Diffusion of antibiotics in intervertebral disc

Delivering charged antibiotics to the intervertebral disc is challenging because of the avascular, negatively charged extracellular matrix (ECM) of the tissue. The purpose of this study was to measure the apparent diffusion coefficient of two clinically relevant, charged antibiotics, vancomycin (positively charged) and oxacillin (negatively charged) in IVD. A one-dimensional steady state diffusion experiment was employed to measure the apparent diffusion coefficient of the two antibiotics in bovine coccygeal annulus fibrosus (AF) tissue. The averaged apparent diffusion coefficient for vancomycin under 20% compressive strain was 7.94 ± 2.00 × 10⁻¹² m²/s (n = 10), while that of oxacillin was 2.26 ± 0.68 × 10⁻¹⁰ m²/s (n = 10). A student’s t-test showed that the diffusivity of vancomycin was significantly lower than that of oxacillin. This finding may be attributed to two factors: solute size and possible binding effects. Vancomycin is approximately 3 times larger in molecular weight than oxacillin, meaning that steric hindrance likely plays a role in the slower transport. Reversible binding between positive vancomycin and the negative ECM could also slow down the rate of diffusion. Therefore, more investigation is necessary to determine the specific relationship between net charge on antibiotic and diffusion coefficients in IVD. This study provides essential quantitative information regarding the transport rates of antibiotics in the IVD, which is critical in using computational modeling to design effective strategies to treat disc infection.     

Transport of neutral solute in articular cartilage: effect of microstructure anisotropy

Due to the avascular nature of articular cartilage, solute transport through its extracellular matrix is critical for the maintenance and the functioning of the tissue. What is more, diffusion of macromolecules may be affected by the microstructure of the extracellular matrix in both undeformed and deformed cartilage and experiments demonstrate diffusion anisotropy in the case of large solute. However, these phenomena have not received sufficient theoretical attention to date. We hypothesize here that the diffusion anisotropy of macromolecules is brought about by the particular microstructure of the cartilage network. Based on this hypothesis, we then propose a mathematical model that correlates the diffusion coefficient tensor with the structural orientation tensor of the network. This model is shown to be successful in describing anisotropic diffusion of macromolecules in undeformed tissue and is capable of clarifying the effects of network reorientation as the tissue deforms under mechanical load. Additionally, our model explains the anomaly that at large strain, in a cylindrical plug under unconfined compression, solute diffusion in the radial direction increases with strain. Our results indicate that in cartilage the degree of diffusion anisotropy is site specific, but depends also on the size of the diffusing molecule. Mechanical loading initiates and/or further exacerbates this anisotropy. At small deformation, solute diffusion is near isotropic in a tissue that is isotropic in its unstressed state, becoming anisotropic as loading progresses. Mechanical loading leads to an attenuation of solute diffusion in all directions when deformation is small. However, loading, if it is high enough, enhances solute transport in the direction perpendicular to the load line, instead of inhibiting it.     

A multiscale theoretical model for diffusive mass transfer in cellular biological media

An integrated methodology is developed for the theoretical analysis of solute transport and reaction in cellular biological media, such as tissues, microbial flocs, and biofilms. First, the method of local spatial averaging with a weight function is used to establish the equation which describes solute conservation at the cellular biological medium scale, starting with a continuum-based formulation of solute transport at finer spatial scales. Second, an effective-medium model is developed for the self-consistent calculation of the local diffusion coefficient in the cellular biological medium, including the effects of the structural heterogeneity of the extra-cellular space and the reversible adsorption to extra-cellular polymers. The final expression for the local effective diffusion coefficient is: D(Abeta)=lambda(beta)D(Aupsilon), where D(Aupsilon) is the diffusion coefficient in water, and lambda(beta) is a function of the composition and fundamental geometric and physicochemical system properties, including the size of solute molecules, the size of extra-cellular polymer fibers, and the mass permeability of the cell membrane. Furthermore, the analysis sheds some light on the function of the extra-cellular hydrogel as a diffusive barrier to solute molecules approaching the cell membrane, and its implications on the transport of chemotherapeutic agents within a cellular biological medium. Finally, the model predicts the qualitative trend as well as the quantitative variability of a large number of published experimental data on the diffusion coefficient of oxygen in cell-entrapping gels, microbial flocs, biofilms, and mammalian tissues.     

Local macromolecule diffusion coefficients in structurally non-uniform bacterial biofilms using fluorescence recovery after photobleaching (FRAP)

Pure culture Pseudomonas putida biofilms were cultivated under controlled conditions to a desired overall biofilm thickness, then employed within classical half-cell diffusion chambers to estimate, from transient solute concentrations, the effective diffusion coefficient for several macromolecules of increasing molecular weight and molecular complexity. Results of traditional half-cell studies were found to be erroneous due to the existence of microscopic water channels or crevasses that perforate the polysaccharidic gel matrix of the biofilm, sometimes completely to the supporting substratum. Thus, half-cell devices measure a composite transfer coefficient that may overestimate the true, local flux of solutes in the biofilm polysaccharide gel matrix. An alternative analytical technique was refined to determine the local diffusion coefficients on a micro-scale to avoid the errors created by the biofilm architectural irregularities. This technique is based upon the Fluorescence Return After Photobleaching (FRAP), which allows image analysis observation of the transport of fluorescently labeled macromolecules as they migrate into a micro-scale photobleached zone. The technique can be computerized and allows one to map the local diffusion coefficients of various solute molecules at different horizontal planes and depths in a biofilm. These mappings also indirectly indicate the distribution of water channels in the biofilm, which was corroborated independently by direct microscopic observation of the settling of fluorescently-labeled latex spheres within the biofilm. Fluorescence return after photobleaching results indicate a significant reduction in the solute transport coefficients in biofilm polymer gel vs. the same value in water, with the reduction being dependent on solute molecule size and shape.     

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.     

Monte Carlo analysis of obstructed diffusion in three dimensions: application to molecular diffusion in organelles.

Molecular transport in the aqueous lumen of organelles involves diffusion in a confined compartment with complex geometry. Monte Carlo simulations of particle diffusion in three dimensions were carried out to evaluate the influence of organelle structure on diffusive transport and to relate experimental photobleaching data to intrinsic diffusion coefficients. Two organelle structures were modeled: a mitochondria-like long closed cylinder containing fixed luminal obstructions of variable number and size, and an endoplasmic reticulum-like network of interconnected cylinders of variable diameter and density. Trajectories were computed in each simulation for >10(5) particles, generally for >10(5) time steps. Computed time-dependent concentration profiles agreed quantitatively with analytical solutions of the diffusion equation for simple geometries. For mitochondria-like cylinders, significant slowing of diffusion required large or wide single obstacles, or multiple obstacles. In simulated spot photobleaching experiments, a approximately 25% decrease in apparent diffusive transport rate (defined by the time to 75% fluorescence recovery) was found for a single thin transverse obstacle occluding 93% of lumen area, a single 53%-occluding obstacle of width 16 lattice points (8% of cylinder length), 10 equally spaced 53% obstacles alternately occluding opposite halves of the cylinder lumen, or particle binding to walls (with mean residence time = 10 time steps). Recovery curve shape with obstacles showed long tails indicating anomalous diffusion. Simulations also demonstrated the utility of measurement of fluorescence depletion at a spot distant from the bleach zone. For a reticulum-like network, particle diffusive transport was mildly reduced from that in unobstructed three-dimensional space. In simulated photobleaching experiments, apparent diffusive transport was decreased by 39-60% in reticular structures in which 90-97% of space was occluded. These computations provide an approach to analyzing photobleaching data in terms of microscopic diffusive properties and support the paradigm that organellar barriers must be quite severe to seriously impede solute diffusion.     

Physiological significance of the structure and components of the apoplast canal system for water absorption in plants

The non-linear differential equation that describes the coupling between water transport and solute transport in the apoplast canal system in plants was proposed by Katou and Furumoto in 1986. In the present paper, we analytically solved the equation in order to find the law describing the canal system. In the canal system, water transport is regulated linearly by solute transport under physiological conditions. The approximate solution of the differential equations defines the conditions of the structure and components of the apoplast canal for optimal water absorption. Water absorption during cell elongation in plants requires that the apoplast canal be composed of a cell wall with an appropriate diffusion coefficient for solute.     

Diffusion in colloidal media

When the molecules of a solute diffuse through a medium containing large colloidal particles, which absorb the diffusing molecules, the latter are transported in the diffusion flow not as free molecules, but as absorbtion compounds: solute+colloid. When the colloidal particle is much larger than the molecule of the solute, and has therefore a much smaller mobility, this results in a reduction of the apparent diffusion coefficient for the solute. The biological implications of this are discussed.     

Mass transport at rotating disk electrodes: effects of synthetic particles and nerve endings

An unstirred layer (USL) exists at the interface of solids with solutions. Thus, the particles in brain tissue preparations possess a USL as well as at the surface of a rotating disk electrode (RDE) used to measure chemical fluxes. Time constraints for observing biological kinetics based on estimated thicknesses of USLs at the membrane surface in real samples of nerve endings were estimated. Liposomes, silica, and Sephadex were used separately to model the tissue preparation particles. Within a solution stirred by the RDE, both diffusion and hydrodynamic boundary layers are formed. It was observed that the number and size of particles decreased the following: the apparent diffusion coefficient excluding Sephadex, boundary layer thicknesses excluding silica, sensitivity excluding diluted liposomes (in agreement with results from other laboratories), limiting current potentially due to an increase in the path distance, and mixing time. They have no effect on the detection limit (6 ± 2 nM). The RDE kinetically resolves transmembrane transport with a timing of approximately 30 ms.     

Analysis of the introduction and removal of glycerol in rabbit kidneys using a Krogh cylinder model

In 1982, Rubinsky and Cravalho described a Krogh cylinder model for the analysis of cryoprotectant transport in a perfused organ. By application of the Kedem-Katchalsky equations, changes in tissue volume caused by movements of water and solute were used to predict changes in capillary radius (Cryobiology 19, 70-82, 1982). We have now measured the changes in vascular resistance that are produced when sucrose or glycerol is introduced into the perfusate flowing through rabbit kidneys at 10 degrees C, and have analyzed these data by means of the Rubinsky-Cravalho semiempirical model. The sucrose data provided an estimate of hydraulic conductivity and the dimensions of the Krogh tissue units. Three rates of addition of glycerol, 10, 30, and 90 mM/min to a final concentration of 3 M, were studied. The vascular resistance fell to approximately 40% of its initial value (radius approximately 128% of initial value) with all three rates of addition, and then returned toward its normal value while the glycerol concentration was still increasing. This behavior could be explained either by a sudden change in solute permeability at that capillary radius, or by an inverse dependence of reflection coefficient upon solute concentration. Evidence is presented that favors the latter interpretation. The best fits for the apparent hydraulic conductivity and apparent solute permeability for glycerol are 1 X 10(-6) cm/sec atm and 6 X 10(-8) cm/sec, respectively, with the reflection coefficient falling from 1.0 when the glycerol concentration is zero to 0.1 when it is 3 M. The model is used to predict tissue concentrations of glycerol throughout each experiment.     

Measurement of Gd-DTPA diffusion through PVA hydrogel using a novel magnetic resonance imaging method.

Polyvinyl alcohol-cryogel (PVA-C) is a hydrogel that is an excellent tissue mimic. In order to characterize mass transfer in this material, as well as to demonstrate in principle the ability to noninvasively measure solute diffusion in tissue, we measured the diffusion coefficient of the magnetic resonance (MR) contrast agent gadolinium diethylene triaminopentaacetic acid (Gd-DTPA) through PVA-C using a clinical MR imager. The method involved filling thick-walled rectangular PVA-C "cups" with known concentrations of Gd-DTPA solutions. Then by using a fast inversion recovery spin echo MR imaging protocol, a signal "null" contour was created in the MR image that corresponded to a second, known concentration of Gd-DTPA. By collecting a series of MR images through the PVA-C wall as a function of time, the displacement of this second known isoconcentration contour could be tracked. Application of Fick's second law of diffusion yielded the diffusion coefficient. Seven separate experiments were performed using various combinations of initial concentrations of Gd-DTPA within the PVA-C cups (3.2, 25.6, or 125 mM) and tracked isoconcentrations contours (0.096, 0.182, or 0.435 mM Gd-DTPA). The experimental results and the predictions of Fick's law were in excellent agreement. The diffusivity of Gd-DTPA through 10% PVA hydrogel was found to be (2.6 +/- 0.04) x 10(-10) m(2)/s (mean +/- s.e.m.). Separate permeability studies showed that the diffusion coefficient of Gd-DTPA through this hydrogel did not change with an applied pressure of up to 7.1 kPa. Accurate measurements could be made within 30 min if suitable Gd-DTPA concentrations were selected. Due to the excellent repeatability and fast data acquisition time, this technique is very promising for future in vivo studies of species transport in tissue.     

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.     

Diffusion and partition of solutes in cartilage under static load

We describe experimental apparatus, methodology and mathematical algorithms to measure diffusion and partition for typical small ionic solutes and inulin (a medium size solute) in statically loaded cartilage. The partition coefficient based on tissue water (K(H(2)O)) of Na(+) increased from 1.8 to 4.5 and for SO(4)(-2) decreased from 0.5 to 0.1, when the applied pressure was raised from zero to 22 atm K(H(2)O) of inulin decreased from 0.3 to 0.05, for an increase in pressure from zero to 11 atm. Our theoretical interpretation of the results is that the partition coefficient can be expressed as a function of fixed charge density (FCD) for both loaded and unloaded cartilage. The partition coefficient shows good agreement with the ideal Gibbs-Donnan equilibrium, particularly when FCD is based on extrafibrillar water (EFW). The diffusion coefficients, D also decreased with an increase in applied pressure; raising the pressure from 0 to 22 atm resulted in the following changes in the values of D: for Na(+) from 2.86 x 10(-6) to 1.51 x 10(-6) cm(2)/s, for SO(4)(-2) from 1.58 x 10(-6) to 7.5 x 10(-7) cm(2)/s, for leucine from 1.69 x 10(-6) to 8.30 x 10(-7) cm(2)/s and for inulin from 1.80 x 10(-7) to 3.30 x 10(-8) cm(2)/s. For the three small solutes (two charged and one neutral) the diffusion coefficient D is highly correlated with the fraction of fluid volume in the tissue. These experimental results show good agreement with the simple model of Mackie and Meares: hence solute charge does not affect the diffusion of small solutes under load. For inulin D & K show some agreement with a modified Ogston model based on two major components, viz., glycosaminoglycans (GAG) and core protein. We conclude that the changes in the partition and diffusion coefficients of small and medium size solutes in statically loaded cartilage can be interpreted as being due to the reduction in hydration and increase in FCD. The change in the latter affects the partition of small ionic solutes and the partition and diffusion of larger molecules. Our results throw light on the ionic environment of chondrocytes in loaded cartilage as well as on the transport of solutes through the matrix.     

Porin channels in Escherichia coli: studies with liposomes reconstituted from purified proteins.

Rates of diffusion of uncharged and charged solute molecules through porin channels were determined by using liposomes reconstituted from egg phosphatidylcholine and purified Escherichia coli porins OmpF (protein 1a), OmpC (protein 1b), and PhoE (protein E). All three porin proteins appeared to produce channels of similar size, although the OmpF channel appeared to be 7 to 9% larger than the OmpC and PhoE channels in an equivalent radius. Hydrophobicity of the solute retarded the penetration through all three channels in a similar manner. The presence of one negative charge on the solute resulted in about a threefold reduction in penetration rates through OmpF and OmpC channels, whereas it produced two- to tenfold acceleration of diffusion through the PhoE channel. The addition of the second negatively charged group to the solutes decreased the diffusion rates through OmpF and OmpC channels further, whereas diffusion through the PhoE channel was not affected much. These results suggest that PhoE specializes in the uptake of negatively charged solutes. At the present level of resolution, no sign of true solute specificity was found in OmpF and OmpC channels; peptides, for example, diffused through both of these channels at rates expected from their molecular size, hydrophobicity, and charge. However, the OmpF porin channel allowed influx of more solute molecules per unit time than did the equivalent weight of the OmpC porin when the flux was driven by a concentration gradient of the same size. This apparent difference in "efficiency" became more pronounced with larger solutes, and it is likely to be the consequence of the difference in the sizes of OmpF and OmpC channels.     

Static compression is associated with decreased diffusivity of dextrans in cartilage explants

The chondrocytes of adult articular cartilage rely upon transport phenomena within their avascular extracellular matrix for many biological activities. Therefore, changes in matrix structure which influence cytokine transport parameters may be an important mechanism involved in the chondrocyte response to tissue compression. With this hypothesis in mind, partitioning and diffusion of 3-, 10-, and 40-kDa dextrans conjugated to tetramethylrhodamine, and 430-Da tetramethylrhodamine itself, were measured within statically compressed bovine articular cartilage explants using a novel experimental apparatus and desorption fluorescence method. Partitioning and diffusion were examined as functions of solute molecular weight and matrix proteoglycan density, and diffusion was measured versus static compression up to 35% volumetric strain. In general, partition coefficients and diffusivities were found to decrease with increasing solute molecular weight. In addition, for a given solute, diffusivities decreased significantly with increasing static compression. Results therefore suggest a possible role for transport limitations of relatively large molecular weight solutes within the extracellular matrix in mediating the biological response of chondrocytes to cartilage compression.     

Predicted permeability parameters of human ovarian tissue cells to various cryoprotectants and water

This study presents a generic numerical model to simulate the coupled solute and solvent transport in human ovarian tissue sections during addition and removal of chemical additives or cryoprotective agents (CPA). The model accounts for the axial and radial diffusion of the solute (CPA) as well as axial convection of the CPA, and a variable vascular surface area (A) during the transport process. In addition, the model also accounts for the radial movement of the solvent (water) into and out of the vascular spaces. Osmotic responses of various cells within an human ovarian tissue section are predicted by the numerical model with three model parameters: permeability of the tissue cell membrane to water (L(p)), permeability of the tissue cell membrane to the solute or CPA (omega) and the diffusion coefficient of the solute or CPA in the vascular space (D). By fitting the model results with published experimental data on solute/water concentrations within an human ovarian tissue section, I was able to determine the permeability parameters of ovarian tissue cells in the presence of 1.5M solutions of each of the following: dimethyl sulphoxide (DMSO), propylene glycol (PROH), ethylene glycol (EG), and glycerol (GLY), at two temperatures (4 degrees C and 27 degrees C). Modeling Approach 1: Assuming a constant value of solute diffusivity (D = 1.0 x 10(-9) m(2)/sec), the best fit values of L(p) ranged from 0.35 x 10(-14) to 1.43 x 10(-14) m(3)/N-sec while omega ranged from 2.57 x 10(-14) to 70.5 x 10(-14) mol/N-sec. Based on these values of L(p) and omega, the solute reflection coefficient, sigma defined as sigma = 1-omega v(CPA)/L(P) ranged from 0.9961 to 0.9996. Modeling Approach 2: The relative values of omega and sigma from our initial modeling suggest that the embedded ovarian tissue cells are relatively impermeable to all the CPAs investigated (or omega approximately 0 and sigma approximately 1.0). Consequently the model was modified and used to predict the values of L(p) and D assuming omega = 0 and sigma = 1.0. The best fit values of L(p) ranged from 0.44 x 10(-14) to 1.2 x 10(-14) m(3)/N-sec while D ranged from 0.85 x 10(-9) to 2.08 x 10(-9) m(2)/sec. Modeling Approach 3: Finally, the best fit values of D from modeling approach 2 were incorporated into model 1 to re-predict the values of L(p) and omega. It is hoped that the ovarian tissue cell parameters reported here will help to optimize chemical loading and unloading procedures for whole ovarian tissue sections and consequently, tissue cryopreservation procedures.