Diffusion-weighted imaging (DWI) is sensitive to both extravascular diffusion and microvascular pseudo-diffusion (cf. perfusion). Intravoxel incoherent motion (IVIM) modelling attempts to separate these factors by fitting a biexponential function to DWI data obtained at many different b-values¹. A major goal of IVIM modelling is to produce pixel-wise maps of the diffusion rate, D, and the pseudo-diffusion rate, D*, and volume fraction, F. However, the pseudo-diffusion parameter estimates are highly sensitive to noise, and in general it is necessary to average data over a region of interest (ROI). Bayesian modelling has been demonstrated recently as a promising alternative to least-squares fitting, by producing clearer parameter maps with much lower estimation uncertainty and therefore arguably greater clinical potential^2,3. In this work we test the robustness of one of these Bayesian approaches, which uses a Gaussian prior², by applying it to simulated noisy data.

Four sets of simulated tissues were considered. The parameter maps associated with each set were chosen to vary either in a discrete fashion (Fig. 1) or a continuous fashion that obeyed a Gaussian-like distribution (Fig 2). The domain of each set was a 128x128 matrix that contained 8 sub-regions representing all the possible high/low combinations of the three IVIM parameters. The parameter values considered were taken from reasonable ranges observed in breast imaging⁴ (Figs. 1-2) and liver imaging⁵ (data not shown). Simulated data were generated at 8 b-values (0, 25, 50, 75, 100, 250, 500, 900 s/mm²) at 3 different Gaussian noise levels (SNR = 80, 50, 20, w.r.t. b₀ image). Pixel-wise least-square fitting was performed using the Matlab^® function lsqcurvefit (trust-region-reflective algorithm).

The Bayesian approach of Orton et al.² was applied to each case. In this approach, successive random perturbations to the parameter estimates are either accepted or rejected based on a probabilistic criterion. This criterion is weighted by both the likelihood of the parameter estimates fitting the data and by an assumed prior distribution for the parameters. Orton et al.² use a multivariate Gaussian prior, which is also updated every iteration by resampling the ROI mean and covariance. Monte Carlo calculations over 20,000 iterations yield the IVIM parameter estimates (mean) and their uncertainties (stdev). In essence, this Bayesian approach results in those estimates with high uncertainty being “shrunk” towards the mean of the ROI. Uncertainties for the least-squares results can be estimated using the same algorithm with a uniform prior.

Fig. 1 displays the parameter maps for the discrete breast tissue example (SNR = 80), obtained using the least-squares and Bayesian approaches and compared to the true values. The grayscale limits were set 1.5 times further from the mean than the max/min values of the tissues within the 8 sub-regions, and red/blue pixels represent estimates outside of these limits. Clearly the Bayesian maps display a much lower level of erroneous heterogeneity compared to the least-squares maps. Note that by definition the least-squares results provide a better fit to the noisy data, and in this case the median relative difference in root-mean-square error between the methods is 11.7% in favour of least-squares. However, the median/mean relative errors between the estimations of each method and the true values are significantly lower for the Bayesian approach (F:15%/20%, D:3%/4%, D*:24%/29%) than for the least-squares approach (F:23%/60%, D:4%/8%, D*:32%/211%). Histograms (Fig. 3) and scatter plots of parameter uncertainties (Fig. 4) give further support to the Bayesian approach. Note that for SNR = 50, the D* maps become heavily corrupted, and for SNR = 20 only the D maps resemble the true values (not shown).

Curiously, two sub-regions in the Bayesian parameter maps for F and D* in Fig. 1 have seemingly disappeared. This is likely due to a large level of uncertainty associated with these parameter combinations, resulting in the Bayesian approach shrinking these estimates to the mean. This feature is also observed in the parameter maps obtained for the Gaussian-like breast tissue example (SNR = 80), displayed in Fig. 2, which rules out the possible explanation that the discrete tissue in Fig. 1 violates the assumption of the Gaussian prior. The results for the more perfuse liver tissue examples (not shown) are much more robust to noise but display a similar disappearance of two sub-regions from the D* maps at SNR ≤ 50.

Bayesian estimation using a Gaussian prior can produce clearer IVIM parameter maps with considerably lower estimation uncertainty and generally better agreement with true IVIM values compared to least-squares fitting. However, it is possible for real features to remain undetected, which demands caution when implementing this technique.