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Table 1. Simulation parameters
Parameter
Value
Effector cells Regulatory Cells
Max. cytokine secretion ( ψ out
max )
2
45.5
Min. stimulation cytokine absorption ( φ i min ) 0.4
0.05
Max. stimulation cytokine absorption ( φ i max )0.8
0.5
Min. regulation cytokine absorption ( φ i min ) .3
-
Max. regulation cytokine absorption ( φ i max )0.5
-
Stimulation constant ( k s )
10
3
Regulation constant ( k r )
10
-
Activation steepness ( σ )
1
2
Additional parameters for both effector and regulatory cells, chosen empirically,
are shown in table 1.
As previously discussed, a key feature of the hypothesis motivating the devel-
opment of the proposed model is the ability of the stimulation cytokine to be ab-
sorbed with different anities by effector and regulatory cells. In order to obtain
the expected system dynamic response (increasing the magnitude of the response,
followed by its decline), it is analysed the case when the effector cell anity for
the stimulation cytokine is greater than the anity by regulatory cells. In this sit-
uation, the regulatory cell would only be activated once a large amount of stimu-
lation cytokine (secreted by activated effector cells) is present in the environment.
The model parameters shown in table 1 were chosen to reflect this assumption.
Special care was taken not to select large diffusion rates, leading to instability
when determining the cytokine diffusion. The activation steepness for effector
cells is twice as low as for regulatory cells, while the stimulation constant for
effector cells is greater than for regulatory cells. Afterwards, the selected pa-
rameters were tuned to lead to a desired characteristic, where the response is
initiated (by the initially stimulated cell), increased (by the recruitment of sur-
rounding effector cells) and terminated (by suppression of the activated cells). It
is important to mention that some combinations of values have lead to oscilla-
tions in the response (data not shown), with the activation level of effector and
regulatory cells increasing and decreasing, without reaching a steady state. This
oscillatory response of the model is undesirable, because there are no reports
from a similar behavior in the natural immune system.
The simulation results obtained for the selected parameters are presented in
figures 6, 7, 8 and 9. By the end of the simulation, the effector cells identified
by numbers 4, 5 and 6, according to figure 5, were not activated, remaining in a
resting state during the simulation. Thus, simulation results for these cells are
not presented. On the other hand, the effector cells identified by the numbers 1
and 3 in figure 5 were successfully recruited for the immune response initiated
by effector cell number 2. Some iterations after the beginning of the simulation,
the regulatory cell (number 7) began to be stimulated, acting, at some time, to
 
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