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WHAT IS IT?
This is a simulation of the control mechanism for the production of white blood cells based on MackeyGlass (1977). Blood cells have a certain halflife but new replacements are produced continuosly. It takes about four days for a new cell to mature so there is a delay in the dynamics governing the number of mature blood cells. The goal of any control system is to keep a certain quantity at a constant level. However, in real systems the control mechanism is not activated inmediately but only after a certain delay corresponding to the period of maduration of the blood cells. The resulting dynamics is characterized by oscillations that for certain delays may turn chaotic. This simulation pretends to ilustrate this process where an initial population of blood cells are generated and new cells are produced in each time step depending of the concentration of mature cells. The "young" cells eventually are old enough to be mature and some of them die in proportion with a decay constant. The model also ilustrate the effect of a delayed mixed feedback in the dynamics of biological systems.
HOW IT WORKS
This simulation is an individual based model (IBM) version of the classic model of MackeyGlass (1977) for phisiological control. The original model is formulated as a delay differential equation (DDE) containing a term corresponding to the production of new blood cells, P(x) and a decay term (alpha) proportional to the number of mature cells in circulation. The simulation was developed using the following procedure:
 Generate inital population of cells  Each cell contain two parameters: an internal clock (ticks) initialized as zero and the initial state of the cells (young or mature). The initialized cells are located randomly in the graphics windows and in each time step the cells "move" in random directions.
 In each time step the internal clock of each cell increases until it reachs the state of maturity where the cell changes from "young" to "mature". The time it takes to reach maturity is given by the delay parameter (in days).
 In each time step we have to types of cells. Mature cells and young cells. The production of new cells are determined by the level of mature cells in circulation. For instance, if the total amount of mature cells is high the production of new cells is low. On the other hand, if the total amount of mature cells is low the production mechanism is high. In any case the model define a production function given by P(x) = (0.2 x)/(1 + x^10) which is the same used in the MackeyGlass delay differential equation model. Since the argument in P(x) is a concentration the NetLogo codes works out the calculation of concentration by dividing the number of cells in each time step by an arbitrary normalization factor (100) that in general could be calibrated with experimental data. The cell production mechanism is represented by the procedure "cellsareborn". Also, and like the original MackeyGlass, the model contains a decay constant (alpha) that correspond to the amount of cells dying in each time step represented by the procedure "cellsdie".
HOW TO USE IT
Button SETUP  Initialize the simulation by cleaning the plot region and the graphics section.
Button START/STOP  Start or stop the simulation. During the pause some of the parameters of the simulation can be modified
Slider initial_concentration  The initial concentration of mature blood cells in arbitrary units.
Slider delay  The period of maduration of the "young" cells in units of days.
Slider alpha  This is a decay constant that determines how many mature cells are going to die in each time step.
Slider ymin  Set the lower limit in the yxis of the plot.
Slider ymax  Set the upper limit in the yaxis of the plot.
The graphics window shows the population of young blood cells (GREEN) and the population of mature cells (RED) in each time step. The young cells are produced in the left side of the graphics window.
THINGS TO NOTICE
If the initial concentration of blood cells is very low (0.2) the production of blood cells increases dramatically in a short period of time. On the other hand, if the initial concentration is relatively high (2.0) the production decreases. This process can be seen in the plot and also by the proportion of RED and GREEN cells in the graphics window.
As in the delay differential equation (DDE) model an increase in the delay parameter implies a transition from periodic oscillations to chaotic dynamics. However, there are certain windows of periodic oscillations.
The simulation is very sensitive to the decay parameter (alpha) if high values are selected the population dies inmediately.
THINGS TO TRY
Watch the dynamics of the simulation by moving back and forth the "delay" slider. In this case we are looking at a more complicated version of the model where we introduce a time dependent modulation in the delay parameter. Something similar may be tried with the decay parameter but in this case the variations have to be smaller.
EXTENDING THE MODEL
Along the lines of the previous "THINGS TO TRY" section an interesting modification could be that the delay parameter or the decay parameter varies periodically or quasiperiodically as a function of time.
Since the model pretends to "solve" a DDE the present model may be calibrated so that the numerical values of the concentrations are fully consistent with standard numerical solutions of DDEs using Euler or RungeKutta methods.
CREDITS AND REFERENCES
[1] Mackey, M.C., and Glass, L. 1977. Oscillation and chaos in physiological control systems. Science 197: 28789.
[2] Glass, L., and Mackey, M.C. 1988. From Clocks to Chaos. New Jersey: Princeton University Press
