Muhammad Usman^{1}, Lebina Kakkar^{2}, Karin Shmueli^{3}, Simon Arridge^{1}, and David Atkinson^{2}

Prostate diffusion EPI scans suffer from geometric distortions, signal pile-up and signal drop-out due to differences in susceptibility values at interface between prostate and rectal-air. In this work, an integrated model based framework is proposed that can correct for signal pile-up in regions of severe distortions and can compensate for any translational offsets that may exist between different scans. In-vivo validation of the proposed method is done in patients.

Introduction

Prostate diffusion EPI scans suffer from geometric distortions, signal pile-up and signal drop-out due to differences in susceptibility values at interface between prostate and rectal-air. Techniques have been proposed that can correct for geometric distortions using an additional B0 map that measures frequency offset from resonance at different spatial locations. In regions of severe signal pile-up, these techniques are prone to failure in recovering the underlying true signal. Furthermore, due to drifts and inaccuracies in the measurement of the center frequency [1], the reconstructed images for the EPI scans can be shifted in the phase encoding direction. These shifts are not present for the B0 scan that has a higher bandwidth, thus making use of a B0 map difficult. In this work, using a set of EPI data acquired in blip-up and blip-down phase encoding directions, an integrated model based framework is proposed that can correct for signal pile-up in regions of severe distortions and compensate for any translational offsets between different scans. Validation of proposed method is done in two patients.

^{}In presence of translational offset along the phase encoding
direction, the distorted k-space **Y** can be related to ideal image **f** via
following model [2]:

$$\mathbf{Y}(k,l)=\sum_{n=0}^{N-1}\sum_{m=0}^{M-1} \mathbf{f}(m,n)e^{-i2\pi(\frac{mk}{M}+\frac{(n-Δn_0)l}{N})}e^{-i2\pi(\triangle \mathbf{B_0} (m,n)t(k,l))} $$

$$\mathbf{Y}=\mathbf{Ef}\space\space\space Eq(1)$$

where **E** is
the encoding operator that contains translational offset and off-resonance
related effects,* m, n* are image
indices and *k, l* are k-space indices,* t(k,l)* is the sample time for location
*(k,l)* in k-space, **ΔB _{0}** is
the B

The proposed method has following two steps:

1) Estimating
the translational offset:
The translational offset Δn_{0}
is estimated as the parameter that maximizes the mutual information (MI) similarity measure between
preliminary model based reconstructions for blip-up and blip-down EPI data by
solving the following unconstrained problem:

$$ \widehat{Δn_0}=arg_{Δn_0}max(MI(\mathbf{E_{1}}^{H}\mathbf{Y_{1}}, \mathbf{E_{2}}^{H}\mathbf{Y_{2}}))\space\space\space Eq(2)$$

where **E _{1}** and

2) Model based Reconstruction: For each b-value, the EPI data acquired in blip-up and blip-down scans can be combined into a single formulation by setting **Y=[Y _{1} Y_{2}]^{T}** and

For patient 1, the
values of optimal translational offset (in pixels) as a function of slice number is
shown in Fig.2a together with convergence plot as function of iteration number in Fig. 2b. The median optimal translational offset was in the range of 3-4
pixels across all the slices. In Fig. 3 and 4, the model based reconstructions from both blip-up and
blip-down data are shown for b-values of 0 and 1000 s/mm^{2}, respectively. Results for patient 2 are summarized in Fig. 5. In the
region around the prostate rectal-air interface, the variation in B_{0} was around 120
Hz that corresponds to shift of 7-8 pixels at bandwidth/pixel of ~15.80 Hz
in phase encoding direction. This resulted in pile-up in regions where B_{0} gradient
magnitude was high and its direction was opposite to that of phase encoding gradient
(see regions pointed out by red arrows in Fig. 4 and Fig.5). The proposed
method corrected for translational offset as well as pile-up artefacts that
cannot be corrected using only one-directional phase encoding. In future works, we will investigate the effects due to phase changes that may occur between diffusion weighted blip-up and blip-down EPI scans.

[1] Andersson et al, NeuroImag 2003

[2] Munger et al, TMI 2000

[3] Stoer et al, Num Math, 1990

[4] Pruessman et al, MRM 2001