Impact of prior distributions and central tendency measures on Bayesian intravoxel incoherent motion model fitting

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Intravoxel incoherent motion (IVIM) imaging has gained increased interest during the last years because of its applicability in abdominal imaging 1. It enables both diffusion and perfusion information to be extracted from MR images completely noninvasively. Commonly, IVIM is described by a biexponential model as follows: JOURNAL/mrim/04.02/01445475-201803000-00047/math_47MM1/v/2018-01-24T161827Z/r/image-png where Si is the measured signal using the ith b‐value, D is the diffusion coefficient, D* is the pseudo diffusion coefficient, f is the perfusion fraction, S0 is the signal intensity without diffusion weighting, and εi is a random deviation from the model following some appropriate distribution, often assumed to be Gaussian 2. Because the signal attenuation caused by perfusion effects is separated from attenuation caused by diffusion in the model, the diffusion coefficient obtained from the IVIM model fitting is less dependent on the choice of b‐values as long as the effect of the dispersion of the diffusion coefficient at high b‐values is avoided 4. However, biexponential model fitting is significantly more difficult than monoexponential fitting, as the former cannot be transformed into a linear model, and because it may have multiple, local optima. Furthermore, in the limit as the perfusion fraction approaches zero, the biexponential model transforms into a monoexponential model and D* cannot be estimated, which may lead to unstable fitting results.
The difficulties concerning the fitting of the IVIM model has resulted in several studies comparing model‐fitting strategies 6. Most of these studies compare different least‐squares methods, but a Bayesian approach was shown early on to be a robust alternative 12. This was also recently reported by Barbieri et al in a comparison among most of the IVIM model fitting strategies found in the literature 6, although the Bayesian approaches used in these studies were slightly different.
The aim of a Bayesian IVIM model fit is to estimate the joint posterior distribution from which the marginal posterior distributions of the IVIM model parameters of interest (i.e., P(D|S), P(D*|S), and P(f|S)) can be derived. The joint posterior distribution is given by Bayes' rule as follows: JOURNAL/mrim/04.02/01445475-201803000-00047/math_47MM2/v/2018-01-24T161827Z/r/image-png where S is the measured data, P(S|D,D*,f,S0) is the likelihood function, and P(D,D*,f,S0) is the joint prior distribution. Apart from choosing an appropriate likelihood function, a prior distribution has to be chosen as well. Most previous studies involving Bayesian IVIM model fitting have used various noninformative or low‐informative priors 6, although some more advanced approaches have been proposed 13. To make the result of a Bayesian fit more comprehensible, the marginalized probability distributions need to be summarized, most importantly in terms of central tendency to describe the center or location of the distribution, but possibly also in terms of width and skewness. In previous studies the mode 6 or mean 13 have been used to describe the central tendency. Estimates of the mode and mean of the marginal posterior distributions are also commonly referred to as the marginal maximum a posteriori estimates and the minimum mean square error estimates, respectively.
Numerous studies concerning IVIM model fitting have been performed, and the Bayesian approach has shown great promise. However, none of the previous studies has, to our knowledge, aimed to assess the impact of the methodology used in the Bayesian model fitting, most importantly including the choices of prior distribution and central tendency measure, which often differ among studies 6. These choices may affect parameter estimation performance, especially when noise limits the information available to the model fitting. Potential estimation bias as a result of these choices must be studied and taken into account when comparing the results of studies with different Bayesian model‐fitting approaches.

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