Optimization and muscle synergy approaches for studying muscle redundancy during walking
Abstract
The human body is an over-actuated multibody system, as each joint degree of freedom can be controlled by more than one muscle. Usually, optimization techniques are used to solve the muscle force sharing problem, that is, finding out how the resultant joint torque is shared among the muscles spanning that joint. The reduction of muscle force redundancy can be achieved in several ways. Although the strategy followed by the central nervous system (CNS) to activate the muscles is not completely clear, one of the most used hypotheses to overcome this redundancy is to consider that the CNS minimizes a physiological variable. In the first study presented in this thesis, the solution to the muscle force sharing problem was approached by minimizing the sum of squared normalized muscle forces. For this purpose, a weighted cost function was designed to evaluate which muscles were more penalized in a subject with anterior cruciate ligament (ACL) deficiency during walking. The results showed that the cost function that best fitted normalized electromyography signals with muscle activations did not treat all muscles equally. Another way to reduce muscle redundancy is using the idea that muscles are activated synergistically when performing a task. In the second study, a muscle synergy analysis was carried out to compare the muscle activation information at two levels: onset-offset activation patterns and muscle synergy components of a sample of 18 ACL-deficient subjects and a sample of 10 healthy subjects. Some differences were found at both levels, what suggests that ACL-deficient subjects alter the muscle activations of their injured leg to stabilize the joint. Finally, in the third study, muscle synergies were used in a two-step optimization method to predict physiologically consistent muscle and knee contact forces, while calibrating muscle parameters. In the outer level, muscle parameters were calibrated; while, in the inner level, muscle activations were calculated using the current muscle parameters. The results showed that a set of muscle parameters were able to reproduce knee contact forces with high accuracy when knee contact forces were used during the calibration process. This study shows the main differences when these forces are available for calibrating muscle parameters and when they are not. The most important differences in the muscle parameter calibration affected lateral muscles. Therefore, this fact suggests that trials where lateral muscles play a more important role should be used to obtain a better calibration when no contact forces are available.
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UNIVERSITAT POLITÈCNICA DE CATALUNYA Biomedical Engineering Doctoral Programme OPTIMIZATION AND MUSCLE SYNERGY APPROACHES FOR STUDYING MUSCLE REDUNDANCY DURING WALKING by Gil Serrancolí Masferrer A thesis submitted for the degree of Doctor for the Universitat Politècnica de Catalunya Supervised by Dr. Josep Maria Font Llagunes1 and Dr. Benjamin Jon Fregly2 1Department of Mechanical Engineering (Universitat Politècnica de Catalunya) 2Department of Mechanical and Aerospace Engineering (University of Florida) Barcelona, February 2015
Agraïments Voldria agrair sincerament a totes aquelles persones que m’han ajudat a fer possible aquesta tesi. A continuació voldria fer una especial menció d’algunes d’elles: En Josep Maria Font, co-supervisor de la tesi des de l’inici, que sempre m’ha encoratjat en tirar endavant la tesi. A part del seu gran suport acadèmic, la seva implicació i compromís al llarg d’aquests quatre anys són d’apreciar. Sempre li agrairé que em donés l’oportunitat de fer l’estada doctoral de sis mesos a Florida. En Benjamin Jon Fregly, co-supervisor de la tesi i cap del Computational Biomechanics Lab de la University of Florida, pel seu excel·lent rigorós treball i dedicació supervisant la tesi. Des del primer dia que vaig arribar al seu departament, m’ha guiat i ajudat, donant-me excel·lents idees i consells d’una manera amable i alhora admirable. Els companys del Computational Biomechanics Lab durant l’estada: Joanathan Walter, Ilan Eskinazi, Allison Kinney, Andrew Meyer, Jennifer Jackson, Laura Li i Romain Leberre. Sempre em van ajudar i em van donar el seu suport durant l’estada a Florida, dins i fora de la universitat. L’Ana Barjau, co-supervisora de la tesi als inicis del doctorat, pel seu suport moral i els seus consells. M’agradaria donar les gràcies tant a l’Ana com a en Joaquim Agulló per haver confiat sempre en mi i donar-me l’oportunitat de donar classes de Mecànica a l’Escola Tècnica Superior d’Enginyeria Industrial de Barcelona (ETSEIB). En Joan Carles Monllau, cirurgià traumatòleg i cap de la Unitat de cirurgia artroscòpia de genoll a l’Institut Català de Traumatologia i Medicina de l’Esport (ICATME), que des que li vaig proposar, va acceptar col·laborar en la tesi i el seu equip va proporcionar els pacients de ruptura del lligament creuat anterior. Espero que aquesta col·laboració pugui continuar en futurs estudis.
L’Amaia Ilzarbe i en Javier Romero, que em van ajudar en fer les mesures biomecàniques pel segon estudi, i a tots els 28 subjectes voluntaris (família, amics i pacients de l’ICATME) que van col·laborar desinteressadament en l’estudi. Els cirurgians Lluís Tresserra, Emilio González i Julio González, i els ortodoncistes Arturo i Marcos Costa, qui indirectament em varen introduir en el fascinant món de l’enginyeria biomèdica des que era petit. La seva gran capacitat de treball, precisió i excel·lent humanitat em van fascinar. La meva mare, que m’ha donat sempre el seu suport i consells per no abandonar, i la resta de la família pel seu incondicional suport moral. I per últim, però no per això menys important, l’Esther, pel seu afecte, per les seves correccions i consells a l’hora d’escriure la tesi i per la seva paciència durant totes les nits, caps de setmana i vacances d’estiu i de Nadal que he hagut d’estar treballant en la tesi.
Acknowledgements There have been several people who paved me the way to make this thesis a reality. I would like to take the opportunity to thank them sincerely. Josep Maria Font Llagunes, who supervised my PhD thesis from the beginning. He always encouraged me to push this thesis forward. Apart from being grateful for his support in an academic level, I really appreciate his personal involvement through this experience. He also gave me the opportunity to do the 6-months PhD stay in Florida, which I will always be thankful for. Benjamin Jon Fregly, head of the Computational Biomechanics Lab at the University of Florida, for his excellent rigorous work supervising my PhD thesis. From the first day he welcomed me to his lab, he has always been providing me with excellent ideas and feedback in a kindly and friendly way that truly motivated me to go on and further. Jonathan Walter, Ilan Eskinazi, Allison Kinney, Andrew Meyer, Jennifer Jackson, Laura Li and Romain Leberre, who helped and gave me their support during my stay in Florida within and outside the Computational Biomechanics Lab. Ana Barjau, who supervised the thesis at the beginning of the PhD, for his moral support and advice. I would like to thank Ana and Joaquim Agulló for relying on me and giving me the opportunity to teach Mechanics lessons at the Barcelona School of Industrial Engineering. Joan Carles Monllau, head of the Knee Surgery Unit at the Catalan Institute of Trauma and Sport Medicine (ICATME), who accepted to collaborate in my thesis and redirected ACLdeficient subjects for my study. I hope that we can continue to collaborate in further studies. Amaia Ilzarbe and Javier Romero, who helped me in the biomechanical measurements for the second study, and all 28 subjects (family, friends and patients from ICATME) who volunteered to participate in the study.
Surgeons Lluís Tresserra, Emilio González and Julio González, and orthodontists Arturo and Marcos Costa, who indirectly introduced me in the fascinating world of biomedical engineering since I was a child. Their hard work, precision and excellent humanity really captivated me. My mother, who gave me all her support and advised me not to abandon, and the rest of my family for their unconditional moral support. And the last but not the least, Esther, for her love, her help with writing and communication skills, and her patience during all my nights, weekends, summer and Christmas holidays spent working on the thesis.
Contents Abstract ................................................................................................................................................ v Resum ................................................................................................................................................. vii List of Tables ....................................................................................................................................... ix List of Figures ..................................................................................................................................... xi List of Symbols .................................................................................................................................. xiii List of Abbreviations ....................................................................................................................... xvii 1 Introduction .................................................................................................................................... 1 2 A weighted cost function to deal with the muscle force sharing problem in injured subjects: a single case study ........................................................................................................................... 5 2.1 Background ............................................................................................................................... 5 2.2 Methods .................................................................................................................................... 7 2.2.1 Modeling of the human gait dynamics ................................................................................ 7 2.2.2 Multi-body formulation ....................................................................................................... 8 2.2.3 Optimization for muscle force estimation ........................................................................... 9 2.3 Results and discussion ............................................................................................................ 13 2.3.1 Experimental methods ....................................................................................................... 13 2.3.2 Kinematic and dynamic data ............................................................................................. 14 2.3.3 Optimization results .......................................................................................................... 16 2.3.4 Muscle forces and activations ........................................................................................... 18 2.3.5 Analysis of the cost functions............................................................................................ 20 2.4 Conclusion .............................................................................................................................. 22
ii Optimization and muscle synergy approaches for studying muscle redundancy during walking 3 Analysis of the activation - deactivation patterns and muscle synergies in ACL-deficient subjects during gait ...................................................................................................................... 25 3.1. Background .............................................................................................................................. 25 3.2. Methods .................................................................................................................................... 27 3.2.1. Subjects ............................................................................................................................. 27 3.2.2. Experimental setup ............................................................................................................ 27 3.2.3. Data Analysis .................................................................................................................... 29 3.3 Results .................................................................................................................................... 30 3.3.1 Activation-deactivation pattern ......................................................................................... 30 3.3.2 Analysis of dimensionality ................................................................................................. 31 3.3.3 Variability intra-groups ..................................................................................................... 34 3.3.4 Variability inter-groups ..................................................................................................... 35 3.4 Discussion ................................................................................................................................. 39 4 Two-step optimization problem formulation for predicting knee muscle and contact forces during gait ..................................................................................................................................... 43 4.1. Background .............................................................................................................................. 43 4.2. Methods .................................................................................................................................... 45 4.2.1 Experimental data .............................................................................................................. 45 4.2.2 Muscle synergy analysis .................................................................................................... 46 4.2.3 Musculoskeletal model analyses ....................................................................................... 46 4.2.4 Optimization problem formulations .................................................................................. 48 4.2.5 Data analysis ..................................................................................................................... 50 4.3. Results ...................................................................................................................................... 51 4.3.1 Muscle parameters ............................................................................................................. 51 4.3.2 Knee contact forces ........................................................................................................... 52 4.3.3 Leg muscle forces .............................................................................................................. 54 4.3.4 Other relevant quantities ................................................................................................... 55 4.4. Discussion ................................................................................................................................ 58
List of Tables Table 2.1. Results using the 4 combinations with the lowest CFA values and the equally weighted cost function (associated with healthy gait). ....................................................................................... 17 Table 3.1. Correlation of the mean NC curves and SV values among modules for all three groups . 35 Table 3.2. Correlation of the mean NC curves and SV values among modules and groups .............. 38 Table 4.1. Similarity of model parameter values obtained in Approach B relative to A, for central, lateral, and all muscles from 6 gait trials ............................................................................................. 52 Table 4.2. Medial, lateral and total contact force accuracy for both approaches. ............................... 53 Table 4.3. Medial, lateral and total contact force similarities. ............................................................ 53 Table 4.4. Similarity of model muscle forces obtained in Approach B relative to A, for medial, central, lateral, and all muscles from 6 gait trials. ............................................................................... 54 Table 4.5. Similarity of model normalized fiber lengths obtained in Approach B relative to A, for medial, central, lateral, and all muscles from 6 gait trials. .................................................................. 55 Table 4.6. Similarity of model passive forces obtained in Approach B relative to A, for medial, central, lateral, and all muscles from 6 gait trials. ............................................................................... 56 Table 4.7. Accuracy of muscle activation predictions for the 16 muscles with associated experimental EMG data relative to their activations reconstructed from synergy components .......... 56 Table 4.8. Accuracy of muscle activation predictions for the 28 muscles without associated experimental EMG data relative to their activations reconstructed from synergy components. ......... 57 Table 4.9. Similarity of muscle activation predictions in Approach B relative to A, for medial, central, lateral, and all muscles from 6 gait trials. ............................................................................... 57 Table A.1. Muscle groups according to their position. ....................................................................... 65 Table A.2. Model muscles which are tracked with their corresponding experimental activity .......... 66 Table A.3. Groups of muscles with similar normalized fiber lengths ................................................ 70 Table A.4 Groups of muscles with similar moment arm deviations ................................................... 70 Table A.5. Mean and standard deviations of percent difference of optimal fiber lengths from literature values. .................................................................................................................................. 71
x Optimization and muscle synergy approaches for studying muscle redundancy during walking Table A.6. Mean and standard deviation of percent difference of slack length of the tendons from literature values. ................................................................................................................................. 71 Table A.7. Mean and standard deviation of moment arm deviations for both approaches. ............... 71 Table A.8. Mean and standard deviation of absolute differences, and percentage differences between parenthesis, of moment arm deviations between approaches. ............................................................. 72
List of Figures Figure 2.1. Representation of muscles used in the model. ................................................................... 8 Figure 2.2. Mechanical characterization of muscle: Hill-type musculo-tendon model. ...................... 8 Figure 2.3. Block diagram of the two-step optimization strategy ...................................................... 13 Figure 2.4. Knee flexion angles for the non-injured knee and the injured knee. ................................ 15 Figure 2.5. Resultant joint moments at the left leg and at the right leg. ............................................. 16 Figure 2.6. CFA values versus JCNS for the 85 combinations of weighting factors. ........................... 17 Figure 2.7. Right knee moment and the forces of the muscles acting at the right knee for the BF cost function and the EW cost function ...................................................................................................... 19 Figure 2.8. Activations of the muscles acting at the right knee for the BF and EW cost functions and the normalized EMG. .......................................................................................................................... 20 Figure 2.9. Components of the cost function , , , m bp bd K K K noK J J J J during the gait cycle for the BF cost function and the EW cost function. Figure 2.10. Values of the JCNS for the BF and EW cost functions .................................................... 21 Figure 3.1. Muscle activation-deactivation patterns during gait cycle for the three groups: Control, Ipsilateral and Contralateral. ............................................................................................................... 31 Figure 3.2. Mean VAF values of all muscles using 4 and 5 modules. ................................................ 32 Figure 3.3. Synergy components from Control group decomposing EMG signals in 4 modules. ..... 33 Figure 3.4. Synergy components from Control group decomposing EMG signals in 5 modules. ..... 34 Figure 3.5. NCs and SVs for Control and Ipsilateral groups. ............................................................. 36 Figure 3.6. NCs and SVs for Control and Contralateral groups. ....................................................... 37 Figure 3.7. NCs and SVs for Ipsilateral and Contralateral groups. .................................................... 38 Figure 4.1. Block-diagram of the two-step optimization formulation. ............................................... 50 Figure 4.2. Experimental knee contact forces and mean knee contact force predictions for each approach (A and B). ............................................................................................................................ 53 Figure 4.3. Muscle force values for muscles with the greatest mean differences between approaches A and B. .............................................................................................................................................. 54
xii Optimization and muscle synergy approaches for studying muscle redundancy during walking Figure 4.4. Normalized fiber lengths for muscles with the greatest differences in mean tendon forces between Approach A and B. ................................................................................................................ 55 Figure 4.5. Activations reconstructed from synergy components and model activations for muscles with associated experimental EMG. .................................................................................................... 57 Figure A.1. Knee contact force predictions (Medial, Lateral and Total) for the three gait trials where these forces were used during the calibration and for other three gait trials using the same set of muscle parameters. .............................................................................................................................. 73
List of Symbols g weighting factors of the cost function p pennation angle d M allowed moment deviation from the one calculated by inverse dynamics a activation m a activation of muscle m m t a activation of muscle m at time frame t res a reserve activation asyn muscle activations reconstructed by synergy components athres threshold for muscle activations C q,q vector of inertia terms of the mechanical system mj d moment arm of muscle m at joint j exp t EMG experimental EMG value at time frame t ,m norm t EMG normalized EMG signal of muscle m at time frame t rec t EMG reconstructed EMG value at time t ce f force of the contractile element of the muscle lat-exp F experimental knee lateral contact force lat-mod F model knee lateral contact force med-exp F experimental knee medial contact force
xiv Optimization and muscle synergy approaches for studying muscle redundancy during walking med-mod F model knee medial contact force m pe F passive force 0m F maximum isometric force f muscle force m f muscle force of muscle m max m f maximum isometric force of muscle m s F strength of the reserve actuator hipadd hip adduction hipflex hip flexion hiprot hip rotation bd k J cost function terms corresponding to the biarticular muscles controlling the knee and the subsequent distal joint (ankle) bp k J cost function terms corresponding to the biarticular muscles controlling the knee and the subsequent proximal joint (hip) CMAES J cost function minimized by CMAES CMAES CNS cons J J J CNS J weighted square sum of normalized forces cons J cost function terms corresponding to the fulfillment of the constraints m k J cost function terms corresponding to the monoarticular muscles controlling the knee noK J cost function terms corresponding to the monoarticular muscles controlling the knee and the ankle kneeadd knee adduction
List of Symbols xv kneeflex knee flexion kneesup-inf knee superior-inferior ce l length of the contractile element of the muscle 0 M l optimal fiber length M l normalized fiber length se l length of the series element of the muscle T s l tendon slack length Mq inertia matrix of the mechanical system j M resultant moment at joint j jk M resultant moment at joint k and time frame j ma moment arm dev ma moment arm deviation thres ma threshold for moment arm deviation MeanD mean of differences nloads number of loads with model moment arm deviations nma number of muscles with non-zero model moment arm deviations p p-value q vector of generalized coordinates q vector of the first derivative of generalized coordinates q vector of the second derivative of generalized coordinates Q q,q vector of generalized forces of the mechanical system
xvi Optimization and muscle synergy approaches for studying muscle redundancy during walking r correlation coefficient sa scale factor of muscle activations for muscles with associated experimental EMG 0 M sl scale factor for optimal fiber length T s sl Scale factor for tendon slack length mod SV synergy vector of a muscle with no available EMG (design variable) on off Threshold EMG threshold to be considered activated 2 R coefficient of determination
List of Abbreviations ACL anterior cruciate ligament addmag adductor magnus addmagDist adductor magnus distal addmagIsch adductor magnus ischial addmagMid adductor magnus middle addmagProx adductor magnus proximal Al left ankle Ar right ankle BF best fit bflh biceps femoris long head bfsh biceps femoris short head CE contractile element CFA cost function accuracy CMA-ES covariance matrix adaptation – evolutionary strategy CNS central nervous system CT computed tomography DOF degree of freedom ED extensor digitorum longus
xviii Optimization and muscle synergy approaches for studying muscle redundancy during walking EMG electromyography EW equally weighted GA gastrocnemius gaslat gastrocnemius lateralis gasmed gastrocnemius medialis GL (in Chapter 2) gluteus GL (in Chapter 3) gastrocnemius lateralis glmed1 gluteus medius anterior glmed2 gluteus medius middle glmed3 gluteus medius posterior GM gluteus maximus GRF ground reaction force HA hamstrings HAT head, arms and trunk Hl left hip Hr right hip ICATME Catalan Institute of Trauma and Sport Medicine IL iliopsoas sup-inf superior-inferior Kl left knee
Chapter 2 2 A weighted cost function to deal with the muscle force sharing problem in injured subjects: a single case study 2.1 Background The force sharing problem in human gait analysis is an open subject of research. It is the consequence of over-actuation: several muscles may control the same joint degree of freedom (DOF), and thus the problem of calculating the force exerted by each muscle from the knowledge of the total wrench (force and moment) at any particular joint is essentially undetermined. Knowing the forces exerted by each single muscle is useful both for planning a rehabilitation treatment [2,3], or for designing assistive devices [4–7]. To face up to this indeterminacy, in vivo measurements of the muscle forces would be helpful, but they are not usual due to their invasiveness. This is where optimization processes come in: it is assumed that the central nervous system (CNS) strategy can be represented through a cost function (associated with a physiological criterion) whose minimum value yields the individual muscle forces. Optimization techniques to solve this problem have been well-known for more than thirty years [8]. Different studies can be found in the literature focused on investigating the strategy followed by the CNS of healthy people to predict the muscle forces from the known motion using static optimization and various cost functions [9]. This approach starts with the motion capture and the calculation of the resultant joint moments, which are shared according to the chosen cost function. Many authors have proposed different cost functions
6 Optimization and muscle synergy approaches for studying muscle redundancy during walking [8,10–12], which differ in the physiological variable they represent (muscle forces, muscle tensions, muscle activations…). In these cost functions all muscles are weighted equally, i.e., the weight factors associated to each muscle in the cost function have all the same value. Muscle forces are related to muscle activations [13]. Thus, muscle forces obtained after solving the muscle force sharing problem should be consistent with experimental activation measurements. Since the normalized electromyography (EMG) signal gives information about the muscle activation, the calculated muscle activations should be compared with EMG measurements [14]. As far as the authors know, the CNS strategy of an injured subject to activate some particular muscles rather than others has not been studied extensively. The present study is devoted to the investigation of that strategy in a single injured subject, in this case suffering anterior cruciate ligament (ACL) rupture. The main function of the ACL is knee stabilization. Therefore its rupture may cause unsteadiness and cartilage damage. During rehabilitation, muscles surrounding the knee supply the function of the ruptured ACL. This suggests that this is a case where the pattern of muscle activations will be different from that of a subject with a healthy knee [15]. The contributions of this work are two-fold: the first one is the assumption that the CNS does not weight equally the muscles when trying to compensate for a lower limb injury during gait. We explored different possibilities where the muscles were weighted differently. In order to determine the cost function that best captured the specific strategy of the subject, we compared computed muscle activations with EMG measurements. The second contribution concerns the numerical implementation of the optimization problem. We solve it in two phases. Firstly, we use an evolutionary algorithm to approach the optimal forces. Secondly, those previous solutions are taken as initial guesses and the optimal forces are obtained through a deterministic algorithm. Finally, muscle activations are calculated from the muscle forces according to a Hill-type model and are compared with the normalized EMG signal. The chapter is organized as follows: Section 2.2 describes the biomechanical model used to study the human gait and presents the optimization algorithm applied to calculate the muscle
A weighted cost function to deal with the muscle force sharing problem in injured subjects 7 forces; then, the experimental method and the results are presented in Section 2.3; and finally, the main conclusions of the work are drawn in Section 2.4. 2.2 Methods 2.2.1 Modeling of the human gait dynamics A musculoskeletal model is used to represent the human body. The biomechanics simulation software OpenSim [16] was used to obtain the joint angles and the resultant joint moments from the motion capture at our biomechanics lab. The musculo-tendon units were modeled through a Hill-type formulation. The OpenSim 3D standard model called gait2392 [17] was scaled to the specific subject. The multi-body system consists of eight segments: two feet, two shanks, two thighs, pelvis and HAT (Head, Arms, Trunk). It has 19 DOF: the pelvis has all 6 DOF, the hips and lumbar joint are modeled as frictionless spherical joints (3x3 DOF) and the ankles and the knees are modeled as revolute joints (4x1 DOF). The length, mass and tensor of inertia of the segments are scaled with the Scale Tool implemented in OpenSim according to the total mass and dimensions of the particular subject whose pathological gait was studied. Following the work by Ackermann [6], eight muscles are considered for each leg which correspond to the main actuators during human gait (Figure 2.1): Iliopsoas –IL–, Gluteus – GL–, Rectus Femoris –RF–, Hamstrings –HA–, Vastus –VA–, Gastrocnemius –GA–, Tibialis Anterior –TA– and Soleus –SO–. A model of the muscle (with a rigid tendon) is used in order to relate the muscle force with the activation of the muscle fibers in a phenomenological way. All musculo-tendon units exhibit an active behavior and a passive viscoelastic one. The active behavior is that of the muscle fibers (or contractile elements, CE), and is usually described through a Hill-type model. The passive behavior can be simply modeled through series (SE) and parallel (PE) springs and dampers, representing the tendons and the connective tissue wrapping the muscle fibers, respectively. Figure 2.2 shows that general model. Angle p stands for the pennation angle (orientation angle between the muscle fibers and the tendon). The pennation
8 Optimization and muscle synergy approaches for studying muscle redundancy during walking angles are obtained from literature [18]. For the sake of simplicity, PE is neglected as in [6]. So, the muscle force is cos m ce p ff . In our study, we have followed the formulation used by Ackermann [6] and Van Soest and Bobbert [19], who give two different expressions according to whether the muscle is undergoing a concentric or an eccentric contraction. Figure 2.1. Representation of the muscles used in the model. The abbreviations coincide with those given in the text. Figure 2.2. Mechanical characterization of muscle: Hill-type musculo-tendon model, adapted from [5]. 2.2.2 Multi-body formulation The application of inverse dynamics to the biomechanical multi-body system yields the resultant joint forces and moments (at the ankle, knee and hip) from the kinematics, the anthropometric parameters and the force plate measurements. The resultant flexor joint moments are associated with the muscles only, whereas the resultant forces include both
A weighted cost function to deal with the muscle force sharing problem in injured subjects 9 muscle and bone-to-bone forces. The biomechanics software OpenSim [16] was used to solve the inverse dynamics problem. The basic equations are: ( ) , , Mq q C q q Q qq (2.1) where ,,qqq are the vectors of generalized coordinates and their derivatives, ()Mq is the inertia matrix of the system, ,C q q is the vector of inertia terms which depend on position and velocity, and ,Q q q is the vector of generalized forces. In this study ,,qqq , ()Mq and ,C q q are known, since the kinematics of the human gait is measured in the laboratory and the inertia terms are scaled. Vector ,Q q q includes terms associated with joint moments and the external wrench applied to the pelvis (our unknowns), the weight of the segments (estimated from measurements) and the foot-ground contact forces and moments (measured). Note that the external wrench applied to the pelvis, also called residual wrench, should be zero if the footground contact wrenches are dynamically consistent with the whole-body motion. In order to achieve musculoskeletal dynamic consistency, the residual wrench has to be minimized. In the present study, we have run an optimization algorithm based on non-linear least squares in order to minimize these residuals while optimizing the global pelvis configuration and fulfilling equation (2.1), similar to [16]. 2.2.3 Optimization for muscle force estimation The selection of the cost function for the optimization problem is crucial. As mentioned before, the cost function represents the physiological variable minimized by the CNS during the gait. Hardt suggested that the sum of muscle forces is minimized during human gait [8]. Crowninshield and Brand proposed that fatigue, which is related to the muscle tension, is the optimized physiological variable [10]. Glitsch and Baumann applied static optimization to calculate the muscle forces for a healthy subject during walking and running comparing the minimization of the two variables [20]. They concluded that the best correspondence with the EMG was obtained when the muscle stresses were minimized. Cost functions for the case of pathological gait have not yet been deeply studied. One reason could be a higher variability: in contrast with those associated to healthy people, gait cost
10 Optimization and muscle synergy approaches for studying muscle redundancy during walking functions in case of injury may vary from one subject to another depending on the specific pathology and subject ability. Our hypothesis is that the CNS of subjects suffering from a joint pathology weights every muscle in a different way and thus cost functions weighting equally all muscles are no longer valid. The particular distribution of muscle weighting factors depends on the particular joint injury. Accordingly, the choice of the cost function was based on the idea that the muscles controlling the injured joint had to be treated differently from the others. Grouping the muscles so that no muscle was counted twice, we came up with four different muscle sets: Monoarticular muscles controlling the injured joint (monoarticular muscles). Biarticular muscles controlling the injured joint and the subsequent proximal joint (biarticular-proximal muscles). Biarticular muscles controlling the injured joint and the subsequent distal joint (biarticulardistal muscles). Muscles not controlling the injured joint (other muscles). Note that for some particular cases the biarticular muscles may not be split into two groups. For our particular application case the injured joint is the knee and the corresponding cost function is the following: 1 2 3 4 m bp bd CNS K K K noK J J J J J (2.2) where 1, ,4 gg are the weighting factors associated to each group, m K J is the cost function term corresponding to the monoarticular muscles controlling the knee, bp K J that of the biarticular muscles controlling the knee and the subsequent proximal joint (hip), bd K J that associated with the muscles controlling the knee and the subsequent distal joint (ankle), and noK J that taking into account all the other muscles: 2 max VA m KVA f Jf (2.3)
A weighted cost function to deal with the muscle force sharing problem in injured subjects 11 22 max max HA RF bp KHA RF ff Jff (2.4) 2 max GA bd KGA f Jf (2.5) 2222 max max max max IL GL TA SO noK IL GL TA SO f f f f Jffff (2.6) In those expressions, m f and max m f stand for the force and maximum isometric force associated with muscle 1, ,8mm . All g combinations were calculated, from 0.1 g to 0.7 g , with intervals of 0.1 and satisfying that 4 1 1 g g . The optimization problem consists of minimizing the cost function at each time frame while guaranteeing the following constraints and boundary conditions: Equality between the moments exerted by the muscles and the resultant joint moments obtained by means of the inverse dynamics: 8 1 , 1,2,3. m mj jm M f d j (2.7) where Mj is the moment at joint j (1-hip, 2-knee, 3-ankle) calculated by inverse dynamics and mj d is the moment arm of the muscle m at the joint j. Muscle force and activation boundary conditions: max 0mm ff (2.8) 01 m a (2.9) where m a is the activation of muscle 1, ,8mm . The Hill’s model described in Section 2.2.1 was used to calculate m a from m f . Each optimization problem was solved in two steps (Figure 2.3). First, following the strategy used in other biomechanical studies [21,22], the evolutionary algorithm called “Covariance Matrix Adaptation Evolution Strategy” (CMA-ES) was used [23], in order that the design
12 Optimization and muscle synergy approaches for studying muscle redundancy during walking variables (muscle forces) approach the optimal solution. Being an evolutionary algorithm, this optimum solution is more likely to correspond to a global minimum than to a local minimum. Since the code does not allow constraints, equation (2.7) was introduced in the cost function: CMAES CNS cons J J J (2.10) where JCNS represents the weighted square sum of the normalized forces defined in equation (2.2) and Jcons represents the fulfillment of the constraints, that is: 2 38 11 m mj cons j d jm J M f d M (2.11) where d M is the allowed deviation from the moment calculated by inverse dynamics. When the sum of the simulated muscle moments becomes higher or lower than these bounds, the cost function term increases, since the term is squared. The boundary conditions for this step are equations (2.8) and (2.9). In the second step, a deterministic algorithm was used. That algorithm uses a sequential quadratic programming (SQP) method to find the minimum of constrained nonlinear multivariable functions. It is implemented in a default optimization library of the Matlab software called fmincon. The problem was slightly different than the previous one, since the constraints were not introduced in the cost function as they were taken into account by the algorithm itself. The cost function has the form of equation (2.2), constraints of equation (2.7) and the boundary conditions of equations (2.8) and (2.9). The reason for using these two algorithms is that although the evolutionary algorithm has a higher computational cost than the deterministic one and does not reach the solution, it is more likely to stop near the global minimum. Alternatively, the deterministic algorithm would be more likely to reach a local minimum for any choice of initial conditions. But with the results of the CMA-ES algorithm as initial guesses, the algorithm is more likely to reach the global minimum.
A weighted cost function to deal with the muscle force sharing problem in injured subjects 13 Figure 2.3. Block diagram of the two-step optimization strategy. 2.3 Results and discussion The results are presented in this section. First, we describe the experimental method that was used to measure the kinematics and the foot-ground contact forces. Then, we introduce a brief review of the kinematic and dynamic results. The third and main subsection contains the optimization results, used to select the cost function that best fits the EMG measurements (which will be called BF cost function, where BF stands for “best fit”). In the fourth subsection, the muscle forces and activations are shown for two strategies, the one considered as the most suitable (BF) and the one using an equally weighted (EW) cost function, which is considered to be followed by a healthy subject. Finally, the values of the cost function terms are analyzed for the mentioned strategies. 2.3.1 Experimental methods The gait of a 59-year-old male subject (weight: 90 kg, height: 1.85 m), suffering an ACL rupture at the right knee, was captured in our biomechanics lab. The subject was asked to walk as naturally as possible without any external aid. The trajectories of 22 markers attached on the skin of the subject were captured. These trajectories were captured at 100 Hz by means of 14 infrared cameras (NaturalPoint, Corvallis, OR), and they were filtered with a third order Butterworth low-pass filter with a cut-off frequency of 6 Hz. Since the skin is a soft tissue and there is a relative movement between it and the bones, kinematic consistency was ensured running the Inverse Kinematics Tool implemented in OpenSim [16] and joint angles were obtained. Then, running the Inverse Dynamics Tool, resultant joint moments were calculated. Dynamic consistency was ensured optimizing the positions of the pelvis as explained in Section 2.2.2.
14 Optimization and muscle synergy approaches for studying muscle redundancy during walking The foot-ground contact forces were measured with two force plates (AMTI, Watertown, MA) with a sample frequency of 100 Hz. These data were also filtered with a third order Butterworth filter with a cut-off frequency of 6 Hz. The walking cycle was defined as the interval between two consecutive heel-strikes for the right leg, and between two toe-offs for the left leg. EMG measurements were also done with an 8-channel surface EMG sensor (Biometrics, Newport, United Kingdom) at 1000 Hz. One sensor was placed on each muscle, following the Surface Electromyography for the Non-Invasive Assessment of Muscles (SENIAM) recommendations [24]. The EMG signal represents the myoelectric signals produced by the physiological variations at the both sides of the muscle fiber membranes when the muscle contracts and relaxes. A higher signal indicates higher muscle activation. The signal has to be filtered, smoothed and normalized to obtain an activation signal [25]. EMG data were filtered using an RMS filter (window length: 30 frames, overlap: 10 frames), then they were filtered again with a third order Butterworth filter at 6 Hz. Finally, they were normalized to the maximum voluntary contraction (MVC) value that was recorded previously. Thus, the resultant EMG signal was kept between 0 to 1. Markers data, foot-ground contact forces and EMG data were synchronized using an own Matlab code. Scaling process, Inverse Kinematics and Dynamics analyses were performed in OpenSim. 2.3.2 Kinematic and dynamic data The injured subject had no difficulties in walking some strides in the lab with a self-selected speed of 1.06 m/s, although he mentioned knee pain when doing certain movements. The kinematic measurements showed that the flexion-extension range of motion of the noninjured knee (62.49º) was higher than the injured knee (53.73º) (Figure 2.4). This is in agreement with the fact that the flexion of the injured knee is lower to prevent a destabilization of the knee, as it reported by Rudolph et al. [26] and Chmielewski et al. [27].
A weighted cost function to deal with the muscle force sharing problem in injured subjects 21 BF cost function. With this cost function the JnoK values were higher while bd K J values were lower. Although the increase of JnoK was high, the effect in the cost function JCNS was low because its weighting factor was 0.1 instead of 0.25 (EW cost function). As shown in Figure 2.10, the total value of JCNS is lower for the BF cost function during most of the cycle. (a) (b) Figure 2.9. Components of the cost function m bp bd K K K noK J ,J ,J ,J during the gait cycle for the BF cost function (a) and the EW cost function (b). Figure 2.10. Values of the JCNS for the BF and EW cost functions.
22 Optimization and muscle synergy approaches for studying muscle redundancy during walking 2.4 Conclusion The main purpose of this study has been to investigate a consistent strategy followed by the CNS of a single injured subject (in this case suffering ACL rupture) to activate the muscles. Two novel contributions have been introduced in this study. The main original contribution is the use of a novel cost function where all muscles are not weighted equally. The muscles are grouped in four sets according to a physiological criterion (monoarticular on the injured joint, biarticular-distal, biarticular-proximal and other muscles). This has led to the simulation of 85 optimization problems. Another original contribution is the use of a twostep optimization method to have a higher probability that a global optimum is found when solving the muscle force sharing problem. The global algorithm CMA-ES gets closer to the minimum and the deterministic algorithm fmincon is used to reach the optimal muscle forces from a good initial estimate. Slight alterations at the level of both joint kinematics and dynamics at the injured leg have been observed which are consistent with the literature studies. A decrease of the knee moment at the injured leg has been detected, which is due to the fact that the contribution to the total support is transferred from the knee to the ankle to avoid knee instabilities. It has been observed that the strategy to activate the muscles which is more consistent with EMG measurements is different from the one that would be followed by a subject that weights equally all muscles. The cost function leading to more realistic muscle activations for our particular subject is the one that penalizes the biarticular muscles acting between the hip and the knee (RF and HA). This is in agreement with the hypothesis that the CNS does not weight all muscles equally. In this study, the optimization formulation based on uniform muscle weightings, i.e. with equally weighted cost function, was associated to healthy subjects since this is the standard approach used in the literature. A further analysis could also have been carried out on healthy individuals to investigate whether or not the equally weighted cost function better fitted predicted muscle activations with normalized EMG compared to a non-uniform weighted cost function, as it was done in the study developed by Botasso et al. [33]. Another limitation is due to the simplicity of the model. As a result of the ACL rupture, the subject can experience small anterior translations between the tibia and the femur and we did not take them into account, since the knee was modeled as a hinge joint (one degree of freedom).
A weighted cost function to deal with the muscle force sharing problem in injured subjects 23 We did that simplification since with the motion capture system used in this work, it is not possible to measure these small relative translations. For this reason, we locked the degree of freedom related to the knee anterior-posterior translation and we assumed that not measuring that translation would not have a great effect neither on the calculation of the knee flexion moment through inverse dynamics nor on the calculation of moment arms. This case study has been based on the kinematic and dynamic data of only one injured subject. The presented methodology to investigate the strategy followed by the CNS to activate the muscles could be also extended to other subjects suffering the same injury or even other joint injuries.
24 Optimization and muscle synergy approaches for studying muscle redundancy during walking
Chapter 3 3 Analysis of the activation - deactivation patterns and muscle synergies in ACL-deficient subjects during gait 3.1. Background During human motion, it is believed that muscles are activated synergistically following a certain pattern depending on the motor task [34–37], that is to say, our central nervous system (CNS) does not activate the muscles independently. Muscle synergies are represented by modules consisting of one neural command (NC), which represents the time activation of a set of muscles, and one synergy vector (SV), which represents the weighting factor of each muscle to its NC. The number of NCs is lower than the number of muscles. Several algorithms have been used in order to calculate the muscle synergy components. The most widely used is the non-negative matrix factorization (NNMF), which ensures that NCs and SVs are both positive, while it minimizes the error between the reconstructed and the original signals [38,39]. It is believed that the number of synergies used by a human being when walking is between 4 and 6 [40–45]. The variance accounted for (VAF) between the reconstructed and the original signals is evaluated to select the proper number of modules to be used when factorizing the signals. Most authors consider that a VAF>0.9 is the threshold to accept the reconstruction [40,45]. It is reported that there are similarities in the muscle synergies when performing the same movement across subjects. Several authors reported muscle synergies when walking [40,41,45–47], walking with perturbations [44] or performing other tasks [48].
26 Optimization and muscle synergy approaches for studying muscle redundancy during walking Clark et al. [40] applied the muscle synergy analysis in post-stroke injured subjects. They observed that, although the patterns were similar among groups, the complexity in poststroke injured subjects was lower than in healthy subjects, i.e., they needed fewer modules to have a good signal reconstruction. It is unclear what synergistic strategy is followed by jointinjured subjects to activate the muscles spanning that joint. Depending on the joint injury, subjects can apply different activation strategies in order to avoid pain or to stabilize the joint. Apart from the clinical evaluation of muscle co-contraction, the use of the factorization can be useful for motion analysis and simulation. There is indeterminacy when calculating the muscle forces, since they cannot be measured experimentally due to invasiveness. The usual method to estimate the forces is with the resolution of an optimization problem [49], which consists of minimizing a cost function (a physiological variable) that represents the strategy of the CNS to activate the muscles. The optimization results can produce multiple physiologically feasible solutions due to the muscle redundancy. Some authors used the muscle synergy components in order to decrease the indeterminacy in the muscle force calculations, either in forward dynamics [43,46] or inverse dynamics [50] approaches. Regarding ACL-deficient subjects, differences have been observed at the joint level as well as at individual EMG signals [1,51–53]. As far as the authors know, the muscle synergy analysis has not been applied yet to subjects with this kind of injury. In consideration of that, this study could be useful at two levels. On the one hand, in a clinical application it would allow the specialist to follow the rehabilitation process of injured subjects. On the other hand, in a motion dynamic analysis, muscle synergies could be used to decrease the indeterminacy in the muscle force calculation of ACL-deficient subjects. In this study, we evaluated and compared the EMG patterns in healthy and injured subjects when walking. EMG data were collected from both legs at the same time in order to avoid the variability that could appear when data are obtained from different gait trials. According to the well-known classification of ACL-deficient subjects, it is considered that they can be divided in three groups [54]: copers, who return to the preinjury level of their daily tasks and sport activities; non-copers, who cannot return to their preinjury level of tasks and sport activities and have episodes of full giving way even in daily tasks; and adapters, who reduce
Analysis of the act-deactivation patterns and muscle synergies in ACL-deficient subjects 27 or modify certain tasks or the sport level to prevent their knee giving way. In our study, all ACL-deficient subjects were considered adapters and the measures were done a few days or weeks before the surgery of the ligament reconstruction. Although muscle synergy patterns can present many similarities among groups, since all of them perform the same task, human gait, our hypothesis was that the pattern of muscle synergy components may have different tendencies. As mentioned, there are studies that evaluate individual muscle activations in ACL-deficient subjects, but the objective of this study is to evaluate the differences in muscle synergies compared to healthy subjects in order to better understand the muscle activation pattern in absence of ACL function. The analysis comprises two steps. The first is a comparison of the activation-deactivation pattern among healthy legs (Control group), injured subjects’ injured legs (Ipsilateral group) and injured subjects’ non-injured legs (Contralateral group). Then, a muscle synergy analysis is reported and compared among the three groups. 3.2. Methods 3.2.1. Subjects Ten healthy subjects, five men and five women (age 31.5±12.9 years), and eighteen ACLdeficient subjects, twelve men and six women (age 32.3±10.99 years), volunteered as participants in this study. No healthy subjects suffered any lower-limb injury. The injured subjects were classified as adapters, according to the medical staff and the widely used classification presented in [54]. All injured subjects reported that they could deal with daily live and they did not suffer pain when normal walking, however, they felt discomfort and pain when they did sports that required knee pivoting, such as football or skiing. The time interval from the injury varied from one month to three years (10.3±12.0 months). All subjects provided their consent to contribute to this study. 3.2.2. Experimental setup All volunteers were asked to walk a minimum of three overground gait cycles at a selfselected speed (0.77±0.12 m/s healthy subjects and 0.80±0.13 m/s injured subjects). One of the gait cycles was selected from the recorded trials and was analyzed.
28 Optimization and muscle synergy approaches for studying muscle redundancy during walking EMG data from sixteen muscles were measured with sixteen surface EMG sensors (Biometrics, Newport, United Kingdom) at 1000 Hz. The signal of eight lower-limb muscles from each leg of the subjects was measured (Tibialis Anterior –TA–, Soleus –SO–, Gastrocnemius Lateralis –GL–, Gluteus Maximus –GM–, Rectus Femoris –RF–, Vastus Lateralis –VL–, Semitendinosus –ST– and Extensor Digitorus Longus –ED–). The EMG data for each subject (right and left leg) came from the same gait trial, which decreases the variability due to the differences that could appear when measuring different gait trials separately. EMG signals were demeaned, rectified and filtered with a Butterworth low-pass filter at 6 Hz. Then, they were normalized by maximum voluntary contraction (MVC) values obtained by MVC exercises previously done. The exercises were selected in order to calculate the maximum muscle excitations [55]. The volunteers were asked to apply force against a resistance along a direction to activate the muscles responsible for: ankle plantar flexion/dorsiflexion (ED, SO, TA, GL), knee flexion/extension and abduction/adduction (RF, ST and VL) and hip flexion/extension and abduction/adduction (GM, ST and RF). Data from these trials were processed in the same way that walking trials (demeaned, rectified and filtered at 6 Hz). The maximum values of EMG were selected among all available trials (MVC exercises and gait trials). These values were verified visually and individually in each subject in order to avoid the acceptance of a wrong maximum value. All MVC exercises, as well as verifications, were carried out by the same technician in order to standardize the comparison. Using this normalization, the signal was constrained to be between 0 (not activated) and 1 (maximum activation). So, an activation close to 1 would mean that the muscle is near to its maximum activation. Ground reaction forces (GRF) and marker trajectories were also measured to identify the events of the gait cycle. The GRF were measured by means of two force plates (AMTI, Watertown, MA) at 100 Hz. Two marker trajectories from each foot (heel and tip of the first metatarsal bone) were captured by fourteen infrared cameras (Naturalpoint, Corvallis, OR). Once the gait cycle was identified for each leg, data was interpolated to 101 frames. EMG, GRF and marker trajectories will be available on the net.
Analysis of the act-deactivation patterns and muscle synergies in ACL-deficient subjects 29 3.2.3. Data Analysis Data analysis was carried out by means of Matlab (Mathworks, Natick, MA) and using the Parallel Computing Toolbox to decompose the signals. All data were divided in three groups: Control, which consists of data from healthy subjects; Ipsilateral, from the ipsilateral leg, which is affected by the ACL injury; and Contralateral, from the non-injured leg of the ACL-deficient subjects. Activation-deactivation pattern An initial analysis of the activation-deactivation pattern for each muscle was carried out to identify the differences in the activation timing between groups. The on-off activation pattern was calculated for each subject, considering EMG signal to be activated when it was higher than the following threshold: min 0.5 max min on off Threshold EMG EMG EMG (3.1) where EMG stands for an EMG signal. The activation pattern was calculated for each group. A muscle was considered to be active when more than 50% of the subjects had this muscle activated at a particular time frame. Non-negative matrix factorization EMG processed signals were factorized applying the non-negative matrix factorization (NNMF) to obtain the muscle synergy components: neural commands (NCs) and synergy vectors (SVs) [39]. The factorization consists of decomposing the matrix containing all EMG signals (nframes x nmuscles) by two matrices: the NCs, which represent time activation of each module (nframes x nmodules) and SVs, which contain the weights of each muscle to each module (nmodules x nmuscles). The algorithm to decompose the signal was based on the non-linear least square algorithm (lsqnonlin), which minimizes the error between the reconstructed and the experimental EMG signals at each iteration step [50]. The maximum value of each SV was constrained to be 1 in order to decrease the indeterminacy of the factorization. The NNMF was applied six times per trial, decomposing the signal in 1 to 6 modules.
30 Optimization and muscle synergy approaches for studying muscle redundancy during walking Comparison of results The match of the reconstructed EMG with the experimental one was evaluated by means of the variance accounted for (VAF) value, calculated as follows: 101 1101 1 1 rec exp tt t exp t t EMG EMG VAF EMG (3.2) where rec t EMG and exp t EMG stand for the reconstructed and the experimental EMG signals at frame t respectively. In order to compare statistically whether two values from two different samples were different (such as the comparison of VAF values), a t-test analysis was carried out and a p-value was calculated. If p-value<0.05, then the values were considered to be statistically different. The differences in the tendency of SVs and NCs were measured by means of the Pearson’s correlation coefficient [40]. If this value was close to 1, it meant that the shape of the compared sets of data was similar. In order to identify whether two NCs or SVs were statistically correlated, the threshold of the p-value was set equal to 0.001. 3.3 Results 3.3.1 Activation-deactivation pattern Figure 3.1 shows the activation-deactivation pattern of the eight analyzed muscles for the three data groups. It is remarkable that Ipsilateral and Contralateral TA showed a longer activation during the early stance phase (0-40%). Moreover, the Ipsilateral’s TA was active during all swing phase whereas Control’s and Contralateral’s TA activation just appeared in the beginning and at the end of this phase. SO and GL were only activated during the stance phase and both were activated earlier in Ipsilateral and Contralateral groups. GM activation was slightly longer in the ipsilateral leg. Regarding the knee muscles, it can be observed that the co-contraction of the injured subjects’ quadriceps (RF and VL) and hamstrings (ST) was longer during the stance phase. Finally, Ipsilateral’s ED activation pattern was slightly different from the other two groups, since both activation and deactivation of the injured subjects’ ED appeared earlier than in the other two groups.
Analysis of the act-deactivation patterns and muscle synergies in ACL-deficient subjects 37 Figure 3.6. NCs and SVs for Control and Contralateral groups. Refer to Figure 3.5 for meaning of thick and dashed lines as well as the error bars. 020 40 60 80 100 0 0.25 0.5 NC 1 Control Contralateral 020 40 60 80 100 0 0.35 0.7 NC 2 020 40 60 80 100 0 0.25 0.5 NC 3 020 40 60 80 100 0 0.25 0.5 NC 4 020 40 60 80 100 0 0.25 0.5 NC 5 % gait cycle 0 0.5 1 SV 1 0 0.5 1 SV 2 0 0.5 1 SV 3 0 0.5 1 SV 4 0 0.5 1 SV 5 TA SO GL GM RF VL ST ED
38 Optimization and muscle synergy approaches for studying muscle redundancy during walking Figure 3.7. NCs and SVs for Ipsilateral and Contralateral groups. Refer to Figure 3.5 for meaning of thick and dashed lines as well as the error bars. Table 3.2. Correlation of the mean NC curves and SV values among modules and groups. NC SV Control Ipsilat. 1 2 3 4 5 Control Ipsilat. 1 2 3 4 5 1 0.77 1 0.80 2 -0.04 0.94 2 -0.36 0.99 3 0.19 -0.63 0.69 3 -0.18 -0.28 0.96 4 0.69 -0.28 0.30 0.86 4 0.28 -0.35 -0.17 0.70 5 -0.42 0.01 0.11 -0.07 0.81 5 -0.32 -0.20 0.03 -0.47 0.96 NC SV Controll Contralat. 1 2 3 4 5 Control Contralat. 1 2 3 4 5 1 0.79 1 0.93 2 0.01 0.92 2 -0.22 0.99 3 0.24 -0.62 0.43 3 0.38 -0.31 0.95 4 0.70 -0.20 0.08 0.84 4 0.10 -0.31 -0.17 0.91 5 -0.44 0.12 -0.36 -0.30 0.73 5 -0.28 -0.17 0.10 -0.43 0.96 020 40 60 80 100 0 0.25 0.5 NC 1 Ipsilateral Contralateral 020 40 60 80 100 0 0.35 0.7 NC 2 020 40 60 80 100 0 0.25 0.5 NC 3 020 40 60 80 100 0 0.25 0.5 NC 4 020 40 60 80 100 0 0.25 0.5 % gait cycle NC 5 0 0.5 1 SV 1 0 0.5 1 SV 2 0 0.5 1 SV 3 0 0.5 1 SV 4 0 0.5 1 SV 5 TA SO GL GM RF VL ST ED
Analysis of the act-deactivation patterns and muscle synergies in ACL-deficient subjects 39 NC SV Ipsilat. Contralat. 1 2 3 4 5 Ipsilat.. Contralat. 1 2 3 4 5 1 0.98 1 0.77 2 0.08 0.96 2 -0.19 0.99 3 0.36 -0.60 0.49 3 0.36 -0.13 0.96 4 0.59 -0.19 0.10 0.69 4 0.07 -0.60 0.23 0.82 5 -0.50 -0.25 -0.26 -0.31 0.72 5 -0.38 -0.28 -0.10 -0.32 0.87 Bold values indicate that there is significant positive correlation (p-value<0.001) 3.4 Discussion This study deals with the investigation of the differences in the muscle activation patterns between healthy subjects (Control group) and ACL-deficient subjects (Ipsilateral and Contralateral groups). The differences were studied at two levels: individual muscle timing patterns and muscle synergies. As far as the authors know, no previous study was carried out calculating the muscle synergy components in ACL-deficient subjects. Courtney et al. [56] reported the onset and offset patterns of the Tibialis Anterior, Gastrocnemius Medialis, Medial Hamstrings and Quadriceps during fast-inclined walking in a treadmill for a Control group and an Adapter ACL-deficient group. Authors found that Tibialis Anterior and Medial Hamstrings had a longer activation period, which is in agreement with the presented results for the TA and ST muscles. They did not observe differences in Quadriceps (represented by RF in our study). Gastrocnemius Medialis pattern of the Adapter group (activated just before and after the heel strike) was different compared to results obtained for the Ipsilateral and Contralateral GL (mainly activated during the mid stance phase). Nevertheless, this difference could be attributed to the fact that the gait was faster than in our study and there was an inclination. Knoll et al. [52] studied the gait adaptations at the level of muscle activations of ACLdeficient subjects before and after the surgery. They measured EMG signals in Vastus Lateralis and Medialis, Adductor Longus and Biceps Femoris during walking. The activation-deactivation patterns obtained in their study are comparable to the ones in Figure 3.1. Overall, Ipsilateral’s VL and Biceps Femoris had a longer activation. In the work presented here, the results of the Biceps Femoris activation pattern are comparable to the ST pattern, which also belongs to Hamstrings. These results show longer co-contraction between Hamstrings and Quadriceps to stabilize the knee joint.
40 Optimization and muscle synergy approaches for studying muscle redundancy during walking These results are in agreement with the fact that both muscles, Hamstrings (knee flexor) and Quadriceps (knee extensor), are stabilizing the knee when the ACL is ruptured [51,57]. These muscles control the joint in order to avoid high displacements of the tibia with respect to the femur. There were also observable differences in muscles which control the ankle (SO, TA, GL and ED). The activation of these muscles was longer for the ACL-deficient subjects. As Chmielewski et al. [57] mentioned, there is a shift of support moment from the knee to the ankle what may suggest that there is a transfer of the leg control away from the injury. There were no statistical differences regarding the dimensionality of the signal factorization. In this case there were no observable differences in the number of modules needed to reconstruct the signals among groups, which represents that the control of the CNS is not more complex in ACL-deficient than in healthy subjects, in contrast with other studies which analyzed post-stroke subjects [40]. The VAF values that evaluated the reconstruction of the EMG signal were similar, either using 4 or 5 modules. These results showed that the pattern of the EMG activity of the ACL-deficient subjects was not more complex than the one of the healthy subjects. However, this fact did not exclude that some differences could be detected between groups. Figure 3.2 showed that in all three groups, the VAF values of all muscles were higher than 0.9 using 5 modules, so the muscle synergy analysis was carried out using 5 synergies [40]. The analysis of the differences intra-groups showed that all modules were overall independent from each other. EMG data were collected at both legs simultaneously. Measuring left and right legs at the same time avoided the differences that could appear when measuring the EMG in two different cycles. EMG was normalized by MVC values, what allowed comparing the differences in the magnitude of the NCs (time-dependent synergy components). In Figures 3.5 to 3.7, it can be observed that NCs 1, 3, 4 and 5 were overall comprised between 0 and 0.4 and NC 2 was slightly higher (comprised between 0 and 0.62). NCs and SVs are comparable to other studies that analyzed the muscle synergy components in healthy subjects. Oliveira et al. [45] carried out a study to compare the influence of EMG processing (averaging, concatenation and the used number of cycles) in the muscle synergy factorization. They reported the results for 5 modules. Our first 4 modules can be identified
Analysis of the act-deactivation patterns and muscle synergies in ACL-deficient subjects 41 in 4 of their modules. The fifth module is different due to the fact that they did not measure ED signal (contained in module 5 in our case). Clark et al. [40] carried out a study of the muscle synergy analysis with paretic and healthy subjects and Neptune et al. [46] simulated gait using muscle activation modules. Both studies reported results for 4 modules that could be identified with our modules 1 to 4. They did not include ED in the muscle synergy analysis, which in our case participated in module 5. In all mentioned studies, EMG was normalized over all trials. In this study, EMG data were normalized by MVC values in order to evaluate differences in the magnitude of NCs. Although the differences were small, SVs tended to be more similar between Control and Contralateral groups, suggesting that in those groups the CNS activates the same groups of muscles synergistically. However, it is not clear which groups presented comparable NCs. Modules 1 and 2 presented similar NCs between Contralateral and Control groups and modules 3 to 5 between Control and Ipsilateral groups. Therefore, the control of both legs of the injured subjects suffered small alterations compared to healthy subjects. Figures 3.5 to 3.7 show that the variability in the NCs among subjects was higher for the injured subjects (Contralateral and Ipsilateral group). However, some tendencies can be observed. The second NC (which activated mainly SO and GL) in the Ipsilateral group presented two peaks during the stance phase and the NCs of modules 1 and 3 were higher at the beginning of early stance compared with the Control group. This could be caused by the fact that the subject tried to stabilize the joints of the ipsilateral leg. In the same line of the observed results in the onset-offset patterns, Quadriceps and Hamstrings presented more cocontraction in the Ipsilateral group than in the Control one. In Figure 3.5, it can be observed that the fourth SV of the Ipsilateral group presented weights higher for the RF and VL (Quadriceps) and lower for the ST (Hamstrings) than in the Control group. This fact yielded similar Quadriceps and Hamstrings weights, indicating higher co-contraction at the knee. Figure 3.7 shows that the peak of the module 5 (mainly related to ED activation) at the transition between the stance to swing phase was higher for the Ipsilateral and Control group than for the Contralateral group. The explanation of this result in module 5 could be twofold: either the control of the injured leg is transferred to the ankle, as mentioned in [57], which is suggested with the comparison of Ipsilateral and Contralateral groups (also
42 Optimization and muscle synergy approaches for studying muscle redundancy during walking observed in onset-offset patterns); or the contralateral leg avoids providing a high acceleration to the body at toe off which would destabilize the injured leg during its initial stance, suggested by the smoother curve of the fifth NC of the contralateral leg compared to the one of the ipsilateral leg. Three main limitations of our study should be recognized. The first one is that, although the identification of the differences from a Control group could be useful to observe objective improvements during a rehabilitation treatment (to reinforce the clinical evaluation), a postsurgery follow-up study should be carried out to see how the muscle synergy components change along time. Some studies in the literature mention that the EMG pattern restores to levels similar to prior the injury [52,58], but authors of these studies analyzed individual EMG signals and no synergy components. The second limitation is the sample size. With a wider sample, a study could be carried out to evaluate whether the ACL-deficient subjects suffer other injuries or not. Comparing the muscle synergies with the observed tendencies from an ACL-deficient population could be useful to detect anomalies. The third limitation is related to the use of only one gait cycle. Due to space limitations, we only analyzed one gait trial. As mentioned, some separated gait trials were performed over the force plates and we picked the one with the cleaner data. Processing more than one trial could reduce intragroups variability yielding to more conclusive results. In conclusion, our initial hypothesis was fulfilled. Despite the similarities among groups, different trends were identified. The analysis of these muscle synergy tendencies can be useful as a follow-up study during a rehabilitation treatment. Another important field of application is in motion simulation. Recently, some studies used muscle synergy components to decrease the indeterminacy when calculating the muscle forces in dynamic simulations of healthy subjects [43,46,50]. Using muscle synergy components extracted from ACLdeficient subjects could be also useful to predict muscle forces in gait dynamic analyses of those subjects.
Chapter 4 4 Two-step optimization problem formulation for predicting knee muscle and contact forces during gait 4.1. Background Disorders affecting walking ability (e.g., osteoarthritis and stroke) are prevalent in society. Worldwide each year, approximately 15 million individuals suffer a stroke [59] and 250 million individuals are diagnosed with knee osteoarthritis [60]. Walking dysfunction from these and other disorders leads to a decreased quality of life and other serious health conditions such as heart disease and diabetes, thereby increasing the risk of death as well [61]. Unfortunately, current clinical interventions are largely ineffective at reversing walking impairments [62,63]. Even total joint replacement frequently does not achieve full normalization of walking function [64]. Improved clinical treatment methods are therefore needed to address this important society problem. Researchers have begun to explore using computational walking models to develop improved clinical interventions for walking impairments [65]. One of the primary challenges with this approach is indeterminacy of model-predicted muscle forces, since more muscle actuators exist than degrees of freedom in the skeleton. Researchers have explored several computational approaches to address the muscle redundancy problem. The most common approach found in the literature is static optimization, where a cost function is minimized one time frame at a time to find muscle forces that reproduce experimental joint moments calculated via inverse dynamics [6,49,66,67]. Musculoskeletal models used in this process are typically scaled versions of generic models available in the literature [68,69]. While this
44 Optimization and muscle synergy approaches for studying muscle redundancy during walking approach can be computationally efficient, it does not take advantage of experimental EMG data when available, and the correct physiological form of the cost function to be minimized is unknown. A growing alternate approach is to use an EMG-driven model whose muscle excitation inputs are taken directly from experimental EMG measurements [70–72]. With this approach, no assumptions are required about the form of the cost function being minimized, plus model parameter values are often calibrated to match the subject’s experimental joint moments from inverse dynamics. However, “flexibility” still remains in the solution process, since the absolute amplitude of each muscle EMG signal is difficult to determine, the number of EMG measurements is often limited, and EMG data are typically unavailable from large deep muscles (e.g., psoas). Regardless of which approach is used, the estimated muscle forces remain sensitive to both the computational method used to resolve muscle force indeterminacy and the parameter values used in the neuromusculoskeletal model. One way to address the dual problem of indeterminate muscle force solutions and unmeasurable model parameter values is to identify additional types of experimental data that could limit muscle force solutions and the model calibration process further. Neural control studies have shown that a large number of muscle EMG signals (8 to 32) collected during walking can be decomposed into three to six neural commands [35,73–75]. Recent work has also shown that neural commands extracted from a subset of experimentally measured EMG signals can reconstruct the shapes of the omitted EMG signals accurately [76]. Thus, it may be physiologically reasonable to use experimentally-derived neural commands as basis functions to construct all model-predicted muscle activations [50]. In addition, instrumented knee studies performed using force-measuring knee replacements have provided internal force data that permit at least two additional inverse dynamic knee loads to be used as constraints in the muscle force estimation and model calibration process [77,78]. While synergy-derived neural commands can limit only muscle activation shapes, knee contact force measurements can limit both the amplitudes and shapes of predicted activations. This study investigates how knowledge of knee contact forces affects calibration of neuromusculoskeletal model parameter values and subsequent prediction of leg muscle and
Two-step optimization formulation for predicting knee muscle and contact forces 45 knee contact forces. The two primary questions investigated were the following. First, can a static optimization that uses a well-calibrated neuromusculoskeletal model predict experimental knee contact forces accurately for multiple gait trials? This question addresses whether poorly calibrated model parameter values may be just as critical as an indeterminate muscle force solution in affecting predicted leg muscle and knee contact forces during walking. Second, which muscle-tendon model parameter values change the most when experimentally measured knee contact forces are not used as part of the model calibration process? This question addresses whether particular types of parameters or particular muscles require more refined calibration methods to achieve accurate knee contact force predictions. The static optimization approach used to address these questions limits predicted activations to be close to scaled experimental EMG signals (for muscles with experimental EMG data) or linear combinations of experimental neural commands (for muscles without experimental EMG data). 4.2. Methods 4.2.1 Experimental data Experimental data for this study were obtained from the Fourth Grand Challenge Competition to Predict In Vivo Knee Loads [77]. Surface marker, ground reaction, electromyographic (EMG), knee contact force, and single-plane fluoroscopic knee motion data were available from a single subject (gender: male, age: 88 years, mass: 65 kg, height: 166 cm) implanted with a force-measuring tibial prosthesis (right knee). The prosthesis possessed four uniaxial load cells located in the four quadrants of the tibial tray [79]. Six normal overground gait trials performed at a self-selected speed (1.26 ± 0.03 m/s) were selected for analysis. Available data included trajectories of 53 surface markers, ground reactions from three force plates, knee contact forces from the instrumented implant, and EMG data from 10 muscles (Adductor Magnus - Addmag; Biceps Femoris Longhead - Bflh; Gastrocnemius Lateralis - GasLat; Gastrocnemius Medialis - GasMed; Peroneus Longus - PerLong; Semimembranosus - Semimem; Soleus - Sol; Tibialis Anterior - TibAnt; Tensor Fascia Latae - TFL; Vastus Lateralis - VasLat).
46 Optimization and muscle synergy approaches for studying muscle redundancy during walking The experimental data were processed using standard methods. The EMG data were highpass filtered (fourth-order zero phase-lag Butterworth at 30 Hz), full-wave rectified, demeaned, and low-pass filtered (fourth-order zero phase-lag Butterworth at 6 Hz). Each processed EMG signal was resampled to 101 data points and normalized to its maximum value over all movement trials from the Fourth Grand Challenge Competition, including trials from other walking conditions (e.g., medial thrust gait). Ground reaction and knee contact force data were also low-pass filtered in a consistent manner (fourth-order zero phase-lag Butterworth at 6 Hz [80]. 4.2.2 Muscle synergy analysis Experimental neural commands were calculated by performing muscle synergy analysis on the 10 processed muscle EMG signals after they were passed through an activation dynamics model. Activation dynamics was modeled using a first-order ordinary differential equation [81], where the activation and deactivation time constants for each muscle were taken from the literature [82]. Muscle synergy analysis was performed on the experimental activations using a non-negative matrix factorization algorithm [39,50,83]. The analysis decomposed the 10 experimental activations into a pre-defined number of synergies (< 10), where each synergy consisted of a single time-varying neural command with a corresponding set of time-invariant weights (called a “synergy vector”) describing how the neural command contributed to each experimental activation. Each synergy vector was normalized to a maximum value of one, and its associated neural command was scaled such that the product of the synergy vector and its neural command did not change. The analysis was performed iteratively with the pre-defined number of synergies incremented by one each time until the total variance accounted surpassed 90%, which required 5 synergies. 4.2.3 Musculoskeletal model analyses A subject-specific musculoskeletal model of the pelvis and right leg (femur, patella, tibia/fibula, and foot) of the subject [77] was constructed in OpenSim [84] and used to calculate joint kinematics and inverse dynamic loads for the six normal walking trials. The model incorporated subject-specific bone and implant geometry and possessed 17 degrees of freedom (DOFs): 3 translations and 3 rotations defining the position and orientation of the pelvis with respect to ground, 3 rotations (flexion, adduction, and rotation) for the hip joint,
Two-step optimization formulation for predicting knee muscle and contact forces 53 total in all cases; see Table 4.2). Mean values of knee contact forces were also different between approaches (p<0.001 for medial, lateral, and total), see Table 4.3. Table 4.2. Medial, lateral and total contact force accuracy for both approaches. Mean and standard deviations of R2, RMSE and r values. Statistically significant differences (p<0.05) between the same quantities from both approaches are indicated by a star (*). Quantity Approach Medial Lateral Total R2 A 0.97±0.01* 0.88±0.05* 0.96±0.02* B 0.10±0.28* -3.43±1.74* -0.36±0.24* RMSE (N) A 52.62±16.37* 56.63±9.54* 92.80±26.03* B 322.76±63.89* 347.79±33.09* 559.90±32.11* r A 0.99±0.01* 0.95±0.02* 0.98±0.01* B 0.90±0.03* -0.10±0.10* 0.74±0.07* Table 4.3. Medial, lateral and total contact force similarities. Mean and standard deviations of mean contact force difference, R2, RMSD and r values between approaches. Statistically significant differences (p<0.05) between mean values from both approaches are indicated by a star (*). Quantity Medial Lateral Total MeanD (N) 223.9±50.15* 143.8±38.8* 367.7±80.3* R2 0.15±0.22 -2.82±1.44 -0.23±0.19 RMSD (N) 315.15±60.25 330.51±25.68 530.47±30.67 r 0.91±0.02 0.05±0.04 0.79±0.09 Figure 4.2. Experimental knee contact forces and mean knee contact force predictions for each approach (A and B). The grey area corresponds to the mean ±1 standard deviation for the experimental forces.
54 Optimization and muscle synergy approaches for studying muscle redundancy during walking 4.3.3 Leg muscle forces Changes in calibrated muscle parameter values lead to differences in muscle force predictions. Most mean muscle forces were different between approaches (29 muscles with p<0.05). Overall they were higher in Approach B by 9.58 N. Lateral muscles had the greatest differences between approaches (p<0.001; see Table 4.4). Eight muscles had mean force differences between approaches that were larger than 40 N and five were lateral muscles (bflh, bfsh, glmed1, glmed3, and vaslat). Some muscle forces were higher in Approach B, while others were higher in Approach A (Figure 4.3). Table 4.4. Similarity of model muscle forces obtained in Approach B relative to A, for medial, central, lateral, and all muscles from 6 gait trials. Mean and standard deviations of mean muscle forces, R2, RMSD and r values between approaches. Statistically significant differences (p<0.05) between mean values from both approaches are indicated by a star (*). Quantity Medial Central Lateral All MeanD (N) -3.13±7.46 13.91±7.05* 24.35±4.11* 9.58±2.66* R2 -4.61±2.00 -1.24±1.42 -90.70±13.73 -36.19±5.24 RMSD (N) 64.5±6.4 86.7±21.88 79.77±7.16 73.6±7.64 r 0.71±0.07 0.63±0.15 0.59±0.03 0.66±0.04 Figure 4.3. Muscle force values for muscles with the greatest mean differences between approaches A and B. The plotted area corresponds to the mean ± 1 standard deviation of all trials.
Two-step optimization formulation for predicting knee muscle and contact forces 55 4.3.4 Other relevant quantities Mean normalized fiber lengths were different for all muscles, though the shapes of the normalized fiber length curves were nearly identical between the two approaches (Table 4.5). In absolute terms, mean differences between approaches were small, and all curves remained within physiological operating ranges (Figure 4.4) [88,89]. Observed variations were consistent with the differences in passive muscle forces between the two approaches (Table 4.6), which were statistically different for 43 muscles (p<0.05). Passive forces were lower than 40 N for most muscles in both approaches. Table 4.5. Similarity of model normalized fiber lengths obtained in Approach B relative to A, for medial, central, lateral, and all muscles from 6 gait trials. Mean and standard deviations of mean normalized fiber forces, R2, RMSD and r values between approaches. Statistically significant differences (p<0.05) between mean values from both approaches are indicated by a star (*). Quantity Medial Central Lateral All MeanD -0.031±0.000* -0.047±0.000* -0.063±0.003* -0.045±0.001* R2 -3.66±0.93 -6.65±0.94 -5.62±1.06 -4.85±0.47 RMSD 0.075±0.000 0.125±0.000 0.128±0.003 0.102±0.001 r 1±0.00 1±0.00 1±0.00 1±0.00 Figure 4.4. Normalized fiber lengths for muscles with the greatest differences in mean tendon forces between approaches A and B. The plotted area corresponds to the mean ± 1 standard deviation of all trials.
56 Optimization and muscle synergy approaches for studying muscle redundancy during walking Table 4.6. Similarity of model passive forces obtained in Approach B relative to A, for medial, central, lateral, and all muscles from 6 gait trials. Mean and standard deviations of mean passive forces, R2, RMSD and r values between approaches. Statistically significant differences (p<0.05) between mean values from both approaches are indicated by a star (*). Quantity Medial Central Lateral All MeanD -4.30±0.61* 1.50±1.57 0.17±1.15 -1.75±0.86* R2 -4.69±1.16 -4.87±0.44 -636.43±152.44 -234.44±55.03 RMSD 12.24±0.81 33.83±1.43 23.83±2.09 19.89±1.06 r 0.91±0.00 1.00±0.00 0.93±0.09 0.93±0.03 Predicted muscle activations demonstrated different trends between the two approaches. For tracking of experimental and synergy-based activations, muscles with experimental EMG data had the lowest magnitude errors (lower mean RMSE values for Approach A) but highest shape errors (lower mean r values for Approach A) (see Tables 4.7 and 4.8). RMS errors were statistically different between the approaches (p<0.05) for all groups (medial, central and lateral muscles) and were generally lower for Approach B, while r values were statistically different between approaches for lateral muscles with associated experimental EMG (p=0.01, Figure 4.5) and medial muscles without associated experimental EMG data (p=0.004). Comparing predicted activations from the two approaches (Table 4.9), activation shapes were most similar for medial muscles (r=0.70) and least similar for lateral muscles (r=0.52). Mean activations were similar for both approaches, with any statistically significant differences and of limited practical importance. Nonetheless, RMS differences between approaches were on the order of 0.1 for all groups. Table 4.7. Accuracy of muscle activation predictions for the 16 muscles with associated experimental EMG data relative to their activations reconstructed from synergy components. Values indicate mean ± standard deviation from 6 gait trials. Statistically significant differences (p < 0.05) between the same quantities from both approaches are indicated by a star (*). Quantity Approach Medial Central Lateral All R2 A -1.51±0.35* -0.20±0.31 -0.85±0.27* -1.10±0.24* B -0.96±0.35* -0.24±0.20 -0.18±0.13* -0.58±0.24* RMSE A 0.05±0.00* 0.14±0.02* 0.11±0.01* 0.09±0.00* B 0.08±0.01* 0.12±0.01* 0.07±0.01* 0.08±0.00* r A 0.37±0.14 0.55±0.19 0.28±0.13* 0.36±0.11* B 0.43±0.18 0.57±0.14 0.44±0.04* 0.45±0.11*
Two-step optimization formulation for predicting knee muscle and contact forces 57 Table 4.8. Accuracy of muscle activation predictions for the 28 muscles without associated experimental EMG data relative to their activations reconstructed from synergy components. Values indicate mean ± standard deviation from 6 gait trials. Statistically significant differences (p < 0.05) between the same quantities from both approaches are indicated by a star (*). Quantity Approach Medial Central Lateral All R2 A -0.15±0.26* -0.10±0.41 0.29±0.20 0.01±0.20* B 0.08±0.21* 0.32±0.22 0.17±0.05 0.15±0.11* RMSE A 0.17±0.02* 0.13±0.03* 0.07±0.01* 0.13±0.02* B 0.12±0.02* 0.08±0.02* 0.10±0.01* 0.11±0.01* r A 0.42±0.14* 0.47±0.31 0.64±0.14 0.51±0.12* B 0.57±0.13* 0.71±0.13 0.56±0.03 0.59±0.07* Table 4.9. Similarity of muscle activation predictions in Approach B relative to A, for medial, central, lateral, and all muscles from 6 gait trials. Values indicate mean ± standard deviation from 6 gait trials. Mean and standard deviations of mean model activations, R2, RMSD and r values between approaches. Statistically significant differences (p<0.05) between mean values from both approaches are indicated by a star (*). Quantity Medial Central Lateral All MeanD -0.000±0.007 0.004±0.011 0.021±0.005* 0.009±0.004* R2 — -1.46±1.30 -47.75±15.57 — RMSD 0.08±0.01 0.09±0.02 0.09±0.01 0.09±0.01 r 0.70±0.07 0.67±0.14 0.52±0.03 0.63±0.05 Figure 4.5. Activations reconstructed from synergy components (aSyn, in solid lines) and model activations (Activations, in dashed lines) for muscles with associated experimental EMG. Asterisk* stands for statistically different r values between approaches A and B. 0 0.5 1addmagProx Activation 0 0.5 1addmagMid* 0 0.5 1addmagDist* 0 0.5 1addmagIsch* 0 0.5 1bflh Activation 0 0.5 1bfsh 0 0.5 1gaslat 0 0.5 1gasmed 0 0.5 1perlong Activation 0 0.5 1semimem 0 0.5 1semiten 0 0.5 1soleus 050 100 0 0.5 1tfl* % gait cycle Activation 050 100 0 0.5 1tibant % gait cycle 050 100 0 0.5 1vasint % gait cycle 050 100 0 0.5 1vaslat* % gait cycle Activation-ASyn Activation-A Activation-BSyn Activation-B
58 Optimization and muscle synergy approaches for studying muscle redundancy during walking 4.4. Discussion The goal of this study was to investigate differences in knee contact and leg muscle force predictions when those forces were used to calibrate the model (Approach A) and when no contact forces were available (Approach B), which is the most usual case. The algorithm to predict those forces consisted of a two-step synergy-based algorithm, which calibrated muscle parameters and predicted knee muscle and contact forces considering that muscle synergies helped to predict knee contact forces [50]. The most significant result was that one set of muscle-tendon parameters allowed static optimization to predict both medial and lateral knee contact forces with high accuracy in terms of magnitude (average RMSE=52.6N medial and RMSE=56.6 lateral) and shape (average r=0.99 medial and r=0.95 lateral). This accuracy was similar or higher than the obtained in other studies that predicted unblinded knee contact forces for one gait trial [50,90]. As expected, when knee contact forces were not used during the calibration process, their prediction was poorer. Medial contact force was overpredicted (average RMSE=322.8N and r=0.90) and lateral contact force had a significantly different shape compared to the experimental one (RMSE=347.8N and r=-0.1). These results indicated that the use of muscle synergies was not sufficient to obtain a good contact force prediction when no contact force information was used to calibrate the model. The differences in the calibration of muscle parameter values between approaches A and B can explain the variations in muscle and knee contact force predictions. The use of knee contact force information when calibrating muscle parameters (Approach A) led to different muscle moment arms, especially in knee flexion and hip rotation moments, which were higher than in Approach B. Moreover, optimal fiber lengths for lateral muscles were statistically higher when no contact forces were used during the calibration. Scale factors of muscle activations for muscles with associated EMG also had differences between approaches. This fact suggests that a better calibration of model parameter values was achieved in Approach A and, therefore, those differences in model parameters between approaches led to different normalized fiber lengths, and thus different activations and muscle forces. It was observed that the main differences between both analyzed approaches came from muscle activations and forces of the lateral muscles. From the eight muscles with greatest
Two-step optimization formulation for predicting knee muscle and contact forces 59 differences in tendon forces (>40N), five of them were from the lateral side (bflh, bfsh, glmed1, glmed3, and vaslat). It is possible that the use of alternate motion patterns (especially ones that preferentially excite lateral muscles) in the calibration process could improve the similarity of calibrated muscle-tendon model parameter values. Our hypothesis was that those muscles would have a higher contribution to hip adduction and subtalar moments (tracked in the inner level) and a better calibration could be achieved. Overfitting is a common issue in optimization problems. In our study, muscle activations were predicted in an inner level for six gait trials at a time, where muscle parameters could not be changed and no knee contact force information was transferred from the outer to the inner level. Therefore, there was no guarantee that another set of muscle parameters made the prediction of knee contact forces as accurate as the obtained one. Nevertheless, we tried the same algorithm (same cost functions and optimization formulations) to calibrate muscle parameters with three overground gait trials (outer and inner level) and then, we predicted knee contact forces with other three overground gait trials (only inner level) with the previously obtained muscle parameters. The results suggested that the obtained set of muscle parameters could predict the knee contact forces with high accuracy (Appendix A.3, Figure A.1). Further validations of the presented formulation are out of the scope of this study. In order to validate the formulation in different motions, other trials should be used when calibrating and predicting. The formulation should also be validated in other subjects wearing a knee instrumented implant. Muscle parameter calibration is critical as suggested by other studies. For example, Manal and Buchanan [91] reported knee contact force predictions for two trials (normal walking and medial thrust gait). However, to obtain accurate knee contact force predictions for both trials, they had to make some model parameter values different from the two trials. In contrast, in the presented study one set of muscle parameters was obtained that predicted accurately knee contact forces for six trials at a time (Approach A). This study had several limitations. They were based on the number of tested subjects and the number of different experimental trials. Data for only one subject was used to calculate knee muscle and contact forces, further analysis in other subjects should be carried out to validate the formulation in other individuals. As suggested, the use of other type of trials in which
60 Optimization and muscle synergy approaches for studying muscle redundancy during walking lateral muscles play a more important role could improve the calibration of these muscle parameters. As future work, the formulation will be tested in other human movements, such as in crouch gait, trunk sway or doing squats in one leg. The lateral ligaments would be more stressed in those trials [92], so, a proper ligament model should be used to reproduce the physiological joint dynamics. Other future work could be focused on investigating whether adding more physiological constrains, when no knee contact force information is used during the calibration, could allow obtaining better knee contact force predictions. On the one hand, less neural commands would decrease the indeterminacy of the muscle activation calculation. And, on the other hand, proper moment arm relations could be used to modify the muscle contributions to the knee superior-inferior force or the muscle adduction moment when no contact forces are available. In conclusion, one set of muscle-tendon parameters that predicts knee contact forces with high accuracy was obtained calibrating six gait trials at a time. It was demonstrated that when no knee contact force information was available, poorer knee contact forces were obtained, especially for the lateral knee contact force. In order to obtain a higher accuracy of the knee contact force predictions when those forces are not available, future work should be focused on calibrating the model with trials which lead to a better calibration of lateral muscles.
Chapter 5 5 Conclusions This thesis comprises three works related to the study of muscle actuation redundancy during walking. In the first one, a simple static optimization approach to predict muscle forces in a single ACL-deficient subject is introduced; secondly, the differences in muscle excitation patterns of ACL-deficient and healthy subjects are investigated through a muscle synergy analysis; and finally, in the last study, a patient-specific model to predict physiologically consistent muscle and knee contact forces is presented. The first study was aimed at analyzing what cost function best fitted model muscle activations and EMG signals in a subject with ACL-deficiency. A generic musculoskeletal model with 8 muscles was scaled in OpenSim to apply inverse kinematics and inverse dynamics. The resultant inverse dynamics loads were consistent with other studies related to ACL-deficient subjects. The results showed that a weighted cost function penalizing biarticular muscles acting on the hip and the knee (Hamstrings and Rectus Femoris) best fitted muscle activations and EMG signals. The hypothesis was fulfilled, since the obtained best cost function lead to different magnitudes of model muscle activations compared to an equally weighted cost function. Some limitations need to be noticed, for example the fact that static optimization itself (only minimizing a cost function) avoids co-contraction of antagonistic muscles. Furthermore, in order to get stronger conclusions, such study should be carried out with more subjects, as well as with different ways to group the muscles in the cost function.
62 Optimization and muscle synergy approaches for studying muscle redundancy during walking With these limitations in mind, the second study was carried out to compare muscle excitations in a group of ACL-deficient subjects and a group of healthy subjects. ACLdeficient subjects were selected by clinicians of the Knee Surgery Unit at the Catalan Institute of Trauma and Sport Medicine (ICATME). Muscle excitations were studied at two levels: EMG signals (onset-offset patterns) and muscle synergies. As far as the author is concerned, no previous muscle synergy analysis in a group of ACL-deficient subjects appears in the literature thus far. EMG data of 8 muscles per leg were measured in 10 healthy subjects and 18 ACL-deficient subjects (before the surgery) when walking. At the level of the onset-offset activation patterns, similar results to the literature were obtained since co-contraction between Quadriceps and Hamstrings in ACL-deficient subjects was longer than in healthy subjects. This fact suggested that these muscles actuated to stabilize the knee joint, that is, they co-contracted to prevent high displacements between the tibia and the femur. At the level of muscle synergies, some differences were observed, like the higher peak in the fifth NC (which mainly activates Extensor Digitorum Longus) for the ipsilateral leg compared to the contralateral leg at the transition between the stance and the swing phases. The major use of ankle muscles in the ipsilateral leg as compared to the contralateral leg was in agreement with the fact that the control of the ipsilateral leg is shifted away from the injury, that is, from the knee to the ankle. Some differences between the Control group and injured subjects (both for ipsilateral and contralateral legs) were also identified: firstly, lower values in the first NC (mainly Gluteus Medius) in the Control group at early stance; secondly, lower values of the third NC (mainly Tibialis Anterior) in the Control group at middle stance; and finally, a greater peak of the second NC (mainly Gastrocnemius Lateralis and Soleus) in the Control group during the stance. This study has the following limitations. In terms of mathematical decomposition of signals, more than one gait trial should be considered in a future work when calculating muscle SVs. In this case, the results would be more accurate. This aspect could not be accomplished due to space limitations of the laboratory. This study was carried out in ACL-deficient subjects before the surgical reconstruction. A future suggested study would be to involve
Supplementary data for the two-step optimization algorithm 69 where i T s sl is the scale factor for the slack length of the tendon of muscle i. It constrains scale factors for slack length of the tendons to be between 0.8 and 1.2. Variable deviation constrain terms 0 10 44 0 0 144 0.2 0.5 ii MT s ii MT s sl sl MT is sl sl Jsl sl (A.11) It penalizes deviations between scale factors of the slack length of tendons and the optimal fiber lengths higher than 20%. 10 6 22 0.1 M l M ngroups MM uv l dev uv ll Jx (A.12) It penalizes deviations of mean normalized fiber lengths higher than 0.1 for the pairs of muscles showed in Table A.3. 10 6 30or6x23 ma ngroups uv maGroupdev uv thres ma ma Jx ma (A.13) It penalizes deviation of mean moment arms higher than 5 mm in moment loads, or 0.015 in the knee superior-inferior force contribution, for the pairs of muscles showed in Table A.4.
70 Optimization and muscle synergy approaches for studying muscle redundancy during walking Table A.3. Groups of muscles with similar normalized fiber length. The first muscle of the group was the reference. adductor magnus proximal, adductor magnus middle, adductor magnus distal, adductor magnus ischial biceps femoris long head, biceps femoris short head extensor digitorum longus, extensor hallucis longus flexor digitorum longus, flexor hallucis longus gastrocnemius lateralis, gastrocnemius medialis gluteus maximus superior, gluteus maximus middle, gluteus maximus inferior gluteus medius superior, gluteus medius middle, gluteus medius inferior gluteus minimus superior, gluteus minimus middle, gluteus minimuz inferior semimembranosus, semitendinosus, biceps femoris long head, gracilis iliacus, psoas peroneus brevis, peroneus longus vastus intermedius, vastus medialis, vastus lateralis, rectus femoris soleus, tibialis posterior Table A.4. Groups of muscles with similar moment arm deviations. The first muscle of the group was the reference. Hip flexion biceps femoris long head, semimembranosus, semitendinosus gluteus minimus anterior, gluteus minimus middle gluteus medius anterior, gluteus medius middle, gluteus medius superior Hip adduction gluteus medius anterior, gluteus medius middle adductor magnus distal, adductor magnus ischial gluteus minimus anterior, gluteus minimus middle, gluteus minimus superior biceps femoris long head, semimembranosus Hip rotation adductor magnus proximal, adductor magnus middle, adductor magnus ischial semimembranosus, semitenendinosus gluteus maximus superior, gluteus maximus middle Knee flexion gracilis, semimembranosus vastus interior, vastus lateralis, vastus medialis Knee adduction semimembranosus, semitendinosus rectus femoris, vastus intermedius Knee superior-inferior biceps femoris long head, semimembranosus, semitendinosus vastus intermedius, vastus medialis, rectus femoris, Sartorius Ankle flexor digitorus longus, flexor hallucis longus gastrocnemius lateralis, gastrocnemius medialis peroneus brevis, peroneus longus Subtalar gastrocnemius lateralis, gastrocnemius medialis flexor digitorus longus, flexor hallucis longus, tibialis posterior
Supplementary data for the two-step optimization algorithm 71 A.3. Supplementary results Muscle parameters: optimal fiber lengths and slack lengths of the tendons. Tables A.5 and A.6 report mean and standard deviation of percent differences between model optimal fiber lengths and slack length of the tendon and literature values. Table A.7 shows the moment arm deviations for both approaches. Table A.8 shows the differences in moment arm deviations between both approaches. Table A.5. Mean and standard deviation of percent difference of optimal fiber lengths from literature values. Approach Medial Lateral Central A 6.5 ± 16.1 1.0 ± 16.9 8.2 ± 17.9 B 10.7 ± 13.8 12.8 ± 15.4 15.9 ± 10.0 Table A.6. Mean and standard deviation of percent difference of slack length of the tendons from literature values. Approach Medial Lateral Central A 3.4 ± 16.3 5.8 ± 14.3 7.6 ± 5.6 B 4.2 ± 15.5 4.6 ± 16.3 5.9 ± 5.5 Table A.7. Mean and standard deviation of moment arm deviations for both approaches (in mm, except for sup-inf force which is dimensionless). In parenthesis, differences in percentage relative to mean moment arm. Asterisk* means statistically different from zero (p<0.05). A B Hip flexion -2 ± 9 (43.7 ± 134.2) -1 ± 12 (64 ± 179.4) Hip adduction 0.08 ± 9 (22.8 ± 78.9) 0.03 ± 11 (40.9 ± 152.1) Hip rotation 3 ± 8 (144.7 ± 421.1) -2 ± 9 (6.1 ± 74.9) Knee flexion 14 ± 9* (33.8 ± 18.4*) -1 ± 5 (-5.5 ± 19.7) Knee adduction -1 ± 11 (20.5 ± 112.7) NA Knee sup-inf force -0.004 ±0.011 (-0.4 ± 1.18) NA Ankle 1 ± 10 (10.9 ± 52.3) 4 ± 10 (16.2 ± 39.6) Subtalar -6 ± 11 (-16.1 ± 59.0) 0.07 ± 8 (90.0 ± 238.7)
72 Optimization and muscle synergy approaches for studying muscle redundancy during walking Table A.8. Mean and standard deviation of absolute differences, and percentage differences between parenthesis, of moment arm deviations between approaches. Quantity Medial Central Lateral All madev hipflex -0.3±11.7 (2.69±55.11) 0.3±0.9 (6.76)† 1.8±3.8 (18.56±26.82) 0.9±0.01 (8.72±45.40) madev hipadd 0.3±7.0 (32.35±104.73) 2.4±6.3 (95.44)† -1.2±5.5 (5.53±38.39) 0.1±8.0 (25.03±85.73) madev hiprot -3.0±6.7 (4.04±139.33) -0.7±1.7 (-31.66)† -2.3±7.3 (-12.36±61.10) -4.0±7.9* (-3.54±111.53) madev kneeflex -3.0±6.6* (-20.16±13.99)* -8.3±14.3 (-41.13±5.83) -3.3±7.3 (-23.71±12.22) -14.2±10.7* (-24.84±13.91)* madev kneeadd 1.6±6.0 (174.22±269.74) -2.5±4.3 (-66.10±16.29) 0.1±5.6 (22.77±76.27) 1.3±10.6 (78.95±204.02) madev kneesup-inf 0.002±0.007 (0.64±1.34) 0.005±0.008 (1.75±0.27) -0.001±0.003 (-0.37±0.54) 0.004±0.011 (0.42±1.20) madev ankle 0.1±4.6 (4.77±45.56) 0.8±8.2 (0.09±27.95) 1.5±3.4 (104.37±141.49) 2.7±9.7 (29.81±83.84) madev subtalar 0.1±0.5 (12.12±5.95) 1.2±8.3 (47.97±96.99) 1.5±6.5 (177.60±320.19) 3.4±10.2 (79.69±178.15) †Only one value A previous study, not presented in this document, was carried out in order to evaluate the ability to predict knee contact forces with gait trials which were not used during the calibration. In this case, three gait trials were used to calibrate muscle parameters and other three gait trials were used to predict knee contact forces. The results are shown in Figure A.1.
Supplementary data for the two-step optimization algorithm 73 Figure A.1. Knee contact force predictions (Medial, Lateral and Total) for the three gait trials where these forces were used during the calibration (Calibration + Prediction) and for other three gait trials using the same set of muscle parameters (Prediction). 050 100 0 900 1800 % gait cycle 0 600 1200 Force (N) R2=0.97 / RMSE=67.9 0 450 900 R2=0.87 / RMSE=59.8 0 900 1800 R2=0.95 / RMSE=114.1 0 600 1200 Force (N) R2=0.93 / RMSE=77.2 0 450 900 R2=0.90 / RMSE=52.8 0 900 1800 R2=0.93 / RMSE=115.5 0 600 1200 Force (N) R2=0.99 / RMSE=33.8 0 450 900 R2=0.84 / RMSE=63.6 0 900 1800 R2=0.97 / RMSE=90.4 0 600 1200 Force (N) R2=0.89 / RMSE=105.6 0 450 900 R2=0.85 / RMSE=67.4 0 900 1800 R2=0.92 / RMSE=130.4 0 600 1200 Force (N) R2=0.90 / RMSE=103.3 0 450 900 R2=0.67 / RMSE=84.5 0 900 1800 R2=0.91 / RMSE=135.4 050 100 0 600 1200 % gait cycle Force (N) R2=0.95 / RMSE=75.4 050 100 0 450 900 % gait cycle R2=0.82 / RMSE=92.3 Calibration + Prediction Prediction LateralMedial Total
74 Optimization and muscle synergy approaches for studying muscle redundancy during walking
Appendix B: Publications Publications derived from the PhD thesis Journal articles Serrancolí G., Kinney A.L., Fregly B.J., Font-Llagunes J.M. Two-step optimization problem formulation for predicting knee muscle and contact forces during gait. To be submitted to the Journal of Biomechanics. Serrancolí G., Monllau J.C., Font-Llagunes J.M. Analysis of the Activation-Deactivation Pattern and Muscle Synergies in ACL-Deficient Subjects during Gait. Submitted to the journal of Clinical Biomechanics. Serrancolí G., Font-Llagunes J.M., Barjau A. A weighted cost function to deal with the muscle force sharing problem in injured subjects: A single case study. Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics 2014, 228:241–251. Congress presentations Serrancolí G., Kinney A.L., Fregly B.J., Font-Llagunes J.M. How do in vivo knee contact force data affect calibration of muscle-tendon model parameter values? VI International Conference on Computational Bioengineering. Barcelona (Catalonia), September 2015. Submitted.
76 Optimization and muscle synergy approaches for studying muscle redundancy during walking Serrancolí G., Kinney A.L., Fregly B.J., Font-Llagunes J.M. What can knee contact force data teach us about calibration of muscle-tendon model parameter values? Summer Biomechanics, Bioengineering and Biotransport Conference. Snowbord, Utah (USA), June 2015. Submitted. Serrancolí G., Kinney A.L., Fregly B.J., Font-Llagunes J.M. Sinergy-based two-level optimization for predicting knee contact forces during walking. 25th Congress of the International Society of Biomechanics. Edinburg (UK), July 2015. Accepted. Serrancolí G., Monllau J.C., Font-Llagunes J.M. A comparative study of the muscle synergy patterns in healthy and ACL-deficient subjects. 25th Congress of the International Society of Biomechanics. Edinburg (UK), July 2015. Accepted. Serrancolí G., Kinney A.L., Fregly B.J., Font-Llagunes J.M. Prediction of knee contact forces using synergy-based two-level optimization. ECCOMAS Thematic Conference on Multibody Dynamics. Barcelona (Catalonia), June 2015. Accepted. Serrancolí, G. Analysis of the Force–Sharing Problem in the Biomechanics of Human Motion. 2a Jornada d’Investigadors Predoctorals Interdisciplinària. Barcelona, February 2014. Serrancolí, G., Walter, J.P., Kinney, A.L., Barjau, A., Fregly, B.J., Font‐Llagunes, J.M. Formulation to predict lower limb muscle forces during gait. III Reunión del Capítulo Español de la Sociedad Europea de Biomecánica (ESB). Barcelona (Catalonia), October 2013. Serrancolí, G., Walter, J.P., Kinney, A.L., Fregly, B.J., Font‐Llagunes, J.M. Optimization problem formulation for predicting knee muscle and contact forces during gait. XIV International Symposium on Computer Simulation in Biomechanics. Natal (Brazil), August 2013. Serrancolí, G., Font‐Llagunes, J.M., Barjau, A. Determination of the strategy for solving the muscle force sharing problem in patients suffering ACL rupture. Congress on Numerical Methods in Engineering. Bilbao (Spain), June 2013.
Supplementary data for the two-step optimization algorithm 77 Serrancolí, G., Font‐Llagunes, J.M., Barjau, A. On the force sharing problem in patients suffering joint pain. 18th Congress of the European Society of Biomechanics. Lisbon (Portugal), July 2012. Other congress presentations in line with the research of the thesis Tassani S., Peiret A., Bosch E., Serrancolí S., Font-Llagunes J.M., Noailly J. A multi-scale study of the hip joint mechanics: influence of inertial forces. 21st Congress of the European Society of Biomechanics. Prague (Czech Republic), July 2015. Submitted. Ilzarbe A., Serrancolí G., Font‐Llagunes J.M. Simulation of active orthosis-assisted gait considering user-device cooperation and interface contact forces. ECCOMAS Thematic Conference on Multibody Dynamics. Barcelona (Catalonia), June 2015. Accepted. Ilzarbe A., Serrancolí G., Font‐Llagunes J.M. A dynamic analysis of an integrated lower limb‐orthosis biomechanical model for interface contact pressures estimation. IV Reunión del Capítulo Español de la Sociedad Europea de Biomecánica (ESB). Valencia (Spain), November 2014. Peiret, A., Bosch, E., Serrancolí, G., Noailly, J., Font‐Llagunes, J.M. Multiscale analysis of the hip joint: translate mechanical information from inverse analyses of body motion into boundary loads for finite element calculations of organ/tissue biomechanics. 7th Annual Symposium on Bioengineering and Nanomedicine. Barcelona (Catalonia), September 2014. Peiret, A., Bosch, E., Serrancolí, G., Noailly, J., Font‐Llagunes, J.M. A multi‐scale study of the hip joint mechanics using rigid‐body inverse dynamics and finite element analysis. WCCM XI ‐ ECCM V ‐ ECFD VI Congress. Barcelona (Catalonia), July 2014. Font‐Llagunes, J.M., Arroyo, G., Serrancolí, G., Romero, F. A powered lower limb orthosis for gait assistance in incomplete spinal cord injured subjects. 4th International Symposium on Applied Sciences in Biomedical and Communication Technologies. Barcelona (Catalonia), October 2011. 050 100 0 900 1800 % gait cycle 0 600 1200 Force (N) Medial Contact Force (N) R2=0.97 / RMSE=67.9 0 450 900 Lateral Contact Force (N) R2=0.87 / RMSE=59.8 0 900 1800 Inf.-Sup. Contact force (N) R2=0.95 / RMSE=114.1 0 600 1200 Force (N) R2=0.93 / RMSE=77.2 0 450 900 R2=0.90 / RMSE=52.8 0 900 1800 R2=0.93 / RMSE=115.5 0 600 1200 Force (N) R2=0.99 / RMSE=33.8 0 450 900 R2=0.84 / RMSE=63.6 0 900 1800 R2=0.97 / RMSE=90.4 0 600 1200 Force (N) R2=0.89 / RMSE=105.6 0 450 900 R2=0.85 / RMSE=67.4 0 900 1800 R2=0.92 / RMSE=130.4 0 600 1200 Force (N) R2=0.90 / RMSE=103.3 0 450 900 R2=0.67 / RMSE=84.5 0 900 1800 R2=0.91 / RMSE=135.4 050 100 0 600 1200 % gait cycle Force (N) R2=0.95 / RMSE=75.4 050 100 0 450 900 % gait cycle R2=0.82 / RMSE=92.3
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