Page 294 - Advances in Biomechanics and Tissue Regeneration

292 15. COMPUTATIONAL SIMULATION OF CELL BEHAVIOR FOR TISSUE REGENERATION

15.2.1.1 Traction Force
Following Mousavi et al. [100], the one-dimensional (1D) model represented in Fig. 15.3B can be particularized to
calculate the cell internal stresses within 3D ECMs. Assuming that the contractile forces exerted by a cell are isotropic
[26], the change of the length of each element is then interpreted as its corresponding volumetric strain [101]. Therefore,
the effective stress transmitted by each cell node located on the cell-ECM interface can be defined by

σ cell ¼ σ act + σ pas (15.1)
i
i
i
act pas
where σ i and σ i are the contractile stresses transmitted through the actin bundles via the ith node on the cell-ECM
interface. The myosin II machinery generates the first internally while the second is produced by the passive resistance
of the cell, basically by the CSK microtubules and the membrane. In Fig. 15.3B, σ cell can also be interpreted as the aver-
age cell stress that bears the submembrane plaque in agreement with the integrin-mediated mechanosensing hypoth-
esis [102]. In addition, E denotes the local volumetric strain. In the model herein, the local strain is computed from the
deformation of the cell external nodes along the direction of the traction force exerted at the corresponding node. E act
stands for the deformation of the active contractile element. This deformation relates to the fact that the real physical
change of the overlap between the myosin and actin filaments occurs when active forces are applied. Finally, E a
represents the deformation of the actin bundles that encourages the transmitted active forces.
Simplifying the cell-ECM structure under moderate cell and ECM strains, we approximate the unidimensional con-
stitutive behavior of the cell by a simple linear-elastic spring [26]. Therefore, for the calculation of the contractile stress,
act
σ , as a function of the contractile element deformation, a simple piecewise linear constitutive model has been used
act
(see Fig. 15.3C). If E min E E max , the active stress, σ , affects cell total stress, σ cell , else it is 0 and σ cell is equal to σ pas .
The stress experienced by the passive contractile elements is directly proportional to the stiffness of the passive ele-
ments and cell deformation as
σ pas ¼ K pas E cell (15.2)

Therefore, by substitution of Eq. (15.2) in Eq. (15.1) and considering the stress curve of the active elements in Fig. 15.3C,
at each point of the cell surface, the cell stress transmitted to the ECM can be calculated as
K pas E E < E min or E
8 cell cell cell
> i i i > E max
>
> K act σ max ðE min E cell
>
K pas E E min E E
> cell i Þ cell
< i + i
σ cell K act E min σ max (15.3)
i ¼
> K act σ max ðE max E cell
K pas E E E
> cell i Þ cell
>
> + E max
> i i
K act E max σ max
:
where E¼ σ max =K act . The characteristic spring constant can be identified with the volumetric stiffness modulus of the

microtubules, K pas , myosin II, K act , and ECM, K subs .
Physically, during the cell movement, the contraction of the actin-myosin apparatus drives forward the transloca-
tion of the cell body and causes traction forces on the ECM [24, 99]. Actually, the movement of cells within a complex
embryo or organism is guided by a complex interplay between chemical and physical signals such as ECM stiffness
[24, 31], boundary conditions, and generated forces due to cell-cell and cell-ECM interactions [24, 99]. Anyway, the
simplified model described earlier is here employed to predict cell migration as a function of cell internal deformation.
The prominent aspect of the presented approach for cell modeling is that the cell can have any morphology, if there
is an interest to consider a variable morphology, and can be represented by any number of finite elements. For this
purpose, an algorithm has been used to track the key parameters required for cell migration at each time step, con-
sidering important processes for cell migration such as the asymmetry of the cell and traction force generation. More-
over, several aspects associated with the ECM such as stiffness, boundary conditions, and their effects on the direction
of cell movement can be considered.
It is considered that a cell first exerts sensing forces to sense its ECM. These forces act at each finite element node of
the membrane toward the cell centroid. The cell deformation due to these sensing forces is shown by the dashed lines in
Fig. 15.4. Therefore, the cell strain at each finite element node of the cell surface (membrane) in the direction of the
corresponding sensing force can be written as
E cell ¼ n i N i (15.4)
i
CN i
where N i is a point on the surface of the undeformed cell (solid line), n i is the same point on the surface of the deformed
cell (dashed line), and C is the cell center. The net traction force of a cell is the result of the local traction forces exerted by
the cell at its front and rear, which can be calculated as [27]

II. MECHANOBIOLOGY AND TISSUE REGENERATION