A mixture theory model of fluid and solute transport in the microvasculature of normal and malignant tissues. I. Theory

In order to better understand the mechanisms governing transport of drugs, nanoparticle-based treatments, and therapeutic biomolecules, and the role of the various physiological parameters, a number of mathematical models have previously been proposed. The limitations of the existing transport models indicate the need for a comprehensive model that includes transport in the vessel lumen, the vessel wall, and the interstitial space and considers the effects of the solute concentration on fluid flow. In this study, a general model to describe the transient distribution of fluid and multiple solutes at the microvascular level was developed using mixture theory. The model captures the experimentally observed dependence of the hydraulic permeability coefficient of the capillary wall on the concentration of solutes present in the capillary wall and the surrounding tissue. Additionally, the model demonstrates that transport phenomena across the capillary wall and in the interstitium are related to the solute concentration as well as the hydrostatic pressure. The model is used in a companion paper to examine fluid and solute transport for the simplified case of an axisymmetric geometry with no solid deformation or interconversion of mass.     

A model for high-pressure ultrafiltration of blood

A semi-analytic model to predict the permeate flux during high-pressure ultrafiltration of blood with highly permeable membranes is proposed. This model explicitly considers the hydraulic resistance of the retained particles that limits the flux. An empirically derived relationship between particle surface concentration and hydraulic resistance is used. This model incorporates the axial variations in blood cell and solute surface concentrations (or concentration polarization), shear-induced diffusion coefficient for the blood cells, effective diffusion coefficient for the blood solutes, hydraulic (lumen) pressure, and flow rate. This model agrees well with experimental results in the pressure-independent filtration flux region.     

Method for determining solutes in the cell walls of leaves

A perfusion method is described whereby large discs of amphistomatous leaves are vacuum-perfused with water so that either successive fractions of perfusate may be analyzed for solutes or the infused water may be displaced and collected after equilibration with the leaf cells. With castor bean leaves, estimates of electrolyte concentration in cell wall water by the two methods were similar. Total electrolytes in leaf cell wall water of castor beans (Ricinus communis), sunflower (Helianthus annuus), and cabbage (Brassica oleracea capitata) from nonsaline cultures were about 2, 2, and 10 milliequivalents per liter, respectively, increasing to 4, 10, and 30 milliequivalents per liter under saline conditions. Electrolytes recovered in successive fractions were similar in composition, and continuous perfusion resulted in a steady release of solutes, the concentration in the perfusate varying inversely with the perfusion rate. Diffusional release of solutes from cells was less than expected at low perfusion rates, suggesting that solute reabsorption may increase as solute concentration in the perfusate increases with decreased perfusion rates. Perfusate concentration and composition were essentially unaffected by temperature (2 and 23 C) or by perfusing with 0.5 mm CaSO₄ rather than with water. Electrolytes in perfusates on an equivalent basis were Ca²⁺, 30%; Mg²⁺, 10%; and Na⁺ + K⁺, 60%, the proportions of sodium increasing from 10 to 50% in leaves (cabbage) that accumulated sodium under saline conditions. Salinity (added NaCl) of the root culture medium caused a 3- to 5-fold increase in total cell wall electrolyte concentration, but this amounted to an increase from less than 1 or a few per cent to no more than 7% (in cabbage) of the cell sap electrolyte concentrations. Solutes in the cell wall appear to be in dynamic equilibrium with intracellular solutes.     

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.     

Scanning gel chromatography: determination of transport parameters from dynamic profiles measured at low solute concentration

Direct optical scanning of solute boundaries in large zone gel Chromatography experiments provides an accurate means of determining boundary profile shapes and rates of motion. A method has been developed for correcting such boundaries to a constant time frame, eliminating the distortion which arises from finite column scanning rate Centroids or the corrected profiles can be used to determine the partition cross section for the solute of interest The partition cross section and flow rate determine translational motion within the column. The axial dispersion coefficient, L, which characterizes rate of boundary spreading may also be calculated from the profiles. In order to explore these procedures a study of four noninteracting solutes was conducted. Partition cross sections determined from rates of motion of boundary centroids were found to be in good agreement with those determined by the equilibrium saturation method on the same column.In order to explore the lowest concentration limits of the technique and to illustrate the boundary characteristics for a selfassociating solute, a study of carboxyhemoglobin was conducted over a wide concentration range. From measurements at 220 nm the lowest concentration where useful data could be obtained was 2 micrograms per ml (0.12 πM heme) These results establish validity of the procedures used in analyzing the rates of boundary transport and in studying solute transport over a wide range of conditions.     

An electrodiffusion model for effects of surface glycocalyx layer on microvessel permeability

To investigate the charge effect of the endothelial surface glycocalyx on microvessel permeability, we extended the three-dimensional model developed by Fu et al. (J Biomech Eng 116: 502-513, 1994) for the interendothelial cleft to include a negatively charged glycocalyx layer at the entrance of the cleft. Both electrostatic and steric exclusions on charged solutes were considered within the glycocalyx layer and at the interfaces. Four charge-density profiles were assumed for the glycocalyx layer. Our model indicates that the overall solute permeability across the microvessel wall including the surface glycocalyx layer and the cleft region is independent of the charge-density profiles as long as they have the same maximum value and the same total charge. On the basis of experimental data, this model predicts that the charge density would be 25-35 meq/l in the glycolcalyx of frog mesenteric capillaries. An intriguing prediction of this model is that when the concentrations of cations and anions are unequal in the lumen due to the presence of negatively charged proteins, the negatively charged glycocalyx would provide more resistance to positively charged solutes than to negatively charged ones.     

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.     

Solute production and net wall synthesis in the growing and non-growing cells of gravistimulated sunflower hypocotyls

Solute generation and cell wall synthesis were examined in sunflower hypocotyl peripheral layers, the growth rate of which had been altered by gravistimulation. Measurements of both the concentrations of the major solutes and the osmotic potential showed that although upper cells stopped growing, the solute levels in these cells continued to increase at rates comparable to those in lower cells. This indicated that altered growth rates, generated during gravicurvature, are not based on solute generation but must result from cell wall changes. Gravimetric and precursor incorporation studies showed that net wall synthesis continued in upper cells despite their lack of growth. An ultrastructural study of the epidermal cells on the uppermost (non-elongating) and lowermost (elongating) surfaces of horizontal cucumber hypocotyls showed that the relative amounts of the various membrane fractions were similar in upper and lower cells despite their very different growth rates.     

Computational pharmacokinetics of solute penetration into human intervertebral discs-Effects of endplate permeability, solute molecular weight and disc size

A finite element model is developed to predict the penetration time-history of three different solutes into the human lumbar disc following intravenous injection. Antibiotics are routinely administered intravenously in spinal surgery to prevent disc infection. Successful prophylaxis requires antibiotics to reach adequate inhibitory levels. Here, the transient diffusion of cephazolin is investigated over 10h post-injection in a human disc model subject to reported concentrations in the blood stream as the prescribed boundary sources. Post-injection variation of cephazolin concentrations in the disc adjacent to supply sources closely followed the decay curve in the blood stream and fell sharply with time. Much lower concentrations were computed in the inner annulus and nucleus; much of the disc (80% at 1h and 49% at 4h) experienced concentrations below required inhibitory level of 1mg/L in agreement with measurements. Changes in endplate permeability, disc size, and solute molecular weight had profound effects on concentration profiles at all times and regions, especially in the disc centre, demonstrating their crucial roles on the adequate delivery of drugs. Larger solutes markedly slow transport into the disc. The failure to reach critical therapeutic levels in the central disc regions, especially when endplates calcify and in larger discs, raises concerns and calls for caution in attempts to extrapolate findings of studies on animals with much smaller and non degenerate discs to the human discs. The current study also demonstrates the capability of computational models in predicting the transport of intravenously injected solutes into the disc.     

Extraction and Quantification of Solutes in Solidified Agar Culture Media

A method is described for determining the concentration of certain solutes in solidified culture media. The method is based upon the finding that under specified conditions the concentration of solute in an agar gel (C_g) is related to the concentration of solute in a centrifugally extracted gel supernatant (C_s) by the ratio, C_g/C_s, which is characteristic for each solute. The method avoids direct assays of the gels and instead involves assaying the supernatants from inoculated and uninoculated (control) gels with conventional liquid assay techniques and then calculating solute concentrations in the inoculated gels by use of the C_g/C_s ratios determined from the controls. Uninoculated agar gels containing known concentrations of various solutes and similar gels inoculated with Neurospora crassa or Escherichia coli were centrifuged at various times, and the supernatants were assayed for solute concentrations. The solute concentrations in the supernatants from the inoculated gels multiplied by the C_g/C_s ratios for those solutes determined at the same times for the uninoculated controls gave calculated solute concentrations in the inoculated gels. The differences between these calculated solute concentrations and those initially present in the inoculated gels indicated the amounts of solutes utilized from the gels by the microorganisms at various incubation times.     

Swelling and viscoelastic properties of osmotically stressed elastin

The swelling and viscoelastic properties of purified elastin were studied in aqueous solutions of superswelling agents or osmotic deswelling agents to develop models to study the behavior of elastin at frequencies not easily accessible by direct measurement. Increasing the concentration of any of the deswelling solutes (glucose, sucrose, sodium chloride, ammonium sulphate, dextran, and polyethylene glycol) increased the tensile storage and loss moduli. The viscoelastic behavior was independent of solute when compared on the basis of swelling behavior. The data collected at various solute concentrations at 37°C could be reduced to one master curve, and the master curves for elastin in each of the deswelling solutes were themselves superposable. The ability to reduce the data indicates that dehydration can be used to model elastin's viscoelastic behavior at high frequencies or over short times. The viscoelastic behavior of elastin in the superswelling agents [potassium thiocyanate (KSCN), dimethyl sulfoxide (DMSO), and ethylene glycol (EG)] depended on the solute and was independent of swelling behavior. In KSCN the behavior of elastin seemed to be a continuation of the pattern established by the deswelling agents in that an increase in swelling was accompanied by a decrease in both moduli, and the viscoelastic spectra were reducible to one master curve. In high concentrations of DMSO and EG the spectra were not reducible. KSCN appears a suitable superswelling solute to model elastin's viscoelastic behavior at low frequencies or over long times. © 1996 John Wiley & Sons, Inc.     

Mathematics of band centrifugation: Concentration-independent sedimentation and diffusion in shallow density gradients

Integral expressions for concentration as a function of time and distance are derived from the continuity equation for centrifugation in a sector-shaped cell for a macro-molecular solute initially contained in a finite upper layer and a solute of low molecular weight in the supporting liquid. Computer patterns based on the sedimentation and diffusion coefficients of sucrose and of spherical and randomly coiled model solutes illustrate: (1) the time course of redistribution of both banded and supporting solutes from initial uniform concentrations; (2) the influence of the initial concentration, width, and solute concentration of the upper band; and (3) the effect of restricted diffusion at the meniscus on subsequent band shape. A Gaussian, approximation to band shape is derived and graphically tested. Rapid methods, not requiring computers, are out lined for the estimation of sedimentation and diffusion coefficients, where their concentration dependence is negligible, by band centrifugtion. The theoretical resolution of mixtures attainable by this technique is compared with moving-boundary centrifugation, with the use of both integral (interferotmetric or absorption) and derivative (schlieren) optics.     

Export of Abscisic Acid, 1-Aminocyclopropane-1-Carboxylic Acid,Phosphate, and Nitrate from Roots to Shoots of Flooded Tomato Plants (Accounting for Effects of Xylem Sap Flow Rate on Concentration and Delivery)

We determined whether root stress alters the output of physiologically active messages passing from roots to shoots in the transpiration stream. Concentrations were not good measures of output. This was because changes in volume flow of xylem sap caused either by sampling procedures or by effects of root stress on rates of whole-plant transpiration modified concentrations simply by dilution. Thus, delivery rate (concentration x sap flow rate) was preferred to concentration as a measure of solute output from roots. To demonstrate these points, 1-aminocyclopropane-1-carboxylic acid (ACC), abscisic acid, phosphate, nitrate, and pH were measured in xylem sap of flooded and well-drained tomato (Lycopersicon esculentum Mill., cv Ailsa Craig) plants expressed at various rates from pressurized detopped roots. Concentrations decreased as sap flow rates were increased. However, dilution of solutes was often less than proportional to flow, especially in flooded plants. Thus, sap flowing through detopped roots at whole-plant transpiration rates was used to estimate solute delivery rates in intact plants. On this basis, delivery of ACC from roots to shoots was 3.1-fold greater in plants flooded for 24 h than in well-drained plants, and delivery of phosphate was 2.3-fold greater. Delivery rates of abscisic acid and nitrate in flooded plants were only 11 and 7%, respectively, of those in well-drained plants.     

Effects of solute concentration on the entrapment of solutes in phospholipid vesicles prepared by freeze-thaw extrusion.

Phospholipid vesicles prepared by the freeze-thaw extrusion method contain internal solute concentrations which are much higher than the external values (entrapment ratios much greater than 1). This concentrating effect is a complex function of the total impermeant solute concentration in the medium used to prepare vesicles, the presence or absence of permeant solutes in the medium and the apparent competitive binding interactions between solutes and phospholipid. Increases in water phase solute concentration during freezing are thought to underlie the concentrating phenomenon, while osmotic pressure driven lysis of vesicles during thawing appears to limit its magnitude. By judicious selection of solute concentration and physical properties, further increases in the entrapment ratio should be obtainable, improving the usefulness of these vesicles as drug delivery vesicles and experimental systems.     

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.     

Analysis of hemodynamic fluid phase mass transport in a separated flow region

The mass transfer behavior in the recirculation region downstream of an axisymmetric sudden expansion was examined. The Reynolds number, 500, and Schmidt number, 3200, were selected to model the mass transfer of molecules, such as ADP, in the arterial system. In a first step the transient mass transport applying zero diffusive flux at the wall was analyzed using experiments and two computational codes. The two codes were FLUENT, a commercially available finite volume method, and FTSP, a finite element code developed at Graz University of Technology. The comparison of the transient wall concentration values determined by the three methods was excellent and provides a measure of confidence for computational mass transfer calculations in convection dominated, separated flows. In a second step the effect of the flow separation on the stationary mass transport applying a permeability boundary condition at the water-permeable wall was analyzed using the finite element code FTSP. The results show an increase of luminal ADP surface concentration in the upstream and in the downstream tube of the sudden expansion geometry in the range of six and twelve percent of the bulk flow concentration. The effect of flow separation in the downstream tube on the wall concentration is a decrease of about ten percent of the difference between wall concentration and bulk concentration occurring at nearly fully developed flow at the downstream region at a distance of 66 downstream tube diameters from the expansion. The decrease of ADP flux into the wall is in the range of three percent of the flux at the downstream region.     

Unexpected spatial stability of water chemistry in headwater stream networks

Understanding how water and solutes enter and propagate through freshwater landscapes in the Anthropocene is critical to protecting and restoring aquatic ecosystems and ensuring human water security. However, high hydrochemical variability in headwater streams, where most carbon and nutrients enter river networks, has hindered effective modelling and management. We developed an analytical framework informed by landscape ecology and catchment hydrology to quantify spatiotemporal variability across scales, which we tested in 56 headwater catchments, sampled periodically over 12 years in western France. Unexpectedly, temporal variability in dissolved carbon, nutrients and major ions was preserved moving downstream and spatial patterns of water chemistry were stable on annual to decadal timescales, partly because of synchronous variation in solute concentrations. These findings suggest that while concentration and flux cannot be extrapolated among subcatchments, periodic sampling of headwaters provides valuable information about solute sources and subcatchment resilience to disturbance.