Elsevier

Medical Image Analysis

Volume 17, Issue 3, April 2013, Pages 325-336
Medical Image Analysis

Reliable estimation of incoherent motion parametric maps from diffusion-weighted MRI using fusion bootstrap moves

https://doi.org/10.1016/j.media.2012.12.001Get rights and content

Abstract

Diffusion-weighted MRI has the potential to provide important new insights into physiological and microstructural properties of the body. The Intra-Voxel Incoherent Motion (IVIM) model relates the observed DW-MRI signal decay to parameters that reflect blood flow in the capillaries (D), capillaries volume fraction (f), and diffusivity (D). However, the commonly used, independent voxel-wise fitting of the IVIM model leads to imprecise parameter estimates, which has hampered their practical usage.

In this work, we improve the precision of estimates by introducing a spatially-constrained Incoherent Motion (IM) model of DW-MRI signal decay. We also introduce an efficient iterative “fusion bootstrap moves” (FBM) solver that enables precise parameter estimates with this new IM model. This solver updates parameter estimates by applying a binary graph-cut solver to fuse the current estimate of parameter values with a new proposal of the parameter values into a new estimate of parameter values that better fits the observed DW-MRI data. The proposals of parameter values are sampled from the independent voxel-wise distributions of the parameter values with a model-based bootstrap resampling of the residuals.

We assessed both the improvement in the precision of the incoherent motion parameter estimates and the characterization of heterogeneous tumor environments by analyzing simulated and in vivo abdominal DW-MRI data of 30 patients, and in vivo DW-MRI data of three patients with musculoskeletal lesions. We found our IM-FBM reduces the relative root mean square error of the D parameter estimates by 80%, and of the f and D parameter estimates by 50% compared to the IVIM model with the simulated data. Similarly, we observed that our IM-FBM method significantly reduces the coefficient of variation of parameter estimates of the D parameter by 43%, the f parameter by 37%, and the D parameter by 17% compared to the IVIM model (paired Student’s t-test, p < 0.0001). In addition, we found our IM-FBM method improved the characterization of heterogeneous musculoskeletal lesions by means of increased contrast-to-noise ratio of 19.3%.

The IM model and FBM solver combined, provide more precise estimate of the physiological model parameter values that describing the DW-MRI signal decay and a better mechanism for characterizing heterogeneous lesions than does the independent voxel-wise IVIM model.

Highlights

► Spatially constrained body DW-MRI signal decay model. ► Fusion bootstrap moves solver to reliably infer the incoherent motion model parameters. ► Increased precision of incoherent motion parameter estimates from in-vivo DW-MRI. ► Improved characterization of heterogeneous tumor environment.

Introduction

Diffusion-weighted MRI (DW-MRI) of the body is a non-invasive imaging technique sensitive to the incoherent motion of water molecules inside the area of interest. This motion is known to be a combination of a slow diffusion component associated with the Brownian motion of water molecules, and a fast diffusion component associated with the bulk motion of intravascular molecules in the micro-capillaries. These phenomena are characterized through the so-called, Intra-Voxel Incoherent Motion (IVIM) model with the slow diffusion (D); the fast diffusion (D) as decay rate parameters; and the fractional contribution (f) of each motion to the DW-MRI signal decay (Le Bihan, 2008, Le Bihan et al., 1988, Koh et al., 2011).

IVIM model parameters have recently shown promise as quantitative imaging biomarkers for various clinical applications in the body including differential analysis of tumors (Sigmund et al., 2011, Re et al., 2011, Klauss et al., 2011, Chandarana et al., 2011, Gloria et al., 2010, Lemke et al., 2009), the assessment of liver cirrhosis (Luciani et al., 2008, Patel et al., 2010), and Crohn’s disease (Freiman et al., 2013).

A key limitation of the IVIM model is that it is an independent voxel-wise model. It models only signal decay related to intra-voxel incoherent motion of the water molecules, while both inter- and intra-voxel incoherent motion of water molecules are related to the DW-MRI signal decay. Moreover, the utility of IVIM parametric imaging with DW-MRI is diminished by a lack of verified methods for producing reliable estimates of both fast and slow diffusion parameters from the DW-MRI signal (Koh et al., 2011).

Specifically, reliable estimates of IVIM model parameters are difficult to obtain because of (1) the non-linearity of the IVIM model; (2) the limited number of DW-MRI images as compared to the number of the IVIM model parameters; and (3) the low signal-to-noise ratio (SNR) observed in body DW-MRI.

In current practice, there are four approaches that will increase the reliability of incoherent motion parameter estimates to varying degrees.

  • 1.

    Approximate the non-linear DW-MRI signal decay by a log-linear model with the apparent diffusion coefficient (ADC) as the decay rate parameter (Stejskal and Tanner, 1965). However, this simplified model precludes the independent characterization of slow diffusion and fast diffusion components – a process essential to accurately quantifying biological phenomena taking place inside the body.

  • 2.

    Increase the DW-MRI SNR by acquiring multiple DW-MRI images from the patient; next, average these results, and then use the averaged DW-MRI signal to estimate IVIM model parameters. However, this requires substantially increased acquisition times – an undesirable outcome, especially in children, who generally have difficulty in remaining still for long periods of time (Koh et al., 2011).

  • 3.

    Increase the DW-MRI SNR by averaging the DW-MRI signal over a region of interest (ROI), effectively yielding more reliable IVIM parameter estimates as used by Zhang et al. in their DW-MRI acquisition optimization study (Zhang et al., 2012). Unfortunately, by averaging the signal over a ROI, the estimated parameters do not reflect critical heterogeneous environments such as the necrotic and viable parts of tumors.

  • 4.

    Bayesian model fitting, proposed by Neil and Bretthorst (1993), and recently used by Koh et al. (2011) aims to increase the reliability of IVIM parameter estimates by calculating the probability distribution function of each parameter rather than by calculating point estimates, as is done using maximum-likelihood estimators. However, this method considers the information at each voxel independently, effectively ignoring its spatial context. Moreover, it requires numerical integration of the marginal posterior probabilities over the possible ranges of parameter values, which are sensitive both to discretization/sampling effects and to the chosen integration limits (Behrens et al., 2003).

In this work, we present a new model of DW-MRI signal decay that accounts for both inter- and intra-voxel incoherent motion of the water molecules (IM) by introducing a model of spatial homogeneity to the IVIM model of DW-MRI signal decay. Essentially, our IM model produces estimates of incoherent motion model parameters for all voxels simultaneously, rather than solving for each voxel independently. As a result, we increase the reliability of the incoherent motion parameter estimates from the DW-MRI data without acquiring additional data or losing spatial sensitivity. Fig. 1 depicts the graphical models previously used to estimate the fast and slow diffusion parameters from DW-MRI data (a–c) compared to the model proposed in this work (d).

Bayesian estimation of Markov Random Field (MRF) models with spatial homogeneity as a prior term has been widely used in computer vision applications since its introduction by Geman and Geman (Geman and Geman, 1984). The equivalence between MRFs and Gibbs distributions established by Hamersley and Clifford (Winkler, 2003) also enabled the modeling of variety of computer vision problems such as energy minimization tasks within the Bayesian framework (Geman and Geman, 1984).

The optimization of MRF-related energy functions is challenging, however, due to the large number of variables that must be optimized simultaneously, especially when compared to the relatively fewer number of variables that are optimized with simple, independent voxel-wise approaches. Besag proposed the iterative conditional modes (ICM) algorithm as an approximation algorithm for discrete MRF optimization (Besag, 1986). That is, the ICM enforces spatial homogeneity by approximating the solution for each voxel independently while fixing the solutions for its neighborhood. Thus, the ICM tends to converge slowly to a sub-optimal solution in the discrete setting (Lempitsky et al., 2010). In the case of a binary field (i.e., the Ising model), graph min-cut techniques are able to pinpoint the globally optimal solution of the energy minimization problem. Further, several combinatorial approximation algorithms were proposed for setting more than two possible labels (i.e., the Potts model). We refer the reader to Szeliski et al. (Szeliski et al., 2008) for a review and comparison of different combinatorial algorithms for the multiple labels case. For inference in continuous MRF models in which each node represents a continuous random variable, the Markov Chain Monte Carlo (MCMC) and the continuous version of the ICM algorithm are commonly used.

In the specific context of parametric MRI, Schmid et al. proposed a Gaussian MRF model with MCMC optimization to increase the reliability of parameter estimates in quantitative, dynamic contrast-enhanced MRI (DCE-MRI) (Schmid et al., 2006). More recently, Kelm et al. proposed a similar Gaussian MRF model with ICM-based optimization in both DCE-MRI (Kelm et al., 2009) and in magnetic resonance spectroscopy (MRS) (Kelm et al., 2012).

To solve the challenging problem of inference of the incoherent motion model parameters imposed by incorporating additional spatial homogeneity prior to the previously used IVIM model, we also introduce the “fusion bootstrap moves” (FBM) solver, an efficient new iterative combinatorial solver that, when applied to our new IM model, generates precise parameter estimates. Our FBM solver iteratively updates parameter estimates by applying binary graph-cut solver to fuse the current estimate of parameter values with a new proposal of the parameter values into a new estimate of parameter values that better fit the observed DW-MRI data (Lempitsky et al., 2010). The proposals of parameter values are sampled from the independent voxel-wise distributions of the parameter values with a model-based bootstrap resampling of the residuals (Davidson and Flachaire, 2008).

This paper extends work previously presented at the MICCAI 2012 conference (Freiman et al., 2012) by offering a more detailed description of the method and additional experiments. Following the Introduction, the paper is organized into six sections (2–7). In Section 2, we briefly describe the DW-MRI signal decay model employed, and we review the conventional approach to IVIM parameter estimation. In Section 3, we introduce the spatial homogeneity prior followed by a description of the FBM solver. In Section 4, we describe the experimental methodology, the DW-MRI data for our simulation, and in vivo experiments. In Section 5, we present results for simulated DW-MRI data as well as in vivo DW-MRI data from normal abdominal organs of 30 subjects and 3 musculoskeletal lesions studies. In Section 6, we discuss study results as well as limitations; and last, we summarize our findings and the impact of our work in Section 7.

Section snippets

The intravoxel incoherent motion model

The Intra-Voxel Incoherent Motion (IVIM) model of DW-MRI signal decay assumes a signal decay function of the form (Le Bihan, 2008, Le Bihan et al., 1988):mv,i=s0,vfvexp-biDv+Dv+(1-fv)exp(-bi(Dv))where mi,v is the expected signal of voxel v at b-value = bi, s0,v is the baseline signal at voxel v; Dv is the slow diffusion decay associated with extravascular water molecules’ motion; Dv is the fast diffusion decay associated with the intravascular water molecules’ motion; and fv is the fraction

Spatial homogeneity prior

Taking the Bayesian perspective, our goal is to find the parametric maps Θ that maximize the posterior probability associated with the maps given the observed signal S and the spatial homogeneity prior knowledge:Θ^=argmaxΘp(Θ|S)p(S|Θ)p(Θ)

Based on the Hammersley–Clifford theorem (Winkler, 2003), by using a spatial prior in the form of a continuous-valued Markov random field, the posterior probability p(SΘ)p(Θ) can be decomposed into the product of node and clique potentials:p(S|Θ)p(Θ)vp(Sv|Θv

Hyper-parameter optimization

The spatial homogeneity prior model defines three hyper-parameters: (1) the standard deviation of the signal noise; (2) the parameter weighting matrix W; and (3) the spatial smoothness prior weight α. The standard deviation of the signal noise is estimated from a pre-defined background region for each dataset. All other parameters were previously determined and have been used subsequently for all experiments. Following the methodology of Kelm et al. (Kelm et al., 2009), we determined the values

Precision and accuracy of the incoherent motion parameter estimates from simulated DW-MRI data

Fig. 6 depicts the parametric maps estimated from the simulated DW-MRI data using the IVIM approach as well as our Bayesian approach with spatial homogeneity prior optimized using (1) the continuous ICM method (IM-ICMC); (2) the discrete ICM method (IM-ICMD); and (3) the FBM method (IM-FBM).

Fig. 7 presents the relative bias; the relative standard deviation of the estimates error; and the relative root mean square error (RRMS) plots of the incoherent motion parameter estimators as a function of

Discussion

Incoherent motion quantification from DW-MRI has a promising role as a quantitative non-invasive imaging biomarker for various clinical applications. However, the commonly used independent, voxel-wise fitting of the IVIM model does not account for inter-voxel interactions. Moreover, the low-quality of the incoherent motion parameter estimates using the IVIM model has hampered its utilization in clinical studies and in patient management (Koh et al., 2011).

In this work, we introduced a new model

Conclusion

The role of incoherent motion parameters as quantitative imaging biomarkers for various clinical applications is becoming increasingly important. However, current techqniqes for estimating the incoherent motion parameters from DW-MRI data do not provide reliable or specific enough parameter estimates.

In this work, we improved the reliability of incoherent motion measurements from DW-MRI data significantly by introducing a model of DW-MRI signal decay that accounts for both inter- and

Acknowledgements

This investigation was supported in part by NIH Grants R01 EB008015, R01 LM010033, R01 EB013248, and P30 HD018655 and by a research grant from the Boston Children’s Hospital Translational Research Program. The authors thank Nancy Drinan for her support in editing this manuscript, and the anonymous reviewers for their insightful comments that helped improve this paper.

References (35)

  • M. Freiman et al.

    Reliable assessment of perfusivity and diffusivity from diffusion imaging of the body

  • M. Freiman et al.

    Quantitative body DW-MRI biomarkers uncertainty estimation using unscented wild-bootstrap

  • S. Geman et al.

    Stochastic relaxation, gibbs distributions, and the bayesian restoration of images

    IEEE Trans. Pattern Anal. Mach. Intell.

    (1984)
  • C. Gloria et al.

    Differentiation of diffusion coefficients to distinguish malignant and benign tumor

    J. Xray Sci. Technol.

    (2010)
  • L. Hedges

    Distribution theory for Glass’s estimator of effect size and related estimators

    J. Educ. Stat.

    (1981)
  • B.M. Kelm et al.

    Using spatial prior knowledge in the spectral fitting of MRS images

    NMR Biomed.

    (2012)
  • B.M. Kelm et al.

    Estimating kinetic parameter maps from dynamic contrast-enhanced MRI using spatial prior knowledge

    IEEE Trans. Med. Imag.

    (2009)
  • Cited by (59)

    • IVIM and Non-Gaussian DWI of the Breast

      2022, Diffusion MRI of the Breast
    • Bayesian inference using hierarchical and spatial priors for intravoxel incoherent motion MR imaging in the brain: Analysis of cancer and acute stroke

      2021, Medical Image Analysis
      Citation Excerpt :

      This method was compared to flat priors in pancreatic patients (Gurney-Champion et al., 2018). In addition to these global approaches, which ignore spatial correlations, local spatial regularization has been exploited by using Markov random fields (MRF) in the liver (Freiman et al., 2013). This approach was compared to the hierarchical Bayesian approach in simulations and in-vivo data of the liver (While, 2017) and in breast cancer (Vidić et al., 2019).

    • Measuring Perfusion: Intravoxel Incoherent Motion MR Imaging

      2021, Magnetic Resonance Imaging Clinics of North America
    • A kernel-based image denoising method for improving parametric image generation

      2019, Medical Image Analysis
      Citation Excerpt :

      We also simulated lesions of different sizes (3, 5 and 7 pixels in radius). The parameter PF for background, liver, pancreas, spleen and kidney was set to 0.1, 0.31, 0.248, 0.13 and 0.187, respectively (Freiman et al., 2013; Kang et al., 2014; Luciani et al., 2008; Sigmund et al., 2012). The parameter D for background, liver, pancreas, spleen and kidney was set to 0.4 × 10−3, 1.0 × 10−3, 1.2 × 10−3, 0.75 × 10−3 and 2.0 × 10−3 mm2/s, respectively (Freiman et al., 2013; Kang et al., 2014; Luciani et al., 2008; Sigmund et al., 2012).

    View all citing articles on Scopus
    View full text