Paradigm I: Ischemia Example (due to SIMULA) Computational results from B.Nielsen‘s group with real data from Rikshospitalet Oslo Tests of robustness with respect to geometry: data of #1 on different geometries #1 #5 #6 #7 #9 #13

Martin Burger Computing Transmembrane Potentials 15 Maastricht, Paradigm I: Ischemia Example (due to SIMULA) Analysis (mb-Nielsen 08 / 09): - At most as unstable as ECGI - No ischemia at all can be detected for simple anisotropies (M i proportional to M e ) - Ischemia can be detected uniquely for certain locations relative to anisotropy directions Conclusion: fiber structure more important than heart geometry !

Martin Burger Computing Transmembrane Potentials 16 Maastricht, Paradigm I: Ischemia Example (due to SIMULA) Major disadvantage of the current approach: Can be used only for ECG data during the resting phase Not the ones with highest sensitivity

Martin Burger Computing Transmembrane Potentials 17 Maastricht, Paradigm II: Infarction Example Example not relevant from a medical point of view, but easiest to illustrate the approach We also use the Fitzhugh-Nagumo model for simplicity, extension is immediate Results from Diploma Thesis of Melanie Schröter

Martin Burger Computing Transmembrane Potentials 18 Maastricht, Paradigm II: Infarction Example Infarction model yield reparametrized bidomain model g is the distributed infarction parameter g equals one in healthy regions g equals almost zero in infarcted regions (stopped ion flow)

Martin Burger Computing Transmembrane Potentials 19 Maastricht, Paradigm II: Infarction Example Infarction model yield reparametrized bidomain model g is the distributed infarction parameter g equals one in healthy regions g equals almost zero in infarcted regions (stopped ion flow)

Martin Burger Computing Transmembrane Potentials 20 Maastricht, Paradigm II: Infarction Example Nonlinear inversion from ECG data to g can be modeled as nonlinear variational problem Descent method, computations of descent directions via an adjoint method (system of linear PDEs)

Martin Burger Computing Transmembrane Potentials 21 Maastricht, Paradigm II: Infarction Example Simulation on simplified quasi-1D example with synthetic data Plots of reconstructed location vs. scaled depth in „heart“

Martin Burger Computing Transmembrane Potentials 22 Maastricht, Paradigm II: Infarction Example Resulting transmembrane potential

Martin Burger Computing Transmembrane Potentials 23 Maastricht, Paradigm II: Infarction Example Advantages: - Temporal correlations inherent in the model - Reduced degree of freedoms (2+1D data, ~ 2 D unknowns) Disadvantages: - Computational effort, needs advanced and efficient schemes since 4D inversion - Needs appropriate models for specific question