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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 Downloaded from http://circres.ahajournals.org/ by guest on June 17, 2017 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 Downloaded from http://circres.ahajournals.org/ by guest on June 17, 2017 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 Downloaded from http://circres.ahajournals.org/ by guest on June 17, 2017 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 Downloaded from http://circres.ahajournals.org/ by guest on June 17, 2017 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) Downloaded from http://circres.ahajournals.org/ by guest on June 17, 2017 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 Downloaded from http://circres.ahajournals.org/ by guest on June 17, 2017 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 ,\ / Downloaded from http://circres.ahajournals.org/ by guest on June 17, 2017 + 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) Downloaded from http://circres.ahajournals.org/ by guest on June 17, 2017 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 Downloaded from http://circres.ahajournals.org/ by guest on June 17, 2017 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) Downloaded from http://circres.ahajournals.org/ by guest on June 17, 2017 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 Downloaded from http://circres.ahajournals.org/ by guest on June 17, 2017 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 Downloaded from http://circres.ahajournals.org/ by guest on June 17, 2017 530 CIRCULATION RESEARCH Downloaded from http://circres.ahajournals.org/ by guest on June 17, 2017 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 Downloaded from http://circres.ahajournals.org/ by guest on June 17, 2017 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 Downloaded from http://circres.ahajournals.org/ by guest on June 17, 2017 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. References 1. Davson H; A textbook of General Physiology. London. J.A. Churchill. Vol. 1, 1970, pp 395-507 2. Johnson JA, Cavert HM, Lifson N: Kinetics concerned with distribution of isotopic water in isolated perfused dog heart and skeletal muscle. Am J Physiol 171: 687-693, 1952 3. Yipintsoi T, Bassingthwaighte JB: Circulatory transport of iodoanti- CAPILLARY AND SARCOLEMMAL BARRIERS IN THE HEART/Rose et al. Downloaded from http://circres.ahajournals.org/ by guest on June 17, 2017 pyrine and water in the isolated dog heart. Circ Res 27: 461-477, 1970 4. Goresky CA: A linear method for determining liver sinusoidal and extravascular volumes. Am J Physiol 204: 626-640, 1963 5. Eichling JO, Raichle ME, Grubb RL Jr, Ter-Pogossian MM: Evidence of the limitations of water as a freely diffusible tracer in brain of the rhesus monkey. Circ Res 35: 358-364, 1974 6. Patlak CS, Fenstermacher JD: Measurement of dog blood-brain transfer constants by cisternal perfusion. Am J Physiol 229: 877-884, 1975 7. Bolwig TG, Lassen NA: The diffusion permeability to water of the rat blood-brain barrier. Acta Physiol Scand 93: 415-422, 1975 8. Ziegler WH, Goresky CA: Transcapillary exchange in the working left ventricle of the dog. Circ Res 29: 181-207, 1971 9. Rose CP, Goresky CA: Vasomotor control of capillary transit time heterogeneity in the coronary circulation. Circ Res 39: 541-554, 1976 10. Levitt DG: Theoretical model of capillary exchange incorporating interactions between capillaries. Am J Physiol 220: 250-255, 1971 11. Yudilevich DL: Serial barriers to blood-tissue transport studied by the single injection indicator diffusion technique. In Capillary Permeability, edited by C Crone and NA Lassen. Copenhagen, Munksgaard, 1970, pp 115-134 12. Ziegler WH, Goresky CA: Kinetics of rubidium uptake in the working dog heart. Circ Res 24: 208-220, 1971 13. Sheehan RM, Renkin EM: Capillary, interstitial, and cell membrane barriers to blood-tissue transport of potassium and rubidium in skeletal muscle. Circ Res 30: 588-607, 1972 14. Tancredi RG, Yipintsoi T, Bassingthwaighte JB: Capillary and cell wall permeability to potassium in isolated dog hearts. Am J Physiol 229: 537-544, 1975 15. Bassingthwaighte JB: A concurrent model for extraction during transcapillary passage. Circ Res 35: 483-503, 1974 16. Goresky CA, Ziegler WH, Bach GG: Capillary exchange modeling: barrier-limited and flow-limited distribution. Circ Res 27: 739-764, 1970 17. Goresky CA, Bach GG, Nadeau BE: On the uptake of materials by the intact liver: the transport and net removal of galactose. J Clin Invest 52: 991-1008, 1973 18. Goresky CA, Nadeau BE: Uptake of materials by the intact liver: the exchange of glucose across cell membranes. J Clin Invest 53: 634-646, 1974 19. Goresky CA, Bach GG, Nadeau BE: On the uptake of materials by the intact liver: the concentrative transport of rubidium-86. J Clin Invest 52: 975-990, 1973 20. Macey RI, Karan DM, Farmer REL: Properties of water channels in human red cells. In Passive Permeability of Cell Membranes, edited by F Kreuzer, JFC Siegers. New York, Plenum Press, 1972 21. Paganelli CV, Solomon AK: Rate of exchange of tritiated water across the human red cell membrane. J Gen Physiol 41: 259-277, 1957 22. Goresky CA, Bach GG, Nadeau BE: Red cell carriage of label; its limiting effect on the exchange of materials in the liver. Circ Res 36: 328-351, 1975 533 23. Goresky CA, Wamica JW, Burgess JH, Nadeau BE: Effect of exercise on dilution estimates of extravascular lung water and on carbon monoxide diffusing capacity in normal adults. Circ Res 37: 379-395, 1975 24. Bassingthwaighte JB, Strandell J, Yipintsoi T: Flow-limited washout of diffusible solutes from the heart. In Capillary Permeability, edited by C Crone and NA Lassen. Copenhagen, Munksgaard, 1970, pp 580-585 25. Bassingthwaighte JB, Yipintsoi T: Organ blood flow, wash-in, washout, and clearance of nutrients and metabolites. Mayo Clin Proc 49: 248-255, 1974 26. Prothero J, Burton AC: The physics of bloodflowin capillaries. I. The nature of the motion. Biophys J 1: 565-579, 1961 27. Polimeni PI: Extracellular space and ionic distribution in rat ventricle. Am J Physiol 227: 676-683, 1974 28. Yipintsoi T, Scanlon PD, Bassingthwaighte JB: Density and water content of dog ventricular myocardium. Proc Soc Exp Biol Med 141: 1032-1035, 1972 29. Perl W, Silverman F, Delea AC, Chinard FP: Permeability of dog lung endothelium to sodium, diols, amides, and water. Am J Physiol 230: 1708-1721, 1976 30. Simionescu N, Simionescu M, Palade GE: Permeability of muscle capillaries to small heme-peptides: evidence for the existence of patent transendothelial channels. J Cell Biol 64: 586-607, 1975 31. Rich GT, Sha'afi RI, Barton TC, Solomon AK: Permeability studies on red cell membranes of dog, cat, and beef. J Gen Physiol 50: 23912405, 1967 32. Villegas R, Villegas GM: Characterization of the membranes in the giant nerve fiber of the squid. J Gen Physiol 43: 73-103, 1960 33. Bunch W, Edwards C: The permeation of non-electrolytes through the single barnacle muscle cell. J Physiol 202: 683-697, 1969 34. 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McLaughlin S: The mechanism of action of DNP on phospholipid bilayer membranes. J Membrane Biol 9: 361-372, 1972 41. Bassingthwaighte JB, Yipintsoi T, Harvey RB: Microvasculature of the dog left ventricular myocardium. Microvasc Res 7: 229-249,1974 The capillary and sarcolemmal barriers in the heart. An exploration of labeled water permeability. C P Rose, C A Goresky and G G Bach Downloaded from http://circres.ahajournals.org/ by guest on June 17, 2017 Circ Res. 1977;41:515-533 doi: 10.1161/01.RES.41.4.515 Circulation Research is published by the American Heart Association, 7272 Greenville Avenue, Dallas, TX 75231 Copyright © 1977 American Heart Association, Inc. All rights reserved. Print ISSN: 0009-7330. 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