Modeling bone regeneration around endosseous implants

28 Chapter 2. Linear stability analysisα m2β m2Ã kn(2,2)(m + , s 2+ ) =(β m2 + s 2+ ) 2 (1 + α p0)m + − A s2 − kAnD 2 s2b=Therefore,β m2= A s2 ( − 1) − k 2 s 2+β m2 + snD s2 = −A s2 − k2+ β m2 + snD 2 s2 .2+Ã kn (m + , s 2+ )(−αm0 m 0 − αms 2+β m+s 2+− knD 2 mχ s 2+β m2 +s 2+−A s2s 2+A s2 α mβ mχ(β m+s 2+ ) (1 − m )+) + knB 2 m2 m +β m2 +s 2+− knD 2 s2Then the characteristic equation for the matrix Ãk n, evaluated at point(m + , s 2+ ), is given by:whereλ 2 (k 2 n) + b(k 2 n)λ(k 2 n) + c(k 2 n) = 0, (2.41)b(k 2 n) = −(Ãk n(1,1)(m + , s 2+ ) + Ãk n(2,2)(m + , s 2+ ))= knD 2 m + α m0 m 0 + α ms 2++ k 2 s 2+β m + snD s2 + A s22+ β m2 + s 2+s 2+= kn(D 2 m + D s2 ) + α m0 m 0 + (α m + A s2 ) ,β m + s 2+.c(k 2 n) =Ãk n(1,1)(m + , s 2+ )Ãk n(2,2)(m + , s 2+ )− Ãk n(1,2)(m + , s 2+ )Ãk n(2,1)(m + , s 2+ )(= knD 2 m + α m0 m 0 + α ) ()ms 2+k 2 s 2+β m + snD s2 + A s22+ β m2 + s 2+(− knB 2 m2 m + +A )s2α m β mχ(β m + s 2+ ) (1 − m s 2++) χ .β m2 + s 2+From equation (2.41) the eigenvalues of Ãk n(m + , s 2+ ) are determinedas:λ 1,2 (kn) 2 = − b(k2 n)± 1 b2 2√ 2 (kn) 2 − 4c(kn). 2 (2.42)It should be mentioned that{s 2+ > 0,m 0 > 0⇒ b(kn) 2 > 0. (2.43)Thus, the following lemma can be formulated.