Determination of diffusion coefficients and diffusion characteristics for chlorferon and diethylthiophosphate in Ca-alginate gel beads

Diffusion characteristics of chlorferon and diethylthiophosphate (DETP) in Ca-alginate gel beads were studied to assist in designing and operating bioreactor systems. Diffusion coefficients for chlorferon and DETP in Ca-alginate gel beads determined at conditions suitable for biodegradation studies were 2.70 x 10(-11) m(2)/s and 4.28 x 10(-11) m(2)/s, respectively. Diffusivities of chlorferon and DETP were influenced by several factors, including viscosity of the bulk solution, agitation speed, and the concentrations of diffusing substrate and immobilized cells. Diffusion coefficients increased with increasing agitation speed, probably due to poor mixing at low speed and some attrition of beads at high speeds. Diffusion coefficients also increased with decreasing substrate concentration. Increased cell concentration in the gel beads caused lower diffusivity. Theoretical models to predict diffusivities as a function of cell weight fraction overestimated the effective diffusivities for both chlorferon and DETP, but linear relations between effective diffusivity and cell weight fraction were derived from experimental data. Calcium-alginate gel beads with radii of 1.65-1.70 mm used in this study were not subject to diffusional limitations: external mass transfer resistances were negligible based on Biot number calculations and effectiveness factors indicated that internal mass transfer resistance was negligible. Therefore, the degradation rates of chlorferon and DETP inside Ca-alginate gel beads were reaction-limited.     

A stirred bath technique for diffusivity measurements in cell matrices

A stirred bath technique was developed for determining effective diffusivities in cell matrices. The technique involves cell immobilization in a dilute gel which has negligible effect on solute diffusion. Agar and collagen were tested as immobilizing gels. Agar gel was shown to have minor interactions with the diffusion of various biological molecules, and was used for immobilization of Ehrlich Ascites Tumor (EAT) cells. Diffusivities of glucose and lactic acid were measured in EAT matrices for cell loadings between 20 and 45 vol %. Treatment with glutaraldehyde was effective in quenching the metabolic activity of the cells while preserving their physical properties and diffusive resistance. The measured data agree favorably with predictions based on Maxwell's equation for effective diffusion in a periodic composite material. The stirred bath technique is useful for diffusivity determinations in immobilized matrices or free slurries, and is applicable to both microbial and mammalian cell systems.     

Diffusion coefficients of glucose and ethanol in cell-free and cell-occupied calcium alginate membranes

The diffusivities of glucose and ethanol in cell-free and cell-occupied membranes of calcium alginate were measured in a diffusion cell. The lag time analysis was used. Diffusivities decreased with increasing alginate concentration and were comparable with those in water for a 2% alginate membrane. Glucose and ethanol concentrations had no effect on the respective diffusion coefficients. The ratio of ethanol diffusivity to glucose diffusivity in 2 and 4% alginate agreed closely with the inverse ratio of the hydrodynamic raii for the two molecules in water, indicating that the hydrodynamic theory of diffusion in liquids may be applicable to diffusion in dilute alginate gels. Also, the presence of 20% dead yeast cells had no effect on the diffusivities. The data reported can be used to study reaction and diffusion in immobilized cell reactors and cell physiology under immobilized conditions.     

An experimental study of mass diffusion and reaction rate in an anaerobic biofilm

An experimental reactor consisting of two chambers, separated by a porous ceramic immobilization matrix, was constructed to measure the effective diffusivity of different compounds and the consumption rates of acetate in developing biofilms. In initial experiments, effective diffusivities for acetate, propionate, isopropanol, and lithium salt through the ceramic immobilization matrix in the absence of biofilm were determined to be 40% to 50% less than in water at infinite dilution. The effective diffusivity of the lithium salt was similar to that of acetate. The effective diffusivity of the lithium salt through biofilms of thickness in the range of 200 to 1200 mum was essentially constant with a value of approximately 7% of that in water at infinite dilution. Acetate consumption in the biofilm was linearly proportional to biofilm thickness up to a biofilm depth of 800 mum. Deviation from linearity appeared in biofilm thicknesses greater than 800 mum. Results of these experiments support previous reports that immobilized cell reactors have significantly higher bioconversion rates than suspended cell systems.     

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.     

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.     

Noninvasive measurement of effective diffusivities in cell immobilization gels through use of near-infrared spectroscopy

Encapsulated cell systems provide some advantages over typical suspension cell cultivations as higher cell densities may be obtained; however, the supply of nutrients to the cells often is a limiting factor in productivity. In this study, we describe the development of a new approach to characterize the effective diffusivity of nutrients in immobilized cell materials. Near-infrared spectroscopy is employed to measure nutrient concentrations within a specially designed diffusion chamber that permits noninvasive sampling at ten spatial positions and multiple timepoints. To demonstrate this technique, we measured the effective diffusivity of glutamine in a cell-free 3% (w/w) agarose gel and determined the effective diffusivity (D(eff)) = 6.46 x 10(-10) m(2)/s, which is in good agreement with theoretical values.     

A novel way to calculate the diffusivity of water in carbon nanotubes

Theoretical studies on water diffusion in carbon nanotubes have been challenged because the diffusivities calculated by molecular simulations are much lower than what experiments show. This paper proposes a new way to estimate transport diffusivity in such systems directly by simply inspecting the long term behaviors of the velocity autocorrelation functions of water molecules. The results from simulation of molecular dynamics based on this method show that the new method can generate much higher diffusivities, in accordance with experimental measurements, and of the correct order of magnitude.     

Effective diffusivity of galactose in calcium alginate gels containing immobilized Zymomonas mobilis.

The effective diffusivity of galactose was measured for calcium alginate gel membranes containing immobilized live Zymomonas mobilis cells at concentrations ranging from 0 to 150 g dry wt/L of gel. Since galactose is not taken up by living Z. mobilis organisms, the diffusion of this representative six-carbon sugar could be studied independently of sugar consumption. Various immobilized biomass loadings were achieved by two different techniques: addition of biomass at known concentrations to the sodium alginate solution before membrane formation and growth of cells in the gel to various biomass concentrations. The highest immobilized cell concentration, attained by in situ growth, corresponds to the maximum of this system, as growth beyond this maximum concentration led to disintegration of the gel membrane. The galactose effective diffusivity measurements for both methods of immobilized cell loading overlap within experimental error and follow the same general monotonic decline with entrapped biomass concentration. Most of the data fall below the upper bound predicted by Hashin and Shtrikman (1962) and show good agreement with the random pore model of Wakao and Smith (1962, 1964). Available effective diffusivity data from the literature provide evidence that the random pore model is an excellent predictor of sugar effective diffusivity in gel immobilized cell systems in general.     

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.     

Particle Diffusivity and Free-Energy Profiles in Hydrogels from Time-Resolved Penetration Data

A combined experimental and theoretical method to simultaneously determine diffusivity and free-energy profiles of particles that penetrate into inhomogeneous hydrogel systems is presented. As the only input, arbitrarily normalized concentration profiles from fluorescence intensity data of labeled tracer particles for different penetration times are needed. The method is applied to dextran molecules of varying size that penetrate into hydrogels of polyethylene-glycol chains with different lengths that are covalently cross-linked by hyperbranched polyglycerol hubs. Extracted dextran bulk diffusivities agree well with fluorescence correlation spectroscopy data obtained separately. Empirical scaling laws for dextran diffusivities and free energies inside the hydrogel are identified as a function of the dextran mass. An elastic free-volume model that includes dextran as well as polyethylene-glycol linker flexibility quantitively describes the repulsive dextran-hydrogel interaction free energy, which is of steric origin, and furthermore suggests that the hydrogel mesh-size distribution is rather broad and particle penetration is dominated by large hydrogel pores. Particle penetration into hydrogels for steric particle-hydrogel interactions is thus suggested to be governed by an elastic size-filtering mechanism that involves the tail of the hydrogel pore-size distribution.     

Diffusion of MRI and CT Contrast Agents in Articular Cartilage under Static Compression

Cartilage has a limited capacity for self-repair and focal damage can eventually lead to complete degradation of the tissue. Early diagnosis of degenerative changes in cartilage is therefore essential. Contrast agent-based computed tomography and magnetic resonance imaging provide promising tools for this purpose. However, the common assumption in clinical applications that contrast agents reach steady-state distributions within the tissue has been of questionable validity. Characterization of nonequilibrium diffusion of contrast agents rather than their equilibrium distributions may therefore be more effective for image-based cartilage assessment. Transport of contrast agent through the extracellular matrix of cartilage can be affected by tissue compression due to matrix structural and compositional changes including reduced pore size and fluid content. We therefore investigate the effects of static compression on diffusion of three common contrast agents: sodium iodide, sodium diatrizoate, and gadolinium diethylenetriamine-pentaacid (Gd-DTPA). Results showed that static compression was associated with significant decreases in diffusivities for sodium iodide and Gd-DTPA, with similar (but not significant) trends for sodium diatrizoate. Molecular mass of contrast agents affected diffusivities as the smallest one tested, sodium iodide, showed higher diffusivity than sodium diatrizoate and Gd-DTPA. Compression-associated cartilage matrix alterations such as glycosaminoglycan and fluid contents were found to correspond with variations in contrast agent diffusivities. Although decreased diffusivity was significantly correlated with increasing glycosaminoglycan content for sodium iodide and Gd-DTPA only, diffusivity significantly increased for all contrast agents by increasing fluid fraction. Because compounds based on iodine and gadolinium are commonly used for computed tomography and magnetic resonance imaging, present findings can be valuable for more accurate image-based assessment of variations in cartilage composition associated with focal injuries.     

Diffusion of MRI and CT Contrast Agents in Articular Cartilage under Static Compression

Cartilage has a limited capacity for self-repair and focal damage can eventually lead to complete degradation of the tissue. Early diagnosis of degenerative changes in cartilage is therefore essential. Contrast agent-based computed tomography and magnetic resonance imaging provide promising tools for this purpose. However, the common assumption in clinical applications that contrast agents reach steady-state distributions within the tissue has been of questionable validity. Characterization of nonequilibrium diffusion of contrast agents rather than their equilibrium distributions may therefore be more effective for image-based cartilage assessment. Transport of contrast agent through the extracellular matrix of cartilage can be affected by tissue compression due to matrix structural and compositional changes including reduced pore size and fluid content. We therefore investigate the effects of static compression on diffusion of three common contrast agents: sodium iodide, sodium diatrizoate, and gadolinium diethylenetriamine-pentaacid (Gd-DTPA). Results showed that static compression was associated with significant decreases in diffusivities for sodium iodide and Gd-DTPA, with similar (but not significant) trends for sodium diatrizoate. Molecular mass of contrast agents affected diffusivities as the smallest one tested, sodium iodide, showed higher diffusivity than sodium diatrizoate and Gd-DTPA. Compression-associated cartilage matrix alterations such as glycosaminoglycan and fluid contents were found to correspond with variations in contrast agent diffusivities. Although decreased diffusivity was significantly correlated with increasing glycosaminoglycan content for sodium iodide and Gd-DTPA only, diffusivity significantly increased for all contrast agents by increasing fluid fraction. Because compounds based on iodine and gadolinium are commonly used for computed tomography and magnetic resonance imaging, present findings can be valuable for more accurate image-based assessment of variations in cartilage composition associated with focal injuries.     

Diffusion of lactose in acidogenic biofilms

Effective diffusivity of lactose in active acidogenic biofilms was measured at 35 degrees C and pH 4.6 with a specially designed diffusion cell. The diffusion cell was designed and operated in such a way that the lactose concentrations on the surface and at the center of a living bacterial aggregate could be measured at steady state. As a model parameter in a widely accepted reaction-diffusion equation which describes lactose distribution in living biofilms, the effective diffusivity of lactose in the biofilms was found to be about 65% of the lactose diffusivity in free solutions. It was experimentally determined that the active biofilms had about 66% void volume made up of channels through which the lactose molecules were transported into the bacterial aggregates. Therefore, the decrease in lactose diffusivity was mainly caused by the biofilm's solid biomass fraction rather than the tortuosity of the channels. (c) 1993 John Wiley & Sons, Inc.     

Continuous glutamate production using an immobilized whole-cell system

For the purpose of saving the energy and raw materials required in glutamate fermentation, an immobilized whole-cell system was prepared and its performance in a continuous reactor system was evaluated. Corynebacterium glutamicum (a mutant strain of ATCC 13058) whole cell was immobilized in K-carrageenan matrix and the gel structure was strengthened by treatment with a hardening agent. The effective diffusivities of carrageenan gel for glucose and oxygen were found to decrease significantly with an increase in carrageenan concentration, while the gel strength showed an increasing trend. Based on the physical and chemical properties of carrageenan gel, the immobilization method was improved and the operation of the continuous reactor system was partially optimized. In an air-stirred fermentor, the continuous production of glutamate was carried out. The effect of the dilution rate on glutamate production and operational stability were investigated. The performance of the continuous whole-cell reactor system was evaluated by measuring glutamate productivity for a period of 30 days; it was found to be far superior to the performance of conventional batch reactor systems using free cells.     

A diffusion model and optimal cell loading for immobilized cell biocatalysts

A diffusion model based on the random pore model is derived for immobilized cell biocatalysts and verified with 19 sets of experimental diffusion data. The predicted effective diffusivity relative to that for the support matrix reflects a quadratic dependence on the cell loading and contains a single parameter that depends on the intracellular diffusivity and the chemical partitioning coefficient. The model is used to predict optimal cell loadings that maximize the total reaction rate in an immobilized cell biocatalyst. A rule of thumb based on the diffusion model is obtained to the effect that the cell loading should be at least (1/3) for single reactions regardless of the kinetics and diffusional resistances. A means of calculating improved lower bounds is provided for cases where the cellular diffusional resistance is known but the kinetics are not. The optimal cell loadings for reversible first-order and for Michaelis-Menten kinetics are presented and demonstrated to be within the range of conditions of practical interest.     

Diffusion of sucrose and yohimbine in calcium alginate gel beads with or without entrapped plant cells

The diffusion characteristics of sucrose, a nutrient, and yohimbine, a secondary metabolite, in alginate gel beads, with or without entrapped periwinkle (Catharanthus roseus) or apple (Malus domestica) cells, were investigated. Effective diffusivities of both solutes in the gel beads were determined by two different methods from transient concentration changes in well-stirred solutions where the beads were suspended. The linear plot method developed in this work is easy to use and requires no data from the initial periods of diffusion experiments. It was found that while the cell-free beads provided only minor diffusional resistance to both solutes, the effective diffusivities of both solutes decreased significantly with the presence of cells in the beads and the amount of reduction was proportional to the amount of cell loading. Further, the effective diffusivity of sucrose appeared to be slightly larger than that of yohimbine under identical conditions. It was also observed that permeabilization of apple cells with dimethyl sulfoxide (DMSO) led to an increase in effective diffusivity with the effect being more significant for yohimbine.     

Effect of cell arrangement and interstitial volume fraction on the diffusivity of monoclonal antibodies in tissue.

We present theoretical calculations relating the effective diffusivity of monoclonal antibodies in tissue (Deff) to the actual diffusivity in the interstitium (Dint) and the interstitial volume fraction phi. Measured diffusivity values are effective values, deduced from concentration profiles with the tissue treated as a continuum. By using homogenization theory, the ratio Deff/Dint is calculated for a range of interstitial volume fractions from 10 to 65%. It is assumed that only diffusion in the interstitial spaces between cells contributes to the effective diffusivity. The geometries considered have cuboidal cells arranged periodically, with uniform gaps between cells. Deff/Dint is found to generally be between (2/3) phi and phi for these geometries. In general, the pathways for diffusion between cells are not straight. The effect of winding pathways on Deff/Dint is examined by varying the arrangement of the cells, and found to be slight. Also, the estimates of Deff/Dint are shown to be insensitive to typical nonuniformities in the widths of gaps between cells. From our calculations and from published experimental measurements of the effective diffusivity of an IgG polyclonal antibody both in water and in tumor tissue, we deduce that the diffusivity of this molecule in the interstitium is one-tenth to one-twentieth its diffusivity in water. We also conclude that exclusion of molecules from cells (an effect independent of molecular weight) contributes as much as interstitial hindrance to the reduction of effective diffusivity, for small interstitial volume fractions (around 20%). This suggests that the increase in the rate of delivery to tissues resulting from the use of smaller molecular-weight molecules (such as antibody fragments or bifunctional antibodies) may be less than expected.     

Immobilized particles in gel matrix-type porous media. Nonhomogeneous cell distribution

The conventional random pore model assumes a homogeneous cell distribution in the gel matrix used to immobilize cells. However, the validity of this model is restricted to values of the exponent alpha, between 1.8 and 2.25, of a model power function relating the diffusivity coefficient in the matrix with the overall cell volume fraction in the system. Based on the analysis of published data for diffusion in gels with immobilized cells and on the homogeneous approach for the random pore model developed in a previous work, a new, nonhomogeneous approach is proposed for alpha values outside the range 1.8-2.25. To explain these data, two main types of nonhomogeneous cell distribution were considered: (1) nonhomogeneous cell distribution in the gel for alpha > 2.25 (type 1) and (2) nonhomogeneity related with anisotropy of cell space orientation when alpha < 1.8 (type 2). In the case of nonhomogeneity of type 1, the cell volume fraction in the layers occupied by cells must be considered in place of the concept previously used for homogeneous distribution, viz., the average cell volume fraction. This model underlines that accumulation of cells in a thin layer close to the surface improves their nutrient intake. For nonhomogeneity of type 2, the tortuosity of such a system is smaller than should be expected if spherical cells were considered, thereby changing the effective diffusion. The model proposed in this work proved to fit into several real cases reported in the literature.