Bone Mineral Metabolism in the Rat 



129 



corresponding values of the function /x([Ca]), expressing the frequency distribution 

 of the population as a function of the calcium content of the elementary volumes. 



i 



^ 2S00 



/500 



J 500 

 ^ 



31 i 



1.6 1.7 1.S 1.9 2.0 2.1 22 2.3 



//5d 



^/d 



J 



a 



-p 



I.S 1.7 1.8 1.920 2.1 2.2 2.3 IS f.7 1.8 1.92.0212.22.3 

 Spec. grav. [mg-/Tum''j 



Fig. 3. Frequency distribution (/x) of the elementary 



for three different ages (/) of the rat. One elementar; 



in mg mm"' . 



umes of bone 

 lume is taken , 

 the age in day 



function of their specific gravity (<3), 

 mm'. The specific gravity is measured 



If we had a mathematical expression for /x([Ca]) as it is the case for /x((o) 

 (Eq. 13), it would be possible to determine the function of calcification [Ca] (oj) in 

 the following way: 



As/x(w)= ^^}^'^ (see Eq. 7) and /x([Ca])= ^^^'\ (see Eq. 15) 



d[Ca] 



it follows that 



d[Ca] 

 d('j 



(16) 



/n(oj) 

 /x([Ca]) 



By integration, Eq. 16 would give the unknown function [Ca] (ej). 



Practically, however, if drawing histograms for /x([Ca]) is possible, there is no 

 obvious linear transformation which would permit to obtain a simple and secure 

 mathematical expression for the corresponding function. Therefore, the function 

 [Ca] (cj) has been established by calculating particular values of the function. This 

 was done as follows. 



By integrating the particular function /x("j) corresponding to any given time t\ , 

 one obtains a new function, N{oj). The known values of /x([Ca]), at given ti , can be 

 used to calculate particular values of the integral of this latter function i. e. 7V([Ca]). 

 In this way, it is possible to find for every particular value of [Ca], the corresponding 

 value of (0. 



Fig. 4 shows for all the animals, the calculated values of [Ca] plotted as a func- 

 tion of (o. Using the classical linear transformation, one finds that [Ca] {(o) can be 

 described by a single hyperbolic function, which can be written 



[Ca] 



: 0.175+ °:2^A- 

 co + 3.6 



(17) 



where [Ca] is the calcium content, expressed in mg present in the elementary volume 

 (taken as 1 mm^ of bone) and oj is the age of the elementary volume expressed in 

 days. The analysis of the adjustement of the hyperbolic function to the experimental 



3'''' Europ. Symp. on Cal. Tissues 



