Tobias J Hoh^{1}, Jonas Walheim^{1}, Mareike Gastl^{1,2}, Alexander Gotschy^{1,2}, and Sebastian Kozerke^{1}

The inflow of a paramagnetic contrast agent (CA) in cardiac dynamic contrast-enhanced (DCE) MRI effects the local phase of magnetization. In this work the impact of phase variations on Partial Fourier (PF) reconstruction is simulated for k-space zero filling, homodyne (HR) and projections onto convex sets (POCS) reconstruction and consequently assessed in in-vivo first-pass perfusion. CA induced phase variations in DCE MRI are seen to compromise HR and POCS reconstruction of PF data to an extent where they do not convey any benefit over simple zero-filling reconstruction.

Partial
Echo (PE) and Partial Fourier (PF) acquisitions schemes^{1} are part of
standard clinical protocols for the assessment of myocardial first-pass perfusion
(FPP) as they shorten echo time (TE) and shot duration of the readout,
respectively^{2}. For both, PE and PF reconstruction, Hermitian symmetry
of k-space is assumed or approximated^{3-5}. In dynamic contrast-enhanced
(DCE) MRI, significant phase variations are induced by the inflow of a
paramagnetic contrast agent (CA)^{6}, violating the Hermitian symmetry
and impairing reconstruction. The
objective of the present work is to assess phase variations in cardiac DCE MRI
and their impact on PE/PF homodyne reconstruction (HR) and the method of
projections onto convex sets (POCS) based on simulated and in-vivo first-pass
myocardial perfusion data.

Since Hermitian symmetry in k-space i.e. $$$\tt \it S\tt \left(\it k\right)=\left[\it S\tt \left(\it k\right)\right]^{*}$$$ is not guaranteed, a low-resolution phase image estimate is extracted from a symmetrically acquired k-space portion centered around DC to correct for phase variations.

The low-resolution object phase $$$\phi_{est}$$$ is subsequently used according to:

$$\it F\left\{s\left(x\right)e^{-i\phi_{est}\left(x\right)}\right\}\left(\it k\right) =\left[\it F\left\{s\left(x\right)e^{-i\phi_{est}\left(x\right)}\right\}\left(\it -k\right)\right]^{*}{}\left(1\right)$$

Real-valued images can be recovered using homodyne reconstruction (HR) according to:

$$ I(x)=\int_{-k_\text{max}}^{k_0}\text{d}k\,\tilde{S}(k)e^{\text{i}kx}+\int_{-k_0}^{k_0}\text{d}k\,\tilde{S}(k)e^{\text{i}kx}+\int_{k_0}^{k_\text{max}}\text{d}k\,\tilde{S}(k)e^{\text{i}kx}=\int_{-k_0}^{k_0}\text{d}k\,\tilde{S}(k)e^{\text{i}kx}+2\text{Re}\left[\int_{k_0}^{k_\text{max}}\text{d}k\,\tilde{S}(k)e^{\text{i}kx}\right]{}\left(2\right)$$

Alternatively, POCS^{4} has been proposed, calculating
projections onto the set $$$\Omega_1$$$ of the images, which agree with the estimated phase $$$\phi_{est}$$$, and projections
onto set $$$\Omega_2$$$ of data, which matches the
acquired k-space data. In the extended, parallel POCS^{7} a weighting $$$\mu$$$ of
these projections is used:

$$\Omega_1=\left\{I(x):\angle{}I(x)=\phi_\text{est}(x)\right\}$$

$$\Omega_2=\left\{I(x):\mathcal{F}\left\{I(x)\right\}(k)=S(k);k\in[k_0;k_\text{max}]\right\}{}\left(3\right)$$

$$x_{n+1}=1-\mu\cdot{}P_{\Omega_1}(x_n)+\mu\cdot{}P_{\Omega_2}(x_n)$$

In DCE MRI, the phase shift induced by influx of CA can be estimated according to

$$\Delta\varphi=\gamma\frac{\Delta{}\overrightarrow{B}\cdot{}\overrightarrow{H_0}}{\left\|\overrightarrow{H_0}\right\|}T_E=4\pi\omega_0X_MF\cdot{}C\cdot{}T_E{}\left(4\right)$$

where $$$\Delta{}\overrightarrow{B}$$$ denotes change in the magnetic induction relative to the static external field $$$\overrightarrow{H_0}$$$ expressed as product of
frequency $$$\omega_0$$$, the molar susceptibility $$$X_M$$$, geometry factor F and the
concentration C of the CA^{6,8}, yielding a linear relationship between $$$\Delta\varphi$$$
and CA
concentration.

The induced phase variation during CA application
was assessed by in-vivo FPP measurements in two healthy volunteers on a Philips
Ingenia 3.0T scanner (Philips Healthcare, Best, The Netherlands) using a
32-channel cardiac coil array. Three measurements were obtained in each subject
separated by 15min with decreasing concentrations of 0.075mmol/kg, 0.05mmol/kg
and 0.025mmol/kg of Gadobutrol (Gadovist, Bayer Schering Pharma, Germany). A
saturation-recovery spoiled gradient-echo sequence with ECG triggering was used
to acquire a fully sampled mid-ventricular slice at mid diastole in a single
breath hold. Imaging parameters were as follows: TE/TR =1.67ms/2.8ms, FOV: 300x290x10mm^{3},
flip angle: 15° and in-plane resolution: 2.5x2.5x10mm^{3}.

A computer model of the left-ventricular short axis was implemented (Figure 2a.) with different phase offsets (0-1.5rad) between blood-pool and myocardium. The complex-valued model was scaled according to in-vivo measured magnitude and phase values at time point of peak blood-pool and myocardium signal.

PF sampling was retrospectly applied to model and in-vivo data for different PF factors (0.625 and 0.75) and the reconstruction compared to the fully sampled reference for HR, POCS, symmetrical (ZF(sym)) and asymmetrical (ZF(asym)) zero-filling reconstructions. The normalized root mean square errors (nRMSE) is reported.

An approximately linear dependency between the CA concentration and the net phase shift in the blood pool was measured in-vivo (Figure 1).

Using the in-vivo phase data as input in the computer model, the performance of ZF(asym), ZF(sym), HR and POCS reconstruction is compared (Figure 2). With increasing phase offsets, HR and POCS lead to an increase in nRMSE. Errors of simulated PF (0.75) reconstructions at peak blood-pool (phase step of 0.81rad) and myocardial enhancement (phase step of 0.67rad) relative to ground truth are shown in Figure 3. Errors of PF reconstructions of in-vivo FPP data relative to the fully sampled reference are summarized in Figure 4. In Figure 5 the corresponding spatial error maps at peak blood-pool and peak myocardial enhancement are shown.

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Figure 1: Dependency of net phase shift on the
administered contrast agent concentration. The location of the region of interests (ROI) in the
bloodpool beneath the papillary muscle (blue) and myocardium (grey) is illustrated in a). The signal magnitude inside the
bloodpool and myocardium are shown as mean ± standard error of the mean across the ROI ( b)
and e) respectively) along with the corresponding signal phase: c) and f) as
function of number of time frame. Phase shifts in myocardium and
bloodpool is presented as function of gadolinium concentration (d). Bars
represent maximal phase change over time.

Figure 2: Performance assessment of homodyne reconstruction (HR),
projections onto convex sets (POCS) and zero-filling reconstruction. The circular model is shown in a) along an
example magnitude profile through the center of the ring. The normalized root
mean squared error (nRMSE) of the different reconstruction methods relative to ground truth is plotted as
function of phase step present at the endocardial border (between black and
grey area) for partial Fourier factors 0.625 b) and 0.75 c).

Figure 3: Errors of simulated partial Fourier (PF=0.75) reconstructions
at maximal blood-pool a) and myocardial b) signal enhancement relative to
ground truth. The two rows depict the signal magnitude for asymmetric (ZFasym)
and symmetric (ZFsym) k-space zero filling, homodyne (HR) and projections onto
convex sets (POCS) reconstruction. The
bottom rows represent a tenfold scaled absolute difference with respect to
ground truth. The respective signal magnitude along a center profile are shown
a’) and b’).

Figure
4: Summary of errors for asymmetric
(ZFasym) and symmetric (ZFsym) k-space zero filling, homodyne (HR) and projections
onto convex sets (POCS) reconstruction of in-vivo first pass perfusion data with a
gadolinium dose of 0.075 mmol/kg. a)
Shows the location of the ROI. The normalized root mean square error (nRMSE) is
shown as function of the number of time frame for a partial Fourier factor of 0.625
b) and 0.75 in c). The nRMSE is computed compared to the fully sampled original
in-vivo acquisition.

Figure 5: Errors of in-vivo partial Fourier
(PF=0.75) reconstructions at maximal blood-pool signal a) and myocardial signal
enhancement b) relative to ground truth (GT) at a Gadolinium concentration of 0.075mmol/kg. The top rows show the
reconstructed magnitude images, center row the tenfold scaled absolute
difference of the signal magnitude to ground truth and the bottom row the corresponding
absolute difference of the signal phase.