Biomedical Engineering Reference
In-Depth Information
4.2. Search Precision
In (7), we introduced
noise
into cellular signal reception, such that the ex-
tent of the noise in the signal is captured by the search precision parameter, de-
noted as
4
, which is positive and between zero and one,
4
[0,1]. Here, the
search precision
4
represents the likelihood with which tumor cells evaluate the
attractiveness of a location without error. As Eq. [2] specifies, the error-free
value of a location is jointly influenced by the onsite levels of nutrient, mechani-
cal confinement, and toxicity. Formally, let
T
j
be the attractiveness of location
j
as evaluated by a tumor cell using its signal receptors, and let
L
j
be the error-free
evaluation of location
j
. The extent of search precision is introduced to the
model such that, due to the noise in the signals, the attractiveness of location
j
is
evaluated without error only
4
proportion of the time:
T
j
=
4
#
L
j
+ (1 -
4
)#F
j
,
[9]
where F
j
~
N
(N,T
2
) is an error term that is normally distributed with mean N and
variance T
2
. As a concrete example, a 70% search precision implies that Prob [
T
j
=
L
j
] = 0.7, i.e., the attractiveness of location
j
is evaluated without error in
seven out of ten trials on average. At one extreme,
4
= 1 represents the case
where tumor cells consistently evaluate the permissibility of a location without
error. At the other extreme,
4
= 0 represents the case when tumor cells always
perform a
random-walk
motion, thus completely ignoring the guidance of the
gradients of environmental variables.
4.3. Structure-Function Relationship
In (8), we investigated the emerging structural patterns of a multicellular
tumor as represented by the
fractal dimensions
of the tumor surface. We then
examine the link between the tumor's fractal dimensions and the cancer system's
dynamic performance (i.e., functionality) as captured by its average velocity of
spatial expansion. The tumor's fractal dimension, which characterizes the irregu-
larities of the tumor surface, is determined using the box-counting method (45).
This method quantitatively measures the extent of surface roughening at the
tumor-stromal border due to both proliferation and migration of malignant tu-
mor cells. The choice of fractal dimensions as a measure of structural pattern is
motivated by the idea that the morphology of a tumor surface depends on the
scale of observation. Let
SA
(
l
) be the entire surface area of the tumor that is
computed by counting the number of boxes,
N
(
l
), each of size
s
, which are
needed to cover the entire area. Then it is the case that
SA
(
l
) =
N
(
l
) #
l
2
. If the
dimension of the tumor surface is indeed fractal, then we will find that:
