Multiscale modeling of the neuromuscular system: Coupling neurophysiology and skeletal muscle mechanics.

Mathematical models and computer simulations have the great potential to substantially increase our understanding of the biophysical behavior of the neuromuscular system. This, however, requires detailed multiscale, and multiphysics models. Once validated, such models allow systematic in silico investigations that are not necessarily feasible within experiments and, therefore, have the ability to provide valuable insights into the complex interrelations within the healthy system and for pathological conditions. Most of the existing models focus on individual parts of the neuromuscular system and do not consider the neuromuscular system as an integrated physiological system. Hence, the aim of this advanced review is to facilitate the prospective development of detailed biophysical models of the entire neuromuscular system. For this purpose, this review is subdivided into three parts. The first part introduces the key anatomical and physiological aspects of the healthy neuromuscular system necessary for modeling the neuromuscular system. The second part provides an overview on state-of-the-art modeling approaches representing all major components of the neuromuscular system on different time and length scales. Within the last part, a specific multiscale neuromuscular system model is introduced. The integrated system model combines existing models of the motor neuron pool, of the sensory system and of a multiscale model describing the mechanical behavior of skeletal muscles. Since many sub-models are based on strictly biophysical modeling approaches, it closely represents the underlying physiological system and thus could be employed as starting point for further improvements and future developments. This article is categorized under: Physiology > Mammalian Physiology in Health and Disease Analytical and Computational Methods > Computational Methods Models of Systems Properties and Processes > Organ, Tissue, and Physiological Models.

A continuum-mechanical skeletal muscle model including actin-titin interaction predicts stable contractions on the descending limb of the force-length relation.

Contractions on the descending limb of the total (active + passive) muscle force-length relationship (i. e. when muscle stiffness is negative) are expected to lead to vast half-sarcomere-length inhomogeneities. This is however not observed in experiments-vast half-sarcomere-length inhomogeneities can be absent in myofibrils contracting in this range, and initial inhomogeneities can even decrease. Here we show that the absence of half-sarcomere-length inhomogeneities can be predicted when considering interactions of the semi-active protein titin with the actin filaments. Including a model of actin-titin interactions within a multi-scale continuum-mechanical model, we demonstrate that stability, accurate forces and nearly homogeneous half-sarcomere lengths can be obtained on the descending limb of the static total force-length relation. This could be a key to durable functioning of the muscle because large local stretches, that might harm, for example, the transverse-tubule system, are avoided.

The Role of Parvalbumin, Sarcoplasmatic Reticulum Calcium Pump Rate, Rates of Cross-Bridge Dynamics, and Ryanodine Receptor Calcium Current on Peripheral Muscle Fatigue: A Simulation Study.

A biophysical model of the excitation-contraction pathway, which has previously been validated for slow-twitch and fast-twitch skeletal muscles, is employed to investigate key biophysical processes leading to peripheral muscle fatigue. Special emphasis hereby is on investigating how the model's original parameter sets can be interpolated such that realistic behaviour with respect to contraction time and fatigue progression can be obtained for a continuous distribution of the model's parameters across the muscle units, as found for the functional properties of muscles. The parameters are divided into 5 groups describing (i) the sarcoplasmatic reticulum calcium pump rate, (ii) the cross-bridge dynamics rates, (iii) the ryanodine receptor calcium current, (iv) the rates of binding of magnesium and calcium ions to parvalbumin and corresponding dissociations, and (v) the remaining processes. The simulations reveal that the first two parameter groups are sensitive to contraction time but not fatigue, the third parameter group affects both considered properties, and the fourth parameter group is only sensitive to fatigue progression. Hence, within the scope of the underlying model, further experimental studies should investigate parvalbumin dynamics and the ryanodine receptor calcium current to enhance the understanding of peripheral muscle fatigue.

A multi-scale continuum model of skeletal muscle mechanics predicting force enhancement based on actin-titin interaction.

Although recent research emphasises the possible role of titin in skeletal muscle force enhancement, this property is commonly ignored in current computational models. This work presents the first biophysically based continuum-mechanical model of skeletal muscle that considers, in addition to actin-myosin interactions, force enhancement based on actin-titin interactions. During activation, titin attaches to actin filaments, which results in a significant reduction in titin's free molecular spring length and therefore results in increased titin forces during a subsequent stretch. The mechanical behaviour of titin is included on the microscopic half-sarcomere level of a multi-scale chemo-electro-mechanical muscle model, which is based on the classic sliding-filament and cross-bridge theories. In addition to titin stress contributions in the muscle fibre direction, the continuum-mechanical constitutive relation accounts for geometrically motivated, titin-induced stresses acting in the muscle's cross-fibre directions. Representative simulations of active stretches under maximal and submaximal activation levels predict realistic magnitudes of force enhancement in fibre direction. For example, stretching the model by 20 % from optimal length increased the isometric force at the target length by about 30 %. Predicted titin-induced stresses in the muscle's cross-fibre directions are rather insignificant. Including the presented development in future continuum-mechanical models of muscle function in dynamic situations will lead to more accurate model predictions during and after lengthening contractions.

Reconstruction of muscle fascicle architecture from iodine-enhanced microCT images: A combined texture mapping and streamline approach.

Skeletal muscle models are used to investigate motion and force generation in both biological and bioengineering research. Yet, they often lack a realistic representation of the muscle's internal architecture which is primarily composed of muscle fibre bundles, known as fascicles. Recently, it has been shown that fascicles can be resolved with micro-computed tomography (µCT) following staining of the muscle tissue with iodine potassium iodide (I2KI). Here, we present the reconstruction of the fascicular spatial arrangement and geometry of the superficial masseter muscle of a dog based on a combination of pattern recognition and streamline computation. A cadaveric head of a dog was incubated in I2KI and µCT-scanned. Following segmentation of the masseter muscle a statistical pattern recognition algorithm was applied to create a vector field of fascicle directions. Streamlines were then used to transform the vector field into a realistic muscle fascicle representation. The lengths of the reconstructed fascicles and the pennation angles in two planes (frontal and sagittal) were extracted and compared against a tracked fascicle field obtained through cadaver dissection. Both fascicle lengths and angles were found to vary substantially within the muscle confirming the complex and heterogeneous nature of skeletal muscle described by previous studies. While there were significant differences in the pennation angle between the experimentally derived and µCT-reconstructed data, there was congruence in the fascicle lengths. We conclude that the presented approach allows for embedding realistic fascicle information into finite element models of skeletal muscles to better understand the functioning of the musculoskeletal system.

Predicting electromyographic signals under realistic conditions using a multiscale chemo-electro-mechanical finite element model.

This paper presents a novel multiscale finite element-based framework for modelling electromyographic (EMG) signals. The framework combines (i) a biophysical description of the excitation-contraction coupling at the half-sarcomere level, (ii) a model of the action potential (AP) propagation along muscle fibres, (iii) a continuum-mechanical formulation of force generation and deformation of the muscle, and (iv) a model for predicting the intramuscular and surface EMG. Owing to the biophysical description of the half-sarcomere, the model inherently accounts for physiological properties of skeletal muscle. To demonstrate this, the influence of membrane fatigue on the EMG signal during sustained contractions is investigated. During a stimulation period of 500 ms at 100 Hz, the predicted EMG amplitude decreases by 40% and the AP propagation velocity decreases by 15%. Further, the model can take into account contraction-induced deformations of the muscle. This is demonstrated by simulating fixed-length contractions of an idealized geometry and a model of the human tibialis anterior muscle (TA). The model of the TA furthermore demonstrates that the proposed finite element model is capable of simulating realistic geometries, complex fibre architectures, and can include different types of heterogeneities. In addition, the TA model accounts for a distributed innervation zone, different fibre types and appeals to motor unit discharge times that are based on a biophysical description of the α motor neurons.

A multiscale chemo-electro-mechanical skeletal muscle model to analyze muscle contraction and force generation for different muscle fiber arrangements.

The presented chemo-electro-mechanical skeletal muscle model relies on a continuum-mechanical formulation describing the muscle's deformation and force generation on the macroscopic muscle level. Unlike other three-dimensional models, the description of the activation-induced behavior of the mechanical model is entirely based on chemo-electro-mechanical principles on the microscopic sarcomere level. Yet, the multiscale model reproduces key characteristics of skeletal muscles such as experimental force-length and force-velocity data on the macroscopic whole muscle level. The paper presents the methodological approaches required to obtain such a multiscale model, and demonstrates the feasibility of using such a model to analyze differences in the mechanical behavior of parallel-fibered muscles, in which the muscle fibers either span the entire length of the fascicles or terminate intrafascicularly. The presented results reveal that muscles, in which the fibers span the entire length of the fascicles, show lower peak forces, more dispersed twitches and fusion of twitches at lower stimulation frequencies. In detail, the model predicted twitch rise times of 38.2 and 17.2 ms for a 12 cm long muscle, in which the fibers span the entire length of the fascicles and with twelve fiber compartments in series, respectively. Further, the twelve-compartment model predicted peak twitch forces that were 19% higher than in the single-compartment model. The analysis of sarcomere lengths during fixed-end single twitch contractions at optimal length predicts rather small sarcomere length changes. The observed lengths range from 75 to 111% of the optimal sarcomere length, which corresponds to a region with maximum filament overlap. This result suggests that stability issues resulting from activation-induced stretches of non-activated sarcomeres are unlikely in muscles with passive forces appearing at short muscle length.

Modeling the chemoelectromechanical behavior of skeletal muscle using the parallel open-source software library OpenCMISS.

An extensible, flexible, multiscale, and multiphysics model for nonisometric skeletal muscle behavior is presented. The skeletal muscle chemoelectromechanical model is based on a bottom-up approach modeling the entire excitation-contraction pathway by strongly coupling a detailed biophysical model of a half-sarcomere to the propagation of action potentials along skeletal muscle fibers and linking cellular parameters to a transversely isotropic continuum-mechanical constitutive equation describing the overall mechanical behavior of skeletal muscle tissue. Since the multiscale model exhibits separable time scales, a special emphasis is placed on employing computationally efficient staggered solution schemes. Further, the implementation builds on the open-source software library OpenCMISS and uses state-of-the-art parallelization techniques taking advantage of the unique anatomical fiber architecture of skeletal muscles. OpenCMISS utilizes standardized data structures for geometrical aspects (FieldML) and cellular models (CellML). Both standards are designed to allow for a maximum flexibility, reproducibility, and extensibility. The results demonstrate the model's capability of simulating different aspects of nonisometric muscle contraction and efficiently simulating the chemoelectromechanical behavior in complex skeletal muscles such as the tibialis anterior muscle.

OpenCMISS: a multi-physics & multi-scale computational infrastructure for the VPH/Physiome project.

The VPH/Physiome Project is developing the model encoding standards CellML (cellml.org) and FieldML (fieldml.org) as well as web-accessible model repositories based on these standards (models.physiome.org). Freely available open source computational modelling software is also being developed to solve the partial differential equations described by the models and to visualise results. The OpenCMISS code (opencmiss.org), described here, has been developed by the authors over the last six years to replace the CMISS code that has supported a number of organ system Physiome projects. OpenCMISS is designed to encompass multiple sets of physical equations and to link subcellular and tissue-level biophysical processes into organ-level processes. In the Heart Physiome project, for example, the large deformation mechanics of the myocardial wall need to be coupled to both ventricular flow and embedded coronary flow, and the reaction-diffusion equations that govern the propagation of electrical waves through myocardial tissue need to be coupled with equations that describe the ion channel currents that flow through the cardiac cell membranes. In this paper we discuss the design principles and distributed memory architecture behind the OpenCMISS code. We also discuss the design of the interfaces that link the sets of physical equations across common boundaries (such as fluid-structure coupling), or between spatial fields over the same domain (such as coupled electromechanics), and the concepts behind CellML and FieldML that are embodied in the OpenCMISS data structures. We show how all of these provide a flexible infrastructure for combining models developed across the VPH/Physiome community.