Synopsis

The generation of synthetic MR images allows the testing of several for postprocessing methodologies under controlled conditions. In this work, we developed an open source Python/C++ library for the generation of synthetic SPAMM, C-SPAMM and DENSE images, based on known magnetization expressions and using finite element geometries.

Introduction

Tagged Magnetic Resonance (TMR) is the most common MR technique for the assessment of the cardiac walls functionality. In this technique, a magnetization grid is generated over the tissue at the beginning of the cardiac cycle, which moves consequently with the cardiac motion. Two of the most popular tagging techniques are Spatial Modulation of Magnetization (SPAMM)¹ and Complementary SPAMM (CSPAMM)², which are based on Stimulated Echoes Acquisition Mode (STEAM). A more recent imaging method is Displacement Encoding with Stimulated Echoes (DENSE)³ which is also based on STEAM sequence and encodes the displacement of the tissue directly in the magnetization phase.

Although there are several validated methods for the postprocessing of SPAMM, C-SPAMM, and DENSE images, new methodologies are rising. For testing purposes, having synthetic images which reproduce the behavior of real MR images would be a very useful tool.

In this work, we developed an efficient open source toolbox for the generation and postprocessing of SPAMM, C-SPAMM and DENSE images, which is based on known expressions of the magnetization and finite element models for the generation of the cardiac geometry and motion.

Methods

Our library uses known SPAMM and DENSE magnetization expressions, obtained from idealized sequence diagrams (see Fig. 1), to generate in-sylico MR images. Thus, the acquired magnetization expressions at time $$$t_n$$$ are:

\[\begin{alignat}{2}M_{\text{SPAMM}}(t_n)&= M\sin(\alpha)\cos^2(\beta) \exp\left(-\frac{t_n}{T_1}\right)+M_0\sin(\alpha)\cos^n(\alpha) \left(1-\exp\left(-\frac{t_n}{T_1}\right)\right)\nonumber\\&-\frac{M}{2}\sin(\alpha)\cos^n(\alpha)\sin^2(\beta)\exp\left(-\frac{t_n}{T_1}\right)\exp\left(-\boldsymbol{i}k_ex\right)\\&-\frac{M}{2}\sin(\alpha)\cos^n(\alpha)\sin^2(\beta) \exp\left(-\frac{t_n}{T_1}\right)\exp\left(+\boldsymbol{i}k_ex\right)\\ \end{alignat}\]

and

\[\begin{alignat}{2}M_{\text{DENSE}}(t_n)&=M_0\sin(\alpha)\cos^n(\alpha)\left\{1-\exp\left(-\frac{t_n}{T_1}\right)\right\} \exp\left\{-\boldsymbol{i}k_e(x+\Delta x)\right\}\\&+\frac{M}{2}\sin(\alpha)\cos^n(\alpha)\exp\left(-\frac{t_n}{T_1}\right)\exp\left(-\boldsymbol{i}k_e\Delta x\right)\\&+\frac{M}{2} \sin(\alpha)\cos^n(\alpha) \exp\left(-\frac{t_n}{T_1}\right)\exp\left\{-\boldsymbol{i}k_e(2x+\Delta x)\right\}\end{alignat}\]

where $$$M_0$$$ represents the thermal equilibrium magnetization, $$$M$$$ the magnetization before the first RF pulse, $$$T_1$$$ the relaxation time, $$$\beta$$$ the flip angle at the times 1 and 3 (see Fig. 1a), $$$\alpha$$$ the flip angle of the acquisition sequence, and $$$x$$$ and $$$\Delta x$$$ the position and the displacement of the tissue, respectively. In the previous expressions, the encoding frequency $$$k_e$$$ controls the motion sensitivity of each imaging modality, and is given by:

$$k_e=\gamma\int_{T_G}G(\tau)~d\tau$$

where $$$\gamma$$$ represents the gyromagnetic constant, and $$$G$$$ and $$$T_G$$$ the strength and duration of the encoding gradient.The generation of images is divided in three main steps: (1) generation of geometry, (2) generation of motion, and (3) magnetization modelling and interpolation. The generation of two dimensional data is based on the work of Glilliam⁵, in which ranges for geometrical and physiological parameters were defined for simulating "healthy" volunteers and patients. Three-dimensional data can also be generated, but steps (2) and (3) needs to be performed outside of the box. A brief summary of the generation process is given in Fig 2. The core parts of PyMRStrain were written in Python⁶ and C++, and parallelized using MPI4py⁷. PyMRStrain runs on Linux, Mac and Windows OS, and can be downloaded from bitbucket.org/hernanmella/pymrstrain. There is also a Matlab-coded part in which self-made implementations of motion estimation techniques such as HARP and SinMod are freely available as well.

One of the advantages of having synthetical data was tested in this work. Using a dataset of 95 two-dimensional synthetic C-SPAMM and DENSE images, we performed a resolution and noise sensitivity analysis to compare semi-automatic postprocessing tools as HARP, SinMod, and DENSEAnalysis^8,9 in the estimation of motion and cardiac strain. Results were compared against exact values, and error was measured using the Normalized Root Square Error (nRMSE) and Directional Error (DE), defined as:

$$nRMSE_a=\sqrt{\frac{\sum_{i=1}^{N}(\lvert a(x)\rvert-\lvert a_e(x)\rvert)^2}{\sum_{i=1}^{N}\lvert a_e(x)\rvert}}$$

$$nRMSE_{\boldsymbol{u}}=\sqrt{\frac{\sum_{i=1}^{N} (\lVert\boldsymbol{u}(x)\rVert_2-\lVert\boldsymbol{u}_e(x)\rVert_2)^2 }{\sum_{i=1}^{N} \lVert\boldsymbol{u}_e(x)\rVert_2}}$$

$$MDE=\frac{1}{N} \sum_{i=1}^{N}\left( 1-\frac{|\boldsymbol{u}(x)\cdot \boldsymbol{u}_e(x)|}{\lVert\boldsymbol{u}(x)\rVert_2\lVert\boldsymbol{u}_e(x)\rVert_2} \right)$$

where $$$N$$$ represents the number pixels in the image and $$$nRMSE_a$$$ (resp. $$$nRMSE_{\boldsymbol{u}}$$$), the $$$nRMSE$$$ for a scalar (resp. vectorial) quantity and subscript $$$()_e$$$ denotes the exact value.

Results

Discussion

Conclusion

Acknowledgements

References

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