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 349

Fig. 4 Cell volume
composition during later
necrosis: Viewed on the long
time scale of necrosis and
calcification, early cell
swelling and lysis are a
fast perturbation on the
longer-term trends of solid
content calcification and
degradation. Figure provided
courtesy of [ 64 ]

Cell volumes throughout necrosis

100

calcification
cytoplasmic solids
cytoplasmic fluids
nuclear fluids
nuclear solids

80

60

40

20

0

0 10 20 30 40 50 60

time (days)

nucleate and grow calcium phosphate crystals [ 47 ]. See Fig. 3 (top: e), the
example in Fig. 1 (right), and Fig. 4 . How this process transpires in vivo is still
poorly understood.

2.2.2 Estimates of Necrosis and Calcification Time Scales

In [ 55 - 57 ], Macklin et al. estimated the various time scales of necrosis and cal-
cification, many of which have not been experimentally measured for carcinoma.
We estimated initial cell swelling and lysis to occur on the order of 2-6 h. Based
upon experimental reports on aortic calcification [ 34 , 42 , 48 ] and our previous
computations [ 55-57 ], we estimate calcification to take on the order of 15-20
days [ 56 ]. Based upon our insights from [ 56 ] (See Sect. 4.3 ) and the existence of
necrotic tissue with intense eosin staining (a sign of cytoplasm with significant
water loss and little calcification) and compact, partly-degraded nuclei, we esti-
mate that water loss occurs more quickly than pyknosis, and that pyknosis is a
faster process than calcification. We have recently hypothesized and found good
evidence that calcifications degrade at a very long time scale (on the order of two
to three months) [ 56 ]. See Sect. 4.5 . Thus, necrosis and calcification have pro-
cesses that operate on time scales ranging from hours to months.

3 Early Continuum Modeling Results: Impact of Necrotic
Core Mechanics on Tumor Progression, Morphology,
and Stability

Following earlier tumor growth models that included necrotic cores [ 12 , 89 , 90 ] and
an earlier non-necrotic free boundary formulation of tumor growth [ 18 ], Macklin and
Lowengrub developed a model of non-symmetric avascular tumor growth in het-
erogeneous tissues which included necrosis [ 51 , 58 - 62 ]. We modeled the tumor as an