Modeling bone regeneration around endosseous implants

16 Chapter 2. Linear stability analysisAccording to Moreo et al. [67], it is defined asp(d) ={0.5(1 − d0.1), if 0 ≤ d < 0.1.0, if d ≥ 0.1∂m∂t = ∇ · [D m∇m − m(B m1 ∇s 1 + B m2 ∇s 2 )](+ α m0 + α ms 1+ α )ms 2m(1 − m) − (α p0 + α mbs 1)m − A m m,β m + s 1 β m + s 2 β mb + s 1(2.2)where the terms in the right-hand side represent random migration, chemotaxis,cell proliferation, differentiation into osteoblasts, and death respectively;∂b∂t = (α p0 + α mbs 1)m − A b b, (2.3)β mb + s 1where A b is the rate of osteoblast death;∂s( 1∂t = ∇ · [D αc1 ps1∇s 1 ] +β c1 + p + α )c2s 1c − A s1 s 1 , (2.4)β c2 + s 1where the terms in the right-hand side model random migration, growthfactor secretion and decay respectively;∂s 2∂t = ∇ · [D s2∇s 2 ] + α m2s 2β m2 + s 2m + α b2s 2β b2 + s 2b − A s2 s 2 , (2.5)where the first term in the right-hand side determines random migration,the second and the third ones – growth factor secretion, and the last one –decay;∂v fn= − α ws 2bv fn (1 − v w ), (2.6)∂t β w + s 2∂v w∂t= α ws 2β w + s 2bv fn (1 − v w ) − γv w (1 − v l ), (2.7)∂v l∂t = γv w(1 − v l ), (2.8)where terms containing coefficients α w , β w and γ model the substitution ofthe fibrin network by woven bone and the remodeling of woven bone intolamellar bone.Moreo et al. [67] proposed the following initial and boundary conditionsfor this equation set. Let Ω be a problem domain with the boundary ∂Ω,