On the feasibility of data-driven estimation of Markov random field parameters for IVIM modelling of abdominal DW-MRI: insights into which parameters can be reliably estimated from clinical data

Matthew R Orton^{1}, Neil P Jerome^{1}, Mihaela Rata^{1}, David J Collins^{1}, Khurum Khan^{2}, Nina Tunariu^{3}, David Cunningham^{2}, Thorsten Feiweier^{4}, Dow-Mu Koh^{3}, and Martin O Leach^{1}

To explore the feasibility of the proposed MRF model and MCMC algorithm, and to show that the convergence properties of the algorithm shed light on which parameters of the IVIM model can be reliably estimated from data similar to that obtained in a clinical setting.

**MRF model:** The data likelihood assumes Gaussian errors between
the log-data and the log-transformed IVIM
model:$$$\small{}\;\log{}S_i=\log{}S_0+\log\left(\,f\mathrm{e}^{-b_iD^*}+(1-f)\mathrm{e}^{-b_iD}\right)+\eta_i$$$,
where $$$\small{}f$$$ is the pseudo-diffusion volume
fraction,$$$\,\small{}D^*$$$ and$$$\,\small{}D$$$ are fast and slow diffusion coefficients
and $$$\small{}\mathrm{var}(\eta_i)=\sigma_S^2$$$ over the whole ROI/image. The smoothness prior is defined by a
product of nearest-neighbour clique terms, each of which is modeled with a Laplace
distribution$$$\,\small{}p\!\left(x_i,\,x_j\right)=\frac{1}{2}w_x\exp\left(-w_x\left|x_i-x_j\right|\right)$$$,
and the scale term $$$\small{}w_x$$$ is the smoothing weight for $$$\small{}x\in\{f,\,D,\,D^*\}$$$. This distribution is the probabilistic
equivalent to the robust L_{1}-norm [4] which has desirable edge-preserving
properties. The posterior distribution
is the product of the data likelihood and the smoothness priors
for$$$\,\small{}f,\,D\,$$$and$$$\,\small{}D^*$$$.
The key innovation introduced here is to estimate the smoothing weights
in addition to the voxel IVIM parameters using an MCMC algorithm.

**Data and evaluation: **Abdominal DWI data were acquired coronally in seven patients with liver tumours on a 1.5T MAGNETOM Avanto (Siemens Healthcare, Erlangen, Germany) using
a prototype single-shot EPI sequence and 20$$$\!\small{}\times$$$5mm$$$\;$$$slices, TR/TE=5000/60ms, matrix=128^{2},
FOV=400mm, 5$$$\;$$$NSA, 3$$$\;$$$orthogonal directions,
phase-partial-Fourier$$$\;$$$7/8, GRAPPA$$$\;$$$2,
b-values=0,20,40,60,120,240,480,900s/mm^{2}.
Individually acquired images were registered [6], and averaged per
b-value. A single slice was
selected covering the liver, and including the kidneys where visible, and a
rectangular portion of the images was cropped to remove the background, see
figure 1. Initial values
were obtained using least-squares, and the MRF/MCMC algorithm used to generate 10,000 posterior samples.

For the Bayesian model given above, estimation of the smoothing weights is essentially driven by a balance between spatial heterogeneity and parameter uncertainty. For these data, the convergence of$$$\small{}\,w_D\,$$$and$$$\small{}\,w_f$$$ in almost all patients indicates that the uncertainty (given the image noise and b-value support) in$$$\small{}\,D\,$$$and$$$\small{}\,f$$$ is sufficiently low that the degree of spatial heterogeity can be determined from the data. However, the divergence of$$$\small{}\,w_{D^*}$$$ leads to very smooth $$$\small{}D^*$$$ maps, which is equivalent to a single value of$$$\small{}\,D^*$$$ for the whole image. This is consistent with the known difficulty in estimating$$$\small{}\,D^*$$$, and is driven by poor data support for the pseudo-diffusion coefficient of the IVIM model. Despite this finding, it is clear that a single value for$$$\small{}\,D^*$$$ is not appropriate when modelling complex anatomy, which motivates the use of a fixed value for$$$\small{}\,w_{D^*}$$$. To account for image acquisition differences (in particular the voxel dimensions), it would be preferable to estimate the smoothing weights independently for every patient, but where this is impractical, weights can be estimated from a subset of patients and the average used for subsequent patients.

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Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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