531/534:[57+61] . . , . . A.A. Zaborskikh, V.M. Tverier Perm National Research Polytechnic University DEVELOPMENT OF MATHEMATICAL DESCRIPTION OF THE INTENAL STRUCTURE OF CANCELLOUS BONE . , . ; (fabric tensor). , . : , ( , ) , , , . Consideration of structural features of the dentofacial system units is one of the main problems of contemporary dental biomechanics. Heterogeneity of spongy structure can be described by methods of quantitative stereology. At the same time, structural bone tissue features are described by means of the fabric tensor . The measuring procedure for stereological investigations is analyzed. Properties of fabric tensor by constructing it for a sample of cancellous bone, carved out of the femoral head of human was investigated. Keywords: biomechanical modelling, Wolff’s law, bone tissue structure, cancellous bone tissue, fabric tensor, ellipse structure. 200 , 158 , . (substantia compacta) . (substantia spongiosa) , , - . . , , - [1]. . , [2]. , , , , , , . - , , , , , - . ( ) . . - : , [2]. . . - , , , [3–9]. , - , , [10–12]. . - Image Tool . . : . ( L. ) . L - . . Underwood [6], – 159 , . L ( . 1. - L , ( . 1). Ab. I( ). – – . [13] , ; , – ) - L b ( θ ) = 2 ∑ AAb , I ( θ) l l– – , Lb( ) ; ; I( ) – Ab – . - Lb( ), . - 2 1 mαα + mββ mαα − mββ + cos 2θ + mαβ sin 2θ, = L θ 2 2 ( ) b g Harrigan ( , . (1). Mann , ) : !) eg e , 1984 . [14], , 2 1 = n ⋅ M ⋅ n, L b ( θ) 160 (1) , - ( Lb - n– – - n = cos θeα + sin θeβ , , . : mgg, m , mg , m11 M = m12 m13 m13 m23 . m33 m12 m22 m23 Whitehouse (1974) [15], Harrigan , [18]. [17] Mann (1984) [16], Turner (1987) , , [12] . ( Lb( ) - ) . 1986 . Cowin , , . , : ( ) 1 2 = M −1 . , H , , . . R def α= mαα + mββ 2 def , β= mαα − mββ 2 def , γ =mαβ , 1 2 α+c R= . α−c 161 , , mαα M = mαβ ) mgg, m Lb( ) mαβ . mββ M ( , (1) mg [19]. (1) f( ), - , 2 1 f ( θ) = . L b ( θ ) . : def - def f l = f ( θl ) , ψ l = 2θl , l = 1, 2, 3, fl – - l. mgg, m mg - : mαα = f 2 (1 − cos ψ1 ) − f1 (1 − cos ψ 2 ) ψ + ψ2 + κ ctg 1 (1 − cos ψ1 ) − sin ψ1 , (2) cos ψ 2 − cos ψ1 2 mββ = f1 (1 + cos ψ 2 ) − f 2 (1 + cos ψ1 ) ψ + ψ2 − κ ctg 1 (1 + cos ψ1 ) + sin ψ1 , (3) cos ψ 2 − cos ψ1 2 mαβ = κ. , def κ= k– , : 162 (4) , f 3 − k ⋅ f 2 + f1 ( k − 1) , sin ψ 3 − k ⋅ sin ψ 2 + sin ψ1 ( k − 1) - cos ψ 3 − cos ψ1 . cos ψ 2 − cos ψ1 def k= sin ( ψ1 − ψ 2 ) + sin ( ψ 2 − ψ 3 ) + sin ( ψ 3 − ψ1 ) ≠ 0. 2 f1 f 2 f 3 : , 1 = 0°, 2 = 120° 3 = 240°. (2)–(4) : mαα = f1 , mββ = 1 ( 2 ⋅ ( f 2 + f3 ) − f1 ) , 3 3 ( f3 − f 2 ) , 3 mαβ = k = 1. f1 > 0, ( ) 2 ( f1 f 2 + f 2 f3 + f1 f3 ) > f12 + f 22 + f32 . ( , ), - , [20]. , ( , , . . ) = H (d, N, d, N , – ( , ) - ), , . , - . 163 , . [21] ( . 2). - 600 ° 4 . - Image Tool. ( . 3). , . . 2. [22] ( ( . 3. 164 - . 5). . 4. . 4), - . 5. – ; – ; – ; : – – ; 165 , , - . , , . – . . - , . , - [23–25]. 1. : . 1 / . . .– : , 1973. – 315 . 2. Martin R.B., Burr D.B., Sharkey N.A. Skeletal tissue mechanics. – 2nd edition. – New York: Springer-Verlag, 1998. – 392 p. 3. . . . – .: , 1958. – 446 . 4. Lloyd E., Hodges D. Quantitative characterisation of bone. A computer analysis of microradiographs // J. Clin. Orthop. – 1971. – Vol. 78. – P. 230–250. 5. Underwood E. Quantitative stereology. – Mass.: Addision Wesley, 1970. – 274 . 6. Whitehouse W.J., Dyson E.D., Jakcson K.C. The scanning electron microscope in studies of trabecular bone in a human vertebral body // J. Anat. – 1971. – Vol. 108. – P. 481–496. 7. Whitehouse W.J. The quantitative morphology of anisotropic trabecular bone // J. Microscopy. – 1974. – Vol. 101. – P. 153–168. 8. Whitehouse W.J. A stereological method for calculating the internal surface areas in structures which have become anisotropic as the result of linear expansions or contractions // J. Microscopy. – 1974. – Vol. 101. – P. 169–176. 9. Whitehouse W.J., Dyson E.D. Scanning electron microscope studies of trabecular bone in the proximal end of the human femur // J. Anat. – 1974. – Vol. 118. – P. 417–444. 10. Cowin S.C. Fabric dependence of an anisotropic strength criterion // J. Mech. Materials. – 1986. – Vol. 5. – P. 251–260. 166 11. Cowin S.C. Bone Mechanics Handbook. – 2nd edition. – New York: CRC Press, 2001. 12. . . . . : . 1. – .: « », 1998. – 463 . 13. Turner C.H., Cowin S.C. On the dependence of elastic constants of an anisotropic porous material upon porosity and fabric // J. Mater. Sci. – 1987. – Vol. 22. – P. 3178–3184. 14. Harrigan T.P., Mann R.W. Characterization of microstructural anisotropy in orthotropic materials using a second rank tensor // J. Mater. Sci. – 1984. – Vol. 19. – P. 761–767. 15. Whitehouse W.J. A stereological method for calculating the internal surface areas in structures which have become anisotropic as the result of linear expansions or contractions // J. Microscopy. – 1974. – Vol. 101. – P. 169–176. 16. Cowin S.C., Mehrabadi M.M. Identification of the elastic symmetry of bone and other materials // J. Biomechanics. – 1989. – Vol. 22. – P. 503–515. 17. Telega J.J., Jemiolo S. Fabric tensor in bone mechanics // J. Engineering Transactions. – 1998. – Vol. 46. – P. 3–26. 18. Fabric and elastic principal directions of cancellous bone are closely related / A. Odgaard, J. Kabel, B. van Rietbergen, M. Dalstra, R. Huiskes // J. Biomechanics. – 1997. – Vol. 30. – P. 487–495. 19. / . . , . . , . . , . . , . . // . – 2007. – . 11, ヽ 1. – . 9–24. 20. Wolff J. Das Gesetz der Transformation der Knochen. – Berlin: Hirshwald, 1892. . ., . ., . . 21. // . – 2010. – . 14, ヽ 4. – . 7–16. . ., . ., . . 22. // . . . – 2007. – ヽ 1. – . 3–11. 23. / . . , . . , . . , . . // . – 2004. – . 8, ヽ 4. – . 15–26. 167 24. , . ., . ., . . . – 2006. – // ヽ 1. – . 9–13. 25. , . ., . ., . . . – 2005. – . 9, ヽ 3. – . 9–15. . 10, // 18.10.2012 – , , , -11, e-mail: kichenko.alex@yandex.ru. , 168 – , -mail: tverier_55@perm.ru. , ,