Hernán Mella^{1,2}, Joquín Mura^{1}, Julio Sotelo^{1,2}, Cristian Montalba^{1}, and Sergio Uribe^{1,3}

The stiffness in the myocardium represents a reliable biomarker of cardiac dysfunction or diseases, such as atherosclerosis or cardiac infarction. Actually, there exist methods to retrieve this indicator using MRI, however, they can be very time-consuming, especially for volumetric evaluations. In this work, we propose a novel method to estimate strains using Phase-Contrast or 4D-flow data, where velocity fields are transformed into strain tensors in a numerically simple but robust manner. Preliminary results are promising and we expect to validate our method in volunteers and patients.

In order to estimate the state of deformations in the myocardium, we need to know with precision the displacement gradients. To this end, velocity images are used to evaluate velocity gradients on the image and then, to integrate over the time variable to recover the strain tensor.Velocity gradients cannot be simply calculated using standard numerical derivation due to images with low signal-to-noise ratio. To tackle this issue, we use the weighted regression method3, obtaining remarkable results with high levels of noise. The Euler-Almansi strain tensor ($$$\textbf{e}$$$) is used to represent large deformations, discarding rigid body motions. The time derivative of $$$\textbf{e}$$$ is defined as

$$\dot{\textbf{e}}=\frac{1}{2}(\nabla\boldsymbol{v} + \nabla\boldsymbol{v}^T - (\nabla\boldsymbol{v})(\nabla\boldsymbol{v})^T),$$

where $$$\boldsymbol{v}$$$ represents the velocity of the body over a certain voxel. Time integration was carried out with the trapezoidal rule to estimate $$$\textbf{e}$$$. High order numerical integration will pollute the solution, considering the poor temporal resolution. Hence, the relation between $$$\dot{textbf{e}}$$$ and $$$\textbf{e}$$$ yields

$$\textbf{e}(t_{k+1})=\int\limits_{t_0}^{t_{k+1}}\dot{\textbf{e}}(\tau)~d\tau \approx \frac{1}{2} \sum\limits_{j=0}^{k+1}(t_{j+1}-t_j)(\dot{\textbf{e}}(t_{j+1}) + \dot{\textbf{e}}(t_j)).$$

Despite gradient and time integration operators can commute in differential calculus, numerically does not occur. The weighted regression gradient can filter noisy data, been more beneficial if it is applied before the integration over the time-domain.

As a preliminary result, 4D-flow data of one healthy volunteer with 25 cardiac phases and a voxel resolution of 2.32x2.32x2.5 mm was acquired in a Clinical 3T Philips Ingenia scanner. All the numerical calculations were performed in Matlab software.

The figures 1 and 2 shows the steps of the segmentation process and the flow work of the mathematical and numerical implementation. The figure 3 show a qualitative comparison between the estimated Von-Mises strain indicator and the tagging acquisition for six cardiac phases. As can be appreciated, our results represents reasonably the deformations of the myocardium, but they have high noise level at the exterior walls. This last behavior is because the meshes obtained after the segmentation process does not couples perfectly with the velocity data, allowing interpolations of values that not represent the cinematic of the cardiac muscle.

1. Reichek, N. (1999), MRI myocardial tagging. J. Magn. Reson. Imaging, 10: 609–616. doi:10.1002/(SICI)1522-2586(199911)10:5<609::AID-JMRI4>3.0.CO;2-2.

2. Kuijer, J. P.A., Hofman, M. B.M., Zwanenburg, J. J.M., Marcus, J. T., van Rossum, A. C. and Heethaar, Rob. M. (2006), DENSE and HARP: Two views on the same technique of phase-based strain imaging. J. Magn. Reson. Imaging, 24: 1432–1438. doi:10.1002/jmri.20749.

3. Correa, C., Hero R. and Ma K-L (2011). A Comparison of Gradient Estimation Methods for Volume Rendering on Unstructured Meshes. IEEE Trans. Visualization and Computer Graphics, 17(3):305-319. doi:10.1109/TVCG.2009.105.

Figure 1: Steps of the proposed segmentation process (A). With the three available images, we use the Balance SSFP image to generate a semiautomatic segmentation of the muscle of the left ventricle. In A) the 4D flow and Balance SSFP were acquired with the same FOV. Also, an interpolation along the time was performed in the tagging images, to obtain the same number of cardiac phases that the others acquisitions. In C) we show the acquisitions parameter used for each data sets acquired.

Figure 2: Flow diagram of the von-mises strain quantification process. First, from the 4D flow MR images we calculate the velocities gradient in tridimensional domain (a). Second, the time derivative of the Euler-Almansi tensor is calculated (b). Third, the Euler-Almansi tensor is estimated using numerical integration. Finally, the results are presented interpolating the Von-Mises norm on a FE mesh of the left ventricle. In the left ventricle mesh we show three different slices where we compare the strain maps with the tagging MR images.

Figure 3: Qualitative comparison between the obtained results and the tagging data for six different cardiac phases and three slices.