128 



L. J. RiCHELLE, C. OnKELINX, J. -P. AuBERT 



The mathematical analysis of the system defined by Eqs. 10, 11 and 12 shows that 

 there exists a series of functions /x(oj), such as, at any time ti , 



h{(o) = 2akN\,,^. [l + ae-'-^'.— ' 



with Q<.(o<.t\. 



Eq. 13 can also be written in a simpler form: 



[1- 



^-kN\ 



(^l-w)]3 



(13) 



/xM 



-co) 



(14) 



2N{t[-oj)N'J^ti 

 N(ti) ' 



where N'{t) is the derivative function of N{t). 



Eqs. 13 and 14 are formally correct only if the total number of elementary 



volumes present at time t\ , does not include elementary volumes, already existing at 



f = and not yet destroyed at t = t\, meaning 

 that the age of these volumes would be greater 

 than ^i. Practically, it is difficult to take this 

 number into account, which is actually small 

 and becomes rapidly negligible when the age 

 of the animal increases. Fig. 2 shows three such 

 functions for ^1 = 31, 61 and 119 days. 



4. Function /x(r>) 

 Separation of bone samples into fractions of 

 increasing specific gravities yield values, which 

 have been expressed in terms of percentage of 

 the total sample calcium or phosphorus content, 

 found for a given interval of specific gravities 

 (RiCHELLE, 1964). In the theoretical represen- 

 tation developed here, such values express the 

 frequency distribution of the population of 

 elementary volumes for finite intervals of spe- 

 cific gravities. For any given age of the animal, 

 e. for any value of N{t), there exists a theoretical function /x(r')), such as 



Fig. 2. Frequency distributions (/.\) of the 

 elementary volumes of bone as a function of 

 their age (w), for three different ages (t) of 

 the rat. One elementary volume is taken as 

 1 mm''. Ages are measured in days 



J h{d)dd = N{t,) 



(15) 



where d is the specific gravity of the elementary volumes expressed in mg/mm^. 



Fig. 3 shows three histograms for ^[ = 31, 61 and 119 days. These histograms could 

 be used to draw the curve expressing jy,{d), if the class interval of specific gravity 

 was sufficiently small, which is experimentally impractical. 



5. Function [Ca] (ru) 



It is the purpose of this paper to establish an expression describing the kinetics 

 of the mineralization process. 



There are several possibilities; one of them is to determine the function of calci- 

 fication [Ca] (oj), i. e. the kinetics of calcium deposition in any elementary volume. 

 As we know particular values of /x(^) and the mean calcium content of a volume of 

 bone, corresponding to a given specific gravity interval, it is possible to calculate the 



