Fisher John P. e.a. (ed.) Tissue Engineering
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6-10 Tissue Engineering
state) [147]. The reconstructed cell trajectories can then be modeled as Markov chains [146,149–151]
and the following cell locomotory parameters can be extracted from the cell trajectory data using the
well-known Markov chain theory [152] (a) transition-state probabilities that quantify the frequency with
which the cells (individually or collectively) move from one state to another and, thus, characterize the
turning and stopping behavior of the cells; (b) waiting times or the average time cells spend in a certain
state (i.e., moving in a certain direction); and (c) steady-state probabilities that provide a measure of the
directionality of cell movement (chemotaxis). The average speed of locomotion can also be estimated from
the reconstructed cell trajectories using the formula:
ν =
N
i=1
d
i
N ·t (6.5)
where d
i
is the displacement of the centroid of a cell between two successive observations, and N is the
numbers of observations made at a fixed time interval t .
The Markov chain model can be used to analyze chemotaxis experiments. If there is a preferred direction
for cell movement, its steady-state probability will be significantly higher than the steady-state probabilities
of the other directions. The Markov chain approach can also provide a more detailed description of cell
migration since it accounts for stops in cell movement and uses more than one descriptor (e.g., transition
state probabilities and average waiting times) to define persistence [148].
A detailed comparison of the random walk and the Markov chain models using experimental data for
migrating endothelial cells has shown [148] that all five locomotory parameters defined above affect the
speed S and persistence time P estimated by the persistent random walk model. The persistent random
walk model can provide a measure of the speed of locomotion S, appropriately weighted to account for
the frequency and duration of cell stops (or slow-downs). The persistence time P, however, is a composite
measure of all five locomotory parameters mentioned above. Hence, it may not be always possible to
extract all the information necessary to describe cell locomotion from the two parameters of the persistent
random walk model. The Markov chain approach, however, also has its drawbacks. Perhaps the most
serious drawback is that the values of cell speeds computed according to Equation 6.5 depend on the time
interval t between cell observations. Accurate estimations of locomotion speeds with Equation 6.5 are
possible only when (a) cell trajectories are not very tortuous and (b) the cell positions are observed at
short time intervals t . This time interval must be carefully chosen using Richardson plots to minimize
errors in the computation of cell speed [148].
6.5 Mathematical Models for Cell Migration and Tissue
Growth
Cell migration influences both the structure and the growth rate of bioartificial tissues built on biomimetic
scaffolds. As we have seen in the previous sections, however, cell migration is a process modulated by a
multitude of system parameters that include the local scaffold properties and nutrient or growth factor
concentrations. Despite the complexity of this process, however, a purely empirical approach is still used
to design scaffolds and select bioreactors appropriate for each application. Noting the limitations of this
approach, Ratcliffe and Niklason [153] proposed an interdisciplinary effort to improve our fundamental
understanding of the dynamic processes characterizing tissue growth and bioreactor operation. This goal
can be achieved with the help of comprehensive computer models that allow for systematic analysis of
tissue regeneration processes. Simulations can shorten the development stage by allowing tissue engineers
to rapidly screen many alternatives on the computer and choose the most promising ones for laboratory
experimentation.
The key challenge here involves the development of computationally efficient algorithms to describe
the dynamics of large cell populations that evolve in a continuously changing environment. Early studies
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Cell Migration 6-11
0
4
8
12
16
0
Normalized number of cells
Time, days
30 ng/ml bFGF
No bFGF
2468
FIGURE 6.2 Normalized cell counts from experiments (symbols) and simulations (lines) for bovine pulmonary
artery endothelial (BPAE) cells seeded uniformly on tissue culture plates and cultures without and with 30 ng/ml of
bFGF. Standard errors are used as error bars. (Adapted from Lee, Y., Mcintire, L.V., and Zygourakis, K., Biochem. Cell
Biol., 1995, 69: 1284–1298. With permission.)
utilized either discrete [154–156] or continuous models [157–161] to treat tissue growth problems. Using
an approach that combined (a) modeling based on cellular automata, (b) computer simulations, and
(c) experimentation (migration and proliferation assays), our group initially focused on the growth
of two-dimensional tissues like vascular endothelium or keratinocyte colonies [145,155,156]. Using
experimental data obtained from long-term tracking and analysis of cell locomotion [162], we developed
a class of discrete models that described the population dynamics of cells migrating and proliferating
on the surface of biomaterials. These models simulated persistent random walks on 2-D square lattices
and accurately accounted for cell–cell collisions. Figure 6.2 shows that model predictions agreed well
with experimentally measured proliferation rates of bovine pulmonary artery endothelial (BPAE) cells
cultured with and without bFGF, a growth factor that upregulates cell motility. Results also showed that
the seeding cell density, the population-average speed of locomotion, and the spatial distribution of the
seed cells are crucial parameters in determining the temporal evolution of cell proliferation rates. More
recently, models were developed to solve 2-D problems involving the aggregation and self-organization
of the cellular slime mold Dictyostelium discoideum [163–166] and to study the interactions between
extracellular matrix and fibroblasts [167]. To account for the effect of transport limitations that restrict
our ability to grow bioartificial tissues [168,169], our group has also developed a hybrid multi-scale model
that combines (a) partial differential equations modeling the simultaneous reaction–convection–diffusion
problems for nutrients and growth factors, (b) discrete models describing the cell-population dynamics,
and (c) single-cell models describing intracellular processes modulating cell function.
To demonstrate the potential of computational models for tissue engineering, we will briefly discuss the
application of a 3-D model we have developed [170] to study a problem in biomimetics. Our objective was
to evaluate the effectiveness of surface modifications in enhancing tissue-growth rates through increased
cell-migration speeds. Such modifications may involve, for example, the immobilization of signaling or
adhesive peptides on the surface of biomaterials [171–173]. To isolate the effect of population dynamics
on tissue growth, simulations on 3-D scaffolds were performed by assuming that nutrient and growth
factor concentrations remained constant throughout the tissue regeneration process. We considered cell-
migration speeds varying across the entire range of observed values (from 0 to 50 µm/h) and used two
different cell seeding modes. In the first mode, cells were seeded uniformly and randomly throughout the
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6-12 Tissue Engineering
0
0.2
0.4
0.6
0.8
1
01012
(a)
50 mm/h
5 mm/h
3 mm/h
0 mm/h
Fractional volume coverage
Time (days)
Cell speed
0
0.2
0.4
0.6
0.8
1
0 12162024
(b)
Fractional volume coverage
Time (days)
Cell speed
246
8
10 mm/h
50 mm/h
5 mm/h
3 mm/h
0 mm/h
48
FIGURE 6.3 Effect of cell-migration speed on tissue-growth rates for two different cell-seeding modes. All simula-
tions were carried out on a cubic domain with side equal to 1.9 mm (128 × 128 ×128 computational sites). (a) Cells
were seeded uniformly and randomly with initial density equal to 0.5%. (b) Seeding simulated wound healing. The
“wound” was a cylinder 0.95 mm in diameter and 1.9 mm in height. (Adapted from Belgacem, B.Y., Markenscoff, P.,
and Zygourakis, K. Proceedings of the 3rd Chemical Engineering Symposium, Athens, Greece, Vol. 2, pp. 1133–1136,
2001. With permission.)
scaffold. The second seeding mode was designed to simulate a “wound-healing” process. We assumed that
a cylindrical wound was created in the center of a 3-D tissue. The wound was then filled with a biomaterial
that allowed cells from the surrounding tissue to move into the wound and proliferate to heal it [170].
For uniform cell seeding, Figure 6.3a shows that increasing migration speeds initially enhanced tissue-
growth rates. As cell speeds increased above 5 µm/h, however, this beneficial effect diminished and
disappeared completely for large-migration speeds. For the wound-healing model, however, the simu-
lations predicted significant enhancements of tissue-growth rates with increasing cell-migration speeds
(Figure 6.3b). A careful analysis of the simulation results for these two cases revealed that the uniform
cell seeding resulted in many more cell–cell collisions that slowed down the migration of cells. These
results point out that the cell-population dynamics, the geometry of the problem, and the initial conditions
can have a profound effect on tissue regeneration rates. The simulations also provide us with invaluable
guidance for the design of experiments [174] that can test the efficacy of surface modifications designed
to enhance cell-migration speeds.
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