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Encyclopedia of Physical Science and Technology EN016J-783 August 1, 2001 10:58

Tissue Engineering 831

FIGURE 10 Definition of parameters that characterize single cell migration.

It is important to note that these parameters are not de- netic” or “haptokinetic” effect, the last term in Eq. (7)
pendent on the geometry of the system used to measure includes a correction for this effect.
them and thus can be used to predict cell migration in other The values of D and χ are the constitutive parameters
geometries.Thismodelchieflyappliestotwo-dimensional describing cell migration in a variety of both in vitro and
surfaces; however, it can be extended to three-dimensional in vivo systems. The expression analogous to Fick’s first
matrices, in which case the effective pore size of the ma- law of diffusion for cell flux is
trix, which may create a hindrance to the migration pro-
∂C dD ∂L
cess, needs to be taken into account. Cell migration in J =−D + C − + χ (8)
∂x 2dL ∂x
a specific direction can be promoted by micropatterning
tracks on a surface, which prevents cells from wander- and a cell concentration profile in any system can be de-
ing away from the desired direction, and in three dimen- rived via the continuity equation:
sions by using materials exhibiting oriented pores and/or
∂C ∂ J
fibers. =− (9)
The above discussion relates to the process of random ∂t ∂x
migration where cells do not move in a preferential di- In real cases, Eq. (9) may need to be solved in con-
rection. It is often the case, however, that soluble and junction with appropariate transport equations for the
insoluble factors causing cells to move in a preferential chemoattractant or haptotactic factor, which may be time
direction are present. Soluble agents that “attract” cells varying.
are called chemotactic, while those immobilized in the As briefly discussed earlier, cells migrating on sub-
extracellular matrix are called haptotactic. In chemotactic strates exert forces that allow them to move. As a result,
or haptotactic migration, a third parameter must be deter- cells on surfaces or inside gels that are compliant can
mined to capture the directional preference of the migra- significantly alter the shape of the material. Quantita-
tion process. This parameter is the chemotactic index (CI), tive analyses and mathematical descriptions of these
which can be determined experimentally from single cell phenomena allow prediction of how the cell-material con-
trajectories by the equation: struct changes shape over time. A well-known example
of cell-mediated contraction is the fibroblast-populated
d
CI = (6) collagen lattice, which forms the basis for some of the
L path
currently used tissue engineered skin grafts. The
where d is the distance of the cell from the point of origin fibroblast-populated collagen lattice is generated by
at the beginning of the experiment, and L path is the length mixing fibroblasts with a chilled solution of collagen
of the path used by the cell to achieve the displacement d . in physiological buffer, followed by exposure to 37 C
◦
The population-relevant parameter that describes chemo- to induce the gellation of the collagen. If the gel is
taxis is the chemotaxis coefficient χ, which is calculated not anchored to any surface, fibroblasts embedded
by the following expression: in a collagen gel cause the contraction of the gel in
an isotropic fashion. The contraction process can be

S · CI 1 dlnP dlnS
χ = − − (7) controlled to a certain extent by mechanically restricting
∇L n dL dL
the motion along certain directions, which also induces a
where L and ∇L are the concentration and spatial gradi- preferential alignment of the collagen fibers as well as the
ent, respectively, of chemoattractant or haptotactic factor. cells within it, which results in a nonisotropic connective
Because chemotactic and haptotactic factors may increase tissue equivalent. Preferential alignment of cells may
cell speed, and thus increase migration via a “chemoki- be important in specific applications, such as in tissue