Trabecular Bone as a Complex System

1
Trabecular Bone as a Complex System

• Iwona Jasiuk
• Dept. of Mechanical Science and Engineering
• UIUC

Understanding Complex Systems, May 16, 2007
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Bone is a Complex System

• Bone is a hierarchical material with complex
  random structures at several different scales
• Bone is a living tissue with continuously
  evolving microstructure (mechanical, biological
  chemical factors)
• loads, diet, medications, genetics, hormones
• Bone is a multi-functional material
• Provides frame
• Protects organs
• Manufactures blood cells
• Stores useful minerals
• Maintains pH in blood, detoxifies, contributes to
  movement

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Hierarchical Structure of Trabecular Bone

• Macrostructure (1-50 cm)
• Whole bone
• Mesostructure (0.510 cm)
• Trabecular network
• Microstructure (10500 mm)
• Single trabecula
• Sub-microstructure (110 mm)
• Single lamella
• Nanostructure (below 1 mm)
• Collagen fibrils
• Apatite crystals

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Macrostructure (1 50 cm)

• Trabecular bone
• High porosity
• 30 to 90
• Skeletal mass
• 20 to 25
• Cortical bone
• Low porosity
• 5 to 30
• Skeletal mass
• 75 to 80

Frontal longitudinal midsection of upper femur
http//www.bartleby.com/107/indexillus.html
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Mesostructure (1 10 cm)
SEM images
Porous random network of trabeculae
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Microstructure (10 - 500 mm)

• Trabecular packets
• 50 mm mean wall thickness
• 600 mm radius

trabecula
Cane et al. 1982
Plywood arrangements
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TEM - Plywood Arrangements of lamellae
P
L
P
Orthogonal plywood motif (0/90)
Twisted or rotated plywood motif
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TEM Atypical lamellar structures

• Unmineralized regions
• Disordered plywood motifs
• Random organization

Bone (2004)
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Sub-microstructure (1 - 10 mm)

• Single lamella level

• single lamella (3 to 7 mm thick)
• branching bundles (1 - 2 mm diameter)
• fibrils show splaying, less than 10o
• ellipsoidal cavities - lacunae (1-2 mm diameter,
  20 mm long)

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Lamellar structure collagen fibrils aligned
Woven bone structure no preferential fibril
arrangement
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Nanostructure (below 1 mm)

• Collagen (Type I) fibrils
• 20 100 nm diameter
• 60 67 nm periodic pattern
• Apatite crystals (calcium phosphorus)
• Shape?
• Irregular Plates
• e.g. Robinson, 1952 Weiner et al,1986
• 50 x 25 x 5 nm
• Arrangement?
• Other proteins, fluids

Rho et al. 1997
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TEM HA crystals in longitudinally-sectioned
fibrils
100 nm
C
Plate-like shape Aligned in
fibril direction
M. Rubin, I. Jasiuk, J. Taylor, I. Rubin. T.
Ganey, R. Apkarian, Bone 33 (2003), 270.
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TEM - Crystal arrangement in cross-sectioned
fibrils
100 nm
M. Rubin, I. Jasiuk, J. Taylor, I. Rubin. T.
Ganey, R. Apkarian (2003) Bone 33, 270-282.
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Mineralized collagen fibrils
SEM images
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Modeling of Trabecular Bone

• Bone is a natural composite material
• (polymer matrix nanocomposite with hierarchical
  structure)
• Nanostructure
• Apatite crystals and collagen fibrils
• Sub-microstructure
• Mineralized collagen fibrils and pores
• Microstructure
• Lamella (distinct anisotropic properties)
• Mesostructure
• Bone tissue and pores
• Bone is stiff, strong, tough and light

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Hierarchical Modeling of Bone Elastic analysis

• Nanostructure
• Micromechanics theories
• Sub-microstructure
• Beam-network model
• Microstructure
• Laminate theory of composite materials
• Mesostructure
• Actual random geometry via FEM
• Output at one scale - input for next scale

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Effective vs. Apparent property
effective property (effective medium, i.e.
RVE- infinite size domain)
apparent property (mesoscale window, finite
sized domain)
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Representative Volume Element (RVE) (Hill, 1963)

• is structurally entirely typical of the whole
  mixture on average
• contains a sufficient number of inclusions for
  the apparent overall moduli to be effectively
  independent of the surface values of traction and
  displacement, as long as these values are
  macroscopically uniform

Hierarchy of bounds (Huet, 1990)
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Nanostructure crystal/collagen level

• Homogeneous, linear elastic and isotropic
  constituents
• Inclusions
• Unidirectionally aligned
• Ellipsoidal shape
• Perfect bonding
• Mori-Tanaka theory
• Eshelbys solution

• Issues/Challenges
• - Continuum?
• - Geometry?
• RVE?
• Properties?
• HA/collagen bonding?
• Collagen crosslinking?

Shear-lag model - Gao
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Sub-microstructure Single lamella

• Beam network model
• Ostoja-Starzewski and Stahl (1999)
• Input parameters
• Fibers with square cross-section
• Rigid (or flexible) connections
• Length to width fiber ratio
• Volume fraction of fibers
• Orientation of fibers
• Output parameters
• Anisotropic stiffness tensor
• Deformations

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Computational mechanics model basic modeling
assumptions

• Fibrils are straight, prismatic
• Distribution function for any fibril property
  (e.g., orientation, length, width)
• Bonds form where fibrils intersect
• Treat each fibril as a series of 3-D rod elements
  with axial extension, torsion, and Timoshenko
  bending in two planes
• FEs are defined as segments between two
  consecutive bonds with other fibrils very short
  FE
• 3-D frame of complex geometry
• Displacement boundary conditions ui eij xj,
  where eij is constant.

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Sub-microstructure

• Beam network model

Under applied horizontal strain Splitting of
network into elongated braids that carry the load

I. Jasiuk and M. Ostoja-Starzewski (2004),
Biomechanics and Modeling in Mechanobiology, 3,
67-74.
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Microstructure Single Trabecula

• Laminate theory for composite materials
• Inputs
• Elastic properties of a single lamella
• Ply orientations
• Outputs
• Elastic properties of laminate
• (trabecular pocket, single trabecula)
• Challenges
• Curvilinear geometry of trabecular pockets
• Actual stacking of trabecular pockets
• Randomness in trabecular pockets geometry
  properties
• Bone remodeling scale constantly changing
  geometry

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Mesostructure Experimental Data

• Goldstein (1987) review
• 0.2 GPalt E lt 3 GPa
• Factor of 10
• Reasons
• Different
• porosities
• anatomical locations
• testing conditions
• loading directions
• methods of storage

Textbook on Histology by Bloom and Fawcett
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Mesostructure - random geometry model

• MicroCT imaging
• Orthopaedic Bioengineering Laboratory
• ScanCO Medical (mCT40)
• Resolution (STL)
• 37 Microns
• 20 microns voxel size
• 173 two dimensional slices
• Contours drawn to obtain 3D image
• Output
• microCT data
• .stl file
• Nodes and 2D tetrahedral surfaces

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Sample Harvesting Digitized Bone Sample
Anisotropy H1 mm 0.674 H2 mm 1.021 H3 mm
0.836 Vol 20.4
Emory Body Donor Program Proximal Tibia Female 75
yrs, normal Cylinders f 5 mm, L
5 mm
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Digital Imaging Issues

• Thresholding
• Baseline value for binary assessment of bone
• Partially filled voxels
• Trabecular strut connections vs. bone
  microstructure distortion
• Mesh cleanup reconnecting struts
• Effect on bone volume fraction f

Threshold 80, f 8 60, f 20
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Meshing

• Meshing
• Hypermesh 6.0
• 6 million elements, 1.5 million nodes
• 4 separate files
• Solid tetrahedral elements
• Solver Post-Processor
• OPTISTRUCT (Linear)
• Volume Stress/Strain/Energy Averages
• Apparent Properties
• Local stress/strain/displacement fields

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Meshing
Close-up view
Side view
Section of finite element mesh from digital
microCT images
Meshed region
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Boundary conditions

• Top surface
• Applied normal displacement
• zero shear traction
• Bottom surface
• zero normal displacement
• zero shear traction
• Side surfaces
• Zero traction boundary conditions

• Challenges
• What boundary conditions
• are applied experimentally?
• Roughness of surface

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Summary (Theory vs. Experiments)









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Applications

• Early Prediction of onset of osteoporosis

Normal bone
Osteoporotic bone
Susan Ott, U Washington http//courses.washington
.edu/bonephys/opmovies.html

• Osteoporosis- Background (http//www.osteo.org)
• Disease (caused by abnormal bone metabolism)
• Low bone mass, Microarchitectural deterioration
  of bone tissue
• Consequent increase in bone fragility,
  susceptibility to fracture
• Affects 44 million Americans No cure, only
  treatment

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• Current Status
• Osteoporosis is diagnosed by
• DEXA (scalar value)
• History of fractures
• Personal data (genetic, lifestyle, diet)
• Open Issues/Challenges
• Identify key factors which contribute to
  mechanical properties of bone
• Obtain these factors using noninvasive techniques
• Use existing techniques and/or develop new ones
• Develop simple algorithms which can be used in
  clinical practice (e.g. neural network approach)
  in terms of parameters which can be measured