Tendon model with exponential force length relation by Rack and Westbury.

A linear force stiffness relationship, as described by Rack and Westbury [2], can be reached by an exponential force strain model. In general the stiffness of a material is defined as (4)

Rack and Westbury could show, based on measurements of the cat soleus tendon, that the stiffness of a mammalian tendon k_T depends linearly on the applied force [2]. (5)

Proske and Morgan concluded that the linear stiffness behaviour of Eq (5) is only valid in the toe region and assumed that the tendon stiffness beyond the toe region is constant [20], thus leading to a linear force strain behaviour beyond the toe region (Fig 2A). Based on these findings, the following differential equation models the stiffness for during the lengthening of the tendon (Δl_T ≥ 0). (6)

Solving the differential equation for each case leads to the following expression, if c₂ ≠ 0 is assumed. (7) Where Δl_toe is the change in tendon length at which the toe region ends (see Fig 2A). A₁ and A₂ are coordinates in the solution space of Eq (7), which also contains physically implausible solutions. In order to gain a physically feasible solution, the condition from Eq (2) is applied and continuity of for Δl_T > 0 is assumed. This results in the following, unique solution. (8) (9)

The substitution of A₁ and A₂ in Eq (7) leads to the following model description. (10)

Here, the tendon parameters are included explicitly in the function definition as parameter vector . (11) Eq (10) is not defined for compression (Δl_T < 0). It is widely assumed that the tendon resists with nearly no force to compression [19, 21]. This compression behaviour, however, is not needed in this study as the tendon will only be observed in extension. Due to the type of data acquired in this study, the inverse of Eq (10) is needed. (12) Eq (12) is used for the tendon model that was used to represent the experimental data as described in the following section.

Manual extraction of the myotendinous junction displacement.

In Fig 5 two B-mode frames in the parasagittal plane are shown (for reference compare with orientation of imaging plane I in Fig 4A). These frames were selected by an experienced ultrasonography operator who decided, based on the movement of the anatomical structures, when the steady states of the myotendinous junction movements were reached. The first steady state was the unloaded state whereas the second state was reached when the subject applied the instructed force level. Due to deformation of the tissue during contraction, the two frames show slightly different viewpoints. During the experiment, the ultrasonography operator chose a fixed reference point and made sure that it stayed within the frame. As reference points, insertion points of brachialis fibres at the humerus were chosen as these are fixed within the coordinate system of the humerus. These points served as reference coordinate system. Fig 5 depicts the bone coordinate frame at that reference point in brown for the unloaded Fig 5A and loaded state Fig 5B. It is oriented such that the abscissa is congruent with the bone surface. The bone surface in the B-mode image is abstracted as a line (Fig 5, drawn in turquoise). The bone line was parametrized by means of of the orientation angle (see Fig 6) and the bone reference point (where the tilde symbol indicates that the respective quantity is linked to the manual data extraction). The orientation angle was derived from the bone reference point and a second marker . These parameters were manually marked for the unloaded state n_u and loaded state n_l.

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Fig 6. Bone line angle calculation in case of (A) the manual and (B) the automatic data extraction.

A sketch which demonstrates how the bone line angle was calculated during manual data extraction () and automatic data extraction (). (A) demonstrates the case of manual bone line labeling in which was derived from the vector as described in Eq (16). Analog (B) demonstrates how was derived from a bone line expressed in Hough-line parameters (cmp. Fig 9) as formulated in Eq (24).

https://doi.org/10.1371/journal.pone.0275128.g006

The point of the myotendinous junction was chosen where the aponeurosis crosses with an imaginary extension of the fascia in both states. The junction point was transformed into the reference coordinate system for each frame. (14) Where is the bone line orientation angle relative to the abscissa of the bone local coordinate system. And R(•) is the rotation matrix for the bone local coordinate system. The bone local coordinate system is a left handed coordinate system. rotates in the clock wise direction for positive angles. Consequently R⁻¹(•) and are described as follows. (15) (16) Where is the vector pointing from the bone reference point to the bone orientation marker as shown in Fig 6A. The operator [•]_x returns the x coordinate and [•]_y the y coordinate of a vector.

Calculation of myotendinous junction movement.

After the myotendinous junction was described in the bone local coordinate system, the tendon lengthening could be calculated for each frame. If the absolute position of the bone reference point in a skeleton model and the path of the tendon were known, the lengthening of the tendon could be calculated accurately. In this work, the lengthening of the tendon was assumed as the movement of the myotendinous junction from unloaded to the loaded state parallel to the bone line. (17) (18)

Data for of subject 0 is presented as an example in Fig 7A. Later the quality of the extracted data is discussed and compared with a (semi-) automatic approach.

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Fig 7. Tendon displacement over wrist force for all elbow angles is shwon in case of (A) manual and (B) (semi-) automatic data extraction, with (C) a comparison of both methods.

The length of the junction movement parallel to the bone of subject 0 is shown for the manually extracted data as depicted in (A) and for the automatically extracted data in (B). In both cases the mytendinous junction movement parallel to the bone is shown over the wrist force F_w for all elbow angles Θ (cmp. Eq (13)). The bone line angle as well as the evaluated frames are adopted from the manual evaluation as described in Eq (31). In (C) the difference between the automatically extracted data and the manually extracted data of subject 0 is shown.

https://doi.org/10.1371/journal.pone.0275128.g007

Computer aided extraction of the myotendinous junction displacement.

In addition to the manual procedure for extracting the myotendinous displacement (see Eqs (17) and (18)) a computer-aided procedure was employed. The computer-assisted method allows a) to minimize any inaccuracies due to the experimenter’s subjective judgment during manual extraction of reference points, b) to ensure reproducibility of results, and c) to achieve higher throughput when processing large data sets.

In order to calculate the junction movement , the computer-aided method extracts the myotendinous junction , the relative orientation of the bone and the bone reference point from B-mode ultrasound frames based on different algorithmic methods as depicted in Fig 8 and described in the following paragraphs. The hat symbol is used as annotation of measurement quantities related to the computer-aided feature extraction.

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Fig 8. Block diagram for each computational method of feature extraction is shown.

The computed features are from left to right: the mytendinous junction, bone line and bone reference point. The mytendinous junction is tracked with a neural network based on the MobileNetV2 algorithm. The bone line is tracked via Hough-Line-Transformation. The bone reference point is tracked by a method based on the Lucas-Kanade algorithm.

https://doi.org/10.1371/journal.pone.0275128.g008

Detection of myotendinous junction using deep neural networks.

In order to detect the myotendinous junction from the data set of B-mode ultrasound frames, a deep learning approach using transfer learning was used (see Fig 8). Automated tracking of the myotendinous junction using convolutional neural networks has recently been shown to provide accurate results comparable to manual tracking methods [24, 25] and to be less error-prone than semi- and fully-automated myotendinous junction tracking approaches based on optic flow [26, 27] or block-matching algorithms [28, 29].

To generate and train a convolutional neural network capable of detecting the myotendinous junction of the biceps brachii from ultrasound images, the markerless pose-estimation framework DeepLabCut [30] was used. The architecture of the underlying convolutional network is based on MobileNetV2 [31] and was pretrained on the ImageNet data set by [32]. A training data set was generated via random selection of 203 video sequences from the data set of B-mode ultrasound images. From each individual video sequence 10 images were extracted in a randomly and temporally uniformly distributed manner resulting in a total of (n =) 2030 frames. In each of the extracted image frames, the myotendinous junction of the biceps brachii was labeled by hand, hence, serving as a ground truth. The labeled images were subsequently split into two data sets for use in training (n_t = 2185 frames) and evaluation (n_e = 115 frames). Training of the neural network was performed using the training data set with a batch size of 20 for 200000 iterations and DeepLabCut’s standard parameters. The performance of the neural network was subsequently evaluated via computation of the the mean average error (MAE) of the Euclidean distance between the predicted and the ground truth location of the myotendinous junction in the evaluation data set. This resulted in an MAE of 4.32 pixels corresponding to 0.50mm. After training, the neural network was used to automatically infer the location of the myotendinous junction from the individual video sequences of B-mode ultrasound images (hat symbol indicates computer-aided feature extraction).

Computation of the bone orientation.

The bone orientation relative to the x-axis was previously calculated from two manually selected points on the bone surface . Due to the lack of explicitly definable features on the bone surface for these points in the B-mode frames, feature tracking analog to the myotendinous junction was not an option. The bone surface appears as a continuous feature in the B-mode image as depicted in Fig 5. However, the bone surface has a high contrast as compared to the surrounding muscle tissue. The transition from muscle tissue to bone can be abstracted as a line. Therefore, the B-mode image was filtered for edges and a line detection algorithm was applied. As the fibers in the muscle tissue also have a line characteristic, the B-mode image was cropped such that the risk of a line feature in the muscle tissue being detected as bone line was minimized. A block diagram of this process is shown in the middle section of Fig 8. In order to confine the algorithm to an image region relevant for the detection of the visual line separating muscle from bone tissue, each ultrasound image frame was cropped using the FFMPEG library [33] resulting in a size of (w_c =) 600 by (h_c =) 414 pixels of the cropped frame. (x_c = 320, y_c = 350) is the horizontal and vertical pixel position of the top left corner of the cropped region within the original frame (see Fig 9A). After cropping, the Canny edge detection algorithm ([34], as implemented in OpenCV [35]), was applied to each frame. This algorithm contains a four-stage process in which 2 of the 4 stages need to be parameterized. First, a noise reduction with a 5x5 Gaussian filter is applied. Second, the intensity gradient is computed. This process is parameterized by the aperture size (a_s = 3) and the norm used in gradient computation. Here, the computationally efficient L₁ norm was used. Third, after gradient computation, a non-maximum suppression is applied. The result is an image in which the value of each pixel describes the gradient of an edge. Finally, in the fourth step, these edges are filtered by hysteresis thresholding. In this work, pixel with a gradient value above grad_max = 220 were accepted as edges. Pixel with a gradient value above grad_min = 50 were considered edges if they were connected to edges above grad_max. All other pixel/edges were filtered out. The used parameters are shown in S2 Table.

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Fig 9. (A) Bone lines detected in a B-mode frame and (B) shown in parameter space.

(A) is a frame of the cropped section of a B-mode video. The detected lines have the same color coding as in the scatter plot of the bone line parameter space. (B) The corresponding scatter plot of detected Hough-Lines in their parameter space. The biggest detected cluster is shown in yellow and its mean is shown in red. All other data points are black.

https://doi.org/10.1371/journal.pone.0275128.g009

The remaining edges were used during the Hough-transformation for line detection. The Hough-lines are defined by a vector in cylindrical coordinates (r, φ). The vector has to be understood as a point on the described line . Where is the parameter form of the bone line, which is defined as follows. (19)

Furthermore, it is assumed that is oriented orthogonal to that line (). Because of these two definitions, the bone line l_b(h) is fully defined by the tuple (r, φ) where r is in pixel and φ in radiant. These parameters span a constrained parameter space of possible lines. (20)

This Hough-line parameter space was transformed into a constrained and discretized Hough-line search space which is described by (21) and (22) where Δr = 1 and are the search resolution parameters.

For each edge pixel in a frame all lines containing this pixel in the discretized parameter space were considered and the counter of the corresponding parameter tuple was increased. After all edge points had been processed, those parameter tuples with counter values larger than a threshold (Threshold_Hough = 40) were considered to be bone line candidates. These candidates were sorted by their counter values. The candidates with the 10 highest counter values were selected for further processing (see Fig 9A). The parameters used for the OpenCV HougLines function were shown in S3 Table in the S3 Table. If a function parameter is not listed, the default value was chosen. The selected Hough-lines were used to calculate a mean bone line. As some structures in the muscle tissue also resulted in detected lines, a clustering algorithm (DBSCAN from SciPy [36]) was applied to filter for bone lines, as shown in Fig 9B. By means of majority voting, the biggest cluster was chosen and its mean parameter vector was used to describe the bone line. In subsequent steps, only the orientation of the bone line was used. Before clustering, the Hough-line parameters were normalized on the expected range for a bone line. (23)

This way distances between parameters in and can be treated equally, which is beneficial for clustering. The function parameter m ∈ {0, 1, …, 9} is the index of a bone line parameter set in frame n. After this normalization, DBSCAN is applied which is a density based clustering algorithm defined by [37]. There are 3 types of data points. 1. a core point, which has at least n_min,s − 1 samples within the distance ϵ_core. 2. a border point, which is within the distance ϵ_core of a core point. 3. noise points / outlier. The latter are not reachable from any point. A point is called reachable, if there is a chain of core points within the distance ϵ_core. Points, which reach each other, are part of the same cluster. For bone line clustering, the amount of core points was set to n_min,s = 2 and ϵ_core = 0.02 was chosen. As distance function, the Euclidean-distance L₂(•) = ‖•‖₂ was used. In S4 Table the parameters are listed with their names as given in the SciPy module [36]. Default values were used for those function parameters that are not listed. The bone line was then calculated as the mean of the biggest cluster (r_b(n), φ_b(n)) where n is the index of the corresponding frame. Based on this findings, the bone line orientation angle relative to the abscissa was calculated as follows. (24)

A derivation of this relationship is shown in Fig 6B. In the used coordinate system, a clock wise rotation is positive. This description of can be used instead of the manual labeled bone orientation .

Tracking of the bone reference point via Lucas-Kanade optic flow estimation.

As compared to the myotendinous junction, the visual texture of bones in B-mode ultrasound images does not provide distinct visual features consistent for all subjects (Fig 4). Hence, the automated inference of reference points using a deep neural network approach based on transfer learning—such as described in the section Detection of myotendinous junction using deep neural networks—is not feasible, as such an approach would require supervised training based on individual features of the bone surface for corresponding subsets of ultrasound images. Instead, for the automated tracking of the bone reference point on the humerus from sequences of ultrasound images, an optic flow based approach was used. Lucas-Kanade optic flow estimation [38] is based on the assumption that the distribution of apparent velocities induced by motion is constant in a local neighbourhood of an image pixel under consideration. This allows for the temporal tracking of visual features in a sequence of images which do not share distinct textural properties across a set of image sequences. Recent studies have used the Lucas-Kanade based estimation of optic flow to temporally track visual features in ultrasound images [26, 27].

To obtain the bone reference point from the data set of B-mode ultrasound images, a region of interest (ROI) representing the bone structure of the humerus was computed for each frame of the data set of ultrasound images. The ROI was defined as the area below the visual line separating muscle from bone tissue within the cropped image frames (see paragraph above; Fig 9A). Within the ROI, visual features were selected for the first frame of each ultrasound image sequence using Shi-Tomasi corner detection [39]; implemented using OpenCV [35]. The maximum number of corners returned via the corner detector was set to N_c,max = 1000 while corners with a quality level of Q_c < 0.25 and a Euclidean distance of d_c,max < 3 image pixels were rejected. The size of an average block for computing a derivative covariance matrix over each pixel neighborhood was set to s_b = 3. This resulted in a mean of corners (standard deviation σ_c = 24.48) detected per analyzed image frame. Subsequently, the detected visual features (or corners, respectively) in the first frame of an image sequence were iteratively tracked in the remaining frames of the corresponding image sequence using a pyramidal implementation of the Lucas-Kanade feature tracking algorithm [40]; implemented using OpenCV [35]. The maximum number of pyramidal levels taken into account was set to N_p,max = 4 and the size of the search window at each pyramid level to . The termination criteria of the iterative search algorithm were set to ϵ_p = 0.03 for the desired change in parameters at which the algorithm stops and i_p,max = 30 for the maximum number of iterations to compute. The bone reference point was subsequently obtained for each frame n in each sequence of ultrasound images via computation of the centroid (i.e. the mean) of the visual features with coordinates [f_x, f_y]^T tracked via Lucas-Kanade optic flow estimation according to (25) whereas features leaving the ROI during tracking within an image sequence due to movement of the bone in relation to the ultrasound sensor were discarded.

Comparison of manual and automatic feature extraction.

The automatic feature extraction was compared to the manually extracted data per feature.

When evaluating the tracking of the myotendinous junction, the absolute movement between states (unloaded, loaded) was compared, (26) as the investigator in the manual method and the automatic method did not necessarily track the same feature of the myotendinous junction. Tracking of the bone line was evaluated separately for the loaded and unloaded state as the angle of the bone line was comparable on a frame basis. (27)

In case of the bone reference point the absolute movement of a manually chosen feature on the bone (see Fig 5) and the absolute movement of an automatically chosen feature by Lucas-Kanade was compared. The comparison of the absolute distance between steady states (28)

The results of these comparisons are shown in Fig 11. For each comparison a mean absolute error (MAE) and the Pearson correlation coefficient (R) is calculated.

Calculation of myotendinous junction displacement.

The myotendinous junction displacement parallel to the bone needed to be calculated analog to the manual data extraction. For that, Eqs (14) and (17) still held true. For each manually extracted feature (, , ) there is an automatically extracted counterpart (, , ). These features are compared in Fig 11. It appears that the error between manually and automatically labeled absolute bone reference point movement (see Eq (8)) correlates with the wrist force F_w (R = −0.58). This finding is later discussed. Therefore, for further steps the bone reference point was used from the manually labeled data set instead. Consequently, the myotendinous junction displacement vector is defined as follows. (29) (30)

Then the displacement parallel to the bone is defined as: (31)

For the steady states n_u and n_l, the same frames were picked as for the manual data extraction. Data for is shown in Fig 7B for subject 0, as example.

Wrist force extraction.

The unfiltered wrist force and data from B-mode ultrasound frames were not synchronized. However, steady states can be identified in . Since it was unknown at which time during the steady state the frames were captured, the mean of the wrist force in each steady state was calculated. The force contributing to a moment at the elbow joint during the unloaded state was assumed to be zero. A measured force during this time window therefore had to be attributed to the preloading of the wrist strap. The measured mean force during the loaded state was therefore offset by the mean force during the unloaded state in order to obtain the actual elbow moment generating force F_w. (32) Where n_l0 is the first sample of the loaded state and n_l1 is the last sample of the loaded state. The same principle applies to the unloaded state (n_u0 first sample—n_u1 last sample).

Torque at the elbow joint.

The moment arm l_w(Θ) changes over the elbow angle due to the construction of the support rack (Fig 3). For this reason, the distance between the elbow joint and the wrist strap was measured for each elbow angle Θ. With this measurement, the corresponding elbow joint torque τ_e was calculated. (33)

This elbow torque was later used to estimate the torque which was generated by the biceps brachii.

Estimation of the tendon force.

In this work, tendon displacement was predicted based on the measured wrist force F_w and an abstraction of the skeletal geometry as shown in Fig 10A. The model results in an effective lever l_eff. From this, the tendon force could be calculated by (35) since the elbow joint was assumed to be a revolute joint. The torque τ_bic is generated by the biceps brachii. It was estimated based on the elbow joint torque, as it was not directly measured. (36)

In this equation, τ_r is the torque generated from other sources (i.e. friction, other muscle groups, surrounding tissue). The forearm, which contains two bones (ulna and radius) as mechanical base structure, was abstracted as a single lever. The distal biceps brachii tendon T₁ inserts into this lever at with the same distance l_i. As distance l_i the distance between the elbow and the insertion into the radius was used. The fascial insertion of the distal biceps brachii aponeurosis was ignored.

The two headed biceps brachii was abstracted to a single head inserting at into the shoulder. With the coordinate systems origin placed within the elbow revolute joint, the points (37) were defined (see Fig 10A). Here, the distance from the distal tendon insertion point to the elbow joint l_i was assumed to be proportional to the length of the ulna (l_i ∝ l_u). (38)

The conversion factor was set based on the proportions depicted in figure. 334 of [41]. The effective lever length can be calculated as follows. (39) Where is a unit vector in the direction of . The course of the effective lever length l_eff over the elbow angle for subject 0 is shown exemplarily in Fig 10B.

With Eqs (35) and (39) the estimation of tendon force is complete. It was used for fitting the tendon model onto the extracted myotendinous junction movement and the wrist force F_w.

Fitting the tendon model to data.

Due to the small amount of data per elbow angle and its high variance in the myotendinous displacement ΔL_mt, it was preferable to fit the model onto all angles in one process instead of breaking it down into several sub-processes—one for each angle. The material properties of the tendon were assumed invariant, also in terms of changes of the elbow angle. However, if the predicted biceps torque is less than the elbow torque (τ_r > 0, see Eq (36)) the tendon parameters are effected. This means that, when τ_r depends on the elbow angle, it seems as if the tendon parameters are changing depending on the angle. The same holds true for the effective moment arm l_eff. These effects are further analyzed in the results and evaluation section. Therefore, it was assumed that only a subset (or sub-range) of the measured elbow angles can be explained by the current model. Within this sub-range, τ_r ≈ 0 was assumed. Additionally, l_eff was assumed to predict the effective lever length for that sub-range. As the valid elbow angle range was not known a priori, it was narrowed down in an optimization process. In this optimization process, fits of the model for decreasing ranges of Θ were performed. Later, the elbow angle range was used as a fit result whose mean cost per sub-range is stable as compared to all angle ranges (cmp. Figs 12A and 13A, example data from three subjects). For each such elbow angle range fit, the error function (40) was minimized. The error function g calculates a sum of differences between the predicted and measured tendon displacement. The function μ(•) was used to distort the error function to cope with variance in the data. In this case, μ(•) = ln(•+ 1) was chosen. The functions Θ[k], F_w[k] and are discrete functions returning the respective values of a data point indexed by k. The set (lb for lower bound) contains all indices of data points where Θ_lb ≤ Θ. Therefore, it is possible to calculate fits over subsets of elbow angles. (41)

This minimization was done by performing fits with (42) as lower boundaries for the elbow angle subsets which resulted in parameter sets for each value of Θ_lb. The mean cost per subset was calculated as follows. (43) Where is the amount of data-points of the corresponding angle subset defined by Θ_lb. As an optimization algorithm, the non linear function optimizer trust region reflective from the Python SciPy package [36] was used. For the optimization process (44) was used as the initial parameter vector. As lower and upper boundaries (45) were used. Here, eps₆₄ is the machine epsilon of the 64 bit float datatype. Results of the fitting process are depicted in Figs 12 and 13 and evaluated in the following section.

Tracking of the myotendinous junction.

The comparison of the absolute myotendinous junction displacement between manual and automatic data extraction resulted in a mean absolute error (MAE) of 0.67 mm ≐ 9:0 pixel. The differences were distributed approximately symmetrical around zero, with two outliers. As for the outliers, the neural network extracted a larger movement than the manual evaluation. The difference of absolute movements correlates with the wrist force by R = −0.07 (see Fig 11, left), where R is the Pearson correlation coefficient.

Tracking of the bone line.

Tracking of the bone line was evaluated separately for the loaded and unloaded state. In both states, the computation of the mean absolute error resulted in MAE < 1^∘ (see Fig 11, middle), which was below the resolution of the Hough-line transformation. Apart from outliers, the mean absolute errors for both states were distributed around zero. This is due to image cropping, which mostly affected the unloaded state due to a smaller distance between bone and ultrasonic probe. Using a higher cropping window would have led to false bone line detection within muscle tissue. In either, the loaded and the unloaded state, the bone line angle γ correlated with the wrist force F_w by R = −0.08 (Pearson correlation coefficient).

Tracking of the bone reference point.

The comparison of the absolute distance between steady states correlated with F_w by R = −0.58 (Pearson correlation coefficient). This resulted in an error distribution around an offset value unequal to zero.

Comparing the (semi-) automatic data extraction with the manual data extraction for subject 0 led to a MAE of 1.15mm (see Fig 7C). Therefore, with a mean absolute error difference close to one millimeter, the results of both methods were assumed to be equivalent.

The effect of l_eff miscalculation.

If the effective lever length of the biceps brachii was miscalculated this would lead to an elbow angle dependent over- or underestimation of the tendon force F_T1. A quantitative analysis was not applicable as l_eff could not be obtained from the participants in the presented measurement setup (see Fig 3). Direct in vitro measurements of the effective lever length of the lower biceps brachii tendon exist. Murray et al. presented data of the peak effective moment arm of the distal biceps brachii tendon over the length of the radius [42]. This data is depicted in Fig 14A. In addition, the peak effective moment arm of subject 0 to 2 over their radius length l_r was entered into the graph. The data of subject 0 to 2 lay within the trend of [42]. Therefore, it was assumed that the maximal moment arm (between 60^∘ and 100^∘, according to [42]) and effective moment arms for angles near the maximum were predicted reasonably well by the presented model.

The effect of τ_r on the measurement.

During model building, τ_r (as defined in Eq (36)) was assumed to be zero as the participant had been advised to flex the elbow joint with no co-contraction. But during the experiment it turned out that subjects had the tendency to co-contract to reach the desired force levels. If co-contraction was noticed during the experiment, it was pointed out to the subject with the request to stop co-contracting even if the requested force level could not be reached exactly. If other than the assumed muscles for co-contraction were dominant, the actual residual torque would be negative (). (46)

Due to Eq (46), co-contraction would lead to an underestimate of the torque generated by the biceps brachii () and therefore also an underestimate of Δl_mt. Analog an overestimate of the torque generated by the biceps brachii () is possible, if additional flexor muscles were engaged during contraction τ_r > 0. Whenever ΔL_mt was overestimated this seemed to be the dominant error source. During the fit process shrinking sub-ranges of Θ were fitted and led to increasing optimal fit results (in case of manual data extraction). Shrinking sub-ranges mean that lower angles were excluded from the fit. Considering that the biceps brachii has its approximate optimal working point at Θ ≈ 115^∘ [43], it seemed to be plausible that the biceps brachii was not the mainly engaged muscle for small elbow angles (arm extended).

The effect of tendon preloading on elbow torque estimation based on wrist force measurement.

The preloading of the tendon is not directly measurable in vivo. A deceptive way to estimate preloading would be to observe the force at the wrist during the unloaded state. However, even if the only physiological torque source was the biceps brachii the preloading would not be distinguishable from preloading of the wrist strap (cmp. Fig 3). If additional torque sources were assumed, the lower tendon of the biceps brachii could be preloaded even if the resulting torque sum at the elbow joint would be zero.

Therefor, tendon pre-loading remains to be unknown. If tendon pre-loading occurs, it would lead to a gradual shift towards the linear region (see Fig 2).

Discussion of the tendon force estimation.

However, during model building it was assumed that the biceps brachii was the only active muscle. This assumption might have led to an overestimation of the tendon force and, as a consequence, to an overestimation of the tendon stiffness. Shukla et al. [4] investigated the properties of the biceps brachii tendon in vitro. They found a mean failure load force of the tendon-radius insertion at 90^∘ elbow flexion of 360N (SE 120N). In this study, a mean tendon force during maximal voluntary contraction at 90^∘ elbow flexion of (SE 84N) was found. This force seems to be too high considering the failure load found by Shukla et al. [4]. However, the comparison with Shukla et al. is difficult because the tearing of the tendon from the bone was measured in vitro. In vivo, the distal biceps brachii tendon not only inserts into the bone but also into the forearm fasciae, which leads to a presumably higher failure force. In addition, other flexor muscles, such as brachialis and radio brachialis, may also contribute to an overestimation of the tendon force. However, the actual force generated by the biceps brachii could be estimated by using the ratio of the maximal cross sectional area of the biceps brachii and the combined maximal area of all elbow flexor muscles. This is based on the assumption that the cross sectional area of a muscle is directly proportional to the force generated during maximal voluntary contraction. According to Erskine et al. [44], the biceps brachii makes up 41.8% (SE 1.2%) of the overall maximal cross sectional areas of all flexor muscles. Following this line of thoughts, the biceps brachiii would exert a force of 580N (SE 50N). If it is assumed that this force was distributed to the bone and fascial insertions, a reasonable maximal voluntary force is reached which is comparable to the findings of Shukla et al.. Without such an estimation of the force generated by the biceps brachii, the tendon force is overestimated. An overestimate of the tendon force leads to an overestimate of the tendon stiffness.

In the captured ultrasonic videos it was noticeable that the brachialis muscle group changes its diameter. This, however, speaks in favour of an increasingly negative τ_r for smaller elbow angles. Therefore, the parameterized tendon model was not used to determine tendon stiffness (change in tendon force per change in tendon length). Rather, the presented model allowed the tendon length change to be determined based on the elbow torque. Further work should focus on the contribution of the remaining flexor muscles on the resulting elbow torque to predict τ_r more accurately.

Smart et al. have conducted a comparable study. In that study, the measured elbow torque was also attributed only to the contraction of the biceps brachii [9]. Their results should therefore be comparable to those in this work. They defined the tendon stiffness as the slope of a linear displacement over force function. This function was defined to include the tendon displacement at 20% and 80% of maximum voluntary contraction. For comparison, the same linear stiffness was calculated for the fit results of this study as demonstrated in Fig 14B and listed in 1. Smart et al. found a stiffness of for 9 young participants [9]. In the present study, for the optimal fit of each subject’s elbow angle subset (Table 1) a mean stiffness of () (in case of the automatic data extraction) and () (for the manually extracted data) was found. These linear stiffness results are considered not to be significantly different (two-sided T-test) as compared to the findings of Smart et al. [9] ( for the automatically extracted data, and for the manually extracted data). A general drawback of the linear stiffness even as medical parameter is, that small variances in 20% of can lead to large variances in c_lin,20-80 due to the non-linearity in the toe region (cmp. Fig 14B).