Oscar Jalnefjord^{1,2}, Mikael Montelius^{1}, Göran Starck^{1,2}, and Maria Ljungberg^{1,2}

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:

$$S(b)=S_0((1-f)e^{-bD}+fe^{-bD^*})\quad[1]$$

where *S*(*b*) is the signal with b-value *b*,
S_{0 }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 omitted^{1,2}. 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:

$$S(b)=S_0((1-f)e^{-bD}+f\delta(b))\quad[2]$$

where δ(*b*=0)=1 and δ(*b*≠0)=0^{2}.

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 IVIM^{3,4}.

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 studies^{5–7}.
The CRLB’s are given by
the diagonal elements of the inverse Fisher matrix:

$$\sigma^2_{\theta_j}\geq(F^{-1})_{jj}\quad[3]$$

where the elements of the Fisher matrix are given by:

$$F_{jk}=\sum_{i=0}^{n}n_i\frac{{\partial}S(b_i)}{\partial\theta_j}\frac{{\partial}S(b_i)}{\partial\theta_k}\quad[4]$$

where *θ*=[*D*,*f*,*S*_{0}], *S*(*b*) is given by Equation 2, *n*+1 is the number of unique b-values and
*n*_{i} is the number of acquisitions
with b-value *b _{i}*.

**Optimization of
b-values**

The error to minimize for a given set of IVIM parameters was formulated as:

$$E=\frac{\sqrt{(F^{-1})_{11}}}{D}+\frac{\sqrt{(F^{-1})_{22}}}{f}\quad[5]$$

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]µm^{2}/ms and *f*∊[0.15 0.30], based on typical parameter values for liver^{8}. Only b-values equal
to 0 or in the range [200 800]s/mm^{2} 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/mm

Simulated data was generated based on Equation 1 for *D*∊[0.5 1.5]µm^{2}/ms and *f*∊[0.05 0.40], and *D**∊{10,20,50}µm^{2}/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 3×3×5mm^{3}, 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/mm

**Simulations**The simulations showed that the use of the optimized b-value scheme reduced the estimation variability by approximately 30% and 20% for

**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.

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