Annelinde Buikema^{1}, Arjan den Dekker^{1,2}, and Jan Sijbers^{1}

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.

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/direction^{4}. 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 T_{2} 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 estimators^{5}.

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.

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.

Figure 1: Parameter estimation from data generated with a constant TE of 200 ms. The spread of estimates is visualised for each parameter (each subplot) as function of SNR. The red dots on the y-axis represent the ground truth value.

Figure 2: Absolute bias for each parameter (each subplot) for constant and variable TE as function of SNR.

Figure 3: Variance for each parameter (each subplot) for constant and variable TE as function of SNR.

Figure 4: MSE for each parameter (each subplot) for constant and variable TE as function of SNR.

Figure 5: MSE for the 6 non-DT-dependent parameters, calculated for the DT-dependent model as well as the non-DT-dependent model with a varying amount of data points.