Age, sex, body anthropometry, and ACL size predict the structural properties of the human anterior cruciate ligament.

Abstract

Anterior cruciate ligament (ACL) injury continues to be at the forefront of sports injury concerns because of its impact on quality of life and joint health prognosis. One strategy is to reduce the occurrence of this injury by identifying at-risk subjects based on key putative risk factors. The purpose of our study was to develop models that predict the structural properties of a subject's ACL based on the combination of known risk factors. We hypothesized that the structural properties of the ACL can be predicted using a multi-linear regression model based on significant covariates that are associated with increased risk of injury, including age, sex, body size, and ACL size. We also hypothesized that ACL size is a significant contributor to the model. The developed models had predictive capabilities for the structural properties of the ACL: load at failure (R2 = 0.914), elongation at failure (R2 = 0.872), energy at failure (R2 = 0.913), and linear stiffness (R2 = 0.756). Furthermore, sex, age, body mass, BMI, and height were contributors (p < 0.05) to all predicted structural properties. ACL minimal area was a contributor to elongation, energy at failure, and linear stiffness (p < 0.05), but not to load at failure. ACL volume was also a contributor to elongation and energy at failure (p < 0.05), but not to linear stiffness and load at failure models. ACL length was not a significant contributor to any structural property. The clinical significance of this research is its potential, after continued development and refinement of the model, for application to prognostic studies that are designed to identify individuals at increased risk for injury to the ligament.

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Age, Sex, Body Anthropometry, and ACL Size Predict the Structural
Properties of the Human Anterior Cruciate Ligament
Javad Hashemi,1,2Hossein Mansouri,3Naveen Chandrashekar,4James R. Slauterbeck,5Daniel M. Hardy,2,6Bruce D. Beynnon5
1Department of Mechanical Engineering, Texas Tech University, Lubbock, Texas 79409-1021,2Department of Mathematics and Statistics, Texas
Tech University, Lubbock, Texas 79409,3Department of Mechanical and Mechatronics Engineering, University of Waterloo, Waterloo, Ontario,
Canada N2L 3G1,4Department of Orthopaedic Rehabilitation, McClure Musculoskeletal Research Center, University of Vermont, Burlington,
Vermont 05405-008,5Department of Cell Biology and Biochemistry, Texas Tech University Health Sciences Center, Lubbock, Texas 79430-6540,
6Department of Orthopaedic Surgery, Texas Tech University Health Sciences Center, Lubbock, Texas 79430-9436
Received 14 January 2010; accepted 15 July 2010
Published online 18 January 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/jor.21245
ABSTRACT:
quality of life and joint health prognosis. One strategy is to reduce the occurrence of this injury by identifying at-risk subjects based on
key putative risk factors. The purpose of our study was to develop models that predict the structural properties of a subject’s ACL based
on the combination of known risk factors. We hypothesized that the structural properties of the ACL can be predicted using a multi-linear
regression model based on significant covariates that are associated with increased risk of injury, including age, sex, body size, and ACL
size. We also hypothesized that ACL size is a significant contributor to the model. The developed models had predictive capabilities for
the structural properties of the ACL: load at failure (R2=0.914), elongation at failure (R2=0.872), energy at failure (R2=0.913), and
linear stiffness (R2=0.756). Furthermore, sex, age, body mass, BMI, and height were contributors (p<0.05) to all predicted structural
properties. ACL minimal area was a contributor to elongation, energy at failure, and linear stiffness (p<0.05), but not to load at failure.
ACL volume was also a contributor to elongation and energy at failure (p<0.05), but not to linear stiffness and load at failure models. ACL
length was not a significant contributor to any structural property. The clinical significance of this research is its potential, after continued
development and refinement of the model, for application to prognostic studies that are designed to identify individuals at increased risk
for injury to the ligament.
Anterior cruciate ligament (ACL) injury continues to be at the forefront of sports injury concerns because of its impact on
© 2011 Orthopaedic Research Society. Published by Wiley Periodicals, Inc. J Orthop Res 29:993–1001, 2011
Keywords:
anterior cruciate ligament; structural properties; ACL size; body size; strength prediction
The high incidence of anterior cruciate ligament (ACL)
injuries suffered by young active individuals, espe-
cially females, and the associated cost of treatment has
been the subject of extensive research.1–4The negative
impact of an ACL tear on a person’s life is exacerbated
by the increased risk of developing osteoarthritis (OA)
when this injury is either treated with surgical recon-
struction or conservatively through rehabilitation and
activity modification.5
Recent research efforts have focused on determin-
ing the risk factors for ACL injury. Risk factor studies
are crucial for developing intervention programs and for
identifying who is at increased risk of suffering ACL
injury so an intervention can be targeted at them. Such
research has produced marked advances over the past
decade.1Although the question regarding sex-based
disparity has not been answered, the susceptibility to
ACL injury is universally believed to be multi-factorial,
with complex interrelationships among key risk factors
and separate risk models for males and females.1Risk
factors associated with increased risk of ACL injury
include: footwear, surface type, subject sex, ACL geom-
etry (size), femoral notch size, menstrual cycle phase,
Q angle, tibial articular surface geometry, knee lax-
ity, knee valgus, foot pronation, and body mass index
Correspondence to: Javad Hashemi (T: 806-742-3563 ext. 247;
F: 806-742-3540; E-mail: javad.hashemi@ttu.edu)
© 2011 Orthopaedic Research Society. Published by Wiley Periodicals, Inc.
(BMI).1–3For the most part, these factors have been
studied in isolation.
The inherent structural properties of the ACL may
also be related to injury of this structure; however,
little is known about this risk because predictive mod-
els of ACL biomechanical properties have not been
established, confirmed, and then applied in prognostic
studies. The ACL’s structural properties are of crucial
importance because the ACL ultimately fails when the
biomechanicalloadplaceduponitexceedsthemaximum
load that it can withstand.6Accordingly, over the past
years, we have focused on anatomical,7–9structural,9,10
and biomechanical11–13aspects of the knee, and specif-
ically the ACL, that influence both the external load
placed upon it and its intrinsic failure load. We showed
that, in cadaveric tissues, sex-based differences in size9
(females have smaller ACLs than males), structural
properties11(females have lower ACL failure load, stiff-
ness, elongation to failure, and energy absorbed at
failure), and ultrastructure12of the ACL (females have
a less densely populated collagen fibril structure) exist
even when considering differences in body size between
sexes.
Here we extend our previous11research by perform-
ing secondary analysis of our data, with a focus on
developing a linear multivariate regression model to
predict the structural properties of a subject’s ACL
based on key covariates related to increased risk of
suffering injury. Our primary hypothesis was that
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HASHEMI ET AL.
structural properties can be predicted based on simul-
taneous consideration of risk factors for injury. Our
secondary hypothesis was that the relative contribu-
tions these risk factors make to predicting properties
depend on whether they are linked directly to the ACL
(length, area, and volume) or the subject (age, sex, body
mass, height, and BMI).
The resulting model will improve our understanding
of the mechanisms by which these risk factors act on
the ACL to cause failure, and will be useful for prog-
nostic studies designed to identify how ACL structural
properties from at risk subjects are related to risk of
injury. Such a model could serve as a tool for future
research in risk management (estimating the probabil-
ity of ACL injury), tissue engineering, and for inclusion
in inverse dynamics simulations to understand subject-
specific responses to ACL loading. Our hope is that the
predictor variables in the model will be easy to measure
using readily available tools. These efforts will allow us
to accomplish our long-term goal of decreasing the mor-
bidity, sequelae, and costs associated with this injury.
MATERIALS AND METHODS
Weperformed
published.11The materials and methods used to acquire and
test the ACLs are presented in detail in previous articles.11
Briefly, 20 unpaired knees (10 male, 10 female) were obtained
from donors within 12h death and frozen at −20◦C until
the day of the experiment. Subjects’ ages ranged from 17 to
50 years (male: 26–50; female: 17–50), heights ranged from
1.52 to 1.8m (male: 1.67–1.8; female: 1.52–1.73), body mass
ranged from 49.48 to 121.47kg (male: 54.02–121.47; female:
49.48–111.62), and BMI ranged from 15.2 to 44.4kg/m2(male:
19.2–37.3; female: 15.2–44.4). Knees with any signs of abnor-
mality were excluded. On the day of the experiment, the knees
were thawed, and all soft tissues except for the ACL were
dissected free. The tibia and femur were cut about 7.5cm
from the ACL attachment sites producing a Femur-ACL-Tibia-
Construct (FATC). The FATC was potted under a minimal
tensile load of 2N, and a 3D scanner was used to create a
3D geometric model of the ACL about its longitudinal axis.10
The minimal cross-sectional area and volume of the ACL were
determined.10The subjects’ ACL lengths ranged from 23.8
to 33.65mm (male: 26.9–33.6; female: 23.7–31.1), ACL min-
imal area ranged from 35.1 to 103.6mm2(male: 56.2–103.6;
female: 35.1–83.8), and ACL volume ranged from 1,108 to
4,718mm3(male: 1,709–4,718; female: 1,108–2,730). The data
for age, body anthropometry, and ACL dimensions are pre-
sented in Table 1. We selected subject characteristics (age and
sex) and anthropometric measurements (body mass, height,
and BMI) as covariates because previous work showed these
to be related to ACL failure.11,14For the same reason,9,10,15we
selectedACLanthropometricmeasurements(length,area,and
volume)ascovariates.Wepreviouslyfoundthatlargersubjects
have larger ACLs9and subjects with larger ACL have stronger
ACLs.11Finally, age16and sex11,12play a role in the structural
properties of the ACL.
The FATC was then mounted in a custom made fixture
attached to a servo-hydraulic testing machine (Instron 8500+)
with the longitudinal axis of the ACL aligned with the load-
ing axis. The FATC was then loaded for 20 cycles between 25
and 150N at 0.25Hz to precondition the tissue. The FATC
secondary analysisondatathatwere
Table 1. Mean and Standard Deviation of Age, Body
Mass, Height, and ACL Size (Length, Minimum Area, and
Volume) for Each Sex
Parameters MaleFemale
Age (years)
Body mass (lb)
Height (ft)
BMI (kg/m2)
ACL length (mm)
Minimum area (mm2)
ACL volume (mm3)
39.0±10.2
172.1±41.36
5.8±0.14
24.8±5.4
29.8±2.5
78.3±23.6
2,967±886
37.7±11.3
158.1±47.1
5.4±0.22
26.1±8.7
26.8±2.8
56.7±14.9
1,954±516
was then loaded to failure at a strain rate of 100%/s. Three
specimens failed prematurely in avulsion and were excluded.
Structural properties (load at failure, linear stiffness, elonga-
tion at failure, and energy at failure) were calculated from
the load-displacement data obtained from the remaining 17
specimens (9 females and 8 males).
A linear model was used to determine whether a general
linear relationship existed between the response variables
(structural properties) and the explanatory variables (age, sex,
height, body mass, BMI, ACL length, minimal area, and vol-
ume).
Response = ?0+ ?1sex + ?2age + ?3height + ?4mass
+?5BMI + ?6ACLMin.area+ ?7ACLVolume
+?8ACLLength+ ε
where the sex term with coefficient ?1distinguishes the sex
effect: (sex=1 if male and 0 if female). This distinction means
that males and females have different model intercepts or
starting points in the equation. The coefficients ?1,...,?8are
the partial slope parameters of the covariates, ?0is the inter-
cept, and ε is the experimental error.
Units used for each covariate included age (years), body
mass(kg),height(m),andBMI(kg/m2),ACLlength(mm),ACL
area (mm2), and ACL volume (mm3). The main criterion used
to assess the predictive capability (goodness of fit) of the linear
model was the magnitude of the coefficient of determination,
R2.
We also devised a graphical technique to present model pre-
dictions and corresponding 95% prediction intervals (Fig. 1).
Using Equation (1), we calculated the predicted values of load
at failure based on the anthropometric data of the 17 subjects.
These loads at failure were then ranked (lowest, 1, to highest,
17), and the percentile rank for each prediction was calcu-
lated as 100×(rank/n+1) where n is the number of subjects
or measurements. For instance, the rank of the median of the
predicted values is 9 and its percentile rank is 100(9/18)=50,
that is, the median is the 50th percentile. We then plotted per-
centile rank (horizontal axis) versus load at failure (vertical
axis). Figure 1 presents the smoothing spline fit to the pre-
dictedloadsatfailure(solidblueline),andtheirrespective95%
prediction curves (solid red lines indicating lower and upper
prediction limits) for randomly selected individual subjects
plottedagainstthepercentileranks.InFigure1,theprediction
interval is the vertical distance between the lower and upper
prediction limits for a given vector of values of the explana-
tory variables. The solid black lines are 95% confidence curves
for the mean load at failure for given values of explanatory
variables.
(1)
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PREDICTED PROPERTIES OF HUMAN ANTERIOR CRUCIATE LIGAMENT
995
Figure 1.
(red), and 95% confidence curves (UCL and LCL) for the mean load at failure for given values of explanatory variables (black). This
graph, in conjunction with Equation (2), can be used to obtain the approximate 95% prediction interval and the approximate 95%
confidence interval for the mean for given values of the explanatory variables (age, sex, body, and ACL anthropometric measurements).
The prediction interval is the vertical distance between the lower (LPL) and upper (UPL) prediction limits. The confidence interval is
the vertical distance between the lower (LCL) and upper (UCL) confidence limits. For example, for a 44-year-old male subject with body
mass=80kg, height=1.7m, and BMI=27.6kg/m2, the corresponding ACL load at failure of 1,500N. This value can be identified on
the vertical axis and from there, a horizontal line (dashed black line) that intersects the blue prediction line can be constructed. Then
a perpendicular line (line B) may be drawn that intersects the 95% confidence lines (solid black lines) showing upper (intersection with
line A) and lower confidence (intersection with line C) limits of 1,800 and 1,200N, correspondingly. Line B also intersects the horizontal
axis showing the rank or percentile of 53. LPL, lower prediction interval; UPL, upper prediction interval; LCL, lower confidence interval;
UCL, upper confidence interval.
Regression curve (blue), 95% prediction curves (UPL and LPL) for load to failure for a randomly selected individual subject
Finally, we evaluated the contribution of ACL geome-
try (length, minimal area, and volume) to prediction of its
structural properties. This analysis was done by excluding
ACL geometry variables from Equation (1), re-evaluating
the model, and then comparing the model R2value without
ACL geometry to the original model R2value that included
geometry.
RESULTS
In our analyses, we have a sample size of 17 (9 males
and 8 females) and 8 eight explanatory variables (age,
sex, height, body mass, BMI, ACL length, minimal area,
and volume). Although we could have benefited from
a larger sample size, the current sample size proved
sufficient for the number of explanatory variables in
producing a good predictive model. In a multiple regres-
sion model, each explanatory variable is judged for its
significance in the presence of other covariates, and
this could create confusion. For instance, moderate and
significant correlations (p<0.05) existed between ACL
length and elongation at failure (r=0.54), ultimate
load (r=0.76), stiffness (r=0.69), and energy at fail-
ure (r=0.72). However, in the presence of the other
covariates, ACL length was neither a significant con-
tributor to the model, nor was it a contributor to the
existing sex differences of any ACL structural property.
As a result, ACL length does not appear in any of the
models.
Load at Failure Prediction
The estimated coefficients along with their respective
standard errors and the p-values for the significance
of the individual explanatory variables are presented
in Table 2. Sex, age, body mass, height, and BMI were
Table 2. Estimates Standard Errors, p-Values, and Standardized Coefficients for Load at Failure
Parameter
Estimate
Standard
Error
Standardized
EstimateLabel
p-Value
Intercept
Sex
Age
Body mass
Height
BMI
Min. area
Volume
29,179
771.5
−50.0
159.3
−15,519
−459.2
14.1
−0.37
10,727
269.8
7.7
65.1
6212.3
192.1
11.1
0.35
0.02
0.01
0.0001
0.03
0.03
0.04
0.23
0.32
0.60
−0.79
4.53
−2.00
−3.71
0.39
−0.39
The standardized estimates indicate that body mass is the most effective covariate in the model, then BMI, height, age, sex, volume, and
area, respectively.
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HASHEMI ET AL.
Figure 2.
(blue diamonds) to the straight line depicting the cumulative normal distribution indicates that the assumption of normality of the
distribution of load to failure is justified.
Normal probability plot of the model residuals. The conformance of the empirical cumulative probabilities of the residuals
significant contributors to the model. The minimal area
andvolumeoftheACLwerenotsignificantcontributors;
nevertheless they were included since these terms were
highly significant for elongation and energy at failure.
Accordingly, the estimated regression model for load at
failure was:
Load at failure = 29,179 + 771sex−50age + 159mass
−15,519height−459BMI
+14ACLMin.area−0.37ACLVolume
The correlation analysis between the model pre-
dicted and experimentally measured load at failure
values revealed a strong positive linear association with
r=0.96. The test for the overall fit of this model was
significant with a p-value <0.0002 and R2=0.914.
The prediction curve (blue) and the corresponding
95% prediction limits (red) are presented in Figure 1.
The datum points represent the actual measured val-
ues.Thisgraph,inconjunctionwithEquation(2),canbe
used to obtain the 95% prediction and confidence inter-
vals for given values of the explanatory variables (age,
sex, body, and ACL anthropometric measurements) for
an individual subject. For example, applying Equation
(2) to a 44-year-old male subject with body mass=80kg,
height=1.7m, and BMI=27.6kg/m2, produces a cor-
responding ACL load at failure of 1,500N. This value
can be identified on the vertical axis of Figure 1 and
from that point a horizontal line (dashed line on the
figure) that intersects the blue prediction line can be
constructed. Then a perpendicular line (line B) can be
drawn that intersects the 95% prediction lines (red
lines) showing upper (intersection with line A) and
lowerprediction(intersectionwithlineC)limitsof2,100
and 800N, correspondingly. Line B also intersects the
horizontal axis showing the rank or percentile of 53,
indicating that the predicted value falls close to the mid-
(2)
dle for the range in the population that we studied. The
same process can be used to find an ∼95% confidence
interval for the mean by using the confidence interval
curves instead of the prediction curves. Finally, in an
effort to assess whether the data were normally dis-
tributed, we constructed a normal probability plot (Fig.
2). The residuals (differences between predicted and
measured values) were plotted against the theoretical
normal. The plot should form a straight line with depar-
tures indicating departures from normality. We found a
close fit between model residuals and no departure from
normality.
RemovalofminimalareaandvolumeoftheACLfrom
themodelhadlittleeffectonoverallfit,slightlyreducing
R2to 0.898; all remaining covariates remained signifi-
cant (p<0.05).
Elongation at Failure Prediction
The estimated coefficients of the regression model and
their respective standard errors and the p-values for
the significance of the individual explanatory variables
are presented in Table 2. Sex, age, body mass, height,
BMI, ACL minimal area, and ACL volume were signifi-
cant contributors. Accordingly, the estimated regression
model for elongation at failure was:
Elongation at failure = 244 + 6.05sex−0.20age
+1.3mass−133height−3.8BMI
−0.0069ACLVolume
+0.15ACLMin.area
(3)
Thecorrelationcoefficientbetweenthemeasuredand
predicted values was r=0.93. The test for the overall fit
was significant with p-value <0.001 and R2=0.872.
Removal of ACL geometry from Equation (3)
impacted the overall fit substantively reducing R2to
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PREDICTED PROPERTIES OF HUMAN ANTERIOR CRUCIATE LIGAMENT
997
0.519, the overall model became borderline significant
(p=0.054), and all remaining covariates except age
became nonsignificant contributors (p>0.05).
Energy Absorbed at Failure Prediction
The estimated coefficients of the regression model for
energy absorbed at failure along with their respective
standard errors and the p-values for the significance of
the individual explanatory variables are presented in
Table 3. Sex, age, body mass, height, BMI, ACL minimal
area, and ACL volume were significant contributors.
Accordingly, the regression model for energy at failure
was:
Energy at failure = 310,927 + 6,603sex−304age
+815mass−52,518height
−5,199BMI−6.5ACLVolume
+150ACLMin.area
(4)
Thecorrelationcoefficientbetweenthemeasuredand
predicted values was r=0.96. The test for the over-
all fit was significant with p-value <0.0002 and R2=
0.913.
Table 3. Estimates Standard Errors, p-Values, and Standardized Coefficients for Elongation at Failure
Parameter
Estimate
Standard
Error
Standardized
Estimate Label
p-Value
Intercept
Sex
Age
Body mass
Height
BMI
Volume
Min. area
244.7
6.0
−0.20
1.3
−133.7
−3.8
−0.0069
0.15
48.07
1.20
0.034
0.29
27.84
0.86
0.0016
0.049
0.0007
0.0007
0.0003
0.001
0.001
0.001
0.002
0.01
1.2
−0.88
10.3
−4.7
−8.4
−2.0
1.1
The standardized estimates indicate that body mass is the most effective covariate in the model, followed by BMI, height, volume, sex,
area, and age, respectively.
Table 4. Estimates Standard Errors, p-Values, and Standardized Coefficients for Energy at Failure
Parameter
Estimate
Standard
Error
Standardized
EstimateLabel
P-value
Intercept
Sex
Age
Body mass
Height
BMI
Volume
Min. area
310,927
6603.9
−304.6
815.9
−52,518
−5199.8
−6.5
150.5
61,462
1545.9
44.6
169.2
10,849
1101.1
2.0
63.8
0.0007
0.002
<0.0001
0.0009
0.0009
0.001
0.01
0.04
0.9
−0.85
8.9
−3.9
−7.3
−1.2
0.74
The standardized estimates indicate that body mass is the most effective covariate in the model, followed by BMI, height, volume, sex,
age, and area, respectively.
Table 5. Estimates Standard Errors, p-Values, and Standardized Coefficients for Stiffness
Parameter
Estimate
Standard
Error
Standardized
EstimateLabel
p-Value
Intercept
Sex
Age
Body mass
Height
BMI
Volume
Min. area
6925.2
212.4
−5.2
16.0
−1183.7
−105.5
−0.13
6.2
2820.3
70.9
2.0
7.7
497.8
50.5
0.094
2.9
0.03
0.01
0.02
0.06
0.04
0.06
0.18
0.06
1.0
−0.54
6.4
−3.2
−5.4
−0.93
1.1
The standardized estimates indicate that body mass is the most effective covariate in the model, followed by BMI, height, area, sex,
volume, and age, respectively.
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HASHEMI ET AL.
RemovalofminimalareaandvolumeoftheACLfrom
Equation (4) had reduced overall fit to R2=0.792, but
the overall model remained significant (p=0.0005), and
all remaining covariates were maintained as significant
contributors to the model (p>0.05).
Linear Stiffness Prediction
The estimated coefficients of the regression model along
with their respective standard errors and the p-values
for the significance of the individual explanatory vari-
ables for linear stiffness are presented in Table 4. Sex,
age, height, and ACL minimal area were found to be sig-
nificant contributors. ACL volume was not significant.
Accordingly, the estimated regression model for linear
stiffness was:
Linear stiffness = 6,925 + 212sex−5.2age + 16mass
−1,183height−105BMI
−0.13ACLVolume+ 6.2ACLMin.area
Figure 3.
of body mass and BMI. (b) Shows the slightly larger region of the permissible values of body mass and height. If the selected height or
BMI of a subject and the corresponding body mass fall outside the ellipses (for instance, in (b), a subject with a body mass of 80kg and a
height of 1.5m—intersection of the dashed lines) the reliability of the prediction will suffer.
Prediction ellipses for collinear variables body mass, BMI, and height. (a) Shows the narrow regions of the permissible values
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PREDICTED PROPERTIES OF HUMAN ANTERIOR CRUCIATE LIGAMENT
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Thecorrelationcoefficientbetweenthemeasuredand
predicted values was r=0.93. The test for the overall fit
was significant with p-value <0.001 and R2=0.872.
If ACL geometry was removed from Equation (5), its
overall fit suffered substantively, reducing R2to 0.575,
but the overall model remained significant (p=0.03),
with only sex and age as significant contributors
(p<0.05).
The range of values for explanatory variables that
can be reliably used with the models (Equations 2–5) is
importanttoappreciate.Iftheexplanatoryvariablesare
not correlated, a simple method is to check whether the
value of each variable falls in the range used in model
estimation. In the case when explanatory variables are
correlated, then their joint range forms a smaller range
of values that are applicable to the model. In our case,
twosetsofvariablesarecorrelated:BMIwithbodymass
andheightwithbodymass.Accordingly,thescatterplots
of the explanatory variables along with their respec-
tive 95% ellipses of concentration (blue solid lines on
Fig. 3) to outline the acceptable ranges of values of the
explanatory variables were provided for the purpose of
prediction. Because of a strong correlation, r=0.915,
between BMI and body mass, the joint area for predic-
tion was quite narrow (Fig. 3a). The correlation between
height and body mass was r=0.519, and hence the pre-
diction range was a larger portion of Figure 3b. These
figures indicate that only combinations of BMI and body
mass (or height and body mass) that fall within the
area defined by the solid blue lines represent combi-
nations that will produce reliable predictions of load at
failure.
DISCUSSION
Our results show that the structural property models
developed in our previous report11have predictive capa-
bilities. Load at failure, elongation at failure, energy
at failure, and linear stiffness can all be reasonably
predicted based on a subset of explanatory variables
associated with increased risk of suffering ACL injury
includingsubjectsex,bodysize(mass,height,andBMI),
and ACL size (minimal area and volume). Although our
analysis does not lend itself to developing distinct mod-
elsforthetwosexes,thebehaviorsformalesandfemales
are nevertheless differentiated through the model inter-
cept (representing the difference in the mean between
males and females adjusted for the covariates). Geomet-
ricallyspeaking,thetwomodelsaredistinctinthesense
thattheyrepresenttwoparallelhyperplanes.Webelieve
that with a larger sample size, two distinct equations
could be developed. Furthermore, when sex differences
in ACL maintenance and repair/remodeling are more
fullyunderstood,6,17molecularcovariatesmaybeadded,
resulting in refined models for male and female ACL
properties that incorporate a spectrum of input risk fac-
tors ranging from genetics to biomechanics and body
morphometry.
In Table 6, we present the prediction of load
at apparent different failure for fictitious male and
female subjects with ages (20, 30, and 40 years), body
mass (ranging from 54.43 to 97.97kg), height (rang-
ing from 1.57 to 1.77m), and BMI (ranging from 21.9
to 30.9kg/m2). All explanatory variables fall within the
acceptable ranges for the correlated predictors defined
by the ellipses in Figure 3. From the data presented in
Table 6. Prediction of Load at Failure for Fictitious Male and Female Subjects with a Variety of Predictor Values (Age,
Body Mass, Height, and BMI)
Body
Mass (kg)
Height
(m)
BMI
(kg/m2)
Predicted
Load (N)
LPL
(N)
UPL
(N)
LCL for
Mean (N)
UCL for
Mean (N)
GenderAge
Female
Female
Female
Female
Female
Female
Female
Female
Female
Male
Male
Male
Male
Male
Male
Male
Male
Male
20
20
20
30
30
30
40
40
40
20
20
20
30
30
30
40
40
40
54.4
66.7
76.7
59.9
73.5
84.4
69.4
85.3
98.0
54.4
66.7
76.7
59.9
73.5
84.4
69.4
85.3
98.0
1.6
1.6
1.6
1.7
1.7
1.7
1.8
1.8
1.8
1.6
1.6
1.6
1.7
1.7
1.7
1.8
1.8
1.8
21.9
26.8
30.9
22.0
27.0
30.9
22.0
27.0
31.0
21.9
26.8
30.9
22.0
27.0
30.9
22.0
27.0
31.0
2398.1
2085.2
1830.3
1576.9
1453.2
1354.2
629.1
853.2
1032.4
3109.4
2796.5
2541.5
2288.2
2164.4
2065.5
1340.3
1564.4
1743.7
1576.7
1307.2
965.0
949.4
807.8
643.0
−208.9
73.4
254.8
1995.9
1744.4
1446.8
1489.9
1373.9
1236.9
656.4
938.5
1109.0
3219.6
2863.3
2472.5
2204.4
2098.5
2065.5
1467.0
1633.0
1514.11
4222.9
3848.5
3636.2
3086.4
2955.0
2894.0
2024.3
2190.4
2378.4
1816.2
1566.4
1188.0
1336.9
1169.7
942.2
24.1
331.6
514.1
2158.7
1918.6
1613.0
1739.4
1626.9
1473.6
977.5
1328.4
1485.3
2980.1
2604.1
2472.5
1816.9
1736.7
1766.2
1234.1
1374.7
1550.7
4060.0
3674.4
3470.1
2836.9
2701.9
2657.4
1703.2
1800.4
2002.1
LPL, lower prediction interval; UPL, upper prediction interval; LCL, lower confidence interval; UCL, upper confidence interval.
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HASHEMI ET AL.
Table 6 (for both sexes), younger, shorter subjects with
lower body mass have ACLs with high ultimate failure
properties. Further, females with the same age, height,
and body mass have a lower ACL load at failure than
males.
Inthe20–30yearsrangebothsexeswillhavereduced
load at failure with increased BMI and body mass.
This observation, combined with the prognostic study
of Uhorchak et al.14that reported increased BMI and
body mass are associated with increased risk of ACL
injury in military cadets, suggests a direct link between
increased BMI/body mass and decreased ultimate fail-
ure load. This suggest the hypothesis that for subjects
with the same height and age, increased body mass
results in inferior structural properties that lowers the
threshold within which the ACL can operate without
injury. This is a concern considering that increased body
mass increases the inter-segmental shear loads at the
tibiofemoral joint that must be resisted by the ACL. It
may be that increased body mass is detrimental to the
ACL in at least two ways: it decreases the structural
properties and produces increased anteriorly directed
shear loads.
Recently, in a case-matched control study of risk
factors for ACL injury, ACL-injured subjects had a
significantly smaller ACL compared to controls.15Previ-
ously, two other reports established that female subjects
had smaller ACLs,10,18even when the differences in
body size between male and females were considered.13
As a result, we assessed the impact of ACL size on
the prediction of structural properties. Excluding size
variables (minimal area and volume) had a negligible
impact on the load at failure model (<2% reduction in
R2) but a detrimental impact elongation at failure (35%
reduction), energy at failure (12% reduction), and linear
stiffness (18% reduction). Thus, ACL size is an impor-
tant contributor to overall model predictability.
The fact that removal of ACL size from the load at
failure model produced a small difference in its overall
effectiveness raises the question that perhaps size is not
important. How can size substantially affect other prop-
erties but not load at failure? We believe this is because
the load distribution across the ACL cross-section dur-
ing failure testing is not uniform due to the differences
in length between the anterior and posterior bundles
andthegeometricallycomplextibialandfemoralattach-
ments. Although we tried to load the ACL along its
longitudinal axis, the failure of the collagen bundles
did not occur uniformly across the cross-section. Only
a proportion of the bundles carried the applied load
during the test. This is why failure always occurred in
quick successive stages with not all fibers failing simul-
taneously. This may have caused enough disturbances
in the failure load prediction that the model did not
show sensitivity to size. Although the minimal area may
not be a good explanatory variable for load at failure, it
is clear that a bigger ACL has more fibrils active when
loaded and should have the capacity to sustain larger
magnitude of applied loads.
Linear stiffness, elongation at failure, and energy at
failure are all calculated with reference to elongation of
the ACL. Elongation appears in the stiffness equation
(change in force over change in length), in the energy
at failure equation (change in force×change in length),
andobviouslyinelongationatfailure(changeinlength).
Although in our tests, we could not control the unifor-
mity of force across the cross-sectional area, we could
better maintain uniform elongation of the fibers by load-
ing the ACL along its longitudinal axis and not allowing
the bony attachments to rotate during loading. This
assured that the fibrils were uniformly elongated, sim-
ilar to a situation in which a group of parallel springs
with different stiffness values are attached to two plates
pulled apart in pure translation, without rotation. Each
spring deforms to the same level, but each will develop
a different force. It is for this reason that stiffness, elon-
gation at failure, linear stiffness, energy at failure may
have shown size sensitivity.
Finally, to address the issue of effectiveness of
the model for obtaining reliable predictions, we must
address the effect of multicollinearity on prediction.
How do strong correlations between height and mass
and BMI and mass impact the predictability of the mod-
els? Multicollinearity is a topic of concern in regression
models because of its adverse effect on variance infla-
tion, which can result in the erroneous conclusion that
a single or a group of effective explanatory variables are
insignificant. In addition, because of multicollinearity,
some of the estimated coefficients may have the wrong
sign;however,thisphenomenondoesnotaffectourmod-
els. We compared variance inflation factors (VIF) for
various explanatory variables to a benchmark. A VIF
>10 indicates variance inflation and depending on its
severity this may or may not influence the significance
of the models. VIF for body mass, height, and BMI
were >10, though these VIFs were not severe enough
to affect the significance of the coefficients. Finally, mul-
ticollinearity makes it difficult to interpret the role of
individual variables in the model. In the final analy-
sis, multicollinearity does not prevent one from using
the model to obtain reliable predictions, as long as the
values of the explanatory variables used for prediction
fall in the joint ranges of the observations used to estab-
lish the model.
In conclusion, we developed an innovative graphical
method to predict the structural properties of the ACL
by depicting a multidimensional model in 2D. Although,
for single-variable regression models, such graphs can
routinely be developed and are generated by statisti-
cal packages, no such graphical relationship has been
discussed in the literature for multivariable regression
models. The graph in Figure 1 presents the predicted
values versus their ranks in ascending order and can
be used to obtain the corresponding prediction intervals
graphically without having to perform complex compu-
tations. The normal probability plot (Fig. 2) shows a
close fit between the model residuals and percentiles
of a normal distribution indicating that the inferences
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PREDICTED PROPERTIES OF HUMAN ANTERIOR CRUCIATE LIGAMENT
1001
based on the assumption are valid. Finally, the limits
of predictability are defined by the areas enclosed by
the dashed ellipses in Figure 3. Our results constitute
the first model to attempt to predict ACL structural
properties based on selected covariates associated with
increased risk of injury. Future models may be sig-
nificantly more complex. Nevertheless, our model is
predictive (because of the high R2value) for a first
attempt.
ACKNOWLEDGMENTS
The corresponding author, Javad Hashemi, acknowledges the
support of Texas Tech College of Engineering and Texas Tech
HealthSciencesCenter.ThesupportoftheNIHRO1AR050421
(BDB) and AR 049767 (DMH)) is also acknowledged.
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