Organ Blood Flow, Wash-in, Washout, and Clearance of Nutrients and Metabolites

Nutrients carried by the blood must traverse a set of barriers before arriving at their destination inside the cells of an organ. These barriers are the capillary endothelial lining, the interstitial space, and the cell membranes. Simplification to this viewpoint is reasonable when exchange between erythrocytes and plasma is either rapid or nonexistent and when intracellular compart-mentalization is negligible.

BARRIERS

The capillary endothelium is the first barrier. Hydrophilic molecules pass mainly through the narrow clefts between endothelial cells, but lipid-soluble molecules can dissolve in endothelial cell membranes and therefore diffuse through the cells to the extra-vascular space.

Diffusion through interstitial fluid takes only a few milliseconds in organs with a dense, well-perfused capillary network, such as the heart and kidney, where intercapillary distances are short and there are not markedly tortuous paths to sequestered regions. When distances between open capillaries are large (mesentery, resting skeletal muscle, cool skin), interstitial diffusion may take many seconds.

These transport processes across capillary membrane and through the interstitium ordinarily are purely passive, with no energy coupling mechanisms being involved; so far, the only known exception to this is glucose transport out of brain capillaries.¹

The major hurdle to entry into the cell is the cell membrane itself, the plasmalemma or, as it is called in muscle, the sarcolemma. Involved here is not only passive diffusional transport, which is influenced by molecular size and solubility, but also active transport (carrier-mediated or facilitated transport). Speaking teleologically, the need for carriers arises because the membrane is nearly impermeable to water-soluble and charged molecules, and attachment to a carrier makes them soluble in the membrane. A carrier can be looked upon as a large molecule, mobile or fixed in the membrane, that has specific affinity for a small class of solute molecules having similar features. For example, a carrier for glucose also may facilitate transport of galactose but have a much decreased or no affinity for fructose.

The interest in carriers stems from a desire to understand disease processes and therapeutic mechanisms. Many poisons inhibit carriers. For example, digitalis inhibits Na⁺,K⁺-activated ATP-ase, a carrier that links sodium extrusion and ATP breakdown; the dramatic effects of digitalis are probably all due to the resultant intracellular buildup of sodium, which is not a restoration to normal but a compensation in part for the abnormal conditions. The carrier-mediated transmembrane transport rates need to be obtained in order to characterize and comprehend the process. Because the membrane process is in series with the extracellular diffusion and capillary membrane permeation processes, it has become important to characterize these quantitatively, in order to distinguish the transfer rates due to each process.

MEASUREMENT OF TRANSPORT RATES

A recently evolved set of methods is based on comparisons of the transport rates for a pair of substances selected so that one enters the cell and the other does not enter the cell but is similar with respect to the extracellular processes. A specific analyzable situation occurs when solute exchange between blood and the extracellular region is flow-limited.

“Flow-limitation to exchange” means that transfer rates across the capillary membrane and due to extracellular diffusion are very rapid compared to movement along the capillary with the flow. When this situation exists for both substances of the pair, then the rate of transfer of the one entering the cell can be determined in “wash-in” experiments, and its rate of exit can be determined in “washout” experiments after the tissue has been loaded with the substance. It is the purpose of this presentation to define “flow-limitation” in experimental situations and to show it in relationship to more complex situations.

CONCEPTS

In order to understand the use of tracers for estimating blood flow or permeability of the blood-tissue exchanging sites in any organ, we have to know the behavior of the tracers. A conceptual approach to blood-tissue exchanges should be all-inclusive, independent of whether one considers movement of tracer from blood into tissue, from tissue into blood, or from the inflow of an organ to the outflow of that organ. In this presentation, in reference to the heart, the approach is simplified by the assumption that throughout the heart the diffusion distance parallel to the flow pathway from inflow to outflow is long compared to radial diffusion per-pendicular to the direction of the flow. With a purely intravascular tracer, this radial distance is the intravascular radius; with substances that permeate the tissue, this will also include the extravascular space occupied by the tracer.

Figure 1 provides a generalized overview. The relationship of clearance to flow is an extension of Renkin's diagram.² (Clearance is the amount of tracer removed from a compartment; it is the product of extraction and plasma flow and has the same units as flow.) This relationship is divided into five parts, depending on the characteristics of the blood-tissue exchange of the tracer. In region I, the clearance equals the flow. Both axial and radial diffusion are so rapid compared to movement by convection that washout is limited completely by the flow. The analog is washout from a completely mixed chamber. In region II, diffusion has two influences: first, it augments the “early” transport from inflow to outflow as tracer enters the organ, and second, it retards the exit of tracer from the depths of the organ in the “late” phase. In region III, “axial” diffusion between inflow and outflow is negligible, but local radial blood-tissue equilibration is rapid, so washout is again flow-limited. Region IV is a transition zone showing that diffusion limits the exchange more and more prominently as the flow increases and capillary transit times decrease. In region V, the rate of blood-tissue exchange is governed solely by dif-fusional processes, either permeation of membranes or diffusion in extravascular space; an increase in flow does not improve the exchange.

Fig. 1.

Diagram of clearance versus flow (F). The Roman numerals denote regions of different flow or of different ratios of intravascular convective velocity to diffusive velocity (or permeation rate) in the extra-vascular tissue. D_L and D_R denote diffusion parallel and perpendicular to the flow stream, respectively.

MEASUREMENT OF FLOW PER UNIT VOLUME

To make this measurement, it is important that the washout be flow-limited, whether one records the concentration-vs.-time curve of the tracer at the outflow or, by external detection, the residual amount of gamma-emitting tracer in the organ. What is essential is the ability to estimate the mean transit time.³ The question is how to define the regions with respect to a particular set of tracers in a given organ: What range of flows is encompassed for a tracer of given diffusibility?

The key to demonstrating flow-limitation to solute exchange is “similarity” between tracer dilution or washout curves at different flows. This is diagrammed for a residue detection experiment in Figure 2.

Fig. 2.

Similarity of a family of residue function curves can be tested by plotting the curves obtained at different flow rates (Left) against time divided by the mean transit time of each curve. Superimposition (Right) of the curves indicates “similarity” and that solute washout is flow-limited.

With Zierler's terminology,³ as recommended by a committee of workers⁴ in the field, after a bolus injection into the inflow to an organ, the residue, R, is given by

$$\mathrm{R}\left(\mathrm{t}\right)=1-{\int }_{\mathrm{o}}^{\mathrm{t}}\mathrm{h}\left(\mathrm{\lambda }\right)\mathrm{d}\mathrm{\lambda }$$ (eq. 1)

in which λ is a substitute variable used in the integration to obtain the area under the curve, h(t). The transport function, h(t), is the impulse response, or the fraction of the dose of injected tracer reaching the outflow per second. When similarity occurs, the form of curves of t·R(t)/t and of t̄²·h(t/t̄) are the same at different flows, which is the same as saying that plots of R(t) or of t·h(t/t) vs. t/f remain superimposed. Why does this demonstrate flow-limitation?

The logic is based on the expectation that, if flow is low enough or diffusion (which includes permeation) is fast enough (or both), there will be local diffusional equilibration (no concentration gradient) and the rate of diffusion will not affect the rate of solute transport out of the organ. Even if there exist barriers that are permeated slowly, or long diffusion distances, then theoretically there must be a flow so low that all concentrations can equilibrate during periods small compared to the transit time of the blood. Then, the washout of tracer is governed completely by flow. This is region I and may only exist theoretically because the conditions may not support life in a biologic system.

If the tracer is highly diffusible (no extravascular concentration gradient) and if there is no source-to-sink intratissue diffusion, then there exists a range of flows such that, again, the washout process is governed completely by flow. This range of flows encompasses those for the superimposed curves in Figure 2 right. For any organ, the highest and lowest flows at which the superimposition fails are the limits for this range of flow-limitations. To utilize tracers in measuring flow per unit volume without first defining this range of flows may introduce avoidable errors. If flow can be made high enough, in spite of the high diffusivity there always will be a stage reached at which diffusion-limitation to efflux will become apparent.

Experimentally, iodoantipyrine (l-Ap) has been shown to be flow-limited in the heart⁶ and in bone:⁶ the residue function curves were super-imposable by plotting R(t) vs. t/t̄. High-molecular-weight tracers that do not leave the vascular bed should be completely flow-limited in their washout because no extravascular diffusion process is involved and intravascular diffusion is too slow to have any effect. Such similarity has been demonstrated by plots of t̄-h(t/t̄) vs. t/t for indocyanine green traversing the human arterial system,^7,8 for Evans blue traversing the renal vascular bed,⁹ and for indocyanine green traversing the lung¹⁰ and the coronary vascular bed.¹¹ An example of this is shown in Figure 3.

Fig. 3.

Transcoronary transport functions for plasma protein-bound indocyanine green obtained at varied flow rates (t̄ from 4.2 to 6.5 seconds) in the dog. The similarity of the curves indicates flow-limited intra-organ distribution of the dye and constancy of the relative distribution of flows.

For similarity to occur, there is another implicit requirement, in addition to the absence of diffusional influence. It is that the relative regional distribution of flow also should remain constant—that is, if the average flow is doubled, then flow in each region should be doubled. Unchanged volume is not a requirement but, if the intravascular or extravascular volume changes or a new section of capillary bed opens up, the distribution of flows through this new region should be similar to that in the section previously open.

SOURCE-TO-SINK DIFFUSIONAL TRANSPORT

At low flows, particularly with substances of high diffusibility, relatively more time is available for diffusion to occur from an inflow region to an outflow region. Perl and Chinard¹² derived equations to cover this possibility in their convection-diffusion model. The phenomena observed are summarily described in region II of Figure 1. On intra-arterial injection of a bolus of tracer, some fraction of it travels via passive diffusion through the tissue from the inflow to the outflow and reaches the outflow region earlier than it would have by flow alone. A sensitive means of testing for this possibility is to examine the fractional escape rates, η(t) (also called the emergence function);¹³ η(t) is calculated from either the outflow dilution curves or escape residue functions. The general fractional escape rate, FER, is the rate of loss of tracer, dq/dt, from an organ divided by the residual amount, q:

$$\mathrm{FER}=\frac{-\mathrm{dq}∕\mathrm{dt}}{\mathrm{q}}.$$ (eq. 2)

η(t) is the specific FER obtained in response to an impulse input or bolus injection into the inflow:

$$\eta \left(\mathrm{t}\right)=\frac{\mathrm{h}\left(\mathrm{t}\right)}{\mathrm{R}\left(\mathrm{t}\right)}$$ (eq. 3)

$$=\frac{-\mathrm{dR}∕\mathrm{dt}}{\mathrm{R}}$$ (eq. 4a)

$$=-\mathrm{d}{\log }_{\mathrm{e}}\mathrm{R}∕\mathrm{dt}$$ (eq. 4b)

$$\eta \left(\mathrm{t}\right)=\frac{\mathrm{h}\left(\mathrm{t}\right)}{1-{\int }_{\mathrm{o}}^{\mathrm{t}}\mathrm{h}\left(\mathrm{\lambda }\right)\mathrm{d}\mathrm{\lambda }}$$ (eq. 5)

$$=\frac{\mathrm{C}\left(\mathrm{t}\right)}{{\int }_{\mathrm{o}}^{\mathrm{\infty }}\mathrm{C}\left(\mathrm{t}\right)\mathrm{dt}-{\int }_{\mathrm{o}}^{\mathrm{t}}\mathrm{C}\left(\mathrm{\lambda }\right)\mathrm{d}\mathrm{\lambda }}.$$ (eq. 6)

Equation 4 is used for residue detection experiments; it has the disadvantage of needing estimation of a derivative from inevitably somewhat noisy data. Equation 6 is used when one records outflow concentration-vs.-time curves, C(t), and necessitates excluding recirculation from the estimation of the area under the curve to time = infinity. A nice way of avoiding the influence of errors concerning the tail of h(t) is to use a combination of techniques, as represented by equation 3. At any time prior to recirculation, the integral of h(t) is 1 − R(t):

$$1-\mathrm{R}\left(\mathrm{t}\right)=\frac{{\int }_{\mathrm{o}}^{\mathrm{t}}\mathrm{C}\left(\mathrm{\lambda }\right)\mathrm{d}\mathrm{\lambda }}{{\int }_{\mathrm{o}}^{\mathrm{\infty }}\mathrm{C}\left(\mathrm{t}\right)\mathrm{dt}}={\int }_{\mathrm{o}}^{\mathrm{t}}\mathrm{h}\left(\mathrm{\lambda }\right)\mathrm{d}\mathrm{\lambda }$$ (eq. 7)

which may then be used to calculate η(t):

$$\eta \left(\mathrm{t}\right)=\frac{\mathrm{C}\left(\mathrm{t}\right)/{\int }_{\mathrm{o}}^{\mathrm{t}}\mathrm{C}\left(\mathrm{\lambda }\right)\mathrm{d}\mathrm{\lambda }/(1-\mathrm{R}\left(\mathrm{t}\right))}{\mathrm{R}\left(\mathrm{t}\right)}=\frac{\mathrm{h}\left(\mathrm{t}\right)}{\mathrm{R}\left(\mathrm{t}\right)}.$$ (eq. 8)

Plots of η(t) are shown in Figure 4. To compare η(t) at different flows, the scales can be normalized as was done for h(t) in Figure 4. In the upper panel, the normalizing procedure of multiplying the time axis by F/W and the ordinate by W/F is essentially the same as using t/t̄ and t̄·η(t/t̄). The reason is that t̄ = W/ρ·F, in which ρ is density (g/ml) and, for the heart, = 1.062 g/ml¹⁴ which is close enough to 1.0 so that the slightly erroneous transformation by F/W is correct enough that the phenomena to be observed have not been obscured. The upper panel shows that η(t) for xenon washout at low flows is relatively high when F/W is low. The corollary is that, if diffusional shunting tends to promote tracer movement from inflow to exit when it is entering, once tracer has penetrated deep into the tissues of the organ then diffusional shunting should promote retention of tracer in the organ and decrease the escape rate. Accordingly, at low flows (short dashes) η(t) is relatively low at later times, Ft/W ≧ 1.

Fig. 4.

Myocardial fractional escape rates after injection of boluses of xenon-133 into coronary inflow at various coronary flows in a dog. F/W is measured total coronary flow (ml/min per g of heart); short dashes = F/W ≦ 1.0 ml/min per g. At early times (Upper) and in the early phase of washout (Lower), the escape rate tends to be higher when F/W is lower (short dashes); at later times (around Ft/W = 1), in a later phase of washout (around H(t) ≈ 0.7), η(t) is relatively low for these curves.

The same impressions can be gained from the lower panel of Figure 4. The abscissa is H(t), which is ${\int }_{\mathrm{o}}^{\mathrm{t}}$ h(t)dt, which is therefore 1 − R(t) or the fraction of tracer that has escaped from the organ at time t. In the early phase (H(t) ≈ 0.3), η(t) is high; in the late phase (H(t) ≈ 0.7), η(t) is relatively low. This plot has the advantage, not only of normalization for comparison of the washout rates but also of time compression so that the whole of the curve is plotted. The experimental methods are those described by Bassingthwaighte and associates.¹⁵

This diffusional shunting of xenon is similar to that seen for tritiated water.⁵ Much more rapid early escape is seen when hot or cold saline is injected onto the coronary artery and the temperature change is recorded via a detector in the coronary sinus. This is to be expected because heat is about an order of magnitude more diffusible than water.

These observations required modification of the original Renkin plot, into that shown in Figure 1, by distinguishing region II. Both the upper and the lower branch of the line are continuously increasing; there is no situation in which an increase in flow leads to a decrease in clearance. But at the upper end of the branch labeled “early” it can be seen that the ratio of clearance to flow decreases as flow increases. During the late phase, an increase in flow increases the ratio of clearance to flow; the clearance increases faster than the flow.

The situation, diagrammed in Figure 5, can be summarized by the following: if the intra-arterial tracer is highly diffusible or if the distance between inflow and outflow is short (counter-current exchange), at early times the tracer can reach the outflow by diffusion. It may add to the tracer arriving there via the flow pathway, or it might even precede it. But at later times, tracer that is in the tissue tends to be retained, by virtue of the same diffusional exchange, which can be demonstrated with a suitable mathematical model.¹⁶

Fig. 5.

Influences of diffusional transfer of tracer between inflow and outflow on clearance.

TRACER EXCHANGE LIMITED BY PERMEABILITY

As Figure 5 implies by the horizontal line at PS = constant, at some very high flow the exchange is completely limited by some diffusional process, usually the permeability of the capillary membrane. In this situation there not only is a concentration gradient across the membrane but there also may be gradients within the extravascular tissue if the diffusion is slow or the intercapillary distances are large.

This is shown in Figure 6 by a mathematical solution of the equations for flow, permeation, and axial and radial intratissue diffusion in a Krogh capillary-tissue cylinder model that has been previously described in detail.¹⁷ These conditions lie within region IV of Figures 1 and 5, the exchange being partly flow-limited and partly diffusion-limited.

Fig. 6.

Capillary-tissue exchange model. Lower, Concentration-vs.-time curve for outflow after a bolus injection into inflow. An initial sharp peak of non-extracted tracer is followed by a slow tail of tracer washing out from the extravascular space. The arrow indicates the time at which the upper panel shows the spatial distribution. Upper, Capillary-tissue region portrayed with extravascular region lying above and capillary below. Greater brightness = higher concentration; the curves are isoconcentration lines. Flow is from left to right, with the intra-arterially injected bolus being about two-thirds of the distance through the capillary. The capillary membrane is represented by the thick horizontal line between capillary and tissue: at the leading edge of the bolus (toward right), capillary concentration is higher than tissue concentration; at the trailing edge of the bolus (left or upstream end), tissue concentration is higher than capillary concentration. At this time, 0.5 second after injection at inflow (left end of capillary), as indicated by the arrow (lower panel), the fastest flowing particles have reached the outflow, producing the upslope and peak of C_v(t) (lower panel), but the major portion of the bolus is spreading from capillary into tissue (upper panel). The trailing wave of concentrated tracer in the tissue has the angle shown here because the time for diffusion from just outside the capillary to the boundary of the tissue region is about the same as the time for convection from capillary inflow to outflow. (This would not apply to the heart, where intratissue diffusion is at least 100 times faster.) In this mathematical solution, a very high value for permeability has been used (about 1,000 times higher than normal values for sodium), which tends to minimize the blood-tissue gradients. This figure exemplifies the situation in which extraction can be limited by slow extravascular diffusion rather than by slow penetration of the capillary membrane.

Experiments have been performed with paired tracer injections into the inflow and calculation of the extraction, E, of a permeating solute by comparison with a nonpermeating reference tracer. Then, when the flow, F, is known, PS can be calculated:

$$\mathrm{PS}=-\mathrm{F}\cdot {\log }_{\mathrm{e}}(1-\mathrm{E}).$$ (eq. 9)

Crone and Thompson¹ used this technique to estimate the permeability of brain capillaries and to document Crone's earlier observation that D-glucose is transported across the capillary barrier by a carrier mechanism. (This implies that much of the blood-brain barrier is at the capillary membrane.) The application of eq. 9 is dependent on E being governed by the unidirectional efflux of tracer from blood into tissue, so that the concentration-vs.-time curve of nonextracted permeable tracer is not deformed by the addition of tracer that has left and then returned to the blood. Ordinarily, this means that estimates of E must be obtained in the first few seconds after injection.

One can obtain estimates of PS also by working in region V and observing the clearance of intratissue depots of tracer. With low permeabilities and high blood flows, capillary blood would have a negligible tracer concentration and escape of tracer would be limited by the membrane only; this is the flat line of permeability, P (cm/s) times S (surface area, cm²), shown in Figure 5. The clearance is thus PS ml/s or PS/V_E ml/s per ml of extravascular space. The same experiment can be done in region IV, but the calculation of PS requires accounting for the partial limitation to washout by the lower flow.

The observations of a nonextracted early peak and a barrier-limited long low tail invite a futher modification of Figure 1 to account for the differences between FER of tracer located intra-arterially and that of tracer located extravascularly. The concept shown in Figure 7 covers all five regions for tracer introduced either into the arterial inflow or into the extravascular region of the tissue. For intra-arterial tracer (continuous line), clearance is never less than F/V_T. Region IV, mixed diffusion- and flow-limitation, shows extraction diminishing with increasing flow until finally, at very high flow or low permeability (region V), the intravascular tracer has not enough time in contact with the capillary membrane to permit permeation, and the extraction falls to zero. Now, three regions of flow-limited washout of intra-arterially introduced tracer can be seen: region I where diffusion gives complete intraorgan equilibration and the washout is monoexponential with clearance = F/V_T (this region is never observed in living hearts); region III where there is local radial blood-tissue equilibration and washout is not monoexponential even while being flow-limited with clearance = F/V_T = 1/t̄ as expected from the general theory; and region V where washout is flow-limited with clearance = F/V_v and where the clearance is higher than for the tracers falling in region III because the vascular volume, V_v, is smaller. Region V applies to intraarterial indocyanine green or albumin.

Fig. 7.

Diagram of clearances at various flows or various ratios of convective to diffusive velocities. Regions defined as in Figure 1. Continuous line = clearance of intra-arterial tracer. Long dashes = clearance of tracer from extravascular region.

Clearance from intratissue depots, either from intratissue injection or from the tail of the washout curve of a tracer that permeated the tissue after intra-arterial injection, is given by the heavy dashed line. Clearance is never greater than F/V. It is decreased further by intratissue diffusional shunting (region II) and by limited diffusibility or permeability (regions IV and V), as discussed above.

SUMMARY

In a broad conceptual approach to the investigation of passive transcapillary exchanges, the following situations are described.

1. Flow-limited exchange, when the curves recorded for a specific tracer are “similar” in shape at different flows, F—that is, the fractional escape rate or clearance at a particular level on the residue function curve is directly proportional to F. “Similarity” occurs in the heart with albumin-bound indocyanine green remaining intravascular¹¹ and for ¹²⁵l-labeled iodoantipyrine distributed throughout tissue water⁵ over a three-fold range of flows (region III).

2. When the ratio of convective velocity (flow) to diffusive velocity (permeability = D_R / Δr = diffusivity radially from the flow stream) is higher, the extraction of intravascular sodium or other partially permeability-limited solutes diminishes at higher flows, so that curves at different flows are dissimilar (region IV).¹⁸

3. At still higher ratios of flow to diffusive velocity, radial diffusivity is the sole limitation (region V).

4. At lower flows or with rapidly diffusible tracers, longitudinal diffusion, D_L, from inflow to outflow or countercurrent exchange is significant relative to flow. After intra-arterial injection, particularly at low flows, highly diffusible heat, ¹³³Xe, and tritiated water have abnormally high “early” escape rates, indicating diffusional shunting, and correspondingly low rates “late” on the washout curve (region II).

5. With the lowest flows, when diffusion is relatively so rapid that tracer concentration throughout the organ is uniform, washout would be monoexponential; this condition apparently does not exist at flows compatible with viability (region I).