Entropy‐based method to quantify limb length discrepancy using inertial sensors
2017; Volume: 8; Issue: 1 Linguagem: Inglês
10.1049/iet-wss.2017.0049
ISSN2043-6394
AutoresSajeewani Karunarathne Maddumage, Saiyi Li, Pubudu N. Pathirana, Gareth Williams,
Tópico(s)Sports injuries and prevention
ResumoIET Wireless Sensor SystemsVolume 8, Issue 1 p. 10-16 Special Issue: Body Sensor NetworksFree Access Entropy-based method to quantify limb length discrepancy using inertial sensors Sajeewani Karunarathne Maddumage, Corresponding Author Sajeewani Karunarathne Maddumage sajeewani@appsc.sab.ac.lk orcid.org/0000-0002-0424-9837 School of Engineering, Deakin University, Waurn Ponds, Australia Faculty of Applied Sciences, Sabaragamuwa University of Sri Lanka, Belihul Oya, Sri LankaSearch for more papers by this authorSaiyi Li, Saiyi Li School of Engineering, Deakin University, Waurn Ponds, AustraliaSearch for more papers by this authorPubudu Pathirana, Pubudu Pathirana School of Engineering, Deakin University, Waurn Ponds, AustraliaSearch for more papers by this authorGareth Williams, Gareth Williams School of Engineering, Deakin University, Waurn Ponds, AustraliaSearch for more papers by this author Sajeewani Karunarathne Maddumage, Corresponding Author Sajeewani Karunarathne Maddumage sajeewani@appsc.sab.ac.lk orcid.org/0000-0002-0424-9837 School of Engineering, Deakin University, Waurn Ponds, Australia Faculty of Applied Sciences, Sabaragamuwa University of Sri Lanka, Belihul Oya, Sri LankaSearch for more papers by this authorSaiyi Li, Saiyi Li School of Engineering, Deakin University, Waurn Ponds, AustraliaSearch for more papers by this authorPubudu Pathirana, Pubudu Pathirana School of Engineering, Deakin University, Waurn Ponds, AustraliaSearch for more papers by this authorGareth Williams, Gareth Williams School of Engineering, Deakin University, Waurn Ponds, AustraliaSearch for more papers by this author First published: 01 February 2018 https://doi.org/10.1049/iet-wss.2017.0049Citations: 3AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Limb length is a useful parameter in the assessment of common musculoskeletal disorders such as limb length discrepancy. The measurement variation among rates adversely affects the quantitative aspect of assessments and introduces a greater subjectivity in the course of treatment. Common practise for measuring limb length is based on radiographic imaging techniques which are inconvenient, costly and require clinical knowledge. Direct instruments are difficult to use with patients due to susceptibility to human error in determining the position of the rotational joint. In this study, the determination of limb length is automated using a contemporary algorithm which applies curvature to the measurements from a low-cost and miniaturised inertial sensor, primarily used in the bio-kinematic research. The motion artefacts contribute to the ultimate estimations and, in this approach, a least noise threshold model is employed to address the robustness. The proposed estimation technique was validated with real-data observed from 14 healthy subjects comparing with radiographic and direct measurements. The experimental results indicate greater accuracy compared with manual measurements with low root mean squared error percentages with values ranging from 5.34 to 5.84%. Additionally, the mean limb length difference between our estimator and both radiographic measurements and direct measurement was 20 mm, a condition which can develop either from childhood or later in life. Consequences associated with LLD include low back pain, osteoarthritis of the hip, stress fractures, aseptic loosening of hip prostheses, standing imbalance, running and walking difficulties. The affected limb LD can be in the range of 9–60 mm [1] and the severity of LLD is determined based on limb length inequalities. According to Reid and Smith [6], treatments are defined as: (i): mild (0–20 mm) – insignificant to treat, (ii) moderate (20–60 mm) should be treated by using shoe lifts, epiphysiodesis or shortening and (iii) severe (60–200 mm) should be treated by combined surgical procedures with prosthetic fitting. However, quantifying the magnitude of LLD for treatments is highly subjective and accurate detection of limb lengths plays a vital role in determining the appropriate course of action [1, 3]. A number of sensor technologies are used to estimate limb lengths. Radiographic technology is considered to be the gold standard. It measures the leg length between markers: femur/pelvis to ankle [1]. There are essentially three commonly used radiographic techniques: orthoroentgenogram, scanogram and computerised digital radio-graph. These technologies have shortcomings such as distortion by parallax error, radiation exposure, cost and the requirement of dedicated laboratory facilities, which restrict use in non-clinical settings. Some investigators have measured limb lengths directly using instruments such as anthropometric callipers and measuring tapes [7–13]. However, they are cumbersome to use and susceptible to human error especially when determining the pelvic bone of LLD patients [1]. Hence, the development of a reliable, accurate, affordable and easy to use limb length measurement technique is a necessity, especially for LLD patients [3]. With the advancement in MicroElectroMechanical Sensors (MEMSs), miniaturised and low-cost sensors which can be packaged as wearable devices are decorously considered for human motion capture [7–11, 13, 14]. More importantly, MEMS sensory devices can be used in non-clinical settings to provide continuous monitoring for an extended period of time. This is considered to be a more cost-effective approach to deliver rehabilitation services [15]. However, usage of MEMS sensors for limb length estimation has not been considered. Furthermore, if this process could be automated by using MEMS sensors, it would not only result in improved accuracy, but also the ability to take measurements in the absence of specialised personnel, thus enhancing the use in non-clinical settings and tele-rehabilitation. To our knowledge, there has not been any attempt to extract limb length using MEMS sensors in the open literature. Thus, this paper aims to provide a novel and systematic approach to estimate limb length using MEMS sensors, especially for applications such as the treatment of LLD. Only one inertial measurement unit (IMU) sensor is required to facilitate the measurement of limb length. The rest of this paper is arranged as follows: Section 2 introduces the novel approach of estimating the limb length based on trajectory curvature ideas followed by signal filtering. The following section discusses the least noise threshold (LNT) method to determine the least noisy portion of a dataset to increase the estimation accuracy. The experimental setup and real-data analysis are explained in Section 4. Finally, this paper is summarised in Section 7. 2 Limb length estimator The underlying approach is based on the following assumptions: Assumption 1.Limb movement starts from a stationary posture. The acceleration required to initiate the movement is negligible with respect to gravity and the accelerating time period is insignificant compared with the overall motion time. Assumption 2.The sensor frame is aligned with the joint coordinate frame. Assumption 3.The limb is rotated in a single plane (either in sagittal, coronal or axial plane) as a rigid body between the rotation joint and the sensor. Assumption 4.The sensor is attached at the distal end of the limb, distance (L) away from the rotating joint. The angular velocities collected from three orthogonally mounted gyroscopes at time are denoted as , where the superscripts indicate the corresponding axes of the readings. The linear acceleration at time t with respect to the sensor coordinate system is denoted as . These measurements of an inertial device are fed into our new estimator. The gyroscope readings are only used for excluding the gravitational impact from accelerometer readings and converting into . In this paper, the superscripts S and E denote the readings with respect to the sensor frame and the earth frame, respectively. Fig. 1 shows the relative sensor coordinate system when the leg is moved by an angle. Initially, the sensor frame S and the earth frame E are well aligned. When the leg is moved by an angle, only the sensor frame is rotated. Fig. 1Open in figure viewerPowerPoint Relative change of sensor coordinate system when leg is moved (a) Before movement, (b) After movement Furthermore, the quaternion () is required to transform the accelerometer readings in the sensor frame to the earth frame at time t. Hence, the quaternion derivative is calculated as (1)where is the pure quaternion of angular rates. The quaternion is calculated using the quaternion propagation equation: [16, 17]. The initial quaternion is considered as [1, 0, 0, 0]. It is necessary to consider the gravity-compensated linear acceleration with respect to the earth frame to calculate the curvature of the motion trajectory. We use the mean value of the accelerometer readings during the static state as the gravitational acceleration . Owing to static conditions, the measurements are entirely based on gravitational force. According to Assumption 2, the gravity readings from sensor are with respect to the earth coordination system. In other words, . When the arm is moved, the sensor frame and earth frame are different. Hence, the accelerometer readings in the sensor frame at time t are transformed to the earth frame as given in (2) (2)Since the accelerometer readings are the resultant acceleration of both kinematic acceleration and g, the kinematic acceleration is calculated using the relationship . As the limb is stationary at the outset, linear velocity () is zero and at each time stamp it is calculated by integrating kinematic accelerations and the trajectory curvature is calculated using (3) [18] (3)The trajectory curvature is constant and equal to the reciprocal of the radius in a circular motion. Hence, the limb length is calculated as follows: (4)We have used accelerometer readings for calculating curvature instead of deriving angular velocity from gyroscope because the accelerometer measures the linear acceleration directly with respect to the earth frame instead of the local frame. The mean of L is considered as limb length. Our technique is illustrated in Fig. 2. However, with the noisy measurements, the accuracy of curvature calculation is significantly affected [19]. The following section describes the LNT approach to computationally remove noisy measurements from the curvature calculation. Fig. 2Open in figure viewerPowerPoint Proposed algorithm 3 Refinement of estimator using LNT Accelerometer readings become noisy and irregular with the presence of noise [20–22]. Noise can mostly be seen at the beginning of movements due to the instantaneous acceleration required to start the limb movement from a static status, as well in the end portion of the measurements due to the decelerating force on the arm required to cease the motion. On the other hand, the resultant acceleration is affected by white noise [22]. However, accelerometer readings are successfully used for physical activity identification based on thresholds [23]. In this paper, noisy portions of acceleration readings are excluded from curvature calculation for accurately estimating limb length. The exclusion of noisy data as in [19] can be conducted by manual observation of initial and terminal phases for exclusion of the uncertainty regions. This can be time-consuming and requires technical know-how. Therefore, a systematic method is required to exclude noisy data to facilitate the overall implementation in a user-friendly manner. Sample entropy (SampEn) is a technique used for determining the regularity of data in complex systems [24, 25]. SampEn produces more consistent outcomes than other entropy-related techniques [26]. In this paper, SampEn is applied to each segment of curvature with and the window size to determine the LNT. Here, j is the index of curvature segments and J is the total number of segments. The value of sampling entropy is calculated with as follows: (5)where dim is the embedded dimension, r is the tolerance used to determine the regularity of two subsets. The least entropy thresholds ( to ) are determined by applying SampEn as pseudocode Algorithm 1 (see Fig. 3) : Determining the mean length. Fig. 3Open in figure viewerPowerPoint Algorithm 1: Determining the mean length In the pseudocode, the variables: entropy as an array, average adult's limb length and the estimated optimal length are denoted as , and L, respectively. Repetitive section A is aimed at capturing to the global minimum and most likely avoiding locals minimum recursively; on the basis of a condition that L should be varied within of . is the corresponding average limb segment length of healthy adults in [27, 28]. We denoted the start index, end index and middle index of local minimum as S, E, M, respectively. If the above condition is not satisfied in current local minimum, the next local minimum will be considered. In Fig. 4, sub-figures (a–d) depict the determined LNT results for limb segments: hip to ankle, hip to knee, shoulder to wrist and shoulder to elbow, respectively. For each limb segment, there are three sub-figures (I–III) showing the sample entropy, kinematic acceleration and trajectory curvature, respectively. In II plot, the simulated linear accelerations without the presence of noise are shown as a dotted line. According to the protocol, the lower limbs were lifted ∼ – within 20–30 s and the upper limbs were moved ∼ – within 60–80 s. The least entropy thresholds ( and ) were determined by applying Algorithm 1. The examples of the least entropy thresholds for various body segments are shown in Fig. 4. We can see that the segment with the least entropies for hip to ankle, hip to knee, shoulder to wrist and shoulder to elbow are 9–12, 14–16, 17–21 and 45–60, respectively. Fig. 4Open in figure viewerPowerPoint Experiment result: Determined LNT using sampled calculated curvature The reasoning behind considering trajectory curvature to determine the sample entropy, but not linear acceleration or linear velocity, is that the curvature is independent from linear acceleration and linear velocity, even though the curvature can be calculated using them. 4 Real-data experiment Computer simulations and feasibility studies were conducted (also in [19]) to estimate limb length using the underline algorithm without LNT. To validate the optimised algorithm using LNT, experimental evidence was taken from 14 healthy subjects (12 males and two females) without any history of orthopaedic or intramuscular impairments. These subjects participated in the experiment after gaining ethics clearance from Deakin University. BioKin [29] wireless inertial sensors were used in the experiment to collect data. According to the experimental protocol, two sensors were attached to two distinct positions on the upper and lower limbs (see Fig. 5). Subsequently, the corresponding distance from the rotation joint to the sensors was manually measured using an anthropometer. Fig. 5Open in figure viewerPowerPoint Experimental setup (a) and (b) Lifting the arm: inertial sensors were attached close to elbow and wrist on the left arm, (c) and (d) Lifting the leg: inertial sensors were attached close to knee and ankle on the right leg (a) Arm down, (b) Lifting arm, (c) Leg down, (d) Lifting leg Even though this study can be conducted wearing one or more sensors in a sensor location of limbs, we have used sensors at the minimum number (only one sensor) in order to render more convenience to participants during the experiments. Further single sensor usage was very practical since it simplifies the experiments and reduces the administrative cost. The placement of sensors on limbs was subject to an attentive consideration, since the position highly affects the accuracy of measurements. According to Mannini et al. [30], common location choices are hip, thigh, upper arm, wrist and ankle for wearable sensor systems with similar applications. In this paper, since we were measuring limb lengths, we always wore the sensors at the distal end of the limbs such as elbow, wrist, knee and ankle under the clothing. Since the sensor was thin and light weighted, the subjects had not any discomfort. The other main concern on selecting wearing position was the even nature of the limb. A little higher to joints (wrist, knee, ankle or elbow), the limb has moderately planar surface, hence the sensors can be easily placed on those locations with least sensor misalignment error [31, 32]. Therefore, the sensors were worn little higher to above joint bones with a planar surface. During the experiment, as for limb movement, each subject was asked to perform limb extension exercises as indicated in Fig. 5. Here, we have investigated the feasibility to accurately determine limb lengths as in four cases such as lengths from shoulder to elbow, shoulder to wrist, hip to knee and hip to ankle. All the participants of the experiment were asked to stretch the upper limb as much as possible and then slowly move in the sagittal plane to their front as in Fig. 5 b from the initial static position (see Fig. 5 a). Then, the subject was asked to slowly lift the lower limb wearing two sensors from the initial position, depicted in Fig. 5 c, to their front in the sagittal plane, shown as Fig. 5 d. 5 Results Four limb sections: shoulder to elbow, shoulder to wrist, hip to knee and hip to ankle were considered for validating the proposed estimator. We refer to these four segments as target limbs in the remaining discussion. The corresponding time duration for least entropy matching for target limb was determined. The limb lengths were estimated with the measurements bounded by the thresholds ( and ) determined using LNT technique. The estimated lengths of the target limbs were then compared with the corresponding measured limb lengths. As evident in Fig. 6, a higher degree of correlation can be observed between the measured and the calculated limb lengths. Fig. 6Open in figure viewerPowerPoint Mean between measured length and calculated length: the x-axis indicates the target limbs such as 1 – shoulder to wrist, 2 – shoulder to elbow, 3 – hip to ankle and 4 – hip to knee The information in Fig. 6 can be represented in terms of limb lengths as shown in Table 1. In this analysis, two types of errors were calculated for each target limb to illustrate the performance of the proposed estimator. First, the root mean squared error (RMSE) between the estimated and actual length of a target limb was calculated. Second, in order to compensate for the varying lengths of the target limbs impacting the error, normalised error percentage was calculated as follows: (6) Table 1. RMSEs, error percentages of the estimates with respect to the actual measurements Limb element RMSE Error percentage, % Mean of measured limb length, m Mean of estimated limb length, m shoulder to elbow 0.0472 9.82 0.2993 0.2796 shoulder to wrist 0.0381 5.34 0.5169 0.5236 hip to knee 0.0449 7.04 0.4610 0.4382 hip to ankle 0.0517 5.84 0.8259 0.8043 The most common method for clinically determining limb length inequalities is radiographic analysis. Furthermore, these two-dimensional (2D) or 3D radiographic measurements are used to determine both functional and anatomical LLDs. Hence, we have compared our estimator with the measured anatomical and functional lower limb lengths using radiographic images in the study [33]. The comparison between the estimator and other measuring methods are as listed in Table 2. Table 2. Comparison of the estimator with direct measurements and radiographic analysis Method Mean, m Maximum, m Minimum, m Standard deviation, m direct measurements – anthropometer based 0.788 0.815 0.75 0.022 2D radiographic measurements – anatomical LLD 0.787 0.884 0.647 0.055 2D radiographic measurements – functional LLD 0.789 0.887 0.651 0.055 3D radiographic measurements – anatomical LLD 0.783 0.879 0.649 0.054 3D radiographic measurements – functional LLD 0.789 0.883 0.64 0.054 limb length estimator 0.804 0.889 0.703 0.047 Furthermore, we have compared our novel LNT-based estimator against other three commonly using noise-reducing mechanisms to validate the performance of the proposed estimator with current practises. The first comparing mechanism is a moving average filter, which averages a number of points from the estimated limb lengths calculated using curvature to produce the noise-reduced limb lengths [32, 34]. This filter is widely used to reduce motion artefacts and the performance of the filter is well proven for limited artefact range similar to our application [34]. In this paper, the window size of the moving average filter was ten points. Second, a first-order butter worth low-pass filter [30, 35, 36] was designed to compare the performance of our estimator. For that, the cut-off frequency was 0.5 Hz. Third, we compared our LNT model with another spectral entropy mechanism which is an approximate entropy. The approximate entropy was introduced before the sample entropy and its lack of regularity measures compared with the sample entropy [35, 37, 38]. We have applied approximate entropy similar to the sample entropy with same window size and variance. Table 3 shows the RMSE of each noise filtering mechanisms compared with direct measurements. Table 3. Performance comparison of the estimator with other noise filtering mechanisms Noise filtering mechanisms Shoulder to elbow RMSE, m shoulder to wrist Hip to knee Hip to ankle moving average filter 0.3519 0.2463 0.2455 0.4182 butter worth low-pass filter 0.2273 0.1444 0.2075 0.2535 approximate entropy 0.1401 0.1189 0.1353 0.1517 sample entropy 0.0472 0.0381 0.0449 0.0517 6 Discussion From Table 1, shoulder to wrist was the most accurately estimated with RMSE of 0.0381 m and the lowest percentage error was observed in shoulder to wrist target limb. However, shoulder to elbow limb had the highest average percentage error (9.82%), though the percentage error for the other limb segments were below 7.04%. This was mainly due to the comparatively shorter limb lengths considered. For example, the average-measured length from shoulder to elbow was 0.2993 m and the estimated length was 0.2796 m. Although the absolute error was only 0.0197 m, the percentage error reached 9.82%. In comparison, the absolute error for hip to ankle was 0.0216 m because the limb length was 0.8259 m, though the percentage error was only 5.84%. Considering the RMSEs for each target limbs, our proposed approach with LNT gave significantly accurate results with a low RMSE (∼0.04 m). However, a lesser percentage error (<7.1%) could be obtained for the longer limb components such as hip to knee, hip to ankle and shoulder to wrist. Furthermore, the mean length and standard deviation of each target limb segments were calculated for the second analysis as shown in Fig. 6. The box plot shown in Fig. 7 compares the average-measured limb length to the estimated limb length of all subjects. According to the statistical distribution shown in Fig. 7, the distributions of measured and estimated limb lengths were similar for each target limb. Furthermore, the estimated and measured values were quite close for the mean limb length. According to Fig. 7 and Table 1, less deviation was observed for the limb from shoulder to wrist. The difference in mean values of measured and estimated shoulder to wrist length was 6.7 mm. The differences in the mean values of measured and estimated limb lengths for shoulder to elbow, hip to knee and hip to ankle were 19.7, 22.8 and 21.6 mm, respectively. From these results, we could see that the automated approach could achieve very close results to the manual method, especially for the longer limb lengths such as shoulder to wrist and hip to ankle. Fig. 7Open in figure viewerPowerPoint Comparison between measured length and calculated length for the targeted limbs (a)–(d) (a) Shoulder joint to elbow, (b) Shoulder joint to wrist, (c) Hip joint to knee, (d) Hip joint to ankle Table 2 compares limb length estimator with alternative measuring methods such as anthropometer-based direct measurements and 2D/3D radiographic measurements for lower limbs. The difference in mean length between the proposed estimator and radiographic analysis was ∼1.7 cm. Furthermore, the standard deviation of the proposed estimator and radiographic analysis was ∼8 cm. Hence, the estimator had significant similarity in accuracy for lower limbs compared with radiographic analysis, the most widely used clinical LLD determination method. Table 3 illustrates a comparison of our proposed LNT model with commonly using noise-reducing mechanism. The LNT model was basically compared with two filtering mechanisms such as moving average filter and butter worth low-pass filter. Then, LNT model was further validated with approximate entropy which is also a spectral entropy technique. According to Table 3, the highest RMSE in each target limbs could be observed in moving average filter, whereas the second highest value was presented in butter worth low-pass filter. Their average RMSEs are around 0.3155 and 0.2082 m, respectively. The second lowest error is stated in approximate entropy-based mechanism which shows 0.1365 m as the average RMSE for each target limbs. The least error was presented in our proposed LNT model which has 0.0454 m of average RMSE. 7 Conclusion In this paper, we introduced a novel method to estimate limb lengths using measurements from inertial sensors. A curvature-based approach is used in the algorithm, which has not yet been presented in the open literature. Additionally, we applied LNT with noisy measurements to systematically determine the curvature to provide an optimised result. The proposed algorithm along with LNT method was evaluated by comparing the estimated limb length with that measured manually and radiographically. Furthermore, the performance of LNT model was validated with available noise filtering mechanisms. The low RMSEs and error percentages confirmed the excellent performance of the approach. 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