ARUP; ISBN: 978-0-9562121-5-3 - CMBBE 2012 - Cardiff University

3 MATERIALS AND METHODS
3.1 Model structure
We implemented an individual-based model (IBM) for cell expansion in the software DEMeter++
[5]. DEMeter++ offers a generic platform for a wide range of particle-based simulations, such as
the discrete element method. Cells are considered as particles that interact by physical forces. At
each timestep the forces are integrated for each particle to update the cell positions.
3.1.1 Equation of motion
Central to the individual-based model is the equation of motion. Because cells live in a low
Reynolds number environment their inertia can be neglected. Therefore we obtain an equation of
motion for each cell that only needs to be solved for the velocity, similar to [4]:
Fi ∑ j + Fi,migration + Fi,division + Fi,Brownian = Γiwvi + ∑ Γi j(vi −v j). (1)
jnni
jnni
At the left hand side of equation 1 we recognize the mechanical forces between next neighbouring
cells jnni, the biological forces of migration and cell division and a force representing Brownian
motion. The viscous terms are at the left hand side of equation 1: the viscous drag force due to
suspension in liquid, Γiwvi, and the cell-cell friction forces between contacting cells jnni.
We use the conjugate gradient method for providing a coupled solution for the velocities. Next, a
forward Euler scheme is used to update the cell positions.
3.1.2 Contact Mechanics
We model the contact mechanics of cells by the Johnson-Kendall-Roberts potential (JKR, [9]).
The JKR potential is considered an appropriate model for describing the elastic and adhesive
forces between contacting cells with an approximately spherical shape. The JKR potential also
incorporates hysteresis: adhesive forces are bigger when cells are pulled away out of contact than
when pushed together into contact, because of already formed molecular bonds (due to adhesion
molecules). In reality, a low number of progenitor cells that attach on a substrate will flatten out
strongly. However, as soon as multiple cell layers emerge and a three-dimensional aggregate is
formed, the cell shape will again start becoming more spherical. An expression for the JKR force
can be derived as:
in which
Ê =
1 − ν 2 i
Ei
Fi j = 4Ê
3 ˆR a3
− 8πσÊa3 , (2)
+ 1 − ν2 j
E j
−1
and ˆR
1
= +
Ri
1
−1 . (3)
R j
Ei and E j are the Young’s moduli, νi and ν j the Poisson numbers, and Ri and R j are the radii of
cells i and j. Finally, a in equation 2 can be obtained iteratively (from [13]):
δ = a2
ˆR −
2πσ
a. (4)
Ê