Application of functional methods for kinematical analysis in biomechanics

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1 Application of functional methods for kinematical analysis in biomechanics Stefano Pastorelli 1, Laura Gastaldi 1, Giulia Lisco 1, Luca Mastrogiacomo 2, Maurizio Galetto 2 1 Dep. of Mechanical and Aerospace Engineering, Politecnico di Torino, Italy stefano.pastorelli@polito.it, laura.gastaldi@polito.it, giulia.lisco@polito.it 2 Dep. of Management and Production Engineering, Politecnico di Torino, Italy luca.mastrogiacomo@polito.it, maurizio.galetto@polito.it Keywords: functional analysis, joint centers, motion capture. SUMMARY. Applied biomechanics is relevant in research fields as gait analysis, where the joint kinematics has become useful for several purposes. Indeed, an accurate assessment of human joint parameters like axes and centers of rotation, is nowadays a critical issue for the quantitative human movement analysis, due to the direct influence on motion patterns. Several methods allow to determine these quantities, based on a different analytical approach. This work aims at proposing and evaluating the feasibility of functional methods for the assessment of knee joint parameters, by means of a polycentric joint model. As well, it underlines how functional approaches turn to be useful and more suitable than the concurrent predictive ones, which may be then replaced. 1 INTRODUCTION Applied biomechanics has become instrumental for human movement analysis, in order to quantify the human body motion and to assess joint kinematics, as well dynamics. Human joint kinematics is hence useful in research areas like gait analysis, prosthetic and rehabilitation, ergonomics, body biodynamics and ageing condition evaluation, due to the proper role in each one, as well to the direct influence on. Indeed, the assessment of joint parameters, like axes or centers of rotation highly condition human motion patterns analysis, due to the direct effects on the lower limbs kinematics and dynamics computation. All these aspects justify the necessity to accurately assess joint parameters. To this end, two main requirements have to be considered: the motion data acquisition and the human body modeling. The first one is made possible throughout the already spread optoelectronic systems used for motion capture, based on the stereophotogrammetric technique and the concurrent use of video-cameras and markers; the second requirement is instead based on the easy assumption of human body made up of rigid segments, connected by mechanical joints, properly identified with adequate degrees of freedom (DoFs). Joints models are hence important to the presented purpose, since they allow to assess parameters and to acquire data not directly measurable on subject. The assessment of joint kinematics, particularly during gait analysis sessions, is often based on the geometrical, as well on motor assumption concerning the point (or the joint) about which two segments are in relative motion. The determination of this point, within its parameters which are the center (CoR) and the axis of rotation (AoR) is often difficult to measure in vivo, representing a crucial issue during movement analysis, especially for people affecting by some pathologies or presenting physical deformities, that directly compromise motion tasks, as well kinematics outcomes [1,2]. 1

2 Joint parameters can be basically assessed according to different main approaches, namely the predictive and the functional method. The former, which is currently adopted by the clinical laboratories for measurements and human movement analysis, is based on regression equations [3,4] between externally palpable bone landmarks, anthropometric dimensions and the joint itself. Thus, this approach requires to adopt specific protocols for markers positioning on anatomical landmarks [5,6], for then using motion data as input for the equations. This approach is simple and immediate, despite it may lack of accuracy. Indeed, the necessity of a proper markers positioning is one of the drawback, due to the not often easily anatomical landmarks identification. As well, this approach is not so suitable for people who present bone-deformities or for nonstandard cases, e.g. in presence of a disease, providing several sources of errors [2,7]. Functional methods have been then introduced to overcome some limitations characterizing the predictive approach. Firstly, they do not refer to the empirical relations and are just based on the relative motion between a couple of bone segments, respect to the identified joint, not imperatively requiring markers positioning on anatomical landmarks. Indeed, in order to define relative segments position and orientation, a group of at least three markers equipped on each rigid body is the only requirement for the further computation. These approaches are hence potentially suitable in cases of motor deficit, becoming adaptable for several people categories. Generally, the assessment of the interested joint parameters is made possible through marker motion data acquisition and technical frame identification to which refer the parameters themselves. Among the second approach, several strategies have been presented in literature, providing another sub-classification in fitting approaches (FA) and transformation techniques (TT), just diversified for the analytical based formalism. The first FA category encompasses variants of the sphere fitting methods, where the center and the radius are optimized to fit the markers position trajectories [8]: each marker can rotates around the same joint axes or center with a separate arc [9,10] without a rigid body assumption. On the contrary, TT consider each body as a rigid one, due to the possibility to assess joint parameters with rigid body transformations, like rotation matrices and translation vectors [11-12]. This enables to define local coordinates system and appropriates transformations into a common reference frame, assuming the joint CoR as a fixed point. Several comparisons in vivo between predictive and functional methods have been discussed, showing how both approaches are influenced from the soft tissue artifact (STA), due to the tissue and the relative motion between the skin and the markers over the body, resulting in errors during the skeletal motion determination [7]. Thus, some authors have proposed a method validation with computer simulation or with some standard mechanical analogs [13,14]. Generally, functional methods provide more accurate results according to what is reported in literature [8,16]. They are nevertheless based on several combinations of kinematical and geometrical constraints, as well optimization techniques, providing good results just in certain conditions [16,17], e.g. with a proper range of motion (RoM). Among human joints, the knee one presents a complex pattern of motion. As a consequence none of the proposed functional methods has ever assessed the real knee joint rototranslation kinematics. A first extension could consider a simplified knee joint model, here presented, according to a polycentric mechanism. This paper attempts to apply three main functional methods on this simplified knee joint model, in order to discuss both the strengths and the limits of application. The main purpose may be in the future that one of promoting functional methods during gait analysis sessions, in order to assess joints parameters and what is correlated during gait trials. 2

3 2 THE CONCEPT OF A FUNCTIONAL APPROACH FOR THE KNEE JOINT MODEL Actually, clinical gait analysis is one of the prominent field of research within the scope of human movement assessment, during which a standard protocol for markers positioning is followed [3,5] in order to compute gait outcomes, like spatio-temporal parameters and joint kinematical curves. Current laboratories of human movement analysis use nowadays a predictive approach for the assessment of joint parameters like AoR and CoR. Concerning the knee joint, this standard practice simply assumes the main knee flexion/extension axis as the line passing through a couple of anatomical markers, placed on the lateral femural epicondyles. The definition of its position with respect to the local reference system of the thigh is carried out only once, during the static acquisition. Thus, just this preliminary information is stored for the entire further movements. As well, the center of knee joint is defined as the midpoint of the segment connecting the two femural epicondyles. Is clear how this represents a too simplified knee joint parameters assessment, owing to the modeling of the knee just as a single fixed axis hinge. A functional approach may overcome this assumption, providing a reconstruction of the knee joint kinematics, using groups of markers placed on both the thigh and the shank in relative motion respect to the knee itself, according to a personalized choice for the positioning protocol. This method hence offers a valid alternative for the knee joint parameters assessment and lays the groundwork for a further description of the real rototranslation kinematics. 2.1 The polycentric system Understanding the real knee joint kinematics is essential to comprehend how it cannot be modeled as a single hinge, with just a fixed axis (or center) of rotation. A different joint modeling is hence required, in order to be able to identify a mobile axis (or center) of rotation, in addition to the fixed one. To this end, a polycentric hinge model was considered. In particular, a mechanical analog has been realized by means of a dummy leg and of a pair of orthopedics polycentric hinges, externally fixed and emulating a human knee joint tutor, as shown in figure 1A and 1B. Figure 1A. Polycentric hinge Figure 1B. Dummy leg and polycentric hinges In particular the polycentric hinge is composed of three roads, connected with hinges plate. Each hinge is then rigidly coupled to a cogwheel, in order to obtain a gear in correspondence to the center of the second road. This mechanism constrains the entire movement of the three roads, allowing the third element to rotate of 2θ angle with respect to the horizontal direction, when the 3

4 second has rotated of θ angle. In particular, the first road within its gear is fixed on the thigh segment of the dummy leg, while the third one on the shank. The mechanical analog allows flexion/extension movement in the sagittal plane, from 0 (dummy extended) up to 120, not considering other movements right now (e.g internal/external rotations and abd/adduction motion). According to this model, reported in figure 2, the relative axis of rotation of the shank with respect to the thigh (geometric AoR) can moves on a circle centered in the thigh gear component, covering an α angle of flexion/extension, according to the RoM. thigh thigh gear plate shank gear faor flex/ext angle geometric AoR maor shank 2.2 Used functional algorithms Figure 2. Polycentric hinge section As explained above, a functional approach can be useful for joint parameters assessment, trying to overcome some limitations currently presented with the standard predictive techniques, prone to errors arising from anatomical landmarks identification and anthropometric regression equations [3,5]. Among several alternatives to the standard methods, three main functional methods [9,12,17] are here discussed, implemented and tested on the dummy mechanical analog, although the presence of a different mechanism respect to the joint with which they were originally processed. A functional FA approach and two different TT of two types, one and double sided, are reviewed. The former, the Halvorsen method (H-method), from the author s name [9], is based on the motion data acquired from a markers set, from which is possible to assess the AoR by minimizing two objective functions. The main features of this algorithm are here reported: a versatility of application to different joint types (single hinge or ball-and-socket one); the absence of a rigid body condition, even if each marker can rotate around the same AoR (or CoR); the use of two objective functions that are minimized to assess the directional cosines of the AoR and its localization respectively, simply indicating a point on it. In particular, the AoR can be assessed as the vector which best fits the rotational axis perpendicular to the plane containing several markers displacement vectors, evaluated between two frames accurately far. The second method, which can be indicated as the Spoor-Veldpaus (SV-method), from the authors names [12] is a one side transformation technique, assuming one segment, therefore the CoR in static condition. This algorithm also takes into account the markers motion data, allowing 4

5 the joint parameters assessment, according to a double process. In particular, this algorithm considers: a rigid body condition, assuming a constant distance of each marker on the relative segment; a rigid motion description with transformations of rototranslation in the space, by means of rotation matrices and translation vectors, which are unknown; the use of a singular value decomposition technique (SVD) in order to solve a problem of eigenvalues and eigenvectors, to approximate a pseudo-inverse matrix; a screw-helical-motion assumption, in order to describe the rototranslation motion as the helical one, considering it between two instant of time (or frames), for the assessment of both the direction and the location of the AoR in the space. Finally, the SARA method [17], according to the acronym of symmetrical axis of rotation approach, is instead a transformation technique type two sided, because it does not require a stationary CoR, allowing to consider both segments in relative motion. The main features in this case are the following: the absence of stationary assumption of one segment (both can move); the lack of the assumption related to the type of joint and the relative motion of the considered segments; the definition of local reference systems for each segment; the assessment of both orientation matrices and translation vectors, in order to compute the transformations between a local reference system into the common global one; the use of a linear least square approach to solve the optimization problem. With this algorithm, two arbitrary points on the joint axis for both local coordinates systems are assessed. Thus, with the hypothesis that each one must be constant relative to both segments, an objective function can be introduced and minimized with a linear least square method. 3 JOINT PARAMETERS ASSESSMENT Once the algorithms have been implemented in Matlab, according to the relative analytical approaches researched in literature [9,12,17], an experimental session was carried out in order to validate the functional methods with the proposed polycentric knee joint model. 3.1 Experimental session A relative swing motion of the shank respect to the thigh segment of the dummy leg was chosen in order to assess the main flexion/extension joint kinematics, as well the interested parameters. All the proposed algorithms required motion data as input and, to this end, a stereophotogrammetric motion capture technique was considered. Indeed, nowadays, among several systems used to achieve the motion capture, the optoelectronic ones are usually chosen. They required both the use of a video-cameras based system, operating in the visible range or in the near infrared, and a set of markers placed on the body segment, according to the standard or to different protocols. For this study, a Vicon 512 motion capture system (Oxford Metrics, Oxford, UK) was used within its 6 infrared cameras, working at 100Hz. Concerning instead the concurrent use of markers, 5

6 reflective ones with 14mm of diameter were considered. A variable number of markers between 10 up to 12 was selected, according to this positioning protocol: 3 markers were placed on the thigh segment, just to verify the static condition; 4 ones were used for the polycentric kinematics identification. In particular, two couple of markers were properly screwed on both lateral hinges termini, according to what is represented in figure 1B, in order to identify both the fixed axis of rotation (faor) and the mobile one (maor). Finally a variable number of 3 up to 5 markers were arbitrary placed on the shank dummy segment, trying to cover the entire body volume, positioning them e.g. on the lateral, medial, frontal and medial-frontal plane of the shank. Several flexion/extension sessions were executed, considering the RoM of about (0-90 ) at 0.3Hz. Markers spatio-temporal trajectories were acquired with respect to the global laboratory reference frame, according to the Vicon system and then a post-processing data analysis was performed for the designated goals. 3.2 Algorithms validation and comparison Two different movement scenarios were considered, in order to validate and compare the aforementioned algorithms with a polycentric knee joint model. In particular, proper intervals were selected in order to cover two RoMs of about 55 (ROM I) and 20 (ROM II) for each flexion, or extension. Concerning the frames selection for the first two methods (H and SV), authors have not considered two subsequent time instants, due to the possibility to compute better results between frames far apart [9]. Thus, a different frames coupling was chosen: starting from the total number n of frames for each swing session, observation sets from (1 to n/2 +1), (2 to n/2+2) up to (n/2 to n) were selected in order to generate two virtual merged initial and final frames of about the first n/2 and the second one respectively. The joint parameters, like the direction and the location of the AoRs have been hence assessed by means of H and SV algorithms, according to this proper frames coupling and by varying the number of markers on the shank from at least 3 up to 5. The third SARA algorithm, instead, just takes into account the minimum number of 3 markers on both segments, in order to define the local reference frames. Moreover, considering just the segments pose variation during the entire movement and not markers trajectories displacements, a standard frames coupling was considered. To better clarify how the algorithms have been used and compared, some details are now described, also considering the presented results. In particular, once both the faor and the maor have been identified as the lines passing through each couple of markers on the relative hinges termini, the direction of the estimated AoR was computed with each algorithm. For what concerning instead its location at the knee joint model level, a point of intersection between the assessed AoR and a plane containing the fixed joint center (fcor) and perpendicular to the axis itself, was assessed. Related instead to the polycentric joint centers of rotation, while the mobile (mcor) and the geometric one (gcor) were always assessed as the intersection points between the considered sagittal plane at the knee level and the relative maor and the geometric axis of rotation, the fixed one (fcor) was identified as the midpoint between the first couple of markers, placed on the hinge termini, relative to the thigh segment road. The dummy segments limit positions, within the described geometric, mobile and fixed entities are shown in figure 3A-3B. According to these figures, the sagittal xz-plane configuration was represented related both to the two RoMs and to the common use of 3 markers placed on the shank for the H, SV and SARA algorithms. 6

7 Figure 3. Limits dummy segments position on the sagittal plane with the assessed mean CoR respect to the gcor for method H (x), SV (o) and SARA (+), according to the RoM I of 55 (A) and RoM II of 20 (B). 3.3 Results The feasibility of the presented functional methods, applied to the dummy knee joint model, was discussed through the evaluation of misalignment (MEs) and distance (DEs) errors for the AoR direction and the CoR location respectively. In particular for each swing session i and the considered frames j, errors are computed as follows: (1) ( ) (2) Concerning the distance errors, these were evaluated considering the mean value of the distribution of the processed knee joint centers. In particular, is possible to derive a distance curve trend, for each swing session. Figures 4 shows this distance trend, considering e.g. one of the swing session for ROM I of the H algorithm, varying also the number of markers (M) on the shank. According to the location of the assessed AoR, at the knee level, along the sagittal direction, is possible to deduce from this trend where the point is located respect to the geometric center of rotation trajectory. Figure 4. Example of distance curve trend between the assessed CoR and the polycentric gcor. 7

8 The accuracy and precision of each analyzed algorithm is evaluated by means of the (mean ± standard deviation) computation, obtained from the MEs and DEs, according to the formulas (1) and (2). Thus, the mean misalignment and distance errors have been obtained from flexion/extension sessions analyzed with each algorithm, for both RoMs and are reported in table 1 and 2 respectively. Related histograms distributions of the averaged MEs and DEs are reported in figures 5A-5B and 6A-6B respectively, for each analyzed motion. Table 1. Averaged misalignment errors for the direction of the assessed AoR. ME [ ] (Accuracy ± Precision) RoM RoM I=55 RoM II=20 Methods H SV SARA H SV SARA N. markers ± ± ± ± ± ± ± ±0.10 x 1.17± ±0.41 x ± ±0.11 x 1.15± ±0.33 x x: for SARA algorithm just a subset of 3 markers was considered. Figure 5. Histogram distribution for the averaged ME [ ]. (A) RoM I and (B) RoM II. Table 2. Averaged distance errors for the location of the assessed AoR. DE [mm] (Accuracy ± Precision) RoM RoM I=55 RoM II=20 Methods H SV SARA H SV SARA N. markers ± ± ± ± ± ± ± ±1.27 x 4.10± ±1.70 x ± ±1.37 x 5.61± ±1.71 x x: for SARA algorithm just a subset of 3 markers was considered. 8

9 Figure 6. Histogram distribution for the averaged DE [mm]. (A) RoM I and (B) RoM II. 4 CONCLUSIONS A functional approach for knee joint parameters assessment has been investigated. An experimental comparison of three main functional algorithms [9,12,17] was performed and validated on a different knee joint model, due to its complex rototranslation motion. A polycentric mechanism was considered, in order to better describe the knee joint kinematics. As a result, just an averaged static AoR of the geometrical one, shown in figure 2, was assessed as the knee flexion/extension axis, considering the different joint model respect to those usually proposed by the reviewed algorithms. As well, none comparison with some references results already presented in literature [14,17] was provided, due to the different knee joint model in use. The obtained results show nevertheless a potential feasibility of application for the declared purposes. Indeed, they remain in the same interval of conventional errors concerning the joint parameters assessment, showing a similar accuracy and precision order. According to what is shown in figure 5 and figure 6, while the MEs decrease with a larger RoM, in all the three presented algorithms, the DEs increase, with the exception of the third method in use [17]. Concerning the misalignment error, this outcome was expected due to the quite constant direction of the geometric AoR, however taking into account some variations due to the realization of the mechanical analog itself and to the polycentric hinges fixation on the dummy leg, involving some accuracy deviations. For the DEs, instead, is clear that with a larger RoM there should be a greater displacement from the geometric AoR, hence a greater error, as it can be seen in figure 3A- 3B. However, the third method provides a better CoR assessment with a larger RoM. Authors also analyzed the influence of the number of markers placed on the shank, for the first two functional methods [9,12]. As it can be seen from the histograms, they have a certain influence on the DEs that increase with more markers, while they do not influence the MEs, despite some slight improvements for the SV case. The study provides a preliminary feasibility assessment of joint parameters with a functional approach, and seems promising for further developments, as well for innovative algorithms joint model-specific, considering the proper polycentric kinematics and also other complex models. 9

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