Download Self-organization and competition in the immune response to cancer

Vol. 16 no. 2 2000
Pages 96–100
BIOINFORMATICS
Self-organization and competition in the immune
response to cancer invasion: a phase-orientated
computational model of oncogenesis
Sorinel Adrian Oprişan 1, Aurel Ardelean 2 and
Petre T. Frangopol 2
1 ‘Al.I.Cuza’
University, Department of Theoretical Physics, Blvd. Carol I, no. 11, 6600
Iaşi, Romania and 2 Western University ‘Vasile Goldis’, Department of Biophysics,
Blvd. Revolutiei no. 81, 2900 Arad, Romania
Received on July 6, 1999; revised on September 15, 1999; accepted on September 17, 1999
Abstract
Motivation: Recent studies indicate that fractal dimensions can uncover aspects of cellular dynamics prior to
pathological manifestation. In this respect we are interested in building a computational model of oncogenesis
able to generate patterns with the same fractal dimension
spectrum as the in vivo tumor.
Results: A new theoretical model incorporating a systemic view of oncogenesis in a computational model was
proposed. The tumor growth is viewed as competition for
resources between the two self-organizing subsystems:
the neoplastic and the immune. Numerical simulations
revealed that tumor escape can be uncovered in some
earlier stage of the immune-system–tumor interaction
using multifractal measures. The described computational
model is able to simulate also the case of immune,
surgical, chemical and radiotherapeutical treatment, as
well as their effects.
Availability: The software used is available on request
from the authors.
Contact: Sorinel Oprisan, University of New Orleans,
Department of Psychology, New Orleans, LA 70148, USA.
[email protected]
Introduction
A theoretical approach of tumor invasion based on the
development of mathematical models appears interesting
from at least two points of view. First, one can expect
that using mathematical models will help to fill the
gap between the quantitative and qualitative information
provided at the level of population by the development
of medical imaging techniques and the description and
comprehension of oncogenesis at the cellular level, with
obvious implications for the optimization of therapeutical
strategies. Second, the morphological differences and
transitions between well and smooth defined benign tumor
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and tentacular malignant tumor suggest a theoretical
analysis of tumor invasion based on the development of
mathematical models exhibiting bifurcations of spatial
patterns in tumor cells density and fractal geometry
(Tracqui, 1995).
A large number of studies have been devoted to the
modeling of tumor growth. The computer modeling of
highly complex biological systems requires a breakdown
of the problem and appropriate simplifications. In fact a
computational model is a ‘model of the model’ (Duchting,
1996). Deterministic models usually consider tumor
growth as a wave propagation phenomenon controlled by
various factors like mitotic inhibitors and nutrient depletion, immune response or competition with normal cells,
and alteration of mitotic rates. Alternatively, stochastic
and cellular automata approaches have been proposed
to describe the spread of tumor cells into the normal
tissue (Keen and Spain, 1992; Meinhard, 1982; Qi et al.,
1993; Smolle and Grimstad, 1992; Smolle and Stettner,
1993) with eventual consideration of cell status with
regard to the cell cycle (Duchting, 1990; Duchting and
Vogelsaenger, 1985). Some of these works have combined
computer simulations and image analysis of histological
tumor sections to characterize the tumor invasion patterns
along a spectrum ranging from broad front invasion to
tentacular invasion (Smolle and Grimstad, 1992). By
comparing experimentally observed patterns with patterns
generated by computer simulations, it was possible to
establish a relationship between tumor cell motility and
the resulting morphological tumor patterns controlled
by the probability of cell division, death and migration
(Smolle and Stettner, 1993; Hogeweg, 1987). One of
the challenges in cancer research is to understand how
cell division, migration, death and effector cells (EC)
attack interplay in the modulation and control of the
spatio-temporal dynamics of tumor growth.
c Oxford University Press 2000
A phase-orientated computational model of oncogenesis
At the cellular level it must be stressed that the natural
immune EC population is highly heterogeneous. Among
other sub-populations with cytotoxic action CTL and
natural killer NK lymphocytes are the most studied. Even
a simple group, like NK cells, is morphologically and
phenotypically heterogeneous. Our present study does not
take into account such diversity and consider the EC
mostly a homogeneous class.
System and methods
Our present work is placed at the cellular level and is
focused on constructing feedback control models that
describe the cell division of normal and tumoral cells.
There is general consensus that cell proliferation plays
an important role in the carcinogenic process (Cohen
and Ellwein, 1990; Moolgavkar and Knudson, 1981;
Moolgavkar and Luebeck, 1995). In our computational
model, malignant cells are clonal, i.e. they arise from a
single progenitor cell that is malignantly transformed. A
malign focus consists of the descendents of a single cell
which acquired the altered phenotype, either spontaneous
or in response to an initiating agent. Altered cells undergo
cell division, migration and cell death. The model includes
apoptosis, or programmed cell death, as an important
feature along the division and migration.
Algorithm
The computational model stands for most representative
and clinically relevant features of oncogenesis viewed as a
fight between two distinct subsystems: the immune system
of the host and the neoplastic system (Oprişan et al.,
1998). For the neoplastic subsystem (S I) the details of
simulation strategy are as follows:
S I.1. (‘active’) if the distance between the tumor cell and
the nutrient medium is less than three cell layers, the
tumor cell is in active [mitosis (M)] stage, with two
different distant-dependent dynamical behaviors
S I.1a. a tumor cell placed on the tumor boundary undergoes cell division phase
A major simplification in our model refers to cell phase
duration that is supposed to take place during a single
time step. We are aware that, for example, cell division is
going on along a series of subphases [G1 in which proteins
and ribonucleic acid are synthesized; S in which DNA
is duplicated; G2 in which the cell synthesizes RNA and
protein molecules required for transit through this phase;
M leading to cell division (Duchting, 1996)]. Our model
combines all the above stages, called M phase, offering
the possibility to the physician or biophysicist of setting a
different duration of the cell division phase. It is important,
from a clinical point of view, to develop a phase orientated
mathematical model of tumor growth since the sensitivity
of tumor cells differs from one cell phase to another. This
is a decisive fact for tumor treatment, because radiation
therapy and chemotherapeutic drugs are killing tumor
cells during distinct cell-cycle phase only .
The second subsystem considered (S II) consists of
cytotoxic active cells EC with a very broad phenotype
ranging from NK cells to CTL cells, macrophages, etc.
These kind of cells are endowed with the following
properties:
S II.1. they can perform a random walk through the
extracellular matrix (ECM)
S II.2. they suffer an aging effect depending on their
interaction history
S II.2a. a free EC increases its age with one unit every
time step
S II.2b. an EC interacting with a tumor cell increases its
age with differing amount, depending on tumor cell
phase
S II.3. an EC will die after a given age. Removal of an EC
can be deterministic or stochastic. In the deterministic
case, all EC older than a given limit age with disappear.
Instead, in a stochastic removal, an EC has a surveillance probability depending on its age:
Psurv =
1
,
1 + exp−γ (t−t0 )
S I.1b. a tumor cell placed on the second layer will force
boundary cells diffusion.
where t0 is the limit age of the immune cells, γ is the
characteristic aging factor of the immune cells.
Some authors (Duchting, 1989) also consider the possibility of tumor cell division even if there is no room left
around the mother cell;
Detailed balance requirements and the Metropolis
Monte Carlo numerical algorithm inspired the above
relationship (Eisen, 1979; Martin et al., 1990). The described complex model was transformed into a computer
program (Borland/Turbo Pascal and MATHEMATICA
language). The computer experiment simulating in vitro
growth of a tumor, starting with a single tumor cell, in
mitotic phase, placed in the center of a nutrient medium
of a two-dimensional cell space. The model is based on
the cell cycle regulation mechanisms described above.
S I.2. (‘dormant’) if the distance between the tumor cell
and the nutrient medium is more than three cell layers,
the tumor cell will enter the resting (dormant) phase
S I.3. (‘necrosis’) a tumor cell resting in the dormant
state for more than a critical number of time steps enters
the necrosis phase (cell death).
97
S.A.Oprişan et al.
Discussion and conclusion
Based on the mathematical model described above it is
possible to evaluate the dynamics of tumor cell growth.
The simulation runs start with a single tumor cell that
resides in the state of mitosis at t = 0. The status of
cells is updated in units of one cell cycle time. Figure 1
indicates the actual cell cycle phase in which every tumor
cell is residing according to computer tomography and
magnetic resonance imaging findings. Our major concern
here addresses not only the qualitative resemblance of
the tumor structures but also the quantitative fit. In this
respect, the scaling exponents of the cross section of
tumor and of the total perimeter of the cross section were
evaluated. Our findings indicate that the cross section of
tumor spheroid scales with tumor cells number as follows:
∝ nα ,
and the perimeter
L ∝ nβ .
These scaling lows resemble quite well the definition
of capacity dimension of the fractal objects and we are
confident that a natural approach of the tumoral pattern
trace form fractal theory. The scaling exponents α and β
include global information on cell cycle phase and they
can be measured from computer generated images using
the algorithm described above.
In the absence of immune response the tumor pattern
is done only by tumor cell division rate and tumor
cell intrinsic diffusion rate. The cross section shown in
Figure 1 has a more regulate structure and than the
contour on Figure 3 (with active immune response). Using
box counting algorithm we found that the corresponding
capacity dimension for the compact structure shown in
Figure 1 is 1.95 ± 0.05. For the case shown in Figure 3
the capacity dimension is 1.15 ± 0.05.
Figure 2 shows the population cell dynamics for three
cell types: A, mitotic phase cells; B, dormant phase; C,
98
Cells number
Fig. 1. The stroboscopic evolution of the tumor cross section followed with a time sampling rate of 100 (in Monte Carlo time steps). The
simulations were performed with the suppressed immune response. The central necrotic compact domain and a general regulate shape of the
tumor characterized by a fractal capacity dimension 1.95 ± 0.05 is shown.
240
220
200
180
160
140
120
100
80
60
40
20
0
–20
–200
A
B
C
0
200
400
600 800 1000 1200 1400 1600
Time
Fig. 2. The time evolution of the cell number in different cell
phase (A, mitotic; B, dormant; C, necrotic phase) without immune
response. It can be observed the saturation of the growth rate for the
cells in the mitotic phase due to the limited carrying capacity of the
surrounding tissue. The initial slope of this curve (prior to saturation
effect) gives the exponent β. The long term behavior indicated a
constant, high, growth rate for the dormant cells. The slope of this
curve gives the exponent α. The fact that α > β results in the tumor
invading the whole tissue.
necrotic phase. As the time elapses the rate of active cell
generation on the tumor boundary decreases and, at the
same time, the rate of necrotic process remains constant
and higher than the mitotic process. Therefore, the final
stage of the system can be easily predicted—necrotic
cells will invade the whole tissue after enough time has
elapsed. Our cross section and perimeter measurements
indicate the smoothness of the topology (regular pattern).
The growth curve of the active cells in Figure 2 can be
described mathematically by the so-called Gompartz
function (Duchting, 1996) and the fractal capacity dimension is high, near the saturation threshold which is given
by the topological dimension of the cross section.
In the presence of the immune response it can be qualitatively observed (see Figure 3 and quantitatively estab-
A phase-orientated computational model of oncogenesis
Fig. 3. The stroboscopic evolution of the tumor cross section followed with a time sampling rate of 100 (in Monte Carlo time steps). The
simulations were performed in the presence of immune response but with a finite, constant, flow of CTL cells and considering the apoptosis
effect. The shape of the tumor is less regular and its fractal capacity dimension is 1.15 ± 0.05.
(a)
Cell number
Active cells
Times
Cell number
(b)
lished that the tumor developed pattern is more ‘fractured’.
In that case the fractal exponents indicate the existence of
a fractal structure. The possible explanation for the observed difference can be that the immune cells attack and
destroy the tumor cells on the boundary. Our assumption
is supported by numerical findings, which provide data on
smoothing of the boundary after suppression of immune
response due to complete destruction of immune cells.
The conclusiveness of our statements must be related
to extrapolation from in vitro to in vivo experiments.
Therefore, a major concern at play is the systematic
reduction of the simplified assumptions of modeling.
Based on the successful correlation between the reliability
of model predictions and experimental observations, it
seems possible to use computer simulations to assist the
evaluation and optimization of clinical treatment schemes.
According to our computer modeling tumor escape can be
uncovered in some earlier stages of the immune system—
tumor interaction by image processing techniques. In this
respect the use of multifractal measures seems to be an
adequate tool.
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Fig. 4. The time evolution of the cell number in different cell phase
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