Toward a robust model of packing and scale-up for chromatographic beds. 1. Mechanical compression

The packing of compressible biochromatographic resins at large scale suffers from a poor understanding of how column packing method, resin properties, and column geometry impact column performance. To improve understanding, we develop and evaluate a one-dimensional, continuum mechanics model of column packing by mechanical compression. We show that the model can quantitatively predict the change in bed height, applied stress, and internal axial porosity profile without adjustable parameters when the modulus and wall friction coefficients are determined independently. The model possesses theoretical relationships for wall support and resin rigidity that should enable it to describe the mechanical compression of any biochromatographic resin for any column diameter. Moreover, this framework could provide a path to analogous models for flow packing and dynamic axial compression.     

Poromechanics of compressible charged porous media using the theory of mixtures

Osmotic, electrostatic, and/or hydrational swellings are essential mechanisms in the deformation behavior of porous media, such as biological tissues, synthetic hydrogels, and clay-rich rocks. Present theories are restricted to incompressible constituents. This assumption typically fails for bone, in which electrokinetic effects are closely coupled to deformation. An electrochemomechanical formulation of quasistatic finite deformation of compressible charged porous media is derived from the theory of mixtures. The model consists of a compressible charged porous solid saturated with a compressible ionic solution. Four constituents following different kinematic paths are identified: a charged solid and three streaming constituents carrying either a positive, negative, or no electrical charge, which are the cations, anions, and fluid, respectively. The finite deformation model is reduced to infinitesimal theory. In the limiting case without ionic effects, the presented model is consistent with Blot's theory. Viscous drag compression is computed under closed circuit and open circuit conditions. Viscous drag compression is shown to be independent of the storage modulus. A compressible version of the electrochemomechanical theory is formulated. Using material parameter values for bone, the theory predicts a substantial influence of density changes on a viscous drag compression simulation. In the context of quasistatic deformations, conflicts between poromechanics and mixture theory are only semantic in nature.     

Subcritical flutter in collapsible tube flow: a model of expiratory flow in the trachea

Using an axisymmetric geometry that retains certain qualitative features of the trachea, we extend one-dimensional modeling of flow in collapsible tubes to include both curved shell effects and, for untethered tubes, wall inertia. A systematic scaling of the finite deformation membrane equations leads to an approximate set which is consistent with the one-dimensional fluid model; axial and normal wall variables are coupled elastically, but only axial inertia is retained. Transverse curvature causes elastic coupling that can give rise to axial wall motion and a flutter instability. The source of instability is the product of a nonzero reference axial curvature with axial tension variation due to axial stretching. The numerical results suggest that this mechanism may be significant even in processes which cannot be assumed one-dimensional.     

Biomechanical properties of human articular cartilage under compressive loads

The function of articular cartilage is to support and distribute loads and to provide lubrication in the diarthrodial joints. Cartilage function is described by proper mechanical and rheological properties, strain and depth-dependent, which are not completely assessed. Unconfined and confined compression are commonly used to evaluate the Young's modulus (E) and the aggregate modulus (H(A)), respectively. The Poisson's ratio (nu) can be calculated indirectly from the equilibrium compression data, or using the biphasic indentation technique; it has recently been optically evaluated by using video microscopy during unconfined compression. The transient response of articular cartilage during confined compression depends on its permeability k; a constant value of k can be easily identified by a simple analytical model of confined compression tests, whereas more complex models or direct measurements (permeation tests) are needed to study the permeability dependence on deformation. A poroelastic finite element model of articular cartilage was developed for this purpose. The elastic parameters (E,nu) of the model were evaluated performing unconfined compression creep tests on human articular cartilage disks, whereas k was identified from the confined test response. Our combined experimental and computational method can be used to identify the parameters that define the permeability dependence on deformation, as a function of depth from articular surface.     

Simulation of the dynamic packing behavior of preparative chromatography columns via discrete particle modeling

Preparative packed‐bed chromatography using polymer‐based, compressible, porous resins is a powerful method for purification of macromolecular bioproducts. During operation, a complex, hysteretic, thus, history‐dependent packed bed behavior is often observed but theoretical understanding of the causes is limited. Therefore, a rigorous modeling approach of the chromatography column on the particle scale has been made which takes into account interparticle micromechanics and fluid–particle interactions for the first time. A three‐dimensional deterministic model was created by applying Computational Fluid Dynamics (CFD) coupled with the Discrete Element Method (DEM). The column packing behavior during either flow or mechanical compression was investigated in‐silico and in laboratory experiments. A pronounced axial compression–relaxation profile was identified that differed for both compression strategies. Void spaces were clearly visible in the packed bed after compression. It was assumed that the observed bed inhomogeneity was because of a force‐chain network at the particle scale. The simulation satisfactorily reproduced the measured behavior regarding packing compression as well as pressure‐flow dependency. Furthermore, the particle Young's modulus and particle–wall friction as well as interparticle friction were identified as crucial parameters affecting packing dynamics. It was concluded that compaction of the chromatographic bed is rather because of particle rearrangement than particle deformation. © 2015 American Institute of Chemical Engineers Biotechnol. Prog., 32:363–371, 2016     

Heterogeneous mechanics of the mouse pulmonary arterial network

Individualized modeling and simulation of blood flow mechanics find applications in both animal research and patient care. Individual animal or patient models for blood vessel mechanics are based on combining measured vascular geometry with a fluid structure model coupling formulations describing dynamics of the fluid and mechanics of the wall. For example, one-dimensional fluid flow modeling requires a constitutive law relating vessel cross-sectional deformation to pressure in the lumen. To investigate means of identifying appropriate constitutive relationships, an automated segmentation algorithm was applied to micro-computerized tomography images from a mouse lung obtained at four different static pressures to identify the static pressure–radius relationship for four generations of vessels in the pulmonary arterial network. A shape-fitting function was parameterized for each vessel in the network to characterize the nonlinear and heterogeneous nature of vessel distensibility in the pulmonary arteries. These data on morphometric and mechanical properties were used to simulate pressure and flow velocity propagation in the network using one-dimensional representations of fluid and vessel wall mechanics. Moreover, wave intensity analysis was used to study effects of wall mechanics on generation and propagation of pressure wave reflections. Simulations were conducted to investigate the role of linear versus nonlinear formulations of wall elasticity and homogeneous versus heterogeneous treatments of vessel wall properties. Accounting for heterogeneity, by parameterizing the pressure/distention equation of state individually for each vessel segment, was found to have little effect on the predicted pressure profiles and wave propagation compared to a homogeneous parameterization based on average behavior. However, substantially different results were obtained using a linear elastic thin-shell model than were obtained using a nonlinear model that has a more physiologically realistic pressure versus radius relationship.     

Improved packing of preparative biochromatography columns by mechanical vibration

The bioprocessing industry relies on packed-bed column chromatography as its primary separation process to attain the required high product purities and fulfill the strict requirements from regulatory bodies. Conventional column packing methods rely on flow packing and/or mechanical compression. In this work, the application of ultrasound and mechanical vibration during packing was studied with respect to packing density and homogeneity. We investigated two widely used biochromatography media, incompressible ceramic hydroxyapatite, and compressible polymethacrylate-based particles, packed in a laboratory-scale column with an inner diameter of 50 mm. It was shown that ultrasonic irradiation led to reduced particle segregation during sedimentation of a homogenized slurry of polymethacrylate particles. However, the application of ultrasound did not lead to an improved microstructure of already packed columns due to the low volumetric energy input (~152 W/L) caused by high acoustic reflection losses. In contrast, the application of pneumatic mechanical vibration led to considerable improvements. Flow-decoupled axial linear vibration was most suitable at a volumetric force output of ~1,190 N/L. In the case of the ceramic hydroxyapatite particles, a 13% further decrease of the packing height was achieved and the reduced height equivalent to a theoretical plate (rHETP) was decreased by 44%. For the polymethacrylate particles, a 18% further packing consolidation was achieved and the rHETP was reduced by 25%. Hence, it was shown that applying mechanical vibration resulted in more efficiently packed columns. The application of vibration furthermore is potentially suitable for in situ elimination of flow channels near the column wall.     

Measuring the Deformation of Cells within a Piece of Compressed Potato Tuber Tissue

For the successful mathematical mechanical modelling of livingplant tissues, relationships between cellular deformations andtissue deformation need to be investigated. In previous workthese relationships have often been assumed. In this paper thedeformation of living cells within potato tuber tissue is measuredusing light microscopy and image analysis and is analysed inrelation to applied tissue deformations. The cell wall deformationwas found to depend upon the orientation of the cell wall faceswith respect to the global axes of the tissue and the appliedtissue deformation. Some faces experienced compression, whichreduced their surface area; others were deformed in bi-axialtension, thus increasing their surface area. These deformationswere successfully related to the global tissue deformations,using a simple constant volume affine deformation model, upto compressive deformations of 20% of specimen height. Somedeviation from the model was observed due to the bending ofcell walls in compression. Copyright 2000 Annals of Botany Company Potato tuber tissue, Solanum tuberosum, mechanical properties, cell walls, strain, re-orientation     

Modelling the mechanical properties of single suspension-cultured tomato cells

BACKGROUND AND AIMS: The relationship between composition and structure of plant primary cell walls, and cell mechanical properties is not fully understood, partly because intrinsic properties of walls such as Young's modulus cannot be obtained readily. The aim of this work is to show that Young's modulus of walls of single suspension-cultured tomato cells can be determined by modelling force-deformation data. METHODS: The model simulates the compression of a cell between two flat surfaces, with the cell treated as a liquid-filled sphere with thin compressible walls. The cell wall and membrane were taken to be permeable, but the compression was so fast that water loss could be neglected in the simulations. Force-deformation data were obtained by compressing the cells in micromanipulation experiments. RESULTS:Good fits were obtained between the model and low-strain experimental data, using the modulus and initial inflation of the cell as adjustable parameters. The mean Young's modulus for 2-week-old cells was found to be 2.3 +/- 0.2 GPa at pH 5. This corresponds to an instantaneous bulk modulus of elasticity of approx. 7 MPa, similar to a value found by the pressure probe method. However, Young's modulus is a better parameter, as it should depend only on the composition and structure of the cell wall, not on bulk cell behaviour. This new method has been used to show that Young's modulus of cultured tomato cell walls is at its lowest at pH 4.5, the pH optimum for expansin activity. CONCLUSIONS:The linear elastic model is very suitable for estimating wall Young's modulus from micromanipulation experiments on single tomato cells. This is a powerful method for determining cell wall material properties.     

Experimental validation of a time-domain-based wave propagation model of blood flow in viscoelastic vessels

Time-domain-based one-dimensional wave propagation models of the arterial system are preferable over one-dimensional wave propagation models in the frequency domain since the latter neglect the non-linear convection forces present in the physiological situation, especially when the vessel is tapered. Moreover, one-dimensional wave propagation models of the arterial system can be used to provide boundary conditions for fully three-dimensional fluid-structure interaction computations that are usually defined in the time domain. In this study, a time-domain-based one-dimensional wave propagation model in a cross-sectional area, flow and pressure (A,q,p)-formulation is developed. Using this formulation, a constitutive law that includes viscoelasticity based on the mechanical behaviour of a Kelvin body, is introduced. The resulting pressure and flow waves travelling through a straight and tapered vessel are compared to experimental data obtained from measurements in an in vitro setup. The model presented shows to be well suited to predict wave propagation through these straight and tapered vessels with viscoelastic wall properties and hereto can serve as a time-domain-based method to model wave propagation in the human arterial system.     

Turgor pressure sensing in plant cell membranes

Experimental evidence is reviewed which shows that the cell membrane is compressible by both mechanical and electrical forces. Calculations are given which show that significant changes in the thickness of cell membranes can occur as a result of (a) direct compression due to the turgor pressure; (b) indirect effects due to the stretching of the cell wall; and (c) the stresses induced by the electric field in the membrane.     

Effect of the fluid-structure interactions on low-density lipoprotein transport within a multi-layered arterial wall

The effects of fluid-structure interactions (FSI) and pulsation on the transport of low-density lipoprotein (LDL) through an arterial wall are analyzed in this work. To this end, a comprehensive multi-layer model for both LDL transport as well as fluid-structure interaction (FSI) is introduced. The constructed model is analyzed and compared with the existing results in the limiting cases. Excellent agreement is found between the presented model and the existing results in the limiting cases. The presented model takes into account the complete multi-layered LDL transport while incorporating the FSI aspects to enable a comprehensive study of the deformation effect on the pertinent parameters of the transport processes within an artery. Since the flow inside an artery is time-dependent, the impact of pulsatile flow is also analyzed with and without FSI. A detailed analysis is presented to illustrate the consequence of different factors on the LDL transport in an artery.     

Cell adhesion mechanisms and stress relaxation in the mechanics of tumours

Tumour cells usually live in an environment formed by other host cells, extra-cellular matrix and extra-cellular liquid. Cells duplicate, reorganise and deform while binding each other due to adhesion molecules exerting forces of measurable strength. In this paper, a macroscopic mechanical model of solid tumour is investigated which takes such adhesion mechanisms into account. The extracellular matrix is treated as an elastic compressible material, while, in order to define the relationship between stress and strain for the cellular constituents, the deformation gradient is decomposed in a multiplicative way distinguishing the contribution due to growth, to cell rearrangement and to elastic deformation. On the basis of experimental results at a cellular level, it is proposed that at a macroscopic level there exists a yield condition separating the elastic and dissipative regimes. Previously proposed models are obtained as limit cases, e.g. fluid-like models are obtained in the limit of fast cell reorganisation and negligible yield stress. A numerical test case shows that the model is able to account for several complex interactions: how tumour growth can be influenced by stress, how and where it can generate cell reorganisation to release the stress level, how it can lead to capsule formation and compression of the surrounding tissue.     

Determination of elastomeric foam parameters for simulations of complex loading

BACKGROUND: Finite element (FE) analysis has shown promise for the evaluation of elastomeric foam personal protection devices. Although appropriate representation of foam materials is necessary in order to obtain realistic simulation results, material definitions used in the literature vary widely and often fail to account for the multi-mode loading experienced by these devices. This study aims to provide a library of elastomeric foam material parameters that can be used in FE simulations of complex loading scenarios. METHOD OF APPROACH: Twelve foam materials used in footwear were tested in uni-axial compression, simple shear and volumetric compression. For each material, parameters for a common compressible hyperelastic material model used in FE analysis were determined using: (a) compression; (b) compression and shear data; and (c) data from all three tests. RESULTS: Material parameters and Drucker stability limits for the best fits are provided with their associated errors. The material model was able to reproduce deformation modes for which data was provided during parameter determination but was unable to predict behavior in other deformation modes. CONCLUSIONS: Simulation results were found to be highly dependent on the extent of the test data used to determine the parameters in the material definition. This finding calls into question the many published results of simulations of complex loading that use foam material parameters obtained from a single mode of testing. The library of foam parameters developed here presents associated errors in three deformation modes that should provide for a more informed selection of material parameters.     

A continuum model for root growth

This paper presents a study of a simple one-dimensional continuum model for growth of the plant root. A fundamental constitutive equation is derived. The model is studied by means of various special cases of increasing complexity. Asymptotic expansions are used to derive approximate solutions to the equation of the model under the fundamental assumption that cell wall thickness is small in comparison with the diameter of the cell. The basic results of the study may be summarized as follows. The observed growth pattern of the root cannot be modelled by a mechanical system whose properties are independent of position on the root. The observed pattern can be modelled by a simple mechanical system in which, for example, cell wall yield stress first decreases and then increases. Two fundamental observations are made based on the modelling study. The first is that any mechanical model must take into account the convective displacement from the tip of points along the root. The second is that in describing growth, data on cell wall mechanical properties are meaningless without corresponding data on cell water potential, and vice versa.     

Numerical simulation of the mechanics of a yeast cell under high hydrostatic pressure

The mechanical effects of the compression of a yeast cell (Saccharomyces cerevisiae) under high hydrostatic pressure used for the processing of food and food ingredients are modelled and simulated with the finite-element method. The cell model consists of a cell wall, cytoplasm a lipid filled vacuole and the nucleus. Material parameters have been taken from literature or have been derived from thermodynamic relationships of water and lipids under high hydrostatic pressure. The model has been validated for a pressure load up to 250 MPa. Comparison of the volume reduction to in situ experimental observations reveals very good agreement. Dimensional analysis of the governing equations shows that transient pressure application in a high-pressure food process does not enhance structural inactivation (mechanical damage), unless pressure oscillation frequencies of 700 MHz are applied. The deformation of the cell under pressure deviates strongly from isotropic volume reduction. Especially, organelle membranes exhibit large effective strain values. Hydrostatic stress conditions are preserved in the interior part of the cell. A pressure load of 400 MPa, which is critical upon disruption of cell organelle membranes, generates an effective strain up to 80%. In the cell wall, the stress state is heterogeneous. Von-Mises stress reaches the critical value upon failure of the cell wall of 70+/-4 MPa at a pressure load between 415 and 460 MPa.     

A microstructural model for the anisotropic drained stiffness of articular cartilage

A constitutive model for articular cartilage is developed to study directional load sharing within the soft biological tissue. Cartilage is idealized as a composite structure whose static mechanical response is dominated by distortion of a sparse fibrous network and by changes in fixed charge density. These histological features of living cartilage are represented in a microstructural analog of the tissue, linking the directionality of mechanical stiffness to the orientation of microstructure. The discretized 'model tissue' is used to define a stiffness tensor relating drained stress and strain over a regime of large deformation. The primary goal of this work was to develop a methodology permitting more complete treatment of anisotropy in the stiffness of cartilage. The results demonstrate that simple oriented microscopic behaviors can combine to produce complicated larger scale response. For the illustrative example of a homogeneous specimen subjected to confined compression, the model predicts a nonlinear anisotropic drained response, with inherent uncertainty at cellular size scales.     

Steady flow and wall compression in stenotic arteries: a three-dimensional thick-wall model with fluid-wall interactions.

Severe stenosis may cause critical flow and wall mechanical conditions related to artery fatigue, artery compression, and plaque rupture, which leads directly to heart attack and stroke. The exact mechanism involved is not well understood. In this paper a nonlinear three-dimensional thick-wall model with fluid-wall interactions is introduced to simulate blood flow in carotid arteries with stenosis and to quantify physiological conditions under which wall compression or even collapse may occur. The mechanical properties of the tube wall were selected to match a thick-wall stenosis model made of PVA hydrogel. The experimentally measured nonlinear stress-strain relationship is implemented in the computational model using an incremental linear elasticity approach. The Navier-Stokes equations are used for the fluid model. An incremental boundary iteration method is used to handle the fluid-wall interactions. Our results indicate that severe stenosis causes considerable compressive stress in the tube wall and critical flow conditions such as negative pressure, high shear stress, and flow separation which may be related to artery compression, plaque cap rupture, platelet activation, and thrombus formation. The stress distribution has a very localized pattern and both maximum tensile stress (five times higher than normal average stress) and maximum compressive stress occur inside the stenotic section. Wall deformation, flow rates, and true severities of the stenosis under different pressure conditions are calculated and compared with experimental measurements and reasonable agreement is found.     

Pressure-flow relationships for packed beds of compressible chromatography media at laboratory and production scale

Pressure drop across chromatography beds employing soft or semirigid media can be a significant problem in the operation of large-scale preparative chromatography columns. The shape or aspect ratio (length/diameter) of a packed bed has a significant effect on column pressure drop due to wall effects, which can result in unexpectedly high pressures in manufacturing. Two types of agarose-based media were packed in chromatography columns at various column aspect ratios, during which pressure drop, bed height, and flow rate were carefully monitored. Compression of the packed beds with increasing flow velocities was observed. An empirical model was developed to correlate pressure drop with the aspect ratio of the packed beds and the superficial velocity. Modeling employed the Blake-Kozeny equation in which empirical relationships were used to predict bed porosity as a function of aspect ratio and flow velocity. Model predictions were in good agreement with observed pressure drops of industrial scale chromatography columns. A protocol was developed to predict compression in industrial chromatography applications by a few laboratory experiments. The protocol is shown to be useful in the development of chromatographic methods and sizing of preparative columns.     

Physical studies on the mechanical stability of columns of calcium alginate gel pellets containing entrapped microbial cells

Constant strain rate and stress relaxation tests on columns of spherical alginate pellets containing entrapped microbial cells demonstrated non-linear viscoelastic behaviour, the columns being relatively resistant to compression over long periods. Compressibility rose with decreases in the alginate concentration used to form the pellets and the soluble sucrose concentration therein, and with increases in temperature and the concentration of cells or other particulate materials; but appeared to be unaffected by the relative dimensions of the column. Pellets were not fractured unless very high pressures were used and deformation was only partially reversible. Over long time periods large creep effects were observed, the rate of compression decreasing exponentially with time. The creep rate increased with the pressure applied, but could be decreased by pumping fluid up the column. Thus the compression of large columns operated continuously for long periods could be modelled by pumping fluid at high flow rates up small columns while applying large pressures to the top of the column. Abbreviation: the ratio of wet weight: dry weight for the yeast cells used in this study is 4:1; weights of cells are always quoted as wet weights.