636 HANDBOOK OF PHYSIOLOGY ^> CIRCULATION I 



which include tlie factor f~^'' are determined by 



(Xi - kr2)Cu + A-12C2, + 0=0 (64) 



isiCi, + (X, - A-,)C5, + kiiCii = o (65) 



o + A-3=Co, + (X - A-3!)C3i = o (66) 

 Equation 64 gives Coi in terms of Cn directly: 



(67) 



„ (h, - Xi) 



Solving equation 66 for C21 in terms of C:n and 

 substituting the result in equation 67 yields 



^32(Xl — A12) 



(68) 



Similarly, the coefficients of the terms containing 



-X.>( 



e - are 



(A-12 — X2) hii'Ki — kn) 

 C22 = ; ^12 3X\(i C32 = ; — 7; ; — - C12 (,09; 



kn 



kn0^2 — "-'32) 



Inclusion of equations expressing the initial condi- 

 tions provides the necessary basis for calculating the 

 absolute values of the C's. For example, if the tracer 

 is initially confined to compartment i in i ^ 2 ;=i 3, 

 at time zero: 



ATIO = A-£ + C'li + C12 



AToo = o = a;£ + Coi + C22 (70) 



Xiti = o = xe + C31 + C32 



Combining equations 67, 68, 69, and 70 leads to 



^^ /^\/A-3,-^A/^^X , 



Xe \ki'.)\kn - X,/\X, - X2/ 



Xe \ ^32 /\X. - Xj/ 



/^32^2\/_J^\^,^, 

 \ A32 /\X2 - X,/ 



Xz , A2 



— = I + ( 



Xe Xi — X2 



+ 1 

 X, 



(7O 



Electronic Computer Application 



Even when only three compartments are inxolved, 

 the relationships between the tracer data and the 

 system parameters arc sufficiently complicated to 

 discourage the use of desk calculations when there 

 are much data to process. Although various kinds of 



analog systems, particularly hydrodynamic systems, 

 are of value in studying compartmented systems and 

 in teaching the principles involved, when it comes 

 down to calculating the results from real systems, the 

 most effective assistance is obtained through the use 

 of electronic computers. 



The two basic types of computers, analog and 

 digital, may be typified by the slide rule and the desk 

 calculator. The former has the virtues of low cost, 

 speed, and convenience, whereas a major advantage 

 of the latter is precision. To a large extent these 

 qualities also hold for the larger electronic counter- 

 parts of each kind of machine. The analog computers 

 are useful for quickly determining which of several 

 possible kinds of systems provides the best explanation 

 for a given set of data. Once the basic model system 

 to be used is established, the digital computers 

 provide a higher precision than can be achieved with 

 analog machines, or if there are ambiguities due to 

 incomplete data available, to establish the range of 

 possible values. 



ANALOG COMPUTER METHODS. Analog Computer 

 methods may be divided into those which employ 

 direct approaches and those which involve indirect 

 approaches. By the direct approach is meant the use 

 of a system in which the electrical components are ar- 

 ranged to give a direct analogy of the biological sys- 

 tem, in contrast with the indirect approach in which 

 the electrical components are arranged to solve the 

 differential equations which describe the behavior of 

 the labeled material in the system. The indirect ap- 

 proach provides greater generality but the direct ap- 

 proach is simpler. Since many of the problems of com- 

 partmental analysis can be treated satisfactorily by the 

 direct approach, its use will be emphasized here. The 

 problems for which the direct approach is inadequate 

 can often be handled by a combination of the tech- 

 niques of the two approaches without the necessity of a 

 complete conversion to the indirect approach. 



The basis for the direct approach is the fact that 

 the equations which describe the specific activity of 

 a tracer in certain compartmental systems are 

 identical in form with those which descrilje the 

 voltages on capacitors in resistor-capacitor (RC) 

 circuits and other first order reactions. 



Figure 4 shows the RC circuit analogs of the four- 

 compartment series and parallel systems. In these 

 systems capacitance is analogous to the size of the 

 compartment and voltage to specific activity. The 

 resistor values are inversely proportional to the flow 

 rates. For pure RC analogies to be valid, it is essential 



