A.I. Liapis

Network modeling of the convective flow and diffusion of molecules adsorbing in monoliths and in porous particles packed in a chromatographic column.

A cubic lattice network of interconnected pores was constructed to represent the porous structure existing in a monolith (continuous bed) or in a column packed with porous chromatographic particles. Expressions were also constructed and utilized to simulate, through the use of the pore network model, the intraparticle interstitial velocity and pore diffusivity of adsorbate molecules in porous chromatographic particles or in monoliths under retained and unretained conditions. The combined effects of steric hindrance at the entrance to the pores and frictional resistance within the pores, as well as the effects of pore size, pore connectivity, nT, of the porous network, molecular size of adsorbate and ligand (active site), and the fractional saturation of adsorption sites (ligands), have been considered. The results for the adsorption systems studied in this work, indicate that the obstruction effects on the intraparticle interstitial velocity, due to (a) the thickness of the immobilized layer of active sites and (b) the thickness of the adsorbed layer, are small and appear to be insignificant when they are compared with the very significant effect that the value of the pore connectivity, nT, has on the magnitude of the intraparticle interstitial velocity. The effective pore diffusion coefficient of the adsorbate molecules was found to decline with increasing molecular size of ligand, with increasing fractional saturation of the active sites or with diminishing pore size, and with decreasing pore connectivity, nT. The results also show that the magnitude of the interstitial fluid velocity is many times larger than the diffusion velocity of the adsorbate molecules within the porous adsorbent particles. Furthermore, the results clearly show that the intraparticle interstitial velocity and the pore diffusivity of the adsorbate increase significantly as the value of the pore connectivity, nT, of the porous medium increases. The results of this work indicate that the pore network model and the expressions presented in this work, could allow one, for a given porous adsorbent, adsorbate, ligand (active site), and interstitial column fluid velocity, to determine in an a priori manner the values of the intraparticle interstitial velocity and pore diffusivity within the monolith or within the porous adsorbent particles as the fractional saturation of the active sites changes. The values of these transport parameters could then be employed in the macroscopic models that could predict the dynamic behavior, scale-up, and design of chromatographic systems. The theoretical results could also have important implications in the selection of a ligand as well as in the selection and construction of an affinity porous matrix.

Theory of perfusion chromatography

Abstract A mathematical model of perfusion chromatography was constructed for column systems. This model could describe the dynamic behavior of single- and multi-component adsorption in columns having perfusive adsorbent particles (the perfusive particles have a non-zero intraparticle fluid velocity). The model expressions for the adsorbent particles include the intraparticle mass transfer mechanisms of convection (intraparticle fluid flow) and diffusion and the mass transfer step involving the interaction between the adsorbate molecules and the active sites on the surface of the porous particles. The continuity expression for the fluid flowing stream in the column includes the mechanism of axial dispersion. When the intraparticle fluid velocity is taken to be equal to zero in the model of perfusion chromatography, the resulting expressions could describe single- and multi-component adsorption in columns having purely diffusive particles. The perfusion chromatographic model was solved and used to study the dynamic behavior of column systems for different particle sizes, column lengths, column fluid superficial velocities, intraparticle fluid velocities and different values of the effective pore diffusivity and of the total number of active sites per volume of adsorbent. Columns with perfusive adsorbent particles and columns with purely diffusive adsorbent particles were considered in this work. The dynamics of the interaction mechanisms of the adsorption step of the systems studies in this work are (a) relatively not fast, (b) relatively fast and (c) infinitely fast. The values of certain variables which could be used to evaluate column performance, and also the breakthrough curves obtained from columns having perfusive adsorbent particles, were compared with those obtained from columns involving purely diffusive adsorbent particles. The results from the systems studied in this work suggest that for adsorption systems having relatively fast or infinitely fast interaction kinetics (that is, the dynamics or the interaction step between the adsorbate molecules and the active sites are relatively fast or infinitely fast), the use of perfusive particles could have the potential to provide improved column performance.

Modeling of the primary and secondary drying stages of the freeze drying of pharmaceutical products in vials: numerical results obtained from the solution of a dynamic and spatially multi-dimensional lyophilization model for different operational policies.

A rigorous unsteady state and spatially multidimensional model is presented and solved to describe the dynamic behavior of the primary and secondary drying stages of the lyophilization of a pharmaceutical product in vials for different operational policies. The results in this work strongly motivate the aggressive control of freeze drying and it is found that heat input control that runs the process close to the melting and scorch temperature constraints yields (i) faster drying times, and (ii) more uniform distributions of temperature and concentration of bound water at the end of the secondary drying stage.

Mathematical Modelling of the Primary and Secondary Drying Stages of Bulk Solution Freeze-Drying in Trays: Parameter Estimation and Model Discrimination by Comparison of Theoretical Results With Experimental Data

ABSTRACT A mathematical model was constructed and solved in order to describe quantitatively the dynamic behavior of the primary and secondary drying stages of the freeze-drying of pharmaceuticals in trays. The theoretical results were compared with the experimental data of the freeze-drying of skim milk, and the agreement between the experimental data and the theoretical results is good. Detailed model calculations have indicated that the contribution of the removal of hound (unfrozen) water to the total mass flux of the water removed during primary drying, is not significant. For this reason, it was found that one could not

Network modeling of the intraparticle convection and diffusion of molecules in porous particles packed in a chromatographic column

A pore network model (cubic lattice network) is constructed to represent the porous structure in a column packed with porous chromatographic particles. Expressions are developed and used to determine, through the utilization of the pore network model, the intraparticle interstitial fluid velocity and pore diffusivity of a solute as the pore connectivity, nT, of the porous medium is varied from 2.6 to 6.0. The results show that the intraparticle interstitial velocity and the pore diffusivity increase significantly as the value of the pore connectivity, nT, increases, and clearly indicate that the pore connectivity, nT, plays a key role in determining the mass transport properties of a porous medium and, therefore, it is an extremely important parameter in the characterization and construction of porous particles. Furthermore, the results show that the intraparticle interstitial fluid velocity, vp,i, is many times larger than the diffusion velocity, vDA, of the solute within the porous medium, and the ratio vp,i/vDA increases significantly as the pore connectivity, nT, increases. The results of this work indicate that the pore network model could allow one, for a given porous medium, solute and interstitial column fluid velocity, to determine the values of the intraparticle interstitial fluid velocity, vp,i, and pore diffusivity, Dp, of the solute in an a priori manner. The values of vp,i and Dp could then be employed in the macroscopic models that describe the dynamic behavior of chromatographic separations in columns packed with porous particles.

Modeling and simulation of the dynamic behavior of monoliths: Effects of pore structure from pore network model analysis and comparison with columns packed with porous spherical particles

A mathematical model is presented that could be used to describe the dynamic behavior, scale-up, and design of monoliths involving the adsorption of a solute of interest. The value of the pore diffusivity of the solute in the pores of the skeletons of the monolith is determined in an a priori manner by employing the pore network modeling theory of Meyers and Liapis [J. Chromatogr. A, 827 (1998) 197 and 852 (1999) 3]. The results clearly show that the pore diffusion coefficient, Dmp, of the solute depends on both the pore size distribution and the pore connectivity, nT, of the pores in the skeletons. It is shown that, for a given type of monolith, the film mass transfer coefficient, Kf, of the solute in the monolith could be determined from experiments based on Eq. (3) which was derived by Liapis [Math. Modelling Sci. Comput., 1 (1993) 397] from the fundamental physics. The mathematical model presented in this work is numerically solved in order to study the dynamic behavior of the adsorption of bovine serum albumin (BSA) in a monolith having skeletons of radius r(o) = 0.75x10(-6) m and through-pores having diameters of 1.5x10(-6)-1.8x10(-6) m [H. Minakuchi et al., J. Chromatogr. A, 762 (1997) 135]. The breakthrough curves of the BSA obtained from the monolith were steeper than those from columns packed with porous spherical particles whose radii ranged from 2.50x10(-6) m to 15.00x10(-6) m. Furthermore, and most importantly, the dynamic adsorptive capacity of the monolith was always greater than that of the packed beds for all values of the superficial fluid velocity, Vtp. The results of this work indicate that since in monoliths the size of through-pores could be controlled independently from the size of the skeletons, then if one could construct monolith structures having (a) relatively large through-pores with high through-pore connectivity that can provide high flow-rates at low pressure drops and (b) small-sized skeletons with mesopores having an appropriate pore size distribution (mesopores having diameters that are relatively large when compared with the diameter of the diffusing solute) and high pore connectivity, nT, the following positive results, which are necessary for obtaining efficient separations, could be realized: (i) the value of the pore diffusion coefficient, Dmp, of the solute would be large, (ii) the diffusion path length in the skeletons would be short, (iii) the diffusion velocity, vD, would be high, and (iv) the diffusional response time, t(drt), would be small. Monoliths with such pore structures could provide more efficient separations with respect to (a) dynamic adsorptive capacity and (b) required pressure drop for a given flow-rate, than columns packed with porous particles.

Evaluation of kinetic models for biospecific adsorption and its implications for finite bath and column performance

Abstract Dynamic models that could describe the adsorption of adsorbate onto ligand immobilized on porous or non-porous particles in batch and column systems, are presented and solved. Two different kinetic models (kinetic models 1 and 2) are used to describe the dynamics of the adsorption mechanism when β-galactosidase is adsorbed onto monoclonal antibody immobilized on porous silica particles. The differences in the theoretical predictions of the concentration of the adsorbate in the fluid of the finite bath obtained from kinetic models 1 and 2, are not significant and the agreement between experiment and theory is good. But the two different kinetic models lead to different estimates for the value of the pore diffusivity, and provide significantly different concentration profiles for the adsorbate in the pore fluid and adsorbed phases of the adsorbent particles of the batch system. The column results indicate that the differences in the breakthrough curves obtained from kinetic models 1 and 2, increase as the column length increases. Also, the concentration profiles of the adsorbate in the adsorbent particles obtained from kinetic models 1 and 2, are significantly different and their differences vary along the axial distance of the column. The results indicate that while it is a necessary condition for a kinetic model to describe properly the experimental overall mass-transfer resistance, this is not also a sufficient condition for the accurate determination of the adsorption mechanism and for the accurate estimation of the values of the rate constants and of the pore diffusivity. Furthermore, the differences in the concentration profiles of the adsorbate in the adsorbent particles, obtained from kinetic models 1 and 2, have important implications on the performance of the adsorption stage, as well as on the performance of the wash and elution stages. Experiments are suggested which could provide information that could significantly improve the model discrimination and parameter estimation studies for the determination of a proper mechanism for the dynamics of the adsorption step and of an accurate estimate for the value of the pore diffusivity. When the estimated value of the pore diffusivity is varied by ± 20%, the effect on the dynamic behavior of the batch and column systems can be appreciable. The effect on the dynamic behavior of the batch and column systems when the estimated value (from a correlation) of the film mass transfer coefficient is varied by ± 20%, is not significant. The batch adsorption of β-galactosidase onto anti-β-galactosidase immobilized on non-porous glass coated beads is found to be controlled by film mass transfer and the dynamics of the adsorption step. The batch model with a second-order reversible interaction mechanism for the adsorption step, provides theoretical predictions such that the agreement between experiment and theory is reasonable. When the estimated value (from a correlation) of the film mass transfer coefficient is varied by ± 20%, the effect on the dynamic behavior of the batch and column systems (having nonporous adsorbent particles) is not significant. Column experiments are suggested which could provide information, in addition to the information obtained from batch experiments, that could improve the model discrimination and parameter estimation studies for the determination of a proper mechanism for the dynamics of the adsorption step, in affinity adsorption systems involving non-porous adsorbent particles.

Perfusion chromatography: Effect of micropore diffusion on column performance in systems utilizing perfusive adsorbent particles with a bidisperse porous structure

Abstract A mathematical model describing single-component and multi-component adsorption in columns with bidisperse perfusive or bidisperse purely diffusive adsorbent particles is constructed and presented. The model is used to study the adsorption of lysozyme onto monocional anti-lysozyme in columns with bidisperse porous adsorbent particles. The influence of the effective pore diffusion coefficient of the adsorbate in the microparticles (microspheres) and the effects of particle size and intraparticle convective flow on column performance are examined. The results for the systems studied indicate that the systems with bidisperse perfusive particles provide a higher dynamic utilization of the adsorptive capacity of the column than the systems having bidisperse purely diffusive particles.

Modeling the velocity field of the electroosmotic flow in charged capillaries and in capillary columns packed with charged particles: interstitial and intraparticle velocities in capillary electrochromatography systems

Mass transfer systems based on electrokinetic phenomena (i.e., capillary electrochromatography (CEC)) have shown practical potential in becoming powerful separation methods for the biotechnology and pharmaceutical industries. A mathematical model has been constructed and solved to describe quantitatively the profiles of the electrostatic potential, pressure, and velocity of the electroosmotic flow (EOF) in charged cylindrical capillaries and in capillary columns packed with charged particles. The results obtained from model simulations (i) provide significant physical insight and understanding with regard to the velocity profile of the EOF in capillary columns packed with charged porous particles which represent systems employed in CEC, (ii) provide the physical explanation for the experimental results which indicate that the velocity of the EOF in capillary columns packed with charged porous particles is a very weak function (it is almost independent) of the diameter of the particles, and (iii) indicate that the intraparticle velocity, nu(p,i), of the EOF can be greater than zero. The intraparticle Peclet number, Pe(int rap), for lysozyme was found to be greater than unity and this intraparticle convective mass transfer mechanism could contribute significantly, if the appropriate chemistry is employed in the mobile liquid phase and in the charged porous particles, in (a) decreasing the intraparticle mass transfer resistance, (b) decreasing the dispersive mass transfer effects, and (c) increasing the intraparticle mass transfer rates so that high column efficiency and resolution can be obtained. Furthermore, the results from model simulations indicate that for a given operationally permissible value of the applied electric potential difference per unit length, Ex, high values for the average velocity of the EOF can be obtained if (1) the zeta potential, zeta(p), at the surface of the particles packed in the column has a large negative magnitude, (2) the value of the viscosity, mu, of the mobile liquid phase is low, (3) the magnitude of the dielectric constant, epsilon, of the mobile liquid phase is reasonably large, and (4) the combination of the values of the concentration, C(infinity), of the electrolyte and of the dielectric constant, epsilon, provide a thin double layer. The theoretical results for the velocity of the EOF obtained from the solution of the model presented in this work were compared with the experimental values of the velocity of the EOF obtained from a fused-silica column packed with charged porous silica C8 particles. Systems with four different particle diameters and three different concentrations of the electrolyte were considered, and the magnitude of the electric field was varied widely. The agreement between theory and experiment was found to be good.

Freeze-Drying of Pharmaceutical Crystalline and Amorphous Solutes in Vials: Dynamic Multi-Dimensional Models of the Primary and Secondary Drying Stages and Qualitative Features of the Moving Interface

ABSTRACT Dynamic and spatially multi-dimensional mathematical models of the primary and secondary drying stages of the freeze-drying of pharmaceutical crystalline and amorphous solutes in vials, are constructed and presented in this work. The models account for the removal of free and bound water and could also provide the geometric shape of the moving interface and its position. It is proved that the temperature of the moving interface can not be constant if the flux of heat flow to the sides of the vial is not zero. It is also proved that the slope of the free surface (moving interface) at the edge of the vial is always curved downward. The numerical solution of the nonlinear partial differential equations of the models would allow model simulations that could indicate design conditions, operating conditions, and control strategies that could provide high drying rates and could lead to a series of novel experiments in freeze-drying.