NSF-JHU Workshop on Probability & Materials Multi-Scale Micro-Mechanical Poroelastic Modeling of Fluid Flow in Cortical Bone Colby C. Swan, HyungJoo Kim Center for Computer-Aided Design The University of Iowa Iowa City, Iowa USA WORKSHOP ON PROBABILITY AND MATERIALS: FROM NANO- TO MACRO-SCALE Johns Hopkins University Baltimore, MD, USA January 5-7, 2005

NSF-JHU Workshop on Probability & Materials BACKGROUND  Bones are adaptive living tissues:  Size & density increase and decrease with the intensity of mechanical loading  Intensified loading within certain frequency range ( Hz) → bone growth  Static loading does not stimulate bone growth  Reduced loading → bone atrophy

NSF-JHU Workshop on Probability & Materials BACKGROUND (CONT’D)  Understanding the precise mechanisms in bone adaptation is an important objective in orthopedic research.  Potentially significant clinical implications  Provide necessary stimulus to maintain bone mass  Loss of bone mass is impediment to human travel in space

NSF-JHU Workshop on Probability & Materials Osteocyte process Fibrous glycocalyx matrix Unit cell of lacuna and canaliculi  m 30  m 0.1  m ~ 1 cm Hierarchical Structure of Whole Bone

NSF-JHU Workshop on Probability & Materials Transverse section of cortical bone

NSF-JHU Workshop on Probability & Materials Transverse section of cortical bone

NSF-JHU Workshop on Probability & Materials Transverse section of cortical bone

NSF-JHU Workshop on Probability & Materials GOALS OF CURRENT RESEARCH  Quantify nature of load-induced fluid flow in lacunar canalicular system;  Magnitude of fluid pressures & shear stresses  Compare pressures & stresses with those known to stimulate osteocyte response in in vitro cell culture studies.  Adequately capture the heterogeneity and hierarchy of cortical bone.  Address relevant multi-scale modeling challenges

NSF-JHU Workshop on Probability & Materials Multi-Scale Modeling X L denotes coordinate scale of lacunae ~ 50  m X C denotes coordinate scale of canaliculus ~ 0.5  m

NSF-JHU Workshop on Probability & Materials Osteocyte process Fibrous glycocalyx matrix MICROMECHANICS & HOMOGENIZATION OF CANALICULAR BONE Temporary idealization solid matrix fluid-filled canal

NSF-JHU Workshop on Probability & Materials SCALE BRIDGING: CANALICULAR TO LACUNAR SCALE KINEMATICS: Total strain rate Rate of fluid content change

NSF-JHU Workshop on Probability & Materials SCALE BRIDGING: CANALICULAR TO LACUNAR SCALE STRESSES: “Solid” stress tensor “Fluid” stress tensor “Total” stress tensor

NSF-JHU Workshop on Probability & Materials Linear Poroelasticity Model (After Biot 1941, 1962) C is the 6x6 symmetric undrained elastic matrix G is the 6x1 strain-pore-pressure coupling matrix Z is the scalar storage modulus Procedure for computing C, G, Z: Impose and  on model; Compute and ;

NSF-JHU Workshop on Probability & Materials Effective poroelastic properties of canalicular bone matrix X 3 direction is taken as aligned with the canaliculus. ; D canaliculi = 0.10  m Storage Modulus (Gpa) Pore-pressure Coupling Moduli (Gpa) Undrained Moduli (Gpa)

NSF-JHU Workshop on Probability & Materials  <<1 where Osteocytic process Bone matrix Fibrous glycocalyx matrix Hydraulic Conductivity of Canalicular System Assumption 1: Canaliculus fully occupied by fluid Assumption 2: Canaliculus occupied by osteocytic process and glycocalyx matrix. (Tsay and Weinbaum, 1991) (Weinbaum et al, 1994)

NSF-JHU Workshop on Probability & Materials Axial permeability (m 2 ) Transverse permeability (m 2 ) Clear canaliculus assumption 6.7· m 2 6.7· m 2 Glycocalyx matrix assumption 7.5· m 2 6.7· m 2 HYDRAULIC CONDUCTIVITIES USED IN COMPUTATIONS

NSF-JHU Workshop on Probability & Materials Poroelastic Modeling at Lacunar Scale Total Medium Momentum Balance Fluid Equation of Motion Strained-Controlled Poroelastic Unit Cell Analysis Method: Impose history of and  on unit cell model; Compute corresponding histories of.

NSF-JHU Workshop on Probability & Materials Modeling of Fluid Lacunar Region (Stokes Flow Model) Interface Conditions: (Between Lacuna and Poroelastic Region)

NSF-JHU Workshop on Probability & Materials Computed fluid pressure distributions under undrained loading of lacunar unit cell model (Stresses are in 10 6 dynes/cm 2 )

NSF-JHU Workshop on Probability & Materials Fluid Pressure Distribution Under Harmonic Shear Loading

NSF-JHU Workshop on Probability & Materials Fluid Pressure Distributions Harmonic Extensional Loading

NSF-JHU Workshop on Probability & Materials Shear loadingAxial loading Local Fluid Flows in Vicinity of Lacuna:

NSF-JHU Workshop on Probability & Materials Energy Considerations Stored and Dissipated Energies Per Unit Volume

NSF-JHU Workshop on Probability & Materials

Modeling of Multiple Lacunae at Osteonal Scale

NSF-JHU Workshop on Probability & Materials OSTEONAL MODEL 4% Haversian porosity 4% Lacunar porosity 2% Canalicular porosity FEM Mesh of Model Canalicular orientations in bone matrix

NSF-JHU Workshop on Probability & Materials Fluid Pressure Distribution Under Harmonic Shear Loading of Osteonal Unit Cell Model  12 =10 -4 ; f = 1 Hz;

NSF-JHU Workshop on Probability & Materials

Summary & Conclusions Significant fluid pressures and flow do occur in the lacunar-canalicular system: At physiologically meaningful frequencies ( 10 0 Hz < f < 10 2 Hz); At physiologically meaningful strains  ~ O(10 -5 ); Remaining challenges (among many): Begin to incorporate osteocyte (cell-mechanics) models; Incorporate chemical transport phenomena;

NSF-JHU Workshop on Probability & Materials Multi-Scale Analysis Process Plain-Weave Textile Composite Hex-arranged fiber Homogenization and Material Modeling of Yarn Composite Aligned Fiber Composite Structure Homogenization and Material Modeling of Textile Composite

NSF-JHU Workshop on Probability & Materials Material Scale Decompositions of Stress and Deformation Decomposition material/ macro stress/ deformation Periodicity of material scale stress and deformation fields:

NSF-JHU Workshop on Probability & Materials Computational Unit-Cell Homogenization Deformation controlled unit-cell analysis - Apply macro-scale deformation field solve for periodic displacement field ; such that satisfies stress equilibrium. Homogenized macro-scale stress / deformation Determine homogenized material constitutive model out of macroscopic stress and deformation relations;

NSF-JHU Workshop on Probability & Materials Computational Unit-Cell Homogenization Deformation controlled unit-cell analysis - Apply macro-scale deformation field solve for periodic displacement field ; such that satisfies stress equilibrium. Homogenized macro-scale stress / deformation Determine homogenized material constitutive model out of macroscopic stress and deformation relations;

NSF-JHU Workshop on Probability & Materials Symmetric and Conjugate Stress, Strain Measures Conjugacy between averaged deformation gradient and averaged nominal stress was demonstrated by Nemat-Nasser (2000). Modified symmetric stress and strain measures  satisfy conjugacy in strain energy Macro 2 nd P-K Stress Tensor; Macro Green-Lagrange Strain Tensor

NSF-JHU Workshop on Probability & Materials Transversely-Isotropic Hyperelastic Model Transversely-Isotropic Hyper-elastic model by Bonet and Burton (1998) where and A is material director of aligned fibers in material configuration A: material director yarn

NSF-JHU Workshop on Probability & Materials Estimation of Coefficients for Aligned Fiber Model Coefficients for Bonet and Burton model Deformation-controlled homogenization where is homogenized stress under k th strain-controlled case is 2 nd PK stress from proposed model.

NSF-JHU Workshop on Probability & Materials Unit-Cell of Hexagonally Packed Aligned Fibers (a) In-plane compression (b) In-plane tension(c) Out-of-plane shear (d) In-plane shear undeformed x1x1 x2x2 x3x3

NSF-JHU Workshop on Probability & Materials Stress-Strain Behavior of Aligned-Fiber Composite E 11 S (GPa) S 11 S 11 Fit S 33 (S 33 Fit) S 22 (S 22 Fit)  11 ≠ 0, others zero

NSF-JHU Workshop on Probability & Materials Unit Cell of Plain-Weave Textile Composite (a) Unit Cell for Plain-Weave Textile Composite (b) Quarter of Unit Cell Periodic Displacement Condition Traction Free x1x1 x2x2 x3x3

NSF-JHU Workshop on Probability & Materials Deformed Shapes of Plain-weave Textile (a) uni-axial compression Loading x1x1 x2x2 x3x3 (b) uni-axial stretch x1x1 x2x2 x3x3 Loading