The effects of flow on the transport of potassium ions through the walls of single perfused frog mesenteric capillaries

Abstract

1. We have investigated the effects of varying flow velocity upon permeability to potassium ions (P_K) of single perfused mesenteric microvessels in pithed frogs.

2. P_K was estimated using a development of the single bolus microperfusion technique at chosen flow velocities (U) in the range of 150–7000 μm s⁻¹.

3. In thirty-seven out of forty-three vessels, there was a strong positive correlation between P_K and U. Average values (median) for P_K (μm s⁻¹) were related to U (μm s⁻¹) by the expression: P_K = 0.0043U+ 4.05 (n = 43).

4. The correlation between P_K and U was independent of microvascular pressure (and hence fluid filtration) over the range of 5–70 cmH₂O.

5. The correlation between P_K and U was independent of the potassium concentration in the bolus over the range of 2–40 mmol l⁻¹ and of the direction of the potassium flux through the capillary walls.

Increasing the blood flow to a tissue not only raises the rate of delivery of solutes dissolved in the blood to exchange vessels of the microcirculation but also increases their transport through the walls of the exchange vessels into the tissue (Renkin, 1959; Alvarez & Yudilevich, 1969; Duran & Yudilevich, 1978; Rippe et al. 1978). This phenomenon of flow-dependent transport has been interpreted in terms of (i) increasing mean concentration of solute within the exchange vessels, thus increasing the gradient down which solute may diffuse between blood and tissue, and (ii) recruiting more exchange vessels as flow rate increases so increasing the area of microvascular wall through which transport can occur. These analyses assume that solute permeability is a constant property of the microvascular wall.

In this paper we present evidence which suggests that permeability itself may increase with blood flow. Our measurements have been made on single mesenteric capillaries which were perfused in situ in pithed frogs and we have estimated the permeability of these vessels to potassium ions (P_K) using a development of the single bolus microperfusion technique of Crone et al. (1978).

Previous workers have suggested that microvascular permeability might increase with microvascular flow. Yuan et al. (1992) found that the permeability to albumin increased with increasing perfusion rate in isolated perfused venules from the pig heart. Two studies on isolated perfused descending vasa recta of the rat from Pallone's laboratory have shown that the permeability to NaCl and raffinose increased with flow (Pallone et al. 1995; Turner & Pallone, 1997). Our findings are also consistent with the report of Friedman & DeRose (1982) that variations in estimates of P_K in frog mesenteric capillaries could be correlated with the variations in flow through these vessels. Because the variations in flow which Friedman & DeRose (1982) observed were those which occurred spontaneously in different vessels of different animals, these authors were unable to demonstrate a causal relation between P_K and flow. The present paper describes how development of the single bolus technique has allowed us to investigate this directly and to eliminate some of the measurement artifacts which might be responsible for the phenomenon.

Preliminary results of some of our data have been presented at the 6th World Congress for Microcirculation (Head et al. 1996) and the British Microcirculation Society (Kajimura et al. 1997a).

METHODS

General preparation

The experiments were carried out on mesenteric microvessels of male frogs (Rana temporaria and Ranapipiens) 6-7.5 cm in length supplied by Blades (Edenbridge, Kent, UK) whose brain and upper parts of the spinal cord were pithed. The mesentery was gently arranged on the surface of a polished Perspex pillar. This allowed transillumination of the mesenteric microvasculature. The upper surface of the mesentery was superfused continuously with frog Ringer solution at 16-18°C. The flow of superfusate was maintained at 3.5-4 ml min⁻¹ and this kept the layer of fluid over the tissue at approximately constant depth. The microvessels chosen for study were mostly venous capillaries (diameters 18–35 μm) though some were true capillaries. The tissue was observed with a stereomicroscope (Wild Heerbrugg M8) with a CCTV camera (Hitachi) attached to the camera tube. The output from the camera was displayed on videomonitors and recorded.

Solutions

Frog Ringer solution was used as the bathing solution for the dissection of the mesentery and for the perfusates, and the superfusates. Its composition was (mmol l⁻¹): 111 NaCl, 2.1 KCl, 1.0 MgCl₂, 1.1 CaCl₂, 0.195 NaHCO₃, 5.5 glucose, buffered with 2.3 Hepes and 2.7 Na-Hepes (Sigma). The pH was adjusted to 7.2 by the ratio of Hepes acid to base. The perfusate contained bovine serum albumin (BSA; A-7638, Fraction V, Sigma) at 50 mg ml⁻¹. In some perfusates, Evans Blue (Sigma) was added (5 mmol l⁻¹) to colour the solution containing BSA (50 mg ml⁻¹). At this concentration 98 % of the dye should be bound to the BSA (Levick & Michel, 1973). Evans Blue perfusates were dialysed in dialysis tubing with a cut-off of molecular weight 8000 (Spectro/Por, Spectrum, CA, USA) against three 2 l changes of Ringer solution of equal osmolarity over a 24 h period at 15°C. High-K⁺ solutions were prepared by replacing NaCl with KCl.

Fabrication of K⁺-sensitive microelectrodes

The electrodes were made according to the method described by Voipio et al. (1994). Single-barrelled pipettes (quartz with filament; o.d., 1.2 mm; i.d., 0.60 mm; Sutter Instrument Co., Navato, CA, USA) were pulled on a micropipette puller (Sutter model P-2000). Micropipettes were mounted horizontally on a brass holder, placed in a Petri dish, and baked at 200°C. After 30 min, approximately 50 μl of N,N-dimethyltrimethylsilylamine (Fluka Chemicals, Gillingham, Dorset, UK) was added to the Petri dish. Baking continued for a further 1 h. This silanization process makes the glass surface hydrophobic and ensures good contact between the glass and the lipophilic ion exchanger. They were then backfilled with a small amount of a liquid ion exchanger (potassium ionophore I - cocktail A, Fluka Chemicals) and filled with an electrolyte solution (0.5 mol l⁻¹ KCl). Electrodes were manufactured on the day of the experiment.

Double-barrelled perfusion pipettes

Double-barrelled microperfusion pipettes similar to those described by Davis & Gore (1987) were made from θ-tubing (o.d., 1.5 mm; Clark Electromedical Instruments, Reading, UK). Two small holes were made on one side of one barrel with a 400 diamond disk (Rx Honing Machine, Mishawaka, IN, USA) at distances of 6 mm and 14 mm from what was to become the open end of the pipette. After being ground, the glass tubes were cleaned with detergent (5 % Decon90, Decon Laboratories Ltd, Sussex, UK) and rinsed in distilled water, methanol and acetone. The hole closest to the open end of the tube and the open end itself were filled with a plug of melted wax. This meant that the lumen of that barrel communicated with the outside through the hole, which was 14 mm from the open end. The pipettes were pulled and then bevelled with a microgrinder (model EG4, Narashige Europe Ltd, London) to give a tip diameter of 18–24 μm. The double-barrelled microperfusion pipette was placed in a holder, which had been manufactured so that the open end of one barrel was separated from the opening in the side of the second barrel by a silicone ring which acted as a water-tight seal between two interior compartments. As pressure could be applied to each of these compartments independently, the ejection or retention of the solution in each barrel could be controlled separately.

Calibration of the K⁺-sensitive microelectrode

All measurements with the K⁺-sensitive microelectrodes were carried out in a Faraday cage. Chlorided silver wires were used to connect the recording apparatus to either the calibration chamber or the preparation. Signals from K⁺-sensitive microelectrodes were amplified using a multipurpose microelectrode amplifier with headstage (HS-2 gain 0.0001M) input resistance above 10¹⁴Ω (Axoprobe-1A, Axon Instruments). The output from the electrometer was fed through an oscilloscope (Tektronix 5A22N differential amplifier, Harpenden, UK) into an AD converter (1401, Cambridge Electronic Design, Cambridge, UK) and into a Pentium 90 computer. Microelectrodes were calibrated before and after the experiments by solutions containing 2.1 and 20 mmol l⁻¹ potassium. The signal measured in the solution containing 2.1 mmol l⁻¹ potassium was used as baseline, i.e. the voltage was set at zero. On some occasions concentrations of 2.1, 3.5, 7, 11, 20 and 40 mmol l⁻¹ potassium were used to check the linearity of the calibration curve (Fig. 1). The time constants of the electrodes were usually less than 50 ms. Electrodes with a time constant greater than 200 ms were rejected. The response time of the recording system itself was less than 25 ms. The electrode typically gave a 55–58 mV change for 2.1 to 20 mmol l⁻¹ change in K⁺ concentration. The response time and calibration of the microelectrodes were estimated at high and low flows through the calibration cell. Both calibration and response time were independent of flow velocity over the range 800–4000 μm s⁻¹.

Figure 1. Calibration of a K⁺-sensitive microelectrode.

The output from the electrode (voltage) is plotted against the K⁺ concentration of five calibration solutions. The output was adjusted to zero with the electrode in contact with normal frog Ringer solution in which [K⁺]= 2.1 mmol l⁻¹. [K⁺] is plotted on a logarithmic scale.

General protocol

Each microvessel was cannulated with a bevelled double-barrelled micropipette (Fig. 2). One barrel of the pipette was filled with a normal-K⁺ solution (2.1 mmol l⁻¹ K⁺) and the other was filled with a high-K⁺ solution (20 mmol l⁻¹ K⁺ unless stated otherwise). The tubes leading from the two barrels of the pipette were connected through an electric rotary valve (Omnifit Ltd, Cambridge, UK) to two water manometers. This arrangement allowed alternate perfusion with the normal-K⁺ solution or the high-K⁺ solution. The heights of the water columns of the two manometers were adjusted initially to the levels at which only the normal-K⁺ solution was being perfused but the high-K⁺ solution was not. To do this, one solution (usually the normal-K⁺ solution) was coloured with Evans Blue (5 mmol l⁻¹), therefore making the interface between the normal- and high-K⁺ solutions visible. The interface between the two solutions at the tip of the perfusion pipette was carefully monitored to prevent either the normal-K⁺ solution from entering the other barrel or the high-K⁺ solution from perfusing the vessel. The pressures of the perfusing and non-perfusing sides represented the pressure applied to the perfusate (P_p) and the effective microvessel pressure at the tip of the pipette (P_e), respectively. Careful and continuous adjustment of P_e was necessary in order to maintain the balance of the interface between the two solutions.

Figure 2. Schematic representation of the measurement of P_K in a single microvessel.

The microvessel was cannulated and perfused by a double-barrelled micropipette, one barrel being filled with normal Ringer solution, the other containing Ringer solution with high [K⁺]. Each barrel of the micropipette was connected through a cross-over tap to a manometer in which pressure was set to achieve either brisk perfusion or zero flow. The vessel was perfused with normal [K⁺] and then the cross-over tap was switched for 2 s so that a bolus of high [K⁺] flowed down the vessel. Changes in [K⁺] inside the vessel downstream from the perfusion pipette were detected by K⁺-sensitive microelectrodes, e₁ and e₂. The control solution was usually coloured blue (Evans Blue + 5 % BSA) whereas the high-K⁺ solution was usually colourless (5 % BSA alone). Adjustments to the heights of the manometers enabled the vessel to be perfused at different flow rates.

After the interface was adjusted, the electric rotary valve, which functioned as a cross-over tap between two manometers, was switched so that the higher pressure was applied to the high-K⁺ solution causing it to flow through the microvessel. After 2 s, the rotary valve was returned to its initial position. Because, in most experiments, the high-K⁺ solution did not contain Evans Blue, its passage through the capillary could be seen as a bolus of clear solution separating columns of blue perfusate. The concentration of K⁺ within the vessel was monitored by K⁺-sensitive microelectrodes at two points, e₁ and e₂, downstream from the perfusion pipette. In forty-three experiments the distance between the perfusion pipette and e₁ was 382 ± 155 μm; e₂ was a further 510 ± 206 μm downstream from e₁. Signals were acquired at the rate of 200 Hz using the Chart software (Cambridge Electronic Design) running on a Pentium 90 computer.

An interval between each measurement of no less than 40 s was allowed to ensure adequate washout of K⁺ from the interstitium surrounding the vessel. The superfusion rate was kept high (3.5-4 ml min⁻¹) to clear K⁺ effectively from the mesothelial surface.

The flow velocity was varied by changing the heights of the two water manometers. For example in one experiment, setting P_p at 12.4 cmH₂O and P_e at 4.1 cmH₂O produced a flow velocity of 861 μm s⁻¹. Raising P_p to 25.1 cmH₂O increased the flow velocity to 1824 μm s⁻¹ and P_e had to be readjusted to the higher pressure, 5.3 cmH₂O, to maintain the balance. The difference between P_p and P_e represented the pressure drop (ΔP) for the flow of perfusate between manometer (and wide region of the perfusion pipette) and the pipette tip. Thus a greater difference between two pressures resulted in a higher flow velocity and vice versa.

Calculation of diffusional potassium permeability (P_K)

The method of Crone et al. (1978) was used to estimate permeability. Briefly, a bolus of high-K⁺ solution flowed along a single microvessel and the intraluminal [K⁺] was recorded at two points by K⁺-sensitive microelectrodes (e₁ and e₂) separated by the length of the vessel over which the permeability was to be determined. If C₁ and C₂ are the K⁺ concentrations at e₁ and e₂, respectively, and C_E is the [K⁺] in the perivascular fluid then,

(1)

where r is the capillary radius and τ is the transit time of the bolus between e₁ and e₂. τ was measured as the difference in appearance time of the high-K⁺ bolus at the two microelectrodes and the mean concentrations of K⁺ at e₁ and e₂ were obtained by integration of the area under the recorded signal from the onset to the peak. This integration was automated using the script language in Spike2 software (Cambridge Electronic Design). It was assumed that C_E was equal to the concentration of K⁺ in the superfusate.

Statistical analysis

Average values are reported as median ± interquartile throughout, unless otherwise specified. The simple linear regression analysis was carried out for the curve fit. To compare and contrast the average values of slopes and intercepts between two groups, the Wilcoxon signed rank test and Mann-Whitney U test were used for paired and unpaired tests, respectively. To compare slopes and intercepts within a single experiment, Student's t test was used. All statistical analyses were carried out using StatView software (Abacus Concepts, Inc., Berkeley, CA, USA). Tests of significance were set at the 5 % level (P < 0.05).

RESULTS

Effect of varying flow on potassium permeability

Figure 3 shows the changes in [K⁺] recorded from two K⁺-sensitive microelectrodes placed in a single microvessel, 560 μm apart, as a bolus initially containing 20 mmol l⁻¹ K⁺ flowed along the vessel. The recordings shown in Fig. 3A and B were made when flow velocities were 4300 and 1150 μm s⁻¹, respectively. The signal at e₂ was seen to rise later than that at e₁; the delay represented the transit time (τ) between the electrodes. The difference in [K⁺] recorded at e₁ and e₂ showed that a significant amount of K⁺ left the vessel between these points.

Figure 3. Time-concentration curves with ‘high’ and ‘low’ flow.

Changes in [K⁺] recorded from two K⁺ microelectrodes separated by a distance of 560 μm, as a bolus of high [K⁺] flowed down the vessel. A, under conditions of high flow (U = 4300μm s⁻¹) when P_K was calculated as 15.6 μm s⁻¹. B, low flow (U = 1150μm s⁻¹) in the same microvessel when P_K was calculated as 10.4 μm s⁻¹. Flow velocities calculated from the transit time, τ, between the electrodes. Vessel radius, 10 μm; distance from perfusion pipette to e₁, 390 μm.

The calculation of P_K from eqn (1) assumes that the K⁺ concentration falls exponentially with time as the bolus flows from e₁ to e₂. Since the bolus is flowing at constant velocity the decline in [K⁺] with distance from the perfusion pipette along the vessel should also be exponential. Figure 4 shows that the data in Fig. 3A are consistent with this prediction of an exponential decline of [K⁺] inside the vessel with distance from the micropipette. The line joining the values of [K⁺] at e₁ and e₂ projects back to a value close to 20 mmol l⁻¹ for the [K⁺] leaving the micropipette. This was found to occur under conditions of high flow in all experiments. At lower flow rates, the intercept was somewhat lower but usually in the range of 12–20 mmol l⁻¹.

Figure 4. Changes in [K⁺] of a K⁺-rich bolus of perfusate with distance from the perfusion pipette.

The [K⁺] has been plotted semi-logarithmically against distance so that a straight line represents an exponential curve. The line drawn through the two points which represent the increment of [K⁺] recorded at e₁ and e₂ (as shown in Fig. 3A) against the distance of e₁ and e₂ from the perfusion pipette extrapolates backwards to a [K⁺] just less than 20 mmol l⁻¹ at the pipette. Since 20 mmol l⁻¹ was [K⁺] in the pipette, [K⁺] declines exponentially with distance along the length of the vessel.

Figure 3A and B shows that as flow was slowed, a larger fraction of K⁺ was lost from the bolus as it flowed between e₁ and e₂. In spite of this, when P_K was calculated from these data, it was found that the recordings in Fig. 3A (when velocity was 4300 μm s⁻¹) yielded a value of 15.6 μm s⁻¹ whereas those in Fig. 3B (when velocity was 1150 μm s⁻¹) gave a value of 10.4 μm s⁻¹. When a series of estimates of P_K were made at different flow velocities (U) on the same microvessel, a clear positive correlation between P_K and U was seen (Fig. 5A; n = 19, r = 0.941, P < 0.0001). This correlation was observed in forty out of forty-three microvessels investigated and in thirty-seven of the forty vessels, it was found to be highly significant. In the three vessels where there was no correlation between P_K and U, the measurements were made in the narrow range of flow velocities (150-1000 μm s⁻¹). Thirty-one of the forty-three vessels were venous capillaries and the small number of measurements made on arterial and true capillaries yielded results which were quantitatively similar. In most experiments, flow velocities were increased to 1000–4000 μm s⁻¹ with a maximum of 7000 μm s⁻¹. Average values (median ± interquartile) for P_K (μm s⁻¹) for all forty-three vessels can be summarized by the expression:

(2)

Figure 5. The relationship between P_K and flow velocity: a single experiment on a venous microvessel.

A, determinations of P_K plotted against corresponding values of flow velocity (U) made on a single venous microvessel. The line is the regression line (n = 19, r = 0.94, P < 0.0001). B, changes in P_K (•) and U (cross) plotted against time of measurement after the start of the experiment. The arrows indicate when the applied pressures were altered to change U.

The relation between P_K and U was independent of the order in which the points were determined. In each experiment the measurements were made at a series of flow velocity steps. Figure 5B shows the sequence of measurements for the data of Fig. 5A where flow velocity was changed between six levels. Here, the initial four measurements were made at a low flow velocity (867 ± 18 μm s⁻¹) and a subsequent three measurements were made at a higher flow velocity (1192 ± 141 μm s⁻¹). The highest flow velocity (1824 ± 41 μm s⁻¹) was then applied for the next three measurements. Then the flow velocities were gradually reduced in three different steps (1216 ± 84, 795 ± 31 and 539 ± 38 μm s⁻¹, respectively). Mean flow velocities of the first and the fifth steps (867 ± 18 and 795 ± 31 μm s⁻¹, respectively) were similar as were their mean P_K values (6.60 ± 0.44 and 6.44 ± 0.56 μm s⁻¹, respectively). This indicates that the effect of increasing the flow velocity to increase P_K was reversible. The changes in P_K occurred relatively rapidly after flow had been changed. P_K was usually measured within 20 s of change in flow when it appeared to have reached a new value.

Changes in K⁺ concentration in capillary and pericapillary fluid

We examined the assumption that [K⁺] in the pericapillary fluid, C_E, was the same as that of the superfusate when a bolus of K⁺-rich solution flows down a single capillary. To do this, one K⁺-sensitive electrode was placed beneath the mesothelium just outside the capillary and directly opposite the other electrode, which was placed inside the capillary lumen. In this way changes in [K⁺] in the capillary (ΔC) and those in the pericapillary fluid (ΔC_E) during brief perfusions of K⁺-rich solutions could be monitored simultaneously at different flow velocities. Figure 6A shows a recording of K⁺ indicator potentials from the two K⁺-sensitive electrodes positioned inside and immediately outside a microvessel when a bolus of high-K⁺ solution flowed past. Figure 6B summarizes the results of three separate experiments of this kind. It can be seen that ΔC_E was linearly related to ΔC as flow was varied. The relation between ΔC_E and ΔC for the three experiments can be summarized by the expression:

(3)

Figure 6. Changes in [K⁺] inside and just outside a microvessel during a 2 s perfusion of a bolus of high [K⁺] at different flow rates.

A, recordings of [K⁺] within (in) and outside (out) the capillary. B, plot of the mean increment of [K⁺] inside the capillary lumen (ΔC) against the mean increment of [K⁺] in the pericapillary fluid (ΔC_E) during the passage of a 2 s bolus of a high-K⁺ solution. Repeated measurements were made on three capillaries at different flow velocities; the different symbols represent different microvessels (n = 29, r = 0.95, P < 0.0001).

The intercept was not different from zero (t test).

‘Partial occlusion’ experiment: separation of effects of microvascular pressure and flow velocity on P_K

We examined whether the increase in P_K is determined by changes in pressure or changes in flow. To do this, P_K was first estimated at different flow velocities under the condition of free flow through a single vessel (control). The vessel was then partially occluded downstream from e₂ and the vessel was reperfused over a range of U comparable with the control but at microvascular pressures 5–70 cmH₂O higher than under control conditions (as estimated from P_e). Data from one of these experiments are shown in Fig. 7A and B. When P_K was plotted against U, the relation appeared as a single correlation with neither a separation of the values of P_K at high and low pressure nor a significant difference between the regression lines drawn through the high and the low pressure data points (t test). The ranges of pressures used to achieve the changes in flow, however, were different. Under control (free flow) conditions, P_K was estimated when P_e was in the range of 5.2-8.6 cmH₂O whereas, after partial occlusion, P_e was in the range of 21.7-43.0 cmH₂O.

Figure 7. ‘Partial occlusion’ experiment: a single experiment.

A, paired measurements of P_K in a single microvessel before and after the ‘partial occlusion’ are shown as a function of U. Seven determinations of P_K were made under the control conditions (•). P_K strongly correlated with U (continuous line, r = 0.87, P < 0.01). The same microvessel was then partially occluded and another 11 determinations of P_K were made (○). The strong positive correlation between P_K and U was maintained (dashed line, r = 0.95, P < 0.01). Neither slopes nor intercepts differ significantly. B, effective microvessel pressure (P_e) is plotted against P_K for the same data as in A (•, control; ○, partial occlusion). *Two data points are overlapped.

For the six paired experiments of this type average values for P_K (μm s⁻¹), under the free flow condition, were related to U by the expression:

(4)

and under conditions of partial occlusion:

(5)

There was no significant change in the slope between P_K and U in any of these six experiments and there was no suggestion that a higher microvascular pressure (over the range 5–70 cmH₂O) resulted in a higher P_K.

Effects of raised potassium concentration on P_K

Figure 3A and B illustrates that the mean [K⁺] in the microvessel is higher with the high flow velocity than it is with the low flow velocity. To test whether the mean capillary [K⁺]per se influences the relation between P_K and U, we increased the [K⁺] in the bolus (the input concentration) from 20 to 40 mmol l⁻¹. Average values of P_K (μm s⁻¹) for six vessels using a raised potassium concentration (40 mmol l⁻¹) in the bolus can be summarized by the expression:

(6)

which was not significantly different from the control relation (Mann-Whitney U test).

‘Reversed gradient’ experiment: P_K measurements when the potassium gradients are reversed

In our standard protocol, P_K was measured from the fall in [K⁺] of a K⁺-rich bolus (20 mmol l⁻¹) of perfusate as it flows along a microvessel between two K⁺-sensitive microelectrodes. To determine whether the direction of the gradient for K⁺ influences the relation between P_K and U, we reversed the concentration gradient for K⁺. This protocol also serves to examine the longer term effects of raised [K⁺] on P_K. The mesentery was superfused and initially perfused with the high-K⁺ solution ([K⁺]= 20 mmol l⁻¹). P_K was measured from the rise in [K⁺] in a bolus of perfusate of normal [K⁺] (2.1 mmol l⁻¹).

The mean interelectrode distance was 287 ± 52 μm and the distance between the perfusion pipette and e₁ was 290 ± 52 μm. These distances were considerably shorter than those in the control experiment in order to acquire a reasonable magnitude of K⁺ indicator potential.

Figure 8 shows the recording from a single run which was an inversion of that obtained with our standard protocol. The relation between P_K and U was maintained in each of eight microvessels investigated in this way. The average value for the slope of the regression line was 0.0031 ± 0.0034, which was not different from the control (Mann-Whitney U test).

Figure 8. The time-concentration curves of the ‘reversed gradient’ experiment: a single run.

The changes in [K⁺] in microvessels recorded from K⁺-sensitive microelectrodes 290 μm apart during the passage of ‘low-K⁺’ bolus. The [K⁺] of the superfusate and perfusate before and after the bolus was maintained at 20 mmol l⁻¹. Estimated P_K= 9.4 μm s⁻¹. Vessel radius, 10.25 μm; distance from perfusion pipette to e₁= 250 μm.

Effects of Evans Blue dye on relation between P_K and U

It has been reported that the dye Evans Blue increases K⁺ channel activity in the smooth muscle of sheep bladder (Cotton et al. 1995; Hollywood et al. 1995). Because our control perfusate contained Evans Blue the possibility arises that the dye might be affecting microvascular permeability in our preparation.

To test this possibility, we added Evans Blue to the high-K⁺ solution while the normal-K⁺ solution was kept clear in colour. We then examined the relations between P_K and U in four microvessels. A strong positive correlation between P_K and U was found with values for P_K similar to those determined when Evans Blue had been in the normal-K⁺ perfusate. Average values of P_K (μm s⁻¹) for the four vessels can be summarized by the expression:

(7)

which was not significantly different from the control relation (Mann-Whitney U test).

DISCUSSION

Our data show a clear positive correlation between the apparent permeability coefficient of frog mesenteric capillaries to K⁺ and the velocity of flow through them. This correlation appears to be independent of microvascular pressure over the range of 5–70 cmH₂O and of [K⁺] over the range of 2–40 mmol l⁻¹.

Before we can be sure that the true permeability of the vessel wall varies with flow velocity, we must consider the assumptions which underlie the single bolus microperfusion technique. To recapitulate, these assumptions are: (i) that the vessels are cylindrical; (ii) that the decay in [K⁺] of high-K⁺ bolus is exponential as the bolus passes down the vessel; and (iii) that the pericapillary [K⁺] approximates to that in the superfusate as the bolus of high-K⁺ flows down the vessel.

(i) Both ultrastructural studies (Mason et al. 1979; Clough & Michel, 1981; Bundgaard & Frøkjær-Jensen, 1982) and more recent investigations using confocal microscopy in living preparations (Adamson et al. 1994) support the assumption that frog mesenteric microvessels are approximately circular in cross-section when they are microperfused. The slightly elliptical profiles which have been seen in some vessels would have a negligible effect on the assumption that the ratio of capillary surface area to its volume is 2/r where r would be the half-axis in the horizontal plane.

(ii) The second assumption, that the [K⁺] in the bolus declines exponentially as it flows along a microvessel, receives justification from the data reported in this paper. If the [K⁺] declines exponentially with the time the bolus spends in the vessel, then the decline with distance from the perfusion pipette should be exponential providing the capillary radius and the flow velocity are constant. The data in Fig. 4 show that when K⁺ concentrations at e₁ and e₂ are plotted against their distance from the tip of the perfusion pipette, the points lay on an exponential curve which extrapolates back to give a [K⁺] close to that of the high-K⁺ solution in the micropipette.

It should be noted that one would predict a lower [K⁺] in the micropipette than that present in the high-K⁺ solution. This is because the high-K⁺ solution is diluted by diffusion from the tip of the micropipette when the vessel is being perfused with the low-K⁺ perfusate. When the bolus of high-K⁺ solution is then despatched, its [K⁺] is lowered by this diluted fraction. The degree of dilution of the bolus is dependent on the volume of the diluted portion of the solution in the pipette tip and magnitude of the flow. Thus when flow is high, the contribution of the diluted portion of the solution to the 2 s bolus of high [K⁺] is less than when flow is low. Consistent with this prediction has been our observation that backward extrapolations of semi-logarithmic plots of [K⁺] at e₁ and e₂ against perfused capillary length give values of [K⁺] in the pipette which are at or just above 20 mmol l⁻¹ when flow is high but 12–20 mmol l⁻¹ when flow is low.

(iii) The third assumption is that the pericapillary [K⁺] does not rise much above the superfusate [K⁺] as a bolus of high [K⁺] passes down the vessel. The data shown in Fig. 6 show that this is not so. Indeed the pericapillary [K⁺], C_E, is approximately 50 % of the mean capillary [K⁺], C. It could be argued that we may have overestimated C_E since these measurements were made at points in the pericapillary space adjacent to where the vessel wall had been penetrated by a second microelectrode to monitor C. Thus it is possible that permeability was locally increased in this region and the rise in C_E was greater here than at distant sites. Against this argument is our failure to observe greater leakage of Evans Blue albumin from vessel in the vicinity of the electrodes than from elsewhere.

If we accept that C_E does increase as much as our measurements indicate, it is necessary for us to consider the relations between our estimates of P_K and the true values of P_K.

Relations between experimental estimates of P_K and the true values of P_K

From the point of view of the relation between P_K and U, the most important feature of Fig. 6 is the linearity between ΔC_E and ΔC. This allows the concentration difference across the vessel wall to be expressed in terms of the concentration within the capillary, and α, the slope of the relation between ΔC_E and ΔC. As shown in the Appendix, the expression for P_K becomes:

(8)

where C₀ is the [K⁺] in the control perfusate and superfusate. Since C₀ is equal to the value of C_E which was used in eqn (1), the difference between our estimates of P_K and its true value is the factor of (1 - α)⁻¹. From Fig. 6 and eqn (3) we see that α has a value of approximately 0.5 in our experiments so that the true values of P_K are twice those estimated using eqn (1). The linearity of the relation between ΔC_E and ΔC means that α is independent of flow and the true P_K should vary with U in a similar manner to our estimates of P_K. Indeed the slopes of the relations between the true P_K and U should be approximately twice the values of the slopes reported in this paper. Thus far from being responsible for a spurious correlation between P_K and U, our underestimation of P_K has led to us minimizing the flow dependence of permeability.

In support of this conclusion, we note that the value of α is reduced if the mean values of C and C_E are estimated over the first second only of the rising phase of K⁺ concentration. This reduction of α is not accompanied by a loss of linearity between ΔC_E and ΔC with variations in flow. Furthermore, when P_K is estimated from values of C₁ and C₂ which are calculated from the first second of the rising phase of K⁺ concentration, higher values are obtained and these show a greater sensitivity to changes in flow than the values of P_K estimated by the standard procedure.

No relation between P_K and mean capillary [K⁺]

We examined the possibility that P_K at a high flow is a simple consequence of the increased delivery of K⁺ or vice versa by doubling the input [K⁺]. If this were the case and if this were also the basis of the flow dependence of P_K, we would expect P_K to increase with increasing input [K⁺] (hence increasing mean capillary [K⁺] or C). We have shown that when the input [K⁺] was raised to 40 mmol l⁻¹ the mean P_K value (eqn (6)) is not different from the control value (eqn (2)) with the normal input [K⁺] (20 mmol l⁻¹). This confirms that the argument that P_K at a high flow is not a simple reflection of increased delivery of K⁺ and that there is no relationship between P_K and C.

Effects of convective component and microvascular pressure on P_K

Our finding that changes in P_K correlate with flow not with microvascular pressure enables us to eliminate two possible hypotheses which could account for our findings. An increase in microvascular pressure (P_e) associated with increased flow might itself increase apparent P_K either (i) by increasing fluid filtration through the vessel wall and thus increasing convective transport of solute; or (ii) by triggering a mechanotransduction cascade which increases microvascular permeability.

The first possibility seems unlikely because our estimate of the Peclet number for K⁺ (the ratio of convection to diffusional transport) is very small (< 0.05) in frog mesenteric capillaries. The demonstration that our estimates of P_K values of the ‘partial occlusion’ experiments are comparable to those of the ‘control’ experiments confirms this calculation. Further confirmation is provided by the ‘reversed gradient’ experiment, for here convection is in the opposite direction to that of K⁺ flux, and the positive correlation between P_K and U is still observed.

Our findings also eliminate the second possibility that an increase in hydrostatic pressure itself increases permeability either by stretching existing ‘pores’ in the vessel wall (Shirley et al. 1957) or by inducing the formation of new openings (e.g. Neal & Michel, 1996). Thus the observed changes in P_K appear to be attributed to ‘flow’ and not to ‘pressure’ at least over the pressure range we have investigated (5-70 cmH₂O).

At this point, we might comment on the absence of a correlation between P_K and P_e for the partial occlusion data shown in Fig. 7B. Since perfusion velocity is expected to be linearly related to P_e, a linear relation between P_K and P_e might be anticipated. This was usually seen for the control (free flow) measurements and sometimes seen for the partial occlusion data. That it was not always seen during partial occlusion probably reflects the variability of the degree of partial occlusion between successive runs. Thus the outflow resistance to flow from the perfused segment of the vessel was variable and the linear relation between pressure and flow was lost.

Other possible artifacts

Taylor diffusion

The phenomenon of diffusion of a solute across the parabolic front of the laminar flow profile of a solvent is known as ‘Taylor diffusion’ (Lassen & Crone, 1969). It is possible that the axial dispersion of K⁺ at the front of the bolus gives rise to the false low [K⁺] at e₂ and hence to the false high extraction and an overestimate of P_K. If the degree of axial dispersion is proportional to the transit time of the bolus, dispersion is increased and P_K will be overestimated when τ is long and flow is low. This is opposite to our observations and leads us to conclude that ‘Taylor diffusion’ does not account for the relationship between P_K and flow velocity.

Unstirred layer effect

Unstirred layers of K⁺ close to the luminal aspect of the endothelium could be a possible source of error in estimating the permeability of a large vessel. In a single capillary, however, the dimensions and the high diffusibility of K⁺ mitigate against such effects. Here such a layer could hardly be thicker than 2 μm and the mean displacement time for K⁺ to diffuse over this distance is 1 ms.

Accumulation of K⁺ in endothelium

It might be argued that K⁺ is entering the endothelial cells but not crossing the capillary wall. A simple calculation, however, reveals that the rate of rise of the [K⁺] within the endothelial cells would have to be greater than 260 mmol l⁻¹ s⁻¹ to account for the efflux rate which is measured in our experiments. Therefore we can discount this possibility. Direct evidence for the passage of K⁺ through vessel walls is shown in Fig. 6A.

Studies by other investigators

In whole organ studies, estimates of the permeability-surface products for various small hydrophilic solutes increase with increasing flow. It has been clearly demonstrated, also, that the clearances of these solutes plateau at high flow rates (Alvarez & Yudilevich, 1969; Duran & Yudilevich, 1978; Watson, 1995). Renkin (1984) evaluated the experimental data of whole organ studies of this kind using a detailed theoretical analysis in which heterogeneous (non-uniformed) distribution of blood flow velocities among recruited exchange vessels is taken into account. He pointed out that it is difficult to fit the experimental data with the theoretical curve for some experiments. Although his analysis assumed that solute permeability stays constant at all flows, it now seems possible that this may not be so.

The idea that permeability is modulated by flow is suggested by several studies using isolated microvessels where surface area for the exchange and the hydrostatic pressure are accurately measured. Friedman & DeRose (1982) suggested that variations in estimates of P_K in frog mesenteric venous capillaries are directly related to the capillary flow rate based on their observation made on different vessels of different animals. If we take an average value of 10 μm for the radius of the capillaries in the study of Friedman & DeRose, their results are summarized by the expression:

(9)

which is consistent with our results. Pallone et al. (1995) demonstrated that permeability to ²²Na⁺ of the microperfused outer medullary descending vasa recta of the rat was positively correlated to the perfusion rate and more recently Turner & Pallone (1997) have shown that permeability coefficient to [³H]raffinose increased 3.2-fold in the flow range from 10 to 30 nl min⁻¹ in the same preparation (Turner & Pallone, 1997). Yuan et al. (1992) showed that permeability coefficient to albumin (P_alb) increased 1.5-fold in the flow range from 7 to 13 mm s⁻¹ in isolated coronary venules of the pig. The authors were able to abolish this phenomenon by a nitric oxide synthase inhibitor, N^G-monomethyl L-arginine (L-NMMA). Based on this, they suggested that the mechanism to modulate P_alb involves nitric oxide. In frog mesenteric capillaries, we have not been able to alter the flow dependence of P_K with nitric oxide synthase inhibitors (Kajimura et al. 1997b). We have, however, been able to reduce or inhibit the effect of flow on P_K in this preparation by raising the levels of cyclic AMP in the vascular endothelium (Kajimura & Michel, 1998). Thus the mechanisms signalling the effects of flow on permeability may differ between species. Further work is required to elucidate the mechanism of this phenomenon.

APPENDIX

Let us consider a cylindrical microvessel of radius, r, perfused at a constant flow, F. Potassium concentration of the perfusate is monitored continuously by K⁺-sensitive microelectrodes at two points 1 and 2, 2 being several hundred micrometres downstream from 1. If a bolus of K⁺-rich solution now flows down the vessel, it takes τ s to pass from point 1 to point 2 and loses K⁺ by diffusion through the vessel walls. As the bolus passes a point A, which can be at any distance between 1 and 2, the mean concentration of K⁺ within the vessel, C, is related to its mean concentration outside the vessel, C_E, through the expression,

(A1)

where C_E(0) and C₍₀₎ are the values of C_E and C before passage of the bolus and α is a constant which is independent of flow velocity (Fig. 6, eqn (3)). Since C_E(0)=C₀, we can write expressions for C_E and (C - C_E) as:

(A2)

As the K⁺-rich bolus flows past A, it loses K⁺ by diffusion through an area of vessel wall, dS, with a consequent fall in concentration, dC. Following the arguments of Renkin (1959) and others we can equate the loss of K⁺ from the capillary as it flows past point A, which is -FdC, to the amount of K⁺ diffusing through the vessel wall, P_KdS(C - C_E),

(A3)

Since A is any point between the two electrodes, a similar expression to eqn (A3) can be written for the K⁺ flux at every point between the two electrodes. The total amount of K⁺ lost from the perfusate as it flows between the electrodes is the integral of eqn (A3) between the limits of C = C₂ and C = C₁ (where C₁ and C₂ are the concentrations recorded by the electrodes inside the vessel at points 1 and 2). Because C_E varies in a way which usually cannot be determined, integration of eqn (A3) usually cannot be carried out without making further assumptions. In our system, however, C - C_E is empirically a linear function of C, and using eqn (A2) to substitute for C - C_E in eqn (A3) results in the expression:

(A4)

Since C₀ is a constant, eqn (A4) can be integrated immediately between points 2 and 1 to give:

(A5)

Thus:

(A6)