Biomedical Engineering Reference
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towards the tumor leading edge. Hence, necrotic cell lysis acts as a mechanical stress
relief, analogously to the mechanical pressure sink terms used in [
58
-
61
].
This can be further confirmed by altering the necrosis model. In [
55
,
57
], we used a
more gradual model of necrotic cell volume loss, where fluid ''leakage'' was spread
over s
C
¼
15 days. The tumor advance accelerated as the viable rim grew, consistent
with exponential growth. In those simulations, the rate of biomechanical stress relief
in the necrotic core was too slow, causing more of the proliferative cell flux to be
directed along the duct, preventing sustained linear growth. When we set s
NL
¼
s
C
¼
15 days, we observed accelerating, exponential-like growth (blue curve after
initial transient dynamics) [
56
]. See Fig.
10
(left). Generally, we found that all
simulations exhibited exponential-like growth for approximately s
NL
time after the
first instance of necrosis. For sufficiently small s
NL
(under 1 day), the brief expo-
nential growth phase could not be detected. This mechanism suggested to us that
because the lumen/necrotic core acts as a ''reservoir'' of mechanical stress relief to
absorb proliferative cell flux, DCIS growth should be fastest in small ducts, and
slowest in larger ducts. In simulations, we found this to be supported [
56
]. See Fig.
10
(right). We found an inverse relationship between duct radius R
duct
and the DCIS
growth rate x
0
V
(the red curve in Fig.
10
(right)):
x
0
V
20
:
52
þ
e
6
:
085
0
:
02584R
duct
lm
=
day
:
ð
14
Þ
Notice that as R
duct
!1
, we find a minimum growth rate of 7.5 mm/year, or a
mammography growth rate (by Eq.
13
) of 6.7 mm/year. Cases with slower growth
would need to be attributed to reduced oxygen or altered cell signaling.
4.3 Proliferative Cell Flux and Multiscale Necrosis Lead
to a Stratified, Age-Structured Necrotic Core
Thus far, we have focused upon the gross macroscopic behavior of DCIS: the
emergent growth rate and the relationship between mammography and pathology.
We now turn our attention to the finer microstructure of the tumor. In Fig.
11
(top),
we highlight several characteristic cross-sections of our DCIS simulation at 45 days.
In Slice a, there is a viable rim of thickness comparable to the remainder of the
tumor, but with little visible evidence of necrosis. Biologically, this section of the
tumor is no different than portions with necrosis (i.e., hypoxia is significant). This
raises the possibility that in cases where too few ducts are sampled, a pathologist
may fail to observe comedonecrosis, potentially (and incorrectly) changing the
patient's Van Nuys Prognostic Index score [
84
] and affecting treatment decisions.
This could be particularly true in cases where
h
PI
i=
s
P
h
AI
i=
s
A
, as little net cell
flux from the viable rim to the necrotic core would be expected [
56
].
Farther from the tumor leading edge in Slice b, a ring of necrotic debris surrounds a
hollow duct lumen. In cross sections like this, there has not yet been sufficient tumor
cell flux from the viable rim to completely fill the lumen with necrotic debris. Farther
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