Xiaopeng Zong^{1} and Weili Lin^{1}

Pathological changes of penetrating arteries (PAs) may be an important contributing factor of cerebral small vessel disease (SVD). Measurement of PA flow velocity and diameter with phase contrast (PC) MRI remains challenging due to the presence of strong partial volume effects. Here we propose model-based analysis of complex difference (MBAC) images to quantify diameter and velocity of PAs. We demonstrated the accuracy of the MBAC method with simulation and phantom studies. In vivo PA diameter and velocity were obtained for the first time. The MBAC method may serve as a useful tool for understanding the etiopathogenesis of SVD.

Assuming vessel perpendicular to the imaging slice, the image from PC-MRI is given by:

I(x,y)=[S_{t}+S_{0}·e(*v*)·exp(i*πv*/VENC)]*sinc(k_{x,max}x)*sinc(k_{y,max}y),
(1)

where * denotes convolution, k_{x,max}
and k_{y,max} are the largest
measured k-space coordinates, S_{t} and S_{0} are the
surrounding tissue signal and blood signal without flow effect, respectively, e(*v*) is flow-induce signal enhancement factor. The velocity spatial distribution is expected
to follow a laminar pattern in PAs.^{3} S_{0} can be inferred from the signal of nearby WM
after correcting for T_{2}^{*}, T_{1}, and water
content differences. In order to
apply MBAC, two images are acquired with the velocity encoding gradient on and
off (VENC=∞),
respectively. Model CD and phase contrast images are calculated based
on Eq. (1), where e(*v*) can be obtained via Bloch simulation. The model images are fitted to measured CD and PC images to extract mean
velocity (*v*_{mean}) and diameter.

MBAC were performed on simulated, phantom, and *in vivo* images to systematically
evaluate its performances. Simulated
images were obtain by adding Gaussian white noise
to images calculated from equation (1).
The noise level and model parameters matched with in vivo condition. Phantom and in vivo images were acquired on a Siemens
7T human MRI scanner with a single slice PC MRI sequence. The phantom consisted
of a water-filled polyimide tube with ID/OD = 0.165/0.305 mm penetrating a water-filled
plastic cylinder. The flow rate was
controlled by a syringe pump to achieve desired *v*_{mean}. In vivo
images were acquired for quantifying PA diameter and velocity in centrum
semiovale in 6 healthy volunteers aged 21–41, with approval by the IRB.
The sequence
parameters were: VENC=4 cm/s; resolution=0.3125×0.3125×2 mm^{3}; TE/TR=15.7/30
ms; NA=15; FA=45^{o}. Images were
interpolated to 0.1563×0.1563 mm^{2} voxel size by zero filling in k-space. The PA orientations were calculated from that
of overlapping PVS visualized with a 3D T_{2}-weighted sequence.^{4}

**RESULTS**

Figure
1 compares noise-related errors of fitted vmean
and diameter for MBAC and MBAP at different *v*_{mean}
and partial volume fraction (PVF). PVF
is calculated relative to the volume of a single voxel. The standard deviation (SD) increases with
decreasing *v*_{mean} and PVF. The SDs for MBAC are generally much smaller
than MBAP at same *v*_{mean}
and PVF.

The
differences between mean fitted and true *v*_{mean}
and D results are shown in Fig. 2(A), showing <10% error when true vmean ≥ 1 cm/s and increased
error at lower velocity of 0.8 cm/s. The
SDs of fitted *v*_{mean} and D
increases monotonically with decreasing velocity, as shown in Fig. 2(B). The tilt of vessel from being perpendicular
has a strong effect on the accuracy, as shown in Fig. 3. The errors increase linearly with tilt angle
until reaching a plateau between 2.5^{o}-5.2^{o}.

Totally 234 PAs were visualized *in vivo* in
the 6 subjects, among which 17 were approximately perpendicular to the slice (tilt<5°). An example of the visualized PAs and fitted
model images are shown in Fig. 4. The
mean (SD) *v*_{mean} and D of the 17 PAs are 1.08 (0.39) cm/s, and 0.17 (0.05) mm,
respectively.

1. Fisher CM. The arterial lesions underlying lacunes. Acta neuropathologica. 1968;12:1-15

2. Hoogeveen RM, Bakker CJ, Viergever MA. Mr phase-contrast flow measurement with limited spatial resolution in small vessels: Value of model-based image analysis. Magn Reson Med. 1999;41:520-528

3. Avila K, Moxey D, de Lozar A, Avila M, Barkley D, Hof B. The onset of turbulence in pipe flow. Science. 2011;333:192-196

4. Zong X, Park SH, Shen D, Lin W. Visualization of perivascular spaces in the human brain at 7t: Sequence optimization and morphology characterization. Neuroimage. 2016;125:895-902

5. Pesce C, Carli F. Allometry of the perivascular spaces of the putamen in aging. Acta neuropathologica. 1988;76:292-294

Figure 1. Color maps of the standard
deviations of fitted *v*_{mean} (A)
and
D (B)
as a
function of true vmean and partial volume fraction. The left and
right panel corresponds to MBAC and MBAP results, respectively. A log scale was used in the color maps to
accommodate the large range of values.
The parameter a in the color bar is equal to 5 cm/s and
2 mm in
(A) and (B), respectively. Overall, the
MBAC method results in smaller standard deviations and systematic errors than
MBAP.

Figure 2. (A) Percent differences between
fitted and true *v*_{mean} and
diameter (*D*) at
different true *v*_{mean}, obtained
with MBAC. (B)
Standard deviations of fitted *v*_{mean} and *D* over repeated scans under the same conditions. The SDs were normalized by corresponding
true values.

Figure
3. Percent differences between fitted and true *v*_{mean} and
diameter (*D*) at
different angles between the vessel and the normal direction of the slice.

Figure 4: (A) shows representative
magnitude image acquired with PC-MRI in CSO.
(B) and (C) are magnified magnitude and phase images, respectively, of
the region enclosed by the rectangle in (A).
Red circles enclose PAs. (D): An
example of the imaginary and real parts of the complex difference images
(left) and corresponding best-fit
MBAC images
(right).