Rositsa Raikova

Sensitivity of predicted muscle forces to parameters of the optimization-based human leg model revealed by analytical and numerical analyses

There are different opinions in the literature on whether the cost functions: the sum of muscle stresses squared and the sum of muscle stresses cubed, can reasonably predict muscle forces in humans. One potential reason for the discrepancy in the results could be that different authors use different sets of model parameters which could substantially affect forces predicted by optimization-based models. In this study, the sensitivity of the optimal solution obtained by minimizing the above cost functions for a planar three degrees-of-freedom (DOF) model of the leg with nine muscles was investigated analytically for the quadratic function and numerically for the cubic function. Analytical results revealed that, generally, the non-zero optimal force of each muscle depends in a very complex non-linear way on moments at all three joints and moment arms and physiological cross-sectional areas (PCSAs) of all muscles. Deviations of the model parameters (moment arms and PCSAs) from their nominal values within a physiologically feasible range affected not only the magnitude of the forces predicted by both criteria, but also the number of non-zero forces in the optimal solution and the combination of muscles with non-zero predicted forces. Muscle force magnitudes calculated by both criteria were similar. They could change several times as model parameters changed, whereas patterns of muscle forces were typically not as sensitive. It is concluded that different opinions in the literature about the behavior of optimization-based models can be potentially explained by differences in employed model parameters.

A GENERAL APPROACH FOR MODELLING AND MATHEMATICAL INVESTIGATION OF THE HUMAN UPPER LIMB

This paper is an attempt to systematize and solve the problems connected with biomechanical modelling and mathematical investigation of the upper limb, when all muscles acting on the shoulder, elbow and wrist joint are included. The basic problem is the assumption of a hypothesis for the unit vectors, collinear with muscle forces. The so-called segment-lines modelling of the muscle force direction is proposed for solving this problem. This approach combines the simplicity of the straight line joining the muscle tendon origin to the insertion point, its application for the analytical description of the model and the real description of the muscle anatomical functions. The principle of the anatomically adequate moment is the general criterion for choosing the segments (straight and curvilinear sections) by which the direction of the muscle action is replaced. The different optimization techniques which are used for solving the statically indeterminate problem, formulated for musculoskeletal biomechanical models, are considered. Aiming to obtain strictly positive and continuous solutions for the muscle forces moduli, the application of the Lagrange multipliers method is proposed. The possibility of applying different optimization criteria is explored. A general conclusion may be formulated: if there exists any criterion which is independent of the joint angles, it follows than that it must be dependent on the joint reactions and, in addition, it must be nonlinearly dependent on the muscle forces moduli.

About weight factors in the non-linear objective functions used for solving indeterminate problems in biomechanics

A lot of non-linear objective criteria are applied for solving the indeterminate problems formulated for different biomechanical models--most of them can be covered by the expression [formula in text]. It might be noted, however, that most of the suggested criteria are not applicable if considerable antagonistic co-contractions exist. This could be an effect of treating the agonistic muscles and their respective antagonists in one and the same manner in the objective function. Using a completely inverse approach (the muscle forces are supposed to be known quantities) and a simple 1DOF model (actuated by three agonistic muscles and one corresponding antagonist) it has been shown which values of the weight factors c(i) may predict different levels of muscle forces from the two antagonistic groups. Three hypothetical border variants for magnitudes of the muscle forces are considered (flexor muscles are only active, extensor muscles are only active, considerable co-contraction of flexors and extensors exists). The main conclusions are: the signs of c(i) at agonistic muscles have to be opposite to the c(i) signs at their antagonists; the signs of the weight factors depend on the direction of the net external joint moment; the closer c(i) to zero, the bigger force will be predicted in the ith muscle.

A MODEL OF THE FLEXION - EXTENSION MOTION IN THE ELBOW JOINT - SOME PROBLEMS CONCERNING MUSCLE FORCES MODELLING AND COMPUTATION

This paper represents an application of a general approach for modelling and mathematical investigation of the human upper limb considered in a previous paper (Raikova, 1992, J. Biomechanics 25, 857-867). Six ways of modelling the muscle force in a plane are suggested and compared to each other. A model of the flexion-extension motion in the elbow joint is proposed, which includes two extensor muscles and three flexor muscles. An optimization task is formulated and solved analytically using the Lagrange multipliers method with an objective function dependent on the n-th power of the muscle forces moduli: sigma c(i)F(i)/(n) (n > 1). The influence of a set of parameters (the power n in the objective function, the weight coefficients c i, the lever arms of the modelled muscle forces, an external force applied to the forearm) on the predicted muscle forces and on the joint reaction is investigated. The results demonstrate that strictly positive and continuous solutions for all muscle forces (for synergistic muscles, also for the antagonistic ones) may be simultaneously predicted if the coefficients C(i) are properly chosen.

Model-generated decomposition of unfused tetani of motor units evoked by random stimulation

Unfused tetani of motor units (MUs) evoked by stimulation at variable interpulse intervals at mean frequencies of 20, 25, 33, 40 and 50Hz were studied using ten functionally isolated fast-type MUs from the medial gastrocnemius muscle of adult Wistar rats. A previously proposed algorithm and computer program for mathematical decomposition of unfused tetani into a series of twitches, representing responses to individual pulses, were used. Analysis of the parameters of the decomposed twitches showed considerable variability in force of successive contractions. These twitches were extremely variable with up to 2-fold higher forces and longer contraction times than a single twitch evoked by one stimulus. However, when the stimulation frequency was decreased, the decomposed twitches became similar to the single twitch with respect to amplitude and contraction time. It was found that the basic contractile parameters of decomposed twitches could be predicted with high accuracy on the basis of the tetanus force level at which the next contraction begins. This analysis of the parameters of decomposed twitches demonstrated that the contractile responses of the muscle fibers to successive action potentials generated by motoneurons are highly variable and depend on the previous MU state.

The Influence of the Way the Muscle Force is Modeled on the Predicted Results Obtained by Solving Indeterminate Problems for a Fast Elbow Flexion

A critical point in models of the human limbs when the aim is to investigate the motor control is the muscle model. More often the mechanical output of a muscle is considered as one musculotendon force that is a design variable in optimization tasks solved predominantly by static optimization. For dynamic conditions, the relationship between the developed force, the length and the contraction velocity of a muscle becomes important and rheological muscle models can be incorporated in the optimization tasks. Here the muscle activation can be a design variable as well. Recently a new muscle model was proposed [22]. A muscle is considered as a mixture of motor units (MUs) with different peculiarities and the muscle force is calculated as a sum of the MUs twitches. The aim of the paper is to compare these three ways for presenting the muscle force. Fast elbow flexion is investigated using a planar model with five muscles. It is concluded that the rheological models are suitable for calculation of the current maximal muscle forces that can be used as weight factors in the objective functions. The model based on MUs has many advantages for precise investigations of motor control. Such muscle presentation can explain the muscle co-contraction and the role of the fast and the slow MUs. The relationship between the MUs activation and the mechanical output is more clear and closer to the reality.

Experimentally verified mathematical approach for the prediction of force developed by motor units at variable frequency stimulation patterns

During normal daily activity, muscle motor units (MUs) develop unfused tetanic contractions evoked by trains of motoneuronal firings at variable interpulse intervals (IPIs). The mechanical responses of a MU to successive impulses are not identical. The aim of this study was to develop a mathematical approach for the prediction of each response within the tetanus as well as the tetanic force itself. Experimental unfused tetani of fast and slow rat MUs, evoked by trains of stimuli at variable IPIs, were decomposed into series of twitch-shaped responses to successive stimuli using a previously described algorithm. The relationships between the parameters of the modeled twitches and the tetanic force level at which the next response begins were examined and regression equations were derived. Using these equations, profiles of force for the same and different stimulation patterns were mathematically predicted by summating modeled twitches. For comparison, force predictions were made by the summation of twitches equal to the first one. The recorded and the predicted tetanic forces were compared. The results revealed that it is possible to predict tetanic force with high accuracy by using regression equations. The force predicted in this way was much closer to the experimental record than the force obtained by the summation of equal twitches, especially for slow MUs. These findings are likely to have an impact on the development of realistic muscle models composed of MUs, and will assist our understanding of the significance of the neuronal code in motor control and the role of biophysical processes during MU contractions.

Prediction of Individual Muscle Forces Using Lagrange Multipliers Method — A Model of the Upper Human Limb in the Sagittal Plane: I. Theoretical Considerations

Abstract Analytical solutions of indeterminate problems formulated for biomechanical models with more than one degree of freedom (DOF7rpar; are rarely found. This paper is an extension of the investigations of a 1 DOF model (Raikova, 1996, J. Biomechanics, 763-772) for a more complex 3 DOF model. The proposed model of the human upper limb is in the sagittal plane and includes ten muscle elements, four of them being two-joint ones. The formulated optimization task is solved analytically using the method of Lagrange multipliers. It is supposed that the optimization function is complex but can be approximated by a weighted sum of the squared muscle forces, where the nature of the weight factors of the muscles are unknown. The proposed computational algorithm for determination of the unknown individual muscle forces and joint reactions is easily implemented and may be extended without difficulties for more DOF and muscles. The aim is to establish a means of investigation of the possible weight coefficients for different modelled situations, which will help in searching for their physiological interpretation and analytical description.

Dynamic changes of twitchlike responses to successive stimuli studied by decomposition of motor unit tetanic contractions in rat medial gastrocnemius

Unfused tetanic contractions evoked by trains of stimuli at variable interpulse intervals (IPIs) were recorded for 10 fast fatigable (FF), 10 fast resistant (FR), and 10 slow (S) motor units (MUs) and subsequently decomposed with a mathematical algorithm into trains of twitch-shape responses to successive stimuli. The mean stimulation frequencies were matched for each MU to evoke tetani of similar fusion degrees, whereas the variability range of IPIs was in each case 50-150% of the mean IPI. Force and time parameters of decomposed twitches were analyzed and related to the first response. Considerable variability of the analyzed twitch parameters was observed in each MU, although the largest range of variability occurred in slow MUs. In general, the decomposed twitch responses had longer duration and higher force than single-twitch contractions, although for nine FF and six FR MUs some of the decomposed responses were slightly weaker (but not faster) than the first twitches of these MUs. Comparison of the strongest decomposed twitch to the first decomposed twitch revealed ratios of forces up to 2.35, 3.33, and 6.89 for FF, FR, and S MUs and ratios of force-time areas up to 3.54, 4.67, and 14.26 for FF, FR, and S MUs, whereas for the contraction times the ratios of the longest decomposed twitch to the first twitch amounted to 2.46, 2.07, and 3.52 for FF, FR, and S MUs, respectively. The results indicate that contractile responses to successive action potentials are considerably variable, especially for slow MUs.

Investigation of the Peculiarities of Two-joint Muscles Using a 3 DOF Model of the Human Upper Limb in the Sagittal Plane: an Optimization Approach

Abstract The purpose of this paper is an investigation of the peculiarities of biarticular muscles by means of modelling and analytical solution of the indeterminate problem. The basic model includes 10 muscle elements performing flexio/extensio in the shoulder, elbow and wrist. Four of them are biarticular muscles. Two modifications of the model with only monoarticular muscles are developed. The indeterminate problem is solved analytically using the objective criterion σciFi 2 where F( is the module of the i-th muscle force and Cj is a weight coefficient. The predicted muscle forces, joint reactions and moments are compared in-between the basic model and its two modifications for different joint angles, external loading and weight coefficients. The main conclusions are: it is impossible to formulate strict advantages of the biarticular muscles under quasistatical conditions, their peculiarities depend on limb position, external loading and neural control; in general, monoarticular muscles are more powerful than biarticular ones; the biarticular muscles fine tune muscle coordination, their control is more precise and graceful; the presence of biarticular muscles leads to an increase of the joint reactions and moments, thus stabilizing the limb.