Medical Physiology, 3rd Edition

Capillary Exchange of Solutes

The exchange of O2 and CO2 across capillaries depends on the diffusional properties of the surrounding tissue

Gases diffuse by a transcellular route across the two cell membranes and cytosol of the endothelial cells of the capillary with the same ease that they diffuse through the surrounding tissue. In this section, we focus primarily on the exchange of O2. Very similar mechanisms exist for the exchange of CO2, but they run in the opposite direction. Arterial blood has a relatively high O2 level. As blood traverses a systemic capillary, the principal site of gas exchange, O2 diffuses across the capillary wall and into the tissue space, which includes the interstitial fluid and the neighboring cells.

The most frequently used model of gas exchange is August Krogh's tissue cylinder, a volume of tissue that a single capillary supplies with O2 (Fig. 20-4A). The cylinder of tissue surrounds a single capillary. According to this model, the properties of the tissue cylinder govern the rate of diffusion of both O2 and CO2. The radius of a tissue cylinder in an organ is typically half the average spacing from one capillary to the next, that is, half the mean intercapillary distance. Capillary density and therefore mean intercapillary distance vary greatly among tissues. Among systemic tissues, capillary density is highest in tissues with high O2 consumption (e.g., myocardium) and lowest in tissues consuming little O2 (e.g., joint cartilage). Capillary density is extraordinarily high in the lungs (see p. 684).

image

FIGURE 20-4 Delivery and diffusion of O2 to systemic tissues. A shows Krogh's tissue cylinder, which consists of a single capillary (radius rc) surrounded by a concentric cylinder of tissue (radius rt) that the capillary supplies with O2 and other nutrients. Blood flow into the capillary is Fin, and blood flow out of the capillary is Fout. The lower panel of A shows the profile of partial pressure of O2 (image) along the longitudinal axis of the capillary and the radial axis of the tissue cylinder.

The Krogh model predicts how the concentration or partial pressure of oxygen (image) within the capillary lumen falls along the length of the capillary as O2 exits for the surrounding tissues (see Fig. 20-4A). The image within the capillary at any site along the length of the capillary depends on several factors:

1. The concentration of free O2 in the arteriolar blood that feeds the capillary. This dissolved [O2], which is the same in the plasma and the cytoplasm of the red blood cells (RBCs), is proportional to the partial pressure of O2 (see p. 647) in the arterioles.

2. The O2 content of the blood. Less than 2% of the total O2 in arterial blood is dissolved; the rest is bound to hemoglobin inside the RBCs. Each 100 mL of arterial blood contains ~20 mL of O2 gas, or 20 volume%—the O2 content (see Table 29-3).

3. The capillary blood flow (F).

4. The radial diffusion coefficient (Dr), which governs the diffusion of O2 out of the capillary lumen. For simplicity, we assume that Dr is the same within the blood, the capillary wall, and the surrounding tissue and that it is the same along the entire length of the capillary.

5. The capillary radius (rc).

6. The radius of tissue cylinder (rt) that the capillary is supplying with O2.

7. The O2 consumption by the surrounding tissues (image).

8. The axial distance (x) along the capillary.

The combination of all these factors accounts for the shape of the concentration profiles within the vessel and the tissue. Although this model appears complicated, it is actually based on many simplifying assumptions. imageN20-1

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Limitations of Krogh's Tissue-Cylinder Model

Contributed by Emile L. Boulpaep

Krogh's tissue-cylinder model (see Fig. 20-4A), which describes O2 and CO2 exchange between the capillary and surrounding tissue, is based on several simplifying—but critical—assumptions.

1. The model assumes that the capillary displays cylindrical symmetry around a central axis, so that only two spatial dimensions must be considered (i.e., x and r in Fig. 20-4A).

2. The model is correct only for the idealized case of capillaries that run in parallel, start and end in the same plane, and carry blood in the same direction.

3. The model neglects any longitudinal diffusion of gas along the x-axis within the tissue and the blood. In other words, Krogh assumes that blood flow is the sole mode for gas to move along the x-axis.

4. The model requires that the capillary wall itself does not constitute a rate-limiting barrier to O2 or CO2 transport; that is, the permeability of the endothelial membranes to these gases is similar to the diffusion properties of the bulk phase. In other words, as stated in point 4 on page 464, the radial diffusion coefficient (Dr) is uniform within the blood vessel, the vessel wall, and the surrounding tissue.

5. Krogh assumes that there is no O2 flow into or out of the tissue cylinder across the cylinder's outer boundary (i.e., beyond the radius, rt). In a regular array of identical tissue cylinders, each neighboring tissue cylinder would have the same image at its outer boundary (see Fig. 20-4C). Therefore, there would be no image difference to drive O2 diffusion from one tissue cylinder to another.

6. The model assumes a steady state. There are no transients; image is a function of position, not of time.

7. The O2 consumption of the tissue must be constant.

8. The upstream image in the capillary must be constant.

Investigators have generated more complicated models that include different geometries of capillary distribution and also incorporate (1) the effects of pH and image changes on the O2 affinity of hemoglobin (see pp. 652 and 653–654, as well as Fig. 29-5); (2) the effects of changes in oxygen solubility on the O2 content of the blood; and (3) the effects of changes in the amount of hemoglobin and its affinity for O2, which even more strongly affects the O2 content of the blood.

The O2 extraction ratio of a whole organ depends primarily on blood flow and metabolic demand

In principle, beginning with a model like Krogh's but more complete, one could sum up the predictions for a single capillary segment and then calculate gas exchange in an entire tissue. However, it is more convenient to pool all the capillaries in an organ and to focus on a single arterial inflow and single venous outflow. The difference in concentration of a substance in the arterial inflow and venous outflow of that organ is the arteriovenous (a-v) difference of that substance. For example, if the arterial O2 content ([O2]a) entering the tissue is 20 mL O2/dL blood and the venous O2 content leaving it ([O2]v) is 15 mL O2/dL blood, the O2 a-v difference for that tissue is 5 mL O2 gas/dL blood.

For a substance like O2, which exits the capillaries, another way of expressing the amount that the tissues remove is the extraction ratio. This parameter is merely the a-v difference normalized to the arterial content of the substance. Thus, the extraction ratio of oxygen (image) is

image

(20-1)

Thus, in our example,

image

(20-2)

In other words, the muscle in this example removes (and burns) 25% of the O2 presented to it by the arterial blood.

What are the factors that determine O2 extraction? To answer this question, we return to the hypothetical Krogh cylinder. The same eight factors that influence the image profiles in Figure 20-4A also determine the whole-organ O2 extraction. Of these factors, the two most important are capillary flow (item 3 in the list above) and metabolic demand (item 7). The O2 extraction ratio decreases with increased flow but increases with increased O2 consumption. These conclusions make intuitive sense. Greater flow supplies more O2, so the tissue needs to extract a smaller fraction of the incoming O2 to satisfy its fixed needs. Conversely, increased metabolic demands require that the tissue extract more of the incoming O2. These conclusions are merely a restatement of the Fick principle (see p. 423), which we can rewrite as

image

(20-3)

The term on the left is the a-v difference. The extraction ratio is merely the a-v difference normalized to [O2]a. Thus, the Fick principle confirms our intuition that the extraction ratio should increase with increasing metabolic demand but decrease with increasing flow.

Another important factor that we have so far ignored is that not all of the capillaries in a tissue may be active at any one time. For example, skeletal muscle contains roughly a half million capillaries per gram of tissue. However, only ~20% are perfused at rest (see Fig. 20-4B). During exercise, when the O2 consumption of the muscle increases, the resistance vessels and precapillary sphincters dilate to meet the increased demand. This vasodilation increases muscle blood flow and the density of perfused capillaries (see Fig. 20-4C). This response is equivalent to decreasing the tissue radius of Krogh's cylinder because each perfused capillary now supplies a smaller region. Other things being equal, reduced diffusion distances cause image in the tissue to increase.

The velocity of blood flow in the capillaries also increases during exercise. All things being equal, this increased velocity would cause image to fall less steeply along the capillary lumen. For example, if the velocity were infinite, image would not fall at all! In fact, because O2 consumption rises during exercise, image actually falls more steeply along the capillary.

According to Fick's law, the diffusion of small water-soluble solutes across a capillary wall depends on both the permeability and the concentration gradient

Although the endothelial cell is freely permeable to O2 and CO2, it offers a significant barrier to the exchange of lipid-insoluble substances. Hydrophilic solutes that are smaller than albumin can traverse the capillary wall by diffusion via a paracellular route (i.e., through the clefts and interendothelial junctions as well as gaps and fenestrae, if these are present).

The amount of solute that crosses a particular surface area of a capillary per unit time is called a flux. It seems intuitive that the flux ought to be proportional to the magnitude of the concentration difference across the capillary wall and that it ought to be bigger in leakier capillaries (Fig. 20-5). These ideas are embodied in a form of Fick's law: imageN20-2

image

FIGURE 20-5 Diffusion of a solute across a capillary wall.

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Fick's Law

Contributed by Emile Boulpaep

The passive movement of a small solute (X) across any surface can be described by Fick's law:

image

(NE 20-1)

where JX is the flux of solute (units: moles · cm−2 · s−1), assuming a positive JX in the direction of increasing distance z. D is the diffusion coefficient in cm2 · s−1, and ∂[X]/∂z is the concentration gradient of X (units: moles · cm−3 · cm−1) along the z axis. In the case of a solute crossing a capillary wall (see Fig. 20-5), we assume that the concentration of the solute in the bulk phase of the capillary ([X]c), as well as in the bulk phase of the interstitial fluid ([X]if), is constant and uniform. We also assume that the diffusion distance along the axis of diffusion (z) is equal to the thickness of the capillary wall (a). We can then rewrite Fick's law as

image

(NE 20-2)

Because the wall thickness (a) is hard to determine, one often combines the terms DX and a into a single permeability coefficient P (units: cm · s−1), defined as PX = DX/a. The permeability coefficient is an expression of the ease with which a solute crosses a membrane, driven by the concentration difference. Therefore, the flux of a solute becomes

image

(NE 20-3)

Equation NE 20-3 is the same as Equation 20-4.

image

(20-4)

In Figure 20-5 and Equation 20-4JX is the flux of the solute X (units: moles/[cm2 s]), assuming a positive JX with flow out of the capillary, into the interstitial fluid. [X]c and [X]if are the dissolved concentrations of the solute in the capillary and interstitial fluid, respectively. Because the capillary wall thickness a (units: cm) is difficult to determine, we combined the diffusion coefficient DX (units: cm2/s) and wall thickness into a single term (DX/a) called PX, the permeability coefficient (units: cm/s). Thus, PX expresses the ease with which the solute crosses a capillary by diffusion.

Because, in practice, the surface area (S) of the capillary is often unknown, it is impossible to compute the flux of a solute, which is expressed per unit area. Rather, it is more common to compute the mass flow (image), which is simply the amount of solute transferred per unit time (units: moles/s):

image

(20-5)

The whole-organ extraction ratio for small hydrophilic solutes provides an estimate of the solute permeability of capillaries

How could we estimate the permeability coefficient for a solute in different capillaries or for different solutes in the same capillary? Unfortunately, it is difficult to determine permeability coefficients in single capillaries. Therefore, investigators use an indirect approach that begins with measurement of the whole-organ extraction ratio for the solute X. As we have already seen for O2 (see Equation 20-1), the extraction ratio (EX) is a normalized a-v difference for X:

image

(20-6)

Thus, EX describes the degree to which an organ removes a solute from the circulation. Unlike the situation for O2, the extraction ratio for small hydrophilic solutes depends not only on total organ blood flow (F) but also on the overall “exchange properties” of all of its capillaries, expressed by the product of permeability and total capillary area (PX · S). The dependence of EX on the PX · S product and F is described by the following equation:

image

(20-7)

Therefore, by knowing the whole-organ extraction ratio for a solute and blood flow through the organ, we can calculate the product PX · S. The second column of Table 20-1 lists the PX · S products for a single solute (inulin), determined from Equation 20-7, for a number of different organs. Armed with independent estimates of the capillary surface area (see Table 20-1, column 3), we can compute PX (column 4). PXincreases by a factor of ~4 from resting skeletal muscle to heart, which reflects a difference in the density of fluid-filled interendothelial clefts. Because a much greater fraction of the capillaries in the heart are open to blood flow (i.e., S is ~10-fold larger), the PX · S product for heart is ~40-fold higher than that for resting skeletal muscle.

TABLE 20-1

PX · S Products for Various Capillary Beds

TISSUE

PXS FOR INULIN* (10−3 cm3/s) (MEASURED)

S, CAPILLARY SURFACE AREA* (cm2) (MEASURED)

PX, PERMEABILITY TO INULIN (× 10−6 cm/s) (CALCULATED)

Heart

4.08

800

5.1

Lung

3.80

950

4.0

Small intestine

1.79

460

3.9

Diaphragm

0.76

400

1.9

Ear

0.34

58

5.9

Skeletal muscle at rest

0.09

75

1.2

*All calculations are normalized for 1 g of tissue from rabbits.

Adapted from Wittmers LE, Barlett M, Johnson JA: Estimation of the capillary permeability coefficients of inulin in various tissues of the rabbit. Microvasc Res 11:67–78, 1976.

The cerebral vessels have unique characteristics that constitute the basis of the blood-brain barrier (see p. 284). The tight junctions of most brain capillaries do not permit any paracellular flow of hydrophilic solutes; therefore, they exhibit a very low permeability to sucrose or inulin, probably because of the abundant presence of CLDN5 and occludin. In contrast, the water permeability of cerebral vessels is similar to that of other organs. Therefore, a large fraction of water exchange in cerebral vessels must occur through the endothelial cells.

Whole-organ PX · S values are not constant. First, arterioles and precapillary sphincters control the number of capillaries being perfused and thus the available surface area (S). Second, in response to a variety of signaling molecules (e.g., cytokines), endothelial cells can reorganize their cytoskeleton, thereby changing their shape. This deformation widens interendothelial clefts and increases PX. One example is the increased leakiness that develops during inflammation in response to the secretion of histamine by mast cells and basophilic granulocytes. imageN20-3

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Effect of Inflammation on Capillary Leakiness

Contributed by Emile Boulpaep

Endothelial tight junctions are regulated by a wide variety of signaling mechanisms, including cytokines; extracellular [Ca2+]; G proteins; intracellular [cAMP] and [Ca2+]; serine, threonine, and tyrosine kinases; and proteases. The increased endothelial permeability induced by the inflammatory response can result from two general mechanisms. First, increased tension caused by actomyosin/cytoskeletal contractility can change the shape of cells and pull individual endothelial cells apart. Second, intercellular adhesion can be decreased by breakdown or modulation of the intercellular junctions.

Histamine increases vascular permeability by causing transitory gaps of 100 to 400 nm between adjacent endothelial cells. These gaps occur without any detectable increased tension within the cells. Instead, histamine alters the adhering junctions between endothelial cells, particularly the adhesions that are based on vascular endothelial cadherin (VE-cadherin; see “Cadherin Terminology” below). Among the cell-cell adhesion molecules (see p. 17), the type I cadherins (i.e., E-, N-, and P-cadherin) associate with cortical actin filaments via α- and β-catenin, whereas VE-cadherin is a type II cadherin that is linked not only to cortical actin by α- and β-catenin, but also to the intermediate filament protein vimentin (see p. 23) via γ-catenin and desmoplakin. Endothelial cells respond to histamine with an increase in intracellular [Ca2+], which stimulates the tyrosine phosphorylation of VE-cadherin and γ-catenin. How these phosphorylation events affect the link of VE-cadherin with the vimentin cytoskeleton is not known.

Cadherin Terminology

E-cadherin (in epithelial cells)

N-cadherin (in nerve and muscle cells)

P-cadherin (in placental and epidermal cells)

VE-cadherin (in vascular endothelial cells)

Small polar molecules have a relatively low permeability because they can traverse the capillary wall only by diffusing through water-filled pores (small-pore effect)

Having compared the permeabilities of a single hydrophilic solute (inulin) in several capillary beds, we can address the selectivity of a single capillary wall to several solutes. Table 20-2 shows that the permeability coefficient falls as molecular radius rises. For lipid-soluble substances such as CO2 and O2, which can diffuse through the entire capillary endothelial cell and not just the water-filled pathways, the permeability is much larger than for the solutes in Table 20-2. Early physiologists had modeled endothelial permeability for hydrophilic solutes on the basis of two sets of pores: imageN20-4 large pores with a diameter of ~10 nm or more and a larger number of small pores with an equivalent radius of 3 nm. Small water-soluble, polar molecules have a relatively low permeability because they can diffuse only by a paracellular path through interendothelial clefts or other water-filled pathways, which constitute only a fraction of the total capillary area. Discontinuities or gaps in tight-junction strands could form the basis for the small pores. Alternatively, the molecular sieving properties of the small pores may reside in a fiber matrix (Fig. 20-6) that consists of either a meshwork of glycoproteins in the paracellular clefts (on the abluminal side of the tight junctions) or the glycocalyx on the surface of the endothelial cell (on the luminal side of the tight junctions). The endothelium-specific, calcium-dependent adhesion molecule VE-cadherin (CDH5; see p. 17) and platelet/endothelial cell adhesion molecule 1 (PECAM1, or CD31 antigen) are important glycoprotein components of the postulated fiber matrix in the paracellular clefts. In fact, the small-pore effect is best explained by an arrangement of discontinuities in the tight junctional strands in series with a fiber matrix on either side of the tight junction.

TABLE 20-2

Permeability Coefficients for Lipid-Insoluble Solutes*

SUBSTANCE

RADIUS OF EQUIVALENT SPHERE (nm)

PERMEABILITY (cm/s)

NaCl

0.14

310 × 10−6

Urea

0.16

230 × 10−6

Glucose

0.36

90 × 10−6

Sucrose

0.44

50 × 10−6

Raffinose

0.56

40 × 10−6

Inulin

1.52

5 × 10−6

*Permeability data are from skeletal muscle of the cat; the capillary surface area is assumed to be 70 cm2/g wet tissue.

Stokes-Einstein radius. imageN20-16

Data from Pappenheimer JR: Passage of molecules through capillary walls. Physiol Rev 33:387–423, 1953.

image

FIGURE 20-6 Model of endothelial junctional complexes. The figure shows two adjacent endothelial cell membranes at the tight junction, with a portion of the membrane of the upper cell cut away. (Data from Firth JA: Endothelial barriers: From hypothetical pores to membrane proteins. J Anat 200:541–548, 2002.)

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Pore Theory

Contributed by Emile Boulpaep

The “pore theory” has been the main model of capillary permeability for a long time. Many studies have attempted to relate the permeability of solutes of various molecular weights to the geometry of hypothetical fluid-filled transendothelial channels, clefts, fenestrations, and gaps (see p. 462). Investigators have used the whole-organ extraction of molecular probes to estimate the size of a pore that would be necessary to allow movement of these probes at the observed rates. The result can then be used to estimate the number or density of pores, assuming that they have a fixed size and geometry. However, structural studies of capillary endothelial cells of various organs have failed to corroborate the initial formulation of the pore theory. Electron microscopic examination reveals that, in tissues such as muscle, transendothelial channels and the junctions between endothelial cells would allow the passage of substances of a molecular radius of 5 nm, exceeding that of inulin. The fenestrae in other tissues—such as the intestine, the kidney, and some glands—have even wider openings, 60 to 80 nm in diameter. However, except in glomerular capillaries, these openings are mostly covered by a thin diaphragm (see p. 727). Thus, the diverse overall geometry, size, and number of the transendothelial channels, clefts, and fenestrae in endothelia do not agree with the limited sets of pores that have been postulated based on pore theory. On the other hand, investigators measuring solute exchange at the level of a single perfused capillary—which is a far simpler system than a whole organ—are beginning to resolve the discrepancy between whole-organ permeability measurements and the images obtained by electron microscopy.

References

Pappenheimer JR. Passage of molecules through capillary walls. Physiol Rev. 1953;33:387–423.

Pappenheimer JR, Renkin EM, Borrero LM. Filtration, diffusion and molecular sieving through peripheral capillary membranes. A contribution to the pore theory of capillary permeability. Am J Physiol. 1951;167:13–46.

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Stokes-Einstein Radius

Contributed by Emile Boulpaep

The Stokes-Einstein radius is the radius of a spherical molecule that would have a diffusion coefficient equivalent to that of the lipid-insoluble substance (which itself may not be spherical).

Interendothelial clefts are wider—and fenestrae are more common—at the venular end of the capillary than at its arteriolar end, so that PX increases along the capillary. Therefore, if the transcapillary concentration difference ([X]c − [X]if) were the same, the solute flux would actually be larger at the venous end of the microcirculation.

Small proteins can also diffuse across interendothelial clefts or through fenestrae. In addition to molecular size, the electrical charge of proteins and other macromolecules is a major determinant of their apparent permeability coefficient. In general, the flux of negatively charged proteins is much smaller than that of neutral macromolecules of equivalent size, whereas positively charged macromolecules have the highest apparent permeability coefficient. Fixed negative charges in the endothelial glycocalyx exclude macromolecules with negative charge and favor the transit of macromolecules with positive charge. Selective permeability based on the electrical charge of the solute is a striking feature of the filtration of proteins across the glomerular barrier of the nephron (see pp. 742–743).

The diffusive movement of solutes is the dominant mode of transcapillary exchange. However, the convective movement of water can also carry solutes. This solvent drag is the flux of a dissolved solute that is swept along by the bulk movement of the solvent. Compared with the diffusive flux of a small solute with a high permeability coefficient (e.g., glucose), the contribution of solvent drag is minor.

The exchange of macromolecules across capillaries can occur by transcytosis (large-pore effect)

Macromolecules with a radius >1 nm (e.g., plasma proteins) can cross the capillary, at a low rate, through wide intercellular clefts, fenestrations, and gaps—when these are present. However, it is caveolae (see p. 461) that are predominantly responsible for the large-pore effect that allows transcellular translocation of macromolecules. The transcytosis of very large macromolecules by vesicular transport involves (1) equilibration of dissolved macromolecules in the capillary lumen with the fluid phase inside the open vesicle; (2) pinching off of the vesicle; (3) vesicle shuttling to the cytoplasm and probably transient fusion with other vesicles within the cytoplasm, allowing intermixing of the vesicular content; (4) fusion of vesicles with the opposite plasma membrane; and (5) equilibration with the opposite extracellular fluid phase.

Although one can express the transcytotic movement of macromolecules as a flux, the laws of diffusion (see Equation 20-4) do not govern transcytosis. Nevertheless, investigators have calculated the “apparent permeability” of typical capillaries to macromolecules (Table 20-3). The resulting “permeability”—which reflects the total movement of the macromolecule, regardless of the pathway—falls off steeply with increases in molecular radius, a feature called sieving. This sieving may be the result of steric hindrance when large macromolecules equilibrate across the neck of nascent vesicles or when a network of glycoproteins in the glycocalyx above the vesicle excludes the large macromolecules. In addition, sieving of macromolecules according to molecular size could occur as macromolecules diffuse through infrequent chains of fused vesicles that span the full width of the endothelial cell. imageN20-5

TABLE 20-3

Capillary Permeability to Macromolecules

MACROMOLECULE

RADIUS OF EQUIVALENT SPHERE (nm)*

APPARENT PERMEABILITY (cm/s)

Myoglobin

1.9

0.5 × 10−6

Plasma albumin

3.5

0.01 × 10−6

Ferritin

6.1

~0

*Stokes-Einstein radius.

Representative value for skeletal muscle.

Data from Pappenheimer JR: Passage of molecules through capillary walls. Physiol Rev 33:387–423, 1953.

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Fast Versus Slow Pathways for Exchange of Macromolecules Across Capillary Walls

Contributed by Emile Boulpaep

Classical transcytosis provides a relatively slow pathway. A fast pathway is provided by the infrequent transient chains of fused vesicles that happen to span the full width of the endothelial cell. Any degree of differential diffusion through this fast pathway will manifest itself as sieving.

Transcytosis is not as simple as the luminal loading and basal unloading of ferryboats—the cell may process some of the cargo. Although the luminal surface of endothelial cells avidly takes up ferritin (750 kDa), only a tiny portion of endocytosed ferritin translocates to the opposite side of the cell (see Table 20-3). The remainder stays for a time in intracellular compartments, where it is finally broken down.

Both transcytosis and chains of fused vesicles are less prominent in brain capillaries. The presence of continuous tight junctions and the low level of transcytosis account for the blood-brain barrier's much lower apparent permeability to macromolecules.