Three-Dimensional Cardiac Tissue Image Registration for Analysis of In Vivo Electrical Mapping

Abstract

A method is presented for registering 3D cardiac tissue images to reference data, for the purpose of analyzing recorded electrical activity. Following left-ventricular in vivo electrical mapping studies in pig hearts, MRI is used to define a reference geometry in the tissue segment around the recording electrodes. The segment is then imaged in 3D using a high-resolution serial imaging microscopy technique. The tissue processing required for this introduces segment-wide distortion. Piecewise-smooth maps are used to correct the tissue distortion and register the 3D images with the reference MRI data. The methods are validated and techniques for identifying the preferred maps are proposed. Recorded electrical activation is shown to map reliably onto cardiac tissue structure using this registration method.

Appendix

To ensure a unique CMap for an underdetermined problem, the admissible space was restricted from the full tricubic-Hermite space by a curvature norm. The norm is constructed by considering the curvature of the iso-surfaces of each normalized coordinate embedded in the CMap. These surfaces are ξ ₁₂, ξ ₁₃, and ξ ₂₃ (Fig. 2). The distance residual of a point before, x, and after, X, applying a CMap is given by:

$$ {\mathbf{u}} = {\mathbf{X}} - {\mathbf{x}} = \Uppsi^{j} \left( {{\hat{\mathbf{X}}}^{j} - {\hat{\mathbf{x}}}^{j} } \right). $$

(A.1)

A Hessian for the second derivatives of the Cartesian coordinates on the ξ ₁₂, ξ ₁₃, and ξ ₂₃ surfaces, based on the distance residual, u, is:

$$ {\hat{\mathbf{H}}}_{ij} = \left[ {\begin{array}{*{20}c} {\left\| {{\frac{{\partial^{2} {\mathbf{u}}}}{{\partial \xi_{i} \partial \xi_{i} }}}} \right\|} & {\left\| {{\frac{{\partial^{2} {\mathbf{u}}}}{{\partial \xi_{i} \partial \xi_{j} }}}} \right\|} \\ {\left\| {{\frac{{\partial^{2} {\mathbf{u}}}}{{\partial \xi_{i} \partial \xi_{j} }}}} \right\|} & {\left\| {{\frac{{\partial^{2} {\mathbf{u}}}}{{\partial \xi_{j} \partial \xi_{j} }}}} \right\|} \\ \end{array} } \right] $$

(A.2)

where $\Vert \cdot \Vert$ denotes the L2 norm of the vector quantity. The first and second eigenvalues of ${\hat{\mathbf{H}}}_{ij}$ are denoted as k ₁ and k ₂. Two indicators of surface shape can be formed from these eigenvalues2:

$$ {\text{Gaussian }}{\text{curvature: }} K_{ij} = k_{1} \times k_{2} = { \det }\left( {{\hat{\mathbf{H}}}_{ij} } \right) $$

$$ {\text{Mean }}{\text{curvature: }} H_{ij} = {\frac{{k_{1} + k_{2} }}{2}} = \frac{1}{2}{\text{tr}}\left( {{\hat{\mathbf{H}}}_{ij} } \right) $$

These two quantities can be related by the deviation from flatness of the surface, C _ij 21:

$$ \begin{aligned} C_{ij} = k_{1}^{2} + k_{2}^{2} & = \left( {2H_{ij} } \right)^{2} - 2K_{ij} \\ & = \left( {{\text{tr}}\left( {\hat{\mathbf{H}}_{ij} } \right)} \right)^{2} - 2{ \det }\left( {\hat{\mathbf{H}}_{ij} } \right) \\ & = \left( {\left\| {{\frac{{\partial^{2} {\mathbf{u}}}}{{\partial \xi_{i} \partial \xi_{i} }}}} \right\| + \left\| {{\frac{{\partial^{2} {\mathbf{u}}}}{{\partial \xi_{j} \partial \xi_{j} }}}} \right\|} \right)^{2} - 2\left\| {{\frac{{\partial^{2} {\mathbf{u}}}}{{\partial \xi_{i} \partial \xi_{i} }}}} \right\|\left\| {{\frac{{\partial^{2} {\mathbf{u}}}}{{\partial \xi_{j} \partial \xi_{j} }}}} \right\| + 2\left\| {{\frac{{\partial^{2} {\mathbf{u}}}}{{\partial \xi_{i} \partial \xi_{j} }}}} \right\|^{2} \\ & = \left\| {{\frac{{\partial^{2} {\mathbf{u}}}}{{\partial \xi_{i} \partial \xi_{i} }}}} \right\|^{2} + 2\left\| {{\frac{{\partial^{2} {\mathbf{u}}}}{{\partial \xi_{i} \partial \xi_{j} }}}} \right\|^{2} + \left\| {{\frac{{\partial^{2} {\mathbf{u}}}}{{\partial \xi_{j} \partial \xi_{j} }}}} \right\|^{2} . \\ \end{aligned} $$

(A.3)

A weighted norm for the curvature in the ij plane of the CMap is:

$$ N_{ij} = \int\limits_{\Upomega } {W_{ij} C_{ij} {\text{d}}\Upomega } . $$

(A.4)

The weighted contribution of all planes to the CMap, Q, is:

$$ \begin{gathered} Q({\mathbf{u}}) = \int\limits_{\Upomega } {\left[ {W_{12} C_{12} + W_{13} C_{13} + W_{23} C_{23} } \right]} {\text{d}}\Upomega \hfill \\ = \int\limits_{\Upomega } {\left[ {\left( {W_{12} + W_{13} } \right)\left\| {{\frac{{\partial^{2} {\mathbf{u}}}}{{\partial \xi_{1}^{2} }}}} \right\|^{2} + \left( {W_{12} + W_{23} } \right)\left\| {{\frac{{\partial^{2} {\mathbf{u}}}}{{\partial \xi_{2}^{2} }}}} \right\|^{2} + \left( {W_{13} + W_{23} } \right)\left\| {{\frac{{\partial^{2} {\mathbf{u}}}}{{\partial \xi_{3}^{2} }}}} \right\|^{2} + } \right.} \hfill \\ \left. {\quad 2W_{12} \left\| {{\frac{{\partial^{2} {\mathbf{u}}}}{{\partial \xi_{1} \partial \xi_{2} }}}} \right\|^{2} + 2W_{13} \left\| {{\frac{{\partial^{2} {\mathbf{u}}}}{{\partial \xi_{1} \partial \xi_{3} }}}} \right\|^{2} + 2W_{23} \left\| {{\frac{{\partial^{2} {\mathbf{u}}}}{{\partial \xi_{2} \partial \xi_{3} }}}} \right\|^{2} } \right]{\text{d}}\Upomega . \hfill \\ \end{gathered} $$

(A.5)

Here the weights are a per-volume weighting, i.e., units of mm⁻³.

The domain of integration of the curvature norm is transformed to the normalized (dimensionless) space of local element coordinates in the normal finite element sense. The element curvature norm is:

$$ \begin{gathered} Q = \sum\limits_{e} {\int\limits_{0}^{1} {\left[ {\left( {W_{12} + W_{13} } \right){\frac{{\partial^{2} u_{i} }}{{\partial \xi_{1}^{2} }}}{\frac{{\partial^{2} u_{i} }}{{\partial \xi_{1}^{2} }}} + \left( {W_{12} + W_{23} } \right){\frac{{\partial^{2} u_{i} }}{{\partial \xi_{2}^{2} }}}{\frac{{\partial^{2} u_{i} }}{{\partial \xi_{2}^{2} }}} + \left( {W_{13} + W_{23} } \right){\frac{{\partial^{2} u_{i} }}{{\partial \xi_{3}^{2} }}}{\frac{{\partial^{2} u_{i} }}{{\partial \xi_{3}^{2} }}}} \right.} } + \hfill \\ \left. {\quad 2W_{12} {\frac{{\partial^{2} u_{i} }}{{\partial \xi_{1} \partial \xi_{2} }}}{\frac{{\partial^{2} u_{i} }}{{\partial \xi_{1} \partial \xi_{2} }}} + 2W_{13} {\frac{{\partial^{2} u_{i} }}{{\partial \xi_{1} \partial \xi_{3} }}}{\frac{{\partial^{2} u_{i} }}{{\partial \xi_{1} \partial \xi_{3} }}} + 2W_{23} {\frac{{\partial^{2} u_{i} }}{{\partial \xi_{2} \partial \xi_{3} }}}{\frac{{\partial^{2} u_{i} }}{{\partial \xi_{2} \partial \xi_{3} }}}} \right]\left| J \right|{\text{d}}{\varvec{\upxi}}. \hfill \\ \end{gathered} $$

(A.6)