Modelling extracellular matrix and cellular contributions to whole muscle mechanics.

Skeletal muscle tissue has a highly complex and heterogeneous structure comprising several physical length scales. In the simplest model of muscle tissue, it can be represented as a one dimensional nonlinear spring in the direction of muscle fibres. However, at the finest level, muscle tissue includes a complex network of collagen fibres, actin and myosin proteins, and other cellular materials. This study shall derive an intermediate physical model which encapsulates the major contributions of the muscle components to the elastic response apart from activation-related along-fibre responses. The micro-mechanical factors in skeletal muscle tissue (eg. connective tissue, fluid, and fibres) can be homogenized into one material aggregate that will capture the behaviour of the combination of material components. In order to do this, the corresponding volume fractions for each type of material need to be determined by comparing the stress-strain relationship for a volume containing each material. This results in a model that accounts for the micro-mechanical features found in muscle and can therefore be used to analyze effects of neuro-muscular diseases such as cerebral palsy or muscular dystrophies. The purpose of this study is to construct a model of muscle tissue that, through choosing the correct material parameters based on experimental data, will accurately capture the mechanical behaviour of whole muscle. This model is then used to look at the impacts of the bulk modulus and material parameters on muscle deformation and strain energy-density distributions.

The Energy of Muscle Contraction. III. Kinetic Energy During Cyclic Contractions.

During muscle contraction, chemical energy is converted to mechanical energy when ATP is hydrolysed during cross-bridge cycling. This mechanical energy is then distributed and stored in the tissue as the muscle deforms or is used to perform external work. We previously showed how energy is distributed through contracting muscle during fixed-end contractions; however, it is not clear how the distribution of tissue energy is altered by the kinetic energy of muscle mass during dynamic contractions. In this study we conducted simulations of a 3D continuum muscle model that accounts for tissue mass, as well as force-velocity effects, in which the muscle underwent sinusoidal work-loop contractions coupled with bursts of excitation. We found that increasing muscle size, and therefore mass, increased the kinetic energy per unit volume of the muscle. In addition to greater relative kinetic energy per cycle, relatively more energy was also stored in the aponeurosis, and less was stored in the base material, which represented the intra and extracellular tissue components apart from the myofibrils. These energy changes in larger muscles due to greater mass were associated lower mass-specific mechanical work output per cycle, and this reduction in mass-specific work was greatest for smaller initial pennation angles. When we compared the effects of mass on the model tissue behaviour to that of in situ muscle with added mass during comparable work-loop trials, we found that greater mass led to lower maximum and higher minimum acceleration in the longitudinal (x) direction near the middle of the muscle compared to at the non-fixed end, which indicates that greater mass contributes to tissue non-uniformity in whole muscle. These comparable results for the simulated and in situ muscle also show that this modelling framework behaves in ways that are consistent with experimental muscle. Overall, the results of this study highlight that muscle mass is an important determinant of whole muscle behaviour.

The Contributions of Extracellular Matrix and Sarcomere Properties to Passive Muscle Stiffness in Cerebral Palsy.

Cerebral palsy results from an upper motor neuron lesion and significantly affects skeletal muscle stiffness. The increased stiffness that occurs is partly a result of changes in the microstructural components of muscle. In particular, alterations in extracellular matrix, sarcomere length, fibre diameter, and fat content have been reported; however, experimental studies have shown wide variability in the degree of alteration. Many studies have reported changes in the extracellular matrix, while others have reported no differences. A consistent finding is increased sarcomere length in cerebral palsy affected muscle. Often many components are altered simultaneously, making it difficult to determine the individual effects on muscle stiffness. In this study, we use a three dimensional modelling approach to isolate individual effects of microstructural alterations typically occurring due to cerebral palsy on whole muscle behaviour; in particular, the effects of extracellular matrix volume fraction, stiffness, and sarcomere length. Causation between the changes to the microstructure and the overall muscle response is difficult to determine experimentally, since components of muscle cannot be manipulated individually; however, utilising a modelling approach allows greater control over each factor. We find that extracellular matrix volume fraction has the largest effect on whole muscle stiffness and mitigates effects from sarcomere length.

The Energy of Muscle Contraction. II. Transverse Compression and Work.

In this study we examined how the strain energies within a muscle are related to changes in longitudinal force when the muscle is exposed to an external transverse load. We implemented a three-dimensional (3D) finite element model of contracting muscle using the principle of minimum total energy and allowing the redistribution of energy through different strain energy-densities. This allowed us to determine the importance of the strain energy-densities to the transverse forces developed by the muscle. We ran a series of in silica experiments on muscle blocks varying in initial pennation angle, muscle length, and external transverse load. As muscle contracts it maintains a near constant volume. As such, any changes in muscle length are balanced by deformations in the transverse directions such as muscle thickness or muscle width. Muscle develops transverse forces as it expands. In many situations external forces act to counteract these transverse forces and the muscle responds to external transverse loads while both passive and active. The muscle blocks used in our simulations decreased in thickness and pennation angle when passively compressed and pushed back on the load when they were activated. Activation of the compressed muscle blocks led either to an increase or decrease in muscle thickness depending on whether the initial pennation angle was less than or greater than 15°, respectively. Furthermore, the strain energy increased and redistributed across the different strain-energy potentials during contraction. The volumetric strain energy-density varied with muscle length and pennation angle and was reduced with greater transverse load for most initial muscle lengths and pennation angles. External transverse load reduced the longitudinal muscle force for initial pennation angles of ß0 = 0°. Whereas for pennate muscle (ß0 > 0°) longitudinal force changed (increase or decrease) depending on the muscle length, pennation angle and the direction of the external load relative to the muscle fibres. For muscle blocks with initial pennation angles ß0 ≤ 20° the reduction in longitudinal muscle force coincided with a reduction in volumetric strain energy-density.

The Energy of Muscle Contraction. I. Tissue Force and Deformation During Fixed-End Contractions.

During contraction the energy of muscle tissue increases due to energy from the hydrolysis of ATP. This energy is distributed across the tissue as strain-energy potentials in the contractile elements, strain-energy potential from the 3D deformation of the base-material tissue (containing cellular and extracellular matrix effects), energy related to changes in the muscle's nearly incompressible volume and external work done at the muscle surface. Thus, energy is redistributed through the muscle's tissue as it contracts, with only a component of this energy being used to do mechanical work and develop forces in the muscle's longitudinal direction. Understanding how the strain-energy potentials are redistributed through the muscle tissue will help enlighten why the mechanical performance of whole muscle in its longitudinal direction does not match the performance that would be expected from the contractile elements alone. Here we demonstrate these physical effects using a 3D muscle model based on the finite element method. The tissue deformations within contracting muscle are large, and so the mechanics of contraction were explained using the principles of continuum mechanics for large deformations. We present simulations of a contracting medial gastrocnemius muscle, showing tissue deformations that mirror observations from magnetic resonance imaging. This paper tracks the redistribution of strain-energy potentials through the muscle tissue during fixed-end contractions, and shows how fibre shortening, pennation angle, transverse bulging and anisotropy in the stress and strain of the muscle tissue are all related to the interaction between the material properties of the muscle and the action of the contractile elements.

Size, History-Dependent, Activation and Three-Dimensional Effects on the Work and Power Produced During Cyclic Muscle Contractions.

Muscles undergo cycles of length change and force development during locomotion, and these contribute to their work and power production to drive body motion. Muscle fibers are typically considered to be linear actuators whose stress depends on their length, velocity, and activation state, and whose properties can be scaled up to explain the function of whole muscles. However, experimental and modeling studies have shown that a muscle's stress additionally depends on inactive and passive tissues within the muscle, the muscle's size, and its previous contraction history. These effects have not been tested under common sets of contraction conditions, especially the cyclic contractions that are typical of locomotion. Here we evaluate the relative effects of size, history-dependent, activation and three-dimensional effects on the work and power produced during cyclic contractions of muscle models. Simulations of muscle contraction were optimized to generate high power outputs: this resulted in the muscle models being largely active during shortening, and inactive during lengthening. As such, the history-dependent effects were dominated by force depression during simulated active shortening rather than force enhancement during active stretch. Internal work must be done to deform the muscle tissue, and to accelerate the internal muscle mass, resulting in reduced power and work that can be done on an external load. The effect of the muscle mass affects the scaling of muscle properties, with the inertial costs of contraction being relatively greater at larger sizes and lower activation levels.

A modelling approach for exploring muscle dynamics during cyclic contractions.

Hill-type muscle models are widely used within the field of biomechanics to predict and understand muscle behaviour, and are often essential where muscle forces cannot be directly measured. However, these models have limited accuracy, particularly during cyclic contractions at the submaximal levels of activation that typically occur during locomotion. To address this issue, recent studies have incorporated effects into Hill-type models that are oftentimes neglected, such as size-dependent, history-dependent, and activation-dependent effects. However, the contribution of these effects on muscle performance has yet to be evaluated under common contractile conditions that reflect the range of activations, strains, and strain rates that occur in vivo. The purpose of this study was to develop a modelling framework to evaluate modifications to Hill-type muscle models when they contract in cyclic loops that are typical of locomotor muscle function. Here we present a modelling framework composed of a damped harmonic oscillator in series with a Hill-type muscle actuator that consists of a contractile element and parallel elastic element. The intrinsic force-length and force-velocity properties are described using Bézier curves where we present a system to relate physiological parameters to the control points for these curves. The muscle-oscillator system can be geometrically scaled while preserving dynamic and kinematic similarity to investigate the muscle size effects while controlling for the dynamics of the harmonic oscillator. The model is driven by time-varying muscle activations that cause the muscle to cyclically contract and drive the dynamics of the harmonic oscillator. Thus, this framework provides a platform to test current and future Hill-type model formulations and explore factors affecting muscle performance in muscles of different sizes under a range of cyclic contractile conditions.

The effect of intramuscular fat on skeletal muscle mechanics: implications for the elderly and obese.

Skeletal muscle accumulates intramuscular fat through age and obesity. Muscle quality, a measure of muscle strength per unit size, decreases in these conditions. It is not clear how fat influences this loss in performance. Changes to structural parameters (e.g. fibre pennation and connective tissue properties) affect the muscle quality. This study investigated the mechanisms that lead to deterioration in muscle performance due to changes in intramuscular fat, pennation and aponeurosis stiffness. A finite-element model of the human gastrocnemius was developed as a fibre-reinforced composite biomaterial containing contractile fibres within the base material. The base-material properties were modified to include intramuscular fat in five different ways. All these models with fat generated lower fibre stress and muscle quality than their lean counterparts. This effect is due to the higher stiffness of the tissue in the fatty models. The fibre deformations influence their interactions with the aponeuroses, and these change with fatty inclusions. Muscles with more compliant aponeuroses generated lower forces. The muscle quality was further reduced for muscles with lower pennation. This study shows that whole-muscle force is dependent on its base-material properties and changes to the base material due to fatty inclusions result in reductions to force and muscle quality.

Regionalizing muscle activity causes changes to the magnitude and direction of the force from whole muscles-a modeling study.

Skeletal muscle can contain neuromuscular compartments that are spatially distinct regions that can receive relatively independent levels of activation. This study tested how the magnitude and direction of the force developed by a whole muscle would change when the muscle activity was regionalized within the muscle. A 3D finite element model of a muscle with its bounding aponeurosis was developed for the lateral gastrocnemius, and isometric contractions were simulated for a series of conditions with either a uniform activation pattern, or regionally distinct activation patterns: in all cases the mean activation from all fibers within the muscle reached 10%. The models showed emergent features of the fiber geometry that matched physiological characteristics: with fibers shortening, rotating to greater pennation, adopting curved trajectories in 3D and changes in the thickness and width of the muscle belly. Simulations were repeated for muscle with compliant, normal and stiff aponeurosis and the aponeurosis stiffness affected the changes to the fiber geometry and the resultant muscle force. Changing the regionalization of the activity resulted to changes in the magnitude, direction and center of the force vector from the whole muscle. Regionalizing the muscle activity resulted in greater muscle force than the simulation with uniform activity across the muscle belly. The study shows how the force from a muscle depends on the complex interactions between the muscle fibers and connective tissues and the region of muscle that is active.

Mathematical modeling of spatio-temporal dynamics of a single bone multicellular unit.

During bone remodeling, bone-resorbing osteoclasts and bone-forming osteoblasts are organized in bone multicellular units (BMUs), which travel at a rate of 20-40 mum/d for 6-12 mo, maintaining a cylindrical structure. However, the interplay of local BMU geometry with biochemical regulation is poorly understood. We developed a mathematical model of BMU describing changes in time and space of the concentrations of proresorptive cytokine RANKL and its inhibitor osteoprotegerin (OPG), in osteoclast and osteoblast numbers, and in bone mass. We assumed that osteocytes surrounding a microfracture produce RANKL, which attracted osteoclasts. OPG and RANKL were produced by osteoblasts and diffused through bone, RANKL was eliminated by binding to OPG and RANK. Osteoblasts were coupled to osteoclasts through paracrine factors. The evolution of the BMU arising from this model was studied using numerical simulations. Our model recapitulated the spatio-temporal dynamics observed in vivo in a cross-section of bone. In response to a RANKL field, osteoclasts moved as a well-confined cutting cone. The coupling of osteoclasts to osteoblasts allowed for sufficient recruitment of osteoblasts to the resorbed surfaces. The RANKL field was the highest at the microfracture in front of the BMU, whereas the OPG field peaked at the back of the BMU, resulting in the formation of a RANKL/OPG gradient, which strongly affected the rate of BMU progression and its size. Thus, the spatial organization of a BMU provides important constraints on the roles of RANKL and OPG as well as possibly other regulators in determining the outcome of remodeling in the BMU.

Complex dynamics of osteoclast formation and death in long-term cultures.

BACKGROUND: Osteoclasts, cells responsible for bone resorption, contribute to the development of degenerative, metabolic and neoplastic bone diseases, which are often characterized by persistent changes in bone microenvironment. We aimed to investigate the dynamics of osteoclast formation and death in cultures that considerably exceeded the length of standard protocol and to design a mathematical model describing osteoclastogenesis. METHODOLOGY/PRINCIPAL FINDINGS: RAW 264.7 monocytic cells fuse to form multinucleated osteoclasts upon treatment with pro-resorptive cytokine RANKL. We have found that in long-term experiments (15-26 days), the dynamics of changes in osteoclast numbers was remarkably complex and qualitatively variable in different experiments. Whereas 19 of 46 experiments exhibited single peak of osteoclast formation, in 27 experiments we observed development of successive waves of osteoclast formation and death. Periodic changes in osteoclast numbers were confirmed in long-term cultures of mouse bone marrow cells treated with M-CSF and RANKL. Because the dynamics of changes in osteoclast numbers was found to be largely independent of monocytes, a two-species model of ordinary differential equations describing the changes in osteoclasts and monocytes was ineffective in recapitulating the oscillations in osteoclast numbers. Following experimental observation that medium collected from mature osteoclasts inhibited osteoclastogenesis in fresh cultures, we introduced a third variable, factor f, to describe osteoclast-derived inhibitor. This model allowed us to simulate the oscillatory changes in osteoclasts, which were coupled to oscillatory changes in the factor f, whereas monocytes changed exponentially. Importantly, to achieve the experimentally observed oscillations with increasing amplitude, we also had to assume that osteoclast presence stimulates osteoclast formation. CONCLUSIONS/SIGNIFICANCE: This study identifies the critical role for osteoclast autocrine regulation in controlling long-term dynamic of osteoclast formation and death and describes the complementary roles for negative and positive feedback mediators in determining the sharp dynamics of activation and inactivation of osteoclasts.

A predictive model of rectal tumor response to preoperative radiotherapy using classification and regression tree methods.

PURPOSE: The ability to predict rectal tumor response to preoperative radiotherapy before treatment would significantly impact patient selection. In this study, classification and regression tree (CART) methods were used to model tumor response to preoperative conformal high-dose rate brachytherapy by assessing the predictive value of vascular endothelial growth factor (VEGF), Bcl-2, p21, p53, and APAF-1. EXPERIMENTAL DESIGN: Immunohistochemistry was used to detect VEGF, Bcl-2, p21, p53, and APAF-1 from 62 pretreatment rectal tumor biopsies. Scores were assigned as percentages of positive tumor cell staining and were used in CART analysis to identify the proteins that best predicted response to radiotherapy. Ten-fold cross-validation was used to prevent overfitting and multiple cross-validation experiments were run to estimate the prediction error. RESULTS: Postoperative pathologic evaluation of the irradiated tumor bed revealed 43 responsive tumors [20 with complete response (T(0)) and 23 with partial response] and 19 nonresponsive tumors. The optimal tree resulting from CART analysis had five terminal nodes with a misclassification rate of 18%. Of the five proteins selected for their predictive value, VEGF and Bcl-2 contributed most to the classification of responsive and nonresponsive tumors. All 10 tumors with no VEGF were completely responsive (T(0)) to radiotherapy; 85% of those with VEGF and negative for Bcl-2 were responsive to therapy. CONCLUSIONS: VEGF and Bcl-2 status in pretreatment rectal tumor biopsies may be predictive of response to preoperative high-dose rate brachytherapy.