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Department of Bioengineering, University of Washington, Seattle, Washington 98195
Submitted 11 July 2002 ; accepted in final form 5 May 2003
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ABSTRACT |
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 TOP ABSTRACT METHODSRESULTSDISCUSSIONAPPENDIXDISCLOSURESREFERENCES |
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28% of the transcapillary water flux going to form lymph was through the
endothelial cell membranes [capillary hydraulic conductivity
(Lp) = 1.8 ± 0.6 x
10–8 cm · s–1
· mmHg–1], presumably mainly through
aquaporin channels. The interendothelial clefts (with Lp =
4.4 ± 1.3 x 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 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 because 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
milliosmolar osmolarity change were 200 times larger than steady-state water
flux. The analysis completely reconciles data from osmotic transient, tracer
dilution, and lymph sampling techniques.
capillary permeability; reflection coefficient; transport modeling; microcirculation; isolated rabbit heart; porous transport
) of small hydrophilic
solutes in the vascular beds of whole organs. Using a single-membrane model of
events during this experiment, Vargas and Johnson
(49) developed a relationship,
=
Jv/(LpS
), where
Lp is capillary hydraulic conductivity, S is
surface area, and is osmotic perturbation, which relates the
volume flux across the capillary wall, Jv, to the solute
reflection coefficient, , immediately following a step change in
perfusate osmolarity. In a whole organ, the total fluid movement across the
capillary wall, JvS, is nearly equal to the rate
of weight change of the organ; knowledge of the magnitude of and
Lp is all that is needed to estimate . Estimates of
using this analysis method and osmotic weight transient measurements
have been obtained in many mammalian tissues including heart
(19,
47,
49,
50), skeletal muscle
(15,
39), and lung
(16,
37). For small solutes like
sucrose, estimates of in skeletal muscle range from 0.08
(39) to 0.41
(54) and in heart are 0.30
(49) and 0.14
(19).
A few investigators have attempted to extract additional information from
the complete time course of the weight transient by using two-compartment
(50), 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 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
obtained by compartmental models tend to increase with increasing flow
through an organ, even though there is no reason to suppose that the capillary
wall morphology is actually changing. Finally, previously described analytical
models depend on the prior knowledge of Lp to obtain
estimates of and/or P.
In the companion paper to this article (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, with 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 Lp and are independent of the flow. Thus our approach marks a significant improvement over previous attempts to analyze osmotic weight transient experiments.
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METHODS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXDISCLOSURESREFERENCES |
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Experimental Methods
Our standard protocol was to induce weight transients in the isolated, Ringer-perfused rabbit heart by 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 4–30 min, with 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 3–4 h (Fig. 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 in this section regarding 1) the standard and osmotically active Ringer solutions, 2) the perfusion system, 3) the surgical procedures, and 4) the data recording devices.
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Solutions. Physiological Krebs-Ringer perfusates were prepared
fresh each morning before the experiments. The control perfusate contained (in
mM) 118 NaCl, 3.8 KCl, 1.2 KH2PO4, 0.7 MgSO4,
2.1 CaCl2, 25 NaHCO3, 0.1 EDTA, 10 dextrose, 5.5
pyruvate, and 0.015 bovine serum albumin (all from Sigma). Perfusate
osmolarity was 280 ± 2 mosM. 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 contained (in
mM), unless otherwise noted, 30 urea, 20 NaCl, 30 glucose, 30 sucrose, 30
raffinose, 5 inulin, and 0.5 albumin. Solution osmolarity was validated with a
freezing-point osmometer for the small solutes (Osmette A, Precision Systems).
For the albumin test solution the concentration, C, was measured by protein
refractometry (accurate to about ±0.2 g/l) and the osmolarity was
calculated by the formula 
= 0.345C + 0.002657C2 + 2.26 x
10–5C3 from McDonald and Levick
(34). The increases in
osmolarity due to the additions of the test solute were (in the same order)
30, 38, 30, 30, 30, 5, and 0.5 mosM. All solutions were filtered through a
1.2-µm filter before use.
The albumin was purified by at least 72 h of dialysis at 4°C with a
membrane with a molecular cut-off mass of 12,000–14,000 Da (Spectra/Por,
Spectrum Laboratories) 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 the
standard Ringer solution and was changed at least three times during dialysis.
The 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 100 g/l, with a total osmolarity 2
mosM higher than that of 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 mass of 2,000 Da (Spectra/Por, Spectrum Laboratories).
Perfusion system. A perfusion system
(Fig. 2) was developed to allow
rapid switching between control and test solutions. Solutions were circulated
continually through membrane oxygenators (Medtronic Minimax Plus) with 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 0.1 ml before the heart
ensured a sharp change between the lines.
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Isolated heart preparation. The protocols for animal use were developed in accordance with National Institutes of Health (NIH) guidelines and were reviewed and approved by the Institutional Animal Care and Use Committee of the University of Washington (IACUC protocol no. 2027-09). Adult female New Zealand White rabbits (2–3 kg) were sedated with acepromazine (1 mg/kg sc) 20–30 min before surgical anesthesia was induced by intramuscular ketamine (40 mg/kg) and xylazine (5 mg/kg). After tracheotomy and start of 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 min, 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 solution until contractions ceased. It was then perfused retrogradely through the aorta while suspended from a T-shaped cannula (Fig. 2). The time from excision to beginning of perfusion was typically <1 min.
The isolated heart was trimmed of excess fat and other tissue. To drain the
left ventricular (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 (RV) 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 – LV drain flow.
Preparations in which flow from the LV drain comprised >5% of the total
flow were discarded because they likely had a damaged aortic valve. The heart
was paced with a stimulator (Harvard Apparatus) at a constant rate of 150
beats/min, and perfusion rate was set at 20 ml/min (2–4
ml·min–1·g–1).
After isolation, the heart was equilibrated for at least 30 min. During this time the heart gained 1–2 g because of 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 gauge of fine needles at the end of the return line, we equalized the pressures and resistances of the two perfusion lines so that switching from one to the other caused a pressure jump of <1 mmHg. One perfusion line was then changed to the test solution, and the experimental protocol was begun.
Data recording. Heart weight W(t) and aortic perfusion pressure were recorded from a force transducer (Transducer Techniques) and a pressure transducer (Statham). The transducer outputs were amplified with a custom-built amplifier, and the weight record was filtered by an analog resister-capacitor filter with a time constant of 0.2 s 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 data points. 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 with 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 in the next section can be downloaded via http://nsr.bioeng.washington.edu/Software/DEMO, under "Blood-Tissue Exchange Models: Whole-organ models: osmotic". This software package was developed by the National Simulation Resource for Circulatory Mass Transport and Exchange.
The model. The model used is described in detail in the companion paper (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. We used the measured dry weight of a heart and the values of water content of vascular, interstitial, and cellular regions for in vivo rabbit hearts (17) to calculate initial values for plasma volume, Vp, interstitial space, Visf, and cell space, Vpc. 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: Visf was increased so that model 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 1% increase in the parameter values. Relative sensitivities
[(
W/W)/(P/P) or ln
W/ ln P] rather than absolute sensitivities
(W/P) were used to provide dimensionless
sensitivity functions. Because 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
RESULTS were generated with 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) Lp,endo, the hydraulic conductivity through the transendothelial pathway, 2) Asp/S, the fractional small-pore area, and 3) rsp, the small-pore radius. The estimates of these parameters are not sensitive to the parameter values for the large-pore system, Alp/S, which was set at 10–6, and rlp, 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 (dW/dt) including both the switch to the test solution and the return to control was used as the data for each optimization procedure. Optimization was performed with 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 (6–30 min/transient at 1 sample/s is 360–1,800 samples per parameter optimization run) and few free parameters (a maximum of 3). The parameters for capillary and parenchymal cell volumes, Vp and Vpc, were those from Ref. 19. The capillary surface area S is 500 cm2/g (6).
The three free parameters are interdependent to some extent and require use
of the full extent of the weight transients and the full set of solutes. The
larger solutes have the greatest influence on estimates of
rsp, because for small solutes the sp is
close to zero. Consequently, it is from the overall dW/dt =
SJV that an overall Lp
(Lp,total) is calculated, thus defining
Lp,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
through the solute effective radius rs and the pore radius
rp (either rsp or
rlp) and the total pore areas. The equations governing
this relationship are given in the APPENDIX (Eqs.
A11–A13).
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RESULTS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXDISCLOSURESREFERENCES |
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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 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 x
10–8 cm · s–1
· mmHg–1 is through a small fraction of the
total capillary surface area, Asp/S = 2.2 x
10–4, equivalent to a pore area per unit path
length of 4.4 cm–1. The small pore has an
effective radius of 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 1%, which is
substantially smaller than the differences between parameter estimates from
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.
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Table 2 shows the parameter estimates obtained with each osmotic agent. Similar estimates were obtained regardless of the solute used, with the most significant disparity being the difference between rsp 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 asterisk in Table 2), the average estimate obtained from solutions with other osmotic agents was used.
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These parameter estimates can be used to calculate solute P and
for the solutes through each pathway
(Table 3). Solute P
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
3.8 to 3.0 cm–1. Inulin and albumin, being
much larger, are significantly hindered compared with the other solutes and
have smaller P/D and larger Jsp.
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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 Fig. 3. After a step increase in perfusate osmolarity with 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 Visf decreases. This is followed by a slower phase where the solute enters the interstitium to partially dissipate its transcapillary concentration gradient; 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 318 mosM would result in a steady-state loss of only 14% of cell water even if the cell was a flaccid bag and slightly less if there was 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.
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The time course of weight change is substantially different for larger
solutes like albumin (Fig. 4).
Because only 0.5 mM albumin was used, the initial rate of weight loss is less
than for NaCl, despite albumin's higher . 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 Jv
causes only a negligible transcapillary gradient in the resident solutes.
Because of the low Jv, the low for the small
solutes, and the very low P for albumin, the Jv
is of water and solute, so the alb persists for a long
time and causes a water shift that is large, decreasing
Visf.
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Sensitivity Analysis
Sensitivity analysis indicated that analysis of transients over a range of molecular weights from NaCl to albumin was needed to provide estimates of the small-pore system parameters Asp/S and rsp and the Lp,endo of the transendothelial pathway. Data spanning a wide range of molecular sizes are crucial to getting reasonable estimates, as there is no single solute size at which all three parameters have substantial and distinctly different sensitivity functions (Fig. 5).
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The values of the large-pore parameters cannot be resolved by osmotic
transient data because the sensitivity functions ln W/
ln rlp and ln W/ ln
Alp 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-to-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 (Fig. 6). This trend was consistently observed over 22 transients in two 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.
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DISCUSSION |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXDISCLOSURESREFERENCES |
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Design of 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
Lp,endo, rsp, and
Asp/S. The parameters
Asp/S and rsp can be
determined from albumin transients because the shapes of ln
W/ ln Asp and ln W/
ln rsp differ greatly and ln W/ ln
Lp,endo and the fluxes across endothelial cells are small.
Lp,endo must then be determined from NaCl transients,
holding Asp/S fixed. Given that the model is an
appropriate descriptor, mid-sized solutes provide redundant information, and
by 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 (mol wt <1,000) only induce a
significant Jv, through the transendothelial pathway,
where it can be assumed that they have a equal to 1 because aquaporin
channels and cell membranes exclude these solutes. Although
Jv is proportional to Lp,endo, the
osmotic driving force for fluid exchange is dissipated as the test solute
permeates through the pores into the interstitium, a flux that is
predominantly diffusive and proportional to
Asp/S. As a first approximation, the magnitude of
the weight transient is governed by the ratio of these processes, or
Lp,endo to Asp/S. As solutes
become larger, sensitivity to Lp,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 Asp/S,
increasing solute size, and therefore increasing , 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
Asp/S is initially negative (increasing
small-pore area results in an increased water efflux). Increasing pore size,
rsp, leads to lower sp, increased water
flux, and reduced solute permeation. The sensitivity to
rsp 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 near 0 for the large pores. Thus varying the
osmolarity with any solute fails to affect Jv,lp or to
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 pisf decreases and interstitial osmotic pressure
isf (exerted by the permanent protein components of the
interstitial matrix) rises to balance the increased osmotic pressure in the
capillaries, that is, p – goes to zero. This result
is dependent on the assumption that the large-pore pathway is in fact in the
neighborhood of Alp/S =
10–6 and rlp = 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 rlp were made smaller and
there were more pores. Transients would be affected if
Alp/S were as large as 5 x
10–6 and rlp 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
Grabowski and Bassingthwaighte
(19) and Vargas and Johnson
(49,
50) used the initial slope of
the osmotic transient weight response, dW/dt =
Jv = LpSRTC,
where R is the gas constant and T is absolute temperature, to estimate the
solute reflection coefficient. Because they assumed that 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 d, the osmotic
reflection coefficient for the membrane as a whole. By itself, the apparent
d for a heterogeneous membrane is insufficient to provide a
complete description of exchange kinetics. Realizing this, Pappenheimer
proposed in 1969 (35) that
50% of transcapillary water exchange occurs through a water-only pathway
to explain early data on hydraulic conductivity and solute permeability in
skeletal muscle measured by his isogravimetric technique
(36) 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 that excludes all solutes (giving
them a of 1) and relatively unrestrictive aqueous pathways with
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 Ref.
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
(Fig. 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 for the test solute. For small solutes like
NaCl, the only pathway with a nonnegligible is the transcellular
pathway. Regardless of its fraction of total hydraulic conductivity, an
increase in capillary NaCl concentration draws water out of the tissues
predominantly across the endothelial cells and out of parenchymal cells,
reduces pisf, and diminishes lymph flow,
JV,L, as shown in Fig.
7. Consequently, the apparent of the capillary wall at
early times in an osmotic transient is larger than later, because the fraction
of Jv 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
apparent approaches a constant value at high
Jv, when solute permeation, Js <
Jv x Cs, where
Js is solute flux and Jv x
Cs is unhindered, expected solute flux.
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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 . The osmotic transient results
presented in Fig. 3 are typical
of small solutes and can be closely matched by 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 x
10–4 cm/s, 50% higher than the multipath model
estimate (6.4 x 10–5 cm/s). Also, of
the single-path model is 0.15, assuming that the Lp of
this single "effective" pathway is equal to the sums of the
Lp values 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 3 nm, incompatible with multiple-indicator dilution 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
Lp,total of 6.5 x 10–8
cm · s–1 ·
mmHg–1 (Table
1) is two-thirds of the Vargas and Johnson
(49) estimate of 1.0 x
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 solution, which artificially
increased transcapillary hydraulic conductivity. A novel result is our
estimation that 28% of the Lp is contributed by the
transendothelial pathway. This value is greater than the 5–10% estimated
in frog mesentery (15) but
less than the 40% predicted by Wolf
(54,
55) in skeletal muscle. Our
estimates of 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 d = 0.7 is higher than
f = 0.59 estimated from filtration-rate independent
CL/Cp measurements
(38), in line with theoretical
expectations (13,
40), but it is lower than the
0.80–0.87 d obtained in other organs
(56). Because 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
(44). Solute permeability is
proportional to Asp/(Sr), where
Asp is the cleft or pore surface area, S is the
capillary surface area, and r is the length of the pathway
from capillary lumen to interstitium. These terms are kinetically inseparable.
In anatomic studies S is 500 cm2/g
(6), and in the geometry of our
model it is 487 cm2/g
(29). From electron microscopy
the distance r is 0.5 µm; thus an estimate of
Asp/(Sr) of 4.4
cm–1 is equivalent to
Asp/S of 0.022% and Asp of
0.11 cm2/g.
For translating from the parameters of our model the traditional
permeability-surface area product PS (in ml ·
g–1 · min–1)
for glucose would be
 | (1) |
Our estimates of Asp/S are compatible with those of Guller et al. (21); for NaCl they used an intrapore diffusion coefficient D = 5 x 10–6 cm2/s, taken from studies of Na diffusion in extracellular space (42, 46) 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 Guller et al. (21) did would suggest a total Asp/S of 2.25 x 10–3, given a 9-nm width. In accordance with Guller et al. (21), because our functional estimate of Ap/S was only 2 x 10–4, we would conclude that <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, d appears to increase
with increasing flow rates
(57). This is because the
Vargas and Johnson (49)
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 d.
Vargas and Johnson (49,
50) 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
(48) argued were small for
their experimental methods. At a flow of 1 ml/s, an extremely high flow for a
rabbit heart of 6 g, Johnson and Bloom
(27) calculated that the
reflection coefficient for NaCl, a solute with relatively high permeability,
would still be underestimated by 16% with the basic Vargas and Johnson model.
The most complete set of flow vs. data available from Wolf and Watson
(57) shows an apparently
increasing 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
(Fig. 6). Estimates of
Lp and 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 (Fig. 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 (57), 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.
|
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 interstitial fluid (ISF) and because of the lowered oncotic pressure of the perfusate. By the end of an experiment, hearts typically had water contents of 82–85%, as measured by wet-to-dry weight ratio, compared with in vivo values of 78 ± 1% (17, 58). Because 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 ISF 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. Whereas 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. On the basis of 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 from 5 to 7 nm when going from blood- to Ringer-perfused preparations (29). Therefore, in the in vivo condition the transendothelial fraction of Lp is probably closer to 40–50% rather than the 28% we found in these experiments (Table 2).
Pressure changes during experiment. Drops in arterial pressure are known to occur during an osmotic transient with a step increase in concentration at constant flow (19, 47, 48). 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 a constantflow 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 with 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 with control and typically <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 cannot 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 with 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 that about two-thirds 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. Stein (45) gives the Post-Jolly equations for Donnan distribution across the cell membrane. The general concept is that if there is a cation that leaks passively into the cells it must be pumped out, otherwise the cell expands. The summarizing equation for cell volume is V = (Ai)/[Na]e x (1 + kleak/kpump), where Ai is moles of intracellular protein, [Na]e is extracellular sodium concentration, and kleak and kpump are first-order rate constants (Eq. B7.1.6 in Ref. 45).
The consequence of this is that if kleak,
kpump, and [Na]e do not 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 and increase intracellular Na content by 10%,
decreasing the leak flux by 10% and increasing the pump flux by 10%, creating
a situation for reducing the intracellular Na content. But neither
kleak nor kpump needs to change for
this to happen. [Adaptive cell behavior is commonly found, however; for
example, kleak changes
(45)].
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 20 min,
although 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
Lp,endo, Asp/S, and
rsp were 33%, 32%, and 23%, respectively. In contrast, the
confidence limits on the free parameter values determined by optimization were
typically 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 three 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 12 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 h of each
experiment because of the effects of isolation and prolonged exposure to
Ringer 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.
In conclusion, 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 that 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 Asp/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 suggest that it is likely
24 nm in radius with an Alp/S of
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 luminal and abluminal
endothelial plasmalemma and myocyte sarcolemma. The analysis of osmotic
transient data we have presented is consistent with our application of the
same model to indicator dilution and steady-state lymph sampling data in the
companion paper (29). Now for
the first time all the various types of observations are brought into
compatibility by a properly comprehensive model.
|
APPENDIX |
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|
TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXDISCLOSURESREFERENCES |
|---|
 | (A1) |
 | (A2) |
 | (A3) |
 | (A4) |
 | (A5) |
 | (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
 | (A7) |
j,k is the
reflection coefficient of the jth solute in the kth pathway,
c,j and
isf,j are the osmotic pressures of the
jth solute in capillary and interstitium, respectively, and
M is the osmotic pressure exerted by interstitial matrix
proteins. Total Jv is the sum of the
Jv values for the individual pathways, and the rate of
weight change is Jv = 
Jv for
k = 1–3, dW(t)/dt =
Jv.
Similarly, the water flux across the parenchymal cell membrane from cells
to interstitium is given by
 | (A8) |
The total solute flux of solute j from capillary to interstitium,
Js,j, is given by the sum of diffusive
and convective transport for each path
 | (A9) |
 | (A10) |
The pore equations of Curry
(12) were used to determine
Lp, P, and through the large- and
small-pore pathways
 | (A11) |
 | (A12) |
 | (A13) |
is the ratio of solute radius to pore radius,
rp is the equivalent pore radius in nanometers; the area
of the kth pathway, Ap = Np
· r2p = pore surface area;
r is the length of the pore from capillary lumen to ISF;
D is the free diffusion coefficient of the solute in the pore;
F() is a factor (0 < F < 1) describing
hindrance to diffusion, given by Curry's Eq. 5.17
(13) as taken from Faxen's
1959 paper
 | (A14) |
) accounts for the difference in solute and water
velocities; 0.5 < G < 1, by Curry's Eq. 5.51
(14)
 | (A15) |
< 0.6 and overestimates G a little
for higher values (lower G values). A pore radius is not
defined for the water-only pathway across the endothelial cell;
Lp,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,
d, can be calculated from for each of the individual
pathways from
 | (A16) |
|
DISCLOSURES |
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TOPABSTRACTMETHODSRESULTS |
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