Computational modeling of the arterial wall based on layer-specific histological data

Arterial walls typically have a heterogeneous structure with three different layers (intima, media, and adventitia). Each layer can be modeled as a fiber-reinforced material with two families of relatively stiff collagenous fibers symmetrically arranged within an isotropic soft ground matrix. In this paper, we present two different modeling approaches, the embedded fiber (EF) approach and the angular integration (AI) approach, to simulate the anisotropic behavior of individual arterial wall layers involving layer-specific data. The EF approach directly incorporates the microscopic arrangement of fibers that are synthetically generated from a random walk algorithm and captures material anisotropy at the element level of the finite element formulation. The AI approach smears fibers in the ground matrix and treats the material as homogeneous, with material anisotropy introduced at the constitutive level by enhancing the isotropic strain energy with two anisotropic terms. Both approaches include the influence of fiber dispersion introduced by fiber angular distribution (departure of individual fibers from the mean orientation) and take into consideration the dispersion caused by fiber waviness, which has not been previously considered. By comparing the numerical results with the published experimental data of different layers of a human aorta, we show that by using histological data both approaches can successfully capture the anisotropic behavior of individual arterial wall layers. Furthermore, through a comprehensive parametric study, we establish the connections between the AI phenomenological material parameters and the EF parameters having straightforward physical or geometrical interpretations. This study provides valuable insight for the calibration of phenomenological parameters used in the homogenized modeling based on the fiber microscopic arrangement. Moreover, it facilitates a better understanding of individual arterial wall layers, which will eventually advance the study of the structure–function relationship of arterial walls as a whole.     

Trabecular bone microdamage and microstructural stresses under uniaxial compression

The balance between local remodeling and accumulation of trabecular bone microdamage is believed to play an important role in the maintenance of skeletal integrity. However, the local mechanical parameters associated with microdamage initiation are not well understood. Using histological damage labeling, micro-CT imaging, and image-based finite element analysis, regions of trabecular bone microdamage were detected and registered to estimated microstructural von Mises effective stresses and strains, maximum principal stresses and strains, and strain energy density (SED). Bovine tibial trabecular bone cores underwent a stepwise uniaxial compression routine in which specimens were micro-CT imaged following each compression step. The results indicate that the mode of trabecular failure observed by micro-CT imaging agreed well with the polarity and distribution of stresses within an individual trabecula. Analysis of on-axis subsections within specimens provided significant positive relationships between microdamage and each estimated tissue stress, strain and SED parameter. In a more localized analysis, individual microdamaged and undamaged trabeculae were extracted from specimens loaded within the elastic region and to the apparent yield point. As expected, damaged trabeculae in both groups possessed significantly higher local stresses and strains than undamaged trabeculae. The results also indicated that microdamage initiation occurred prior to apparent yield at local principal stresses in the range of 88-121 MPa for compression and 35-43 MPa for tension and local principal strains of 0.46-0.63% in compression and 0.18-0.24% in tension. These data provide an important step towards understanding factors contributing to microdamage initiation and establishing local failure criteria for normal and diseased trabecular bone.     

Relationships of loading history and structural and material characteristics of bone: development of the mule deer calcaneus

If a bone's morphologic organization exhibits the accumulated effects of its strain history, then the relative contributions of a given strain stimulus to a bone's development may be inferred from a bone's hierarchical organization. The artiodactyl calcaneus is a short cantilever, loaded habitually in bending, with prevalent compression in the cranial (Cr) cortex, tension in the caudal (Cd) cortex, and shear in the medial and lateral cortices (i.e., neutral axis). Artiodactyl calcanei demonstrate unusually heterogeneous structural and material organization between these cortices. This study examines potential relationships between developmental morphologic variations and the functional strain distribution of the deer calcaneus. One calcaneus was obtained from each of 36 (fetus to adult) wild deer. Predominant collagen fiber orientation (CFO), microstructural characteristics, mineral content (% ash), and geometric parameters were determined from transversely cut segments. Radiographs were examined for arched trabeculae, which may reflect tension/compression stress trajectories. Results showed that cross-sectional shape changes with age from quasi-circular to quasi-elliptical, with the long axis in the cranial-caudal direction of habitual bending. Cranial ("compression") cortical thickness increased at a greater rate than the Cd ("tension") cortex. Fetal bones exhibited arched trabeculae. Percent ash was not uniform (Cr > Cd), and this disparity increased with age (absolute differences: 2.5% fetuses, 4.3% adults). Subadult bones showed progressively more secondary osteons and osteocyte lacunae in the Cr cortex, but the Cd cortex tended to have more active remodeling in the subadult and adult bones. Nonuniform Cr:Cd CFO patterns first consistently appear in the subadults, and are correlated with secondary bone formation and habitual strain mode. Medial and lateral cortices in these groups exhibited elongated secondary osteons. These variations may represent "strain-mode-specific" (i.e., tension, compression, shear) adaptations. The heterogeneous organization may also be influenced by variations in longitudinal strain magnitude (highest in the Cr cortex) and principal strain direction-oblique in medial-lateral cortices (where shear strains also predominate). Other factors such as local reductions in longitudinal strain may influence the increased remodeling activity of the Cd cortex. Some structural variations, such as arched trabeculae, that are established early in ontogeny may be strongly influenced by genetic- or epigenetic-derived processes. Material variations, such as secondary osteon population densities and CFO, which appear later, may be products of extragenetic factors, including microdamage.     

A Gauss-Kronrod-Trapezoidal integration scheme for modeling biological tissues with continuous fiber distributions

Fibrous biological tissues may be modeled using a continuous fiber distribution (CFD) to capture tension–compression nonlinearity, anisotropic fiber distributions, and load-induced anisotropy. The CFD framework requires spherical integration of weighted individual fiber responses, with fibers contributing to the stress response only when they are in tension. The common method for performing this integration employs the discretization of the unit sphere into a polyhedron with nearly uniform triangular faces (finite element integration or FEI scheme). Although FEI has proven to be more accurate and efficient than integration using spherical coordinates, it presents three major drawbacks: First, the number of elements on the unit sphere needed to achieve satisfactory accuracy becomes a significant computational cost in a finite element (FE) analysis. Second, fibers may not be in tension in some regions on the unit sphere, where the integration becomes a waste. Third, if tensed fiber bundles span a small region compared to the area of the elements on the sphere, a significant discretization error arises. This study presents an integration scheme specialized to the CFD framework, which significantly mitigates the first drawback of the FEI scheme, while eliminating the second and third completely. Here, integration is performed only over the regions of the unit sphere where fibers are in tension. Gauss–Kronrod quadrature is used across latitudes and the trapezoidal scheme across longitudes. Over a wide range of strain states, fiber material properties, and fiber angular distributions, results demonstrate that this new scheme always outperforms FEI, sometimes by orders of magnitude in the number of computational steps and relative accuracy of the stress calculation.     

A model of stress and strain in the interosseous ligament of the forearm based on fiber network theory

Fiber network theory was developed to describe cloth, a thin material with strength in the fiber directions. The interosseous ligament (IOL) of the forearm is a broad, thin ligament with highly aligned fibers. The objectives of this study were to develop a model of the stress and strain distributions in the IOL, based on fiber network theory, to compare the strains from the model with the experimentally measured strains, and to evaluate the force distribution across the ligament fibers from the model. The geometries of the radius, ulna, and IOL were reconstructed from CT scans. Position and orientation of IOL insertion sites and force in the IOL were measured during a forearm compression experiment in pronation, neutral rotation, and supination. An optical image-based technique was used to directly measure strain in two regions of the IOL in neutral rotation. For the network model, the IOL was represented as a parametric ruled three-dimensional surface, with rulings along local fiber directions. Fiber strains were calculated from the deformation field, and fiber stresses were calculated from the strains using average IOL tensile properties from a previous study. The in situ strain in the IOL was assumed uniform and was calculated so that the net force predicted by the network model in neutral rotation matched the experimental result. The net force in the IOL was comparable to experimental results in supination and pronation. The model predicted higher stress and strain in fibers near the elbow in neutral rotation, and higher stresses in fibers near the wrist in supination. Strains in neutral forearm rotation followed the same trends as those measured experimentally. In this study, a model of stress and strain in the IOL utilizing fiber network theory was successfully implemented. The model illustrates variations in the stress and strain distribution in the IOL. This model can be used to show surgeons how different fibers are taut in different forearm rotation positions-this information is important for understanding the biomechanical role of the IOL and for planning an IOL reconstruction.     

Contribution of collagen fibers to the compressive stiffness of cartilaginous tissues

Cartilaginous tissues such as the intervertebral disk are predominantly loaded under compression. Yet, they contain abundant collagen fibers, which are generally assumed to contribute to tensile loading only. Fiber tension is thought to originate from swelling of the proteoglycan-rich nucleus. However, in aged or degenerate disk, proteoglycans are depleted, whereas collagen content changes little. The question then rises to which extend the collagen may contribute to the compressive stiffness of the tissue. We hypothesized that this contribution is significant at high strain magnitudes and that the effect depends on fiber orientation. In addition, we aimed to determine the compression of the matrix. Bovine inner and outer annulus fibrosus specimens were subjected to incremental confined compression tests up to 60 % strain in radial and circumferential direction. The compressive aggregate modulus was determined per 10 % strain increment. The biochemical composition of the compressed specimens and uncompressed adjacent tissue was determined to compute solid matrix compression. The stiffness of all specimens increased nonlinearly with strain. The collagen-rich outer annulus was significantly stiffer than the inner annulus above 20 % compressive strain. Orientation influenced the modulus in the collagen-rich outer annulus. Finally, it was shown that the solid matrix was significantly compressed above 30 % strain. Therefore, we concluded that collagen fibers significantly contribute to the compressive stiffness of the intervertebral disk at high strains. This is valuable for understanding the compressive behavior of collagen-reinforced tissues in general, and may be particularly relevant for aging or degenerate disks, which become more fibrous and less hydrated.     

Characterizing the mechanical contribution of fiber angular distribution in connective tissue: comparison of two modeling approaches

Modeling of connective tissues often includes collagen fibers explicitly as one of the components. These fibers can be oriented in many directions; therefore, several studies have considered statistical distributions to describe the fiber arrangement. One approach to formulate a constitutive framework for distributed fibers is to express the mechanical parameters, such as strain energy and stresses, in terms of angular integrals. These integrals represent the addition of the contribution of infinitesimal fractions of fibers oriented in a given direction. This approach leads to accurate results; however, it requires lengthy calculations. Recently, the use of generalized structure tensors has been proposed to represent the angular distribution in the constitutive equations of the fibers. Although this formulation is much simpler and fewer calculations are required, such structure tensors can only be used when all the fibers are in tension and the angular distribution is small. However, the amount of error introduced in these cases of non-tensile fiber loading and large angular distributions have not been quantified. Therefore, the objective of this study is to determine the range of values of angular distribution for which acceptable differences (less than 10%) between these two formulations are obtained. It was found, analytically and numerically, that both formulations are equivalent for planar distributions under equal-biaxial stretch. The comparison also showed, for other loading conditions, that the differences decrease when the fiber distribution is very small. Differences of less than 10% were usually obtained when the fiber distribution was very low (κ ≈ 0.03; κ ranges between 0 and 1/3, for aligned and isotropic distributed fibers, respectively). This range of angular distribution greatly limits the types of tissue that can be accurately analyzed using generalized structure tensors. It is expected that the results from this study guide the selection of a proper approach to analyze a particular tissue under a particular loading condition.     

Micromechanical Modeling Study of Mechanical Inhibition of Enzymatic Degradation of Collagen Tissues

This study investigates how the collagen fiber structure influences the enzymatic degradation of collagen tissues. We developed a micromechanical model of a fibrous collagen tissue undergoing enzymatic degradation based on two central hypotheses. The collagen fibers are crimped in the undeformed configuration. Enzymatic degradation is an energy activated process and the activation energy is increased by the axial strain energy density of the fiber. We determined the intrinsic degradation rate and characteristic energy for mechanical inhibition from fibril-level degradation experiments and applied the parameters to predict the effect of the crimped fiber structure and fiber properties on the degradation of bovine cornea and pericardium tissues under controlled tension. We then applied the model to examine the effect of the tissue stress state on the rate of tissue degradation and the anisotropic fiber structures that developed from enzymatic degradation.     

The influence of fiber orientation on the equilibrium properties of neutral and charged biphasic tissues

Constitutive models facilitate investigation into load bearing mechanisms of biological tissues and may aid attempts to engineer tissue replacements. In soft tissue models, a commonly made assumption is that collagen fibers can only bear tensile loads. Previous computational studies have demonstrated that radially aligned fibers stiffen a material in unconfined compression most by limiting lateral expansion while vertically aligned fibers buckle under the compressive loads. In this short communication, we show that in conjunction with swelling, these intuitive statements can be violated at small strains. Under such conditions, a tissue with fibers aligned parallel to the direction of load initially provides the greatest resistance to compression. The results are further put into the context of a Benninghoff architecture for articular cartilage. The predictions of this computational study demonstrate the effects of varying fiber orientations and an initial tare strain on the apparent material parameters obtained from unconfined compression tests of charged tissues.     

Biaxial tensile testing and constitutive modeling of human supraspinatus tendon

The heterogeneous composition and mechanical properties of the supraspinatus tendon offer an opportunity for studying the structure-function relationships of fibrous musculoskeletal connective tissues. Previous uniaxial testing has demonstrated a correlation between the collagen fiber angle distribution and tendon mechanics in response to tensile loading both parallel and transverse to the tendon longitudinal axis. However, the planar mechanics of the supraspinatus tendon may be more appropriately characterized through biaxial tensile testing, which avoids the limitation of nonphysiologic traction-free boundary conditions present during uniaxial testing. Combined with a structural constitutive model, biaxial testing can help identify the specific structural mechanisms underlying the tendon's two-dimensional mechanical behavior. Therefore, the objective of this study was to evaluate the contribution of collagen fiber organization to the planar tensile mechanics of the human supraspinatus tendon by fitting biaxial tensile data with a structural constitutive model that incorporates a sample-specific angular distribution of nonlinear fibers. Regional samples were tested under several biaxial boundary conditions while simultaneously measuring the collagen fiber orientations via polarized light imaging. The histograms of fiber angles were fit with a von Mises probability distribution and input into a hyperelastic constitutive model incorporating the contributions of the uncrimped fibers. Samples with a wide fiber angle distribution produced greater transverse stresses than more highly aligned samples. The structural model fit the longitudinal stresses well (median R(2) ≥ 0.96) and was validated by successfully predicting the stress response to a mechanical protocol not used for parameter estimation. The transverse stresses were fit less well with greater errors observed for less aligned samples. Sensitivity analyses and relatively affine fiber kinematics suggest that these errors are not due to inaccuracies in measuring the collagen fiber organization. More likely, additional strain energy terms representing fiber-fiber interactions are necessary to provide a closer approximation of the transverse stresses. Nevertheless, this approach demonstrated that the longitudinal tensile mechanics of the supraspinatus tendon are primarily dependent on the moduli, crimp, and angular distribution of its collagen fibers. These results add to the existing knowledge of structure-function relationships in fibrous musculoskeletal tissue, which is valuable for understanding the etiology of degenerative disease, developing effective tissue engineering design strategies, and predicting outcomes of tissue repair.     

Transfer of macroscale tissue strain to microscale cell regions in the deformed meniscus

Cells within fibrocartilaginous tissues, including chondrocytes and fibroblasts of the meniscus, ligament, and tendon, regulate cell biosynthesis in response to local mechanical stimuli. The processes by which an applied mechanical load is transferred through the extracellular matrix to the environment of a cell are not fully understood. To better understand the role of mechanics in controlling cell phenotype and biosynthetic activity, this study was conducted to measure strain at different length scales in tissue of the fibrocartilaginous meniscus of the knee joint, and to define a quantitative parameter that describes the strain transferred from the far-field tissue to a microenvironment surrounding a cell. Experiments were performed to apply a controlled uniaxial tensile deformation to explants of porcine meniscus containing live cells. Using texture correlation analyses of confocal microscopy images, two-dimensional Lagrangian and principal strains were measured at length scales representative of the tissue (macroscale) and microenvironment in the region of a cell (microscale) to yield a strain transfer ratio as a measure of median microscale to macroscale strain. The data demonstrate that principal strains at the microscale are coupled to and amplified from macroscale principal strains for a majority of cell microenvironments located across diverse microstructural regions, with average strain transfer ratios of 1.6 and 2.9 for the maximum and minimum principal strains, respectively. Lagrangian strain components calculated along the experimental axes of applied deformations exhibited considerable spatial heterogeneity and intersample variability, and suggest the existence of both strain amplification and attenuation. This feature is consistent with an in-plane rotation of the principal strain axes relative to the experimental axes at the microscale that may result from fiber sliding, fiber twisting, and fiber-matrix interactions that are believed to be important for regulating deformation in other fibrocartilaginous tissues. The findings for consistent amplification of macroscale to microscale principal strains suggest a coordinated pattern of strain transfer from applied deformation to the microscale environment of a cell that is largely independent of these microstructural features in the fibrocartilaginous meniscus.     

A Transversely Isotropic Hyperelastic Constitutive Model of the PDL. Analytical and Computational Aspects

This study describes the development of a constitutive law for the modelling of the periodontal ligament (PDL) and its practical implementation into a commercial finite element code. The constitutive equations encompass the essential mechanical features of this biological soft tissue: non-linear behaviour, large deformations, anisotropy, distinct behaviour in tension and compression and the fibrous characteristics. The approach is based on the theory of continuum fibre-reinforced composites at finite strain where a compressible transversely isotropic hyperelastic strain energy function is defined. This strain energy density function is further split into volumetric and deviatoric contributions separating the bulk and shear responses of the material. Explicit expressions of the stress tensors in the material and spatial configurations are first established followed by original expressions of the elasticity tensors in the material and spatial configurations. As a simple application of the constitutive model, two finite element analyses simulating the mechanical behaviour of the PDL are performed. The results highlight the significance of integrating the fibrous architecture of the PDL as this feature is shown to be responsible for the complex strain distribution observed.     

A transversely isotropic hyperelastic constitutive model of the PDL. Analytical and computational aspects

This study describes the development of a constitutive law for the modelling of the periodontal ligament (PDL) and its practical implementation into a commercial finite element code. The constitutive equations encompass the essential mechanical features of this biological soft tissue: non-linear behaviour, large deformations, anisotropy, distinct behaviour in tension and compression and the fibrous characteristics. The approach is based on the theory of continuum fibre-reinforced composites at finite strain where a compressible transversely isotropic hyperelastic strain energy function is defined. This strain energy density function is further split into volumetric and deviatoric contributions separating the bulk and shear responses of the material. Explicit expressions of the stress tensors in the material and spatial configurations are first established followed by original expressions of the elasticity tensors in the material and spatial configurations. As a simple application of the constitutive model, two finite element analyses simulating the mechanical behaviour of the PDL are performed. The results highlight the significance of integrating the fibrous architecture of the PDL as this feature is shown to be responsible for the complex strain distribution observed.     

A new constitutive formulation for characterizing the mechanical behavior of soft tissues.

We present a new constitutive formulation that combines certain desirable features of two previously used approaches (phenomenological and microstructural). Specifically, we assume that certain soft tissues can be idealized as composed of various families of noninteracting fibers and a homogeneous matrix. Both the fibers and the matrix are assumed to follow the gross deformation. Within the usual framework of pseudoelasticity, incompressibility, homogeneity, and the continuum hypothesis, a pseudostrain-energy function (W) is proposed wherein W is expressed in terms of matrix and fibrous contributions. Unlike phenomenological approaches where a W is usually chosen in an ad hoc manner, the present approach can be used to postulate reasonable forms of W based on limited structural information and multiaxial stress-strain data. Illustrative applications of the theory are discussed for visceral pleura and myocardium. Concise structurally motivated constitutive relations result, wherein load-dependent anisotropy, nonlinear material behavior, finite deformations, and incompressibility are accounted for.     

Simulation of the behaviour of the L1 vertebra for different material properties and loading conditions

Three-dimensional finite element models of the thoracolumbar junction (T12–L2) and isolated L1 vertebra were developed to investigate the role of material properties and loading conditions on vertebral stresses and strains to predict fracture risk. The geometry of the vertebrae was obtained from computed tomography images. The isolated vertebra model included an L1 vertebra loaded through polymethylmethacrylate plates located at the top and bottom of the vertebra, and the segment model included T12 to L2 vertebrae and seven ligaments, fibrous intervertebral discs and facet joints. Each model was examined with both homogeneous and spatially varying bone tissue properties. Stresses and strains were compared for uniform compression and flexion. Including material heterogeneity remarkably reduced the stiffness of the isolated L1 vertebra and increased the magnitudes of the minimum principal strains and stresses in the mid-transverse section. The stress and strain distributions further changed when physiological loading was applied to the L1 vertebra. In the segment models, including heterogeneous material properties increased the magnitude of the minimum principal strain by 158% in the centre of the mid-transverse section. Overall, the inclusion of heterogeneity and physiological loading increased the magnitude of the strains up to 346% in flexion and 273% in compression.     

Nonlinear Mechanical Properties of Prestressed Branched Fibrous Networks

Random fiber networks constitute the solid skeleton of many biological materials such as the cytoskeleton of cells and extracellular matrix of soft tissues. These random networks show unique mechanical properties such as nonlinear shear strain-stiffening and strain softening when subjected to preextension and precompression, respectively. In this study, we perform numerical simulations to characterize the influence of axial prestress on the nonlinear mechanical response of random network structures as a function of their micromechanical and geometrical properties. We build our numerical network models using the microstructure of disordered hexagonal lattices and quantify their nonlinear shear response as a function of uniaxial prestress strain. We consider three different material models for individual fibers and fully characterize their influence on the mechanical response of prestressed networks. Moreover, we investigate both the influence of geometric disorder keeping the network connectivity constant and the influence of the randomness in the stiffness of individual fibers keeping their mean stiffness constant. The effects of network connectivity and bending rigidity of fibers are also determined. Several important conclusions are made, including that the tensile and compressive prestress strains, respectively, increase and decrease the initial network shear stiffness but have no effect on the maximal shear modulus. We discuss the findings in terms of microstructural properties such as the local strain energy distribution.     

Strain transfer through the aortic valve

The complex structural organization of the aortic valve (AV) extracellular matrix (ECM) enables large and highly nonlinear tissue level deformations. The collagen and elastin (elastic) fibers within the ECM form an interconnected fibrous network (FN) and are known to be the main load-bearing elements of the AV matrix. The role of the FN in enabling deformation has been investigated and documented. However, there is little data on the correlation between tissue level and FN-level strains. Investigating this correlation will help establish the mode of strain transfer (affine or nonaffine) through the AV tissue as a key feature in microstructural modeling and will also help characterize the local FN deformation across the AV sample in response to applied tissue level strains. In this study, the correlation between applied strains at tissue level, macrostrains across the tissue surface, and local FN strains were investigated. Results showed that the FN strain distribution across AV samples was inhomogeneous and nonuniform, as well as anisotropic. There was no direct transfer of the deformation applied at tissue level to the fibrous network. Loading modes induced in the FN are different than those applied at the tissue as a result of different local strains in the valve layers. This nonuniformity of local strains induced internal shearing within the FN of the AV, possibly exposing the aortic valve interstitial cells (AVICs) to shear strains and stresses.     

Craniofacial sutures: morphology, growth, and in vivo masticatory strains.

The growth and morphology of craniofacial sutures are thought to reflect their functional environment. However, little is known about in vivo sutural mechanics. The present study investigates the strains experienced by the internasal, nasofrontal, and anterior interfrontal sutures during masticatory activity in 4-6-month-old miniature swine (Sus scrofa). Measurements of the bony/fibrous arrangements and growth rates of these sutures were then examined in the context of their mechanical environment. Large tensile strains were measured in the interfrontal suture (1,036 microepsilon +/- 400 SD), whereas the posterior internasal suture was under moderate compression (-440 microepsilon +/- 238) and the nasofrontal suture experienced large compression (-1,583 microepsilon +/- 506). Sutural interdigitation was associated with compressive strain. The collagen fibers of the internasal and interfrontal sutures were clearly arranged to resist compression and tension, respectively, whereas those of the nasofrontal suture could not be readily characterized as either compression or tension resisting. The average linear rate of growth over a 1-week period at the nasofrontal suture (133.8 micrometer, +/- 50.9 S.D) was significantly greater than that of both the internasal and interfrontal sutures (39.2 micrometer +/- 11.4 and 65. 5 micrometer +/- 14.0, respectively). Histological observations suggest that the nasofrontal suture contains chondroid tissue, which may explain the unexpected combination of high compressive loading and rapid growth in this suture.     

Orientation of spin-labeled myosin heads in glycerinated muscle fibers.

We have used electron paramagnetic resonance (EPR) spectra to study spin labels selectively and rigidly attached to myosin heads in glycerinated rabbit psoas muscle fibers. Because the angle between the magnetic field and the principal axis of the probe determines the position of the EPR absorption line, spectra from labeled fibers oriented parallel to the magnetic field yielded directly the distribution of spin label orientations relative to the fiber axis. Two spin labels, having reactivities resembling iodoacetamide (IASL) and maleimide (MSL), were used. In rigor fibers with complete filament overlap, both labels displayed a narrow angular distribution, full width at half maximum approximately 15 degrees, centered at angles of 68 degrees (IASL) and 82 degrees (MSL). Myosin subfragments (heavy meromyosin and subfragment-1) were labeled and allowed to diffuse into fibers. The resulting spectra showed the same sharp angular distribution that was found for the labeled fibers. Thus is appears that virtually all myosin heads in a rigor fiber have the same orientation relative to the fiber axis, and this orientation is determined by the actomyosin bond. Experiments with stretched fibers indicated that the spin labels on the fraction of heads not interacting with actin filaments had a broad angular distribution. Addition of ATP to unstretched fibers under relaxing conditions produced orientational disorder, resulting in a spectrum almost indistinguishable from that of an isotropic distribution of probes. Addition of either an ATP analog (AMPPNP) or pyrophosphate produced partial disorder. That is a fraction of the probes remained sharply oriented as in rigor while a second fraction was in a disordered distribution similar to that of relaxed fibers.