Dafang Wang

Inverse electrocardiographic source localization of ischemia: An optimization framework and finite element solution

With the goal of non-invasively localizing cardiac ischemic disease using body-surface potential recordings, we attempted to reconstruct the transmembrane potential (TMP) throughout the myocardium with the bidomain heart model. The task is an inverse source problem governed by partial differential equations (PDE). Our main contribution is solving the inverse problem within a PDE-constrained optimization framework that enables various physically-based constraints in both equality and inequality forms. We formulated the optimality conditions rigorously in the continuum before deriving finite element discretization, thereby making the optimization independent of discretization choice. Such a formulation was derived for the L2-norm Tikhonov regularization and the total variation minimization. The subsequent numerical optimization was fulfilled by a primal-dual interior-point method tailored to our problems specific structure. Our simulations used realistic, fiber-included heart models consisting of up to 18,000 nodes, much finer than any inverse models previously reported. With synthetic ischemia data we localized ischemic regions with roughly a 10% false-negative rate or a 20% false-positive rate under conditions up to 5% input noise. With ischemia data measured from animal experiments, we reconstructed TMPs with roughly 0.9 correlation with the ground truth. While precisely estimating the TMP in general cases remains an open problem, our study shows the feasibility of reconstructing TMP during the ST interval as a means of ischemia localization.

Finite-Element-Based Discretization and Regularization Strategies for 3-D Inverse Electrocardiography

We consider the inverse electrocardiographic problem of computing epicardial potentials from a body-surface potential map. We study how to improve numerical approximation of the inverse problem when the finite-element method is used. Being ill-posed, the inverse problem requires different discretization strategies from its corresponding forward problem. We propose refinement guidelines that specifically address the ill-posedness of the problem. The resulting guidelines necessitate the use of hybrid finite elements composed of tetrahedra and prism elements. Also, in order to maintain consistent numerical quality when the inverse problem is discretized into different scales, we propose a new family of regularizers using the variational principle underlying finite-element methods. These variational-formed regularizers serve as an alternative to the traditional Tikhonov regularizers, but preserves the L2 norm and thereby achieves consistent regularization in multiscale simulations. The variational formulation also enables a simple construction of the discrete gradient operator over irregular meshes, which is difficult to define in traditional discretization schemes. We validated our hybrid element technique and the variational regularizers by simulations on a realistic 3-D torso/heart model with empirical heart data. Results show that discretization based on our proposed strategies mitigates the ill-conditioning and improves the inverse solution, and that the variational formulation may benefit a broader range of potential-based bioelectric problems.

A toolkit for forward/inverse problems in electrocardiography within the SCIRun problem solving environment

Computational modeling in electrocardiography often requires the examination of cardiac forward and inverse problems in order to non-invasively analyze physiological events that are otherwise inaccessible or unethical to explore. The study of these models can be performed in the open-source SCIRun problem solving environment developed at the Center for Integrative Biomedical Computing (CIBC). A new toolkit within SCIRun provides researchers with essential frameworks for constructing and manipulating electrocardiographic forward and inverse models in a highly efficient and interactive way. The toolkit contains sample networks, tutorials and documentation which direct users through SCIRun-specific approaches in the assembly and execution of these specific problems.

Resolution Strategies for the Finite-Element-Based Solution of the ECG Inverse Problem

Successful employment of numerical techniques for the solution of forward and inverse ECG problems requires the ability to both quantify and minimize approximation errors introduced as part of the discretization process. Our objective is to develop discretization and refinement strategies involving hybrid-shaped finite elements so as to minimize approximation errors for the ECG inverse problem. We examine both the ill-posedness of the mathematical inverse problem and the ill-conditioning of the discretized system in order to propose strategies specifically designed for the ECG inverse problem. We demonstrate that previous discretization and approximation strategies may worsen the properties of the inverse problem approximation. We then demonstrate the efficacy of our strategies on both a simplified and a realistic 2-D torso model.

Identifying myocardial ischemia by inversely computing transmembrane potentials from body-surface potential maps

We attempted to solve the inverse electrocardio-graphic problem of computing the transmembrane potentials (TMPs) throughout the myocardium from a body-surface potential map, and then used the recovered potentials to estimate the size and location of myocardial ischemia. We modeled the bioelectric process by combining a static bidomain heart model with a torso conduction model. Although the task of computing myocardial TMPs at an arbitrary time instance is still an open problem, we showed that it is possible to obtain TMPs with moderate accuracy during the ST segment by assuming all cardiac cells are at the plateau phase. Moreover, the inverse solutions yielded a good estimate of ischemic regions, which is of more clinical interest than merely reporting the voltage values. We formulated the inverse problem as a minimization problem constrained by a partial differential equation that models the forward problem. This framework greatly reduces the computational costs compared with the traditional approach of building the lead-field matrix. It also enables one to flexibly set different discretization resolutions for the source variables and other state variables, a desirable feature for solving ill-posed inverse problems. We conducted finite element simulations of a phantom experiment over a 2D torso model with synthetic ischemic data. Preliminary results indicated that our approach is feasible and suitably accurate for the common case of transmural myocardial ischemia.

An optimization framework for inversely estimating myocardial transmembrane potentials and localizing ischemia

By combining a static bidomain heart model with a torso conduction model, we studied the inverse elec-trocardiographic problem of computing the transmembrane potentials (TMPs) throughout the myocardium from a body-surface potential map, and then used the recovered potentials to localize myocardial ischemia. Our main contribution is solving the inverse problem within a constrained optimization framework, which is a generalization of previous methods for calculating transmembrane potentials. The framework offers ample flexibility for users to apply various physiologically-based constraints, and is well supported by mature algorithms and solvers developed by the optimization community. By avoiding the traditional inverse ECG approach of building the lead-field matrix, the framework greatly reduces computation cost and, by setting the associated forward problem as a constraint, the framework enables one to flexibly set individualized resolutions for each physical variable, a desirable feature for balancing model accuracy, ill-conditioning and computation tractability. Although the task of computing myocardial TMPs at an arbitrary time instance remains an open problem, we showed that it is possible to obtain TMPs with moderate accuracy during the ST segment by assuming all cardiac cells are at the plateau phase. Moreover, the calculated TMPs yielded a good estimate of ischemic regions, which was of more clinical interest than the voltage values themselves. We conducted finite element simulations of a phantom experiment over a 2D torso model with synthetic ischemic data. Preliminary results indicated that our approach is feasible and suitably accurate for the common case of transmural myocardial ischemia.

Finite Element Discretization Strategies for the Inverse Electrocardiographic (ECG) Problem

Successful employment of numerical techniques for the forward and inverse electrocardiographic (ECG) problems requires the ability to both quantify and minimize approximation errors introduced as part of the discretization process. Conventional finite element discretization and refinement strategies effective for the forward problem may become inappropriate for the inverse problem because of its ill-posed nature. This conjecture leads us to develop discretization strategies specifically for the inverse ECG problem. By quantitatively analyzing the connection between the ill-posedness of the continuum inverse problem and the illconditioning of its discretized version, we propose strategies involving hybrid-shaped finite elements to discretize the inverse ECG problem effectively and efficiently. We also propose the criteria for evaluating the quality of the resultant discrete system. The efficacy of the strategies are demonstrated on a realistic torso model in both two and three dimensions.