The blood velocity profile–fitting model described in equation 3
differs from the commonly used blood velocity profile equation
) is the velocity at radial position r
is the centerline velocity, and R
is the radius of the blood vessel cross section, K
is the bluntness index without any scale factor involved.17,18
Equation 7 forces the velocity at the wall to be 0, according to the nonslip boundary condition.37
Because equation 7
does not include a scale factor β, the bluntness index K
works differently from our bluntness index B
, defined in equation 3
, in describing the flatness of the blood velocity profile. Although the model of equation 7
worked for our larger vessels (lumen size, >100 μm) as reported by others,17,19,35
there were systematic fitting errors for medium or smaller vessels, including systematic underestimation of velocities near the vessel wall and overestimation of velocities near the lumen center. presents an example with a 72-μm artery. The dashed curve in A is the best fit for equation 7
and the solid curve for equation 3
, together with the residual errors in B. Fitting according to equation 3
(solid curve) generates significantly smaller mean residuals than does equation 7
= 0.0012 for a one-tailed t
-test with 95% CI). Equation 3 therefore fits both small and large vessel velocity profiles, since for the large vessels the scale factor β is small (). When the scale factor is 0, equation 3
becomes mathematically equivalent to equation 7
. However, for medium or smaller blood vessels, equation 3
offers a more accurate model of the blood velocity profile ().
Comparison between fitting models to an instantaneous blood velocity profile measured in a medium-sized retinal artery (72 μm).
The success of fitting by equation 3
does not imply that the nonslip boundary condition37
is incorrect. We are fitting the velocity distribution in the central core of the lumen, and because the plasma layer has no information on the vessel-wall–limiting condition. With no cells present the fluid viscosity of the near-wall region is presumably different from the rest of the lumen. As a result, the velocity distribution in the near-wall region should be modeled separately to satisfy the nonslip condition, especially in medium-sized or smaller vessels and not be modeled across the entire lumen as equation 7
does. Tangelder et al.38
have proposed a linear decline model for the blood velocity distribution in the near-wall region,38
which will fulfill the nonslip boundary condition. Our fitting model for the central core region, along with the linear decline model in the near-wall region, can constitute a two-tiered modeling method that may provide a better theoretical description of the velocity distribution across the whole lumen. However, for calculating the average cross-sectional velocity, using our fitting method described in equation 3
across the whole lumen does not introduce a large error, because the volume of blood in the near-wall region is small compared with the central volume. Even for the smallest vessel we studied (32 μm), the error is less than 5%, and the error is smaller in larger vessels.
In contrast, for calculation of wall shear, the nature of the velocity variation in the cell-poor layer near the vessel wall is important and necessitates separate modeling. The present model helps establish the boundary condition for the velocities at the interface between the near-wall region and central core of the blood vessel, which is essential for the linear decline modeling in the near-wall region.
The concept of scale factor β was first introduced by Tangelder et al.15
to address the problem described in . By assuming a symmetric blood velocity profile, their method of fitting can be expressed as:
They originally introduced β as a mathematical parameter with no physical meaning. We slightly modified their model, to link the scale factor β directly to the intercept value of the fitting curve against the vessel wall, as described in equation 3
. In this way, the scale factor can directly describe how much the velocity decreases from the centerline to the near-wall region, in other words, how “flat” the blood velocity profile is. When it comes to small retinal blood vessels, the scale factor plays an increasingly important role in influencing the flatness of blood velocity profile. The data in show that for veins smaller than 50 μm, although the bluntness index B
decreases, the scale factor β increases markedly with decreasing lumen size. Influenced by the two parameters, the velocity ratio η decreases with the lumen size, which indicates that the blood velocity profile gets flatter when the lumen size decreases, as reported by Gaehtgens et al.39
Our calculated velocity ratio, the profile flatness indicator, differs from the empiric velocity ratio introduced by Baker et al.40
The purpose of their velocity ratio is to adjust for the signal averaging artifact in double-slit photometric measurements. Their ratio of 1.6 is actually for Poiseuille flow, with a parabolic profile, and does not suggest flatter profiles. Our data are consistently flatter than predicted by Poiseuille flow suggesting that Poiseuille (laminar) flow does not apply in retinal blood vessels, although the deviation from parabolic profiles is smaller in large retinal vessels than in smaller retinal vessels. In retinal arteries, the blood velocity profile of at the diastolic cardiac phase is the flattest, which may result from more erythrocyte aggregation when the centerline velocity decreases, as reported by Bishop et al.18
The velocity ratio function described in equation 6
cannot directly apply to retinal arteries, because it does not consider the influence of cardiac pulsatility on velocity profile flatness.
We determined the blood velocity in blood vessels as small as 32 μm. For even smaller vessels the concept of a profile becomes difficult. Although we can measure velocity in these smaller vessels, individual blood cells occupy a significant proportion of the lumen and other constraints emerge for describing blood velocity including cell–cell interactions.
For the current system the line scan frequency is 20 kHz. This scan frequency represents a fundamental sampling rate for this technique, just as it does for the similar approach used by the Heidelberg flowmeter (Heidelberg Engineering Heidelberg, Germany),41
although the line scan frequencies for the two techniques are different. Directly related to this fundamental sampling rate, the upper limit of the measurable blood velocities for the current system is approximately 100 mm/s. However, this upper limit is only for the velocity components in the direction of the scanning line—in other words, the Vp
described in equation 2
. Since we can rotate our line scan to adjust the angle α between the scanning line and the vessel axis, the upper limit in our system is adequate for even the largest retinal arteries. Another simpler solution for improving the upper limit is to install a faster line scanner, which will help more directly in facilitating high-velocity profile measurements.
If the target blood vessel were oriented at an angle in depth, we would underestimate the absolute velocity. However, since the underestimation will apply to all data points in a blood velocity profile, this underestimation factor will be eliminated after the velocity measurements are normalized against the profile centerline velocity. For our data, the underestimation was small, because most of the vessels in the size range of interest lay approximately in the plane of the retina,42
and we chose regions where the vessels remained in good focus over a considerable lateral extent.