Citation: Rasmussen, J.; Skejø, S.;
Waagepetersen, R.P. Predicting Tissue
Loads in Running from Inertial
Measurement Units. Sensors 2023, 23,
9836. https://doi.org/10.3390/
s23249836
Academic Editor: Georg Fischer
Received: 19 October 2023
Revised: 28 November 2023
Accepted: 13 December 2023
Published: 15 December 2023
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© 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
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conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
sensors
Article
Predicting Tissue Loads in Running from Inertial
Measurement Units
John Rasmussen 1,*
, Sebastian Skejø 2,3
and Rasmus Plenge Waagepetersen 4
1
Department of Materials and Production, Aalborg University, Fibigerstraede 16, 9220 Aalborg East, Denmark
2
Department of Public Health, Aarhus University, Bartholins Allé 2, 8000 Aarhus, Denmark; sdsk@ph.au.dk
3
Research Unit for General Practice, Aarhus University, Bartholins Allé 2, 8000 Aarhus, Denmark
4
Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, 9220 Aalborg East, Denmark;
rw@math.aau.dk
*
Correspondence: jr@mp.aau.dk
Abstract: Background: Runners have high incidence of repetitive load injuries, and habitual runners
often use smartwatches with embedded IMU sensors to track their performance and training. If
accelerometer information from such IMUs can provide information about individual tissue loads,
then running watches may be used to prevent injuries. Methods: We investigate a combined physics-
based simulation and data-based method. A total of 285 running trials from 76 real runners are
subjected to physics-based simulation to recover forces in the Achilles tendon and patella ligament,
and the collected data are used to train and test a data-based model using elastic net and gradient
boosting methods. Results: Correlations of up to 0.95 and 0.71 for the patella ligament and Achilles
tendon forces, respectively, are obtained, but no single best predictive algorithm can be identiﬁed.
Conclusions: Prediction of tissues loads based on body-mounted IMUs appears promising but
requires further investigation before deployment as a general option for users of running watches to
reduce running-related injuries.
Keywords: running; injuries; Achilles tendon; patella ligament; IMU; data science; biomechanics;
public health
1. Introduction
Running is a popular physical activity not only for recreational purposes, but also
among elite athletes. For instance, a survey covering the years 2015 through 2022 in
England [1] showed that about 10% of the population, corresponding to roughly 6 million
people, regularly engage in running. Running thus engages enough participants to have
an impact on the overall activity level of the population, and the health beneﬁts of active
lifestyles are well documented [2,3].
Unfortunately, running is also associated with a high risk of sustaining a running-
related injury, and injury incidence between 8 and 18 injuries per 1000 h of running has
been reported [4]. Especially the knee and the ankle are susceptible to injuries with Achilles
tendinopathy being the most incident injury and patellofemoral pain syndrome being the
most prevalent injury [5]. Apart from the immediate discomfort and long recovery time
associated with running-related injuries [6,7], injuries are also the most common reason for
stopping running [8,9]. Therefore, it is imperative to mitigate the risk of running-related
injuries, which requires a profound understanding of why these injuries occur [10].
Running-related injuries are commonly assumed to be caused by the repetitive loading
of the affected tissues, resulting in inﬂammation and tissue failure over time [11]. Loads
on muscles, tendons, and bones are, however, difﬁcult—bordering on impossible—to
measure in vivo, even in advanced laboratories [12]. Advances in model-based simulation
of musculoskeletal forces over the past two decades have provided algorithms [13,14] and
models [15] that veriﬁably predict internal musculoskeletal forces, but these methods rely
Sensors 2023, 23, 9836. https://doi.org/10.3390/s23249836
https://www.mdpi.com/journal/sensors
Sensors 2023, 23, 9836
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on complete kinematic input, typically obtained from optical motion capture systems [16]
or motion capture suits [17]. Optical systems are impractical in ﬁeld settings, and both types
are too comprehensive for research with large cohorts as well as to inform the individual
runner. Therefore, there is a need for light-weight technologies that can accurately estimate
tissue loads and injury risk factors in ﬁeld settings.
Such technology could be running watches, which many recreational and elite runners
rely on to track their training efforts and performance [18,19], and whose popularity has
also been attributed to independence from organized coaching [20]. Running watches
typically record biometrics, such as pulse, and kinematic data, such as accelerations and
GPS positions, from which running speed, step frequency and step length can be derived.
Recent developments in data science have inspired data-based models linking inertial
measurement unit (IMU) kinematics, complete kinematics [21] and musculoskeletal kinet-
ics [22–28]. Such models can be trained on musculoskeletal simulations based on optical
motion capture data and have shown promising results for modeling of a variety of tasks,
but on relatively small and uniform cohorts of test subjects.
The data space of all possible movements performed by all possible people is very
large, and the connection between kinematics and kinetics expressed by the laws of dynam-
ics is highly nonlinear, so it is unlikely that machine learning will outcompete physics-based
models in general. However, running is a small subset of human movements, and given
runners’ need for fast and lightweight feedback from IMUs, it is worth investigating
whether reliable kinetic estimations, i.e., tissue loads, are possible from the combination
of IMU data, a database of veriﬁed running kinematics for a medium-size heterogeneous
population, and simple anthropometrics for the individual runner.
When choosing predictors, there are a few considerations to keep in mind. First and
foremost, the predictors should be feasible and inexpensive to collect in ﬁeld-based settings
and, therefore, require as little additional equipment as possible. Secondly, to be relevant
for performance and injury mechanisms, the predictors should carry as much information
as possible about running biomechanics. Thirdly, the total of number of predictors should
be curbed to avoid overﬁtting.
IMUs, such as those embedded in running watches and smartphones, are inexpensive
and feasible to use in ﬁeld-based settings. They provide kinematic data only for their
attachment points, but they typically sample at frequencies of 400–500 Hz [29] and therefore
accumulate a large amount of data over a short time. In the interest of data reduction, some
studies extract a few features from the raw signal’s time domain, such as the maximum
angle of a segment or peak accelerations, which capture some, but not all, aspects of running
kinematics. An alternative approach is to transform the entire signal to the frequency
domain using discrete Fourier transformation. Given the periodic nature of the data, this
provides an accurate, yet compressed, description of the entire running kinematics [30].
We investigate the prediction of Achilles tendon and patella ligament loads based on data
from IMUs positioned on easily accessible anatomical locations, i.e., wrists, ankles, and the
sternum, and we compare IMU positions and sets of predictors to optimize the results.
2. Materials and Methods
Figure 1 illustrates the data ﬂow and computational investigations.
2.1. Experimental Data
The experimental and biomechanical simulation procedures were described previously
in detail [30], and are brieﬂy summarized here for completeness: a nine-camera Qualisys
Miqus system (Qualisys AB, Gothenburg, Sweden) was used to collect full-body optical
marker data for treadmill running for 78 runners (30 female, 48 male) in 285 trials (180 male
trials and 105 female trials) with speeds ranging between 6 and 20 km/h at 300 frames per
second and a resolution of 2 megapixels. The subjects were asymptomatic and ranged in
skill level from beginner to elite. The optical marker data were transferred to a physics-
based simulation in the AnyBody Modeling System version 7.2 (AnyBody Technology
Sensors 2023, 23, 9836
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A/S, Aalborg, Denmark) [13], which converted the marker data to anthropometrical data,
i.e., individual segment dimensions, and anatomical joint angle time series [31]. Using
the Twente Lower Extremity Model version 2.0 [32], this process also simulates internal
biomechanical kinetics. For the purposes of this paper, simulated maximum values of the
patella ligament force and the Achilles tendon force over the running cycle were stored for
each recorded trial and normalized by body mass.
Sensors 2023, 23, x FOR PEER REVIEW 
3 of 13 
 
 
Figure 1. Data ﬂow and computational methods. Figures in parentheses designate numbers of var-
iables. The collected data set contains a total of 1229 variables of which 43 are virtual accelerometer 
data and 132 are anthropometric parameters. The prediction methods are, respectively, Elastic Net 
with two tuning parameters and XGB with three tuning parameters. The experimental data are ran-
domly split into training and validation set. 
2.1. Experimental Data 
The experimental and biomechanical simulation procedures were described previ-
ously in detail [30], and are brieﬂy summarized here for completeness: a nine-camera 
Qualisys Miqus system (Qualisys AB, Gothenburg, Sweden) was used to collect full-body 
optical marker data for treadmill running for 78 runners (30 female, 48 male) in 285 trials 
(180 male trials and 105 female trials) with speeds ranging between 6 and 20 km/h at 300 
frames per second and a resolution of 2 megapixels. The subjects were asymptomatic and 
ranged in skill level from beginner to elite. The optical marker data were transferred to a 
physics-based simulation in the AnyBody Modeling System version 7.2 (AnyBody Tech-
nology A/S, Aalborg, Denmark) [13], which converted the marker data to anthropomet-
rical data, i.e., individual segment dimensions, and anatomical joint angle time series [31]. 
Using the Twente Lower Extremity Model version 2.0 [32], this process also simulates in-
ternal biomechanical kinetics. For the purposes of this paper, simulated maximum values 
of the patella ligament force and the Achilles tendon force over the running cycle were 
stored for each recorded trial and normalized by body mass. 
The time series of anatomical joint angles were segmented into strides, which were 
transferred to the frequency domain by Fast Fourier Transform (FFT), retaining ﬁve sine 
and ﬁve cosine terms and the DC (constant) component, i.e., 11 coeﬃcients to describe the 
motion of each of the 88 independent anatomical degrees-of-freedom. This procedure was 
described in detail previously [30]. 
Using the model, we emulated ﬁve virtual, three-axis accelerometers located on the 
two wrists, the two ankles and the sternum. The accelerometers were constructed in the 
model as body segment-ﬁxed local reference frames, whose acceleration vectors could be 
extracted. The placements and local coordinate systems are illustrated on Figure 2. Gyro-
scopic information was not included The virtual accelerations were combined after the
Methods
10%
90%
PREDICTION
Anthropometrics (43)
Methods
Elastic net
XGB
Parameters:
α and λ
Parameters:
m, d, η
Random split
Validation data
Training data
Experimental  
data (1229)
Accelerations (132)
Max. Achilles tendon force (1)
Max. patella ligament force (1)
Max. Achilles tendon force (1)
Max. patella ligament force (1)
Max. Achilles  
tendon force
Max. patella 
ligament force
Validation
Figure 1. Data ﬂow and computational methods. Figures in parentheses designate numbers of
variables. The collected data set contains a total of 1229 variables of which 43 are virtual accelerometer
data and 132 are anthropometric parameters. The prediction methods are, respectively, Elastic Net
with two tuning parameters and XGB with three tuning parameters. The experimental data are
randomly split into training and validation set.
The time series of anatomical joint angles were segmented into strides, which were
transferred to the frequency domain by Fast Fourier Transform (FFT), retaining ﬁve sine
and ﬁve cosine terms and the DC (constant) component, i.e., 11 coefﬁcients to describe the
motion of each of the 88 independent anatomical degrees-of-freedom. This procedure was
described in detail previously [30].
Using the model, we emulated ﬁve virtual, three-axis accelerometers located on the
two wrists, the two ankles and the sternum. The accelerometers were constructed in the
model as body segment-ﬁxed local reference frames, whose acceleration vectors could
be extracted. The placements and local coordinate systems are illustrated on Figure 2.
Gyroscopic information was not included. The virtual accelerations were combined, after
the appropriate coordinate transform, with the gravity component to mimic the accel-
erations including gravity that would have been measured by body-ﬁxed IMUs. Time
series for the resulting virtual accelerations in segment-ﬁxed x, y, and z coordinates along
with the magnitude of the acceleration vectors were also subjected to FFT transformation,
retaining 11 Fourier coefﬁcients for each direction and the magnitude, leading to a total of
44 coefﬁcients for each accelerometer.
Sensors 2023, 23, 9836
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including gravity that would have been measured by body-ﬁxed IMUs. Time series for 
the resulting virtual accelerations in segment-ﬁxed x, y, and z coordinates along with the 
magnitude of the acceleration vectors were also subjected to FFT transformation, retaining 
11 Fourier coeﬃcients for each direction and the magnitude, leading to a total of 44 coef-
ﬁcients for each accelerometer. 
 
Figure 2. Placement of virtual IMUs on the left wrist, sternum and left ankle, and their respective 
local coordinate systems. The IMU on the right wrist is placed similarly to the IMU on the left wrist. 
The purple dots are ﬂoor contact points, and the red dots are spinal joint centers. 
This study used a previously anonymized version of the data resulting from the data 
processing described above. The process results in a matrix with 285 rows corresponding 
to the recorded trials. The number of columns is 1229, of which 968 contain Fourier coef-
ﬁcients for the model’s kinematic degrees-of-freedom, and 220 = 5 × 44 columns contain 
Fourier coeﬃcients for the virtual accelerations of the ﬁve selected placements. The max-
imal forces over the cycle for the right and left patella ligaments and Achilles tendons were 
extracted from the kinetic analysis, yielding four additional columns. Furthermore, 35 col-
umns contain anthropometric measurements such as segment lengths, body weight, stat-
ure, and gender. Finally, the table contains running speed and angular step frequency. 
Initial investigations of virtual accelerometer data revealed a high degree of sym-
metry between the left and right ankles. It was therefore decided to discard the right ankle 
and include only accelerations of the two wrists, the left ankle, and the sternum, in total 
176 columns of Fourier coeﬃcients of acceleration data. It was furthermore decided that 
combinations of more than two IMUs would be impractical for most runners and, conse-
quently, the following combinations were selected for further investigation: 
1. 
left wrist 
2. 
left wrist and right wrist 
3. 
left wrist and sternum 
4. 
left wrist and left ankle 
5. 
sternum and left ankle 
6. 
left ankle 
7. 
sternum. 
Single-IMU options 1, 6 and 7 lead to inclusion of 1 × 4 × 11 = 44 columns of Fourier 
coeﬃcients, and double-IMU options 2–5 each lead to inclusion of 88 columns. In the in-
terest of simplicity of the procedure for the runner and to minimize the risk of overﬁtting, 
only the step frequency, sex, age, thigh length, shank length, foot length, body mass, 
Figure 2. Placement of virtual IMUs on the left wrist, sternum and left ankle, and their respective
local coordinate systems. The IMU on the right wrist is placed similarly to the IMU on the left wrist.
The purple dots are ﬂoor contact points, and the red dots are spinal joint centers.
This study used a previously anonymized version of the data resulting from the data
processing described above. The process results in a matrix with 285 rows corresponding to
the recorded trials. The number of columns is 1229, of which 968 contain Fourier coefﬁcients
for the model’s kinematic degrees-of-freedom, and 220 = 5 × 44 columns contain Fourier
coefﬁcients for the virtual accelerations of the ﬁve selected placements. The maximal forces
over the cycle for the right and left patella ligaments and Achilles tendons were extracted
from the kinetic analysis, yielding four additional columns. Furthermore, 35 columns
contain anthropometric measurements such as segment lengths, body weight, stature, and
gender. Finally, the table contains running speed and angular step frequency.
Initial investigations of virtual accelerometer data revealed a high degree of symmetry
between the left and right ankles. It was therefore decided to discard the right ankle and
include only accelerations of the two wrists, the left ankle, and the sternum, in total
176 columns of Fourier coefﬁcients of acceleration data. It was furthermore decided
that combinations of more than two IMUs would be impractical for most runners and,
consequently, the following combinations were selected for further investigation:
1.
left wrist
2.
left wrist and right wrist
3.
left wrist and sternum
4.
left wrist and left ankle
5.
sternum and left ankle
6.
left ankle
7.
sternum.
Single-IMU options 1, 6 and 7 lead to inclusion of 1 × 4 × 11 = 44 columns of Fourier
coefﬁcients, and double-IMU options 2–5 each lead to inclusion of 88 columns. In the
interest of simplicity of the procedure for the runner and to minimize the risk of overﬁtting,
only the step frequency, sex, age, thigh length, shank length, foot length, body mass, stature,
BMI and running speed were included as additional variables, leading to 54 or 98 predictor
variables for single and double-IMU conﬁgurations, respectively.
Sensors 2023, 23, 9836
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2.2. Prediction Algorithms
We are in a setting with a relatively low number of observations (285 running trials)
compared to the number of predictor variables (54 or 98, depending on whether a single-
IMU or double-IMU conﬁguration is used). A standard multivariate linear regression
model is therefore not suitable since it will be prone to overﬁtting. Thus, we consider
regularized versions of multivariate linear regression using the elastic net approach [33],
which encompasses ridge regression, LASSO (Least Absolute Shrinkage and Selection
Operator), and combinations of ridge regression and LASSO. The elastic net further enables
variable selection that discards weak predictors. This has the potential to reduce the
number of variables needed for prediction, which can make the method more feasible for
future users.
A regularized linear regression is ”easy” to interpret, but clearly has shortcomings
in case of nonlinear relationships or interaction effects (unless the latter are explicitly
included in the model). Therefore, we also consider a ﬂexible tree-based machine learning
method based on gradient boosting using speciﬁcally the computationally efﬁcient XGB
algorithm [34]. In this case, the prediction is obtained from a sequence of relatively shallow
trees. These trees are ﬁtted sequentially, where each new tree is ﬁtted to residuals arising
from prediction by the current sequence of trees. The splits employed when constructing
prediction trees are very useful for handling nonlinearities and interactions.
2.2.1. Tuning Parameters
The elastic net approach relies on two tuning parameters, α and λ. The ﬁrst of
them, α, ranges between 0 and 1, where 0 gives ridge regression, 1 gives LASSO, and
intermediate values provide combinations of ridge and LASSO. For α, we consider the
values 0, 0.5 and 1. The other parameter, λ, determines the strength of regularization
and is determined by minimizing a cross validation score. Following the implementation
in the glmnet package [35], the data is partitioned into 10 subsets. For each subset, the
model is trained on the remaining data and is then used to predict the observations in the
subset. The resulting mean square prediction errors over each subset are averaged to get
the cross-validation score.
For XGB, there is a wide range of tuning parameters. Here, we restrict attention to
the important number of sequentially ﬁtted trees m, tree depth d and learning rate η and
leave the remaining tuning parameters at their default values. The number of trees m is
chosen by cross-validation as implemented in the XGB package (with 5 subsets, referring
to the explanation above of cross-validation). Following recommendations in the machine
learning literature, we further consider rather shallow trees, d = 3, 6 and η = 0.1, 0.3. There
is a trade-off between the latter two parameters, so that deeper trees should in general be
combined with a lower learning rate and vice versa.
2.2.2. Model Evaluation and Selection
The models with varying values of tuning parameters are trained on a random subset
of the data containing 90% of the observations (the training set) and the prediction perfor-
mance is next evaluated on the remaining 10% of the observations (the test set). To avoid
sensitivity to the random splitting into training and test data sets, we consider 2500 in-
dependent random splits (with replacement), carry out the training and test procedure
on each resulting training/test data set, and average test results over the 2500 splits. We
measure prediction performance in terms of average correlation and normalized root mean
square error (nRMSE). For each test set, the correlation is between the physics-based values
of the dependent variables and the predicted values obtained from the model trained on
the training data set. The root mean square error (RMSE) is the square root of the average
of squared differences between physics-based values in the test data set and corresponding
predicted values. The RMSE is normalized by dividing by the average of the dependent
variable over the full data set. In a few cases, the elastic net method produced a constant
Sensors 2023, 23, 9836
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predictor. Then, the correlation is not well-deﬁned due to division by zero. We hence
ultimately use nRMSE for identifying the best models.
3. Results
3.1. Comparison of Prediction Methods and Conﬁgurations of Accelerometers
Figures 3 and 4 show performance measures (correlation and normalized RMSE)
when patella ligament forces and Achilles tendon forces, respectively, are predicted us-
ing elastic net and XGB with different settings of tuning parameters and conﬁgurations
of accelerometers.
 
 
average of squared diﬀerences between physics-based values in the test data set and cor-
responding predicted values. The RMSE is normalized by dividing by the average of the 
dependent variable over the full data set. In a few cases, the elastic net method produced 
a constant predictor. Then, the correlation is not well-deﬁned due to division by zero. We 
hence ultimately use nRMSE for identifying the best models. 
3. Results 
3.1. Comparison of Prediction Methods and Conﬁgurations of Accelerometers 
Figures 3 and 4 show performance measures (correlation and normalized RMSE) 
when patella ligament forces and Achilles tendon forces , respectively, are predicted using 
elastic net and XGB with diﬀerent settings of tuning parameters and conﬁgurations of 
accelerometers. 
 
Figure 3. Performance measures for prediction of patella ligament forces by elastic net and XGB, 
respectively. The left frame shows the normalized root mean square error, and the right frame shows 
the correlation. Conditions: lw = left wrist, rw = right wrist, ste = sternum, ank = ankle. 
Figure 3. Performance measures for prediction of patella ligament forces by elastic net and XGB,
respectively. The left frame shows the normalized root mean square error, and the right frame shows
the correlation. Conditions: lw = left wrist, rw = right wrist, ste = sternum, ank = ankle.
Sensors 2023, 23, x FOR PEER REVIEW 
7 of 13 
 
 
Figure 4. Performance measures for prediction of Achilles tendon forces by elastic net and XGB, 
respectively. The left frame shows the normalized root mean square error, and the right frame shows 
the correlation. Conditions: lw = left wrist, rw = right wrist, ste = sternum, ank = ankle. 
Considering patella ligament force, best prediction results were obtained when at 
least one measurement unit was placed at the ankle. In that case, elastic net is superior to 
XGB and correlations up to 0.96 are achieved. For the Achilles tendon force, XGB generally 
outperforms elastic net, and the best results were obtained with at least one measurement 
Figure 4. Performance measures for prediction of Achilles tendon forces by elastic net and XGB,
respectively. The left frame shows the normalized root mean square error, and the right frame shows
the correlation. Conditions: lw = left wrist, rw = right wrist, ste = sternum, ank = ankle.
Sensors 2023, 23, 9836
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Considering patella ligament force, best prediction results were obtained when at
least one measurement unit was placed at the ankle. In that case, elastic net is superior to
XGB and correlations up to 0.96 are achieved. For the Achilles tendon force, XGB generally
outperforms elastic net, and the best results were obtained with at least one measurement
unit placed at the sternum. For the Achilles tendon force, correlations up to 0.72 were
achieved. Generally, both for the patella ligament force and the Achilles tendon force, and
over the seven conﬁgurations of accelerometers, alpha equal to 0.5 or 1 (elastic net) and
shallow trees of depth 3 (XGB) gave the best results in terms of nRMSE.
For the patella ligament force and Achilles tendon force alike, the best results were
obtained when just one measurement unit was used (at the ankle for the patella ligament
force and at the sternum for Achilles tendon force).
3.2. Detailed Inspection of Best Prediction Methods and Measurement Unit Conﬁgurations
To closer inspect the prediction methodology, we considered the best conﬁgurations
of prediction algorithms and accelerometer conﬁgurations, i.e., elastic net with one mea-
surement unit at the ankle and α = 1 for the patella ligament force, and XGB with one unit
at the sternum, d = 3 and η = 0.3 for the Achilles tendon force. Figures 5 and 6 show scatter
plots (so-called calibration plots) of physics-based values and predictions from the test
data sets. To avoid too dense scatterplots, we only included 250 randomly sampled points
from the test data sets. The solid and dashed lines are the identity and the least squares
lines, respectively. The normalized RMSE and correlation are 0.12 and 0.95, respectively for
the patella ligament scatter plot and 0.18 and 0.71 for the Achilles tendon scatter plot. A
moderate bias in the predictions for the Achilles tendon force is revealed by the discrepancy
between the identity and least squares lines, whereas bias is essentially absent for patella
ligament force predictions.
Sensors 2023, 23, x FOR PEER REVIEW 
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Figure 5. Correlation between predicted and physics-based simulated patella ligament forces. The 
legend shows intercepts and slopes of identity line (solid) and least squares line (dashed). 
 
Figure 6. Correlation between predicted and physics-based simulated Achilles tendon forces. The 
legend shows intercepts and slopes of identity line (solid) and least squares line (dashed). 
3.3. Variable Importance 
The importance of the various predictor values was assessed for the patella ligament 
force with elastic net and one measurement unit at the ankle. With α = 1, elastic net may 
Figure 5. Correlation between predicted and physics-based simulated patella ligament forces. The
legend shows intercepts and slopes of identity line (solid) and least squares line (dashed).
Sensors 2023, 23, x FOR PEER REVIEW 
8 of 13 
 
 
Figure 5. Correlation between predicted and physics-based simulated patella ligament forces. The 
legend shows intercepts and slopes of identity line (solid) and least squares line (dashed). 
 
Figure 6. Correlation between predicted and physics-based simulated Achilles tendon forces. The 
legend shows intercepts and slopes of identity line (solid) and least squares line (dashed). 
3.3. Variable Importance 
The importance of the various predictor values was assessed for the patella ligament 
force with elastic net and one measurement unit at the ankle. With α = 1, elastic net may 
estimate some predictor coeﬃcients to be exactly zero, so that the corresponding predictor 
Figure 6. Correlation between predicted and physics-based simulated Achilles tendon forces. The
legend shows intercepts and slopes of identity line (solid) and least squares line (dashed).
Sensors 2023, 23, 9836
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3.3. Variable Importance
The importance of the various predictor values was assessed for the patella ligament
force with elastic net and one measurement unit at the ankle. With α = 1, elastic net may
estimate some predictor coefﬁcients to be exactly zero, so that the corresponding predictor
has no effect on the prediction. Thirteen variables were included in less than 10% of the
test/training data sets. These are age, body mass, stature and ten acceleration measurement
coefﬁcients. Nineteen variables were included in at least 90% of the cases. These are the
angular frequency, gender, shank length and 16 acceleration coefﬁcients. The predictors
can also be ranked according to their estimated effect sizes. The 10 variables with highest
average rank over test/training data sets were (in rank order):
1.
the average acceleration magnitude of the left ankle
2.
the second sine coefﬁcient of the acceleration magnitude of the left ankle
3.
the ﬁrst cosine coefﬁcient of the acceleration magnitude of the left ankle
4.
shank length
5.
the ﬁrst sine coefﬁcient of the acceleration magnitude of the left ankle
6.
the third cosine coefﬁcient of the anterior/posterior acceleration of the left ankle
7.
the ﬁrst sine coefﬁcient of the anterior/posterior acceleration of the left ankle
8.
the third sine coefﬁcient of the acceleration magnitude of the left ankle
9.
the fourth cosine coefﬁcient of the acceleration magnitude of the left ankle
10.
the second sine coefﬁcient of the anterior/posterior acceleration of the left ankle
The angular frequency of the Fourier series and gender have ranks 15 and 22, respec-
tively. Overall, the data do not support reducing the number of predictors for the patella
ligament force signiﬁcantly.
In case of the XGB predictions for the Achilles tendon force using sternum acceleration
measurements, the importance of variables may be assessed by their so-called gain, which
measures the contribution of a variable to the prediction. We assessed the average gains
and average ranks of variables according to their gains over the 2500 test/training data sets.
The ten variables with the highest average gains are:
1.
The fourth sine coefﬁcient of the vertical acceleration of the sternum;
2.
The third sine coefﬁcient of the lateral acceleration of the sternum;
3.
BMI;
4.
The third cosine coefﬁcient of the lateral acceleration of the sternum;
5.
The second sine coefﬁcient of the anterior/posterior sternum acceleration;
6.
Shank length;
7.
The second sine coefﬁcient of the magnitude of the sternum acceleration;
8.
Foot length;
9.
The ﬁrst cosine coefﬁcient of the anterior/posterior sternum acceleration;
10.
The second sine coefﬁcient of the vertical acceleration of the sternum.
It is worth noticing that the gains taper off rapidly, with variables from rank 8 and
upwards sharing similar, small gains with variables outside the list. The variables speed
and gender have lowest and third lowest average ranks, respectively. The remaining
additional (non-acceleration) variables are among the 50% variables with the highest rank;
except perhaps for speed and gender, there does not seem to be an obvious opportunity to
eliminate additional variables.
4. Discussion
Running can be performed at a wide range of speeds, and runners exhibit different
styles depending on their anthropometry and other physiological preconditions. The
participants in the present study are all able-bodied runners, but the data represent speeds
between 6 and 20 km/h and skill levels between beginner and elite, and the resulting
biomechanical loads vary considerably. Nonetheless, despite the complexities of the data,
we obtain satisfactory correlations and prediction errors, and the presented method is a
promising approach to predicting running biomechanics in ﬁeld settings.
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The present study differs from most previously reported results by targeting so-called
structure-speciﬁc loads, i.e., the force on the Achilles tendon and patella ligament. To our
knowledge, only one other study has attempted to predict these structure-speciﬁc loads
from wearable devices [36]. In this study, Brund et al. found mean absolute percentage
errors ranging from 13 to 30%, which appear similar to the normalized root mean squares
error of 15–20% we report for the best performing models. However, the predictions by
Brund et al. show markedly worse calibration upon visual inspection, which may be a
result of the much lower number of predictors (ﬁve) and less sophisticated prediction
models (simple multiple linear regression) compared to the present study.
A couple of studies have predicted tibial bone forces from wearable devices and found
accurate predictions with mean absolute percentage errors ranging from 2.6 to 17.9% [28,37].
However, these studies are dependent on measurements from pressure-sensing insoles,
which are not common equipment for runners currently. Most other prediction studies
using wearable devices have not predicted structure-speciﬁc loads but rather net joint
moments, ground reaction forces (and derivatives thereof), or kinematics such as joint
angles or stride length [38]. While such measures might be interesting in other contexts, an
injury prevention context begs for structure-speciﬁc loads as these are closest to the actual
injury mechanism [39].
In terms of correlations, the patella ligament loads are clearly better predicted than
Achilles tendon loads for all IMU conﬁgurations. Our predictions are further inferior to
those reported by Long et al. [40], who, however, considered a much smaller and more
homogeneous cohort of four male basketball players. The Patella ligament force is strongly
related to quadriceps activity, which again is related to the exerted knee extension moment.
The latter is given by the product of the vertical acceleration of the masses above the
knee and the moment arm, which is given by the knee ﬂexion. IMU data reﬂecting these
properties would convey the necessary information to accurately assess patella ligament
loads, as shown by the results.
Prediction of the Achilles tendon force, on the other hand, appears to be more challeng-
ing. The explanation might be found in the binary nature of the foot contact mechanism,
where small motion differences determine contact or non-contact of speciﬁc parts of the
foot. Indeed, forefoot versus heel strike running styles cause different ground reaction
force patterns and different load patterns of the Achilles tendon, even though the kinematic
differences in terms of heel position can be quite small. Possibly for this reason, correlations
for the Achilles tendon force peak around 0.70. It is possible that inclusion of a categorical
variable for heel versus forefoot strike would improve the predictions.
The normalized root mean square errors for both output variables, i.e., patella ligament
force and Achilles tendon force, appear to be more similar than the correlations, with best
results for both cases in the range of 15–20%. The non-negligible root mean square errors
mean that it is commendable to present, e.g., 95% prediction intervals to users rather than
just one number. Correlations, on the other hand, do not contain information about the
precision of the absolute numbers but rather about the ability of the model to capture
changes correctly. This can be useful for an individual runner who considers a change in
running style and wants to know whether that would inﬂuence the load on a given tissue
positively or negatively.
The ten ﬁrst predictors in terms of rank contain several sine and cosine terms of
higher order, which might be surprising given that the higher-order Fourier coefﬁcients
are supposed to diminish. However, a running cycle comprises the time from right heel
strike to right heel strike, which causes the leg and arm movements to have their dominant
frequency at the step cycle frequency, while torso motions tend to repeat with double
frequency, where terms from second order and upwards dominate.
The results do not support identiﬁcation of a single best type of algorithm. Elastic
net performs better than XGB for patella ligament force and vice versa for Achilles tendon
force. However, XGB may have some merit over elastic net, in the sense that prediction
results for XGB are more stable over IMU placement while, especially for Achilles tendon
Sensors 2023, 23, 9836
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force, elastic net results are quite sensitive to IMU placement. For both algorithms and
forces to be predicted, using a single IMU gave better results than using two. The reason
may be that the resulting lower number of Fourier acceleration coefﬁcients (44 rather than
88) guards against overﬁtting.
The high correlations for especially the patella ligament force suggest that the method
has the potential to evolve into a reliable information source for injury prevention ef-
forts. Speculatively, our method could be used to estimate whether a change in running
biomechanics would lead to changes in injury risk for a given runner. However, running
biomechanics is complex, and an attempt to ofﬂoad a given anatomical structure might
increase the load of other structures. Running habits, such as speed and weekly distance,
might also be affected by a person’s running biomechanics and affect injury risk indirectly.
In this study, the two structure-speciﬁc loads were included as their respective max-
imum values over the running cycle. These values may occur at different times in the
running cycle depending on the running style, for instance heel or forefoot strike. Future
studies may consider predicting the entire cycle of the forces as a more informative measure
of the load exposure of the tissues.
It should be noted that the accelerations, from which the loads are predicted, are
perfect virtual accelerations from treadmill running that are not affected by measurement
noise, soft tissue artefacts, uncertain positioning, and variable terrain circumstances, which
will be confounding factors for real measurements. A similar approach was previously
employed [28,37] by ﬁrst developing a tibial bone force prediction model using virtual
instruments in [28] and subsequently validating the prediction model using physical
instruments in [37].
5. Conclusions
Data-based methods for predicting structure loads in running from a small number
of accelerometers appear to have merit in the case of running. However, further inves-
tigations of data-based methods against physics-based models and in vivo experimental
data, including injury registration, and the inclusion of accelerations measured in ﬁeld
conditions, are commendable.
Author Contributions: Conceptualization and physics-based models, J.R.; development of statistical
models and machine learning protocols, R.P.W.; running injury state-of-the-art, statistics and etiology,
S.S. All authors contributed equally to the writing and reviewing process. All authors have read and
agreed to the published version of the manuscript.
Funding: This research received no external funding.
Institutional Review Board Statement: Ethical review and approval were waived for this study
because the underlying data were not collected as a part of the study but was insourced in an
anonymized form from previous studies.
Informed Consent Statement: Not applicable.
Data Availability Statement: Anonymized data can be made available by personal contact to the
authors.
Acknowledgments: The authors thank Kaiser Sport og Ortopædi A/S for their contributions to data
collection.
Conﬂicts of Interest: The authors declare no conﬂict of interest.
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Predicting Tissue Loads in Running from Inertial Measurement Units.
Rasmussen, John,Skejø, Sebastian,Waagepetersen, Rasmus Plenge