Fitting a Round Peg Fitting a Round Peg to a Square Holeto a Square Hole

Janna Balling Janna Balling

and Andrew Andersonand Andrew Anderson

A Method for Physical Validation of Finite Element Pressure ModelsA Method for Physical Validation of Finite Element Pressure Models

IntroductionIntroduction

Validate subject-Validate subject-specific FE models specific FE models of the hip using of the hip using experimental dataexperimental data

Single-leg-stance Single-leg-stance and stair-climbingand stair-climbing

Cartilage contact Cartilage contact stress measured stress measured using pressure using pressure sensitive filmsensitive film

IntroductionIntroduction

Pressure sensitive filmPressure sensitive film– Cut into rosette patternCut into rosette pattern– Fit to femoral headFit to femoral head– Film scanned in 2DFilm scanned in 2D– CalibratedCalibrated– Color intensity is Color intensity is

proportional to applied proportional to applied pressurepressure

ObjectiveObjective

Validate FE model predictions of Validate FE model predictions of contact stress with pressure film contact stress with pressure film measurementsmeasurements– Convert 3D FE model pressure plot into a Convert 3D FE model pressure plot into a

2D synthetic image2D synthetic image– Compare synthetic image with pressure film Compare synthetic image with pressure film

imageimage

?

Fit A SphereFit A Sphere

GivenGivend={x,y,z} for n pointsd={x,y,z} for n points

Least-Squares FitLeast-Squares Fit

WhereWhere

LoopLoop

r

(a,b,c)

a= -0.000422081b= 0.000270402c= 0.146251r= 20.5007 only 317 iterations

n

i i

ion

ii

n

ii L

xa

nL

nx

na

111

)(1*

11

n

i i

ion

ii

n

ii L

yb

nL

ny

nb

111

)(1*

11

n

i i

ion

ii

n

ii L

zc

nL

nz

nc

111

)(1*

11

n

iiLn

r1

1

aaoo-a ≈ 0 b-a ≈ 0 bo-b ≈-b ≈ 0 c0 co-c≈-c≈ 00

else aelse ao=a b=a bo=b c=b co=c=c

0 approximation = 2.2204460492503131e-0160 approximation = 2.2204460492503131e-016

222 )()()( oioioii czbyaxL

origin:

radius:

Transform to Sphere Transform to Sphere CoordinatesCoordinates

GivenGivend={x,y,z} origin=(a,b,c) d={x,y,z} origin=(a,b,c)

radius=rradius=r Surface Point along vector to Surface Point along vector to

originorigin(x,y,z)

(a,b,c)

(x’,y’,z’)

(0,0,0)x

y

r

x’-ax-a

L

x’

r

top

centerorigin

Transform to Femur Transform to Femur CoordinatesCoordinates

GivenGivend={x,y,z} origin=(a,b,c) radius=rd={x,y,z} origin=(a,b,c) radius=r

center=(cx,cy,cz) top=(tx,ty,tz)center=(cx,cy,cz) top=(tx,ty,tz) Find Coordinate Transform to Find Coordinate Transform to

originorigin

Apply To Each PointApply To Each Point

vxvzvy vzvyvx origincentervz

topcentervy

)](,,[][][]'[ rcbadRd

y

-z

origin top

]0,0,1[][][ vxR

]1,0,0[][][ vzR]0,1,0[][][ vyR

Transform to Plane Transform to Plane CoordinatesCoordinates

GivenGivend={x,y,z}d={x,y,z}

origin=(a,b,c)origin=(a,b,c)

radius=rradius=r

center=(0,0,0)center=(0,0,0)

top=(tx,ty,tz)top=(tx,ty,tz) Preserve arc length and x-y orientationPreserve arc length and x-y orientation

..

*rd

d(0,0,0)

(x,y,z)r

Lσ

x

z

y

(0,0,0)

(x,y,z)

xβ

(tx,ty,tz)

(0,0,0)

(x’,y’)d

x’

y’

(tx,ty,tz)

β)/(tan 1 yx

)cos(*' dx )sin(*' dy

)/2(tan*2 1 rL

Map Plane to JPEGMap Plane to JPEG

GivenGivend={x,y} pressure={p(x,y)}d={x,y} pressure={p(x,y)}

resolution=(xres,yres)resolution=(xres,yres)

Determine Bins and Determine Bins and SampleSample

bwx = dx/xres bwy= dy/yresbwx = dx/xres bwy= dy/yres

Pressure = average of 3 nearest nodesPressure = average of 3 nearest nodes

bwx

bwy

dy

dx

Future WorkFuture Work

JPEG Bin assignmentJPEG Bin assignment– 3 node average3 node average– all nodes averageall nodes average– average of 4 cornersaverage of 4 corners– transfer transfer

function/smoothingfunction/smoothing

Comparison of JPEG values Comparison of JPEG values to film JPEGto film JPEG– bitwisebitwise– regionregion

Real Femur FE MeshReal Femur FE Mesh– increased complexityincreased complexity– sphere fittingsphere fitting