Experiments and mechanochemical modeling of smooth muscle contraction: Significance of filament overlap

Mechano-chemo-biological model of atherosclerosis formation based on the outside-in theory

Atherosclerosis is a disease in blood vessels that often results in plaque formation and lumen narrowing. It is an inflammatory response of the tissue caused by disruptions in the vessel wall nourishment. Blood vessels are nourished by nutrients originating from the blood of the lumen. In medium-sized and larger vessels, nutrients are additionally provided from outside through a network of capillaries called vasa vasorum. It has recently been hypothesized (Haverich in Circulation 135:205-207, 2017) that the root of atherosclerotic diseases is the malfunction of the vasa vasorum. This, so-called outside-in theory, is supported by a recently developed numerical model (Soleimani et al. in Arch Comput Methods Eng 28:4263-4282, 2021) accounting for the inflammation initiation in the adventitial layer of the blood vessel. Building on the previous findings, this work proposes an extended material model for atherosclerosis formation that is based on the outside-in theory. Beside the description of growth kinematics and nutrient diffusion, the roles of monocytes, macrophages, foam cells, smooth muscle cells and collagen are accounted for in a nonlinear continuum mechanics framework. Cells are activated due to a lack of vessel wall nourishment and proliferate, migrate, differentiate and synthesize collagen, leading to the formation of a plaque. Numerical studies show that the onset of atherosclerosis can qualitatively be reproduced and back the new theory.

A new model for evaluating pressure-induced vascular tone in small cerebral arteries

The capacity of small cerebral arteries (SCAs) to adapt to pressure fluctuations has a fundamental physiological role and appears to be relevant in different pathological conditions. Here, we present a new computational model for quantifying the link, and its contributors, between luminal pressure and vascular tone generation in SCAs. This is assembled by combining a chemical sub-model, representing pressure-induced smooth muscle cell (SMC) signalling, with a mechanical sub-model for the tone generation and its transduction at tissue level. The devised model can accurately reproduce the impact of luminal pressure on different cytoplasmic components involved in myogenic signalling, both in the control case and when combined with some specific pharmacological interventions. Furthermore, the model is also able to capture and predict experimentally recorded pressure-outer diameter relationships obtained for vessels under control conditions, both in a Ca2 + -free bath and under drug inhibition. The modularity of the proposed framework allows the integration of new components for the study of a broad range of processes involved in the vascular function.

A computational bridge between traction force microscopy and tissue contraction

Arterial wall active mechanics are driven by resident smooth muscle cells, which respond to biological, chemical, and mechanical stimuli and activate their cytoskeletal machinery to generate contractile stresses. The cellular mechanoresponse is sensitive to environmental perturbations, often leading to maladaptation and disease progression. When investigated at the single cell scale, however, these perturbations do not consistently result in phenotypes observed at the tissue scale. Here, a multiscale model is introduced that translates microscale contractility signaling into a macroscale, tissue-level response. The microscale framework incorporates a biochemical signaling network along with characterization of fiber networks that govern the anisotropic mechanics of vascular tissue. By incorporating both biochemical and mechanical components, the model is more flexible and more broadly applicable to physiological and pathological conditions. The model can be applied to both cell and tissue scale systems, allowing for the analysis of in vitro, traction force microscopy and ex vivo, isometric contraction experiments in parallel. When applied to aortic explant rings and isolated smooth muscle cells, the model predicts that active contractility is not a function of stretch at intermediate strain. The model also successfully predicts cell-scale and tissue-scale contractility and matches experimentally observed behaviors, including the hypercontractile phenotype caused by chronic hyperglycemia. The connection of the microscale framework to the macroscale through the multiscale model presents a framework that can translate the wealth of information already collected at the cell scale to tissue scale phenotypes, potentially easing the development of smooth muscle cell-targeting therapeutics.

A review on the biomechanical behaviour of the aorta

Large aortic aneurysm and acute and chronic aortic dissection are pathologies of the aorta requiring surgery. Recent advances in medical intervention have improved patient outcomes; however, a clear understanding of the mechanisms leading to aortic failure and, hence, a better understanding of failure risk, is still missing. Biomechanical analysis of the aorta could provide insights into the development and progression of aortic abnormalities, giving clinicians a powerful tool in risk stratification. The complexity of the aortic system presents significant challenges for a biomechanical study and requires various approaches to analyse the aorta. To address this, here we present a holistic review of the biomechanical studies of the aorta by categorising articles into four broad approaches, namely theoretical, in vivo, experimental and combined investigations. Experimental studies that focus on identifying mechanical properties of the aortic tissue are also included. By reviewing the literature and discussing drawbacks, limitations and future challenges in each area, we hope to present a more complete picture of the state-of-the-art of aortic biomechanics to stimulate research on critical topics. Combining experimental modalities and computational approaches could lead to more comprehensive results in risk prediction for the aortic system.

Stiffness sensing by smooth muscle cells: Continuum mechanics modeling of the acto-myosin role

Aortic smooth muscle cells (SMCs) play a vital role in maintaining homeostasis in the aorta by sensing and responding to mechanical stimuli. However, the mechanisms that underlie the ability of SMCs to sense and respond to stiffness change in their environment are still partially unclear. In this study, we focus on the role of acto-myosin contractility in stiffness sensing and introduce a novel continuum mechanics approach based on the principles of thermal strains. Each stress fiber satisfies a universal stress-strain relationship driven by a Young's modulus, a contraction coefficient scaling the fictitious thermal strain, a maximum contraction stress and a softening parameter describing the sliding effects between actin and myosin filaments. To account for the inherent variability of cellular responses, large populations of SMCs are modeled with the finite-element method, each cell having a random number and a random arrangement of stress fibers. Moreover, the level of myosin activation in each stress fiber satisfies a Weibull probability density function. Model predictions are compared to traction force measurements on different SMC lineages. It is demonstrated that the model not only predicts well the effects of substrate stiffness on cellular traction, but it can also successfully approximate the statistical variations of cellular tractions induced by intercellular variability. Finally, stresses in the nuclear envelope and in the nucleus are computed with the model, showing that the variations of cytoskeletal forces induced by substrate stiffness directly induce deformations of the nucleus which can potentially alter gene expression. The predictability of the model combined to its relative simplicity are promising assets for further investigation of stiffness sensing in 3D environments. Eventually, this could contribute to decipher the effects of mechanosensitivity impairment, which are known to be at the root of aortic aneurysms.

Chemo-mechanical modeling of smooth muscle cell activation for the simulation of arterial walls under changing blood pressure

In this paper, a novel chemo-mechanical model is proposed for the description of the stretch-dependent chemical processes known as Bayliss effect and their impact on the active contraction in vascular smooth muscle. These processes are responsible for the adaptive reaction of arterial walls to changing blood pressure by which the blood vessels actively support the heart in providing sufficient blood supply for varying demands in the supplied tissues. The model is designed to describe two different stretch-dependent mechanisms observed in smooth muscle cells (SMCs): a calcium-dependent and a calcium-independent contraction. For the first one, stretch of the SMCs leads to an inlet of calcium ions which activates the myosin light chain kinase (MLCK). The increased activity of MLCK triggers the contractile units of the cells resulting in the contraction on a comparatively short time scale. For the calcium-independent contraction mechanism, stretch-dependent receptors of the cell membrane stimulate an intracellular reaction leading to an inhibition of the antagonist of MLCK, the myosin light chain phosphatase resulting in a contraction on a comparatively long time scale. An algorithmic framework for the implementation of the model in finite element programs is derived. Based thereon, it is shown that the proposed approach agrees well with experimental data. Furthermore, the individual aspects of the model are analyzed in numerical simulations of idealized arteries subject to internal pressure waves with changing intensities. The simulations show that the proposed model is able to describe the experimentally observed contraction of the artery as a reaction to increased internal pressure, which can be considered a crucial aspect of the regulatory mechanism of muscular arteries.

Modeling Reactive Hyperemia to Better Understand and Assess Microvascular Function: A Review of Techniques

Reactive hyperemia is a well-established technique for the non-invasive evaluation of the peripheral microcirculatory function, measured as the magnitude of limb re-perfusion after a brief period of ischemia. Despite widespread adoption by researchers and clinicians alike, many uncertainties remain surrounding interpretation, compounded by patient-specific confounding factors (such as blood pressure or the metabolic rate of the ischemic limb). Mathematical modeling can accelerate our understanding of the physiology underlying the reactive hyperemia response and guide in the estimation of quantities which are difficult to measure experimentally. In this work, we aim to provide a comprehensive guide for mathematical modeling techniques that can be used for describing the key phenomena involved in the reactive hyperemia response, alongside their limitations and advantages. The reported methodologies can be used for investigating specific reactive hyperemia aspects alone, or can be combined into a computational framework to be used in (pre-)clinical settings.

Characterization of the active response of a guinea pig carotid artery

This work presents a characterization of the active response of the carotid artery of guinea pig fetuses through a methodology that encompasses experiments, modeling and numerical simulation. To this end, the isometric contraction test is carried out in ring samples subjected to different levels of KCl concentrations and pre-stretching. Then, a coupled mechanochemical model, aimed at describing the smooth cell behavior and its influence on the passive and active mechanical response of the vascular tissue, is calibrated from the experimental measurements. Due to the complex stress and strain fields developed in the artery, a finite element numerical simulation of the test is performed to fit the model parameters, where those related to the phosphorylation and dephosphorylation activity along with the load-bearing capacity of the myosin cross-bridges are found to be the most predominant when sensitizing the active response. The main strengths of the model are associated with the prediction of the stationary state of the active mechanical response of the tissue through a realistic description of the mechanochemical process carried out at its cellular level.

Thermodynamically-motivated chemo-mechanical models and multicellular simulation to provide new insight into active cell and tumour remodelling

Computational models can shape our understanding of cell and tissue remodelling, from cell spreading, to active force generation, adhesion, and growth. In this mini-review, we discuss recent progress in modelling of chemo-mechanical cell behaviour and the evolution of multicellular systems. In particular, we highlight recent advances in (i) free-energy based single cell models that can provide new fundamental insight into cell spreading, cancer cell invasion, stem cell differentiation, and remodelling in disease, and (ii) mechanical agent-based models to simulate large numbers of discrete interacting cells in proliferative tumours. We describe how new biological understanding has emerged from such theoretical models, and the trade-offs and constraints associated with current approaches. Ultimately, we aim to make a case for why theory should be integrated with an experimental workflow to optimise new in-vitro studies, to predict feedback between cells and their microenvironment, and to deepen understanding of active cell behaviour.

Role of smooth muscle activation in the static and dynamic mechanical characterization of human aortas

The rupture of aortic aneurysms causes around 10,000 deaths each year in the United States. Prosthetic tubes for aortic repair present a large mismatch of mechanical properties with the natural aorta, which has negative consequences for perfusion. This motivates research into the mechanical characterization of human aortas to develop a new generation of mechanically compatible aortic grafts. Experimental data and a suitable material model for human aortas with vascular smooth muscle (VSM) activation are not available. Hence, the present study provides experimental data that are needed. These data made it possible to develop a precise structure-based model of active aortic tissue. The results show the importance of VSM activation on the static and dynamic mechanical response of human aortas.

Experimental data and a suitable material model for human aortas with smooth muscle activation are not available in the literature despite the need for developing advanced grafts; the present study closes this gap. Mechanical characterization of human descending thoracic aortas was performed with and without vascular smooth muscle (VSM) activation. Specimens were taken from 13 heart-beating donors. The aortic segments were cooled in Belzer UW solution during transport and tested within a few hours after explantation. VSM activation was achieved through the use of potassium depolarization and noradrenaline as vasoactive agents. In addition to isometric activation experiments, the quasistatic passive and active stress–strain curves were obtained for circumferential and longitudinal strips of the aortic material. This characterization made it possible to create an original mechanical model of the active aortic material that accurately fits the experimental data. The dynamic mechanical characterization was executed using cyclic strain at different frequencies of physiological interest. An initial prestretch, which corresponded to the physiological conditions, was applied before cyclic loading. Dynamic tests made it possible to identify the differences in the viscoelastic behavior of the passive and active tissue. This work illustrates the importance of VSM activation for the static and dynamic mechanical response of human aortas. Most importantly, this study provides material data and a material model for the development of a future generation of active aortic grafts that mimic natural behavior and help regulate blood pressure.

Efficient computational modelling of smooth muscle orientation and function in the aorta

Understanding the mechanical effects of smooth muscle cell (SMC) contraction on the initiation and the propagation of cardiovascular diseases such as aortic dissection is critical. Framed by elastic lamellar sheets in the lamellar unit, there are SMCs in the media with a distinct radial tilt, which indicates their contribution to the radial strength. However, the mechanical effects of this type of anisotropy have not been fully discussed. Therefore, in this study, we propose a constitutive framework that models the passive and active mechanics of the aorta, taking into account the dispersed nature of the aortic constituents by applying the discrete fibre dispersion method. We suggest an isoparametric approach by evaluating various numerical integration methods and introducing a non-uniform discretization of the unit hemisphere to increase its computational efficiency. Finally, the constitutive parameters are fitted to layer-specific experimental data and initial computational results are briefly presented. The radial tilt of SMCs is also analysed, which has a noticeable influence on the mechanical behaviour of the aorta. In the absence of sufficient experimental data, the results indicate that the active contribution of SMCs has a remarkable impact on the mechanics of the healthy aorta.

In vivo parameter identification in arteries considering multiple levels of smooth muscle activity

In this paper an existing in vivo parameter identification method for arteries is extended to account for smooth muscle activity. Within this method a continuum-mechanical model, whose parameters relate to the mechanical properties of the artery, is fit to clinical data by solving a minimization problem. Including smooth muscle activity in the model increases the number of parameters. This may lead to overparameterization, implying that several parameter combinations solve the minimization problem equally well and it is therefore not possible to determine which set of parameters represents the mechanical properties of the artery best. To prevent overparameterization the model is fit to clinical data measured at different levels of smooth muscle activity. Three conditions are considered for the human abdominal aorta: basal during rest; constricted, induced by lower-body negative pressure; and dilated, induced by physical exercise. By fitting the model to these three arterial conditions simultaneously a unique set of model parameters is identified and the model prediction agrees well with the clinical data.

A computational framework for modeling cell-matrix interactions in soft biological tissues

Living soft tissues appear to promote the development and maintenance of a preferred mechanical state within a defined tolerance around a so-called set point. This phenomenon is often referred to as mechanical homeostasis. In contradiction to the prominent role of mechanical homeostasis in various (patho)physiological processes, its underlying micromechanical mechanisms acting on the level of individual cells and fibers remain poorly understood, especially how these mechanisms on the microscale lead to what we macroscopically call mechanical homeostasis. Here, we present a novel computational framework based on the finite element method that is constructed bottom up, that is, it models key mechanobiological mechanisms such as actin cytoskeleton contraction and molecular clutch behavior of individual cells interacting with a reconstructed three-dimensional extracellular fiber matrix. The framework reproduces many experimental observations regarding mechanical homeostasis on short time scales (hours), in which the deposition and degradation of extracellular matrix can largely be neglected. This model can serve as a systematic tool for future in silico studies of the origin of the numerous still unexplained experimental observations about mechanical homeostasis.

A Chemomechanobiological Model of the Long-Term Healing Response of Arterial Tissue to a Clamping Injury

Vascular clamping often causes injury to arterial tissue, leading to a cascade of cellular and extracellular events. A reliable in silico prediction of these processes following vascular injury could help us to increase our understanding thereof, and eventually optimize surgical techniques or drug delivery to minimize the amount of long-term damage. However, the complexity and interdependency of these events make translation into constitutive laws and their numerical implementation particularly challenging. We introduce a finite element simulation of arterial clamping taking into account acute endothelial denudation, damage to extracellular matrix, and smooth muscle cell loss. The model captures how this causes tissue inflammation and deviation from mechanical homeostasis, both triggering vascular remodeling. A number of cellular processes are modeled, aiming at restoring this homeostasis, i.e., smooth muscle cell phenotype switching, proliferation, migration, and the production of extracellular matrix. We calibrated these damage and remodeling laws by comparing our numerical results to in vivo experimental data of clamping and healing experiments. In these same experiments, the functional integrity of the tissue was assessed through myograph tests, which were also reproduced in the present study through a novel model for vasodilator and -constrictor dependent smooth muscle contraction. The simulation results show a good agreement with the in vivo experiments. The computational model was then also used to simulate healing beyond the duration of the experiments in order to exploit the benefits of computational model predictions. These results showed a significant sensitivity to model parameters related to smooth muscle cell phenotypes, highlighting the pressing need to further elucidate the biological processes of smooth muscle cell phenotypic switching in the future.

Computational Study of Growth and Remodeling in Ascending Thoracic Aortic Aneurysms Considering Variations of Smooth Muscle Cell Basal Tone

In this paper, we investigate the progression of Ascending Thoracic Aortic Aneurysms (ATAA) using a computational model of Growth and Remodeling (G&R) taking into account the composite (elastin, four collagen fiber families and Smooth Muscle Cells—SMCs) and multi-layered (media and adventitia) nature of the aorta. The G&R model, which is based on the homogenized Constrained Mixture theory, is implemented as a UMAT in the Abaqus finite-element package. Each component of the mixture is assigned a strain energy density function: nearly-incompressible neo-Hookean for elastin and Fung-type for collagen and SMCs. Active SMCs tension is additionally considered, through a length-tension relationship having a classic inverted parabola shape, in order to investigate its effects on the progression of ATAA in a patient-specific model. A sensitivity analysis is performed to evaluate the potential impact of variations in the parameters of the length-tension relationships. These variations reflect in variations of SMCs normal tone during ATAA progression, with active stress contributions ranging between 30% (best case scenario) and 0% (worst case scenario) of the total wall circumferential stress. Low SMCs active stress in the worst case scenarios, in fact, affect the rates of collagen deposition by which the elastin loss is gradually compensated by collagen deposition in the simulated ATAA progression, resulting eventually in larger aneurysm diameters. The types of length-tension relationships leading to a drop of SMCs active stress in our simulations reveal a critical condition which could also result in SMCs apoptosis.

Force distribution and multi-scale mechanics in smooth muscle tissues

The mechanical role of smooth muscle tissue in many physiological processes is vital to their healthy function. In this work, we provide a deeper understanding of the underlying mechanisms that govern the smooth muscle tissue response. Specifically, we model and investigate the distribution and the transmission of passive and active forces throughout the microstructure. Broadly, smooth muscle cells contain a structural network with two types of load carrying structures: (1) contractile units made of actin and myosin filaments, which are capable of generating force, and (2) intermediate filaments. The extracellular matrix comprises elastin and collagen fibers that can sustain stress. We argue that all of the load carrying constituents in the tissue participate in the generation and the transmission of passive and active forces. We begin by modeling the response of the elements in the smooth muscle cell and defining a network of contractile units and intermediate filaments through which forces are transferred. This allows to derive an expression for the stress that develops in the cell. Next, we assume a hyperelastic behavior for the extracellular matrix and determine the stress in the tissue. With appropriate kinematic constraints and equilibrium considerations, we relate the macroscopic deformation to the stretch of the individual load carrying structures. Consequently, the stress on each element in the tissue can be computed. To validate the framework, we consider a simple microstructure of a smooth muscle tissue and fit the model parameters to experimental findings. The framework is also used to delineate experimental evidence which suggests that the suppression of intermediate filaments reduces the active and passive forces in a tissue. We show that the degradation and the reduction of the number of intermediate filaments in the cell fully explains this observation.

Fundamental Roles of Axial Stretch in Isometric and Isobaric Evaluations of Vascular Contractility

Vascular smooth muscle cells (VSMCs) can regulate arterial mechanics via contractile activity in response to changing mechanical and chemical signals. Contractility is traditionally evaluated via uniaxial isometric testing of isolated rings despite the in vivo environment being very different. Most blood vessels maintain a locally preferred value of in vivo axial stretch while subjected to changes in distending pressure, but both of these phenomena are obscured in uniaxial isometric testing. Few studies have rigorously analyzed the role of in vivo loading conditions in smooth muscle function. Thus, we evaluated effects of uniaxial versus biaxial deformations on smooth muscle contractility by stimulating two regions of the mouse aorta with different vasoconstrictors using one of three testing protocols: (i) uniaxial isometric testing, (ii) biaxial isometric testing, and (iii) axially isometric plus isobaric testing. Comparison of methods (i) and (ii) revealed increased sensitivity and contractile capacity to potassium chloride and phenylephrine (PE) with biaxial isometric testing, and comparison of methods (ii) and (iii) revealed a further increase in contractile capacity with isometric plus isobaric testing. Importantly, regional differences in estimated in vivo axial stretch suggest locally distinct optimal biaxial configurations for achieving maximal smooth muscle contraction, which can only be revealed with biaxial testing. Such differences highlight the importance of considering in vivo loading and geometric configurations when evaluating smooth muscle function. Given the physiologic relevance of axial extension and luminal pressurization, we submit that, when possible, axially isometric plus isobaric testing should be employed to evaluate vascular smooth muscle contractile function.

Modeling the effects of muscle contraction on the mechanical response and circumferential stability of coronary arteries

Smooth muscle contraction regulates the size of the blood vessel lumen which directly affects the mechanical response of the vessel. Folding in arteries has been observed in arteries during excessive contraction, known as a coronary artery spasm. The interplay of muscle contraction, geometry, and material responses and their effects on stability can be understood through mathematical models. Here, we consider a three-layer cross-sectional model of a coronary artery with anisotropic properties and intimal thickening, and perform a linear stability analysis to investigate the circumferential folding patterns that emerge due to muscle contraction. Our model shows that a critical level of contractile activity yields a uniform strain distribution across the arterial wall. When the muscle is contracted above this critical level, the tissue behaves isotropically and it is more prone to circumferential instability. This theoretical framework could serve as a valuable tool to understand the relationship between arterial lumen morphology and wall contraction in health and disease.

A multiscale active structural model of the arterial wall accounting for smooth muscle dynamics

Arterial wall dynamics arise from the synergy of passive mechano-elastic properties of the vascular tissue and the active contractile behaviour of smooth muscle cells (SMCs) that form the media layer of vessels. We have developed a computational framework that incorporates both these components to account for vascular responses to mechanical and pharmacological stimuli. To validate the proposed framework and demonstrate its potential for testing hypotheses on the pathogenesis of vascular disease, we have employed a number of pharmacological probes that modulate the arterial wall contractile machinery by selectively inhibiting a range of intracellular signalling pathways. Experimental probes used on ring segments from the rabbit central ear artery are: phenylephrine, a selective α1-adrenergic receptor agonist that induces vasoconstriction; cyclopiazonic acid (CPA), a specific inhibitor of sarcoplasmic/endoplasmic reticulum Ca²⁺-ATPase; and ryanodine, a diterpenoid that modulates Ca²⁺ release from the sarcoplasmic reticulum. These interventions were able to delineate the role of membrane versus intracellular signalling, previously identified as main factors in smooth muscle contraction and the generation of vessel tone. Each SMC was modelled by a system of nonlinear differential equations that account for intracellular ionic signalling, and in particular Ca²⁺ dynamics. Cytosolic Ca²⁺ concentrations formed the catalytic input to a cross-bridge kinetics model. Contractile output from these cellular components forms the input to the finite-element model of the arterial rings under isometric conditions that reproduces the experimental conditions. The model does not account for the role of the endothelium, as the nitric oxide production was suppressed by the action of L-NAME, and also due to the absence of shear stress on the arterial ring, as the experimental set-up did not involve flow. Simulations generated by the integrated model closely matched experimental observations qualitatively, as well as quantitatively within a range of physiological parametric values. The model also illustrated how increased intercellular coupling led to smooth muscle coordination and the genesis of vascular tone.

Computational Multiscale Solvers for Continuum Approaches

Computational multiscale analyses are currently ubiquitous in science and technology. Different problems of interest-e.g., mechanical, fluid, thermal, or electromagnetic-involving a domain with two or more clearly distinguished spatial or temporal scales, are candidates to be solved by using this technique. Moreover, the predictable capability and potential of multiscale analysis may result in an interesting tool for the development of new concept materials, with desired macroscopic or apparent properties through the design of their microstructure, which is now even more possible with the combination of nanotechnology and additive manufacturing. Indeed, the information in terms of field variables at a finer scale is available by solving its associated localization problem. In this work, a review on the algorithmic treatment of multiscale analyses of several problems with a technological interest is presented. The paper collects both classical and modern techniques of multiscale simulation such as those based on the proper generalized decomposition (PGD) approach. Moreover, an overview of available software for the implementation of such numerical schemes is also carried out. The availability and usefulness of this technique in the design of complex microstructural systems are highlighted along the text. In this review, the fine, and hence the coarse scale, are associated with continuum variables so atomistic approaches and coarse-graining transfer techniques are out of the scope of this paper.

Maladaptive aortic remodeling in hypertension associates with dysfunctional smooth muscle contractility

Intramural cells are responsible for establishing, maintaining, and restoring the functional capability and structural integrity of the aortic wall. In response to hypertensive loading, these cells tend to increase wall content via extracellular matrix turnover in an attempt to return wall stress and/or material stiffness toward homeostatic values despite the elevated pressure. Using a common rodent model of induced hypertension, we found marked mouse-to-mouse differences in thoracic aortic remodeling over 2-4 wk of pressure elevation, with mechanoadaptation in some but gross maladaptation in most mice despite the same experimental conditions and overall genetic background. Consistent with our hypothesis, we also found a strong correlation between maladaptive aortic remodeling and a dysfunctional ability of the vessel to vasoconstrict, with maladaptation often evidenced by marked adventitial fibrosis. Remarkably, mouse-to-mouse variability did not correlate with the degree or duration of pressure elevation over the 2- to 4-wk study period. These findings suggest both a need to study together the structure, mechanical properties, and function across layers of the wall when assessing aortic health and a need for caution in using common statistical comparisons across small seemingly well-defined groups that may mask important underlying individual responses, an area of investigation that demands increasing attention as we move toward an era of precision diagnosis and patient care. NEW & NOTEWORTHY There are three primary findings. Marked mouse-to-mouse differences exist in large vessel hypertensive remodeling in an otherwise equivalent cohort of animals. The degree of maladaptation correlates strongly with decreases in smooth muscle contractile capacity. Finally, short-term maladaptive remodeling is independent of the precise degree or duration of the pressure elevation provided that thresholds are exceeded. Therapeutic targets should thus be personalized and focus on both layer-to-layer interactions and early interventions.

Biomechanical relevance of the microstructure in artery walls with a focus on passive and active components

The microstructure of arteries, consisting, in particular, of collagen, elastin, and vascular smooth muscle cells, plays a very significant role in their biomechanical response during a cardiac cycle. In this article, we highlight the microstructure and the contributions of each of its components to the overall mechanical behavior. We also describe the changes of the microstructure that occur as a result of abdominal aortic aneurysms and disease, such as atherosclerosis. We also focus on how the passive and active constituents are incorporated into a mathematical model without going into detail of the mathematical formulation. We conclude by mentioning open problems toward a better characterization of the biomechanical aspects of arteries that will be beneficial for a better understanding of cardiovascular pathophysiology.

UNDERSTANDING THE VISCOELASTIC PROPERTIES OF RABBIT CORNEA BASED ON STRESS RELAXATION TESTS AND CYCLIC UNIAXIAL TESTS

Background: Determining the viscoelastic properties of cornea is important in the fields of understanding of the tissue’s response to mechanical actions and the accurate numerical simulation of corneal biomechanical behavior under the effects of keratoconus and refractive surgery. To address this need, we present an approach to model the viscoelastic response of rabbit cornea from uniaxial test data. Methods: The corneal strip samples from six rabbits were obtained to perform cyclic uniaxial tension tests and stress relaxation tests. We investigated the suitability of six constitutive models, including empirical models and hyperelastic models, by a quasi-linear viscoelastic law. Applying non-linear optimization techniques, we found material parameters for each different strip sample. Results and conclusions: The model gave a better fit to loading data with [Formula: see text], and predicted the unloading data in the cyclic uniaxial tests with errors-of-fit ranging from 0.03 to 0.06. The results indicate that the best model is the power of the first invariant of strain with Prony form relaxation model, and that the method to identify the material parameters are valid for modeling the visoelastic response of cornea from uniaxial test data.

Multiscale and Multiaxial Mechanics of Vascular Smooth Muscle

Mathematical models can facilitate an integrative understanding of the complexity underlying biological structure and function, but they must be informed and validated by empirical data. Uniaxial contraction of an arterial ring is a well-used in vitro approach for studying characteristics of smooth muscle contractility even though this experimental arrangement does not mimic the in vivo vascular geometry or loading. In contrast, biaxial contraction of an inflated and axially extended excised vessel provides broader information, both passive and active, under more realistic conditions. Few investigations have compared these two in vitro approaches directly, namely how their results overlap, how they differ, or if each provides unique complementary information. Toward this end, we present, to our knowledge, a new multiscale mathematical model of arterial contractility accounting for structural and functional constituents at molecular, cellular, and tissue levels. The artery is assumed to be a thick-walled incompressible cylinder described by an anisotropic model of the extracellular matrix and, to our knowledge, novel model of smooth muscle contractility. The latter includes a 3D structural sensitivity to deformation, including microscale muscle filament overlap and filament lattice spacing. The overall model captures uniaxial and biaxial experimental contraction data, which was not possible when accounting for filament overlap alone. The model also enables parameter sensitivity studies, which confirmed that uniaxial contraction tests are not as efficient as biaxial tests for identifying changes in vascular smooth muscle function.

Experimental characterization of the biaxial mechanical properties of porcine gastric tissue

Health problems related to the stomach are among the most important sources of morbidity in industrialized countries. There is evidence that mechanics may play an important role in various such pathologies. However, so far experimental data characterizing the mechanical properties of gastric tissue remain scarce, which significantly limits our understanding of the mechanics of the stomach. To help close this gap, we performed biaxial mechanical tests of porcine gastric tissue patches. Our experiments reveal a considerable anisotropy and different mechanical properties in the three major regions of the stomach (fundus, corpus, antrum). Moreover, they demonstrate that the mechanical properties of the gastric wall and the physiological function of the different regions of the stomach are closely related. This finding suggests that further examination of the mechanics of the gastric wall may indeed be a promising avenue of research towards a better understanding of the organic causes of frequent health problems related to the stomach.

Identification of the mechanical parameters for the human uterus in vivo using intrauterine pressure measurements

There are limited experimental data to characterize the mechanical response of human myometrium. A method is presented in this work to identify mechanical parameters describing the active response of human myometrium from the in vivo intrauterine pressure measurements. The human uterine contraction during labor is simulated by implementing a coupled model in a finite element scheme, and the intrauterine pressure is evaluated as the outcome. A meta-model is developed to approximate the finite element simulation response with a lower computational cost and used to identify model parameters through fitting its prediction to the in vivo measurements. Copyright © 2016 John Wiley & Sons, Ltd.

The biomechanical significance of pulley on binocular vision

Background

Pulleys have been reported as the functional origins of the rectus extraocular muscles (EOMs). However, biomechanical significance of pulleys on binocular vision has not been reported.

Methods

Three eye movement models, i.e., non-pulley model, passive-pulley model, and active-pulley model, are used to simulate the horizontal movement of the eyes from the primary position to the left direction in the range of 1°–30°. The resultant forces of six EOMs along both orthogonal directions (i.e., the x-axis and y-axis defined in this paper) in the horizontal plane are calculated using the three models.

Results

The resultant force along the y-axis of the left eye for non-pulley model are significantly larger than that of the other two pulley models. The difference of the force, between the left eye and the right eye in non-pulley model, is larger than those in the other two pulley models along x-axis and y-axis.

Conclusion

The pulley models present more biomechanical advantage on the horizontally binocular vision than the non-pulley model. Combining with the previous imaging evidences of pulleys, the results show that pulley model coincides well with the real physiological conditions.

Reduced Biaxial Contractility in the Descending Thoracic Aorta of Fibulin-5 Deficient Mice

The precise role of smooth muscle cell contractility in elastic arteries remains unclear, but accumulating evidence suggests that smooth muscle dysfunction plays an important role in the development of thoracic aortic aneurysms and dissections (TAADs). Given the increasing availability of mouse models of these conditions, there is a special opportunity to study roles of contractility ex vivo in intact vessels subjected to different mechanical loads. In parallel, of course, there is a similar need to study smooth muscle contractility in models that do not predispose to TAADs, particularly in cases where disease might be expected. Multiple mouse models having compromised glycoproteins that normally associate with elastin to form medial elastic fibers present with TAADs, yet those with fibulin-5 deficiency do not. In this paper, we show that deletion of the fibulin-5 gene results in a significantly diminished contractility of the thoracic aorta in response to potassium loading despite otherwise preserved characteristic active behaviors, including axial force generation and rates of contraction and relaxation. Interestingly, this diminished response manifests around an altered passive state that is defined primarily by a reduced in vivo axial stretch. Given this significant coupling between passive and active properties, a lack of significant changes in passive material stiffness may help to offset the diminished contractility and thereby protect the wall from detrimental mechanosensing and its sequelae.

Length adaptation of smooth muscle contractile filaments in response to sustained activation

Airway and bladder smooth muscles are known to undergo length adaptation under sustained contraction. This adaptation process entails a remodelling of the intracellular actin and myosin filaments which shifts the peak of the active force-length curve towards the current length. Smooth muscles are therefore able to generate the maximum force over a wide range of lengths. In contrast, length adaptation of vascular smooth muscle has attracted very little attention and only a handful of studies have been reported. Although their results are conflicting on the existence of a length adaptation process in vascular smooth muscle, it seems that, at least, peripheral arteries and arterioles undergo such adaptation. This is of interest since peripheral vessels are responsible for pressure regulation, and a length adaptation will affect the function of the cardiovascular system. It has, e.g., been suggested that the inward remodelling of resistance vessels associated with hypertension disorders may be related to smooth muscle adaptation. In this study we develop a continuum mechanical model for vascular smooth muscle length adaptation by assuming that the muscle cells remodel the actomyosin network such that the peak of the active stress-stretch curve is shifted towards the operating point. The model is specialised to hamster cheek pouch arterioles and the simulated response to stepwise length changes under contraction. The results show that the model is able to recover the salient features of length adaptation reported in the literature.

Mechanics of Vascular Smooth Muscle

Vascular smooth muscle (VSM; see Table 1 for a list of abbreviations) is a heterogeneous biomaterial comprised of cells and extracellular matrix. By surrounding tubes of endothelial cells, VSM forms a regulated network, the vasculature, through which oxygenated blood supplies specialized organs, permitting the development of large multicellular organisms. VSM cells, the engine of the vasculature, house a set of regulated nanomotors that permit rapid stress-development, sustained stress-maintenance and vessel constriction. Viscoelastic materials within, surrounding and attached to VSM cells, comprised largely of polymeric proteins with complex mechanical characteristics, assist the engine with countering loads imposed by the heart pump, and with control of relengthening after constriction. The complexity of this smart material can be reduced by classical mechanical studies combined with circuit modeling using spring and dashpot elements. Evaluation of the mechanical characteristics of VSM requires a more complete understanding of the mechanics and regulation of its biochemical parts, and ultimately, an understanding of how these parts work together to form the machinery of the vascular tree. Current molecular studies provide detailed mechanical data about single polymeric molecules, revealing viscoelasticity and plasticity at the protein domain level, the unique biological slip-catch bond, and a regulated two-step actomyosin power stroke. At the tissue level, new insight into acutely dynamic stress-strain behavior reveals smooth muscle to exhibit adaptive plasticity. At its core, physiology aims to describe the complex interactions of molecular systems, clarifying structure-function relationships and regulation of biological machines. The intent of this review is to provide a comprehensive presentation of one biomachine, VSM.

Adaptation of active tone in the mouse descending thoracic aorta under acute changes in loading

Arteries can adapt to sustained changes in blood pressure and flow, and it is thought that these adaptive processes often begin with an altered smooth muscle cell activity that precedes any detectable changes in the passive wall components. Yet, due to the intrinsic coupling between the active and passive properties of the arterial wall, it has been difficult to delineate the adaptive contributions of active smooth muscle. To address this need, we used a novel experimental-computational approach to quantify adaptive functions of active smooth muscle in arterial rings excised from the proximal descending thoracic aorta of mice and subjected to short-term sustained circumferential stretches while stimulated with various agonists. A new mathematical model of the adaptive processes was derived and fit to data to describe and predict the effects of active tone adaptation. It was found that active tone was maintained when the artery was adapted close to the optimal stretch for maximal active force production, but it was reduced when adapted below the optimal stretch; there was no significant change in passive behavior in either case. Such active adaptations occurred only upon smooth muscle stimulation with phenylephrine, however, not stimulation with KCl or angiotensin II. Numerical simulations using the proposed model suggested further that active tone adaptation in vascular smooth muscle could play a stabilizing role for wall stress in large elastic arteries.

Simulating uterine contraction by using an electro-chemo-mechanical model

Contractions of uterine smooth muscle cells consist of a chain of physiological processes. These contractions provide the required force to expel the fetus from the uterus. The inclusion of these physiological processes is, therefore, imperative when studying uterine contractions. In this study, an electro-chemo-mechanical model to replicate the excitation, activation, and contraction of uterine smooth muscle cells is developed. The presented modeling strategy enables efficient integration of knowledge about physiological processes at the cellular level to the organ level. The model is implemented in a three-dimensional finite element setting to simulate uterus contraction during labor in response to electrical discharges generated by pacemaker cells and propagated within the myometrium via gap junctions. Important clinical factors, such as uterine electrical activity and intrauterine pressure, are predicted using this simulation. The predictions are in agreement with clinically measured data reported in the literature. A parameter study is also carried out to investigate the impact of physiologically related parameters on the uterine contractility.

Biaxial Contractile Mechanics of Common Carotid Arteries of Rabbit

Few multiaxial constitutive laws under the vasoactive condition have been proposed as compared with those under the passive condition. The biaxial isometric properties of vasoactive rabbit arteries were studied, although the constitutive law was not proposed. The purpose of the present study is also to describe the multiaxial active mechanical properties of arteries. A novel strain energy function for the active stress has been proposed. This function is simple and may describe the multiaxial characteristics of constricted vessels. Although this study used mean stress and mean stretch ratio to determine the mechanical properties of vessels, a triaxial constitutive law of constricted vessels may be developed. There remains the subject of residual strains under active condition. If this problem will be solved, the accurate stress analysis under vasoactive conditions is possible.

Bio-Chemo-Mechanical Models of Vascular Mechanics

Models of vascular mechanics are necessary to predict the response of an artery under a variety of loads, for complex geometries, and in pathological adaptation. Classic constitutive models for arteries are phenomenological and the fitted parameters are not associated with physical components of the wall. Recently, microstructurally-linked models have been developed that associate structural information about the wall components with tissue-level mechanics. Microstructurally-linked models are useful for correlating changes in specific components with pathological outcomes, so that targeted treatments may be developed to prevent or reverse the physical changes. However, most treatments, and many causes, of vascular disease have chemical components. Chemical signaling within cells, between cells, and between cells and matrix constituents affects the biology and mechanics of the arterial wall in the short- and long-term. Hence, bio-chemo-mechanical models that include chemical signaling are critical for robust models of vascular mechanics. This review summarizes bio-mechanical and bio-chemo-mechanical models with a focus on large elastic arteries. We provide applications of these models and challenges for future work.

On high heels and short muscles: a multiscale model for sarcomere loss in the gastrocnemius muscle

High heels are a major source of chronic lower limb pain. Yet, more than one third of all women compromise health for looks and wear high heels on a daily basis. Changing from flat footwear to high heels induces chronic muscle shortening associated with discomfort, fatigue, reduced shock absorption, and increased injury risk. However, the long-term effects of high-heeled footwear on the musculoskeletal kinematics of the lower extremities remain poorly understood. Here we create a multiscale computational model for chronic muscle adaptation to characterize the acute and chronic effects of global muscle shortening on local sarcomere lengths. We perform a case study of a healthy female subject and show that raising the heel by 13cm shortens the gastrocnemius muscle by 5% while the Achilles tendon remains virtually unaffected. Our computational simulation indicates that muscle shortening displays significant regional variations with extreme values of 22% in the central gastrocnemius. Our model suggests that the muscle gradually adjusts to its new functional length by a chronic loss of sarcomeres in series. Sarcomere loss varies significantly across the muscle with an average loss of 9%, virtually no loss at the proximal and distal ends, and a maximum loss of 39% in the central region. These changes reposition the remaining sarcomeres back into their optimal operating regime. Computational modeling of chronic muscle shortening provides a valuable tool to shape our understanding of the underlying mechanisms of muscle adaptation. Our study could open new avenues in orthopedic surgery and enhance treatment for patients with muscle contracture caused by other conditions than high heel wear such as paralysis, muscular atrophy, and muscular dystrophy.

The Living Heart Project: A robust and integrative simulator for human heart function

The heart is not only our most vital, but also our most complex organ: Precisely controlled by the interplay of electrical and mechanical fields, it consists of four chambers and four valves, which act in concert to regulate its filling, ejection, and overall pump function. While numerous computational models exist to study either the electrical or the mechanical response of its individual chambers, the integrative electro-mechanical response of the whole heart remains poorly understood. Here we present a proof-of-concept simulator for a four-chamber human heart model created from computer topography and magnetic resonance images. We illustrate the governing equations of excitation-contraction coupling and discretize them using a single, unified finite element environment. To illustrate the basic features of our model, we visualize the electrical potential and the mechanical deformation across the human heart throughout its cardiac cycle. To compare our simulation against common metrics of cardiac function, we extract the pressure-volume relationship and show that it agrees well with clinical observations. Our prototype model allows us to explore and understand the key features, physics, and technologies to create an integrative, predictive model of the living human heart. Ultimately, our simulator will open opportunities to probe landscapes of clinical parameters, and guide device design and treatment planning in cardiac diseases such as stenosis, regurgitation, or prolapse of the aortic, pulmonary, tricuspid, or mitral valve.

Human thoracic and abdominal aortic aneurysmal tissues: Damage experiments, statistical analysis and constitutive modeling

Development of aortic aneurysms includes significant morphological changes within the tissue: collagen content increases, elastin content reduces and smooth muscle cells degenerate. We seek to quantify the impact of these changes on the passive mechanical response of aneurysms in the supra-physiological loading range via mechanical testing and constitutive modeling. We perform uniaxial extension tests on circumferentially and axially oriented strips from five thoracic (65.6 years ± 13.4, mean ± SD) and eight abdominal (63.9 years ± 11.4) aortic fusiform aneurysms to investigate both continuous and discontinuous softening during supra-physiological loading. We determine the significance of the differences between the fitted model parameters: diseased thoracic versus abdominal tissues, and healthy (Weisbecker et al., J. Mech. Behav. Biomed. Mater. 12, 93-106, 2012) versus diseased tissues. We also test correlations among these parameters and age, Body Mass Index (BMI) and preoperative aneurysm diameter, and investigate histological cuts. Tissue response is anisotropic for all tests and the anisotropic pseudo-elastic damage model fits the data well for both primary loading and discontinuous softening which we interpret as damage. We found statistically relevant differences between model parameters fitted to diseased thoracic versus abdominal tissues, as well as between those fitted to healthy versus diseased tissues. Only BMI correlated with fitted model parameters in abdominal aortic aneurysmal tissues.

Use it or lose it: multiscale skeletal muscle adaptation to mechanical stimuli

Skeletal muscle undergoes continuous turnover to adapt to changes in its mechanical environment. Overload increases muscle mass, whereas underload decreases muscle mass. These changes are correlated with, and enabled by, structural alterations across the molecular, subcellular, cellular, tissue, and organ scales. Despite extensive research on muscle adaptation at the individual scales, the interaction of the underlying mechanisms across the scales remains poorly understood. Here, we present a thorough review and a broad classification of multiscale muscle adaptation in response to a variety of mechanical stimuli. From this classification, we suggest that a mathematical model for skeletal muscle adaptation should include the four major stimuli, overstretch, understretch, overload, and underload, and the five key players in skeletal muscle adaptation, myosin heavy chain isoform, serial sarcomere number, parallel sarcomere number, pennation angle, and extracellular matrix composition. Including this information in multiscale computational models of muscle will shape our understanding of the interacting mechanisms of skeletal muscle adaptation across the scales. Ultimately, this will allow us to rationalize the design of exercise and rehabilitation programs, and improve the long-term success of interventional treatment in musculoskeletal disease.

Experimental evidence and mathematical modeling of thermal effects on human colonic smooth muscle contractility

It has been shown, in animal models, that gastrointestinal tract (GIT) motility is influenced by temperature; nevertheless, the basic mechanism governing thermal GIT smooth muscle responses has not been fully investigated. Studies based on physiologically tuned mathematical models have predicted that thermal inhomogeneity may induce an electrochemical destabilization of peristaltic activity. In the present study, the effect of thermal cooling on human colonic muscle strip (HCMS) contractility was studied. HCMSs were obtained from disease-free margins of resected segments for cancer. After removal of the mucosa and serosa layers, strips were mounted in separate chambers. After 30 min, spontaneous contractions developed, which were measured using force displacement transducers. Temperature was changed every hour (37, 34, and 31°C). The effect of cooling was analyzed on mean contractile activity, oscillation amplitude, frequency, and contraction to ACh (10(-5) M). At 37°C, HCMSs developed a stable phasic contraction (~0.02 Hz) with a significant ACh-elicited mean contractile response (31% and 22% compared with baseline in the circular and longitudinal axis, respectively). At a lower bath temperature, higher mean contractile amplitude was observed, and it increased in the presence of ACh (78% and 43% higher than the basal tone in the circular and longitudinal axis, respectively, at 31°C). A simplified thermochemomechanical model was tuned on experimental data characterizing the stress state coupling the intracellular Ca(2+) concentration to tissue temperature. In conclusion, acute thermal cooling affects colonic muscular function. Further studies are needed to establish the exact mechanisms involved to better understand clinical consequences of hypothermia on intestinal contractile activity.

Investigating the role of smooth muscle cells in large elastic arteries: a finite element analysis

Physiological loading in large elastic arteries is considered to be mainly carried by the passive components of the media but it is not known how much the contraction of the smooth muscle cells is actually involved in the load carrying. Smooth muscle contraction is considered to occur in a relatively slow time domain but the contraction is able to produce significant tension. In the present work the role of smooth muscle contraction in large elastic arteries is investigated by analyzing how changes in the intracellular calcium, and thereby the active tone of smooth muscle cells, influence the deformation and stress behavior; different intracellular calcium functions and medial wall thicknesses with cycling internal pressure are studied. In particular, a recently proposed mechanochemical model (Murtada et al., 2012. J. Theor. Biol. 297, 176-186), which links intracellular calcium with mechanical contraction and an anisotropic model representing the elastin/collagen composite, was implemented into a 3D finite element framework. Details of the implementation procedure are described and a verification of the model implementation is provided by means of the isometric contraction/relaxation analysis of a medial strip at optimal muscle length. In addition, numerically obtained pressure-radius relationships of arterial rings modeled with one and two layers are analyzed with different geometries and at different calcium levels; a comparison with the Laplace equation is provided. Finally, a two-layer arterial ring is loaded with a realistic pressure wave and with various intracellular calcium functions (different amplitudes and mean values) and medial wall thicknesses; residual stresses are considered. The finite element results show that changes in the calcium amplitudes hardly have an influence on the current inner ring radius and the circumferential stress. However, an increase in the mean intracellular calcium value and the medial wall thickness leads to a clear influence on the deformation and the stress behavior.

A constitutive model for cytoskeletal contractility of smooth muscle cells

The constitutive model presented in this article aims to describe the main bio-chemo-mechanical features involved in the contractile response of smooth muscle cells, in which the biochemical response is modelled by extending the four-state Hai–Murphy model to isotonic contraction of the cells and the mechanical response is mainly modelled based on the phosphorylation-dependent hyperbolic relation between isotonic shortening strain rate and tension. The one-dimensional version of the model is used to simulate shortening-induced deactivation with good agreement with selected experimental measurements. The results suggest that the Hai–Murphy biochemical model neglects the strain rate effect on the kinetics of cross-bridge interactions with actin filaments in the isotonic contractions. The two-dimensional version and three-dimensional versions of the model are developed using the homogenization method under finite strain continuum mechanics framework. The two-dimensional constitutive model is used to simulate swine carotid media strips under electrical field stimulation, experimentally investigated by Singer and Murphy, and contraction of a hollow airway and a hollow arteriole buried in a soft matrix subjected to multiple calcium ion stimulations. It is found that the transverse deformation may have significant influence on the response of the swine carotid medium. In both cases, the orientation of the maximal value of attached myosin is aligned with the orientation of maximum principal stress.

A continuum model for excitation-contraction of smooth muscle under finite deformations

The main focus in most of the continuum based muscle models is the mechanics of muscle contraction while other physiological processes governing muscle contraction, e.g., cell membrane excitation and activation, are ignored. These latter processes are essential to initiate contraction and to determine the amount of generated force, and by excluding them, the developed model cannot replicate the true behavior of the muscle in question. The aim of this study is to establish a thermodynamically and physiologically consistent framework which allows us to model smooth muscle contraction by including cell membrane excitability and kinetics of myosin phosphorylation, along with dynamics of smooth muscle contraction. The model accounts for these processes through a set of coupled dissipative constitutive equations derived by applying first principles. To show the performance of the derived model, it is evaluated for two different cases: a chemo-mechanical study of pig taenia coli cells where the excitation process is excluded, and an electro-chemo-mechanical study of rat myometrium. The results show that the model is able to replicate important aspects of the smooth muscle excitation-contraction process.

A three-dimensional chemo-mechanical continuum model for smooth muscle contraction

Based on two fields, namely the placement and the calcium concentration, a chemo-mechanically coupled three-dimensional model, describing the contractile behaviour of smooth muscles, is presented by means of a strain energy function. The strain energy function (Schmitz and Böl, 2011) is additively decomposed into a passive part, relating to elastin and collagen, and an active calcium-driven part related to the chemical contraction of the smooth muscle cells. For the description of the calcium phase the four state cross-bridge model of Hai and Murphy (Hai and Murphy, 1988) has been implemented into the finite element method. Beside three-dimensional illustrative boundary-value problems demonstrating the features of the presented modelling concept, simulations on an idealised artery document the applicability of the model to more realistic geometries.