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Fig. 1. Steps for the simulation of the proposed model
“more” activated and immunocompetent a given cell can be considered to be,
in contrast to a resting condition, represented by an activation level close to
zero. The anity between a cell and a cytokine, a key point of the motivating
hypothesis, is modelled by constants used to update the cell activation level,
based on the cytokine absorption, that will be described in greater detail. This
cytokine anity is proportional to the increase in the cell activation level, so
that cells with a large anity will be highly stimulated upon absorption of a
given stimulation cytokine. This approach to the simulation is very similar to
the proposal of [18], where a cellular automaton is used to simulate the dynamics
of the immune system during immunization.
Due to the complexity involved, each distinct step in the simulation is pre-
sented separately, in the following sub-sections.
4.1
Cytokine Decay and Diffusion in the Environment
Updating the cytokine concentration in the environment is conducted in accor-
dance with the discrete two-dimensional diffusion equation [19], using equations
1 for diffusion and 2 for decay, where ψ ( x, y, t ) is the cytokine concentration at
the point defined by the coordinates ( x, y )atthetimeinstant t , k d is the cytokine
diffusion rate, Δt is the simulation time step, ζ is the decay constant, n ( x, y )is
the number of valid slots surrounding the position defined by points ( x, y )(rep-
resenting the tissue boundary conditions) and h x and h y are the environment
dimensions. The artificial tissue has been modelled as a compartment isolated
from the body, so that there's no cytokine flux coming in or out of the simulation
environment. Therefore, all cytokines secreted by the cells in the tissue remain
confined to the environment, without taking the decay into consideration.
ψ ( x, y, t + Δt )= ψ ( x, y, t )+ k d ·
Δt
h y ·
( ψ ( x
1 ,y,t )+
h x ·
ψ ( x +1 ,y,t )+ ψ ( x, y
1 ,t )+ ψ ( x, y +1 ,t ))
n ( x, y )
·
ψ ( x, y, t ))
1
x
h x , 1
y
h y
(1)
ψ ( x, y, t + Δt )= ψ ( x, y, t )
·
(1
ζ ) , 0
ζ
1
(2)
 
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