Equation, cell population balance

Kumar, J., Peglow, M., Warnecke, G. Heinrich, S. 2008a The cell average technique for solving multi-dimensional aggregation population balance equations. Computers and Chemical Engineering 32,1810-1830. 

With this description, then, the standard population balance equation describing the cell kinetics is ... 

The properties of microbial populations result from the characteristics of metabolism and Its control at the level of the Individual cell. Useful descriptions of population behavior may therefore be constructed based on understanding of single-cell metabolism, and, conversely, studies of population properties may be employed to extract Information about single-cell operation. Population balance equations provide the primary mathematical tool for these purposes, and flow cytometry allows experimental access to the distribution of cell states In the population. Application of these methods to bacteria and yeasts Is Illustrated using three exaaq>le systems. 

Adoption of this approach to microbial process development cannot occur until methods exist for determining the influence of reactor design and operating parameters on single-cell metabolic control actions and reaction rates. If this Information is available, population balance equations and associated medium conservation equations provide the required bases for reactor analysis (1, ). For example, for a well-mixed, continuous-flow Isothermal mTcroblal reactor at steady-state, the population balance equation may be written ... 

Quantitative analysis of these measurements has been tmder-taken by two different strategies based upon the governing population balance equations which describe the system (25). One method employed the Colllns-Richmond equation (26), the Integrated form of the population balance equation, with results which are summarized In detail elsewhere (25). In the second approach, a simple functional form was chosen for the fission frequency function which contained two adjustable parameters (the results of Harvey, Marr and Painter (27) Indicate that the results of population balance model solutions are not highly sensitive to the functional form of this function In the model). Then, four different function forms for the single-cell rate of protein synthesis fimctlon r were examined. For the following four functional forms. 

The state (py, pg, pj) = (p, 0,0) is the final state, is stable, and represents a completely invaded state where all the host cells were killed producing new viruses. We are interested in studying a front of invasion connecting both states. Since lysis of bacteria results in a precipitous dechne in the culture turbidity, virus population growth produces circular clearings called plaques. The growth of these plaques in size is not only a result of the kinetics but also due to the dispersal movement of the viruses within the bacterial culture. To derive an equation for vims dispersal in agar, we start with the mesoscopic balance equation for the vims density at point x at time t obtained from Model C (see Sect. 3.4.3) [121] ... 

Initially, the cells of total population density are assumed to be all of age zero. The number density function / (t, t) must satisfy the population balance equation... 

We consider a population of cells distributed according to cell mass x with growth rate JT(x). The cells further undergo binary division at the rate b(x) into daughter cells whose mass distribution is given by P x y) where y represents the mass of the mother cell and x is the mass of either daughter cell. The population balance equation may be written as... 

The unstructured and segregated model considers the cell culture as a heterogeneous population, with individual cells characterized by age, mass, or size. The variation of properties follows a statistical distribution. The most concise way to deal with such population heterogeneity is through a population balance model (PBM). These models can become mathematically complex because the equations that describe the population have to consider that the properties vary in the population and... 

The derivation of regular patterns of cellular behavior that can be assigned to certain defined subpopulations (or single cells) from measured data necessarily involves modeling approaches that allow assignment of mechanistic (e.g., metabohc) models to subpopulations with individual parameters or other variations. Further, such models must allow the tracking of individual cell s dynamic behavior even when they fluctuate between different populations. Two principal approaches can be distinguished systems of partial different equations (population balance systems, PBE) and stochastic cell ensemble models (CEMs)... 

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