Biomedical Engineering Reference
In-Depth Information
l
d
Nl
()~
,
[10]
f
where
d
=
lim[ln ( ) / ln (1 / )]
N l
l
stands for the fractal dimension of the tumor
f
s
l
0
surface.
4.4. Molecular Level Dynamics
In (9), we augment our 2D agent-based model with the molecular level dy-
namics of alternating
gene expression
profiles. Specifically, in that study we
analyze the impact of environmental factors on gene expression changes, which
in turn have been found to accompany the phenotypic cellular "switch" from
proliferation to migration. For reasons of tractability, we focus on the behavior
of two genes, namely Tenascin C and PCNA, which have been chosen on the
basis of their reportedly active roles during proliferation and migration of
glioma cells. Tenascin C is an extracellular matrix glycoprotein overexpressed
in malignant in-vivo gliomas. (46) has shown that tumor cell motility is stimu-
lated when human SF-767 glioma cells are placed on Tenascin C. On the other
hand, PCNA stands for "proliferating cell nuclear antigen," whose gene expres-
sion markedly rises during neuroepithelial cell proliferation (47). Experimental
findings have shown that an increase in the gene expression of Tenascin (hereaf-
ter
gTenascin
) is associated with an increase in both levels of nutrients and tis-
sue hypoxia, or (for our purposes, in more general terms) toxicity. Accordingly,
in our model,
gTenascin
is computed as a simple continuous positive function of
both the normalized levels of nutrients
ˆ
G
and toxic metabolites
ˆ
U
:
gTenascin
ˆ
ˆ
Tj
=
b GU
. On the other hand, the literature suggests that an increase in the gene
expression of PCNA (hereafter
gPCNA
) is associated with an increase in nutri-
ents and a
decrease
in toxicity. Accordingly, in our model we compute
gPCNA
as a positive function of nutrients yet negatively affected by toxicity,
gPCNA
=
j
ˆ
ˆ
b GU
. The molecular modules of both
gTenascin
and
gPCNA
enable our
agent-based model to generate a virtual
time-series profile
of both gene expres-
sions as they relate to the proliferative, migratory and quiescent tumor cell phe-
notype. A particular aim of a time-series analysis (see also Part II, chapter 1, by
Shalizi, this volume) is to ascertain whether a dynamic series exhibits intertem-
poral long-range autocorrelations. The presence of such autocorrelations indi-
cates the potential use of past historical values to forecast future outcomes. For
that purpose, we applied
detrended fluctuation analysis
(DFA), which Peng et
al. (48) developed as a robust method to detect long-range correlations in vari-
ous DNA sequences. If the statistical properties of a time series exhibits a
ran-
dom walk
(no autocorrelations across time), then DFA would yield an
autocorrelation measure B = 0.5. In contrast, DFA would detect a long-range
autocorrelation by a value of B that significantly deviates from the random walk
value, i.e., Bg 0.5.
/
Tj
j
