Cell models suspensions

Because the expert system was not connected to a real reactor, we built a small table-driven simulation to model the growth of cells in suspension. The graphical interface includes images representing the reactor itself, several feed bins and associated valves. Also shown in Figure 1 are several types of gauges, including a strip chart, monitors of various states and alarm conditions, temperature, and the on/off state of heaters and coolers. 

Techne sell a bioreactor for growing cells in suspension in volumes up to 3 1 (Fig. 3.10) and New Brunswick has a model which will take 5 1. The Techne bioreactor maintains the cells or microcarriers in suspension with a floating stirrer but such a mechanism becomes increasingly less efficient up to 20 1. pH, p02 and temperature are monitored and controlled and the reactor can be used for batch cultures or in continuous mode. 

Cell models constitute a second major class of empirical developments. Among these, only two will be mentioned here as constituting the most successful and widely used. The first, due to Happel (1957,1958), is useful for estimating the effective viscosity and settling velocity of suspensions. Here, the suspension is envisioned as being composed of fictitious identical cells, each containing a single spherical particle of radius a surrounded by a concentric spherical envelope of fluid. The radius b of the cell is chosen to reproduce the suspension s volume fraction

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Monte Carlo techniques were first applied to colloidal dispersions by van Megen and Snook (1975). Included in their analysis was Brownian motion as well as van der Waals and double-layer forces, although hydrodynamic interactions were not incorporated in this first study. Order-disorder transitions, arising from the existence of these forces, were calculated. Approximate methods, such as first-order perturbation theory for the disordered state and the so-called cell model for the ordered state, were used to calculate the latter transition, exhibiting relatively good agreement with the exact Monte Carlo computations. Other quantities of interest, such as the radial distribution function and the excess pressure, were also calculated. This type of approach appears attractive for future studies of suspension properties. 

In this chapter, we extend the electrokinetic theory of soft particles (Chapter 21), which is applicable for dilute suspensions, to cover the case of concentrated suspensions [1-3] on the basis of Kuwabara s cell model [4], which has been applied to theoretical studies of various electrokinetic phenomena in concentrated suspensions of hard colloidal particles [5-23]. 

Consider a concentrated suspension of charged spherical soft particles moving with a velocity 17 in a liquid containing a general electrolyte in an applied electric field E. We assume that the particle core of radius a is coated with an ion-penetrable layer of polyelectrolytes with a thickness d. The polyelectrolyte-coated particle has thus an inner radius a and an outer radius b = a + d. We employ a cell model [4] in which each particle is surrounded by a concentric spherical shell of an electrolyte solution, having an outer radius c such that the particle/cell volume ratio in the unit cell is equal to the particle volume fraction 4> throughout the entire dispersion (Fig. 22.1), namely. 

Electrokinetic equations describing the electrical conductivity of a suspension of colloidal particles are the same as those for the electrophoretic mobility of colloidal particles and thus conductivity measurements can provide us with essentially the same information as that from electrophoretic mobihty measurements. Several theoretical studies have been made on dilute suspensions of hard particles [1-3], mercury drops [4], and spherical polyelectrolytes (charged porous spheres) [5], and on concentrated suspensions of hard spherical particles [6] and mercury drops [7] on the basis of Kuwabara s cell model [8], which was originally applied to electrophoresis problem [9,10]. In this chapter, we develop a theory of conductivity of a concentrated suspension of soft particles [11]. The results cover those for the dilute case in the limit of very low particle volume fractions. We confine ourselves to the case where the overlapping of the electrical double layers of adjacent particles is negligible. 

For concentrated suspensions, hydrodynamic interactions among particles must be considered. The hydrodynamic interactions between spherical particles can be taken into account by means of a cell model, which assumes that each sphere of radius a is surrounded by a virtual shell of outer radius b and the particle volume... 

Consider a concentrated suspension of porous spheres of radius a in a liquid of viscosity rj [27]. We adopt a cell model that assumes that each sphere of radius a is surrounded by a virtual shell of outer radius b and the particle volume fraction 4> is given by Eq. (27.2) (Eig. 27.3). The origin of the spherical polar coordinate system (r, 6, cp) is held fixed at the center of one sphere. According to Simha [2], we the following additional boundary condition to be satisfied at the cell surface r = b ... 

The specific standard methods of a new perfusion culture will now be described for growth and maintenance of mammalian cells in suspension cultures at high density. The biofermenter was used for high density culture of Namalwa cells with serum-free medium as the model. In 1980, the parent Namalwa cells were obtained fi-om Mr. F. Klein of Frederick Cancer Research Center, Frederick, Maryland, U. S. A. In our laboratories, we were able to adapt the cells to a serum and albumin-fi ee medium and named the cells KJM-1. ITPSG and ITPSG+F68 used a serum-free medium containing insulin, 3 g/ml Transferrin, 5 g/ml sodium pyruvate, 5 mM seienious acid,... 

The steady-state velocities of the gravitational sedimentation in suspensions obtained with the help of cell models were compared with numerous experimental data in [450], It was shown that the most precise results can be obtained by using the Slobodov-Chepura model [450], in which the drag forces lead to formula (2.9.1), where the correction coefficient can be calculated as... 

For more concentrated suspensions, one can estimate p by using the cell model. 

Figure 2. Diagram showing the equivalent circuit model for a cell in suspension. The electrical double layer (EDL) on the surface of the electrodes is model as a capacitor C i.

Boogaard PJ, Nagelkerke JF, Mulder GJ. Renal proximal tubular cells in suspension or in primary culture as in vitro models to study nephrotoxisity. Chemico Biological Interactions 1990 76 251-292. 

Cell models have also been applied to the approximate determination of the properties of concentrated suspensions. This is touched upon in our discussion of suspension viscosity at high shear rate in Section 9.3. 

Cell models akin to those discussed in Section 8.5 have also been applied to the determination of the properties of concentrated suspensions (Happel Brenner 1983, van de Ven 1989). Although it is another method which has been used to obtaining approximate expressions for the high shear relative viscosity, we choose not to expand upon it here, instead referring the reader to the references cited. One of the difficulties is that the determination of the boundary conditions at the cell surface is somewhat arbitrary. Furthermore, expressions obtained by this approach indicate that the cell model is inappropriate for highly concentrated suspensions and is most satisfactory only at low to moderate concentrations. 

J. Val, Modelling the physiology of plant cells in suspension culture. Ph.D. Thesis, Leiden University, 1993. 

TA Strout. Attenuation of Sound in High Concentration Suspensions Development and Application of an Oscillatory Cell Model. Thesis. The University of Maine, 1991. 

The a-dispersion exhibited by cell suspensions could basically be explained by rather complementary mechanisms, formally described by different microscopic models of cell systems, as the ones based either on displacement of counter-ions (Gheorghiu, 1993, 1994) or on shape effects (e.g., exhibited by clusters of interconnected cells as shown by Vrinceanu and Gheorghiu, 1996 Asami et al., 1999 Gheorghiu et al., 2002, 2010). These studies described both a and P dispersions based on unitary microscopic models. The reports emphasizing shape effect on the impedance spectra of (non)spheroidal living cells in suspension have also supported application of time-based impedance spectroscopy assays to noninvasively assess cell dynamics (e.g., cell-cycle progression). 

Dukhin et al. [83-85] have performed the direct calculation of the CVI in the situation of concentrated systems. In fact, it must be mentioned here that one of the most promising potential applicabilities of these methods is their usefiilness with concentrated systems (high volume fractions of solids, 4>) because the effect to be measured is also in this case a collective one. The first generalizations of the dynamic mobility theory to concentrated suspensions made use of the Levine and Neale cell model [86,87] to account for particle-particle interactions. An alternative method estimated the first-order volume fraction corrections to the mobility by detailed consideration of pair interactions between particles at all possible different orientations [88-90]. A comparison between these approaches and calculations based on the cell model of Zharkikh and Shilov [91] has been carried out in Refs. [92,93],... 

There is, of course, an analogy between the study of a fixed and regular suspension and that of a spatially periodic medium. Dilute suspensions of particles have been intensively investigated by O Brien and coworkers (cf., for instance. Ref. 10 for a recent review of these works). Only a few contributions deal with nondilute suspensions an alternative and efficient approach is to use a cell model as Levine and Neale [11] did in order to take into account the effect of the finite solid volume void fraction. 

Rigid random arrays have generally been simulated by cell models that have not been limited to dilute suspensions. An early example of a cell model is that of Brinkman (1947), who eonsidered flow past a single sphere in a porous medium of permeability k. The flow is deseribed by an equation that collapses to Darcy s (1856) law (in its post-Darcy form, which includes viscosity) for low values of and to the creeping flow version of the Navier Stokes equation for high values of K. His solution is... 

A comparable cell model is that of Kuwabara (1959), the only difference being that the spherical cell surface is in this case assumed to be at zero vorticity rather than at zero shear stress. The coefficient of c / in Eq. (28a) for dilute suspensions becomes 1.8 in Kuwabara s solution, instead of 1.5. 

It is notable that in the equations for both the ordered arrangement of spheres and the cell models, l-(U/Uo) for dilute suspension is directly proportional to c / (or c / in the case of Brinkman s model) rather than to c as in Eq. (25), a fact that would appear to render those equations inadequate for fluidization or sedimentation purposes (Batchelor, 1972 Saffman, 1973). If we compare the actual values of 1 — (U/Uo) predicted by the above equations at c = 0.01, we find that 1 — (E//C/o) = 0.0655, 0.379, 0.212, and 0.323 from Eqs. (25), (26), (27a), and (28a), respectively, which, combined with experimental findings that values of 1 — (U/Uq) for uneharged settling spheres at c = 0.01 are even lower than 0.0655 (Buscall et al., 1982 Tackie et al., 1983), confirms this inadequacy, especially for dilute suspensions. 

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