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Circ Res. Author manuscript; available in PMC 2010 November 16.

Published in final edited form as:

PMCID: PMC2982781

NIHMSID: NIHMS233297

ESTIMATION OF CAPILLARY PERMEABILITY BY RESIDUE AND OUTFLOW DETECTION

Departments of Physiology and Biophysics, Pediatric Cardiology, and Therapeutic Radiology, Mayo Graduate School of Medicine, Rochester, Minnesota 55901.

Please address reprint requests to Dr. J. B. Bassingthwaighte, School of Medicine, University of Washington, Seattle, Washington 98105.

Sudden injections of boluses containing both ^{131}I-albumin and ^{24}NaCl 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, *F _{s}*, ranging from 0.3 to 1.8 ml/g min

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* (cm^{2}/g), using the standard formula (1, 3, 4):

$$\mathit{PS}=-{F}_{s}\phantom{\rule{thinmathspace}{0ex}}{\mathrm{log}}_{e}(1-E),$$

(1)

where *F _{s}* is the flow (ml/g sec

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, *t _{p}*, of

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, *F _{s}*. 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 , the mean transit time, which is a little after the time of the peak,

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.

The technique of the present experiments was to inject a short bolus containing a nonpermeating reference tracer, *N* (^{131}I-albumin), and a permeating or diffusible tracer, *D* (^{24}NaCl), into the inflow to the heart and to record the concentration-time curves in the venous outflow, *C _{N}*(

$$R\left(t\right)={C}^{\ast}\left(t\right)\u2215{C}^{\ast}\left(0\right),$$

(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:

$$h\left(t\right)=C\left(t\right)/{\int}_{0}^{\infty}C\left(t\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}t.$$

(3a)

This procedure was usually adequate for albumin, but the tails of the curves for ^{24}Na 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 ${\int}_{T}^{\infty}h\left(t\right)$ *h*(*t*) d*t* is exactly *R*(*T*) when there is no delay between exit from the organ and arrival at the venous sampling site, then

$$\begin{array}{cc}\hfill h\left(t\right)& =\frac{C\left(t\right)}{{\int}_{0}^{\infty}C\left(t\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}t}=\frac{C\left(t\right)\phantom{\rule{thickmathspace}{0ex}}{\int}_{0}^{T}C\left(t\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}t}{{\int}_{0}^{T}C\left(t\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}t\cdot {\int}_{0}^{\infty}C\left(t\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}t}\hfill \\ \hfill h\left(t\right)& =\frac{C\left(t\right)}{{\int}_{0}^{T}C\left(t\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}t}\phantom{\rule{thickmathspace}{0ex}}{\int}_{0}^{T}h\left(t\right)\mathrm{d}t=\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\frac{C\left(t\right)}{{\int}_{0}^{T}C\left(t\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}t}\cdot \left[1-{\int}_{T}^{\infty}h\left(t\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}t\right],\hfill \\ \hfill h\left(t\right)& =\frac{C\left(t\right)}{{\int}_{0}^{T}C\left(t\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}t}\cdot [1-R\left(T\right)].\hfill \end{array}$$

(3b)

This equation provides values of *h*(*t*) for *t* ≤ *T*, 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 *h _{D}*(

Methods of estimating extractions. E. Broken lines = diffusible indicator (subscript D) and solid lines = reference indicator (subscript N). **Left:** The transport function, h_{D}(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*, *E*_{1}(*t*) uses the whole of the curves obtained up to time *t*; for outflow sampling

$${E}_{1v}\left(t\right)={\int}_{0}^{t}[{h}_{N}\left(\tau \right)-{h}_{D}\left(\tau \right)]\mathrm{d}\tau ,$$

(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, *E*_{1H} (*t*), theoretically identical to *E*_{1V}(*t*), can be obtained from the externally monitored residue functions:

$${E}_{1H}\left(t\right)={R}_{D}\left(t\right)-{R}_{N}\left(t\right),$$

(4b)

where the subscript *H* denotes external monitoring of the organ (heart). The equivalence to *E*_{1V} follows from the definitions given in the legend to Figure 1:

$${E}_{1H}\left(t\right)={R}_{D}\left(t\right)-{R}_{N}\left(t\right)={\int}_{0}^{t}{h}_{N}\left(\tau \right)\mathrm{d}\tau -{\int}_{0}^{t}{h}_{D}\left(\tau \right)\mathrm{d}\tau ={E}_{1v}\left(t\right).$$

The integral net extraction (i.e., *E*_{1V} or *E*_{1H}) reaches a plateau when *h _{N}*(

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

$${E}_{2v}\phantom{\rule{thinmathspace}{0ex}}\left(t\right)=[{h}_{N}\phantom{\rule{thinmathspace}{0ex}}\left(t\right)-{h}_{D}\phantom{\rule{thinmathspace}{0ex}}\left(t\right)]\u2215{h}_{N}\phantom{\rule{thinmathspace}{0ex}}\left(t\right).$$

(5a)

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

$$\begin{array}{cc}\hfill {E}_{2H}\left(t\right)& =\frac{-\mathrm{d}{R}_{N}\left(t\right)\u2215\mathrm{d}t+\mathrm{d}{R}_{D}\left(t\right)\u2215\mathrm{d}t}{-\mathrm{d}{R}_{N}\left(t\right)\u2215\mathrm{d}t}\hfill \\ \hfill & =1-\frac{\mathrm{d}{R}_{D}\left(t\right)}{\mathrm{d}{R}_{N}\left(t\right)}.\hfill \end{array}$$

(5b)

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

$${E}_{2H}\left(t+\frac{\mathrm{\Delta}t}{2}\right)=1-\frac{{R}_{D}(t+\mathrm{\Delta}t)-{R}_{D}\left(t\right)}{{R}_{N}(t+\mathrm{\Delta}t)-{R}_{N}\left(t\right)}.$$

(5c)

When Δ*t* 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:

$${E}_{2H}\left(t+\frac{\mathrm{\Delta}t}{2}\right)=1-\frac{{R}_{D}(t+2\mathrm{\Delta}t)+{R}_{D}(t+\mathrm{\Delta}t)-{R}_{D}\left(t\right)-{R}_{D}(t-\mathrm{\Delta}t)}{{R}_{N}(t+2\mathrm{\Delta}t)+{R}_{N}(t+\mathrm{\Delta}t)-{R}_{N}\left(t\right)-{R}_{N}(t-\mathrm{\Delta}t)}.$$

(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 *E*_{2}(*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 *E*_{2}(*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, *E*_{2M}, is approximately that occurring with the average flow, *F _{s}*, as one sees with mathematical models (5, 7).

*The area-weighted extraction, E*_{3}(*t*), is conceptually similar to *E*_{2}(*t*) in that one expects the early parts of the outflow dilution curves to have similar shapes; using integrals causes *E*_{3}(*t*) to be smoother than *E*_{2}(*t*). *E*_{3}(*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.

$$\begin{array}{cc}\hfill & {E}_{3V}\left(t\right)={\int}_{0}^{t}[{h}_{N}\left(\tau \right)-{h}_{D}\left(\tau \right)]\mathrm{d}\tau \u2215{\int}_{0}^{t}{h}_{N}\left(\tau \right)d\tau \hfill \\ \hfill & \text{or}\hfill \\ \hfill & {E}_{3V}\left(t\right)=1.0-{\int}_{{t}_{a}}^{t}{h}_{D}\left(\tau \right)d\tau \u2215{\int}_{{t}_{a}}^{t}{h}_{N}\left(\tau \right)d\tau ,\hfill \end{array}$$

(6a)

where *t _{a}* is the appearance time of the indicators in the venous outflow (see Fig. 1). The values for

$${E}_{3H}\left(t\right)=\frac{{R}_{D}\left(t\right)-{R}_{N}\left(t\right)}{1-{R}_{N}\left(t\right)}.$$

(6b)

Lassen and Crone (3) expressed a preference for using the areas up to the time of the peak, *t _{p}*, of

The maximum values of these various *E*(*t*)'s will be denoted by *E*_{1VM}, *E*_{1HM}, *E*_{2VM}, *E*_{2HM}, *E*_{3VM}, and *E*_{3HM}

*PS products* were calculated from the instantaneous extraction *E*_{2V}(*t*). The first and simplest method was the straightforward application of Eq. 1, using *E*_{2V}(*t _{p}*) or

The second method was to fit the observed curves of *h _{N}*(

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 (*D _{CD}* and

Venous outflow dilution curves fitted by the Krogh model solutions for two hearts. The experimental albumin curve. h_{N}(t). is used as input to the model. The experimental sodium curve. h_{D}(t), is the dotted line. The observed extraction. E_{2V}(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 *V*′_{E} 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; *V*′_{E} was permitted to range freely for the fitting procedure.

To fit the curves, the observed *h _{N}*(

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 *t _{p}*, the time of the peak of

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}. *F _{s}* was plasma flow, (1 –

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 R_{N}(i) and R_{D}(t) and from the venous outflow for h_{N}(t) and h_{N}(t): the **...**

^{24}NaCl solution (Cambridge Nuclear Corporation), a sterile solution of 99% radiochemical purity, was diluted to approximately 100 *μ*c/ml. Human serum ^{131}I-albumin (RISA-131, Abbott Laboratories) having an activity of 100 *μ*c/ml was supplied in sterile isotonic saline solution. Boluses containing approximately 0.2 ml of the ^{24}Na and 0.4 ml (0.1–0.8 ml) of the ^{131}I-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.

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 ^{24}Na counted pulses over 1 Mev and was uninfluenced by ^{131}I emissions. The ^{131}I 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 ^{24}Na window when only ^{24}Na 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.

The maximum values for *EWM _{1VM}*,

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 *h _{N}*,

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 (*F _{s}* = 1.23 ml/g min

Concentration-time curves of ^{2}Na (labeled D, diffusible) and ^{131}I-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, *E*_{2H}(*t*) reached zero faster than did *E*_{2V}(*t*), which was recorded some distance downstream. Accordingly, the rate of diminution of *E*_{2H}(*t*) was faster than that of *E*_{2V}(*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 *E*_{2V}(*t*) compared with that of *E*_{2H}(*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.

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 *E _{2V}(t)* obtained from one isolated heart at blood flows ranging from 0.47 to 2.04 ml/g min

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 *E _{1HM}/E_{1VM}* was 0.96 ± 0.20 (n = 11), that of

As one anticipates from Figure 1, the net extraction, *E _{1}(t)*, was smaller initially, at

The differences between *E _{2}* and

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 *F _{s}*, hematocrit. and heart weight were obtained experimentally; capillary radius (

Particular values of *E′ _{2V}* from the Krogh cylinder modeling are given in columns 14 and 15 of Table 2, choosing

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

Table 2 lists the three estimates of PS obtained from *E*_{2}*( t).* For the sake of brevity and simplicity of comparison, we emphasized the

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 − 0.042*F _{s}* with

A simpler method was developed using a semilog plot of 1 − *E′ _{2V}(t_{p})* versus

$$\begin{array}{cc}\hfill & {\mathit{PS}}^{\prime}=-{F}_{s}\phantom{\rule{thinmathspace}{0ex}}{\mathrm{log}}_{e}\phantom{\rule{thinmathspace}{0ex}}\{[0.88-{{E}^{\prime}}_{2V}\left({t}_{p}\right)]\u22150.88\}\hfill \\ \hfill & \text{or}\hfill \\ \hfill & {\mathit{PS}}^{\prime}=-{F}_{s}\phantom{\rule{thinmathspace}{0ex}}{\mathrm{log}}_{e}\phantom{\rule{thinmathspace}{0ex}}[1-1.14{{E}^{\prime}}_{2V}\left({t}_{p}\right)].\hfill \end{array}$$

(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, *E _{2VM}* and

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 *E*_{2VM} and 0.94 ± 0.38 ml/g min^{−1} (N = 25) for those calculated from *E*_{2HM} . 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 *F _{s}* > 1.0 ml/g min

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

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).

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 *E*_{2}(*t*) and *E*_{3}(*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 *E*_{2}(*t*) and *E*_{3}(*t*) would be expected to increase from the initial value. (The form of *E*_{1}(*t*) would also change but not so obviously.) Another source of initially rising *E*_{2}(*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/F _{s}* 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

Having the preceding considerations in mind, we conclude that *E*_{2}(*t*) and *E*_{3}(*t*) are most readily interpretable and therefore preferable to other techniques. The *E*_{1} technique is strongly influenced by back diffusion by the time *E*_{1}(*t*) reaches its maximum. Clearly, one could derive a correction factor based on a capillary-tissue model, as we have done for *E*_{2V}, 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 *E*_{3}, since it is clearly less susceptible to noise on the curves of *h*(*t*). In addition, *E*_{3}(*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. *E*_{2V}(*t*) is a continuous estimate of the instantaneous extraction, hiding nothing and having a sensitivity that the smoothing integration removes from *E*_{3}. Therefore, for venous sampling, we have a preference for *E*_{2}(*t*). For data acquired by external monitoring, *E*_{2H}(*t*) is relatively noisy because of the need to find the derivative (Eq. 5b), and *E*_{3H}(*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 *R _{D}*(

Lassen and Crone (3) observed that in experimental situations in which there is a long collecting system the earliest values of *E*_{2}(*t*) and *E*_{3}(*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 *E*_{2}(*t*) and *E*_{3}(*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.

The calculations of *PS* are dependent primarily on the measurement of *E* and *F _{s}* 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

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 ^{24}Na is quite high, about 5 × 10^{−6} cm^{2}/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* · *F _{s}* vs. flow,

Clearance, E·F_{s}, as a function of flow, F_{s}. 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 (*E*_{2VM} or *E*_{3VM}) 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 *E _{2VM}·F_{S}* ml/g min

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 (*F _{s}* from 0.23 to 0.50 and from 0.44 to 0.81 ml/g min

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

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 ^{24}Na 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 ^{24}NaCl need be also considered as a diffusing species. These factors probably have little influence on the estimates.

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).

If permeation were by free unrestricted diffusion through clefts between endothelial cells where the average cleft depth *L* was 0.4*μ*, then a *PS* of 0.88 ml/g min^{−1} could be interpreted as *D _{Na}A_{s}/L*, where

If the slit width, *w _{s}*, is 100 Å (27) and

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.

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.

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.

^{1}Mr. T. J. Knopp devised this system. The 1-*μ*Sec output pulses from each pulse-height analyzer were used as input to a 1-decade counter which, after each ten pulses, generated a 200-*μ*sec, 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 *μ*sec 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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