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Diffusion time dependence in the mid-time regime: a simulation study using PGSE.1imec-Visionlab, Department of Physics, University of Antwerp, Antwerp, Belgium, 2Delft Center for Systems and Control, Delft University of Technology, Delft, Netherlands
Synopsis
The purpose of this work is to study the effect of varying the diffusion time on the estimation of the parameters of the two-compartment diffusion tensor model in the mid-time regime. Simulation results show that the precision of the diffusion time-dependent compartmental parameter estimates increases when a variable echo time acquisition scheme is used. At low SNR, however, including diffusion time-dependence may lead to a high bias and variance compared to the more conventional non diffusion time-dependent model.
Introduction
Recently, Fieremans et al modelled the longitudinal and transverse time-dependence of diffusion tensor eigenvalues in white matter regions1. The model describes the time-dependence of the diffusion signal in both longitudinal and transverse directions, which can be interpreted as an effect of structural disorder at the mesoscopic scale. It reveals the non-Gaussian nature of diffusion in one of the compartments in the longitudinal direction and in the extra-axonal compartment in the transverse direction1-3. This model has been studied in long time ranges ($$$\Delta=45-600$$$ ms) using STEAM-DTI1. Recently, Lee et al presented an extension of this model to DKI, and showed that the addition of the temporal dimension lifts the notorious degeneracy in parameter estimation of multi-compartment models, but with a remaining challenge in terms of precision4. However, to our knowledge, this diffusion time-dependent (DT-dependent) model has not yet been studied in the mid-time regime with varying echo times, which would be valuable for clinical practice. In this simulation study, we use the pulsed-gradient spin-echo (PGSE) sequence and study the effects of varying the diffusion time ($$$\Delta=20-180$$$ ms) on the parameter estimation in a mid-time regime.Methods
The model consists of two compartments, axonal and extra-axonal, each represented with a specific diffusion tensor. The axonal compartment is modelled as a stick with no radius (no radial diffusivity) and the extra-axonal compartment as a tensor with axial symmetry:
$$S=S_0\cdot(f_a\cdot{e^{-TE/T_{2,a}}}e^{-bg[V{A_a}V']g'}+(1-f_a)\cdot{e^{-TE/T_{2,e}}}e^{-bg[V{A_e}V']g'})$$
with $$$A_a$$$ and $$$A_e$$$ the eigenvalue matrices, with eigenvalues
$$\lambda_{1,a}={{D_{a,\infty}}}+\frac{{c_a}}{\sqrt{\Delta}}$$
$$\lambda_{1,e}={{D_{e,\infty}^{\parallel}}}+\frac{{{c_e^{\parallel}}}}{\sqrt{\Delta}}$$
$$\lambda_{2,e}=\lambda_{3,e}={{D_{e,\infty}^{\perp}}}+{{c_e^{\perp}}}\cdot\frac{log(\Delta/\delta)}{\Delta}$$
with $$$f_a$$$ the intra-axonal water fraction, {$$$D_{a,\infty}$$$,$$$D_{e,\infty}^{\parallel}$$$,$$$D_{e,\infty}^{\perp}$$$} the bulk (turtuosity-limit) diffusivities and {$$$c_a$$$,$$$c_e^{\parallel}$$$,$$$c_e^{\perp}$$$} strengths of restrictions in each compartment/direction4. V is the rotation matrix from the Euler angles, leading in two additional parameters $$$\psi$$$ and $$$\theta$$$. When the time-dependence of diffusion is ignored, the model simplifies to the more conventional non-DT-dependent model, where the terms with the c-parameters are ignored.
In a simulation study, the accuracy and precision with which the parameters can be estimated is evaluated. First, we compared estimation from simulated data with a constant echo time (TE) on one hand and data with variable echo times on the other hand, employing the DT-dependent model for both data generation and parameter estimation. Second, we simulated data using the DT-dependent model and compared estimation from these data when using either the DT-dependent model or the non-DT-dependent model for parameter estimation. Both approaches were performed over a range of SNR values from 10 to 100, where SNR is defined based on the non-diffusion weighted signal at TE=200 ms.
The data were generated with 33 equidistant diffusion times in the range $$$\Delta=20-180$$$ ms, $$$\delta=15$$$ ms in 60 directions (uniformly distributed over the half-sphere) and with either TE=200 ms or TE={50,100,150,200} ms (equally distributed), after which physically impossible combinations of TE and $$$\Delta$$$ were excluded.
In the first comparison, sufficient repetitions of the constant TE signal were generated to guarantee an equal number of data points in both data sets. From the sampled signal, Rician distributed data was generated with varying noise levels. Furthermore, we assume the T2 values to be known. All simulations were based on 500 noise realisations.
Parameter estimation is performed using a maximum likelihood approach with a lower bound of 0 for all parameters and upper bounds {$$$D_a$$$,$$$D_{e,par}$$$,$$$D_{e,\perp}$$$}$$$\leq{3}\mu{m}^2/ms$$$, $$$f_a{\leq}1$$$ and {$$$\psi$$$,$$$\theta$$$}$$${\leq}2\pi$$$.
Bias, variance and mean squared error (MSE) are calculated to statistically compare the different estimators5.
Results
Fig.1 shows the estimates of each model parameter for constant and variable TE, as well as the ground truth parameters (red dots).
Fig.2, Fig.3 and Fig.4 show respectively the bias, variance and MSE for constant and variable TE. As can be observed from Fig.4, the MSE for estimation with variable TE is consistently lower compared to estimation with constant TE. This is mainly due to the availability of data points at shorter echo times (and hence higher SNR) resulting in a gain in precision for each SNR.
Fig.5 shows a dependence on the amount of data points as well as a dependence on SNR. At low SNR values, the non-DT-dependent model outperforms the DT-dependent model and vice-versa at higher SNR values.
Discussion and conclusions
Based on this simulation study, estimation of the parameters of the DT-dependent model improves in terms of accuracy and precision with a variable TE acquisition scheme, using clinical ranges of TE and $$$\Delta$$$, also for low SNR. To use this in clinical practice, the amount of data points should be reduced. Optimal experimental design should be used to maximize precision in a clinical time frame. A possible future advantage of varying TE can be the simultaneous estimation of T2 relaxometry parameters. Finally, at low SNR, including diffusion time-dependence can lead to a high bias and variance compared to the more conventional non-DT-dependent model.Acknowledgements
The authors acknowledge financial support from the Research Foundation Flanders (FWO, Belgium) through Project No. G0D7216N.References
1. Fieremans, E., et al. In vivo observation and biophysical interpretation of time-dependent diffusion in human white matter. NeuroImage 2016;129:414-427
2. Novikov, D.S., Jensen, J.H., Helpern, J.A., Fieremans, E. Revealing mesoscopic structural universality with diffusion. PNAS 2014;111(14):5088-5093
3. Burcaw, L.M., Fieremans, E., and Novikov, DS. Mesoscopic structure of neuronal tracts from time-dependent diffusion. NeuroImage 2015;114:18–37
4. Lee, HH., Fieremans, E. and Novikov D.S. LEMONADE(t): Exact relation of time-dependent diffusion signal moments to neuronal microstructure. Proc. Intl. Soc. Mag. Reson. Med. 2018;26:0884
5. van den Bos A. Parameter estimation for scientists and engineers. Hoboken, NJ: Wiley-Interscience, 2007.
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