Bone Mineral Metabolism in the Rat 127 



3. Function fy{(o) 



In a population, the elements of which are continuously renewed by processes of 

 appearance and disappearance, it is possible, at a given time, to define an age for 

 every element of the population and to draw the curve expressing the frequency 

 distribution of the population as a function of the age of the elements. 



In the kinetic analysis of bone metabolism, it is necessary, therefore, to distin- 

 guish two types of time functions: in the first case, the time is measured by the age 

 of the animal and is refered to as t\ in the second case, the time is measured by the 

 age of the elementary volumes and is refered to as co. Hence, for an animal of age ti , 

 the range of ages for the elementary volumes will extend from up to t\ . 



For any given age of the animal, i. e. for every particular value of N{t), there 

 exists a corresponding function of frequency distribution /x(('j), so that 



J/x(oj)dw = N(fi) (7) 







As N{t) is known (Eq. 5), in order to calculate /x(oj), it is necessary to know either 

 the function of formation or the function of destruction of the elementary volumes. 

 Because, as mentionned earlier, the process of bone destruction appears to bear syn- 

 chronously on all the constituents of bone, the experimental values of the intensity of 

 calcium removal from bone obtained by the kinetic analysis of calcium metabolism 

 in vivo can be used to establish the mathematical expression of the function of 

 destruction. This transformation is done by the same procedure as used to obtain 

 Eq. 5. One gets Eq. 8: 



^■'W-I + hE-Wt (8) 



Eq. 8 expresses the evolution with the age of the animal (t), of the total number of 

 elementary volumes which have been destroyed, expressed in number of mm-''. Hence 

 the rate of destruction can be written: 



' '' = 1.199X10-5 A? (4452 _N] (9) 



Comparison of Eqs. 6 and 9 shows that the rate of growth and the rate of destruc- 

 tion are given by functions which, for all practical purposes, can be considered 

 identical. Identity of Eqs. 6 and 9 simplifies the mathematical treatment. A more 

 general solution when these equations are not identical is possible, but will not be 

 presented here. 



The behaviour of the population of so-called elementary volumes can, therefore, 

 be described in a general manner by the following equations: 



a) the rate of growth is given by 



'Jj^ =kN{N„,,,,-N) (10) 



b) the rate of destruction is given by 



df 



c) it follows that the rate of appearance is given by 



with ki = 2k. 



^^^'' =^N(N,„„.-N) (11) 



^=k,N{N,,,..~N) (12) 



