MEGN 536 – Computational Biomechanics

Prof. Anthony J. Petrella

Bone Material Properties

Bone Macrostructure

Long bone Epiphysis Diaphysis Compact bone (cortical) Spongy bone (cancellous)

1www.agen.ufl.edu/~chyn/age2062/lect/lect_19/lect_19.htm2webschoolsolutions.com/patts/systems/skeleton.htm

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1academic.wsc.edu/faculty/jatodd1/351/ch4outline.html2castlefordschools.com/Kent/Lessons/Advanced%20Biology%20Lessons/chapter%2037/…Advanced%20Biology%20Chapter%2037%20Introduction%20to%20Body%20Structure_files/image026.jpg

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Bone Microstructure

Cortical bone Note circumferential layers Structure influences the material properties

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Bone Microstructure

Cancellous Bone Trabeculae – struts Notice axial alignment Some plate-like structures

40x1academic.wsc.edu/faculty/jatodd1/351/ch4outline.html2www.gla.ac.uk/ibls/fab/public/docs/xbone1x.html

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Bone Constituents

Red marrow Red blood cells, platelets, most white blood cells arise in red

marrow Found in flat bones (sternum, pelvis) and epiphyses

Yellow marrow Some white blood cells arise here Color comes from much higher fat content Found in medullary canals of diaphyses in long bones

Both types of marrow contain numerous vessels

Lots of “squishy” stuff here

Bone Properties

Like many biological tissues with “squishy” stuff, bone can behave viscoelastically -- Guedes et al., J.Biomech, 2006

Some studies have shown tensile and compressive behavior similar and linear elastic -- Keaveny et al., J.Biomech, 1994

Many studies have shown that bone is inhomogeneousand anisotropic Inhomogeneous – properties vary with location Anisotropic – properties vary with direction of loading

Modulus for cortical bone usually in the 15-20 GPa range, cancellous bone in the 100-500 MPa range

Example: Inhomogeneous Strength

Inhomogeneity

The inhomogenous nature of bone suggests that it’s important to model the material properties with correct spatial variation

A recent study shows that patient-specific models are inaccurate without a correct inhomogeneous mapping of material properties -- Taddei et al., J.Biomech, 2006

One of the advantages of Simpleware… the software can automate this inhomogeneous mapping

Hooke’s Law

Recall Hooke’s law for a linear elastic, isotropic material: s = E e

We also need to know Poisson’s ratio: n Isotropic elastic requires only two constants: E, n

Many studies have shown that bone is transversely isotropic, which means the axial direction behaves differently than the radial direction

Transverse isotropic materials exhibit properties that are invariant under axial rotation

Recall axial alignment ofbone structure…

Constitutive Models for Bone

A transverse isotropic model requiresfive elastic constants:

Ez, Exy, nxz = nyz, nxy, Gxz = Gyz

These constants can be found experimentally, but most basic bone models in the literature still use an isotropic model for simplicity

A transverse isotropic model also cannot be easily parameterized using CT data

z xy

Bone Density

Bone contains many internal structures/spaces and constituents besides calcified tissue

Some density metrics try to account for this

Apparent density (range: 0.05 – 2.0 g/cm3) Your usual density measure Mass of sample divided by total volume of sample

Ash density (range: 0.03 – 1.2 g/cm3) Seeks to eliminate non-calcified tissue Mass of bone ash divided by volume of bone only Bone ash created by drying out bone and incinerating

Modulus Relationship to Density

Density can be expressed as linear function of Hounsfield units

r = a + b * HU (g/cm3)

Modulus and strength have been shown to obey a power-law relationship to density

E = c + d * re (GPa)S = f + g * rh (MPa)

Coefficients vary among different studies, but exponents are usually in the 1-3 range

-- Keller, J.Biomech, 1994

Modulus-Density Relation also Inhomogeneous

Mapping Properties with Simpleware

For simplicity, we stay with a linear elastic, isotropic constitutive model

Use Simpleware automatic mapping to account for inhomogeneity

Necessary number of materials depends on the specific model How much density variation is there? How large is the domain?

Typical numbers of distinct materials in validation studies are in the 100-500 range --Taddei, J.Biomech, 2006

For simplicity, we will use 10 materials

A Note about Density and Units in FEA

Abaqus does not use explicit units Unit system must be consistent…

We are using N, mm, sec Stiffness is in terms of N/mm2 → MPa Similarly… mass * accel → must give Newtons Density must be units of tonne/mm3 where tonne = 1000 kg,

this compensates for accel in units of mm/s2 rather than m/s2

(𝑚𝑚𝑠2 )( 1𝑘𝑔𝑚𝑚3 )𝑚𝑚3=

𝑘𝑔 ∙𝑚𝑚𝑠2

=10− 3𝑁𝑒𝑤𝑡𝑜𝑛𝑠

(𝑚𝑚𝑠2 )( 1000𝑘𝑔𝑚𝑚3 )𝑚𝑚3=

𝑘𝑔 ∙𝑚𝑠2

=𝑁𝑒𝑤𝑡𝑜𝑛𝑠

A Note about Density and Units in FEA

Density of tissue similar to water… ~ 1 g/cm3

Stable time increment of FE simulation approx…

where h is an element edge length and c is the wave speed in the relevant material, but note that

∆ 𝑡 ≅h𝑚𝑖𝑛

𝑐

𝑐=√ 𝐸𝜌

1𝑔𝑐𝑚3=

10− 3𝑘𝑔𝑐𝑚3 =10

−6𝑘𝑔𝑚𝑚3 =

10− 9(1000𝑘𝑔)𝑚𝑚3

A Note about Density and Units in FEA

Estimate the following…

It is common to artificially increase density in order to increase the stable time increment and decrease total solution time

h𝑚𝑖𝑛=1𝑚𝑚 𝐸=1000𝑀𝑃𝑎 𝜌=10−9(1000𝑘𝑔)

𝑚𝑚3

𝑐=√ 𝐸𝜌=106𝑚/ 𝑠

∆ 𝑡 ≅h𝑚𝑖𝑛

𝑐 = 10−3𝑚106𝑚 /𝑠

=10− 9𝑠