624 



HANDBOOK OF PHYSIOLOGY 



CIRCULATION I 



To work part a this way, we would have written 

 dP dt = K. In this example dP and dt may iae sepa- 

 rated and treated as if they were ordinary algebraic 

 entities, so by rearrangement we have dP = Kdt. 

 The equation can now be solved by integration, 

 which for the present purposes may be regarded as a 

 method for achieving the summation of many small 

 increments. (A rigorous explanation of integration 

 cannot be given here, but is available in the standard 

 texts on calculus.) For the present problem it must 

 sufhce simply to state that the integral of dP is P 

 and that of Kdt is Kt, if A' is a constant. Integration 

 also introduces another terin called the constant of 

 integration, which in this e.xample is the starting 

 value of P, or Pf, . We thus have, as expected, the 

 solution, P = Po + Kt 



It is apparent that any value of Po could be used 

 in this equation, and in general it is true that the 

 constant of integration may have any of a wide 

 range of possible values. Thus another characteristic 

 of differential equations is the infinitude of the number 

 of solutions. A solution expressed in a form in which 

 arbitrary values may be assigned to the constants of 

 integration is called the general solution; a solution 

 having definite values assigned to the constants of 

 integration is called a particular solution. 



The solution of case b follows the same procedure, 

 and we begin with dP/dt = KP, which upon separa- 

 tion of the variables becomes dPP = Kdt. Again 

 the integral of Kdt is A7, but the integral of dP/P 

 happens to be log P, with the constant of integration 

 being expressible as log Pn , so the solution for b is: 

 log P = log Pa -+- Kt. In this branch of mathematics 

 it is "understood" that log means the logarithm to 

 the base e (2.718 •■■), or the natural logarithm, 

 rather than the common logarithm using the JDase 

 10. Any number could be used as the base of a 

 logarithm scale, but e is preferred because its func- 

 tions have the simplest results when subjected to the 

 processes of differentiation and integration, not 

 requiring a conversion constant. With this under- 

 standing, the above answer may be converted to the 

 form : P = Poe'^K When the rate of decrease is pro- 

 portional to the amount present, —A' is used and 

 we have: P = PoC'^'. The latter equation appears 

 frequently, as it describes the rate of radioactive 

 decay and similar processes. It also describes the 

 disappearance of a label from a one-compartment 

 open system when Pu of the label is initially present, 

 the inflow is not labeled, the inflow and outflow 

 rates of the substance being traced are equal, and 

 the flow rates are constant. 



In problems involving more than one compartment, 

 a system of differential equations is used, with one 

 equation lor each compartment. In general, it is 

 possible to reduce such a systein to a single equation 

 which will inx'olve the second and higher derivatives 

 of a single variable, d'-P/dt-, d'P/dt', etc. Those 

 differential equations which involve onlv the first 

 powers of these derivatives are called linear equations. 

 The equations useful in describing the behavior of 

 tracers in compartmcnted systems in the steady 

 state are generally linear differential equations with 

 constant coefficients and are relatively easy to solve, 

 as differential equations go. One form of the solutions 

 of such equations is a series of exponential terms. 

 For example, an equation of the form: 



d^.x d-x d.x 



\- a 1- A — -f- f.v = </ 



di' df' dt 



has a solution of the form: 



X = Ae-^i' + Be-'^i> + Ce~'^3' + D 



The three exponents, Xi, Xj, and X3 are the three 

 roots of the auxiliary equation: ( — X)^ + a( — \)- -\- 

 b{ — \) -|- f = o. The coefficients. A, B, C, and D 

 depend not only upon a, b, c, and d, but upon the 

 "boundary" or starting conditions and are not readily 

 expressed in a general form. Of course this is only 

 one example of the differential equations method, 

 but it is one with direct applications in tracer theory. 

 Further examples appear in the treatment of com- 

 partmcnted systems which follows. Among the 

 several textbooks which can be recommended for 

 further studv is Ford (25). 



MATRICES AND DETERMINANTS. Both matrices and 

 determinants are arrays of numbers arranged in 

 rows and columns, but quite different properties are 

 assigned to them and their uses are correspondingly 

 different. 



We can mention some of the most useful features 

 here but can really hardly begin to explore the 

 subject. For those who wish to study the methods 

 further, an excellent but abstract approach is avail- 

 able in Finkbeiner (23) and a more easily followed, 

 if somewhat chatty, approach is found in Stigant 

 (69). A thorough treatment of determinants is avail- 

 able in Muir & Metzler (44). 



The fundamental distinction between a determi- 

 nant and a matrix is that the former represents one 

 number and can be reduced according to an es- 

 tablished set of rules to a single numerical value, 



