SERIES OR CATENATED 



INDICATOR SUBSTANCES AND FLOW ANALYSIS 637 



PARALLEL OR MAMMILLARY 



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FIG. 4. Kinetic analysis of compart- 

 mental systems using electrical analogs. 

 The analogy between the basic elements 

 of electrical resistor-capacitor (RC) cir- 

 cuits and two basic types of biological 

 compartmented systems is shown. 



that the flow rates in the two opposing directions 

 between compartments be equal. This requirement is 

 satisfied for open systems in which the flows into and 

 out of the system involve the same compartment. 

 Some of the basic compartmented systems which are 

 analyzable with RC analog circuits and typical 

 curves obtained with an analog computer are shown 

 in figure 5. In each case the tracer is initially present 

 in the compartment indicated by an asterisk. It 

 should be noted that in the two-compartment 

 system, in wliich the initially labeled compartment is 

 open, the specific activity curves cross, whereas when 

 the second compartment is open the specific activity 

 in the initially labeled compartment always remains 

 the higher of the two. 



The most satisfactory method for using an analog 

 computer to obtain data fitting curves is to generate 

 the data points electronically and to display them 

 and the analog curves on the same oscilloscope 

 screen, as is illustrated in figure 6, for a hypothetical 

 four-compartment system. Once an acceptable fit is 

 obtained, the values of the capacitor settings used 

 are proportional to the sizes of the biological com- 

 partments. Interpretation of the resistor settings 

 depends upon the time scale used. The electrical time 

 units and real time units are connected through the 

 relationship that the product of the resistance in 

 ohms and the capacitance in farads gives the time 

 constant in seconds, and this time constant is analo- 

 gous to the average time, or the reciprocal of the 

 turnover rate constant for the biological system. The 

 conversion factor, a, may be determined by calcula- 

 tion or may be determined empirically for a given 

 setting of the oscilloscope controls from the half-time 

 of a single exponential curve, which in turn may be 

 generated from the analog of a one-compartment 



open system, and using the relationship k = log a/TU 

 = i/a RC 



It is not necessary that the tracer be injected in- 

 stantaneously. Figure 7 shows a curve fitted with an 

 analog computer to data obtained when creatinine 

 was injected in a dog by an intravenous infusion 

 taking 30 min. In this ca.se a constant rate of infusion 

 was assumed, and this was simulated by a "'pump" 

 circuit giving a constant current. For more compli- 

 cated inflow patterns a function generator may be 

 needed. 



The key feature of analog circuits designed for 

 solving diff"erential equations is the operational 

 amplifier, which is a very high gain (100,000 and 

 more) DC amplifier. Stacy (67) discusses the basic 

 principles of these circuits. 



The advantage of operational amplifiers is that 

 they can be used to reproduce a given voltage at 

 another point without drawing current from the 

 original source. Depending upon the choice of 

 electrical elements used in their feed-back loops, 

 operational amplifiers may be used to add, subtract, 

 multiply, divide, integrate, and differentiate, thus 

 providing all of the operations needed for solving 

 differential equations. 



The use of operational amplifiers in conjunction 

 with RC circuits makes it possible to maintain the 

 basic simplicity of the direct analog method without 

 the restrictions which the pure RC method imposes on 

 flow rates. For example, Macdonald el al. (43) 

 utilize a circuit for the simulation of unequal forward 

 and reverse flow rates between two compartments in 

 an otherwise basically RC circuit computer. 



Among the illustrative examples which may be 

 cited concerning applications of analog computers in 



