INDICATOR SUBSTANCES AND FLOW ANALYSIS 



623 



ionizing radiation will perturb the system. In general, 

 this turns out not to be a real problem when tracers 

 are used but in some situations, as in the use of 

 labeled substances which concentrate in a particular 

 organ or in a sensitive location such as cell nuclei, the 

 possibility of getting radiation eflfects must be care- 

 fully evaluated. 



Finally, it is essential that the material introduced 

 as a tracer not have toxic or pharmacological prop- 

 erties which would disturb the system. When the 

 substances being studied are normal constituents 

 of the body, this is no problem, but when labeled 

 drugs are being studied, there may be undesirable 

 effects. 



HOMOGENEITY WITHIN COMPARTMENTS. For simplicity 



of the mathematical considerations, it is usually 

 necessary to regard the system being studied as 

 consisting of a number of compartments, within 

 each of which mixing is regarded as being immediate 

 and perfect, giving homogeneity at all times within 

 a given compartment. Those who know what really 

 happens in biological systems are somewhat justified 

 in regarding this as the most outrageous a.ssumption 

 made. No account is taken of the known concentra- 

 tion gradients within cells and across membranes. 

 Because of the fact that it does take a definite time, 

 measured in seconds, for the blood to complete a 

 loop of its path from the heart to a tissue and back 

 to the heart, mi.xing within the circulation obviously 

 cannot be instantaneous. Ne%'ertheless, analyses 

 based on the assumption of perfect mixing have been 

 successful as the basis for explaining the results of 

 many tracer experiments, and the model system 

 constructed using this assumption does provide a 

 basis for evaluating the importance of factors which 

 require modification of the assumptions. 



MATHEMATICAL B.^SIS 



Synopsis of Mathematical Techniques 



Because many biologists who are interested in 

 tracer methods have not had training in some of the 

 branches of mathematics of use in developing the 

 theoretical basis for the interpretation of the behavior 

 of tracers, it seems appropriate to include at this 

 point a brief introduction to pertinent selected topics 

 in these mathematical methods. It is hoped that 

 workers who have heretofore avoided such subjects 

 may gain some insight into the power of the applica- 

 tions and be encouraged to deeper study, mathe- 



matics being one of the few subjects which can be 

 mastered through self-study. In general, however, 

 textbooks of applied mathematics are written for 

 physicists, chemists, engineers, etc., and, excepting 

 the field of statistics, there is a dearth of mathematics 

 books concentrating on topics chiefly of interest to 

 biologists. A recently published book by Defares & 

 Sneddon (20) does emphasize biological applications 

 and is recommended for further study. 



LINEAR DIFFERENTIAL EQ^UATioNS. Differential equa- 

 tions are in many respects like any of the familiar 

 algebraic equations, but there are, of course, some 

 important differences. The distinguishing char- 

 acteristic is that the rate of change of a variable is 

 expressed in terms of the values of the variable itself. 

 Such an equation cannot be solved by the usual 

 algebraic techniques. In fact, the term "solved"' 

 acquires a somewhat different meaning. Whereas in 

 algebra, solving an equation involves finding nu- 

 merical or symbolic values for constants which 

 satisfy the equation, the solution of a differential 

 equation is usually an algebraic equation or a family 

 of algebraic equations describing the functional 

 relationships of the variables involved. 



For example, suppose that we are asked to describe 

 the growth of a cell population, given the information 



a) that the rate of growth is constant or, alternatively, 



b) that the rate of growth at any moment is pro- 

 portional to the population at that moment. Using 

 the symbols P for population, R for rate of growth, t 

 for time, and A' for the constant, we can answer 

 part a immediately; but part b is more difficult. 

 For part a we need only assume some starting value 

 for P, Pa, at time zero, and note that since the rate of 

 growth is constant, i.e., R = K, the number added to 

 P in time t is simply Rt = Kt, so we have P = Po + 

 Kt. It should be noted that Kt may be thought of as 

 the result of summation of the growth which occurs 

 in all of the shorter intervals of time which add up 

 to t. For part b we may write R = KP, but if we 

 are restricted to algebraic methods, this is the end 

 of our road. In this case the rate is changing con- 

 tinuously, and it is not correct to use Rt to sum up 

 the addition to the population, as doing so implies 

 that R is constant, as it was in part a. 



Instead, we replace i? by a special symbol, dP/dt, 

 the first derivative of P, which may be thought of as 

 the ratio of an infinitesimally small change in P, dP, 

 to a correspondingly very small change in t, dt, a 

 concept consistent with our usual notion of the 

 meaning of rate of change at any instant. 



