Interstitial diffusion of macromolecules in the rat mesentery

MICROVASCULAR

RESEARCH

18,

Interstitial JAMES California

255-276 (1979)

Diffusion of Macromolecules in the Rat Mesentery’ R. Fox AND HAROLD

Institute

of Technology,

Pasadena,

Received

12, 1977

July

WAYLAND California

91125

Apparent diffusion coefficients of fluorescein isothiocyanate (FIT(Z)-labeled dextrans (M, 3400-41,200) and rat serum albumin (RSA-FITC) were measured in the interstitial space of the rat mesentery. Average ratios of apparent diffusion coefficient to free diffusion coefficient for the dextrans ranged from 0.26 for dextran 3400 to 0.033 for dextran 41,200 and 0.073 for RSA-FITC. These values are lower than those reported for loose connective tissue (umbilical cord) but greater than those from dense connective tissue (cartilage). The apparent diffusion coefficient for RSA-FITC was approximately one-half the apparent diffusion coefficient of FITC-dextran having the same free diffusion coefficient (and thus the same Stokes radius). Mathematical models describing solute diffusion in gels or glycosaminoglycan solutions as a function of solute size did not give a good fit to the data but indicated that the presence of a relatively dense interstitial matrix is required in order to explain the data.

The physical and chemical properties of the components of the interstitial space are important in determining transvascular fluid movement and tissue hydration (Wiederhielm, 1968; Guyton et al., 1971) and also interstitial solute transport (Comper and Laurent, 1978; Wayland and Silberberg, 1978; Preston and Snowden, 1973; Laurent, 1972; Ogston, 1970). Studies of solute diffusion in various tissues or in model systems indicate a significant barrier may exist to interstitial solute movement, especially for large molecules (Maroudas, 1970; Preston and Snowden, 1973). The relationship between interstitial solute diffusibility and solute size, shape, and charge is important in understanding transport of solutes in tissue as well as shedding additional light on the composition and physicalchemical properties of the interstitial components. To examine the relationship between interstitial transport and molecular size we have quantitated the movement of serum albumin and dextrans of various molecular weights in a simple two-dimensional connective tissue, the rat mesentery, in vivo. METHODS Fluorescent tracer preparation. Narrow-fraction dextrans labeled with fluorescein isothiocyanate (FITC) were obtained from Pharmacia AB (Uppsala, Sweden) (de Belder and Granath, 1973; Schroder et al., 1976). Degree of substitution (FITC molecules per glucose subunit) ranged from 8 x lop3 to 6 x 10p4. Dextrans were dissolved in physiological solution to give a final concentration of 300 mgml. 1 Supported by USPHS Grant HL 08977. 255 OOZC2862/79/050255-22102.0010 Copyright @ 1979 by Academic Prcra. Inc. All rights of reproduction in any form reserved. Printed in U.S.A.

256

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Rat serum albumin (fraction V powder, Sigma Chemical Co., St. Louis, MO.) was labeled with FITC using the method described by Landel (1976). Albumin dimers and other high molecular weight impurities were removed from the RSAFITC by elution from a Sepharose 6B column with Nairn’s phosphate-buffered saline, pH 7.4. The final concentration was approximately 5 g % protein, and the degree of substitution was approximately 1.0 molecules of FITC per molecule of albumin (Landel, 1976). Animal preparation. Sixteen female Sprague-Dawley rats, age 3-4 months (Hilltop Labs, Inc., Chatsworth, Calif.) were anesthetized with sodium pentobarbital, 30 mg/kg ip, and tracheotomized, and a cannula was placed in the external jugular vein for tracer injection. In most experiments a supplemental dose of anesthetic was given 20 min prior to tracer injection. Loops of small intestine were exteriorized through a midline abdominal incision into a bath of mineral oil maintained at 37”. After a suitable area of mesentery was selected, the membrane was gently spread over a circular glass coverslip (12 mm diameter) supported by a hollow cylindrical pedestal, and was covered by a second coverslip held in place by a hollow cylindrical weight. If circulation was visibly impaired, a new mesenteric fan was selected. A drip of 37” mineral oil was maintained above the upper coverslip. Rat rectal temperature was automatically maintained at 37” by a heating pad. Optics and instrumentation. The mesentery was epi-illuminated with the 488-nm line of an argon laser, with the tissue at the front principal focus of a Leitz UM 32/.30 objective modified for immersion (Wayland, 1975). The image was formed at the face of a silicon intensifier target television tube (SIT) with a 300-mm focal length transfer lens (total optical magnification 52x), and the resulting video signal recorded on a video tape recorder (VTR 1200, Sanyo Electric, Inc.). Selected single frames of videotape were later digitized into 256 x 256 arrays of eight-bit picture elements and stored on digital tape (Wayland et al., 1975) for subsequent computer analysis by a PDP 8f (Digital Equipment Corp., Maynard, Mass.). The spacing between adjacent picture elements (relative to the tissue) was 1.1 pm. Computer analysis permitted a correction of the data for each picture element to yield a value which ranged from 0 to 255 and which was directly proportional to fluorescence intensity at the corresponding location in the picture (see Appendix II). Isointensity contour plots can then be made by a digital plotter from the corrected picture element values, or these values can be printed from selected picture subregions (Wayland and Fox, 1978). Experimental procedures. It was found that a fluid film existed on the serosal surface of the mesothelium when the mesentery was placed in mineral oil (Fox et al., 1977), giving an alternative high-permeability pathway for tracer transport. Placing glass coverslips above and below the mesentery, however, caused separation of the oil and water phases, and allowed selection of regions where it appeared that only oil was in contact with the tissue. It was assumed that in these regions tracer movement was confined to the mesenteric membrane. Handling of the tissue generally produced local areas of increased microvascular permeability. These areas were selected as sites where the intravascularly injected fluorescent tracer would rapidly reach the interstitial space. To locate these sites and to determine regions where only oil was in contact with the tissue, a small amount (0.1 ml) of tracer was injected and the pattern of movement

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FIG. 1. Rat mesentery 5 min after intravenous injection of FITC-dextran (ji?, cence appears white against a dark background. (a) Low magnification (41 x) shows is in contact with the tissue, appearing as dark areas. (b) High magnification (336~) oil is in contact with the tissue and surrounded by an aqueous region (white). Under tracer is restricted to the tissue, giving a mottled and in some places filamentous

257

19,100). Fluoresregions where oil of a region where the oil phase the appearance.

observed for several minutes. A fairly rapid movement of tracer through the aqueous regions adjacent to the tissue was observed, outlining areas where only oil was in contact with the tissue (Fig. 1). Five minutes after the first tracer injection, 1.0-1.5 ml of the tracer solution was injected during a 30-set interval and its movement through the selected region was recorded on videotape for 2 min after it first appeared in the microscopic field of view. The oil temperature directly above the tissue was then measured. System calibration was performed by placing

258

FOX

AND

FIG.

WAYLAND

l-conrinued

various dilutions of the tracer solution in a hemocytometer under the microscope (see Appendix II and Wayland and Fox (1978)). If any movement of the tissue or change of focus occurred during the period of data collection the experiment was discarded. Calculation of apparent diffusion coefficients. Isointensity contour plots generally showed regions where contours were straight and parallel (Fig. 2). Assuming uniform concentration in the direction perpendicular to the membrane (see Appendix I), a one-dimensional transport model can be used in these regions. A portion of the picture was selected from one of these regions and a computer printout made of the corrected values of picture elements in a direction perpendicular to the contours (defined as the x-direction). Thus, assuming tissue of

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OF

t = 10 set

MACROMOLECULES

259

t =40 set

FIG. 2. Isointensity contours from a portion of a videoframe taken at 10 and 40 set after appearance of FITC-dextran (;i?, 19,100) at the edge of an oil-covered region. The subregion chosen for analysis is indicated by a rectangle, with increasing concentration toward the left.

uniform thickness, values proportional to tissue tracer concentration were obtained as a function of x for various times. In the example illustrated in Fig. 2, the subregion chosen for analysis (indicated by a rectangle) was a 6 x 40 array of picture elements. Averages of the six values from the columns of data parallel to the contours were computed to reduce variations due to electrical noise and local differences in tissue properties, and these values were plotted as a function of distance from the left edge of the rectangle (Fig. 3). Values at the various times at x = 0 and values at the various locations at t = 10 set were used as boundary and initial conditions, respectively, for the numerical solution (see Appendix III) of the one-dimensional diffusion equation: Wx,t)

at

= D

a”c(x,t)

ax2

’

where C is solute concentration as a function of distance, x, and time, t, and D is the diffusion coefficient (assumed constant). The value of diffusion coefficient that gave the best fit to the data was called D’, the apparent diffusion coefficient for the tracer in mesentery, to distinguish it from the free diffusion coefficient in distilled water extrapolated to infinite dilution, D,.

260

FOX

AND

WAYLAND

CONCENTRATION

b\

OF D,,,,,

VS. x

1

Experimental

Theoretical -d=l.5~IO-~cm~/sec

IO

20 x (w)

30

FIG. 3. Concentration (arbitrary units) of FITC-dextran (aw 19,100) at various times as a function of distance from the left edge of the rectangle in Fig. 2. The solid lines are solutions to the diffusion equation (Eq. 1) for a value of D that gave a good fit to the data.

To investigate the possible role of convection in tracer transport, experimental data were also compared to solutions of an equation describing diffusion and convection in an ideal one-dimensional binary system:

where V is solvent velocity. Solutions to this equation, obtained in the same manner as described above, were computed for various values of D and V until the best fit to the data was obtained (Wayland and Fox, 1978). RESULTS Injection of RSA-FITC usually gave no observable change in blood flow in the mesenteric microvasculature or in the systemic blood pressure. On the other hand, injection of dextran-FITC generally caused a decrease in systemic arterial blood pressure, and visibly slowed the flow in the mesenteric microvessels, presumably due to the histamine-releasing action of dextran in the rat. Although dextran modified the microcirculatory flow and even the transmural transport of the tracer, it is unlikely that these changes affected the data reported here, which are involved only with measurements of molecular movement within the interstitial space. Local heterogeneities in tracer distribution in the mesentery were observed, giving the tissue a mottled or sometimes filamentous appearance (Fig. l), which

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261

was more apparent for the high molecular weight tracers. This uneven tracer distribution, however, was not resolved in the digitized picture due to the coarse picture element spacing (1.1 pm) and to spatial filtering, and was manifesfested as “noise” in the contours and concentration profiles. Occasionally larger areas (5-10 pm diameter) of tracer exclusion were seen. These were assumed to be regions occupied by cells (fibroblasts, mast cells) and were avoided when choosing subregions for analysis (Fig. 4). Regional variations in tracer distribution, as would be expected for regional variation in tissue thickness or volume available to the tracer, were rarely observed over the areas studied and were not tolerated in subregions chosen for analysis. Contour plots from a typical experiment are shown in Fig. 2 and the concentration profiles from the selected subregion are illustrated in Fig. 3. Solid lines are solutions to the diffusion equation for a value of diffusion coefficient that gave the best fit to the data. In 13 of 16 experiments the diffusion equation satisfactorily fit all of the data for the low molecular weight tracers and fit the data for early times for the high molecular weight tracers. However, for the high molecular weight tracers there was systematic deviation from the diffusion equation at later times. In addition, in three experiments it was impossible to obtain any satisfactory fit to the data. It was suspected that in these three experiments mineral oil was only in contact on one side of the membrane, giving a parallel pathway for tracer transport in the aqueous phase. To illustrate the sensitivity of the solution of Eq. (1) to values of D’, Fig. 5 shows the experimental data at 40 set (from Fig. 3) and solutions of Eq. (1) for three values of D’. Typically the experimental data fell within an envelope which was formed by curves with theoretical values of D’ which were 85 and 115% of our best estimate for the interstitial D’. Since the initial condition used for the solution of Eq. (1) was empirically determined for each experiment, no meaningful statistical test could be applied to the calculated values for D’. Accepting the maximum uncertainty in the D’ values which were calculated from the experimental data does not alter our observation that there is a strong molecular weight dependence for the apparent diffusion coefficient in the interstitium of the rat mesentery. Values obtained for the apparent diffusion coefficient of the various molecular weight dextrans and for rat serum albumin from the 13 experiments in which a value of D’ could be obtained giving an acceptable fit to the data are listed in Table 1. Each value of D’ represents a single experiment, each in a different animal. Values in Table 1 for free diffusion coefficients for dextrans were obtained by extrapolation or interpolation from values of Granath and Kvist (1967) converted to 37” using absolute temperature and viscosity of distilled water.

DISCUSSION Application of the diffusion equation (Eq. (1)) to a system as complex as the interstitial space requires some caution, since this equation was derived for an infinitely dilute, homogeneous, isotropic binary solute-solvent system. Although steric and possibly electrostatic interaction between tracer molecules and other solutes and with the interstitial matrix undoubtedly occur, and although local variation in tracer distribution was observed (similar to patterns reported by

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WAYLAND

INTERSTITIAL

DIFFUSION

60

OF

I

CONCENTRATION

01 0

263

MACROMOLECULES

IO

OF D,9,,,,,o VS. x

20

30

40

x (pL)

FIG. 5. Data from Fig. 3 at 40 set (filled circles) and solutions to the diffusion equation (Eq. 1) for three values of D.

McMaster and Parsons (1939) and by Wiederhielm (1966)), Eq. (1) was applied to a sufficiently large area (generally 5.5 x 44 pm) that any random spatial variation of tissue properties tended to average out, and systematic deviations from ideality were manifested in the value obtained for the diffusion coefficient. Thus the apparent diffusion coefficient, D’, must be considered simply a descriptive parameter that results from the particular model chosen, and more sophisticated models must be considered (see below) before this parameter can be related to actual molecular phenomena. However, since transport of saccharides and serum albumin through tissue seems to fit the diffusion equation, the apparent diffusion coefficient is a useful way to compare data from the various types of tracers and data from other laboratories. There are several possible sources of error in our experiments. First, since transport measurements are made near regions of increased microvascular permeability, tissue hydration could be greater than normal. This would tend to increase the apparent diffusion coefficient for the larger tracers (Granger et al., 1975). Second, a convective component of tracer transport (which would tend to be in the same direction as the diffusive flux) could lead to an overestimate of D’. This would also be most pronounced for the larger tracers. Third, our data are biased toward tracer molecules with the greatest fluorescent emission and thus the greatest degree of substitution. This, in turn, would bias our data toward tracer molecules with the greatest deviation in molecular size, shape, charge, and other properties from the unlabeled molecule. With a degree of substitution of O.Ol0.001 FITC molecules per glucose subunit, the relative effect of the label on the dextran molecule should be greatest for the low molecular weights. Fourth, since

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TABLE

1

APPARENT DIFFUSION COEFFICIENTS OF FLUORESCENT TRACERS IN MESENTERY”

DO Tracer Dextran-FITC

RSA-FITC

(cm%ec x 10’)

;i?,

D’

(cm*/sec x 10’)

D’ID,

3,400

2,500

14.3

22.9

5.6 5.9

0.25 0.26

19,100

17,300

31.2

10.6

1.5

1.5

0.14 0.14

19,000

17,500

26,350 35,200 41,200

26,000 28,200

68,000

32.2 36.4 42.1 44.9

10.6 9.1 7.9 7.3

2.6 1.15 0.74 0.18 0.29

0.25 0.13 0.094 0.025 0.040

34.5

9.5

0.38 0.45 0.50 1.4

0.040 0.047 0.053 0.15

.;i;i, = weight average molecular weight; li?, = number average molecular weight; a = StokesEinstein radius (Eq. (3)); Do = free diiusion coefficient; D’ = apparent diffusion coefficient in mesentery. D, for dextran from Granath and Kvist (1967); D, for RSA from Creeth (1952). b li?, used to compute D, for dextran 3400 included an additional 389 for one FITC molecule.

data are taken from the leading edge of a diffusing tracer front, and since even the narrow dextran fractions which we used contained a range of molecular weights, our data are probably biased toward the lower molecular weight components of the fraction, leading to an overestimate in D’. Fifth, a thin aqueous film could still be present between the mesentery and the mineral oil. This would lead to a low-resistance parallel pathway for tracer diffusion and thus to an overestimate in D’, and a greater similarity to the molecular weight dependence in free diffusion. The only factor which could lead to an underestimate of D’ would be if the mineral oil absorbed enough moisture from the tissue to reduce its level of hydration. Except for the work from this laboratory, there are virtually no detailed measurements on the molecular weight dependence of interstitial transport in vivo at a local level. Most published data on interstitial transport are for experiments with small molecules (micro-ions, mono- and disaccharides, gases). Maroudas ( 1970)) however, measured transport of various saccharides ranging from glucose to 40,000 molecular weight dextran across cartilage slices placed in a diffusion cell. These data, corrected to 37” (by assuming that the reported ratios of apparent diffusion coefficient to free diffusion coefficient would obtain at 37”) are plotted in Fig. 6 along with the data from Table 1 and the free diffusion coefficients of the tracers. Data from the mesentery show a sharp decrease in the apparent diffusion coefficient for the higher molecular weight dextrans. The data from the cartilage slices show a similar relationship between apparent diffusion coefficient and molecular weight. The data for the 10,000 and 40,000 molecular weight dextrans in cartilage should probably be shifted to the left as indicated by the arrows, since wide dextran fractions were used, and since the average molecular weight of the dextrans crossing the cartilage slice was probably considerably lower than that of the original sample placed in the diffusion cell. Ratios of apparent diffusion coefficient to free diffusion coefficient, D/Do,

INTERSTITIAL

DIFFUSION DIFFUSION

0k10-7 I I)free

10-4

265

MACROMOLECULES

COEFFICIENTS

VS

R,

dIffusEon

a

dextrans

x

saccharides (Maraudas.

100

OF

I” mesentery in caridage 1970)

I 1000

CP I I0.000

I I00.000

h.4

FIG. 6. Free diffusion coefficients mesentery and cartilage as a function

of various saccharides of A,,,. For explanation

and apparent diffusion of arrows, see text.

coefficients

in

obtained for serum albumin in the mesentery ranged from 0.04 to 0.15. Data on serum albumin diffusion in tissue and in model systems have been reported by a number of other investigators. Granger and Taylor (1975) reported a value of D’/DO of 0.35 in internal linings of implanted capsules, and Granger et al. (1975) obtained a ratio of 0.25 in normally hydrated human umbilical cord. Schultz (1976) obtained a ratio of 1.0 in the interstitium of the rat diaphragm. Ogston and Sherman (l%l) reported ratios ranging from 0.07 to 0.16 for serum albumin diffusion in 0.6% hyaluronate solutions, although more recent experiments give considerably higher values (Laurent er al., 1963; Ogston et al., 1973). Preston and Snowden (1973) obtained ratios of 0.5 in 1.8% hyaluronate solutionsconcentrations which are much higher than found in most tissues. Rasio (1970) measured permeability of the rat mesentery to serum albumin in a direction perpendicular to the membrane, obtaining a value of 6.6 x 10m5cm2/sec at 37”. Assuming an average membrane thickness of 20 pm and assuming that, due to handling, the permeability of the mesothelium was very high compared to the connective tissue layer (Frasher and Marcus, 1976), we calculate a value of 0.14 for D’lD, of the connective tissue layer. If the mesothelium presented a significant barrier to albumin in these experiments, and since membrane thickness often exceeds 20 pm (Fig. 4), the value of 0.14 for D’lD, of the connective tissue layer is probably an underestimate. Nakamura and Wayland (1975) obtained apparent diffusion coefficients for FITC-dextrans and FITC-serum albumin in cat mesentery that were nearly as great as the free diffusion coefficients. However, in their experiments, it was likely that the tracers passed through the mesothelium and diffused in the water film above and below the tissue. Since these aqueous layers were considerably greater in thickness than the tissue, the fluorescent emission from tracers in the aqueous layer would dominate and values of apparent diffusion coefficients would approach the values for free diffusion coefficients.

FOX AND WAYLAND

266

It is difficult to compare diffusion data for serum albumin with those of dextran on the basis of molecular weight because there are differences in shape, compactness, density, and charge between these two types of molecules. To provide a comparison on the basis of transport properties, we plotted the apparent diffusion coefficient in the mesentery as a function of the free diffusion coefficient (Fig. 7). It can be seen in Fig. 7 that D’ for serum albumin is lower than D’ for dextran having the same D, (and therefore the same Stokes radius). This phenomenon has also been reported in glomerular transport studies (Renkin, 1970) but the opposite relationship has been reported in blood-to-lymph studies (Carter et al., 1974). Solute charge has been suggested as an important factor in serum albumin transport. Since conjugation of FITC to dextran adds one negative charge to the dextran molecule, dextran tracers will have a net molecular charge of - 1 or greater, depending on the degree of substitution. Serum albumin, however, is significantly more negative in charge (Landis and Pappenheimer, 1963), and thus might be restricted to a significantly smaller volume (reducing the effective “pore” size) in a negatively charged matrix (Meares, 1968; Renkin and Gilmore, 1973). The work of Rennke et al. (1975) supports this hypothesis by suggesting that the negative charge of the ferritin molecule reduces its penetration of the glomerular basement membrane. However, in studies of sedimentation of serum albumin through hyaluronic acid solutions, variation of pH from 5.5 to 8.5 and ionic strength from 0.02 to 0.5 did not alter sedimentation rates (Laurent and Pietruszkiewiez, 1961; Laurent and Persson, 1964), indicating that under these conditions the change in the charge of serum albumin (and any change in the hyaluronate) did not significantly affect serum albumin transport through the negatively charged hyaluronate matrix. Most mathematical models of transport through porous or fibrous matrices incorporate only one or two types of solute-matrix interaction, and are only applicable for situations where the matrix is relatively dilute and where the solute dimensions are small relative to the average pore (or interfiber) dimensions (Mackie and Meares, 1955; Laufer, 1961). However, the equation derived by Ogston et APPARENT DIFFUSION IN MESENTERY, D’ VS COEFFICIENT,

COEFFICIENT FREE DIFFUSION 0,

(37%)

6-

d

5"0 x

4-

-z I3> E

02 b %

0 6

11, IO

I 14 Do (cm’/sec)

I I6 x IO7

FIG. 7. Apparent diffusion coefficients of FITC-dextran of reported free diffusion coefficients.

I 22

I 26

and RSA-FITC in mesentery as a function

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267

MACROMOLECULES

al. (1973) for diffusion of spheres through a suspension of randomly oriented and positioned straight fibers has been applied successfully to hyahrronate solutions of concentrations up to 1.8% (Preston and Snowden, 1973). This equation is: D’/D,,

= exp - %kL

VI/Z CI/Z~

(3)

rf

where = = rf v = Cf = a

solute radius fiber radius specific volume of fibers concentration of fibers

cm, cm, cm3/g, g/cm3.

Solutions to this equation, expressing D’/Do as a function of solute radius, a, are shown in Fig. 8, along with the experimental data from Table 1. These solutions used values for rf (5.4 x lo-* cm) and V (0.65 cm3/g) obtained by Ogston et al. (1973) for hyaluronate. The Stokes-Einstein relationship was used to compute solute radius for the tracer molecules: RT

10-j (cm),

3.28;”

(4)

a=w=

where = = r) = N = Do = R T

gas constant, absolute temperature, viscosity of water, Avogadro’s number, free diffusion coefficient

D’/Do

0.01 1 0

FIG. 8. RSA-FITC for diffusion

1 IO

VS

STOKES

RADIUS.

I

1

20 Stakes

30 radius,

in cm*/sec.

a

1 40

t 50

a (A)

Ratios of apparent diffusion coefficient to free diffusion coefficient as a function of Stokes radius. Solid lines are solutions to the model in hyaluronate solutions (Eq. 3).

for FITC-dextran and of Ogston et al. (1973)

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It can be seen that for this simplified model, hyaluronate concentrations ranging from 6 to 28% would be needed to explain the experimental data, values which are considerably higher than hyaluronate concentrations found in tissue. However, other connective tissue components such as collagen (Wiederhielm and Black, 1976), elastin (Partridge, 1967), and chondroitin sulfate (Gerber and Schubert, 1964; Shaw, 1976) are capable of excluding macromolecules and therefore could also be important in restricting macromolecular diffusion. The possible effect of these components and, probably even more important, their physical and chemical interactions should also be considered in models of interstitial transport (Laurent, 1977; Comper and Laurent, 1978; Wayland and Silberberg, 1978). An equation proposed by Renkin (1954) which includes an exclusion factor and the frictional drag equation derived by Fax&n (1922) has been successfully applied to diffusion in gels (Ackers and Steere, 1962; Newsom and Gilbert, 1964), and various other porous membranes. This exclusion term does not apply for our data since concentrations and boundary conditions are all taken within the membrane and the Faxen equation is inaccurate for the ratios of solute radius to pore radius which are necessary to explain our experimental data. An expression for frictional drag in right circular cylindrical pores modified to include solute-to-pore radius ratios up to 0.5 has been proposed by Haberman and Sayre (Verniory et al., 1973): 3 5 6 1 - 2.105; + 2.085 - 1.7068 0; + 0.72603 0; -=D’ 7 (5) Dll

where r = pore radius. Solutions of this equation for various values of r are shown in Fig. 9. According to this equation, pore radii ranging from 35 to 75 A are necessary to explain our experimental data. D’/D,

0.01

0

I IO

VS

STOKES

I 20 Stokes

RADIUS,

I 30 rodws.

a

I 40

I 50

o

FIG. 9. Ratios of apparent diffusion coefficient to free diffusion coefficient for FWC-dextran and RSA-FITC as a function of Stokes radius. Solid lines are solutions to an equation expressing effects of frictional drag on spheres in right circular cylindrical pores (Eq. 5).

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269

Although comparisons of the experimental data and calculations using Eq. (3) or Eq. (5) do not show impressive agreement, these comparisons suggest that the mesenteric interstitial matrix is extremely concentrated and that the dimensions of the larger tracers approached the average pore (or interfiber) dimensions. In addition, the above models assume that the matrix is a rigid, unchanging barrier. However, the molecules composing the interstitial matrix undoubtedly have some freedom of movement and exhibit Brownian motion. As solute molecules approach matrix pore dimensions, the rate of formation of spaces between fiber segments may become important (Preston and Snowden, 1973). This would lead to deviations from the Fick diffusion equation in a number of ways (Frisch, 1965) and would add an additional temperature-dependent term to the diffusion equation. Thus D’D, would be expected to increase with temperature for large solutes but remain relatively constant for small solutes (Frisch, 1965; Lieb and Stein, 1971). The possible magnitude of this effect is, however, difficult to predict. This relationship between D’D, and solute size and temperature was seen by Rasio (1970) for transport across the rat mesenteric membrane. He interpreted this relationship as evidence for pinocytotic transport of serum albumin in mesothelial cells. In the above discussion the possible role of convection in interstitial solute transport has not been treated. In most tissues, there is a net movement of water from the vascular system through the tissue and into the lymphatics. This might have been a significant factor in our experiments since our data were taken near regions of high molecular permeability. To include convection in the mathematical description of transport in an ideal binary system merely requires the addition of a single term to the Fick diffusion equation (Eq. (2)). When we obtained solutions to this equation and compared these solutions to our experimental data we could neither determine unique values for D and V nor observe any improvement over the use of Eq. (1) (Wayland and Fox, 1978). Since the inclusion of a convection term actually decreases the calculated value of D’, this would accentuate the molecular weight dependence of interstitial diffusion. A full treatment of the role of convection in solute transport through a porous medium would require consideration of the effects of solute dispersion, solute-matrix interaction (frictional drag and sieving), solute orientation, and solute deformation. Thus, an even more complex model would be needed to predict transport phenomena in a porous or fibrous network on the basis of mechanisms at the molecular level. An alternative model of interstitial transport has been suggested (Wiederhielm, 1968, 1972) where transport occurs through continuous unobstructed fluid-filled channels between regions of a dense glycosaminoglycan matrix (the gel phase). Small solutes would be expected to penetrate the gel phase easily, but large solutes would be sterically restricted to the channels between gel regions. Thus, in this model, interstitial solute transport would not only depend on diffusion and convection in the fluid-filled channels, but would also depend on the volume available to the solute in the gel phase and on the rate of equilibration between the two compartments. This situation has been simulated mathematically for networks with dead-end pores (Goodknight ef al., 1960), for gel chromatography (Ackers, 1970), and for the interstitial space of the dog paw (Watson, 1975). The model of Watson also fits the transient data of Garlick and Renkin (1970) for blood-lymph transport of various-sized solutes. However, we were not able to make a quantitative comparison of Watson’s model with our experimental data

270

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because this comparison would require numerical values for a number of parameters which cannot readily be obtained for the mesentery. CONCLUSIONS Our data suggest that the interstitial matrix of the rat mesentery is extremely dense, and offers a significant barrier to diffusion of macromolecules. Although there are several possible sources of bias in our experiments, all of the potential biases would overestimate the apparent diffusion coefficients of the tracers, accentuating the molecular weight dependence reported. The low apparent diffusion coefficients for the larger tracers in our experiments suggest that, under some conditions, the interstitium should not be considered as a well-stirred compartment. For example, in transient phenomena such as a sudden change in microvascular macromolecular permeability or for an intravascular injection of a macromolecular tracer, interstitial concentration gradients would be expected and thus transport would be affected by properties of the interstitial matrix. Spatial variations in macromolecular permeability within the microvascular bed could also lead to interstitial protein concentration gradients. Steady-state interstitial protein concentration gradients in the rat mesentery have been reported by Witte and Zenzes-Geprags (1977), although in their experiments the tissue was exposed to an aqueous solution which could have considerably modified interstitial gradients, and tissue thickness variation near the vessels was not taken into account. Comparison of our data to various models which relate apparent diffusion coefficients to phenomena at a molecular level failed to give much insight into mechanisms of interaction between the tracers and the interstitial matrix or into the relative roles of convection and diffusion. Further experimentation and improvement of transport models would be necessary to meaningfully apply our results to draw more detailed conclusions about the physical-chemical properties of the interstitial space. However, our data indicate that the role of the interstitial matrix in solute transport may be considerably greater than indicated by steadystate whole-organ studies. APPENDIX

I

A mathematical model was used to investigate the assumption of zero concentration gradient in the mesentery in a direction perpendicular to the membrane. The equation for diffusion in a two-dimensional isotropic homogeneous medium with a constant diffusion coefficient was used:

where C = C(x, y, t) = solute concentration, D = solute diffusion coefficient to simulate diffusion from a straight, infinitely long, g-pm-diameter cylindrical source in an infinite slab 30 pm thick. This equation was solved numerically for a

INTERSTITIAL

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271

MACROMOLECULES

rectangular net with l-pm spacing between points. The capillary cross section was approximated by an octagon 8 pm in diameter (Fig. Al). Due to geometrical symmetry, solutions were needed only for one quadrant, and it was assumed that the surfaces of the slab were impermeable to the solute. The following finite difference equation was used to approximate Eq. (Al) (Kunz, 1957):

Ci,j,m+l - Ci>,m _- 7D

[(Ci+lj,m

h

-

2Ci.Am

+

Ci-lj,m)

+ (Ci,j+m - 2Ci,j,m + Ci,j-l,m)l, where Ci,j,m = C(~i7YjJm)~

100, 515,

Xi = pi,

05 i I

Yj = pj, ttn = hm, p = 1.0 pm,

O~rn~3oo0,

0
h = 3.33 . . . msec,

and i, j, and m are integers. The boundary acwJ) ay

= 0

achy4

=

and initial

’ o

ax

wd-4t) ay

= 0

wo,ht)

=

7 =

used were:

’

ax

C(X,Y,O)

C(capillary

conditions

o

’

0,

boundary,

t) = 1.0,

where a = lOOpurn, b = 15 ,um. The width of the slab, a, chosen was large enough that it had no effect on the solution in the region of interest. The boundary and initial conditions written as difference equations were: 13 = 6.4 x IO-'

t = IO set

cm’/sec

----

15

-z ?O

1.0 .9

-15 lg'l 0

.a

.7

.6

5

.4

3

2

: i0

40

I ---1~ 60

I

x (pm)

FIG. Al. concentration

A solution to Eq. A2 expressed at the boundary of an octagonal

as isoconcentration vessel in a slab.

contours for a step increase D . t = 6.4 x 10” cme.

in

272

FOX

Ci,ls,m Ci,l,* CIOIAm ClJ,?ll

C 0.4.m

-

Ci,~4,m Ci,-l,m CW,j,m C-lj,m

= = = =

cid,O

= 07

= c

1.4.m

AND

WAYLAND

09 09 09 0,

(A3)

- c2,4,?n = G3,m

=

C4,m

=

C4,lJn

= C4,0,m = 1.

Equation

(A2) was solved for C.tJ,m+l and used to compute successive values of from values of Ci,j,m, C+ld,m, Ci,j-l,m, Ci+lj,m, and Ci,j+.l,m. This “marching” technique (Kunz, 1957) computes new points in time from values at the previous time. The boundary conditions (Eqs. (A3)) were substituted into Eq. (A2) at the appropriate locations (Carslaw and Jaeger, 1959). A typical solution, shown as isoconcentration contours for D * c = 6.4 x lop6 cm2, is illustrated in Fig. Al. It was found that concentration gradients were essentially parallel to the slab surface at distances greater than 20 pm from the center of the source, and could be adequately described by a one-dimensional model.

Ci,j,m+l

APPENDIX

II

It was found by interposing calibrated neutral density filters in the optical path that the value of any given picture element, Vij, was linearly related to light intensity, I, at the corresponding location (xi,yj) in the picture. However, the slope, A, and intercept, B, varied with location. Thus: Vij = Ai,jZ(xi,yj)

+ Bi>,

where i andj are integers, 0 5 i 5 255, 0 ‘j 5 255. The intercept, Bij, represents an electrical baseline. To examine the relationship between fluorescent emission and tracer concentration, solutions containing various tracer concentrations were placed in a hemocytometer under the microscope. It was found that the value of any given picture element was linearly related to the tracer concentration, C, and excitation, I,, at the corresponding location in the picture: Vi,j = Ai,jZ~(x,,yj)kC(x,,yj)

+ Ze(xt,YJM(xt,Yj) + B~J,

(A4)

where k is proportional to the quantum efficiency of the tracer and optical pathlength through the tracer solution (100 pm for the hemocytometer) and M(niyj) represents autofluorescence and backscattering of the exciting light from the object under observation. To correct experimental data for these nonuniformities in illumination and sensitivity, an array of picture elements, V$, was computed by subtracting values for an image of a hemocytometer containing zero tracer concentration from values for an image of a hemocytometer containing a uniformly thin film of tracer of concentration C,,. Thus:

where kh is proportional to tracer quantum efficiency in the physiological solution and to the thickness of the film of tracer. To correct experimental data for background due to autofluorescence and backscattering from the tissue and for electrical baseline an array of picture elements, Vf,,, was computed by subtracting

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MACROMOLECULES

values for an image of the tissue before injection of the fluorescent tracer from the values of images taken at the same location with the same excitation at the various times after tracer injection. Thus, at some time t,: where k, is proportional to the quantum efficiency of the tracer in the tissue, to tissue thickness, and to the fraction of the tissue volume available to the tracer. The final corrected array of picture elements, Pi,j,m, is computed by dividing each element in Vd by the corresponding element in VP:

APPENDIX Expressed as a finite difference

equation

Ci,m+l - Ci,m _- 7D h

III (Kunz, -

[Ci+l,m

2Ci,7n

1957), Eq. (1) becomes: +

Ci-l.mlr

where ci,, = C(Xi,fA xi = x, + Pi, t,=t,+hm, p = 1.1 pm,

OIi5 OIm,

100,

and i and m are integers. The value for h was chosen to insure stability solution:

of the

h+ The following

boundary

and initial

conditions

were used:

C(xo,d = At), aa&l

+ 4

ax

axJo)

= 03

= g(x),

where a chosen was large enough (I IO CL)that the solution in the region of interest was not affected. Expressed as finite difference equations these become: C 0.m = ALA c 101,m

-

Gwn

=

2P

0

,

Go = gw . Both the boundary condition,f(t,), and the initial condition, g(x*), were obtained from experimental data. The value of p used was the same as the actual spacing of picture elements (relative to the tissue). Since the time intervals between pictures,

274

FOX

AND

WAYLAND

however, ranged from 2 to 15 set (depending on the molecular weight of the tracer) linear interpolation was used to obtain values off(fm) close enough together to fulfill the stability criterion (Eq. (A6)). Equation (A5) was solved for Ci,,+, and used to compute successive values of from values of Ci,m, Ci-l,m, and Ci+l,, (Kunz, 1957). The boundary condiCi,m+* tions were substituted into Eq. (AS) at the appropriate locations (Carslaw and Jaeger, 1959). Since the value of each picture element is related to actual tissue tracer concentration by an unknown proportionality constant (Appendix II) the units of the initial and boundary conditions are unknown. However, since C appears in every term of Eq. (A5) and in each term of the initial and boundary conditions, the solution to Eq. (A5) will have the same units as the experimental data. Thus, although absolute tissue concentration was not computed, values for D are not affected. A typical set of solutions of Eq. (A5) is shown as solid lines in Fig. 3. ACKNOWLEDGMENTS We are grateful for the excellent technical assistance of Francis Waiyaki, Paul Knust-Graichen, and Dale Elmore and sincerely thank Dr. Aurora Landel for the preparation of the fluorescent rat serum albumin.

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