Brief Review
An Examination of the Measurement of Flow
Heterogeneity in Striated Muscle
Brian R. Duling and Deborah H. Damon
T
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ranscapillary exchange of small solutes is dependent on blood flow; exchange is also dependent on the uniformity with which the inflowing blood is distributed among the exchange
vessels. A nonuniform or "heterogeneous" distribution
of flow among the vessels of an organ has been invoked to explain such diverse phenomena as flowlimited muscular performance,1 suboptimal capillary
transport of small solutes,2 pathology of hypotension, 34 and heterogeneous tissue Po 2 distributions.5
Thus, it is apparent that a clear delineation of the
amount of microvascular flow heterogeneity and its
control in the microcirculation is essential to our understanding of the interplay between tissue function
and vascular perfusion.
Measurements of heterogeneity are derived from a
variety of experimental approaches, and, unfortunately, the nature and limitations of methodology used in
one type of experiment are often not fully appreciated
by investigators using other methods. The purpose of
this review is to summarize briefly the present state of
quantitative information on flow heterogeneity in the
microcirculation of striated muscle. Our most important aim will be to compare the various methods for
assessment of flow heterogeneity and to relate them so
that nonexperts in the use of the various techniques can
perceive what is measured in each case. The report is
not meant to be a comprehensive examination of heterogeneity, whether spacial, segmental, or temporal.
Rather, we focus on one aspect of heterogeneity in the
peripheral vasculature: flow heterogeneity. It will be
shown that confusion in terminology exists, that the
different methods used to measure heterogeneity are
often not equivalent either quantitatively or qualitatively, and that a number of key measurements are
lacking if we are to specify the occurrence and significance of flow heterogeneity.
The review is divided into three sections: 1) exploration of definitions of flow heterogeneity and examinations of methods used in its study; 2) summarization of
existing measurements of heterogeneity in striated
muscle; and 3) analysis of quantitative relations among
From the Department of Physiology, University of Virginia,
School of Medicine, Charlottesville, Virginia.
This work was supported by NIH Grant HL 12792.
Address for reprints: Dr. Brian R. Duling, Department of Physiology, University of Virginia, School of Medicine, Box 449, Charlottesville, VA 22908.
Received January 8, 1985; accepted October 6, 1986.
microvessel flow heterogeneity and capillary transport
of small solutes.
Measurement and Potential Sources of Flow
Heterogeneity in Striated Muscle
DEFINITIONS. Flow is the convective movement of
blood, red cells, or plasma through the vasculature
(often referred to as bulk flow). The vasculature is
composed of many parallel flow pathways between
artery and vein, and flow heterogeneity means simply
that the total inflow is not distributed identically
among the perfused vessels. In the case of perfectly
homogeneous perfusion, all vessels would receive \ln
of the total inflow, where n is the number of any class
of vessels arranged in parallel. To the extent that some
vessels receive more and some less flow than their
appropriate fraction of the total, the flow is judged to
be heterogeneous. Following prior discussion, we refer to such heterogeneity as the relative dispersion of
the flow.6
The relative dispersion can only be accurately assessed using a combination of two statistical parameters: an index of the central tendency of the data and
an index of the dispersion. The mean flow is a measure
of the former, and the standard deviation is a measure
of the latter. The quotient of the two, the coefficient of
variation, is a convenient measure of the relative dispersion or the true heterogeneity of flow.
It must be appreciated that the relative dispersion
will not accurately reflect the range of flows among
vessels in a heterogeneously perfused tissue. The
range or total dispersion of flow is subject to two
determinants: the relative dispersion (defined above)
and the absolute value of the total flow entering the
vasculature. The combined magnitude of the two
sources of flow dispersion is often expressed by a
single measurement, the standard deviation, which we
refer to as the absolute dispersion of the flows.
The importance of making the definitions and distinctions between the two aspects of dispersion and
heterogeneity can be appreciated by considering the
result of increasing total flow into a vascular bed with
constant relative dispersion (true heterogeneity). The
standard deviation of the flow will increase in direct
proportion to the mean flow, and the increase in standard deviation might be misinterpreted as an increase
in the inequality of flow distribution among the vessels
of the network. Such a misinterpretation is forestalled,
however, by computing the coefficient of variation.
Circulation Research
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Since both mean and standard deviation increase in
proportion to the change in total flow, the coefficient of
variation will remain constant if there is a constant
fractional distribution of flow among the microvessels,
i.e., constant relative dispersion. Alternatively, at a
constant level of mean flow, fractional flow distribution might be altered by redistributing flow among
patent vessels, and the coefficient of variation will
show the change in relative dispersion.
Other important information regarding flow dispersion may be gained through an analysis of the histograms describing the sample distribution. In principle,
it is possible that the coefficient of variation could
remain constant during an experimental intervention
while the shape of the sample distribution would
change, indicative of a redistribution of flow. This
possibility would be manifested as a change in the
shape of the histograms. 578
The importance of distinguishing between the absolute and the relative dispersions in an examination of
microvascular flow stems from the fact that the three
variables—total flow through the organ, the number of
vessels perfused, and the fractional distribution of flow
Vol 60, No 1, January 1987
among perfused channels—may be altered as semiindependent variables by physiological controls and
pathological events each with different implications
for function and each requiring separate analysis if
vascular control is to be fully understood.
We turn now to a comparison of the various methods
used to assess flow heterogeneity and to some comments on their strengths and weaknesses. Many of the
comparisons are summarized schematically in Figure
1, which shows the relation of the various flow measurements to each other; the relations among such factors as transit times (Sections I and II), volume, and
flow in the various vessel segments (Sections I and II);
and to observations using in vivo microscopy (Section
HI).
TISSUE DEPOSITION INDICATORS. Several different indicators are deposited in the tissue in proportion to the
rate at which they are convected into a region by the
inflowing blood, e.g., tritiated water, antipyrene, and
microspheres. A common means of assessing perfusion heterogeneity is based on the use of such indicators followed by careful dissection of the tissue into
small samples and subsequent estimation of flow in
Microsphere
—£gjr f ^""^^r^ir
t=
(Eqn.
FIGURE 1. Schematic illustration of different aspects of the measurement of flow heterogeneity. Section I illustrates a group of
capillaries in which the capillary perfusion heterogeneity is estimated from the transit time of an intravascular reference indicator;
tfor each capillary is proportional to V/F (volume/flow) and the mean transit time for the group is the flow-weighted average transit
time. Section II illustrates the paired use of an intravascular reference indicator and a diffusible indicator to determine the capillary
exchange capacity, PS. Note that the region in which exchange occurs (stippled area) is not confined to the same portions of the
vasculature as is traced by the reference indicator (hatched area). Most important, the transit time for the reference indicator is
influenced both by the flow heterogeneity and by the volume heterogeneity in precapillary and postcapillary volumes (Equation 1 and
text). Section III shows a measurement that might be made with in vivo microscopy. Red cell transit times can be converted to transit
times measured in Section I only by flow weighting the individual transit times. A microsphere is shown lodged in an arteriole to
emphasize that the sphere distribution needjiot be equivalent to either the distribution described by the tracers or that described by
direct observation.
Duling and Damon
Flow Heterogeneity in Striated Muscle
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each bit of tissue from the quantity of indicator
trapped. Although these techniques provide a direct
measurement of flow heterogeneity, they are limited in
spatial resolution. Counting statistics make accuracy
inversely proportional to the tissue sample size; the
size of tissue in which heterogeneity can be detected is
dependent on the specific activity of the indicator and
the amount of indicator that can be placed in the tissues. This limit is typically reached with a mass of
tissue that contains many capillaries, arterioles, and
venules, thus making it difficult to know how accurately microvessel flow may be traced by the tissue deposition method. Note also that microspheres9 probably
trace flow heterogeneity in arterioles, where they are
lodged in Figure 1,9 but not necessarily the flow heterogeneity of the exchange vessels.
VISUAL OBSERVATION OF FLOW PATTERNS. The flow
distribution of blood within an organ can be directly
visualized using x-ray observation of contrast media or
in vivo microscopy. Because of our interest in microvessel function and flow heterogeneity, we focus
on the latter technique. Estimates of microvessel flow
are derived from the observation of red cells or the
motion of labelled plasma or other visible indicators
placed within the vasculature. The most important attribute of these techniques is that the main exchange
microvessels can be placed under direct observation
and that flow heterogeneity can be assigned to specific
microvascular segments (Figure 1, Section III).
While providing precise information about capillary
function and perfusion, in vivo microscopy is also
limited in many ways. Even though velocity dispersion
in individual microvessels can be accurately measured, flow can be estimated only by assuming that red
ceil velocity has a constant relation to the mean velocity and to the capillary cross section (Figure 1, Section
III), assumptions that have yet to be validated in the in
vivo situation and which may be in doubt.10"13
Perhaps the most limiting aspect of the direct observation method is that it is usually applied to specialized
tissues that are either superficial or quite thin and may
not be representative of tissues studied by other techniques. These tissues are exposed and often supervised
with solutions of poorly defined gas composition. As a
result, experiments based on in vivo microscopy may
be more likely to be associated with tissue disturbance
and altered vascular perfusion patterns. 1314 Whereas it
has been shown that total muscle flow need not be
markedly altered if the tissue is prepared with adequate
care,14"16 there is no such demonstration for possible
changes in flow distribution or heterogeneity.
In addition to flow heterogeneity, transit-time heterogeneity can also be measured using in vivo microscopy in conjunction with labelled cells or plasma. This
is of some significance, since the indicator dilution
techniques (described below) are based on the evaluation of vascular transit-time measurements (Figure 1;
Section III, Equation 1). Since the indicator dilution
techniques can be carried out on intact undisturbed
tissues, the possibility of making transit-time measurements on a microvessel level provides an opportunity
for making direct comparisons between the two techniques; however, discretion must be exercised in relating the two types of measurement. Measurement of the
average time required for red cells to pass from an
arteriole to a venule and of the statistical distribution of
such individual transit times17 is not the same as the
mean transit time determined with an intravascular
dilution indicator, even if data are obtained on the
microcirculation.18 This follows from the fact that the
average of the transit times for individual cells is not
flow weighted. In this study, the time the red cells are
in the capillaries is designated the sojourn time. 19 To
convert the average sojourn time to the more commonly used mean transit time, i.e., to a measure comparable to that obtained by studying indicator washout patterns, the red cell sojourn times must be flow weighted
for each flow path, a calculation only rarely made.17
Microcirculatory data have also been used to estimate transit time by dividing the total flow by the
estimated number of capillaries among which the flow
is distributed and then assuming values for the capillary size and red cell velocity.2021 Such a calculation
assumes that the number of parallel paths through
which blood flows is well defined. This is not always
true, however, since even the definition of capillary is
equivocal.8 Furthermore, the computation is likely to
yield improbable results. 1020
Perhaps most important in attempting to reconcile
data obtained by microvascular observation with those
obtained by indicator dilution measurements is the fact
that, even if the flow paths were accurately defined,
microvessel observation can only yield data on the
heterogeneity of transit times among the smallest vessels, and this need not correspond to a measurement of
transit time through the organ, which includes passage
through many classes of vessels of all sizes.
An indirect estimate of microvessel flow heterogeneity, yielding a measure intermediate in character
between that obtained with the tissue-deposition indicators and with in viVo microscopy, has been made by
studying the sequence of capillary filling of microvessels in fixed tissue sections following timed injections
of a visible indicator such as carbon black or fluorescent dye. 2223 Inherent in this measurement is the
thought that the indicators accurately trace flow patterns by their appearance times. This technique extends the spatial limitation of the deposition indicators
to smaller tissue regions and can be performed on a
wide variety of tissues where direct observation is impossible. The usefulness of the method is severely
limited, however, in that it is highly dependent upon
preservation of the geometry during preparation of tissue slices and upon the thickness of the histological
section.24
The most important concern regarding this method
is that it is impossible to distinguish true flow heterogeneity from spatial heterogeneity. The method depends on selecting an appropriate sample time after
injection. At short times, only the vessels with high
flows will be filled. Only as sample times are extended
will all vessels patent at a given time be perfused.
Circulation Research
However, as the sample times become long to ensure
complete filling, vessels that were unperfused at the
time of injection will have time to open and, thus, the
measurement will inaccurately reflect the instantaneous flow distribution in the organ. The relation of
these facts to cyclical vasomotor phenomena will be
discussed below.
INDICATOR DILUTION AND INTRAVASCULAR TRACERS . Di-
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rect measurement of regional or microvessel flow heterogeneity is often impossible, and one must therefore
resort to indirect estimates of flow heterogeneity or to
inferences about heterogeneity based on measurements
of extraction fraction or transit times for various tracers. The most appropriate means for the evaluation of
flow heterogeneity is through an examination of the
venous outflow pattern of an indicator confined to intravascular space. A bolus of such a tracer is smeared
during passage through the vasculature by several processes, including intravascular dispersion due to velocity gradients within the vessels and passage of portions of the bolus through flow paths with different
transit times. The net result produces a complex venous outflow curve that contains historical information
on all the processes that led to dispersion of the indicator. If the outflow curves can be deconvoluted, it is
theoretically possible to obtain measures of the magnitude of heterogeneities in flow.
There are three fundamental problems, often unappreciated by the nonexperts in the field, that limit the
ability of these methods to describe flow heterogeneity
in the microvessels in a way that might be related to
more direct measurements of flow dispersion. First, a
critical oversight is commonly made by ignoring the
fact that heterogeneities in transit time for a reference
indicator are the composite of heterogeneities in both
flow and volume, not flow alone (Figure 1, Equation
1). Since there is no independent method to assess the
volume of the vascular compartments in which the
flow heterogeneity occurs, the transit-time measurement can never be used to estimate directly anything
other than the combined heterogeneity of the two parameters. Obviously this issue provides a potential focus for important interactions between investigators
studying the microcirculation directly and those using
the indicator techniques. Until regional variations in
vascular volume are more fully explored, estimates of
flow heterogeneity from transit-time data must be accepted with serious reservations.
DIFFUSIBLE INDICATORS. In view of the comments to
follow, it is reasonable to assert that measurement of
the capillary exchange of diffusible indicators is not a
currently acceptable means of assessing the degree of
flow heterogeneity. However, this type of measurement has provided much of the historical basis for
assuming the existence of flow heterogeneity, and thus
its significance and limits should be discussed. Theoretically, the degree of heterogeneity can be estimated
from the flow dependence of the capillary exchange of
small solutes.326"28 One of the earliest assertions that
flow heterogeneity existed was derived from the theoretical prediction that the capillary exchange capacity
Vol 60, No J, January 1987
for small solutes in a heterogeneously perfused organ
should asymptotically approach a maximum as the
flow was increased. It is now clear, however, that
understanding the significance of these measurements
and relating them to the microcirculation depends on
the realization that even under optimal conditions the
measurement of the exchange of diffusible indicators
yields no estimate of the flow heterogeneity per se.
Rather, some estimate of the ratio of the permeabilitysurface-area product (PS) to the flow (F) can be determined. Unfortunately, as will be discussed below,
there is no unequivocal means by which the individual
components of the ratio can be independently analyzed. One would hope that the three microvascular
parameters P, S, and F might be distinguished by direct measurement using in vivo microscopy but, to
date, technical limitations have completely prevented
this.
A second problem related to the analysis of microvascular heterogeneities with diffusible tracers arises
from the fact that the specific vessels across which
solute flux occurs and among which one wishes to
study flow heterogeneity cannot easily be related to the
transit-time distribution. The capillaries are not the
only exchange vessels, and differences in transit times
for the intravascular indicator are not solely caused by
differences in the flow through the capillaries. Rather,
transit time is the aggregate of all the delays incurred in
all vessel segments (arterioles, capillaries, and venules) from inflow to outflow (Figure 1, Equation 1).
The volume of the venous compartment is larger than
that of the other segments, and therefore the transit
time through this portion of the vasculature will dominate the characteristics of the outflow pattern. This
problem has been dealt with in the case of intravascular
indicators by assuming a proportionality between large
vessel transit time and capillary transit time,29 but there
is no evidence that such a proportionality exists. Our
unpublished observation of the flow patterns in a number of capillary beds with in vivo microscopy suggests
that it does not. A second means of handling this problem assumes no volume heterogeneity and that all heterogeneity is in the flow parameter, an equally unlikely
assumption.30 Hence, the heterogeneities in transit
time detected at a venous sampling site cannot be presumed to yield specific information about the heterogeneities in capillary flow per se.
Even if an experimental measurement of transit
times could be restricted to a measurement of the capillary transit time, there is an inescapable limitation to
the use of the indicator dilution techniques in that all
three determinants of capillary exchange—PS, F, and
vascular volume (V)—may be heterogeneously distributed within the capillaries. Transit time for an intravascular indicator is determined by the ratio of V to F
(see Figure 1, Section I), whereas transcapillary exchange is determined by the ratio of PS to F. Were all
capillaries homogeneous with respect to V/F, they
need not be homogeneous with respect to PS/F because
the surface area of a capillary varies linearly with its
diameter, whereas its volume varies with the square of
Duling and Damon
Flow Heterogeneity in Striated Muscle
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its diameter. Consequently, precise measurement of
capillary transit time of a diffusible indicator in comparison with the transit time of a reference indicator
need not yield a unique deconvolution of the components of the transit times into heterogeneities of PS and
V, let alone of F. These problems are discussed more
fully by Bassingthwaighte and Goresky26 and Bass and
Robinson.25
Heterogeneity—Spatial and Temporal
We have discussed issues related to largely steadystate analysis of the heterogeneity of flow distribution
among perfused vessels extant within a tissue. However, not all vessels within a tissue are perfused at any
moment, thus leading to spatial heterogeneity of perfusion. Such heterogeneity will have a major impact on
the microvascular transport capacity, but it must be
maintained distinct from flow heterogeneity and analyzed separately.
While the subject of this review is flow heterogeneity, it should be recognized that spatial heterogeneity
also contributes importantly to estimates of flow heterogeneities measured with the different methods. The
volume of tissue in which flow is detected may be quite
different and, in some cases, is undetermined (Table
1). This raises an important issue regarding sampling
and comparison of data between laboratories, since the
heterogeneities that are observed with one sampling
technique may be undetectable with another. For example, heterogeneities in uptake of diffusible indicators would likely pass undetected in most in vivo microcirculatory experiments since the sampling window
for an average microscopic observation is usually 0.4
mm x 0.7 mm and only 50-100 jum deep, while the
sample size used to study uptake of tracer water by
Paradise et al was 0.2 ml or roughly equal to a sample
6-mm square.31 Only by exhaustive, repeated sampling
of the muscle surface would a microcirculatory obser-
vation ever delineate even this relatively small-scale
heterogeneity detected by the tracer methodology.
The distinction between spatial heterogeneity and
flow heterogeneity may become blurred by the presence of spontaneous cyclic vasomotion or temporal
heterogeneity. Since all vessels need not oscillate in
synchrony, the degree of heterogeneity may shift in
time. The effect of cyclic vasomotion on exchange is
complex and consideration is beyond the scope of this
work. However, the presence of cyclic vasomotion
will have different effects on various measurement
techniques. This is related to the fact that each of the
methods used in the assessment of flow distribution is
subject to significant and different sampling errors as a
result of temporal and spatial variations in flow.30 Vasomotion, which is common in the microcirculation,
leads to cyclic variations in flow and its distribution.
Therefore, the sample interval used to obtain data will
influence the degree of heterogeneity recorded.32 As
the sample interval becomes long relative to the period of oscillation, flow will appear to become more
uniform.
Without details on the frequency of vasomotion and
the sampling interval in most experiments, the importance of this cannot be fully assessed. However, a few
observations are pertinent. The in vivo observation
techniques involve essentially instantaneous samples,
since they are commonly obtained from stop-motion
playback of video recordings. The indicator dilution
methods and the microsphere technique integrate the
flow over a period that is usually several seconds, as
determined by the dispersion of the indicator between
the injection and sample sites. When using the visible
indicators of capillary flow and subsequent histological examination, equilibration times must be long
enough to ensure that the indicator traces all patent
flow paths. This requirement implies a long equilibra-
Table 1. Measured Perfusion Heterogeneities in Literature by Sample Size, Method, Index, and Species
Index of
Tissue sample size
(mm3)
Method
Species tissue
Reference
heterogeneity
None given
5xl03
Number of microspheres
Cat triceps
33
in histological cross-section
Unknown
Tissue clearance of tracer
Cat gastroc. and sartorius
36
None given
Unknown
A-V transit times
Dog heart
26
c.v. = 1.07*
(vasoconstricted)
c.v.=0.62*
(vasodilated)
2
1.5 xlO
A-V transit times
25
c.v.=0.36
Dog heart
39
c.v.=0.29
9xlO 2
Microspheres
2x10' t o 2 x l 0 2
Tissue [THO]
Dog EDL
31
c.v.=0.24
(unisolated)
c.v.=0.59
(isolated)
lxlO2
Microspheres
Rat skeletal muscle
40
c.v. =0.56
2X10" 1
Number of ink-filled
Rabbit leg muscle
22
c.v.=0.28t
capillaries in histological
sections
1X10" 1
Rat heart
NADH fluorescence
42
None given
in myocardium
3
3X1O"
Hamster cremaster
54
Direct observation
c.v. =0.56
of capillary rbc's
•Standard deviation estimated using width of outflow fraction curves at 50% of peak height.6
tEstimated at t = 30 seconds; c.v. (coefficient of variation) = 50/x .
Circulation Research
tion time, which means the elements of the vasculature
that were closed at the time of the initial injection may
open, thereby yielding artificially high perfusion fractions and apparent uniformity.22 Careful experimentation is required to determine how cyclic vasomotion
contributes to overall flow heterogeneity.
SUMMARY. The foregoing comments indicate that
there is no perfect method for measuring flow heterogeneity; each has its own strengths and weaknesses.
Each may measure a different aspect of vascular function, and only by integrating the results obtained by
several methods and by appreciating their unique characteristics can a meaningful analysis of vascular heterogeneity be conducted.
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Heterogeneity—Magnitude and Control
The following is a compilation of a variety of reports
on the nature and control of heterogeneous perfusion in
the vasculature. The sampling is not meant to be exhaustive, but to indicate the sources of our understanding of the existence of heterogeneity and the nature of
the data in light of the foregoing comments on the
strengths and limitations of the methodology.
LENGTH SCALE OF HETEROGENEITY MEASUREMENTS. A
sampling of reports of flow heterogeneity in striated
muscle is presented in Table 1, with the data presented
in decreasing order of the probable size of the vascular
element in which the flow heterogeneity was detected.
The table indicates the method of measurement and,
where reported, a quantitative index of the variability
in flow. This organization allows us to present measurements of flow heterogeneities in tissue samples
ranging over 7 orders of magnitude. We assume that
as the problem of flow heterogeneity is more clearly
understood it may be possible to establish quantitative
relations between the size of a tissue element perfused
and a particular vascular element or metabolic process
responsible for the heterogeneity.
The inhomogeneities with the largest scale reflect
variations in bulk flow to different parts of the tissue,
e.g., between proximal and distal portions of a skeletal
muscle33 or from apex to base and endocardium to
epicardium in cardiac muscle. 3435 Although the causes
of heterogeneities at this level are undetermined, they
are likely to reflect differences in the anatomy or behavior of the larger feed vessels and may also reflect
regional differences in metabolism.
Flow heterogeneities are also reported among different tissue components in the muscle. Barlow et al,36
Lindbom,37 and Sparks and Mohrman38 all report that
flow is distributed heterogeneously between connective tissue and myocytes. Furthermore, there appear to
be differences in the flow control in the two tissue
types.37 Such heterogeneities are likely to be partially
responsible for multicomponent indicator washout
curves seen in this tissue. As a result, flows to connective tissue will be difficult to distinguish from those to
heterogeneously perfused portions of the muscle.
Note in Table 1 that the measurement of heterogeneity based on indicator washout cannot be related to a
particular element of tissue. 2936 This is because both
Vol 60, No 1, January 1987
volume and flow heterogeneities from all parts of the
tissue contribute to the time course of the washout, and
only aggregate heterogeneity can be assessed.
Perfusion heterogeneities with dimensions from a
few cubic millimeters to a cubic centimeter are commonly reported. Paradise et al31 found that the uptake
of tritiated water by the dog extensor digitorum longus
muscle was nonuniform, an observation which may be
partly due to preparative trauma. Heterogeneities of
similar magnitude were found using radioactive microspheres.32'39"41
In observations that might be related to flow heterogeneity, Steenbergen et al42 observed patterns of hypoxic foci that devejoped when the oxygen content of
the perfusate supplying Langendorff hearts was reduced. These foci, whose presence were demonstrated
by an increase in NADH fluorescence on the surface of
the heart, appeared in discrete patches (approximately
0.5 mm), suggesting a heterogeneous perfusion or, at
least, a heterogeneous distribution of the inflowing
oxygen. However, their results are also compatible
with small-scale heterogeneities in tissue oxygen
consumption.
A smaller scale of flow heterogeneity (10" 1 mm) is
suggested by data obtained with histological examination of sections made from muscles fixed following
timed injection of visible indicators. Capillary filling is
patchy, and small groups of capillaries fill more rapidly with indicator than their neighbors, suggesting heterogeneous inflow of the dye. 2223
There is anatomical and functional evidence to support our opinion that the regions of heterogeneous perfusion with dimensions of cubic millimeters are attributable to the behavior of single arterioles that, by
opening and closing, can control flow simultaneously
in groups of capillaries.2943*48 This view of micro vessel
organization, based on the regulation of the behavior
of groups of capillaries, suggests a lower limit to the
size of tissue whose perfusion can be regulated and to
the degree of control of perfusion heterogeneity. However, only preliminary studies of the participation of
these units in control have been carried out.47 (This
view of capillary flow control and heterogeneity may
have some clinical significance. If flow to a small
group of capillaries is controlled by a single arteriole,
then occlusion of the vessel might be expected to produce a local area of tissue necrosis. Factor and Sonnenblick49 report that small localized foci of tissue
necrosis occur in certain types of cardiomyopathies,
with a size similar to that of the groups of capillaries
filled by injection indicators. Furthermore, it has been
proposed that Duchenne's muscular dystrophy may be
based on a similar vascular disorder.50)
The smallest scale upon which flow heterogeneities
might be examined is at the level of single microvessels (Table 1, bottom). In the past 20 years, there have
been many observations on red cell flow patterns in
microcirculation, but relatively few have been conducted in a manner that allows quantitative assessment
of the magnitude of perfusion heterogeneity. Capillary
flow has not been measured directly but has been in-
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Flow Heterogeneity in Striated Muscle
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ferred from red cell velocity. The coefficients of variation of measured red cell velocity that have been
reported suggest a fairly large degree of microvessel
flow heterogeneity, with values ranging from 3 0 50%.8.51-55 w n e r e velocity histograms have been published, they show substantial dispersion and are often
skewed toward low velocities. 515354
Flow need not be homogeneous along the length of
an individual capillary because there are frequent
cross-connections between adjacent capillary seg-'
ments through which blood can flow.8204356 As blood
moves along a capillary and passes from one segment
to another, there should be step changes in flow at the
bifurcations where cross-connections originate that
would cause longitudinal heterogeneity in capillary
segment flow. No one has yet made a systematic study
of this possibility by making serial measurements
along the length of the various flow paths in striated
muscle.
CONTROL OF FLOW HETEROGENEITY. IS heterogeneity
under physiological control by local, neural, or endocrine mechanisms? This possibility has not been investigated in detail, but there is abundant evidence that
has been interpreted to indicate that bulk flow into a
tissue and the functional distribution of flow within the
tissue may be affected independently by physiological
control processes. Little effort has been devoted to an
evaluation of these reports in the light of current understanding of the accuracy of the measurement techniques. There are few data showing control of heterogeneity based on direct observation of the vasculature,
assessment of the behavior of microspheres or tissue
deposition indicators, or use of intravascular indicators.
The potential for differential control of total flow
and flow distribution (heterogeneity) is important because in principle it is possible to alter the capillary
perfusion heterogeneity and the net capillary exchange
while maintaining a constant total flow into an organ.57
The efficiency of materials used in the inflowing blood
could be modified to meet physiological demands.
Evidence suggests that capillary exchange of diffusible
indicators, capillary oxygen transport, and distribution
of red blood cells among the capillaries may all be
influenced by flow heterogeneities and controlled by
the microvessels.
In the simplest case, vasodilation would be expected
to increase blood flow, capillary density, and exchange capacity in parallel, and that this would be
correlated with improved tissue function. However, a
variety of experimental maneuvers produce opposite
effects on flow and capillary exchange. For example,
infusion of vasodilators may increase vascular conductance but at the same time reduce capillary transport of
diffusible indicators.357"60
Similar indirect evidence for control of heterogeneity is based on the results obtained in parallel measurements of capillary filtration coefficient (CFC) and PS
product (two measures of capillary exchange capacity). It has been found that CFC can be increased
during vasodilation at a time when PS product is either
decreased or unchanged.57 Theoretically, the CFC
measurement should be relatively independent of flow
heterogeneity, whereas the PS product measurement
should be sensitive to heterogeneity. Therefore, the
simplest interpretation of the disparity between the
CFC measurement and the PS product is that the flow
heterogeneity was altered by vasodilation. For the purposes of this discussion, the possible role of back diffusion, mismatches between flow and PS, and oversimplified analytical models in inducing errors in the
estimation of the PS product have been ignored (e.g.,
see Bassingthwaighte and Goresky26).
Vasoconstrictor stimuli may also produce complex
effects on flow and exchange. Catecholamine infusion
or sympathetic nerve stimulation can induce changes
in capillary exchange suggestive of various modulations of perfusion heterogeneity.60"63
Surprisingly, vasodilation may also be associated
with increases in flow heterogeneity. These may be
large enough to limit cellular oxygen availability in
some portions of the tissue and thus limit oxygen consumption or tissue function in the organ as a whole.
For example, the contractile force of a striated muscle
perfused at constant flow has been shown to decrease
during vasodilator infusion in spite of the fact that flow
resistance was reduced and functional capillary density
was presumably increased.64 Similarly, there are reports that oxygen consumption is reduced during vasodilation, both with and without an increase in blood
flow
.,62,65-67
Tissue and microvessel Po 2 should reflect the net
result of heterogeneities in metabolism and flow, and,
as such, Po 2 histograms might provide an index of
changes in distribution of oxygen during vasomotor
responses. Schroeder and Rathscheck66 observed that
vasodilation in guinea pig striated muscle resulted in a
decrease in the average tissue Po 2 , rather than the expected increase, and a broadening of Po 2 histograms.
Further support for the idea that oxygen inflow to tissue need not be distributed uniformly is the observation that an increase in arterial oxygen tension in human subjects, induced by elevating the fraction of
oxygen in the inspired gas, produced little change in
mean tissue Po 2 while broadening the tissue Po 2 histogram and producing a fall in tissue Po2 in a significant
number of locations. 568 To the extent that venous
blood oxygen saturation can be taken as an index of
local flow, the observation by Monroe et al69 that vasodilation causes a decrease in the heterogeneity of
venular oxygen saturation in the myocardium supports
the proposal that heterogeneity is physiologically
controlled.
In a number of experimental preparations, raising
the total oxygen input to a tissue by elevating perfusion
pressure or arterial oxygen content has been reported to
increase oxygen consumption. The cause for this increase is not clear, but it has been observed under
conditions in which the venous oxygen saturation and
blood flow are high enough to make it unlikely that
capillary blood flow would have been limiting if the
flow were uniformly distributed.70"73
8
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Measurements of capillary red cell flow by microscopic observation provide the most direct means of
assessing microvascular flow dispersion, but such
measurements have only rarely been made during experimental interventions, when vascular control
mechanisms could be studied. Vetterlein and
Schmidt74 showed that infusion of acetylcholine
caused a decrease in the velocity of red blood cells in
marginal capillaries in striated muscle at a time when
total flow into the muscle increased.Their finding suggests that vasodilation induced either massive capillary
recruitment or increased capillary flow heterogeneity.
The significance of this observation is compromised
by the fact that marginal portions of the tissue were
observed, and thus the vessels studied may have been
damaged or not representative of the tissue as a
whole.1331 Burton and Johnson75 found that the velocity distributions of red cells in capillaries of cat sartorius muscle were broadened during reactive hyperemia,
and Damon and Duling51 reported that the standard
deviation of velocities sampled in capillaries of the
hamster anterior tibialis muscle increased during muscular contraction. Sarelius et al8 also reported that
maximum dilation of the capillary bed caused an increase in the dispersion of capillary red cell velocities.
The results obtained during elevation of the superfusion solution oxygen tension are conflicting, with reports that the heterogeneity of red cell velocity increased,54 decreased,55 or did not change.51
In an attempt to clarify the relation between vasomotor state and capillary perfusion heterogeneity, as well
as to test the commonly held view that perfusion heterogeneity is regulated in response to altered oxygen
availability or muscle work, 13 ' 26 ' 2846656776 we have assembled from a variety of published sources, estimates
of perfusion heterogeneity at the level of the capillaries
(Table 2). Several of the measurements are from a
single set of experiments on the tibialis anterior observed during vasoconstriction induced by elevating
the oxygen concentration from 0 to 5% in a superfusion solution, during vasodilation induced either by
striated muscle stimulation at 1 Hz and 4 Hz, or by
application of supramaximal doses of adenosine.51 Red
cell velocity was measured as an index of capillary
flow. Table 2 presents the mean velocities, the standard deviation of the velocity as a measure of the
absolute dispersion, and the coefficient of variation as
a measure of the relative dispersion. The data are arranged in order of increasing mean velocity.
Examination of Table 2 shows a high degree of
parallelism between the mean velocity and the standard deviation of the velocity; over a sevenfold change
in mean velocity, the coefficient of variation varies by
less than 2. The constancy of the coefficient of variation suggests that the relative dispersion of the flow
among the microvessels was invariant in the face of
these changes in vascular tone. The simple interpretation is that the increased standard deviation was predominantly the result of an increase in total flow
through the system, not an alteration in the distribution
of flow among the vessels.
Circulation Research
Vol 60, No 1, January 1987
Table 2. Means, Standard Deviations, and Coefficients of
Variation of Capillary Red Blood Cell Velocities in Striated
Muscles*
Mean
(/xm/sec)
90
145
152
157
160
180
210
214
223
230
240
274
283
290
290
384
431
548
675
Standard
deviation
(/Mm/sec)
53
82
88
97
145
60
120
148
161
130
130
161
218
140
154
250
284
412
302
Coefficient
of variation
Reference
51
0.59
51
0.57
0.58
54
8
0.62
8
0.91
0.33
55
55
0.57
8
0.69
8
0.72
55
0.57
55
0.54
21
0.59
0.77
8
55
0.48
0.53
51
21
0.65
8
0.66
51
0.75
51
0.45
0.67+0.03 (mean ± SD)
•Tibialis anterior, hamster51; cremaster, hamster854; tenuissimus, rabbit55; and spinotrapezius, rat.21
These findings suggest that for the metabolic stimuli
reported here, capillary flow heterogeneity may not be
under physiological control. Rather, it appears that the
fractional distribution of flow among the capillaries is
stable. The constancy of flow dispersion, even with
maximal vasodilation, suggests that flow distribution
may depend on undefined aspects of vascular function
that are inherent in the structure and rheology of microcirculation. It is possible that a 60% coefficient of
variation represents a minimal value for perfusion heterogeneity in capillaries perfused with blood and that
the heterogeneity is determined by basic patterns of red
cell flow in capillaries. It is interesting to note in Table
1 that a similar trend is suggested in whole organ
experiments. The data show rather small ranges for the
coefficient of variation, much smaller than the range of
blood flows.
Data in Table 2 were chosen to reflect heterogeneities at the level of the capillaries, and since they were
all obtained using in vivo microscopy, they reflect the
behavior of relatively small sample volumes. Other
nonuniformities in perfusion observed in whole organ
experiments may reflect nonuniform perfusion on a
larger scale than the capillary. This possibility would
imply the existence of integrative vasomotor control in
relatively large vessels. We must emphasize that our
focus here is on flow heterogeneity and does not address possible heterogeneity of spatial distribution of
perfused capillaries. However, it is apparent that spatial heterogeneity must also contribute to the limits of
exchange.
An interesting aspect of this problem is that functional dilation is usually accompanied by an increase in
the number of capillaries containing flowing red cells.
Since the process of capillary recruitment takes place
Duling and Damon Flow Heterogeneity in Striated Muscle
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while maintaining a constant coefficient of variation in
capillary flow (Table 2), we infer that the capillary
recruitment during vasodilation is accomplished in
such a way that the flow through the population of
newly recruited capillaries experiences the same fractionation as does that passing through the capillaries
perfused in the control condition. It is unknown whether recruitment of capillaries with a constant degree of
flow dispersion occurs or how this is accomplished.
The possibility that capillary recruitment is related to
arteriolar closure was deduced from observations of
capillary recruitment patterns by Honig et al.46 Their
data, as well as ours, are consistent with the idea that
capillaries are recruited in small functional groups or
units and that the heterogeneity of flow among these
groups is the source of the relatively constant dispersion.47 This possibility could be easily explored in striated muscle by direct measurement of flow at the origin of these units, a measurement yet to be made under
well-defined conditions.
CAPILLARY HETEROGENEITY AND OXYGEN TRANSPORT.
We can also make a very simple estimate of the effect
of capillary perfusion heterogeneity on tissue oxygenation with an analysis similar to one proposed some
years ago by Kety,78 taking advantage of apparent constancy of capillary flow heterogeneity (Table 2). If we
assume that the capillary can be treated as a cylinder
supplying a uniform annulus of tissue, that the affinity
of the tissue enzymes for oxygen is high relative to the
oxygen tension in the tissue and blood, and that blood
flow velocity is constant along the length of the vessel,
then oxygen content will fall linearly from the arteriolar to the venular end of the capillary, with a slope
proportional to the ratio of the metabolic rate to the
blood flow (Figure 2). This linear fall in capillary
oxygen content will continue until tissue Po2 falls
0.20
0.15
o
S3
0.10
0.05
C 0 =C A - (M/F)(L/L 0 )
LU
o
0.20
o
CM
O
\\T---..
4 Hz
S
^ " - ^
0.15
- ^F__+ SDf
X
_l
0.10
\
\
lid
Effect of Heterogeneity on Microvessel Function
EFFECT ON SOLUTE EXCHANGE. While the foregoing
suggests that perfusion heterogeneity of the capillaries
may not be metabolically regulated, it does not detail
how a given level of heterogeneity might effect capillary exchange. We have used the fact that the coefficients of variation shown in Table 2 are constant over
a variety of experimental circumstances to obtain a
quantitative estimate of the functional significance of
changes in flow, capillary density, and absolute dispersion.51 We assumed that the same degree of flow
heterogeneity would be reflected in the capillaries of
the dog gracilis, a tissue in which all the necessary data
were available. Based on published values for PS and
capillary surface area at rest and during exercise,7-77 we
made a simple calculation to assess the effects of a
constant level of heterogeneity on capillary transport at
resting blood flow and during muscle stimulation at 4
Hz. We used the heterogeneity estimate to predict the
spread of clearances around the average values. For
simplicity, clearances (C) were normalized to the existing PS. 57 On transition from resting to 4 Hz contraction, the calculation predicted that the average C/PS
would increase by 156% when all capillaries were perfused identically. Assuming a heterogeneous distribution of flow with a coefficient of variation of 60% we
computed a reduction in the capillary exchange capacity of 8% from the ideal case at rest, but only 1%
following 4 Hz stimulation. Thus, the effect of flow
heterogeneity did appear to be less during muscle
work, but this is the result of the nonlinearities in the
relation between flow and exchange. More important
is the fact that the observed heterogeneity seems to
have a rather small effect on capillary exchange.
Renkin28 has utilized a more complex model of capillary exchange, as described by Bass and Robinson,25
to compute the degree of heterogeneity of PS/F that
would be required to explain the flow dependence of
extraction observed in striated muscle. His computation indicates that a coefficient of variation of about
90% would be required, i.e., greater by 30% than we
observed in microcirculation. The inequality between
the coefficient of variation estimated by Renkin28 and
that observed in in vivo studies suggests that the heterogeneities traced out by the indicator diffusion methods may well reflect the combined results of both macroscale and microscale heterogeneities. However, we
recognize that our assessment is a very simple one and
that final conclusions will require fitting the presently
available data on capillary flow heterogeneity to more
complex exchange models.26 '
^
\
\
\
o
\
0.05
N
sF - SDf
\
0
1
|
0.2
0.4
N
,
1
1
0.6
0.8
1.0
NORMALIZED CAPILLARY LENGTH (L/L o )
FIGURE 2. Estimate of the effect of heterogeneous flow on
capillary oxygenation in resting (top) and working (bottom)
striated muscle. It was assumed that the capillary oxygen content falls linearly along the capillary with a slope equal to the
ratio of the tissue oxygen consumption to the flow. The coefficient of variation of capillary flow was assumed to be 0.60.
Capillary oxygen contents for flow 1 standard deviation above
and below the mean are shown by the dotted lines.
Circulation Research
10
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below that required for mitochondrial oxidative phosphorylation. For the sake of this discussion, we assume
that the oxygen consumption is a zero-order process.
We used literature data for the dog gracilis muscle
for flows varying from 5 ml/min/100 g at rest to 75
ml/min/100 g during 4 Hz stimulation. We chose oxygen consumptions of .22 and 12.8 ml/min/100 g at rest
and during 4 Hz stimulation.7 Assuming that the coefficient of variation of capillary blood flow is equal to
0.60, we examine the fall in oxygen content in capillaries perfused at the mean blood flow and those perfused at the mean flow plus or minus one standard
deviation of flow. The predicted diminution of capillary blood oxygen content along the length of the capillary is shown for the resting muscle in Figure 2 (top
panel) and for a muscle contracting at 4 Hz in Figure 2
(bottom panel). The lower panel shows that, for the
working muscle, the oxygen content falls to zero in
capillaries that are perfused at mean flow less one
standard deviation of flow. Thus, even at one standard
deviation from the mean velocities that we observed, it
appears that oxygen supply could be rate limiting for
oxygen uptake by the tissue. Many capillaries would
naturally have even slower velocities.
It must be emphasized that both this analysis of
capillary oxygen gradients and the preceding one on
clearance heterogeneities are based only on an examination of the expected effects of flow heterogeneities
in capillaries. The critical factor in the clearance calculation is the ratio of PS to F in a given capillary. In our
analysis, we have ignored systematic variations in permeability and/or capillary geometry, which clearly
could alter our predictions. There are no systematic
studies comparing velocity and capillary permeability,
the absence of which must provide a spur for new
measurements. Similarly, heterogeneities in capillary
length or in flow path could in principle offset changes
in flow heterogeneity and obviate the analysis carried
out above. Length measurements in the microcirculation do show a fair degree of heterogeneity.205479 The
critical issue is the ratio of the path length to the flow
velocity, and Klitzman and Johnson54 report that there
is no correlation between the two variables. Also of
importance is data on path length change during vasodilation. The available data20 suggest that the change in
flow path is minimal, but there is a great need for more
detailed exploration of the relations between various
heterogeneities in microvessels.
RELATIONS BETWEEN FLOW AND HEMATOCRIT. Our
ini-
tial interest in the problem of microvessel flow heterogeneity was prompted by the often ignored fact that
flow dispersion may indirectly influence tissue oxygenation by inducing heterogeneities in red cell distribution and capillary hematocrit. Knowledge of capillary hematocrit is also critical to the interpretation of
tracer estimates of heterogeneity.26-29 The potential importance of changes in capillary hematocrit may be
appreciated from the fact that microvessel hematocrits
in striated muscle are consistently less than systemic
and vary from 5 to 37% as a function of vasomotor
state. 816 Thus, if the reported capillary hematocrits
Vol 60, No 1, January 1987
accurately reflect the red cell flux, the delivery of red
cells to the capillaries may be increased by as much as
sevenfold through a redistribution of blood flow and
red cells within the tissue.
CONTRIBUTION OF CAPILLARY FLOW HETEROGENEITY TO
LOW CAPILLARY HEMATOCRIT. While it is clear from in
vitro measurements that heterogeneous flow can induce heterogeneities in hematocrit, none of the efforts
to relate heterogeneous capillary flow to low capillary
hematocrit have been entirely successful. 101680 ^ 2 Kanzow et al81 attempted to use histograms to explain the
low number of microvessel hematocrits within microcirculation, with the discovery that hematocrit reductions as a result of heterogeneous branching were inadequate to explain the data. Some amount of red cell
shunting had to be added to the model to reconcile
theory and observation, but no red cell shunts have
been observed in striated muscle. 1083
Cokelet80 pointed out that the effects of heterogeneities on microvessel tube hematocrit might be compounded over successive bifurcations leading to large
effects on capillary hematocrit. He concluded that heterogeneous flows might thereby explain the findings
on capillary tube hematocrit. However, his model predicted that there would be a significant population of
microvessels with zero hematocrit, and, in spite of
efforts to detect such vessels, they do not appear to
exist in significant numbers in tissues in which low
capillary hematocrits are reported.24
The data shown in Table 2 provide a unique opportunity to evaluate the hypothesis that capillary hematocrit may be related to dispersion of capillary flow. As
pointed out above, capillary hematocrit increases with
vasodilation, 8I684 ' 85 which leads to the expectation
that, if the heterogeneity hypothesis is valid, greater
flow heterogeneity should be associated with a reduction in hematocrit. The data in Table 2 make it obvious
that the absolute dispersion of capillary velocity increases and that the relative dispersion in velocity is
unchanged during vasodilation, a time when the hematocrit increases. Thus, the measurements of velocity dispersions in the microvasculature are inconsistent
with the hypothesis that the capillary hematocrit
changes because of altered flow dispersion.
A pivotal measurement that should be taken in order
to deepen our understanding of the relations among
flow heterogeneity, capillary rheology, and microvessel hematocrit is the discharge hematocrit of individual
microvessels. Of particular interest would be the heterogeneity in discharge hematocrit. The need for this
measurement stems from the fact that it will provide
the unique link between flow heterogeneity, red cell
flux, and tube hematocrit. Because of the small size,
reactivity, and flow patterns in microvessels, it has
been difficult to make such measurements, but the
literature does contain one report in which undefined
renal "microvessels" were punctured with micropipettes and blood samples were withdrawn,86 yielding
an average value for the microvessel samples equal to
the systemic hematocrit. Similar measurements have
been made in this laboratory on vessels as small as
Flow Heterogeneity in Striated Muscle
11
9 /am in diameter87 with an equivalent result, i.e., that
discharge hematocrits in both arterial and venous microvessels are equal to arterial hematocrits. If these
measurements prove to be representative of microcirculation as a whole, they strongly imply that low capillary hematocrits are not the result of flow heterogeneity within the microcirculation but rather that low
capillary hematocrit results from some other aspect of
microvascular function.1087
Further studies must concentrate on three aspects of
the measurement of capillary flow heterogeneity: 1)
additional quantitative studies on the degree of perfusion heterogeneity within the vasculature, both on the
macroscopic and microscopic scale; 2) complementary
simultaneous measurements of flow, vessel geometry,
permeability, and volume to describe precisely the
process of capillary exchange; and 3) assessment of
flow heterogeneity at microvessel bifurcations with
concurrent measurements of blood flow and red cell
flux.
Duling and Damon
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Summary
This review leads us to a number of conclusions and
suggestions for further study. First, we find wide differences in the meaning of flow heterogeneity, arising
as a result of the different methods used. These differences will have to be reconciled to form a comprehensive view of the role of heterogeneity in determining
vascular function.
Second, in the future, the meaning of heterogeneity
must be clearly defined and related to a particular microvascular component, and it is imperative that the
differences in scale of heterogeneity be appreciated
when comparing data from various laboratories. These
heterogeneities have different implications for function, and failure to distinguish among them leads to
confusion.
Third, the degree to which perfusion heterogeneity
is regulated in the microcirculation remains in doubt.
Reports of variations in flow heterogeneity in response
to physiological stimuli are for the most part based on
highly questionable indirect methods.
Fourth, the heterogeneity that can be demonstrated
at the capillary level within striated muscle does not
appear to be large relative to the capacity for the microcirculation to exchange most diffusible solutes. Thus,
the inferences regarding heterogeneity, as evidenced
by diffusible indicators, are likely to be the result of
different preparations, damage to the preparations, or
perhaps large-scale heterogeneities in the tissue. An
alternate possibility would be that the heterogeneity
occurs at the microvascular level but reflects some
other aspect of microcirculatpry function, such as
length or hematocrit heterogeneities, but not flow heterogeneities.
Fifth, flow heterogeneity within microvessels implies important consequences for capillary exchange
and tissue oxygenation. Heterogeneities of velocity of
a magnitude comparable to those observed by direct
visualization of microcirculation can clearly produce
reductions in oxygen supply to small tissue regions of a
degree that may limit oxygen delivery, and thereby,
tissue function.
Sixth, flow heterogeneity may also influence capillary hematocrit and/or red cell spacing by producing
cell separation at bifurcations and a resultant reduction
in mean capillary tube hematocrit. There is as yet no
agreement on why and how these hematocrits influence tissue oxygenation and function. Although several hypotheses are advanced to explain the distribution
of blood flow and red cells within microcirculation,
each lacks a critical experimental test at present.
Acknowledgments
The scientific criticisms of Drs. Gore, Jackson, Sarelius, Segal, and Wieringa have made a major contribution to this work, and we gratefully acknowledge
their assistance. Due to space limitations, many important works in this field have been omitted, and we
apologize to those who feel their contributions may
have been overlooked.
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KEY WORDS • capillary • hematocrit • red cell • tracer • microsphere
An examination of the measurement of flow heterogeneity in striated muscle.
B R Duling and D H Damon
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Circ Res. 1987;60:1-13
doi: 10.1161/01.RES.60.1.1
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