Can Cramer-Ráo Lower Bound be used to find optimal b-values for IVIM?

Oscar Gustafsson^{1,2}, Maria Ljungberg^{1,2}, and Göran Starck^{1,2}

Constrained minimization was performed in
MATLAB for three sets of tissue parameters (table 1) separately and combined ($$$
\gamma = [\frac{1}{200^2}, \frac{1}{30^2}, \frac{1}{20^2}] $$$ weights from Lemke^{4})
to obtain a total of four optimal b-value schemes. To assess the effects of
boundary conditions the optimizations was performed for multiple lower and
upper limits of b-values (b_{lower} = [0,10,20], b_{upper} = [600,1000,2000]).

To assess the effect of b-value scheme on the model
parameter uncertainties data was simulated for all tissue parameter sets by
adding Rician noise to signal values given by the IVIM model. The model
parameters were obtained by weighted non-linear least squares fitting. The
uncertainties were characterized by IQR normalized to the true tissue parameter
values. This was compared to simulated data using a b-value scheme comparable
to those commonly seen in the literature (b_{lit}=[0,10,20,30,40,50,75,100,200,300,400,600]).
To match the total number of acquisitions the proportions of the optimal
schemes were multiplied by 12 and b-values were rounded to the nearest multiple
of ten.

For all sets of tissue parameters the optimal
b-value scheme included no more than four unique b-values with non-zero
proportion (up to 10 unique b-values were allowed) (table 2). This
is in agreement with previous observations^{3}. The same result
was produced when the optimization was performed over a weighted sum of multiple
tissue parameter sets ($$$b_{sum} = [0,30,220,600],\alpha_{sum}=[0.18,0.32,0.32,0.18]
$$$). Limits had small effect on the α:s for most settings. Increased lower
limit/decreased upper limit mainly affected the lower/higher b-values
respectively (table 2).

The CRLB b-value schemes gave smaller model parameter uncertainties in 11 of the 18 cases (table 3). The best gain in using the CRLB scheme was seen for the medium perfusion parameter set, which is characterized by relatively high f and low D*.

1. Le Bihan et al. 1988 Separation of Diffusion and Perfusion in Intravoxel Incoherent Motion MR Imaging. Radiology

2. Alexander. 2008 A General Framework for Experiment Design in Diffusion MRI and Its Application in Measuring Direct Tissue-Microstructure Features. Magnetic Resonance in Medicine

3. Leporq et al. 2014 Optimization of Intra-voxel Incoherent Motion Imaging at 3.0 Tesla for Fast Liver Examination. Journal of Magnetic Resonance Imaging

4. Lemke et al. 2011 Toward an optimal distribution of b values for intravoxel incoherent motion imaging. Magnetic Resonance Imaging

Table 1. Tissue perfusion parameters used
in this study and previously by Lemke^{4}

Table 2. The effect of b-value limits on
optimal b-values and proportionality coefficients (b and α). The first three rows show the effect of
changing the upper limit whereas the last three rows show the effect of
changing the lower limit. b-values were rounded for better readability (nearest 5
for b < 50 s/mm^{2} and nearest 10 for b > 50 s/mm^{2}) Note: in some cases the
displayed proportions do not add up to 1 due to rounding error.

Table 3. Relative uncertainties for the model
parameters, from fitting of simulated data, calculated as IQR/true value. Diff was
calculated as (IQR(Literature)/IQR(CRLB) – 1). As a result, positive numbers
indicate better performance for the CRLB scheme and vice versa.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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