Synopsis

A novel approach to estimate noise and Rician signal bias in diffusion MRI magnitude data is proposed. Rather than relying on repeat measurements for estimation of noise and expected signal, the methods uses multi-b measurements and non-monoexponential signal fits. In addition to noise and bias estimation this approach also provides signal averaging over all b-factors and permits determination of non-monoexponential tissue water diffusion signal decay. Testing was performed with Monte-Carlo simulations and on diffusion-weighted high-b prostate image data obtained with an external coil array.

Introduction

Methods

We propose a novel approach, where all measurements contribute meaningful information and at the same time allow the determination of <S> and $$$\sigma_g$$$ for each pixel position. We use the fact that model functions describe the signal decay well (Fig. 2) and that residual differences (at least up to a certain b-factor) can be considered predominantly noise related. Except for the kurtosis model with cumulant expansion, model fits are strictly monotonously decaying. Thus, there is no risk that with increasing number of parameters the residuals vanish.The fit conducted over N samples provides fitted signals $$$\hat{s}_i$$$ that are as good or even better estimates of the expected signal <$$$s_i$$$> than could be obtained with N averages (Fig. 3). The absolute residuals between fitted signals $$$\hat{s}_i$$$ and measured signals $$$s_i$$$, i.e., $$$r_i$$$ $$$=|\hat{s}_i-s_i$$$| are proportional to the SD of noise. Considering the distribution of $$$r_i$$$ at each b-factor, one can conjecture that the mean or expected $$$r_i$$$, i.e., <$$$r_i$$$> = $$$\alpha\cdot\sigma_g$$$,where $$$\alpha$$$ is a function of SNR. To find $$$\alpha$$$, we ran MonteCarlo simulations and fitted a function to the simulated data (Fig. 1C).

Now we resort to the realistic case of having a single realization of the diffusion signal decay. Again, assuming a perfect fit, an unbiased estimator of $$$\sigma_g$$$ at each b-factor would be $$$r_i/\alpha(\theta_i)$$$, where $$$\theta_i$$$ equals the SNR at $$$\hat{s}_i$$$.If we sum up all N estimations, we obtain a single unbiased estimator of $$$\sigma_g$$$:

\begin{equation}\sigma_g =\frac{1}{N}\sum_{i=1}^{N}r_i/\alpha(\theta_i)\end{equation}

At this point, we do not know any of the $$$\theta_i$$$ values. But as an initial guess, we can assume a Gaussian signal distribution for all $$$s_i$$$, where $$$r_i$$$ follows a half-folded normal distribution with $$$\alpha=\sqrt{2/\pi}$$$. We now have initial guesses of $$$\sigma_g$$$ and of N expected signals <$$$s_i$$$> at respective b-factors. For each <$$$s_i$$$> the Rician bias that would arise can then be calculated analytically. We subtract this bias value from our measured signal $$$s_i$$$ and finally obtain a bias corrected signal value $$$s_i^c$$$ to which we again can apply our model fit. This process is performed iteratively to obtain increasingly better estimates of $$$\sigma_g$$$ and of N $$$s_i^c$$$ values with associated bias. The standard deviation of the noise distribution varies with SNR from $$$\sigma_g$$$ when measured signals are Gaussian distributed to $$$\sqrt{(4-\pi)/2}\cdot\sigma_g$$$ when measured signals are Rayleigh and, therefore, much more narrowly distributed. The standard deviation of noise as a function of SNR can be determined analytically and can be accounted for by using different weights to each measurement. Multi-coil reconstruction can introduce signals that follow a non-central $$$\chi$$$ distribution [4,5]. With parallel imaging, which was used here, however, signals can be considered Rician distributed [6].

Results

The process converges after a few iterations and the estimation of fit parameters is reliable, as long as SNR does not fall below 1 over an extended range of b-factors, e.g., with unsuitable experiment design. Bias correction results from Monte-Carlo simulations are presented in Fig. 4. The virtual averaging effect of a 21 b-factor protocol was estimated to be 4-fold at b0 and 16-fold at 3000 s/mm². Results of bias correction of in vivo scans are presented in Fig. 5.

Discussion and Conclusion

Acknowledgements

ALFGBG-449801, CAN2017/558, NIH R01 CA160902

References

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