Different Morphologies of Tumor Invasion Fronts - Cellular Potts Models: Multiscale Extensions and Biological Applications - page 155

sis of external cells is only able to balance the loss of volume of the interior
necrotic region. This limits the size of the tumor spheroid, which reaches a
well-characterized steady state [88, 316]. Indeed, in order to grow further, the
cancer mass will have to coopt the existing vasculature, or to stimulate the
formation of new capillaries. .
The above-described diffusional limitations of growth can be overcome by
the carcinoma with morphological instabilities which, as a result of many
factors, including the downregulation of intercellular adhesion, specific me-
chanical stresses, or the enhancement in cell motility, increment the surface
area of its interface with the host, thereby allowing internal individuals to
have greater access to vital molecules.

8.9 Mathematical Model
The previous model is indeed generalized to three-dimensions, focusing on the
growth of a multicellular spheroid and on the detachment of invading cells from
it. In this case, the cancer cells are uncompartmentalized standard individuals
of type = C. This choice is done to avoid unnecessary computational
costs given the fact that intracellular units do not play a fundamental role
in the early stage of spheroid tumor growth. This means that most of the
parameters previously used are formally transferred to this model dropping
the subscript or substituting it with , e.g., in Equation (8.3) and in
the similar ones s ; becomes simply s , as now the internal state vector
characterizes the entire cell.
With respect to the two-dimensional case treated in the previous section,
an essential feature of three-dimensional growth is volume change because
of growth and death. Growth was already introduced in Section 8.8, where a
probability of undergoing mitosis (now P ) was introduced in Equation (8.9).
Necrosis is here introduced as a consequence of lack of suitable growth factors.
In fact, in the presence of an adequate intracellular level of growth factors,
malignant cells keep almost the same volume during tumor growth: therefore,
for any , A volume = A volume

C

is their initial volume and volume

= volume

C
is a high constant value. On the contrary, when the internal level of growth
factors drops below a certain threshold, here defined as nl, l , a cell enters in
a irreversible necrotic state, and starts to lose volume at a constant rate.
This process, the model counterpart of the biological lysing of death cells, is
modelled by setting:

dA volume
C
dt

= k (8.10)

if n( ;t) < n l . Zero volume cells are finally removed.
Moreover, given that A surface

= A surface
C

is the initial cell surface, because