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Time-dependent and anisotropic diffusion in the heart: linear and spherical tensor encoding with varying degree of motion compensation

Samo Lasic^1,2, Henrik Lundell², Filip Szczepankiewicz^3,4,5, Markus Nilsson³, Jürgen E. Schneider⁶, and Irvin Teh⁶
¹Random Walk Imaging, Lund, Sweden, ²Danish Research Centre for Magnetic Resonance, Centre for Functional and Diagnostic Imaging and Research, Copenhagen University Hospital Hvidovre, Copenhagen, Denmark, ³Clinical Sciences, Lund University, Lund, Sweden, ⁴Harvard Medical School, Boston, MA, United States, ⁵Brigham and Women's Hospital, Boston, MA, United States, ⁶Leeds Institute of Cardiovascular and Metabolic Medicine, University of Leeds, Leeds, United Kingdom

Synopsis

Spherical tensor encoding (STE) can potentially shorten acquisition of mean diffusivity (MD) compared to the traditional linear tensor encoding (LTE). To avoid negative effects of motion, e.g. in the heart, motion compensation is needed. However, motion compensation requires altering diffusion gradient waveforms and their sensitivities to time-dependent diffusion. To exclude motion, we first investigated LTE and STE with different degrees of motion compensation in ex vivo pig hearts. We observed significantly different MD, which can be attributed to time-dependent diffusion and microscopic diffusion anisotropy. Our analysis suggests that time-dependent diffusion is a critical determinant of MD in the myocardium.

Introduction

Isotropic diffusion weighting^1-3, also known as spherical tensor encoding (STE)^4,5, could potentially accelerate the measurement of mean diffusivity (MD) at least two-fold compared to the traditional linear tensor encoding (LTE) approach⁶. Motion compensated diffusion encoding waveforms^7-9 are often used to avoid motion artefacts, which are very challenging particularly in cardiac imaging. We have recently proposed a new STE strategy with simultaneous nulling of arbitrary diffusion encoding gradient moments and concomitant gradients¹⁰. Significantly different MD values could be obtained from different encoding protocols. Here we considered two sources affecting the estimated MD value: (1) the effect of time-dependent diffusion^11-17, e.g. caused by restrictions and (2) the effects of microscopic diffusional anisotropy^18-20. Effects of time-dependent diffusion depend on size of restrictions and spectral characteristics of encoding waveforms (diffusion times), even when equal encoding times are used^15-17. The pronounced non mono-exponential signal decay due to microscopic anisotropy^1,4,5,20,21 could negatively bias MD when relatively large b-values are used. Our analysis suggests that due to relatively large cardiomyocytes²², time-dependent diffusion is expected to be the dominant source of MD bias on clinical scanners. Restricted diffusion model fits well to the data across 13 different waveforms and can be used to estimate cell sizes.

Restricted diffusion model

To fit data, we used the model of diffusion restricted in impermeable cylinders^14,17. Diagonal components of diffusion spectra $$$D_{ii}(\omega,R)$$$ were computed for cylinders of varying radii $$$R$$$ (validated by Monte-Carlo simulations¹⁷. MD was computed from $$b\cdot\text{MD}(R)=\sum_{i=1}^3\int_{\infty}^{\infty}s_{ii}(\omega)\,D_{ii}(\omega,R)\,d\omega.$$ Here $$$s_{ii}(\omega)$$$ are diagonal elements of the dephasing cross power spectral density given by Fourier transforms of dephasing waveforms $$$q_{i}(t)$$$ as $$ s_{ij}(\omega)=q_{i}(\omega)\overline{q}_{j}(\omega).$$ The total encoding power, known as b-value, is given by the integral of spectral trace $$$s(\omega)$$$ over all frequencies $$b=\sum_{i=1}^3\int_{-\infty}^{\infty}s_{ii}(\omega)d\omega\equiv\int_{-\infty}^{\infty}s(\omega)d\omega.$$ The spectral trace, shown in Figure 1 for all waveforms, is thus the key parameter affecting MD.

Data fitting

A two-compartment model was fitted globally to the data from 13 different encoding waveforms: $$\text{MD}_{\text{calc}}=f_{\text{res}}\text{MD}_{\text{res}}(R)+(1-f_{\text{res}})D_{\text{h}},$$ where $$$f_{\text{res}}$$$ is an apparent signal fraction from the restricted compartment and $$$D_{\text{h}}$$$ is an apparent diffusion coefficient for the compartment with hindered Gaussian diffusion. $$$D_0 = 2 \, \mu \text{m}^2/\text{ms}$$$ was used for $$$\text{MD}_{\text{res}}$$$.

Experiments

Two pig hearts were obtained from a local butcher, embedded in 1% agarose PBS gel, and scanned at room temperature. MRI data were acquired on a 3T clinical scanner (Siemens Healthcare, Erlangen, Germany) using a prototype single-shot spin-echo EPI: TR = 5 s, TE = $$$\tau$$$ + 22 ms, partial-Fourier=6/8, in-plane resolution=3×3 mm², slice thickness=8 mm, FOV = 320×118 mm², with ZOOM-IT excitation²¹. Six different diffusion encoding waveforms, i.e. LTE and STE with all gradient moments nulled up to zeroth, first and second order (M0, M1, M2) were applied with three different encoding times ($$$\tau$$$ = 46, 57, 67 ms) and b-values of 0.1 and 0.45 ms/µm² (three orthogonal directions and 4 repetitions for LTE, and 12 repetitions for STE). See Table 1 and Figure 1. Average MD within the left ventricular myocardium in the middle slice was estimated, denoted MD_exp in Figures 2 and 3.

Results and discussion

The two-compartment model fits experimental data well (see Figures 2 and 3) and yields comparable results for the two examined hearts. For heart 1, we get: R = 6.1 µm, f_res = 0.31, D_h = 0.42 µm²/ms. The root mean square error (RMSE) was 0.010 µm²/ms and the sum of residuals was 0.002 µm²/ms. For heart 2, we get: R = 7.5 µm, f_res = 0.33, D_h = 0.32 µm²/ms. RMSE and sum of residuals were 0.014 s and 0.001 µm²/ms, respectively. The fitted size corresponds well to the sizes observed in human hearts with microscopy and Coulter Channelyzer²², while a relatively low signal fractions f_res could be a consequence of cell degradation in the non-fixed samples.

A small tendency toward negative fit residuals was observed from LTE data (see Figure 2). Fitting only STE data (see Figure 3) but accounting for residuals from the entire dataset yields comparable RMSE values (0.013 and 0.026 µm²/ms) but large negative sums of residuals (-0.055 and -0.164 µm²/ms). This could potentially be a consequence of diffusion anisotropy, which causes higher signals for LTE compared to STE at larger b-values (see Figure 4) leading to negatively biased MD values. The effect of time-dependent diffusion becomes dominant for larger restrictions, where different MD estimates are expected from LTE and STE data. Our results suggest that the time-dependent diffusion effect is significant.

As we have demonstrated, strong time-dependent diffusion effects in the heart may lead to different MD (in the range 0.5-0.8 µm²/ms) estimated with different diffusion encoding waveforms even when equal encoding times are used (see Figure 5). This can be particularly significant if LTE and STE with different degrees of motion compensation are compared.

Acknowledgements

SL and HL have received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 804746). This work was supported by the British Heart Foundation, UK (SI/14/1/30718). SL is supported also by Random Walk Imaging.

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Figures

Figure 1: Effective gradient waveforms g(t), dephasing waveforms q(t) and normalized spectral traces s(ω). Upper row is for spherical tensor encoding (STE) and lower row is for linear tensor encoding (LTE). Columns from left to right are for gradient waveforms with all moments nulled up to zeroth, first and second order (M0, M1, M2). Note the shift of spectra from zero frequency to higher frequencies, which alters sensitivity to time-dependent diffusion.

Figure 2: Average mean diffusivity (MD_exp) within an ROI across the left ventricle myocardium in the central slice (markers, upper row), theoretical prediction (MD_calc, lines, upper row) and residuals (MD_exp-MD_calc) vs. MD_exp (lower row). Data for all waveforms are fitted globally by the hindered-restricted model to yield estimates for cell diameter (2R), hindered diffusion coefficient (D_h) and an apparent fraction of restricted compartment (f_res). The model fits well to all data. A small negative bias from LTE is observed (see crosses leaning toward negative values).

Figure 3: Similar to Figure 2 but here only STE data is fitted.

Figure 4: Simulated MD bias due to anisotropy and time-dependent diffusion (slopes on signal vs b). Theoretical signal vs b for randomly oriented cylinders (D₀ = 2 µm²/ms) and diameter (2R) from STE (solid line) and LTE (dashed line). The initial slope from LTE (dotted line) corresponds to the predicted MD from LTE. A negative bias is expected when MD is estimated from relatively large LTE b-values (see curving lines caused by anisotropy). For larger diameters, the anisotropy bias vanishes, and the difference arises between LTE and STE estimates due to time-dependent diffusion.

Figure 5: Similar to Figures 2 and 3 but here only data at equal encoding time (67 ms) is fitted. Resulting fit parameters are comparable to the global fit.

Proc. Intl. Soc. Mag. Reson. Med. 28 (2020)

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