ELASTIC GROWTH IN TISSUES Program in Applied Mathematics ELASTIC GROWTH IN TISSUES Rebecca Vandiver Program in Applied Mathematics University of Arizona Los Alamos Days, February 2007 Advisor: Alain Goriely

Outline Basic problem I. Introduction II. Theory and Methods To study the mechanics of growing tissues Obtain a general formulation of volumetric growth Generate a description of cumulative growth I. Introduction Volumetric growth Differential growth Residual stress II. Theory and Methods Mechanics of soft tissues Kinematics of growth Cauchy stress Equilibrium equations III. Application Growing cylinder Future work http://insects.tamu.edu/images/insects/color/cotton/fig1.jpg

Part I I. Introduction Volumetric growth Differential growth Residual growth

Growth Surface growth: Volumetric growth: Occurs by accretion, deposition of new material on a preexisting surface Volumetric growth: Growth takes place in the bulk of the material

Bulk Growth: Plant Stems Orange grows faster than green, so green is in tension. This stress is called residual stress. Historical note: Tissue tension and differential growth first investigated by Hofmeister (1859) and Sachs (1865).

Residual stress Rhubarb Simple experiment reveals residual stress Differential growth causes residual stress Residual stress may increase rigidity

Bulk Growth: Arteries Stress in arteries No Residual stress Stress in arteries Residual stress in unstressed arteries (Fung, 1984) Create compressive and tensile layers Believed to reduce stress gradient Residual stress Hozapfel-Ogden

Part II. II. Theory and Methods Mechanics of soft tissues Kinematics of growth Cauchy stress Equilibrium equations

MECHANICAL PROPERTIES OF BIOLOGICAL TISSUES BONE AORTA bone aorta σ (MPa) σ (MPa) σs 1.75 70 Non-linear stress-strain curve ε ε .004 6 Large deformation

KINEMATICS Let X denote the position vectors in the reference configuration of an elastic body and x define the current configuration after growth. x=c(X,t) X The deformation tensor F = Grad(c) relates a material segment in the reference configuration to the same segment in the current configuration

KINEMATICS Decomposition: F= AG where A: pure elastic response Rodriguez, et al. (1994) Decomposition: F= AG where A: pure elastic response G: pure growth

Incremental Growth F(k) = FkFk-1…F2F1 G(k) = GkGk-1…G2G1 A(k) F(k) = FkFk-1…F2F1 = AkGkAk-1Gk-1…A1G1 G(k) = GkGk-1…G2G1 A(k) = AkAk-1…A2A1

Cauchy Stress We assume that the body is hyperelastic. That is, the material can be described by a strain energy function W=W(A) and the Cauchy stress tensor, T, can be written as If there is one (or more) constraint such as incompressibility then, we introduce a Lagrange multiplier p where p is the hydrostatic pressure

EQUILIBRIUM EQUATIONS The balance of linear momentum leads to Cauchy’s first law of motion: The balance of angular momentum leads to Cauchy’s second law of motion: If the body is at rest and body forces are absent, With given boundary conditions: (e.g. )

Part III III. Applications Growing cylinder Future work

Growing cylinder Consider the elastic growth of an incompressible cylindrical tube ri = bi ri = ai Ri = B Ri = A

Growing cylinder Growth along the z-axis: Gi = diag(1,1,ginc(ri))  g(R) = ginc(ri) Consider a linear incremental growth function ginc = 1+μ(bi-ri) bi ai Initial values: A=1, B=2 Calculate μ at each iteration such that the volume increases by 1%

STRESSES IN THE CYLINDER trr (radial stress) tθθ (hoop stress) tzz (axial stress)

Axial stress curves Longitudinal stress plotted in the current configuration Tensile Compressive

Cumulative growth Cumulative growth function plotted in the current configuration

Cumulative growth

Cumulative growth

Conclusions and perspectives Elastic growth Decomposition of growth Formulation of incremental growth applied to a growing cylinder Description of cumulative growth Ongoing projects Stress-dependent growth Cavitation Buckling formula for general 3-D case F

References Ben Amar, M., and Goriely, A., Growth and instabilities in elastic tissues, J. Mech. Phys. Solids, 53:2284-2319 (2005). Goriely, A., and Ben Amar, M. On the definition and modeling of incremental, cumulative, and continuous growth laws in morphoelasticity, Biomechanics and Modeling in Mechanobiology (2006). Haughton, D. M., and Orr, A., On the eversion of incompresible elastic cylinders, Int. J. Non-Linear Mechanics 30:81-95 (1995). Rodriguez, E. K., A. Hoger, and McCulloch, A. D., Stress-dependent finite growth in soft elastic tissues, Journal of Biomechanics, 27:455-467 (1994).

FUTURE WORK Euler showed there was a critical load for buckling of a slender column No comparable formula for a more general geometry F Goal: a) Work on a general theory of buckling of a cylinder under a load in which we could relate parameters in a more general case b) Explore how growth affects the buckling parameter

CUMULATIVE GROWTH Cumulative deformation gradient2: F(k) = FkFk-1…F2F1 = AkGkAk-1Gk1…A2G2A1G1 If growth and elastic tensors commute: A(k) = AkAk-1…A2A1, G(k) = GkGk-1…G2G1 Stress in Bk: where

CYLINDER: MODELING CUMULATIVE GROWTH Incompressibility constraint  det(A)=1  det(FG-1)=1  det(F)=det(G)  Integrate to obtain

CYLINDER: MODELING CUMULATIVE GROWTH Radial stress: We want surface of cylinder to be free of any traction, but t3=t3(R). Instead, impose a zero resultant load: Solve for ai and λi at each iteration