INDICATOR SUBSTANCES AND FLOW ANALYSIS 



633 



/: = (4o)(i0 



Ai 4 

 Ai —2 

 ^3 -5 



D 



3.4 1 - 9-4; + 6.43 



264 



h = (40) 



I 4 ^1 



I -18 aJ, 



I 15 All 3.4i — A«. - 2.43 



(25) 



(26) 



D 264 



In this method the coefficients of the .-l"s form the 

 inverse matrix, giving 



15 10 8 

 1320 1320 1320 



[C]-' = 



3-96 

 264 264 264 



3 -1-2 

 _ 264 264 264 



(27) 



and 



(28) 



[k] 



Comparison of [A] in equation 28 with equation 20 

 shows that this method does generate the k's (and 

 if's) as it should. The power of the matrix method is 

 shown by the fact that all the A's are solved for simul- 

 taneously, whereas other methods generate them one 

 at a time. 



Access to all compartments is often not feasible, 

 however, and it is usually necessary to work with 

 data from only a few compartments. Useful solutions 

 for the A's may be obtained by manipulating the 

 equalities obtained in equations 16 and 1 7 to eliminate 

 the Cs for experimentally inaccessible compartments. 

 For the three-compartment systems (including some 

 open ones), it is possible to get all of the k's and .S"s, 

 and hence the p's when it is assumed that only one 

 compartment is accessible (62). This is important in 

 circulation studies because often the only data 

 obtainable are the tracer concentrations in the blood 

 (or a blood fraction) and direct measurements in 

 other body organs or tissues of interest difficult or 

 impossible to obtain in human beings. When the 



tracer is a gamma-ray emitter, in vivo counting data 

 may be valuable [Conn (17)]. 



In a provocative article, Bergner (5) asserts that 

 data from a single compartment are generally 

 inadequate for the analysis of three-compartment 

 systems. This is partially refuted by the analysis of 

 Skinner et al. (62) giving explicit formulas for these 

 systems. In an exchange of notes between Herman & 

 Schoenfeld (8) and Bergner (6), the disagreement 

 seems to come down to the question of whether it is 

 acceptable to use additional available information to 

 associate compartment numbers and flow rates; 

 in multicompartment systems involving nonlinear 

 equations, Bergner (6) points out that ambiguities may 

 remain. 



If the tracer is initially confined to one compart- 

 ment, the total rate constant, A'l, for the injected 

 compartment (assumed to be compartment i ) is 

 simply (from equations 14 and 16) 



A. = 



c„ X, -I- 



Cin X,i 



(29) 



Other rate constants generally require more compu- 

 tation. The procedures for several three-compartment 

 systems are recapitulated here in the present notation. 

 In I ?=i 2 ^ 3, when it is assumed that the label is 

 initially confined to compartment i and that only 

 compartment i is accessible, numerical values for 

 5i ,vio (the amount of activity injected), atio, Xe, Cn, 

 C12, X], and \y are known or obtainable by experi- 

 ment. The formulas for the rate constants are [Cohn 

 & Brues (15), Robertson (52)]. 



fcr = 



Cn Xi -|- C12 X, 



^■i2(X, -I- X. - ka) - Xi X.. 



A--,, = 



0-2) 



h, = 



A-=.= 



■V£ Xi X2 



-Vio A'21 

 XlX-2 



^,7 



•Vin 



A-32 



(30) 



(3O 



(32) 

 (33) 



Once the A:"s are known, the p's are quickly computed 



(^1 -Vio) 



^10 



P12 

 P23 — P32 — 7- ^23 



-kn 



(34) 

 (35) 



In I 



3, when the label is initially confined to 



