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300 15. COMPUTATIONAL SIMULATION OF CELL BEHAVIOR FOR TISSUE REGENERATION

undergoes maturation and growth. Once cell maturation is achieved, based on the strength of the mechanical signal
sensed by the cell, it enters into the M phase and forms two nonmature daughter cells. Consequently, in the present
model, each cell is in the quiescence phase unless it delivers two daughter cells. In other words, a cell is either under
maturation or in the proliferation stage. This process can be represented by

8 prof
1 Mother cell ! 2 Daughter cells γ γ
i
<
and MI ¼ 1 (15.38)
Cell proliferation ¼
:
No cell division Otherwise
where i 2{m, s, c, l} and γ prof < γ is the mechanical signal defining the proliferation limit of the ith cell [76]. After divi-
i u
sion of a mother cell, the position of two daughter cells is assumed as
x ð1Þ
daut ¼ x moth (15.39)
x ð2Þ ¼ x moth +2re rand
daut
where “moth” and “daut” subscripts represent mother and daughter cells, respectively.

15.3 NUMERICAL IMPLEMENTATION AND APPLICATIONS

The described model is implemented in the commercial software ABAQUS coupled with a user-defined element
subroutine (UEL) [142]. The parameters of the model are listed in Table 15.1. An ECM is mesh using 16,000 3D
hexahedral elements and with 18,081 nodes. The initial cell radius is assumed to be 15 μm with a total number of
24 nodes on its surface. In the following sections, we will illustrate some interesting applications of the model.

15.3.1 Effect of ECM Depth on Cell Mechanosensing and Migration

In this application, we study the cell migration due to the effect of the ECM depth when a sloped surface is con-
strained while the rest of the boundaries are free, as shown in Fig. 15.8. The elastic modulus of the ECM is assumed to
be 100 kPa and the cell is initially located at the maximum depth of the ECM. The results show that, as shown in
Fig. 15.8A, the cell tends to migrate on the ECM surface in a random directional trajectory toward locations with lower
depth because, during the cell mechanosensing process, the cell senses less internal deformation in the lower depth
direction, which is more rigid due to a constrained sloped surface [24, 100]. Fig. 15.8B shows the deformations gen-
erated in the ECM due to the sensing forces.

15.3.2 Cell Behavior Within a Multisignaling ECM

To consider the effect of different signals on cell behavior, we assume that there is a stiffness gradient in the
x-direction within the ECM (from 10 kPa at x ¼ 0 to 100 kPa at x ¼ 800 in Fig. 15.9). Afterward, we add other cues
such as thermal gradient (T ¼ 35°Cat x ¼ 0 and T ¼ 39°Cat x ¼ 800μm), chemical gradient (C ¼ 10 4 Mat x ¼
800), and dcEF (E ¼ 100mV/mm with cathode located at x ¼ 800μm) in the ECM to evaluate the effect of different
stimuli on the cell behavior compared with pure mechanotaxis. Next, two cases of multicell migration and cellular
morphology changes will be studied.

15.3.2.1 Multicell Migration Within a Multisignaling ECM
Experimental observations show that cell populations migrate directionally toward stiffer regions of the ECM in the
presence of a stiffness gradient (mechanotaxis) [24, 31], toward warmer sites in the presence of a thermal gradient
(thermotaxis) [146, 147], toward higher concentrations of nutrients when there is a chemotactic attraction [148],
and/or toward the cathode or anode (depending on cell type) in the presence of dcEF (electrotaxis) [47, 48, 50,
122–124].
To apply the present model, initially the cells are distributed in the first quarter of the ECM. The results show that,
first, the cells tend to migrate toward each other, even when they are located in the stiffer regions, stimulated by the
stretched regions between cells. However, the final destination of the cells is dominated by the migration along the
existent gradients or toward the cathode, regardless of the primary distribution of the cells (see Fig. 15.9). In the case
of pure stiffness gradient (Fig. 15.9A), despite the maximum stiffness being located at the end of the ECM, the cells do

II. MECHANOBIOLOGY AND TISSUE REGENERATION