Predictions of co-contraction depend critically on degrees-of-freedom in the musculoskeletal model

In biomechanics, the calculation of individual muscle forces during movements is based on a model of the musculoskeletal system and a method for extracting a unique set of muscle forces. To obtain a unique set of muscle forces, non-linear, static optimisation is commonly used. However, the optimal solution is dependent on the musculoskeletal geometry, and single joints may be represented using one, two or three degrees-of-freedom. Frequently, a system with multiple degrees-of-freedom is replaced with a system that contains a subset of all the possible degrees-of-freedom. For example, the cat ankle joint is typically modelled as a planar joint with its primary degree-of-freedom (plantar-dorsiflexion), whereas, the actual joint has three rotational degrees-of-freedom. Typically, such simplifications are justified by the idea that the reduced case is contained as a specific solution of the more general case. However, here we demonstrate that the force-sharing solution space of a general, three degrees-of-freedom musculoskeletal system does not necessarily contain the solutions from the corresponding one or two degrees-of-freedom systems. Therefore, solutions of a reduced system, in general, are not sub-set solutions of the actual three degrees-of-freedom system, but are independent solutions that are often incompatible with solutions of the actual system. This result shows that representing a three degrees-of-freedom system as a one or two degrees-of-freedom system gives force-sharing solutions that cannot be extrapolated to the actual system, and vice-versa. These results imply that general solutions cannot be extracted from models with fewer degrees-of-freedom than the actual system. They further emphasise the need for precise geometric representation of the musculoskeletal system, if general force-sharing rules are to be derived.     

Analytic analysis of the force sharing among synergistic muscles in one- and two-degree-of-freedom models

Mathematical optimization of specific cost functions has been used in theoretical models to calculate individual muscle forces. Measurements of individual muscle forces and force sharing among individual muscles show an intensity-dependent, non-linear behavior. It has been demonstrated that the force sharing between the cat Gastrocnemius, Plantaris and Soleus shows distinct loops that change orientation systematically depending on the intensity of the movement. The purpose of this study was to prove whether or not static, non-linear optimization could inherently predict force sharing loops between agonistic muscles. Using joint moment data from a step cycle of cat locomotion, the forces in three cat ankle plantar flexors (Gastrocnemius, Plantaris and Soleus) were calculated using two popular optimization algorithms and two musculo-skeletal models. The two musculo-skeletal models included a one-degree-of-freedom model that considered the ankle joint exclusively and a two-degree-of-freedom model that included the ankle and the knee joint. The main conclusion of this study was that solutions of the one-degree-of-freedom model do not guarantee force-sharing loops, but the two-degree-of-freedom model predicts force-sharing loops independent of the specific values of the input parameters for the muscles and the musculo-skeletal geometry. The predicted force-sharing loops were found to be a direct result of the loops formed by the knee and ankle moments in a moment-moment graph.     

Finite element analysis of the impact of bone nanostructure on its piezoelectric response

Biomechanics and Modeling in Mechanobiology - The piezoelectric response of bone at the submicron scale is analyzed under mechanical loadings using the finite element (FE) method. A new algorithm...     

Multi-functionality of the cat medical gastrocnemius during locomotion

The functional role of biarticular muscles was investigated based on direct force measurement in the cat medial gastrocnemius (MG) and analysis of hindlimb kinematics and kinetics for the stance phase of level, uphill, and downhill walking. Four primary functional roles of biarticular muscles have been proposed in the past. These functional roles have typically been discussed independently of each other, and biarticular muscles have rarely been assigned more than one functional roles for different phases of the work cycle. The purpose of this study was to elucidate the functional role of the biarticular cat MG during locomotion. It was found that MG forces were primarily associated with the moment requirements at the ankle for most of the stance phase, but also helped to satisfy the moments at the knee in the initial phase of stance. In the second half of stance, MG transferred mechanical energy from the knee to the ankle from the knee to the ankle, while simultaneously producing a substantial amount of mechanical work. Based on these results, we hypothesize that MG's primary function is that of an ankle extensor. However, because of the coupling of the ankle extensor moment with a knee flexor moment in the initial, and a knee extensor moment in the final phase of stance, MG satisfies two joint moments in early stance, and transfers mechanical energy from the knee to the ankle in late stance. We conclude that cat MG has multiple functional roles during the stance phase of locomotion, and speculate that such multi-functionality also exists in other bi- and multi-articular muscles.     

A task-specific validation of homogeneous non-linear optimisation approaches

In biomechanics, musculoskeletal models are typically redundant. This situation is referred to as the distribution problem. Often, static, non-linear optimisation methods of the form “min: φ(f) subject to mechanical and muscular constraints” have been used to extract a unique set of muscle forces. Here, we present a method for validating this class of non-linear optimisation approaches where the homogeneous cost function, φ(f), is used to solve the distribution problem. We show that the predicted muscle forces for different loading conditions are scaled versions of each other if the joint loading conditions are just scaled versions. Therefore, we can calculate the theoretical muscle forces for different experimental conditions based on the measured muscle forces and joint loadings taken from one experimental condition and assuming that all input into the optimisation (e.g., moment arms, muscle attachment sites, size, fibre type distribution) and the optimisation approach are perfectly correct. Thus predictions of muscle force for other experimental conditions are accurate if the optimisation approach is appropriate, independent of the musculoskeletal geometry and other input required for the optimisation procedure. By comparing the muscle forces predicted in this way to the actual muscle forces obtained experimentally, we conclude that convex homogeneous non-linear optimisation approaches cannot predict individual muscle forces properly, as force-sharing among synergistic muscles obtained experimentally are not just scaled versions of joint loading, not even in a first approximation.     

Mapping the classical cross-bridge theory and backward steps in a three bead laser trap setup

According to the cross-bridge theory (Huxley, 1957) [1], the interaction between myosin and actin is governed by a deterministic process where the myosin molecule pulls the actin filament in one specific direction only. However, studies on single myosin-actin interactions produced displacements of actin not only in the preferred but also in the opposite direction. This phenomenon is typically referred to as backward steps by the myosin head. Molloy et al. (1995) [2] speculated that these backward steps are not caused by the molecular interactions of actin with myosin but are an artifact of the Brownian motion associated with these molecular level experiments. The aim of this study was to investigate, whether a theoretical model can support Molloy’s speculation. We therefore developed a theoretical model of actin-myosin based muscle contraction that was strictly based on Huxley’s assumption of one stepping direction only, but incorporated Brownian motion, as observed in single cross-bridge-actin interactions. The mathematical model is based on Langevin equations describing the classical three-bead laser trap setup and uses a novel semi-analytical approach to study the percentage of backward steps. We analyzed the effects of different initial actin attachment site distribution and laser trap stiffness on the ratio of forward to backward steps. Our results demonstrate that backward steps and the classical cross-bridge theory are perfectly compatible in a three-bead laser trap setup.     

A Common Variant of NGEF Is Associated with Abdominal Visceral Fat in Korean Men

Central adiposity, rather than body mass index (BMI), is a key pathophysiological feature of the development of obesity-related diseases. Although genetic studies by anthropometric measures such as waist circumference have been widely conducted, genetic studies for abdominal fat deposition measured by computed tomography (CT) have been rarely performed. A total of 1,243 participants who were recruited from two health check-up centers were included in this study. We selected four and three single-nucleotide polymorphisms (SNPs) in NGEF and RGS6, respectively, and analyzed the associations between the seven SNPs and central adiposity measured by CT using an additive, dominant, or recessive model. The participants were generally healthy middle-aged men (50.7 ± 5.3 years). In the additive model, the rs11678490 A allele of NGEF was significantly associated with total adipose tissue, visceral adipose tissue (VAT), and subcutaneous adipose tissue (all P < 0.05). The AA genotype of this SNP in the recessive model showed a more significant association with all adiposity traits, and its association with VAT remained significant even after adjustment for BMI (P = 0.005). In the overall or visceral obesity group analysis, the AA genotype of rs11678490 showed no association with overall obesity (P = 0.148), whereas it was significantly associated with visceral obesity both before (P = 0.010) and after (P = 0.029) adjustment for BMI. In particular, an AA genotype effect was conspicuous between lower and upper groups with 5% extreme VAT phenotypes (OR = 9.59, 95% CI = 1.50–61.31). However, we found no significant association between SNPs of RGS6 and central adiposity. We identified a visceral-fat-associated SNP, rs11678490 of NGEF, in Korean men. This study suggests that the genetic background of central adiposity and BMI is different, and that additional efforts should be made to find the unique genetic architecture of intra-abdominal fat accumulation.     

Are titin properties reflected in single myofibrils?

Titin is a structural protein in muscle that spans the half sarcomere from Z-band to M-line. Although there are selected studies on titin's mechanical properties from tests on isolated molecules or titin fragments, little is known about its behavior within the structural confines of a sarcomere. Here, we tested the hypothesis that titin properties might be reflected well in single myofibrils. Single myofibrils from rabbit psoas were prepared for measurement of passive stretch-shortening cycles at lengths where passive titin forces occur. Three repeat stretch-shortening cycles with magnitudes between 1.0 and 3.0μm/sarcomere were performed at a speed of 0.1μm/s·sarcomere and repeated after a ten minute rest at zero force. These tests were performed in a relaxation solution (passive) and an activation solution (active) where cross-bridge attachment was inhibited with 2,3 butanedionemonoxime. Myofibrils behaved viscoelastically producing an increased efficiency with repeat stretch-shortening cycles, but a decreased efficiency with increasing stretch magnitudes. Furthermore, we observed a first distinct inflection point in the force-elongation curve at an average sarcomere length of 3.5μm that was associated with an average force of 68±5nN/mm. This inflection point was thought to reflect the onset of Ig domain unfolding and was missing after a ten minute rest at zero force, suggesting a lack of spontaneous Ig domain refolding. These passive myofibrillar properties observed here are consistent with those observed in isolated titin molecules, suggesting that the mechanics of titin are well preserved in isolated myofibrils, and thus, can be studied readily in myofibrils, rather than in the extremely difficult and labile single titin preparations.