Ziwei Zhao^{1,2}, Kai Wang^{1}, Danny JJ Wang^{1}, and Lirong Yan^{1}

Quantification of hemodynamics benefits clinical diagnosis. Non-contrast enhanced MRA with golden-angle radial acquisition has capability of characterization of dynamic flow with high spatiotemporal resolution within a short scan time. Here, we proposed a direct reconstruction framework of arterial blood flow (aBF) from undersampled radial dMRA K-t space data, which mitigated streaking artifacts induced by image-based reconstruction. Both simulation and experimental data suggested that direct optimization method provides reliable aBF under different undersampling rates while preserving detailed delineation of vascular structures, compared to the conventional post-processing singular value decomposition (SVD) method.

Introduction

Evaluation of flow dynamics with quantitative hemodynamics
provides important information in the diagnosis of cerebrovascular disease.
Recently, a rapid non-contrast enhanced dynamic 4D MR angiography (NCE-dMRA)
using golden-angle radial sampling has shown promise to depict dynamic flow
information with good image quality.^{1, 2} The modern techniques
for arterial blood flow(aBF) quantification mainly rely on post-processing
methods after image reconstruction, e.g. truncated singular value decomposition
(t-SVD). However, the quantification accuracy is highly affected by the
deconvolution procedure accompanied by potential noise and artifacts introduced
during reconstruction. The purpose of this study is to develop and evaluate a
direct reconstruction method by which aBF map is directly estimated from
under-sampled golden-angle radial NCE-dMRA K-t space data.

The forward flowchart of aBF reconstruction using under-sampled K-t space data can be illustrated by three steps(Figure 1). Specifically, the vectors $$$\mathbf{r} \in (x,y,z)$$$ and $$$\mathbf{k} \in ( k_x, k_y, k_z)$$$ represent image domain spatial coordinates and K-space frequency domain coordinates, respectively. $$$\it t$$$, $$$\it c$$$ are time and coil dimensions.

Step1, the time series of
subtracted blood signals $$$\Delta M(\mathbf{r},t_j)$$$ can be generated
by the convolution of arterial input function (AIF) and residual function (R)
using a general kinetic model ^{3 }(Eq.[1]): $$\Delta M(\mathbf{r},t_j) =
\Delta TI \cdot aBF(\mathbf{r})\cdot\sum_{i=0}^j AIF(t_j)\cdot
R(\mathbf{r},t_{j-i}) \tag{1}$$ where
$$$aBF(\mathbf{r})$$$ is arterial blood flow of each voxel, and $$$\Delta
TI$$$ is the time interval between phases. By setting $$$R'(t) = aBF
\cdot R(t)$$$, the formula can be rewritten as (Eq.[2]): $$\Delta
M(\mathbf{r},t_j) = \Delta TI \cdot\sum_{i=0}^j AIF(t_j)\cdot
R'(\mathbf{r},t_{j-i}) \tag{2}$$ Step2,
dynamic labeled images are calculated by subtracting from fully-sampled control
image (Eq.[3]): $$M_{label}(\mathbf{r},t) = M_{control}(\mathbf{r}) - \Delta
M(\mathbf{r},t) \tag{3}$$ Step3, the under-sampled raw (k-t) data is obtained
from labeled images $$$M_{label}(\mathbf{r},t)$$$ combining with coil
sensitivities $$$C(\mathbf{r},c)$$$ and under-sampling Fourier
operator $$$F_u$$$ (Eq.[4]): $$S(\mathbf{k},t,c) =
F_uC(\mathbf{r},c)M_{label}(\mathbf{r},t) \tag{4}$$ Combining the 3 steps
above, the overall transformation function, denoted by $$$f$$$, shows a general
relationship between $$$aBF(\mathbf{r})$$$ and under-sampled K-t space
data $$$S(\mathbf{r},t,c)$$$ (Eq.[5]):
$$S(\mathbf{r},t,c)=f(aBF(\mathbf{r}),R(\mathbf{r},t);AIF(t),\Delta TI,M_{control}(\mathbf{r}),C(\mathbf{r},c))
\tag{5}$$ Instead of solving $$$aBF(\mathbf{r})$$$, we solve residue function
$$$R'(\mathbf{r},t)$$$ by least-square optimization using compressed
sensing method with additional sparsity constrains, after which $$$aBF(\mathbf{r})$$$ parameters
can be achieved by taking the maximum value of $$$R'(\mathbf{r},t)$$$.^{4} The optimization problem with
additional regularization parameters is formulated as follows (Eq.[6]):
$$\underset{R'(\mathbf{r},t)}{\mathrm{argmin}} \| S’(\mathbf{k},t,c) - F_u
C(\mathbf{k},c)(M_{control}(\mathbf{r})-\Delta TI \cdot AIF(t) \ast
R'(\mathbf{r},t))\|^2_2 +\lambda _1\|\psi R'(\mathbf{r},t)\|_1+\lambda
_2\|\psi' R'(\mathbf{r},t)\|_1 \tag{6}$$ where $$$S’(\mathbf{k},t,c)$$$ is
raw (k-t)-space data of labeled images. For sparsity terms, we use spatial
$$$(\psi)$$$ and temporal $$$(\psi ')$$$ total variation(TV) terms with
corresponding regularization parameters $$$\lambda _1$$$ and $$$\lambda
_2$$$. This nonlinear optimization problem is solved by Nonlinear Conjugate
Gradient method.

*
Simulation
*

Radial NCE-dMRA data was
simulated with known aBF on a virtual phantom generated from TOF images to
evaluate the robustness of aBF quantification under different under-sampling
rates and noise levels. A global AIF was simulated by a gamma-variate function.^{5} Residual
function of each voxel was an exponential decay by incorporating blood arrival
time $$$\tau _r$$$. aBF values were calculated by directed reconstruction with
under-sampling rate R = 10, 20, 40, 80 at SNR = 5, 10, 15, 20 and 25,
respectively.

*
In-vivo MRI scan
*

All experiments were performed
on a Siemens Prisma 3T scanner with a 20-channel head coil. Golden-angle radial
NCE-dMRA data were collected on two healthy volunteers (FOV=256x256mm^{2}, voxel size=1x1x1.5mm^{3}, 500 views per slice). No pre-saturation
was applied before ASL inversion pulse.

Figure 2 shows simulated aBF reconstruction and corresponding error maps (RMSE = 0.4730, 0.7431, 1.2019 and 1.7839) with different under-sampling rates (R = 10, 20, 40, 80). Directed reconstruction method preserved reliable estimation accuracy even under large under-sampling rates up to 80, while maintaining an average low noise level.

Figure 3 shows simulated average aBF with different under-sampling rates and different noise levels. Underestimated aBF is noticed when the under-sampling rate increases due to the energy spreading out. Nevertheless, the aBF quantification is more tolerant to different noise levels.

Figure 4 shows aBF maps from two subjects with direct reconstruction compared with the indirect t-SVD method(threshold: 1%, 10%, 25%). Significant intensity variation of aBF with different thresholds was observed in SVD results. However, the evaluation from proposed method is more reliable and robust, overcoming the uncertainty and noise introduced by the indirect method.

Figure 5 displays the reconstructed aBF maps from another subject along three orthogonal directions. One can appreciate a good vessel delineation with quantitative flow information.

[1]. Song HK, Yan L, Smith RX, et al. Noncontrast enhanced four‐dimensional dynamic MRA with golden angle radial acquisition and k‐space weighted image contrast (KWIC) reconstruction. Magn Reson Med. 2014; 72(6): 1541-1551.

[2]. Zhou Z, Han F, Yu S, et al. Accelerated noncontrast‐enhanced 4‐dimensional intracranial MR angiography using golden‐angle stack‐of‐stars trajectory and compressed sensing with magnitude subtraction. Magn Reson Med. 2018; 79(2): 867-878.

[3]. Buxton RB, Frank LR, Wong EC, et al. A general kinetic model for quantitative perfusion imaging with arterial spin labeling. Magn Reson Med 2005; 40(3): 383-396.

[4]. Petersen ET, Lim T, Golay X. Model‐free arterial spin labeling quantification approach for perfusion MRI. Magn Reson Med 2005; 55(2): 219-232.

[5]. Calamante F, Gadian DG, Connelly A. Delay and dispersion effects in dynamic susceptibility contrast MRI: simulations using singular value decomposition. Magn Reson Med 2000; 44(3): 466-473.

Figure 1 shows a forward model of directed estimation for aBF, illustrating the conversion from aBF map to under-sampled (K-t)
space. A general kinetic model is used to convert aBF map to
magnetization
differences, which is subtracted from fully-sampled control images generating dynamic anatomic labeled
images. With Fourier transform, sensitivity maps, and radial sampling pattern,
the labeled images are transformed to multi-coil K-t space measurements.

Figure 2 shows simulated aBF reconstruction results
with
different under-sampling rates (R = 10, 20, 40, 80) and corresponding
error maps, with reference
to a fully-sampled and noise-free aBF map. The root mean square error (RMSE) values are 0.4730, 0.7431, 1.2019 and 1.7839, respectively. It shows that the direct reconstruction
method preserves reliable
estimation accuracy even under high under-sampling rate up to 80, while maintaining
an
average low noise level.

Figure 3 shows simulated average aBF value with different under-sampling rates(10x, 20x, 40x, 80x) and different noise levels (SNR = 5, 10, 15, 20, 25). Dashed
line indicates the reference aBF value (53.16 ml/100 ml/min) of fully sampled noise-free aBF map. The value of aBF is the average of whole cerebral vessels. Although the aBF value slightly changes with different under-sampling rates due to the energy spreading out, it shows that the average aBF is more tolerant to different noise levels.

Figure 4 shows aBF maps from one healthy subject
with the t-SVD (a-c) and proposed
(d)
methods, and
maximal-intensity projection (MIP) of 3D aBF volumes was performed towards
axial, sagittal
and coronal
planes. From SVD results, small vessels in posterior cerebral arteries (PCA) being merged into background signals in dynamic MRA images was recovered in aBF result by direct reconstruction method. Significant
variation in image intensities was observed in t-SVD method at different
thresholds of singular values, whereas results from proposed method are more
reliable and robust, preserving detailed patterns as well as lower level
backgrounds noise.

Figure 5 shows three-plane views of five-time
frames of reconstructed NCE-dMRA images and corresponding aBF maps reconstructed from K-t space data.