Use of the stereological counting measurements in testing theories of growth rates

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city r e q u i r e s s t e r e o l o g i c a l e s t i m a t i o n of the s u r f a c e a r e a of the moving b o u n d a r y p e r unit volume of s t r u c t u r e , SV, in a d d i t i o n to the v o l u m e f r a c t i o n . If the l o c a l v e l o c i t y of a m o v i n g s u r f a c e e l e m e n t is v, then t h i s " s u r f a c e a r e a a v e r a g e d i n t e r f a c e v e l o c i t y " is given b y : f f vdS s 1 dVv [11 Vs=- f f dS - S v dt S w h e r e the i n t e g r a t i o n is o v e r the a r e a of the b o u n d a r y . T h i s r e l a t i o n i s g e n e r a l , and r e q u i r e s no s i m p l i f y i n g g e o m e t r i c a s s u m p t i o n s about the n a t u r e of the p a r t i c l e s involved in the p r o c e s s . T h i s m e a s u r e of the a v e r a g e growth r a t e has b e e n a p p l i e d to a s s e s s g r o w t h r a t e s in a n u m b e r of p r o c e s s e s ; s o m e t i m e s the r e s u l t s a r e confusing, or, w o r s e , m i s l e a d i n g . ~ It is the p u r p o s e of this p a p e r to d e m o n s t r a t e that, while Eq. [1] p r o v i d e s a v a l i d m e a s u r e of the a v e r a g e i n t e r f a c e v e l o c i t y in an e v o l v i n g m i c r o s t r u c t u r e , t h i s i n f o r m a t i o n m a y be only i n d i r e c t l y r e l a t e d to r a t e s of change of c h a r a c t e r i s t i c p a r t i c l e d i m e n s i o n s , which a r e the t y p i c a l d e s c r i p t o r s d e r i v e d in m e c h a n i s t i c m o d e l s f o r growth. Indeed, in s o m e c a s e s , v s contains no i n f o r m a t i o n at a l l about such growth r a t e s . Thus, while ~-~ is a v a l i d m e a s u r e of a v e r a g e i n t e r f a c e v e l o city, it is not n e c e s s a r i l y a u s e f u l one. In p a r t i c u l a r , it will be shown that ~ is an i n a p p r o p r i a t e d e s c r i p t o r of growth (in the s e n s e that it is not r e l a t e d to r a t e s of change of c h a r a c t e r i s t i c d i m e n s i o n s ) in plate growth and r o d growth. Two o t h e r a v e r a g e int e r f a c e v e l o c i t i e s , weighted s o m e w h a t d i f f e r e n t l y , have b e e n defined, ~ and m a y be d e t e r m i n e d f r o m global p r o p e r t i e s that a r e a c c e s s i b l e to s t e r e o l o g i c a l e s t i m a t i o n . T h e mean c u r v a t u r e weighted a v e r a g e i n t e r f a c e v e l o city, VH, i s defined by weighting the l o c a l v e l o c i t y with the l o c a l value of the m e a n c u r v a t u r e of the s u r f a c e , H, d e f i n e d a s the a v e r a g e of the two p r i n c i p a l n o r m a l c u r v a t u r e s for e a c h s u r f a c e e l e m e n t [H =- 1/2(1/71 + l / r 2 ) ] . It can be shown that this v e l o c i t y a v e r a g e i s r e l a t e d to the s u r f a c e a r e a p e r unit v o l u m e , SV, and the i n t e g r a l m e a n c u r v a t u r e p e r unit v o l u m e , M V , which is the i n t e g r a l of the m e a n c u r v a t u r e o v e r the s u r f a c e --

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