Intravoxel incoherent motion (IVIM) imaging¹ has gained renewed attention during the last years due to its potential of simultaneously monitoring tissue diffusion and perfusion completely non-invasively. However, to this date there is no consensus with regard to which b-values should be used. In this study we investigate the potential of Cramer-Ráo Lower Bound (CRLB), which is a commonly used tool for optimal experiment design². The effect of averaging is considered and generalized compared to previous studies³.IVIM is described by a biexponential model¹: $$ S=S_0((1-f)e^{-bD}+fe^{-b(D+D^{*})})$$ where D is the diffusion coefficient, D* is the pseudodiffusion coefficient, f is the perfusion fraction and S₀ is the signal without diffusion weighting. Given a model the CRLB sets a lower limit of the parameter uncertainties: $$\sigma ^2_{w_i} \leq (F^{-1})_{ii}$$ where $$$\sigma^2_{w_i}$$$ is the actual parameter uncertainty, $$$w \in\{D,D^*,f,S_0\}$$$ and F is Fisher's information matrix. Given Gaussian noise and considering the effect of averaging, F has elements: $$ F_{ij} = \sigma^{-2}\sum_{k=1}^K n_k \frac{\partial S_k}{\partial w_i}\frac{\partial S_k}{\partial w_j}$$ where $$$\sigma^2$$$ is the noise variance, n_k is the number of averages for the k:th b-value, S_k is the expected signal for the k:th b-value given the IVIM model. The noise level and the total number of measurements $$$(N=\sum^K_{k=1}n_k)$$$ are constants, therefore a matrix M may be used in the optimization instead of F: $$ M_{ij}=\frac{\sigma^2}{N}F_{ij}=\frac{\sigma^2}{N}\sigma^{-2}\sum_{k=1}^K n_k \frac{\partial S_k}{\partial w_i}\frac{\partial S_k}{\partial w_j}=\sum_{k=1}^K \alpha_k \frac{\partial S_k}{\partial w_i}\frac{\partial S_k}{\partial w_j}$$ where $$$\alpha_k = \frac{n_k}{N}$$$, which implies that $$$0\leq\alpha_k\leq1$$$ and $$$\sum^K_{k=1}\alpha_k=1$$$. Considering multiple sets of tissue parameters and adjusting for the different magnitude of the model parameters the total relative uncertainty, which is to be minimized, is given by: $$C = \sum^S_{s=1}\gamma_s\sum_{i=1}^3\frac{(M_s^{-1})_{ii}}{w_{i,s}^2}$$ where $$$\gamma_s$$$ is a weighting factor. The uncertainty of S₀ is omitted in the inner sum as it is not of primary interest.

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 Lemke⁴) 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 observations³. 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*.

CRLB optimization favors averaging of the minimum number of unique b-values rather than adding more unique ones. In practice the optimized b-value schemes were found to decrease the model parameter uncertainty in simulated data with mid-range perfusion.