Modeling bone regeneration around endosseous implants

20 Chapter 2. Linear stability analysisIf α p0 ≠ 0, then differentiation takes place also when s 1 is zero. Thisallows to obtain the solution for osteoblasts, which does not converge tozero, and hence, is more realistic from biological point of view. For thisreason, the parameter values in equation (2.12), as proposed by Moreo et al.[67], and the alternative values in equation (2.13) are considered.Moreo et al. [67] investigated the linear stability of the constant solutionsof the system, which is similar to system (2.14)–(2.16), against purelytemporal perturbations. In this paper, the system stability with respect toarbitrary perturbations (including non-homogeneous perturbations) is studied.Constant solutions z ′ = (m ′ , s ′ , b ′ ) of system (2.14)–(2.16) are derivedfrom the algebraic system:⎧ (α m0 + α ms ′ )2⎪⎨ β m + s ′ m ′ (1 − m ′ ) − (α p0 + A m )m ′ = 0,2α m2 s ′ 2β m2 + s⎪⎩′ (m ′ + b ′ ) − A s2 s ′ (2.17)2 = 0,2α p0 m ′ − A b b ′ = 0.Two solutions of the above system have been denoted by Moreo et al. [67]as:where• “Chronic non healing state”: z t = (0, 0, 0),• “Low density state”: z 0 = (m 0 , 0, b 0 ),m 0 = 1 − α p0 + A mα m0, b 0 = α p0A bm 0 . (2.18)From system (2.17), it follows, that root s ′ 2 can not be equal to −β m < 0or to −β m2 < 0. Vectors z − = (m − , s 2− , b − ) and z + = (m + , s 2+ , b + ) aredefined ass 2± = −a 1 ± √ a 2 1 − 4a 2a 02a 2, (2.19)m ± = A bA s2 (s 2± + β m2 )α m2 (A b + α p0 )= A s2(s 2± + β m2 ), b ± = α p0m ± , (2.20)χA bwhere⎧ (a 2 = A s2 1 + α )m,⎪⎨α m0(a 1 = 1 + α )m(β m2 A s2 − χm 0 ) + α mχ(m 0 − 1) + β m A s2 ,α m0 α m0⎪⎩a 0 = β m (β m2 A s2 − χm 0 ),(2.21)