## Permeability and Exchange of Fluid and Solute between Blood and Tissues under Physiological Conditions

Fluid Exchange

Fluid transport is determined by the differences in hydrostatic and the effective osmotic pressures across microvascular walls. This is Starling's principle and when written as an equation it is:

where JV is net fluid transport (filtration) from blood to tissue, S is surface area of microvascular wall through which filtration is occurring, AP is the mean hydrostatic pressure difference across the walls of the exchange vessels, and S„o„An„ is the summation of the effective osmotic pressures set up across the exchange vessel walls by all the solutes present in the plasma. Note that when fluid is being reabsorbed into the blood, JV has a negative sign, indicating that the tissues are losing fluid.

Starling, in 1896, first appreciated that the effective osmotic pressure difference was determined by macromolecules. He pointed out that while these solutes make a small contribution to the total osmotic pressure of the plasma, they represent the only solutes that are normally present at dif ferent concentrations in the plasma and in the interstitial fluids. For all other solutes, An is zero across most microvascular walls. Furthermore, we now know that unlike the other solutes of the plasma, the macromolecules normally exert more than 85 percent of their full osmotic pressure (o > 0.85).

Microvascular walls are slightly but measurably permeable to all the macromolecules that contribute to the effective osmotic pressure term. This means that the difference in macromolecular concentration between the plasma and the tissue fluid (and hence the effective osmotic pressure term) is itself a function of the permeability and the fluid filtration rate. In most tissues, the concentration difference depends on fluid entering the tissue spaces faster than macromole-cules, diluting the pericapillary macromolecules to concentrations well below those of the plasma. Drainage of tissue fluid by the lymph at the same rate as it is being formed by microvascular ultrafiltration keeps the interstitial fluid volume constant. Net fluid uptake from the tissues into the plasma (reabsorption) accelerates the equilibration of macromolecules across microvascular walls, diminishing their effective osmotic pressure. Thus fluid reabsorption can occur for only limited periods in most tissues before the effective osmotic pressure difference across the microvessel walls is reduced to less than the hydrostatic pressure difference. Reabsorption then ceases and a low level of filtration develops. One important consequence of this is that the popular textbook diagram illustrating Starling's principle of microvascular fluid balance, where fluid is filtered from the arterial end of a capillary and is reabsorbed at the venous end, bears little relation to reality.

Continuous reabsorption of fluid does occur in tissues such as the intestinal mucosa and the kidney. Here the interstitial concentration of macromolecules is kept low by the continuous secretion of protein-free fluid into the intersti-tium by the adjacent epithelial cells. For a full discussion of microvascular fluid exchange, the reader should consult the references by Levick (1991) or Michel (1997), listed at the end of this chapter.

Solute Exchange

### Convection and Diffusion

The transport of solute through microvascular walls is the sum of transport of the solute by diffusion and by convection. This addition is slightly complicated for when solute and fluid are being carried in the same channels through microvascular walls; convection of solute in the same direction as diffusion reduces the concentration gradient at the entry to the channels so that the mean gradient is no longer the difference in concentration across the wall divided by wall thickness. Convection in the opposite direction to diffusion increases the gradient. This effect of convection on the diffusion gradient is expressed in terms of the ratio of the velocity of solute transport by convection to its velocity by diffusion through the membrane. This ratio is the Péclet number, Pe, and here it is defined as:

Using Pe to modify the expression for the diffusional component, the total solute transport by diffusion and convection from micro-vessel to tissues becomes (see Curry 1984):

where (CC - C) is the mean concentration difference across the walls of the exchange vessels. Taylor, Renkin, and their colleagues have used expressions of the general form of Eq. (7) to analyze macromolecular transport between the plasma and lymph.

### Mean Concentration in the Exchange Vessels

In many microvascular beds, small highly diffusible molecules have very low Péclet numbers and convective solute transport is negligible. Under these conditions, JS may be described by the simple diffusion expression

CC, the mean concentration in the microvessels, lies between the arterial concentration, Ca, and the venous concentration, CV, by an amount that is determined by the blood flow, F, and the permeability-surface area product, PJS. Writing this as an equation we find

When Ci is constant, Renkin showed that Cc falls exponentially between Ca and Cv. For the special case of Ci = 0, Cv is related to Ca by the expression

Various forms of Eqs. (9) and (10) have been used to develop methods for measuring permeability to small molecules. Because microvascular permeability to macromole-cules is usually low, their concentrations fall negligibly between entry and exit of exchange vessels and for these solutes, CC approximates to Ca. This approximation may not be valid when permeability is increased.

### Clearance and Extraction

Renkin also pioneered the use of Eq. (10) for examining how F and PJS influence blood-tissue exchange. At the same time he clarified the terms solute clearance and solute extraction showing how they would vary with blood flow and solute permeability.

The clearance of a solute from the blood into the tissues supplied by a microcirculation is the net rate of solute transport divided by its arterial concentration, that is, JS/Cct Clearance from the tissues to blood is J/Ci. Renkin argued that blood-tissue clearance could be thought of being the product of blood flow and extraction where the extraction was the arteriovenous concentration difference divided by either Ca (for clearance blood to tissue) or Ci for clearance between tissues and blood. Using Eq. (10), he argued that

where the extraction is (1 - e~PdSSF) = (Ca - Cv)/Ca. From this it follows that when PJS is large compared with F, clearance is determined largely by blood flow, but when F exceeds PJS, clearance approximates more and more closely to PJS as F increases. These predictions are consistent with experiments where the blood and tissue clearances of diffusible solutes have been investigated over a wide range of blood flows.

The permeability of microvascular walls to macromole-cules under normal conditions is very low compared with F. The unidirectional clearance of macromolecules from blood to tissue does not, however, yield a value of PJS. From Eq. (7), unidirectional clearance (i.e., clearance when Ci is close to zero) is

Only when JV is small relative to PdS does clearance approximate to PJS.

Clearance is sometimes used as a measure of microvascular permeability, but Eqs. (11) and (12) show that this is valid for small molecules only when blood flow is large relative to PJS and for large molecules only when filtration rate is small relative to PJS. Furthermore, it is important to remember that PJS is not permeability but the product of permeability and surface area. This means that when looking for changes in permeability, Pd and S have to be separated before a change in permeability can be inferred from a change in PdS.

### Glossary

Clearance of a solute from blood to tissues: The net transport of solute per unit time (flux) divided by the arterial (inflowing) concentration. Clearance of a solute from tissues to blood is the net flux divided by the tissue concentration. Clearance is also the product of extraction and blood flow.

Extraction, E, of a solute from blood into tissues: The difference between the arterial and venous blood (or plasma) concentrations divided by the arterial concentration. Extraction of a solute from tissues to blood (when the arterial concentration is zero) is the venous concentration of that solute divided by the tissue concentration.

Hydraulic permeability, LP: Net fluid flow per unit area of microvascular wall divided by the difference in hydrostatic pressure across the wall. It is determined by both the frequency and the dimensions of the channels that transmit fluid through the wall.

Solute diffusional permeability, Pd: Net transport per unit time of a specific solute through unit area of microvascular wall divided by the concentration difference of that solute across the wall. For hydrophilic solutes, Pd is determined by both the number of channels per unit area of wall and the molecular radius of the solute relative to the dimensions of the channels.

Solute reflection coefficient, O: Has two definitions. It is either the fraction of molecules of a specific solute that are reflected at (i.e., not carried through) the microvascular wall during ultrafiltration of fluid through the wall, or it is the ratio of the effective osmotic pressure that a given concentration difference of the solute exerts across the microvascular wall to the osmotic pressure that the same concentration difference would exert across a membrane that was impermeable to the solute but permeable to water. For hydrophilic molecules, it is determined by the size of the channels to solute and to water and is independent of the frequency of channels.

### Bibliography

Bates, D., Lodwick, D., and Williams, B. (1999). Vascular endothelial growth factor and microvascular permeability. Microcirculation 6, 83-96. This has an excellent introductory section on the assessment of changes in permeability. Curry, F. E. (1984). Mechanics and thermodynamics of transcapillary exchange. In Handbook of Physiology (E. M. Renkin and C. C. Michel, eds.), Section 2, Vol. 4, pp. 309-374. Bethesda, MD: American Physiological Society. This is still the best comprehensive review of the biophysics of microvascular permeability. Levick, J. R. (1991). Capillary filtration-absorption balance reconsidered in the light of dynamic extravascular factors. Exp. Physiol. 76, 825-857. An excellent compilation of the experimental measurements of hydrostatic and effective osmotic pressure differences across microvascular walls with important conclusions. Michel, C. C. (1997). Starling: The formulation of his hypothesis of microvascular fluid exchange and its significance after 100 years. Exp. Physiol. 82, 1-30. This is partly historical but also addresses recent work and recent questions about microvascular fluid exchange arising after Levick's review. Michel, C. C., and Curry, F. E. (1999). Microvascular permeability. Physiol. Rev. 79, 703-761. This is the most comprehensive modern review of the ultrastructural basis of microvascular permeability and of recent work on the regulation of permeability.

Renkin, E. M. (1984). The control of the microcirculation and blood tissue exchange. In Handbook of Physiology (E. M. Renkin and C. C. Michel, eds.), Section 2, Vol. 4, pp. 627-687. Bethesda, MD: American Physiological Society. Although this review does not include some new hypotheses for the regulation of microvascular blood flow, it is the best treatment of the principles of blood tissue exchange.

Rippe, B., Rosengren, B. I., Carlsson, O., and Venturoli, D. (2002). Transendothelial transport: The vesicle controversy. J. Vasc. Res. 39, 375-390. This is an excellent review interpreting recent evidence concerning the mechanisms of transport of macromolecules through microvascular walls.

Taylor, A. E., and Granger, D. N. (1984). Exchange of macromolecules across the microcirculation. In Handbook of Physiology (E. M. Renkin and C. C. Michel, eds.), Section 2, Vol. 4, pp. 467-520. Bethesda, MD: American Physiological Society. Many applications of convection diffusion equations in the interpretation of macromolecular transport.

### Capsule Biography

C. Charles Michel is Emeritus Professor of Physiology and Senior Research Investigator at Imperial College, London. His main research interest for over 30 years has been microvascular permeability and microvascular exchange and his contributions have been recognized by various awards including the Malpighi Medal of the European Society for Microcirculation (!984), the Annual Prize Lecture of The Physiological Society (1987), and his election to Honorary Membership of the American Physiological Society (1993), The Physiological Society (2001), and the British Society for Microcirculation (2001).

## Essentials of Human Physiology

This ebook provides an introductory explanation of the workings of the human body, with an effort to draw connections between the body systems and explain their interdependencies. A framework for the book is homeostasis and how the body maintains balance within each system. This is intended as a first introduction to physiology for a college-level course.

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