Modelling intravascular delivery from drug-eluting stents with biodurable coating: investigation of anisotropic vascular drug diffusivity and arterial drug distribution

In-stent restenosis occurs in coronary arteries after implantation of drug-eluting stents with non-uniform restenosis thickness distribution in the artery cross section. Knowledge of the spatio-temporal drug uptake in the arterial wall is useful for investigating restenosis growth but may often be very expensive/difficult to acquire experimentally. In this study, local delivery of a hydrophobic drug from a drug-eluting stent implanted in a coronary artery is mathematically modelled to investigate the drug release and spatio-temporal drug distribution in the arterial wall. The model integrates drug diffusion in the coating and drug diffusion with reversible binding in the arterial wall. The model is solved by the finite volume method for both high and low drug loadings relative to its solubility in the stent coating with varied isotropic–anisotropic vascular drug diffusivities. Drug release profiles in the coating are observed to depend not only on the coating drug diffusivity but also on the properties of the surrounding arterial wall. Time dependencies of the spatially averaged free- and bound-drug levels in the arterial wall on the coating and vascular drug diffusivities are discussed. Anisotropic vascular drug diffusivities result in slightly different average drug levels in the arterial wall but with very different spatial distributions. Higher circumferential vascular diffusivity results in more uniform drug loading in the upper layers and is potentially beneficial in reducing in-stent restenosis. An analytical expression is derived which can be used to determine regions in the arterial with higher free-drug concentration than bound-drug concentration.     

Determination of cell membrane permeability in concentrated cell ensembles

The method of volume averaging is used to analyze the process of diffusion in concentrated cell ensembles in which significant resistance to mass transfer is caused by the cellular membrane. A general closure scheme is given that allows for direct theoretical prediction of effective diffusivities for any cellular geometry. Numerical results are presented for the classical parallelepiped arrangement used to model cellular systems, and these results are used in conjunction with experimental studies of concentrated cell ensembles to determine membrane permeabilities for solute diffusion in several cellular systems. Membrane permeabilities are compared with predictions from other models of diffusion in cellular systems.     

Unravelling the Turing bifurcation using spatially varying diffusion coefficients

. The Turing bifurcation is the basic bifurcation generating spatial pattern, and lies at the heart of almost all mathematical models for patterning in biology and chemistry. In this paper the authors determine the structure of this bifurcation for two coupled reaction diffusion equations on a two-dimensional square spatial domain when the diffusion coefficients have a small explicit variation in space across the domain. In the case of homogeneous diffusivities, the Turing bifurcation is highly degenerate. Using a two variable perturbation method, the authors show that the small explicit spatial inhomogeneity splits the bifurcation into two separate primary and two separate secondary bifurcations, with all solution branches distinct. This splitting of the bifurcation is more effective than that given by making the domain slightly rectangular, and shows clearly the structure of the Turing bifurcation and the way in which the! var ious solution branches collapse together as the spatial variation is reduced. The authors determine the stability of the solution branches, which indicates that several new phenomena are introduced by the spatial variation, including stable subcritical striped patterns, and the possibility that stable stripes lose stability supercritically to give stable spotted patterns.. Received: 10 January 1996/Revised version: 3 July 1996     

Dissipative pattern formation in ternary non-linear reaction-electrodiffusion systems with concentration-dependent diffusivities

Conditions for the emergence of ionic dissipative structures after Turing instability of a spatially and temporaly homogeneous stationary concentration distribution are derived for electroneutral ternary ionic reaction systems with concentration-dependent diffusivities. The inhomogeneous solution branch first bifurcating while crossing a critical ratio of diffusivities is given as a series of eigenfunctions of the Laplace operator. The results are specified for a model reaction system and polynomial concentration dependence of diffusion. It is shown that the inclusion of concentration dependence controls the amplitudes of the spatial distribution.     

Analysis of a model of a vertically transmitted disease

A model is presented of a disease that can be transmitted directly from parent to offspring (vertical transmission) as well as through contact with infectives. A global stability analysis is given for the basic model and the epidemiological effects of vertical transmission are discussed. The effects of the addition of maturation and incubation delays as well as spatial diffusion are analyzed in some special cases.     

Modeling and analysis of diffusion and reaction in legume nodules

Diffusion of gases through legume nodules is important for nitrogen fixation. A mathematical model is presented for diffusion and enzymatic reaction for legume nodules with a reactive core and an inert shell. The transient model is solved numerically for spherical geometry for acetylene reduction by nitrogenase enzyme. The results are used to estimate the diffusivities of acetylene and ethylene in the nodules by comparing predicted and experimental lag times. The experimental results are also analyzed using an effectiveness factor plot for spherical nodules with inert shells and reactive cores. The results show that the diffusivities are slightly higher than those for acetylene and ethylene in water because of some contribution of gas phase diffusion. Applications to oxygen diffusion through nodule tissue are suggested.     

Computer Simulation of Isothermal Mass Transport in Graphite Slit Pores

Abstract

Results are presented from a simulation study of the mass transport of oxygen and nitrogen through graphite slit pores. The work is motivated by an attempt to understand the molecular origins of the kinetic selectivity displayed when air is separated into its major components using pressure swing adsorption. A combination of non-equilibrium molecular dynamics (NEMD), equilibrium molecular dynamics (EMD) and grand canonical Monte Carlo methods has been employed in our study to extract the maximum information. Transport diffusivities, self-diffusivities, permeabilities and Darken thermodynamic factors have been calculated as a function of pore width and temperature for pure component oxygen and nitrogen. In addition, new EMD simulation data for an 80:20 mixture of nitrogen and oxygen is reported, including a direct calculation of the Stefan-Maxwell coefficients. The results are discussed in terms of the oxygen selectivity and the possible mechanisms, which increase or decrease this quantity.

We find that the pore width behaviour of the diffusion coefficients consists of three distinct regimes: a regime at larger pore widths in which single component diffusion coefficients are largely independent of pore width, an optimum pore width at which both diffusivities increase substantially but the slit pore is selective towards nitrogen, and a regime at very low pore widths at which the diffusivities decrease sharply, but the slits are selective towards oxygen. The mechanism behind each of these regimes is discussed in terms of “entropic” effects and potential barrier heights.

We have also found that permeability selectivity is substantially reduced in a mixture of the two gases with a composition similar to that of air. Cross diffusion coefficients in the mixture have been calculated and shown to be non-negligible.     

Transport diffusivities of fluids in nanopores by non-equilibrium molecular dynamics simulation

We present a method to study fluid transport through nanoporous materials using highly efficient non-equilibrium molecular dynamics simulations. A steady flow is induced by applying an external field to the fluid particles within a small slab of the simulation cell. The external field generates a density gradient between both sides of the porous material, which in turn triggers a convective flux through the porous medium. The heat dissipated by the fluid flow is released by a Gaussian thermostat applied to the wall particles. This method is effective for studying diffusivities in a slit pore as well as more natural, complex wall geometries. The dependence of the diffusive flux on the external field sheds light on the transport diffusivities and allows a direct calculation of effective diffusivities. Both pore and fluid particle interactions are represented by coarse-grained molecular models in order to present a proof-of-concept and to retain computational efficiency in the simulations. The application of the method is demonstrated in two different scenarios, namely the effective mass transport through a slit pore and the calculation of the effective self-diffusion through this system. The method allows for a distinction between diffusive and convective contributions of the mass transport.     

Methods to determine slow diffusion coefficients of biomolecules. Applications to Engrailed 2, a partially disordered protein

We present new NMR methods to measure slow translational diffusion coefficients of biomolecules. Like the heteronuclear stimulated echo experiment (XSTE), these new methods rely on the storage of information about spatial localization during the diffusion delay as longitudinal polarization of nuclei with long T₁ such as nitrogen-15. The new BEST-XSTE sequence combines features of Band-selective Excitation Short-Transient (BEST) and XSTE methods. By avoiding the saturation of all protons except those of amide groups, one can increase the sensitivity by 45% in small proteins. The new experiment which combines band-Selective Optimized Flip-Angle Short-Transient with XSTE (SOFAST-XSTE) offers an alternative when very short recovery delays are desired. A modification of the HSQC-edited version of the XSTE experiment offers enhanced sensitivity and access to higher resolution in the indirect dimension. These new methods have been applied to detect changes in diffusion coefficients due to dimerization or proteolysis of Engrailed 2, a partially disordered protein.     

Smoldyn on graphics processing units: massively parallel Brownian dynamics simulations

Space is a very important aspect in the simulation of biochemical systems; recently, the need for simulation algorithms able to cope with space is becoming more and more compelling. Complex and detailed models of biochemical systems need to deal with the movement of single molecules and particles, taking into consideration localized fluctuations, transportation phenomena, and diffusion. A common drawback of spatial models lies in their complexity: models can become very large, and their simulation could be time consuming, especially if we want to capture the systems behavior in a reliable way using stochastic methods in conjunction with a high spatial resolution. In order to deliver the promise done by systems biology to be able to understand a system as whole, we need to scale up the size of models we are able to simulate, moving from sequential to parallel simulation algorithms. In this paper, we analyze Smoldyn, a widely diffused algorithm for stochastic simulation of chemical reactions with spatial resolution and single molecule detail, and we propose an alternative, innovative implementation that exploits the parallelism of Graphics Processing Units (GPUs). The implementation executes the most computational demanding steps (computation of diffusion, unimolecular, and bimolecular reaction, as well as the most common cases of molecule-surface interaction) on the GPU, computing them in parallel on each molecule of the system. The implementation offers good speed-ups and real time, high quality graphics output     

A comparison of three models for the diffusion of oxygen in electrolyte solutions

Diffusion of oxygen through aqueous solutions is of great importance in biological systems. In this work, three models for the diffusion of oxygen through aqueous salt solutions are compared. One model uses mole fraction as the driving force (Fick's Law) and another uses chemical potential. The third model uses the gradient in oxygen activity as the driving force. This new model was chosen because of the availability of oxygen electrodes which directly measure oxygen activity in aqueous solution. These models have been used to reevaluate the technique of measuring O(2) diffusivities. We show that Pick's Law diffusion coefficients do not vary strongly with salt concentration as was erroneously reported in the literature. In addition, we compare the predicted O(2) fluxes of the three models over a wide range in O(2) concentrations. For oxygen concentrations of biological interest, the three models give identical predictions of the flux.     

Application of Quasi-Steady-State Methods to Nonlinear Models of Intracellular Transport by Molecular Motors

Molecular motors such as kinesin and dynein are responsible for transporting material along microtubule networks in cells. In many contexts, motor dynamics can be modelled by a system of reaction–advection–diffusion partial differential equations (PDEs). Recently, quasi-steady-state (QSS) methods have been applied to models with linear reactions to approximate the behaviour of the full PDE system. Here, we extend this QSS reduction methodology to certain nonlinear reaction models. The QSS method relies on the assumption that the nonlinear binding and unbinding interactions of the cellular motors occur on a faster timescale than the spatial diffusion and advection processes. The full system dynamics are shown to be well approximated by the dynamics on the slow manifold. The slow manifold is parametrized by a single scalar quantity that satisfies a scalar nonlinear PDE, called the QSS PDE. We apply the QSS method to several specific nonlinear models for the binding and unbinding of molecular motors, and we use the resulting approximations to draw conclusions regarding the parameter dependence of the spatial distribution of motors for these models.     

Diffusion of acetophenone and phenethyl alcohol in the calcium-alginate-bakers' yeast-hexane system

The intrabead diffusion coefficients of acetophenone and phenethyl alcohol were measured at 30 degrees C in the triphasic immobilized yeast-water-hexane system. Saccharomyces cerevisiae cells were deactivated with hydrochloric acid and entrapped in calcium-alginate beads. Measurements of dry cell loss during deactivation, shrinkage of the beads during deactivation and the final porosity of the beads were made for various cell loadings. Final concentrations of wet cells in the beads ranged from approximately 0.25 to 0.30 g/mL. Mass transfer in the hexane phase, external to the beads, was eliminated experimentally. The estimated error of 5% to 10% in the diffusion coefficients is within the experimental error associated with the bead method. The effect of significant sampling volumes on the diffusivities was estimated theoretically and accounted for experimentally. The intrabead concentration of acetophenone and phenethyl alcohol was 150 to 800 ppm. The deactivated cells were shown to be impervious to acetophenone so that the measured diffusivities are extracellular parameters. The cell volume fraction in the beads ranged from 0.70 to 0.90, significantly higher than previously reported data. The effective diffusion coefficients conform to the random pore model. No diffusional interaction between acetophenone and phenethyl alcohol was observed. The addition of 2 vol% ethanol or methanol slightly increased the diffusivities. The thermodynamic partition coefficients were measured in the bead-free water-organic system and found to be an order of magnitude lower than the values calculated from the analysis of the diffusion data for the organic-bead system, suggesting that bead-free equilibrium data cannot be used in triphasic systems. (c) 1994 John Wiley & Sons, Inc.     

The theoretical distributions and diffusivities of small ions in chondroitin sulphate and hyaluronate

The electrostatic interactions between polyionic glycosaminoglycans and small mobile ions are investigated using the Poisson-Boltzmann equation and a rod-in-cell model of the polyelectrolyte. Calculations are made for the range of polyelectrolyte concentrations and buffer compositions for which measurements of ion distributions and diffusivities are reported in a companion paper (Maroudas et al., Biophys. Chem. 32 (1988) 257). We conclude that the distribution of mobile ions is largely determined by the 'far-field' potential and is adequately described by the Poisson-Boltzmann theory and also by more approximate theories such as ideal Donnan or 'condensation' theory. The measured variations in cation diffusivities, particularly the increase in diffusivity with increasing matrix concentration at low ionic strengths, are predicted qualitatively using an approximate diffusion theory together with the calculated potential fields. However, the same theory applied to anion diffusion gives qualitatively wrong results.     

Homogenization Theory for the Prediction of Obstructed Solute Diffusivity in Macromolecular Solutions

The study of diffusion in macromolecular solutions is important in many biomedical applications such as separations, drug delivery, and cell encapsulation, and key for many biological processes such as protein assembly and interstitial transport. Not surprisingly, multiple models for the a-priori prediction of diffusion in macromolecular environments have been proposed. However, most models include parameters that are not readily measurable, are specific to the polymer-solute-solvent system, or are fitted and do not have a physical meaning. Here, for the first time, we develop a homogenization theory framework for the prediction of effective solute diffusivity in macromolecular environments based on physical parameters that are easily measurable and not specific to the macromolecule-solute-solvent system. Homogenization theory is useful for situations where knowledge of fine-scale parameters is used to predict bulk system behavior. As a first approximation, we focus on a model where the solute is subjected to obstructed diffusion via stationary spherical obstacles. We find that the homogenization theory results agree well with computationally more expensive Monte Carlo simulations. Moreover, the homogenization theory agrees with effective diffusivities of a solute in dilute and semi-dilute polymer solutions measured using fluorescence correlation spectroscopy. Lastly, we provide a mathematical formula for the effective diffusivity in terms of a non-dimensional and easily measurable geometric system parameter.     

A novel technique for measuring solute diffusivities in entrapment matrices used in immobilization

A novel technique has been developed for measuring effective solute diffusivities in entrapment matrices used for cell immobilization. In this technique radiotracers were used to measure effective diffusivities and equilibrium partition coefficients of the solute between the liquid and solid matrix. Ca-alginate was used in this study, because it is one of the most commonly employed matrices for the immobilization of microbial, plant and mammalian cells. The experimental apparatus consisted of a single spherical Ca-alginate bead which was attached to a rotating rod and immersed in water containing C(14)-glucose. The rotational speed of the spherical bead was controlled and resulted in excellent mixing, and negligible external film mass transfer resistance, which allowed the measurement of true effective solute diffusivity within the solid matrix. The rates of C(14)-glucose diffusion within the Ca-alginate sphere were measured using a scintillation spectrometer. A mathematical model of unsteady-state diffusion in a sphere was used with appropriate boundary conditions, and the effective diffusivity of glucose was found from the best fit of the experimental data using a computer regression analysis method. Using 2% (w/v) Ca-alginate beads in this new radiotracer technique the effective diffusivity and partition coefficient of glucose were found to be 6.62 x 10(-10) m(2)/s and 0.98, respectively. The accuracy, advantages, and simplicity of this new method for diffusivity measurements are also compared to other existing methods.     

A memory diffusion model for molecular anisotropic diffusion in siliceous β-zeolite

A memory diffusion model of molecules on β-zeolite is proposed. In the model, molecular diffusion in β-zeolites is treated as jumping from one adsorption site to its neighbors and the jumping probability is a compound probability which includes that provided by the transitional state theory as well as that derived from the information about which direction the target molecule comes from. The proposed approach reveals that the diffusivities along two crystal axes on β-zeolite are correlated. The model is tested by molecular dynamics simulations on diffusion of benzene and other simple molecules in β-zeolites. The results show that the molecules with larger diameters fit the prediction much better and that the “memory effects” are important in all cases.     

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.     

A simple correlation for predicting effective diffusivities in immobilized cell systems

A simple correlation method has been developed to predict effective diffusivities of small molecules in heterogeneous materials such as immobilized cell systems. This correlation uses a single diffusivity measurement at one cell volume fraction to predict diffusivities for any other volume fraction of cell. The method has been applied to 20 sets of published diffusivity measurements in immobilized cell systems and accurately predicts affective diffusivities of molecules for the full range of cell fractions. It may also be used to predict effective diffusivities in heterogeneous materials in which the diffusivity of a molecule in each phase and the volume fraction of each phase are known. (c) 1996 John Wiley & Sons, Inc.