Hyperparameter optimisation and validation of registration algorithms for measuring regional ventricular deformation using retrospective gated computed tomography images

Recent dose reduction techniques have made retrospective computed tomography (CT) scans more applicable and extracting myocardial function from cardiac computed tomography (CCT) images feasible. However, hyperparameters of generic image intensity-based registration techniques, which are used for tracking motion, have not been systematically optimised for this modality. There is limited work on their validation for measuring regional strains from retrospective gated CCT images and open-source software for motion analysis is not widely available. We calculated strain using our open-source platform by applying an image registration warping field to a triangulated mesh of the left ventricular endocardium. We optimised hyperparameters of two registration methods to track the wall motion. Both methods required a single semi-automated segmentation of the left ventricle cavity at end-diastolic phase. The motion was characterised by the circumferential and longitudinal strains, as well as local area change throughout the cardiac cycle from a dataset of 24 patients. The derived motion was validated against manually annotated anatomical landmarks and the calculation of strains were verified using idealised problems. Optimising hyperparameters of registration methods allowed tracking of anatomical measurements with a mean error of 6.63% across frames, landmarks, and patients, comparable to an intra-observer error of 7.98%. Both registration methods differentiated between normal and dyssynchronous contraction patterns based on circumferential strain (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_1=0.0065$$\end{document}p1=0.0065, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_2=0.0011$$\end{document}p2=0.0011). To test whether a typical 10 temporal frames sampling of retrospective gated CCT datasets affects measuring cardiac mechanics, we compared motion tracking results from 10 and 20 frames datasets and found a maximum error of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$8.51\pm 0.8\%$$\end{document}8.51±0.8%. Our findings show that intensity-based registration techniques with optimal hyperparameters are able to accurately measure regional strains from CCT in a very short amount of time. Furthermore, sufficient sensitivity can be achieved to identify heart failure patients and left ventricle mechanics can be quantified with 10 reconstructed temporal frames. Our open-source platform will support increased use of CCT for quantifying cardiac mechanics.

. Isoparametric mapping of the deformation between the simplex Ω , the reference configuration Ω 0 , and the current configuration Ω induced by the deformation.
Rearrangement of this relation to express the deformation gradient leads to: F = j J −1 . Finally, the Green-Lagrange strain tensor E is calculated by equation: where I is the identity tensor. Both tensors quantities, F and E, are constant throughout the 3-node triangular element because of the linear shape functions.

Strain Tests
Computation of the Green-Lagrange strain tensor E is necessary for calculation of circumferential and longitudinal strains. We verified the correctness of the implementation using idealised problems. The triangular meshes deployed in these tests were generated to represent the respective geometry in the reference configuration and artificial finite deformations were applied to the mesh vertices to obtain the current configuration. The results provided for verification include the deformation gradient tensor F and the Green-Lagrange strain tensor E in the global and the element coordinate system, respectively. Both measures are provided for the last element in each problem.

Tensile Test
A square (3 x 3 cm) was meshed using triangular elements of approximately 1 mm edge length resulting in 1235 vertices and 2348 elements. A biaxial deformation field (see Figure A2(a)) was applied to verify the mathematical description derived in the method's section by: x 1 = 1.300 · X 1 , where x i and X i , with i = 1, 2, 3, are the coordinates in the current and reference configuration, respectively. The implementation of the Green-Lagrange strain tensor provided: where F is in the global Cartesian coordinate system and E e is in the cylindrical element coordinate system with v ab = [0 1 0] T . The results suggest the correct implementation under conditions where the element's normal remained fixed and strains were aligned with the rectangular Cartesian coordinates.

Shear Test
A square (3 x 3 cm) was meshed using triangular elements of approximately 1 mm edge length resulting in 1235 vertices and 2348 elements. A simple shear deformation field (see Figure A2(b)) was applied to verify the mathematical description derived in the method's section by: where x i and X i , with i = 1, 2, 3, are the coordinates in the current and reference configuration, respectively. The implementation of the Green-Lagrange strain tensor provided: where F is in the global Cartesian coordinate system and E e is in the cylindrical element coordinate system with v ab = [0 1 0] T . The results suggest the correct implementation under conditions where shear strains were aligned with the rectangular Cartesian coordinates.

Tube Test
A tube (3 x 3 cm) was meshed using triangular elements of approximately 1 mm edge length resulting in 3416 vertices and 6640 elements. A multi dimensional deformation field (see Figure A2(c)) was applied to verify the mathematical description derived in the method's section by: where x i and X i , with i = 1, 2, 3, are the coordinates in the current and reference configuration, respectively. The implementation of the Green-Lagrange strain tensor provided: where F is in the global Cartesian coordinate system and E e is in the cylindrical element coordinate system with v ab = [0 0 1] T . The results suggest the correct implementation under conditions where strains on surfaces are not aligned with the Cartesian coordinate system. As an additional test, the surface of the same tube was labelled with standard AHA segments and an identical multi dimensional deformation field was applied to the 5 th and 6 th time frames in the sequence to verify the mathematical derivation. It is clear from Figure A3 that all the AHA curves corresponding to different regions of the tube expand simultaneously and illustrate a correct value of strain at the given time point in the sequence. Figure A2. Reference (grey) and current (red) configuration of all described test problems implemented for verification purposes. The triangular mesh is indicated in the current configuration. The coordinate origin of all geometries is the lower left corner, except for the tube tests, which is located at the centre.

Left Ventricular Mesh Test
Previous verifications provide confidence of the correct derivation and implementation of the Green-Lagrange strain tensor E for triangular elements. The following additional test was implemented to provide special cases of the presented problem. Endocardium of an LV was meshed using 32952 vertices and 65900 triangular elements. A multi dimensional deformation field was applied to all the 10 input meshes in the sequence to verify the mathematical description derived in the previous section by: x 1 = 0.800 · X 1 , where x i and X i , with i = 1, 2, 3, are the coordinates in the middle time frames configurations of the sequence and the reference configuration, respectively. A monotonic increase and decrease of these values was also applied to the rest of the configurations corresponding to all time points in the sequence. The implementation of the Green-Lagrange strain tensor E provided a monotonic contraction of all AHA segments in the LV suggesting a correct implementation (see Figure A4).  Figure A5 illustrates the results of an exhaustive grid search for the optimal values of bending energy and sparsity weight in TSFFD registration framework. Dual optimisation of sparsity weight does not converge to a local minima. Similarly, Figure  A6 demonstrates the effect of optimising Θ from DEEDS on 10 training sets. This hyperparameter can be increased to obtain smoother transforms and decreased to make the registration more aggressive. The search in the hyperparameter's space does not reveal a significant trend.

Registration Hyperparameters
Below is the full list of default hyperparameters for both TSFFD and DEEDS registration methods.   Figure A6. Optimisation of the registration hyperparameter Θ on the training sets and its effect on endocardial tracking accuracy. The accuracy was measured at selected anatomical sites. X axis is the examined range for the registration hyperparameter, whereas Y axis displays the average error in percentage. Standard error is illustrated as a shaded region.