Download The Capillary and Sarcolemmal Barriers in the Heart

515
The Capillary and Sarcolemmal Barriers in the
Heart
An Exploration of Labeled Water Permeability
COLIN P. ROSE, CARL A. GORESKY, AND GLEN G.
BACH
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SUMMARY Although the exchange of labeled water between blood and tissue in the heart has usually been
assumed to be flow-limited, the outflow patterns of labeled water, relative to intravascular references, in a multiple
indicator dilution experiment, have appeared to be anomalous in terms of the models used to explain the transport of
less permeable substances. Data showing a change in the shape of the labeled water outflow curve after vasodilation
and after the infusion of toxic doses of 2,4-dinitrophenol led us to propose a new model for labeled water permeation
which includes barriers at both the capillary wall and the sarcolemmal membrane. This model explains adequately the
form of the outflow curve, provides parameters related to the permeability at the two barriers, and gives an estimate
of the ratio of the intracellular to interstitial space. Dinitrophenol infused intra-arterially in a dose sufficient to cause
S-T elevation in the electrocardiogram is found to reduce the sarcolemmal water permeability by an order of
magnitude, but to have no effect on capillary water permeability. We conclude that water transport in the heart is
barrier-limited at both the capillary and sarcolemmal membranes and that sarcolemmal water permeability is probably
mediated at least in part by a structure sensitive to the effects of dinitrophenol, presumably a protein channel. Since
the outflow patterns of inert gases resemble that of labeled water, it is possible that oxygen distribution is also barrierlimited.
ALTHOUGH cell membranes are known to be highly
permeable to water, they do reduce the rate of diffusion of
water molecules to about a hundred thousandth that of
free diffusion.1 Despite this fact it has been hypothesized
that the membrane permeability is high enough and blood
flow is slow enough that the distribution of labeled water
in an organ such as the heart, where intercapillary distances are small, is flow-limited at physiologic rates of
perfusion.213 At this point in time, however, it has been
possible to quantitatively corroborate this assumption only
for the liver .4 This organ is very specialized, in terms of the
structure of its basic microcirculatory unit, the hepatic
sinusoid. There is no significant barrier to small molecules
corresponding to the capillary membrane, and the plasma
membranes of the hepatocytes are massively expanded in
area by virtue of their innumerable microvillous processes.
In addition, the mean sinusoidal transit time is much
longer than that in the capillaries of a visceral organ
perfused at arterial pressure.
On the other hand, there is evidence that the distribution of labeled water in the brain is barrier-limited at high
but physiologic perfusion rates.5"7 In the heart, the organ
we wish to consider here, Ziegler and Goresky8 have
shown that the shape of the outflow concentration-time
From the McGill University Medical Clinic in the Montreal General
Hospital, Montreal, Quebec, Canada H3G 1A4; and the Departments of
Physiology, Medicine, and Mechanical Engineering of McGill University.
Supported by the Medical Research Council of Canada and the Quebec
Heart Foundation.
Dr. Rose is a Fellow of the Canadian Heart Foundation, and Dr.
Goresky is a Medical Research Associate of the Medical Research Council
of Canada.
Address for reprints: Carl A. Goresky, M.D., Montreal General Hospital, 1650 Cedar Avenue, Montreal, Quebec, Canada H3G 1A4.
Received July 7, 1976; accepted for publication March 9, 1977.
curve for labeled water in a multiple indicator dilution
experiment could not be explained by the assumption of
flow-limited exchange unless the additional assumption of
random diffusional capillary interaction was made. In contrast, Rose and Goresky9 more recently have shown that
for sucrose, a small molecule confined to the extracellular
space in the heart, there is no necessity to assume that this
interaction occurs. The outflow form of the dilution curves
for this substance can be explained quite precisely by the
assumption of noninteracting large vessel-capillary units in
the coronary circulation.
In view of the foregoing we have once again attempted
to characterize the factors underlying the shape of a labeled water outflow curve from the heart and have explored an alternate explanation for the shape, i.e., the
hypothesis that there are constraints on the movement of
labeled water both at the capillary and sarcolemmal membranes. We have developed a model incorporating both
these barriers. This model adequately explains the shape
of the water outflow curve and gives reasonable estimates
of the permeabilities of the membranes and the relative
sizes of the interstitial and intracellular spaces. The hypothesis fits the data even when we also assume that the
large-vessel capillary units in the heart are independently
regulated and noninteracting. It does away with the need
to adduce the presence of diffusional interaction between
capillaries with a random scattering of entrances and exits,9- l0 a hypothesis difficult to support for the heart from
the morphological point of view.
At the same time we have characterized another phenomenon. While investigating the effect of 2,4-dinitrophenol on metabolite transport, we discovered that, when
infused intra-arterially in a dose sufficient to cause S-T
segment elevation, dinitrophenol causes a reversible
516
CIRCULATION RESEARCH
change in the shape of the labeled water outflow curve. In
terms of our modeling, we found this change to be due to a
measurable and fairly marked reduction in the permeability of the sarcolemma to labeled water.
These discoveries may be of practical significance in
understanding sarcolemmal membrane structure and the
processes underlying the distribution of oxygen and carbon dioxide.
Methods
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The left main coronary artery or its circumflex branch
was isolated in mongrel dogs as described previously.9 An
injection mixture was made up as follows: 6 ml of the
mixture, adjusted to the same hematocrit as the dog,
contained 5lCr-labeled red cells, 0.05 mCi of 125I-human
albumin (Charles E. Frosst), 0.1 mCi of 14C-sucrose (New
England Nuclear), and 4 mCi of tritiated water (New
England Nuclear). The red cells were labeled by adding
0.125 mCi of sodium 5lCr-chromate (Charles E. Frosst) to
10 ml of whole blood, incubating at 37°C for 30 min,
reducing to the chromic ion with ascorbic acid, and then
washing the red cells twice in saline containing a small
proportion of plasma as an antisphering agent. Finally, the
red cells were suspended in plasma. At the time of a run,
0.5 ml of this mixture was injected into the perfusion
cannula and samples were collected from the coronary
sinus at a rate of 0.6-0.8 sec/sample.
A volume of 0.1 ml of each sample (usually 42/run) was
diluted in 1.5 ml of saline and assayed for radioactivity in a
two-channel gamma ray spectrometer set for the photopeaks of 51Cr and I25I. The proteins were then precipitated
with 0.2 ml of trichloroacetic acid, and 0.2 ml of the
supernatant fluid was pipetted into 10 ml of a scintillation
cocktail (Aquasol, New England Nuclear) and assayed for
H
C and 3H beta activity in a liquid scintillation counter.
Samples from the injection mixture and cross-over standards were treated identically. Total activity in each energy range was corrected for cross-over from the other
isotopes with the appropriate simultaneous equations. The
corrected activity per ml for each isotope in each sample
was then divided by the corresponding activity injected.
This value, the outflow fraction per milliliter of venous
blood, was plotted against time for each tracer to give a set
of simultaneous normalized outflow dilution curves.
In some experiments, after a control run, an infusion of
2,4-dinitrophenol (1 g dissolved in 30 ml of 95% ethanol)
was made directly into the perfusion tubing at a rate of
about 0.05 ml/min until marked S-T elevation was apparent in the electrocardiogram, and then another run was
carried out. In the later experiments, the infusion was
stopped and another run was performed after the S-T
segment had returned to the isoelectric level. It was found
that, at this level of infusion, coronary vasodilation occurred and persisted for many minutes. It was during this
latter interval that the last run was done. Larger rates of
infusion resulted in massive necrosis of the myocardium.
With this there is a large increase in vascular resistance
after the initial vasodilation. Initial experiments, in which
this happened, were rejected.
VOL. 41, No. 4, OCTOBER 1977
THE TWO-BARRIER MODEL
The problem we are about to examine is a classical one.
It has been called, by Yudilevich,11 the two-barrier problem, and has previously been explored only for potassium
or rubidium uptake, where the initial cellular uptake can
be treated as a unidirectional flux.12-''1 It is characterized
by a further degree of complexity than the case in which
there is only a single barrier, that at the level of the
capillary.
In order to examine this problem, we will first develop
the expression for the output from a single capillary and its
associated tissue and then will combine this with the model
for the whole coronary circulation, which we developed to
account for the changes in the labeled sucrose curve which
are found to occur with changes in the degree of vascular
resistance.9 The latter will be the bridge that will enable us
to take the single capillary model and apply it to outflow
curves from the whole heart. In developing this model we
have also included, in addition to the rate constants across
the capillary and sarcolemmal barriers, an intracellular
irreversible sequestration rate constant, k5, in anticipation
that the same model will be applicable to metabolized
substances. For labeled water k5 is, of course, zero.
SINGLE CAPILLARY MODEL
Definitions
Figure 1 shows the relationship between the surfaces,
spaces, and transport parameters in the model. The following symbols are used:
A, B, and C are the cross-sectional areas of the capillary, extracellular (not including vascular) and intracellular spaces, respectively
W is the velocity of blood flow in the capillary
u(x, t), v(x, t), and z(x, t) are the concentrations of the
substance in the capillary, extracellular, and cellular
spaces, respectively, at some point x along the length
of the unit at the time t
4>i and fa are the surface areas per unit length of the
capillary and tissue cells, respectively
k', and k'2 are the permeabilities for flux out of and into
the capillary, respectively
FIGURE 1 Schematic diagram of the relationship between spaces,
surfaces, and transport parameters in the model. The assumption
of infinitely rapid lateral diffusion implies that the cylindrical
geometry of the capillary can be ignored.
CAPILLARY AND SARCOLEMMAL BARRIERS IN THE HEART/Rose et al.
k'3 and k'4 are the permeabilities for flux into and out of
the tissue cells, respectively
k5 is the rate of irreversible sequestration per unit accessible intracellular space.
K,=1.0 . 1 K 4 =10
517
Assumptions
In the following development we assume that (1) diffusion in the direction perpendicular to the capillary is infinitely rapid, i.e., that the perfusion is rich enough and the
K,=1.0 . |K,=0.)
ICT3
*•(.
3
0.741
K,=K,=0.1 T
Z
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o
10*
o
u
o
IMOUC«*UI Aft* 3 10'
10"
K,= K,= 10 T"
10"*
'HtOUGHPUf & • ( * = . O S
101
T'
to
10
IS
20
FIGURE 2 Outflow profile for substances undergoing passive barrier-limited distribution at the capillary, and concomitant exchange with the
cell at the tissue cell membrane, in response to unit input. The profiles are normalized so that the area under each is unity. The abscissa is
normalized to th, and the ordinate correspondingly becomes Fc r u(L, t). In all cases, the capillary membrane has been assumed to be
equilibrative (k, =ki)andy
=3.0. The panels on the left are for an equilibrative tissue cell membrane [k3 = k4 = / .0] with (y/6) = 1.0; and
those on the right are for a concentrative tissue cell membrane [k3 = 1.0 and k4 = 0.1], with (y/6) = 1.0; or they are for the equivalent case,
with the cell volume expanded by a factor of 10 (that is, (y/d) = 0.1 ] and an equilibrative cell membrane (k3 = k4 = 1.0]. The abscissa scale is
linear; the ordinate scale, logarithmic. In each case, capillaries with equal (PSJFJ values are displayed on the two sides of the plots, and the
panels are arranged in order of increasing membrane permeability from top to bottom. Dashed lines represent the output if no cells were
present (k3 •= kt =0). Solid lines represent the output which occurs with the tissue cells present but no intracellular sequestration and the dotdashed lines represent the output for different degrees of intracellular metabolic sequestration (that is, varying values ofk-). The first part of
the illustrated output, in the cases with low permeability, is an impulse function with a normalized area, exp( —ktyr), or expf—PSJFJ. It is
difficult to illustrate this form, which theoretically has an infinitesimally small duration. We have simply placed a vertical line at the site of the
function and have placed on the illustration a number representing its normalized area. When the spike area is less than 0.01 of the total, we
have used a broken rather than a solid line. It should be noted that, in the bottom panel, when the capillary permeability has been allowed to
become infinite, the throughput area refers to the label emerging in the delayed impulse function.
518
CIRCULATION RESEARCH
spaces are small enough that there is no concentration
gradient perpendicular to the capillary in the extracellular
or cellular spaces,15 (2) diffusion parallel to the capillary is
insignificant, and (3) there is bolus or plug flow in the
capillary.
The Basic Partial Differential Equations
Consideration of the events that occur at each element
in space and time leads to three partial differential equations, which must be solved simultaneously. The first, the
VOL. 41, NO. 4, OCTOBER 1977
equation for conservation of matter, is
(i)
where 8 = C/A and y = B/A.
The second, the rate equation for accumulation in the
extracellular space is
at
B
B
B
B
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XT
1.0
DISTANCE ( V L )
FIGURE 3 Longitudinal concentration profiles for intravascular (solid line), extracellular (dashed line), and intracellular (mixed dashed line)
spaces. The distance along the capillary is normalized to x/L. Again the capillary is equilibrative, k, = A:2 = 0.5 (time units)'1, and y =3.0.
The transit time is one time unit. Panels on the left are for a substance that does not enter the tissue cell [k3 = kt = 0], and are included for
comparison. They are the same as those in Figure 2 of Reference 16. Those in the middle are for tissue cells with an equilibrative membrane
[k3 = k4 = 1.0], and those on the right are for tissue cells with a concentrative membrane [k3 = 1.0 and kt = 0.1]. Note that in order to
calculate the concentration in the cellular space it is necessary to assume a value for the space ratio (y/8). Here we have computed the profiles
for the space ratio (yld) = 1.0. In each case a "snapshot" of the profiles has been taken at 10 normalized elapsed times. The vertical lines
leading the profiles again represent the propagating impulse function, within the vascular space, and the numbers beside them, thefractionof
the material still associated with this throughput. The throughput areas in the middle and righthand panels are the same as that on the left. The
sequestration process, kif does not have major influence on the profiles until late in time (thus, in Figure 2, the effects are seen to be large at
times much beyond those illustrated here). This serves to emphasize that the patterns of metabolism in tissues supplied by capillaries are
dependent in major fashion on capillary and cell membrane permeabilities, relative space sizes, and transit times.
CAPILLARY AND SARCOLEMMAL BARRIERS IN THE HEART/Rose et al.
519
the time t = 0. Equations 1 to 3 must then be solved
according to the initial conditions
or
| ^ = k,u - k2v - k3v + k4z,
dt
(2)
u(0, x) = c6(x)
v(0, x) = z(0, x) = 0,
where the permeabilities have been replaced by their respective permeability-surface products per unit extracellular space.
Finally, the rate equation for accumulation in the intracellular space (with the intracellular removal process, in
addition) is
at
_
C
where c is a constant with the dimensions (amount/cm2).
Now apply to Equations 1-3 the Laplace operator with
respect to time
L£«x,t)]=
f(x,t)e- s 'dt =
whence
5
C
+ sy V + S0Z + k50Z = c5(x)
sU + W ~
or
sV = k,U - k2V - k3V
(3)
(4)
(5)
(6)
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Simultaneous Solution of the Equations for the Single
Capillary-Time Unit
Now introduce at the origin (x = 0) of the initially
empty capillary with flow F c , the quantity of material q0 at
and
(6A)
s + (y/O)^ + k5
12
18
20
.2
A
A
B
U>
0
.
2
.
4
4
.
8
1
0
0
7
A
a
to
DISTANCE! V I )
FIGURE 4 Longitudinal concentration profiles for intravascular (solid line), extracellular (dashed line), and intracellular (mixed dashed line)
spaces when the capillary permeability has been increased by a factor of 10. In this case k, = k2 = 5.0 (time units)'1.
520
CIRCULATION RESEARCH
V=
V O L . 4 1 , No. 4, OCTOBER 1977
*£.
\S ' k 2 i KgJ
~
and
(6B)
Substituting Equations 5A and 6B into Equation 4, we
find
dx
s + (y/6)kt + k5
•=o8(x).
(7)
[s
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Rearranging,
dO
dx
k!ys[s + (y/fljkj + kj + -yk,k3( s + k5)
(s + k2 + kJLs + (•y/0)k4 + k j -
0 /
WV
\
=
c_
Subtracting and adding-yki[k2s + (-y/0)k2k4 + k2k5] in the
numerator of the second term in the brackets and rearranging the terms in the numerator gives
dO
0 /
dx
W V
yk.kjs + (y/ffjk. + kj
yKl
s2 + [k 2 + k 3 + (y/e)k4
\
+ kjs + ( y / f l j k ^ + k 2 k 5 + k 3 k 5 /
=
±
W°
w
"
K
° >
The last term in the bracket can be broken into two partial
fractions:
dU
_U /
dlT
WlS
, _ yk,k 2 A' _ -ykik2B'\
y
'
s- d
~T-rT"/'
= ^ 8(x) (8B)
where d and f are the roots of the quadratic equation,
s2 + [k2 + k3 + (•y/0)k.( + k5]s
+ [(y/0)k 2 k 4 + k2k5 + k 3 k 5 ] = 0
. , _ d + [(y/^llct + k5]
and
Integrating Equation 8B, we find
= ^ -ee w
w
ss
k,k,y(x/W)A'
ee
"'''w
"'''we
s
-d
e
s-f
S(x)
(9)
where S(x) is a step function at x = 0.
Using the Taylor series expansion of e"
= ^ e
wse
k y
' w+ ^
e
w s e k'rvv
n ^,
(s-d)-n!
(9A)
CAPILLARY AND SARCOLEMMAL BARRIERS IN THE HEART/Rose et al.
521
where c/W = qo/Fc, as before.16 It should be noted that,
when fc, and k5 are zero, the transform converges to that
developed by Goresky and Ziegler to describe the virtually
unidirectional sarcolemmal uptake of labeled rubidium
from the extracellular space of the heart;12 that, when kj
and k2 become infinite, it converges to the transform
developed to describe the hepatocyte uptake and sequestration of galactose from the freely accessible space of
Disse, adjacent to the liver sinusoidal lumen;17 and when
k] and k2 become infinite, but k5 is very small, it converges
to the transform describing glucose exchange between the
space of Disse and the hepatic parenchymal cells.18
Inverting Equation 9A into the time domain, we find
q0
^ \-\
di-
\
**
n! (n + 1)!
x R ,\
/
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+ e'
X
Y _x
n! (n + 1)!
/ ik ik 2 y x
( '
n+
AAn+1/i.
x
_£) + r
W
/
Jx/
w A 'j ( h - w ) f(t_h
\"
e
n!(n+D!
n! (n + 1)!
-dh
(10)
where sit — — j is an impulse function at t = x/W and
SI t — rrrl is a step function beginning at t = x/W. From
this expression we find, in alternate form that, at the end
of a capillary of length L, and transit time L/W = T
T)
u(L, t) = 35-
+ 95.e-k
1
' [2Vyk,k2rA'(t - r)]
'(t - r)]J S(t - r)
Jf e""">
(10A)
I, [2Vyk,k2rB'(t - h)] dh
x e"
where I^p) is a first order modified Bessel function with
argument p.
Again substituting Equation 9 into 5A gives
V(x, s) = pr
k,U[s + (y/fl)k4 + k j
k2k5 + k3k5]
[k,
_ k,U[s + (y/8)k4 + k5]
(s - d)(s - f)
_ k,A'O
(s - d)
k,BT)
(s - f)
(s - f)n n!
522
CIRCULATION RESEARCH
n^o
VOL. 41, No. 4, OCTOBER
1977
(s-f)cnn!
(s - f)=n n!
n tl
^o
(s - f)- "+1 n!
(s-f) n + 1 n!
n ^,
(s-d)-n!
(11)
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In turn, we find that the distribution of tracer in the
extracellular space is
v(x, t) =
J e""
w
i 4 7 i ) ] <• - £
[
J" e
[2^k,k2y^B'(t - h)]
(12)
" w
where lo(p) is the zero order modified Bessel function with
argument p.
Likewise, substituting Equation 9 into 6B gives
Z(x, s) =
s2 + [k2 + k3 + (y/6>)k4]s
k2k5 + k3k5]
(s - d)(s - f)
B 4i
(s - d) n + 1 n!
(13)
so that, for the intracellular space,
z(x, t) = *
Jf e*'-"> Io [2^k,k 2 y^A'(t - h)]
(14)
CAPILLARY AND SARCOLEMMAL BARRIERS IN THE HEART/Rose et al.
Note that for both the capillary and cell membranes it is
impossible to distinguish concentrative transport (k,/k2 >
1 or k^k^ > 1) by model analysis from equilibrative
transport with an expanded space size. In order to extract
information on intrinsic membrane transport parameters
from this type of modeling, one must have independent
data on the relative size of the compartments, i.e., y and
y/8. In addition, it is appropriate to note that partitioning
effects due to protein binding, steric exclusion, or Donnan
equilibria will be reflected in the relative space sizes.
Changes in the Outflow Concentration-Time Curves from a
Single Capillary as Substances with Increasing Permeability
are Examined
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In order to appreciate the possibilities inherent in the
model, it is appropriate to compute representative single
capillary outflows while varying the parameters in a systematic manner. In Figure 2 we have plotted the outflow
profiles of five cases, with progressively increasing capillary permeabilities (k, = k2 = 0, 0.1, 1.0, 10.0, and °°),
both for the case in which the substance, at equilibrium,
reaches the same concentration in the tissue cell and the
plasma, and for the case in which it would be, at equilibrium, concentrated within the cell by a factor of 10, or,
alternately, the case in which the cell is 10 times as large.
As the capillary permeability increases, the magnitude of
the throughput or nonexchanging component in the capillary decreases. The cases selected correspond to those
previously explored, in modeling capillary exchange with a
noncompartmented extra vascular space .16 It is appropriate
to note that, at infinite capillary permeability (bottom
panels), the throughput becomes a delayed impulse function emerging at t = (1 + y)r = 4r, and damped by
cellular uptake. The material in the extracellular space
becomes part of a common bolus of material which is
moving within both the plasma and the extracellular space,
and which is delayed by virtue of its propagation along the
length within this larger space. This last phenomenon
corresponds to that which we have encountered in the
liver, where the hepatic sinusoid lining is freely permeable
to soluble materials.17-19
The addition to the model of the cell space with finite
cell membrane permeability results (in the illustration) in
two apparent outflow components, the earlier apparently
corresponding to return from the extracellular space and
the later corresponding to return from the intracellular
space. With the larger or more concentrative cell, the later
component is lower and flatter, as expected. When intracellular irreversible sequestration is present, only this latter part is affected. It is thus not possible, even with
extremely high values of k5, to completely obliterate the
returning component for finite membrane permeabilities.
Some material continues to return to the capillary from the
extracellular space without ever encountering the irreversible metabolic uptake mechanism. It is also obvious from
the illustration that the sequestration process, k5, is much
more efficient when the cell is large compared to the
accessible extracellular space or when the cell membrane
is concentrative.
The two returning components in the modeling correspond qualitatively to those described on the downslope of
523
labeled water curves from the heart3-" and to their increasing prominence at higher flow rates.8
Form of the Concentration Profile within the Capillary, the
Extracellular Space, and the Tissue Cell
The outflow profile is the result of what has occurred
previously in time and space along the length of the capillary-tissue unit. To gain an understanding of these events,
it is appropriate to examine the concentration profiles
within the capillary, the extracellular space, and the tissue
cells, as a function of time. Figures 3 and 4 show how the
addition of the sarcolemmal barrier and cell space affects
concentration profiles parallel to the capillary axis. In
comparison with the case with no cell present, the concentrations of material in the capillary and extracellular
spaces, behind the impulse function, are reduced. Since
the tissue cells are not adjacent to a space in which flow is
occurring, the concentrations there usually decrease monotonically from the beginning to the end of the capillary,
except at the higher capillary permeability, when a peak in
the profile tends to emerge (Fig. 4).
Steady State Solutions
The basic differential equations can be reduced to their
steady state form by eliminating the time-varying terms.
Thus we find
(15)
W ^ + k50z = 0
k,u - k2v - k3v + k4Z = 0
(y/6)k v - (y/6)k<z - ksz = 0
(16)
(17)
For water, when k5 = 0, these equations are trivial. For
sequestered substances, where k5 is finite, their solution
yields a description of the steady state lengthwise concentration profile in each space,
u( x ,oo) = u(0, t)e (y/e)k,k. + k,k, + k,k,
(18)
where u(0, t) is a constant concentration at x = 0.
x u(x,oo)
Z(x,oo) =
(20)
RECOVERY AND MEAN TRANSIT TIME
Recovery = Fc
r
Jo
u(L, t) dt
F c | u(L,t)tdt
Mean transit time = f = —^—
Fc J u(L, t) dt
Instead of attempting to evaluate these expressions by
integrating the outflow concentration directly, we can use
the following property of the transformed equation:
U(L, s)
f u(L, t)e-sldt
= a, -
CIRCULATION
524
RESEARCH
where
VOL. 41, No. 4, OCTOBER
= T,m + (1 - b ) ( t - T , . - T € J ,
a, = f u(L, t)dt = U(L, s)|, = ,
and
-a* =
Thus, from Equation 9,
and when k5 = 0, that is, when no material is lost,
Jf u(L, t)dt = | -
(21A)
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and the recovery
Fc j " u ( L , t ) d t = q0.
(21B)
1977
(23B)
and w[Tc(t) + r/t)] dt is the number of capillary-large
vessel units with total transit times between [rc(t) + T^t)]
and [Tc(t) + T((t) + dt]. Equation 10A can be substituted
in Equation 23 to give the predicted output, remembering
that t in Equation 10A becomes [t - TAX)] and (t — T)
becomes [t — TC — T X O ] .
The dimensionless parameter b can vary between 0 and
1 and partitions the dispersion of total transit times
through the microcirculation between the nonexchanging
and exchanging vessels.9 When the arteriolar or precapillary sphincters are dilated, b approaches 0 (the capillary
transit times become uniform), and when they are constricted, b approaches 1 (the large vessel transit times then
become uniform). Practically, b cannot be obtained unambiguously when it is less than 1. It is reported as a relative
parameter, b' = b(ki"y). Likewise TCm is not explicit except
when b = 1. It is reported here as a' = Tcjkiy)All else being equal, variations in a' and b' are reflections of changes in the flow patterns in the microcirculation and might not be expected to affect steady state
parameters such as the mean transit time. In practice, it is
Now
3U(L, s)
ds
=
k.k.yrB 'i
f2
J
%
Fc
LK2K5 "+" ^ K g +
Thus the mean transit time measured at the capillary
outflow is
(22)
[k 2 k 5 + k 3 k 5
impossible to change the resistance of a vascular bed
without secondary changes in either the perfusion pressure
at constant flow or the flow at constant perfusion pressure.
Analysis of Data
When k5 is finite, this accounts for only part of the total
transit time, the other part being the outflow at the sequestration site. When kb = 0,
t = T + (k./kjyr + [(k I kJ/(k I k 4 )]0r. (22A)
The last expression is the one we would have intuitively
expected.
THE ORGAN MODEL
Consider, as we have proposed previously, that the
heart is an organ composed of parallel capillary-large vessel units with no bypass interactions.7 Then the outflow
concentration of tracer is given by
(23)
w[Tc(t)
dt,
where rCm and i>_ are the minimum capillary and large
vessel transit times, respectively, TC and rf are the capillary
and large vessel transit times, q is the amount of injected
tracer, F is the total flow through the organ,
b(t - T,. - TCJ
(23A)
COMPUTATION OF THE APPROPRIATE REFERENCE
FOR LABELED WATER
Unlike sucrose, labeled water is transported in red cells
as well as plasma and, if it were confined to the vessels,
would emerge with an outflow pattern different than that
of the plasma tracer, labeled albumin. Since the half-time
for exchange of water across the red cell membrane is of
the order of milliseconds,20' 21 we can assume that the
presence of red cells results in a transit time corresponding
to the flow in both phases22 but does not hinder the
movement of water out of the capillary. Given this assumption, we can calculate the appropriate reference for
water from the red cell and albumin curves23
C(t)refw = j ~ [Hct frbc Qt) r h e
(24)
where fb^, frbc, and fp^ are the volume fractions of water in
whole blood, red cells, and plasma, respectively, and
C(t)rbc and C(t) a]b are the outflow fraction-time curves for
labeled red cells and albumin, respectively. Here we assume f^ = 0.7 ml/ml, and fp, = 0.94 ml/ml. Thus
fb,
0.7 Hct + 0.94(1 - Hct
CAPILLARY AND SARCOLEMMAL BARRIERS IN THE HEART/Rose el al.
PARAMETERS DERIVED FROM THE SUCROSE CURVE
Using the albumin curve as a reference in each case, we
fitted the sucrose curves to the varying capillary transit
time parallel capillary-large vessel organ model which we
have described previously.9 The fit provides four unambiguous parameters, two related to the permeability of the
capillary and the extravascular space of distribution and
two describing the degree of capillary transit time heterogeneity:
k;
= the capillary permeability-surface product
per unit extravascular space,
4>s
= the flow p e r unit accessible extravascular
space
^* w2J —— where -ys is the ratio of extravascu-
525
display sets of outflow curves from the heart, with the
coronary circulation in a relatively vasoconstricted and
then in a relatively vasodilated state. A large change in the
curve shape occurs with the change in vascular resistance.
In the vasoconstricted state, there is an early and relatively
well defined peak. In the vasodilated state, the peak of the
curve is lower and later and the early downslope declines
less quickly. The curve shape change illustrated here is
characteristic and the change occurs in a relatively predictable fashion, as the coronary vascular resistance is varied.
The effect of the dinitrophenol infusion is even more
striking (Fig. 6). With dinitrophenol infusion, S-T segment
elevation occurs in the electrocardiogram and the resistance measurements show that coronary vasodilation has
occurred. The water curve changes in a characteristic fash-
i-l TsTs,
Downloaded from http://circres.ahajournals.org/ by guest on June 17, 2017
lar to intravascular space for sucrose, T^ is
the reference capillary transit time for sucrose of a given capillary-large vessel unit,
and Wi is the fraction of the total number of
units represented by index i. In general,
only the -y^s, products can be optimized.
The two elements in the product become
explicitly separable only when the previously described parameter b equals one,
when the large vessel transit times become
uniform. And
a' s andb' s = the intercept and slope, respectively, of the
initial linear portion of the log ratio-time
plot, the graph of ln[C(t)alb/C(t)sucrosc] vs.
t.
a's and b' s can then be used in the model for the water
curve. Note that these parameters are based on the use of
albumin as a reference. In order to use these parameters to
describe the behavior of labeled water (which, in contrast
to the labeled sucrose, enters the red cells) we must develop suitable corrections. These are described below.
RUN 1
VASOCONSTRICTED
S-T-»
RUN 2
DNP INFUSION
VASODILATED
S-T t
VARIATION IN SHAPE OF WATER OUTFLOW CURVES
The labeled water outflow curves from the heart form
the data base for our subsequent analysis. In Figure 5 we
VASODILATED
VASOCONSTRICTED
itf
RUN 3
VASODILATED
S-T-
lrf
lrf
10
15
20
0
TIME.SEC.
FIGURE 5 Example of change in shape of the water outflow curve
with vasodilation (data for experiment 2, runs 2 and 3). A small
amount of redrculation was present in this experiment since only
the circumflex branch of the left coronary artery was cannulated.
Dashed lines are exponential extrapolations.
X)
lOli.
20
0
20
TIME (SEC.)
FIGURE 6 Data from experiment 6, runs 1, 2 and 3 plotted
arithmetically on the left and semilogarithmically on the right.
526
CIRCULATION RESEARCH
ion which is different from that illustrated previously,
where the stimulus has produced vasodilation alone. The
peak of water curve is increased in magnitude, and the
early downslope declines more quickly. In the last run of
'he figure, the dinitrophenol infusion has been stopped.
Despite this, the vasodilation continues for many minutes.
A dilution study carried out at this point in time shows that
the shape of the labeled water curve has. returned to what
we would have expected, on the basis of Figure 5. The
peak of the labeled water curve has become lower (it is
about half that found with the dinitrophenol infusion), and
the early downslope of the curve again declines less rapidly. It is appropriate to note that the values for the
coronary vascular reference in runs 2 and 3 of this figure
are essentially the same. The toxic dose of dinitrophenol
has produced an effect on the shape of the labeled water
curve which appears to be relatively independent, in this
instance, of the state of the coronary vascular resistance.
Downloaded from http://circres.ahajournals.org/ by guest on June 17, 2017
Labeled water has generally been considered to undergo
flow-limited distribution into the myocardium, in the past.
Bassingthwaighte et al.24 have carried out a set of studies
in the isolated perfused heart, in which the shapes of
outflow curves for labeled water and antipyrine were observed, at varying flow rates. On the basis of their analysis
of these data, they suggested that when flow-limited distribution is taking place and when the relative heterogeneity
of flow remains constant, plots of fC(t) vs. t/t are invariant
with flow, and that this invariance can, in the converse
fashion, be used in a practical way to indicate the presence
of flow-limited distribution .24- 25 Here the changes in curve
shape appear not to conform to the above definition of
flow limitation. It could be argued, then, that the changes
observed are due to a complicated change in the diffusional interaction between unlike capillaries, accompanying the alteration in vascular resistance. This could be so,
but this explanation cannot be extended to the dinitrophenol infusion results. The unexpected change in the shape
of the labeled water curve, with the infusion of dinitrophenol, and its reversion to the expected form during the later
period of continuing vasodilation, lead one to infer that, in
at least this instance, the curve shape change has occurred
as the result of a change in the permeability of one of the
biological barriers in the heart to labeled water. One might
then also ask whether, under ordinary circumstances, the
biological barriers in the heart act as an effective kinetic
constraint on the distribution of labeled water, in an indicator dilution experiment.
APPLICATION OF THE TWO-BARRIER MODEL TO
THE PROBLEM OF THE EXCHANGE OF LABELED
WATER
In its distribution into the myocardium, labeled water
encounters two surfaces, the capillary endothelium and
the sarcolemmal membrane. It seemed likely that with the
two-barrier model we could quantitate the degree of constraint imposed at each surface. Further, since the sarcolemmal membrane offers the more complete surface, it
also seemed likely that the dinitrophenol effect would be
found to be mediated through effects at that surface.
The combination of the two-barrier single-capillary
model with the parallel capillary-large vessel unit organ
VOL. 41, No. 4, OCTOBER
1977
model entails the optimization of six unambiguous parameters:
kc
= the capillary permeability-surface product
per unit extravascular-extracellular (interstitial) space for water (equal to k, or
k2 of the model),
kmw
= the
sarcolemmal permeability-surface
p r o d u c t per unit interstitial space for water (k 3 or k4 of the m o d e l ) ,
= the ratio of interstitial to intracellular
spaces for w a t e r ,
yw/0w
<1> W
= the flow per unit interstitial space for water, and
a'w and b' w = the structural parameters corresponding
to those for the sucrose model but in
general not equal to them.
Although it is theoretically possible to optimize all of
these parameters, the task would require many hours of
central processing unit (CPU) time and would probably
not give unique values for each parameter. Therefore we
reduced the number of parameters to be fitted to four by
using a's and b' s from the sucrose fit instead of a'w and b' w .
The use of this simplification makes it impossible to reach
a valid estimate of the parameter y w . Instead, an alternate
parameter y' w arises. The parameters a's and b' s , derived
from the model analysis of the labeled albumin and sucrose curves, implicitly define the capillary transit times
for a plasma label in the myocardium. The vascular reference label transit times suitable for reconstruction of the
events underlying the distribution of labeled water differ
from these, however.23 The labeled water enters the red
cells which bypass a small peripheral layer of plasma,
during their flow through the capillaries,26 and thus travel
slightly faster than the plasma. The fit to the water curve
will provide values for
*w, = 1/CywTw.)
In this case, since the values for a given sucrose capillary
transit time TS have been used, the fit provides the alternate equivalent
so that
w =
TW/TS,
where y' w /y w is smaller than one. It should be noted that
the procedure developed for fitting the sucrose curve9
results only in values for the products ysrSl, that specific
values for ys or Tj, are accessible only in the case in which
the large vessel transit times are uniform, and that; similarly, the fitting procedure to the water curve results in
values only for the products y'KTH. Hence the working
relation arising from the fits is
Finally, the fit to the water curve provides an estimate of
the ratio yw/0w. The value obtained appears both valid and
appropriate. It is independent of any assumptions concerning the transit times.
CAPILLARY AND SARCOLEMMAL BARRIERS IN THE HEART/Rose et al.
The space ratio yvlys, although inaccessible, also deserves some mention. If we define
_ red cell water space in the capillary
plasma water space in the capillary
then
y./y.
=
Now if Hcte is the hematocrit of blood in the capillary, fr is
the volume fraction of water in red cells, and fp is the
volume fraction of water in plasma,
= Hctefr/(1 and
(1 (1 Downloaded from http://circres.ahajournals.org/ by guest on June 17, 2017
FITTING PROCEDURES
In order to evaluate the integrals in the expressions for
the second or returning components of Equation 10A, it is
necessary to interpolate between data points on the reference curve. This is done most conveniently and unprejudiciously by fitting a cubic spline through the data points and
then using the spline coefficients as required. The model
was then translated into Fortran and automatically fitted,
by the technique of steepest descent, to the sucrose and
water data, utilizing, in each case, the appropriate vascular
reference curve. The coefficient of variation was used as
the criterion for goodness of fit. The fit to the sucrose
curve was relatively fast, requiring only 2 to 3 minutes of
527
CPU time, but that for the water curve required about 30
minutes of CPU time.
Computing was done on a remote terminal connected to
an IBM 360 digital computer. The initial fit to a water
curve, which entailed a trial-and-error search for the appropriate domain in four-dimensional parameter space,
was greatly facilitated by the use of a Tektronix 4012
Computer Display Terminal which allowed immediate
output of the estimate in graphical form. Without this tool
it is unlikely that the explanation for the dinitrophenol
effect would have been found. In dealing with models of
this complexity, it is almost essential to have immediate
and rapid graphical output in order to explore and appreciate the possibilities of the model without having to laboriously plot every curve by hand. The differences between
columns of numbers are almost unintelligible unless
viewed in two dimensions.
Results
Table 1 shows the hemodynamic parameters for the
eight experiments reported here, and Table 2, the corresponding best fit model parameters. As we observed previously,9 a reduction in coronary resistance at constant
flow is accompanied by a decrease in capillary transit time
heterogeneity (b' decreases). In spite of these changes in
flow pattern, the best fit value for k ^ remains constant;
k<.w decreases proportionately to ks with a reduction in
perfusion pressure due to reduction in the total number of
perfused capillaries. The ratio of kCw to ks remains about
20. With the infusion of dinitrophenol, kmw is reduced by a
factor of 6 to 15 and returns toward normal when the
TABLE 1 Hemodynamic and Electrophysiological Parameters
Experiment
Run
Perfused
heart weight
(g)
Flow
(ml/min)
Hct
Heart rate
(beats/min)
Systolic
pressure
(torr)
Perfusion
pressure
(torr)
Resistance
(torr • min/ml)
1
1
2*
56
76
0.37
162
156
95
105
230
60
3.03
0.79
2
1
2
3*
70
84
0.48
174
180
120
110
115
50
195
195
55
2.32
2.32
0.66
1
2
98
138
144
114
90
85
20
130
135
75
1.35
1.41
0.78
3
96
0.37
3t
4
1
2
5
1
2
6
1
121
107
1
119
1
58
2t
3
• Dilated by dipyridamole.
t Intracoronary infusion of 2.4-dinilrophenol.
T
0.41
126
126
120
120
140
135
1.61
1.55
—
83
0.40
150
138
70
65
90
60
1.08
0.72
->
98
0.61
222
210
150
100
105
110
225
160
140
2.30
1.63
1.43
T
156
162
80
70
210
80
2.47
1.06
174
156
150
70
65
95
145
65
65
2.20
0.98
0.98
85
0.37
2t
8
^
87
2t
3
7
S-T
66
0.37
t
T
CIRCULATION RESEARCH
528
TABLE
2 Best Fit Model Parameters
Run
(sec-)
(sec-)
a'
b'
(sec-)
CV S '
kc.
(sec-)
y'Jy*
(sec-)
y/e
1
1
2
0.0291
0.0288
0.0516
0.0599
0.1090
0.1990
0.0824
0.0599
0.0484
0.0300
1.1900
0.6130
0.3760
0.6200
0.4070
0.3710
0.2780
0.5050
0.0630
0.0649
2
1
2
3
0.0387
0.0389
0.0402
0.0728
0.0890
0.0984
0.2100
0.0877
0.2900
0.0857
0.1040
0.0313
0.0413
0.0612
0.0590
0.7680
2.1300
0.7180
0.4430
0.4810
0.8780
0.4950
0.4770
0.4960
0.3310
0.3060
0.3950
0.0579
0.0902
0.0868
3
1
2
3
0.0963
0.0702
0.0123
0.1190
0.0863
0.0360
0.2180
0.2260
0.3420
0.1320
0.1070
0.0000
0.0679
0.0540
0.0764
1.8600
1.1600
0.2130
0.7190
0.6440
0.5080
0.4620
0.4340
0.0256
0.2450
0.2560
0.3800
0.0400
0.0417
0.0797
4
1
2
0.1010
0.1070
0.0926
0.0943
0.2160
0.2070
0.1390
0.1580
0.0764
0.0648
1.6200
1.3900
0.6410
0.6610
0.4780
0.4680
0.1790
0.2090
0.0837
0.0463
5
1
2
0.0478
0.0431
0.0730
0.0799
0.3380
0.3150
0.0678
0.0503
0.0741
0.0313
0.7430
0.6720
0.6390
0.7360
0.4360
0.4350
0.4130
0.4160
0.0496
0.0490
6
1
2
3
0.0861
0.0116
0.0056
0.0980
0.0515
0.0193
0.1400
0.2170
0.2610
0.1370
0.0021
0.0075
0.0700
0.0433
0.0435
1.3600
0.2960
0.4760
0.6300
0.4620
0.1820
0.4060
0.0639
0.1930
0.2540
0.1750
0.6010
0.0397
0.1161
0.0594
7
1
2
0.0304
0.0102
0.0720
0.0483
0.0944
0.1810
0.0666
0.0065
0.0739
0.0627
0.8630
0.2490
0.4840
0.6410
0.2860
0.0530
0.3230
1.1470
0.1280
0.0772
8
1
2
3
0.0470
0.0183
0.0233
0.0860
0.0897
0.0931
0.2045
0.2040
0.2290
0.0648
0.0000
0.0046
0.0440
0.0355
0.0467
0.8880
0.3000
0.5021
0.6970
0.9470
0.7220
0.4710
0.0678
0.3850
0.3030
0.1790
0.6080
0.0484
0.0711
0.0809
Experiment
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1
VOL. 41, No. 4, OCTOBER 1977
CVW*
CV, and CVW are coefficients of variation for fits of the model to the sucrose and water curves, respectively.
infusion is stopped. Dinitrophenol does not affect the
capillary permeabilty independently of changes in surface
area as reflected in the sucrose kg. The change in kmw with
the dinitrophenol infusion is independent of the associated
dinitrophenol-induced vasodilation, since the preparation
remains maximally dilated for a long time despite the
return of kmw toward normal and the return of an isoelectric S-T segment. The average value for kmw in the control
experiments is 0.41 sec"'.
y'vily* ' s always less than one, since labeled water has a
larger intravascular space of distribution than does the
labeled sucrose.
The average value for y/6 is 0.24, when those experiments where kn,w is reduced are excluded from the calculations.
Figure 7 (upper left panel) shows the best fit to the data
of run 1 of experiment 4. Although the throughput component is very small (less than 1 % of the total outflow)
and thus would be undetectable by unreferenced residue
detection, it has a profound influence on the shape of the
labeled water outflow pattern. In addition to the best fit,
three other cases have been illustrated. On the upper
right, we have illustrated the effect of assuming that the
sarcolemma has become impermeable, but all other parameters are the same. It is seen that this case closely
resembles the data found during dinitrophenol infusion.
The peak and early downslope are much increased in
magnitude. On the lower left, we have illustrated the
effect of assuming that the sarcolemma has become, in
contrast, very permeable. The peak has become quite
delayed. Finally, in the lower right, we have illustrated the
effect of assuming that both the capillary and sarcolemma
are very permeable. The appearance is delayed, the upslope increases very slowly, and the peak is very delayed.
The inference arising both from this illustration and
from the values for k,,w and kmw in Table 2 is that labeled
water is not distributed into the myocardium in a flowlimited manner.
Discussion
VALIDITY OF THE MODELING
If this model is a reasonable first approximation to the
true behavior of water in the heart, we should arrive at an
estimate of y/6, the ratio of interstitial to intracellular
spaces consistent with morphometric and gravimetric data.
The total water content of the heart is 0.76 ml/g 3 and the
total volume of plasma and interstitial space is 0.19 ml/
ml.27 Since the specific gravity of dog ventricle is 1.06,28
the intracellular water occupies 0.56 ml/g. We know that
the total intravascular sucrose space is about 0.067 ml/g.8
Unfortunately, we do not know exactly what fraction of
the total intravascular space is occupied by capillaries. If
we assume for the sake of argument that two-thirds of the
intravascular space is capillaries, then the size of the interstitial space is 0.19 - 0.04 = 0.15. The ratio of the
interstitial space to intracellular space is then 0.15/0.56 =
0.27. This estimate is not exact, since we do not know the
exact capillary plasma volume. The average best fit value
for y/6 of 0.24, arising from the dilution curves, is very
close to this estimate. With this agreement as a criterion,
the model provides as good a description of the behavior
of labeled water as the data permit. It should be noted that
CAPILLARY AND SARCOLEMMAL BARRIERS IN THE HEART/Rose et al.
529
brain capillaries would be expected to be less permeable
than muscle capillaries, because of the presence of tight
junctions between the endothelial cells and the lack of endothelial vesicles. Our model analysis gives a value for cardiac capillary water permeability that is about 20 times
that for sucrose. Since the sucrose capillary permeablity is
approximately 3 x 10~5 cm/sec,9 the corresponding
permeability of the cardiac capillaries to labeled water is
about 6 x 10~4 cm/sec. This value is comparable to that
recently found for lung capillaries by Perl et al.29 of (15 ±
5) x 10~4 cm/sec. Eichling et al.5 have estimated the
labeled water permeability of the brain capillaries to be
1.9 x 10~4 cm/sec; and Bolwig and Lassen, 0.4 x 10~4
cm/sec.7 The higher permeability of the heart endothelium
is likely due both to the absence of interendothelial cell
tight junctions in this tissue and to the presence of vesicular transendothelial channels.30 If the permeability of the
endothelial sheet, apart from the communicating pathways, is the same in the heart as the brain, the labeled
water flux across the capillary wall through the pathways is
only a little greater than through the cells.
Since k ^ = Pmw x Smw/Ve we can calculate the water
permeability of the sarcolemma, Pmw, if we know the
sarcolemmal surface area, Sn,, and the size of the interstitial space, Ve, in addition to kmw. Using the value of 0.15
ml/g for Ve, calculated earlier, and 4200 cm2/g for Sm,,,14
we then find Pmw = 0.41 x 0.15/4200 = 0.15 x 10"4 cm/
20
0
sec. This value is surprisingly low, compared with the dog
TIME.(SEC)
erythrocyte water permeability of 44 x 10~4 cm/sec,31 but
FIGURE 7 Fit of the model to the data of run I of experiment 4
with that of the squid axon, 1.4 x 10"4
with predicted outflow curves for altered capillary or sarcolemmal is not incompatible
32
cm/sec, the barnacle muscle cell, 2.6 x 10~4 cm/sec,33
permeabilities: 5lCr-RBC, solid line; '2hI-albumin, large dashed
4
34
l4
3
line; C-sucrose, small dashed line; H2O, dot-dashed line; model, the amoeba, 0.23 x 10" cm/sec, or the Bolwig4 and
Lassen
estimate
for
the
brain
capillary,
0.4 x 10~ cm/
dotted lines. The throughput or nonexchanging component of the
sec.7
model is shaded. The total water outflow (upper dotted line) is the
sum of this component and the returning component.
This limited permeability of the sarcolemma to water
probably explains the inability of Suenson et al.35 to explain the transient diffusion of water through a sheet of
right ventricle. It should be possible to obtain an indewhen k,,^ decreases greatly, the best fit value for y/d
pendent confirmation of our estimate of sarcolemmal
becomes inaccurate, since most of the tracer returns from
permeability by extending the standard plane sheet tranthe interstitial space and very little enters the cell in a
sient diffusion equation to include this barrier.
single passage. If there is also a significant barrier to water
In the presence of toxic doses of dinitrophenol, the
diffusion at the level of the mitochondrial membrane, it
sarcolemmal permeability is reduced by an order of magseems unlikely that we will be able to dissect it out with
nitude. A search of the literature failed to reveal any
these dilution studies.
previous report of such an effect. Abood et al.36 have
WATER PERMEABILITIES OF THE CAPILLARY AND
reported that dinitrophenol reduces the membrane potenSARCOLEMMA
tial of the frog sartorious muscle, independent of its effect
on oxidative phosphorylation. Tasaki et al.37 have atOur analysis indicates that both the capillary and sarcotempted unsuccessfully to alter the water permeability of
lemmal membranes are significant barriers to the diffusion
the squid axon but apparently did not try dinitrophenol.
of labeled water, and that the water permeability of the
Macey et al.20 have discovered that sulfhydryl reagents
sarcolemma can be reduced by dinitrophenol, in concert
such
as p-chloromercuribenzene sulfonate (PCMBS) dewith changes in the electrical properties of the membrane.
crease the water permeability of the red cell membrane by
There are data in the literature concerning the permeabilian order of magnitude at millimolar concentrations, and
ties of the two barriers which tend to complement the
these investigators proposed that a large fraction of the
present study. The observation of a reduction in the perwater flux across the membrane is mediated by a protein
meablity of the sarcolemma to labeled water, with dinitrocarrier or "pore." If water does cross biological memphenol infusion, appears to be original. We could find no
branes only via randomly opening "cracks" in the lipid
previous record of a similar observation.
bilayer, it would be very difficult to explain the large effect
In recent studies of labeled water exchange in the
5 7
of
small concentrations of dinitrophenol and PCMBS on
brain, " it has been reported that the distribution of tracer
water permeability.
is, in part, barrier-limited. The endothelium lining the
10
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530
CIRCULATION RESEARCH
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For many years, dinitrophenol and related compounds
have been known to affect the ion permeability of red cell
and artificial membranes. Shulachev et al.38 first showed
that dinitrophenol and other uncoupling agents caused an
increase in the conductivity of artificial phospholipid
membranes due to an increase in the selective permeability of hydrogen ions. Rothstein et al.39 have reviewed the
effects of sulfhydryl- and amino-reactive agents on red cell
membrane permeability. l-Fluoro-2,4-dinitrobenzene
(FDNB), an amino-reactive agent, causes an increase in
potassium permeabilty but a decrease in sulfate permeability. McLaughlin40 has proposed that dinitrophenol is
unlike other uncouplers, in that it binds to the surface of
the bilayer to cause a negative surface potential. While
these studies show that uncoupling agents and substances
reacting with proteins can change the properties of membranes by different mechanisms, there is at present no
unifying hypothesis accounting for all these effects. The
change in water permeability induced by dinitrophenol
appears to be another facet of the total effect.
Apart from its importance in the theory of membrane
structure, the exact magnitude of the water permeability
of membranes is of limited practical importance. However, since the washout of inert gases and antipyrine from
the heart is similar to that of water,3 the understanding of
the factors governing water exchange is of direct relevance
to the transport of respiratory gases. In the case of oxygen,
the combination of limited capillary permeability with a
large intravascular binding capacity (y<S 1) would imply
that a large fraction of the coronary venous oxygen content has not exchanged with the tissue at all.
VOL. 41, No. 4, OCTOBER 1977
distributed over the whole heterogeneity of transit times,
is the kind of labeled water curve illustrated in the lefthand
panel of Figure 5, with a relatively abrupt rise and relatively high and early peak.
When vasodilation occurs, the spectrum of capillary
transit times condenses and converges toward a common
value, larger than a large proportion of the transit times
found in the vasoconstricted state.9 This results in a slower
initial rise in the upslope of the labeled water curve, with a
later and lower peak, the result of the greater equilibration
time to which the earliest outflow elements have been
exposed. The analysis thus leads one, in a qualitative
sense, to expect the change in shape of the labeled water
curve, displayed in Figure 5.
The distribution of labeled water in the myocardium is
not flow-limited. It is limited by barriers both at the
capillary and the sarcolemmal cell surface. It is appropriate to note, for future reference, that metabolites consumed by the heart will be similarly affected by both
phenomena at these barriers and the heterogeneity of
capillary transit times. The proportion of material presented to the intracellular metabolic machinery will depend both on the permeabilities of the capillary and sarcolemmal cell surface and on the capillary transit time distribution. Knowledge of the parameters describing both sets
of phenomena will be essential.
TRANSIT TIME = 2 SEC.
1 5
TRANSIT TIME = 10SEC.
r-
0 006 7
THE FORM OF THE LABELED WATER OUTFLOW
CURVE
In the vasoconstricted situation, it is especially instructive to consider the effect of the capillary transit time
heterogeneity on the distribution of labeled water in the
vascular, interstitial, and cellular spaces. To exemplify
these effects, we have picked a capillary transit time corresponding to the upslope of the outflow curve (2 seconds),
and one corresponding to the downslope (10 seconds),
and then have used the permeabilities and space sizes
obtained from the best fits to the labeled water curve to
generate the two sets of panels in Figure 8. With the
shorter transit time (the lefthand panels), there is a significant loss of material into the interstitial space but, early in
time, little of the tracer penetrates the sarcolemmal barrier
to enter the cell. A throughput component reaches the
outflow. With the longer transit time (the righthand
panels), a larger proportion of tracer is lost to the interstitium, and a greater proportion enters the muscle cell
(although, even here, the tracer entry does not occur
rapidly enough for tracer equilibration to occur). No significant proportion of the throughput arrives at the outflow. Thus, early in genesis of the outflow curve, when the
coronary vasculature is vasoconstricted, more of the tracer
emerging has come through as throughput or has entered
only the interstitial space, and returned to the outflow
without entering the cells. Later in the genesis of the
curve, a larger proportion of the tracer has been delayed
by cellular entry. The result of events such as these,
X
z
O 05
£ °
z '
UJ
1.5
u
u
x/L
FIGURE 8 77ie e/jfecf o / capillary transit time on the theoretical
distribution of labeled water in the intravascular, interstitial, and
intracellular spaces. The plots are analogous to those of Figures 3
and 4, but here we have used the approximate values for the
parameters obtained from the actual fit to the water curve: k, = kt
= kCic = 1.0 sec'1, y = 1.0, km<r = 0.4 sec'', and y/6 = 0.25. Note
that the difference in transit time can be due either to a change in
capillary length at constant velocity of flow or to a change in
velocity along a constant capillary length. With the longer transit
time, more of the tracer enters the cellular space and the tracer is
retained in all spaces for a longer time.
CAPILLARY AND SARCOLEMMAL BARRIERS IN THE HEART/Rose et al.
TABLE 3
531
Outflow Fraction/ml x JO3 for Experiment Illustrated in Figure 6*
Run 1
Run 2
Run 3
51
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Samplet
"Cr-RBC
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
1.38
11.00
33.20
61.20
85.00
93.60
84.60
64.00
43.90
29.50
21.20
15.30
11.00
7.65
6.34
4.88
3.90
2.94
2.28
1.77
1.44
1.04
1IS
I-Alb
0.80
6.24
19.10
36.90
53.00
64.90
70.90
68.30
53.30
45.10
35.80
27.10
21.60
16.00
12.80
10.30
8.66
6.64
5.42
3.97
3.17
2.42
14
3
C-Sucr
0.66
5.12
14.30
26.80
34.00
36.20
35.20
32.00
26.10
22.30
18.80
16.40
14.80
13.20
12.40
11.20
10.60
9.60
8.80
8.00
7.50
6.90
H,O
0.15
0.90
3.26
6.31
9.30
10.90
12.50
12.60
12.20
11.60
11.10
10.40
10.00
9.40
9.00
8.60
8.30
8.10
7.80
7.50
7.20
6.90
51
Cr-RBC
10.40
63.40
114.00
121.00
99.00
65.00
41.50
24.80
17.00
11.20
8.00
6.04
4.30
3.41
2.75
2.25
1.79
1.56
1.33
1.15
0.98
0.88
0.79
1B
I-Alb
4.46
38.40
83.70
104.00
98.00
75.80
53.58
34.20
26.40
18.20
13.00
10.40
7.79
6.57
5.40
4.35
3.70
3.29
2.90
2.59
2.23
2.00
1.75
"C-Sucr
THO
3.53
30.40
67.80
86.00
82.20
62.70
43.30
27.80
21.30
15.20
11.10
8.65
6.73
5.69
4.91
4.34
3.69
3.31
3.03
2.75
2.50
2.35
2.15
0.31
2.96
8.33
14.50
20.20
23.00
23.20
22.50
21.40
19.60
17.50
16.50
14.70
13.60
12.70
12.00
11.00
10.20
9.40
9.00
8.30
8.03
7.50
CrRBC
ls
I-Alb
1.12
15.40
52.20
81.90
71.10
55.90
38.30
26.90
17.90
12.80
9.66
7.65
5.75
4.58
3.90
3.06
2.50
2.11
3.82
33.10
83.70
99.50
65.10
43.40
25.30
16.30
10.20
6.97
5.08
3.39
2.64
2.08
1.65
1.42
1.23
1.05
14
C-Sucr
0.79
11.20
40.20
62.30
55.00
42.50
29.00
19.40
13.80
9.61
7.06
5.64
4.54
3.68
2.88
2.54
2.02
1.82
3
H,O
0.08
0.71
2.31
4.60
5.83
7.41
8.16
8.77
9.24
9.30
9.30
9.20
8.90
8.62
8.30
7.90
7.60
7.20
* Data for experiment 6. Dinitrophenol was infused during run 2.
t Each sample represents 0.74, 0.70, and 0.71 seconds in runs 1, 2, and 3, respectively.
THE CONCEPT OF FLOW-LIMITATION
In the light of the present findings of significant barriers
to the distribution of labeled water at the capillary and
sarcolemmal membranes, it is appropriate to examine the
basis for the hypothesis that water is a flow-limited substance in the heart. 2 ' 3 - 24r 25 We have already noted that
water does not behave precisely as expected, according to
the practical definition of flow-limitation proposed by Bassingthwaighte et al. 24 - 25 Ziegler and Goresky 8 showed that
the two components of the downslope of labeled water
outflow curves become more emphasized when flow is
increased in a vasodilated preparation, and we have shown
here that, even at constant flow rate, the shape of the
outflow curve can be altered in a major way by vasodilation and dinitrophenol infusion. On closer examination it
turns out that plots of t-C(t) or residue function vs. t/I
are rather insensitive to flow changes when the barriers to
diffusion are of the magnitude encountered for water. In
Figure 9 we have plotted the predicted residue function for
labeled water for a single capillary, as a function of normalized time, over a 4-fold range of transit times (or flows
if the capillary has a constant length). These curves resemble in form the normalized whole-organ experimental residue curves obtained by Yipintsoi et al .3 in an isolated heart
perfused at a variety of flows. In the case of the whole
organ, the variation in the form of the outflow data was
similar to that illustrated here, but the data showed a
somewhat random rather than systematic change in form
as a function of flow. The authors used the relative concurrence of their data to adduce the hypothesis of flowlimited distribution of labeled water in the heart (in the
absence of any specific way of accounting for the shape of
the curve). Simple extrapolation from the single capillary
outflow profiles would lead one to expect a systematic
ordering of the small differences in shape of the wholeorgan residue profiles. More specific consideration of the
phenomena underlying the whole-organ outflow profiles
1.0
R(t|
0.5
12
6
T
FIGURE 9
/r
Plots of residue junction, R(t) = (1 — }u(L,
t)), vs. th
for a single capillary, utilizing the same values for capillary and cell
membrane permeabilities as in Figure 8. The function is shown for
capillary transit times of 2, 5, 7, and 10 seconds. There is remarkably little change in the curves when normalized in this manner.
These curves should be compared with those in Figure 1 of Reference 3. These curves in that reference are from a perfused whole
heart and include effects due to changes in the heterogeneity of
transit times and the capillary surface area. The first factor would
not be expected to result in a systematic deviation since, in this
preparation, the whole-heart curves will be heavily weighted for a
small range of transit times at the peak of the reference curve. The
second factor will tend to cause faster capillary transit times than
expected al low flow rates because of a decreased vascular volume
at low perfusion pressures.
532
CIRCULATION RESEARCH
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leads one to expect that these will, in addition, be affected
by the heterogeneity of transit times in large vessels and
capillaries, and by the number of capillaries open at a
given vascular resistance and pressure. The resulting effects will tend to blur the small systematic change expected
on the basis of the single capillary modeling. At the level
of the whole organ, outflow observations can be predicted
only for a given state of coronary vascular resistance, and
predictions concerning their change with flow in a reactive
coronary circulation can be made only if there is firm
knowledge of the accompanying change in vascular resistance and hence of transit times. In a passive nonreactive
coronary circulation, expectations of the changes with flow
will again differ.
The conclusion of the foregoing is clear. Our analysis of
the forms of the labeled water outflow curves indicates
that the distribution of labeled water is barrier-limited,
whereas that of Yipintsoi et al.3 would indicate that it is
flow-limited. More precise experimental data obtained
with well defined preparations would help resolve the
issue, but an unequivocal method of measuring the permeability of a capillary of the type found in the heart, which
resolves the issue, would be even more informative.
A second phenomenological definition of flow-limitation was proposed by Ziegler and Goresky.8 Since the
outflow curves of water, ethanol and antipyrine are almost
superimposible they defined flow-limited behavior as the
outflow pattern that appears to be independent of the
tracer diffusion coefficient. However, this behavior can
also arise if the membrane permeabilities of this group of
substances are equal and if the diffusion distances between
membranes are small enough that virtually instantaneous
equilibration occurs radial to the axis of the capillary for
all of the soluble molecules, or if the profiles belong to a
regime in which the changes in form consequent to
changes in permeability are small. These authors also
noted that the form of the outflow profiles for this group of
substances does not correspond to the form expected for a
delayed wave flow-limited output from a multicapillary
organ.16 Given the assumptions used in the present analysis, the predicted flow-limited outflow would be predicted
to vary between the two asymptotic forms shown in Figure
10. In the vasoconstricted case, the predicted outflow is
analogous to that observed for labeled water outflow profiles from the liver, where there is a large heterogeneity of
capillary transit times.4 It should be noted that, in that
case, the water curve can be superimposed on the reference curve by use of the normalizing procedure suggested
by Bassingthwaighte et al. However, in the vasodilated
case, where the capillary transit times are constant, this
normalization will not result in superimposition. In neither
the vasoconstricted nor the vasodilated case is the predicted coronary outflow profile for labeled water anywhere near the observed outflow profile. It is impossible
to force the flow-limited model to fit the water data by
manipulating the transit time heterogeneity parameters.
Thus, if water, ethanol, and antipyrine are really distributed in a flow-limited manner in the heart, it is necessary
to introduce the additional hypothesis that the higher
membrane permeability of these substances results in a
large degree of diffusional interaction between unlike cap-
VOL. 41, No. 4, OCTOBER
1977
VASOCONSTRICTED
10 -
id2
id3
ltf YZ-t-L-
z
o
INTRAVASCULAR REFERENCE
-THO
PREDICTED FLOW-LIMITED OUTFLOW
VASODILATED
30
40
TIME (SEC.)
50
60
70
80
FIGURE 10 Predicted flow-limited behavior using the data from
runs 1 and 2 of experiment 8. The extravascular volume available
to labeled water was assumed to be four times that for sucrose. The
reference curve is a composite of the red cell and albumin curves, as
explained in the text.
illaries. However, the magnitude of the effect needed to
explain the data is such that it amounts to assuming that
the tissue behaves as a single well-stirred compartment.
This could occur in two ways: either by such rapid longitudinal diffusion that the whole behaves as a single wellmixed compartment, or by virtue of a systematic architectural staggering of entrances and exits such that massive
diffusional intercapillary connections occur. The calculations of Bassingthwaighte15 indicate that the former is
unlikely, and the silicone reconstructions of Yipintsoi et
al.41 virtually exclude the latter. We therefore were faced
with a paradox. The barrier-limited modeling was found to
fit the sucrose data within experimental error, yet we were
unable to fit the labeled water outflow curves with similar
assumptions. Our development of the two-barrier model
represents our resolution of this paradox.
Acknowledgments
We thank Louise Gagnon and Kim McMillan for their technical assistance; Margaret Mulherin for typing the manuscript; Dr. W. D. Thorpe and
the McGill Computing Center especially for permission to use the computer for a nominal rate after 6:00 p.m.; and the Montreal General
Hospital Research Institute for the funds to aid in the purchase of a
Tektronix display terminal.
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C P Rose, C A Goresky and G G Bach
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Circ Res. 1977;41:515-533
doi: 10.1161/01.RES.41.4.515
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