# Influence of Spatial Resolution and Compressed SENSE Acceleration Factor on Flow Quantification with 4D Flow MRI at 3 Tesla

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## Abstract

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^{3}) to 2D flow MRI and a flow sensor. Additionally, the velocity field in an aneurysm model acquired with 4D flow MRI was compared to the one simulated with computational fluid dynamics (CFD). A strong correlation was observed between flow sensor, 2D flow MRI, and 4D flow MRI (rho > 0.94). The use of fewer than seven voxels per vessel diameter (nROI) resulted in an overestimation of flow in more than 5% of flow measured with 2D flow MRI. A negative correlation (rho = −0.81) between flow error and nROI were found for CS = 2.5 and 4.5. No statistically significant impact of CS factor on differences in flow rates was observed. However, a trend of increased flow error with increased CS factor was observed. In an aneurysm model, the peak velocity and stagnation zone were detected by CFD and all 4D flow MRI variants. The velocity difference error in the aneurysm sac did not exceed 11% for CS = 4.5 in comparison to CS = 2.5 for all spatial resolutions. Therefore, CS factors from 2.5–4.5 can appear suitable to improve spatial or temporal resolution for accurate quantification of flow rate and velocity. We encourage reporting the number of voxels per vessel diameter to standardize 4D flow MRI protocols.

## 1. Introduction

^{3}) and four compressed SENSE (CS) acceleration factors (2.5, 4.5, 6.5, and 13) were considered. We investigated the matter in vitro using silicone tubes of different diameters (2, 3, 4, and 5 mm) and a patient-derived 3D-printed aneurysm model. The results were compared to flow sensor, 2D flow MRI, and CFD simulations.

## 2. Materials and Methods

#### 2.1. Flow Models and Circulation Setup

#### 2.2. Magnetic Resonance Imaging

^{3}) and CS acceleration factor were varied by the operator; all other parameters were kept the same or were set automatically by the system. TR and TE were automatically set to the shortest values allowed by the MRI system. MRI protocol parameters are summarized in Table 1.

#### 2.3. Time-Resolved Hemodynamic Simulations

^{3}, respectively. This resulted in Re ≈ 1120; therefore, laminar flow conditions were considered.

#### 2.4. Data Processing

- The linear offset phase correction was conducted on each slice individually to correct for the presence of eddy currents. The fit was calculated at the reference heart phase (at the peak flow time) and then applied to all heart phases. A phase correction to compensate for concomitant gradients (Maxwell terms) and geometry correction to compensate for inhomogeneities of the main magnetic field and non-linearity of the gradient fields was performed on MR systems as part of the standard phase-contrast MR image reconstruction.
- Velocities in voxels outside of the flow lumen were nulled based on a magnitude intensity threshold.
- The data were inspected against phase-aliasing and manually corrected if necessary.

- ROIs were created manually on MRI magnitude data. First, contours around the tube’s lumen were drawn using a b-spline curve (feature in GTflow) on 2D flow MRI. Note that the 2D flow MRI acquisition planes are already perpendicular to the flow direction. Next, the resulting 2D flow ROIs were translated to the 4D flow MRI data, ensuring identical placement of ROIs on the 2D and 4D flow datasets.
- In a given ROI, the flow of all voxels was summed up for each time point $f\left(t\right)={\sum}_{i}{f}_{i}\left(t\right)$, where i indicates the voxel and t the temporal point. The flow was spatially averaged over ROI A-C, as follows: $f\left(t\right)=\frac{1}{3}\left(f{\left(t\right)}_{ROIA}+f{\left(t\right)}_{ROIB}+f{\left(t\right)}_{ROIC}\right)$.
- The number of voxels per ROI diameter (nROI) was calculated to obtain a measurement not depending on the voxel size and vessel diameter as $nROI=\frac{ROI\text{}diameter\text{}\left[mm\right]}{1D\text{}voxel\text{}size\text{}\left[mm\right]}$.
- The time-dependent difference between flow values obtained with 4D and 2D flow MRI was calculated as $difference\left(t\right)\text{}[\%]=\frac{f{\left(t\right)}^{4Dflow}-f{\left(t\right)}^{2D\text{}flow}\text{}}{mean(flow\left(2Dflow\right)}\xb7100\%$. Similarly, the difference between flow values obtained with 4D flow MRI and US sensor was calculated.
- The normalized root-mean-square (RMS) error was used to assess the accuracy of flow quantification. RMS was calculated as the sum of squared differences between 2D and 4D flow MRI data over the time steps and normalized by the time-averaged flow acquired with 2D flow MRI: $RMS=\frac{\sqrt{{\sum}_{t}^{Num}{\left(f{\left(t\right)}^{4D\text{}flow}-f{\left(t\right)}^{2D\text{}\mathrm{flow}\text{}}\right)}^{2}}/Num}{mean\text{}flow\text{}\left(2D\text{}flow\right)}$, where t indicates the temporal measurement point and Num is the number of temporal points. Similarly, RMS between flow values obtained with 4D flow MRI and US sensor was calculated.
- The velocity magnitude for each voxel in the aneurysm 3D ROI (Figure 1d, right) was calculated and averaged over time ${v}_{i}^{mag}=\frac{{\sum}_{t}^{Num}\sqrt{{\left({v}_{i}^{x}\text{}\left(t\right)\right)}^{2}+{\left({v}_{i}^{y}\left(t\right)\right)}^{2}+{({v}_{i}^{z}\left(t\right))}^{2}}}{Num}$, where i indicates voxel, t temporal point, and Num the number of temporal points. Aneurysm velocity distributions were presented using histograms similar to Schnell et al. [38].
- Time-averaged velocity magnitude in the evaluation plane across the aneurysm was visualized pixel-wise on a color-coded representation (MATLAB R2019a, MathWork, Natick, MA, USA).
- Repeatability of flow measurements with 2D flow MRI was assessed with repeatability coefficient (RC) as follows: (1) 2D flow MRI was measured five times in parental vessel and at the aneurysm sac; (2) RC was calculated for each time point using the equation adapted from Raunig et al. [39] $RC\left(t\right)=1.96\ast \sqrt{2SD{\left(t\right)}^{2}}/{f}^{mean}\left(t\right)\ast 100\%$, where SD is the standard deviation and ${f}^{mean}$ is the mean flow rate over five measurements; (3) time-depended RC were time-averaged as $RC={\sum}_{t}^{Num}RC\left(t\right)/Num$, where t indicates the temporal measurement point and Num is the number of temporal points.

#### 2.5. Statistical Analysis

## 3. Results

#### 3.1. Flow in Silicone Tubes

^{2}> 0.96, Table 2). The linear slope was close to 1 for 0.5 and 1 mm

^{3}voxel size, regardless of the CS factor. For a voxel size of 1.5 mm

^{3}, the linear slope was close to 1 only for CS = 2.5. At higher acceleration factors (CS = 4.5–13), the flow values with 4D flow MRI were overestimated in comparison to 2D flow MRI (1.21 < slope < 1.29).

^{3}and CS = 2.5 and 4.5 for all tubes (p < 0.05). Summary statistics of flow rate values for each tube ID, voxel size, and CS factor are summarized in the supporting information (Table S3).

^{3}, CS = 6.5) to +18.31% (tube ID = 2 mm, voxel size = 1.5 mm

^{3}, CS = 2.5). However, the median of flow difference calculated between 4D and 2D flow MRI was statistically higher than 10% only for voxel size of 1.5 mm

^{3}and CS = 2.5 for tube ID = 2, 4, and 5 mm (p < 0.001), and for a voxel size of 0.5 mm

^{3}and CS = 6.5 for tube ID = 4 mm (p < 0.001, Figure 4 right). Summary statistics of flow differences for each tube ID, voxel size, and CS factor is summarized in supporting information (Tables S4 and S5). Flow values and flow differences were obtained by 4D flow MRI in comparison to US sensor presented in Figure S4.

^{3}and CS = 2.5–13, 1.0 mm

^{3}and CS = 4.5 and 13, and 1.5 mm

^{3}and CS = 6.5. Flow calculated in ROI C at the tube with ID = 2 mm was most affected, while ROI A and B were almost intact.

#### 3.2. Velocity in an Aneurysm Model

^{3}, respectively (Figure 8a, left). However, the velocity distributions were statistically different (Kruskal–Wallis test, p-value << 0.01, Figure 8a, right). The decreasing median velocity value was observed for an increasing acceleration factor while keeping the voxel size the same (Figure 8b). In particular, (1) a voxel size of 0.5 mm

^{3}yielded a median velocity of 11.2, 10.5, and 9.4 cm/s for acceleration factors 2.5, 4.5, and 6.5, respectively; (2) a voxel size of 1.0 mm

^{3}resulted in a median velocity of 10.9, 9.8, and 9.8 cm/s; and (3) a voxel size of 1.5 mm

^{3}yielded a median velocity equal to 10.7, 9.7, and 8.9 cm/s. Note that the CS factor affected median values stronger than voxel size. For each spatial resolution, the velocity obtained with CS = 4.5 and 6.5 were compared against the one obtained with CS = 2.5. Statistical difference was observed for all cases (Wilcoxon rank-sum test, p << 0.01, Table S7). However, the changes did not exceed 11% for CS = 4.5 and 17% for CS = 6.5 in comparison to CS = 2.5 (Table S7). In addition, the median of velocity distribution obtained using 4D flow MRI with voxel size 0.5 mm

^{3}and CS = 2.5 was compared to all other combinations of CS factors and voxel sizes. In this case, the difference between medians ranged from −20.82% (voxel size = 1.5 mm

^{3}, CS = 6.5) to −3.09 % (voxel size = 1 mm

^{3}, CS = 2.5). Overall, the median velocity obtained with voxel size 0.5 and CS = 2.5 was underestimated by average on −11.42 ± 6.1% in comparison to all other combinations of voxel size and CS factors used.

## 4. Discussion

#### 4.1. Effect of Spatial Resolution and MR Acceleration on the Flow in Silicone Tubes

^{3}. However, flow was overestimated in comparison to 2D flow MRI for a greater voxel size (1.5 mm

^{3}), where the linear slope varied from 1.05 to 1.29. The flow was overestimated in all experiments by an average of 1.33 ± 8.31% and 0.24 ± 10.73% in comparison to 2D flow MRI and the US sensor, respectively, for all voxel sizes and CS acceleration factors used for 4D flow MRI. Flow was overestimated by 16%, as reported previously in a swine study (ID at different segments = 11.6, 16.55, and 18.1 mm, voxel size: 2.1 × 1.7 × 2.8 mm

^{3}, nROI = 5, 7, and 8) [43]. In another study, the flow was overestimated by 5% in a pulsatile phantom (ID = 20 mm) [22] compared to the transonic sensor. In some cases, the difference was eliminated when a high resolution was used (voxel size: 1.9 and 1.5 mm

^{3}instead of 2.8 × 2.8 × 2.2 mm

^{3}, nROI = 10 and 13 instead of 7, respectively) [22]. However, another study reports that no statistical difference was found for flow estimated with 2D flow MRI obtained with 1 × 1 mm

^{2}and 2 × 2 mm

^{2}(single-slice acquisition) compared to the flow sensor (ID = 3.5 ± 0.7 mm, nROI = 2–4 instead of 1–2) [28]. The spatial resolution itself without a context on vessel diameters does not allow to estimate the flow quantification error. It might be no effect of spatial resolution seen because it was already sufficient resolution and its increase did not change anything, or it might not be sufficient in both cases, and the flow is calculated with error in both cases. In the recent review [41], the authors showed that the spatial resolution did not have a statistical impact on the agreement between 2D flow and 4D flow MRI; likely some other parameters had a stronger impact, or the spatial resolution was sufficient to eliminate an effect on the flow quantification error.

^{3}). Moreover, the smallest linear slope (1.05) was yielded when 1.5 mm

^{3}voxels were used. Additionally, a higher acceleration factor (CS = 6.5) resulted in statistically higher RMS values than the ones obtained with a smaller acceleration factor (CS = 2.5); however, no statistical difference was found between CS = 2.5 and CS = 4.5. Additionally, the dependence of flow error from nROI was observed only for CS = 2.5 and 4.5; for CS = 6.5 and 13, some other factors likely play the dominant role. Previously, the feasibility of using acceleration factors of 4 to 10 [50] and even 10 to 30 [51] was demonstrated for in vivo aortic applications. Therefore, the use of higher CS factors might be appropriate. Yet, in the present study, an acceleration factor of 13 resulted in severe image distortions. No significant impact of CS = 13 on flow quantification was seen here likely due to the consistent placement of ROIs used for flow quantification. Still, peak flow was previously underestimated in the order of 25% [51] for an acceleration factor of 20, and 4% and 5% for acceleration factors of 8 and 13, respectively [42]. However, previously, Aristova et al. [45] showed no significant relationship between acceleration factor and flow measurement error for dual-venc 4D flow MRI accelerated with PEAK-GRAPPA. Overall, the number of studies considering the effect of acceleration of flow quantification even in the big vessels is limited. Moreover, it is worth noting that according to the review on 4D flow MRI in the heart and great vessels [41], most of the clinical centers do not use any acceleration technique at all. Recently, Pathrose et al. [52] studied CS 4D flow acquisitions in patients with aortic disease in comparison to 4D flow accelerated with GRAPPA. On average, the net flow was underestimated in the aortic arch by −7.1 ± 10.5%, −6.8 ± 13.8%, and −5.0 ± 10.7% for CS = 5.7, 7.7, and 10.2, respectively. However, at descending aorta, it was underestimated by −5.3 ± 11.8% only when CS = 10.2 was used. Moreover, at the ascending aorta, no significant impact of CS factor was found. Moreover, for some patients, the flow values were in good agreement between all 4D flow acquisition protocols. That being said, likely some other factors lead to the flow quantification error in combination with acceleration. In our study, on the contrary, flow overestimation was observed with increased CS factor. We hypothesize that the accelerated 4D flow acquisition is more capable of measuring more accurately peak flow values, and thus, better agreement between flow measured with CS = 6.5 and flow sensor was observed in Figure 2.

#### 4.2. Velocity in an Aneurysm Model

^{3}) compared to tomographic PIV (isotropic voxel size 0.23 mm

^{3}). However, when the PIV was downsampled to 4D flow MRI voxel size, the difference was reduced to 0.02 cm/s. Furthermore, further downsampling of the PIV data (isotropic voxel size 0.83 mm

^{3}) resulted in an average velocity magnitude reduction of 9.8 cm/s compared to the original data. The large voxel size likely represents a major factor in underestimating the velocity magnitude, which might be independent from the measurement technique. In another study, significant variation between velocity fields was observed in a comparison of 4D flow MRI (isotropic voxel size 0.5 mm

^{3}) and PIV (isotropic voxel size 0.17 mm

^{3}), which was eliminated after downsampling the spatial resolution of PIV [54].

^{3}, respectively. Thus, spatial resolution was likely already sufficient for all voxel sizes. For smaller aneurysms with additional morphological features, a higher resolution is recommended.

#### 4.3. Limitation

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic view of model 1 ((

**a**) silicone tubes) and model 2 ((

**b**) patient-derived aneurysm). (

**a**) Silicone tubes were wound around a falcon tube (

**a**top) and connected sequentially to the pump (

**a**bottom). A US sensor was placed at the outflow of the tubes (blue line). The flow was analyzed at the plane’s ROI A-C. (

**b**) Volume rendering of the patient-derived vascular model. A US sensor was placed at the inlet of the model. (

**c**) The velocity field was visualized at the 2D evaluation plane (

**c**left) and calculated in the aneurysm 3D ROI (

**c**right).

**Figure 2.**Time-resolved flow rate curves measured with US sensor, 2D flow MRI, and 4D flow MRI in silicone tubes with an inner diameter (ID) of 2, 3, 4, and 5 mm, respectively. The flow rates were calculated in three ROI A-C, as shown in Figure 1, and then averaged. Qualitatively, the peak flow values were overestimated for lower resolution 4D flow MRI data in comparison to 2D flow MRI.

**Figure 3.**Scatter plots and Spearman-rank correlation coefficient between time-resolved net flow rate values obtained by 2D and 4D flow MRI in silicone tubes. A strong correlation was observed for all voxel sizes and acceleration factors (rho > 0.97).

**Figure 4.**Box plots representing flow values obtained with 4D flow MRI (left) and flow differences calculated between 4D and 2D flow MRI (right). A voxel size of 1.5 mm

^{3}and CS = 2.5 and 4.5 resulted in statistically different flow median values (p < 0.001, star sign, left, a two-sided paired Wilcoxon signed-rank test). However, the difference was significantly higher than 10% only for four datasets (p < 0.001, red circle, right, a two-sided paired Wilcoxon signed-rank test).

**Figure 5.**Scatter (left) and box plots (right) for experiments performed on silicone tubes. (

**a**) The median difference between flow rate values measured with 4D and 2D flow tended to decrease with increasing voxel number per diameter. (

**b**) Median flow difference tended to decrease with an increased CS factor. (

**c**) RMS tended to decrease with increasing voxel number per diameter. (

**d**) RMS tended to increase with increased CS factor from 2.5 to 6.5. RMS obtained with CS = 2.5 was significantly different from CS = 6.5. For creating box plots in (

**b**,

**d**), RMS and flow difference values, obtained by 4D flow MRI with various nROIs but with the same CS factor, were joined together to make one dataset. Therefore, the dependency of data on nROI was excluded.

**Figure 6.**A representative cross-section of the silicone tubes with a diameter of 4 and 5 mm that were imaged with 4D flow MRI. The sequence was accelerated factors of 2.5-, 4.5-, 6.5-, and 13 (from left to right), and an isotropic voxel size of 1 mm

^{3}was used. Note the severe image distortion when an acceleration factor of 13 is applied. Phase image with velocity encoding in foot to head direction, which is collinear with a flow direction, is shown.

**Figure 7.**Effect of resolution and acceleration on velocity in an aneurysm flow model. (

**a**) Visualization of velocity streamlines of the flow in a patient-derived aneurysm model, obtained by 4D flow MRI with an isotropic voxel size of 1 mm

^{3}and a CS factor of 4.5. (

**b**) Velocity field in patient-specific aneurysm model measured with 4D flow MRI and calculated with CFD. Note that a strong inflow jet from the parent artery into the aneurysm sac (gray asterisk) and residual flow on the opposite aneurysm wall (black arrows) were observed with all combinations of voxel sizes and CS factor of 4D flow MRI.

**Figure 8.**Quantitative analysis of velocity distribution within an aneurysm sac of the aneurysm model. (

**a**) Boxplot and histogram of velocity magnitude distribution obtained with different voxel sizes regardless of the CS acceleration factor. (

**b**) Histograms have a similar shape for the 4D flow MRI parameters considered. Moreover, the median velocity values were similar among the voxel sizes but were statistically different. Boxplot and histograms of velocity distribution were obtained with varying CS acceleration factors but with constant resolution. The trend of decreasing velocity with increasing CS acceleration factor was observed.

**Table 1.**Protocol parameters for the in-house 2D and 4D flow and TOF MRI sequences modified from the vendor protocol.

MRI Protocol | P1 | P2 | P3 | P4 | P5 | P6 | P7 | P8 |
---|---|---|---|---|---|---|---|---|

MRI sequence | 2D flow | 4D flow | TOF | |||||

model | tubes, aneurysm | tubes | aneurysm | aneurysm | ||||

TR/TE [ms] | 9.4/6.2 | 10/6.3 | 7.5/4.6 | 6.6/4.0 | 10.6/6.4 | 7.2/4.4 | 6.5/3.9 | 25/5.8 |

vox. size [mm ^{3}] | 0.5 × 0.5 × 4 | 0.5 × 0.5 × 0.5 | 1 × 1 × 1 | 1.5 × 1.5 × 1.5 | 0.5 × 0.5 × 0.5 | 1 × 1 × 1 | 1.5 × 1.5 × 1.5 | 0.25 × 0.40 × 050 |

FOV [mm ^{3}] | 180 × 180 | 110 × 78 × 30 | 100 × 100 × 20 | 180 × 180 × 160 | ||||

CS factor | 2.5 | 2.5; 4.5; 6.5; 13 | 2.5; 4.5; 6.5 | 4.7 | ||||

acq. Time [min] | 2 | 11.2–57.5 | 2.7–14.2 | 1.2–6.2 | 28.5–73.2 | 7.4–18.8 | 3.2–8.2 | 20 |

card. Phase | 24 | - | ||||||

Venc | 60, 80 | 60 | 80 | - |

**Table 2.**Parameters of linear fit calculated for flow values acquired with 2D and 4D flow MRI data in silicone tubes. The linear slope was close to 1 for most 4D flow MRI parameters. The flow was overestimated (linear slope > 1.21) by 4D flow MRI acquired with 1.5 mm

^{3}spatial resolution and accelerated by factors of 4.5 to 13.

Acquisition Parameters of 4D Flow MRI | A Linear Fit | ||
---|---|---|---|

Spatial Resolution [mm^{3}] | Acceleration Factor | Linear Slope | R^{2} |

0.5 | 2.5 | 1.08 | 0.99 |

4.5 | 1.05 | 0.99 | |

6.5 | 1.07 | 0.97 | |

13 | 1.01 | 0.96 | |

1.0 | 2.5 | 0.93 | 0.97 |

4.5 | 1.06 | 0.97 | |

6.5 | 1.08 | 0.97 | |

13 | 1.09 | 0.97 | |

1.5 | 2.5 | 1.05 | 0.97 |

4.5 | 1.25 | 0.97 | |

6.5 | 1.21 | 0.97 | |

13 | 1.29 | 0.97 |

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**MDPI and ACS Style**

Pravdivtseva, M.S.; Gaidzik, F.; Berg, P.; Ulloa, P.; Larsen, N.; Jansen, O.; Hövener, J.-B.; Salehi Ravesh, M.
Influence of Spatial Resolution and Compressed SENSE Acceleration Factor on Flow Quantification with 4D Flow MRI at 3 Tesla. *Tomography* **2022**, *8*, 457-478.
https://doi.org/10.3390/tomography8010038

**AMA Style**

Pravdivtseva MS, Gaidzik F, Berg P, Ulloa P, Larsen N, Jansen O, Hövener J-B, Salehi Ravesh M.
Influence of Spatial Resolution and Compressed SENSE Acceleration Factor on Flow Quantification with 4D Flow MRI at 3 Tesla. *Tomography*. 2022; 8(1):457-478.
https://doi.org/10.3390/tomography8010038

**Chicago/Turabian Style**

Pravdivtseva, Mariya S., Franziska Gaidzik, Philipp Berg, Patricia Ulloa, Naomi Larsen, Olav Jansen, Jan-Bernd Hövener, and Mona Salehi Ravesh.
2022. "Influence of Spatial Resolution and Compressed SENSE Acceleration Factor on Flow Quantification with 4D Flow MRI at 3 Tesla" *Tomography* 8, no. 1: 457-478.
https://doi.org/10.3390/tomography8010038