Transient transcapillary exchange of water driven by osmotic forces in the heart

Abstract

Osmotic transient responses in organ weight following changes in perfusate osmolarity have implied steric hindrance to small-molecule transcapillary exchange, but tracer methods do not. We obtained osmotic weight transient data in isolated Ringer-perfused rabbit hearts using NaCl, urea, glucose, sucrose, raffinose, inulin, and albumin, and analyzed the data with a new anatomically and physico-chemically based model which accounted for: (1) transendothelial water flux, (2) two sizes of porous passages across the capillary wall, (3) axial intracapillary concentration gradients, and (4) water fluxes between myocytes and interstitium. During steady-state conditions about 28% of the transcapillary water flux going to form lymph was through the endothelial cell membranes (L_p = [1.8 ± 0.6] × 10⁻⁸ cm s⁻¹ mmHg⁻¹), presumably mainly through aquaporin channels. The interendothelial clefts (with L_p = [4.4 ± 1.3] × 10⁻⁸ cm s⁻¹ mmHg⁻¹) account for 67% of the water flux; clefts are so wide (equivalent pore radius was 7 ± 0.2 nm, covering about 0.02% of the capillary surface area) that there is no apparent hindrance for molecules as large as raffinose. Infrequent large pores account for the remaining 5% of the flux. During osmotic transients due to 30 mM increases in concentrations of small solutes, the transendothelial water flux was in the opposite direction and almost 800 times as large, and was entirely transendothelial since no solute gradient forms across the pores. During albumin transients, gradients persisted for long times because albumin does not permeate small pores; the water fluxes per mM osmolarity change were 200 times larger. The analysis completely reconciles data from osmotic transient, tracer dilution, and lymph sampling techniques.

INTRODUCTION

The osmotic weight transient method is one of the few techniques available to measure the reflection coefficient (σ) of small hydrophilic solutes in the vascular beds of whole organs. Using a single-membrane model of events during this experiment, Vargas and Johnson (47) developed a relationship, σ = J_v/(L_pSΔπ), that relates the volume flux across the capillary wall, J_v, to the solute reflection coefficient, σ, immediately following a step change in perfusate osmolarity. In a whole organ, the total fluid movement across the capillary wall, J_vS, is nearly equal to the rate of weight change of the organ; knowledge of the magnitude of the osmotic perturbation, Δπ, and the capillary hydraulic conductivity, L_p, is all that is needed to estimate σ. Estimates of σ using this analysis method and osmotic weight transient measurements have been obtained in many mammalian tissues including heart (19, 47-49), skeletal muscle (15, 39), and lung (37). For small solutes like sucrose, estimates of σ in skeletal muscle range from 0.08 (39) to 0.41 (55), and in heart are 0.30 (47) and 0.14 (19).

A few investigators have attempted to extract additional information from the complete time course of the weight transient using two compartment (48), three compartment (7), or three region axially-distributed models (19). The intended use of these models was to provide estimates of the solute permeability (P), but it is not clear that any of these models have captured the underlying physiology in enough detail to provide meaningful parameter estimates. The standard methods for analyzing osmotic weight transient data have several shortcomings. First, although the necessity of viewing transport across the capillary wall as a process occurring simultaneously through several disparate pathways is now recognized (40), this concept has not been built into any of the models used to analyze osmotic weight transient data. Therefore, the σ obtained from all previously developed methods of analysis is only a functional description of the behavior of the capillary wall as a whole, not information on specific pathways. Second, estimates of σ obtained by compartmental models tend to increase with increasing flow through an organ, even there is no reason to suppose the capillary wall morphology is actually changing. Finally, previously described analytical models depend on the prior knowledge of the hydraulic conductivity (L_p) in order to obtain estimates of σ and/or P.

In a previous paper (29), we developed an anatomically and physiologically realistic model of the transcapillary exchange process, and showed that the model was consistent with data from three different sources: multiple indicator dilution, osmotic weight transient, and lymph sampling experiments. Although previous models had integrated two of the three types of data sets (23, 41), this was the first model to be applied to all three of the major methods of investigating transcapillary exchange rates. The objective of the present paper is to determine key parameter values in our model (29), particularly those that govern water and solute exchange, using osmotic weight transient data obtained from isolated Ringer-perfused rabbit hearts. This approach has allowed us to extract more complete and more accurate information from the weight transient record than previous methods of analysis. The parameters obtained from this analysis do not require prior measurement of total L_p, and are independent of the flow. Thus, our approach marks a significant improvement over previous attempts to analyze osmotic weight transient experiments.

METHODS

An osmotic weight transient experiment consists of the gravimetric measurement of the net transcapillary flow (J_v) induced by a step change in perfusate osmolarity. We based our experimental protocols on the basic experimental methodology developed by Vargas and Johnson (47), but used a new model (29) for the analysis of the record of weight changes. We used sensitivity analysis on preliminary data and parameters obtained from one heart to determine the model parameters identifiable from the data. We then optimized the model parameters governing transcapillary exchange using a larger data set of 111 transients on 12 different hearts.

Experimental Methods

Our standard protocol was to induce weight transients in the isolated Ringer-perfused rabbit heart using as the osmotic agent one of a series of hydrophilic solutes with molecular weights ranging between 58 (NaCl) and 68,000 (albumin). Steps were made in osmolarity, from control to a hyperosmotic solution. The step durations were 4 to 30 minutes, using longer transients for larger solutes. The perfusate was then switched back to control for a reequilibration period that lasted at least as long as the original transient. Typically, four to five test solutes were used in one experiment. For each solute, one to three transients were recorded; the total time for one experiment was three to four hours (Figure 1). In a second protocol, we conducted a set of transients using sucrose as the osmotic agent while varying the flow through a range from 1.75 to 3.9 mL min⁻¹ g⁻¹. Details of the experiment are given below regarding: (1) the standard and osmotically-active Ringer’s solutions, (2) the perfusion system, (3) surgical procedures, and (4) the data recording devices.

Fig. 1.

Weight transient responses to osmolarity steps (Expt. 121202). Timing for step changes in osmolarity are indicated at the top. The discontinuity in weight at the beginning of the albumin transient is an artifact arising from the increased perfusate viscosity which causes an increased aortic volume due to an elevated aortic pressure. Prior to t = 0 there was an equilibration period in which weight rose to a steady level.

Solutions

Physiological Krebs-Ringer perfusates were prepared fresh each morning prior to the experiments. The control perfusate was (in mM): NaCl 118, KCl 3.8, KH₂PO₄ 1.2, MgSO₄ 0.7, CaCl₂ 2.1, NaHCO₃ 25, EDTA 0.1, dextrose 10, pyruvate 5.5, bovine serum albumin 0.015 (all from Sigma Chemical Co.). Perfusate osmolarity was 280 2 mOsmol. Papaverine at 5 mg/L was also used in all solutions to ensure vasodilation of the preparation. Osmotic test solutions were identical to the control except for the presence of one additional solute that served as the osmotic agent during the experiment. Test solute concentrations added to the perfusate concentrations given above were, unless otherwise noted, urea 30 mM, NaCl 20 mM, glucose 30 mM, sucrose 30 mM, raffinose 30 mM, inulin 5 mM, and albumin 0.5 mM. Solution osmolarity was validated using a freezing-point osmometer for the small solutes (Osmette A, Precision Systems, Inc.). For the albumin test solution the concentration, C, was measured by protein refractometry (accurate to about ± 0.2 g/l) and the osmolarity calculated by the formula π = 0.345 C + 0.002657 C² + 2.26 × 10⁻⁵ C³ from McDonald and Levick (34). The increase in osmolarity due to the additions of the test solute were (in the same order): 30, 38, 30, 30, 30, 5 and 0.5 mOsmol. All solutions were filtered through a 1.2 μm filter prior to use.

The albumin was purified by at least 72 hours of dialysis at 4 °C using a membrane with a molecular cut-off weight of 12,000-14,000 daltons (Spectra/Por, Spectrum Laboratories, Inc.) to remove small vasoactive compounds and control the ionic composition of the solution. The dialysis buffer was a physiological salt solution containing the same concentrations of NaCl, KCl, KH₂PO₄, MgSO₄, and CaCl₂ as in the standard Ringer’s solution, and was changed at least 3 times during dialysis. The volume volume of the buffer was at least nine times that of the albumin solution. The final concentration of albumin in the dialyzed solution was determined by refractometry; it was typically about 100 g/L with a total osmolarity about 2 mOsmol higher than the buffer. This solution was then diluted to the experimental concentration of 34 g/L on the day of the experiment. Inulin was similarly purified by dialysis against deionized water using a membrane with a molecular cut-off weight of 2000 D (Spectra/Por, Spectrum Laboratories, Inc.).

Perfusion System

A perfusion system (Figure 2) was developed to allow rapid switching between control and test solutions. Solutions were circulated continually through membrane oxygenators (Medtronic Minimax Plus) using a 95% O₂ 5% CO₂ gas mixture and a 10 μm in-line filter. A constant-flow perfusion pump (Gilson Minipuls II) was used to send fluid through a heat exchanger, consisting of a stainless steel tubing coil submersed in a temperature controlled water bath and a temperature controlled windkessel on its way to the heart. Two identical perfusion lines were kept running continuously. A switching valve located about 0.1 mL before the heart ensured a sharp change between the lines.

Fig. 2.

Isolated perfused heart system with continuous measurement of weight. Left panel: retrograde aortic flow. Coronary flow is the sum of pulmonary arterial flow and right ventricular drainage. The LV vent flow was small, and considered to be leakage across the aortic valve. Right panel: A dual perfusion system allows for rapid switching between different perfusates over the course of an experiment. The oxygenation circuit is shown for only one of the perfusates.

Isolated heart preparation

Adult female New Zealand White rabbits (2-3 kg) were sedated with acepromazine (1 mg/kg, subcutaneously) 20 to 30 minutes before surgical anesthesia was induced by intramuscular ketamine (40 mg/kg) and xylazine (5 mg/kg).¹ After tracheotomy and starting mechanical ventilation, the chest was opened with a midline incision through the sternum and a ligature was placed around the aorta. The rabbit was then given 300 U of heparin through the marginal ear vein. After 2 minutes, the aorta was occluded with the ligature to prevent air from entering the coronary arteries. The heart was removed immediately and temporarily immersed in iced Ringer’s solution until contractions ceased. It was then perfused retrogradely through the aorta while suspended from a T-shaped cannula (Figure 2). The time from excision to beginning perfusion was typically less than a minute.

The isolated heart was trimmed of excess fat and other tissue. To drain the LV cavity of leakage across the aortic valve a cannula was inserted through the thin apical myocardium (introducing it via a pulmonary vein). The right ventricle was drained similarly (introducing the cannula via the pulmonary artery). Coronary flow was taken to be the RV drain flow, and was checked by calculation of perfusion pump flow minus LV drain flow. Preparations in which flow from the left ventricular drain comprised more than 5% of the total flow were discarded because they likely had a damaged aortic valve. The heart was paced with a stimulator (Harvard Apparatus Co.) a constant rate of about 150 bpm, and perfusion rate was set at approximately 20 mL/min (2 to 4 mL min⁻¹ g⁻¹).

After isolation, the heart was equilibrated for at least 30 minutes. During this time the heart gained 1 to 2 g due to the low colloid osmotic pressure of the perfusate, but reached a steady baseline weight by the end of the equilibration period. At the end of the equilibration period, a series of switches was made between the two perfusion lines, both of which contained the control solution at the same flow. By adjusting height and the gage of fine needles at the end of the return line, the pressures and resistances of the two perfusion lines were equalized, so that switching from one to the other caused a pressure jump of less than 1 mm Hg. One perfusion line was then changed to the test solution and the experimental protocol begun.

Data recording

Heart weight and aortic perfusion pressure were recorded from a force transducer (Transducer Techniques) and pressure transducer (Statham). The transducer outputs were amplified using a custom-built amplifier and the weight record was filtered by an analog RC filter with a time constant of 0.2 sec. to remove high frequency noise. Signals were digitized and acquired by a Macintosh Power Mac 7100 running LabVIEW4 (National Instruments) at 250 Hz. Data points were reduced to one per second by taking the average of each 250 beats. Both weight and pressure signals were recorded continuously from the beginning of perfusion to the completion of the experiment. At the conclusion of all experiments, the heart was quickly removed from the experimental equipment, blotted, weighed, and then sectioned and dried at 100 °C to a constant dry weight.

Analytical Techniques

The analysis of these experiments using sensitivity analysis and parameter optimization was based on a novel model of microcirculatory solute and water exchange. All analysis was performed on a LINUX workstation using the XSIM modeling environment. The XSIM simulation interface and the model described next can be downloaded via http://nsr.bioeng.washington.edu/Software/DEMO, under “Blood–Tissue Exchange Models: Whole-organ models: osmotic”.

The model

The model used has been described in detail by Kellen and Bassingthwaighte (29) and is summarized in the Appendix. The standard parameters were adjusted to match the conditions of each experiment. Before the start of the experiment the model’s flow and aortic pressure were set to the values measured experimentally; venous outflow and lymph back-pressure were at atmospheric pressure. From the measured dry weight of a heart we used the values of water content and of vascular, interstitial, and cellular regions for in vivo rabbit hearts (17) to calculate initial values for plasma volume, V_p, interstitial space, V_isf, and cell space, V_pc. To correct for edema we assumed that background weight gained during the equilibration period and later in the experiment was due exclusively to fluid accumulation in the interstitium: V_isf was increased so that model heart weight W(t) matched the weight measured experimentally.

Sensitivity analysis

Sensitivity analysis provides insight into complex models and is an aid to experimental design. The sensitivity functions, defined as the instantaneous fractional change in a model output (in this case organ weight, W) induced per fractional change in a model parameter, P, were approximated by comparing model solutions produced with a one-percent increase in the parameter values. Relative sensitivities ((∂W/W)/(∂P/P) or ∂ln W/∂ lnP) rather than absolute sensitivities (∂W/∂P) were used to provide dimensionless sensitivity functions. Since the sensitivity functions are dependent on the parameter values (8), sensitivity analysis was performed after an initial manual fit to a small number of preliminary data sets. The sensitivity curves shown in the Results section were generated using the mean parameter values obtained from optimization against the full data set, but do not differ significantly from the preliminary results used for experimental design.

Parameter estimation

Optimization of model fits to data was used to determine three model parameters: (1) L_p,endo, the hydraulic conductivity through the transendothelial pathway, (2) A_sp/S, the fractional small pore area, and (3) r_sp, the small pore radius. The estimates of these parameters are not sensitive to the parameter values for the large pore system, A_lp/S, which was set at 10⁻⁶, and r_lp, which was set at 24 nm. These values were determined from the observations of lymph to plasma concentration ratios in the heart (32,38) and assumed to apply to our hearts. The complete weight record including both the switch to the test solution and the return to control were used as the data for each optimization procedure. Optimization was performed using SENSOP (11), which uses sensitivity functions to optimize parameter sets and provide confidence ranges from the covariance matrix. Local minima in the sum of squares of differences between model solutions and data were not observed; this is to be expected when there are many data points per weight transient (6 to 30 minutes per transient, at one sample per second is 360 to 1800 samples per parameter optimization run) and few free parameters (a maximum of three). The parameters for capillary and parenchymal cell volumes V_p and V_pc were those from (19). The capillary surface area S is 500 cm² g⁻¹ (5).

The three free parameters are interdependent to some extent, and require using the full extent of the weight transients, and the full set of solutes. The larger solutes have the greatest influence on estimates of r_sp, since for small solutes the σ_sp is close to zero. Consequently, it is from the overall dw/dt = SJ_V that an overall L_p is calculated, so defining L_p,endo through the iterative parameter adjustments to account for the observed balances of solute and water fluxes that fit the rate of water flux and dissipation of solute gradients.

Calculation of derived parameters

The model parameters optimized to fit the data are explicitly related to the phenomenological coefficients of Kedem and Katchalsky (28), namely solute permeabilities, P, and reflection coefficients, σ, through the solute effective radius r_s and the pore radius r_p (either r_sp or r_lp) and the total pore areas. The equations governing this relationship are given in the Appendix (A11 to A13).

RESULTS

We obtained estimates of the parameters L_p,endo, A_sp/S, and r_sp from a total of 111 high-quality osmotic weight transient data sets in 12 different Ringer-perfused rabbit hearts. Coronary flows during the experiment ranged from 1.5 to 4 mL min⁻¹ g⁻¹; aortic pressures were typically between 20 and 30 mmHg in these vasodilated hearts. Heart weights at the conclusion of the experiments were 7.5 to 10 g, and 82-87% water, compared to initial weights of 5.5-6.5 g estimated from the final dry weight and the known wet-to-dry ratio of freshly excised rabbit heart (17). Edema during the experiments was therefore substantial, but, most of this fluid gain occurred during the initial 20-30 minute equilibration period; the baseline weight was relatively stable over the time of a single transient, but a gradual weight gain occurred throughout the duration of the complete experimental period, such as can be seen in Figure 1.

Parameter Estimates

The means of the parameter estimates obtained from the complete set of osmotic weight transient data are presented in Table 1. The endothelial pathway for transcapillary water-only exchange accounts for about 28% of the total transcapillary hydraulic conductivity, the large pore pathway accounts for 5% of the hydraulic conductivity, and the majority, 67%, is via the small pore pathway. The small pore hydraulic conductivity of 4.4 × 10⁻⁸ cm s⁻¹ mmHg⁻¹ is through a small fraction of the total capillary surface area, A_sp/S = 2.2 × 10⁻⁴, equivalent to a pore area per unit path length of 4.4 cm⁻¹. The small pore has an effective radius of approximately 6.9 nm, almost one third of that of the large pore, assumed to be 24 nm in radius from prior model fits to lymph sampling data (32, 38). The coefficient of variation for individual parameter estimates was only around 1%, which is substantially smaller than the differences between different data sets. Table 1 therefore treats each transient as providing an independent set of parameter estimates, and gives the mean and the standard deviation of the set of parameter estimates.

Table 1.

Summary of capillary pathway parameter estimates

Pathway	Ap/S	rpk, nm	Lpk, cm·s−1·mmHg−1	Lpk/Lp,total
Endothelial k = 1	0.9998	Aquaporin channel	Lp,endo 1.8±0.6×10−8	0.28
Small pore k = 2	2.2×10−4±0.68×10−4	6.9±0.16	Lp,sp 4.4±1.3×10−8	0.67
Large pore k = 3	1×10−6 (from Refs. 32 and 38)	24 (from Refs. 32 and 38)	Lp,lp 3.3×10−9	0.05

Values are means ± SD. A_p/S, fractional pore area; r_p, pore radius; L_p, capillary hydraulic conductivity; L_p,endo, capillary hydraulic conductivity through transendothelial pathway; L_p,sp, small-pore capillary hydraulic conductivity; L_p,lp, large-pore capillary hydraulic conductivity. L_p,total = 6.5±1.5×10⁻⁸. With mean capillary pressure $\left({\stackrel{‒}{P}}_{\mathrm{cap}}\right)=5.0\mathrm{mmHg}$, mean interstitial fluid pressure $\left({\stackrel{‒}{P}}_{\mathrm{isf}}\right)=4.6\mathrm{mmHg}$, and with lymphatic pressure (p_lymph) = 0, the capillary-to-lymph flux was 0.003 ml·g⁻¹·min⁻¹.

Table 2 shows the parameter estimates obtained using each osmotic agent. Similar estimates were obtained regardless of the solute used, with the most significant disparity being the difference between small pore radius by inulin vs. albumin osmotic transients. Each individual transient could not be used to obtain estimates of all three free parameters, either because there was too much correlation between pairs of free parameters, or because there was too little sensitivity of the weight response to the values of the parameter (see discussion). When a given parameter could not be determined for a transient induced by a given osmotic agent (indicated by an * in Table 2), the average estimate obtained from solutions using other osmotic agents was used.

Table 2.

Parameter estimates by solute

Osmotic Agent	No. of Transients	No. of Hearts	Asp/S	rsp, nm	Lp,endo×108, cm·s−1·mmHg−1
20 mM NaCl	23	8	2.19×10−4*	6.9*	1.51±0.36
30 mM urea	15	5	2.19×10−4*	6.9*	2.17±0.97
30 mM glucose	11	4	2.0±0.59×10−4	6.9*	1.67±0.43
30 mM sucrose	34	8	2.0±0.77×10−4	6.9*	1.75±0.52
30 mM raffinose	15	6	2.4±0.4×10−4	6.9*	2.27±0.48
5 mM inulin	5	3	1.8±0.3×10−4	5.20±0.26	1.45±0.49
0.5 mM albumin	8	7	2.7±0.94×10−4	8.24±1.54	1.8*

Values are means ± SD.

^*Not estimated by this solute; simulations used aggregate values.

These parameter estimates can be used to calculate solute permeabilities and reflection coefficients for the solutes through each pathway (Table 3). Solute permeabilities for small solutes (NaCl through raffinose) are dominated by diffusion through the small pore, and are almost proportional to their free molecular diffusion coefficients in water, although there is the expected consistent decrease in the permeability to diffusion coefficient ratio (P/D) from about 3.8 to 3.0 cm⁻¹. Inulin and albumin, being much larger, are significantly hindered compared to the other solutes, and have smaller P/D and larger σ_sp.

Table 3.

Solute Phenomenological Transport Parameters

Solute	${\mathrm{D}}_{\mathrm{w}}^{\mathrm{o}}$ (cm2/sec)	rs (Å)	σ sp	Psp (cm/s)	σ lp	Plp (cm/s)	(Psp+Plp)/D cm−1	σ d
NaCl	1.72×10−5	2.5	0.0038	6.4 × 10−5	0.0003	3.3×10−7	3.72	0.28
Urea	1.50×10−5	2.2	0.0029	5.7×10−5	0.0002	2.9×10−7	3.80	0.28
Glucose	7.79×10−6	4.4	0.012	2.6×10−5	0.0008	1.5×10−7	3.33	0.29
Sucrose	6.03×10−6	5.2	0.017	1.9×10−5	0.0010	1.1×10−7	3.15	0.29
Raffinose	5.02×10−6	6.0	0.022	1.5×10−5	0.0014	9.2×10−8	2.99	0.29
Inulin	1.85×10−6	16.6	0.17	2.5×10−6	0.011	2.8×10−8	1.35	0.39
Albumin	7.87×10−7	36	0.62	1.4×10−6	0.052	8.8×10−8	1.78	0.70

${\mathrm{D}}_{\mathrm{w}}^{\mathrm{o}}$ = diffusion coefficients in water at 37C. Equations for σ_sp, σ_lp, P_sp, P_lp, and σ_d are A12, A13, and A16 in the Appendix.

Fitting the model to data on W(t)

A representative fit to an osmotic weight transient experiment on an isolated rabbit heart is shown in Figure 3. Following a step increase in perfusate osmolarity using 20 mM NaCl, there is an initial rapid loss of water from the interstitium and cells to the plasma as osmotic equilibrium is restored. This rapid shift of water diminishes the osmotic pressure gradient in the test solute, and also creates opposing concentration gradients in the resident solutes previously at equilibrium (osmotic buffering). Interstitial hydrostatic pressure falls as V_isf decreases. This is followed by a slower phase where the solute enters the interstitium to partially dissipate its transcapillary concentration gradient, and mechanical elastic and secondary osmotic forces act to partially restore interstitial volume. Water loss from cardiomyocytes is small, and the rise of perfusate osmolarity from 280 to 300 mOsm would result in a steady-state loss of only 7% of cell water even if the cell was a flaccid bag, and slightly less if there is resistance to deformation. A steady-state interstitial-to-plasma concentration ratio is achieved when the rate of solute entry into the interstitium is equal to the rate of removal by lymph.

Fig. 3.

Weight transient response to an osmotic step increase of 38 mOsm due to 20 mM NaCl. Flow was 13 mL/min, aortic pressure was 24 mmHg, the initial heart weight was 8.6 g. The basic parameter set was adjusted by increasing d_myo to 3 μm and decreasing interstitial collagen and interstitial matrix (29) Q_col and Q_im to 0.021 and 5.4×10⁻⁴ g/mL respectively to account for edema present in the experimental preparation. Panel A: Fit of model (solid line) to organ weight data. Panel B: Predicted volume loss from interstitial space and parenchymal (myocardial) cells. Panel C: Predicted time course of NaCl concentration at upstream (x = 0) and downstream (x = L) ends of capillary and ISF. Panel D: Time course of concentration of resident solutes in the capillary and ISF (line types as defined for panel C). The capillary transit time was 0.75 seconds: the sharp downward deflection in the resident solute concentration in the capillary outflow, C_cap(x = L), at t = 110 s is due to the water flux from ISF into capillary during capillary transit.

The time course of weight change is substantially different for larger solutes like albumin (Figure 4). Because only 0.5 mM albumin was used, the initial rate of weight loss is less than for NaCl, despite albumin’s higher reflection coefficient. However, the duration of the weight transient is longer because transcapillary concentration gradients persist as the test solute penetrates the interstitium only very slowly. Osmotic buffering effects are minimal because the low J_v causes only a negligible transcapillary gradient in the resident solutes. Because of the low J_v, and the low σ for the small solutes, and the very low P for albumin, the J_v is of water and solute, so the Δπ×_alb persists for a long time and causes a water shift that is large.

Fig. 4.

Response to 0.5 mM albumin in the same heart as in Figure 3, with the same control conditions. Panel A: The rate of weight loss is lower, but longer-lasting than with smaller solutes. The discontinuities at the beginning and end of the transient in the weight record are experimental artifacts arising from the differences in control and test solution viscosity and consequent aortic pressure and volume steps. Panel B: Predicted volume loss from interstitial and parenchymal cells. There is at t = 28 min a diminution of myocardial cell volume of 0.5/280, although this is too small to show on the graph. Panel C: Albumin penetration of the interstitium which is through the large pore system (with a uniform distribution of pore locations). Panel D: There is only a minimal rise in interstitial resident solute concentration.

Sensitivity Analysis

Sensitivity analysis indicated that analysis of transients over a range of molecular weights for NaCl to albumin was needed to provide estimates of the small pore system parameters A_sp/S and r_sp, and the hydraulic conductivity (L_p,endo) of the transendothelial pathway. Data spanning a wide range of molecular sizes is crucial to getting reasonable estimates, as there is no single solute size at which all three parameters have substantial and distinctly different sensitivity functions (Figure 5).

Fig. 5.

Sensitivity functions ∂ln W/∂ln (parameter value) for the three fitting parameters A_sp/(SΔr), r_sp, and L_p,endo, for osmotic transients induced by 20 mM NaCl (= 38 mOsm Δπ) (panel A), sucrose (panel B), inulin (panel C), and albumin (panel D). The weight response to cases A and D are given in Figure 3 and Figure 4 respectively.

The values of the large pore parameters cannot be resolved by osmotic transient data because the sensitivity functions ∂lnW/∂lnr_lp and ∂lnW/∂ln A_lp with NaCl, sucrose, inulin, and albumin as the osmotic agent are all nearly zero over the time course of an osmotic transient experiment, therefore these parameters are estimated from lymph/plasma concentration ratios for a set of solutes (32,38).

Osmotic Transients at Variable Flows

We found that flow had no effect on the parameter estimates produced by our analysis. Osmotic transients with 30 mM sucrose as the osmotic agent produced weight responses with an increasing slope as flow increased from 1.75 to 3.9 mL min⁻¹ g⁻¹ (Figure 6). This trend was consistently observed over 22 transients in 2 different hearts. The parameters estimated from these data sets did not have a variability higher than that observed in the standard study, and showed no apparent trend as a function of flow.

Fig. 6.

Effect of perfusate flow F_p on the form of the initial part of the weight transient in response to a step in perfusate of 30 mM sucrose. Increasing initial rate of weight loss. The figure shows the model fits the original data obtained at different flows in the same heart. Original data are noisy, as shown in Figures 3 and 4, and differences are more difficult to discern. Organ weight was around 7 g, and flows ranged from 1.8 to 3.9 mL min⁻¹ g⁻¹. The tails of the curves do not equilibrate at the same total weight because the volume of the interstitium relative to the other compartments changes from transient to transient.

DISCUSSION

The use of an anatomically detailed model of the coronary microcirculation has enabled us to design osmotic weight transient protocols that are effective in revealing the physiological parameters. The osmotic transient methods provides a means of distinguishing and estimating the parameters governing exchange through the small pore and transendothelial pathways. The results provide for a more complete description of the exchange process than has previously been obtained in the heart.

Design of the Experiment

Sensitivity analysis provides a tool to develop strategies for parameter optimization. When applied to our osmotic transient protocol, sensitivity analysis suggested a means to estimate the parameters L_p,endo, r_sp, and A_sp/S. The parameters A_sp/S and r_sp can be determined from albumin transients since the shapes of ∂lnW/∂ln A_sp and ∂lnW/∂lnr_sp differ greatly, while ∂lnW/∂ln L_p,endo and the flux across endothelial cells are small. L_p,endo must then be determined from NaCl transients, holding A_sp/S fixed. Given that the model is an appropriate descriptor, midsized solutes provide redundant information, and using their known molecular sizes, the fits to their transients should only require minor adjustments of parameter estimates.

This strategy makes sense given our understanding of the microcirculatory exchange model. Small hydrophilic solutes (MW < 1,000 daltons) only induce a significant J_v, through the transendothelial pathway, where it can be assumed they have a σ equal to one because aquaporin channels and cell membranes exclude these solutes. While J_v is proportional to L_p,endo, the osmotic driving force for fluid exchange is dissipated as the test solute permeates through the pores into the interstitium, a flux which is predominantly diffusive and proportional to A_sp/S. As a first approximation, the magnitude of the weight transient is governed by the ratio of these processes, or L_{p endo} to A_sp/S. As solutes become larger, sensitivity to L_p,endo per unit osmolarity change stays the same; experimentally, because smaller osmotic step changes are used for larger solutes, the absolute magnitude of the maximum weight is usually smaller. For any specific A_sp/S, increasing solute size, and therefore increasing σ, reduces the rate of solute permeation into the interstitium, and increases water flux out of the interstitium. For solutes larger than NaCl the sensitivity to A_sp/S is initially negative (increasing small pore area results in an increased water efflux). Increasing pore size, r_sp, leads to lower σ_sp, increased water flux and reduced solute permeation. The sensitivity to r_sp is greater the larger the solute.

Sensitivity analysis shows that the parameters governing transport through the large pores cannot be determined from the experimental data: even the largest solutes have a σ near 0 for the large pores. Thus, varying the osmolarity with any solute fails to affect J_v,lp or influence the pattern of weight change. Although permeation of large solutes through the large pore path is a significant fraction of the total diffusive permeability, the tail of the weight transient response is also independent of the parameterization of the large pore pathway because their area is so small that flux of large solutes into the interstitium is too slow to affect the osmotic driving force, even late in the osmotic transient. Weight loss, −dW/dt, goes to zero when interstitial hydrostatic pressure p_isf decreases and interstitial osmotic pressure π_isf (exerted by the permanent protein components of the interstitial matrix) rises to balance the increased osmotic pressure in the capillaries, that is, Δp – Δπ goes to zero. This result is dependent on the assumption that the large pore pathway is in fact in the neighborhood of A_lp/S = 10⁻⁶ and r_lp = 24 nm or larger. Our model simulations show that a putative large pore pathway would not make a measurable (>5%) contribution to the total osmotically induced water flux by albumin unless r_lp was made smaller and there were more pores. Transients would be affected if A_lp/S were as large as 5×10⁻⁶ and r_lp were as small as 12 nm, but these values are well outside of the range compatible with the lymph to plasma concentration ratios of large plasma proteins (32 , 38).

Interpretation of the weight transient curve

Bassingthwaighte and Grabowski (19) and Vargas and Johnson (47,48) used the initial slope of the osmotic transient weight response, dW/dt = J_v = σL_pSRT ΔC , to estimate the solute reflection coefficient. Since they assumed coupled transport of water and solute occurred through a single pathway, they estimated relatively narrow pore dimensions by hydrodynamic theory. In an actual heteroporous capillary, this measurement provides σ_d, the osmotic reflection coefficient for the membrane as a whole,. By itself, the apparent σ_d for a heterogeneous membrane is insufficient to provide a complete description of exchange kinetics. Realizing this, Pappenheimer in 1969 (36) proposed that about 50% of transcapillary water exchange occurs through a water-only pathway in order to explain early data on hydraulic conductivity and solute permeability in skeletal muscle measured by his isogravimetric technique (35) and the indicator-dilution work of Alvarez and Yudilevich (3). More recently, Watson (53) and Wolf (54-57) have proposed three pathway pore models for solute exchange in mammalian skeletal muscle, with 41% of steady-state flow through a water-only pathway, 17% through a large-pore pathway of radius 28.5 nm, and 42% through a small-pore pathway of radius 4.57 nm. The high reflection coefficients obtained from osmotic transient experiments can then be understood as an averaging between a transendothelial pathway which excludes all solutes (giving them a σ of 1), and relatively unrestrictive aqueous pathways with σ close to 0 for at least the smaller hydrophilic solutes.

Here, we use a model that explicitly accounts for the fluxes across each transcapillary pathway to directly estimate the parameters governing exchange through each. This approach allows us to extract more information from an osmotic weight transient curve than the simple Vargas and Johnson analysis. On the other hand, we have not taken into account many of the details of interstitial transport, e.g., differential protein velocities due to convection combined with molecular exclusion (see 33), and therefore might not represent the transit times of proteins from blood to lymph accurately. In a system with two or more pathways, the distribution of fluxes changes during the course of an osmotic transient experiment (Figure 7), as appreciated by Rippe and Haraldsson (40). A step increase in capillary osmotic pressure induces water flux only through pathways that have a high σ for the test solute. For small solutes like NaCl, the only pathway with a non-negligible σ is the transcellular pathway. Regardless of its fraction of total hydraulic conductivity, an increase in capillary NaCl concentration draws water out of the tissues predominately across the endothelial cells and out of parenchymal cells, reduces p_isf and diminishes lymph flow, J_V,L, as shown in the right panels of Fig. 7. Consequently the apparent σ of the capillary wall at early times in an osmotic transient is larger than later, because the fraction of J_v through the transendothelial cellular pathway is greater. In contrast, albumin can induce osmotically-driven fluid movements through the small pore system, as well as across the endothelial cells. In contrast, changes in hydrostatic pressures, commonly used in lymph sampling studies produce the same change in driving force for all pathways, and thus σ_apparent approaches a constant value at high J_v, when solute permeation, J_s << J_v × C_s, where J_s is solute flux and J_v × C_s is unhindered, expected solute flux.

Fig. 7.

Left panel: The steady-state flux distribution was 81% through the small pore, 4% through the large pore, and 15% through the transendothelial path. Total steady-state transcapillary fluid flux (equal to lymph flow at steady-state) was 0.0030 ml g⁻¹ min⁻¹. These model predictions are substantially different than those given in (29) because of the lack of a colloid osmotic pressure in our preparation. Right upper panel: Flux magnitudes during an osmotic transient due to 30 mM glucose relative to their steady-state levels. The initial peak transendothelial flux was opposite to and almost 800 times larger than the steady-state J_v,endo of 0.0024 ml g⁻¹ min⁻¹. Note that lymph flow changes also. Right lower panel: Responses to a 1 mM albumin transient.

We investigated the significance of explicitly modeling fluxes through each pathway individually by fitting model results generated by the three-path model with a single-path model that lumped all transcapillary exchange into a single effective P and σ. The osmotic transient results presented in Figure 3 are typical of small solutes, and can be closely matched using a parametrically reduced form of the model with a single lumped pathway for transcapillary exchange. The estimated NaCl permeability by the reduced model is 1×10⁻⁴ cm/s, 50% higher than the multipath model estimate (6.4 × 10⁻⁵ cm/s). Also, σ of the single-path model is 0.15, assuming the L_p of this single “effective” pathway is equal to the sums of the L_p’s of the three-path description. The higher apparent solute permeability predicted by the single-path model at least partially explains the historic discrepancy between indicator dilution and osmotic transient measurements of small solute permeability. The actual permeability estimates obtained by Vargas and Johnson are higher than modern estimates even accounting for this error, presumably because they used no albumin in their perfusate. A single path reflection coefficient of 0.15 is incompatible with the observed lymph-to-plasma concentration ratio of unity for small solutes at even the highest filtration rates. It also implies a channel width of about 3 nm, incompatible with MID and electron microscopic observations.

Parameter Values

Our estimates of the general parameters governing exchange in the microvasculature of the heart are compatible with previous results. Our total L_p, L_ptotal, of 6.5 × 10⁻⁸ cm s⁻¹ mmHg⁻¹ (Table 1) is two-thirds of the Vargas and Johnson (47) estimate of 1.0×10⁻⁷ cm s⁻¹ mmHg⁻¹ in isolated Ringer-perfused rabbit heart. The most likely reason for our lower estimate is that Vargas and Johnson used a completely protein-free Ringer’s solution, which artificially increased transcapillary hydraulic conductivity. A novel result is our estimation that 28% of the L_p is contributed by the transendothelial pathway. This value is greater than the 5-10% estimated in frog mesentery (13), but less than the 40% predicted by Wolf (55, 56) in skeletal muscle. Our estimates of σ_d for small solutes are just over 0.28, a consequence of a reflection coefficient of 1 in the transcellular pathway, and a reflection coefficient close to zero through the large and small pores. For albumin, our estimate of σ_d = 0.7 is higher than σ_f, = 0.59 estimated from filtration-rate independent C_L/C_p measurements (38), in line with theoretical expectations (14, 40), but it is lower than the 0.80-0.87 σ_d obtained in other organs (57). Since permeabilities are known to be artificially elevated in the Ringer-perfused heart, it is possible that the reflection coefficient is slightly higher than our value in vivo.

The solute permeabilities obtained in this study are also consistent with estimates obtained from indicator dilution studies in Ringer-perfused rabbit and guinea pig heart preparations (43). Solute permeability is proportional to A_sp/(SΔr). A_sp is the cleft or pore surface area, S the capillary surface area, and Δr is the length of the pathway from capillary lumen to interstitium. These terms are kinetically inseparable. In anatomic studies, S is about 500 cm²/g (5), and in the geometry of our model is 487 cm²/g (29). From electron microscopy the distance Δr is about 0.5 μm; thus an estimate of A_sp/(SΔr) of 4.4 cm⁻¹ is equivalent to A_sp/S of 0.022% and Asp of 0.11 cm²/g.

For translating from the parameters of our model the traditional PS (in ml g⁻¹ min⁻¹) for glucose would be

$$\begin{array}{cc}\hfill \mathrm{PS}& =\frac{D{\mathrm{cm}}^2∕\mathrm{s}}{\Delta r\mathrm{cm}}\cdot \frac{(A_{\mathrm{sp}}+A_{\mathrm{lp}})}{S}\cdot S\left({\mathrm{cm}}^2{\mathrm{g}}^{-1}\right)\cdot 60\mathrm{s}∕\min \hfill \\ \hfill & =\frac{0.8\times {10}^{-5}}{0.5\times {10}^{-4}}\cdot 2.2\times {10}^{-4}\cdot 500\cdot 60\hfill \\ \hfill & =1.1\mathrm{ml}{\mathrm{g}}^{-1}{\min }^{-1}\hfill \end{array}$$ (1)

For glucose, multiple indicator dilution methods in Ringer-perfused hearts (rabbits, except where specified otherwise) gave PS equal to 2.3 ± 0.7 (6), 1.2 ± 0.16 (31), and 2.3 ± 0.3 in guinea pig (30). For sucrose, indicator dilution estimates were 2.3 ± 0.9 (18), 2.0 ± 0.42 (43), 1.9 ± 0.6 in guinea pig (44). Rather high values of 5.1 ± 1.4 ml g⁻¹ min⁻¹ (10) were obtained in rat hearts using albumin-free perfusate and are therefore not directly comparable to our studies reported here. Permeabilities are lower normally in vivo. In blood-perfused dog hearts the PS’s are lower: for glucose, PS_c = 0.3 ± 0.1 ml g⁻¹ min⁻¹ (3, 59); for sucrose, PS = 0.24 ± 0.07 ml g⁻¹ min⁻¹ (3), 0.22 ± 0.1 ml g⁻¹ min⁻¹ (59) and 0.26 ± 0.02 ml g⁻¹ min⁻¹ (22).

Our estimates of A_sp/S are compatible with those of Guller et al. (21); for NaCl they used an intrapore diffusion coefficient D = 5 × 10⁻⁶ cm²/s, taken from studies of Na diffusion in extracellular space (46, 42) on the basis that the diffusion in the cleft between endothelial cells is reduced by the presence of glycocalyx. Their values of D within the cleft are probably more nearly correct than assuming one equivalent to that in water, as we did for Table 3. Estimating the total length of interendothelial cleft per unit capillary surface area as did Guller et al. (21) would suggest a total A_sp/S of 2.25 × 10⁻³, given 9 nm width. In accordance with Guller et al. (21), since our functional estimate of A_p/S was only 2 × 10⁻⁴, we would conclude that less than 10% of the cleft is open, and that diffusion through the other 90% is blocked by interendothelial gap junctions (2, 9).

Flow Effects, An Artifact of Compartmental Analysis

Investigators using the osmotic transient approach observed that flow had an effect on the weight response. Our results confirmed the observation that the initial rate of weight loss during an osmotic response increases with increasing flow. Consequently, when compartmental analysis techniques are used to obtain reflection coefficients, σ_d appears to increase with increasing flow rates (54). This is because the Vargas and Johnson (47) estimate of reflection coefficient is based on the prediction that the initial volume flux is proportional to a step change in concentration along the whole length of the capillary. Because in actuality the concentration step travels along the capillary with the fluid velocity and is being dissipated as it travels, the initial water fluxes are smaller in reality than the compartmental model assumes, with the result that the compartmental model underestimates the apparent σ_d.

Vargas and Johnson (47, 48) handled the flow-limitation problem by performing experiments at increasing flows until constant estimates of phenomenological coefficients were obtained. The three-compartment transcapillary exchange model of Bloom and Johnson (7) also neglected flow effects, which Vargas and Blackshear argued were small for their experimental methods (50). At a flow of 1 mL/s, an extremely high flow for a rabbit heart of around 6 grams, Johnson and Bloom (27) calculated that the reflection coefficient for NaCl, a solute with relatively high permeability, would still be underestimated by 16% using the basic Vargas and Johnson model. The most complete set of flow vs. σ data available from Wolf and Watson (54) show an apparently increasing σ without a plateau as flow increases, an artifact of using compartmental analysis.

In contrast, axially distributed models properly account for the sharper initial weight changes at higher flows and show steeper initial slopes of W(t) at higher flows (Figure 6). Estimates of L_p and σ are independent of flow when the axial concentration gradients are accounted for. The only previous axially-distributed capillary model applied to the osmotic transient perturbation, developed by Grabowski and Bassingthwaighte (19), accounted for axial concentration gradients in the capillaries although they did not examine the effect of flow on their model solutions.

The fundamental flaw in the compartmental representation is that an instantaneously mixed chamber cannot account for internal gradients or for the considerable time lag for the venous end of the capillary to respond to the step change at the capillary inlet (Figure 8). The concentration gradients in a long capillary develop not only because solute leaves the capillary as it moves downstream, but also because the influx of solute-free fluid into the capillary from the surrounding tissues dilutes test solute in the capillary as material moves downstream. This previously unrecognized mechanism for the establishment of axial concentration gradients is maximal during the initial phase of an osmotic transient experiments, independent of the solute’s PS/F, and is occurring when transcapillary water fluxes are the highest. Although our data, like those of Wolf and Watson (54), show an increasing initial rate of weight loss at higher flows, our axially-distributed model fits these observations over the whole range of flows without altering the parameters governing transcapillary exchange.

Fig. 8.

Capillary concentration of the osmotic agent (NaCl) as a function of time and location, with a fractional capillary position (x/l) of 0 corresponding to the arterial end of the capillary.

Errors and Limitations

Low oncotic pressure preparations

The isolated Ringer-perfused heart is a convenient preparation for osmotic transient studies, but differs from the in vivo blood-perfused heart. Ringer-perfused hearts beat less strongly than blood-perfused hearts and so develop more edema because of poorer squeezing of the ISF, and because of the lowered oncotic pressure of the perfusate. By the end of an experiment, hearts typically had water contents of 82 to 85%, as measured by wet-to-dry weight ratio, compared to in vivo values of 78 ± 1% (17, 58). Since total perfusate osmolarity is kept at normal levels, it seems likely that much of this change is due to an increased interstitial volume. In our analysis, we have assumed that all weight gain during the baseline equilibration period is caused by expansion of the interstitial fluid volume, almost a doubling of interstitial volume.

A problem with using low protein perfusates is that capillary permeability tends to rise (19, 24, 25), so the parameters estimated for the Ringer-perfused heart are not necessarily representative of the in vivo values. While we used a background level of 1 g/L albumin in all solutions to help maintain normal permeability properties, it is possible that other serum proteins are also necessary to maintain completely normal permeabilities. There is possibly some degradation of the capillary glycocalyx, which contributes some of the resistance to exchange. As interpreted by our pore theory model, such degradation would be observed by increases in both pore radius and pore area. Based on comparisons of multiple indicator dilution experiments in different heart preparations, we expect roughly a doubling of the pore area and an increase in effective radius form 5 to 7 nm when going from blood to Ringer-perfused preparations (29). Therefore, in the in vivo condition the transendothelial fraction of L_p is probably closer to 40 to 50% rather than the 28% we found in these experiments (Table 2).

Pressure changes during the experiment

Drops in arterial pressure are known to occur during an osmotic transient with a step increase in concentration at constant flow (19, 49, 50). If these pressure changes propagated to the capillaries they would partially offset the increase in perfusate osmolarity, leading to a smaller than expected change in the Starling forces for a measured rate of fluid exchange. However, in our constant-flow preparation, capillary pressures will be reduced only if capillary and postcapillary venular resistances are selectively reduced.

Coronary arterial smooth muscle is known to dilate in response to osmolarity changes (4, 51). To test the hypothesis that the pressure changes are indeed caused by arteriolar coronary dilation, we fully dilated the arterial vessels using papaverine, a smooth muscle relaxant, causing a reduction in perfusion pressure (data not shown). This intervention also kept capillary pressures low and partially offset the initial weight gain on the low oncotic pressure perfusate. In the presence of papaverine, the pressure transient following an osmotic change was diminished compared to control and typically less than 2 mmHg. This number represents an upper bound on the variation in hydrostatic pressure that could occur at the capillary level during an experiment.

Source of fluid loss

Gravimetric measurements can not distinguish whether the fluid lost from an organ originates from the cellular or interstitial spaces. Regardless of its original origin, the model predicts that this fluid is always hypotonic compared to interstitial fluid because at least a portion of the flux is pure water coming through the transendothelial pathway. The smaller the molecular size of the osmotic agent, the more hypotonic is the volume flux because it is localized more exclusively to the transcellular pathway. This prediction is in agreement with the observations of Effros (16). Our model predicts about 2/3 of the fluid loss during a small solute transient in the heart originates in the parenchymal cells. Wangensteen et al. (52) used morphometric measurements in lungs to show that most of the water extracted during a shift in perfusate osmolarity originated from the intracellular volumes. Morphometric or indicator dilution studies in heart to determine cellular and interstitial fluid volumes before and after an osmotic transient could provide additional validation of model predictions.

Potential Influence of Donnan forces on intracellular volumes

W. Stein (45), gives the Post-Jolly equations for Donnan distribution across the cell membrane. The general concept is that if there is a cation which leaks passively into the cells, it must be pumped out, otherwise the cell expands. The summarizing equation for cell volume is V = (A_i)/[Na]_e × (1 + k_leak/k_pump), where A_i is moles of intracellular protein, [Na]_e is extracellular sodium, and k_leak and k_pump are first order rate constants (Stein’s Eq. B7.1.6).

The consequence of this is that if neither k_leak, nor k_pump, nor [Na]_e change during an osmotic transient, the volume V is not affected. This is to say that V is controlled by the osmotic concentrations and not by Donnan effects. To examine this in more detail, one would see that a 30 mM sucrose step increase would cause cell shrinkage, increase intracellular Na_i by about 10%, decreasing the leak flux by 10% and increasing the pump flux by 10%, and creating a situation for reducing the intracellular Na content. But neither k_leak nor k_pump need to change for this to happen. (Adaptive cell behavior is commonly found, however; for example, k_leak changes [45, p. 276]). The time frame for adaptation by changes in leak and pump rates is not known for the heart. Lymphocytes adapt to a new steady state in about 20 minutes, though they never return to baseline (20). If cardiac cell adaptation is at the same rate this would not affect the interpretation of the responses to small solute molecules, but would influence our thinking about albumin-induced transients. More research is needed here.

Errors in Parameter Estimates

Much of the variation in parameter estimates in this study likely arose in the real variability of capillary permeability between hearts, and within the same heart over the course of a series of experiments. The coefficients of variation (standard deviation of parameter estimates divided by the mean parameter estimates) for L_p,endo, A_sp/S, and r_sp were 33, 32, and 23% respectively. In contrast, the confidence limits on the free parameter values determined by optimization were typically around 1% for fitting of our model to the individual experimental data sets. This is an underestimate of the true uncertainty in the parameter estimates because having only 3 free parameters in the optimization does not account for variation in other model parameters in estimating the uncertainty in the parameter estimates. The variation in parameter estimates varied by nearly as much over the course of a given heart experiment as they did in all twelve hearts. This was so even when the same osmotic agent was used at different times during the experiment. However, transients very close to each other in time produced quite similar parameter estimates. It is likely that the permeability of our preparations increased over the 3-4 hours of each experiment because of the effects of isolation and prolonged exposure to Ringer’s solution. The overall error in the study is comparable to that achieved by other methods for measuring capillary permeability properties in whole organs. The osmotic transient method has the advantage of providing more complete information on transport across the capillary wall than either the multiple indicator dilution or lymph sampling methods.

Conclusions

We have gained insight into the processes of water and solute across the capillary and cell membranes through extensive, carefully controlled experiments using the osmotic weight transient method in isolated perfused hearts, and quantitative analysis via a comprehensive and detailed model of the underlying physiology. By using this more precise model, we have extracted more information from the weight transient record than was previously possible. Our results suggest the small pore system is well-represented by a population of pores with radius of 6.9 ± 1.7 nm and a fractional pore area A_sp/S of 0.022 ± 0.007%. The size and density of the large pore pathway cannot be determined by osmotic transient methods, but lymph sampling data suggests it is likely around 24 nm in radius with an A_lp/S of about 0.0001%. There is a significant pathway for solute-free water exchange in myocardial microvessels, accounting for about a quarter of the transcapillary hydraulic conductivity. This measurement is too large to result from the permeability of water through pure lipid bilayers, so aquaporin water channels presumably play a role as water traverses both lumenal and ablumenal endothelial plasmalemma and myocyte sarcolemma. The analysis of osmotic transient data we have presented is consistent with our previous application of the same model to indicator dilution and steady-state lymph sampling data (29). Now for the first time all the various types of observations are brought into compatibility by a properly comprehensive model.

Appendix

The model used has been described in detail by Kellen and Bassingthwaighte (29). Briefly, it consists of an axially-distributed blood tissue exchange region in which fluid and solutes exchange between vascular, interstitial, and parenchymal cell volumes. The following equations describe the coupled transcapillary exchange of fluid (J_Vc) and solutes (J_s), lymphatic drainage of interstitial fluid (F_L), and water exchange across the parenchymal cell membrane (J_Vpc). These fluxes determine changes in the fluid volumes of interstitium (V_f,isf) and parenchymal cells (V_f,pc), perfusate velocities (u) in a constant-volume capillary, and solute quantities of N_s different solutes in all three regions (n_c,j, n_isf,j, and n_pc,j):

$$\frac{\partial u}{\partial x}=\frac{-S_{\mathrm{c}}{J}_{\mathrm{Vc}}}{V_{\mathrm{c}}},$$ (A1)

$$\frac{\partial V_{f,\mathrm{isf}}}{\partial t}=S_{\mathrm{c}}{J}_{\mathrm{Vc}}+S_{\mathrm{pc}}{J}_{\mathrm{Vpc}}-F_{\mathrm{L}},$$ (A2)

$$\frac{\partial V_{f,\mathrm{pc}}}{\partial t}=-S_{\mathrm{pc}}{J}_{\mathrm{Vpc}},$$ (A3)

$$\frac{\partial n_{c,j}}{\partial t}=-\frac{\partial }{\partial x}(u\cdot n_{c,j})-S_{\mathrm{c}}{J}_{s,j}\text{for}j=1\text{to}{N}_{\mathrm{s}},$$ (A4)

$$\frac{\partial n_{\mathrm{isf},j}}{\partial t}=S_{\mathrm{c}}{J}_{s,j}-F_L{C^{\prime }}_{\mathrm{isf},j}\text{for}j=1\text{to}{N}_{\mathrm{s}},$$ (A5)

$$\frac{\partial n_{\mathrm{pc},j}}{\partial t}=0\text{for}j=1\text{to}{N}_{\mathrm{s}},$$ (A6)

Three separate pathways exist for the coupled exchange of water and solute across the capillary: one pathway for solute-free water exchange, and small pore and large pore pathways for coupled fluid and solute exchange. The fluid flux through each transcapillary pathway is given by

$$J_{\mathrm{Vk}}=L_{\mathrm{pk}}\left\{(p_{\mathrm{c}}-p_{\mathrm{isf}})-\sum _j\left[{\sigma }_{j,k}\right({\Pi }_{c,j}-{\Pi }_{\mathrm{isf},j}\left)\right]+{\Pi }_{\mathrm{M}}\right\},$$ (A7)

where L_pk is the hydraulic conductivity through the kth pathway, σ_j,k is the reflection coefficient of the jth solute in the kth pathway, and π_c,j and π_isf,j are the osmotic pressures of the jth solute in capillary and interstitium, and π_M is the osmotic pressure exerted by interstitial matrix proteins. Total J_v is the sum of the J_v’s for the individual pathways, and the rate of weight change is $J_{\mathrm{V}}={\sum }_k{J}_{\mathrm{V}}$ for k = 1 to 3, dW(t)/dt = J_v.

Similarly, the water flux across the parenchymal cell membrane from cells to interstitium is given by

$$J_{V,\mathrm{pc}}=L_{p,\mathrm{pc}}\left((p_{\mathrm{pc}}-p_{\mathrm{isf}})-\left({\Pi }_{\mathrm{pc}}-{\Pi }_{\mathrm{M}}-\sum _{\mathrm{j}}{\Pi }_{\mathrm{isf},j}\right)\right),$$ (A8)

The total solute flux of solute j from capillary to interstitium, J_s,j, is given by the sum of diffusive and convective transport for each path:

$$J_{s,j}=\sum _{\mathrm{k}}\left[J_{\mathrm{Vk}}(1-{\sigma }_{j,k})C_{c,j}+P_{j,k}(C_{c,j}-C_{\mathrm{isf},j})\left(\frac{{\mathrm{Pe}}_{j,k}}{e^{{\mathrm{Pe}}_{j,k}}-1}\right)\right].$$ (A9)

Pe_j,k is the Péclet number for the jth solute traveling through the kth pathway, defined by

$${\mathrm{Pe}}_{j,k}=\frac{J_{\mathrm{Vk}}(1-{\sigma }_{j,k})}{P_{j,k}}.$$ (A10)

The pore equations of Curry (12) were used to determine L_p, P, and σ through the large and small pore pathways:

$$L_{\mathrm{p}}=\left(\frac{A_{\mathrm{p}}}{S_c\Delta r}\right)\frac{r_{\mathrm{p}}^2}{8\eta }$$ (A11)

$$P=\left(\frac{A_{\mathrm{p}}}{S_c\Delta r}\right)(1-\alpha )F\left(\alpha \right)D$$ (A12)

$$\sigma =1-[1-{(1-{(1-\alpha )}^2)}^2]G\left(\alpha \right)+\frac{16}{9}{\alpha }^2{(1-\alpha )}^{2}F\left(\alpha \right),$$ (A13)

where α = ratio of solute radius to pore radius, r_p = equivalent pore radius, nm; area of kth pathway, $A_{\mathrm{p}}=N_{\mathrm{p}}\cdot \Pi r_{\mathrm{p}}^2=\text{pore surface area}$. Δr = length of the pore from capillary lumen to ISF. D is the free diffusion coefficient of the solute in the pore. F(α) is a factor (0 < F < 1), describing hindrance to diffusion, given by Curry’s Eq. 5.17 (14) as taken from Faxen’s 1959 paper:

$$F\left(\alpha \right)=1-2.10444\alpha +2.08877{\alpha }^3-0.94813{\alpha }^5-1.372{\alpha }^6+3.87{\alpha }^8-4.19{\alpha }^9,$$ (A14)

and G(α) accounts for the difference in solute and water velocities, and 0.5 < G < 1, using the equation from Curry’s Eq. 5.51 (14):

$$G\left(\alpha \right)=\frac{1-2{\alpha }^2∕3-0.20217{\alpha }^5}{1-0.75851{\alpha }^5},$$ (A15)

which is accurate for α < 0.6 and overestimates G a little for higher α’s (lower G’s). A pore radius is not defined for the water-only pathway across the endothelial cell; L_p,endo is estimated directly from the weight transients, and neglecting the change that occurs in the small volume of the endothelial cells, represents conductance across the luminal and abluminal surfaces in series.

The apparent osmotic reflection coefficient for the membrane, σ_d, can be calculated from σ for each of the individual pathways from

$${\sigma }_d=\sum \frac{L_{p,i}{\sigma }_i}{L_{p,\text{total}}}$$ (A16)