Modeling of Bioreactor Systems for Tissue Engineering

The appropriate molecular and macroscopic architecture of 3D tissue constructs is essential when producing a phenotypically appropriate tissue [8]. The exact local conditions experienced by the cells must be understood, yet in many cases the culture systems and bioreactors used for 3D tissue culture have not been optimized in this respect. Several parameters, such as perfusion rate, flow conditions, shear stress, and compression magnitude, have been varied, quite often by using a trial-and-error approach. Furthermore, different conditions must be examined accurately with regard to their effect. For example, hydrostatic pressure applied during cartilage culture can lead to an improved mass transfer of small and large molecules into the cartilage matrix, but can also induce a mechanical stimulation of embedded cells.

As an example, mass transfer effects within a bioreactor designed for the cultivation of artificial blood vessels will be discussed in order to provide a deeper

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understanding of the physiological situation of the cells, especially with regards to oxygen supply within the reactor system. Oxygen can be supplied to the cells within the vessel matrix only by diffusion, as the vessel wall is not vascularized. As a direct measurement of oxygen concentration within the vessel matrix is not possible, the only way to obtain a better understanding ofthe oxygen concentration profile within the vessel matrix is to develop a mathematical model considering the mass transfer limitations on both sides of the vessel, as well as within the vessel matrix. The model assumptions are described in Figure 2.5a, while Figure 2.5b shows the radial oxygen profile within the vessel wall for different cell numbers. This indicates that, over the radius of the vessel, the oxygen concentration decreases rapidly, depending on the number of immobilized cells.

For cell densities of approximately 4 X 107 mL-1, severe oxygen limitation must be expected, whereas for cell densities of 108 mL-1 (a tissue-like cell density) the penetration depth for oxygen is less than 100 ^m. As the thickness ofvessels used experimentally at present is approximately 1 mm, an appropriate cell density with sufficient oxygen supply is about 107 mL-1 of the vessel matrix. Further detailed simulations showed that the flow rate had no significant effect on the oxygen profile within the vessel wall, as the main mass transfer resistance is not in the boundary layer medium/vessel wall, but rather is within the vessel wall. Other parameters (e.g., vessel thickness, vessel inner radius and/or length) were also of minor importance. It can be concluded from these results that the number of cells within the vessel matrix and/or the thickness of the vessel matrix, which is not perfused, are strongly limited. These findings agree well with those of investigations into the cell penetration depth in macroporous carriers (using a NMR technique), and were in the range of 100-200 ^m [36, 85].

To prove further how much oxygen is supplied from the outer medium (which is not flowing), a corresponding model was formulated. The vessel wall was described by a plate geometry, with the further assumption of one-dimensional diffusion without reaction. From Figure 2.5c it can be concluded that the oxygen concentration in the outer vicinity of the vessel decreases very rapidly, so that the oxygen contribution from this side may be neglected.

This small example underlines the importance of theoretical considerations regarding mass transfer effects in 3D tissue cultures. Therefore, experimental studies should always be supported by simulation methods such as computational fluid dynamics (CFD), or the finite-element approach. Several examples underline the potential of an integrated study of mechanical and biomechanical factors that control the functional development of tissue-engineered constructs [86-92], and this approach will undoubtedly significantly improve bioreactor design in the near future.

Fig. 2.5 (legend see p. 65)

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