Scientific Computing and Imaging (SCI) Institute and the School of Computing, University of Utah, Salt Lake City, UT 84112, USA.
IEEE Trans Biomed Eng. 2011 Jun;58(6):1827-38. doi: 10.1109/TBME.2011.2122305. Epub 2011 Mar 3.
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 L(2) 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.
我们考虑从体表电位图计算心外膜电位的逆心电图问题。我们研究了当使用有限元方法时如何改进逆问题的数值逼近。由于逆问题不适定，因此需要与相应的正问题不同的离散化策略。我们提出了细化准则，专门解决问题的不适定性。所得到的准则需要使用由四面体和棱柱单元组成的混合有限元。此外，为了在将逆问题离散化为不同尺度时保持一致的数值质量，我们使用基于有限元方法的变分原理提出了一种新的正则化器家族。这些变分形成的正则化器可作为传统 Tikhonov 正则化器的替代方法，但保留了 L(2)范数，从而在多尺度模拟中实现了一致的正则化。变分公式还可以在不规则网格上简单地构建离散梯度算子，这在传统离散化方案中很难定义。我们通过使用经验心脏数据的真实 3-D 躯干/心脏模型进行模拟验证了我们的混合元素技术和变分正则化器。结果表明，基于我们提出的策略的离散化可以减轻不适定性并改善逆解，并且变分公式可能有益于更广泛的基于势的生物电问题。
