628 



HANDBOOK OF PHVSIOLOGV 



CIRCULATION I 



s. 



= amount of substance (labeled plus nonlabeled) in uh 

 compartment 



= rate of flow of substance (labeled plus nonlabeled) 

 into (th compartment from ;th compartment 



= specific activity (e.g., microcuries per gram) of radio- 

 active label in Si 



= Xi at time zero 



= Xi at equilibrium (infinite time) 



— ' = fractional turnover rate constant of .S', due to flow 

 Si 



from jth compartment 

 K, = 3"" — = 52 — ~ total turnover rate constant for .S', in 



Xio 



xe 



ki, = 





j=0 



Si 



steady rate 

 coefficients of exponential terms in equations describing 



specific activity curves in ;th compartment 

 X„ = exponential constant in nth exponential term 

 \C\ = determinant of C's 

 [C] = matrix of C's 



The One-Compartment Open System and the 

 Two-Compartment Closed System 



Some of the mathematics of the one-compartment 

 open system has already been developed as an ex- 

 ample under the explanation given of dififerential 

 equations and, of course, the one-compartment 

 system is of fundamental importance. Since, how- 

 ever, the mathematics of the two-compartment 

 closed system is only slightly more difficult, and 

 since the one-compartment system may be regarded 

 as a degenerate case of the two-compartment closed 

 system, with the second compartment being infinite, 

 the two systems may be considered together. Although 

 the object of these considerations is to develop a 

 procedure for deducing the properties of the system 

 from the behavior of a tracer in it, we begin with 

 the inverse problem of predicting the behavior of a 

 tracer in a given system. 



The two-compartment closed system may be 

 represented by the model: i i=; 2, where the numbers 

 designate the compartments and the arrows indicate 

 flow rates between the compartments in the two 

 opposing directions. In the steady state these two 

 flow rates are equal. 



If Si and ^2 represent the amounts of the substance 

 being traced present in compartments i and 2, and 

 p is the rate of flow of S, the turnover rate constants, 

 or the fractional rates of flow for the two compart- 

 ments, arep/.S'i and p/Si. It is assumed that the same 

 fractional rates of flow apply to the tracer so that, 

 in the terminology of radioactive tracers, the specific 

 activity in the outflow from each compartment at 

 any time is equal to the specific activity within the 

 compartment at that time. The differential equations 



for a tracer in a two-compartment closed system may 

 be written as 



I d {S, X,) 

 dt 



I d(S2 ATg) 



dt 



= —pXi + p.Vn 



= PATI 



pXi 



(0 

 (2) 



where in addition to the symbols defined above 



X = specific activity, or concentration of the labeled form of 

 5 in 5. 



(The equations for total substance are analogous to 

 equations i and 2.) 



Each of the above equations states that the rate of 

 change {d/di) of the amount of tracer present iSx) is 

 the difference between the rate of flow of tracer in, and 

 the rate of flow of tracer out of the compartment, in 

 conformance with the law of conservation of matter. 

 It will be noted that the sum of equations i and 2 is 

 zero, indicating that there is no change in the total 

 tracer in the system. Equations i and 2 are solved as 

 simultaneous differential equations. Several methods 

 are applicable. One which yields the general solutions 

 in an efficient manner involves the use of the Laplace 

 transform. With x\a and xm being the initial, or time 

 zero, values of .vi and .V2, respectively, the Laplace 

 transforms for equations i and 2 are : 



(-1)" 



•Vl(^) — — X-.{s) = .Vic 



- - -^M -\- 



(-0 



X,{s) = X20 



(3) 

 (4) 



Solving equations 3 and 4 algebraicalh' for .vi(.y) 

 where .vi(j-) = L [.vi(/)] 



x,{s) 



p p 



02 oi 



X2-F 



ik-l) 



which may be broken into the simpler fractions: 



-ViO — .v^ 



I 



I 



*.w = - + 



S -\ 



Si s~. 



where xe (for "equilibrium" specific activity) is 



