INDICATOR SUBSTANCES AND FLOW ANALYSIS 



635 



Si 



= as 



Pis Ctiiai Oj 06 — 02 04^ — O5) 



Sj 04(01 03 04 — O3 O5 — 0,2) 



P32 _ Oi Os O4 — 03 Oj 

 ^2 O3 O4 



— n,2 



(56) 



'57) 



(58) 



Once the rate constants have been calculated, the 

 conversion to rates (p) is readily achieved from the 

 appropriate relationships above. 



Similar analyses of two other three-compartment 

 open systems, one having flow into the center compart- 

 ment and out one of the end compartments, the other 

 ha\ing flow into and out of the third compartment 

 as well, are given in detail, with numerical examples, 

 by Skinner et al. (62). 



If data are available from additional compartments 

 and/or the outflows, the equations used in the above 

 procedures are simpler, and if all three compartments 

 are accessible, all of the rate constant equations 

 become linear. 



CALCULATION OF EXPONENTIAL CONSTANTS. At the 



stage of model analysis, it is sometimes desirable to 

 predict the tracer behavior from the assumed param- 

 eters of the system. The X's in equation 15 may be 

 calculated independently by substituting — X.v, for 

 dx,/dt in equation 14, and setting the determinant 

 of the coefficients of the x's equal to zero." 



Thus, for the complete three-compartment system 

 (fig. 3), this substitution gives: 



X — Al A 12 A 13 



A'21 X — Ao ftog 



A*3i k^^ \ — A3 



(59) 



E.xpansion of equation 59 shows that the X's are 

 the three roots of a third order algebraic equation: 



X3 - (A'l -I- Ao -I- As) X2 



-I- (A,A2 -I- A,A:, -I- KoKs - k.2,ki2 - ksiky, - k32k23)\ 



— AiA'aA'a -|- A"32^"23A'i -f- k^iki^K^ -{- k^iki-^hs 



+ k2ili32k,3 + k3iki2k32 = o (60) 



- The basis for this substitution is the fact tliat the coefficients 

 of successive derivatives, d/dl, <P-/dt-, etc. of <"" are a, a-, etc. 



,^^!S 



FIG. 3. Complete three-compartment open system. 



The imposition of constraints such as eliminating 

 inflows and outflows or the exchange between two 

 compartments modifies equation 59 by eliminating 

 some A's and by removing some pa/Si terms included 

 in the K^s. For the closed three-compartment system, 

 the last six terms in equation 60 add up to zero, and 

 X may be factored out of the remaining terms, leaving, 

 for the non-zero X's 



\- - (A, -I- A., -I- A-,)X -f- A'lA'o 



-f- AiAs -f- AjA.i — k2\ki2 ~ A'3iA'i3 — A'a^A'^s 



(61) 



Further reduction of the system to i ^ 2 ^ 3 by 

 eliminating pis and pn automatically eliminates ki^ 

 and ^31 and imposes the requirement that pn = pi> 

 and P32 = P23, giving A'l = ^12 and A'.! = A' 33, reducing 

 equation 61 to 



V - (kr2 + A., -I- k32) X -I- ^,2^23 -I- ^32*21 "j- knK32 = O (62) 



The X's for the two-compartment open systems are 

 also obtained as a reduced form of equation 6i : 



X2 - (A, 4- A2)X -I- AiA, 



kiik 21 = o 



(63) 



In general the X's for an //-compartment open system 

 are the n roots of an nth order algebraic equation, and 

 those for an n-compartment closed system are the 

 (n — i) roots of an (ri — i )th order equation. Itwill be 

 noted that the X's are determined entirely by the 

 kinetic characteristics of the system and are not 

 affected by variations in the initial distribution of the 

 tracer. 



CALCULATION OF COEFFICIENTS. In Contrast to the 

 case for the X's, the ab.solute values of the coefficients, 

 the C's in equation 1 5 do depend upon the boundary 

 conditions such as whether or not the inflow is labeled 

 and which compartments are labeled initially. The 

 relative values, however, may be expressed inde- 

 pendently of the boundary conditions. A procedure 

 for generating the relative values of the C's is to 

 regard the entries in equation 59 or its counterpart 

 for other systems as the coefKcients, not of the .v's, but 

 of the C's associated with a particular value of X. The 

 same relationships are obtained by equating corre- 

 sponding terms in equation 14 and the derivative of 

 equation 15. Thus, for the constrained three-compart- 

 ment system i ^ 2 ^ 3, the coefficients of the terms 



