Trabecular Bone Locations

Epiphyses and metaphyses

Vertebral bodies

within flat bones

Trabecular vs Cortical Bone

(Volume Fraction, Surface to Volume Ratio)

Typical bone volume fraction: 0.85-0.95 (Cortical), 0.05-0.6 (trabecular)

Surface to Volume Ratio: 2.5 (Cortical), 20 (trabecular)

Trabecular Rods and Plates

Network of Rods-open cell structure, lower density, regions of lower stress, located between condyles

Network of plates- closed cell structure, higher density, regions of high stress, located under the patellar groove, on the condyles

trabecular remodeling

coupled bone resorption followed by bone formation at the same site

osteoblasts

bone forming cells

synthesize and deposit unmineralized organic matrix components of bone (osteoids)

round cells with a single nucleus

osteoclasts

bone resorbing cells

round cells with a bottom ruffled border that dissolved bone materials and removes organic components and has multiple nuclei

osteocytes

inactive osteoblasts surrounded by bone matrix

located in lucunae

looks similar to neuron

osteoid

flat single nucleus unmineralized organic matrix components of bone that are deposited osteoblasts

mechanical properties of trabecular bone (stress-Strain)

Compression of a bone cube is similar to a cellular solid

Initial linear portion (modulus)

Constant stress region

Final failure (consolidation)

Studies to determine mechanical properties of trabecular bone

Gibson (1985)- Compression of a bone cube is similar to a cellular solid, stress-strain curve

Goldstein review (1987)- reviewed previous studies and gave huge modulus and ultimate strengths

Ciarelli (1991)- compression bone cubes controlled experiment, gave more accurate range

compression properties of trabecular bone dependence

results of compression testing are dependent on:

Age

Testing conditions

Location, function

Strain rate

Loading direction

Storage

wolff's law

Bone of high stiffness and strength is found in regions of high loads

Bone structure is oriented in direction of predominant loading (Trajectorial Theory)

explains anisotropy in bone (inferior/superior is higher modulus than other directions)

Fracture risk equation

Risk=expected load from and activity/(load causing failure experimentally)

risk> 1.0 High risk of failure

risk< 1.0 low risk of failure

regression analysis

Predictive power depends on strength of correlation between two variables

Least-squares approach minimizes the sum of the squared differences between original data and predicted relationship

r^2 defines the strength of the relationship (expresses the proportion of variance in the dependent variable that is explained by the independent variable

linear or non-linear

variables considered to predict fracture risk

scalar variables-density (apparent), bone volume fraction, porosity

clinical density measures- DEXA, QCT density

architectural variables- trabecular plate number, connectivity, mean intercept length

If relationship exists between stiffness/strength and variable, then it may be used to predict failure load

scalar variables to predict fracture risk

“Density”

Apparent Density- Mass of bone tissue / apparent volume of space including marrow and bone

Bone Volume Fraction- Volume of bone / apparent volume

Porosity- Volume of marrow / apparent volume

clinical density measures

'bone mineral density" (BMD)

Dual-energy radiography (DEXA)

QCT Density

Dual-energy radiography (DEXA)

Reduced x-ray exposure

Limited accuracy in determining BMD

Images include cortical shell

error associated with densities due to bone thickness

correlates well with whole bone strength, since it reflects both bone size and mass

not a true volumetric density (over projected area, not volume)

QCT density

Computed tomography

Series of planar x-rays reconstructed to form 3D image

Can assess trabecular bone alone

Must be calibrated with a phantom (s piece of material with known density that is used to compare to bone)

relatively good predictor of BMD and strength

bone mineral density (BMD)

can be measured via QCT density and DEXA scans

among the strongest predictors of fracture risk

useful in the diagnosis and monitoring of treatment

operational definition of osteoporosis

BMD measured by DEXA is the refernece standard for osteoporosis diagnosis: T-score < -2.5 (2.5 SD below mean of premenopausal women)

for each SD decrease in BMD, fracture risk doubles

factors that increase the risk of fracture

age

low body weight

high bone turnover

lifestyle (risk of falls, smoking, excessive alcohol)

medical history (prior fracture, family history)

secondary causes (medications)

fracture prevention strategies

Applied load/ bone strength

reduce applied load (reduce fall frequency/severity, proper lifting techniques, hip protectors)

maintain or increase bone strength (exercise, diet (Ca2+, vitamin D), pharmacological intervention)

influences on bone strength

Bone mass

Bone turnover

Bone geometry

Bone architecture

Bone mineralization

Bone composition (HAP, collagen)

effects of again/osteopenia on trabecular structure

Decreases in

Bone mineral density

Number of trabeculae

Thickness of trabeculae

Connectivity

Number of transverse trabeculae?

Therefore - degree of anisotropy may change

Fewer, thinner, longer trabeculae more prone to buckling

microarchitectural changes in osteoporosis

osteoporotic bone has a decrease in mass and number of trabeculae, and abnormalities in architecture compared to normal

architectural variables

trabecular plate number (trabeculae per unit length, always more vertical than horizontal)

connectivity- measure of branching nodes, reduced in osteoporosis

mean intercept length- measure of thickness in particular direction, quantifies anisotropy, can identify principal material directions

trajectorial theory (1866)

calculation of principal stresses in a crane, similar to orientations of trabecular bone in the proximal femur

bone structure

bone structures are "optimized" for maximum strength with minimal weight

trabeculae are aligned with principal stress directions

bone structure is regulated by cells responding to a mechanical signal

straightening long bones

abnormally curved long bones can straighten through remodeling

requires apposition on concave side (bone growth on side bent in)

requires resorption on convex side

centralized canal remains

axial loads applies to long bones(bending)

axial loading on curved bone creates:

compression on concave side- leads to bone apposition (growth)

tension on convex side- leads to bone resorption (decay)

axial loads applies to long bones(gradients)

Axial loading on curved bone creates:

Stress gradient from tensile (positive) to less tensile (negative) on convex side

Stress gradient from less compressive (positive)to more compressive (negative) on concave side

positive gradient leads to bone resorption

negative stress gradient leads to bone formation

positive from tensile to less tensile

remodeling and repair facts

Turnover rate: ~ 5% each year

i.e. every 20 yrs almost all bone is “new”

In vitro testing of bone fatigue can’t take this into account

Disease and aging may impact remodeling rate

Age related fractures often appear to be “spontaneous”, may be damage accumulation

10% hip fractures, 50% spine fractures

methods of bone remodeling evaluation

External bone dimensions (diameter, etc.)

Bone density

QCT, DEXA, µCT

Local bone apposition rate

Fluorochromes

The ‘Law of Bone Transformation’: A case of crying Wolff? Bertram & Swartz, 1991

testing to determine if mechanical influences can cause transformation of bone (Wolff's law)

confounding factors (surgical procedure, growing animal, invasive procedures)

Evidence may not reflect bone response that would occur in normal healthy adult without previous injury!

applications of remodeling theories

Ability to predict responses to

Implants (stress shielding)

Altered loading

Ability to improve and control

Corrections of deformities

Fracture healing

Age-related bone loss

Is bone density distribution “optimal” ?

Can we predict shape changes

Can we predict normal variations in bone density

Can we predict changes in bone density

bone remodeling- feedback loops, Beaupre and carter (1990)

daily stress stimulus equation

Daily Stress stimulus (gamma= "expected" level of stress

remodeling is driven by difference (e) between actual stress and "expected stress"

assume linear relationship between error and apposition rate (M)

n is number of cycles of “i” type of loading

m is constant (reflecting relative importance of magnitude vs. number of cycles)

sigma is some scalar value of “stress” or strain or energy

Strain energy density, Von mises stress, Hydrostatic pressure

bone density remodeling equation

Assume linear relationship between error and apposition rate

M = c1 + c2e

Apposition may be on internal trabecular surfaces

M = apposition rate (µm/day)

Bv = bone volume

k = fraction of surfaces actively remodeling

Sv is the surface area / unit volume

bone remodeling procedure

Create starting model, apply loading conditions, material properties

Calculate stress stimulus for each element

Calculate error and change in BVF, then change in modulus

Repeat analysis with changed modulus

Continue until loads change or steady-state

remodeling simulations- Carter (1987)

Start with arbitrary uniform modulus, then iterate until steady state

Low and high density regions are well predicted

as long as multiple loading conditions are used

“lazy zone” is needed to maintain bone with low loads

trabecular remodeling simulations- Mullender (1994)

Start with arbitrary uniform modulus, then iterate until steady state

Trabecular structure is predicted

Need to use decay function or lazy zone to prevent “checkerboarding”

remodeling cautions

Constants needed

At least 5: m, C1, C2, k, Sv, power law constants

Loading conditions are critical and simulations are very sensitive to them!

Ignoring biological response can lead to nonsensical results (Disuse will reduce, not eliminate bone; Lazy zone; Baseline biological activity; May differ between normal and healing)

bone remodeling graph

factors

variables whose influence you want to study

(ex. Temperature, pressure)

denoted by X1, X2

levels

specific values given to a factor during experiments

initial limit- 2 levels

(ex. 20 degrees C, 1 Pa)

use -1, +1 to designate levels

treatment condition

one running of the experiment

ex. set T=50 degrees C, P- 1Pa and measure volume

response

result measured for a treatment condition

ex. measured volume

denoted by y

effect

calculated from responses (y1, y2)

m=(y1+y2)/2

effect of X1= (ave at +1)-(ave at -1)=(m+1)-(m-1)=y2-y1= change (delta) X1

difference in the average value at the 2 levels

degrees of freedom (DOF)

counter for information

experimental data= number of data points (y1+y2=2 DOF)

analysis- number of results

information is conserved

2^(number of factors)=responses=DOF

one at a time experiment (OAT)

change one factor at a time

ex. y1=baseline= X1=-1, X2=-1

y2= X1= +1, X2= -1

y3= X1=-1, X2= +1

choice of baseline is critical

inefficient use of data- only use part of data to calculate each effect

does not account for factors in combinations

full factorial approach

test all combinations of factors

effect of X1- averaged over both X2 levels- no baseline

all results-> all effects- use all data in each calculation= efficient use of data

order

number of factors in the interaction term of a factorial analysis

ex. X1X2= second order

randomization

experiments can be influenced by time related changes (ex. temperature changes over the course of the day)

randomize the order in which treatment conditions are run (reduces the chance that time related changes will be misattributed to factor effects)

continuous factor

involves something that can be quantified on a continuous scale (ex. temperature, pressure, time, voltage, etc.)

discrete factor

nominal, categorical

cannot be put on a continuous time scale

ex. supplier, country, gender, etc

advantages and disadvantages of full factorial designs

advantages-

not dependent on a baseline choice

all data is used to calculate each effect ("efficient")

can measure interactions between factors

convert easily to a multi-factor model

disadvantages-

work best with only 2 (maybe 3) levels for a factor

many DOF used to measure higher-order interactions

ANalysis Of Means (ANOM)

most basic level of analysis

which effects are largest (which factors/interactions are most important?

based off of delta (change) values (ave at level +1- ave at level -1)

magnitude of change= importance

positive change= +1 increases response, -1 decreases

negative change= -1 increases response, +1 decreases

ANalysis Of VAriance (ANOVA)

second level of analysis

which observed effects are statistically significant?

based on comparing observed effects against an estimate of error

compares delta^2 for factors and interactions with delta^2 for error

actually comparing "mean square"

replication error

Repeat (replicate) each of the treatment conditions

Independent experimental runs (not multiple measurements from the same TC)

Differences in the responses measured for identical TCs run at different times provide error

“Pure error” not dependent on modeling assumptions

Best way to estimate error, but greatly increases effort

error- pooling of higher-order interactions

Assume that higher-order interactions are unimportant/zero

Must choose these interactions upfront (before examining results)

these form a “pool” for error

Effects measured for pooled interactions are used to estimate error

“Error” - includes modeling error (i.e. assumptions about interactions)

Requires less experimental effort, but error estimate is not as good

replication vs pooling

replication-

"pure error"- no modeling assumptions needed

best for- small number of factors, systems with large uncertainties

pooling-

assess more factors for the same effort (or same number of factors for less effort)

best for- large numbers of factors, systems with strong time/cost constraints on experimental size

anova table

typical way of presenting ANOVA results

source- factors interactions and the error (total includes everything except m*,find error values from subtraction from total)

SS=Sum of Squares=measures the variance from each effect,factor,interaction= delta^2*(# of TC)/4

DOF= total= 1 DOF less than # of responses/TCs

MS=Mean Square= SS/DOF (SS normalized against DOF)

F=MS (effect)/MS (error) larger F, more likely the effect is "real"

p= chance that the value of F occured randomly, depends on F and DOF (effect and error)

%SS= measures "importance" of each factor/interaction- how much variance from the total does each factor attribute

judging statistical significance

if F>Fcritical (found from a table) or p>pcritical the factor/interaction is significant

alpha= the chance that an F/p value larger than the critical value could occur randomly

Treatment conditions (TC)

required number of TC is the product of levels for each factor (ex. a 2 level factor and a 3 level factor need 6 TC)

each TC has a difference set of levels (ex. TC1= +1 -1, TC2= +1 +1)

why use more than 2-levels?

continuous factor- test non-linear effects

discrete factor- test more than 2 possibilities (DOF= # levels- 1 in 3+ levels)

continuous vs. discrete- effect analysis

fractional factorials

"fractions" of full factorials (2 level designs: 1/2 or 1/4 the # of TC)

smaller experiment, but with the same number of effects

effects are "confounded" with each other and cannot be separated

use sparsity of effects- assume interaction terms are zero

design of fractional factorials

must preserve symmetry of design

can produce different confounding patterns

"resolution" is one measure of the severity of confounding

resolution in fractional factorials

higher values indicate less severe confounding

one measure of the severity of confounding

III is the lowest (worst) level

in practive III, IV, and V resolutions are common

model verification

"solving the equations right"

numerically accurate

model validation

"solving the right equations"

accurately predicts the physical phenomenon it was designed to replicate

validation/verification requirements for theoretical use

Requirements for numerical studies for theoretical use:

define model selection (relevant geometry, physics, boundary conditions, material properties)

verification (mesh refinement, code validation, non-linear convergence)

proper parameter identification (able to be duplicated, reasonably unique)

validation/verification requirements for "applied biomechanics" use

Requirements for numerical studies for "applied biomechanics" use:

Sensitivity study to understand uncertainty in inputs-Define range for comparison to experiments, Show how uncertainty impacts conclusions

Inter-subject variability assessment- May be parameterized, Perhaps several subjects, rather than single model

validation/verification requirements for clinical use

Requirements for numerical studies for clinical use:

Validation against in vitro experiments

Risk-benefit analysis- Collaboration with clinical researchers

Retrospective studies- Provide sensitivity and specificity

Prospective studies

types of fractures

closed fracture- crack/break through bones but they stay in place

open fracture- bones break through the skin (soft tissue damage)

transverse fracture- caused by tension

greenstick- due to bending and mostly occurs in children because their bones are more ductile and will bend more before fracture (can occur in adults if it is at low speed)

comminuted fracture- bone broken into small fragments

what determines the type of fracture?

type of loads (tension, compression, bending, torsion)

energy of impact

age

healing process

hematoma (bringing in blood supple, bruising)

inflammation

soft callus- either intramembranous formation (bone growth within membranous tissue at the periosteum and endosteum) or endrochondral ossification (cartilage formation, bone is formed on top, cartilage is replaced)

hard callus- mineralization of bone that begins away from the fracture and changes the diameter an material properties of the bone and affects rigidity

remodeling

"Ilizarov type" external fracture fixation

metal external splint with wires and rods to hold the bone in place and can gradually straighten the bone with screws

can also length the bone if the fracture is stretched on both sides to allow for more soft callus formation

useful for comminuted fractures

things that change external fixator stiffness

screw length

screw diameter

number of screws or wires

distance of screws from fracture

size of the connecting bar

number of connecting bars

distance between connecting bars

intramedullary pins

insertion of a pin through the intramedullary space in the center of the bone

reamed options (drilling through the canal) may rely on friction

unreamed pins may require screw fixation

may have a lot of movement if the pin is too narrow for the canal

compression plates

Plate provides compression across transverse fracture or fragments

May use a tension device or specially designed holes to pull ends together as plate is tightened in place

Contact of plate with bone may be critical in altering blood supply and therefore healing

biomechanics of fracture healing

Interfragmentary movement

Small movements stimulate healing

Too much movement leads to non-union

Detection of movements can also characterize healing process

Blood supply is critical

Unstable fixation may disrupt vascular supply

High hydrostatic pressure may limit vascularization (ex. cartilage)

bone fatigue behavior

S-N curve (left)

failure occurs instantly if ultimate stress is applied

"theoretical endurance limit"- theoretical stress that can always be applied and the bone will never fail

right graph- if the same stress is causing greater strains over time then the modulus must be decreasing

Miner-Palmgren's Rule

combines stress states for multiple cycles of multiple types of loading

Dmp = Damage according to Miner-Palmgren Rule

Ni = # of cycles at a particular load level

Nfi = # of cycles to failure at particular load level i

m = different load levels considered

For a linear model, Dmp = 1.0 at failure

mechanisms of fractures and crack initiators in bone

loading -> crack initiation (with repair prevents fatigue)-> crack growth -> failure

crack initiators- haversian canals, lacunae, canaliculi

crack initiation increase= decrease in crack growth