Modification of the Kedem-Katchalsky equations

The presented modification of the transport equations of Kedem-Katchalsky resulted in the introduction of (omega s/omega) and omega/(omega-Lp sigma[(1-sigma)C1-(1-sigma s)C2]) factors into the Kedem-Katchalsky equations. The above factors determine the influence of boundary layers on transport across the membrane. The modified Kedem-Katchalsky equations were verified for synthetic membranes and it was shown that the value of the (omega s/omega) factor depended on the type of membrane and the membrane configuration system. This modification facilitated a wider range of application of the Kedem-Katchalsky equations to systems in which the solutions were stirred or unstirred.     

Mechanistic equations for membrane substance transport and their identity with Kedem-Katchalsky equations

Since the physical interpretation of practical Kedem-Katchalsky (KK) equations is not clear, we consider an alternative, mechanistic approach to membrane transport generated by osmotic and hydraulic pressure. We study a porous membrane with randomly distributed pore sizes (radii). We postulate that reflection coefficient (sigma(p)) of a single pore may equal 1 or 0. From this postulate we derive new (mechanistic) transport equations. Their advantage is in clear physical interpretation and since we show they are equivalent to the KK equations, the interpretation of the latter became clearer as well. Hence the equations allow clearer and more detailed interpretation of results concerning membrane substances transport.     

Some remarks on the Kedem-Katchalsky equations for non-electrolytes

The Kedem-Katchalsky equation for the flow of a non-electrolyte through a homogeneous membrane is shown to be a first order expansion of an exact integral of the Spiegler-Bearman-Kirkwood frictional equations under the assumption that the partial frictional coefficients, ζ _ij , are concentration independent. The equations are solved in terms of volume flow; there are no water-to-volume flow correction terms for the permeability, ω, or the reflection coefficient, σ. The precision of the expansion depends upon the magnitude of the water flow. The frictional coefficientsf _sm andf _sw are given as functions of the experimentally determined parameters ω and σ; the frictions, are shown to be independent ofL _p .     

On the derivation of the Kargol's mechanistic transport equations from the Kedem-Katchalsky phenomenological equations

In the present article, it was demonstrated that--by starting from the so-called adjusted Kedem-Katchalsky (KK) phenomenological equations (Suchanek et al. 2004), i.e. the equations: Jv=LpDeltaP-LpDDeltaPi. JD=-LDpDeltaP+LDDeltaPi it is possible to derive practical transport equations (for the volume flow and the solute flow) in the form of the Kargol s mechanistic transport equations (Kargol and Kargol 2000, 2001, 2003a,b,c, 2004; Kargol 2002). On this basis, it has been found that the KK thermodynamic formalism for membrane transport (practical equations) is in general identical with the mechanistic equations for membrane transport.     

Effect of heteroporosity on flux equations for membranes

An investigation is made of the possible errors in simple integrated equations for solute flux across both non-pieving and sieving porous membranes that can result from variations in the membrane structure. Detailed structural models are used, beginning with a membrane consisting of a parallel array of pores and progressing to series-parallel combinations of pore segments of various lengths and cross-sectional areas, with internal cross connections among pore segments allowed. It is shown that there are both upper and lower mathematical bounds on the possible variations that can be produced in a curve of solute flux versus volume flow by arbitrary variation in the membrane structure, subject only to certain general conditions. In particular, the flux equation for a homoporous membrane is a lower bound- The maximum deviations from this lower bound for a membrane of arbitrary structure are only moderately large, and require rather extreme pore size distributions; most distributions introduce only small errors. Implications of these results in studies of real membrane structure and in the design of experiments are discussed.     

Derivation of unstirred-layer transport number equations from the Nernst-Planck flux equations.

Since the late 1960s it has been known that the passage of current across a membrane can give rise to local changes in salt concentration in unstirred layers or regions adjacent to that membrane, which in turn give rise to the development of slow transient diffusion potentials and osmotic flows across those membranes. These effects have been successfully explained in terms of transport number discontinuities at the membrane-solution interface, the transport number of an ion reflecting the proportion of current carried by that ion. Using the standard definitions for transport numbers and the regular diffusion equations, these polarization or transport number effects have been analyzed and modeled in a number of papers. Recently, the validity of these equations has been questioned. This paper has demonstrated that, by going back to the Nernst-Planck flux equations, exactly the same resultant equations can be derived and therefore that the equations derived directly from the transport number definitions and standard diffusion equations are indeed valid.     

Slow phase of transvascular fluid flux reviewed

In six circuit experiments using a clinical hemofiltration device, we validated a colorimetric technique to measure transvascular volume exchange (VE). In 12 isolated excised canine left lower lobes, continuous colorimetric measurements of VE correlated well with calculations of VE from changes in microhematocrit obtained simultaneously. We introduced step increases in microvascular hydrostatic pressure (Pc) of 9 +/- 4.8 (SD) cmH2O and followed the time course of weight and continuous hematocrit changes measured colorimetrically for 40 min, after which Pc was returned to base line, while measurements were continuously obtained. This procedure was repeated for an additional 30 min. VE was calculated from the hematocrit signals and compared with the time course of the weight signal. After increases in Pc, followed by a rapid weight gain, weight signals followed a slow exponential time course, whereas the calculated VE changed linearly. VE reflected approximately 60% of the slow weight gain. When Pc was decreased, weight signals decreased exponentially, whereas VE continued to increase linearly at a slower rate. These results suggest that a significant component of the slow weight signal represents slow vascular volume changes. Contrary to what the weight signal suggested, edema was never reabsorbed over the range of Pc measured.     

Colorimetric device for measurement of transvascular fluid flux in blood-perfused organs

The aim of this study was to develop a device capable of measuring transvascular fluid flux in blood-perfused organs. For any given blood flow through the organ (QT), transvascular flux (QF) can be considered as the fraction of QT exchange. Presumably, QF would change the background concentration of an impermeable tracer residing in the perfusate. Thus QF could be calculated from the relative changes in tracer concentration for any given QT. We have used Blue Dextran (1 g/l of blood) as the reference tracer. Because the minimum molecular weight of Blue Dextran is 2 X 10(6), we anticipated it to behave as an impermeable tracer in most organs. QF was simulated with continuous infusions of plasma, normal saline solution, and a 50% mixture of both. Changes in Blue Dextran concentration were continuously followed colorimetrically by changes in transmission of specific light at a wavelength of 632 nm. Because 632-nm light is affected by hematocrit and O2 saturation changes, two additional wavelengths were used: 815-nm, which is not affected by saturation or Blue Dextran concentration changes, was used to account for changes in hematocrit, and 887-nm specific light, which is not affected by Blue Dextran, served to correct for saturation changes. Red cells could not be used as the reference tracer because of the possibility of hematocrit changes independent of fluid flux (Fahraeus effect). The device so constructed proved capable of measuring rates of fluid infusion in the order of 0.1% of QT with a variability of 10% around the mean.     

Fast-phase transvascular fluid flux and the Fahraeus effect

Transvascular fluid flux was induced in six isolated blood-perfused canine lobes by increasing and decreasing hydrostatic inflow pressure (Pi). Fluid flux was followed against the change in concentration of an impermeable tracer (Blue Dextran) measured directly with a colorimetric device. The time course of fluid flux was biphasic with an initial fast transient followed by a slow phase. Hematocrit changes unrelated to fluid flux occurred due to the Fahraeus effect, and their contribution to the total color signal was subtracted to determine the rate of fast fluid flux (Qf). Qf was related to Pi to derive fast-phase conductance (Kf). Slow-phase Kf was calculated from the constant rate of change of lobe weight. For a mean change in Pi of 7 cmH2O, 40% of the color signal was due to fluid flux. Fast- and slow-phase Kf's were 0.86 +/- 0.15 and 0.27 +/- 0.05 ml X min-1. cmH2O-1 X 100 g dry wt-1. The fast-phase Kf is smaller than that reported for plasma-perfused lobes. Possible explanations discussed are the nature of the perfusate, the mechanical properties of the interstitium, and the slow rate of rise of the driving pressure at the filtration site on the basis of a distributed model of pulmonary vascular compliance.     

Effect of lung inflation on fluid flux in zone 1 lungs

Because of conflicting data in the literature, we studied the effect of positive-pressure inflation on transvascular fluid filtration in zone 1 lungs. Lungs from New Zealand White rabbits (n = 10) were excised, perfused with saline and autologous whole blood (1:1), ventilated, and continuously weighed. Pulmonary arterial and venous pressures (Pvas) were referenced to the most dependent part of the lung. A change in vascular volume (delta Vvas) and a fluid filtration rate (FFR) were calculated from the change in lung weight that occurred from 0 to 30 s and from 3 to 5 and 5 to 10 min, respectively, after changing alveolar pressure (PA). FFR's and delta Vvas's were measured with Pvas equal to 2 or 10 cmH2O and PA changing from 15 to 30 cmH2O when the lungs were normal and after they were made edematous. When Pvas = 2 cmH2O, increasing PA increased the Vvas and the FFR in both normal and edematous lungs. However, when Pvas = 10 cmH2O, increasing PA only slightly changed the Vvas and reduced the FFR in the normal lungs, and decreased Vvas and markedly decreased the FFR in the presence of edema. Inflating zone 1 lungs by positive pressure has an effect on transvascular fluid flux that depends on the Pvas. The results suggest that the sites of leakage in zone 1 also vary depending on Pvas and PA.     

Kinetics of fluid flux in the rat diaphragmatic submesothelial lymphatic lacunae

The specific role of loops and/or linear segments in pleural diaphragmatic submesothelial lymphatics was investigated in seven anesthetized, paralyzed, and mechanically ventilated rats. Lymphatic loops lay peripherally above the diaphragmatic muscular plane, whereas linear vessels run over both the muscular and central tendineous regions. Lymph vessel diameter, measured by automatic software analysis, was significantly greater (P < 0.01) in linear vessels [103.4 +/- 8.5 microm (mean +/- SE), n = 18] than in loops (54.6 +/- 3.3 microm, n = 21). Conversely, the geometric mean of intraluminal flow velocity, obtained from the speed of distribution of a bolus of fluorescent dextrans injected into the vessel, was lower (P < 0.01) in linear vessels (26.3 +/- 1.4 microm/s) compared with loops (51.3 +/- 3.2 microm/s). Lymph flow, calculated as the product of flow velocity by vessel cross-sectional area, was similar in linear vessels and in individual vessels of a loop, averaging 8.6 +/- 1.6 nl/min. Flow was always directed from the diaphragm periphery toward the medial tendineous region in linear vessels, whereas it was more complex and evidently controlled by intraluminal unidirectional valves in loops. The results suggest that loops might be the preferential site of lymph formation, whereas linear vessels would be mainly involved in the progression of newly formed lymph toward deeper collecting diaphragmatic ducts. Within the same hierarchic order of diaphragmatic lymphatic vessels, the spatial organization and geometrical arrangement of the submesothelial lacunae seem to be finalized at exploiting the alternate contraction/relaxation phases of diaphragmatic muscle fibers to optimize fluid removal from serosal cavities.     

Single Water Channels of Aquaporin-1 Do not Obey the Kedem-Katchalsky Equations

The Kedem-Katchalsky (KK) equations are often used to obtain information about the osmotic properties and conductance of channels to water. Using human red cell membranes, in which the osmotic flow is dominated by Aquaporin-1, we show here that compared to NaCl the reflexion coefficient of the channel for methylurea, when corrected for solute volume exchange and for the water permeability of the lipid membrane, is 0.54. The channels are impermeable to these two solutes which would seem to rule out flow interaction and require a reflexion coefficient close to 1.0 for both. Thus, two solutes can give very different osmotic flow rates through a semi-permeable pore, a result at variance with both classical theory and the KK formulation. The use of KK equations to analyze osmotic volume changes, which results in a single hybrid reflexion coefficient for each solute, may explain the discrepancy in the literature between such results and those where the equations have not been employed. Osmotic reflexion coefficients substantially different from 1.0 cannot be ascribed to the participation of other 'hidden' parallel aqueous channels consistently with known properties of the membrane. Furthermore, we show that this difference cannot be due to second-order effects, such as a solute-specific interaction with water in only part of the channel, because the osmosis is linear with driving force down to zero solute concentration, a finding which also rules out the involvement of unstirred-layer effects. Reflexion coefficients smaller than 1.0 do not necessitate water-solute flow interaction in permeable aqueous channels; rather, the osmotic behaviour of impermeable molecular-sized pores can be explained by differences in the fundamental nature of water flow in regions either accessible or inaccessible to solute, created by a varying cross-section of the channel.