Synopsis

The strain in the left ventricle is a well-known biomarker for cardiac diseases. Nowadays, several acquisition techniques have been developed to improve the diagnose of this kind of conditions. Usually, strain biomarkers are obtained by mean of image post-processing techniques using different deformation metrics. In this work we present a numerical framework for the generation of left-ventricle strain phantoms using three different acquisition sequences in order to provide a broad database of patients and volunteers with different types of diseases. Our library provides a robust image generation tool to compare and develop new post-processing methods for quantifying strain phantoms

Introduction

The strain in the left ventricle is a well-known biomarker for the assessment of different cardiac diseases. Nowadays, several acquisition techniques have been developed to improve the estimation of strain parameters. Two of the most important approaches are Tagging¹ and Displacement Encoding with Stimulated Echoes² (DENSE), the actual and the next gold standard for strain estimation³. There are also some reports using tissue velocity mapping⁴ (TVM), which comes from phase contrast MR⁵ measurements.

Methods

The direct quantification of the strain in the left ventricle considers the displacement field $$$\vec{\boldsymbol{u}}$$$, acquired using tagging or DENSE techniques. Alternatively, 2D or 3D velocity fields $$$\vec{\boldsymbol{v}}$$$ from TVM can also provide qualitative strain measures⁴ from strain rate calculations.

The library is composed by multiple functions (written in Python programming language) that can be divided in three modules: (I) the generation of displacement fields, (II) post-processing and interpolation of data to images and (III) reading and writing. The first module is performed by solving the hyperelastic equations using an open source finite elements library called FEniCS⁶ or with the generation of analytical phantoms⁷. The second module uses the numpy⁸ and FEniCS libraries to mimic the three previously described sequences. The image construction is performed knowing the expression of the magnetization for the three acquisition sequences. The tagging images are generated using the SPAMM⁹ sequence, from which the tag-lines are distributed spatially according to

$$ M_{xy} = M_0 e^{-t/T_1}(\cos^2\alpha + \sin^2\alpha\cos^2(k_e x_i) - 1) + M_0 $$

where $$$M_0$$$ is thermal equilibrium magnetization, $$$\alpha$$$ the flip angle, $$$k_e$$$ the encoding frequency and $$$x_i$$$ the position in the direction $$$i$$$. In the case of DENSE and PC-MR the relation between the phase $$$\Delta\phi$$$ and the displacements and velocities (respectively) has been deeply studied^5,10 and took the form

$$ \Delta\phi_{x_i} = k_e u_i, \qquad \Delta\phi_{x_i} = \frac{v_i \pi}{\text{venc}} $$

where the encoding frequency $$$k_e$$$ (also for SPAMM) is given by

$$ k_e=\gamma\int_{\Delta T} G_{x_i}(\tau)d\tau $$

and where $$$\Delta T$$$ and $$$G_{x_i}$$$ are the encoding time and gradient, respectively.

The last module consist of reading and writing functions that can export the phantom simulations into DICOM images and VTK files (displacement and velocity fields).

We used this library to simulate numerical 2D and 3D dynamic phantoms that provide a FE mesh with a displacement field. The 2D phantoms consist of two concentric circles that moves with a prescribed displacement as explained in Gilliam et al⁸. The 3D phantom was obtained by solving the hyperelasticity equations over an idealized geometry of the left-ventricle¹¹. The 2D and 3D ventricle geometries are then deformed using the obtained field displacements. This deformed geometry is finally used to build Tagging, DENSE and TVM images with the equations described above.

Results

Figures 1 and 2 shows the FE results obtained with our library for 2D and 3D phantoms. The resulting 2D and 3D images are appreciated in Figures 3 and 4. In both figures, we show the resulting images for one simulated volunteer and one patient. For both phantoms the displacements and velocities are between physiological ranges.

In Figure 3 we can appreciate differences in the displacements fields, i.e, in how the ventricle is being deformed, whereas in Figure 4 places with higher and different displacements and velocities are indicated in blue circles. For the last case, the phantom simulating a patient condition was obtained changing the stiffness of the free wall at the basal level.

Discussion and Conclussion

Acknowledgements

We thank to CONICYT-PIA-Anillo ACT1416, CONICYT FONDEF/I Concurso IDeA en dos etapas ID15/10284 and FONDECYT #1141036, FONDECYT Postdoctorado 2017 #3170737 and National PhD scholarship N3455/2017 grants.

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