Optimization of b-value sampling using error propagation methods for intravoxel incoherent motion imaging for various organs

Suguru Yokosawa^{1}, Hisaaki Ochi^{1}, and Yoshitaka Bito^{2}

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

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

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

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: *S _{0}*, 1.0; σ

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/mm^{2}).

1. Le Bihan D, Breton E, Lallemand D, et al. MR imaging of intravoxel incoherent motions: application to diffusion and perfusion in neurologic disorders. Radiology. 1986;161(2):401-407.

2. Patel J, Sigmund EE, Rusinek H, et al. Diagnosis of cirrhosis with intravoxel incoherent motion diffusion MRI and dynamic contrast-enhanced MRI alone and in combination: preliminary experience. J Magn Reson Imaging. 2010;31(3):589-600.

3. Notohamiprodjo M, Chandarana H, Mikheev A, et al. Combined intravoxel incoherent motion and diffusion tensor imaging of renal diffusion and flow anisotropy. Magn Reson Med. 2015;73(4):1526-1532.

4. Riches SF, Hawtin K, Chares-Edward EM, et al. Diffusion-weighted imaging of the prostate and rectal wall: comparison of biexponential and monoexponential modelled diffusion associated prefusion coefficient. NMR Biomed. 2009;22(3):318-325.

Table
1. Typical values of IVIM parameters for three organs

Table 2. Comparison of standard deviations estimated with error propagation methods and Monte Carlo simulations

Table 3. Optimized set of b-values for three organs

Figure
1. Comparison of SNRs of IVIM parameters calculated
from (a) optimized b-value sampling and (b) conventional b-value sampling. The
SNRs were calculated by error propagation methods, and images were obtained
from set of simulation data with added random noise.

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

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