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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
 
Circ Res. Author manuscript; available in PMC 2010 November 16.
Published in final edited form as:
PMCID: PMC2982781
NIHMSID: NIHMS233297

Myocardial Sodium Extraction at Varied Coronary Flows in the Dog

ESTIMATION OF CAPILLARY PERMEABILITY BY RESIDUE AND OUTFLOW DETECTION

Abstract

Sudden injections of boluses containing both 131I-albumin and 24NaCl were made into the coronary artery inflow of isolated blood-perfused dog hearts. Indicator dilution curves were recorded using gamma emissions from both the intact heart and the coronary sinus outflow, with plasma flows, Fs, ranging from 0.3 to 1.8 ml/g min−1 Three measures of sodium extraction, E, during transcapillary passage were obtained from each site by comparison of the sodium and albumin curves. The most useful estimates of E were “instantaneous extractions” obtained from the later part of the upslope and the peak of the venous dilution curves (coronary sinus) or from the corresponding early phase of washout of the externally monitored curves (intact organ). Extractions were lower at higher flows. Permeability-surface area products, PS, were computed (1) by the formula PS = −Fsloge (1 – E), (2) by fitting the observed dilution curves with a Krogh capillary-tissue cylinder model, and (3) by the approximating formula PS = −Fsloge (1 – 1.14E). The two latter approaches provided a correction for back diffusion of tracer from tissue to blood. For sodium, the values of PS averaged 0.88 ± 0.36 (SD) ml/g min−1, (N = 52). At high flows, with Fs > 1.0 ml/g min−1, the values of PS averaged 1.01 ± 0.38 ml/g min−1 (N = 11). Assuming S = 500 cm2/g and plasma to be 93% water, our findings suggest capillary permeabilities for sodium of about 3.1 × 10−5 cm/sec.

The transcapillary exchange of a nonmetabolized diffusible substance is dependent on the surface areas and the rates of permeation through the capillary and cell membranes, the rates of diffusion in the extravascular tissue, and the capillary blood flow. When there is some diffusional resistance in the capillary membrane or the tissue, the amount of the substance escaping from the capillary blood to enter the tissue is less at high flow (with high blood velocities in the capillary) than it is at low flow. One approach to quantifying this diffusional resistance is to calculate the fraction of a diffusible substance leaving the capillary during one passage. Crone (1) originally calculated an instantaneous extraction of a permeating tracer by comparing its concentration in the venous outflow with that of a nonpermeating reference indicator after both had been injected simultaneously into the inflowing arterial blood. Zierler (2) used the time integral of the difference between the curves, which has its maximum at the time of crossover of the curves for the reference and the permeating tracer. Lassen and Crone (3) preferred an average extraction calculated from the areas of the curves up to the peak of the outflow dilution curve. We will compare these three methods. In addition, we will introduce and demonstrate the validity of the calculation of instantaneous extractions from the residue functions obtained by external monitoring. A set of three estimates of extraction, E, conceptually analogous to the three estimates from the outflow curves, are provided by the residue function curves, R(t), permitting comparison with each other and with the outflow extractions.

When the extraction E is attributable to a unidirectional efflux from the plasma across the capillary membrane and into the tissue and is not reduced by a flux in the opposite direction, then one can estimate PS, the product of permeability, P (cm/sec), and capillary surface area, S (cm2/g), using the standard formula (1, 3, 4):

PS=Fsloge(1E),
(1)

where Fs is the flow (ml/g sec−1) of solute-carrying mother fluid, which for sodium is simply the plasma flow. This formula is based on the assumptions that (1) there is no differential intravascular dispersion leading to the separation of permeant and nonpermeant tracers along the stream, (2) there is no return of tracer (“back diffusion”) from tissue to blood at the moment of estimation, and (3) only unextracted “throughput” material reaches the outflow.

Tracer back diffusion inevitably soon occurs, since the extravascular volume is limited. Yudilevich and Martin de Julian (4), assuming that the organ was homogeneously perfused and that the earliest extraction gave the truest estimate, have proposed a correction based on the assumption that the permeating tracer escapes into a first-order mixing chamber and returns. The idea is reasonable, but its applicability is questionable when there is intravascular dispersion or heterogeneous perfusion. Arguments based on Krogh cylinder modeling with a dispersed input and intravascular diffusion and dispersion (5) support the use of an extraction occurring at or near the time of the peak, tp, of hN(t), the transport function, the frequency function of transit times, of the nonpermeating tracer through the organ from the arterial inflow to the venous sampling point.

Heterogeneity of regional perfusion in the heart can be substantial (6), adding to the complexity of the situation. Eq. 1 illustrates that with a given PS the extraction, E, is lower in regions of higher flow; thus, one should design the experiment to provide E from a region whose local perfusion (per gram of tissue) is the same as the average over the whole organ, Fs. In a compact organ like the heart without great disparities in the volumes of inflow and outflow vessels in different regions, the outflows from regions of average flow will occur with average transit times, that is, near t, the mean transit time, which is a little after the time of the peak, tp. The modeling by Goresky et al. (7) of a group of capillaries with different transit times shows that the maximum values of an instantaneous extraction give values closest to the correct estimate of PS. From the experimental data of the present study, we will show that sodium extraction is maximum very near the time of the peak, tp, and therefore that a compromise between tp, t, and the time of maximum extraction is very easy, the minor differences being of no practical and little theoretical importance quantitatively. These differences are also small compared with the differences in estimates of PS due to correction for back diffusion.

Our objectives are to estimate the PS of myocardial capillaries for sodium and to ascertain the influences of total coronary blood flow on the estimates of PS for sodium in the heart. The data of the study permitted transcapillary extraction to be estimated by different methods using both outflow and residue detection.

ANALYTICAL APPROACH TO THE EXPERIMENTATION

The technique of the present experiments was to inject a short bolus containing a nonpermeating reference tracer, N (131I-albumin), and a permeating or diffusible tracer, D (24NaCl), into the inflow to the heart and to record the concentration-time curves in the venous outflow, CN(t) and CD(t), and, via an external gamma detector, the organ's residual content of tracer, C*N(t) and C*D(t). From these residual contents, C*N(t) and C*D(t), RN(t) and RD(t), the residue functions, were calculated directly:

R(t)=C(t)C(0),
(2)

where C*(0) is the peak net count rate obtained 2 or 3 seconds after injection. (The implicit assumptions are that all of the tracer is in the organ, none having yet escaped, and that the time required to achieve the peak is negligible compared with the time course of washout. The validity of the calculation of E is scarcely affected by minor inaccuracies in these assumptions, which are made primarily to simplify the conceptual formalism of the equations that follow.)

When the data were recorded for very long times without recirculation, the outflow response could be normalized to unity area as follows:

h(t)=C(t)/0C(t)dt.
(3a)

This procedure was usually adequate for albumin, but the tails of the curves for 24Na were very much longer, and it was therefore important to account for the residual tracer in the heart at a final time T. Since the area Th(t) h(t) dt is exactly R(T) when there is no delay between exit from the organ and arrival at the venous sampling site, then

h(t)=C(t)0C(t)dt=C(t)0TC(t)dt0TC(t)dt0C(t)dth(t)=C(t)0TC(t)dt0Th(t)dt=C(t)0TC(t)dt[1Th(t)dt],h(t)=C(t)0TC(t)dt[1R(T)].
(3b)

This equation provides values of h(t) for tT, but beyond this time the shape remains undetermined even though the area is known. The terminology is that of Zierler (2) and of a symposium volume (8).

Fractional extractions can be calculated either from the paired h(t)'s from the venous outflow or from the residue function curves, the R(t)'s, obtained by external gamma detection. Two situations are diagramed in Figure 1: the left sections portray a hypothetical situation in which tracer that has escaped from blood into tissue does not return to the blood until all of the nonextracted tracer has passed the outflow sampling point—a clear separation of extracted from nonextracted components—and the right sections portray a situation in which the two components of hD(t) have shapes identical to those of the left sections but, instead of being widely separated, have a degree of overlap close to what can be expected in most organs and, therefore, summate. In practice, there is some delay and distortion introduced by tubing or veins between the exit from the organ and the site of detection of the outflow curves so that Eq. 3b is not exact. Complete correction for the difference would require accounting for the transport function of the sampling system, either by convoluting it with the R(t)'s or by deconvoluting the h(t)'s. The major part of the correction was obtained simply by accounting for the delay, using the mean transit time of the sampling system, and ignoring the dispersion.

FIGURE 1
Methods of estimating extractions. E. Broken lines = diffusible indicator (subscript D) and solid lines = reference indicator (subscript N). Left: The transport function, hD(t), the probability density function of arrival times of the permeating tracer ...

The integral formula of Zierler (2) for the net or the integral extraction, E1(t) uses the whole of the curves obtained up to time t; for outflow sampling

E1v(t)=0t[hN(τ)hD(τ)]dτ,
(4a)

where the subscript 1 refers to this approach, v indicates that the experimental data are from the venous outflow, and τ is a variable used only for the integration.

By performing the integration in Eq. 4a and using the information that the amount of indicator not yet arrived at the venous site is that which remains in the organ, an extraction, E1H (t), theoretically identical to E1V(t), can be obtained from the externally monitored residue functions:

E1H(t)=RD(t)RN(t),
(4b)

where the subscript H denotes external monitoring of the organ (heart). The equivalence to E1V follows from the definitions given in the legend to Figure 1:

E1H(t)=RD(t)RN(t)=0thN(τ)dτ0thD(τ)dτ=E1v(t).

The integral net extraction (i.e., E1V or E1H) reaches a plateau when hN(t) washes out prior to the washout of the late fraction of hD(t) which traversed a partially extravascular route, as in the left sections of Figure 1; however, when the two components overlap, as in the right sections of Figure 1, E1(t) fails to achieve the same maximum. In this latter case, the maximum, reached as hD(t) exceeds hN(t), provides an underestimate of the fraction traversing an extravascular pathway. Although we will show that this problem inhibits the use of this technique in estimating sodium extraction in the heart, the method is useful when tracer return is slow, as it is for calcium or strontium extraction in bone.

The second method is the instantaneous extraction proposed by Crone (1):

E2v(t)=[hN(t)hD(t)]hN(t).
(5a)

The subscript 2 denotes this “instant-by-instant” extraction. E2(t) can also be calculated from the derivatives of the curves obtained by external monitoring over the heart:

E2H(t)=dRN(t)dt+dRD(t)dtdRN(t)dt=1dRD(t)dRN(t).
(5b)

Using finite precordial counting periods of duration At, the simplest finite difference approximation to Eq. 5b is:

E2H(t+Δt2)=1RD(t+Δt)RD(t)RN(t+Δt)RN(t).
(5c)

When Δt was short, 1 second, we used another linear approximation to Eq. 5b that employed more data points and helped to reduce the variation due to the Gaussian scatter of the count rates:

E2H(t+Δt2)=1RD(t+2Δt)+RD(t+Δt)RD(t)RD(tΔt)RN(t+2Δt)+RN(t+Δt)RN(t)RN(tΔt).
(5d)

So long as regional blood flows in the organ are uniform and the shortest transit time for a tracer traversing an extravascular region is longer than the shortest intravascular transit time, as is the case for the functions diagramed in Figure 1, then the initial values of E2(t) are uninfluenced by back diffusion from tissue to blood. When there is heterogeneity of regional flows, then one naturally expects the earliest values of E2(t) to represent the extraction in high-flow regions, but in our experiments only the mean flow is known. Our analysis is based on the argument that the maximum extraction, E2M, is approximately that occurring with the average flow, Fs, as one sees with mathematical models (5, 7).

The area-weighted extraction, E3(t), is conceptually similar to E2(t) in that one expects the early parts of the outflow dilution curves to have similar shapes; using integrals causes E3(t) to be smoother than E2(t). E3(t) is an area-weighted mean extraction where the value is influenced by the whole curve up to time t; it is the area between the curves up to time t divided by the area of the reference curve up to the same time.

E3V(t)=0t[hN(τ)hD(τ)]dτ0thN(τ)dτorE3V(t)=1.0tathD(τ)dτtathN(τ)dτ,
(6a)

where ta is the appearance time of the indicators in the venous outflow (see Fig. 1). The values for E3 are equal to E10thN(τ)dτ that is, E1, divided by the fraction of tracer which has escaped from the heart. This equality is in accord with the definition 0th(τ)dτ=1R(t). Using this definition for RN (t) and RD (t), we can derive an expression analogous to Eq. 6a providing the extraction curve, E3H (t), directly from recorded residue functions:

E3H(t)=RD(t)RN(t)1RN(t).
(6b)

Lassen and Crone (3) expressed a preference for using the areas up to the time of the peak, tp, of hN(t). We will see that the E3 (tP)'s and the maximums differ very little. The first point of E3 is identical to that of E2, but thereafter they differ when the extraction is variable.

The maximum values of these various E(t)'s will be denoted by E1VM, E1HM, E2VM, E2HM, E3VM, and E3HM

PS products were calculated from the instantaneous extraction E2V(t). The first and simplest method was the straightforward application of Eq. 1, using E2V(tp) or E2VM as suggested from Krogh cylinder modeling (5). This method provides no correction for back diffusion.

The second method was to fit the observed curves of hN(t), hD(t), and E2V(t) using a fully defined Krogh cylinder single capillary model (5) and adjusting the parameters to achieve a best fit, which automatically corrects for back diffusion in so far as the model is applicable. The model contains several parameters, but most of them are not freely variable. Best fits were obtained by adjusting principally only two parameters, the permeability, P(cm/sec), and the extravascular volume of distribution for sodium, VE(ml/g). The main kinetic events can be expressed in terms of two dimensionless groupings of parameters which contain P and VE; these groupings are PS/Fs for instantaneous extraction and PStcap/VE for back diffusion from tissue to blood (where tcap is the capillary transit time, the capillary volume divided by the blood flow). The surface area S and the capillary volume (and therefore tcap and the ratio of extravascular to intravascular volumes of distribution) have narrowly limited variability; the maximum values were obtained from anatomic studies in dog myocardium (19) providing an average capillary radius of about 2.5μ and an average intercapillary distance (from the capillary density) of about l6μ. (These values give, by Eq. 12 of ref. 5, a maximum surface area, S, of 667 cm2/g. The average value of S used was that with an intercapillary distance of 18μ or 527 cm2/g, which is close to the anatomically estimated average of 500 cm2/g [9]. The variation in S is taken into account in our reporting PS rather than P.)

Axial dispersion of permeant and nonpermeant tracers in the capillary blood also influences the apparent extraction, particularly in the earliest samples during the initial part of the upslope of h(t). Over a large range of possible effective dispersion coefficients (DCD and DCN of ref. 5), the influence on Emax and E′(tp) was small, the most obvious effect being a smoothing of the early portion of E2V(t), as can be seen in Figure 6. Because of the long sampling intervals (1 second or more) and the potential errors due to the geometry of the detection systems, the data of this study cannot provide a critical test of the influences of intravascular dispersion on the spread of131I-albumin and 24NaCl. Values of 0.5–1 × 10−5 cm2/sec were used for these axial dispersion coefficients to give an appropriate rising curve in the early portion of E′(t), as in Figure 6, but there was no discernible effect on Emax or E′(tp).

FIGURE 6
Venous outflow dilution curves fitted by the Krogh model solutions for two hearts. The experimental albumin curve. hN(t). is used as input to the model. The experimental sodium curve. hD(t), is the dotted line. The observed extraction. E2V(t) (crosses) ...

The myocardial water content of hearts perfused with blood in the fashion of these experiments is about 0.79 ml/g (10), which is higher than that for normal dog hearts, 0.777 ml/g (11), but less than that for hearts perfused with Ringer's or Tyrode's solution in our laboratory, 0.83–0.85 ml/g. Myocardial density, ρ, was calculated from its known relationship to water content (11), 1/π = 0.69 + 0.322 × water content, giving ρ = 1.06 g/ml for calculating the myocardial volume from the measured weight. Although values for VE could have been considered to be bounded below by the 0.19 ml/g estimated for sulfate space in hearts from intact rats by Polimeni (12) and above by the 0.4 ml/g obtained by Johnson and Simonds (13) for sucrose space in Ringer's-perfused rabbit hearts, such restrictions were not imposed; VE was permitted to range freely for the fitting procedure.

To fit the curves, the observed hN(t) was shifted forward in time by tcap and used as the input function to the model (5); then, P and VE were adjusted so that the computed transport function, hD(t), and the computed extraction, E2V(t), matched the experimental curves, hD (t) and E2V(t), as closely as possible as judged by viewing them displayed on an oscilloscope. (The prime is used to indicate a function or a value resulting from the Krogh cylinder modeling.)

In addition, we derived a simple approximate method of estimating PS that also accounts for back diffusion. The method can be applied by using either a graph or a simple equation; it does not require a computer. This method based on the analyses using the Krogh model and on the assumptions, argued previously (5), that the regions with average flows per unit volume have transit times near tp, the time of the peak of hN(t), and that the extractions measured at tp are representative of regions with average flow. The derivation and the application of the method are given together under Results.

Experimental Methods

Experiments were performed on 12 isolated beating hearts taken from dogs weighing 6–9 kg and perfused with blood from support dogs weighing 20–25 kg as described previously (10, 11). Figure 2 is a diagram of the experiment. The perfusion rate, F/W (the measured blood flow divided by the heart weight), was controlled by a roller pump and varied from 0.5 to 3.1 ml/g min−1. Fs was plasma flow, (1 – Hct)F/W, since negligible amounts of sodium enter the erythrocytes during the passage of the bolus. The highest flows were generally attained with nonpulsatile perfusion pressures of less than 170 mm Hg by decreasing the coronary vascular resistance with intra-arterially administered epinephrine (5–15 μg/min) or dipyridamole (Persantin, 0.1–0.2 mg/min). Perfusion pressure, blood flow, blood temperature, heart weight, and heart rate were monitored throughout the experiment. Coronary sinus outflow discharged through a Silastic tube (2.5 mm, i.d.) inserted into the right ventricle through the pulmonary artery (PA in Fig. 2). Recirculation of blood containing the indicator was prevented or delayed by permanently or temporarily withholding the coronary venous return during the first 3 minutes after injection. Less than 3% of the tracer was retained at this time so there was little need for semilogarithmic extrapolation of the curves.

FIGURE 2
Experimental setup for the detection of tracer clearance from the isolated blood-perfused heart. Nal-crystal gamma detectors recorded emissions from the heart itself (LV crystal) for RN(i) and RD(t) and from the venous outflow for hN(t) and hN(t): the ...

Indicators

24NaCl solution (Cambridge Nuclear Corporation), a sterile solution of 99% radiochemical purity, was diluted to approximately 100 μc/ml. Human serum 131I-albumin (RISA-131, Abbott Laboratories) having an activity of 100 μc/ml was supplied in sterile isotonic saline solution. Boluses containing approximately 0.2 ml of the 24Na and 0.4 ml (0.1–0.8 ml) of the 131I-albumin solution were injected into the aortic cannula in about 0.5 seconds. The volume of the aortic root was variable (from 2 to 5 ml) and allowed some expansion with the injection (and with each cardiac contraction) so that changes in perfusion pressure were negligible. The peak count rates for each isotope over the heart were 3000–8000 counts/sec; over the venous outflow they were 2000–5000 counts/sec.

Isotope Detector-Counter Systems

Each of two systems (detector, photomultiplier, preamplifier, amplifier, and two pulse-height analyzers and counters) was similar to that used previously (10). One 5-cm (diameter) Nal crystal detector probe, surrounded by a steel and lead collimator, faced the posterolateral wall of the left ventricle. The other, facing the venous outflow tubing, was uncollimated but surrounded by 10-cm thick lead shielding (Fig. 2). Each amplifier output went to two pulse-height analyzers (Hamner Electronics) and counters (Picker digital dual rate computer). The energy window for 24Na counted pulses over 1 Mev and was uninfluenced by 131I emissions. The 131I window was very narrow, 0.35–0.38 Mev, accepting only emissions around the primary gamma peak; count rates in this window were corrected for a spillover of 6–9% of the rates obtained simultaneously in the 24Na window when only 24Na was present. This percent, K, depended on probe position, the window settings, and the distance of the heart from the probe and was calculated from observations made before and after each experiment using the same instrument settings and isotope source position as were used during the experiment. Variation in the position of the heart comparable to that occurring during an experiment and absorption by heart tissue and water-filled phantoms were observed to have only insignificant effects on values of K. The systems showed no losses in counts and no spectral distortion at rates up to 120,000/sec. Counting periods were 1 second at high flows and 3 seconds at low flows. Albumin curves were terminated at 0.25% of peak and sodium curves at 1–2%.

Because four channels of simultaneous isotope counts were required during the experiment and only two could be put on punched paper tape at once, the data were temporarily recorded and stored on analog FM magnetic tape (Hewlett Packard model 3900) with the other data such as pressures and timing signals.1 For convenience in analysis, the data were stored on IBM punched cards.

Estimation of Extractions

The maximum values for EWM1VM, E1HM, E2HM, E2HM, E3HM, and E3HM, were obtained from their respective continuous functions by manual smoothing through the three to five points around the maximum and taking the maximum value of the smooth line. In general the maximums of E2 and E3 occurred near tp, but when the flow was low their maximums tended to be somewhat earlier, as can be seen in Figure 4.

FIGURE 4
Extractions, E2V(t), for sodium in the venous outflow at various plasma flows, Fs. At high flow, transit times were short. Lower flows resulted in larger extractions, as might be expected from longer transcapillary transport time or contact time. At all ...

Results

Responses to 41 injections were obtained in 12 isolated hearts. Thirteen pairs of simultaneous albumin and sodium curves of both h(t) and R(t) were recorded by detection of tracer in both the venous outflow and the heart. Eight pairs of curves were recorded from the heart only, and the remaining 20 pairs were recorded from the venous outflow only. Responses recorded following single injections in two hearts are provided in Table 1. Following the injection, both outflow and residue responses were recorded; for these two experiments, the simultaneous values of hN, hD, RN, and RD are provided so that they will be available for analysis by others. Some time delay between tracer seen by the residue function detector and the venous outflow detector was evident: the appearance of tracer in the outflow occurred in each case about two sample intervals after the peak of the residue function. For the same reason, each integral 0Th(t)dt was less than 1 – R(T), but, for example, for experiment 26018-1 it was approximately equal to 1 – R(T – 5 seconds), thus providing a rough estimate of 5 seconds for the mean transit time of tracer from the heart to the venous outflow detector. The volume calculated to account for such a delay is 1.6 ml/sec × 5 seconds = 8 ml, which is twice as large as the volume of the actual tubing and perhaps implies that some of the tracer contained in the right ventricle and the outflow tract was not seen by the residue detector. These points are brought out only to show that Eq. 3b cannot be applied directly at any time without considering the transit time delay. However, during the tails of the curves, the delay is of little consequence, and Eq. 3b can be applied with less than 2% error.

TABLE 1
Responses Recorded following Single Injections in Two Hearts

Typical curves shown in Figure 3 are from one heart at two flows. The h(t)'s calculated from the venous dilution curves are shown in the left top section. At high flow (Fs = 1.23 ml/g min−1, Fig. 3, left, top section of top group) the early portion of hD(t) had a shape similar to hN(t), but at low flows (Fs = 0.48 ml/g min−1, Fig. 3, left, top section of bottom group) the sodium curve, hD(t), had a low blunt peak. The extractions, E1V(t), E2V(t), and E3V(t), shown in the left bottom section of each group are typical in that (1) the early portions of the instantaneous E2V(t) and the area-weighted E3V(t) curves were similar to each other and (2) the maximum of the net integral extraction, E1V(t), was later and lower than those of either E2V(t) or E3V(t). Since, by definition, the maximum value of E1V(t) occurs at the time of the crossover of hN(t) and hD(t), significant back diffusion must have occurred, thus emphasizing that E1V(t) cannot reach values representing extraction due to unidirectional efflux from capillary blood to tissue but is useful for providing a measure of the net retention of tracer in the tissue. E2VM, the maximum of the instantaneous extraction, E2V(t), occurred synchronously with or a bit later than tp, the time of the peak of the reference tracer curve, and therefore E2V(tp) and E2VM were similar.

FIGURE 3
Concentration-time curves of 2Na (labeled D, diffusible) and 131I-albumin (labeled N, nonpermeating) were recorded by continuous detection over the venous outflow (labeled V. normalized to give h[t], left) and simultaneously over the heart (labeled H ...

In general, the E(t) curves obtained from the residue functions (Fig. 3, right) had shapes and peak values similar to those obtained from the venous outflow curves. However, some small consistent differences were evident. For example, E2H(t) reached zero faster than did E2V(t), which was recorded some distance downstream. Accordingly, the rate of diminution of E2H(t) was faster than that of E2V(t); the difference was accentuated at lower flows, as can be seen in the lower sections of Figure 3, right. In this experiment, the volume of tubing between the observation sites was π(0.15)2 × 20 or 1.4 ml; some of the slowing in the downslope of E2V(t) compared with that of E2H(t) can be attributed to the dispersion in the venous system accompanying the transport delay.

The data from individual experiments, showing the experimental conditions, the various calculations of the maximum extraction, and the derived values for PS are given in Table 2. The symbols and the experiment numbers identify the data shown in the figures and in Table 1.

TABLE 2
Sodium Extractions and Estimates of PS

Effect of Flow on Extraction

In the seven dogs in which more than two sets of dilution curves were recorded, maximum flows ranged from 1.3 to 5.7 times the minimum. Figure 4 shows curves of E2V(t) obtained from one isolated heart at blood flows ranging from 0.47 to 2.04 ml/g min−1 (Fs = 0.28 to 1.23 ml/g min−1). The curves of E2V(t) consistently showed low early values rising to the maximum, and there was some tendency to more prolonged plateaus at lower flows. More importantly, the maximum values, not only of E2V but also of Eiv and E3V, decreased at higher flows (see Table 2).

Comparisons of Maximum Extractions

The maximums of the several E(t)'s from both heart and venous outflow are given in Table 2. The simultaneous estimates from venous and heart curves obtained by each method were comparable: the mean ± sd of E1HM/E1VM was 0.96 ± 0.20 (n = 11), that of E2HM/E2VM was 0.94 ± 0.13 (n = 12) (Fig. 5, right), and that of E3HM/E3VM was 0.96 ± 0.11 (n = 11). None was significantly different from unity, suggesting that the dispersion in the tubing between the heart and the venous observation point did not produce a detectable diminution in the values of Ev compared with those of EH. This finding is in line with the predictions from the model study (Fig. 8 of ref. 5) which showed that, when the extravascular volume of distribution of the solute was moderately large (20–30% of the tissue space) and the sampling system volume small, the dispersion resulted in very little reduction in E2VM.

FIGURE 5
Comparisons of the maximum of E2V(t) with the maximums, EIVM, E3VM, and E2HM. Values of E1VM underestimated E2VM. E2VM differed little from E2HM or E3VM. Straight lines are lines of identity. Symbols are the same as they are in Table 2.
FIGURE 8
Effect of flow on PS for sodium. The PS values are PS(graph) taken from columns 18 and 19 of Table 2 or the average PS(graph) when both E2VM and E2HM were obtained experimentally. Symbols are defined in Table 2. The increase at high flows presumably reflects ...

As one anticipates from Figure 1, the net extraction, E1(t), was smaller initially, at tp, and at its maximum than was either E2(t) or E3(t). The ratio E1VM/E2VM was 0.66 ± 0.13 (n = 27), as shown in Figure 5, left. The other ratios were: E1VM/E3VM = 0.67 ± 0.13 (n = 27), E1HM/E2HM = 0.68 ± 0.09 (n = 25), and E1HM/E3HM = 0.67 ± 0.11 (n = 25). Thus, E1M substantially underestimated the maximums of E2 or E3, demonstrating that the back diffusion from tissue to blood occurred without clear separation between tracer which remained in the blood and tracer which entered the tissue and returned. Again, this finding is in accordance with the model studies (5), and we conclude that E1, should not be used in Eq. 1 to calculate PS. It is, however, specifically useful in providing the net extraction by an organ at each time t and is more valuable than the residue RD(t), since it takes into account the fraction of intravascular tracer which has not washed out. The ratios of E1/E2 and the relationship of E1/E2 to flow were both too scattered to allow correction factors to be used to predict the unidirectional flux from E1 alone, particularly from E1HM, although this prediction might be possible in theory.

The differences between E2 and E3 were not great (Fig. 5, center). Numerically, E3(t) must have the same initial value as E2(t), but thereafter it is free to differ, as it will whenever there is heterogeneity of regional flows or extractions. Because E3(t) continues to be influenced by initial values and E2(t) does not, when the experimental values of E2(t) rise in the first few seconds, the maximum values for E3(t) will be slightly smaller than those for E2(t). However, the differences were negligibly small; E3VM/E2VM averaged 0.99 ± 0.12 (n = 27) (Fig. 5, center), and E3HM/E2HM averaged 1.03 ± 0.14 (n = 25).

Application of the Krogh Cylinder Model

Figure 6 shows two sets of experimental dilution curves fitted by theoretical curves derived from our version of the Krogh cylinder model (5). The experimental albumin curve was used as the input function forcing the response of the model: values for Fs, hematocrit. and heart weight were obtained experimentally; capillary radius (Rc = 2.5μ) and length (500μ) were chosen in accordance with data on dog hearts (9). To obtain a best fit, as judged by eye and quantified by the coefficient of variation, manual adjustments were made in the permeability, P (cm/sec), the extravascular volume of distribution, V′E (ml/g) (via adjustments in the fractional extravascular volume available to sodium and the intercapillary distance, over the range from 16 to 20μ, which influences the capillary surface area, S[cm2/g], and the relative volumes of capillary and tissue). (Changes in intercapillary distance from 15 to 19μ, giving values of S from 667 to 416 cm2/g, resulted in variations of less than 0.6% in the value of PS obtained from the fitting procedure.) The two freely adjustable fitting parameters, PS and V′E , are reported in columns 17 and 20 of Table 2 for the experiments on which we recorded outflow concentration-time curves. The goodness of fit was highly sensitive to PS, particularly in the regions of the maximum of hD(t) and in the part of E2V(t) preceding its rapid diminution. V′E was strongly governed by the shape of the tail of hD(t) and most particularly by the phase of rapid diminution of E2V(t). Adjustments in V′E and PS interacted, since both govern the amount of escaped tracer returning from tissue to blood (back diffusion), which raises h′D(t) and reduces E′2V(t) compared with the shapes they would have if no back diffusion occurred. The shape of h′D(t) that would be obtained without back diffusion is given by the dash-dot curves in Figure 6 obtained with the same parameters except for V′E = ∞ . The shape of the curve of tracer returning from the extravascular region is given by the difference between the dotted and the dash-dot curves and differs a little from those shown by Ziegler and Goresky (14) because of the axial diffusion. (Since Eq. 1 assumes no back diffusion, the importance of its neglect can be examined by comparing the estimates of PS from Eq. 1 with those determined from model fitting, columns 16 and 17 of Table 2.)

Particular values of E′2V from the Krogh cylinder modeling are given in columns 14 and 15 of Table 2, choosing E′2VM, the maximum, and E′2V(tp), the extraction at the peak of hN(t). In the experimental data, E2VM preceded and was usually higher than E2V(tp), but there was substantial variation in both of these. Figure 6 shows the most common situation where E′2VM and E2VM occurred before tp and exceeded E2V(tp). Experimentally, E2VM was obtained by averaging through a few points near its maximum, which did not precede tp by more than a few seconds; therefore, E′2V(tp) (column 15) was not much smaller than E2VM (column 9). The ratio E′2V(tp)/E2VM averaged 0.95 ± 0.10 (n = 27), which is not statistically different from unity. This finding can be regarded as empirical support for the recommendation (5) to use an averaged value of E2V near its peak. Even so, one can now recognize the degree to which E2V is reduced by back diffusion; the reduction was significant even in the experiments at high flow, for E′2V(tp) was still up to 13% less than E′2VM at plasma flows greater than 1.0 ml/g min−1. If they are used in Eq. 1, both E′2VM and E′2V(tp) give underestimates of the PS given by the model parameters.

There is one fairly consistent difference between the model and the experimental curves: the model's E′2V(t) rises very rapidly to its peak (the rise would be instantaneous if it were not for axial intracapillary dispersion), as seen in Figure 6, but the experimental E2V(t)'s, seen in Figure 4 particularly, tend to rise more slowly. It is likely that the slow rise is due to heterogeneity of regional blood flow; the tracer first emerges from regions of high Fs and low extraction. In these isolated heart preparations, the distributions of local flows have standard deviations of about 40% of the mean (6), which has not been accounted for in the present analysis but which probably accounts for the shape of the experimental curves.

Permeability-Surface Area Products for Sodium

Table 2 lists the three estimates of PS obtained from E2(t). For the sake of brevity and simplicity of comparison, we emphasized the E2(t)'s, particularly E2V(t) and E2VM, but we should point out that when one has only RN(t) and RD(t) that E3H(t) is a better function to work with. The first estimate of PS (column 16) involves the use of E2VM, E2HM, or their average in the classical equation PS = −Fsloge(l − E), (our Eq. 1), which does not account for back diffusion and can be expected to give underestimates of PS. The second estimate (column 17) is directly from the best fit of the model to each pair of venous outflow curves, hN(t) and hD(t), which provide PS′ and V′E, the primes indicating that the values come from the model analysis. The third estimate (columns 18 and 19) is from a graphical analysis to be described, which is an approximation to the modeling.

The model gave PS′ values higher than those from Eq. 1. Considering the model to give the best answer, PS from Eq. 1 (column 16) underestimated PS′ (column 17) by 23%; the ratio PS′/PS = 1.29 ± 0.19 (n = 27) with the probability of a chance difference from unity being less than 0.01. The degree of underestimation was not apparently different at different flows, in spite of our expectation that back diffusion would cause greater underestimation at low flows. (The regression relationship was PS′/PS = 1.33 − 0.042Fs with SDyx = 0.19, n = 27, and r = 0.99). Some correction factor would obviously be desirable; since E2VM (column 9 of Table 2) differed only slightly from the model's values of E′2VM and E′2V(tp) (columns 14 and 15), a simplified method of obtaining values equivalent to the model's PS′ should be attainable without the need for computer solutions.

A simpler method was developed using a semilog plot of 1 − E′2V(tp) versus PS−/FS (Fig. 7, left). A best-fitting curve through the data was obtained from the model with V′E = 0.21 ml/g. An empirical, but quite good, approximation to this curve is given by the equation:

PS=Fsloge{[0.88E2V(tp)]0.88}orPS=Fsloge[11.14E2V(tp)].
(7)

The 0.88 is derived from the best-fitting exponential, as described in the legend to Figure 7. The derivation of the approximation is given in Figure 7, right, on the basis that it may be useful for others to derive a similarly simple approximation suitable for other organs with variations in extravascular volumes of distribution or other features. For each experimental extraction, E2VM and E2HM, this line provided values for PS/FS, which, multiplied by the known solute flows, Fs, gave the values for PS(graph) in columns 18 and 19 of Table 2. Since PS(graph) differed little and only randomly from PS′ from the modeling (column 17), the approximation appears valid, and it has the virtue of not requiring a computer. Values for PS(graph) obtained using E2HM are listed in column 19 and do not differ significantly from those calculated from E2VM (column 18).

FIGURE 7
Development of an empirical relationship to simplify the estimation of PS/Fs from the instantaneous extraction. Left: E′2V(tp) values from the solutions of the model fitted to experimental curves, as in Figure 6, are plotted versus the PS′/F ...

Thus, arbitrarily taking the PS(graph) values to summarize, capillaries of the dog ventricular myocardium have PS products for sodium ranging from 0.32 to 1.67 ml/g.min−1 and averaging 0.83 ± 0.35 ml/g min−1 (N = 27) for those calculated from E2VM and 0.94 ± 0.38 ml/g min−1 (N = 25) for those calculated from E2HM . The mean of the estimates for 41 dilution curves (average of the two values or the sole value) was 0.91 ± 0.37 ml/g min−1. Figure 8 shows that PS(graph) increased with increasing flow; the number used for PS(graph) is the average of columns 18 and 19 when both are available. Assuming that the permeability characteristics of the capillary membrane do not change with flow or intracapillary pressure, this finding suggests that the surface area is increased, that is, that there is recruitment of more capillaries at higher flow rates. At high flows, with Fs > 1.0 ml/g min−1, the PS(graph) values averaged 1.01 ± 0.38 ml/g min−1(N = 11). Using S = 500 cm2/g for a fully vasodilated bed in ventricular myocardium (9) and taking the plasma water content to be 93% (which implies that the flow of solute-containing water is really 0.93Fs), one can estimate the myocardial capillary permeability to be 0.93 × (1.01 ± 0.38)/(500 × 60) or P = (3.1 ± 1.2) × 10−5 cm/sec.

Volumes of Distribution

The model fitting gave estimates of V′E (column 20 of Table 2). The total volume of distribution, capillary plus extravascular, is given in column 21, and the difference between columns 20 and 21 is the intravascular plasma volume.

The variation in V′E is quite large. Experiment 7047 went well technically, and yet the mean ± SD of V′E was 0.23 ± 0.056 ml/g; experiments 8038 and 15038 showed even more variability. We feel that fitting the venous outflow curves is not a very accurate method of estimating the sodium volume of distribution, but it does provide approximate values in a situation where estimates of the mean transit time volume of sodium are almost useless because of inaccuracy in separating the very long low tails of the sodium curves from background radiation. (Backgrounds are high in these experiments using a relatively exposed flow-through NaI crystal detector, and the statistical variation in the counts due to 24Na within the outflow detector is large, since the count rates are low [small amounts of tracer in the approximately 1 ml of contained blood] and the counting periods short, 1–3 seconds. Multiple sample collection would provide statistically more accurate data.)

Discussion

The primary purpose of this study was to test the tracer diffusion technique of Crone (1) in an organ with relatively homogeneous regional flows and a small volume in the drainage vessels and to obtain estimates of the PS product of myocardial capillaries for sodium. In the course of achieving these aims, comparisons had to be made of various techniques for measuring the extraction, and satisfactory analysis had to be preceded by development of a satisfactory descriptive model (5). The testing provided an assessment of the effects of back diffusion on apparent extraction and led to a method of correction for back diffusion by a simple approximating equation (Eq. 7), which closely resembles the classical equation (Eq. 1).

Extractions

The curves of E(t) can be significantly influenced by heterogeneity of regional blood flows, which causes separation of the fractions of tracer that have passed through regions having short and long transit times. The diagrams of Figure 1 are drawn as if the extraction from vascular pathways having short transit times was the same as that from pathways having long transit times, so that E2(t) and E3(t) are identical and initially flat. This situation could occur despite heterogeneities in P and S in different regions. If rapid-transit pathways had lower extractions than did slow-transit pathways, then E2(t) and E3(t) would be expected to increase from the initial value. (The form of E1(t) would also change but not so obviously.) Another source of initially rising E2(t)'s is relatively greater axial intravascular diffusion of sodium compared with albumin, especially in small vessels where the velocity profile is flattened; however, this phenomenon could at most have a miniscule effect. Yet another possibility is intratissue diffusion between arterial and venous ends of the capillary-tissue region or diffusion between neighboring capillaries as Levitt (15) has proposed or between closely approximated arterial and venous ends of the exchange region (counter-current exchange) as occurs for tracer water and for heat (16). For antipyrine in the heart such an effect has been demonstrated to be negligible (10); it can therefore be pretty well ruled out as an influence on dilution curves for sodium, whose diffusion coefficient in extracellular space in myocardium is smaller (17). Other effects are possible if intracapillary radial diffusion is slow (2), but such is not the case. Therefore, we conclude that heterogeneity of flow is the principal cause of the rising E(t)'s.

The influence of this flow heterogeneity on the adequacy of Eqs. 1 and 7 has not been thoroughly tested even in models. However, examination of the figures of Goresky et al. (7) from a multicapillary model with a variation in capillary lengths indicates that when PS/Fs is small the error introduced by flow heterogeneity is small. Figures Figures3,3, ,7,7, and and88 in the paper of Goresky et al. (7), provide evidence supporting our preference for using E2VM or E2V (tp) in calculating PS. Their average PS/Fs values were 0.3 and 1.2 (their K = 0.1 and 0.4), and even with the extravascular volume only three times the capillary volume (γ = 3.0), which exaggerates back diffusion, their observed maximums of E2(t) were about 0.3 and 0.6 (left top two sections of their Fig. 7). These values can be compared with the expected values of 0.26 and 0.70 from a uniform system without back diffusion. In general, it may be said that the greater the heterogeneity in flow the greater the error. (The modeling of Goresky et al. [7] at PS/Fs values higher than 3.0 has no relevance to our problem of obtaining estimates of PS from the extraction of small hydrophilic solutes, since our PS/Fs values did not exceed 1.4; however, it is pertinent to an understanding of the shapes of dilution curves of more rapidly permeating solutes.)

Having the preceding considerations in mind, we conclude that E2(t) and E3(t) are most readily interpretable and therefore preferable to other techniques. The E1 technique is strongly influenced by back diffusion by the time E1(t) reaches its maximum. Clearly, one could derive a correction factor based on a capillary-tissue model, as we have done for E2V, but it does not seem to be very profitable to do so, because the correction would have to be very large and therefore more susceptible to errors due to disparities between the model and the tissue studied.

There is some justification for the preference of Crone (1) and of Lassen and Crone (3) for E3, since it is clearly less susceptible to noise on the curves of h(t). In addition, E3(t) has the advantage that it is readily calculable from venous outflow or residue curves (Eq. 6a and b) and is almost unaffected by interlaminar diffusion effects. E2V(t) is a continuous estimate of the instantaneous extraction, hiding nothing and having a sensitivity that the smoothing integration removes from E3. Therefore, for venous sampling, we have a preference for E2(t). For data acquired by external monitoring, E2H(t) is relatively noisy because of the need to find the derivative (Eq. 5b), and E3H(t) is preferable.

Sejrsen (18) has used another method of measuring E by external detection; it is dependent on the assumption that escape from the tissue results in multiexponential washouts, the fastest component of which is slow compared with intravascular washout. Analysis of RD(t), residue curves for sodium reported in this study, provided unreliable and inconsistent estimates of extraction and PS; therefore, we conclude that the method is not useful for estimating sodium permeability, although its possible use for estimating albumin extraction was not tested.

Lassen and Crone (3) observed that in experimental situations in which there is a long collecting system the earliest values of E2(t) and E3(t) are often higher than the later values; they attributed this fact to more rapid interlaminar diffusion of the permeating solute (Taylor's effect). Observationally, this phenomenon means simply that the intravascular bolus of the permeating tracer is less dispersed during transorgan passage than is the bolus of nonpermeating tracer. We did not see such effects, a result that is explicable on three bases: (1) our sampling system and venous outflow volumes were small, (2) convective mixing in the outflow stream, as in the right ventricle, tends to mask and reduce any effects of interlaminar diffusion, and (3) a heterogeneity of flow always produces rising early values of E2(t) and E3(t), and, although the rate of rise might be reduced by any interlaminar diffusion effect, it is not possible to recognize this fact without using several tracers of different diffusibility simultaneously.

Permeability-Surface Area Products for Sodium

The calculations of PS are dependent primarily on the measurement of E and Fs and on the applicability of the Krogh cylinder model and of the simplified models represented by Eqs. 1 and 7 to the experimental situation. The most questionable feature is whether a single capillary model is sufficient in the face of a recognizable heterogeneity of flow; although it is clear that realistic multicapillary models must be tested, we have argued earlier in this paper that the error in E and, therefore, in PS is not likely to be very great.

If, in addition to the permeability barrier, intratissue diffusion impedes tracer equilibration in the extravascular space, then the indicator concentration in the immediate pericapillary region will rise rapidly and back diffusion will be more rapid, reducing the apparent E. We think this problem is negligible, because the intratissue diffusion coefficient for 24Na is quite high, about 5 × 10−6 cm2/sec (17) and intercapillary distances are so short that intratissue diffusion must be rapid and gradients negligible (5).

Renkin's diagram (Fig. 3 of ref. 19) showing “clearance” or E · Fs vs. flow, Fs, for various PS products is quite useful but requires modification when there is back diffusion. Figure 9 shows a plot of theoretical and experimental clearance against plasma flow, Fs. Theoretical clearance lines of constant PS are provided by Eq. 1 (continuous lines) and Eq. 7 (broken lines), which was derived from the single capillary tissue model (5) as described for Fig. 7 using V′E = 0.21 ml/g over the entire range of the data. At the lowest flows, the solid line for PS = ∞ shows that clearance flow, a concept of completely flow-limited exchange. The broken line for PS = ∞ indicates the same concept modified by the idea that the maximum extraction is 88% no matter what the flow is, which implies that the maximum is determined by the ratio of extravascular to intravascular volumes of distribution and by the interactions between capillaries or the role of diffusion relative to that of convection in moving tracer from the tissue into the outflow from the capillaries. At high flows, the extraction becomes limited by the rate of transport across the capillary membrane. At the highest flows, the transmembrane transport is the sole limiting process, and the lines of constant PS become horizontal. In the intermediate region of mixed flow and diffusion limitation, where the sodium data lie, the clearance-flow relationship is curvilinear even if there is no back diffusion, as is indicated by Eq. 1 and the solid lines of Figure 9. When the extravascular volume, V′E in the model, is small, then back diffusion reduces the extraction or clearance somewhat more, particularly in the range about PS/Fs of 1.0 or greater; the amount of back diffusion is inversely proportional to the extravascular distribution space, V′E. Presumably when Fs/PS is 2 or greater, the Eq. 7 approximation might begin to fail: one would expect a system with constant PS to behave as if the broken lines continue upward as flow increases to join the continuous lines asymptotically.

FIGURE 9
Clearance, E·Fs, as a function of flow, Fs. The line of identity (solid line with PS = ∞) indicates flow-limited washout for which local blood-tissue equilibration is essentially instantaneous. The solid lines are given by the equation ...

This approximation, Eq. 7, was derived completely empirically, without implications as to a logical basis. Another viewpoint is to interpret it as a modification of the Krogh model (5) based not on the accounting for back diffusion but on the idea that extraction cannot be 100% for even the most diffusible tracers. This interpretation would be in accordance with the observations of Yipintsoi and Bassingthwaighte (10) that tritiated water extraction is about 90% and does not vary systematically with coronary blood flow. The idea is most clearly expressed by Perl (20) in his consideration of an “interpolation model” in which he suggests that the pertinent extraction is the observed extraction (E2VM or E3VM) divided by the extraction of tracer water. Although this procedure gives a directionally and, perhaps, even a quantitatively appropriate correction, incomplete water extraction is due to a combination of completely flow-limited blood-tissue exchange locally and diffusional shunting (10); therefore, it does not necessarily indicate that the maximum extraction of sodium cannot exceed 90%. To be sure, the diffusional shunting could be occurring at a capillary level, as Levitt (15) has suggested. If it is there rather than between arterioles and venules, then diffusion of sodium between neighboring capillary regions might make its maximum extraction as low as 90% or conceivably even less, since its volume of distribution is less than that of water. In this study, sodium extractions did not exceed 72%, fortunately, for there is no doubt that the estimation of PS is quite inaccurate when extraction is high (see Fig. 3 of ref. 5). Probably the most realistic concept will be one that includes both single-capillary blood-tissue exchanges, which gave us the empirical value of 88% extraction from which to subtract the observed extraction to compensate for back diffusion, and capillary interactions, which combine with back diffusion to give a 90% water extraction.

The experimental data (clearance is E2VM·FS ml/g min−1) from a given heart, well shown, for example, by the crosses or solid circles in Figure 9, have a tendency to cross through the theoretical lines of constant PS (continuous lines), supporting the idea that PS for sodium is higher at higher flows. It is not known whether this increase in PS is due to the recruitment of capillaries ordinarily closed, to actual increases in P without a change in S, which seems less likely but might accompany increased capillary distending pressures, or possibly to residual underestimation of PS at low flows because of the inability of the model to correct completely for back diffusion. We think recruitment is responsible for the increase in PS.

Comparison of PS Estimates with Those in the Literature

Our estimates of PS for sodium increased with flow, showing a trend toward values around 1 ml/g min−1. The values and the tendency to increase with flow are compatible with the PS estimates of 0.28–1.03 ml/g min−1 obtained by Yudilevich et al (4, 21) at rather low flows (Fs from 0.23 to 0.50 and from 0.44 to 0.81 ml/g min−1) using the same double-tracer technique. Their analysis employed an estimate of extraction obtained by extrapolation back to the time of appearance of the indicators, the E(0) technique which they developed; we believe this technique underestimates the average extraction to some extent. Subsequently, Alvarez and Yudilevich (22) reported on a larger series of sodium extraction data obtained from blood-perfused dog hearts with Fs up to 2.0 ml/g min−1 (our estimation, based on an approximate hematocrit of 0.20) in which the estimated PS increased from about 0.23 ml/g min−1 at the lowest flow to about 0.91 ml/g min−1 at the highest flow. Ziegler and Goresky (14, 23) in dog hearts in situ with Fs of about 0.2−1.3 ml/g min−1 showed rising PS values (corrected for back diffusion) asymptoting at about 0.8–0.9 ml/g min−1.

These values are not very different from those that we can estimate from the data of Conn and Wood (24) on analysis of sodium washout from dog hearts. These authors obtained an average tissue-blood exchange rate of 1.8 min−1, which when interpreted as being PS/V′E gives a PS of 1.8 × 0.21 or 0.38 ml/g min−1, or perhaps 0.5 ml/g min−1 when an approximate correction for back diffusion is applied. The later portions of the downslopes of our sodium curves give similar values and suggest that extravascular diffusional hindrance beyond the capillary membrane may contribute to retarding the washout.

Thus, our experiments and methods of analysis, although providing a thorough test of the instantaneous extraction technique, have led to only modest increases in the estimates of PS. Our study should therefore be regarded as confirming the approach pioneered by Crone, setting the idea on a firmer basis.

The estimates of PS would be 10% lower if they were calculated on the basis that F, should be the flow of the water in the blood rather than the plasma flow. Since the tracer 24Na is dissolved in the water of the plasma, the concept is reasonable. Additional factors to be considered are the activity coefficient, the degree of ionization, and whether the 24NaCl need be also considered as a diffusing species. These factors probably have little influence on the estimates.

Measurement of Blood Flow from Sodium Washout

One result of this study is that it is abundantly clear that sodium washout is very far from being flow limited but is markedly barrier limited. If the permeabilities were only one order of magnitude higher, there would be substantial flow limitation (5) but still not enough to justify the use of sodium washout curves for the estimation of myocardial blood flow. Its applicability in skeletal muscle or other situations can be tested by observing whether the time for washout is exactly inversely related to the flow measured by another method, as has been shown for antipyrine in the heart (10).

Interpretation of the Values for PS

If permeation were by free unrestricted diffusion through clefts between endothelial cells where the average cleft depth L was 0.4μ, then a PS of 0.88 ml/g min−1 could be interpreted as DNaAs/L, where DNa is the diffusion coefficient for sodium in extracellular fluid (about 5.1 × 10−6 cmVsec) (17). Thus, a value of 0.26 cm2/g for As, the surface area of interendothelial clefts, results. If S = 500 cm2/g, the fraction of membrane available would be As/S or 0.26/500 or 5.2 × 10−4, lower than the 14 × 10−4 estimated for muscle capillaries by Pappenheimer (25, p 409) using osmotic water fluxes but an order of magnitude higher than Perl's (26, p 246) estimate of 3.3 × 10−5. The difference from Perl's estimate is due almost entirely to his assumption of a cleft depth L of 238 Å = 0.024μ rather than our 0.4μ. Pappenheimer used L = 0.3μ, which we consider to be too short, since the slits take a rather oblique course from capillary lumen to basement membrane. In actuality, the fundamental estimates, As(SL), are not very different, being 13.0 cm−1 from our data, 13.9 cm−1 by Perl's calculation, and 46.7 cm−1 by Pappenheimer's estimation. The latter value is representative of data obtained by osmotic transient methods, which have usually shown relatively high values for permeability. In this discussion As has been considered to be the area (of slit) available for solute transfer not water transfer, and so long as the slits are not very narrow (less than 20–30 Å) deeper consideration of relative areas and of reflection coefficients contributes little to the arguments which follow.

If the slit width, ws, is 100 Å (27) and As is the product wslT, where lT is the total length of slits on the endothelial surfaces in one gram of myocardium, then the ratio lT/S is approximately = 500 cm−1. This value is small compared with the length expected, for, if the diameter of endothelial cells were 8μ (arrayed hexagonally on the cylindrical surface), then, without any tortuosity in the slits, the ratio lT/S would be 4μ or about 2500 cm−1. The junctional areas certainly are markedly tortuous so that the true length must be greater than this value; therefore, it seems reasonable to suggest that only a small fraction of the total real slit length is available for solute passage, something substantially less than 500/2500 or 20%. If with tortuosity the real slit length were four times greater, we would guess that only 20/4 or 5% of the length of the circumferential clefts is open to solute passage. Such a guess, however rough, is compatible with Karnovsky's (28) observation that most (95%) of the sections through interendothelial cell clefts show tight or gap junctions, electron dense regions of close apposition of neighboring cells. These junctional regions might be totally impermeant to ions. If this were true, then we would think of permeation as occurring through 100-Å wide slits whose lengths were short, not encircling the whole of an endothelial cell. This view is an alternative to considering that solutes enter the clefts freely and then must penetrate narrow junctional areas (about 40 Å wide) circumscribing every cell, as Perl has outlined so well in his analysis (26). The multiple solute experiments of Yudilevich and Alvarez (21), Yipintsoi et al. (29), and Duran et al. (30) do distinguish between these possibilities. The data may not be entirely conclusive, but they suggest that the ratios of PS over diffusion coefficient are the same for the various solutes, which indicates that the clefts or pores for transcapillary exchange are large compared with the diameters of the solute molecules, that is, the cleft widths are not likely to be 40 Å but are at least 70–100 Å.

Volumes of Distribution

The mean value for the total volume of distribution for sodium, 0.37 ml/g (column 21 of Table 2), is substantially larger than the 0.19 ml/g for sulfate and 0.18 ml/g for sucrose observed by Polimeni (12) in hearts from intact rats but is only a little smaller than the 0.4 ml/g found for sucrose by Johnson and Simonds (13) in Ringer's solution-perfused rabbit hearts and the 0.44 ml/g and 0.40 ml/g found for the rapidly equilibrating fractions of sucrose and sul-fate space in rat ventricle perfused with a physiological salt solution by Page and Page (31). That sodium space should be larger than sulfate space is reasonable, since it almost certainly enters the sarcoplasmic reticulum, as argued on logical grounds (32) and as persuasively expressed by Rogus and Zierler (33) from their analysis of tracer sodium washout from rat skeletal muscle. The principal conclusion to be drawn from these data on V′E, which is only a secondary consideration in this study, is that the sodium space in these hearts is probably somewhat larger than normal. Since this phenomenon is apparently occurring in spite of perfusion with whole blood at 37 °C, it seems likely that the trauma of the isolation procedure, the lack of development of intraventricular and intramyocardial pressure to extrude lymph, and perhaps the presence of another dog's plasma proteins combine to cause the isolated nonworking heart to develop a large extracellular space and perhaps an increased myocardial cell permeability to sodium.

Acknowledgment

Mr. A. R. Wanek, Mr. T. J. Knopp, and Mr. C. Field performed much of the analysis. Mr. Tom Brimijoin prepared the illustrations, and Ms. Jane Irving and Ms. Ardith Benjegerdes prepared the manuscript. Dr. C. A. Goresky, Dr. K. L. Zierler, and Dr. W. Perl contributed helpful criticisms of the content and the presentation of this work.

This investigation was supported in part by U. S. Public Health Service Grants HL 9719 and RR7 from the National Institutes of Health. Dr. Yipintsoi was supported by NIH Special Fellowship HL43128; Dr. Bassingthwaighte holds NIH Career Development Award K3HL22649.

Footnotes

Dr. Guller's present address is Louisiana State University Medical Center, 1542 Tulane Avenue, New Orleans, Louisiana 70112. Dr. Yipintsoi's present address is Department of Medicine. Montefiore Hospital, Bronx, New York, and Department of Physiology, Albert Einstein College of Medicine.

1Mr. T. J. Knopp devised this system. The 1-μSec output pulses from each pulse-height analyzer were used as input to a 1-decade counter which, after each ten pulses, generated a 200-μsec, 1.5 v pulse that was recorded on analog tape running at 30 inches/sec. To count the stored pulses on playback, the output from the analog tape was put into a pulse shaper which, for each pulse in, generated an output pulse (1.5 v, 0.5 μsec width) that was used as input to the pulse counter. This system was accurate for input count rates up to 25,000/sec on each of four channels and resulted in no significant time skewing or loss of resolution in count rate with rates greater than 200/sec.

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