# Noninvasive Determination of Shear-Rate Distribution Across the Arterial Lumen

## Abstract

*Abstract *In vitro experiments have shown that the shear stress exerted by flowing blood on the endothelial surface affects the morphology of the vascular wall and the release of vasoactive substances and growth factors by that wall. It is believed that the caliber of a vessel adjusts to the local shear stress to maintain a specific value of the shear stress. The local shear stress follows from local shear rate by multiplying shear rate by the local blood viscosity. The present article describes a method in which ultrasound techniques are used to assess transcutaneously the time-dependent wall shear rate in vivo in arteries. This method is applied to the assessment of wall shear rate in the common carotid artery of volunteers, presumed to be healthy, in two age categories (young age group, 20 to 30 years old, n=8; old age group, 60 to 70 years old, n=6). Although the peak shear rate in the young age group is markedly higher than in the old age group, the mean shear rate averaged over a cardiac cycle has the same value of 210 s^{−1} for both groups, corroborating earlier observations that mean shear rate and, hence, mean shear stress are maintained at a particular value. Conversion of the observed shear rates to shear stresses, assuming a blood viscosity of 3.5 mPa.s for both age groups, gives shear stresses of approximately 0.7 Pa. This is a factor of two lower than the shear stresses estimated from the relation between volume flow and artery caliber (1.5 Pa).

Consequent to the hemodynamic situation, a blood vessel is exposed to two main stress components. One is the local (pulsatile) blood pressure, which is expressed as a circumferential stress, and the other is the tangential stress exerted by the flowing blood on the endothelial cells of the arterial wall, a mechanical entity known as shear stress. The first stress can be derived directly by measuring blood pressure and vessel caliber (diameter and wall thickness), but reliable assessment of shear stress is quite difficult. For newtonian fluids, shear stress equals the local viscosity times the local wall shear rate, which can be derived from the measured or estimated shape of the instantaneous velocity distribution across the lumen.

There is increasing evidence that shear rate or shear stress is an important determinant of endothelial cell structure and function. Shear rate or shear stress makes the endothelial cells orient in the prevailing direction of flow^{1} ^{2} ; induces the production of endothelium-derived relaxing factor^{3} and prostacyclin^{4} ; and activates K^{+} channels.^{5} Moreover, the efficiency of the biochemical processes in endothelial cells depends to a high degree on the actual (pulsatile) wall shear rate or stress.^{6} It is interesting to note that endothelial cells can distinguish between cyclic strain and shear stress where the production of autacoids is concerned. For example, cyclic strain enhances the production of endothelin-1,^{7} whereas shear stress suppresses its production.^{8} Also, the dimensions and structure of the arterial wall are modified in relation to the prevailing shear stress.^{9} ^{10} ^{11} ^{12} ^{13} ^{14}

Most of the information available about the effect of shear rate or shear stress on the structure and function of the arterial wall or its components has been obtained through in vitro and animal studies, hampering definite conclusions regarding the role of these mechanical factors in vascular disorders, as seen in hypertension and atherosclerosis. Studies on shear rate in humans are scarce, mainly because adequate, noninvasive methods to assess wall shear rate in humans are lacking. In this study, we present a new approach to the noninvasive assessment of wall shear rate in humans.

To estimate the local wall shear rate, the velocity distribution v(r) as function of the radial position r generally is modeled as follows:

where v_{m} is maximum velocity at the center position r=0 and R is the radius of the artery. The bluntness factor n controls the shape of the velocity distribution. For n=2, the result will be a parabolic velocity profile, as may be observed for fully developed flow under stationary conditions (constant flow velocity). For increasing n, the velocity profile will become more and more blunt, as may happen in early systole and midsystole. The shear rate SR at the wall is given by the derivative of v(r) with respect to r at r=R:

Equation 2 can be given as function of volume flow *Q=v̅A* deriving from Equation 1 the velocity averaged over the cross-sectional area *A=πR*^{2}:

Substituting (3b) into (2) yields

For n=2 (parabolic velocity distribution), Equation 4 converts to the well-known Hagen-Poiseuille relation, which is applicable for steady flow through a rigid tube. The inverse relation of shear rate to the cubed power of the radius implies a high sensitivity: any change in mean volume flow rate can be compensated for by a relatively small change in vessel radius. The (mean) wall shear stress is the wall shear rate multiplied by the local blood viscosity.

In the past, the Hagen-Poiseuille relation frequently was used to estimate wall shear rate in the circulation. This requires the assessment of the volume flow Q and the local vessel radius R. Alternatively, one may use the law of conservation of blood volume at branches to estimate wall shear rate. In both cases, assumptions are made regarding the shape of the velocity distribution (parabolic velocity profile). Moreover, neither approach takes into account the effect of elasticity. It has been shown that the distensibility of the wall markedly reduces local wall shear rate.^{15} A direct method based on assessment of the radial derivative of the velocity distribution with a high-resolution multigate pulsed Doppler ultrasound system has the advantage that no assumptions are involved. This would require the development of an ultrasound system capable of measuring low blood flow velocities close to the wall with high precision and high spatial resolution.

Multigate pulsed Doppler systems can assess instantaneous velocity distribution along a single line of observation known as M-mode.^{16} High resolution along the ultrasound beam can be achieved if the duration of the emitted ultrasound bursts is short and the processing is adjusted to these short emissions (wide signal bandwidth with respect to the carrier frequency). Because of the wide signal bandwidth, combined with the frequency-dependent attenuation, the carrier frequency of the signal returned by a scatterer or reflector at some depth will deviate in an unpredictable way from the emitted center frequency, rendering conventional ultrasound Doppler systems inadequate. To reduce the inherent effect of the width of the ultrasound beam on the axial resolution, the angle of observation should be obtuse (greater than 60°). This has the adverse effect of causing more reverberations, resulting in spurious echo signals (clutter) that apparently originate from within the lumen. Moreover, the vessel walls will return signals with a higher amplitude, which will add to the clutter and severely mask the signals returned by the red blood cells close to the (anterior) wall. By use of a high-pass filter with a cutoff frequency related to the maximal anticipated Doppler frequency induced by the clutter, the blood Doppler signals at a relatively high Doppler frequency can be regained at the expense of a reduction of the available Doppler frequency (velocity) range.

Appropriate processing of the received radio frequency signal reduces the shortcomings of conventional pulsed Doppler systems, enabling the high-precision assessment of low-blood-flow velocities close to the wall.^{17} ^{18} The mean of the maximum of the radial derivatives of the velocity distributions at the anterior and posterior walls is used as an estimate for the local axial wall shear rate. The method, as described in “Methods,” was used in experiments to assess the (variations in) wall shear rate in the common carotid artery of young (20- to 30-year-old) and old (60- to 70-year-old) volunteers presumed to be healthy. Repeated measurements of wall shear rate indicate the within-session variability, whereas comparison of the results obtained for both age groups may lead to preliminary conclusions about the effect of age on wall shear rate and wall shear stress.

## Methods

### Assessment of Wall Shear Rate

As mentioned before, the basic requirements for reliable estimation of the velocity distribution in an artery, which enable shear-rate assessment, are (1) a large velocity range extending to low velocities, (2) low sensitivity for vessel-wall artifacts, (3) high spatial resolution, and (4) high-velocity resolution. Direct processing of the radio frequency signal based on the cross-correlation of corresponding segments (depth window) of the radio frequency signal over subsequent observations makes the velocity estimation insensitive to the carrier frequency. Moreover, the sensitivity to noise (velocity resolution) can be made low arbitrarily by extending the length of the depth window or of the window in time (number of subsequent observations or radio frequency lines). However, a long window in depth reduces the depth resolution and therefore reduces the assessment of velocities close to the wall, whereas a long window in time reduces the time resolution.

The location of the peak of the cross-correlation function, representing the mean velocities of the scatterers within the sample window, can be estimated by exploring the cross-correlation function over some depth shift interval. This procedure can be accelerated while at the same time the precision of the result is made, independent of the sample frequency of the radio frequency signal, by modeling the spectral power density distribution of the radio frequency signal.^{17} If one assumes that the spectral distribution has a gaussian shape, the shape of the cross-correlation function is known. A few points of the cross-correlation function in the vicinity of a depth and a time lag of zero sample points suffice to solve for the mean velocity and for the signal-to-noise ratio. To give the estimated velocity distribution a smooth and detailed shape, the length of the sample windows should match the resolution of the ultrasound system in echo mode (0.2 to 0.3 mm) along the ultrasound beam, while subsequent windows may partially (50%) overlap.

The cross-correlation model method is presently executed off-line in software on a depth and time matrix of radio frequency data points. In M-mode, conventional echo-Doppler systems do not provide a high pulse repetition frequency in combination with a large signal bandwidth to achieve a wide range to measure velocity with a good axial resolution. Therefore, during data acquisition, the emitter/receiver of the echo system is replaced by a separate combined emitter/receiver, which also supplies the reference signal for sampling. This implies that there is no (visual) control over the beam direction and signal level during acquisition. Sampling in depth is synchronized with the time of emission at a sample frequency of approximately four times the emission frequency. The dynamic range of the sample points is 10 bits (60 dB). To allow for the assessment of low amplitude values, a signal overload of the analog-to-digital converter of 12 to 18 dB (2 to 3 bits) is tolerated. Conversion starts synchronously with an electrocardiographic trigger. The size of the internal memory of the data acquisition system (presently four M samples) limits data acquisition to a depth range of 10 mm and a time range of 1 second at a 6-kHz pulse repetition frequency of the ultrasound system (emission frequency, 5 MHz).

To extend the velocity range to low velocities, the following processing scheme is used. First, the signal-to-noise ratio and the velocity of the raw signal are estimated.^{18} If the signal-to-noise ratio is high and the observed velocity is low, then a significant clutter component is present within the signal window. In that case, signals in time direction are corrected for the observed velocity, resulting in a phase alignment of all signal segments. A high-pass filter with a low cutoff frequency is then used to eliminate the clutter component. Because of the phase alignment, the cutoff frequency is now dictated by the bandwidth of the clutter component rather than by the maximal anticipated velocity of the slowly moving structures, allowing for a relatively low cutoff frequency independent of the phase of the cardiac cycle. The apparent result is an extension of the velocity range to low velocities of the order of a few centimeters per second. After clutter removal, the signal-to-noise ratio and mean velocity again are estimated by use of the cross-correlation model method. If the signal-to-noise ratio is sufficiently high (>−6 dB), the remaining signal originates from moving blood cells and the estimated velocity (after correction for the imposed frequency correction related to clutter suppression) is accepted; otherwise it is set to zero. This procedure is repeated for all partially overlapping sample windows, each with a length of 0.3 mm, spaced along the ultrasound beam at 0.15-mm range intervals, resulting in a reliable and detailed estimate of the instantaneous velocity distribution. Also, subsequent windows in time (with a length of 5 or 10 milliseconds) overlap by 50%. The resulting two-dimensional velocity distribution is smoothed by a 3×3 median filter to remove the occasional extreme velocity value. To further reduce fluctuations of the velocity estimates, the median filter is followed by a 3×3 sliding-window averaging filter (see Fig 1⇓).

The time-dependent shear-rate distribution directly follows from the observed velocity distribution by taking the absolute value of the radial derivative (Fig 2⇓). This provides high spatial peaks at sites with a high spatial velocity gradient. The shear-rate distribution shown is corrected for the angle of observation (60° to 70°, estimated from the B-mode) affecting both the velocity and the spacing along the beam. For each line in depth, the mean of the peak values at the anterior and posterior luminal sites is considered to be the estimate of the wall shear rate. Repeating the procedure for all available lines results in the shear-rate waveform (Fig 3⇓).

The shear-rate assessment can be extended with the assessment of the change in lumen diameter over time by manually placing sample windows at the anterior and posterior walls. On the basis of the cross-correlation model method the displacement of structures within these sample windows is estimated so that the position of the sample windows is adjusted to the observed velocity (tracking windows) in discrete steps set by the spatial sample distance. This procedure provides the displacement waveforms of both walls, whereas the difference between both waveforms reflects the change in luminal diameter over time (Fig 3⇑). The phase lag between wall shear rate and distension waveform is indicative of the compliance at the insonated section.

### Subject Population

To validate the shear-rate assessment through experimentation, two groups of male volunteers, presumed to be healthy, were recruited. Group 1 consisted of volunteers 20 to 30 years old (n=8) and group 2 of volunteers 60 to 70 years old (n=6). All volunteers gave written informed consent. The research protocol was approved by the ethics committee. Both groups were normotensive and had no evidence of cardiovascular disease.

After an acclimatization period of 20 minutes, during which the subject was in a supine resting position, the common carotid artery was visualized in B-mode (ATL Mark IV, 5-MHz mechanical sector scanner, Advanced Technology Labs) with the subject’s head tilted in the contralateral direction. The probe position was manipulated until a line of sight crossing the common carotid artery at an angle of 60° or 70°, 2 to 3 cm proximal to the tip of the flow divider, could be selected. After positioning the M-line, the system was switched to M-mode, and a pulsed Doppler registration was made with the sample volume located midstream. Subsequently the radio frequency data acquisition was activated. Before accepting and repeating a measurement, a quick check of the data (taking approximately 5 minutes) was executed. The probe was removed from the subject in between measurements, which led to the inherent problem of how to replace the probe in the same position and orientation (see “Discussion”). Basic requirements for acceptance were agreement of the peak systolic center stream velocities, obtained with the conventional pulsed Doppler system (ATL) and with the shear-rate system, and consistent and corresponding behavior of the shear rate and velocity waveforms.

For each measurement, the shear-rate distribution, the wall shear-rate waveform, and the distension waveform were computed. On the order of 20% to 30% of the registrations made had to be excluded because of radio frequency interference, motion artifacts (movements of the artery with respect to the probe), or an inadequate signal-to-noise ratio. All the examinations were performed by the second author (S.K.S.).

## Results

For each subject, the peak systolic velocity and shear rate; the mean of the maximum velocity and of the shear rate averaged over a cardiac cycle; and the end-diastolic diameter, velocity, and shear rate were assessed for both the left and right common carotid arteries at 2 to 3 cm proximal to the tip of the flow divider. Repeating the measurements enabled the coefficients of variation, defined as the ratio of the standard deviation and the mean of the estimate expressed as a percentage, to be estimated.

Tables 1⇓ and 2⇓ list the maximum velocity and its mean averaged over a cardiac cycle in the left and right common carotid arteries, respectively, for the young age group. The numbers in the “Subject” column refer to the year each volunteer was born. There is no substantial difference between the observed values for both arteries, although the coefficient of variation for the right artery is somewhat smaller than for the left artery.

Tables 3⇓ and 4⇓ give the observed shear rates in the left and right common carotid artery, respectively, of the young age group. “Minimum shear rate” indicates the end-diastolic value of the shear rate and is not necessarily the minimum value in the cardiac cycle. Again, both arteries show the same behavior.

Tables 5⇓ and 6⇓ give the velocities in the left and right common carotid artery, respectively, for the old age group, and the results presented may be compared with those given in Tables 1⇑ and 2⇑. In the old age group, both the maximum and mean velocities are lower than those of the young age group, although the decrease in mean velocity is only marginal.

Tables 7⇓ and 8⇓ list the observed shear rates in the left and right common carotid artery, respectively, for the old age group. Comparing the results to those given in Tables 3⇑ and 4⇑, it is evident that although the maximum shear rate decreases with age, the mean shear rate is maintained at the same level. This inference also follows from Table 9⇓, which summarizes the results.

## Discussion

Despite the widespread interest in (mean) wall shear rate and wall shear stress, relatively little is known about the values of these mechanical entities in vivo.^{13} ^{19} The basic problem in the noninvasive determination of wall shear rate in vivo concerns measurement of the velocity gradient close to the wall with a minimum of interference. Recently an article was published on the assessment of the instantaneous wall shear rate in the abdominal aorta of dogs, in which hot-film anemometry was used.^{20} However, the implantation of the transducers involves a far-reaching surgical procedure, making the method unsuitable for application to humans. Moreover, the measurement setup greatly affects the physiological condition, as may be inferred from the low relative change in aortic diameter observed in the present study.

With the use of dedicated ultrasound techniques, it is possible to assess in humans in a noninvasive way the instantaneous velocity distribution over time. High velocity and spatial resolution is achieved by capturing the received radio frequency signals over a limited time window (one cardiac cycle) and processing the signals off-line, paying specific attention to selective clutter suppression, ie, the rejection of signals induced by stationary and slowly moving reflectors such as vessel walls, resulting in an extension of the estimation range to lower velocities.

The resolution perpendicular to the axis of the artery is governed by the system bandwidth in combination with the angle of observation and the local width of the ultrasound beam. Since the present study used an angle of 70°, the major restriction for the radial resolution originates from the duration of the emitted ultrasound burst (here, two periods at 5 MHz each), the bandwidth of the transducer, and the smoothing procedures used to reduce the variance of the velocity estimates. Matching the sample windows to the burst duration gives windows corresponding to 0.3 mm and spaced at 0.15-mm intervals along the beam. Considering the angle of observation, the interspacing converts to radial increments of 0.14 mm. An observed wall shear rate of, for example, 1000 s^{−1} then corresponds to a peak velocity gradient between successive sample gates of 140 mm/s. This implies that starting from the wall at this shear rate a maximum velocity of 700 mm/s (young age group) can be reached within five sample gates or 0.75 mm, supporting the present claim about radial resolution. Some subjects with a relatively high systolic center stream flow velocity indeed showed peak shear rates of up to 1200 s^{−1} and mean wall shear rates of up to 300 s^{−1}, indicating that the system can resolve high spatial velocity gradients. Whether the two-dimensional smoothing filters have a limiting effect on the spatial resolution needs further investigation.

The importance of radial resolution for velocity measurements close to the wall can be inferred from the ratio of the shear rate observed at some arbitrary radial position r=βR(0<β≤1) and the theoretical wall shear rate. This ratio can be derived from Equation 2 and equals β^{n−1}. Now let us consider the situation at peak systole with bluntness factor n=3 at β=0.8 (corresponding to a measurement at 0.6 mm off the wall for a vessel with a diameter of 6 mm) the wall shear rate is underestimated by 36%. If β is increased to 0.9, however, the underestimation related to the site of measurement is reduced to 19%.

A disadvantage of the use of the hot-film anemometer for wall shear-rate assessment is its nondirectionality, which makes obtaining the time-averaged wall shear rate for biphasic flow patterns problematic.^{20} The wall shear-rate measurement system, based on processing of radio frequency ultrasound signals as presented in this study, has the same drawback. It is almost impossible to conclude from the observed instantaneous velocity distribution the wall positions and, hence, to measure wall shear rate. We therefore considered wall shear rate to be the maximum absolute value of the radial gradient of the axial velocity close to the wall-lumen boundary (within 2 mm). To eliminate the effect of radial blood flow velocities due to the compliant nature of the wall, the instantaneous axial wall shear rate can be obtained from the shear rates observed near both walls. Consequent to the absolute value operation, the shear rate should be unidirectional to allow for averaging over time. Since the method is presently applied to the flow behavior in the common carotid artery, the restriction on velocity direction does not create problems because no backflow is normally observed in this artery at 2 to 3 cm proximal to the flow divider.^{21}

Despite the large percentage of registrations that had to be rejected and the relatively high coefficients of variation, the results obtained are encouraging. For both sides (left and right common carotid artery), the same values are observed (Table 9⇑). By comparison of the old and young age groups, it can be concluded that the decreased maximum and mean velocities and maximum shear rate at older age does not result in a decreased mean shear rate. The reduced relative change in diameter, associated with a reduced elasticity of the arterial wall at older age, does not result in a higher mean wall shear rate because of the increase in diameter. This corroborates earlier observations about the possible interaction between mean wall shear rate and the caliber of the artery.^{9} ^{10} ^{14}

It is surprising to note that for both age categories the shear rates are quite low. Conversion of the observed mean shear rates to shear stresses, assuming a whole blood viscosity of 3.5 mPa.s, results in shear stresses of 0.7 Pa (7 dynes/cm^{2}), which is a factor of two lower than that obtained by use of the Hagen-Poiseuille approach, in which volume flow and vessel diameter were obtained separately by invasive methods. As follows from Equation 4, the Hagen-Poiseuille approximation may give an underestimation for not fully developed flow because the bluntness factor is assumed to be two (parabolic flow). In reality, especially in the systolic phase, the velocity profile appears to be quite blunted (bluntness factor in the range from three to four, unpublished results). For a fully developed flow distribution, as occurs far from the artery entrance or from bends, the time average will converge to a parabolic shape. Under that condition, the assumption about the bluntness factor is acceptable. On the other hand, the pulsatile nature of the volume flow in combination with the distensible artery walls may give a reduction of wall shear rate with a factor of 30% compared with rigid arteries.^{15} ^{22} Actually the observed shear rates are of the same order as those observed in in vitro simulations of the hemodynamics in the aortoiliac bifurcation.^{22} ^{23}

An explanation for the discrepancy between the observed shear rates and those reported in the literature can be related to the extrapolated use of Equation 4, derived for a stationary flow through a rigid tube, to a pulsatile flow through a distensible artery. In the latter case, the ratio of the temporal mean of the flow and the cubed power of the radius is not equal to the mean of the instantaneous ratios. The mean wall shear rate, as estimated by use of the technique in this study, should then be comparable to the mean of the ratios, which will be lower than the ratio of the means. Further detailed analysis of the results in a larger group of subjects is required to ascertain whether the computation method used explains fully the observed discrepancy.

The relatively large errors can be attributed to the replacement during data acquisition of the linear array system, because of its poor imaging quality, by a commercial B-mode ultrasound system in combination with a separate emitter/receiver. The external connections between the echo system, the emitter/receiver module, and the probe are susceptible to signal interference and affect the impedance matching of the probe to the system. In the present system, switching to the separate emitter/receiver while capturing data does not provide feedback to the examiner about the signal level and the quality of the registration. Only after signal processing is complete can the examiner decide whether the waveforms assessed are consistent with each other and whether the registration is acceptable.

### Conclusion

A method is described to noninvasively assess the instantaneous and mean wall shear rate in the human common carotid artery. By use of high-resolution ultrasound techniques and dedicated signal processing with respect to the rejection of signals originating from stationary and slow-moving reflectors and to blood velocity estimation, a detailed time-dependent velocity distribution across the arterial lumen is obtained. The maximum of the radial velocity gradients at the anterior and posterior walls yields the instantaneous shear rate and averaging over time yields the mean shear rate. Assessment of the wall shear rate in the common carotid artery of normotensive young and old volunteers presumed to be healthy revealed that although the peak shear rate decreases with age the mean shear rate is the same for both age categories, corroborating the hypothesis about the regulation of the mean wall shear rate and stress. However, the observed mean shear rate is a factor of two lower than that reported in earlier studies based on the extrapolation from steady flow through a rigid tube to pulsatile flow through a distensible tube. Apparently this extrapolation leads to overestimation of the mean shear rate and hence the mean shear stress exerted on the endothelial cells. On the other hand, it may still be possible that the present methods of signal processing, especially the smoothing procedures involved, have a limiting effect on the axial resolution, resulting in an underestimated velocity gradient at the walls.

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- Noninvasive Determination of Shear-Rate Distribution Across the Arterial LumenArnold P. G. Hoeks, Steven K. Samijo, Peter J. Brands and Robert S. RenemanHypertension. 1995;26:26-33, originally published July 1, 1995https://doi.org/10.1161/01.HYP.26.1.26
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