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294 15. COMPUTATIONAL SIMULATION OF CELL BEHAVIOR FOR TISSUE REGENERATION

corresponding deformation and consequently the nodal traction forces are not zero due to deformation compatibility.
This means that all the cell nodes, including the nodes that are in contact (n 1 , n 2 , n 3 , and n 4 in Fig. 15.5), deform together
[100, 105].

15.2.1.2 Protrusion Force
During cell migration, cells send out local protrusions to probe their environment by exerting a protrusion force.
This force is generated by actin polymerization and must be distinguished from the cytoskeletal contractile force [27]. It
arises from cell-matrix attachments at the new sites of lamellipodia and filopodia development, which have a stochas-
tic nature during cell migration [106]. This causes cells to move along a directed random path toward the effective
signals. The direction and magnitude of the protrusion force are chosen randomly at each time step. It is remarkable
that the order of the protrusion force magnitude is the same as that of the traction force, but with lower amplitude [27,
100, 107–109]. Therefore, we randomly estimate it as
F prot ¼ κF trac e rand (15.8)
net
trac
where e rand represents a random unit vector and F net is the modulus of the net traction force while 0 κ < 1 is a ran-
dom number [100].

15.2.1.3 Drag Force
By contrast, the drag force comes from the viscous resistance to the cell movement. In a Maxwell solid, the needed
force for deforming the ECM depends on the deformation rate and, accordingly, the velocity. The main objective here is
to imply a velocity-dependent opposing force to associate the viscoelastic character of the cell surrounding the ECM.
To define a velocity-dependent opposing force associated with the linear viscoelastic character of the ECM surround-
ing the cell, we have assumed a simplification that the ECM is a viscoelastic medium [27]. At a microscale, the viscous
resistance dominates while the inertial resistance of a viscous fluid is small enough to be ignored. Stokes [110]
described the drag force of a sphere at the limit of negligible convection as
s
F ¼ βv (15.9)
D
where v is the relative velocity and β is often referred to as the Stokes’ drag regime for a small spherical object moving
slowly through a viscous fluid expressed as
(15.10)
β ¼ 6πrηðE sub Þ
where r is the object radius and η(E sub ) is the effective medium viscosity. In an ECM with a stiffness gradient, we
assume that it is linearly proportional to the ECM stiffness, E sub , at each point. Therefore, it can be calculated as
ηðE sub Þ¼ η min + λE sub (15.11)
where η min is the minimum viscosity of the ECM corresponding to minimum stiffness and λ is the proportionality coef-
ficient. The viscosity coefficient may eventually be saturated with ECM rigidity; however, this saturation occurs out-
side the ECM rigidity range suitable for the cell phenotypes considered here [111].
The drag of nonspherical solid particles will depend on the degree of nonsphericity as well as their orientation to the
flow (the drag will generally be anisotropic with respect to direction). When cell morphology changes during migra-
tion, Eq. (15.9) will not deliver an accurate representation of the drag force. Its definition in the case of irregular par-
ticles is further complicated by the randomness of the shapes and dynamics. However, a review of experimental
studies of the mean drag of irregularly shaped particles suggests that it is reasonable and appropriate to use a shape
factor, f shape , to moderate the Stokes expression as [112, 113]

F drag ¼ f shape F s (15.12)
D
Nevertheless, it is expected that only approximate and probabilistic predictions are possible for highly irregular par-
ticles. A wide variety of shape-characterizing parameters has been suggested for irregular particles, the most common
and most successful being the Corey Shape Factor (CSF). This shape factor employs three lengths of a particle in mutu-
ally perpendicular directions, being representative of cell surface area changes [112]; the cell’s longest dimension, l max ,
the shortest dimension, l min , and the intermediate or medium dimension, l med . Herein, we take advantage of a
convenient version of CSF used by Loth [112] to estimate the shape factor as

II. MECHANOBIOLOGY AND TISSUE REGENERATION