## Permeability Coefficients

For practical purposes, there are three permeability coefficients. Two of these describe the permeability to a particular solute (the diffusional permeability and the solute reflection coefficients) and the third describes fluid permeability (the hydraulic permeability).

### The Solute Diffusional Permeability Coefficient

The solute diffusional permeability, Pd, is defined under conditions where there are no net movements of fluid through microvascular walls. It is the transport of the solute through unit area of microvascular wall divided by the concentration difference of that solute between the fluid inside and that immediately outside the microvessel. Pd has units of velocity (cm sec-1). Writing the definition of Pd as an equation:

where JSd is the net transport of solute from the microvascular blood into the tissues in the absence of net fluid flux through the vessel wall, S is the area of microvascular wall through which transport occurs, Cc is the mean solute concentration inside, and Ci is the mean solute concentration immediately outside the microvessel. The condition that net fluid movements between the blood and the tissue should be zero is especially important when Pd is being estimated for macromolecules. It is of less importance when permeability is high, as it is for small molecules (see later discussion). The number of perfused capillaries in a tissue determines S, and this varies independently of permeability. Ci is often unknown and estimates of Pd are often made under conditions where Ci can be assumed to be approximately zero. Cc is a function of the arterial and venous concentrations and also of microvascular blood flow and of the permeability itself (see later discussion).

Pd is dependent on the lipid solubility, molecular size, and charge of the solute. Lipid-soluble molecules have high permeabilities, as they can pass through cell membranes and so can exchange through the entire area of the microvascu-lar walls. Oxygen, nitrogen, and carbon dioxide have high enough fat solubilities to fall into this category. The Pd values of lipid-soluble molecules may be greater that 0.1cm sec-1 Water-soluble molecules are confined to specialized pathways through the endothelium and their values for Pd are nearly always less than 10-3cmsec-1. In most tissues (e.g., skin, connective tissues, muscle, nerves) the specialized pathways are located between the endothelial cells and pass through occasional breaks in the junctional strands. In tissues associated with secretory and absorptive epithelia (e.g., intestinal mucosae, kidney, exocrine and endocrine glands) the permeability pathways are fenestrations, disklike openings (400 to 700 A in diameter) through thinned areas of endothelium covered with a fine diaphragm with an overlying layer of glycocalyx (cell coat). In intact microves-sels, values for Pd of water-soluble molecules fall more rapidly than their values for diffusion coefficient as molecular size increases (see Figure 1A).

### Solute Reflection Coefficient

Solutes are also transported through microvascular walls by fluid filtration and reabsorption. This convective component of solute transport is proportional to net fluid flow and to the solute reflection coefficient, a. If net fluid filtration from plasma to tissues is JV and the mean plasma concentration of the solute inside the microvessel is Cc, the con-vective component of transport, JSc, is

The reflection coefficient is therefore the fraction of solute that is reflected or rejected at microvascular walls during ultrafiltration and (1 - a) is the fraction of the solute molecules that is transported. The reflection coefficient also determines the effective osmotic pressure that the solution of a particular solute can exert across a particular

Figure 1 Microvascular permeability to fluid and hydrophilic molecules. (A) Diffusional permeability (PJ) to solutes of varying molecular radii in mesenteric capillaries (open circles) and muscle capillaries (closed squares). Data are plotted as log10 values to cover the range of Pd. Note how Pd for the the two capillary types differ by > 10-fold for the smallest molecules but converge as log radius approaches 1.6 (~40 A). (B) Log Pd to NaCl (open triangles) and inulin (closed circles) plotted against log LP. The slopes of the relations do not differ significantly from direct proportionality. [Modified from Michel and Curry (1999). Physiol. Rev. 79, 703-761.] (C) Reflection coefficient (O) for hydrophilic molecules for capillaries of cat skeletal muscle (open squares), rat limb capillaries (crosses), and frog mesentery (solid circles) plotted against molecular radius. The high values of c to the smallest molecules differ in cat muscle capillaries suggest a large aquaporin pathway in these vessels [data of Wolf and Watson (1989). Am. J. Physiol. 256, H282-H290]. (D) Reflection coefficient to serum albumin (O^b) against log10LP for capillaries of different tissues. The lowest values of LP are for skeletal muscle and the highest are for glomerular capillaries [based on Michel, C. C. (1988). J. Physiol. 404, 1-29].

Figure 1 Microvascular permeability to fluid and hydrophilic molecules. (A) Diffusional permeability (PJ) to solutes of varying molecular radii in mesenteric capillaries (open circles) and muscle capillaries (closed squares). Data are plotted as log10 values to cover the range of Pd. Note how Pd for the the two capillary types differ by > 10-fold for the smallest molecules but converge as log radius approaches 1.6 (~40 A). (B) Log Pd to NaCl (open triangles) and inulin (closed circles) plotted against log LP. The slopes of the relations do not differ significantly from direct proportionality. [Modified from Michel and Curry (1999). Physiol. Rev. 79, 703-761.] (C) Reflection coefficient (O) for hydrophilic molecules for capillaries of cat skeletal muscle (open squares), rat limb capillaries (crosses), and frog mesentery (solid circles) plotted against molecular radius. The high values of c to the smallest molecules differ in cat muscle capillaries suggest a large aquaporin pathway in these vessels [data of Wolf and Watson (1989). Am. J. Physiol. 256, H282-H290]. (D) Reflection coefficient to serum albumin (O^b) against log10LP for capillaries of different tissues. The lowest values of LP are for skeletal muscle and the highest are for glomerular capillaries [based on Michel, C. C. (1988). J. Physiol. 404, 1-29].

membrane. Thus the osmotic pressure of a 5-mM solution of glucose may be 127 cm H2O in an osmometer fitted with a membrane impermeable to glucose molecules in agreement with van't Hoff's law. A 5-mM difference in glucose concentration across the walls of a mesenteric capillary, however, exerts an effective osmotic pressure less than one tenth of this. This is because 90 percent of the glucose molecules pass relatively freely through capillary walls and only 10 percent are reflected, that is, o for glucose is just less than 0.1. This leads to an alternative definition of o in terms of the total (or van't Hoff) osmotic pressure and the effective osmotic pressure that a solution of a particular solute exerts across a particular membrane, that is,

Effective osmotic pressure Theoretical osmotic pressure

Strictly speaking, o defined in Eq. (3) is the osmotic reflection of the solute (od) and that defined in Eq. (2) is the solvent drag reflection coefficient (of). The values of od and of are identical for solutions where the relations between concentration and osmotic pressure are linear. Small differences are present for of and od of macromolecules where osmotic pressure rises as a second order power function of concentration. For all solutes at all membranes, o has a maximum value of 1.0 when the membrane is completely impermeable to solute but permeable to solvent. For water-soluble molecules of biological solutions, o has a minimum value of zero. Because o describes a fraction or a ratio, it has no units.

### Hydraulic Permeability

The hydraulic permeability, LP, is also referred to as the hydraulic conductivity, hydraulic conductance, water permeability, and filtration coefficient. It is defined as the net fluid flow through unit area of microvascular wall per unit difference in pressure between the lumen of the vessel and its abluminal surface. Thus if net fluid filtration or reabsorption, JV, occurs between blood and tissue through an area of capillary wall, S, then:

Jv/S

where DP and oAn are the differences in hydrostatic pressure and effective osmotic pressure, respectively, between the inside and outside of the microvessel. The hydraulic permeability has units of velocity per unit pressure difference, such as cmsec-1 cm H2O-1. In microvessels with fenestrated endothelia, water passes through the fenestrations with water-soluble molecules. Similarly, in vessels with nonfen-estrated endothelia, water shares the pathways between the cells with hydrophilic solutes, but here there is an additional (water only) pathway through cell membranes via aquaporin channels. In most nonfenestrated vessels the water-only pathways contribute 10 percent or less to LP, though there is evidence in some skeletal muscle capillaries their contribution may be as high as 40 percent.