Euler-Lagarange Modeling

Detailed mathematical models capturing the variation in both the extracellular environment and the metabolism of a segregated biophase promise to aid significantly in describing the behavior of cell populations in bioreactors. This requires a combination of both approaches outlined above in other words, an Euler-Lagrange simulation. For the first time, this interaction between the intracellular state of the individual cells of the population and the turbulent flow field in the bioreactor was tackled by Lapin et al. [69-71]. 

Slip between cells and their environment can be neglected, because the fluid velocity can be shown to surpass the sHp velocity by several orders of magnitude. Momentum transfer between the particles and the fluid phase also does not require explicit consideration because the suspension can be treated as a quasisingle phase for particles smaller than the mesh spacing, as shown previously [72]. 

Thus it can be assumed that the movement of the single cells can be modeled by following the individual fluid parcel as it moves through time and space. 

Within a finite volume element given by the Eulerian simulation of the extracellular liquid phase, the velocities of the liquid phase in each spatial coordinate can be linearly approximated, which allows one to obtain an analytical solution of Eq. (3.21). The random movement caused by turbulent dispersion is superimposed on the convective flow represented by the velocity field v. In the Lagrangian 

Prediction of the intracellular state of a single cell along the trajectory is performed by incorporating the system of intracellular balance equations into the model  

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