Magnetic Resonance Velocimetry for porous media: sources and reduction of measurement errors for improved accuracy

Flow velocities can be measured using PC-MRV. Here velocity encoding gradients are applied which cause a phase evolution of moving spins proportional to the velocity while for static spins the initial phase is retained after encoding. The phase \(\Phi\) of the Nuclear Magnetic Resonance (NMR) signal is described bywhere \(\gamma\) is the gyromagnetic ratio and \(\vec{v}\) is the velocity. \(\vec{M}_{1}\) is the first moment of the gradient applied for the duration T defined by:\(\Phi _{0}\) results from background phase effects due to eddy currents, magnetic susceptibility and concomitant gradients. To filter out such effects at one-directional velocity, two measurements are done with different first moments \(\vec{M}_{1}^{{(1)}}\) and \(\vec{M}_{1}^{(2)}\) (e.g. using inverted gradient polarities) but under otherwise same conditions. Subtraction of the signal phases results in \(\Delta \Phi\), which is dependent on velocity \(\vec{v}\) and \(\Delta \vec{M}_{1}=\vec{M}_{1}^{(1)}-\vec{M}_{1}^{(2)}\). Nonetheless, eddy current errors originating from the velocity encoding gradients are not eliminated. Therefore, an additional measurement is done with same parameters but without flow and the phase images are subtracted.

In this work the four-step balanced encoding was used (Pelc 1991). \(Z_{1}\) to \(Z_{4}\) are the complex images acquired for individual encoding steps. The velocity v is determined asThe phase can only be uniquely measured over an interval of \(2\pi\). If there is flow in positive and negative direction, it is common that the VENC is thus defined as the velocity that produces a phase shift \(\Delta \Phi =\pi\), i.e.For too high velocities, the velocity dependent phase shift can exceed \(\pm \pi\) and phase aliasing occurs. The phase shift can be calculated as \(\Delta \Phi =\arg (Z_{1})-\arg (Z_{2})=\Phi _{1}-\Phi _{2}\), which is computationally costly because of two \(\arg\) operations. Also additional aliasings may be introduced. Instead, the phase shift is calculated by \(\Delta \Phi =\arg (Z_{1}/Z_{2})=\arg (Z_{1}\cdot {\bar{Z}}_{2})\) (Bernstein 2004). The phases in x-, y- and z-direction are calculated as follows:Analogously to Lee (1995) the standard deviation of the velocity is determined aswhere N is the number of encoding steps. For each voxel the Signal-to-Noise Ratio (SNR) is calculated from the magnitude I of the complex signal and the standard deviation \(\sigma\):In magnitude images the noise distribution in regions without NMR signal is governed by a Rayleigh distribution (Gudbjartsson 1995). If the standard deviation \(\sigma _{0}\) is determined in the signal free part of a magnitude image, the standard deviation \(\sigma\), that is used to characterise the VNR, is given by \(\sigma =1.5264\cdot \sigma _{0}\). In Eq. (8), it is visible that the VENC value should be set as low as possible to achieve a low noise value \(\sigma _{v}\) (i.e. good VNR). However, it must be noted that for \(\text{VENC}<v_{\text{true}}\) aliasing occurs. For correction of the aliasing a reference scan with a higher VENC value is used and difference \(D=v_{\text{high}}-v_{\text{low}}\) is determined. The number of wrappings w is then determined aswhere \(\text{N.I.}\) is the nearest integer function. The corrected (unwrapped) velocity is calculated as:In case of three-directional velocity, an eight-step measurement can be done (four steps for low- and four steps for high-VENC measurement). This doubles the number of measurements in comparison with a single-VENC measurement but results in a VNR improvement by a factorIt should be noted that for a large R unwrapping may fail due to large noise values in the difference D. Neglecting pulsation effects and assuming stable conditions (temperature, electrical components) during the whole measurement operation, the condition from Lee (1995) for a low number of wrong unwrappings can be written asRegarding Eq. (12) we formulate the noise value at each voxel by \(\text{noise}(D)=m\cdot \sqrt{\frac{1}{R^2}+1}\cdot \sigma _{v-\text{high}}\) where m is the number of standard deviations that can be tolerated. With Eq. (8) it followsFor standard normal distribution, the probability of a result outside of the interval \([-m\cdot \sigma ;m\cdot \sigma ]\) can be calculated bywhere f(x) is the probabilty density function and \(\text{erf}\) the Gaussian error function. For a given ratio R and SNR, the fraction P of wrongly unwrapped voxels can be calculated with Eqs. (14) and (15).

An additional VNR-improvement is achieved by weighted averaging of the unwrapped low-VENC and the high-VENC maps:

A SE and PC-based MRV pulse sequence was implemented (Fig. 1). Spatial resolution was achieved by one read and two phase encoding gradients. The spatial encoding by the readout gradients was sufficient to avoid signal wrap-around for the measured samples and the chosen Field Of View (FOV). Therefore, non-selective rectangular RF pulses were used for excitation and refocusing to reduce the interecho delay and avoid signal losses. To suppress unwanted signals, spoiler gradients were applied before and after the refocusing pulses. In Shiko (2012) and Huang (2017) the velocity encoding was done before the multi-echo loop. However, in this case measurements are limited to slow velocities. Otherwise large displacement errors would arise for the later echoes. To avoid this, velocity encoding was done similar to the FLIESSEN sequence (Amar 2010). Before and after each echo velocity encoding pairs with same strength but opposite direction were applied to set the phase shift to zero before the next Radio Frequency (RF) refocusing pulse. This increases the echo time but reduces displacement effects. In total eight different velocity encoding gradients were applied to achieve a full dual-VENC encoding scheme for acquisition of three-directional velocity in one measurement. For every echo pair consisting of two consecutive echoes (odd and even) the same velocity encoding was done. Datasets from odd and even echoes are combined to suppress additional phase differences caused by different coherence pathways (for details see Shiko 2012; Huang 2017). Between velocity encoding and decoding, spins may be accelerated or decelerated (e.g. for spins moving near a narrowing). In this case, the signal phase is not completely eliminated after decoding and an acceleration-dependent phase shift remains. Combination of odd and even echoes cancels this effect (Amar 2010) (provided that the acceleration remains constant for both echoes). The velocity with \(\text{VENC}_{\text{low}}\) was reconstructed from the first eight echoes and the velocity with \(\text{VENC}_{\text{high}}\) was reconstructed from the last eight echoes. Associated velocities are called \(v_{1}\) and \(v_{2}\). Hence, in total 16 echoes were acquired. As discussed by John (2020), the maximum displacement in number of voxels is given bywhere \(t_{\text{displ}}\) is the time between the centre of velocity encoding and the centre of acquisition. Even though \(t_{\text{displ}}\) should be chosen as short as possible it shall be noted that the gradient amplitude is limited and therefore for a given VENC the duration \(\delta\) and delay \(\Delta\) cannot be chosen arbitrarily short. Since gradients with large amplitude may produce eddy currents which worsen the image quality, these should only be applied if compensation is provided to correct for eddy currents. If the compensation is not perfect, which is usually the case for experimental data, the remaining artefacts scale with the gradient amplitude. Additionally, doing measurements with large gradient amplitude may shorten the lifespan of the gradient system. Using \(\tau =6.0\,\text{ms}\), a duration of \(192\,\text{ms}\) results from excitation pulse until acquisition of the last of the 16 echoes. Experimentally a transversal relaxation time \(T_{2}=0.41\,\text{s}\) was determined. Thus, due to transversal relaxation the SNR of the last echo is reduced by \(36.5\%\) in comparison with the first echo. The SNR of the final (unwrapped) velocity map is mainly determined by the \(\text{VENC}_{\text{low}}\). Therefore, the \(\text{VENC}_{\text{high}}\) encoding steps are measured in the latter echoes. The acquisition time per echo is \(1.5\,\text{ms}\). The rectangular flow encoding gradients have a \(\delta =0.78\,\text{ms}\) and a delay of \(\Delta =2.71\,\text{ms}\). The time between the centre of velocity encoding and the centre of acquisition was \(t_{\text{displ}}=3.58\,\text{ms}\). It should be noted that this value may differ for other sequences, since it depends on several factors (hardware, acquisition time, voxel size, VENC, flow velocity, etc.) (Table 1).

For a given sequence, the SNR can be improved by increasing the number of accumulations or increasing the repetition time \(T_{\text{R}}\). For a constant measurement time, it shall be analysed at which ratio between accumulations and \(T_{\text{R}}\) the optimal SNR improvement is achieved. Therefore, the longitudinal magnetisation is considered. After applying the \(90^{\circ }\) excitation pulse at \(t_{0}=0\) the longitudinal magnetisation is \(M_{\text{z0}}=M_{\text{z}}(t_{0})=0\). Due to longitudinal relaxation the magnetisation after applying the first \(180^{\circ }\) pulse (at \(t_{1}\)) isIn total 16 inversion pulses are applied in the echo train. The longitudinal magnetisation directly after the k-th inversion pulse is calculated as follows:For \(T_{1}=2\,\text{s}\), which was experimentally determined, a quite low \(M_{\text{z16}}=-0.003\cdot M_{0}\) results. Therefore, \(M_{\text{z}16}\) is neglected in the following. We consider the time between the last inversion pulse and the excitation pulse where longitudinal relaxation takes place. With \(M_{0}=1\) and normalisation to unit time, the longitudinal magnetisation is approximated by (Fig. 2)

Plots for Eq. (20) at different \(T_{1}\) values are shown in Fig. 3. For long \(T_{1}\) a broad maximum is visible. However, an optimal \(T_{\text{R}}\) value should be chosen to avoid SNR-losses, which is of particular importance for short \(T_{1}\).

Flow measurements were done for two different samples: A flow phantom and an OCF. The flow phantom was manufactured in the university workshop by heating and turning a glass tube (inner diameter \(5\,\text{mm}\)) in helical shape. Since more heat was added from one side during the manufacturing process, the helix is flattened on the outside of the windings. Also the cross section was narrowed. The cross-sectional area is decreased by \(32\%\) in comparison with the straight section (the value was determined on the basis of the 3D NMR data). A picture of the flow phantom is shown in the appendix (Fig. 10). The OCF with 10 pores per inch (\(25\,\text{mm}\) in diameter) consists of polylactic acid and was 3D-printed by a Digital Light Processing printer (Elegoo Mars, Shenzhen, China) with \(2\,\text{k}\) resolution (\(12.5\,\mu \text{m}\times 12.5\,\mu \text{m}\)) in XY direction and layer height (Z) of \(50\,\mu \text{m}\). To prevent unwanted bypass flow, seal tape was first wrapped around the OCF before it was pushed into the glass vessel. The properties of the OCF are given in Table 5 in the supplementary material. To cover similar velocity ranges, flow rates of \(9.9 \pm 0.1\frac{{{\text{ml}}}}{{{\text{min}}}}\) (flow phantom) and \(206 \pm 2{\kern 1pt} \frac{{{\text{ml}}}}{{{\text{min}}}}\) (OCF) were realised using a peristaltic pump (Standard Digital Drive with Easy-Load II Pump Head; Masterflex, Vernon Hills, USA) with pulse dampener and pump tubing also from Masterflex. PVC-tubing was used for connection to the flow phantom and the glass vessel with the OCF (Fig. 4). Tap water was used as the operating fluid since this is widely used as flow model for MRV measurements. The appropriate entrance length (\(L_{\text{e}}\)) was considered to provide full development of flow before reaching the flow phantom or OCF:Here \(D_{\text{in}}\) is the inner tube diameter, Re is the Reynolds number, \(\rho _{\text{f}}\) is the fluid density and \(\mu _{\text{f}}\) is the dynamic viscosity of the fluid. The average inlet velocity \(u_{\text{z}}\) was calculated depending on the Volumetric Flow Rate (VFR). Reynolds numbers were \(Re=42\) (flow phantom) and \(Re=175\) (glass vessel with OCF). Entrance length was \(L_{\text{e}}=12\,\text{mm}\) (flow phantom) and \(L_{\text{e}}=245\,\text{mm}\) (OCF). For calculating the pore-Re number, the pore diameter \(5.8\,\text{mm}\) (Table 5) was used instead of the tube diameter. Since it is difficult to estimate the average velocity in a pore, the high-VENC was chosen for \(u_{\text{z}}\) leading to a pore-Re of 261. Therefore, laminar flow can be assumed.

All measurements were performed on a horizontal \(7\,\text{T}\) scanner (BioSpec 70/20 USR, Bruker BioSpin MRI, Ettlingen, Germany), which is equipped with a \(114\,\text{mm}\) bore gradient system (BGA 12S2, maximum gradient strength of \(440{\kern 1pt} \frac{{{\text{mT}}}}{{\text{m}}}\), maximum slew rate of \(3440\,\frac{\text{mT}}{\text{m}\cdot \text{ms}}\)). A horizontal 72-mm bore birdcage \(^{1}\text{H}\) quadrature transceiver RF coil (Bruker BioSpin MRI, Ettlingen, Germany) was used in all measurements. ParaVision 5.1 was used as the user interface to perform the measurements. The repetition time was \(T_{\text{R}}=2.9\,\text{s}\) (according to the maximum in Fig. 3). The gradient amplitude was set depending on the desired VENC value. Further measurement parameters are listed in Table 2.

The measurements for the flow phantom were done at \({\text{VENC}}_{1} = {\text{VENC}}_{2} = 40{\kern 1pt} \frac{{{\text{mm}}}}{{\text{s}}}\) to analyse the reproducibility of the measured velocity \(v_{1}\) and \(v_{2}\). For improving the VNR, the dual-VENC method was used for the OCF. VENC values were \({\text{VENC}}_{1} = 20{\kern 1pt} \frac{{{\text{mm}}}}{{\text{s}}}\) and \({\text{VENC}}_{2} = 45{\kern 1pt} \frac{{{\text{mm}}}}{{\text{s}}}\). The final (unwrapped) velocity is mainly determined by the VNR of the low-VENC data. Thus, the low-VENC value was chosen for \(\text{VENC}_{1}\) because the first echoes experience a lower SNR-loss (respectively VNR-loss) due to transversal relaxation. Measurements were done with and without flow. Considering the no-flow data, noise parameters were determined for the high-VENC data (last eight echoes) after averaging the two accumulations. For each pair of consecutive odd and even echo the magnitude images were combined (averaging) and the respective SNR was calculated according to Eq. (9). The mean value was \(\text{SNR}=199.6\) which was used to calculate \(\sigma _{v} = 0.144{\kern 1pt} \frac{{{\text{mm}}}}{{\text{s}}}\) (Eq. (8), number of encoding steps \(N=4\)). Post-processing of the data was performed using a self-developed MATLAB script (R2021a MathWorks, Natick, MA, USA). Binary masking was based on Otsu’s thresholding (Otsu 1979) for the corresponding magnitude images.

If offset correction is done by subtracting the no-flow velocity map, the VNR is lowered by \(\sqrt{2}\). For reduced VNR-loss another correction method was used. Analogously to Lingamneni (1995) a polynomial fit was applied to the no-flow dataset for reducing the effect of random noise. The following algorithm was used:

In Fig. 5, the velocity maps (\(v_{1}^{\text{z}}\)) for axial slice 61 (out of 120) of the flow phantom are shown. Corresponding velocity maps for \(v_{2}^{z}\) are shown in Fig. 11 in the supplementary material. For the maps without flow an offset is visible (depending on the spatial position up to \(10\%\) of the VENC). For a quantitative analysis for each of these maps the VFR was calculated in a Region Of Interest (ROI) (straight tube) bywhere \(L_{\text{x}}\) and \(L_{\text{y}}\) are the voxel size in x- and y-direction. The bar plot with the resulting VFR for \(v_{1}^{\text{z}}\) and \(v_{2}^{\text{z}}\) is shown in Fig. 6. For all cases without offset correction (odd, even, combined) the VFR is between 12 and \(14{\kern 1pt} \frac{{{\text{ml}}}}{{{\text{min}}}}\) and therefore does not fit the expected \(9.9 \pm 0.1{\kern 1pt} \frac{{{\text{ml}}}}{{{\text{min}}}}\). Better agreement is visible for the data that are corrected by subtraction of a reference scan without flow. Similar results were obtained for all axial slices. Considering only odd or only even echoes, the VFR is still a bit too low or too high. Best agreement is obtained for the combination (averaging) of odd and even echoes and correction with the no-flow measurement. However, these values are always between 10.1 and \(10.2{\kern 1pt} \frac{{{\text{ml}}}}{{{\text{min}}}}\), thus overestimating the expected VFR of \(9.9 \pm 0.1{\kern 1pt} \frac{{{\text{ml}}}}{{{\text{min}}}}\) by 1 to \(4\%\). An analogue analysis was done for velocities \(v_{1}\) and \(v_{2}\) in x- and y-direction (see appendix). A plot with the VFR calculated for each axial slice can be found in the appendix Figs. 13 and 14. Also here the best agreement is visible for combination of odd and even echoes and correction with the no-flow measurement. Considering the whole 3D dataset, maximum velocities were found in the centre of the glass tube and had values of \(23.5{\kern 1pt} \frac{{{\text{mm}}}}{{\text{s}}}\) (x-/y-direction) and \(17.3{\kern 1pt} \frac{{{\text{mm}}}}{{\text{s}}}\) (z-direction). Since the cross section is slightly flattened due to the manufacture process, it is not easy to predict the flow behaviour. However, for the straight tube with a round cross section the velocity profile can be predicted. For a laminar tube flow it is \(v_{\text{max}}=2\cdot v_{\text{mean}}\). Regarding the tube diameter and the VFR given in Sect. 3.3, it was therefore calculated \(v_{{\max }} = 16.8\frac{{{\text{mm}}}}{{\text{s}}}\). This value is \(3\%\) lower than the measured one. For laminar flow, the mean axial velocity \(v_{r}\) at radius r can be calculated as a function of the maximum velocity \(v_{\text{max}}\) and the tube radius a:For one exemplary axial slice, the calculated axial velocity is compared with the measured one (Fig. 7).

A dual-VENC measurement was done for the OCF. Complex valued images for each encoding step were combined (averaging odd and even echoes), before velocity values in x-, y- and z-direction were calculated. In a measurement under stable conditions, the velocity maps of two accumulations would only differ by their noise values. In the following, the reproducibility of a measurement is analysed by comparing two accumulations using the example of the high-VENC velocity maps. After subtracting the two accumulations from each other, the standard deviation within the pores was determined and compared with the value calculated according to Eq. (8) (for details see Sect. 3.4). The results with and without flow are compared to determine if flow effects (such as unwanted pulsation) have an additional effect. Corresponding values are listed in Table 3. Without flow (left column), similar values are visible in x-, y- and z-direction which are also roughly twice as high as the value \(0.144{\kern 1pt} \frac{{{\text{mm}}}}{{\text{s}}}\) calculated according to Eq. (8). This corresponds to the expectations because this \(0.144{\kern 1pt} \frac{{{\text{mm}}}}{{\text{s}}}\) was determined after averaging both accumulations (noise reduction by \(\sqrt{2}\)) and the values shown in Table 3 were calculated by subtracting both accumulations (noise increased by \(\sqrt{2}\)). Values resulting from the velocity maps with flow (right column) are larger, between 2.2 and 3.7 fold increased.

Offset correction was done by direct subtraction of the no-flow velocity map and by using the algorithm described in Sect. 3.5. As an example, the procedure is shown for one axial slice and one velocity component (\(v_{1}^{\text{y}}\)), which was calculated from the first accumulation of the measurement. The corresponding binary masks \(\text{BM}_{1}\) and \(\text{BM}_{2}\) are shown in Fig. 8a and b. For the no-flow velocity map an offset up to \(10.5\%\) of the VENC is visible depending on the spatial position (Fig. 8c). After applying the binary mask \(\text{BM}_{2}\), polynomial fitting for the no-flow velocity map was done by using an open-source MATLAB toolbox (D’Errico 2022). The polynomial fits with first and second order are shown in Fig. 8d and e, respectively. To quantify the goodness of the fits, the Root-Mean-Square Error (RMSE) was calculated. Using the first order fit to correct the no-flow velocity map results in Fig. 8f. Areas with similar nonzero values can be observed locally. In contrast, the values in Fig. 8g (second order fit) are more randomly distributed around zero. Calculating the no-flow velocity map from each of the two accumulations and subtract the two from each other, results in Fig. 8h. The standard deviation of the polynomially fitted maps is normalised to the standard deviation of Fig. 8h. Values can be found in Table 4. From first to second order fit a significant reduction can be observed for both, standard deviation and RMSE. For higher order fits only a slight reduction is visible. For increased VNR, both accumulations were averaged and velocity maps calculated. Polynomial fits (first to sixth order) were done for each velocity component. Corresponding RMSE values are shown in Fig. 18 in the supplementary material. The offset of the velocity maps were corrected by the third order polynomial fit. A higher order fit was not used since it does not deliver a significantly larger accuracy and to avoid the risk of overfitting. The dual-VENC method was used for unwrapping. For an additional VNR-gain, unwrapped low-VENC and high-VENC data were weighted averaged based on the ratio \(R=2.25\). Velocity maps are shown in Fig. 9.

A flow phantom with predictable flow behaviour was measured to test the new multi-echo sequence. An offset was observed and the measured VFR was overestimated if only odd or only even echoes are considered. This confirms that a combination (averaging) of odd and even echoes is necessary to correct for different coherence pathways. An offset was also observed if only the velocity maps with flow are considered. This confirms the assumption that the phase difference image (Eqs. (5) to (7)) does not eliminate all phase errors. These findings are consistent with Huang (2017). Eddy currents depend on the strength and direction of the applied magnetic gradients. Thus, the eddy currents originating from the velocity encoding gradients behave additive in the phase difference image. Therefore, it was necessary to subtract flow and no-flow measurements for correction of remaining offsets. The corresponding VFR was slightly overestimated by 1 to \(4\%\). It can be assumed that the slow velocities, which are located at the surface of the flow phantom, were slightly overestimated due to partial volume effects. Details about such effects are described in Tang (1993); Bernstein (2004).

The measured axial velocity was compared with theoretically predicted values for laminar flow. A good agreement was visible for the flow profile. The maximum velocity determined from the measured axial velocity was slightly larger than the predicted value due to superimposed noise.

After one excitation pulse, several echoes were acquired. For the transversal relaxation time \(T_{2}=0.41\,\text{s}\) it was possible to measure 16 echoes with acceptable signal loss to achieve full dual-VENC encoding. Velocity maps \(v_{1}\) and \(v_{2}\) were reconstructed from the first and last eight echoes, respectively. For setting \(\text{VENC}_{1}=\text{VENC}_{2}\), reproducibility was observed, even though \(v_{2}\) has lower SNR due to longer echo times.

Velocity encoding and decoding gradients were applied before and after each echo. Thus, the echo time was increased, respectively the SNR decreased. In Huang (2017) the interecho delay was shorter because the velocity encoding gradients were applied before the echo train. However, this approach increases displacement errors. Thus, if the same echo time is assumed for both sequences, the displacement error for the last echo differs by a factor 50. Single-point imaging sequences, as proposed by Bruschewski (2019), are able to reach a great reduction in displacement errors but in comparison with the sequence shown here the measurement time is strongly increased and not suitable for 3D imaging. The reduction in displacement errors by updating the flow encoding for each echo was already proposed in Amar (2010) for the FLIESSEN sequence. Since this sequence aimed at short total measurement time, each echo of the long echo train was measured with different spatial phase encoding. In contrast, the sequence proposed in the present work uses all echoes for flow encoding. Therefore, the minimum total measurement time is longer than in FLIESSEN or other sequences that have a long echo train length with short \(T_{\text{R}}\). However, for the sequence shown here a reduced spatial resolution in phase encoding directions is avoided as each image is measured at a fixed echo time.

The proposed approach assumes constant velocity and does not compensate for nonzero acceleration. To minimise the displacement errors, the time \(t_{\text{displ}}\) between velocity encoding and acquisition was kept as short as possible. Thus, also the influence of acceleration on the signal phase is limited. Most importantly, only two consecutive echoes (odd, even) are used for the calculation of the same image as different velocity encoding steps are performed along the echo train. Thus, summation of acceleration effects (via the consecutive echoes) would not lead to incorrect velocity encoding, but to reduced signal due to intra-voxel dephasing. However, a comparison of the data measured with and without flow yielded no notable differences in the signal intensity, neither for the flow phantom nor the OCF. Therefore, acceleration effects, although being not suppressed, were negligible in the experiments. This was also proven by the fact that the measured VFR corresponds well with the expected values.

For the proposed sequence a good compromise was found to achieve reduced displacement errors, acceptable total measurement time and sufficient VNR.

For the flow phantom maximum velocities of \(23.5{\kern 1pt} \frac{{{\text{mm}}}}{{\text{s}}}\) (x-/y-direction) and \(17.3{\kern 1pt} \frac{{{\text{mm}}}}{{\text{s}}}\) (z-direction) in the centre of the glass tube were determined. Higher velocities in x-/y-direction occurred due to the reduced cross-sectional area at the windings. Displacement errors are always a problem if the direction or magnitude of the velocity vectors change. This is generally the case for turbulent flow. Since for the measurement with the flow phantom the Reynolds number was \(Re=42\), laminar flow can be assumed. For the special case that only the straight tube of the flow phantom is considered (only axial velocity), in each axial slice a similar velocity should be present and therefore displacement errors would not have a large impact on the velocity maps. However, large displacement errors would negatively influence the measured velocity fields in the windings of the flow phantom.

In this measurement a voxel size of \(0.30\,\text{mm}\) was chosen. Considering the maximum velocity in x-/y-direction and the time between velocity encoding and readout, with Eq. (17) a \({\rm displacement}=0.28\,{\rm voxels}\) can be calculated, which corresponds to \(0.08\,\text{mm}\). Even though this value is below one voxel, displacement errors may occur, namely for spins located at the border of a voxel. However, it should be noted that the estimated displacement error occurs for the maximum velocity and therefore in most voxels the displacement error is much lower.

For the OCF, the maximum velocity was \(44.7{\kern 1pt} \frac{{{\text{mm}}}}{{\text{s}}}\) and the voxel size \(0.45\,\text{mm}\). Therefore, displacement errors up to \({\rm displacement}=0.36\,{\rm voxels}\) (corresponding to \(0.16\,\text{mm}\)) were present. This displacement error is small in comparison with the pore size of \(5.8\pm 1.9\,\text{mm}\). Additionally, the displacement error will be lower in most regions of the OCF as the displacement value of \(0.36\,{\rm voxels}\) was calculated for the maximum velocity.

When applying the dual-VENC technique, often a ratio R between two and three is chosen (Schnell 2017; Aristova 2019; Callahan 2020; Nakaza 2022; Mahinrad 2022). In the present study, an optimal R with negligible wrong unwrappings (\(m=5\), \(P=6\cdot 10^{-7}\)) was calculated. Therefore, the optimal \(R=22\) could be chosen (Eqs. (14) and (15), \(N=8\) due to subtraction of flow and no-flow velocity maps). For this calculation the mean SNR was assumed. However, voxels located at the edge areas are only partly filled with fluid and therefore have a lower SNR. For example, if a voxel is only filled to a quarter, the calculated optimal ratio would be \(R=5.4\). Due to large specific surface area, especially in porous media such edge voxels should be taken into account. Moreover, the calculation for such optimal R is only true for fully reproducible measurements. For a real measurement changes may occur due to fluctuations of the fluid field or changes of the temperature or electrical components of the NMR system. Especially for 3D measurements, which usually run over longer time (the measurement for the OCF needed \(20.6\,\text{h}\)), such variations may be a significant factor. In Table 3, the standard deviation of the velocity maps with flow was up to 3.7 fold increased in comparison with the ones without flow. This indicates the presence of larger variations during the measurements with flow, which are presumably due to pulsation effect caused by the peristaltic pump used. Therefore, a rather low \(R=2.25\) was chosen. However, a more systematic analysis of these effects should be done to achieve a better understanding and reduce flow induced phase fluctuations to such a level that similar \(\sigma _{v}\) values are obtained as in no-flow measurements. This should preferably be done by improving the pulsation dampening (Blythe 2015) or using a pulsation-free pump system. Alternatively, triggering techniques could be used that are widely applied in medical applications of MRV (Selby 1992; Brown 2014). Then larger R values could be chosen to use the full potential of dual-VENC encoding. Each slice (and each velocity component) was visually inspected to determine if wrong unwrapping was present. Therefore, inside of the pores it was searched for voxels, which show a large difference (about \(2\cdot \text{VENC}\)) in comparison with the surrounding ones. None of such voxels was found.

Flow through porous media, like the used OCF, usually has a large dynamic range. Therefore, the dual-VENC approach is of great use to prevent aliasing in fast flow regions but still realise a sufficient VNR in slow flow regions. For the chosen ratio between high- and low-VENC, the VNR was improved by \(R=2.25\) in comparison with a single-VENC measurement. Instead of doing a dual-VENC measurement, in the same time two accumulations of the high-VENC measurement could be done and averaged, which increases the VNR by \(\sqrt{2}\). Therefore, the effective VNR-improvement of the dual-VENC technique with \(R=2.25\), is \(\frac{2.25}{\sqrt{2}}\)=1.59. An additional VNR-improvement was achieved by weighted averaging unwrapped and high-VENC data, as described by Eq. (16). For \(R=2.25\) this is \(9.4\%\).

For both samples (flow phantom and OCF) an offset was observed in the velocity images underlining the necessity of using a no-flow measurement for corrections. However, if velocity maps of flow and no-flow measurements are subtracted, this decreases the VNR by \(\sqrt{2}\) if the same \(\sigma _v\) values occur in datasets measured with and without flow. Instead, a polynomial fit can be done for the no-flow map and be subtracted from the flow map to decrease the influence of noise. However, it should be noted that for inaccurate fitting systematic errors may be introduced. Porous media, like the used OCF, usually have a large specific surface area. An algorithm was used (see Sect. 3.5) to ignore edge voxels, which could worsen the quality of the fit (e.g. due to partial volume effects). Thus, only voxels inside of the pores were considered for the polynomial fit. For the correction of \(v_{1}^{\text{y}}\) by a first order polynomial fit, locally significant systematic errors occur (Fig. 8f). In contrast, when using a second order polynomial fit, velocity values are more equally distributed without large areas that could indicate systematic fitting errors (fig. 8g). Analogously, also the RMSE decreased for higher order fits (see Table 4 and Fig. 18). Both offset correction methods - polynomial fit and direct subtraction - were evaluated in relation to the standard deviation. For correction with the first order polynomial fit the standard deviation was \(6\%\) larger in comparison with direct offset correction for each voxel. This can be explained by the systematic errors introduced by the fit. For higher order fits the influence of such systematic offsets was lower. However, the maximum improvement of \(\sqrt{2}\) was not achieved. For the second order fit the standard deviation was \(9\%\) lower than for the direct offset correction and for the sixth order even \(12\%\). Therefore, the VNR can be improved if the phase offset is corrected by using at least a second order polynomial fit. One should note that for a too high order overfitting could take place and have a negative impact.

The fact that the VNR improvement was considerably lower than the improvement by \(\sqrt{2}\) expected from theory is consistent with the findings of larger \(\sigma _v\) values for flow data than for no-flow data (see Table 3). Since the VNR was thus dominated by the data measured with flow, the VNR improvement by polynomial fitting of the no-flow data will increase if the flow induced phase fluctuations are reduced to a minimum, either by better pulsation dampening or avoiding the use of a peristaltic pump.