Critical analysis of musculoskeletal modelling complexity in multibody biomechanical models of the upper limb

The inverse dynamics technique applied to musculoskeletal models, and supported by optimisation techniques, is used extensively to estimate muscle and joint reaction forces. However, the solutions of the redundant muscle force sharing problem are sensitive to the detail and modelling assumptions of the models used. This study presents four alternative biomechanical models of the upper limb with different levels of discretisation of muscles by bundles and muscle paths, and their consequences on the estimation of the muscle and joint reaction forces. The muscle force sharing problem is solved for the motions of abduction and anterior flexion, acquired using video imaging, through the minimisation of an objective function describing muscle metabolic energy consumption. While looking for the optimal solution, not only the equations of motion are satisfied but also the stability of the glenohumeral and scapulothoracic joints is preserved. The results show that a lower level of muscle discretisation provides worse estimations regarding the muscle forces. Moreover, the poor discretisation of muscles relevant to the joint in analysis limits the applicability of the biomechanical model. In this study, the biomechanical model of the upper limb describing the infraspinatus by a single bundle could not solve the complete motion of anterior flexion. Despite the small differences in the magnitude of the forces predicted by the biomechanical models with more complex muscular systems, in general, there are no significant variations in the muscular activity of equivalent muscles.     

A multi-phase optimal control technique for the simulation of a human vertical jump

A multi-phase optimal control technique is presented that can be used to solve dynamic optimization problems involving musculoskeletal systems. The biomechanical model consists of a set of differential equations describing the dynamics of the multi-body system and the generation of the dynamic forces of the human muscles. Within the optimization technique, subintervals can be defined in which the differential equations are continuous. At the boundaries the dimension of the state- and control vector as well as the dimension of the right-hand side may change. The problem is solved by a multiple shooting approach which converts the problem into a non-linear program. The method is applied to simulate a human jump movement.     

Modelling, simulation and optimisation of a human vertical jump.

This paper describes an efficient biomechanical model of the human lower limb with the aim of simulating a real human jump movement consisting of an upword propulsion, a flying and a landing phase. A multiphase optimal control technique is used to solve the muscle force sharing problem. To understand how intermuscular control coordinates limb muscle excitations, the human body is reduced to a single lower limb consisting of three rigid bodies. The biomechanical system is activated by nine muscle-tendon actuators representing the basic properties of muscles during force generation. For the calculation of the minimal muscle excitations of the jump movement, the trajectory of the hip joint is given as a rheonomic constraint and the contact forces (ground reaction forces) are determined by force plates. Based on the designed musculoskeletal model and on the differential equations of the multibody system, muscle excitations and muscle forces necessary for a vertical jump movement are calculated. The validity of the system is assessed comparing the calculated muscle excitations with the registered surface electromyogramm (EMG) of the muscles. The achieved results indicate a close relationship between the predicted and the measured parameters.     

A Novel Application of Musculoskeletal Ultrasound Imaging

Ultrasound is an attractive modality for imaging muscle and tendon motion during dynamic tasks and can provide a complementary methodological approach for biomechanical studies in a clinical or laboratory setting. Towards this goal, methods for quantification of muscle kinematics from ultrasound imagery are being developed based on image processing. The temporal resolution of these methods is typically not sufficient for highly dynamic tasks, such as drop-landing. We propose a new approach that utilizes a Doppler method for quantifying muscle kinematics. We have developed a novel vector tissue Doppler imaging (vTDI) technique that can be used to measure musculoskeletal contraction velocity, strain and strain rate with sub-millisecond temporal resolution during dynamic activities using ultrasound. The goal of this preliminary study was to investigate the repeatability and potential applicability of the vTDI technique in measuring musculoskeletal velocities during a drop-landing task, in healthy subjects. The vTDI measurements can be performed concurrently with other biomechanical techniques, such as 3D motion capture for joint kinematics and kinetics, electromyography for timing of muscle activation and force plates for ground reaction force. Integration of these complementary techniques could lead to a better understanding of dynamic muscle function and dysfunction underlying the pathogenesis and pathophysiology of musculoskeletal disorders.     

Deterministic global optimization algorithm based on outer approximation for the parameter estima- tion of nonlinear dynamic biological systems

ABSTRACT: BACKGROUND: The estimation of parameter values for mathematical models of biological systems is an optimization problem that is particularly challenging due to the nonlinearities involved. One major difficulty is the existence of multiple minima in which standard optimization methods may fall during the search. Deterministic global optimization methods overcome this limitation, ensuring convergence to the global optimum within a desired tolerance. Global optimization techniques are typically classified into stochastic and deterministic. The former typically lead to lower CPU times but offer no guarantee of convergence to the global minimum in a finite number of iterations. In contrast, deterministic methods provide solutions of a given quality (i.e., optimality gap), but tend to lead to large computational burdens. RESULTS: This work presents a deterministic outer approximation-based algorithm for the global optimization of dynamic problems arising in the parameter estimation of models of biological systems. Our approach, which offers a theoretical guarantee of convergence to the global minimum, reformulating the set of ordinary differential equations into an equivalent set of algebraic equations through the use of orthogonal collocation methods, giving rise to a nonconvex nonlinear programming (NLP) problem. This nonconvex NLP is decomposed into two hierarchical levels: a master mixed-integer linear programming problem (MILP) that provides a rigorous lower bound on the optimal solution, and a reduced-space slave NLP that yields an upper bound. The algorithm iterates between these two levels until a termination criterion is satisfied. CONCLUSION: The capabilities of our approach were tested in two benchmark problems, in which the performance of our algorithm was compared with that of the commercial global optimization package BARON. The proposed strategy produced near optimal solutions (i.e., within a desired tolerance) in a fraction of the CPU time required by BARON.     

A complete set of control equations for the human musculo-skeletal system

A mathematical model of the total human musculo-skeletal system is presented. The model comprises a link-mechanical and a musculo-mechanical set of ordinary first-order differential equations which describe the dynamics of the segment model and muscle model respectively. The interdependence of the two sets of equations is demonstrated. The set of musculo-mechanical equations contains the two neuromuscular control parameters motor unit recruitment and stimulation rate, and the significance of such a representation for a control-theoretical treatment of musculo-skeletal systems is discussed. Finally, after a short discussion of the successful application of the present model in the prediction of an optimal human motion, further possibilities are indicated of the use of the model for investigations into the control behaviour of musculo-skeletal systems.     

The numerical solution of stiff differential equations

This paper first discusses the conditions in which a set of differential equations should give stable solutions, starting with linear systems assuming that these do not differ greatly in this respect from non-linear systems. Methods of investigating the stability of particular systems are briefly discussed. Most real biochemical systems are known from observation to be stable, but little is known of the regions over which stability persists; moreover, models of biochemical systems may not be stable, because of inaccurate choice of parameter values.The separate problem of stability and accuracy in numerical methods of approximating the solution of systems of non-linear equations is then treated. Stress is laid on the consistently unsatisfactory results given by explicit methods for systems containing "stiff" equations, and implicit multistep methods are particularly recommended for this class of problem, which is likely to include many biochemical model systems. Finally, an iteration procedure likely to give convergence both in multistep methods and in the steady-state approach is recommended, and areas in which improvement in methods is likely to occur are outlined.     

A dynamic model of the forearm including fatigue

A biomechanical model of the forearm, consisting of 61 muscle-tendon systems or tendons and 8 sections, is presented. The model can be used to calculate the muscle forces when resultant of the external forces and the motion is known. Calculations are based on constraints of muscle forces, joint forces, contact forces, and tendon junctions, and a load sharing principle telling which of the feasible solutions are likely and which are not. Fatigue is accounted for by updating the upper limits of the muscle forces according to the loading history. As an example, the model is used to predict the load sharing between the fingers when they are pressed against a table with a given total force.     

Spatial reconstruction of the human motion based on images of a single camera

The inverse dynamic analysis procedures used in the study of the human gait require that the kinematics of the supporting biomechanical model is known beforehand. The first step to obtain the kinematic data is the reconstruction of human spatial motion, i.e., the evaluation of the anatomic points positions that enables to uniquely define the position of all anatomical segments. In photogrammetry, the projection of each anatomical point is described by two linear equations relating its three spatial coordinates with the two coordinates of the projected point. The need for the image of two cameras arises from the fact that three equations are necessary to find the original spatial position of the anatomical point. It is shown here that the kinematic constraint equations associated with a biomechanical model can be used as the extra set of equations required for the reconstruction process, instead of the equations associated with the second camera. With this methodology, the system of equations arising from the point projections and biomechanical model kinematic constraints are solved simultaneously. Since the system of equations has multiple solutions for each image, a strategy based on the minimization of the cost function associated to the smoothness of the reconstructed motion is devised, leading to an automated computer procedure enabling a unique reconstruction.     

Human leg model predicts ankle muscle-tendon morphology, state, roles and energetics in walking

A common feature in biological neuromuscular systems is the redundancy in joint actuation. Understanding how these redundancies are resolved in typical joint movements has been a long-standing problem in biomechanics, neuroscience and prosthetics. Many empirical studies have uncovered neural, mechanical and energetic aspects of how humans resolve these degrees of freedom to actuate leg joints for common tasks like walking. However, a unifying theoretical framework that explains the many independent empirical observations and predicts individual muscle and tendon contributions to joint actuation is yet to be established. Here we develop a computational framework to address how the ankle joint actuation problem is resolved by the neuromuscular system in walking. Our framework is founded upon the proposal that a consideration of both neural control and leg muscle-tendon morphology is critical to obtain predictive, mechanistic insight into individual muscle and tendon contributions to joint actuation. We examine kinetic, kinematic and electromyographic data from healthy walking subjects to find that human leg muscle-tendon morphology and neural activations enable a metabolically optimal realization of biological ankle mechanics in walking. This optimal realization (a) corresponds to independent empirical observations of operation and performance of the soleus and gastrocnemius muscles, (b) gives rise to an efficient load-sharing amongst ankle muscle-tendon units and (c) causes soleus and gastrocnemius muscle fibers to take on distinct mechanical roles of force generation and power production at the end of stance phase in walking. The framework outlined here suggests that the dynamical interplay between leg structure and neural control may be key to the high walking economy of humans, and has implications as a means to obtain insight into empirically inaccessible features of individual muscle and tendons in biomechanical tasks.     

Simulated parallel annealing within a neighborhood for optimization of biomechanical systems

Optimization problems for biomechanical systems have become extremely complex. Simulated annealing (SA) algorithms have performed well in a variety of test problems and biomechanical applications; however, despite advances in computer speed, convergence to optimal solutions for systems of even moderate complexity has remained prohibitive. The objective of this study was to develop a portable parallel version of a SA algorithm for solving optimization problems in biomechanics. The algorithm for simulated parallel annealing within a neighborhood (SPAN) was designed to minimize interprocessor communication time and closely retain the heuristics of the serial SA algorithm. The computational speed of the SPAN algorithm scaled linearly with the number of processors on different computer platforms for a simple quadratic test problem and for a more complex forward dynamic simulation of human pedaling.     

MusIC method enhancement by a sensitivity study of its performance: Application to a lower limbs musculoskeletal model

Providing a biomechanical feedback during experimental sessions is a real outcome for rehabilitation, ergonomics or training applications. However, such applications imply a fast computation of the biomechanical quantities to be observed. The MusIC method has been designed to solve quickly the muscle forces estimation problem, thanks to a database interpolation. The current paper aims at enhancing its performance. Without generating any database, the method allows to identify optimal densities (number of samples contained in the database) with respect to the method accuracy and the off-line computation time needed to generate the database. On a lower limbs model (12 degrees of freedom, 82 muscles), thanks to this work, the MusIC method exhibits an accuracy error of 0.1% with an off-line computation time lower than 10?minutes. The on-line computation frequency (number of samples computed per second) is about 58?Hz. Thanks to these improvements, the MusIC method can be used to produce a feedback during an experimentation with a wide variety of musculoskeletal models or cost functions (used to share forces into muscles). The interaction between the subject, the experimenter (e.g. trainer, ergonomist or clinician) and the biomechanical data (e.g. muscle forces) in experimental sessions is a promising way to enhance rehabilitation, training or design techniques.     

Dynamic motion planning for the design of robotic gait rehabilitation

In this paper we examine a method to control the stepping motion of a paralyzed person suspended over a treadmill using a robot attached to the pelvis. A leg swing motion is created by moving the pelvis without contact with the legs. The problem is formulated as an optimal control problem for an underactuated articulated chain. The optimal control problem is converted into a discrete parameter optimization and an efficient gradient-based algorithm is used to solve it. Motion capture data from an unimpaired human subject is compared to the simulation results from the dynamic motion optimization. Our results suggest that it is feasible to drive repetitive stepping on a treadmill by a paralyzed person by assisting in torso movement alone. The optimized, pelvic motion strategies are comparable to "hip-hiking" gait strategies used by people with lower limb prostheses or hemiparesis. The resulting motions can be found at the web site http://ww.eng.uci.edu/-chwang/project/stepper/stepper.html.     

Optimization of muscle activity for task-level goals predicts complex changes in limb forces across biomechanical contexts

Optimality principles have been proposed as a general framework for understanding motor control in animals and humans largely based on their ability to predict general features movement in idealized motor tasks. However, generalizing these concepts past proof-of-principle to understand the neuromechanical transformation from task-level control to detailed execution-level muscle activity and forces during behaviorally-relevant motor tasks has proved difficult. In an unrestrained balance task in cats, we demonstrate that achieving task-level constraints center of mass forces and moments while minimizing control effort predicts detailed patterns of muscle activity and ground reaction forces in an anatomically-realistic musculoskeletal model. Whereas optimization is typically used to resolve redundancy at a single level of the motor hierarchy, we simultaneously resolved redundancy across both muscles and limbs and directly compared predictions to experimental measures across multiple perturbation directions that elicit different intra- and interlimb coordination patterns. Further, although some candidate task-level variables and cost functions generated indistinguishable predictions in a single biomechanical context, we identified a common optimization framework that could predict up to 48 experimental conditions per animal (n = 3) across both perturbation directions and different biomechanical contexts created by altering animals' postural configuration. Predictions were further improved by imposing experimentally-derived muscle synergy constraints, suggesting additional task variables or costs that may be relevant to the neural control of balance. These results suggested that reduced-dimension neural control mechanisms such as muscle synergies can achieve similar kinetics to the optimal solution, but with increased control effort (≈2×) compared to individual muscle control. Our results are consistent with the idea that hierarchical, task-level neural control mechanisms previously associated with voluntary tasks may also be used in automatic brainstem-mediated pathways for balance.     

The influence of muscle model complexity in musculoskeletal motion modeling

A comparative study of four different muscle models in a musculoskeletal motion problem is made. The models vary in complexity from the simple input-output model to the more complex model of Hatze [1]. These models are used to solve a minimum time kicking problem using an optimal control algorithm. The results demonstrate the strong influence of the model choice on the various predicted kinematic and kinetic parameters in the problem. The study illustrates some of the advantages and disadvantages involved in trade-offs between model complexity and practicability in musculoskeletal motion studies. The results also illustrate the importance of appropriate detailed parameter estimation studies in the mathematical modeling of the musculoskeletal system.     

Development of software for human muscle force estimation

Muscle force estimation (MFE) has become more and more important in exploring principles of pathological movement, studying functions of artificial muscles, making surgery plan for artificial joint replacement, improving the biomechanical effects of treatments and so on. At present, existing software are complex for professionals, so we have developed a new software named as concise MFE (CMFE). CMFE which provides us a platform to analyse muscle force in various actions includes two MFE methods (static optimisation method and electromyographic-based method). Common features between these two methods have been found and used to improve CMFE. A case studying the major muscles of lower limb of a healthy subject walking at normal speed has been presented. The results are well explained from the effect of the motion produced by muscles during movement. The development of this software can improve the accuracy of the motion simulations and can provide a more extensive and deeper insight in to muscle study.     

Development of software for human muscle force estimation

Muscle force estimation (MFE) has become more and more important in exploring principles of pathological movement, studying functions of artificial muscles, making surgery plan for artificial joint replacement, improving the biomechanical effects of treatments and so on. At present, existing software are complex for professionals, so we have developed a new software named as concise MFE (CMFE). CMFE which provides us a platform to analyse muscle force in various actions includes two MFE methods (static optimisation method and electromyographic-based method). Common features between these two methods have been found and used to improve CMFE. A case studying the major muscles of lower limb of a healthy subject walking at normal speed has been presented. The results are well explained from the effect of the motion produced by muscles during movement. The development of this software can improve the accuracy of the motion simulations and can provide a more extensive and deeper insight in to muscle study.     

Modeling and inverse simulation of somersaults on the trampoline

This paper describes a biomechanical model for numerical simulation of front and back somersaults, without twist, performed on the trampoline. The developed mathematical formulation is used to solve an inverse dynamics problem, in which the moments of muscle forces at the joints that result in a given (measured) motion are determined. The nature of the stunts and the way the human body is maneuvered and controlled can be studied. The calculated torques can then be used as control signals for a dynamic simulation. This provides a way to check the inverse dynamics procedures, and influence of typical control errors on somersault performance can be studied. To achieve these goals, the nonlinear dynamical model of the trampolinist and the interacting trampoline bed has been identified, and a methodology for recording the actual somersault performances was proposed. Some results of numerical simulations are reported.     

Modeling the biomechanics of swine mastication – An inverse dynamics approach

A novel reconstructive alternative for patients with severe facial structural deformity is Le Fort-based, face-jaw-teeth transplantation (FJTT). To date, however, only ten surgeries have included underlying skeletal and jaw-teeth components, all yielding sub-optimal results and a need for a subsequent revision surgery, due to size mismatch and lack of precise planning. Numerous studies have proven swine to be appropriate candidates for translational studies including pre-operative planning of transplantation. An important aspect of planning FJTT is determining the optimal muscle attachment sites on the recipient?s jaw, which requires a clear understanding of mastication and bite mechanics in relation to the new donated upper and/or lower jaw. A segmented CT scan coupled with data taken from literature defined a biomechanical model of mandible and jaw muscles of a swine. The model was driven using tracked motion and external force data of one cycle of chewing published earlier, and predicted the muscle activation patterns as well as temporomandibular joint (TMJ) reaction forces and condylar motions. Two methods, polynomial and min/max optimization, were used for solving the muscle recruitment problem. Similar performances were observed between the two methods. On average, there was a mean absolute error (MAE) of <0.08 between the predicted and measured activation levels of all muscles, and an MAE of <7 N for TMJ reaction forces. Simulated activations qualitatively followed the same patterns as the reference data and there was very good agreement for simulated TMJ forces. The polynomial optimization produced a smoother output, suggesting that it is more suitable for studying such motions. Average MAE for condylar motion was 1.2 mm, which reduced to 0.37 mm when the input incisor motion was scaled to reflect the possible size mismatch between the current and original swine models. Results support the hypothesis that the model can be used for planning of facial transplantation.