Biomedical Engineering Reference
In-Depth Information
assuming the cells grow in response to the nutrient field, the growth rate of the
construct was determined.
The theoretical study led to the important biological insight that if the construct
position is stationary when viewed from the laboratory frame, then a nutrient
depletion zone will form in the neighbourhood of the construct. Thus it is more
desirable to operate the bioreactor in a regime in which the construct undergoes a
time-dependent trajectory when viewed from the laboratory frame, mixing the
nutrient concentration as it goes, and eliminating the development of nutrient
depletion zones.
Still motivated by tissue growth in rotating bioreactors, Waters et al. ( 2006 )
considered a two-region, single-phase model to determine the morphology of a
tissue construct formed from a single-cell suspension in culture media, within
a rotating bioreactor. Experimental results indicated that at rotation rates below a
critical value, the cells 'self-assemble' to form smooth nodules which are
approximately elliptical in cross-section. However, at higher rotation rates, an
amorphous construct forms with a highly irregular boundary. Histological studies
show that the construct consists of a fluid interior which is a mix of apoptotic cells
and culture media, surrounded by a membrane of proliferating cells and collagen.
Motivated by these observations, Waters et al. ( 2006 ) consider a more dense
viscous fluid drop surrounded by an extensible membrane of constant membrane
tension in a less dense immiscible viscous fluid within a rotating bioreactor sys-
tem. Both thin-disc and slender-pipe bioreactors (for which the bioreactor aspect
ratio is small or large respectively) were considered so that, similarly to Cummings
and Waters ( 2006 ), the fluid-dynamical equations may be simplified leading to a
series of spatially 2D problems. The authors consider the interfacial stability of the
initially circular fluid-fluid interface to small-amplitude, oscillatory perturbations;
the instability arises as a result of the competition between the destabilising effects
of centrifugal forces, and the stabilising effects of surface tension. The theoretical
results indicate that culture within thin-disc bioreactors is more likely to result in
irregular shape constructs than culture within slender-pipe bioreactors, and that in
the thin-disc regime the wave number of the most unstable mode increased as the
rotation rate increases, in line with experimental observations. Of course, the
instability mechanism examined in Waters et al. ( 2006 ) is not the only one by
which irregular constructs might be expected to arise, and the modelling high-
lighted here does not include any biochemical features, e.g. growth in response to
nutrient fields, or cell proliferation, in the model.
In Cummings and Waters ( 2006 ) bioactive processes that modify the porous
scaffold are neglected—rather the construct is modelled as an impermeable solid
object. However, a number of studies have explicitly accounted for the properties
of the porous scaffold and their effect on the fluid flow and nutrient transport. In
Whittaker et al. ( 2009 ) a simple mathematical model is developed for forced flow
of culture medium through a porous scaffold. Flow is forced through the scaffold
via inlet and outlet pipes, and also through identical porous-walled fibres inserted
through the scaffold: fluid is pumped into one end of each fibre and the other end is
sealed so that the fluid is forced to travel through the porous fibre walls into the
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