Optimization of b-values for estimation of IVIM D and f
Oscar Jalnefjord1,2, Mikael Montelius1, Göran Starck1,2, and Maria Ljungberg1,2

1Department of Radiation Physics, University of Gothenburg, Gothenburg, Sweden, 2Department of Medical Physics and Biomedical Engineering, Sahlgrenska University Hospital, Gothenburg, Sweden


IVIM parameter estimation restricted to D and f (avoiding D*) has gained increased popularity. In this study we propose a framework for optimization of b-value schemes for this application. We show that optimized b-values schemes contain exactly three unique b-value, regardless of the total number of acquisitions, and that parameter estimation uncertainty can be substantially reduced by the use of the optimized b-value schemes.


The intravoxel incoherent motion (IVIM) model enables extraction of diffusion and perfusion information from diffusion weighted images (DWI)1. The signal model commonly used for IVIM analysis is given by:


where S(b) is the signal with b-value b, S0 is the signal without diffusion weighting, f is the perfusion fraction, D is the diffusion coefficient and D* is the pseudo-diffusion coefficient.

To conform to a clinical setting with limited scan time and image quality, it has been proposed to acquire and analyze data such that only D and f are obtained, while the noise sensitive D* is omitted1,2. This is accomplished by the used of b-values either equal to zero or large enough to consider the signal from the vascular compartment negligible. Under these circumstances, the IVIM model (Eq. 1) simplifies to:


where δ(b=0)=1 and δ(b≠0)=02.

Although the interest for IVIM analysis limited to D and f is increasing, only little work has been done regarding optimization of b-value schemes used for this approach to IVIM3,4.

The aim of this study was to develop and evaluate a framework for optimization of b-value schemes for DWI data used to estimate the IVIM parameters D and f.


The optimization framework proposed in this study is based on Cramer-Rao lower bounds (CRLB’s), which set lower bounds for the parameter estimation variance and have been used for optimization of diffusion MRI in several previous studies5–7. The CRLB’s are given by the diagonal elements of the inverse Fisher matrix:


where the elements of the Fisher matrix are given by:


where θ=[D,f,S0], S(b) is given by Equation 2, n+1 is the number of unique b-values and ni is the number of acquisitions with b-value bi.


Optimization of b-values
The error to minimize for a given set of IVIM parameters was formulated as:


The objective function used in the optimization was calculated as the average error over a range of typical parameter values for the tissue of interest.

To test the optimization framework, b-value schemes were generated for examination of the liver with 3-12 acquisitions and in the limit of an infinite number of acquisitions. The objective function was evaluated over the ranges D∊[1.0 1.2]µm2/ms and f∊[0.15 0.30], based on typical parameter values for liver8. Only b-values equal to 0 or in the range [200 800]s/mm2 were allowed in the optimized b-values scheme in order to avoid bias from perfusion or kurtosis effects.

Assessment of optimized b-value scheme
To assess the potential gain in using the proposed optimization, the optimized b-value scheme with 8 acquisitions (2×0,3×200,3×800s/mm2; Table 1) was compared with a scheme with linearly distributed b-values (0,200,300,…,800s/mm2).

Simulated data was generated based on Equation 1 for D∊[0.5 1.5]µm2/ms and f∊[0.05 0.40], and D*∊{10,20,50}µm2/ms with SNR=20. D and f were estimated by segmented model fitting where D was estimated by a monoexponential model fit with b>0 and f was calculated from the difference between the measured S(b=0) and the signal at b=0 extrapolated from the monoexponential fit.

In vivo data was acquired by scanning seven healthy volunteers with a Philips 3T Achieva. DWI’s with b-values as given by the compared schemes were acquired four times without moving the subject to assess the repeatability of each b-value scheme (TE=55ms, voxel size 3×3×5mm3, SNR≈20). Parameter maps were obtained by applying the segmented model fit in each voxel as described for the simulated data.


Optimization of b-values
The optimized b-value schemes contained exactly three unique b-values (0, 200 and 800s/mm2). If more than three acquisitions were considered, repeated measurements were favored by the optimization. For larger number of acquisitions, the repetitions were distributed approximately as 1:2:2 (Table 1).

The simulations showed that the use of the optimized b-value scheme reduced the estimation variability by approximately 30% and 20% for D and f, respectively, compared with the use of the linear b-value scheme (Figure 1).

In vivo measurements
Estimates of D and f obtained in vivo were approximately the same for the two b-value schemes, but the variability was smaller for the optimized scheme (parameter maps of example subject in Fig. 2, summary of results from all subjects in Fig. 3). The improvement in repeatability related to choice of b-value scheme was similar to that predicted by the simulations.


The b-value schemes obtained from optimization based on Cramer-Rao lower bound for estimation of IVIM D and f comprises of three unique b-values, which are repeated if more than three acquisitions are used. The use of an optimized b-value scheme can substantially reduce the estimation uncertainty.


The study was supported by grants from the Swedish Cancer Society and the King Gustav V Jubilee Clinic Cancer Research Foundation


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Figure 1. Relative difference [%] in estimation variability (standard deviation of parameter estimates) between optimized and linear b-value scheme. Negative numbers correspond to lower estimation variability with the optimized b-value scheme. The value for each combination of D and f (shown as pixels) is based on 10,000 data series. The results shown are based on data simulated with D*=20µm2/ms, similar results were seen for the other magnitudes of D*

Figure 2. Parameter maps of repeated acquisitions with the two compared b-value schemes along with maps of the voxelwise mean and standard deviation of parameter estimates. The parameter maps (color) are superimposed on a b=0 image (grayscale)

Figure 3. Comparison of b-values schemes for in vivo data with respect to standard deviation of parameter estimates (top plot), and with respect to the estimated diffusion coefficient (middle plot) and the estimated perfusion fraction (bottom plot). The boxplots show the distribution across subjects of ROI median of either parameter estimate variability or parameter estimate, depending on the context. The whiskers show the minimum and maximum values. Note that negative values in the top plot imply less variability with the optimized b-value scheme

Table 1. Optimum number of acquisitions of each optimum b-value for different total number of acquisitions (ntot) and the proportions for the limiting case of an infinite total number of acquisitions. The optimal b-values were b0=0, b1=200 and b2=800s/mm2 regardless of the value of ntot

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)