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A simple method for determining the homoporous solute-membrane permeability from plasma-to-lymph measurements
MICROVASCULAR
RESEARCH
30, 235-241 (1985)
BRIEF COMMUNICATIONS A Simple Method for Determining the Homoporous Solute-Membrane Permeability from Plasma-to-Lymph Measurements LYNN J. GROOME* AND GARY T. KINASEWITZ~ *Department of Physiology and Biophysics and TDepartment of Medicine, Louisiana State University School of Medicine, Shreveport, Louisiana 71130 Received August 29, 1983 The exact equation for ultratiltration of a neutral solute across a homoporous membrane contains only two adjustable parameters, the reflection coefficient (a,) and the permeability coefficient (PJ-surface area (S) product, P$.S. Specifying (T, and PJS therefore defines completely the dependence of the protein ratio (R) on the volume flow (JJ for a homoporous membrane. The reflection coefficient is determined from the high J, limit of R: lim J,+m [R] = (1 - u,). The purpose of this paper is to present a simple procedure for estimating P;S from the slope of R as J, approaches zero: lim J, + 0 [dR/dlJ = -u,/(P;S). Both relations are exact limits of the nonlinear neutral solute/homoporous membrane transport equation and thus provide a simple, yet rigorous, method for estimating W$and P%4’. o 1985 Academic
Press, Inc.
INTRODUCTION In the case of plasma-to-lymph ultrafiltration, plasma is forced across a heteroporous capillary membrane in response to a hydrostatic pressure difference and the newly formed lymph is collected downstream; steady state is achieved once the lymph flow (J,) and the plasma (C,) and lymph (CL) solute concentrations are no longer changing with respect to time. A series of such measurements, obtained by changing the hydrostatic pressure difference and allowing J, and R (=C,/C,,) to become constant over a period of time, provides a relationship between R and J, at steady state. The dependence of R on J, observed experimentally is then used to estimate the effective solute-membrane reflection coefficient (a,) and the permeability coefficient (P&surface area (S) product, P;S. The values found for a, and P;S, however, depend on the theoretical equation relating R to J, which, in effect, defines these coefficients. Moreover, because of the complex nature of the vascular membrane, it is often necessary to use an approximate equation for R. However, for the special case of a neutral solute and a homoporous membrane, steady-state ultrafiltration can be described by an exact equation. Consider a membrane composed of identical pores separating well-stirred solutions of a single neutral solute at different concentrations (CL, C,). The local form of the 235 00262862’85 $3.00 Copyright 8 1985 by Academic Press, Inc. All rights of reproduction in any form resewed. Printed in U.S.A.
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steady-state equation for the solute flux across a single pore is j, = -D, (dc,/dx) + (1 - CT& CS,
(la)
where c, is the concentration of solute at a distance x into the pore and dc,/dx is the concentration gradient at position x; j, and j, denote the single-pore solute and volume fluxes, respectively. Equation (la) states that the single-pore solute flux at point x is the sum of the diffusive [ -D,(dc,/dx)] and convective [(1 a&c,] fluxes at that point. For a specified set of membrane boundary conditions, the steady-state mass balance requires that bothj, andj, be constant with respect to position within the pore. Integrating Eq. (la) across the length of the pore (for constant j, and j,) yields J, = (1 - ~#,[c,e-”
- C,]/(e-K - I),
(lb)
where K = (1 - (r,)J,/(P;S) for the membrane as a whole. Equation (lb) was first derived by Patlak and co-workers (1963) and subsequently by Levitt (1975). Due to the nature of the ultrafiltration process, in which the downstream lymph concentration is simply CL = J,/J,, it follows from Eq. (lb) that (Granger and Taylor, 1980) R = (1 - cr,)/(l - a,e?). UC) This equation is exact for the steady-state ultrafiltration of a neutral solute by convection and diffusion across a common homoporous pathway. That Eq. (lc) is exact, if only for the case of ultrafiltration across a homoporous membrane, is not insignificant, since it provides a simple, well-defined standard to which experimental measurements can be compared. Any other equation describing ultrahltration of a neutral solute across a homoporous membrane, e.g., the arithmetic-mean equation R = (1 - v,)/2 + (Ps * W/J” (1 + us)/2 + (P, * WJ”’
(14
would not only approximate the true charged solute/heteroporous membrane, but would also be an approximation to Eq. (1~). This introduces further uncertainty into the estimation of the set {us, P;S). For example, if the arithmetic-mean approximation is used, deviations from Eq. (lc) become significant at relatively low lymph flows, and the estimate for us, and hence P;S, is therefore incorrect (Granger and Taylor, 1980). Although Eq. (lc) is an “exact approximation” which provides a good fit of the experimental data, the exponential dependence of R on J, detracts from its usefulness and is in part responsible for the popularity of Eq. (Id). For example, a trial-and-error search for the set {us, P;S} which best fits the data is necessary if Eq. (lc) is used: an initial guess for us and P;S is made, the sum-of-squares error is calculated, and, if not of sufficient accuracy, a new guess is made for uS and P;S and the process is repeated. By linearizing Eq. (lc), e.g., by using Eq. (Id), the trial-and-error procedure is bypassed and an approximate set {a,, Ps*S} is determined directly without iteration. Ideally, combining the simplicity of the linear least-squares analysis with the rigor of Eq. (lc) provides an optimal solution for the set {us, P;S}. This is, in fact, possible. Since the precise form of the ultrafiltration curve for sieving of a neutral solute across a homoporous membrane is determined entirely by us
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and P;S, the behavior of R in certain limits of J, yields these two coefficients directly, without having to first linearize Eq. (1~). A unique value for (T, can be found from measurements of R at high lymph flows (Yablonski and Lifson, 1976; Granger and Taylor, 1980). The purpose of this paper is to present a simple procedure for estimating P;S from the slope of R as J, approaches zero. Because Eq. (lc) has only two degrees of freedom, choosing crs and P;S based on the limiting behavior of R not only circumvents the nonlinear data reduction posed by Eq. (lc), but also defines completely the J, dependence of R for ultrafiltration of a neutral solute across a homoporous membrane. ANALYTIC
METHOD
The motive for choosing the slope of R as J, approaches zero as a basis for estimating P;S is not readily apparent on examination of Eq. (1~). Here a simple physical argument suggests the appropriate limits for evaluating both g‘, and P;S. Consider, for example, a volume of fluid moving down a pore with a mean velocity Jv/S. Since crs can be thought of as the fraction of solute molecules which are reflected at the pore entrance and hence do not penetrate the membrane by convection, the fraction which is in the volume element when it enters the pore is (1 - (T,). As the control volume travels down the pore it will exchange solute molecules by diffusion with the adjacent fluid. The net exchange, however, depends on the length of time that the volume element remains in the pore. Since an increase in J, decreases the time available for diffusion, a value for J, exists above which ditfusion is negligible. In this limit the difference in solute concentration across the membrane is the greatest, and R approaches (1 - a,) as J, becomes infinitely large: lim J, + a [R] = 1 - u,.
(24
This limit for estimating u, has been suggested elsewhere (Yablonski and Lifson, 1976; Granger and Taylor, 1980), and can be derived by simply taking the high J, limit of Eq. (1~). In contrast, when the volume flow is small, sufficient time exists for diffusion to significantly reduce the concentration difference between volume elements; R therefore approaches unity as J, approaches zero. Although the concentration difference across the membrane is particularly sensitive to P;S in the region, R will always equal one at zero volume flow, regardless of the value of P;S. However, as J, is increased slightly above zero, a solute with a high permeability will remain nearer to equilibrium relative to a solute with a low permeability. Therefore, the appropriate limit to estimate P;S is the rate at which R approaches unity as J, approaches zero. This can be expressed mathematically by using Eq. (lc) to determine the slope of R, dR/dJ,, in the limit of zero volume flow (see Appendix): lim J, + 0 [dR/dl,]
= - cr&P;S).
(2b)
Equations (2a) and (2b) are exact limits of Eq. (1~). DISCUSSION For a given set {us, P;S}, the concentration difference across the membrane is a function of J, only. Therefore, determining us and P;S from the experimental
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data at the limits given by Eqs. (2a) and (2b) completely defines the J, dependence of R for a neutral solute across a homoporous membrane. Use of these relations, however, requires that lymph-to-plasma protein ratios are measured at the two extremes of high and low lymph flows. Any fluctuations in the data in these regions can lead to errors in us and P;S. While this is not too severe a constraint on (T,, P;S is particularly dependent on the accuracy of the experimental measurements, since this coefficient is determined from the slope of R at a single point, i.e., zero J,. In estimating P;S, therefore, it is desirable to use as many of the data points as possible. One approach is to determine dR/dJ, for various volume flows and extrapolate this to zero J,. However, fluctuations in the data on either side of a given J, can lead to significant errors in dR/dJ, at J,. To minimize this error, we define a slope referenced to the origin (J, = 0, R = I), thereby removing the dependence of the estimate on data points neighboring .lV: 1-R
(34
u=oy where u has the property lim J, + 0 [u] = - (r,/(P;S).
(3b)
This definition for u is suggested by the equality between u and dR/dJ, at zero J V’ That a reference slope, u, can be determined for each data point raises the question: What is the correlation between u and J, which yields the best estimate of u at zero J,? This relationship is not readily apparent due to the nonlinear nature of Eq. (1~). However, since the exponential form is the result of convection and diffusion occurring across a common pathway, in this case a pore, we can avoid this nonlinearity by considering a membrane for which convection and diffusion occur across separate pathways. For this approximation the diffusive and convective contributions are strictly additive: J,C, = (1 - u,)JvCp + P;S(C, - C,).
W-4
An equation of this form was first proposed by Pappenheimer (1953) and subsequently written in terms of u, by Parker et al. (1981). It follows from Eq. (4a) that R = 1 - agv/(P;S
+ Jv).
(4b)
Equation (4b) satisfies exactly the high and low J, limits as expressed by Eqs. (2a) and (2b). This is in contrast to Eq. (Id): although the limit of dR/dJ, as J,+ 0 is correctly predicted, the high J, limit of R is (1 - cr,)/(l + us), thus leading to errors in the estimation of (T, and hence P;S. Agreement between Eq. (4b) and the exact equation at the limits of high and low volume flow occurs because coupling between convection and diffusion is significant only at intermediate volume flows. Equation (4~) therefore provides a useful guide for correlating u and Jv: -U/.4
= W~,)Jv + (P,Wu,.
(4c)
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100
20 0 0
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40
J,,
60
80
100
/.dlmin
FIG. 1. A plot of (- l/u) as a function of the lymph flow (J,); II is calculated using Eq. (3a) for
the experimental data (0) of Rutili er al. (1982). The solid line is the linear least-squares fit of the data.
When u is determined from the experimental data according to Eq. (3a), Eq. (4~) suggests that a plot of (l/u) vs J, will be more or less linear, thereby facilitating extrapolation to zero volume flow. The important point here is that Eq. (lc) is not approximated by a linear equation; rather, a linear equation is used only to suggest a means for correlating the experimental data to allow for an accurate estimation of the slope of R at zero J,. The utility of this correlation rests in the equality [u] = [dR/dJ,] = -(r,/(P;S) at J, = 0. Figure 1 illustrates this procedure using the plasma-to-lymph measurements of Rutili et al. (1982). The J, = 0 intercept, and hence (P;S)/o,, is determined to be -3.2 ,ul/min from a linear least-squares fit of the data (- l/u) vs J, for J, < -20 pl/min; (T, is found to be -0.90 from the data for R at high lymph flows, i.e., for J, > -60 pl/min. In Fig. 2 Eq. (lc) is compared with the experimental 1.0
8.0
6.0 R 4.0 2.0 0
%k-d+-G
0
100
J,, /d/m~n FIG. 2.
The lymph-to-plasma total protein ratio (R) as a function of lymph flow (JJ. The open circles are the experimental data of Rutili et al. (1982); the solid line is the predicted curve for crp = 0.90 and P;S = 2.9 pl/min using Eq. (1~).
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data. It is apparent that this simple technique, which circumvents the nonlinear data reduction posed by Eq. (Ic), provides a good fit of the exact homoporous equation to the experimental data at all lymph flows. However, solute flux measurements suggest the existence of at least a two-pore pathway across most vascular membranes and that solute charge may indeed be a factor in determining solute transport. Therefore, the set {us, Ps*,S}thus found is only approximate since Eq. (lc) does not account for heteroporosity and charge effects (Deen et al., 1980; Groome et al., 1983).As discussed by Rut& et al. (1982), the discrepancy between their data and the theoretical homoporous equation at low lymph flows may be due to experimental error and a delay in solute equilibration, in addition to heteroporosity and charge effects. CONCLUDING
REMARKS
Regardless of the method used to evaluate u, and P;S, it is possible that phenomena not accounted for by Eq. (lc) may be encountered at the two extremes in J,: “pore stretching” may occur due to the large intravascular pressure needed to maintain a high lymph flow, and the lymphatic concentration may be altered by the interstitium before collection, particularly when the lymph flow is low. If “pore stretching” and the interstitial contributions do exist, it is probable that such effects do not occur at discrete values for J,, but rather exist to varying degrees over the entire range of lymph flows. Since it is not possible to establish upper (“pore stretching”) and lower (“interstitial modification”) limits for J, which minimize these effects, it is best to assume a model for solute transport and determine the set {gs, Ps.S} which best describes (with this model) the experimental data over the entire region of J,. If one is to assume a neutral solute-homoporous membrane model, then parameter estimation is based on Eq. (1~). Since this equation has only two degrees of freedom, once us and P;S are specified the J, dependence of R is completely defined. It is shown [Eqs. (2a) and (2b)] that these two (homoporous) parameters can be simply estimated from appropriate limits of R. Moreover, if these two limits are not satisfied, then the set {crs,,P;S} is inconsistent with Eq. (1~). Of course, one must always consider the effects of heteroporosity and solute charge on the values found for u, and P;S when using Eq. (lc), regardless of the method used to estimate these coefficients. APPENDIX Here details of the derivation of Eq. (2b) are presented. Differentiating Eq. (lc) with respect to J, yields dR/dJ, = - [(I - us)/( 1 - cr,e-K)2] . d(1 - usewK)/dlv. Since K = (1 - u,)J,/(P;S),
(Al)
it follows that
d(1 - u,eeK)/dJv = u&l - u,)emK/(P;S),
WI
or, on substituting Eq. (A2) into Eq. (Al), dR/dl,
= - [u,/(P;S)lYK
((1 - a,)/(1 - use-“)}‘.
(A3)
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However, the term in { . . . } is simply R. Equation (2b) is thus obtained by taking the low J, limit of Eq. (A3), noting that R + 1 as J, + 0. ACKNOWLEDGMENTS The authors thank Cathy Couvillon for secretarial assistance and W. Faye Boykins for drawing the figures. They also thank Dr. John N. Diana and Dr. Aubrey E. Taylor for their thoughtful comments. This work was supported in part by the American Heart Association-Louisiana Fellowship, National Heart Lung and Blood Institute New Investigator Research Award HL 2799, and a research grant from the American Lung Association.
REFERENCES DEEN, W. M., SATVAT, B., AND JAMIESON, J. M. (1980). Theoretical mode1 for glomerular filtration
of charged solutes. Amer. J. Physiol. 238, F120-F139. GRANGER, D. N., AND TAYLOR, A. E., (1980). Permeability of intestinal capillaries to endogenous
macromolecules. Amer. J. Physiol. 238, H457-H464. GROOME, L. J., KINASEWITZ, G. T., AND DIANA, J. N. (1983). Diffusion and convection across
heteroporous membranes: A simple macroscopic equation. Microvasc. Res. 26, 307-322. LEVITT, D. G. (1975). General continuum analysis of transport through pores. I. Proof of Onsager’s
reciprocity for a uniform pore. Biophys. J. 15, 533-551. PAPPENHEIMER, J. R. (1953). Passage of molecules through capillary walls. Physiol. Rev. 33, 387423.
PARKER,J. C., GRAIN, M., G~~MBERT,F., RUTILI, G., AND TAYLOR,A. E. (1981). Total lung lumph flow and fluid compartmentation in edematous dog lungs. J. Appl. Physiol. 51, 1268-1277. PATLAK, C. S., GOLDSTEIN, D. A., AND HOFFMAN,J. F. (1963). The flow of solute and solvent across a two-membrane system. J. Z’heor. Bid. 5, 426-442. RUTILI, G., GRANGER, D. N., TAYLOR, A. E., PARKER, J. C., AND MORTILLARO, N. A. (1982). Analysis of lymphatic protein data. IV. Comparison of the different methods to estimate reflection coefficients and permeability-surface area products. Microvasc. Res. 23, 347-360. YABLONSKI, M. E., AND LIFSON,N. (1976). Mechanism of production of intestinal secretion by elevated venous pressure. J. Clin. Invest. 57, 904-915.
RESEARCH
30, 235-241 (1985)
BRIEF COMMUNICATIONS A Simple Method for Determining the Homoporous Solute-Membrane Permeability from Plasma-to-Lymph Measurements LYNN J. GROOME* AND GARY T. KINASEWITZ~ *Department of Physiology and Biophysics and TDepartment of Medicine, Louisiana State University School of Medicine, Shreveport, Louisiana 71130 Received August 29, 1983 The exact equation for ultratiltration of a neutral solute across a homoporous membrane contains only two adjustable parameters, the reflection coefficient (a,) and the permeability coefficient (PJ-surface area (S) product, P$.S. Specifying (T, and PJS therefore defines completely the dependence of the protein ratio (R) on the volume flow (JJ for a homoporous membrane. The reflection coefficient is determined from the high J, limit of R: lim J,+m [R] = (1 - u,). The purpose of this paper is to present a simple procedure for estimating P;S from the slope of R as J, approaches zero: lim J, + 0 [dR/dlJ = -u,/(P;S). Both relations are exact limits of the nonlinear neutral solute/homoporous membrane transport equation and thus provide a simple, yet rigorous, method for estimating W$and P%4’. o 1985 Academic
Press, Inc.
INTRODUCTION In the case of plasma-to-lymph ultrafiltration, plasma is forced across a heteroporous capillary membrane in response to a hydrostatic pressure difference and the newly formed lymph is collected downstream; steady state is achieved once the lymph flow (J,) and the plasma (C,) and lymph (CL) solute concentrations are no longer changing with respect to time. A series of such measurements, obtained by changing the hydrostatic pressure difference and allowing J, and R (=C,/C,,) to become constant over a period of time, provides a relationship between R and J, at steady state. The dependence of R on J, observed experimentally is then used to estimate the effective solute-membrane reflection coefficient (a,) and the permeability coefficient (P&surface area (S) product, P;S. The values found for a, and P;S, however, depend on the theoretical equation relating R to J, which, in effect, defines these coefficients. Moreover, because of the complex nature of the vascular membrane, it is often necessary to use an approximate equation for R. However, for the special case of a neutral solute and a homoporous membrane, steady-state ultrafiltration can be described by an exact equation. Consider a membrane composed of identical pores separating well-stirred solutions of a single neutral solute at different concentrations (CL, C,). The local form of the 235 00262862’85 $3.00 Copyright 8 1985 by Academic Press, Inc. All rights of reproduction in any form resewed. Printed in U.S.A.
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steady-state equation for the solute flux across a single pore is j, = -D, (dc,/dx) + (1 - CT& CS,
(la)
where c, is the concentration of solute at a distance x into the pore and dc,/dx is the concentration gradient at position x; j, and j, denote the single-pore solute and volume fluxes, respectively. Equation (la) states that the single-pore solute flux at point x is the sum of the diffusive [ -D,(dc,/dx)] and convective [(1 a&c,] fluxes at that point. For a specified set of membrane boundary conditions, the steady-state mass balance requires that bothj, andj, be constant with respect to position within the pore. Integrating Eq. (la) across the length of the pore (for constant j, and j,) yields J, = (1 - ~#,[c,e-”
- C,]/(e-K - I),
(lb)
where K = (1 - (r,)J,/(P;S) for the membrane as a whole. Equation (lb) was first derived by Patlak and co-workers (1963) and subsequently by Levitt (1975). Due to the nature of the ultrafiltration process, in which the downstream lymph concentration is simply CL = J,/J,, it follows from Eq. (lb) that (Granger and Taylor, 1980) R = (1 - cr,)/(l - a,e?). UC) This equation is exact for the steady-state ultrafiltration of a neutral solute by convection and diffusion across a common homoporous pathway. That Eq. (lc) is exact, if only for the case of ultrafiltration across a homoporous membrane, is not insignificant, since it provides a simple, well-defined standard to which experimental measurements can be compared. Any other equation describing ultrahltration of a neutral solute across a homoporous membrane, e.g., the arithmetic-mean equation R = (1 - v,)/2 + (Ps * W/J” (1 + us)/2 + (P, * WJ”’
(14
would not only approximate the true charged solute/heteroporous membrane, but would also be an approximation to Eq. (1~). This introduces further uncertainty into the estimation of the set {us, P;S). For example, if the arithmetic-mean approximation is used, deviations from Eq. (lc) become significant at relatively low lymph flows, and the estimate for us, and hence P;S, is therefore incorrect (Granger and Taylor, 1980). Although Eq. (lc) is an “exact approximation” which provides a good fit of the experimental data, the exponential dependence of R on J, detracts from its usefulness and is in part responsible for the popularity of Eq. (Id). For example, a trial-and-error search for the set {us, P;S} which best fits the data is necessary if Eq. (lc) is used: an initial guess for us and P;S is made, the sum-of-squares error is calculated, and, if not of sufficient accuracy, a new guess is made for uS and P;S and the process is repeated. By linearizing Eq. (lc), e.g., by using Eq. (Id), the trial-and-error procedure is bypassed and an approximate set {a,, Ps*S} is determined directly without iteration. Ideally, combining the simplicity of the linear least-squares analysis with the rigor of Eq. (lc) provides an optimal solution for the set {us, P;S}. This is, in fact, possible. Since the precise form of the ultrafiltration curve for sieving of a neutral solute across a homoporous membrane is determined entirely by us
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and P;S, the behavior of R in certain limits of J, yields these two coefficients directly, without having to first linearize Eq. (1~). A unique value for (T, can be found from measurements of R at high lymph flows (Yablonski and Lifson, 1976; Granger and Taylor, 1980). The purpose of this paper is to present a simple procedure for estimating P;S from the slope of R as J, approaches zero. Because Eq. (lc) has only two degrees of freedom, choosing crs and P;S based on the limiting behavior of R not only circumvents the nonlinear data reduction posed by Eq. (lc), but also defines completely the J, dependence of R for ultrafiltration of a neutral solute across a homoporous membrane. ANALYTIC
METHOD
The motive for choosing the slope of R as J, approaches zero as a basis for estimating P;S is not readily apparent on examination of Eq. (1~). Here a simple physical argument suggests the appropriate limits for evaluating both g‘, and P;S. Consider, for example, a volume of fluid moving down a pore with a mean velocity Jv/S. Since crs can be thought of as the fraction of solute molecules which are reflected at the pore entrance and hence do not penetrate the membrane by convection, the fraction which is in the volume element when it enters the pore is (1 - (T,). As the control volume travels down the pore it will exchange solute molecules by diffusion with the adjacent fluid. The net exchange, however, depends on the length of time that the volume element remains in the pore. Since an increase in J, decreases the time available for diffusion, a value for J, exists above which ditfusion is negligible. In this limit the difference in solute concentration across the membrane is the greatest, and R approaches (1 - a,) as J, becomes infinitely large: lim J, + a [R] = 1 - u,.
(24
This limit for estimating u, has been suggested elsewhere (Yablonski and Lifson, 1976; Granger and Taylor, 1980), and can be derived by simply taking the high J, limit of Eq. (1~). In contrast, when the volume flow is small, sufficient time exists for diffusion to significantly reduce the concentration difference between volume elements; R therefore approaches unity as J, approaches zero. Although the concentration difference across the membrane is particularly sensitive to P;S in the region, R will always equal one at zero volume flow, regardless of the value of P;S. However, as J, is increased slightly above zero, a solute with a high permeability will remain nearer to equilibrium relative to a solute with a low permeability. Therefore, the appropriate limit to estimate P;S is the rate at which R approaches unity as J, approaches zero. This can be expressed mathematically by using Eq. (lc) to determine the slope of R, dR/dJ,, in the limit of zero volume flow (see Appendix): lim J, + 0 [dR/dl,]
= - cr&P;S).
(2b)
Equations (2a) and (2b) are exact limits of Eq. (1~). DISCUSSION For a given set {us, P;S}, the concentration difference across the membrane is a function of J, only. Therefore, determining us and P;S from the experimental
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data at the limits given by Eqs. (2a) and (2b) completely defines the J, dependence of R for a neutral solute across a homoporous membrane. Use of these relations, however, requires that lymph-to-plasma protein ratios are measured at the two extremes of high and low lymph flows. Any fluctuations in the data in these regions can lead to errors in us and P;S. While this is not too severe a constraint on (T,, P;S is particularly dependent on the accuracy of the experimental measurements, since this coefficient is determined from the slope of R at a single point, i.e., zero J,. In estimating P;S, therefore, it is desirable to use as many of the data points as possible. One approach is to determine dR/dJ, for various volume flows and extrapolate this to zero J,. However, fluctuations in the data on either side of a given J, can lead to significant errors in dR/dJ, at J,. To minimize this error, we define a slope referenced to the origin (J, = 0, R = I), thereby removing the dependence of the estimate on data points neighboring .lV: 1-R
(34
u=oy where u has the property lim J, + 0 [u] = - (r,/(P;S).
(3b)
This definition for u is suggested by the equality between u and dR/dJ, at zero J V’ That a reference slope, u, can be determined for each data point raises the question: What is the correlation between u and J, which yields the best estimate of u at zero J,? This relationship is not readily apparent due to the nonlinear nature of Eq. (1~). However, since the exponential form is the result of convection and diffusion occurring across a common pathway, in this case a pore, we can avoid this nonlinearity by considering a membrane for which convection and diffusion occur across separate pathways. For this approximation the diffusive and convective contributions are strictly additive: J,C, = (1 - u,)JvCp + P;S(C, - C,).
W-4
An equation of this form was first proposed by Pappenheimer (1953) and subsequently written in terms of u, by Parker et al. (1981). It follows from Eq. (4a) that R = 1 - agv/(P;S
+ Jv).
(4b)
Equation (4b) satisfies exactly the high and low J, limits as expressed by Eqs. (2a) and (2b). This is in contrast to Eq. (Id): although the limit of dR/dJ, as J,+ 0 is correctly predicted, the high J, limit of R is (1 - cr,)/(l + us), thus leading to errors in the estimation of (T, and hence P;S. Agreement between Eq. (4b) and the exact equation at the limits of high and low volume flow occurs because coupling between convection and diffusion is significant only at intermediate volume flows. Equation (4~) therefore provides a useful guide for correlating u and Jv: -U/.4
= W~,)Jv + (P,Wu,.
(4c)
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120
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20 0 0
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40
J,,
60
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100
/.dlmin
FIG. 1. A plot of (- l/u) as a function of the lymph flow (J,); II is calculated using Eq. (3a) for
the experimental data (0) of Rutili er al. (1982). The solid line is the linear least-squares fit of the data.
When u is determined from the experimental data according to Eq. (3a), Eq. (4~) suggests that a plot of (l/u) vs J, will be more or less linear, thereby facilitating extrapolation to zero volume flow. The important point here is that Eq. (lc) is not approximated by a linear equation; rather, a linear equation is used only to suggest a means for correlating the experimental data to allow for an accurate estimation of the slope of R at zero J,. The utility of this correlation rests in the equality [u] = [dR/dJ,] = -(r,/(P;S) at J, = 0. Figure 1 illustrates this procedure using the plasma-to-lymph measurements of Rutili et al. (1982). The J, = 0 intercept, and hence (P;S)/o,, is determined to be -3.2 ,ul/min from a linear least-squares fit of the data (- l/u) vs J, for J, < -20 pl/min; (T, is found to be -0.90 from the data for R at high lymph flows, i.e., for J, > -60 pl/min. In Fig. 2 Eq. (lc) is compared with the experimental 1.0
8.0
6.0 R 4.0 2.0 0
%k-d+-G
0
100
J,, /d/m~n FIG. 2.
The lymph-to-plasma total protein ratio (R) as a function of lymph flow (JJ. The open circles are the experimental data of Rutili et al. (1982); the solid line is the predicted curve for crp = 0.90 and P;S = 2.9 pl/min using Eq. (1~).
240
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data. It is apparent that this simple technique, which circumvents the nonlinear data reduction posed by Eq. (Ic), provides a good fit of the exact homoporous equation to the experimental data at all lymph flows. However, solute flux measurements suggest the existence of at least a two-pore pathway across most vascular membranes and that solute charge may indeed be a factor in determining solute transport. Therefore, the set {us, Ps*,S}thus found is only approximate since Eq. (lc) does not account for heteroporosity and charge effects (Deen et al., 1980; Groome et al., 1983).As discussed by Rut& et al. (1982), the discrepancy between their data and the theoretical homoporous equation at low lymph flows may be due to experimental error and a delay in solute equilibration, in addition to heteroporosity and charge effects. CONCLUDING
REMARKS
Regardless of the method used to evaluate u, and P;S, it is possible that phenomena not accounted for by Eq. (lc) may be encountered at the two extremes in J,: “pore stretching” may occur due to the large intravascular pressure needed to maintain a high lymph flow, and the lymphatic concentration may be altered by the interstitium before collection, particularly when the lymph flow is low. If “pore stretching” and the interstitial contributions do exist, it is probable that such effects do not occur at discrete values for J,, but rather exist to varying degrees over the entire range of lymph flows. Since it is not possible to establish upper (“pore stretching”) and lower (“interstitial modification”) limits for J, which minimize these effects, it is best to assume a model for solute transport and determine the set {gs, Ps.S} which best describes (with this model) the experimental data over the entire region of J,. If one is to assume a neutral solute-homoporous membrane model, then parameter estimation is based on Eq. (1~). Since this equation has only two degrees of freedom, once us and P;S are specified the J, dependence of R is completely defined. It is shown [Eqs. (2a) and (2b)] that these two (homoporous) parameters can be simply estimated from appropriate limits of R. Moreover, if these two limits are not satisfied, then the set {crs,,P;S} is inconsistent with Eq. (1~). Of course, one must always consider the effects of heteroporosity and solute charge on the values found for u, and P;S when using Eq. (lc), regardless of the method used to estimate these coefficients. APPENDIX Here details of the derivation of Eq. (2b) are presented. Differentiating Eq. (lc) with respect to J, yields dR/dJ, = - [(I - us)/( 1 - cr,e-K)2] . d(1 - usewK)/dlv. Since K = (1 - u,)J,/(P;S),
(Al)
it follows that
d(1 - u,eeK)/dJv = u&l - u,)emK/(P;S),
WI
or, on substituting Eq. (A2) into Eq. (Al), dR/dl,
= - [u,/(P;S)lYK
((1 - a,)/(1 - use-“)}‘.
(A3)
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241
However, the term in { . . . } is simply R. Equation (2b) is thus obtained by taking the low J, limit of Eq. (A3), noting that R + 1 as J, + 0. ACKNOWLEDGMENTS The authors thank Cathy Couvillon for secretarial assistance and W. Faye Boykins for drawing the figures. They also thank Dr. John N. Diana and Dr. Aubrey E. Taylor for their thoughtful comments. This work was supported in part by the American Heart Association-Louisiana Fellowship, National Heart Lung and Blood Institute New Investigator Research Award HL 2799, and a research grant from the American Lung Association.
REFERENCES DEEN, W. M., SATVAT, B., AND JAMIESON, J. M. (1980). Theoretical mode1 for glomerular filtration
of charged solutes. Amer. J. Physiol. 238, F120-F139. GRANGER, D. N., AND TAYLOR, A. E., (1980). Permeability of intestinal capillaries to endogenous
macromolecules. Amer. J. Physiol. 238, H457-H464. GROOME, L. J., KINASEWITZ, G. T., AND DIANA, J. N. (1983). Diffusion and convection across
heteroporous membranes: A simple macroscopic equation. Microvasc. Res. 26, 307-322. LEVITT, D. G. (1975). General continuum analysis of transport through pores. I. Proof of Onsager’s
reciprocity for a uniform pore. Biophys. J. 15, 533-551. PAPPENHEIMER, J. R. (1953). Passage of molecules through capillary walls. Physiol. Rev. 33, 387423.
PARKER,J. C., GRAIN, M., G~~MBERT,F., RUTILI, G., AND TAYLOR,A. E. (1981). Total lung lumph flow and fluid compartmentation in edematous dog lungs. J. Appl. Physiol. 51, 1268-1277. PATLAK, C. S., GOLDSTEIN, D. A., AND HOFFMAN,J. F. (1963). The flow of solute and solvent across a two-membrane system. J. Z’heor. Bid. 5, 426-442. RUTILI, G., GRANGER, D. N., TAYLOR, A. E., PARKER, J. C., AND MORTILLARO, N. A. (1982). Analysis of lymphatic protein data. IV. Comparison of the different methods to estimate reflection coefficients and permeability-surface area products. Microvasc. Res. 23, 347-360. YABLONSKI, M. E., AND LIFSON,N. (1976). Mechanism of production of intestinal secretion by elevated venous pressure. J. Clin. Invest. 57, 904-915.