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 352

Fig. 5 Left: Growth of a necrotic avascular tumor into well-perfused, mechanically-stiff tissue.
The invasive ''fingers'' and necrotic regions acquire relatively fixed, characteristic sizes. Right:
Impact of the rate of cell proliferation (G) and necrotic volume loss (G N ) on invasive fingering
growth. G acts primarily as a time scale (tumor morphologies are the same but evolve more
quickly with increased G), whereas larger values of G N can destabilize the morphology (seen here
as changing rounded protrusions into invasive fingers). Legend: Viable (gray) and necrotic tissue
(black) grow in host tissue (white). Figures adapted with permission from [ 60 ]

volume loss in the necrotic core, leading tumor spheroids to grow to a steady size.
Our work had an additional insight: even during growth (and overall morpho-
logical instability), tumor proliferation and necrotic volume loss could equilibrate
locally, leading to (1) a near-constant necrotic volume fraction, and (2) the
emergence of characteristic feature sizes and shapes. For example, a tumor
growing into well-perfused (D [ 1), mechanically-stiff (l 1) tissue develops
invasive fingers with a characteristic width. See Fig. 5 (left) for such an example.
The qualitative tumor behavior (classified as fragmenting, fingering, or hollow/
compact growth) was primarily dependent upon the microenvironmental param-
eters D and l. However, the quantitative behavior—viable rim thick-ness, necrotic
volume fraction, overall growth rate, etc.—was strongly dependent upon tumor
cell characteristics, particularly the necrosis parameters r N and G N . The viable rim
size was determined by the balance of nutrient penetration into the host tissue (D),
apoptosis (A), and the tumor cells' resistance to hypoxia (r N ). The size of the
necrotic core was primarily determined by the rate of volume loss in necrotic
tissue (G N ). See Fig. 5 (right), where we show how the tumor varied with G N for
several values of G. A key finding was that while moderate rates of necrotic
volume loss indeed contribute to the emergence of a steady state size for the
spherical case, fast necrotic volume loss (large G N ) can destabilize the tumor
morphology.
This work revealed a few outstanding problems with continuum necrosis
models of the time. First, defining the necrotic region implicitly through r as in
Eq. ( 6 )

could cause unexpected behavior for complex tumor morphologies.