Purpose

Estimation of Intravoxel Incoherent Motion (IVIM) parameter maps from a set of diffusion-weighted (DW) images acquired at multiple b-values often suffers from low SNR, which may increase the variance of the estimated maps. Unfortunately, there is currently no consensus on the optimal b-values needed to minimize the estimation variance (i.e., maximize precision and noise performance) of IVIM parameter maps. Therefore, the purpose of this work is to optimize the set of b-values for IVIM parameter mapping utilizing the Cramér-Rao Lower Bound¹ (CRLB) analysis under the realistic MRI assumption of Rician distributed data².

Methods

Determination of optimal b-values: The IVIM signal model is $$$S(b)~=~S_0[fe^{-b\cdot{D}^*}~+~(1-f)e^{-b\cdot{D}}]$$$, where f, D*, D, and S₀ are the perfusion fraction, pseudo-diffusion, diffusion, and signal with no diffusion-weighting, respectively. In order to reduce the estimation variance of IVIM parameters we optimize the b-value set via the CRLB approach. Similar approaches were previously employed for the monoexponential model^3,4 and for the IVIM model⁵ with Gaussian noise. Given a set of independent Rician distributed observations of the IVIM signal model, S(b_k) where k∈[1,...,K], with the same noise level, the CRLB of each of the IVIM parameters (CRLB_S0, CRLB_D, CRLB_D*, and CRLB_f) is on the diagonal of the inverse of the Fisher Information Matrix¹ (Eq. 1), where I_n(S,σ) is described in Ref⁶. Therefore, for a given target set of IVIM parameters (S₀, D, f, and D*), noise level (σ), and number of b-values (K), the determination of the set of K b-values that minimizes the estimation variance of the IVIM parameters is performed by a Four b-value Search algorithm. With this algorithm, the optimal set is restricted to be composed of four different b-values repeated k₀, k₁, k₂, and k₃ times, respectively. The first b-value is fixed to be b=0s/mm² to have measurements with maximum signal-to-noise-ratio (SNR), whereas the other three (low, middle, and high) are chosen among a large pool of b-value candidates. Then, an exhaustive search is performed to find the optimal set of K b-values (k₀+k₁+k₂+k₃=K). Finally, the [zero(k₀), low(k₁), middle(k₂), high(k₃)] combination of b-values that achieves minimum CRLB_D, CRLB_f, CRLB_D*, or CRLB of the figure of merit $$$\Gamma~=\frac{\sqrt{\text{CRLB}_\text{D}}}{D}~+~\frac{\sqrt{\text{CRLB}_\text{f}}}{f}~+~\frac{\sqrt{\text{CRLB}_{\text{D}^*}}}{D^*}$$$ (CRLB_𝛤) is considered optimal.

Furthermore, optimal b-value sets obtained with the proposed approach were computed from parameters shown in Table 1 to study their distribution and properties.

Eq. 1: $$\begin{bmatrix}\sum\limits_{k=1}^{K}[fe^{-b_k\cdot\text{D}^*}+(1-f)e^{-b_k\cdot\text{D}}]^2\mathcal{I}_n(S(b_k),\sigma)&-\sum\limits_{k=1}^{K}S(b_k)(1-f)b_ke^{-b_k\cdot\text{D}}\mathcal{I}_n(S(b_k),\sigma)&\sum\limits_{k=1}^{K}S(b_k)[e^{-b_k\cdot\text{D}^*}-e^{-b_k\cdot\text{D}}]\mathcal{I}_n(S(b_k),\sigma)&-\sum\limits_{k=1}^{K}S(b_k)fb_ke^{-b_k\cdot\text{D}^*}\mathcal{I}_n(S(b_k),\sigma)\\-\sum\limits_{k=1}^{K}S(b_k)(1-f)b_ke^{-b_k\cdot\text{D}}\mathcal{I}_n(S(b_k),\sigma)&\sum\limits_{k=1}^{K}S^2_0(1-f)^2b^2_ke^{-2\cdot{b}_k\cdot\text{D}}\mathcal{I}_n(S(b_k),\sigma)&-\sum\limits_{k=1}^{K}S^2_0(1-f)b_k[e^{-b_k\cdot\text{D}^*}-e^{-b_k\cdot\text{D}}]e^{-b_k\cdot\text{D}}\mathcal{I}_n(S(b_k),\sigma)&\sum\limits_{k=1}^{K}S^2_0(f-f^2)b^2_ke^{-b_k\cdot(\text{D}+\text{D}^*)}\mathcal{I}_n(S(b_k),\sigma)\\\sum\limits_{k=1}^{K}S(b_k)[e^{-b_k\cdot\text{D}^*}-e^{-b_k\cdot\text{D}}]\mathcal{I}_n(S(b_k),\sigma)&-\sum\limits_{k=1}^{K}S^2_0(1-f)b_k[e^{-b_k\cdot\text{D}^*}-e^{-b_k\cdot\text{D}}]e^{-b_k\cdot\text{D}}\mathcal{I}_n(S(b_k),\sigma)&\sum\limits_{k=1}^{K}S^2_0[e^{-b_k\cdot\text{D}^*}-e^{-b_k\cdot\text{D}}]^2\mathcal{I}_n(S(b_k),\sigma)&-\sum\limits_{k=1}^{K}S^2_0fb_k[e^{-b_k\cdot\text{D}^*}-e^{-b_k\cdot\text{D}}]e^{-b_k\cdot\text{D}^*}\mathcal{I}_n(S(b_k),\sigma)\\-\sum\limits_{k=1}^{K}S(b_k)fb_ke^{-b_k\cdot\text{D}^*}\mathcal{I}_n(S(b_k),\sigma)&\sum\limits_{k=1}^{K}S^2_0(f-f^2)b^2_ke^{-b_k\cdot(\text{D}+\text{D}^*)}\mathcal{I}_n(S(b_k),\sigma)&-\sum\limits_{k=1}^{K}S^2_0fb_k[e^{-b_k\cdot\text{D}^*}-e^{-b_k\cdot\text{D}}]e^{-b_k\cdot\text{D}^*}\mathcal{I}_n(S(b_k),\sigma)&\sum\limits_{k=1}^{K}S^2_0f^2b^2_ke^{-2\cdot{b}_k\cdot\text{D}^*}\mathcal{I}_n(S(b_k),\sigma)\\\end{bmatrix}$$

Validation of optimal b-values: With IRB approval and informed written consent, full liver DWI (see Table 2) was acquired in a healthy volunteer to compare the IVIM parameter maps of b-value sets obtained with the proposed approach to those proposed in the literature^7,8,9 for IVIM liver DWI (see Table 2). The whole acquisition was repeated 16 consecutive times to enable voxel-wise determination of IVIM parameter and SNR statistics. IVIM parameter maps were obtained with a maximum likelihood estimator using all DWI available. SNR was computed voxel-wise on the S₀ images through the ratio between the signal intensity and the standard deviation across the 16 repetitions. Liver IVIM parameters and SNR values were obtained averaging the values of a 30mm² ROI drawn on each slice of the liver acquisition (see Table 2).

Additionally, given a b-value set, corresponding DWI of an acquisition were selected for IVIM parameter estimation with a maximum likelihood estimator. However, if a b-value was repeated, the equivalent number of DWI of the consecutive repetitions were selected. This method was repeated 16 consecutive times, one per acquisition, obtaining a set of 16 repeated IVIM parameter maps. Then, we obtained the voxel-wise standard deviation statistics across all the estimated maps (σ_D, σ_f, and σ_D*). This experiment is a procedure to compare the estimation variance of various b-value sets.

Results

Figure 1 shows the ratio between the number of times each b-value is repeated (k_i) in the optimal set and total number of b-values (K), when minimizing CRLB_D, CRLB_f, CRLB_D*, and CRLB_𝛤 (D=7x10^-3mm²/s, f=0.3, D*=20x10^-3mm²/s, SNR=[200,50,10]). Figure 2 shows the b-value distribution of the optimal K=13 b-value sets obtained for parameters of Table 1. Finally, Figure 3 shows the performance comparison in terms of voxel-wise standard deviation reductions of each estimated IVIM parameter map (i.e., $$$\frac{\sigma_\text{D}-\sigma_\text{D-Rician}}{\sigma_\text{D}}$$$, $$$\frac{\sigma_\text{D*}-\sigma_\text{D*-Rician}}{\sigma_\text{D*}}$$$, $$$\frac{\sigma_\text{f}-\sigma_\text{f-Rician}}{\sigma_\text{f}}$$$) between the proposed optimal b-value set and three sets proposed in the literature^6,7,8.

Discussion

Simulations show the optimized b-value set depends on the D, f, D*, SNR, and on the minimization parameter (i.e., CRLB_D, CRLB_f, CRLB_D*, and CRLB_𝛤). Further, optimal b-values can be grouped as the color-coded representation of Figure 2, indicating that to reduce the estimation variance of IVIM parameter maps the low, middle and high b-values should be 10-60s/mm², 60-200s/mm², >200s/mm², respectively. Finally, the optimal b-values obtained with the proposed approach minimizing the figure of merit (CRLB_𝛤) achieve lower estimation variance (higher noise performance) on the estimated liver IVIM parameter maps than b-values from the literature. However, in order to develop a procedure suitable for clinical settings and to study its discrimination ability, further validation is still required including the T₂ relaxation effect and more in-vivo healthy and diseased patients.

Conclusion

b-Value optimization is crucial to reduce the estimation variance (i.e., maximize precision and noise performance) of IVIM parameters estimation in DW-MRI. The proposed approach may help optimize and standardize liver DW-MRI acquisitions.

Acknowledgements

The authors acknowledge grant RTI2018-094569-B-I00 from the Ministerio de Ciencia e Innovación of Spain. The main author would also like to acknowledge the Consejería de Educación of Junta de Castilla y León, and the Fondo Social Europeo for his predoctoral grant.