INTRODUCTION
There is many interesting researches about cancer and Tumor which has been
done till now (Looi et al., 2006; Kumar
et al., 2006; El-Naggar et al., 2011).
Tumor development occurs in two distinct stages. Initially, tumor cells in small
avascular tumors’ gain the oxygen and nutrients they need for survival
and growth by diffusion from the existing vasculature in the normal tissue that
surrounds the tumor. The amount of oxygen thus supplied is limited by the surface
area of the growing tumor while the amount of oxygen required by the cells is
proportional to their total volume. In consequence, the tumor reaches a diffusion-limited
size, where the amount of oxygen and nutrients entering balances the amount
consumed by the live tumors cells (that exist near the rim of the tumor). The
size of such a tumor is limited to 1-2 mm (Holmgren et
al., 1995). The cells near the centre of such tumors are subjected to
hypoxic stress (that is, low oxygen tensions which gradually ‘suffocate’
the cells). Exposure to this stress slows their rate of proliferation and stimulates
them to express diffusible factors such as Vascular Endothelial Growth Factor
(VEGF), Tumor Necrosis Factor (TNF-α), Transforming Growth Factor (TGF-β)
which travel outwards from the tumor, towards the surrounding vasculature (Yazdi
et al., 2011). Once these factors reach the surrounding blood vessels,
they stimulate endothelial cells lining the walls of the blood vessels to proliferate,
migrate and differentiate to form new blood vessels which grow towards the tumor.
Formation of new blood vessels by this process is called tumor angiogenesis
(Bicknell et al., 1997; Homaei-Shandiz
et al., 2009). Once the vessels have reached the tumor and circulatory
loops have formed, blood can flow through the vessels and supply the tumor cells
with additional oxygen and nutrients. Vascular tumor growth then commences.
During vascular tumor growth, previously dormant cells that are close to the
neovas-culature are able to proliferate once again and the growth of the tumor
as a whole can recommence. This may become life threatening for the host, since,
both the tumor and any metastases (secondary tumors that form when tumor cells
escape from the primary tumor into the general blood or lymph circulation and
establish colonies in other tissues) may cause malfunction of vital organs (Folkman,
1950). The tumor structure is heterogeneous: tumor cells in close proximity
to blood vessels proliferate rapidly in the presence of abundant oxygen and
nutrients (Hahnfeldt et al., 1999; Sodde
et al., 2011) and where the rate of proliferation outstrips the rate
of new blood vessel formation, transient areas of hypoxia form. Vascular tumors
consist of a mixture of cell types, including live and dead tumor cells, macrophages
and endothelial cells lining the blood vessels, all embedded in an extracellular
matrix. Cells attach themselves to the extracellular matrix by expressing proteins
and other molecules on their surface which bind to complementary molecules in
the matrix (Alberts et al., 1994). There have been
numerous mathematical models of avascular tumor growth (Byrne
and Chaplain, 1995) for example. Such models normally prescribe the oxygen
tension at the proliferating rim and may involve ‘compartmentalizing’
the tumor into proliferating, quiescent and necrotic regions. There have also
been several studies of angiogenesis (Byrne and Chaplain,
1995). However, only a small amount of (mathematical) literature concerns
the growth of vascular tumors. Macro scale models (Byrne
and Chaplain, 1995; Hahnfeldt et al., 1999)
have been generated to describe various tumor phenomena: generate a model that
describes how fluid moves out through the tumor’s periphery; (Folkman,
1950; Ali et al., 2011) allows cell motion
by diffusion and by taxis up gradients of capillary vessels and they calculate
the density of live tumor cells and the surface area of the capillary vessels
per unit volume. In this study, we present a mathematical model to describe
the behavior of tumor development stages in time by use of porous medium equations
and solve these by means of artificial neural network which help us to detect
the time of beginning of each stage. The main differences between our model
and numerical method with other are: fast convergence- the final solution is
a function which we can use it to calculate other values of function at every
point in training interval-some small errors, we can also change the neural
network model or use heuristic optimization algorithms.
MATHEMATICAL MODEL
We assume that the microenvironment within a tumor consists of a mixture of
live and dead tumor cells, macrophages, extracellular water and endothelial
cells (forming the tumor vasculature) embedded in a tissue matrix. The live
tumor cells proliferate if the local oxygen tension is high enough and the rate
at which proliferation occurs is assumed to be proportional to both the local
oxygen tension and the (number) density of live cells. We assume that live cells
can die by two processes. The first, apoptosis (Alberts et
al., 1994) is programmed cell death (‘old age’) and occurs
at all oxygen tensions. The rate of apoptosis may increase with decreasing oxygen
tension (Hahnfeldt et al., 1999) but in this
model we assume it is proportional to the local density of live cells and independent
of the local oxygen tension The second death mechanism we consider is oxygen-induced-necrosis
(henceforth termed necrosis) which we assume here to be cell death induced by
oxygen starvation (suffocation).
|
| Fig. 1: |
Disrupting the immune response by tumor cells |
The rate of such necrosis is assumed to be proportional to the local density
of live cells and to be induced only when the oxygen tension drops below a threshold
value. Other forms of necrotic cell death occur at all oxygen tensions, for
example, necrosis due to low pH (Bicknell et al., 1997),
we stress that we do not consider such mechanisms in this study. When live cells
die by apoptosis, they become inactive but retain their form until they are
either degraded by enzymes in the matrix or phagocytosed by macrophages (a type
of stromal cell that moves around the tumor and accumulates in hypoxic areas)
(Alberts et al., 1994). We suppose that these processes
(which together we call degradation) occur at a rate proportional to the local
density of dead cells Fig. 1. Cell proliferation and degradation
in local sites throughout the tumor mass cause pressure gradients to be established.
These, in turn, cause the blood vessel to open close and drive the movement
of the live and dead cells. In this way the supply oxygen tension may vary due
to changes in the local densities of the two cell types. We denote the density
of live tumor cells by n, the density of dead tumor cells by and the oxygen
tension by C. We let the live cells move with velocity vn and the
dead cells move with velocity vm. Using conservation of mass, we
formulate the following equation to describe the evolution of the densities
of live and dead tumor cells:
which proliferation does not occur (H is the Heaviside function), A is the
apoptosis rate, N is the necrosis rate, C2 is a threshold oxygen
tension below which necrosis occurs, F is the dead cell degradation rate, y
denotes the distance ‘outwards’ from the blood vessel and t denotes
time. In Eq. 1 and 2, we have also assumed
that a live cell and a dead cell have the same volume. We note that when C>C1
cells undergo proliferation and apoptosis.
When C1>C>C2 cells undergo apoptosis only and when C<C2 cells undergo apoptosis and necrosis. We remark further that, since, the proliferation rate depends on the local nutrient concentration and since, the local nutrient concentration depends on the density of live cells Eq. 4, (1) does not automatically produce exponential tumor growth. Since, the total cell density remains constant, we have a further equation relating n and m, namely:
where, M is the (constant) density of live and dead cells at an arbitrary point within the tumor mass. A conservation of mass equation could also be formulated for the evolution of the density of the Struma. Since, we assume that this density is constant throughout the tumor mass, the equation would tell us the speed with which this component moved. We use conservation of mass to generate the equation governing the evolution of the oxygen tension C in the tumor. We assume that changes in C are due to advection, diffusion and consumption and so the resulting equation reads:
In Eq. 4, D is the diffusivity, E1 is the rate at which live tumor cells consume oxygen and E2 is the rate at which stromal cells consume oxygen. We have assumed, for simplicity, that oxygen advects with the (linear) phase averaged velocity, vav, where:
The precise form for the advection velocity is immaterial, since, as we will see later in this section, in situations of practical interest the nondimensionalised leading order version of Eq. 4 is dominated by diffusion. We impose the following initial and boundary conditions on the tumor tissue. The initial density of live cells (and hence the density of dead cells) is prescribed. We assume the average cell speed vav at the vessel wall y = h matches the wall speed and no flow of cells across the line of symmetry out at y = L. We also assume symmetry of the oxygen tension about the symmetry line. These conditions are equivalent to prescribing:
where, n0 is the initial density of live cells. We are now in a position to nondimensionalise the model. Since, we are interested in interactions between tumor cells and the blood vessel, we scale time with the tumor cell proliferation timescale (λC0). We scale lengths with half the intravascular distance (i.e., with L), the live and dead cell densities with the initial total density of cells (M) and we scale the oxygen tension with the oxygen that would be supplied by a vessel of width:
We eliminate vn and vm from Eq. 1,
2 and 4 and the resulting nondimensionalised
system of equations and boundary conditions reads (dropping primes):
with:
The non-dimensional parameter groups introduced into Eq. 11-19
are defined below as:
We now estimate that two of the parameters are small:
| • |
ε~0.02«1, for blood vessels of width hB
= 10 μm and a tumor intravascular distance of 1 mm (so that L = 5x10-4
m) |
| • |
Pe~0.03«1, for diffusivity D~10-10 m2 sec-1
and with a proliferation timescale of 1 day (λC0~105
sec) |
Thus, we take the limits ε→0, Pe→0, with consequences that the blood vessel boundary conditions are applied on y = 0, the speed of the cells at the blood vessel wall is zero and the problem for the oxygen tension becomes quasi-steady.
Numerical solutions by means of artificial neural network: Consider
the equation Eq. 11 with assumed boundary conditions (Myers,
1996). The suggested trial solution can be as (Lagaris
et al., 1998; Al-Daoud, 2009; Effati
et al., 2010; Tortoe et al., 2011):
where, N is the neural network and P contains the weights of neural network
model (Asadollahi-Baboli, 2011; Senol
and Ozturan, 2008; Olanrewaju et al., 2011;
Khalaf et al., 2011; Yu et
al., 2011). The neural network architect is similar to Fig.
2:
Indeed the weight vector P contains the weights of input layer (W), bias vector (b) and the weights of output layer (V). “sigmoid” is the activation function of the network:
Sigmoid transfer function is plotted in Fig. 3.
It is easy to check that the trial solution Eq. 21 satisfies
the conditions Eq. 15-19. Suppose that
xε[x0, x1] and tε[t0, t1].
Now putting Eq. 21 in 11 we have:
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| Fig. 2: |
Neural network architect |
|
| Fig. 3: |
Sigmoid activation function |
We define the following optimization problem:
Equivalently if we define:
Then we have the following optimization problem:
|
| Fig. 4: |
Exact and approximated solutions of the development stages
of tumor |
To solve Eq. 24 we can use any optimization algorithms such
as the steepest descent method and the conjugate gradient or quasi-Newton methods.
The Newton method is one of the important algorithms in nonlinear optimization.
The main disadvantage of the Newton method is that it is necessary to evaluate
the second derivative matrix (Hessian matrix). In this paper we use quasi-Newton
BFGS method, however, we can use heuristic algorithms such as PSO,GA and ACO.
The most fundamental idea in quasi-Newton methods is the requirement to calculate
an approximation of the Hessian matrix (Ahmadian and Afsharinafar,
2011). After the optimization step, we replace the optimal values of weight
vector P in to the Eq. 21 which is the final solution. The
exact and approximated solution can be found in Fig. 4 and
5. To solve this problem we used MATLAB 7 optimization toolbox
using the BFGS hessian updater.
CONCLUSION
We have presented a mathematical model to describe micro scale interactions
between tumor cells and a compliant blood vessel within a vascular tumor. When
developing he model, we assumed that the tumor mass comprised a mixture of live
and dead cells. Analytical and numerical technique were utilized to illustrate
the qualitative behavior of the model solutions and the system was observed
to settle to a steady state (Fig. 4).