# Accuracy and kinematics consistency of marker-based scaling approaches on a lower limb model: a comparative study with imagery data

Pierre Puchaud, Christophe Sauret, Antoine Muller, Nicolas Bideau, Georges Dumont, Hélène Pillet, Charles Pontonnier

<figure><img src="image_1.png" /></figure>

To cite this version:

Pierre Puchaud, Christophe Sauret, Antoine Muller, Nicolas Bideau, Georges Dumont, et al.. Accuracy and kinematics consistency of marker-based scaling approaches on a lower limb model: a comparative study with imagery data. Computer Methods in Biomechanics and Biomedical Engineering, 2019, 23 (3), pp.1-12. 10.1080/10255842.2019.1705798. hal-02546021

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# Accuracy and kinematics consistency of marker-based scaling approaches on a lower limb model: a comparative study with imagery data

P. Puchaud<sup>a,b,c</sup> Ⓓ, C. Sauret<sup>d</sup> Ⓓ, A. Muller<sup>a,e</sup> Ⓓ, N. Bideau<sup>b</sup> Ⓓ, G. Dumont<sup>a</sup> Ⓓ,
H. Pillet<sup>d</sup> Ⓓ and C. Pontonnier<sup>a,c</sup> Ⓓ

<sup>a</sup>Univ Rennes, CNRS, Inria, IRISA - UMR, Rennes, France; <sup>b</sup>Univ Rennes, Inria, Rennes, France; <sup>c</sup>Centre de Recherche des Écoles de St-Cyr Coëtquidan (CREC), Guer, France; <sup>d</sup>Institut de Biomécanique Humaine Georges Charpak, Arts et Métiers ParisTech, France; <sup>e</sup>Institut de Recherche Robert-Sauvé en Santé et en Sécurité du Travail (IRSST), Montréal, QC, Canada

## ABSTRACT

Medical images are not typically included in protocol of motion laboratories. Thus, accurate scaling of musculoskeletal models from optoelectronic data are important for any biomechanical analysis. The aim of the current study was to identify a scaling method based on optoelectronic data, inspired from literature, which could offer the best trade-off between accurate geometrical parameters (segment lengths, orientation of joint axes, marker coordinates) and consistent inverse kinematics outputs (kinematic error, joint angles). The methods were applied on 26 subjects and assessed with medical imagery building EOS-based models, considered as a reference. The main contribution of this paper is to show that the marker-based scaling followed by an optimisation of orientation joint axes and markers local coordinates, gives the most consistent scaling and joint angles with EOS-based models. Thus, when a non-invasive mean with an optoelectronic system is considered, a marker-based scaling is preliminary needed to get accurate segment lengths and to optimise joint axes and marker local coordinates to reduce kinematic errors.

**Abbreviations:** AJC: Ankle joint centre; CKE: cumulative kinematic error; DoF: degree of freedom; EB: EOS-based; HB: height-based; HJC: hip joint centre; KJC: knee joint centre; MB: marker-based; MSM: musculoskeletal models; SPM: statistical parametric mapping; STA: soft tissue artifact; EB<sub>a,m*</sub>: EOS-based with optimised joint axes, and all model markers coordinates; MB<sub>a,m*</sub>: marker-based with optimised joint axes, and all model markers coordinates; MB<sub>l,a,m</sub>: marker-based with optimised segment lengths, joint axes, and selected model markers coordinates; ASIS: anterior superior illiac spine; PSIS: posterior superior illiac spine

## KEYWORDS

Scaling; motion capture;
EOS; non-invasive;
numerical kinematics;
optimisation

## 1. Introduction

Accuracy of musculoskeletal analyses relies on multiple factors, in which the scaling of the musculoskeletal model (MSM) to the subject is of primary importance. The scaling is generally performed at multiple layers: muscle, inertial, and geometrical. The latter is fundamental to compute accurate joint angles and geometrical parameters (segment lengths, orientations of joint axes, and anatomical positions corresponding to marker local coordinates or muscle insertions) that are necessary to compute muscle paths and moment arms (van den Bogert et al. 2013). Therefore, a scaling method should provide accurate geometrical parameters to ensure a non-negative impact on the inverse kinematics outputs.

Models based on medical images are the current gold standard, they provide the most accurate

geometrical parameters (Scheys et al. 2006; Blemker et al. 2007; Valente et al. 2014). In such methods, data from MRI (Kainz et al. 2016, 2017; Halonen et al. 2017), EOS® (Clément et al. 2015), or CT-scans (Bartels et al. 2015; Marra et al. 2015) was used to reconstruct 3D bone geometries through image segmentation manually (Valente et al. 2014) or semi-automatically (Scheys et al. 2005). Generally, imagery data acquisition and post-processing is time-consuming. It limits its use to small cohorts (Handsfield et al. 2014) and prevents any routine protocol. More recently, several authors proposed to use anthropometric similarities to find the closest model within a database of models extracted from MRIs and scaling it proportionally to the subject (Ding et al. 2019; Klemt et al. 2019).

Scaling methods based on optoelectronic data are used in many studies (Hamner and Delp, 2013;

<figure><img src="image_2.png" /></figure>
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Subject equipped with 30 reflective markers (14 single markers and 16 markers on technical clusters placed on both thighs and shanks). A technical marker was placed on the left iliac crest only for labelling purpose. Foot markers were not included in this study. Markers were placed by skilled experimenters who received a training in palpatory anatomy.</figcaption></figure>

Dupré et al. 2018; Muller et al. 2019a) since they require less required time, knowledge and money.

Many computational methods were developed, including linear scaling (proportional scaling of the segment lengths based on the marker placements, Rassmussen et al. 2005), non-linear scaling (e.g., morphing, radial-basis functions, ..., Lund et al. 2015; Zhang et al. 2016; Nolte et al. 2019), and optimisation-based scaling (van den Bogert et al. 1994; Reinbolt et al. 2005, 2007; Andersen et al. 2010b; Lund et al. 2015; Muller et al. 2015a) The optimisation-based scaling relies on dynamic trials according to Lund et al. 2015 and consists in minimising over a selected time interval the least-squares error between experimental markers and their positions on the pre-defined kinematic model while adjusting geometrical parameters. This approach tends to distribute the experimental (soft tissue artefacts (STA), marker errors, ...) and model errors on the optimised geometrical parameters. Softwares such as OpenSim (Delp et al. 2007), AnyBody (Damsgaard et al. 2006) and CusToM (Muller et al. 2019b) use this geometrical scaling method. This optimisation ensures a low kinematic error (mean distance between experimental and model markers over all the frames of a given trial) after running inverse kinematics. However, the optimised geometrical parameters cannot be directly validated without medical images.

A few studies proposed direct validations of segment lengths scaled from optoelectronic data with medical images. For example, in Kainz et al. (2017) and Bartels et al. (2015), the Delp leg model (Delp et al. 1990) was linearly scaled with the Opensim software and segment lengths were compared to CT-scans and MRI extractions. The studies highlighted differences between linear scaling and medical imagery up to respectively 30 and 100 mm for the femur length. However, these studies only relied on one scaling approach and they did not investigate the geometrical scaling effects on joint angles.

Therefore, the aim of the current study was to identify a scaling method based on optoelectronic data offering the best trade-off between accurate geometrical parameters (segment lengths, orientation of joint axes, marker coordinates) and consistent inverse kinematics outputs (joint angles and kinematic error). For this purpose, 5 scaling methods were applied to a lower limb model on 26 subjects: two image-based scaling methods, one linear scaling methods, and two optimisation-based scaling methods. Kinematic errors and joint angles were evaluated on hip- and knee-joint functional movements and gait. The primary assumption of the study was that optimisation-based scaling methods should provide the best trade-off between geometrical parameters accuracy with respect to image-based methods and ensure the best inverse kinematics consistency by reducing the kinematic error.

# 2. Material and methods

## 2.1. Experimental data

Optoelectronic and biplanar radiographic data was collected from several studies, all of which received ethical approvals (Comité de Protection des Personnes, 2006-A00386-45, 2015-A01760-49, 2018-A00173-52). Twenty-six subjects (four females and twenty-two males, age: 24.3 ± 11.1 years old, height: 176.2 ± 7.6 cm, weight: 67.3 ± 9.4 kg, BMI: 21.6 ± 2.14 m/kg²) were equipped with a set of 30 reflective markers: 14 were placed on anatomical landmarks of the lower limbs and adapted from ISB recommendations (Wu et al. 2002); four technical clusters composed of four plate-mounted markers were strapped on both shanks and thighs (Figure 1). These clusters were only used in a solidification procedure to reconstruct incomplete trajectories of hidden markers (Söderkvist and Wedin 1993). They were not used for inverse kinematics since it has been shown to have a minor influence on computed joint angles (Kainz et al. 2016).
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Summary of scaling steps applied to each model. The reader can refer to sections 2.2.2, 2.2.3 and 2.2.4 for more details.</figcaption></figure>

Subjects equipped with markers underwent EOS® biplanar radiographs (EOS® Imaging, Paris, France), allowing three-dimensional reconstruction of the pelvis, femurs, tibiae and fibulae following a validated protocol (Chaibi et al. 2012)— see Figure 3. This data was used to compute the reference models (image-based models) presented in section 2.2.3. Hip- and knee-joint functional movements in upright position were recorded for the right and left legs by a 12-cameras motion capture system (Vicon™ system; Nexus 2 software; Oxford Metrics, UK) at 100 Hz: flexion/ extension of the knee; flexion/extension, abduction/ adduction and internal and external rotation of the hip (Camomilla et al. 2006). Among the 26 subjects, 6 subjects additionally performed two gait cycles. Markers trajectories were smoothed with a five-frame central window moving average filter. Two passes were done in reverse direction to minimise the shifting effect. Gaps in trajectories were filled using a C2-spline interpolation (gaps shorter than 15 frames, i.e., 0.15 s) or using a solidification procedure based on the other markers of the same body segment (gaps longer than 15 frames).

## 2.2. Geometrical scaling

For each subject, five geometrical scaling methods issued from the literature were applied on a generic model resulting in five scaled models: two image-based models (raw and optimised) that were

considered as a reference, one linear models and two optimised models. Figure 2 provides acronyms and a sum up of the scaling methods applied. Linear and optimised models were calibrated using algorithms implemented in CusToM (Muller et al. 2019b), an open-source Matlab toolbox dedicated to inverse dynamics based musculoskeletal simulation.

### 2.2.1. Generic model

A predefined 14 Degrees of Freedom (DoF) lower limbs kinematic model was used as a generic model for motion capture data and EOS data. It is adapted from a leg musculoskeletal model available in the AnyBody Managed Model Repository (Delp et al. 1990; Lund et al. 2018) and implemented in CusToM. The lower limbs model had 6 DoF between the ground frame and the pelvis reference frame, 3 rotational DoF at the hip joint centres (HJCs), and 1 rotational DoF (pure hinge joint) at the knee joint centres (KJCs) as in (Reinbolt et al. 2005; Andersen et al. 2010b). The kinematic model was composed of 5 segments (pelvis, both femurs and both shanks), with no foot-ending at the ankle joint centres (AJCs). ISB conventions (Wu et al. 2002) were followed for segment frame orientations. The Z-axis of the thigh was used as the knee joint axis. In this model, scalable geometrical parameters were the segment lengths, the orientations of joint axes, and the model marker positions. Depending on the type of scaling method
<figure><img src="image_5.png" /><figcaption>**Figure 3.** X-rays of a subject equipped with 30 markers and with four four-marker plates. X-rays were acquired with EOS. Reconstruction of bones (pelvis, femurs, tibiae and fibulae) is also shown. The subject is positioned in a 'shifted-feet' standing position to simplify bony structures recognition. This standing position has been previously validated (Chaibi et al. 2012).</figcaption></figure>

applied thereafter, all part of these parameters
were scaled.

**2.2.2. Linear scaling method**

One linear-scaled models were computed from the
generic model called marker-based linear scaling.
Homothetic factors were computed from the ratio
between the experimental lengths and the model
lengths. The experimental segment lengths were esti-
mated from experimental markers and the generic
model segment lengths were estimated from model
markers. Pelvis experimental and model lengths were
computed with the right and left posterior superior
iliac spines (RPSIS, LPSIS) and the right and left
anterior superior iliac spines (RASIS, LASIS). Four

lengths were computed: RASIS-to-LASIS, RPSIS-to-
LPSIS, RASIS-to-RPSIS and LASIS-to-LPSIS. The
mean of the fourth ratios between experimental and
model lengths was used as the homothetic factor of
the pelvis. Experimental lengths of the femurs were
computed from HJCs (Harrington et al. 2007) and
the middle of the medial and lateral epicondyles
markers. For the shank, experimental markers placed
on knee epicondyles and malleoli were used to esti-
mate knee and ankle joint positions. It resulted in a
Marker-Based model (MB) presented in figure 2.

**2.2.3. Image-based scaling method**

An image-based scaling method was applied with
EOS data. An EOS-based model (EB) was built and
<table><caption>**Table 1.** Markers local coordinates allowed to be optimised (Wu et al. 2002) with MB<sub>l.a.m.</sub>, scaling. The seven markers are placed on the right and left sides. x, y, z coordinates are respectively antero-posterior, longitudinal and medio-lateral. With MB<sub>a.m.</sub>, and EB<sub>a.m.</sub>, every coordinates are optimised. Marker clusters on thighs and shanks were not considered in geometrical scaling and inverse kinematics. The choice of the coordinates to optimise was based on the analysis of the biomechanical model ensuring a proper convergence of the algorithm.</caption><thead><tr><th>Palpated landmarks for marker placement</th><th>Body segment</th><th>optimised x-coordinate</th><th>optimised y-coordinate</th><th>optimised z-coordinate</th></tr></thead><tbody><tr><td>Posterior superior iliac spine</td><td>Pelvis</td><td>Yes</td><td>No</td><td>Yes</td></tr><tr><td>Anterior superior iliac spine</td><td>Pelvis</td><td>Yes</td><td>No</td><td>Yes</td></tr><tr><td>Lateral femur epicondyle</td><td>Thigh</td><td>No</td><td>No</td><td>No</td></tr><tr><td>Medial femur epicondyle</td><td>Thigh</td><td>Yes</td><td>No</td><td>Yes</td></tr><tr><td>Fibula head</td><td>Shank</td><td>Yes</td><td>No</td><td>Yes</td></tr><tr><td>Lateral malleolus</td><td>Shank</td><td>No</td><td>No</td><td>No</td></tr><tr><td>Medial malleolus</td><td>Shank</td><td>Yes</td><td>No</td><td>No</td></tr></tbody></table>

considered as a reference for each subject (Melhem et al. 2016). The 3D reconstructions of the lower limb bones were extracted from the biplanar radiographs. Femurs, tibiae and fibulae were reconstructed based on a parametric model and on statistical inferences (Chaibi et al. 2012). Regions were then automatically segmented on the bones. The HJCs were identified with a least-squares sphere fitting on the femoral heads regions of the 3D mesh (Pillet et al. 2014). Spheres were least-squares fitted on medial and lateral posterior aspects of femoral condyles. The KJCs were defined as the mid-points between the two spheres centres and the knee joint axis by the line passing through them (Sauret et al. 2016). Finally, malleoli regions of the fibulae and the tibiae were selected on the parametric bone models. Their barycentres were used to create middle points which were considered as the AJCs.

Moreover, EOS enabled to place the markers in the segment frame directly from the reconstruction. Markers' locations were computed by adjusting manually 14 mm diameter sphere models on the biplanar radiographs to match markers contours. The reproductibility of this procedure was previously determined at 0.35 mm (Sauret et al. 2016). HJCs locations were expressed in the pelvic coordinate system based on the external markers of the pelvis seen on the radiographs (Wu et al. 2002). Markers positions placed on femurs and tibiae were expressed in segments frames, following ISB recommendations.

### 2.2.4. Optimisation-based scaling methods

To obtain the three last models of the study, two kinds of optimisation-based scaling methods were applied. Firstly, the optimisation scaling method consisted in optimising homothetic factors of segments, knee joint axes orientations and model markers local coordinates (Table 1). It resulted in the computation of the optimised marker-based model denoted as MB<sub>l.a.m.</sub>, with optimised parameters denoted as

subscripts: segment lengths (l), joint axes (a) and model markers (m).

Secondly, another optimisation-based scaling method was applied on MB and EB models. It consisted in only optimising joint axes orientations and every model markers local coordinates specified in Table 1. It resulted in the computation of the optimised marker-based model denoted as MB<sub>a.m.</sub>, and EB<sub>a.m.</sub>, with optimised parameters denoted as subscripts: joint axes (a) and every model markers local coordinates (m*).

For all methods, hip- and knee-joint functional movements recorded as presented in section 2.1 were used as an input. According to Lund et al. (2015), dynamic trials are relevant for geometrical scaling, especially to study motions involving large joint amplitudes.

The MB<sub>l.a.m.</sub> method was performed using the MB model as initial guess (see section 2.2.2). The MB<sub>a.m.</sub>, and EB<sub>a.m.</sub>, methods used the MB and EB models as initial guesses respectively (see Section 2.2.3).

The optimisation-based scaling methods consisted in the following steps (Muller et al. 2015a): $N_f$ frames equally spaced in time were extracted from hip- and knee-joint functional movements data. It was set to $N_f = 100$ after prior experiments as a good trade-off between convergence and computation time. A first inverse kinematics step (Lu and O'Connor 1999) was performed over the selected frames (with MB or EB models) using an interior-point algorithm to get an initial guess of joint angles $q$. Then, a parameter optimisation step was performed to identify homothetic factors $k$, rotations $\alpha$ of joint axes and variations of marker local coordinates $\Delta p$. All variables were normalised between [-1,1] (Reinbolt et al. 2005). These parameters were optimised by minimising the cumulative kinematic error (CKE), denoted as $\Phi$, in Equation 1. The CKE is the cumulative quadratic sum of the Euclidean distances between experimental markers positions $X_{exp,m}$ and model markers positions $X_{mod,m}$ over the $N_f$ selected frames:
$$ \Phi = \sum_{f}^{N_f} \sum_{m}^{N_m} \left\| \mathbf{X}_{exp, m}(t_f) - \mathbf{X}_{mod, m}^{Rglobal}(\mathbf{q}(t_f), \mathbf{k}, \mathbf{\alpha}, \mathbf{\Delta p}) \right\|^2 \quad (1) $$

where $N_m$ is the number of markers in the model. The minimisation was bounded with constraints as shown in Equation (2):

$$ \begin{array}{ll} \min_{\mathbf{k}, \mathbf{\alpha}, \mathbf{\Delta p}} & \Phi(\mathbf{q}(t_f), \mathbf{k}, \mathbf{\alpha}, \mathbf{\Delta p}) \\ \text{s.t.} & \forall s \in [[1; N_s]], \quad \left| \frac{k_s}{k_s^0} - 1 \right| < 20\% \\ & \forall a \in [[1; N_\alpha]], \quad \alpha_{a,min} < \alpha_a < \alpha_{a,max} \\ & \forall lc \in [[1; N_{lc}]], \quad |\Delta p_{lc}| < 0.05m \end{array} \quad (2) $$

where $N_s$ is the number of solids, $N_\alpha$ is the number of parameters associated to the rotation of joint axes and $N_{lc}$ is the number optimised marker local coordinates. Markers local coordinates to be optimised are adapted from (Andersen et al. 2010b) and ensured the equation system to be determined for a single frame. Homothetic factors were constrained to vary by ±20% in the $MB_{l,a.m.}$ scaling and they did not vary in $MB_{a.m.}$ and $EB_{a.m.}$. Marker local coordinates variations were constrained to vary by ±5 cm. Knee joint axes (Z-oriented) were optimised with a X-Y rotation sequence in the thigh local frame. Rotations were constrained to ±20° around X-axis and to ±30° around Y-axis.

Finally, inverse kinematics and parameters optimisation were successively executed in a loop until the mean variation of CKE between $(i-1)^{th}$ and $i^{th}$ loops was less than 1%.

## 2.3. Model evaluation & statistical analysis

Models were evaluated thanks to three quantities: geometrical parameters (segment lengths, rotations of joint axes, markers local coordinates), kinematic errors and joint angles.

Segment lengths of $MB/MB_{a.m.}$ and $MB_{l,a.m.}$ models were compared to the EB/EB$_{a.m.}$ reference models and to the literature (Bartels et al. 2015; Kainz et al. 2017): namely inter-hip distance (right to left HJC), femur lengths (HJC to KJC) and shank lengths (KJC to AJC). A non-parametric Friedman's test verified if the segment lengths of the five models among subjects were significantly different ($p < 0.05$). A post-hoc Tukey's HSD test assessed significant differences among models compared to EB models. The X and Y rotations of joint axes and variations of markers local coordinates were quantified on optimisation-based models $MB_{l,a.m.}$, $MB_{a.m.}$ and $EB_{a.m.}$.

Joint angles of hip- and knee-joint functional movements and gait cycles were computed using inverse kinematics, also called multibody kinematic optimisation (Lu and O'Connor 1999; Begon et al. 2017), with markers mentioned in Table 1. They had a weight factor of 1. Resulting kinematic errors of functional movements were compared for the five models. As for the segment lengths, the same statistical tests were applied for kinematic errors. For gait cycles, joint angles of hips and knees obtained with all models were compared to $EB_{a.m.}$ results, considered as a reference. One-way ANOVA tests using statistical parametric mapping (SPM) (Pataky et al. 2015) over the duration of the gait trials were computed to compare each model condition as a repeated measure. In case of significant difference, two-tailed t-tests ($p < 0.05$) between models and the reference $EB_{a.m.}$ were applied to identify when these differences occurred during the gait cycle.

# 3. Results

## 3.1. Geometrical parameters

Segment lengths are presented in Table 2. Distances with respect to the EB model are presented in Figure 4. Significant differences were found among models for each length ($p < 0.05$ and $p < 0.001$). Indeed, for the right femur, the mean distances between MB and $MB_{l,a.m.}$ models with the EB model were respectively, $4.0 \pm 13.8$ mm and $8.4 \pm 12.5$ mm. For the left femur, MB and $MB_{l,a.m.}$ models exhibited mean distances of $0.88 \pm 16.3$ mm and $5.1 \pm 13.3$ mm.

Mean distance of inter-hip distance with the EB model ranged between $-10.8 \pm 19.9$ mm (MB) and $4.0 \pm 18.5$ mm ($MB_{l,a.m.}$). Right and left shanks lengths exhibited errors similar to the femurs. Tukey's HSD revealed that right femur and tibia lengths from $MB_{l,a.m.}$ models were significantly different from the EB model. However, inter-hip distance from MB/ $MB_{a.m.}$ and $MB_{l,a.m.}$ were not significantly different from the EB model.

Initial medio-lateral knee axes orientations were optimised in three models: $MB_{l,a.m.}$, $MB_{a.m.}$ and $EB_{a.m.}$. Mean and standard deviation associated to successive X and Y rotations are presented in Table 3. Both orientation adjustments were about $-2°$ to $2°$ Standard deviations of $MB_{l,a.m.}$ models were higher than other models ($3.22°$ to $4.06°$).

In average, variations of markers local coordinates of $MB_{l,a.m.}$, $MB_{a.m.}$, $EB_{a.m.}$ models were respectively of $5.6 \pm 12.7$ mm, $12.3 \pm 9.8$ mm, $6.3 \pm 7.8$ mm.
<table><caption>**Table 2.** Segment lengths (mean and standard deviation) for five models: MB, $MB_{l.a.m.}$, $MB_{a.m.}$, EB and $EB_{a.m.}$. Friedman's test investigated significant differences between models and Tukey's HSD investigated significant differences with the EB model. *, **, *** indicated respective *-values < 0.05, < 0.005, < 0.001.</caption><thead><tr><th></th><th>Friedmans' test significance</th><th>MB / MB<sub>a.m.</sub></th><th>MB<sub>l.a.m.</sub></th><th>EB / EB<sub>a.m.</sub></th></tr></thead><tbody><tr><td>Right Femur</td><td>*</td><td>435.8</td><td>440.2**</td><td>431.8</td></tr><tr><td>Left Femur</td><td>*</td><td>434.0</td><td>438.2*</td><td>433.1</td></tr><tr><td>Inter-Hip</td><td>***</td><td>162.7</td><td>177.4</td><td>173.5</td></tr><tr><td>Right Shank</td><td>***</td><td>419.6</td><td>431.5***</td><td>415.7</td></tr><tr><td>Left Shank</td><td>***</td><td>418.5</td><td>431.1***</td><td>414.7</td></tr></tbody></table>

## 3.2. Kinematic errors

Overall, inverse kinematics on hip- and knee-joint functional movements resulted in different kinematic errors (Figure 5). MB models exhibited the largest kinematic errors (27.4 ± 3.7 mm). $MB_{l.a.m.}$ and $MB_{a.m.}$ models showed lower kinematic errors (11.0 ± 1.6 mm and 6.5 ± 1.2 mm). EB model showed higher kinematic errors than optimised models (8.9 ± 3.1 mm) but $EB_{a.m.}$ models showed the lowest kinematic error (5.4 ± 0.8 mm).

Friedman's test revealed significant differences between the models. The kinematic errors of MB models were significantly higher than these of $MB_{l.a.m.}$, $MB_{a.m.}$, EB and $EB_{a.m.}$ models. Also, kinematic errors from $EB_{a.m.}$ models were significantly lower than $MB_{l.a.m.}$ and EB models. However, no significant difference was found for kinematic errors from $MB_{a.m.}$ models with respect to EB and $EB_{a.m.}$ models. Finally, $MB_{a.m.}$, EB and $EB_{a.m.}$ models had similar kinematic errors.

## 3.3. Joint angles

Among the six subjects who performed two gait cycles each, significant differences in joint angles were identified between models, particularly at the hip joint (Figure 6). Right and left hip flexion angles computed with $EB_{a.m.}$ model were significantly different from these resulting from the MB, $MB_{l.a.m.}$ and EB models. Also EB models were significantly different from $EB_{a.m.}$ models for right and left knee flexion. Differences between angles arose at different instants of the stride, depending on the considered joint.

## 4. Discussion

The primary assumption of the study was that optimisation-based scaling methods should provide the best trade-off between geometrical parameters accuracy with respect to image-based methods and ensure the best inverse kinematics consistency by reducing the kinematic error. To investigate this assumption, 5 scaling methods were applied to a lower limb model on 26 subjects with a

14-DoF generic kinematic model of the lower limbs. The EOS reconstructions of the cohort, considered as the image-based ground truth (Melhem et al. 2016), were a valuable opportunity to assess geometrical scaling methods based on optoelectronic data. The EOS-based models (EB and $EB_{a.m.}$) have been considered as a reference for the geometrical parameters and only $EB_{a.m.}$ models were considered for the joint angles.

## 4.1. Geometrical parameters

Optimising lengths and markers concomitantly induce overfitting for geometrical parameters. Indeed, left and right femur lengths were best evaluated by the marker-based (MB) method which does not optimise segment lengths. However, several studies have claimed that optimising all geometrical parameters is better at estimating segment lengths (Reinbolt et al. 2005; Andersen et al. 2010b; Lund et al. 2015). They reported up to 80, 67.1, and 10 mm in segment lengths variations between optimisation-based and linear scaling approaches - see Table 4. However, none of these studies validated the optimised segment lengths directly.

The current study showed significantly lower differences between MB scaled femur lengths and the EB reference compared to the literature (Bartels et al. 2015; Kainz et al. 2017), which exhibited up to 100 mm differences with CT-scan results. It can be explained by the cohorts of the these studies which were partly pathological (osteoarthritis and cerebral palsy). In the current study, the 26 subjects were healthy and practiced regular physical activity.

Optimisation of joint axes was required because such anatomical aspects differ for every subject (Eckhoff et al. 2003). It led to changes in the orientations of knee joint axes below 5°. These low changes may indicate that the kinematic errors were mostly due to inadequate lengths.

Optimisation of model markers local coordinates was supposed to limit the effects of STA and of the uncertainty of marker placement, even if this last source of error was mitigated by the training of the experimenters (Della Croce et al. 2005). In the last paragraph of the section 3.1, the changes in marker
<figure><img src="image_6.png" /><figcaption>**Figure 4.** Distance (mean and standard deviation) between segment lengths (femurs, shanks, inter-hip distances) issued from the reference model (EB) and segment lengths issued from MB, MB<sub>a.m.</sub>, and MB<sub>i.a.m.</sub> models. Results from the literature are presented (Bartels et al. 2015; Kainz et al. 2017). Results from Kainz et al. (2017) refer to Figure 1 of the article, in the accuracy column, typically developed participants.</figcaption></figure>

<table><caption>**Table 3.** Frontal plane rotations X and transverse plane rotations Y (mean and standard deviation) of right and left knee rotation axes for three models: MB<sub>i.a.m.</sub>, MB<sub>a.m.</sub>, EB<sub>a.m.</sub>.</caption><thead><tr><th rowspan="2">Optimisation-based models<br>Rotations (')</th><th colspan="2">MB<sub>a.m.</sub></th><th colspan="2">MB<sub>i.a.m.</sub></th><th colspan="2">EB<sub>a.m.</sub></th></tr><tr><th>x</th><th>y</th><th>x</th><th>y</th><th>x</th><th>y</th></tr></thead><tbody><tr><td>Right Knee Axis</td><td>0.44<br>(2.26)</td><td>-2.06<br>(2.68)</td><td>1.21<br>(3.59)</td><td>-0.54<br>(4.06)</td><td>1.94<br>(1.83)</td><td>-2.02<br>(2.1)</td></tr><tr><td>Left Knee Axis</td><td>-0.54<br>(2.14)</td><td>1.85<br>(2.6)</td><td>-1.84<br>(3.22)</td><td>0.87<br>(3.57)</td><td>-0.81<br>(1.84)</td><td>0.81<br>(2.16)</td></tr></tbody></table>

<figure><img src="image_7.png" /><figcaption>**Figure 5.** Kinematic errors (mean and standard deviation) on hip- and knee-joint functional movements with five models: HB, MB, MB<sub>i.a.m.</sub>, MB<sub>a.m.</sub>, EB and EB<sub>a.m.</sub>. *, **, *** indicated respective p-values < 0.05, < 0.005, < 0.001 with respect to Tukey's honest significant difference criterion.</figcaption></figure>

local coordinates after optimisation were consistent with respect to the literature because they had the same order of magnitude as the reported STA: 20 mm for the shank, 30 mm for the thigh and 9 mm for the pelvis (Leardini et al. 2005; Camomilla et al. 2017). The choice of dynamic trials as an input of the optimisation procedures had an impact on the results, since they are supposed to be subject to STA through the body motion. Indeed, static trials would have given relevant results close to the static configuration but may have suffered from larger kinematics errors for bigger joint angle magnitudes of the motions to be analysed. Therefore, in this study, dynamic trials were preferred to static trials since the motions to be evaluated were gaits, involving large joint amplitudes.

## 4.2. Kinematic errors

Resulting kinematic errors of MB<sub>i.a.m.</sub>, MB<sub>a.m.</sub>, EB and EB<sub>a.m.</sub> models were in accordance with literature results of optimisation-based approaches (Reinbolt
<figure><img src="image_8.png" /><figcaption>**Figure 6.** Hip, knee joint angle means over 12 gait trials performed by 6 subjects with a standard deviation cloud for each of the 5 models for one stride. Dark blue line represents the MB model. Light blue and green lines represent $MB_{l,a.m.}$, $MB_{a.m.}$ models respectively. The EB and $EB_{a.m.}$ models are in orange and yellow. Shaded rectangles along the x-axis show incidences of significant differences between at least 2 model conditions (p < 0.05) after one-way ANOVA and SPM-Bonferroni correction. Vertical bars represent the timing of the gait events. CTO: Contra-lateral toe-off. CHS: Contra-lateral heel-strike. TO: Toe-off.</figcaption></figure>

<table><caption>**Table 4.** Maximal length variations between scaling methods in the literature.</caption><thead><tr><th>Maximal length variations reported</th><th>Reinbolt et al. 2005</th><th>Andersen et al. 2010b</th><th>Lund et al. 2015</th><th>Present study</th></tr></thead><tbody><tr><td>Value (mm)</td><td>67.1</td><td>10</td><td>80</td><td>14.7</td></tr><tr><td>Geometrical parameter</td><td>Right hip location</td><td>Left foot length</td><td>Right hip location</td><td>inter-hip distances</td></tr></tbody></table>

et al. 2005, 2007; Andersen et al. 2010b; Lund et al. 2015; Muller et al. 2015a) - see Table 5. However, MB models exhibited larger kinematic errors than the literature. So, tuning only segment lengths is not enough to scale the model, mainly because of misplacements of experimental markers and STA.

On the contrary, the $MB_{l,a.m.}$ and $MB_{a.m.}$ methods were robust to the minimisation of kinematic errors over the 26 subjects, despite various sources of errors (Begon et al. 2017): measurement errors, segment lengths, DoF of the model. However, $MB_{l,a.m.}$ model misestimated segment lengths compared to MB and $MB_{a.m.}$ models which were closer to the $EB_{a.m.}$ reference models.

Therefore, the most appropriate method to be applied in marker-based geometrical scaling is an optimisation of marker local coordinates which enables a better location of markers and minimises STA effects. These results are in accordance with the Opensim scaling method (Reinbolt et al. 2005).

### 4.3. Joint angles

The $MB_{a.m.}$ and $EB_{a.m.}$ methods exhibited similar joint angles. This is of interest since in these models,

<table><caption>**Table 5.** Mean kinematic errors of scaled models after inverse kinematics.</caption><thead><tr><th>Study/methods</th><th>Mean kinematic errors (mm)</th></tr></thead><tbody><tr><td>Reinbolt et al. (2005)</td><td>7.8</td></tr><tr><td>Andersen et al. (2010b)</td><td>5.5</td></tr><tr><td>Muller et al. (2015a)</td><td>8.6</td></tr><tr><td>Lund et al. (2015)</td><td>4.5</td></tr><tr><td>Present study – $MB_{l,a.m.}$</td><td>10.8</td></tr><tr><td>Present study – $MB_{a.m.}$</td><td>6.5</td></tr><tr><td>Present study – EB</td><td>8.8</td></tr><tr><td>Present study – $EB_{a.m.}$</td><td>5.3</td></tr></tbody></table>

the segments lengths were very similar and supposed to be accurate. It indicates that the optimisation of the marker local coordinates resulted in a similar effect on the joint angles computation for both methods, whereas $MB_{l,a.m.}$ led to different results. Low kinematic errors of $MB_{a.m.}$ and $EB_{a.m.}$ does not necessarily mean that joint angles were accurate or that STA were fully compensated (Bonnet et al. 2017). Kinematic accuracy can only really be assessed with intra-cortical pins (Andersen et al. 2010a) or biplanar fluoroscopic data (Gasparutto et al. 2015; Richard et al. 2017). However, the consistency between both results and the low kinematic error suggests that the primary assumption of the study, i.e., that
optimisation-based scaling methods should provide
the best trade-off between geometrical parameters
accuracy with respect to image-based methods and
ensure the best inverse kinematics consistency, is not
supported. It seems more relevant to use MB<sub>a.m.</sub>.
instead of MB<sub>l.a.m.</sub> to scale the model in a musculo-
skeletal study.

4.4. Methodological limitations

The study has several limitations that should be
noted. First of all, optimisation-based methods may
lead to overfitting. It tends to spread the experimental
and model errors on the optimised geometrical
parameters, namely the segment lengths, the marker
local coordinates and the joint axes. To ensure a good
balance between correct segment lengths and low
kinematic errors, well-chosen constraints and correct
initial guesses are required. The optimisation-based
methods are robust and guarantee to reduce kine-
matic errors but the results of the study show that it
is preferable to exclude the segment lengths of the
lower limbs from any optimisation-based scaling.

Computationally, a subject was scaled in less than
4 minutes for Nf = 100 frames (3.10 GHz laptop, 32
Go RAM). This computation time may be reduced by
working on the choice and the number of frames to
include in the optimisation (Muller et al. 2015b). The
study was also limited by the way the scaling was
defined: the homothetic factor was uniformly applied
in all directions of a given segment. It has consequen-
ces on the scaling of large bones such as the pelvis. A
solution may be to declare multiple scaling factors for
each segment and optimise them separately
(Rassmussen et al. 2005; Delp et al. 2007) with well-
chosen markers weights in the optimisation (Trinler
and Baker 2018). Non-linear functions could also be
used to enhance the scaling properties before an opti-
misation as in MB<sub>a.m.</sub>*, as it has been done in Lund
et al. (2015); Zhang et al. (2016); Nolte et al. (2019).
Another enhancement may be to use a database of
models extracted from MRIs and to get a closer initial
guess, as proposed in (Ding et al. 2019; Klemt
et al. 2019).

Feet scaling is another issue that was not handled
in the current study. Feet data from the EOS biplanar
images have to be considered for a future
extended protocol.

Last, the knee model was limited to one rotational
DoF and its validity is still discussed in the literature
(Gasparutto et al. 2015; Clément et al. 2015); accord-
ing to Kainz et al. 2016, knee models have an

influence on computed joint angles. In the future,
more complex models could be evaluated to confirm
the use of MB<sub>a.m.</sub> for scaling.

5. Conclusion

The current study aimed at identifying a scaling
method based on optoelectronic data offering the best
trade-off between accurate geometrical parameters
(segment lengths, orientation of joint axes, marker
coordinates) and consistent inverse kinematics out-
puts (joint angles and kinematic error). The primary
assumption of the study was that optimisation-based
scaling methods should provide the best trade-off
between geometrical parameters accuracy with respect
to image-based methods and ensure the best inverse
kinematics consistency. It has not been supported by
the results, since the optimised segment lengths
obtained with MB<sub>l.a.m.</sub> were significantly different
from the these obtained from the EOS reference
measures, and led to an higher kinematic error.
Finally, the marker-based scaling with optimised
model marker coordinates and joint axes in MB<sub>a.m.</sub>.
gave the most consistent scaling and joint angles with
regard to these obtained from the EOS measures.
This study is of interest since this scaling is the first
step to be applied to a model in any musculoskeletal
study. An interesting future work may be to investi-
gate the effect of such scaling on the muscle paths
and the moment arms of the model that are funda-
mental to run accurate musculoskeletal analyses.

Disclosure statement

No potential conflict of interest was reported by
the author(s).

Funding

The present study was partly funded by the chair of the
enhanced soldier in the digital battlefield from the St-
Cyr Foundation.

ORCID

P. Puchaud http://orcid.org/0000-0002-9335-630X

C. Sauret http://orcid.org/0000-0002-6196-2698

A. Muller http://orcid.org/0000-0003-2606-1429

N. Bideau http://orcid.org/0000-0001-5333-7991

G. Dumont http://orcid.org/0000-0002-0709-0921

H. Pillet http://orcid.org/0000-0002-0263-5910

C. Pontonnier http://orcid.org/0000-0003-1140-3772
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