V GROWTH IN TIME OF THE TOTAL ORGANISM 207 



differences like that between mouse and rat, as mentioned, would remain as 

 unexplained as they are otherwise on the basis of available physiological knowledge. 

 The simplest assumption for growth reaching a terminal value in finite time 

 is that the exponent k is not constant but decreases monotonically with time 

 (Haardick, 1956) : 



with ^ = A' at the beginning of growth, and T = duration of growth. This gives 

 for length growth : 



d (/*-/) __ KT 



~dt T-t 



which is a modification of equation (5.26, 5.29) 



d (/*-/) 



(/*-/) (5-46) 



By integration we obtain : 



= —kil*-l) 



KT 

 I ^l*-l 



(i-^) (5-47) 



which, for T — > 00 , gives equation (5.29) as a special case. 



This equation takes account of the terminal length (/*) as well as the finite 

 duration of growth (T). 



A case calculated with the generalized formula (5.47) is length growth of 

 vertebrae in the rabbit (Haardick, 1956). It appears that application of equation 

 (5.29) is not satisfactory because the value of k changes, but equation (5.47) 

 with KT = 3 gives satisfactory fit. However, numerical calculation according 

 to equation (5.47) is somewhat elaborate, and decrease of k according to this 

 equation has no physiological proof at present. 



5. 77?^ most severe oversimplification in Bertalanffy's model is the lumping together 

 of growth-promoting and growth-inhibiting processes under terms of "anabolism" 

 and "catabolism of building materials". This is justified insofar as the model 

 yields considerable "dividends", the mathematical structure remaining simple 

 enough for easy implementation; combined with the principle of Occam's razor 

 that no factors should be introduced into a theory that cannot be empirically 

 tested. However, this does not preclude but rather implies that the model should 

 be further elaborated both by refining the mathematical structure and further 

 analysis of the overall terms used, if and when complicating factors appear and 

 are amenable to experimentation. 



This criticism particularly applies to the identification of the parameter of 

 "catabolism of building material" (x) with protein metabolism. Experience shows 

 that this simplification applies to a range of "model organisms" where the growth 

 curves as well as the numerical value of the parameter in question are predicted 

 by the theory with considerable accuracy. There is no doubt, however, and it 

 has been strongly emphasized (Bertalanffy, 1934, 1951a; Racine, 1953) that this is 

 a first approximation. 



Literature p. 253 



