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Perturbationbased estimation of withinstride cycle metabolic cost
Journal of NeuroEngineering and Rehabilitation volume 21, Article number: 131 (2024)
Abstract
Metabolic cost greatly impacts tradeoffs within a variety of human movements. Standard respiratory measurements only obtain the mean cost of a movement cycle, preventing understanding of the contributions of different phases in, for example, walking. We present a method that estimates the withinstride cost of walking by leveraging measurements under different force perturbations. The method reproduces time series with greater consistency (r = 0.55 and 0.80 in two datasets) than previous modelbased estimations (r = 0.29). This perturbationbased method reveals how the cost of pushoff (10%) is much smaller than would be expected from positive mechanical work (~ 70%). This work elucidates the costliest phases during walking, offering new targets for assistive devices and rehabilitation strategies.
Introduction
Metabolic cost is a critical measure used to characterize movement behavior [1,2,3]. Healthy walkers naturally adopt an energetically optimal stride cycle, for example, by walking with a step length [4] and knee flexion angle [5] that minimizes metabolic cost. Pathologies like stroke and cerebral palsy alter patients’ walking stride resulting in increases to metabolic cost by 60 to 300% [6, 7]. Such increases in metabolic cost correlate to drastic reductions in people’s mobility and overall quality of life [8, 9]. If we understand how stride cycle phases contribute to metabolic cost, therapies and devices may be better optimized to improve mobility (Fig. 1A).
Measurements of metabolic cost are too slow to detect the contributions of different stride phases. Current methods to calculate energy from oxidative reactions include measuring respiratory CO_{2} production by ingesting water with a radioisotope (‘doubly labelled water method’), measuring oxidative heat production using a chamber (‘direct calorimetry’), and measuring O_{2} consumption from respiration (‘indirect calorimetry’) [10]. Indirect calorimetry is the fastest and most commonly used method for measuring metabolic cost during locomotion; however, it still requires averaging several minutes of breaths to be reliable [11,12,13]. A typical walking stride lasts about one second meaning current methods can only measure the mean metabolic cost following a bout of steadystate walking. Experiments that approximated the cost of the swing phase by recording cyclical leg swinging [14] and by measuring blood flow from injected microspheres in animals that are then sacrificed [15] suggest that the stridemean metabolic cost does not necessarily represent the contributions of individual phases (‘withinstride metabolic cost’).
Several modelbased methods of estimating withinstride metabolic cost have been proposed but remain inconclusive. Umberger developed a set of equations to estimate metabolic cost from muscle parameters and used this to produce the first estimation of withinstride metabolic cost from a forward simulation of walking [16]. Other groups used EMGdriven simulations [17] or equations based on joint kinetics instead of muscle parameters [18]. However, when comparing those methods to each other, their estimations of withinstride metabolic cost are relatively inconsistent (Pearson correlation: r = 0.29, n = 6 estimations, Fig. 1B) [19]. Currently, there is no way to validate these modelbased estimations for withinstride metabolic cost since measurements from indirect calorimetry only obtain a stride mean. This motivates the development of an alternative method to estimate withinstride metabolic cost that is supported by indirect validation approaches.
We hypothesized that applying a set of perturbations creates a set of instances of the behavior where the differences in the time series between each perturbed instance can be attributed to the different magnitudes and timings of the applied perturbation. By applying perturbations repeatedly to a specific part of the gait cycle for several minutes, we can induce changes in the stridemean metabolic cost as well as in the biomechanical time series (e.g., kinematics, kinetics, and muscle activations) [19,20,21]. We postulated the variation across the set of perturbed walking strides would be representative of the fluctuations in metabolic cost within the stride cycle so long as the set contained a large number of different perturbations. If true, this would enable a method to extract key features of withinstride metabolic cost. Our approach is inspired by prior studies that utilized ankle perturbations to assess time series of joint impedance during the stance phase [22, 23] as well as studies that used elastic bands and added mass to estimate the cost of stance and swing phases [24, 25]. To the best of our knowledge, using of a perturbationbased approach for estimating withinstride metabolic cost time series is novel.
Using this concept, extraction of withinstride behaviour from a collection of perturbed instances, we developed an alternative method to estimate withinstride metabolic cost that we refer to as our ‘perturbationbased method’. Our method estimates withinstride metabolic cost using measurements from a set of perturbed walking strides. We then evaluated our method’s ability to consistently reproduce modelbased estimates of withinstride metabolic cost.
Materials and methods
Overview
We created two datasets; a dataset generated using a neuromechanical simulation and a dataset collected from human experiments [20, 26, 27]. Each of these datasets contain walking kinematics, kinetics, and muscle activation time series during 35 different perturbed walking conditions and one unperturbed, normal walking condition (36 walking conditions per dataset). The ‘perturbation’ applied during each of the 35 perturbed walking conditions was a force profile applied to the COM.
Our perturbationbased method was initially developed and tuned using the dataset from the neuromechanical simulation [26, 27]. Tuning consisted of adjusting which kinematic, kinetic, and muscle activation time series from the neuromechanical simulation were used for estimating withinstride metabolic cost. The timeseries included from different combinations of kinematic, kinetic, and muscle activation data will be referred to as ‘derived time series’. The choices from tuning were made based on the perturbationbased method’s performance at estimating other kinematic, kinetic, and muscle activation time series.
We indirectly validated our perturbationbased method by evaluating its ability at estimating 10model based metabolic cost timeseries. First we inputted the kinematic, kinetic, and/or muscle activation data into 10 modelbased methods in order to generate withinstride metabolic cost timeseries (five models per dataset). We then indirectly validated our perturbationbased method based on its ability at reproducing the five modelbased metabolic time series from the neuromechanical dataset and the five modelbased metabolic time series from the human experimental dataset. Reevaluating our method in two distinct datasets avoids dataset bias [28]. Finally, we assessed the perturbationbased method’s estimate of withinstride metabolic cost when using \(\:\dot{V}{O}_{2}\) and \(\:\dot{V}{CO}_{2}\) data from the human experiment.
Simulation dataset
We adapted a neuromechanical simulation from Song and Geyer to walk under force perturbations from a waist tether [26, 27]. Specifically, we used a twodimensional variant that restricts motion to the sagittal plane [26]. We simulated perturbations with forward forces applied at the hip of a model with seven rigid segments in Simscape First Generation Multibody (MathWorks, Natick, MA). In this framework, we simulated 32 sinusoidal force profiles with peak timings covering the entire gait cycle and peak forces ranging from 0 to 24% of body weight, three constant force profiles, and an unperturbed walking condition.
The neuromechanical model’s walking control strategy was optimized for each perturbed walking condition (cf. Supplementary: Neuromechanical simulation dataset for tuning and in silico evaluation). Briefly, the cost function was piecewise with two different functions based on a conditional of a walking pattern that reached 20 s without a fall. Firstly, the optimization would search for control parameters that increased the distance travelled with consistent stepping, the number of steps taken, and the time the simulation successfully walked without falling. Once a set of control parameters achieved a walking pattern that could walk for a simulated 20 s, the control parameters would be refined to match a target walking velocity of 1.25 ms^{− 1}, minimize muscle activations, maintain a consistent walking pattern, and penalize unnatural range of motion.
Time series data (simulated kinematics, kinetics, and muscle activations) were extracted for each of the optimized control strategies to constitute the neuromechanical dataset. We then constructed 100 time series to serve as test data for tuning our perturbationbased method. These test time series were random linear combinations of the different biomechanical time series, so they were distinct from the modelbased estimates that would be used later for evaluation.
Experimental dataset
We used biomechanical and indirect calorimetry data from previous human experiments [20] with a robotic waist tether [21] for the in vivo evaluation and application of our perturbationbased method (Supplementary Data 1). Ten healthy participants (age: 28.0 ± 4.7 years, body mass: 83.2 ± 12.2 kg, height: 1.80 ± 0.05 m; mean ± SD) walked under the same perturbations as in the neuromechanical simulation dataset. In this case, the perturbations were generated by a robotic waist tether controlled by a temporal algorithm that enables pulling during a specific portion of the gait cycle with high consistency.
Perturbationbased method input signals
Our perturbationbased estimation method uses the stridemean metabolic cost as well as withinstride biomechanical time series to estimate withinstride metabolic cost (Fig. 2C and F. Methods: Perturbationbased method). The biomechanical time series as well as additional mathematically derived combinations of those time series are considered potential estimates of withinstride metabolic cost (cf. Methods: Additional input signals and algorithm tuning). Our perturbationbased method first calculates the mean cycle from 0 to 100% of the stride for each biomechanical time series for each perturbation condition. Then each stridenormalized biomechanical time series is reduced to one scalar for each perturbation condition using a custom standardization method based on the deviation from unperturbed walking (Fig. 2, cf. Methods: Custom standardization method). A collection of these standardized scalar values of biomechanical data across all perturbations form a perturbed biomechanical set. Finally, we select the biomechanical set that matches the perturbed set of the stridemean metabolic cost (cf. Methods: Time series estimation procedure). The original biomechanical time series that most closely matched the standardized set for the stridemean metabolic cost is used as the estimate of withinstride metabolic cost.
We chose to estimate the metabolic cost of one side of the body rather than the whole body’s metabolic cost. The withinstride metabolic cost of one side of the body provides more descriptive and potentially useful information for interventions, such as assistive devices, than wholebody cost, which cannot be attributed to a specific leg. Using modelbased methods, we generated a set of five estimates of the withinstride metabolic cost to indirectly validate our perturbationbased method’s performance which were distinct from the five evaluations that were used in the neuromechanical dataset (cf. Supplementary: Modelbased metabolic costs used; [18, 29,30,31,32,33,34,35]).
All kinematic, kinetic, and muscle activation time series as well as the derived signals (cf. Methods: Additional derived input time series and algorithm tuning) are stridenormalized and organized in matrices with one row for each percent of the stride cycle and one column for each of the 36 perturbation conditions.
Each perturbation’s force profile was repeated over multiple stride cycles for a sufficient duration to obtain steadystate metabolic cost (40 s to obtain ten sufficiently stable strides in the neuromechanical simulations and 2 min to estimate the steadystate metabolic cost in the human experiments) [11].
The stride mean metabolic cost for every condition is also used as an input in our perturbationbased method.
This stride mean can be estimated from modelbased metabolic costs as well as from respiratory \(\:\dot{V}{O}_{2}\) and \(\:\dot{V}C{O}_{2}\) measurements; hence this input is available when estimating the withinstride metabolic cost in human experiments.
Custom standardization method
Each time series is standardized using a custom method (Supplementary Data 1). First, we take the stride mean of each biomechanical time series for every perturbation condition.
Next, we calculate the deviation of each perturbed walking condition from the unperturbed walking condition.
where \(\:{\Delta\:}{\stackrel{}{X}}_{bts}\) is the set of deviations from the unperturbed condition and \(\:{\stackrel{}{X}}_{bts,0}\) is the stride mean of the unperturbed condition.
Each set of deviations is then normalized by its range of deviations from unperturbed walking.
where \(\:{\stackrel{}{X}}_{stand}\:\)is the standardized set of deviations from unperturbed walking for each biomechanical time series and n_{bins} is the number of bins. The standardized set is enumerated to reduce the effects of floatingpoint differences between biomechanical measurements. The number of bins was set to 80 based on tuning (cf. Methods: Tuning of available data for metabolic cost estimation, Supplementary Data 2). This process is similar to Slade et al., (2022) [36].
In summary, this procedure converted the stride means of biomechanical time series to a range of standardized values ranging from 1 to 80. We also applied the same standardization procedure (Eqs. 4, 5) to the stride means of derived biomechanical time series as well as to the stride mean metabolic cost (\(\:\stackrel{}{Y}\)).
Time series estimation procedure
We ran a minimization procedure that evaluates which standardized biomechanical time series best matches the standardized metabolic cost. First, we evaluate how well the standardized set of each biomechanical time series and each derived time series matches the standardized set of metabolic cost using a sum of square comparison
where \(\:SS\) is the sum of squares and \(\:c\) represents each perturbation condition.
Then, we conduct a stepwise optimization procedure whereby we evaluate if adding another standardized biomechanical time series or derived signals to the previous standardized set improves the \(\:SS\)
where \(\:{\stackrel{}{X}}_{stand,c,\:prev\:opt\:SS}\) is the standardized set that produced the best SS in the previous iteration and j represents a new biomechanical measurement or derived signal that is evaluated.
Finally, the time series of the biomechanical measurement, derived signal, or combination of signals with the lowest \(\:SS\) is then used to estimate withinstride metabolic cost (Fig. 3). If the lowest SS results from one single biomechanical measurement or derived signal, the corresponding unperturbed time series is used to estimate withinstride metabolic cost.
where \(\:{Y}_{estimated}\) is the estimated withinstride metabolic cost, \(\:{X}_{SS\:opt}\)is the time series of the biomechanical measurement or derived signal that resulted in the lowest \(\:SS\). In the event the lowest \(\:SS\) is from a combination of biomechanical measurements and derived signals, we normalize each signal by its range and sum to serve as the estimate of withinstride metabolic cost
where \(\:i\) is the index of the biomechanical signals used to achieve the lowest sum of squares.
The approach of leveraging perturbations constitutes a paradigm shift compared to previous iterative improvements of modelbased methods. Our procedure of using data from the perturbed conditions to estimate the unperturbed condition intrinsically involves estimating (just) outside of test data, and it is known that overfitting can be an issue in such a procedure. Some features of the perturbationbased method likely helped avoid this overfitting. We limited the number of inputs by using a standardization that converted each time series to a scalar (Eqs. 3–5). We also generated a very large number of derived signals.
Additional derived input signals and algorithm tuning
We tuned two features of our perturbationbased method: the selection of which mathematical derived time series would be available for creating the estimation of withinstride metabolic cost and the number of bins in the custom standardization procedure (cf. Methods: Additional derived input signals and algorithm tuning). During the tuning, we evaluated which settings improved the lowerbound, 95% confidence interval of Pearson’s correlations between the estimated and the test time series. After tuning, the mean Pearson’s correlation between our perturbationbased method’s estimate and time series within the test set was 0.41 (95% CI = 0.33–0.50). We evaluated the impact of the following options:

Options 1–2: The separation of positive and negative regions of the original biomechanical time series.

Options 3–5: The square, cube, or inverse of the original biomechanical time series.

Options 6–8: The subtraction, addition, or multiplication of all pairs of the biomechanical time series.

Option 9: An additional set of additions and multiplication of pairs of the mathematically derived time series (generated from options 1–8).
We restricted option 9 to stop after generating 4000 combinations because considering all the combination permutations was not feasible. We also tuned the number of bins for standardizing biomechanical time series (Eq. 5). This tuning is similar to the sensor selection and bin optimization in Slade et al. [36].
The tuning criterion was correlation performance against 100 test time series. The test time series used were distinct from the modelbased metabolic costs to avoid biasing the evaluation of our method [28]. As test time series for tuning, we generated 100 time series based on random combinations of the biomechanical time series from the neuromechanical simulation dataset.
where \(\:{Y}_{tuning,k}\) represents one of the 100 test time series,\(\:\:{c}_{1}\) to n are random coefficients between 0 and 1, \(\:{X}_{bts,1}\) to \(\:{X}_{bts,n}\) are the positive or negative portions of a randomly chosen number of biomechanical measurement time series.
The perturbationbased method’s correlation with the 100 test time series was evaluated for each of 512 (2^{9}) combinations of mathematically derived time series for bin numbers ranging from 10 to 100 (Supplementary Data 2).
Statistical analysis
As a measure of the uncertainty in the literature, we generated a crosstable with pairwise Pearson correlations between six previously reported plots of withinstride metabolic cost in the literature [19], and we calculated the mean and 95% confidence interval of the correlations (Fig. 1b). Due to the limits of a Pearson correlation at − 1 and 1, we converted each rvalue to a Zscore using Fisher’s Ztransformation. Average Zscores and zscore confidence intervals across the correlations in literature, between perturbationbased and neuromechanical modelbased, and between perturbationbased human experimental modelbased were converted back to Pearson rvalues for easier interpretation [37]. All analyses were conducted in MATLAB 2021b.
Results
Once tuning was completed, and our perturbationbased method was finalized, we evaluated its performance at reproducing a variety of modelbased estimates of withinstride metabolic cost. We calculated five withinstride metabolic costs using modelbased methods (cf. Supplementary: Modelbased metabolic costs used in neuromuscular simulation dataset; [30, 32,33,34,35]). The mean Pearson’s correlation between the five different modelbased withinstride metabolic costs and our estimations of those using the perturbationbased method was 0.55 (95% CI = 0.22–0.77). This evaluation performance constitutes an improvement of at least 50% compared to the mutual consistency between modelbased estimations in the literature for four out of five estimations (Fig. 4AE; Table 1).
We also indirectly validated our perturbationbased method in data from human experiments. In vivo, human walking experiments were conducted with a perturbation from a robotic waist tether applied to the COM (cf. Supplementary: Human experimental dataset for invivo evaluation and application) [20]. In each condition, the tether applied pulling forces with a specific profile repeatedly to stride cycles for a sufficient duration to induce a different steadystate gait. We applied the same perturbationbased method to our human experimental dataset without any additional tuning or changes. Our estimation reproduced the abovementioned five independent modelbased estimations of metabolic cost with a mean Pearson’s correlation of 0.80 between the modelbased metabolic costs and their estimations using the perturbationbased method (95% CI = 0.57–0.91, Table 2). This result is also greater than the correlation between modelbased estimations currently in literature with an improvement of at least 75% (Fig. 4FJ) [19, 38].
After successfully completing the indirect validations, we applied our perturbationbased method to estimate withinstride metabolic cost based on \(\dot V{O_2}\) and \(\dot VC{O_2}\) data from the human experiment (Fig. 5). When we divide the stride into the first double stance (1–15% of the stride), single stance (16–50%), pushoff (51–65%), and swing (66–100%), their metabolic cost respectively accounted for 20, 49, 10 and 21% of the total. The estimated cost of pushoff is considerably lower than that of single stance. This is markedly different from the evolution of positive mechanical work performed by the leg onto the COM, which is about three times as much during pushoff compared to single stance. As such, our perturbationbased estimation confirms that metabolic cost can be related to sources other than mechanical work [39, 40].
Our estimation that pushoff accounts for about onetenth of the total metabolic cost is similar to the first estimation using a forwarddynamics musculoskeletal modelbased approach (8% [16]) but is low compared to estimations from modelbased methods that use only jointbased equations (39% [18] and 49% [19]). Our estimation of the cost of the swing phase (21%) is close to the mean from previous modelbased studies (24%, 95% CI = 19–28% [16, 17, 19, 38, 41, 42]). This also supports previous estimations from experimental studies with perturbations to the swing or stance phase that suggest that the swing phase substantially contributes to the metabolic cost of walking (swing phase contribution to metabolic cost reported as 10, 12.5 and 17% [14, 24, 25]).
While our approach of using perturbations is innovative and yields results consistent with existing literature, we acknowledge some limitations in our methods, results, and the application. One methodological limitation is that our method solely relied on lower limb signals for estimating metabolic costs. Our evaluation replicated modelbased costs using lowerlimb data and a simplified neuromuscular model. Notably, we did not directly account for metabolic contributions from trunk and arm muscles [43]. Another methodological constraint is the tuning of the derived time series and the number of perturbations required to create the datasets. Adapting this method for other datasets might require expanding the types of derived time series. In terms of the results, we recognized that our perturbationbased method for estimating withinstride metabolic cost is empirical. While this offers the advantage of being less biased than modelbased methods, this is not favorable for understanding causal relationships, such as the impact of altering a specific gait impairment [27, 44, 45]. Applicationwise, a drawback of our method is its reliance on datasets of walking under various perturbations which can be timeconsuming and physically demanding for participants.
To advance perturbationbased withinstride metabolic cost estimation’s practicality, future research needs to tackle challenges concerning tuning, time efficiency, and validation. Developing algorithms with greater generality, such as neural networks, could mitigate reliance on specific tuned options. Investigating perturbation types yielding the most valuable data will streamline data collection efforts. Finally, exploring innovative indirect validation methods could bolster confidence in the methodology.
Conclusion
The present work describes a perturbationbased method that can reproduce a wide variety of modelbased, withinstride metabolic costs in two different datasets using a collection of perturbed conditions. The result suggests that the metabolic cost of pushoff is lower than the preceding single stance phase and that the swing phase has a nonnegligible metabolic cost. These findings may have important applications for designing rehabilitation strategies and assistive devices. For example, the finding of a large cost of single stance may help explain how an unpowered ankle exoskeleton that primarily provides torque during single stance could reduce metabolic cost despite increasing plantar flexor activation during pushoff [46]. The trajectory of community research has incrementally reduced the time to estimate steadystate metabolic cost from several minutes using Douglas bag, mixing chamber, to 1–2 min with breathbybreath systems [47] and fitted approximation methods [11, 48, 49], and finally, to a matter of seconds via a combination of sensors and fitting methods [50, 51]. The present work grants greater understanding of metabolic cost beyond what was previously possible by presenting within movement cycle interpretability instead of more rapid interpretation of steadystate metabolic cost.
Data availability
All data are available in the main text or the supplementary materials.
Abbreviations
 EMG:

Electromyography
 COM:

Center of Mass
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Acknowledgements
We would like to thank Eric Perault, Daniel P. Ferris, Kota Takahashi, Rodger Kram, and Steven H. Collins for helpful suggestions on this project. We would like to thank Ben Senderling, and HuMoTech for support with the experimental data collection. We would also like to thank Keegan Moore, Nathaniel Hunt, and Mukul Mukherjee for feedback on the initial draft.
Funding
This work was supported by National Science Foundation Grant No. 2203143, NU Collaboration Grant No. 27102, National Institutes of Health Grants No. R00AG065524 and P20GM109090, and the Center for Research in Human Movement Variability of the University of Nebraska at Omaha. The conclusions in this article are only those of the authors and do not necessarily reflect the views of the funders.
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Conceptualization: ACD, PM; Methodology: ACD, SS, PM; Investigation: ACD, PA, AMG; Visualization: ACD, PM; Funding acquisition: PM; Project administration: PM; Supervision: PM; Writing – original draft: ACD, SS, PM; Writing – review & editing: ACD, PA, AMG, SS, PM.
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The study protocol was approved by the University of Nebraska Medical Center’s Institutional Review Board in accordance with the Declaration of Helsinki. Informed consent was obtained from all participants prior to their participation in the study.
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PA, AMG, and PM submitted a provisional patent application (serial number: 63/320,303; docket number 22057P) on the waist tether used for the human experimental dataset.
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Dzewaltowski, A.C., Antonellis, P., Mohammadzadeh Gonabadi, A. et al. Perturbationbased estimation of withinstride cycle metabolic cost. J NeuroEngineering Rehabil 21, 131 (2024). https://doi.org/10.1186/s12984024014248
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DOI: https://doi.org/10.1186/s12984024014248