Computational Modelling of Mechanics and Transport in Growing Tissue

Transcription

1 Computational Modelling of Mechanics and Transport in Growing Tissue H. Narayanan, K. Garikipati, E. M. Arruda & K. Grosh University of Michigan Eighth U.S. National Congress on Computational Mechanics July 25 th, 2005 Austin, TX

2 Motivation and definition Growth/Resorption An addition or loss of mass Engineered tendon constructs [Calve et al, 2004] Increasing collagen concentration with age

3 Motivation and definition Growth/Resorption An addition or loss of mass Engineered tendon constructs [Calve et al, 2004] Increasing collagen concentration with age Open system with multiple species inter-converting and interacting

4 Modelling approach Classical balance laws enhanced via fluxes and sources Solid Collagen, proteoglycans, cells Extra cellular fluid Undergoes transport relative to the solid phase Dissolved solutes (sugars, proteins,...) Undergo transport relative to fluid

5 Modelling approach Classical balance laws enhanced via fluxes and sources Solid Collagen, proteoglycans, cells Extra cellular fluid Undergoes transport relative to the solid phase Dissolved solutes (sugars, proteins,...) Undergo transport relative to fluid

6 Modelling approach Classical balance laws enhanced via fluxes and sources Solid Collagen, proteoglycans, cells Extra cellular fluid Undergoes transport relative to the solid phase Dissolved solutes (sugars, proteins,...) Undergo transport relative to fluid Brief subset of related literature: Cowin and Hegedus [1976] Kuhl and Steinmann [2002] Sengers, Oomens and Baaijens [2004] Garikipati et al. Journal of the mechanics and physics of solids (52) [2004]

7 Balance of mass N M ι ϕ Π ι X n m ι π ι x ρι 0 Species concentration Π ι Species production M ι Species flux Ω 0 ρ For a species: ι 0 t = Π ι X M ι Solid No flux; No boundary conditions Fluid No source; Concentration or flux boundary conditions Solute Flux and source; Concentration boundary condition Ω t

8 Balance of mass N M ι ϕ Π ι X n m ι π ι x ρι 0 Species concentration Π ι Species production M ι Species flux Ω 0 ρ For a species: ι 0 t = Π ι X M ι Solid No flux; No boundary conditions Fluid No source; Concentration or flux boundary conditions Solute Flux and source; Concentration boundary condition Ω t

9 Configuration and physical boundary conditions dρ i dt + ρi x v = x m i + π i ρ ι Current species concentration π ι Current species production m ι Current species flux v Solid velocity Boundary condition specification

10 Balance of momentum N M ι q ι X g Ω 0 ϕ n m ι P N x σn ρ ι 0 Species concentration V Solid velocity V ι Species relative velocity g Body force q ι Interaction force P ι Partial stress Ω t For a species, velocity relative to the solid: V ι = (1/ρ ι 0 )F Mι ρ ι 0 t (V + V ι ) = ρ ι 0 (g + qι ) + X P ι ( X (V + V ι ))M ι Negligible contribution to mechanics from dissolved solutes

11 Balance of momentum N M ι q ι X g Ω 0 ϕ n m ι P N x σn ρ ι 0 Species concentration V Solid velocity V ι Species relative velocity g Body force q ι Interaction force P ι Partial stress Ω t For a species, velocity relative to the solid: V ι = (1/ρ ι 0 )F Mι ρ ι 0 t (V + V ι ) = ρ ι 0 (g + qι ) + X P ι ( X (V + V ι ))M ι Negligible contribution to mechanics from dissolved solutes

12 Balance of momentum N M ι q ι X g Ω 0 ϕ n m ι P N x σn ρ ι 0 Species concentration V Solid velocity V ι Species relative velocity g Body force q ι Interaction force P ι Partial stress Ω t For a species, velocity relative to the solid: V ι = (1/ρ ι 0 )F Mι ρ ι 0 t (V + V ι ) = ρ ι 0 (g + qι ) + X P ι ( X (V + V ι ))M ι Negligible contribution to mechanics from dissolved solutes

13 Growth kinematics ϕ X F Ω 0 Ω t x F gι F F eι κ F e u X Ω F = F e F e ι F gι ; F eι = F e F e ι ; Internal stress due to Isotropic swelling due to growth: F gι = ρι 0 ρ ι 0 ini 1 Saturation and swelling F eι

14 Growth kinematics ϕ X F Ω 0 Ω t x F gι F F eι κ F e u X Ω F = F e F e ι F gι ; F eι = F e F e ι ; Internal stress due to Isotropic swelling due to growth: F gι = ρι 0 ρ ι 0 ini 1 Saturation and swelling F eι

15 Growth kinematics ϕ X F Ω 0 Ω t x F gι F F eι κ F e u X Ω F = F e F e ι F gι ; F eι = F e F e ι ; Internal stress due to Isotropic swelling due to growth: F gι = ρι 0 ρ ι 0 ini 1 Saturation and swelling F eι

16 Energy balance and entropy inequality N Q ẽ ι r ι X ρ ι 0 Species concentration e ι Specific internal energy P ι Partial stress F Deformation gradient V ι Species relative velocity Q ι Partial heat flux r ι Species heat supply ẽ ι Energy transfer M ι Species flux Ω 0 P N ρ ι 0 eι t = P ι : Ḟ +P ι : X V ι X Q ι +r ι +ρ ι 0ẽι X e ι (M ι )

17 Energy balance and entropy inequality N Q ẽ ι r ι X Ω 0 P N ρ ι 0 Species concentration e ι Specific internal energy P ι Partial stress F Deformation gradient V ι Species relative velocity Q ι Partial heat flux r ι Species heat supply ẽ ι Energy transfer M ι Species flux η ι Species entropy θ Temperature ρ ι 0 eι t = P ι : Ḟ +P ι : X V ι X Q ι +r ι +ρ ι 0ẽι X e ι (M ι ) ω ρ ι η ι 0 t ω ( r ι θ Xη ι M ι X Q ι θ + ) Xθ Q ι θ ι=α ι=α 2

18 Constitutive relations for fluxes Combine first and second laws to get dissipation inequality Constitutive hypothesis e ι = ê ι (F eι,ρ ι 0,ηι ) Consistent constitutive relations Fluid flux relative ( to collagen ) M f = D f ρ f 0 F T g + F T X P f X (e f θη f ) Solute flux (proteins, sugars, nutrients,...) relative to fluid Ṽ s = V s V f M s = D s ( X (e s θη s )) D f and D s Positive semi-definite mobility tensors Magnitudes from literature, e.g. Mauck et al. [2003]

19 Constitutive relations for fluxes Combine first and second laws to get dissipation inequality Constitutive hypothesis e ι = ê ι (F eι,ρ ι 0,ηι ) Consistent constitutive relations Fluid flux relative ( to collagen ) M f = D f ρ f 0 F T g + F T X P f X (e f θη f ) Solute flux (proteins, sugars, nutrients,...) relative to fluid Ṽ s = V s V f M s = D s ( X (e s θη s )) D f and D s Positive semi-definite mobility tensors Magnitudes from literature, e.g. Mauck et al. [2003]

20 Constitutive relations for fluxes Combine first and second laws to get dissipation inequality Constitutive hypothesis e ι = ê ι (F eι,ρ ι 0,ηι ) Consistent constitutive relations Fluid flux relative ( to collagen ) M f = D f ρ f 0 F T g + F T X P f X (e f θη f ) Solute flux (proteins, sugars, nutrients,...) relative to fluid Ṽ s = V s V f M s = D s ( X (e s θη s )) D f and D s Positive semi-definite mobility tensors Magnitudes from literature, e.g. Mauck et al. [2003]

21 Saturation and Fickian diffusion Configuration 1 Configuration 2 Change in configurational entropy with distribution of solute particles... if solvent is not saturated with solute

22 Saturation and Fickian diffusion Only possible configuration Saturated Single configuration No Fickian diffusion Still have concentration-gradient driven transport due to stress gradient contribution to flux

23 Worm-like chain model based internal energy density ρ 0cê c (F ec, ρ c 0 ) c b = Nkθ ( r 2 4A 2L + L 4(1 r/l) r ) 4 ( ) Nkθ 2A 4 2L/A L + 1 4(1 2A/L) 1 log(λ a2 1 4 λb2 2 λc2 3 ) a + γ β (Jeι 2β 1) + 2γ1: E ec embed in multi chain model [Bischoff et al., 2002] r = 1 2 a 2 λ e2 1 + b2 λ e2 2 + c2 λ e2 3 λ e I elastic stretches along a, b, c λ e I = N I C e N I

24 Computational formulation details Implementation in FEAP Coupled implementation; Staggered scheme (Armero [1999], Garikipati et al. [2001]) Nonlinear projection methods to treat incompressibility (Simo et al. [1985]) Energy-momentum conserving algorithm for dynamics (Simo & Tarnow [1992a,b]) Backward Euler for time-dependent mass balance Mixed method for stress/strain gradient-driven fluxes (Garikipati et al. [2001]) Large advective terms require stabilization

25 Examples of coupled computation Constriction N M f P N P N Simulating a tendon immersed in a bath Constrict it to force fluid and dissolved nutrient flow guided tendon growth Biphasic model Worm-like chain model for collagen Ideal nearly incompressible fluid ρ f ê f = 1 2 κ(det(f ef ) 1) 2 Fluid mobility D f ij = δ ij, Han et al. [2000] First order rate law: Π f = k f (ρ f ρ f 0 ini ), Π c = Π f N M f

26 Examples of coupled computation Constriction N M f P N P N Simulating a tendon immersed in a bath Constrict it to force fluid and dissolved nutrient flow guided tendon growth Biphasic model Worm-like chain model for collagen Ideal nearly incompressible fluid ρ f ê f = 1 2 κ(det(f ef ) 1) 2 Fluid mobility D f ij = δ ij, Han et al. [2000] First order rate law: Π f = k f (ρ f ρ f 0 ini ), Π c = Π f N M f

27 Examples of coupled computation Constriction N M f P N P N Simulating a tendon immersed in a bath Constrict it to force fluid and dissolved nutrient flow guided tendon growth Biphasic model Worm-like chain model for collagen Ideal nearly incompressible fluid ρ f ê f = 1 2 κ(det(f ef ) 1) 2 Fluid mobility D f ij = δ ij, Han et al. [2000] First order rate law: Π f = k f (ρ f ρ f 0 ini ), Π c = Π f N M f

28 Results and inferences Total flux in the vertical direction Stress driven diffusion

29 Results and inferences Regions of high fluid concentration Faster growth Relaxation after constriction concludes

30 Swelling of a tendon immersed in a bath Volume evolution curve 1.245e-08 Volume of the Cylinder (m^3) 1.24e e e e e e e e e Stress-extension curves Before Growth After Growth Collagen concentration evolution

31 Summary and further work Physiologically relevant continuum formulation describing growth in an open system consistent with mixture theory Relevant driving forces arise from thermodynamics coupling with mechanics Gained insights into the problem Issues of saturation and growth Saturation and Fickian diffusion Configurations and physical boundary conditions More careful treatment of biochemistry nature of sources Formulated a theoretical framework for remodelling Engineering and characterization of growing, functional biological tissue to drive and validate modelling

32 Summary and further work Physiologically relevant continuum formulation describing growth in an open system consistent with mixture theory Relevant driving forces arise from thermodynamics coupling with mechanics Gained insights into the problem Issues of saturation and growth Saturation and Fickian diffusion Configurations and physical boundary conditions More careful treatment of biochemistry nature of sources Formulated a theoretical framework for remodelling Engineering and characterization of growing, functional biological tissue to drive and validate modelling

33 Summary and further work Physiologically relevant continuum formulation describing growth in an open system consistent with mixture theory Relevant driving forces arise from thermodynamics coupling with mechanics Gained insights into the problem Issues of saturation and growth Saturation and Fickian diffusion Configurations and physical boundary conditions More careful treatment of biochemistry nature of sources Formulated a theoretical framework for remodelling Engineering and characterization of growing, functional biological tissue to drive and validate modelling