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 of (de)acylated antibiotic A40926 in alginate and carrageenan beads with or without cells and/or soybean meal

Effective diffusion coefficients (D(e)) of antibiotic A40926 and its deacylated derivative were determined in Ca-alginate (2% wt/wt) and kappa-carrageenan (2.6% wt/wt) gel beads with or without immobilized Actinoplanes teichomyceticus cells and/or soybean meal (SBM). The method used was based on transient concentration changes in a well-stirred antibiotic solution in which gel beads, initially free of solute, were suspended. Unsteady-state diffusion in a sphere was applied and D(e) determined from the best fit of experimental data. A40926 showed markedly different diffusion characteristics than its deacylated derivative. Diffusivity of deacyl-A40926 in alginate or carrageenan gel beads was six to seven times that of A40926. Large differences in partition coefficients (Kp) were also found. In case of beads without additions, A40926, in contrast to deacyl-A40926, strongly partitioned to the liquid phase. Introduction of SBM and/or mycelium in the gel beads decreased the effective diffusivity of deacyl-A40926, but increased its partitioning to the solid phase. Our findings indicate that a relatively moderate structural change of a lipoglycopeptide molecule could lead to a major change in its diffusion/partition characteristics.     

Oxygen diffusivity in gel beads containing viable cells

This article proposes a simple steady-state method for measuring the effective diffusion coefficient of oxygen (D(e)) in gel beads entrapping viable cells. We applied this method to the measurement of D(e) in Ca- and Ba-alginate gel beads entrapping Saccharomyces cerevisiae and Pseudomonas ovalis. The diffusivity of oxygen through gel beads containing viable cells was measured within an accuracy of +/-7% and found not to be influenced by cell density (0-30 g/L gel), cell type, and cell viability in gel beads. The oxygen diffusivity in the Ca-alginate gel beads was superior to that of the Ba-alginate gel beads, and the D(e) in the Ca-alginate gel beads nearly equalled the molecular diffusion coefficient in the liquid containing the gel beads. The oxygen concentration profile in a single Ca-alginate gel bead was calculated and compared to the distribution of mycelia of Aspergillus awamori grown in that gel bead. This procedure indicated that the oxygen concentration profile is useful for the estimation of the thickness of the cell layer in a gel bead. Numerical investigation revealed that high effectiveness factors, greater than 0.8, could be obtained using microgel beads with a radius of 0.25 mm.     

Diffusion of lactose in kappa-carrageenan/locust bean gum gel beads with or without entrapped growing lactic acid bacteria

Effective diffusion coefficients (De) of lactose in kappa-carrageenan (2.75% wt/wt)/locust bean gum (0.25% wt/wt) (LBG) gel beads (1.5-2.0-mm diameter)with or without entrapped lactic acid bacteria (LAB) were determined at 40 degrees C. The effects of lactose concentration, bacteria strain (Streptococcus salivarius subsp. thermophilus and Lactobacillus casei subsp. casei) and cell content at various steps of the fermentation process (after immobilization, pre-incubation of the beads and successive fermentations) were measured on De as a first step for process modelling. Results were obtained from transiend concentration changes n well-stirred lactose solutions in which the beads were suspended. A mathematical model of unsteady-state diffusion in a sphere was used, and De was obtained from the best fit of the experimental data. Diffusivity of lactose in cell-tree beads was significantly lower than in pure water mainly because of the obstruction effect of the polymer chains and the hydration region. Furthermore, effective diffusivity and equilibrium partition factor were independent of lactose concentration in the range from 12.5 to 50 g/L. No significant difference was found for De (effective diffusivity) and Kp (partition) coefficients between beads entrapping S. thermophilus (approximately 5 x 10(9) CFU/mL) and cell-free beads. On the other hand higher cell counts obtained with L. casei (close to 1.8 x 10(11) CFU/mL) increased mass transfer resistance resulting in lower effective diffusivities and Kp. Finally, the effects of the type of bacteria and their distribution in the beads on the diffusivity were also discussed.     

Characterisation of small molecules diffusion in hydrogel-membrane liquid-core capsules

The overall diffusion coefficients for several low molecular weight solutes, such as glucose, fructose, sucrose, lactose, and vitamin B(12) have been determined in Ca-alginate membrane liquid-core capsules using the unsteady-state method following the release of solutes from the capsules to a well-stirred solution of limited volume. The diffusion coefficients obtained for saccharides were 5-20% lower than the corresponding diffusivity in water while for vitamin B(12) about 50% that of water. The diffusion coefficients of the investigated capsules were not influenced by the change in alginate concentration in the capsule membrane from 0.5 to 1.0%. Lower diffusivities and higher deviations from the diffusivity in water were obtained for higher molecular weight solutes.     

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.     

Limitations of methods of determining effective diffusion coefficients in cell immobilization matrices

Various techniques are available for determining the effective diffusivity D(e) of solutes such as glucose in cell immobilization matrices. Nearly all, if not all, are subject to errors and limitations as regards the ranges of temperature, pressure, and/or concentration over which they give reliable results. It is the purpose of this article to compare three of these methods, designated (a) thin disc, (b) cylinder, and (c) beads types, and to show by means of a sensitivity and error analysis of the equation used in each method that the thin-disc and cylindrical techniques give more accurate results of D(e) than does the bead method.     

Diffusion of sucrose and dextran through agar gel membranes

Mass transfer limitations severely impede the performance of bioreactions involving large molecules by gel-entrapped microorganisms. This paper describes a quantitative investigation of such diffusional limitations in agar gel membranes. Sucrose and commercial dextran fractions with (weight-average) molecular weights ranging from 10,000 to 2,000,000 Da were used as standard diffusants. For all tested solutes but sucrose, the values of the agar/water partition coefficients highlighted steric hindrance at the entrance of the membrane pores. The effective diffusivity of sucrose in agar was similar to that in water. All dextran fractions, however, displayed restricted diffusion in the agar membranes. Their effective diffusivities were a decreasing function of the agar content of the gel membrane (0.5, 1.0, or 1.5% w/v). The effective diffusivity in a given membrane decreased as the molecular weight of the diffusing molecule increased. T500 (ucbar|M_w = 470,000 Da) and ucbar|M_w = 1,950,000 Da) fractions were unable to diffuse through 1.0 or 1.5% agar membranes. The diffusion data did not agree with the classical (Renkin) model for a hard sphere diffusing through a cylindrical pore. These results are discussed in terms of gel and diffusant characteristics.     

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.     

Determination of glucose and ethanol effective diffusion coefficients in Ca-alginate gel.

Glucose and ethanol diffusion coefficients in 2% Ca-alginate gel were measured using the experimental technique based on solute diffusion into or out of gel beads in a well-stirred solution. The aim of the study was to make the measurements under typical conditions found in alcoholic fermentations, such as the concentrations of glucose (100 g l-1) and ethanol (50 g l-1), the simultaneous counter-diffusion of glucose and ethanol, and the presence of cells in the gel beads at a level of 10(9) cells g-1 of beads. Previously, an evaluation of the error associated with the methodology used indicated how the experimental procedure would minimize the error. The individual measurement of glucose and ethanol coefficients in 2% Ca-alginate with no cells gave values of 5.1 and 9.6 x 10(-6) cm2 s-1, respectively, which are lower than those in water. When the effect of counter-diffusion was investigated, both coefficients decreased: glucose by 14% and ethanol by 28%. When cells were incorporated into the beads, only the ethanol coefficient decreased significantly, while the glucose coefficient apparently increased its value to 6.9 10(-6) cm2 s-1.     

The effective diffusion coefficient and the distribution constant for small molecules in calcium-alginate gel beads

The effective diffusion coefficient, D(e), and the distribution constant, K(i), for selected mono- and disaccharides and organic acids were determined in homogeneous calcium-alginate gel with and without entrapped bacteria. Results were obtained from transient concentration changes in well-stirred solutions of limited volume, in which the gel beads were suspended. The effective diffusioncoefficients and the distribution constants were estimated by fitting mathematical model predictions to the experimental data using a nonlinear model fitting program (MODFIT). Both single solute diffusion and multiple solute diffusion were performed. A small positive effect was obtained onthe values of D(e) for the system of multiple solute diffusion; however, the values of K(i) were not significantly influenced. For the nine solutes tested, D(e) for 2% Ca-alginate gel beads was found to be approximately 85% of the diffusivity measured in water. The effects on D(e) and K(i), for lactose and lactic acid were determined for variations of alginate concentration, pH, temperature, and biomass content in the beads. D(e) decreased linearly for both lactose and lactic acid with increasing cell concentration in the Ca-alginate gel. K(i), was constant for both lactose and lactic acid with increasing cell concentration. D(e) was significantly lower at pH 4.5 than at pH 5.5 and 6.5 for both lactose and lactic acid. Furthermore, D(e) seemed to decrease with increased alginate concentration in the range of 1% to 4%. The diffusion rate increased with increasing temperature, and the activation energy for the diffusion process for both lactose and lactic acid was constant in the temperature range tested. (c) 1995 John Wiley & Sons Inc.     

Diffusion in kappa-carrageenan gel beads

Using a well-mixed and temperature-led vessel, the diffusion characteristics of various solutes into spherical kappa-carrageenan gel beads were experimentally investigated. The diffusion coefficient of glucose was markedly affected by the glucose concentration and the operating temperature. In all cases the diffusivity obtained was noticeably smaller than that of glucose in pure water. The experimental data also indicated an inverse relationship between the diffusivity and the polymer concentration used in the gel preparation. As well, the glucose diffusivity was affected by the presence of other solutes in the glucose solution. Electrolytes such as ammonium sulfate, KCl, and CaCl(2) were observed to enhance the diffusion coefficient. On the other hand, the addition of arginine or bovine serum albumin had an adverse effect on the diffusivity. No diffusion of albumin into the gel beads was observed, and such a solute created a significant mass transfer resistance during the diffusion process.     

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.     

Diffusivity of oxygen into carriers entrapping whole cells

The effective diffusivity of oxygen, D(e), in Ca-alginate and PVA-SbQ gels was measured using a two-chamber vessel with a membrane between the two chambers. The effect of cell density, C(c), on D(e) in Ca-alginate gels was studied. The effective diffusivity of oxygen decreased with increasing cell density, to C(c) = 170 kg dry cells/m(3) gel. The dependency of D(e) on cell density was discussed in terms of a random-pore model. The model correlated well with experimental data, i.e., kD(e)/D(0) = 0.86(1 - 1.47 x 10(-3) C(c))(2). Here, k is the partition coefficient, and D(0) is diffusivity in water.     

Diffusion in cell‐free and cell immobilising κ‐carrageenan gel beads with and without chemical reaction

Diffusion into and from κ‐carrageenan gel beads was studied, both in the absence and presence of bacterial cells, both with and without biochemical reaction. The solutes were indole, L ‐serine, and L ‐tryptophan. The reaction was that of indole and L ‐serine to give L ‐tryptophan. Established theory concerning diffusion of a single solute in cell‐free gels was found to describe well the effect of the gel on diffusivity. Simultaneous diffusion of the three solutes resulted in lower diffusivities than those for individual solutes, suggesting the need to use multicomponent diffusion theory. The effect of cells on diffusion could only be accounted for by models assuming permeable cells. Diffusion with chemical reaction was reasonably well described by an effectiveness factor calculated using an effective diffusivity estimated from diffusion data without reaction. © 1999 John Wiley & Sons, Inc. Biotechnol Bioeng 63: 625–631, 1999.     

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.     

Diffusion characteristics of substrates in Ca-alginate gel beads

The diffusion characteristics of several substrates of varying molecular sizes into and from Ca-alginate gel beads in well-stirred solutions were investigated. The values of the diffusion coefficient (D) of substrates such as glucose, L-tryptophan, and alpha-lactoalbumin [with molecular weight (MW) less than 2 x 10(4)] into and from the gel beads agreed with those in the water system. Their substrates could diffuse freely into and from the gel beads without disturbance by the pores in the gel beads. The diffusion of their substrates into and from the gel beads was also not disturbed by increasing the Ca-alginate concentration in the beads and the CaCl(2) concentration used in the gel preparation. In the case of higher molecular weight substances such as albumin (MW = 6.9 x 10(4)), gamma-globulin (MW = 1.54 x 10(5)) and fibrinogen (MW = 3.41 x 10(5)), the diffusion behaviors of the substrates into and from the gel beads were very different. No diffusion of their substrates into the gel beads from solutions was observed, and only albumin was partly absorbed on the surface of the gel beads. The values of D of their substrates from the gel beads into their solutions were smaller than their values in the water system, but all their substrates could diffuse from the gel beads. The diffusion of high molecular weight substrates was limited more strongly by the increase of Ca-alginate concentration in the gel beads than by the increase of the CaCl(2) concentration used in the gel preparation. Using these results, the capacity of Ca-alginate gel as a matrix of immobilization was discussed.     

Effect of solvent concentration on the extraction kinetics and diffusivity of Cyclosporin A in the fungus Tolypocladium inflatum

The kinetics of solid-liquid extraction and extraction yields of the immunosuppressant drug Cyclosporin A (CyA) from the mycelia of Tolypocladium inflatum were examined in this study. A 2 L stirred, baffled vessel was used to extract CyA from wet mycelia mass. Three different organic solvents were used, namely, methanol, acetone, and isopropanol at different concentrations in aqueous mixtures at room temperature. It was found that the best solvent was acetone at 50% v/v concentration achieving 100% extraction of CyA from the mycelia of T. inflatum. Although acetone proved to be the better solvent for CyA extraction, further studies were performed using methanol. A linear relationship was found between extraction yield of CyA and methanol concentration with 100% CyA extraction at 90% v/v methanol. The partition coefficients of CyA between the solid mycelia phase and the aqueous solvent phase were found to decrease exponentially with increasing methanol concentration. A liquid extraction model was developed based on the diffusion equation to correlate the kinetic data of CyA extraction from the solid mycelia of T. inflatum. Non-linear regression analysis of experimental data was used with the diffusion equation in order to calculate the effective diffusivities of CyA in the mycelia of T. inflatum. For all three organic solvents used, the effective diffusivities of CyA were found to be between 4.41 x 10(-15) and 6.18 x 10(-14) m(2)/s. This is the first time CyA effective diffusivities in T. inflatum are reported in the literature.     

Effective diffusivity and tortuosity in a porous glass immobilization matrix

The effective diffusivity of glucose in porous glass beads was determined using a transient method. Predictions for the intraparticle and surface concentrations were made by an analytical solution of the mass balance. The value of the diffusivity was expected to be lower than the value of the corresponding diffusion coefficient in water, but the opposite was observed. This effect results from intraparticle fluid flow, leading to high values of the apparent effective glucose diffusivity. To measure diffusion only and to prevent any internal convection during the diffusion experiment, the pores of the porous glass beads were filled with Ca-alginate gel. For these glass beads (internal porosity, , equal to 0.56), we found an effective glucose diffusivity of 2.2×10^–10 m²/s at 30°C. Using the relationship to effective intraparticle diffusivity (D_eff)=effective diffusivity in 1% Ca-alginate beads (D_gel) / (with the tortuosity factor) this gives =1.7. For known and measuring by the method described, the D_eff can be calculated for other porous materials or diffusing substances. Knowledge of the exact value of the effective diffusivity is a necessity in bioreactor modelling and was demonstrated by prediction of the residence time distribution profiles in a packed-bed bioreactor containing immobilized yeast cells.     

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.