INDICATOR SUBSTANCES AND FLOW ANALYSIS 



63. 



f = 



t = 00 



• • 



• • • 



• • • 



• • • 



• • • 



AAA 



AAA 

 AAA 

 AAA 



AAA 

 AAA 



A A 



■ A A 

 AAA 



A A 

 A A 



■ ■ A 

 AAA 

 AAA 

 A A • 



• • • 



• • • 



activity in the kh compartment of an n-compartment 

 system is tlien 



_i — I 1 1 — I 1 — J 



2 4 6 8 10 12 14 



FIG. 2. Redistribution in a three-compartment system. Tlie 

 circles, triangles, and squares each represent a unit amount of 

 the same substance according to whether it is initially in the 

 top, middle, or lower compartment. The pattern at the right 

 indicates how the substance is redistributed at equilibrium. 

 The cur\es indicate the sequence by which this redistribution 

 takes place and correspond to the curves which would be 

 exhibited by a label introduced into the compartment indi- 

 cated by the shajie of the symbol and sampled in the compart- 

 ment indicated by the position in the diagram. Thus the lower- 

 most curve indicates the rate of appearance in the bottom 

 compartment of a tracer introduced at time zero in the top 

 compartment. A numerical analysis of this system illustrating 

 how one set of curves yields, by matrix algebra, the flow rate 

 constants for the system, is presented in the text. 



factors in tlie equations wlien multicompartment 

 systems are analyzed, and the form of the equations 

 is simplified if each p/S factor is replaced by a single 

 symbol. For this purpose the fractional turnover rate 

 (or flow rate) constant, k,j, is defined in terms of the 

 rate of flow into the ilh compartment by the formula : 

 kij = pij/Si. Because perfect mixing is assumed, the 

 specific activity is the same for all outflows from a 

 given compartment, and notational complexity may 

 be avoided by collecting all of the outflow fractions, 

 the Pji/Si terms, in a single symbol, Ki, the total 

 turnover rate constant, defined by Ki = y^,- pg/Sj. 

 Of course. A', = ^y pa/Si also, by definition of the 

 steady state. In nonsteady states the turnover rate is 

 not constant and A' is not used. 



The basic differential equation for the specific 







(14) 



where 7 = o is used to designate the inflow in open 

 systems when the inflow is labeled and when the 

 specific activity in the inflow is constant. For a system 

 of n compartments there are n equation 14's, each 

 having up to n -{- i terms. Equation 14 is a linear 

 differential equation with constant coefficients and a 

 set of n such equations has as a solution a set of n 

 equations of the following type: 



•v.- = *£ + C,i «-M' -f C,n e-^2' 



+ C„ 



(■5) 



Equation 15 may be derived from equation 14 by 

 the Laplace transform method, which yields the 

 C's and X"s in terms of the k's, as is illustrated above 

 for the two-compartment system, or equation 15 may 

 be taken as an assumed solution and the C's and 

 X's calculated independently by methods described 

 below. 



Open ^-compartment systems have n exponential 

 terms in each equation 15, and xe (the equilibrium 

 or infinite time value of all the .v's) is equal to the 

 specific activity in the inflow, which is assumed to be 

 constant or zero for the present discussion. In closed 

 systems there are only (n — i) exponential terms in 

 addition to xe- As is illustrated above for the two- 

 compartment closed system, the Laplace transform 

 method may be used to derive equation 15's from 

 equation 14's. 



CALCULATION OF RATE CONSTANTS. As the Starting 



point for this phase of the analysis it is assumed that 

 the experimental data have been processed to the 

 point where they are described by a set of equations 

 of the equation 15 type. In other words, it is assumed 

 that experimentally determined values of the C's 

 and X's are available. If data are available from all 

 compartments or for closed systems, the following 

 general procedures provide a method for calculating 

 the rate constants, and hence the flow rates and 

 quantities which characterize the system. The 

 handling of incomplete data will be considered 

 subsequently. 



The relationships among the I's in equation 14 and 

 the C's and X's in equation 15 may be established by 

 equating the first derivative of equation 1 5 to equation 

 14, with the right-hand members of equation 15 

 being substituted for the .v"s in equation 14. Thus the 



