632 



HANDBOOK OF PHYSIOLOGY 



CIRCULATION I 



first derivatives of equation 1 5 have the form 



dt 



= -Ca Xi f-^i' + C„. X,. e-Ki 



(.6) 



whereas substitution of the right-hand members of 

 the equation 15's in the equation 14's and collecting 

 the coefficients of the exponential factors gives 



dx. 



(kn Cn ■ 



K,Cn + A:.„C„i)f-H' 



dt (,7) 



+ + (/.-,i C,„ A'. C,„ + ki„ C„„)c-*-.' 



Not only is equation 16 = equation 17, but the 

 coefficient of each exponential factor {e~ ' ) in 

 equation 16 equals that of the corresponding term in 

 equation 17. These relationships provide a basis for 

 expressing the values of the rate constants, the k's, in 

 terms of the Cs and X's, and for expressing the C's in 

 terms of the k's and X's. 



The most general technique involves the use of 

 matrix algebra. The matrix of CX products in equation 

 16 is regarded as being generated by post-multiplica- 

 tion of a matrix composed of the C terms in equation 

 15, [C], by the diagonal matrix of the exponents 

 [ — X]. Thus [C] [ — X] represents 



C„ 



C,„ 



C„. 



C,„n 



-X„ 



Similarly, the kC products in equation i 7 are gener- 

 ated by pre-multiplication of [C] by the matrix of 

 k^s, [k], from equation 14, arranged by columns of 

 Xi, Xi x„ : 



-A', k,, 

 k-n -K. 



ku, 



k-.n 



-A'„ 



[C] 



The equality between equation 16 and equation 17 

 thus gives 



mm = [c][-x] 



(18) 



Post-multiplication of both sides of equation 18 by 

 the inverse of [C], \CY > solves equation 18 for \_k\ 

 since by definition 



[C] [C]-' = [i], 

 the identity matrix, and 



[k] [C] [C]-' = [k] = [C] [-X] [C]-i 



(19) 



The catch to this simple-appearing method is the 

 difficulty of generating [C]~' when real data invoh'ing 

 decimal fractions are used. (Matrix inversion pro- 

 grams are routine for modern digital computers, so 

 this is not a proljlem if such a machine is available.) 

 Also, in principle, [C] requires data from all n com- 

 partments although Berman & Schoenfeld (7) and 

 Berman et al. (9) have developed techniques for 

 working with incomplete data. 



A numerical example for the hypothetical model 

 system illustrated in figure 2 will serve to illustrate 

 the method implied in equation ig. 



In the system i ?=i 2 ^ 3 having Si = 15, 6*2 = 

 10, Si = 8, pio = P21 = 30 and P23 = paa = 40, the 

 differential equations for specific activities are 



— 2.V1 + 2. v.; 4- O 



dxi 



'dt 



dx. 



-:r = 3*1 - (3 + 4) *2 -f- 4^i 



at 



(20) 



dxz 

 dt 



= 0-1- 5.V, - 5*3 



If in this system .vjo = 88 and .Voo = x-m = o, the 

 'data" will be described bv 



xi = 40 -h 4 <■ '" -f 44 e" 



ATj = 40 — 18 f~'" — 22 



ATa = 40 -f 15 «-" 



(21) 



55 « 



And 

 [C][-X] = 



(22) 



The inverse matrix [C]~' is generated as follows. 



Using the method given in BirkhoflF & Mac Lane 

 (12, p. 210), in which the inverse matrix is generated 

 by determinants, and using D for the denominator 

 determinant, we have 



D = (4o)(ii) 



h= (II) 



il 



15 



= (4o)(i 0(264) (23) 



D 



15.4, -f loAi -f 8/I3 

 1320 



(24) 



