ANALYSIS OF TIBIAL CURVATURE, FIBULAR …...ANALYSIS OF TIBIAL CURVATURE, FIBULAR LOADING, AND THE...

ANALYSIS OF TIBIAL CURVATURE, FIBULAR LOADING, AND THE TIBIA INDEX

J.R. Funk – Biodynamic Research Corporation (USA)

R.W. Rudd, J.R. Kerrigan, J.R. Crandall – University of Virginia Center for Applied Biomechanics

ABSTRACT

The tibia index (TI) is commonly used to predict leg injury based on measurements taken by an

anthropomorphic test device (ATD). The TI consists of an interaction formula that combines axial

loading and bending plus a supplemental compressive force criterion. Current ATD lower limbs lack

geometric biofidelity with regard to tibial curvature and fibular load-sharing. Due to differences in

tibial curvature, the midshaft moments induced by axial loading are different in humans and ATDs.

Midshaft tibial loading in the human is also reduced by load-sharing through the fibula, which is not

replicated in current ATDs. In this study, tibial curvature and fibular load-sharing are quantified

through CT imaging and biomechanical testing, and equations are presented to correct ATD

measurements to reflect the loading that would be experienced by a human tibia.

Key words: Biomechanics, Biofidelity, Dummies, Injury Criteria, Legs

TIBIAL FRACTURE is a common and disabling injury in motor vehicle crashes. Fractures of the

tibia comprise approximately 20% of AIS 2+ lower extremity injuries sustained in frontal crashes,

with about one third of fractures occurring at the diaphysis (Taylor et al., 1997). Midshaft tibial

fractures are problematic due to the anatomy of the tibia and surrounding tissue. The tibia is

relatively slender and unprotected anteromedially, making it vulnerable to open injury, which carries a

high risk of infection. The tibia is also prone to developing non-union due to compromise of its

vascular supply during injury or surgical reconstruction. Epiphyseal fractures are potentially even

more problematic than midshaft fractures. Tibial plateau and pilon fractures are generally associated

with significant impairment due to disruption of the articular cartilage, which can lead to long-term

osteoarthritis and joint degeneration.

To predict leg injuries and provide a framework for evaluating countermeasures, Mertz (1993)

developed the tibia index (TI). The TI addresses combined axial compression and bending

experienced at the midshaft of the leg:

cc M

M

F

F+=TI (1)

where F is axial compressive force, M is the sagittal plane bending moment, and Fc and Mc are critical

force and moment values, respectively. Mertz (1993) suggested that injury was unlikely when TI < 1.

The critical force and moment values were based on quasistatic biomechanical tests reported by

Yamada (1970). The critical force value was based on the compressive strength of the middle fifth

segment of the tibia, and the critical bending value was based on anteroposterior bending tests of

isolated whole tibiae. Yamada (1970) reported that bone strength depended strongly on the age and

gender of the donor. Although not stated explicitly, it appears that Mertz (1993) chose critical values

for the Hybrid III 50th percentile male dummy that were representative of young male donors.

In addition to the above formulation (eq. 1), Mertz (1993) proposed a supplemental compressive

force criterion of 8 kN for the 50th percentile male dummy. This limit was meant to protect against

fracture of the tibial plateau, which is weaker in compression than the diaphysis. It is based on tests

conducted by Hirsch and Sullivan (1965), who reported an average failure load of 800 kg for

quasistatic compression tests of isolated knee joints tested in 0° - 20° of flexion. Having established

injury assessment reference values (IARVs) for the midsize male, Mertz (1993) applied mass and

IRCOBI Conference – Lisbon (Portugal), September 2003 135

length scaling techniques to obtain IARVs for various sized occupants (Table 1). Recently, several

testing agencies have broadened the TI by applying the criterion to the resultant tibial bending

moment, rather than just the component of the bending moment that is in the sagittal plane.

Table 1. Injury assessment reference values for the tibia proposed by Mertz (1993).

Parameter Small female Midsize male Large male

Axial compression (N) 5104 8000 9840

Tibia index (eq. 1) 1 1 1

Mc - critical bending moment (Nm) 115 225 307

Fc - critical compressive force (kN) 22.9 35.9 44.2

The TI is based on a combined stress analysis of a beam. As formulated (eq. 1), the TI predicts

injury based on loading experienced by the tibia at its centroid in the transverse plane. It is important

to note that the internal moment experienced at the centroid of the bone is not necessarily the same as

a moment applied externally through the ankle and knee joints or by transverse forces applied to the

leg. A substantial moment can be induced by pure axial loading in the midshaft of the tibia due

simply to the curvature of the bone. The magnitude of this induced moment is equal to the axial load

multiplied by the perpendicular distance between the axial load path and the centroid of the bone.

Although it is generally agreed that the Hybrid III leg and Thor-Lx do not match the curvature of the

human tibia (Welbourne and Shewchenko, 1998; Zuby et al., 2001), the average curvature of the

human tibia has not been quantified.

Another factor that affects tibial loading in the human is the presence of the fibula. Researchers

have generally reported that the fibula bears anywhere from 5%-15% of the total leg load, with a

greater percentage of axial load being borne when the ankle is dorsiflexed or everted (Lambert et al.,

1971; Takebe et al., 1984, Goh et al., 1992; Crandall et al., 1996). The injury tolerance of the entire

leg is therefore greater than the injury tolerance of the tibia alone in cases where the fibula does not

fracture first. Because dummy legs do not incorporate a fibula, they measure the load sustained by the

entire leg. However, the TI is based on the injury tolerance of isolated tibiae.

In order for an injury criterion to be effective, it must be able predict injury based on

measurements made by an ATD. An injury criterion based on human tolerance values requires a

biofidelic ATD response. There are many aspects of the human leg that should be matched in order

to create a biofidelic ATD, such as mass and inertial properties, dynamic joint stiffness, and active

and passive muscle tension. These issues are not addresses in this paper. Instead, this paper focuses

on the unrealistic curvature of the dummy leg, as well as its lack of a fibula, which can also be

considered a lack of biofidelity. Fortunately, this particular problem can be addressed analytically,

either by correcting the unbiofidelic ATD measurements before applying them to the TI, or by

reformulating the TI to accommodate unbiofidelic ATD measurements. This paper will explore both

approaches based on biomechanical analyses of tibial curvature and fibular load-sharing.

METHODS

QUASISTATIC LEG COMPRESSION TESTS: In order to evaluate the midshaft loading of the

tibia and fibula in pure compression, quasistatic compression tests were performed on five (5)

cadaveric leg specimens from four (4) individuals (Table 1). Human lower limb specimens were

obtained from medical cadavers in accordance with ethical guidelines and research protocol approved

by the Human Usage Review Panel and a University of Virginia institutional review board. Prior to

testing, all specimens were screened for HIV and hepatitis, and pre-test x-rays and CT scans were

checked for signs of pre-existing bone and joint pathology. Specimens were fresh frozen and allowed

to thaw at room temperature for at least 24 hours prior to testing. Each specimen was sectioned mid-

femur and instrumented with implanted 6-axis tibial and fibular load cells. A custom-made jig

ensured that the tibial bone ends maintained their original length, rotation and alignment during

implantation of the load cell (Funk et al., 2002). The foot of each specimen was potted in a foot box

using a quick-hardening urethane pour foam (4lb, U.S. Composites, West Palm Beach, FL). Care was

taken to pot the heel well below the ankle joint so that the malleoli would remain unconstrained.


Table 2. Cadaver specimen donor information.

ID # Age (yrs) Gender Height (cm) Weight (kg) Use in present study

153 60 Male 178 105 Compression tests

171 67 Male 180 92 Compression tests

173 67 Female 160 57 Compression tests

176 85 Female 157 58 Compression tests

122 63 Male 175 79 CT geometry




After preparation, specimens were mounted vertically in a universal test machine (Tinius-Olson

Locap Test Machine, Willow Grove, PA) (Fig. 1). The knee was positioned in 90° of flexion and tied

to a molded knee block with a strap. This knee orientation was chosen to simulate a typical driving

posture (Viano, 1978). The foot box was fixed to a 6-axis load cell on the base of the test fixture in

nine (9) orientations: neutral, 20° dorsiflexed, 20° dorsiflexed/everted, 20° everted, 20°

plantarflexed/everted, 20° plantarflexed, 20° plantarflexed/inverted, 20° inverted, and 20°

dorsiflexed/inverted. Non-neutral ankle orientations were obtained by fixing the foot box to a 20°

wedge that was successively positioned about the long axis of the leg in 45° increments. When

positioned 45° to the anatomical axes, the 20° wedge created a tilt of 14.4° in the direction of each

anatomical axis. Each leg was tested in all nine (9) ankle orientations. Before each test, the leg

specimen was positioned in the desired ankle orientation and preloaded to 10-20 N. Specimens were

loaded quasistatically (25 mm/min) to 1000 N of axial compression, then unloaded. Data were

collected at 100 Hz using a DSP TRAQ-P data acquisition system and debiased using offsets recorded

in an unloaded state immediately after each test.

Fig. 1. Photograph of an axial compression test.

After testing, CT scans were obtained of each specimen to obtain the precise location and

orientation of the tibial and fibular load cells. A three-dimensional coordinate transformation was

applied to the raw tibial and fibular load cell data in order to obtain measurements in the leg

coordinate frame as defined by the knee and ankle centers (Funk et al. 2002). The portion of the total

axial load borne by the fibula was calculated as a function of ankle orientation and level of axial load:

( ) %100% ⋅+

=zz

z

FFibulaFTibia

FFibulaloadFibular (2)

The effect of ankle orientation on the percentage of the axial load borne by the tibia was characterized

by a multiple least-squares regression analysis. A second-order model characterizing the effect of


dorsiflexion/plantarflexion angle and inversion/eversion angle was fit to the data. The model was

then reduced until only statistically significant (p < 0.05) parameters remained.

ANALYSIS OF BONE GEOMETRY FROM COMPUTED TOMOGRAPHY: Computed

tomography (CT) scans were taken of eight (8) leg specimens from four (4) individuals (Table 2).

The resolution was 664 microns and the slice thickness was 2 mm. The coordinate frame for the leg

was defined according to the SAE sign convention (x-axis anterior, y-axis right, z-axis down) using

CT images of the tibial plateau and tibial plafond. For the purpose of this study, the origin of the leg

coordinate system was defined as the centroid of the tibial plateau in the transverse plane. This

location was determined using an in-house program that calculated the centroid of an arbitrary shape

based on 15- 20 points that were manually chosen to define the perimeter (Fig. 2). The superior-

inferior axis (z-axis) was defined as the line extending from the centroid of the tibial plateau to the

centroid of the articular surface of the tibial plafond, which was defined in a similar manner (Fig. 2).

The medial-lateral axis (y-axis) was defined as the line passing through the centroid of the tibial

plateau and parallel to the two most posterior extensions of the medial and lateral condyles of the the

tibial plateau (Fig. 2). The anterior-posterior axis (x-axis) was defined to be perpendicular to the

medial-lateral axis.

x

y

Fig. 2. CT images of the tibial plateau (left) and tibial plafond (right) showing the points used to

define the perimeter of each. The centroid (x), x- and y-axes, and points used to define the medial-

lateral direction (+) are also shown.

Tibial curvature was defined by the location of the centroid of each cross-sectional image in the

tibial coordinate frame. An in-house computer program was written to automate the calculation of

area properties of bone slices using the CT data. The program first distinguished cortical bone from

the surrounding tissue by thresholding the grayscale value of the signal based on Hounsfield number.

Images were cropped to isolate the tibia from the fibula. The program then calculated the cross-

sectional area, centroid, and moments of inertia (Ixx, Iyy, and Ixy) for each section. Coordinate

transformations were applied to the program output in the CT frame to obtain the location of the

centroid in the tibial coordinate frame based. In order to determine an average tibial geometry, all

centroid coordinates were linearly scaled to a 50% percentile male based on tibial length:

( ) ( )[ 416/' mmlengthTibiammxx centroidcentroid ⋅= ] (3a)

( ) ( )[ 416/' mmlengthTibiammyy centroidcentroid ⋅= ] (3b)

where xcentroid’ and ycentroid’ denote centroid coordinates in the tibial frame scaled to the length of the

tibia in the Hybrid III 50th percentile male ATD, which is 416 mm. Measurements for each specimen

were interpolated to a common length scale expressed as a percentage of tibial length. Point-by-point

averages and standard deviations were calculated. In addition, a fourth order polynomial was fit to

the point by point averages of both the x and y centroid coordinates in order to obtain a convenient

expression for determining the average location of the centroid of the tibia at any point vertically

along its diaphysis.

REFORMULATION OF TIBIA INDEX: In order to meaningfully compare the injury boundaries

predicted by the TI as it is currently used with ATDs, it was desired to express the TI in terms of an

externally applied moment. The tibia is initially analyzed as a curved beam subjected to a static

compressive force and externally applied bending moment in the sagittal plane (Fig. 3). In

accordance with the SAE sign convention, compression of the leg is considered a negative z-axis

force. The degree of curvature of the tibia is characterized by the eccentricity e, which is the


perpendicular distance from the line of the compressive loading path to the centroid of the tibia in the

anterior direction.

Fig. 3. Static free body diagram of the leg in the sagittal plane.

MYext MYext

-FZ

eY

Z

X

-FZ

Mechanically, a slightly curved beam behaves like a straight beam that is eccentrically loaded.

The eccentric axial loading vector can be decomposed into a concentric loading component and an

induced bending moment (Minduced) equal to the axial force (Fz) times the eccentricity (e):

(4) eFM zinduced ⋅=Thus, the tibia can be modeled as a straight beam subjected to three loading components: a concentric

axial force, an externally applied bending moment, and an induced bending moment (Fig. 4).

-Minduced -Fz

Mext

-Fz

Mext

Compressive

stress

Tensile stress -Minduced

Fig. 4. Free body diagram of compressive loading of a straight beam (equivalent to Fig. 3).

At any point x along the cross-section of the beam, the stress due to each loading component can

be calculated from basic equilibrium equations. The stress due to the concentric axial force

component of the loading (σF) is simply:

A

FzF =σ (5)

where Fz is the axial force and A is the cross-sectional area of the beam. σF is the same at all locations

in the x-y plane. The stress due to the externally applied moment (σMext) is:

yy

extMext

I

xMx

⋅−=)(σ (6)

where Mext is the externally applied moment, x is the anterior distance from the centroid of the beam,

and Iyy is the cross-sectional area moment of inertia of the beam. σMext varies according to the

transverse distance x from the centroid of the beam and is negative (compression) on the anterior

surface when Mext is positive. The stress in the beam due to the induced moment (σMind) can likewise

be calculated by substituting equation (4) into equation (6):

yy

z

yy

inducedM

I

xeF

I

xMx

ind

⋅⋅−=

⋅−=)(σ (7)

The stress in the beam due to the combined effects of all three loading components is the sum of the

stresses due to each individual loading component:

yy

z

yy

extzMMF

I

xeF

I

xM

A

Fx

indext

⋅⋅−

⋅−=++= σσσσ )( (8)

Applying this analysis to the tibia, failure will occur when the ultimate compressive stress (σultcomp)

or the ultimate tensile stress (σulttens) of the bone is exceeded. The peak stress will always occur on the

edge of the bone, at either the anterior surface or the posterior surface (Fig. 4). For this simplified


analysis, the cross-section of the bone is assumed to have an outer radius r from the centroid that is

the same for both the anterior (xant) and posterior surface (xpost):

rxrx postant −== ; (9)

This implies four possible modes of failure for the human tibia subjected to combined axial loading

and bending in the sagittal plane: local compressive failure on the anterior surface of the bone, local

tensile failure on the anterior surface of the bone, local compressive failure on the posterior surface of

the bone, and local tensile failure on the posterior surface of the bone (Table 3).

Table 3. Failure criteria for a tibia subjected to combined axial loading and bending in the sagittal

plane (equation numbers are listed in parentheses). Compressive stresses are defined as negative.

Failure mode

Compression Tension

Anterior surface 1)(≥

=

compult

rx

σσ

(10a) 1)(≥

=

tensult

rx

σσ

(10b)

Failure

site

Posterior surface 1)(≥

−=

compult

rx

σσ

(10c) 1)(≥

−=

tensult

rx

σσ

(10d)

It is desired to couch the injury criterion in terms of structural parameters, such as force and

moment, rather than the material parameters of stress and strain. The equilibrium equations (5) and

(6) can be used to estimate the ultimate stress of bone based on the critical force and moment values

chosen by Mertz (1993). The critical force (Fc) provides a surrogate marker for the ultimate

compressive stress of bone:

A

Fcultcomp

=σ (11)

where Fc is negative, because it represents compression. Because bone is actually stronger in

compression than tension (Yamada, 1970), pure bending generally results in a local tensile failure.

Therefore, the critical bending moment (Mc) provides a surrogate measure of the ultimate tensile

strength of bone:

yy

cult

I

rMtens

⋅=σ (12)

where Mc is positive, because it represents tension. Mertz (1993) effectively assumed that the

ultimate strength of bone is the same in compression and tension. However, the TI may be formulated

assuming unequal ultimate strengths for compression and tension (Welbourne and Shewchenko,

1998). The ultimate compressive and tensile stress of bone can be related to each other in terms of

their ratio:

A

Fc

ult

ult

ult

comp

tens

tens⋅

=

σσ

σ (13)

yy

c

ult

ult

ultI

rM

tens

comp

comp

⋅⋅

=

σ

σσ (14)

where the ratio of the ultimate compressive stress to the ultimate tensile stress is negative.

Each of the four possible failure modes (Table 3) can be expanded by substituting equations (8),

(11), and (12) into equations (10a)-(10d). In order to do this, it is necessary to define stress at the

bone surfaces using equation (8):

( )

yy

ext

I

reFM

A

Frx

⋅⋅+−== )(σ (15a)

( )

yy

ext

I

reFM

A

Frx

⋅⋅++=−= )(σ (15b)


For the case in which the bone fails due to local compression on the anterior surface, the failure

boundary is defined by substituting equations (11), (14), and (15a) into equation (10a):

( )

1≥

⋅

⋅+

−

c

ult

ult

ext

cM

eFM

F

F

tens

comp

σ

σ (16a)

This equation can be rearranged to resemble the TI formulation (eq. 1) proposed by Mertz (1993):

1≥

⋅

−

+

⋅−⋅

⋅⋅

c

ult

ult

ext

cc

ult

ult

cc

ult

ultM

M

FeM

MF

F

tens

comp

tens

comp

tens

comp

σ

σ

σ

σ

σ

σ (16b)

Similar mathematical operations were performed to evaluate the other three failure criteria (Table 3)

by substituting equations (11-14) into equations (10b-10d) as appropriate. For each failure criteria,

the equation can be rearranged as in the example of equation (16b) to obtain a reformulated TI of the

form:

1''≥+

c

ext

c M

M

F

F (17)

where F is the axial force, Mext is an externally applied bending moment, and F’c and M’c are revised

critical values (Table 4). Thus, in this reformulation, four equations are required to predict tibial

fracture due to combined axial loading and bending in the sagittal plane, one for each possible failure

mode. A compressive force criterion to address epiphyseal fractures would be supplemental to these

four equations. These equations can be extended to the coronal plane if medial and lateral are

substituted for anterior and posterior, respectively.

Table 4. Matrix of revised critical values F’c and M’

c for the reformulated tibia index (eq. 17).

Failure mode F’c M’

c

Compressive failure on

Anterior surface

cc

ult

ult

cc

ult

ult

FeM

MF

tens

comp

tens

comp

⋅−⋅

⋅⋅

σ

σ

σ

σ

c

ult

ultM

tens

comp ⋅

−

σ

σ

Tensile failure on

Anterior surface cc

ult

ult

cc

FeM

MF

tens

comp ⋅−⋅

⋅

σ

σ

cM−

Compressive failure on

Posterior surface

cc

ult

ult

cc

ult

ult

FeM

MF

tens

comp

tens

comp

⋅+⋅

⋅⋅

σ

σ

σ

σ

c

ult

ultM

tens

comp ⋅

σ

σ

Tensile failure on

Posterior surface cc

ult

ult

cc

FeM

MF

tens

comp ⋅+⋅

⋅

σ

σ

cM


RESULTS

QUASISTATIC LEG COMPRESSION TESTS: In general, the percentage of axial load borne by

the fibula (eq. 2) in each test remained fairly constant during loading and unloading of the leg (Fig. 5).

The percentage of the overall axial load borne by the fibula ranged from -8% to 19% (Table 5). In

most specimens, the fibula experienced the greatest compressive load when the ankle was everted, and

the least compressive load (sometimes tension) when the ankle was inverted. Ankle flexion angle had

little effect on fibular load-bearing. Although individual limbs generally exhibited similar patterns of

fibular loading as a function of ankle orientation, there was substantial inter-specimen variability,

including variability between the left and right legs of the same individual. A multiple linear

regression analysis performed on all 45 data points failed to demonstrated a statistically significant

correlation between inversion/eversion angle and fibular loading (Fig. 6):

( ) ( )deg29.048.7% eversionloadFibular ⋅+= (18)

-1200

-1000

-800

-600

-400

-200

0

0 10 20 30 40

Time (sec)

Fo

rce

(N

)

0

1

2

3

4

5

6

7

8

9

10

Fib

ula

Lo

ad

(%

)

Tibia Fz (N)

Fibula Fz (N)

Fibula Load (%)

Fig. 5. Sample plot of axial loads versus time.

Table 5. Percentage of the axial load borne by the fibula at the time of peak load in all tests

(D = dorsflexion, P = plantarflexion, E = eversion, I = inversion, N = neutral).

Specimen Ankle orientation

ID DE DN DI NE NN NI PE PN PI

171-R 8.9% 5.1% -6.7% 17.0% 5.0% -7.7% 7.5% -0.6% -2.1%

153-R 8.6% 5.4% 0.3% 12.2% 1.3% -0.1% 7.0% 3.4% 2.3%

173-L 12.5% 10.4% 11.7% 13.3% 10.2% 10.3% 13.5% 12.1% 12.6%

176-L 11.2% 5.0% 0.9% 18.7% 2.1% 1.3% 11.5% 1.3% 0.8%

176-R 14.9% 12.3% 9.1% 18.2% 11.2% 8.2% 14.4% 11.2% 11.1%

Mean 11.2% 7.6% 3.1% 15.9% 6.0% 2.4% 10.8% 5.5% 4.9%

s.d. 2.6% 3.5% 7.4% 3.0% 4.6% 7.2% 3.4% 5.8% 6.5%

-10

-5

0

5

10

15

20

-30 -20 -10 0 10 20 30

Eversion Angle (deg)

Fib

ula

r lo

ad

(%

)

171-R 153-R

173-L 176-L

176-R Curve fit

Fig. 6. Fibular load as a function of eversion angle.


This model was statistically significant (p < 0.0001), but did not explain much of the variation in the

data (R2 = 0.39). Additional parameters were investigated, but not found to be significant.

ANALYSIS OF BONE GEOMETRY FROM COMPUTED TOMOGRAPHY: CT analysis

proved robust when analyzing the diaphyseal region of the bone (approximately the middle 70% of

the bone length). However, the tibial epiphyses could not be effectively analyzed due to the

predominance of trabecular bone and the proximity of the fibula. Therefore, the path of the centroid

of the tibial diaphyseal cross-section was calculated for each bone only between 15% and 85% of the

bone length. The centroid of the tibia was found to be generally anterior and medial to the line

connecting the centroids of the tibial plateau and tibial plafond (Fig. 7). The offset was generally

largest in the anterior direction in the proximal tibia and decreased towards the distal tibia. In the

eight tibiae studied, left and right tibiae from the same individual showed a similar degree of

geometric variation to each other as to tibiae from other individuals. The mean values of the

coordinates of the tibial centroid were fit to a fourth order polynomial:

( ) ( ) ( ) ( ) 26.5%137-%495%760-%390234

' +⋅⋅+⋅⋅= tibtibtibtibcentroid zzzzx (19a)

( ) ( ) ( ) ( ) .591%134-%553%392-%149234

' +⋅⋅+⋅⋅= tibtibtibtibcentroid zzzzy (19b)

where xcentroid’ and ycentroid’ denote centroid coordinates in the tibial frame scaled to a 50th percentile

male (eq. 3ab) and ztib denotes the vertical position as a percentage of tibial length, where 0% is the

top of the tibial plateau, and 100% is the tip of the medial malleolus. Equations (19ab) are only valid

for the region of the tibia between 15% and 85% of the bone length (R2 > 0.98, error < 0.5 mm).

-5

0

5

10

15

20

15% 25% 35% 45% 55% 65% 75% 85%

Tibial Length

Cen

tro

id c

oo

rdin

ate

s (

mm

)

X (anterior)

Y (medial)

+/- 1 s.d.

Fig. 7. Mean location of the centroid of the tibia in the transverse plane as a function of bone length.

REFORMULATION OF TIBIA INDEX: The reformulated TI (eq. 17 and Table 4) demonstrates

that the injury boundary imposed by the TI is different for the human, Hybrid III leg, and Thor-Lx

(Fig. 8). This is due to differences in bone curvature. While the Thor-Lx is perfectly straight, the

Hybrid III leg and human tibia are curved, though to different degrees and sometimes in different

directions. At the proximal tibial load cell, the Hybrid III leg extends 28 mm anterior to the

compressive load path, while the average human tibia curves approximately 13 mm anteriorly and 3

mm medially (Fig. 8). At the distal tibial load cell, the Hybrid III leg curves posteriorly 6 mm,

whereas the human tibia curves anteriorly about 5 mm. If the fact that the ultimate strength of bone is

greater in compression than tension is factored into the reformulated TI, the injury boundary expands

relative to injury boundary predicted under the assumption that bone strength is equal in tension and

compression (Fig. 9).


-12

-10

-8

-6

-4

-2

0

-300 -200 -100 0 100 200 300 400

External Y-Axis Moment (Nm)

Ax

ial

Fo

rce (

kN

)

Human

Hybrid III

Thor-LX

No injury

Injury

Fig. 8. Injury boundaries predicted by the reformulated TI (eq. 17 and Table 4)

at the location of the proximal tibial load cell in the Hybrid III.

-12

-10

-8

-6

-4

-2

0

-300 -200 -100 0 100 200 300

External Y-Axis Moment (Nm)

Ax

ial

Fo

rce

(k

N)

comp = t

comp > ten

No injury

Injuryσc / σt = -1

σc / σt = -1.2

Fig. 9. The effect of considering differences in bone strength in tension and compression on the

injury boundaries defined by the reformulated TI for the human at the midshaft.

DISCUSSION

The TI was originally formulated using a number of simplifying assumptions to facilitate injury

prediction based on available biomechanical data. The TI (eq. 1) is derived from a combined stress

analysis of a beam, and the critical force and moment values proposed by Mertz (1993) (Table 1) are

meant to be surrogate measures of the ultimate stress of bone. The TI predicts injury based on the

axial force and internal bending moment experienced by the centroid of the tibia. The reformulated

TI (eq. 17) is based on a combined stress analysis of an eccentrically loaded beam, and the revised

critical values (Table 4) are meant to represent structural tolerances of the entire tibia in various

failure modes. The reformulated TI defines the injury boundaries imposed by the TI in terms of axial

force and externally applied bending moments. The reformulated TI demonstrates that the effect of

the interaction between axial force and external bending on predicted injury has a strong directional

dependence that is proportional to the curvature of the bone. Because the geometry of the Hybrid III

leg and Thor-Lx are different from the geometry of the human leg, the TI, as it is currently used,

actually predicts different injury boundaries for each structure given equivalent external loading

conditions.

The importance of the effect of tibial curvature at high levels of axial load has been appreciated

previously. Messerer (in Nyquist, 1986) noted that pure axial compression of whole tibiae resulted in

bending failure at an average axial load of 10.36 kN in males, which is far lower than 35.9 kN

necessary to compressively fail midshaft tibial segments (Yamada, 1970). This observation led

Kuppa et al. (2001) to propose the Revised Tibia Index (RTI) with critical values of Fc = 12 kN and

Mc = 240 Nm. However, this reduction in critical force value is inappropriate because the critical


force value is meant to reflect the material strength of the bone in pure compression (eq. 11), not the

structural strength of the whole leg. Welbourne and Shewchenko (1998) noted that due to the large

offset of the proximal tibial load cell in the Hybrid III leg (28 mm anterior to the axial load path), the

TI reaches a value of 1 in pure axial compression at an axial load level of only 6.5 kN, which is

reflected in the reformulated TI (Fig. 8). Zuby et al. (2001) proposed a correction of tibial y-axis

bending moments in the Hybrid III leg that involved entirely removing the effect of the offset distance

between the load cells and the axial load path. This correction would accurately represent bending

experienced by the human tibia if it were a straight beam. However, the human tibia actually is

curved (Fig. 7). ATD measurements (MATD) may be corrected so that they represent the bending

moment a human leg would experience under the same external loading conditions (Mhuman) using the

following equations (Funk et al., 2002):

( )humanyATDyzATDxhumanx eeFMM ____ −⋅+= (20a)

( )humanxATDxzATDyhumany eeFMM ____ −⋅−= (20b)

where Fz is the axial force and e is the perpendicular distance from the line of action of the axial force

and the tibial centroid in the transverse plane. According to the SAE sign convention, Fz is negative

in compression, anterior bowing of the tibia is represented by a positive ex, and medial bowing of the

tibia is represented by a positive ey for a left leg and a negative ey for a right leg. Once bending

moments measured by the ATD have been transformed to their human equivalents, their resultant

value may be applied directly to the original TI (eq. 1) to predict injury.

The average tibial curvature documented here (Fig. 8 and eq. 19ab) is based on bone geometry

measured from CT scans, and should only be regarded as a first approximation of the eccentricity e

associated with compressive loading of the tibia. During compression of the leg, the axial load path

acts between the centers of pressure of the knee and ankle joints, which may or may not correspond to

the centroids of the articular surfaces. No attempt was made to determine joint centers of pressure in

this study. Previous studies suggest that centers of pressure of both the knee and ankle joints move as

a function of ankle orientation (Calhoun et al., 1994, Driscoll et al., 1994) and knee flexion angle

(Fukubayashi and Kurosawa, 1980; Ahmed and Burke, 1983; Nisell, 1985; Wretenberg et al., 2002).

In addition, knee bolster loading during a frontal crash may cause the tibia to translate anteriorly or

posteriorly relative to the femoral condyles, depending on whether contact is made with the patella or

anterior tibial tubercle.

In addition to the geometric issues described above, there are several assumptions inherent in the

formulation of the TI that require evaluation. First, the contribution of the fibula is omitted from the

ultimate strength criteria. However, results from the present study suggest that the fibula does not

bear much load, except when the ankle is strongly everted. The finding that the fibula bears only 6%

of the axial load with the ankle neutrally oriented is in agreement with previous studies that have

measured fibular loading directly with load cells (Takebe et al., 1984; Goh et al., 1992). Other studies

have estimated fibular load-sharing using strain gauges, but these studies have either ignored the

effect of bone bending (Lambert et al., 1971; Wang et al., 1996) or failed to explain how fibular loads

were calculated (Segal et al., 1981). The accuracy of the TI may be slightly improved by correcting

the ATD measurement (Fz_ATD) to represent the axial load that a human leg would experience

(Fz_human) by estimating fibular loads based on the results of this study (eq. 18):

( )[ ] ( )[ ]deg0029.0925.0%100/%1 ___ evFloadFibFF ATDzATDzhumanz ⋅−⋅=−⋅= (21)

Another assumption inherent in the TI formulation is that the strength of the bone is equal in

tension and compression. However, Yamada (1970) reported that the ultimate stress of bone is 140

MPa in tension and -167 MPa in compression, which suggests that the ratio σult_comp/σult_tens = -1.2. As

noted by Welbourne and Shewchenko (1998), incorporating this effect into the TI expands the injury

boundary somewhat (Fig. 9). The injury boundary is further expanded raising the TI injury threshold

from 1 to 1.3, as is currently the practice of the European Enhanced Vehicle-safety Committee

(EEVC) for the Euro NCAP (Kuppa et al., 2001). This modification is inappropriate, because the

threshold of 1 is based on standard engineering failure criteria (Table 3). In fact, this modification has

the paradoxical effect of eliminating the need to consider the TI interaction formula (eq. 1) at all,

because individual force or moment limits will always be exceeded before the TI reaches 1.3

(Welbourne and Shewchenko, 1998).


The purpose of this study was to elucidate the theoretical basis for the TI and assess the effect of

differing leg geometries between humans and ATDs on the injury boundaries predicted by the TI.

The TI, like any injury criterion based on human tolerance values, requires that ATD measurements

represent a biofidelic response. This study addressed dummy biofidelity in terms of tibial curvature

and fibular load-sharing. There are numerous other dummy biofidelity issues not addressed in this

paper, including mass and inertial properties, dynamic axial stiffness, appropriate ankle and knee joint

response, and active and passive muscle tension. In addition, this study has not evaluated the IARVs

proposed by Mertz (1993) in light of more recent biomechanical data. It should further be noted that

these IARVs are meant to predict tibial fracture, and may not be appropriate for predicting ankle

injuries.

CONCLUSIONS

1.) The tibia index (eq. 1) is based on a combined loading analysis of a beam. The injury boundary of

1 is an engineering failure criterion. The critical force and moment values (Table 1) are actually

surrogate measures of ultimate stress. They are not meant to represent structural tolerances to external

loading modes, although the TI may be reformulated in those terms (eq. 17 and Table 4).

1.) The tibia index imposes different injury thresholds on the Hybrid III leg, Thor-Lx, and human leg

for the same set of external loading conditions. This is due to the different tibial curvatures in the

three structures.

2.) The human tibial diaphysis generally bows anteriorly and medially, especially at the proximal

tibia. The accuracy of the TI may be improved by analytically transforming the tibial moments

measured by the Hybrid III leg and Thor-Lx to their human equivalents.

3.) The fibula bears an average of 6% of the axial leg load when the ankle is neutrally oriented. The

proportion of axial load borne by the fibula increases with eversion, decreases with inversion, and is

generally unchanged by dorsiflexion and plantarflexion.

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