228
I Erythropoietin I
J. KIRK, and J. S. ORR
Red cell
population
,..--_--1 1 Chalone!-- L_--.
1
1
r-------,
~
I
1
1
'---___ ----.,1- ___ :!em c:~ ____ !--____ J "'.1'--_--'
population
~
I
1
A
I
I
I
,
L _ _ _ _ _ _ _ _ _ _ .J
Fig. 1. Block diagram of the oscillatory system
The Mathematical Formulation
The system is described mathematically by four simultaneous nonlinear difference-differential equations in the four variables; the red cell
population (R), the stem cell population (5), erythropoietin (E) and chalone
(C). Since these equations cannot be solved analytically, the daily values of
the above variables are calculated by numerical methods using a digital
computer. The evaluation was carried out using an integration step of one
day. The equations representing the addition of the daily changes of the
variables to their previous values are:
a
Er = 1 RZ /0 + {3E r - 1
+ r-l
C, = y [5r - 5r- 1 (1 - m C~_I)] + P C,-1
5r = 5r- 1 {2 - mC~-I[1 + (w + p bE~_I)/mC:])
Rr = Rr- 1 + kPbE~_x5r-xq_x/C: - F (R)r
The constants in these equations are determined from a wide range of
experimental data.