Anisotropic poroelastic model of the osteon (sample version)

Universita’ of Naples – Federico II

MSc. In Materials Engieneering

AN ANISOTROPIC POROELASTIC MODEL OF THE OSTEON

Supervisor:Prof. Massimiliano FraldiAssistant Supervisor:Eng. Angelo R. Carotenuto

Academic Year2013/2014

Departement of structures for engineering and architeture

Emanuele ZappiaM68/87

SAMPLE VERSION

An Anisotropic Poroelastic Model of the Osteon The Bone Tissue

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STATIC FEATURES

•Load alignment•Improved stiffness•Stress delocalization

DYNAMIC FEATURES

•Continuous loads response•Impulsive loads response•Remodeling/Self-healing

Hierarchical Structure

The OsteonAn Anisotropic Poroelastic Model of the Osteon

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The primary histological unit of bone structure

Transverse LongitudinalAlternated

• Lamellar structure with central fluid filled lumen (Havers’s canal) and capillaries network in radial direction (Volkamnn’s canals)

• Fiber reinforced lamellae (hydroxiapatite and collagen) are present in alternating orientations (clockwise and anticlockwise)

• Different kinds of fibers orientation patterns could be present inside each lamella

Ascenzi et al. 1967 - The Anatomical recordGiaraud-Guille 1988 - Calcified tissue internationalSchrof et al. 2014 - Journal of structural biology

An Anisotropic Poroelastic Model of the Osteon

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Crack Arresting Effect

How the cracks are able to nucleate and

be confined in the osteon structure at

the same time?

BridgingFormation of uncracked ligaments in the crack wake. These bridges sustain a portion of applied load.

Crack DeflectionCrack deflection due to interaction with the harvesian system and microcracks at osteonal level.

BluntingDue to microcracks that surround a bigger crack there is a crack tip blunting with relative crack arrest.

MICROCRACKSIN THE

OSTEON

CRACK ARREST MECHANISMS

Nalla et al. 2005 - Nature MaterialsKoester et al. 2008 - Biomaterials

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An Anisotropic Poroelastic Model of the Osteon Bone Remodeling

OsteoblastsSynthesize

several specialized proteins to

form new bone

OsteocytesControl the

BMU activity and detect

areas have to be remodeled

OsteoclastsResorb bone by releasing a powerful acid

and an enzyme

REMODELING MECHANISM

When the remodeling mechanism occurs?

New mechanical stimuli

Everyday turnover

Bone tissue growth period

Living bones can be seen as a system of continual damage

and repair

OSTEOCYTES

DAMAGED AREAS

BMUosteoblastsosteoclasts

ACTIVATION

DETEC

T

REPAIR

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An Anisotropic Poroelastic Model of the Osteon Bone Remodeling

MECHANOTRANSDUCTION PROCESS

OSTEONMICROCRACKS

PHYSIOLOGICAL MECHANICAL

STIMULIpFLUID

WALKWAY

OSTEONSTRUCTURE

OSTEOCYTES

NUTRIENTSOVERSUPPLY MECHANICAL

SIGNAL(shear stress)

• How the osteon is able to confine such cracks?

• How the osteon structure is able to translate axial loads into shear stresses detectable by the osteocyte?

AXIAL LOADS

Taylor et al. 2007 - Nature materials

Thesis GoalAn Anisotropic Poroelastic Model of the Osteon

MECHANICAL EXPLANATION• Quantitative explanation of the coexistence of microcracks nucleation and crack

arresting phenomena at the osteonal level

• Quantitative explanation of how the osteon structure is able to translate different kinds of mechanical stimuli in a mechanical signal for the osteocyte

POROELASTIC MODELINGAnisotropic poroelastic model of the osteon and consecutive developing of ad hoc

analytical steady state solutions under prescribed boundary conditions

BONE FEATURES• Crack arresting phenomena in which the osteon (microcracks) plays an important role

• Remodeling processes in which the osteon structure appears to play a key role

Poroelastic ModelAn Anisotropic Poroelastic Model of the Osteon

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POROELASTIC THEORY

POROELASTIC MEDIUM• POROUS ELASTIC SOLID MATRIX FILLED

WITH FLUID• MECHANICAL STRESS TRANSFERRED ALSO

TO THE FLUID PHASE

CONSTITUTIVE RELATIONS

DRAINED ELASTIC TENSORS

SOLID MATRIX ELASTIC TENSORS

Cowin - Tissue Mechanics (2007)

, ,

Governing EquationsAn Anisotropic Poroelastic Model of the Osteon

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SOLUTIONS OF INTEREST

HOOP STRESS sjj

Linked with the crack confinement and nucleation effect

IN PLANE SHEAR STRESS szj

Mechanical signal detected by the osteocyte

Elastic SolutionsAn Anisotropic Poroelastic Model of the Osteon

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TWO PHASE SYSTEM

BOUNDARY CONDITIONS

FIBER ORIENTATION

PROBLEMVALUES

ELASTICCONSTANTS

45°/-45° R0=50 mm E=2.3 GPa

45°/-50° R1=75 mm =0.3n

45°/-40° R2=100 mm nt=0.3

70°/-70° L=500 mm a=16.2

70°/-75° - h=9.48

70°/-65° - -

• ABSENCE OF KINEMATIC TORSION IMPOSED

• PRESCRIBED AXIAL DISPLACEMENT APPLIED (e0=-1/1000)


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ELASTIC RESULTS - sjj HOOP STRESS

• PRESENCE OF A STRESS (sjj) GRADIENT TROUGH THE RADIUS OF THE PHASES• DIFFERENT SIGN OF THE AVERAGE VALUES IN THE TWO PHASES• GAP (ALSO IN SIGN) AT THE PHASE BOUNDARY WHEN THE SHIFT ANGLES ARE INTRODUCED


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• PRESENCE OF A STRESS (sjj) GRADIENT TROUGH THE RADIUS OF THE PHASES• DIFFERENT SIGN OF THE AVERAGE VALUES IN THE TWO PHASES• GAP (ALSO IN SIGN) AT THE PHASE BOUNDARY WHEN THE SHIFT ANGLES ARE INTRODUCED

ELASTIC RESULTS - sjj HOOP STRESS


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ELASTIC RESULTS - sjz IN PLANE SHEAR STRESS

• PRESENCE OF A TORSIONAL STRESS (sjz) WITH AXIAL DISPLACEMENT IMPOSED• DIFFERENT SIGN OF THE STRESS VALUES IN THE TWO PHASES• GAP (ALSO IN SIGN) AT THE PHASE BOUNDARY

Poroelastic SolutionsAn Anisotropic Poroelastic Model of the Osteon

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BOUNDARY CONDITIONS

FIBER ORIENTATION

PROBLEMVALUES

PROBLEMVALUES

ELASTICCONSTANTS

45°/-45° R0=50 mm PHAVERS=5 Kpa E=2.3 GPa

45°/-50° R1=75 mm h1=1000 =0.3n

45°/-40° R2=100 mm

k1=0.3 nt=0.3

70°/-70° L=500 mm k2=0.1 a=16.2

70°/-75° - - h=9.48

70°/-65° - - -

• PARAMETRICL ANALYSIS AT DIFFERENT LEAKAGE COEFFICIENTS h2

• DIFFERENT AVERAGE GRADIENT PRESSURE MODULUS IMPOSED (>0 and <0)

• PRESCRIBED AXIAL DISPLACEMENT (e0=-1/1000) and ABSENCE OF KINEMATIC TORSION IMPOSED

TWO PHASE SYSTEM

Cowin et al. 1994 – Journal of the mechanics and physics of solids


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POROELASTIC RESULTS - sjj HOOP STRESS [45°/-45°]

ELASTIC CASE RETRIEVED FOR HIGH

LEAKAGE COEFFICIENT

GAP ARISING WITH LEAKAGE COEFFFICIENT

DECREASING EVEN WITHOUT SHIFT ANGLE

PRESENCE

• PRESENCE OF A STRESS (sjj) GRADIENT TROUGH THE RADIUS OF THE PHASES• DIFFERENT SIGN OF THE AVERAGE VALUES IN THE TWO PHASES


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POROELASTIC RESULTS - sjj HOOP STRESS [45°/-50°]

ELASTIC CASE RETRIEVED FOR HIGH

LEAKAGE COEFFICIENT

GAP ARISING WITH LEAKAGE COEFFFICIENT

DECREASING

• PRESENCE OF A STRESS (sjj) GRADIENT TROUGH THE RADIUS OF THE PHASES• DIFFERENT SIGN OF THE AVERAGE VALUES IN THE TWO PHASES• GAP AT THE PHASE BOUNDARY


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POROELASTIC RESULTS Vs ELASTIC RESULTS (HOOP STRESS)

ELAS

TIC

RESU

LTS

PORO

ELAS

TIC

RESU

LTS

The poroelastic model confirmed the features of the elastic analysis. In addition, for low leakage coefficient:• AMPLIFICATION OF THE

AVERAGE STRESS DIFFERENCE BETWEEN ADJCENT PHASES

• GAP AMPLIFICATION• STRESS GRADIENT

AMPLIFICATION


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POROELASTIC RESULTS - sjz IN PLANE SHEAR STRESS [45°/-45°]

• PRESENCE OF A TORSIONAL STRESS (sjz) WITH AXIAL DISPLACEMENT IMPOSED• DIFFERENT SIGN OF THE STRESS VALUES IN THE TWO PHASES• GAP (ALSO IN SIGN) AT THE PHASE BOUNDARY

ARISING SHIFT WITH LEAKAGE COEFFFICIENT

DECREASING

DECRISING SLOPE WITH LEAKAGE COEFFFICIENT

DECREASING

ConclusionsAn Anisotropic Poroelastic Model of the Osteon

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OSTEON CRACK NUCLEATION & CONFINEMENT EFFECT

+

+-

-

-+

+-

-

-

• PRESENCE OF A STRESS (sjj) GRADIENT THROUGH THE RADIUS OF THE SYSTEM• GAP (ALSO IN SIGN due TO SHIFT ANGLE PRESENCE) AT THE PHASE BOUNDARY

BETWEEN ADJACENT LAMELLAE• DIFFERENT SIGN OF AVERAGE STRESS (sjj) VALUE FOR ADJACENT PHASES (load sign

independent feature)• POROELASTICITY CONFIRMS THE LIMIT CASE OF ELASTICITY AND SHOWS FURTHER

AMPLIFICATION EFFECTS


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STRESS-MEDIATED MECHANICAL EFFECT

LOADS &MICROCRACKS

OSTEONMONOCLINICSTRUCTURE

OSTEOCYTES

PRESENCE OF TORSIONAL STRESS (sjz)

BMUosteoblastsosteoclasts

ACTIVATION

• PRESENCE OF TORSIONAL STRESS (sjz) INDUCED BY AXIAL COMPRESSION/TENSION DUE TO THE PECULIAR MONOCLINIC STRUCTURE OF THE OSTEON

• ACCORDING TO LITERATURE SUCH STRESSES ARE THE MECHANICAL SIGNALS NECESSARY TO ACTIVATE THE REMODELING PROCESS


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CONCLUSION

FUTURE DEVELOPMENTS

The present analytical model has managed to give a mechanical explanation of the coexistence of microcracks nucleation and crack arresting phenomena at the osteonal level, as well as the way in which the osteon structure produces torsional stress under axial loads.Finally, it could be said that the osteon is a smart microstructure that combines several exceptional features in order to realize an optimized biomechanical machine.

• MORE ACCURATE NUMERICAL SIMULATION AND RELATIVE COMPARISON• EXPERIMENTAL VALIDATION OF THE PRESENT MODEL• TIME DEPENDENT ANALYSIS• IMPLEMENTING FRACTURE/DAMAGE MECHANICS IN THE MODEL• NEW BIOINSPIRED MATERIALS

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Anisotropic poroelastic model of the osteon (sample version)