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Optimization of b-values for estimation of IVIM D and f1Department of Radiation Physics, University of Gothenburg, Gothenburg, Sweden, 2Department of Medical Physics and Biomedical Engineering, Sahlgrenska University Hospital, Gothenburg, Sweden
Synopsis
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.
Introduction
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, 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:
$$S(b)=S_0((1-f)e^{-bD}+f\delta(b))\quad[2]$$
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.
Theory
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:
$$\sigma^2_{\theta_j}\geq(F^{-1})_{jj}\quad[3]$$
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,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.
Methods
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]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.
Results
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).
Simulations
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.
Conclusions
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.Acknowledgements
The study was supported by grants from the Swedish Cancer Society and the King Gustav V Jubilee Clinic Cancer Research FoundationReferences
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