globals [number concentration]              ;; global variables are the numbe of cells and concentration 
breeds [cells]                              ;; let's call them cells instead of turtles 
cells-own [state ticks]                     ;; each cell contain its current set ("young" or "mature") 
                                            ;; and "internal" clock 
                                            
to setup 
  clear-all                                         
  set-default-shape cells "circle"            ;; define cells as circles 
  set number initial_concentration * 100.0    ;; initial number of cells from given conc. (in %)
  create-custom-cells number [                ;; create initial population of mature cells 
                                 set ticks 0.0
                                 set state "mature"  
                                 set color red
                                 random-pos  
                             ]
  setup-plot 
end 


to move-cells 
  locals [dt] 
  set dt 1                                                    ;; time step
  ask cells [
                fd random-float 1.0                           ;; move in random direction 
                if ticks >= (delay * dt) and state = "young"  ;; after some delay cells become mature 
                [
                   set state "mature"                       
                   set color red                              ;; mature cells are red 
                ]
                set ticks ticks + dt                          ;; time increment 
              ]             
  cells-die                                                   ;; call cells-die procedure 
  set number (count cells with [state = "mature"])            ;; count the number of mature cells  
  cells-are-born                                              ;; call cells-are-born procedure 
  compute-concentration                                       
  do-plot                                                     ;; plot concentration 
end


to cells-die         ;; cells die in proportion with decay constant alpha 
  ask random-n-of (int (alpha * number)) cells with [state = "mature"] 
  [
     die
  ]   
end 


to cells-are-born    ;; cells are produced depending on the number 
  locals [n]
  set n int (prod number)
  create-custom-cells n  [                           ;; generate cells from production function 
                           set color green           ;; young cells are green 
                           set state "young"  
                           set ticks 0.0             ;; internal clock initialization 
                           setxy (- screen-edge-x) 0 ;; cells are produced in the left side of the window
                           set heading (random 180)  ;; moving from left to right 
                         ]   
end 


to compute-concentration  ;; compute concentrarion of blood cells 
  set concentration (count cells with [state = "mature"]) / (initial_concentration * 100)     
end 

to random-pos        ;; setup cells in random positions moving in random directions 
   setxy (random-float 2 * screen-edge-x) (random-float 2 * screen-edge-y) 
   set heading (random 360)
end 


to-report prod [n]   ;; production function of blood cells from Mackey-Glass (1977)
  locals [x]
  set x n / 100.0    ;; convert from number to concentration 
  report (100 * (0.2 * x) / (1 + x ^ 10))
end 


to do-plot           ;; do the plot 
  set-current-plot "Concentration of Blood Cells"
  set-current-plot-pen "mature"
  plot concentration 
end


to setup-plot        ;; set up plotting
  set-current-plot "Concentration of Blood Cells"
  set-plot-x-range 0 600 
  set-plot-y-range ymin ymax
end


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@#$#@#$#@
WHAT IS IT?
-----------

This is a simulation of the control mechanism for the production 
of white blood cells based on Mackey-Glass (1977).  Blood cells 
have a certain half-life 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 Mackey-Glass (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 Mackey-Glass 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 "cells-are-born". Also, and like the  
  original Mackey-Glass, the model contains a decay constant (alpha) that 
  correspond to the amount of cells dying in each time step represented 
  by the procedure "cells-die". 
 

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 y-xis of the plot.  

Slider ymax - Set the upper limit in the y-axis 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 quasi-periodically 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 Runge-Kutta methods. 

CREDITS AND REFERENCES
----------------------

[1] Mackey, M.C., and Glass, L. 1977. Oscillation and chaos in physiological control systems. Science 197: 287-89.

[2] Glass, L., and Mackey, M.C. 1988. From Clocks to Chaos. New Jersey: Princeton University Press  

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NetLogo 2.0.0
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