Effect of blood flow on near-the-wall mass transport of drugs and other bioactive agents: a simple formula to estimate boundary layer concentrations

Transport of bioactive agents through the blood is essential for cardiovascular regulatory processes and drug delivery. Bioactive agents and other solutes infused into the blood through the wall of a blood vessel or released into the blood from an area in the vessel wall spread downstream of the infusion/release region and form a thin boundary layer in which solute concentration is higher than in the rest of the blood. Bioactive agents distributed along the vessel wall affect endothelial cells and regulate biological processes, such as thrombus formation, atherogenesis, and vascular remodeling. To calculate the concentration of solutes in the boundary layer, researchers have generally used numerical simulations. However, to investigate the effect of blood flow, infusion rate, and vessel geometry on the concentration of different solutes, many simulations are needed, leading to a time-consuming effort. In this paper, a relatively simple formula to quantify concentrations in a tube downstream of an infusion/release region is presented. Given known blood-flow rates, tube radius, solute diffusivity, and the length of the infusion region, this formula can be used to quickly estimate solute concentrations when infusion rates are known or to estimate infusion rates when solute concentrations at a point downstream of the infusion region are known. The developed formula is based on boundary layer theory and physical principles. The formula is an approximate solution of the advection-diffusion equations in the boundary layer region when solute concentration is small (dilute solution), infusion rate is modeled as a mass flux, and there is no transport of solute through the wall or chemical reactions downstream of the infusion region. Wall concentrations calculated using the formula developed in this paper were compared to the results from finite element models. Agreement between the results was within 10%. The developed formula could be used in experimental procedures to evaluate drug efficacy, in the design of drug-eluting stents, and to calculate rates of release of bioactive substances at active surfaces using downstream concentration measurements. In addition to being simple and fast to use, the formula gives accurate quantifications of concentrations and infusion rates under steady-state and oscillatory flow conditions, and therefore can be used to estimate boundary layer concentrations under physiological conditions.     

Transport of ions across the blood-brain barrier

Capillaries in the brain are formed by a uniquely specialized endothelial cell that regulates the movement of substances between blood and brain. Although they provide an impermeable barrier to some solutes, brain capillary endothelial cells facilitate the transcapillary exchange of others. In addition, they contain specific enzymes that contribute to a metabolic blood-brain barrier by limiting the movement of compounds such as neurotransmitters across the capillary wall. Studies of sodium and potassium transport by brain capillaries indicate that the endothelial cell contains distinct types of ion transport systems on the two sides of the capillary wall, i.e., the luminal and antiluminal membranes of the endothelial cell. As a result, specific solutes can be pumped across the capillary against an electrochemical gradient. These transport systems are likely to play a role in the active secretion of fluid from blood to brain and in maintaining a constant concentration of ions in the brain's interstitial fluid. In this way, the brain capillary endothelium is structurally and functionally related to an epithelium.     

Fluid flow and convective transport of solutes within the intervertebral disc

Previous experimental and analytical studies of solute transport in the intervertebral disc have demonstrated that for small molecules diffusive transport alone fulfils the nutritional needs of disc cells. It has been often suggested that fluid flow into and within the disc may enhance the transport of larger molecules. The goal of the study was to predict the influence of load-induced interstitial fluid flow on mass transport in the intervertebral disc.An iterative procedure was used to predict the convective transport of physiologically relevant molecules within the disc. An axisymmetric, poroelastic finite-element structural model of the disc was developed. The diurnal loading was divided into discrete time steps. At each time step, the fluid flow within the disc due to compression or swelling was calculated. A sequentially coupled diffusion/convection model was then employed to calculate solute transport, with a constant concentration of solute being provided at the vascularised endplates and outer annulus. Loading was simulated for a complete diurnal cycle, and the relative convective and diffusive transport was compared for solutes with molecular weights ranging from 400 Da to 40 kDa.Consistent with previous studies, fluid flow did not enhance the transport of low-weight solutes. During swelling, interstitial fluid flow increased the unidirectional penetration of large solutes by approximately 100%. Due to the bi-directional temporal nature of disc loading, however, the net effect of convective transport over a full diurnal cycle was more limited (30% increase). Further study is required to determine the significance of large solutes and the timing of their delivery for disc physiology.     

Permselectivity of the glomerular capillary wall to macromolecules. I. Theoretical considerations.

The transport of macromolecules across the renal glomerular capillary wall has been described theoretically using flux equations based on (a) restricted transport through small pores, and (b) the Kedem-Katchalsky formulation. The various assumptions and limitations inherent in these two approaches are discussed. To examine the coupling between macromolecular solute transport and the determinants of glomerular filtration rate, these flux equations were combined with mass balance relations which allow for variations in the transmembrane driving forces along a glomerular capillary. It was predicted, using both pore theory and the Kedem-Katchalsky equations, that fractional solute clearance should be strongly dependent on the determinants of glomerular filtration rate when convection and diffusion both contribute to solute transport. When convection becomes the sole mechanism for transcapillary solute transport, however, fractional solute clearance is essentially independent of changes in the determinants of glomerular filtration rate. Consequently, unless diffusion is absent, fractional solute clearances alone are insufficient to characterize the permselective properties of the glomerular capillary wall, since these values may be altered by changes in glomerular pressures and flows as well as changes in the properties of the capillary wall per se.     

Molecular modeling and simulation of Mycobacterium tuberculosis cell wall permeability

The low permeability of the mycobacterial cell wall is thought to contribute to the intrinsic drug resistance of mycobacteria. In this study, the permeability of the Mycobacterium tuberculosis cell wall is studied by computer simulation. Thirteen known drugs with diverse chemical structures were modeled as solutes undergoing transport across a model for the M. tuberculosis cell wall. The properties of the solute-membrane complexes were investigated by means of molecular dynamics simulation, especially the diffusion coefficients of the solute molecules inside the cell wall. The molecular shape of the solute was found to be an important factor for permeation through the M. tuberculosis cell wall. Predominant lateral diffusion within, as opposed to transverse diffusion across, the membrane/cell wall system was observed for some solutes. The extent of lateral diffusion relative to transverse diffusion of a solute within a biological cell membrane may be an important finding with respect to absorption distribution, metabolism, elimination, and toxicity properties of drug candidates. Molecular similarity measures among the solutes were computed, and the results suggest that compounds having high molecular similarity will display similar transport behavior in a common membrane/cell wall environment. In addition, the diffusion coefficients of the solute molecules across the M. tuberculosis cell wall model were compared to those across the monolayers of dipalmitoylphosphatidylethanolamine and dimyristoylphosphatidylcholine, are two common phospholipids in bacterial and animal membranes. The differences among these three groups of diffusion coefficients were observed and analyzed.     

Analysis of coupled intra- and extraluminal flows for single and multiple capillaries

Matched asymptotic expansions are used to study a model of the coupled fluid flow in the capillaries and tissue of the microcirculation. These capillaries are long, narrow cylindrical tubes embedded in a uniform tissue space. The capillary, or intraluminal, flow is assumed to be that of an incompressible Navier-Stokes fluid wherein colloids are represented as dilute solute; the extraluminal flow in the tissue is according to Darcy's law. Central to this fluid exchange is the boundary condition on the fluid radial velocity at the semipermeable wall of the capillary. This boundary condition, involving the local hydrostatic and colloidal osmotic pressures in both the capillary and the tissue, together with the radial gradient of the tissue hydrostatic pressure, couples the intra- and extraluminal flow fields. With this model we investigate the relationship between transport properties, hydrostatic pressures, and flow exchange for a single capillary, and describe the fluid transport in the tissue space produced by an array of such capillaries.     

Contribution of solvent drag through intercellular junctions to absorption of nutrients by the small intestine of the rat

The lumen of the small intestine in anesthetized rats was recirculated with 50 ml perfusion fluid containing normal salts, 25 mM glucose and low concentrations of hydrophilic solutes ranging in size from creatinine (mol wt 113) to Inulin (mol wt 5500). Ferrocyanide, a nontoxic, quadrupally charged anion was not absorbed; it could therefore be used as an osmotically active solute with reflection coefficient of 1.0 to adjust rates of fluid absorption, Jv, and to measure the coefficient of osmotic flow, Lp. The clearances from the perfusion fluid of all other test solutes were approximately proportional to Jv. From Lp and rates of clearances as a function of Jv and molecular size we estimate (a) the fraction of fluid absorption which passes paracellularly (approx. 50%), (b) coefficients of solvent drag of various solutes within intercellular junctions, (c) the equivalent pore radius of intercellular junctions (50 A) and their cross sectional area per unit path length (4.3 cm per cm length of intestine). Glucose absorption also varied as a function of Jv. From this relationship and the clearances of inert markers we calculate the rate of active transport of glucose, the amount of glucose carried paracellularly by solvent drag or back-diffusion at any given Jv and luminal glucose concentration and the concentration of glucose in the absorbate. The results indicate that solvent drag through paracellular channels is the principal route for intestinal transport of glucose or amino acids at physiological rates of fluid absorption and concentration. In the absence of luminal glucose the rate of fluid absorption and the clearances of all inert hydrophilic solutes were greatly reduced. It is proposed that Na-coupled transport of organic solutes from lumen to intercellular spaces provides the principal osmotic force for fluid absorption and triggers widening of intercellular junctions, thus promoting bulk absorption of nutrients by solvent drag. Further evidence for regulation of channel width is provided in accompanying papers on changes in electrical impedance and ultrastructure of junctions during Na-coupled solute transport.     

A Biomechanical Triphasic Approach to the Transport of Nondilute Solutions in Articular Cartilage

Biomechanical models for biological tissues such as articular cartilage generally contain an ideal, dilute solution assumption. In this article, a biomechanical triphasic model of cartilage is described that includes nondilute treatment of concentrated solutions such as those applied in vitrification of biological tissues. The chemical potential equations of the triphasic model are modified and the transport equations are adjusted for the volume fraction and frictional coefficients of the solutes that are not negligible in such solutions. Four transport parameters, i.e., water permeability, solute permeability, diffusion coefficient of solute in solvent within the cartilage, and the cartilage stiffness modulus, are defined as four degrees of freedom for the model. Water and solute transport in cartilage were simulated using the model and predictions of average concentration increase and cartilage weight were fit to experimental data to obtain the values of the four transport parameters. As far as we know, this is the first study to formulate the solvent and solute transport equations of nondilute solutions in the cartilage matrix. It is shown that the values obtained for the transport parameters are within the ranges reported in the available literature, which confirms the proposed model approach.     

The variance of the distribution of traversal times in a capillary bed

A random walk model of capillary tracer transit times is developed that treats simulataneously: plug flow in the capillary, radial and axial diffusion in the capillary cylinder and tissue annulus, and endothelial barriers to solute transport. The mean transit time is simply the volume of distribution divided by blood flow. Variance of transit times has additive terms for radial, axial, and barrier influences that are reduceable to variances of simpler models of capillary exchange. The dependence of variance on the solute diffusion coefficient is not monotonic, but has a minimum near 0·5 × 10?6 cm²/s for reasonable parameters and no barrier, Small molecules like inert gases are expected to have larger variances with higher diffusion coefficients, while larger molecules and barrier limited solutes will have the reverse dependence. Available literature data indicates that capillary heterogeneity will have a major influence on whole-body variance of transit times.     

Preservation and analysis of nonequilibrium solute concentration distributions within mechanically compressed cartilage explants

Solute transport within articular cartilage is of central importance to tissue physiology, and may mediate effects of mechanical compression on cell metabolism. We therefore developed and applied a freeze-substitution method for fixation of cartilage explant disks which had been compressed axially during radial solute desorption. Dextrans were used as model solutes. Explant morphology was well preserved and nonequilibrium solute concentration distributions were stable for several hours at room temperature. For desorption from explants compressed statically to 0-46% strain, analysis of laser confocal images and comparison to a theoretical model permitted measurement of effective diffusivities. Results were consistent with previous studies suggesting a role for transport limitations in mediating the decreases of chondrocyte metabolic rates associated with static compression. In explants compressed dynamically (23+/-5% strain at 0.001 Hz), evidence was obtained for the augmentation of effective transport rate of 3 kDa dextrans by oscillatory interstitial fluid flows. This suggests that augmented solute transport may play a role in mediating the increases of chondrocyte metabolic rates associated with dynamic compression. Methods appear suitable for quantitative studies of transport within mechanically compressed cartilage-like tissues, and may be valuable for identification of loading environments which optimize solute transport in tissue engineering applications.     

Modeling of neutral solute transport in a dynamically loaded porous permeable gel: implications for articular cartilage biosynthesis and tissue engineering

A primary mechanism of solute transport in articular cartilage is believed to occur through passive diffusion across the articular surface, but cyclical loading has been shown experimentally to enhance the transport of large solutes. The objective of this study is to examine the effect of dynamic loading within a theoretical context, and to investigate the circumstances under which convective transport induced by dynamic loading might supplement diffusive transport. The theory of incompressible mixtures was used to model the tissue (gel) as a mixture of a gel solid matrix (extracellular matrix/scaffold), and two fluid phases (interstitial fluid solvent and neutral solute), to solve the problem of solute transport through the lateral surface of a cylindrical sample loaded dynamically in unconfined compression with frictionless impermeable platens in a bathing solution containing an excess of solute. The resulting equations are governed by nondimensional parameters, the most significant of which are the ratio of the diffusive velocity of the interstitial fluid in the gel to the solute diffusivity in the gel (Rg), the ratio of actual to ideal solute diffusive velocities inside the gel (Rd), the ratio of loading frequency to the characteristic frequency of the gel (f), and the compressive strain amplitude (epsilon 0). Results show that when Rg > 1, Rd < 1, and f > 1, dynamic loading can significantly enhance solute transport into the gel, and that this effect is enhanced as epsilon 0 increases. Based on representative material properties of cartilage and agarose gels, and diffusivities of various solutes in these gels, it is found that the ranges Rg > 1, Rd < 1, correspond to large solutes, whereas f > 1 is in the range of physiological loading frequencies. These theoretical predictions are thus in agreement with the limited experimental data available in the literature. The results of this study apply to any porous hydrated tissue or material, and it is therefore plausible to hypothesize that dynamic loading may serve to enhance solute transport in a variety of physiological processes.     

A compartmental capillary, convolution integration model to investigate nutrient transport and metabolism in vivo from paired indicator/nutrient dilution curves.

Thirty-three paired indicator/nutrient dilution curves across the mammary glands of four cows were obtained after rapid injection of para-aminohippuric acid (PAH) plus glucose into the external iliac artery. For the measurement of extracellular volume and kinetics of nutrient uptake from indicator dilution curves, several models of solute dispersion and disappearance have been proposed. The Crone-Renkin models of exchange in a single capillary assume negligible washout of solutes from the extracellular space and do not describe entire dilution curves. The Goresky models include a distribution of capillary transit times to generate whole system outflow profiles but require two indicators to parametize extracellular behavior. A compartmental capillary, convolution integration model is proposed that uses one indicator to account for the extracellular behavior of the nutrient after a paired indicator/nutrient injection. With the use of an iterative approach to least squares, unique solutions for nonexchanging vessel transit time t(mu) and its variance sigma were obtained from all 33 PAH curves. The average of heterogeneous vascular transit times was approximated as 2sigma = 8.5 s. The remainder of indicator dispersion was considered to be due to washout from a well-mixed compartment representing extracellular space that had an estimated volume of 5.5 liters or 24% of mammary gland weight. More than 99% of the variation in the time course of venous PAH concentration after rapid injection into the arterial supply of the mammary glands was explained in an unbiased manner by partitioning the organ into a heterogeneous nonexchanging vessel subsystem and a well-mixed compartmental capillary subsystem.     

Determination of microvessel permeability and tissue diffusion coefficient of solutes by laser scanning confocal microscopy

Interstitium contains a matrix of fibrous molecules that creates considerable resistance to water and solutes in series with the microvessel wall. On the basis of our preliminary studies, by using laser-scanning confocal microscopy and a theoretical model for interstitial transport, we determined both microvessel solute permeability (P) and solute tissue diffusion coefficient (D) of alpha-lactalbumin (Stokes radius 2.01 nm) from the rate of tissue solute accumulation and the radial concentration gradient around individually perfused microvessel in frog mesentery. P(alpha-lactalbumin) is 1.7 +/- 0.7(SD) x 10(-6) cm/s (n = 6). D(t)/D(free) for alpha-lactalbumin is 27% +/- 5% (SD) (n = 6). This value of D(t)/D(free) is comparable to that for small solute sodium fluorescein (Stokes radius 0.45 nm), while p(alpha-lactalbumin) is only 3.4% of p(sodium fluorescein). Our results suggest that frog mesenteric tissue is much less selective to solutes than the microvessel wall.     

Nonradioactive monitoring of organic and inorganic solute transport into single Xenopus oocytes by capillary zone electrophoresis.

Transport of organic and inorganic solutes into and out of cells requires specialized transport proteins. Given a sufficiently sensitive analytical method for measuring cellular solute concentrations, it should be possible to monitor solute transport across the plasma membrane at the level of single cells. We report a capillary zone electrophoresis approach that is generally applicable to monitor solute transport into Xenopus laevis oocytes, requires only nanoliters of sample, and involves no radioactive materials. The sensitivity of capillary electrophoresis with UV detection is typically on the order of 10(-5)-10(-6) M, resulting in the mass detection limits in the low femtomole range. We show that capillary zone electrophoresis serves as a simple technique to measure solute transport into oocytes. Studies of the mammalian oligopeptide transporter PepT1 and the Na(+)- and K(+)-coupled epithelial and neuronal glutamate transporter EAAC1 expressed in oocytes demonstrate that transport of the dipeptide Trp-Gly via PepT1 and transport of Na+ and K+ via EAAC1 across the oocyte plasma membrane can be monitored by measuring intracellular tryptophan absorption and by indirect UV detection of inorganic ions, respectively. The CZE method allowed the simultaneous detection of changes of intracellular Na+ and K+ concentrations in response to EAAC1-mediated Na+ cotransport and K+ countertransport. This is the first report of a capillary zone electrophoresis-based quantitative analysis of intracellular components of a single cell in response to transport activity.     

Some compartmental models of the root: steady-state behavior

Plots of the pressure difference (DeltaP) applied to plant roots vs. the resulting volume flow rate (Q(v)) often exhibit an anomalous offset that has been difficult to explain. The present analysis suggests that solute build-up in two- and three-compartment models of the root cannot account for this offset. The Ginsburg-Newman three-compartment model explains the offset in terms of differing reflection coefficients for the membranes bounding the intermediate compartment. This model appears more promising, but it predicts a minimum in the plot of xylem-sap osmotic pressure vs. Q(v)which is not observed in practice. Fiscus hypothesized that an internal asymmetric distribution of non-mobile solutes is responsible for the offset. In the present study, this hypothesis is incorporated into a four-compartment model of the root that is conceptually related to the three-compartment model of Miller. But according to the four-compartment model, the asymmetric solute distribution does not arise because of solvent drag. Rather the anomalous offset is associated with a concentration gradient of photoassimilates (the non-mobile solutes) that exists in the absence of volume flow, and which drives the diffusive transport of these solutes from the stele to the cortex via endodermal plasmodesmata. This model is consistent with the existence of radial symplastic osmotic-pressure gradients, and it appears to have greater explanatory power than the Ginsburg-Newman model. In particular, it suggests explanations for diurnal variations in DeltaP-Q(v)curves, as well as the effects of changing external solute concentrations. It also shows how the overall root reflection coefficient can be less than unity, even when the cell membranes are effectively ideally semipermeable, and there is negligible extracellular transport of water and solutes. The model makes a number of experimentally testable predictions.     

Apparent Diffusivity and Taylor Dispersion of Water and Solutes in Capillary Beds

A physical theory explaining the anisotropic dispersion of water and solutes in biological tissues is introduced based on the phenomena of Taylor dispersion, in which highly diffusive solutes cycle between flowing and stagnant regions in the tissue, enhancing dispersion in the direction of microvascular flow. An effective diffusion equation is derived, for which the coefficient of dispersion in the axial direction (direction of capillary orientation) depends on the molecular diffusion coefficient, tissue perfusion, and vessel density. This analysis provides a homogenization that represents three-dimensional transport in capillary beds as an effectively one-dimensional phenomenon. The derived dispersion equation may be used to simulate the transport of solutes in tissues, such as in pharmacokinetic modeling. In addition, the analysis provides a physically based hypothesis for explaining dispersion anisotropy observed in diffusion-weighted imaging (DWI) and diffusion-tensor magnetic resonance imaging (DTMRI) and suggests the means of obtaining quantitative functional information on capillary vessel density from measurements of dispersion coefficients. It is shown that a failure to account for flow-mediated dispersion in vascular tissues may lead to misinterpretations of imaging data and significant overestimates of directional bias in molecular diffusivity in biological tissues. Measurement of the ratio of axial to transverse diffusivity may be combined with an independent measurement of perfusion to provide an estimate of capillary vessel density in the tissue.     

A critical parameter for transcapillary exchange of small solutes in countercurrent systems

Small solute transport by a countercurrent capillary loop was studied using a theoretical model. In the model, the afferent and the efferent limbs of the loop share a common interstitial space, with which exchange of solute occurs. Sources of solute, epithelial cells, exist near capillaries and secret solute into the interstitial fluid. Parameters based on experimental measurements on young Sprague-Dawley rats were used in the model, and asymptotic solutions were derived. Comparison of the solute distribution in the interstitium between a capillary loop and a single capillary reveals that the ratio of the product of permeability (P(1)) and surface area (A(1)) to flow (F(1)) of the afferent limb, gamma(1)=P(1)A(1)/F(1) is a critical parameter for the countercurrent exchange system. It alone determines whether the countercurrent arrangement of capillaries facilitates clearance of solute from the interstitial fluid, a greater axial gradient of solute in the interstitium from the base to the tip of the capillary loop and a greater effect of flow, F, upon this gradient. The properties of the efferent limb affect the results, but it is gamma(1) that determines the characteristic difference between a capillary loop and a single capillary.     

Concentrating Engines and the Kidney: II. Multisolute Central Core Systems

The analysis of the central core model of the renal medulla is extended to multisolute systems. It is shown that total solute concentration obeys the same differential equations for core and ascending limb as in a single solute system. Equations are derived for the concentration of individual solutes. Application of these equations to a two solute system shows that a central core system can concentrate with all transport being down a concentration gradient. This analysis applied to the renal medulla shows that mixing of urea from the collecting duct (CD) and salt from the loop of Henle in the central core of the inner medulla contributes to the concentration of urine during antidiuresis. It also sets criteria for completely passive function of the loop in the inner medulla, but whether these are satisfied cannot be determined from present experimental data.     

Solute exchange between the plasma and epithelial lining fluid of rat lungs.

Although the transport of solutes from air spaces to plasma has been extensively studied, comparatively little information is available concerning solute equilibration between the plasma and the epithelial lining fluid (ELF) of air-filled lungs. In the present study, 11 lipophobic indicators varying in molecular mass between 22 and 80,000 Da were injected intravenously and/or intramuscularly into anesthetized rats in a manner designed to keep blood concentrations constant. The animals were killed by rapid lavage of their lungs at various intervals up to 120 min after the injections had been made. Indicator concentrations in the bronchoalveolar lavage (BAL) fluid and plasma were determined, and BAL-to-plasma concentration ratios were calculated for indicators that were injected (exogenous: [14C]urea, 22Na+, [3H]mannitol, 99mTc-diethylenetriaminepentaacetate (a chelate), 51Cr-(ethylene dinitrilo)tetraacetate (a chelate), 113mIn-transferrin, human albumin, and Evans blue-labeled rat albumin) and those that were already present from the plasma and ELF (unlabeled urea, rat albumin, and rat transferrin). Leakage of exogenous indicators in the blood into the BAL fluid was observed during the lavage procedure. Leakage of [14C]urea, 22Na+, and [3H]mannitol exceeded that of the heavier solute molecules. Diffusion of proteins and the labeled chelates into the ELF before lavage occurred at similar rates, suggesting vesicular transport. Use of rapidly diffusible solutes such as urea for determining dilution of ELF by BAL should be accompanied by intravascular injections of labeled solutes to correct for diffusion from the blood during lavage. Alternatively, labeled chelates or serum proteins can be used to estimate dilution of ELF by BAL. Interstitial sampling may be inevitable if the epithelium has been injured before lavage.