System identification: a feasible, reliable and valid way to quantify upper limb motor impairments

Participants

We recruited adult patients from the outpatient clinic of the Rijndam Rehabilitation Center that presented upper limb motor impairments due to stroke and CP. Patients were only included if they: (1) had a clinically-confirmed upper limb impairment, (2) were able to achieve active shoulder abduction (up to 80°), (3) were able to achieve visible active elbow extension, and (4) had a minimal passive range-of-movement for shoulder abduction of 80° and horizontal shoulder adduction of 45°. Exclusion criteria were: (1) hemiplegic shoulder pain, (2) a history of pre-existing neuromuscular disorders affecting upper limb function, (3) fixed upper limb contractures, and (4) the inability to understand verbal instructions. We only included stroke patients when they were considered chronic, suffering the stroke at least six months before study inclusion.

For comparison, we recruited a group of age-matched healthy controls without a known history of neurological or orthopedic disorders. The study was approved by the Medical Ethics Committee of the Erasmus Medical Center, Rotterdam, (protocol number NL64660.078.18), and conducted following the declaration of Helsinki.

Experimental setupShoulder elbow perturbator

The Shoulder-Elbow-Perturbator (SEP—Hankamp Rehab, Enschede, The Netherlands) was used to assess participants' elbow dynamics. The design of the SEP enables independent manipulation of elbow angle and weight support of the human arm [12]. A direct-drive servo motor (HIWIN TMS3C, Offenburg, Germany) attached to a lever supporting the lower arm controlled the elbow angle by aligning its axis to the medial epicondyle of the humerus (Fig. 1). A computer with Etherlab and MATLAB Simulink was used to control the SEP and capture elbow torque and angle with a sample rate of 1 kHz.

Fig. 1figure 1

The Shoulder Elbow Perturbator (SEP), a robotic device to quantify multiple upper limb impairments. a Participant positioned in the SEP with the shoulder abducted in 80°, the forearm strapped (1) to the lever arm of the SEP, and the medial epicondyle of the humerus aligned with the motor rotation axis. b Internals of the SEP showing that the motor (4) transmits a torque through the torque link (3), allowing elbow rotation (2). c Internals of the SEP showing the shoulder abductor manipulation mechanism. The sarrus linkages (5) allow vertical displacement of the arm, and the arm is supported by two springs (6) with an upward force. With the cable routing and pulley configuration, the upward force can be manipulated independently of the linkage position (7). (Modified – with permission from [20])

Measurement protocol

All ninety-four participants underwent the same measurement protocol. Fifty-four participants performed the measurement protocol twice, separated by at least 7 days, to assess the test–retest reliability of the extracted parameters from each task.

Participant profiling

Before the SEP experiments, the age, sex, dominant arm, and side of the paretic arm were registered and body mass and length were measured. For stroke patients, the date and type of stroke were obtained from their medical records. For CP patients, the level of the Gross Motor Function Classification System (GMFCS) and the Manual Ability Classification (MACS) were obtained from their medical records [21, 22]. Finally, for each patient, synergy was quantified by either the score on the upper limb section of the Fugl-Meyer assessment (for stroke patients) or the Test of Arm Selective Control (TASC, for CP patients) [23, 24]. Spasticity was quantified using the MTS. All clinical tests were performed by an experienced assessor (LL).

SEP measurements

Participants were seated next to the SEP on a custom-made chair. Straps were used to limit torso displacement by immobilizing the torso against the back of the chair. The chair and the height of SEP were adjusted to achieve 80° shoulder abduction and 30° horizontal adduction. To ensure the axis of the servo motor remained aligned with the medial epicondyle of the humerus, the lower arm and wrist were fixed to the SEP lever using Velcro straps and a custom clamp mechanism with safety pins. The wrist remained in a ± 10° dorsiflexion position throughout the experiment using a cock-up cast. Before commencing the tests, the SEP ROM limits for the arm were determined manually.

Each participant was measured according to the ‘Re-Arm protocol’ [12] and the ‘System identification protocol’. All performed tests were presented in random order to avoid an order effect. Moreover, a 5-min rest period was adopted between each test to prevent fatigue.

System identification protocol

A system identification protocol was used as an all-encompassing way to quantify upper limb impairments using an active and passive elbow perturbation task. Participants were instructed to maintain the 80° shoulder abduction angle during the recordings during either a DNI or a resist task while receiving continuous torque perturbations to the elbow. The DNI task was performed once for five different arm weight support levels (100%, 75%, 50%, 25%, and 0% support). In contrast, the resist task was performed only once at two weight support levels (100 and 0% support) to limit fatigue and participant burden. The 100% weight support level was determined by gradually increasing the upward force of the support mechanism until the participant could fully relax the arm while the support surface remained at the same height. To aid participants in minimizing elbow movements during the resist task conditions, a vertical bar was mounted to the SEP near the hand to highlight the reference position where the hand had to be kept. Each trial lasted 55 s.

The torque perturbation signal was designed as a random-phase multisine torque signal (i.e., the sum of several sinusoids with random phase) to challenge the sensorimotor system with a frequency content relevant to study the dynamics of the elbow (natural frequency of the elbow is ~ 1–2 Hz). The period of the disturbance signal was set to 5 s to accommodate low frequencies but still enable sufficient periods in the given measurement time. All frequencies between 0.2 and 12 Hz were included in the perturbation signal (frequency resolution of 1/5 s = 0.2 Hz). The perturbation was scaled such to result in a root mean square elbow rotation during a trial of ~ 0.03 rad for each participant. Hence, each 55 s trial consisted of 11 consecutive periods of a perturbation period. The recorded elbow angle and torque were used to quantify elbow dynamics (Fig. 2).

Fig. 2figure 2

The torque perturbation signal in a frequency—and b time-domain was composed of frequencies between 0.2–12 Hz. c Typical elbow torque and d elbow angle throughout a single 5 s recording period

To quantify protocol feasibility, for all participants, the total time required for the system identification protocol was noted and afterwards participants were asked to rate the pain of the protocol on a 0–10 scale (zero representing ‘no pain’). Moreover, forty participants (20 stroke patients and 20 healthy controls) were asked to rate the burden of the protocol on a 0–10 scale (zero representing ‘no burden’).

Re-Arm protocol

The results of the system identification protocol were compared with results obtained using the Re-Arm protocol presented previously [12, 20] to establish the construct validity of the system identification parameters. The Re-Arm protocol was designed to objectify manual performed clinical tests, thereby obtaining a comprehensive description of four characteristics that describe upper limb function: muscle weakness, abnormal synergy, spasticity, and viscoelasticity of the elbow. Three are used in this study:

1.

To quantify abnormal synergy, participants were positioned to start at maximum elbow flexion and instructed to actively and slowly extend the elbow as far as possible. This was done once for five different levels of arm weight support (100%, 75%, 50%, 25%, and 0% support).

2.

To quantify spasticity, participants were instructed to fully relax their arm and shoulder while the SEP passively moved their arm from maximum flexion to maximum extension at 100°/s. This was performed three times under full weight support of the arm and with a 5 s rest between repetitions.

3.

To quantify viscoelasticity, participants were instructed to fully relax their arm and shoulder while the SEP passively moved their arm from maximum flexion to maximum extension and back at 6°/s. This was performed three times under full weight support of the arm and with a 5-s rest between repetitions.

Data analysisSystem identification analysis

Data from the system identification protocol was first analyzed using a non-parametric analysis to estimate elbow admittance (i.e., the resistance of the elbow to an external torque and thereby the inverse of impedance). The elbow admittance is represented by a frequency response function calculated by dividing the cross-spectral densities between the elbow angle (\(\theta\)) and elbow torque (\(T\)) with the perturbation torque (\(d\)):

$$_\left(f\right)= \frac_\left(f\right)}_\left(f\right)}$$

Cross-spectral densities \(_\left(f\right)\) and \(_\left(f\right)\) were estimated using Welch’s method [25] with a rectangular window corresponding to the length of the period in the perturbation signal (5 s). Before calculating the elbow admittance, the cross-spectral densities were smoothed using 3-point frequency averaging to reduce the variance of the estimations [26]. In addition to the elbow admittance, the coherence \(}_^(f)\) for the elbow angle was estimated following estimation of the auto-spectral densities \(_\left(f\right)\) and \(_\left(f\right)\):

$$}_^\left(f\right)= \frac_\left(f\right)\right|}^}_\left(f\right)_\left(f\right)}$$

The coherence indicates whether the relation between perturbation torque and elbow angle is linear and noise-free.

Physiological meaningful parameters were obtained by fitting a neuromuscular model. This model represents the mechanical elbow admittance and the contact dynamics, i.e., dynamics associated with the interface between the lower arm and robotic lever. Therefore, the neuromuscular model is described by:

$$_(s)=\frac^+bs+k}+\frac_s+_}$$

where \(I\) is the lower arm inertia, \(b\) the elbow muscle viscosity, \(k\) the elbow stiffness, \(_\) the contact viscosity and \(_\) the contact stiffness. The best model fit to the data was sought by minimizing the following criterion function [27]:

$$L\left(p\right)= \sum \frac}_^(f)}_\left(f\right)}_(f)}\right) \right|}^$$

where we only included frequencies that had power in the perturbation signal following 3-point frequency averaging. A least-squares criterion with a logarithmic difference was used because of the large range in magnitude of the frequency response function. Moreover, the least-squares criterion was weighted with the coherence to limit the effect of the less reliable frequencies and with \(^\) to prevent emphasis on higher frequencies [27, 28]. Hence, both the DNI and resist conditions were fitted simultaneously with the same inertia. In addition, the contact dynamics were considered independent of the arm weight support level but different between the DNI and resist task conditions. For the DNI task, contact viscosity was taken as \(_=2 Ns/m\) while contact stiffness was \(_=340 N/m\), while for the resist task a contact viscosity of \(_=4 Ns/m\) and contact stiffness of \(_=500 N/m\) were used. These values provided an appropriate fit of the contact dynamics for all participants. In total, 15 parameters were estimated. Table 1 summarizes all model parameters to be estimated in this study.

Table 1 System identification parameters to be estimated

Following fitting the neuromuscular model and extracting the system identification parameter values, two additional parameters, \(_\) and \(_,\) were determined by performing a first-order polynomial least squares fit to the estimated elbow viscosity and stiffness across all weight support levels during the DNI task.

The goodness of the fit of the neuromuscular model, and thus quality of parametric estimates, was expressed using the variance-accounted-for (VAF) between the estimated and measured elbow angle. The estimated angle was calculated by multiplying the estimated mechanical admittance model with the perturbation torque in frequency domain, and subsequently taking the inverse Fourier transform. Any individual conditions were excluded from further analysis when the VAF < 0%, indicating a poor model estimate. When a participant had two or more missing DNI levels or a VAFs < 0%, the participant was excluded from further analysis.

Re-Arm protocol analysis

Data and results from the Re-Arm protocol were presented previously [20]. Here, we used the same data processing pipeline to extract synergy, spasticity, and viscoelasticity and thereby establish validity of the system identification technique:

1.

To quantify abnormal synergy, the maximum elbow extension angle was determined for each level of weight support. The extension angle at 100% weight support was defined as zero. Subsequently, a linear regression line was estimated to relate weight support level to the maximum extension angle. The slope of the regression line was adopted as a measure of synergy, with 0 indicating an absence of synergy. A negative slope of the regression line would indicate the presence of synergy, since that would represent that the maximum elbow extension angle decreases with decreased arm weight support [29].

2.

To quantify spasticity, the maximum torque during the three passive elbow extension movements was averaged.

3.

To quantify viscoelasticity, the mean elbow torque at ten evenly-spaced positions during the ROM was extracted. The relationship between elbow angle and torque data was parameterized by fitting a regression line, of which the slope was taken as a measure of elbow viscoelasticity.

Statistical analysis

The analysis was separated in four parts:

(1)

Protocol feasibility and quality of the parametric estimates: To establish feasibility of the system identification protocol and determine the quality of parametric estimates in patients we compared the total test time, perceived burden, pain score and quality of parametric estimates (using the VAF) between healthy controls and patients using an independent t-test or Mann–Whitney Test when data was non-normally distributed or had unequal variances across groups. Groups were considered statistically significantly different when p < 0.05.

(2)

Test–retest reliability of the system identification parameters: The test–retest reliability of the system identification parameters was determined by exploring Bland–Altman plots with the limits of agreement (LOA) (LOA = mean difference ± 1.96*standard deviation of the difference between the two measurements) and calculating intraclass correlation coefficients (ICCs). ICCs were calculated with two-way random effects, absolute agreement, single rater formula, ICC(2,1), where the ratio of the variance between participants to the variance between participants plus error variance was calculated. Values less than 0.4 indicated poor reliability, between 0.4 and 0.75, fair to good reliability, and higher than 0.75 are indicative of excellent reliability [30]. In addition to the ICC, the Smallest Detectable Change (SDC) was calculated as \(SDC = 1.96 * SEM * \surd 2\) [31]. Different scores between two measurements larger than the SDC can be interpreted as true differences at an individual subject level.

(3)

Healthy controls vs. patients: We performed an exploratory analysis to test differences between groups in the mean and variance of the extracted system identification parameters. All 15 system identification parameters, and the two slope parameters, were treated as independent variables. Normality and equality of variances were tested using the Shapiro–Wilk test and Levene’s test for homogeneity of variances, respectively. Subsequently, a one-way ANOVA was performed to test for differences between groups. In case a parameter was found to be not normally distributed or presented unequal variance between groups, a Kruskal–Wallis test was performed. Any significant differences (p < 0.05) were further scrutinized using a (Dunn-)Bonferroni post hoc test.

(4)

Construct validity of system identification parameters: Construct validity was tested by comparing the parameters of the system identification protocol to the parameters extracted from the Re-Arm protocol. We used Pearson correlations to assess whether parameters from both protocols can be related. The following comparisons were made:

Synergy: The system identification protocol does not involve any active movement, making a direct quantification of synergy by studying range of movement in different arm weight conditions as done routinely in clinical practice or during the Re-Arm protocol difficult. However, we assume that the abnormal synergy patterns causing impaired arm extension when reducing the arm’s weight support also results in a higher elbow viscosity and stiffness. This is based on observations that when reducing the arm’s weight support elbow extension is limited. The reduced reaching workspace has been suggested to be caused by an altered passive joint stiffness [32, 33]. Therefore, we correlated the change in elbow viscosity and stiffness across weight support levels \(_, k}_)\) with the slope of the regression line relating weight support level and maximum extension angle as obtained during the Re-Arm protocol.

Spasticity: The system identification protocol does not administer fast passive elbow movements to quantify spasticity under full weight support of the arm. However, we assumed spasticity would become evident from the measures of elbow viscosity and stiffness when the elbow is fully relaxed and arm weight is fully supported during the DNI task which imposes small elbow movements at varying velocities. This assumption is based on the observation that spasticity is primarily reported to limit passive movement [34], while full arm weight support removes the potential confounding effect of synergy. Therefore, we correlated the elbow viscosity and elbow stiffness during the DNI task under full arm weight support \(_, k}_)\) with the Re-Arm parameter of spasticity (maximum torque during fast passive elbow extension).

Viscoelasticity: The system identification protocol separates viscoelasticity, determined during slow passive elbow movement during full arm weight support, in elbow viscosity and stiffness. We assumed changes in viscoelasticity would be reflected in both elbow viscosity and elbow stiffness as determined during the DNI task under full arm weight support. Therefore, we correlated both the elbow viscosity and elbow stiffness during the DNI task under full weight support \(_, k}_)\) with the Re-Arm parameter of viscoelasticity (slope of the regression line relating elbow torque and angle).

The correlations coefficients were calculated using the corrcoef function in MATLAB2020a with the significance threshold set at \(p<0.05.\)

Statistical tests were conducted in SPSS version 25 for Windows (IBM, Armonk, NY, US) and values will be reported as mean (SD), unless stated otherwise.

Sample size estimationSample size estimation for the test–retest reliability

Following Walter, Eliasziw [35], we tested whether the expected ICC (ρ1) was equal (null hypothesis) or higher (alternative hypothesis) than the acceptable ICC (ρ0). A ρ0 value of 0.60 was used, based on the literature, and a ρ1 of 0.85 [3, 36]. The number of observations was fixed at 2. Using a significance level (α) of 0.05 and a power (1–β) of 0.80, a sample size of 21 participants per group (patient and healthy control group) is required. To account for an expected dropout of 10%, the current study aimed at sample sizes of 24–30 participants who participated twice.

Sample size estimation for healthy controls vs. patients

We based our sample size calculation on the test of whether stiffness at a 100% weight support level was equal (null hypothesis) or different (alternative hypothesis) for the healthy controls and patients. Based on previous literature and preliminary findings we estimated the difference between the means to be \(3\, \text\) with a standard deviation of \(5\, \text\). Using a significance level (α) of 0.05 and a power (1–β) of 0.80, a sample size of 44 participants per group (patient and healthy control group) is required. To account for an expected dropout of 10%, the current study aimed at sample sizes of 45–50 participants.

Sample size estimation for the construct validity

We hypothesized that the system identification outcomes would correlate moderately (between 0.50 and 0.70) with their Re-Arm counterparts. Using a significance level (α) of 0.05 power (1–β) of 0.80 and an expected correlation coefficient of 0.50 a sample size of at least 29 participants per group (patient and healthy control group) is required. To account for an expected dropout of 10%, the current study aimed at sample sizes of at least 32 participants.

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