Modeling bone regeneration around endosseous implants

26 Chapter 2. Linear stability analysisis made, instead of equation (2.16). The system is defined as:⎧∂m∂t ⎪⎨=∇ · [D m∇m − B m2 m∇s 2 )](+ α m0 + α )ms 2m(1 − m) − (α p0 + A m )m,β m + s 2∂s 2 ⎪⎩∂t =∇ · [D s2∇s 2 ] + α m2s 2(1 + α p0)m − A s2 s 2 .β m2 + s 2 A b(2.38)Substitution of (2.37) into equation (2.16), yields the condition ∂b∂t= 0,which is not true in general case. Therefore, system (2.14)–(2.16) and system(2.38) are not equivalent, and their stability properties are different ingeneral. However, it will be shown in Section 2.3.4, that there is a certainsimilarity (or correspondence) between the stability properties of the twosystems. This similarity is sufficient, to transfer important results, obtainedfrom the stability analysis for the system of two equations (2.38), onto thesystem of three equations (2.14)–(2.16).System (2.38) has constant solutions, that are analogous to those of system(2.14)–(2.16). They are: ˜z t = (0, 0), ˜z 0 = (m 0 , 0), ˜z + = (m + , s 2+ ),˜z − = (m − , s 2− ). Linearizing the system near the point (m ′ , s ′ 2 ), withm(x, t) = m ′ + εm p (x, t) and s 2 (x, t) = s ′ 2 + εs 2p(x, t), yields:⎧⎪⎨⎪⎩∂m p∂t∂s 2p∂t[(=D m ∇ 2 m p − m ′ B m2 ∇ 2 s 2p + α m0 + α ms ′ 2β m + s ′ 2]− (α p0 + A m ) m p +=D s2 ∇ 2 s 2p + α m2s ′ 2β m2 + s ′ 2+α mβ m(β m + s ′ 2 )2 m′ (1 − m ′ )s 2p ,(1 + α p0A b)m p[αm2 β m2(β m2 + s ′ 2 )2 (1 + α p0A b)m ′ − A s2]s 2p .Considering solutions of the form⎧m ⎪⎨ p (x, t) =⎪⎩ s 2p (x, t) =∞∑Cn m (t)φ n (x),n=0∞∑n=0C s2n (t)φ n (x),)(1 − 2m ′ )(2.39)and substituting them into system (2.39), for each n = 0, 1, . . . we arrive at:⎡dC m ⎤n (t) [ ]⎢ dtCm⎥⎣ dCn s2 (t) ⎦ = n (t)Ãk nCn s2 ,(t)dt