Material Forces in the Context ofBiological Tissue Remodelling

H. Narayanan, K. Garikipati, E. M. Arruda, K. Grosh

University of Michigan, Ann Arbor

Seventh U.S. National Congress on Computational Mechanics

Albuquerque, New Mexico, July 27–31, 2003

Development of Biological Tissue

Growth and Remodelling

Growth is a change in density due to mass transport(Epstein & Maugin [2000], Tao et al. [2001], Taber &Humphrey [2001], Humphrey & Rajagopal [2002],Lubarda & Hoger [2002], Kuhl & Steinmann [2002], KGet al. [2003]) Tissue is open with respect to mass Multiple species, treated by mixture theory

Remodelling is an evolution of the microstructure (Taber& Humphrey [2001], Ambrosi & Mollica [2002],Humphrey & Rajagopal [2002]) Local reconfiguration of material: self-assembly Evolution of “reference” configuration: remodelled

configuration

Development of Biological Tissue

Growth of tendon constructs

A B

C D

Calve et al. 2003

Development of Biological Tissue

Remodelling of collagen during growth

Calve et al. 2003

Development of Biological Tissue

Remodelling during growth

Hirsch et al. 1998

Development of Biological Tissue

Remodelling of bone

University of Wisconsin, Dept. of Anatomy

The tissue reconfigures by changing its microstructurewhen stressed (Wolff [1892])

Development of Biological Tissue

Remodelling of collagen due to load while healing

Provenzano et al. 2003

Development of Biological Tissue

Remodelling is the reconfiguration of the material Stress-driven “Preferred” configuration that varies pointwise and is

in general incompatible. A further configurationalchange can occur, resulting in a compatibleconfiguration.

Biological tissue is capable of changes in configurationby motion of particles relative to ambient material Motion in material space/Configurational change

Continuum Field Formulation

F

X

x

Kr

Kc

KF*

X*

Ω0

Ω∗

Ωtϕ

u∗κ

Kr is given. κ(X, t) =? (motion in material space)

Continuum Field Formulation

F

X

x

Kg

Ke

KF*

X*

Ω0

Ω∗

Ωtϕ

u∗κ

Kg is a kinematic “growth” tensor , K

e and F∗ are

elastic deformation gradients—internal stress problem

A Variational Method

F

X

x

Kr

Kc

KF*

X*

Ω0

Ω∗

Ωtϕ

u∗κ

Π[u∗, κ] :=

∫

Ω∗

ψ∗(F∗, Kc, X∗)dV ∗−

∫

Ω∗

f∗· (u∗ + κ)dV ∗

−

∫

∂Ω∗

t∗ · (u∗ + κ)dA∗

A Variational Method

Variation in spatial position: u∗

ε= u

∗ + εδu∗

Equilibrium with respect to u∗:

d

dεΠ[u∗

ε,κ]

∣

∣

∣

ε=0= 0

Euler-Lagrange equations:

Div∗P∗ + f∗ = 0, in Ω∗; P∗N∗ = t∗ on ∂Ω∗; where P∗ :=∂ψ∗

∂F∗

Quasistatic balance of linear momentum in remodelledconfiguration, Ω∗

A Variational Method

Equilibrium with respect to material motion:κε = κ + εδκ

d

dεΠ[u∗,κε]

∣

∣

∣

ε=0= 0

Euler-Lagrange equations:

−Div∗(ψ∗1 − F

∗TP

∗ + Σ∗) +

∂ψ∗

∂X∗

= 0 in Ω∗,

−

(

ψ∗1 − F

∗TP

∗ + Σ∗

)

N∗ = 0 on ∂Ω∗

where Σ∗ :=

∂ψ∗

∂KcK

cT

Eshelby stress: ψ∗1−F

∗TP

∗; configurational stress: Σ∗

Remodelling Examples

κ = L∗− L, u∗ = l − L∗

Π[u∗, κ] =1

2k∗(κ + L − Lr

1)2 +1

2k∗(κ + L − Lr

2)2 + 2 ·1

2ku∗2

− T (u∗ + κ)

∂Π

∂u∗

= 0 ⇒ 2ku∗ = T ;∂Π

∂κ= 0 ⇒ κ =

k

k∗

u∗−

(

L −

Lr1

+ Lr2

2

)

Remodelling Examples

Kr

Kc

F*

t

u*κκκ

Kr =

1 + β 0 0

0 1 + γ 0

0 0 1 + γ

, t∗ = δeR

• ψ∗(F∗, Kc, X∗) = ψ∗

1(F∗) + ψ∗

2(Kc), (compressible neo-Hookean)

Remodelling Examples

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−4

−2

0

2

4

6

8

10

12x 10

−13 Sf−>0 (β=0.0, γ=0.0)

κ(r

adia

l)

R

Kr =

1 0 0

0 1 0

0 0 1

, t∗ = 0 Pa

Remodelling Examples

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45Sf −> 0 (β=0.2, γ=0.2−>0.6)

κ(r

adia

l)

R

Kr =

1 + β 0 0

0 1 + γ 0

0 0 1 + γ

, β = 0.2, γ = 0.2 − 0.6; t∗ = 0 Pa

Remodelling Examples

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045Sf ≠ 0 (β=0.0, γ=0.0)

κ(r

adia

l)

R

Kr =

1 0 0

0 1 0

0 0 1

, t∗ ≈ 109eR Pa

Remodelling Examples

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35Sf ≠ 0 (β=0.2, γ=0.2)

κ(r

adia

l)

R

Kr =

1.2 0 0

0 1.2 0

0 0 1.2

, t∗ ≈ 109eR Pa

Remodelling Examples

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Sf ≠ 0 (β=0.2, γ=0.5)

κ(r

adia

l)

R

Kr =

1.2 0 0

0 1.5 0

0 0 1.5

, t∗ ≈ 109eR Pa

Remarks

Remodelling is coupled with growth—separatetreatment for conceptual clarity

The remodelled configuration, κ depends uponψ∗(•,Kc, •)

Remodelled configuration is assumed to be anequilibrium state Perturb conditions—new equilibrium

Self-assembly processes in materials are similarlydescribed by minimizing the Gibbs free energy of thesystems with respect to the configuration