Acoustics Ultrasonic guided waves in cortical bone modeled as a functionnally graded anisotropic tube: characterization of ageing process

Transcription

1 Aim of the study Acoustics 2012 Ultrasonic guided waves in cortical bone modeled as a functionnally graded anisotropic tube: characterization of ageing process Cécile Baron UMR CNRS 7287, Institut des Sciences du Mouvement - Etienne-Jules Marey, F Marseille, France. 23 rd -27 th april Nantes 1/16

2 Outline Aim of the study 1 Aim of the study 2 A functionnally graded structure Bone ageing process Guided waves propagation 3 Dispersion curves Cut-off frequencies 4 5 2/16

3 Aim of the study Aim of the study Variation in cortical porosity along the thickness of the bone as an ageing process parameter Variation in cortical porosity elastic properties gradient (C ij and ρ) Ageing Periosteum Endosteum Young Elder FGM Aim of the study : To investigate the sensitivity of guided waves to this gradient of material properties in human cortical bone Watanabe et al. (1998) FGM Young Elder Cooper at al. (2003) 3/16

4 Bone model Aim of the study A functionnally graded structure Bone ageing process Guided waves propagation Geometry : tube in vacuum Anisotropy : transverse isotropy Heterogeneity : radially graded Long bone functionnally graded anisotropic tube Hypothesis : the outer diameter (a q ) remains the same after 30 years t (mm) a 0 (mm) a q (mm) t/a q [30-39] [60-69] [80-99] t e θ e z e r a a q 0 4/16

5 Aim of the study Gradient of porosity : evolution with ageing A functionnally graded structure Bone ageing process Guided waves propagation From previous studies [1,2] Variation in porosity accross the cortical thickness : 3-point measurements (periosteal, mid-cortical and endosteal regions) 3 age ranges : [30-39], [60-69] and [80-99] p(%) [30 39] [60 69] [80 99] t [80 99] t [60 69] t [30 39] periosteum endosteum p(r)=αr+β grad(%/mm) [30-39] [60-69] [80-99] Gradient of cortical porosity increases with age [1] Bousson et al. Radiology (217), 2000 [2] Bousson et al. JBMR (16), /16

6 Aim of the study Gradient of material properties A functionnally graded structure Bone ageing process Guided waves propagation Mass density ρ Hypothesis : pores full of water Classical mixture law : ρ(r)=ρ bone (1 p(r))+ρ water p(r);p(r)=αr+β 6/16

7 Aim of the study Gradient of material properties A functionnally graded structure Bone ageing process Guided waves propagation Mass density ρ Hypothesis : pores full of water Classical mixture law : ρ(r)=ρ bone (1 p(r))+ρ water p(r);p(r)=αr+β with p the porosity, ρ bone =1.9 g/cm 3 and ρ water =1 g/cm 3. ρ(r)=γr+δ 6/16

8 Aim of the study Gradient of material properties A functionnally graded structure Bone ageing process Guided waves propagation Mass density ρ Hypothesis : pores full of water Classical mixture law : ρ(r)=ρ bone (1 p(r))+ρ water p(r);p(r)=αr+β with p the porosity, ρ bone =1.9 g/cm 3 and ρ water =1 g/cm 3. ρ(r)=γr+δ Elastic propertiesc ij C ij (p)=ap+b [3] [3] Grimal et al. Biomech Model Mechanobiol (10), /16

9 Aim of the study Gradient of material properties A functionnally graded structure Bone ageing process Guided waves propagation Mass density ρ Hypothesis : pores full of water Classical mixture law : ρ(r)=ρ bone (1 p(r))+ρ water p(r);p(r)=αr+β with p the porosity, ρ bone =1.9 g/cm 3 and ρ water =1 g/cm 3. ρ(r)=γr+δ Elastic propertiesc ij C ij (p)=ap+b [3] and p(r)=αr+β Cij (GPa) p(%) C 33 C 11 =C 22 C 13 =C 23 C 12 C 44 =C 55 C 66 [3] Grimal et al. Biomech Model Mechanobiol (10), /16

10 Aim of the study Gradient of material properties A functionnally graded structure Bone ageing process Guided waves propagation Mass density ρ Hypothesis : pores full of water Classical mixture law : ρ(r)=ρ bone (1 p(r))+ρ water p(r);p(r)=αr+β with p the porosity, ρ bone =1.9 g/cm 3 and ρ water =1 g/cm 3. ρ(r)=γr+δ Elastic propertiesc ij C ij (p)=ap+b [3] and p(r)=αr+β Cij (r)=ar+b Cij (GPa) p(%) C 33 C 11 =C 22 C 13 =C 23 C 12 C 44 =C 55 C 66 [3] Grimal et al. Biomech Model Mechanobiol (10), /16

11 Aim of the study Gradient of material properties A functionnally graded structure Bone ageing process Guided waves propagation Mass density ρ Hypothesis : pores full of water Classical mixture law : ρ(r)=ρ bone (1 p(r))+ρ water p(r);p(r)=αr+β with p the porosity, ρ bone =1.9 g/cm 3 and ρ water =1 g/cm 3. ρ(r)=γr+δ Elastic propertiesc ij C ij (p)=ap+b [3] and p(r)=αr+β Cij (r)=ar+b Cij (GPa) p(%) C 33 C 11 =C 22 C 13 =C 23 C 12 C 44 =C 55 C 66 [3] Grimal et al. Biomech Model Mechanobiol (10), /16

12 Aim of the study Gradient of material properties A functionnally graded structure Bone ageing process Guided waves propagation C ij (r)=ar+b with A= Cper ij C end ij t and B= Cper ij (t 1)+C end ij t [30-39] [60-69] [80-99] C 11 per (GPa) end C 12 per (GPa) end C 13 per (GPa) end C 33 per (GPa) end C 44 per (GPa) end C 66 per (GPa) end ρ per (g/cm 3 ) end /16

13 Aim of the study Wave propagation in radially graded tube A functionnally graded structure Bone ageing process Guided waves propagation System equations divσ = ρ 2 u t 2, σ = 1 2 C(gradu+gradT u), σ : stress tensor ρ = ρ(r) : mass density u : displacement vector C=C(r) : stiffness tensor Sought solutions u(r,θ,z;t)=u (n) (r)expı(nθ +k z z ωt) σ r (r,θ,z;t)=t (n) (r)expı(nθ +k z z ωt) k z : axial wavenumber n : circumferential wave-number σ r = σ.e r : radial traction vector Two types of modes circumferential waves axial waves : n=0,1,2,... L(0,m) ; T(0,m) and F(n,m)n 1 8/16

14 Aim of the study Wave propagation in radially graded tube A functionnally graded structure Bone ageing process Guided waves propagation System equations divσ = ρ 2 u t 2, σ = 1 2 C(gradu+gradT u), σ : stress tensor ρ = ρ(r) : mass density u : displacement vector C=C(r) : stiffness tensor Sought solutions u(r,θ,z;t)=u (n) (r)expı(nθ +k z z ωt) σ r (r,θ,z;t)=t (n) (r)expı(nθ +k z z ωt) k z : axial wavenumber n : circumferential wave-number σ r = σ.e r : radial traction vector Two types of modes circumferential waves axial waves : n=0,1,2,... L(0,m) ; T(0,m) and F(n,m)n 1 8/16

15 Aim of the study Wave propagation in radially graded tube A functionnally graded structure Bone ageing process Guided waves propagation System equations divσ = ρ 2 u t 2, σ = 1 2 C(gradu+gradT u), σ : stress tensor ρ = ρ(r) : mass density u : displacement vector C=C(r) : stiffness tensor Sought solutions u(r,θ,z;t)=u (n) (r)expı(nθ +k z z ωt) σ r (r,θ,z;t)=t (n) (r)expı(nθ +k z z ωt) k z : axial wavenumber n : circumferential wave-number σ r = σ.e r : radial traction vector Two types of modes circumferential waves axial waves : n=0,1,2,... L(0,m) ; T(0,m) and F(1,m) 8/16

16 Aim of the study Solution of the wave equation A functionnally graded structure Bone ageing process Guided waves propagation Hamiltonian form of the wave equation (Fourrier domain) : sextic Stroh s formalism d dr η(r)= 1 r Q(r)η(r) η(r) : state vector Q(r) : heterogeneity (6,6)-matrix (C ij (r),k z,ω) Peano expansion of the matricant r r ς M(r,r 0 )=I+ Q(ς)dς+ Q(ς) Q(ς 1 )dς 1 dς+..., r 0 r 0 r 0 I : identity matrix (6, 6) Free boundary conditions ( ) ( u(r=aq ) M1 M = 2 0 M 3 M 4 )( u(r=a0 ) 0 ) detm3 =0 kz,ω : dispersion curves 9/16

17 US in long bones Aim of the study Dispersion curves Cut-off frequencies Typical frequency range : 50 khz khz (λ t) Frequency-thickness product Age range ft (MHz.mm) [30-39] [ ] [60-69] [ ] [80-99] [ ] 10/16

18 Dispersion curves Aim of the study Dispersion curves Cut-off frequencies Longitudinal waves L(0, m) ft (MHz.mm) [30-39] [60-69] [80-99] kt 11/16

19 Dispersion curves Aim of the study Dispersion curves Cut-off frequencies Flexural waves F(1, m) ft (MHz.mm) [30-39] [60-69] [80-99] kt 12/16

20 Cut-off frequencies Aim of the study Dispersion curves Cut-off frequencies f 30/60 f 60/80 f 30/80 (khz) (khz) (khz) L(0, 2) L(0, 3) F(1, 2) F(1, 3) F(1, 4) F(1, 5) TABLE: Variations of cut-off frequencies for longitudinal and flexural modes with aging. 13/16

21 Aim of the study Stroh s formalism and bone model approximation of the exact solution of wave equation in continuously varying media artificially stratified media 14/16

22 Aim of the study Stroh s formalism and bone model approximation of the exact solution of wave equation in continuously varying media artificially stratified media anisotropy & tubular geometry 14/16

23 Aim of the study Stroh s formalism and bone model approximation of the exact solution of wave equation in continuously varying media artificially stratified media anisotropy & tubular geometry Material properties gradient as a determinant of bone quality 14/16

24 Aim of the study Stroh s formalism and bone model approximation of the exact solution of wave equation in continuously varying media artificially stratified media anisotropy & tubular geometry Material properties gradient as a determinant of bone quality evolution of porosity across the cortical thickness induced material properties gradient 14/16

25 Aim of the study Stroh s formalism and bone model approximation of the exact solution of wave equation in continuously varying media artificially stratified media anisotropy & tubular geometry Material properties gradient as a determinant of bone quality evolution of porosity across the cortical thickness induced material properties gradient geometry-structure-material 14/16

26 Aim of the study Stroh s formalism and bone model approximation of the exact solution of wave equation in continuously varying media artificially stratified media anisotropy & tubular geometry Material properties gradient as a determinant of bone quality evolution of porosity across the cortical thickness induced material properties gradient geometry-structure-material homegeneization 14/16

27 Aim of the study Stroh s formalism and bone model approximation of the exact solution of wave equation in continuously varying media artificially stratified media anisotropy & tubular geometry Material properties gradient as a determinant of bone quality evolution of porosity across the cortical thickness induced material properties gradient geometry-structure-material homegeneization sensitivity of guided waves 14/16

28 Aim of the study Stroh s formalism and bone model approximation of the exact solution of wave equation in continuously varying media artificially stratified media anisotropy & tubular geometry Material properties gradient as a determinant of bone quality evolution of porosity across the cortical thickness induced material properties gradient geometry-structure-material homegeneization sensitivity of guided waves Extension of the model 14/16

29 Aim of the study Stroh s formalism and bone model approximation of the exact solution of wave equation in continuously varying media artificially stratified media anisotropy & tubular geometry Material properties gradient as a determinant of bone quality evolution of porosity across the cortical thickness induced material properties gradient geometry-structure-material homegeneization sensitivity of guided waves Extension of the model soft tissue : fluid loading 14/16

30 Aim of the study Stroh s formalism and bone model approximation of the exact solution of wave equation in continuously varying media artificially stratified media anisotropy & tubular geometry Material properties gradient as a determinant of bone quality evolution of porosity across the cortical thickness induced material properties gradient geometry-structure-material homegeneization sensitivity of guided waves Extension of the model soft tissue : fluid loading 14/16

31 Aim of the study Stroh s formalism and bone model approximation of the exact solution of wave equation in continuously varying media artificially stratified media anisotropy & tubular geometry Material properties gradient as a determinant of bone quality evolution of porosity across the cortical thickness induced material properties gradient geometry-structure-material homegeneization sensitivity of guided waves Extension of the model soft tissue : fluid loading gradual variation of bone matrix properties 14/16

32 Aim of the study Stroh s formalism and bone model approximation of the exact solution of wave equation in continuously varying media artificially stratified media anisotropy & tubular geometry Material properties gradient as a determinant of bone quality evolution of porosity across the cortical thickness induced material properties gradient geometry-structure-material homegeneization sensitivity of guided waves Extension of the model soft tissue : fluid loading gradual variation of bone matrix properties 14/16

33 Aim of the study & Perspectives Bone : functionnally graded anisotropic tube Original method to solve wave equation in functionnally graded anisotropic tube : Peano series of the matricant (Stroh s formalism) Perspectives In-vitro experimental programm Inverse problem 15/16

34 Aim of the study & Perspectives Bone : functionnally graded anisotropic tube Original method to solve wave equation in functionnally graded anisotropic tube : Peano series of the matricant (Stroh s formalism) Perspectives In-vitro experimental programm Inverse problem 15/16

35 Aim of the study Thank you for your attention 16/16

36 Porosity gradient and ageing 30 p (%) [10 19] [20 29] [30 39] [40 49] [50 59] [60 69] [70 79] [80 99] 0 periosteum endosteum FIGURE: Variation of the porosity along the cortical thickness : linear regression for each range of age (R 2 0.9). 17/16

37 Porosity gradient and ageing porosity gradient (%/mm) [10-19][20-29][30-39][40-49][50-59][60-69][70-79][80-99] Age (years) FIGURE: Age-related evolution of the gradient of porosity : exponential regression (R 2 = 0.93). 18/16

38 Equation Q(r)= 1 r c 12 c 11 ın c 12 c 11 ın 1 ık z r 0 ı(γ 12 r 2 ρω 2 ) nγ 12 nγ 12 ın 2 γ 12 +ır 2 (kz 2c 44 ρω 2 ) k z rγ 23 ınk z r(γ 23 +c 44 )... with... ık z r c 13 c 11 ı c ı c 66 0 ı c 55 k z rγ 23 c 12 c 11 ın ık z r ınk z r(γ 123 +c 44 ) ın c 12 c ın 2 c 44 +ır 2 (kz 2 γ 13 ρω 2 ) ık z r c 13 c γ 12 =c 22 c2 12 c 11 ; γ 13 =c 33 c2 13 c 11 ; γ 23 =c 23 c 12 c 13 c /16