Introduction

Cardiac Diffusion Tensor Imaging (cDTI) provides invaluable information about the state of myocardial microstructure in both healthy and diseased conditions^1–4. To address signal loss due to contractile motion during the diffusion encoding process, second-order motion-compensated (M012) diffusion waveforms with spin-echo sequences have been established⁵. The signal loss occurs due to the spatial gradient of phase directly relating to myocardial strain⁶, which potentially can serve as indicator of pathological cardiac function^7–9. For myocardial material point trajectories with motion terms higher than second order, the phase of cDTI data contains surrogate motion related to macroscopic strain. At the same time, macroscopic strain during diffusion encoding affects diffusion tensor estimation⁶ and hence needs to be taken into account. Therefore, it is appealing to aim at deriving both diffusion and strain information from a single cDTI exam.
In this study we investigate the feasibility of jointly deriving strain and diffusion tensors from complex-valued M012-SE cDTI data. To this end, we present a theoretical framework for a simplified analytical motion model as well as general contractile motion. Furthermore, we show how our simulation allows the direct evaluation of the strain tensor for an analytic trajectory as well as the generation of labeled data for realistic contractile motion.

Methods

Cardiac model
The digital phantom used in this work, is based on a biophysical mesh model of a contracting left ventricle (LV)¹⁰. Prior to simulation the mesh was refined, and a subset of nodes was selected forming a slice of 8 mm thickness perpendicular to the LV-long-axis. Random smoothly varying diffusion tensors were assigned to each mesh node¹¹. Two trajectories $$$\mathbf{r}_{i,analytic}(t)$$$ and $$$\mathbf{r}_{i, realistic}(t)$$$ were assigned to each node, where $$$\mathbf{r}_{i,realistic}(t)$$$ was obtained from biophysical simulation and $$$\mathbf{r}_{i,analytic}(t)=\mathbf{r}_{i,0}+\mathbf{k(r_{i,0})}\sin(ct)$$$ describes planar, radial pseudo contractile motion.

Theory
Assuming separation of the scales for contractile macroscopic motion and microscopic tissue diffusion, one can describe the diffusion effect as magnitude attenuation and the contractile motion as a phase effect. Further assuming a material point representing an ensemble of spins is assigned a diffusion tensor and a motion trajectory, then the phase of a material point, after applying diffusion gradient $$$g_j(t)$$$ evaluates as:

$$\phi_j(\tau,\mathbf{r_0})=\gamma\int_0^Tg_j(t)\mathbf{r}(t-\tau,\mathbf{r_0})\cdot\mathbf{n}_jdt,$$

where $$$\tau$$$ denotes the trigger delay. Furthermore, the displacement $$$\mathbf{u}(\tau, \mathbf{r_0};T)$$$ of a material point at $$$\mathbf{r_0}$$$ during the encoding period $$$T$$$ is given as:
$$\mathbf{u}(\tau,\mathbf{r_0};T)=\mathbf{r}(\tau+T,\mathbf{r_0})-\mathbf{r}(\tau,\mathbf{r_0}).$$

Simplified analytic motion model
For $$$\mathbf{r}_{i, analytic}(t)$$$, the relation between phase and displacement can be derived directly with following steps:

1. Replace $$$\mathbf{r}(t-\tau,\mathbf{r}{0})(t)$$$by its Taylor expansion in equation (1)
2. All terms of the integral with quadratic or lower order in $$$t$$$ evaluate to zero, yielding a direct relation between the phase and the spatially varying amplitude $$$\mathbf{k(r_0)}$$$
3. Insert $$$\mathbf{r}_{analytic}(t)$$$ into equation (2) and replace $$$\mathbf{k(r_0)}$$$ with the expression obtained fro the previous step, which yields:

$$\mathbf{n}_j\cdot\mathbf{u}(\tau,\mathbf{r_0})=\phi_j(\tau,\mathbf{r_0})\frac{\sin(c(\tau+T))-\sin(c\tau)}{c^3\gamma\int_0^Tg(t)(t-\tau)^3dt}\\\rightarrow\hat{\mathbf{u}}(\tau,\mathbf{r_0})=\arg\min_u\sum_j\left|\mathbf{n}_j\cdot\mathbf{u}(\tau,\mathbf{r_0})-\phi_j(\tau,\mathbf{r_0})\frac{\sin(c(\tau+T))-\sin(c\tau)}{c^3\gamma\int_0^Tg(t)(t-\tau)^3dt}\right|^2$$
General contractile motion
A relation between the displacement and the phase of diffusion encoded images for non-parameterized motion can be obtained by the following steps:

1. Assume the Taylor expansion $$$\mathbf{r}(t,\mathbf{r_0})\approx\mathbf{r_0}+\mathbf{v(r_0)}t+\mathbf{a{r_0}}t^2+\mathbf{j(r_0)}t^3$$$ describes the trajectory of material points in the myocardium. Again, using the second order motion compensation of the waveform, the residual phase encodes the jerk $$$\mathbf{j_0}$$$ according to: $$\phi_j(\tau,\mathbf{r_0})=\gamma\mathbf{n}_j\cdot\mathbf{j_0}\int_0^Tg(t)(t-\tau)^3dt$$
2. Evaluate the derivative of displacement (2) with respect to the trigger delay $$$\tau$$$, where $$$\mathbf{r}(\tau+T,\mathbf{r_0})$$$ and $$$\mathbf{r}(\tau,\mathbf{r_0})$$$ are approximated using the Taylor expansion: $$\frac{\partial\mathbf{u}(\tau,\mathbf{r_0};T)}{\partial\tau}\approx2T\mathbf{a(r_0)}+\mathbf{j(r_0)}[3T^2+6T\tau]$$
3. From combining the two previous expression follows: $$\mathbf{n}_j\cdot\mathbf{u}(\tau,\mathbf{r_0})=2T\tau(\mathbf{n}_j\cdot\mathbf{a(r_0)})+(\mathbf{n}_j\cdot\mathbf{c(r_0)})+\int\phi_j(\tau,\mathbf{r_0})\frac{3T^2+6T\tau}{\gamma\int_0^Tg(t)(t-\tau)^3 dt}d\tau$$
   

Simulation
A graphical summary of the simulation experiments for $$$\mathbf{r}_{i,analytic}(t)$$$ as well as $$$\mathbf{r}_{i,realistic}$$$ is given in Fig. 1. Simulation was performed by evaluating (1) on a temporal grid with $$$\Delta t=0.1ms$$$ for all mesh points, where the diffusion gradient was defined according to Stoeck et al.⁵ Furthermore, for each particle with the associated diffusion tensor $$$\mathbf{D}$$$, the factor $$$\exp(-b\mathbf{n^T}_j\mathbf{D}\mathbf{n}_j)$$$ was evaluated. This is followed by a single shot echo planar imaging readout. To obtain reference displacements on image resolution, a displacement encoding simulation was performed in the same fashion. Both simulations were repeated for varying trigger-delays for $$$\mathbf{r}_{i,analytic}(t)$$$ as well as $$$\mathbf{r}_{i,realistic}$$$. Additionally, the direct inversion according to equation (4) was performed using varying SNR values during simulation.

Results

Figure 2 illustrates the simulated motion trajectories. Simulation results for displacement encoding and diffusion weighted images are shown in Fig. 3. The displacement fields derived from displacement encoding and the ones from solving equation (4) for for multiple SNRs are presented in Fig. 4. Figure 5 shows a qualitative comparison of in-vivo data and the simulated signal with biophysically modelled contractile motion.

Discussion

The simulation results for analytic motion show the feasibility of deriving both macroscopic strain and microscopic tissue diffusion from the same cDTI imaging protocol. Furthermore, the qualitative comparison of image phase across the LV for in vivo data and our simulation using the biophysical LV model yields qualitative agreement. The relation of displacements and residual phase in cDTI data is very complex as the provided theory demonstrates. Therefore, directly inferring the displacements from real data might be challenging especially when including respiratory motion and assuming low SNR. Training neural networks to perform such inversions on synthetic data was shown to be feasible^11,12. Hence, training a neural network on our simulated data appears a promising next step.

Acknowledgements

No acknowledgement found.

References