634 



HANDBOOK OF PH'iSIOLOGV -^ CIRCULATION I 



compartment 2, and only compartment 2 is accessible, 

 the input data are 



(iS'2.V2((), .V20, xe, C-21, C22, Xi and X-.> 



The rate constants are deduced by using the 

 following sequence of calculations [Gellhorn et al. 

 (27), Robertson (52)]: 



ka= }i 



Xi + X-; - A'2 



Xi Cn + X; C;; 

 Xw 



(36) 



-/|/(x.. 



^■21 = 



, + X-2 - K-,)' - 4X, 



Xi + X; - /v. 



+ /i/ (^1 + ^= - f^i)' - 4^ 



Xi X2 — A12 /r^o — ^'12 -^2 





(37) 



(38) 



"-32 — ^12 



^'23 = ^'2 — ^'21 



(39) 



(40) 



Again the rates are readily calculated from the 

 rate constants: 



(■^2 .Vzo) 

 P12 — P21 — A'21 



(S2 Xid) 



•V20 



^-23 



(4O 

 (42) 



Another set of formulas is given by Solomon (65) 

 for the case in which the paired rate constants are 

 equal. 



The method of Skinner ft al. (62) is applicable to 

 open three-compartment systems, and the following 

 example is based upon their analysis. It is assumed 

 that only compartment i is accessible, that at time 

 zero the activity is confined to compartment i, and 

 that the inflow is not labeled. Their method is first to 

 establish three equations witich express the efi~ects of 

 the restrictions on the system and five constants 

 derived from relationships among the coefficients, 

 exponents, and rate constants. These eight equations 

 are then solved for the rate constants in terms of the 

 five constants. 



Equations by Skinner el al. (62) are in terms of 

 amounts rather than specific activities. This does not 

 aflfect the exponents but, of course, the coefficients in 

 equation 13 must he in amount units. For consistency 

 with the present notation, (Ci„ .^i) will be used for 

 //„ in the original version. The eight relationships 



used are in the present notation: 



PTfi + P\l + P13 = K\S\ 



(43) 

 (44) 

 (45) 



P'jd + P-Jl + P23 = K2S2 

 P30 ~r P3l "T P32 = KzSz 



A'o + A'3 = Xo + X3 



+ (X, - X2) (Cr,S,) + (X, - X3) (Cu5,) = a, 



A2A3 — A'23^'32 



= (X2 - X,) (X2 - X.,) (Ci:5'i) - \.} + (CuS^) X2 = flo 



A'l = Xi + Xo + X3 — ai =03 (48) 



kinkii + kinkii = 02 + "301 — X, X2 



(46) 



(47) 



ii3 (A-32 A-2, + A'o k:n) 



-\- ky>(k-i:i A'31 + A'3 k-2] 



X-i X3 — X3 Xi = (3i 



— Xl '^■>\i + ^3 «2 = ^5 



(49) 



(50) 



The above equations can Y)e used to solve for the 

 rate constants defined in terms of inflow rates (p , ,7 S ,) 

 but in the original version (62) the outflow rate 

 constants {pji/S,) are used, and to facilitate com- 

 parison with the original, the following is a direct 

 conversion to the present notation for the case — > i 

 ^ 2 «=i 3 — >. Equations 46, 47, and 48 are as given 

 above, whereas the others reduce to 



P12P21 



P12P21 



s-.s, 



A'3 = as 



(45') 

 (49') 

 (50') 



In this case the rate constants are then calculated 

 from the combinations of the a's indicated below ; 



S,/Si = 



s,/s, = 



Pm/Si — 



02 



