Bone Mineral Metabolism in the Rat 131 



Because of the complexity of the system and, moreover, of the existence of iso- 

 ionic exchange, it is evident that no simple description of the radioactive calcium 

 incorporation to the elementary volumes can be proposed. It is possible however to 

 test the validity of the function of calcification by another experimental approach. 

 Eq. 18 can be used to calculate the age of an elementary volume as a function of its 

 calcium content, which is given by 



[Ca]-[Ca]„ 

 [Ca],.ax.-[Ca] ^ ^ 



On the other hand, the age of the elementary volumes can be determined experi- 

 mentally by labelling collagen with radioactive proline. In such an experiment, at the 

 time of the injection, only the collagen of the forming elementary volumes is labelled. 

 Later, the maximum specific activity of the collagen hydroxyproline indicates the 

 calcium content of the elementary volumes formed at the time of the injection. This 

 measurement, done at different times after the injection, permits to establish on other 

 experimental data, the function (o [Ca]. The actual data (Lapiere, 1965) are in 

 agreement with the values calculated from Eq. 21. 



Conclusions 



This report represents an attempt to develop a formal analysis taking into account 

 the heterogeneity of bone tissue In this analysis, bone is considered as a heterogeneous 

 population of elementary units. The evolution of the system and of its elements has 

 been described by a series of time functions. 



A set of these functions (Eqs. 10, 11 and 12) describe the evolution of the popula- 

 tion as a whole. They are obviously simplified expressions of the underlying phe- 

 nomena, which involve the complex processes of cellular activity. Such expressions 

 would be of little help if one attempted to study specifically these processes in growth 

 or aging. 



However, general expressions of this type, which describe the evolution of the 

 whole system are nevertheless useful as operational definitions. They make it possible 

 to give numerical values for the rate of growth, the rate of appearance and the rate 

 of destruction of the elements of the population. Indeed, these values are necessary in 

 order to analyse the kinetic events which take place in individual elements of the 

 population. 



Among the various events taking place, we have limited ourselves in this paper to 

 the study of the calcification. A hyperbolic function has been found to describe 

 adequately the kinetics of this process. It remains to be shown what are the actual 

 phenomena responsible for this type of kinetic behaviour. 



References 



AuBERT, J.-P., et G. Milhaud: Methode de mesures des principales voies du metabolisme 



calcique chez I'homme. Biochim. biophys. Acta 39, 122 (1960). 

 Deakins, M.: Changes in ash, water, and organic content of pig enamel during calcification. 



J. dent. Res. 21, 429 (1942). 

 Herman, H., et L. Richelle: Le calcium echangeable de la substance mincrale de I'os 



etudiee a I'aide du Ca-45. VII. Activite comparee de fractions d'os total de densite dif- 



ferente. Bull. Soc. Chim. biol. 43, 273 (1961). 

 Lapiere, Ch. M.: The remodelling of the bone matrix. This book, p. 20. 



9=:- 



