Velocity‐compensated intravoxel incoherent motion of the human calf muscle

To determine whether intravoxel incoherent motion (IVIM) describes the blood perfusion in muscles better, assuming pseudo diffusion (Bihan Model 1) or ballistic motion (Bihan Model 2).


INTRODUCTION
Intravoxel incoherent motion (IVIM) imaging is an important tool in clinical and preclinical MRI, which is used to assess tissue perfusion and diffusion information simultaneously with DWI sequences. 1IVIM imaging is useful in assessments of human skeletal muscle, as diffusion and perfusion parameters are critical indicators in the diagnosis of diseases affecting muscles [2][3][4] as well as recovery after training or injury. 5,6][8] In their paper from 1988, Le Bihan et al. introduced IVIM as a means to separate diffusion and perfusion. 9n the most commonly used limit, the pseudo-diffusion limit, the signal is described as a sum of two exponential functions: where the first summand represents the signal of the perfusion compartment and is characterized by the perfusion fraction f and the pseudo-diffusion coefficient D * .The second summand represents the diffusion compartment with the diffusion coefficient D. The compartments can be separated, as D * and D usually differ by at least one order of magnitude. 9he exponential signal decay is a good approximation if the vessels contributing to the perfusion signal change the flow direction multiple times on the timescale of the diffusion encoding.However, it has been shown that using diffusion gradients, which compensate signal decay from blood moving with constant velocity (i.e., velocity compensation) leads to a diminished IVIM effect, [10][11][12][13][14][15][16] specifically in the human liver 15,16 and pancreas, 15 and the human 14 and cat brain. 13This reduced IVIM effect indicates that the pseudo-diffusion limit is not met in the organs considered in these studies.Instead, the ballistic limit seems more appropriate.That is, blood does not change flow direction during diffusion encoding, leading to a linear dependence of the pseudo-diffusion coefficient on diffusion encoding time. 17To our knowledge, such an investigation has not been performed yet for skeletal muscles. 18herefore, this study aimed to assess whether the pseudo-diffusion model (Bihan Model 1) or the ballistic limit (Bihan model 2) is more appropriate to describe IVIM exams in the human skeletal muscle.We used velocity-compensated diffusion gradients and conventional diffusion-encoding gradients with two different encoding times for this purpose.

Data acquisition
Diffusion-weighted images were acquired for both calves of 16 healthy subjects (seven females and nine males).The volunteers were aged between 18 and 26 years (mean: 22.8 years).Written informed consent was obtained from all volunteers.This study was approved by the local ethics committee.
All measurements were performed on a 3T MRI scanner (Magnetom Prisma; Siemens Healthineers, Erlangen, Germany) using the vendor-provided receive coils Body-18 and Spine-32.An in-house-developed single-refocused diffusion-weighted EPI sequence was used, 19,20 which allows an arbitrary diffusion gradient time profile (henceforth "gradient profile") specified in a text file to be applied (measurements from a free diffusion phantom can be found in Figure S1).The b-values were selected to be similar to a recent study 3 but with additional nominal b-values 1 and 5 s/mm 2 and without the b-value 800 s/mm 2 due to gradient amplitude constraints with the desired TE.In total, 24 DWIs were acquired with 17 different b-values (Table 1).Three different gradient profiles (short, long, and

F I G U R E 1
The short, long, and velocity-compensated (VC) diffusion gradient profiles used.velocity-compensated [VC]) were used to generate DWIs with a short (i.e., 19.7 ms) and long (i.e., 44.7 ms) diffusion gradient separation and with velocity compensation (Figure 1).The diffusion gradients were applied in the slice direction.Diffusion-weighted images were acquired once per b-value, with the exception of b = 100, 200, 400, and 600 s/mm 2 , which were acquired, 2, 2, 3, and 4 times, respectively.All other sequence parameters are provided in Table 2. Additionally, the vendor-provided 2D distortion correction option was selected to mitigate image distortions arising from gradient nonlinearities and a raw filter of medium strength to reduce Gibbs ringing.The prescan normalize option was used to compensate for surface coil flare.All images were acquired in transversal orientation.DWI sequences at rest and after activation were acquired in the following order: short, long, VC.Each sequence took 1:38 min.The total acquisition time for all three gradient profiles was 4:54 min.
The diffusion sequence was simulated with the vendor-provided tool POET to improve the accuracy of the b-values.These simulations provided the actual gradient-time profiles, which were used to compute the applied b-values, explicitly accounting for imaging and crusher gradients.The corrected b-values for each gradient profile are given in Table 1.
Volunteers were asked to forego sports for at least 48 h before the scan to avoid sore muscles.First, a set of baseline images was acquired.Next, volunteers were instructed to perform jumping jacks for 1 min to activate the calf muscles.Then, a second set of images was acquired.The time between the end of the jumping jack exercise and the start of IVIM data acquisition was approximately 10 min.

Data evaluation
All images were inspected visually for image quality and apparent artifacts.For both legs, gastrocnemius medialis (GM) and tibialis anterior (TA) muscles were segmented into regions of interest (ROIs) using the Medical Imaging Interaction Toolkit. 21The respective ROI was placed in the lowest b-value image and confirmed to match for all b-values.Due to the repositioning, separate ROIs were created for the measurements at rest and after activation.
To achieve better segmentation results, the T1w and T2w images were overlaid onto the DWIs and used for anatomical guidance.Major blood vessels and fascia were excluded from the ROI.The mean signal values of all voxels within an ROI were calculated for every slice and b-value.Images and ROIs before and after muscle activation were registered manually in the slice direction to compensate for repositioning using the anatomical reference images.Four middle slices were selected from the matched slices for IVIM parameter estimation (a comparison between evaluating two and four middle slices is provided in the Supporting Information as per request by the peer-review Figure S2).

IVIM parameter estimation
3][24][25][26][27][28] First, the mean signal values of the ROI were normalized to S(b 0 ), and the natural logarithm was applied.From data acquired with b-values 200-600 s/mm 2 , the diffusion coefficient D and perfusion fraction f temp were approximated with a linear fit.b-values that were acquired multiple times were treated as separate datapoints in the fit, as follows: ln An initial estimate of the pseudo-diffusion coefficient D * and S 0,fit was obtained from the data acquired with nominal b-values from 0 to 40 s/mm 2 with a linear fit, The sequence parameters for the different gradient profiles differ only in the gradient separation time.

Short Long Velocity Compensated
Sequence In the second fitting stage, the initial values were used as starting points for a Trust Region Reflective algorithm implemented in Python 3.9.13(scipy.optimize.least_squares()).The diffusion coefficient was held constant in the second stage.Bounds were set for the fitted signal intensity S 0,fit (45-150 a.u.), the perfusion fraction f (0-1), and the pseudo-diffusion coefficient D * (0.0001-3 mm 2 /s).
The mean effective flow velocity was estimated from the fitted pseudo-diffusion coefficients under the assumption of the ballistic limit and the Gaussian phase approximation, as follows 17 : This equation can be adapted for the velocity-compensated gradient profile to estimate a mean effective flow acceleration, under the assumption that spins which move with a constant velocity v do not contribute to the signal attenuation, as follows: where M i is the ith gradient moment of the gradient profile, and ⟨ v 2 ⟩ and ⟨ a 2 ⟩ are the quadratic expectation values of spin velocity and acceleration, respectively.Here, D * does not represent a true pseudo-diffusion coefficient but originates from a cumulant expansion of the phase distribution P() of the water molecules in the blood; and D * belongs to the  2 term.In principle, this expansion is always applicable, but the question of which b-values from the higher-order terms dominate the expansion is more intricate.Such an analysis is beyond the scope of this study, however.

Statistical analyses
Group averages were compared between gradient profiles with the nonparametric Friedman test, with a post hoc Tukey test to assess pairwise mean differences.Group averages were compared between the TA and GM and between rest and activation within a given muscle with the Wilcoxon signed-rank test.All analyses were performed in Python 3.9.12 using scipy (version 1.9.1) and statsmodels (version 0.13.2).A p-value of 0.05 was set as the statistical significance threshold.

RESULTS
Figure 2 shows representative diffusion-weighted images for all b-values of 1 volunteer.Image quality was generally good, and no relevant imaging artifacts were visible.A few images (e.g., image of b-value 25.45 s/mm 2 [green arrow in Figure 2]) show a signal dropout in the muscle tissue, of which most are located in the soleus muscle and hence not relevant to this study.Figure 3 shows the normalized, logarithmic signal attenuation curves averaged over all volunteers in the GM and TA after muscle activation.In the GM, the long and short gradient profiles showed a strong initial signal decay at low b-values (0-20 s/mm 2 ), whereas the VC gradient profile showed a much-reduced initial signal attenuation.
In the TA, the signal decay at low b-values was smaller for all gradient profiles.
Figure 4 shows boxplots of the diffusion coefficients for the GM and TA before and after activation.The median diffusion coefficient at rest was higher for the TA (median ≈ 1.88 μm 2 /ms) than for the GM (median ≈ 1.74 μm 2 /ms).The median values for the TA did not differ significantly among the different gradient profiles.The median diffusion coefficient increased from D ≈ 1.88 μm 2 /ms at rest to D ≈ 1.91 μm 2 /ms after activation.In the GM at rest, the median diffusion coefficient was higher for the VC gradient profile (D ≈ 1.77 μm 2 /ms) than for short and long gradient profiles (D ≈ 1.74 μm 2 /ms).The Friedman test indicated a significant difference, although this was not confirmed by the post hoc tests.In the GM, all three gradient profiles resulted in approximately equal diffusion coefficients after activation (median ≈ 1.87 μm 2 /ms).No significant difference was found.The diffusion coefficient increased by approximately 0.13 μm 2 /ms after activation.
Figure 5 shows boxplots of the perfusion fractions.In the TA at rest, the VC gradient profile had the lowest median perfusion fraction of f ≈ 2.3%, which was significantly lower than the f ≈ 3.2% of the long gradient profile (p < 0.05).In the activated TA, the perfusion fraction increased by approximately 2% for the short and long gradient profiles and by approximately 1% for the VC gradient profile.The Friedman test indicated a significant difference among gradient profiles, although this was not confirmed by the post hoc tests.The single slice values of the perfusion fraction showed a higher median absolute deviation in the TA than in the GM.The median perfusion fractions at rest did not differ significantly between the GM and the TA.In the GM at rest, the Friedman test suggested a significant difference among gradient profiles, although this was not supported by the post hoc tests.After activation, f increased more in the GM than in the TA.The increase was significantly smaller for the VC gradient profile (f ≈ 4.4%) than for the short (f ≈ 6.8%) and long (f ≈ 6.9%) gradient profiles.
Figure 6 shows boxplots of the pseudo-diffusion coefficients, which appear visually similar for the TA at rest, TA activated, and GM at rest: The short gradient profile had a lower D * than the long gradient profile, and the VC gradient profile had the lowest D * and interquartile range.In all three gradient profiles, the Friedman test indicated a significant difference, although this was not supported by the post hoc tests.D* showed a significant increase in the activated GM compared with the resting GM in the short (median D * ≈ 0.012 → 0.047 mm 2 /s) and Table 3 lists all group-averaged medians for the diffusion coefficient D, perfusion fraction f, and pseudo-diffusion coefficient D* and their respective median absolute deviations.Additionally, Table 3 displays the estimated velocities √ ⟨v 2 ⟩ according to Eq. ( 4), the estimated accelerations √ ⟨a 2 ⟩ according to Eq. ( 5), and the change in velocity Δv VC , which was calculated by multiplying √ ⟨a 2 ⟩ with the total duration of the VC gradient profile (50 ms).

DISCUSSION
We investigated the influence of velocity compensation of the diffusion-encoding gradient profile on IVIM parameters for the calf muscles.We found an overall reduced IVIM effect when using velocity compensation.This reduction was more pronounced after inducing an activated muscle state (i.e., there was a larger difference in the perfusion fraction f and the pseudo-diffusion coefficient D* between VC and non-VC encodings).The pseudo-diffusion coefficient D* also differed among the non-VC diffusion encodings of different durations.][16] The velocity compensation of the diffusion gradients leads to the rephasing of the magnetization of the water molecules moving with a constant velocity.Therefore, blood flowing coherently through straight vessels does not contribute to the initial signal decay at low b-values, resulting in a reduced IVIM effect compared with the conventional encodings, leading to lower fit parameters for the perfusion fraction f and the pseudo-diffusion coefficient D* (Figure 3).
The perfusion fraction f in IVIM has been interpreted to represent the T 2 -weighted blood volume fraction, 29,30 meaning that f is understood to be a tissue parameter and, as such, should not depend on the gradient profile.Therefore, it is important to differentiate between the true perfusion fraction f and the fit parameters f VC , f Short , and f Long .Although the f Short and f Long measured in this study agree with literature values and can be interpreted as T2w blood volume fractions, f VC was significantly lower and thus cannot be interpreted in this way.Rather, unlike f Short and f Long , it presumably contains information about the proportion of blood in the ballistic limit.For example, a perfect ballistic regime would be present for f VC = 0 and f Short > 0, [14][15][16] and a perfect pseudo-diffusion regime would be present for Long ⟩ are within physiologically reasonable limits. 31The calculated changes in velocity Δv from √ ⟨ a 2 VC ⟩ are of the same order of magnitude as the estimates for the velocity, which we deem somewhat larger than expected.For example, in the GM after activation, the estimated velocity for the long gradient profile is 1.59 mm/s.According to √ ⟨ a 2 VC ⟩ , the change in velocity in this case equals 1.46 mm/s.This might indicate pulsatile flow.Another possible explanation could be that higher-order terms of the cumulant expansion affect the IVIM signal decay, which is not captured in this estimate.
The effect of velocity compensation has been previously investigated.Notably, Ahn et al. 10 already reported in 1987 a difference in the amplitude of even and odd Boxplots of the perfusion fraction f.Each data point represents one slice of 1 volunteer.Not all outliers are shown.A red asterisk indicates a significant difference among the three gradient profiles (p < 0.05).A brace indicates a significant difference in the Tukey's honest significant difference (HSD) post hoc test.GM, gastrocnemius medialis; TA, tibialis anterior; VC, velocity-compensated.
spin echoes when flow is present.Although not acquiring spatially encoded images nor fitting attenuation curves to b-values, they were able to quantize the difference between flow and flow-compensated diffusion measurements and showed that only spins in the ballistic limit rephase for even echoes, hence effectively measuring f VC .
More recent studies often rely on more advanced modeling approaches to interpret velocity-compensated data.Wetscherek et al. investigated the effect of diffusion-encoding time on velocity-compensated encodings.They found that the efficacy of velocity compensation to reduce the low b-value signal drop decreased for longer diffusion-encoding times and used a random walk phase distribution model to interpret their data.With this model, they inferred correlation times of the blood flow.Ahlgren et al. 14 used another approach.They assumed the ballistic limit to be valid, effectively setting the correlation time to infinity.They found this approach to be valid in the human brain and could infer blood flow velocities.A limitation may be that the reduced perfusion fraction in the brain presumably makes it more difficult to assess the validity limits (2.4% in the brain 14 vs. >30% in the liver 15 ).Moulin et al. also used the Ahlgren model to infer blood flow velocities in the liver. 16They used gradients that continuously interpolate between bipolar and velocity-compensated diffusion encoding ("slider gradients" 32 ).In the spirit of multidimensional diffusion experiments, 33,34 this has the advantage that one can span two-dimensional space with the velocity-weighting and diffusion-weighting dimensions, allowing for a detailed analysis of the two effects.
Unlike these advanced models, we chose a more basic approach to establish whether the pseudo-diffusion limit or the ballistic limit is valid in the human calf muscles.Therefore, we avoided the common problem of these advanced models that they might not be entirely appropriate for the system under investigation (e.g., velocity distributions are known to be present, 35 but have previously often been ignored 35 ).
We clearly observed that velocity compensation reduced the small b-value signal decay-a hallmark of IVIM exams in the ballistic limit.Another hallmark of the ballistic limit is that the pseudo-diffusion coefficient should increase linearly with the diffusion-encoding time. 17We observed some increase in D* with diffusion time, but it was smaller than linear (i.e., D* Long /D* Short

F I G U R E 6
Boxplots of the pseudo-diffusion coefficient D*.Not all outliers are shown.A red asterisk indicates a significant difference among the three gradient profiles (p < 0.05).A brace indicates a significant difference in Tukey's honest significant difference (HSD) post hoc test.GM, gastrocnemius medialis; TA, tibialis anterior; VC, velocity-compensated. ≈ 1.47 instead of the expected value of ≈ 2 17 ).This finding would argue against the full validity of the ballistic limit.However, we believe that the assumption behind this linearity hallmark is unmet in our experiments.In particular, a wide spread of blood velocities, which may be expected in actual tissues, may strongly limit the b-value regime in which the cumulant expansion, which underlies the D* definition, is valid.Moreover, this observation may be explained by a larger proportion of the blood deviating from the ballistic limit at longer diffusion-encoding times.
We also investigated the change in the perfusion fraction f and the pseudo-diffusion coefficient D* after muscle activation.We showed that the jumping jacks exercise activated the GM more than the TA (i.e., a larger change in f and D * ).This particular exercise is not usually used in studies investigating muscle activation.However, we found that other common exercises, such as calf raises, while offering better controllability on paper (i.e., subjects performed a certain number of repetitions), differ greatly in perceived strain on subjects to a point where barely any activation was visible for some subjects, whereas others reported multiple days of strained muscle.This effect is no longer hyperemia.Instead, changes to the muscle microstructure (i.e., the diffusion compartment) might be present.Therefore, we opted for an exercise in which subjects controlled the perceived strain.The activation was visible in all investigated diffusion parameters and gradient profiles.The perfusion fraction, which is used commonly as a measure of activation, showed an increase in all muscles and gradient profiles, whereas the pseudo-diffusion coefficient only increased in the muscle targeted by the exercise (i.e., the GM), suggesting that D* might be a better indicator of changes to the perfusion in the muscle than the perfusion fraction f .This difference can be appreciated in Figure 6, where the activated GM showed visually different characteristics than all the other plots in Figure 6.
The main assumption this paper relies on is that there is no change in the IVIM parameters during the acquisition of all three gradient profiles.Otherwise, a time-dependence effect might be overlayed over the gradient profile effects.Other studies investigated the time dependence of the activation state of the muscle.Notably, Adelnia et al. 3 (thigh) and Filli et al. 6 (forearm) investigated IVIM at rest and at multiple time points after  .This difference might reflect a higher D* in the thigh muscles compared with the calf muscles, the fitting algorithm used, or the method for creating ROIs in the muscles.Another point to consider is the muscle fiber direction relative to the gradient direction.However, these reasons were not investigated in our study, and it is impossible to narrow down the cause of this difference without further investigation.
IVIM parameters might depend on the ambient and/or tissue temperature.Although the scanner room is air conditioned and kept under constant conditions, the temperature in the muscle tissue might change due to the muscle activation.We observed an increase of diffusivity in the GM by approximately 7%, which would correspond to a 3 • C temperature increase for free water. 36Yanagisawa and Fukubayashi 37 measured the ADC in the muscle after cooling of the whole calf for 30 min and reported the corresponding skin temperatures.Interpolating from their results, we again found that a temperature increase of 3 • C-4 • C could explain our increase in D. These temperature differences are of reasonable magnitude and could explain the observed changes of D.
Our interpretation for the increased D * and f is that, due to the exercise, an increased perfusion is achieved in the muscle.This can be explained by increasing volume and throughput of the blood in the targeted muscle (i.e., increasing f and D * ).
The statistical analysis for the diffusion coefficient suggested a significant difference to be present among the gradient profiles in the GM at rest.Although significant, we deem this result not fully reliable.We observed a change of median D-values of approximately 2% between the different gradient profiles, whereas within-subject coefficients of correlation for different muscles ranged from approximately 2% to 6% for DTI 38 and were larger than 10% for IVIM exams of the pterygoid muscle. 39nglund et al. published a review of 37 articles on IVIM in the skeletal muscle. 18These studies included several field strengths, muscle groups, and acquisition parameters, and some investigated diseased muscles.Englund et al. also proposed investigating velocity compensation and diffusion time dependence in muscle IVIM, as it proved insightful for other organs.They hypothesized that this might yield insight into the muscle's capillary segment length and flow velocity.These suggestions were partly investigated in our study.We were able to show a time dependence in D* in the activated muscle for the non-VC gradient profiles, indicating ballistic properties of the IVIM in the muscle at the investigated time scales.This finding indicates that Englund's hypothesis is correct and suggests that modeling of capillary segment lengths can be feasible.
Our study had some limitations.It was conducted using a custom-written sequence, which enabled the easy application of arbitrary gradient profiles, originally intended to investigate multidimensional diffusion encodings. 19,20Although allowing arbitrary gradient waveforms, the sequence had a minimum b-value of 1.44 s/mm 2 due to imaging and spoiler gradients not being optimized for IVIM imaging.Therefore, we were not able to resolve the quickly decaying components of the IVIM effect.To acquire all gradient profiles within a reasonable time frame, the sequence was adjusted so that it took only 1:30 min per gradient profile.The highest achievable b-value for our setup within that time frame was a nominal b-value of 600 s/mm 2 .
High-performance gradient systems, such as the one used for this study, can suffer from gradient nonlinearity effects, when acquiring large imaging volumes. 40,41We did not correct for such nonlinearity effects.However, the data evaluation in this study was performed on the slices closest to the isocenter, so the nonlinearity was kept as small as possible.
In some cases, the pseudo-diffusion coefficient was not separated from the diffusion coefficient by one order of magnitude (i.e., the VC gradient profile for the TA at rest).Although the blood diffusion coefficient has been included in the IVIM model in similar previous cases, 42 this was not done in our study to keep the model as simple as possible.
We could reproduce a selective increase in the IVIM parameters for all investigated gradient profiles.Although the increase was larger in the muscles targeted by the exercise (GM), an increase in the perfusion fraction was also observed in the antagonist muscle (TA) not targeted by the exercise.

CONCLUSIONS
The IVIM effect of the human calf muscle can be suppressed significantly using velocity compensation, suggesting that the ballistic limit is approximately-but not fully-reached.A time dependence was detected for the pseudo-diffusion coefficient, but theoretical estimates predicted a larger dependence if the ballistic limit was fully reached.Physiologically reasonable mean blood flow velocities v Short and v Long can be estimated for the respective gradient profiles from the pseudo-diffusion coefficient D * , suggesting that the model is not only a signal representation 43 but provides insight into muscular physiology.

F I G U R E 3
Group-average signal attenuation for the tibialis anterior (TA) and gastrocnemius medialis (GM) muscles after activation with jumping jacks.The behavior for small b-values is shown on the left.The entire b-value range is shown on the right.The velocity-compensated (VC) gradient waveform compensates for the initial signal attenuation in both muscles.The dashed lines show the mono-exponential extrapolation of the diffusion compartment.
long (median D* ≈ 0.015 → 0.069 mm 2 /s) gradient profiles (p < 0.05).However, it did not show a significant change in the VC gradient profile.Comparing D* Long and D* Short suggests a diffusion-encoding time dependence (D* Long / D* Short ≈ 1.47).

F I G U R E 4
Boxplots of the diffusivity D. Each data point represents one slice of 1 volunteer.A red asterisk indicates a statistical difference between the three gradient profiles in the Friedman test (p < 0.05).
Group medians and the median of the absolute deviations of all data points in the boxplots are reported.Abbreviations: GM, gastrocnemius medialis; IVIM, Intravoxel incoherent motion; TA, tibialis anterior; VC, velocity-compensated. between our study and Adelnia et al. is found in the pseudo-diffusion coefficient.For all measurements, Adelnia et al. reported pseudo-diffusion coefficients that are 1-2 orders of magnitude larger than those in our study.Although Adelnia et al. did not specify the gradient profile they used, all other acquisition parameters were comparable (specifically TR, TE, b-values, gradient direction Note: