Intravoxel incoherent motion (IVIM)¹ is widely used in clinical examination of various organs. Although the diffusion property of tissue varies with the organ, a fixed set of b-values, which is 10 to 16 different b-values including more number of low b-values, are generally used regardless of examination sites. IVIM is more sensitive to data outliers than mono-exponential analysis due to the difficulty in bi-exponential model fitting. A set of b-values should be more carefully chosen depending on the organ to better estimate the IVIM parameters. In this study, a method for optimizing b-value sampling using error propagation methods for IVIM is proposed. We investigated the difference in an optimal set of b-values depending on the organ and the effect of b-value sampling on the precision of IVIM parameters.

The proposed method optimizes a b-values sampling by minimizing the standard deviation (SD) of IVIM parameters estimated with error propagation methods. The IVIM signal intensity model is described with the following equation.

$$ \frac{S_{b}}{S_{0}}=f\exp\left[-bD^{*}\right]+\left(1-f\right)\exp\left[-bD\right], $$

where S_b is the diffusion weighted image signal at a b-value b, S₀ is the signal at b = 0, f is the perfusion fraction, D* is the pseudo-diffusion coefficient, and D is the tissue diffusion coefficient. The IVIM parameters f, D*, and D can be determined using the non-linear least square method. In the model-fitting process, random noise in the image signal σ_s propagates into the estimates of the IVIM parameters. Given a set of IVIM parameters and that of b-values, the SD for each IVIM parameter σ_f, σ_D*, and σ_D is calculated with the following equation using error propagation methods.

$$ \left(\begin{array}{c}\sigma_{f}&\sigma_{D^*}&\sigma_{D}\end{array}\right)=\sigma_{s}\left(\begin{array}{c}\sqrt{\sum_i^n(L_{1,i})^2(\frac{1}{S_{0}^2}+\frac{S_{bi}^2}{S_{0}^2})}&\sqrt{\sum_i^n(L_{2,i})^2(\frac{1}{S_{0}^2}+\frac{S_{bi}^2}{S_{0}^2})}&\sqrt{\sum_i^n(L_{3,i})^2(\frac{1}{S_{0}^2}+\frac{S_{bi}^2}{S_{0}^2})}\end{array}\right). $$

$$ J= \begin{bmatrix}\frac{\partial g_{1}}{\partial f} & \frac{\partial g_{1}}{\partial D^*}&\frac{\partial g_{1}}{\partial D} \\: &: &: \\\frac{\partial g_{n}}{\partial f} & \frac{\partial g_{n}}{\partial D^*}&\frac{\partial g_{n}}{\partial D} \end{bmatrix}, g_{i}=f\exp\left[-b_{i}D^{*}\right]+\left(1-f\right)\exp\left[-b_{i}D\right], (J^{\top}J)^{-1}J^{\top}=\begin{bmatrix}L_{1,1} & \cdot\cdot& L_{1,n} \\L_{2,1} & \cdot\cdot& L_{2,n}\\L_{3,1} & \cdot\cdot& L_{3,n} \end{bmatrix},$$

where n is the number of sampling points of the b-value. The σ_s is assumed to be a normal distribution whose average is 0 and the same value regardless of the b-value. For optimization, the typical IVIM parameters for various organs obtained from the literature are listed in Table 1. A set of b-values for maximizing the signal-to-noise ratio (SNR) of the IVIM parameters was explored for each organ (liver, kidney, and prostrate) by minimizing ξ given in following equation.

$$ \xi=\frac{\sigma_{f}}{f}+\frac{\sigma_{D^*}}{D^*}+\frac{\sigma_{D}}{D}.$$

The other calculation conditions were as follows: S₀, 1.0; σ_s, 0.01; maximal b-value, 1000 s/mm²; and n, 10 (at least one sampling point at b = 0).

We used Monte Carlo simulations to evaluate σ_f, σ_D*, and σ_D calculated with the error propagation methods. To assess the benefits of b-value optimization, we compared the SNRs of the IVIM parameters obtained from the optimized set of b-values with those obtained from conventional b-value sampling (b = 0, 10, 20, 30, 40, 50, 100, 200, 400, 800 s/mm²).

The results of comparing the SDs estimated from error propagation methods and Monte Carlo simulations are listed in Table 2. The SDs of the IVIM parameters were almost the same between error propagation methods and Monte Carlo simulations. The results of the optimized set of b-values for each organ are listed in Table 3. In each case, the optimal b-values were divided into four b-values. However, the combination of b-values varied with the organ. The comparison of the SNRs of the IVIM parameters between optimized b-value sampling and conventional b-value sampling is shown in Figure 1. The results show that the noise of IVIM maps was reduced using optimized b-value sampling and that the SNR increased 1.2–1.6 times by optimization. Comparison of the SDs estimated with the Monte Carlo simulations and error propagation methods demonstrated that error propagation methods were appropriate for optimization of b-value sampling. By comparing optimized b-value sampling with conventional b-value sampling, we found that the precision of IVIM parameters is dramatically affected by a combination of b-values. According to the optimization of the set of b-values, increasing the number of signal averages at four b-values improves the precision of IVIM parameters more than increasing different sampling points of those b-values. The optimal set of b-values varied with each organ. In particular, the maximal b-value for the kidney was interestingly different from other organs. These results suggest that b-value sampling should be optimized depending on the examination site. The limitation of this study was that we did not evaluate the range of IVIM parameters and the pathological lesions. We revealed that the optimization of b-value sampling corresponding to an organ can improve the fitting precision of IVIM parameters.