This document presents a velocity profile equation for blood flow in small arterioles and venules of small mammals in vivo. The equation includes two bluntness parameters, κ1 and κ2, which allow for better fitting of experimental velocity profile data compared to equations with a single parameter. The equation satisfies the assumptions of axisymmetric flow in cylindrical tubes, zero velocity at the wall, and a blunter than parabolic profile. The equation is evaluated based on 17 previously published velocity profiles from mice, rats and rabbits, with diameters ranging from 17 to 38.6 μm. Correlation coefficients for fitting the profiles were over 0.96, and estimated volume flow errors were low, indicating the equation can accurately describe microvascular velocity profiles

1. Clinical Hemorheology and Microcirculation 43 (2009) 321–334 321
DOI 10.3233/CH-2009-1243
IOS Press
A velocity profile equation for blood flow
in small arterioles and venules of small
mammals in vivo and an evaluation
based on literature data
Aristotle G. Koutsiaris a,b,∗
a
Hemodynamics Laboratory, Department of Vascular Surgery, Faculty of Medicine, University of
Thessaly, Larissa, Greece
b
Department of Medical Laboratories, School of Health Sciences, Technological Educational Institute
of Larissa, Larissa, Greece
Abstract. An empirical parametric equation with 2 bluntness parameters was introduced for describing the velocity profile
of blood in the small arterioles and venules of small mammals, in vivo, with the basic approximations of the axisymmetric
flow in cylindrical geometry, zero velocity at the wall and a blunter than parabolic flow profile. The purpose was to evaluate
the usefulness of this equation in describing the velocity profile and in estimating the volume flow when only one velocity
measurement is available near the vessel axis. The equation was tested on 17 velocity profiles (9 arteriolar and 8 venular)
previously measured by particle image velocimetry (PIV) techniques, at diameters ranging from 17 to 38.6 µm. The correlation
coefficients of each experimental profile were higher than 0.96. The average relative error-bias measured at 10 radial segments
ranged between −5% to 1%, leading to an average relative volume flow estimation error for all the 17 velocity profiles of −1.8%
with a standard deviation of 4.3%.
Keywords: Velocity profile, in vivo, microvascular hemodynamics
1. Introduction
The bluntness of the velocity profile of blood in comparison to a parabolic profile was observed and
quantified many years ago, first in vitro [2,6,8] and later in vivo [5,11–13,15,17]. The wall shear rates
(WSRs) by virtue of the blunt velocity profile, are higher than those expected by a parabolic one.
Pittman and Ellswotrh [13] proposed an oblate parabola equation with one bluntness parameter B, for
the description of the red blood cell (RBC) velocity profile in the microvessels of the hamster retractor
muscle. This equation cannot be used for the description of the velocity profile of blood, since the zero
slip condition on the vessel wall is not satisfied. The same is true for the Roevros’s equation [14] used
by Tangelder et al. [19].
*
Corresponding author: Dr. Aristotle Koutsiaris, Hemodynamics Laboratory, Department of Vascular Surgery, Faculty of
Medicine, University of Thessaly, 9 Miauli St., Larissa 41223, Greece. Fax: +30 2410 555378; E-mails: ariskout@otenet.gr,
ariskout@teilar.gr.
1386-0291/09/$17.00 © 2009 – IOS Press and the authors. All rights reserved

2. 322 A.G. Koutsiaris / Microvascular velocity profile of blood in vivo
A simplified form of the Roevros’s equation, satisfying the zero slip condition and employing only
one bluntness parameter κ, was used more recently [1,12]. However, it had already been shown [19] that
this equation tended to underestimate RBC velocities near the vessel axis.
Recently [5], a velocity profile equation was proposed, fitting very well particle image velocimetry
data from mouse venules. Neglecting the term responsible for the velocity description of blood on the
hydrodynamic interface between the glycocalyx and plasma, this equation comprises three different
terms (each depending non-linearly on another two independent parameters and on vessel diameter) and
a hyperbolic cosine term.
Here, an alternative empirical velocity profile equation is proposed depending directly on two blunt-
ness parameters κ1 and κ2.
The purpose of the present work was to use the proposed equation when only one experimental veloc-
ity point near the vessel axis is available and see to what degree of accuracy the velocity profile of blood
can be described and the volume flow can be estimated.
A relatively accurate profile description might be proved a useful tool for the experimentalists since a
full velocity profile measurement in vivo remains a difficult task. When only one velocity measurement
near the vessel axis is required, the measurement accuracy of the radial position is not so important
because of the local velocity profile flatness. In addition, axial velocity can be approximated by any
velocity measurement near the vessel axis. Also, flow tracing can be avoided since axial velocity can be
measured non-invasively by using the Doppler effect.
The efficiency of describing the actual velocity profile of blood, was tested using the criteria of the
correlation coefficient, the velocity relative error at 10 different radial segments and the volume flow
relative error, on 17 previously published velocity profile data by established researchers from mice
[5,11], rats [17] and rabbits [18,19].
2. Methods
2.1. The proposed equation
The analytical description of a fluid velocity profile necessitates the existence of a continuum, which
for the case of blood is true for microvessel diameters greater than ∼20 µm [4].
A velocity profile equation refering to blood flow in microvessels (arterioles and venules) should
satisfy the following requirements (or approximations): (1) blood behaves as a continuum, (2) the mi-
crovessels are cylindrical with a radius R, (3) the time averaged blood flow is axisymmetric with its
maximum value Vm on the vessel axis, (4) the blood velocity is zero on the vessel wall (zero slip condi-
tion), (5) the velocity profile is blunter than a parabola with the same Vm, i.e.: V (r) > Vp(r) ∀0 < r < R
and V (0) = Vp(0) = Vm and V (R) = Vp(R) = 0, where V (r) and Vp(r) are the proposed and the
parabolic flow velocity distributions respectively and r is the vertical distance from the vessel axis, and
(6) the term velocity V (r) refers to the average value of many cardiac cycles at the same phase.
The 5th requirement is based on experimental evidence collected in the past 5 decades (as men-
tioned in the Introduction) and can be satisfied by an equation using one independent parameter [1,12].
However, for a better description of the profile, it would seem logical to assume that 2 parameters are
needed: one affecting the bluntness near the vessel axis and one affecting the bluntness near the vessel
wall.

3. A.G. Koutsiaris / Microvascular velocity profile of blood in vivo 323
The general form of such an equation can be:
V (r) = Vm 1 − κ1
r
R
2
1 −
r
R
κ2
, (1)
where κ1 and κ2 are two parameters affecting the velocity profile shape with 0 < κ1 < 1 (first condition)
and κ2 > 2 (second condition). The profile shape reduces to the parabolic (Newtonian) when κ1 = 0
and κ2 = 2 which case here is excluded.
As κ1 approaches zero and assuming that κ2 > 2, the profile becomes flatter than a parabola near
the vessel axis (Fig. 1(a)). As κ2 takes values higher than 2 and assuming that 0 < κ1 < 1, the profile
becomes flatter near the vessel wall (Fig. 1(b)).
(a)
(b)
Fig. 1. The behavior of the proposed equation V (r) with respect to parameters κ1 and κ2 is shown in dashed black line. In
part (a), where κ2 = 5 (arbitrary number greater than 2), the flattening of the velocity profile near the vessel axis increases, as
κ1 is reduced from 0.8 down to 0.1. In part (b), where κ1 = 0.8 (arbitrary number less than 1), the flattening of the velocity
profile near the vessel wall increases, as κ2 increases from 5 up to 100. In part (c), the coordinates of the points between the
ordinate and the curve (1 − κ1)κ2 = 2, define the parametric pairs (κ1, κ2) which give V (r) > V p(r) ∀0 < r < R. In part (d),
the proposed equation with κ1 = 0.58 and κ2 = 22 is shown in dashed line. The standard deviation limits are shown in dashed
lines with black rectangles. The corresponding parabolic profile with the same axial velocity is presented with a solid black line
and is clearly sited outside the standard deviation limits.

4. 324 A.G. Koutsiaris / Microvascular velocity profile of blood in vivo
(c)
(d)
Fig. 1. (Continued.)
However, in some cases (for example κ1 = 0.8 and κ2 = 5, in Fig. 1(a and b)), the profile is not flatter
than the parabolic profile near the vessel wall and the 5th requirement is not satisfied. Therefore, a third
condition will be introduced in the following section.
2.2. WSR
The distribution of shear rates over the cross sectional area of the vessel SR(r) is given by differenti-
ating Eq. (1) with respect to the radial distance r:
SR(r) = −
Vm
R
κ2
r
R
κ2−1
+ 2κ1
r
R
− κ1(κ2 + 2)
r
R
κ2+1
. (2)
The WSR is the value of SR(r), at r = R:
WSR = −
Vm
R
(1 − κ1)κ2. (3)

5. A.G. Koutsiaris / Microvascular velocity profile of blood in vivo 325
An important ratio generally used as a near wall bluntness index, to indicate the deviation of a velocity
profile from a parabolic one near the vessel wall, is the ratio of the corresponding WSRs, namely Λ1:
Λ1 =
WSR
WSRp
=
(1 − κ1)κ2
2
, (4)
where WSRp is the wall shear rate of a parabolic velocity profile with the same maximum velocity of
Eq. (1). For a velocity profile blunter than a parabolic one, it is required that Λ1 > 1 or (1 − κ1)κ2 > 2,
which is the third condition for the 5th requirement of Eq. (1).
In summary, all three conditions for the fulfillment of the 5th requirement, are satisfied by the set of
points (κ1, κ2) sited between ordinate and the curve (1 − κ1)κ2 = 2 shown in Fig. 1(c). The points
located on the ordinate and on the curve (1 − κ1)κ2 = 2 are not included in this set.
2.3. Mean cross sectional velocity Vs
The mean cross sectional velocity is defined as:
Vs =
1
S S
V (r) ds, (5)
where S = πR2
. Solving this integral for Vs, with V (r) given by Eq. (1):
Vs = Vm
κ2[(1 − κ1/2)κ2 − κ1 + 4]
(κ2 + 2)(κ2 + 4)
. (6)
From the above equation, the ratio Λ2 of the maximum velocity Vm over the mean cross sectional veloc-
ity Vs, can be calculated as:
Λ2 =
Vm
Vs
=
(κ2 + 2)(κ2 + 4)
κ2[(1 − κ1/2)κ2 − κ1 + 4]
. (7)
The ratio Λ2 can be considered as a near axis bluntness index showing how the profile bluntness affects
the relationship between axial and mean cross sectional velocity.
2.4. Velocity profile data
The velocity profile data were taken from the sources shown in Table 1 and were photocopied and
magnified 4 times. A fine grid was plotted on each photocopy through which the velocity values corre-
sponding to the original data points were acquired. Then, all velocity data points were filtered using the
Damiano et al. [5] criterion according to which, velocity decreases monotonically with increasing r. In
this way, an optimal subset of the data was found, constituting the fluid velocity profile of the midsagittal
plane (Appendix).
In the velocity profile data taking into account the glycocalyx thickness [5,11], the free lumen defined
by the internal surface of the glycocalyx was considered as diameter.

6. 326 A.G. Koutsiaris / Microvascular velocity profile of blood in vivo
Table 1
Experimental velocity profile data
Profile D (µm) Data source (in vivo) Animal DC (µm) Microvessel Cardiac Measuring Flow
number tissue type phase technique markers
1 17 Tangelder et al. [19], Fig. 4 RBM 7 Arteriole D PIV FP
2 21.5 Damiano et al. [5], Fig. 2(b) MCM 5.7 Venule – PIV FMS
3 23 Tangelder et al. [18], Fig. 2 RBM 7 Arteriole D PIV FP
4 23.3 Long et al. [11], Fig. 21 MCM 5.7 Venule – PIV FMS
5 24 Tangelder et al. [19], Fig. 5A RBM 7 Arteriole D PIV FP
6 24 Tangelder et al. [19], Fig. 5B RBM 7 Arteriole S PIV FP
7 24.7 Sugii et al. [17], Fig. 7(c) RTM 6.5 Arteriole A APIV RBC
8 25 Tangelder et al. [19], Fig. 2A RBM 7 Arteriole D PIV FP
9 25 Tangelder et al. [19], Fig. 2B RBM 7 Arteriole S PIV FP
10 25.7 Sugii et al. [17], Fig. 7(d) RTM 6.5 Arteriole A APIV RBC
11 31.6 Long et al. [11], Fig. 24 MCM 5.7 Venule – PIV FMS
12 31.8 Long et al. [11], Fig. 17 MCM 5.7 Venule – PIV FMS
13 32 Tangelder et al. [19], Fig. 3 RBM 7 Arteriole S PIV FP
14 33.3 Long et al. [11], Fig. 15 MCM 5.7 Venule – PIV FMS
15 35.6 Long et al. [11], Fig. 26 MCM 5.7 Venule – PIV FMS
16 36.6 Long et al. [11], Fig. 16 MCM 5.7 Venule – PIV FMS
17 38.6 Long et al. [11], Fig. 3 MCM 5.7 Venule – PIV FMS
Notes: Experimental velocity profile data and the corresponding sources (third column from the left) were ordered according
to the microvessel diameter D (second column from the left). In the rest of the columns, more details are shown, like animal
tissue (RBM: RaBbit mesentery, MCM: mouse cremaster muscle, RTM: RaT mesentery), RBC diameter (DC), microvessel
type (arteriole or venule), arteriolar cardiac phase (D: diastolic, S: systolic and A: average from ≈13 cardiac cycles), measuring
technique (PIV: particle image velocimetry with resolution depending on marker size, APIV: automated PIV with resolution
depending on interrogation window size, here 1.8 × 1.8 µm) and flow marker type (FP: fluorescent platelet, FMS: 0.47 µm
Fluoresbright MicroSpheres and RBC: red blood cell). The mean RBC diameter was considered equal to 5.7 µm for mice [16],
6.5 µm for rats [3,9] and 7 µm for rabbits [7]. The mean diameter of 6.9 µm for rat RBCs measured in Ringer solution [3] was
reduced to 6.5 µm to take into account the presence of blood protein [9].
2.5. Estimation of average κ1 and κ2
Given that all the profile data of Table 1 refer to diameters between 17 and 40 µm, average values of
κ1 and κ2 were estimated as described in the following paragraphs.
Damiano et al. [5] and Long et al. [11] measured the ratio Λ1 in mouse venules of the cremaster
muscle, using a particle image velocimetry technique. The first group measured the velocity profiles in
9 diameters ranging between 19 and 31 microns and reported Λ1 = 4.2 ± 0.6 (standard deviation). The
second group measured the velocity profiles in 12 diameters ranging between 24 and 42.9 microns and
reported Λ1 = 4.9 ± 1.69. Here, Λ1 was considered equal to the average value (4.6 ± 1.36) of the ratios
reported by the two groups.
In another work [10], a profile factor function (PFF) was used for rabbits, to calculate the ratio Λ2 for
any microvessel to RBC diameter ratio (D/DC) greater than 0.6:
Λ2 = 1.58 1 − e−
√
2D/DC
. (8)
Here, it was assumed that the PFF holds for mice and rats as well provided that the correct RBC
diameter is used. As it is shown in Table 2, the ratios Λ2 given by the PFF, corresponding to the D/DC

7. A.G. Koutsiaris / Microvascular velocity profile of blood in vivo 327
Table 2
Velocity profile parameters and results
Profile number Λ2 (r/R)CPA VCPA (µm/s) Vm (µm/s) rp Qre (%)
1 1.41 0.080 2260 2268 0.985 −3.7
2 1.48 0.107 1025 1032 0.998 −3.7
3 1.46 0.233 4776 4931 0.981 −9.0
4 1.49 0.073 1885 1891 0.993 −0.9
5 1.46 0.190 5050 5158 0.962 −4.0
6 1.46 0.220 4375 4501 0.975 −4.3
7 1.48 0.060 3140 3147 0.968 −3.2
8 1.47 0.190 1610 1644 0.976 −3.7
9 1.47 0.045 2920 2923 0.994 −5.6
10 1.49 0.035 3080 3082 0.973 −4.6
11 1.52 0.049 800 801 0.988 5.4
12 1.52 0.120 1841 1856 0.988 −4.7
13 1.50 0.165 5100 5182 0.987 4.6
14 1.53 0.090 1950 1959 0.994 −0.7
15 1.53 0.011 863 863 0.984 3.5
16 1.54 0.027 1533 1534 0.983 −3.2
17 1.54 0.052 2227 2230 0.986 6.6
Average 0.983 −1.8
Standard deviation 0.01 4.3
Notes: Velocity profile parameters: the near axis bluntness index Λ2, the position (r/R)CPA and velocity
VCPA of the closest experimental point to the vessel axis. The maximum velocity Vm, the correlation
coefficients rp and the relative volume flow errors Qre for the experimental velocity profile data of
Table 1, are shown in the last three columns from the left. The average value and standard deviation of
the rp and Qre for all the 17 velocity profiles are shown in the last 2 lines of the table.
ratios taken from Table 1, did not vary much (1.41–1.54). Therefore, their average value (1.49 ± 0.04)
was used.
Putting the aforementioned average values of Λ1 = 4.6 and Λ2 = 1.49, into Eqs (4) and (7) respec-
tively, the solution of the corresponding pair of equations is κ1 = 0.58 and κ2 = 22 and thus Eq. (1)
becomes:
VF (r) = Vm 1 − 0.58
r
R
2
1 −
r
R
22
. (9)
Equation (9) is shown in Fig. 1(d) together with a parabolic profile with the same Vm for comparison.
Using the range of the Λ1 and Λ2 values determined by their standard deviations, the corresponding
standard deviation ranges of κ1 and κ2 were determined (0.44 κ1 0.65 and 12 κ2 34). In
consequence, the standard deviation velocity profile limits of Eq. (9) were drawn (Fig. 1(d)).
2.6. Correction of Vm
The estimated average values of κ1 and κ2 were treated as static parameters defining Eq. (9). The only
parameters left in this equation are measurable quantities: the vessel radius R and the maximum or axial
velocity Vm.

8. 328 A.G. Koutsiaris / Microvascular velocity profile of blood in vivo
In case there is no velocity measurement exactly on the axis, given the relative flatness of the profile,
Vm can be approximated by the velocity VCPA of the closest experimental point to the vessel axis, using
Eq. (9):
Vm =
VCPA
[1 − 0.58(r/R)2
CPA][1 − (r/R)22
CPA]
, (10)
where (r/R)CPA is the normalized radial position of the same point.
In this way, the filtering criterion is always satisfied. The coordinate pairs [(r/R)CPA, VCPA] for all the
velocity profiles of Table 1, are shown in Table 2.
2.7. Correlation coefficient
The correlation efficiency was evaluated for each velocity profile separately by the classic correlation
coefficient (Pearson) rp which was used as an approximate first order evaluation index.
2.8. Error evaluation
2.8.1. Velocity relative error RE
Because of the high variance in the absolute velocities among the various velocity profile data, the
relative error (RE) for all the experimental points of each experimental profile was estimated:
RE(r) =
VF (r) − Experimental Value
Experimental Value
100%. (11)
In order to see if and how the RE(r) changes along the profile line, the normalized radius was divided
into 10 equal segments:
j − 1
10
r
R
<
j
10
, (12)
where j is an integer (1 j 10). By selecting not more than 10 segments, the number of experimental
points Nj was higher or equal than 10 for every j, as it is shown in Fig. 2. The maximum number of
points taken from each velocity profile was 2 at radial segments with j 5 and 4 at radial segments
with j > 5. The total sum of the velocity points of all the velocity profiles was 227. One point of the
velocity profile number 10 was repositioned closer to the vessel wall because of the very low velocity
(100 µm/s) occurring in the vicinity of the internal surface of the glycocalyx [5].
2.8.2. Volume flow relative error Qre
The relative volume flow error Qre for each velocity profile was calculated by the following equa-
tion:
Qre =
Qe
Q
=
10
j=1 REjVFj Sj
10
j=1 VFj Sj
, (13)
where Q is the reference volume flow, estimated from the velocity profile equation normalized
with respect to the maximum velocity Vm and Qe is the volume flow error. For each radial seg-

9. A.G. Koutsiaris / Microvascular velocity profile of blood in vivo 329
Fig. 2. The number of experimental velocity points N per radial segment j from all the 17 velocity profiles of Table 1.
ment j of each velocity profile, the following terms were used: (1) the cross sectional area: Sj =
π0.12
[j2
− (j − 1)2
], (2) the relative error REj defined as the average RE of each experimen-
tal profile, at segment j, and (3) the velocity at the center of each segment: VFj = VF [0.05 +
(j − 1)0.1].
3. Results
Nine arteriolar velocity profiles were taken from seven different arterioles with diameters ranging
from 17 to 32 µm and 8 venular velocity profiles were taken from eight different venules with diameters
ranging from 21.5 to 38.6 µm.
For each of the velocity profile data of Table 1, the maximum velocity Vm and the correlation coeffi-
cients are shown in Table 2. The correlation coefficients ranged between a minimum of 0.962 (profile 5)
and a maximum of 0.998 (profile 2) with an average value ( rp ) of 0.983 ± 0.01. The profiles 5 and 2
together with the proposed equation are shown in Fig. 3.
In each radial segment j, the average and standard deviation of RE ( RE j and RESDj
, respectively)
of all the 17 velocity profiles of Table 1 was estimated. The RE j and RESDj
are shown in Fig. 4, where
the RE j ranged between 1% and −5% and the RESDj
11% except for the 10th segment where it
reached the value of 25%.
The relative volume flow error Qre of each velocity profile is shown in Table 2. The average value
and standard deviation of all the 17 experimental profiles are shown at the end of the Table 2: −1.8 ±
4.3%.
4. Discussion and conclusions
The purpose of the paper was to introduce an equation for the description of the velocity profile and
the estimation of volume flow of blood when only one velocity measurement is available near the vessel

10. 330 A.G. Koutsiaris / Microvascular velocity profile of blood in vivo
(a)
(b)
Fig. 3. A graphical presentation of the correlation of the proposed equation with velocity experimental points which are shown
as black dots. In part (a) the points of profile number 5 are shown, whose correlation coefficient rp was minimum (see Table 2)
and in part (b) the points of profile number 2 are shown, whose correlation coefficient rp was maximum (see Table 2).
axis. This equation was tested on 17 previously published experimental velocity profile data [5,11,17–
19] ranging in diameter from 17 to 38.6 µm.
The correlation coefficients of the proposed equation with the experimental velocity profile data of
Table 1, were higher than 0.962 and the average relative volume flow error was −1.8% with a standard
deviation of 4.3%, as it is shown in Table 2.
The average RE ( RE ) of the experimental velocity profiles of Table 1, shown in Fig. 4, ranged
between +1 and −5% at all radial segments. In seven out of the 10 segments, the average RE was
negative, resulting in a slight total negative bias which was the cause of the aforementioned volume flow
error.

11. A.G. Koutsiaris / Microvascular velocity profile of blood in vivo 331
Fig. 4. The average (gray columns) and standard deviation (black bars) of the velocity relative error RE of all the velocity
profiles of Table 1, at each radial segment j.
The RE was minimum in the first segment (j = 1) because Vm was approximated using the velocity
point closest to the vessel axis (Eq. (8)). The distance from the vessel axis was less than 0.24R for all
these points, as it is shown in Table 2.
The RESDj
measuring the scatter of the RE increases to a value of 25% at the 10th radial segment. This
rise of the RE scatter at the radial segment closest to the vessel wall was presumably expected because
of the very steep velocity gradient and the presence of glycocalyx contributing to the random motion of
the flow tracers.
A limitation in the application of Eq. (9) to the microcirculation of different mammal species may
arise from the fact that some of them, called ‘athletic’ species, such as humans and horses, exhibit much
higher RBC aggregation than small mammals do [1,20]. However, RBC aggregation is a phenomenon
occurring at very low or zero shear rate conditions (SR 3 s−1
[20]), meaning that it is a region limited
phenomenon near the vessel axis.
Once a description of the velocity profile exists, many other blood flow characteristics, such as the
shear rate profile, the pressure gradient, the shear stress profile and the viscosity profile can easily be
estimated under certain assumptions [5]. Then, the wall shear stress and the relative apparent viscosity
can also be estimated which are physical quantities with clinical importance.
The axial velocity required for the subsequent velocity profile description has the advantage that it can
be measured non-invasively in the microcirculation by using the laser Doppler effect.
In conclusion, an alternative velocity profile expression was proposed for blood flow, in small arteri-
oles and venules of small mammals, in the case of a reliable axial velocity measurement. The proposed
equation can describe velocity with a maximum positive bias along the diameter of +1% and a maxi-
mum negative of −5%. The maximum RE scatter of 25% occurs near the vessel wall and the average
relative volume flow error is −1.8 ± 4.3%.

12. 332 A.G. Koutsiaris / Microvascular velocity profile of blood in vivo
Appendix. Filtered velocity profile data
Table A
Profile number
1 2 3 4 5 6
r/R V r/R V r/R V r/R V r/R V r/R V
0.080 2260 0.107 1025 0.233 4776 0.073 1885 0.190 5050 0.220 4375
0.165 2175 0.172 1010 0.378 4670 0.103 1880 0.470 5020 0.330 4300
0.240 2160 0.414 965 0.483 4610 0.225 1770 0.515 4420 0.510 4025
0.275 2075 0.502 915 0.529 4540 0.469 1760 0.610 4185 0.730 3690
0.360 2025 0.572 840 0.644 4190 0.688 1495 0.664 3600 0.775 3165
0.410 1950 0.712 740 0.694 3970 0.810 1247 0.715 3530 0.785 2990
0.700 1845 0.823 665 0.706 3620 0.858 1080 0.790 3275 0.875 2780
0.750 1750 0.860 630 0.747 3560 0.883 950 0.850 3025 0.890 2725
0.790 1660 0.916 525 0.792 3380 0.933 725 0.940 2425 0.900 2200
0.865 1435 0.930 475 0.865 2930 0.956 470 0.955 2330 0.912 2125
0.875 1290 0.947 410 0.888 2756 0.973 318 0.925 1775
0.918 1215 0.963 350 0.894 2588 0.950 1320
0.930 1060 0.967 275 0.953 2300
0.940 925 0.977 250
0.960 800
Table B
Profile number
7 8 9 10 11 12
r/R V r/R V r/R V r/R V r/R V r/R V
0.060 3140 0.190 1610 0.045 2920 0.035 3080 0.049 800 0.120 1841
0.125 3125 0.525 1560 0.285 2810 0.275 3075 0.209 786 0.157 1711
0.272 3120 0.600 1480 0.535 2640 0.330 3050 0.355 731 0.281 1670
0.330 3075 0.740 1345 0.538 2520 0.410 3020 0.507 667 0.353 1664
0.395 3030 0.750 1275 0.570 2420 0.465 3010 0.587 630 0.531 1617
0.465 2940 0.780 1150 0.630 2360 0.525 2950 0.619 595 0.658 1555
0.530 2860 0.785 1100 0.675 2310 0.590 2840 0.692 588 0.765 1227
0.600 2800 0.870 1030 0.780 2075 0.650 2715 0.704 513 0.854 1195
0.670 2650 0.920 860 0.820 1950 0.715 2625 0.774 492 0.909 939
0.725 2455 0.930 750 0.830 1835 0.770 2475 0.784 412 0.959 750
0.790 2190 0.940 565 0.860 1675 0.840 2090 0.877 363 0.962 554
0.850 1760 0.950 450 0.910 1490 0.933 855 0.901 323 0.969 500
0.920 1075 0.960 378 0.925 1260 0.993 100 0.947 259 0.982 356
0.962 1085 0.975 154
0.970 980

13. A.G. Koutsiaris / Microvascular velocity profile of blood in vivo 333
Table C
Profile number
13 14 15 16 17
r/R V r/R V r/R V r/R V r/R V
0.165 5100 0.090 1950 0.011 863 0.027 1533 0.052 2227
0.330 4775 0.138 1925 0.297 841 0.328 1525 0.317 2190
0.400 4630 0.195 1910 0.410 770 0.437 1450 0.481 2030
0.490 4550 0.378 1825 0.495 726 0.618 1300 0.505 1757
0.550 4475 0.489 1680 0.567 677 0.662 1225 0.539 1692
0.755 4000 0.604 1560 0.640 654 0.745 972 0.644 1618
0.765 3240 0.637 1470 0.724 619 0.791 960 0.712 1610
0.820 2780 0.640 1425 0.731 519 0.839 860 0.741 1300
0.870 2625 0.745 1375 0.774 511 0.882 779 0.849 1216
0.880 2375 0.796 1190 0.779 435 0.902 764 0.894 860
0.890 2080 0.859 1025 0.882 412 0.925 705 0.955 518
0.925 1850 0.909 920 0.908 344 0.954 700 0.985 220
0.950 1220 0.946 760 0.953 315 0.974 407
0.960 850 0.970 640 0.966 162 0.984 300
0.986 325 0.981 146
0.987 115
Note: Filtered velocity profile data of the velocity profiles of Table 1, where r/R is the normalized radial position and V is the
velocity in µm/s.
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