Modeling Multiscale Necrotic and Calcified Tissue Biomechanics in Cancer Patients: Application to Ductal Carcinoma In Situ (DCIS) - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 351

The tissue moved with velocity u in response to forces generated by cell prolifer-
ation and death. Under the incompressibility and constant cell density assumptions,
the local rate of volume change is given by r u. In dimensionless form,

r u ¼ 0 x 2 X H
r u ¼ G r ð Þ x 2 X V
r u ¼ GG N

ð 7 Þ

x 2 X N ;

where G, A, and G N are dimensionless parameters characterizing the rates of cell
proliferation, apoptosis, and necrotic tissue volume loss relative to the time scale
k R . See [ 60 ] for greater detail on the nondimensionalization and these parameters.
We introduced a dimensionless proliferation-generated mechanical pressure p
as a simplified model of tissue stress, and assumed a Darcy flow response: cells
can respond to the pressure by overcoming cell-cell and cell-ECM adhesion and
moving through the porous medium (the ECM) supporting the cells. Moreover, the
ECM itself can deform in response to p. Hence, u ¼ l r p, where l is the tissue
mobility (its ability to respond to pressure gradients). Assuming constant cell-cell
adhesive forces and cell density throughout X V , cell-cell adhesion can be modeled
as a surface tension proportional to the curvature j along R ð t Þ . Thus, as in [ 18 ],

r l H r p ð Þ 0 x 2 X H

r l T rð Þ¼ G ð r A Þ

x 2 X V
x 2 X N

ð 8 Þ

GG N

subject to boundary and matching conditions

½ R ¼ jl r p n ½ R ¼ 0

p ð x Þ o X H [ X

ð 9 Þ

Þ ¼ 0 :

ð

In [ 60 ], l T ¼ 1 as result of nondimensionalization.
We implicitly tracked the moving boundary position using the level set method:
an auxiliary distance function / satisfies /\0inX, / [ 0inX H , / ¼ 0onR, the
outward normal vector is given by n ¼r /, and j ¼r n. The outward normal
velocity of R ð t Þ is obtained by evaluating u n ¼ lim X 3 y ! x l T r p ð y Þ n for any
x 2 R. The motion of R then becomes an advection equation for / [ 51 , 58 - 62 ].
We solved Eqs. ( 3 - 4 ) and ( 8 - 9 ) using a second-order accurate ghost fluid method
[ 51 , 58 - 62 ]. We let D ¼ D H = D T and l ¼ l H = l T denote the relative oxygenation
and mechanical compliance of the surrounding host tissue, respectively.

3.1 Impact of Necrotic Core Biomechanics: Key Results

As in earlier tumor spheroid models [ 12 , 89 , 90 ] and early non-symmetric necrotic
tumor simulations in [ 93 ], our theoretical and numerical analyses [ 51 ] showed that
even with A ¼ 0, volume creation in the proliferative rim could balance with