Arterial remodeling in response to increased blood flow using a constituent-based model

Previous theoretical models of arterial remodeling in response to changes in blood flow were based on the assumption that material properties of the arterial wall remain unchanged during the remodeling process. According to experimental findings, however, remodeling due to increased flow is accompanied by alteration in the structural properties of elastin, which results in a decrease in its effective elastic stiffness. To account for these effects, we propose a predictive model of arterial remodeling hypothesizing that the variation in mechanical properties of elastin is initiated and driven by the deviation of the intimal shear stress from its baseline value. Geometrical remodeling restores the wall stress distribution as it was under normal flow conditions. A constrained mixture approach is followed. Artery is modeled as a thick-walled cylindrical tube made of non-linear, elastic, anisotropic and incompressible material. Data for a rabbit thoracic aorta have been employed. At the final adapted state, the model predicts a non-monotonic dependence of arterial compliance on the magnitude of flow. This result is in agreement with available experimental data in the literature.     

Uniform stress state in bone structure with residual stress.

Residual stress and strain in living tissues have been investigated from the viewpoint of mechanical optimality maintained by adaptive remodeling. In this study, the residual stresses in the cortical-cancellous bone complex of bovine coccygeal vertebrae were examined. Biaxial strain gages were bonded onto the cortical surface, so that the gage axes were aligned in the cephalocaudal and circumferential directions. Strains induced by removal of the end-plate and the cancellous bone were recorded sequentially. The results revealed the existence of compressive residual stress in the cortical bone and tensile residual stress in the cancellous bone in both the cephalocaudal and the circumferential direction. The observed strains were examined on the basis of the uniform stress hypothesis using a three-bar model for the cephalocaudal direction and a three-layered cylinder model for the circumferential direction. In this model study, the magnitude of effective stresses, which is defined as the macroscopic stress divided by the area fraction of bone material, was found not to differ significantly between cephalocaudal and circumferential directions, although they were evaluated using independent models. These results demonstrate that the uniform stress state of the cortical-cancellous bone structure is consistent with results obtained in the cutting experiment when the existence of residual stress is taken into account.     

Devolution of inhomogeneities in bone structure--predictions of adaptive elasticity theory

In this paper we consider an initially inhomogeneous adaptive elastic body subjected to a steady homogeneous stress state. The adaptive elastic body, which is a model for living bone tissue, is inhomogeneous in both its anisotropic elastic properties and its density. The principal result of the paper is the determination of the devolution of the initially inhomogeneous body to a homogeneous body under the influence of the steady homogeneous stress state. A cylindrical body that is inhomogeneous along the axis of the cylinder, but homogeneous in each transverse plane of the cylinder, is used as an example. This cylindrical body is loaded by a steady uniform stress directed along the cylindrical axis. The temporal devolution of an inhomogeneity in the initial shape of a sine wave is illustrated. As time progresses the amplitude of the sine wave decreases, rapidly at first and then more slowly. As time becomes very large the sine wave becomes a straight line signifying that the cylinder has become homogeneous.     

Magnetic resonance elastography in nonlinear viscoelastic materials under load

Characterisation of soft tissue mechanical properties is a topic of increasing interest in translational and clinical research. Magnetic resonance elastography (MRE) has been used in this context to assess the mechanical properties of tissues in vivo noninvasively. Typically, these analyses rely on linear viscoelastic wave equations to assess material properties from measured wave dynamics. However, deformations that occur in some tissues (e.g. liver during respiration, heart during the cardiac cycle, or external compression during a breast exam) can yield loading bias, complicating the interpretation of tissue stiffness from MRE measurements. In this paper, it is shown how combined knowledge of a material’s rheology and loading state can be used to eliminate loading bias and enable interpretation of intrinsic (unloaded) stiffness properties. Equations are derived utilising perturbation theory and Cauchy’s equations of motion to demonstrate the impact of loading state on periodic steady-state wave behaviour in nonlinear viscoelastic materials. These equations demonstrate how loading bias yields apparent material stiffening, softening and anisotropy. MRE sensitivity to deformation is demonstrated in an experimental phantom, showing a loading bias of up to twofold. From an unbiased stiffness of \(4910.4 \pm 635.8\) Pa in unloaded state, the biased stiffness increases to 9767.5 \(\pm \,\)1949.9 Pa under a load of \(\approx \) 34% uniaxial compression. Integrating knowledge of phantom loading and rheology into a novel MRE reconstruction, it is shown that it is possible to characterise intrinsic material characteristics, eliminating the loading bias from MRE data. The framework introduced and demonstrated in phantoms illustrates a pathway that can be translated and applied to MRE in complex deforming tissues. This would contribute to a better assessment of material properties in soft tissues employing elastography.

     

Passive material properties of intact ventricular myocardium determined from a cylindrical model

The equatorial region of the canine left ventricle was modeled as a thick-walled cylinder consisting of an incompressible hyperelastic material with homogeneous exponential properties. The anisotropic properties of the passive myocardium were assumed to be locally transversely isotropic with respect to a fiber axis whose orientation varied linearly across the wall. Simultaneous inflation, extension, and torsion were applied to the cylinder to produce epicardial strains that were measured previously in the potassium-arrested dog heart. Residual stress in the unloaded state was included by considering the stress-free configuration to be a warped cylindrical arc. In the special case of isotropic material properties, torsion and residual stress both significantly reduced the high circumferential stress peaks predicted at the endocardium by previous models. However, a resultant axial force and moment were necessary to cause the observed epicardial deformations. Therefore, the anisotropic material parameters were found that minimized these resultants and allowed the prescribed displacements to occur subject to the known ventricular pressure loads. The global minimum solution of this parameter optimization problem indicated that the stiffness of passive myocardium (defined for a 20 percent equibiaxial extension) would be 2.4 to 6.6 times greater in the fiber direction than in the transverse plane for a broad range of assumed fiber angle distributions and residual stresses. This agrees with the results of biaxial tissue testing. The predicted transmural distributions of fiber stress were relatively flat with slight peaks in the subepicardium, and the fiber strain profiles agreed closely with experimentally observed sarcomere length distributions. The results indicate that torsion, residual stress and material anisotropy associated with the fiber architecture all can act to reduce endocardial stress gradients in the passive left ventricle.     

Regional distribution of axial strain and circumferential residual strain in the layered rabbit oesophagus

The oesophagus is subjected to large axial strains in vivo and the zero-stress state is not a closed cylinder but an open circular cylindrical sector. The closed cylinder with no external loads applied is called the no-load state and residual strain is the difference in strain between the no-load state and zero-stress state. To understand oesophageal physiology and pathophysiology, it is necessary to know the distribution of axial strain, the zero-stress state, the stress-strain relations of oesophageal tissue, and the changes of these states and relationships due to biological remodeling of the tissue under stress. This study is addressed to such biomechanical properties in normal rabbits. The oesophagi were marked on the surface in vivo, photographed, excised (in vitro state), photographed again, and sectioned into rings (no-load state) in an organ bath containing calcium-free Kreb's solution with dextran and EGTA added. The rings were cut radially to obtain the zero-stress state for the non-separated wall and further dissected to separate the muscle and submucosa layers. Equilibrium was awaited for 30min in each state and the specimens were photographed in no-load and the zero-stress states. The oesophageal length, circumferences, layer thicknesses and areas, and openings angle were measured from the digitised images. The oesophagus shortened axially by 35% after excision. The in vivo axial strain showed a significant variation with the highest values in the mid-oesophagus (p<0.001). Luminal area, circumferences, and wall and layer thicknesses and areas varied in axial direction (in all tests p<0.05). The residual strain was compressive at the mucosal surface and tensile at the serosal surface. The dissection studies demonstrated shear forces between the two layers in the non-separated wall in the no-load and zero-stress states. In conclusion, our data show significant axial variation in passive morphometric and biomechanical properties of the oesophagus. The oesophagus is a layered composite structure with nonlinear and anisotropic mechanical behaviour.     

Built-in mechanical stress in viral shells

Mechanical properties of biological molecular aggregates are essential to their function. A remarkable example are double-stranded DNA viruses such as the φ29 bacteriophage, that not only has to withstand pressures of tens of atmospheres exerted by the confined DNA, but also uses this stored elastic energy during DNA translocation into the host. Here we show that empty prolated φ29 bacteriophage proheads exhibit an intriguing anisotropic stiffness which behaves counterintuitively different from standard continuum elasticity predictions. By using atomic force microscopy, we find that the φ29 shells are approximately two-times stiffer along the short than along the long axis. This result can be attributed to the existence of a residual stress, a hypothesis that we confirm by coarse-grained simulations. This built-in stress of the virus prohead could be a strategy to provide extra mechanical strength to withstand the DNA compaction during and after packing and a variety of extracellular conditions, such as osmotic shocks or dehydration.     

Quantifying whole bladder biomechanics using the novel pentaplanar reflected image macroscopy system

Optimal bladder compliance is essential to urinary bladder storage and voiding functions. Calculated as the change in filling volume per change in pressure, bladder compliance is used clinically to characterize changes in bladder wall biomechanical properties that associate with lower urinary tract dysfunction. But because this method calculates compliance without regard to wall structure or wall volume, it gives little insight into the mechanical properties of the bladder wall during filling. Thus, we developed Pentaplanar Reflected Image Macroscopy (PRIM): a novel ex vivo imaging method to accurately calculate bladder wall stress and stretch in real time during bladder filling. The PRIM system simultaneously records intravesical pressure, infused volume, and an image of the bladder in five distinct visual planes. Wall thickness and volume were then measured and used to calculate stress and stretch during filling. As predicted, wall stress was nonlinear; only when intravesical pressure exceeded ~ 15 mmHg did bladder wall stress rapidly increase with respect to stretch. This method of calculating compliance as stress vs stretch also showed that the mechanical properties of the bladder wall remain similar in bladders of varying capacity. This study demonstrates how wall tension, stress and stretch can be measured, quantified, and used to accurately define bladder wall biomechanics in terms of actual material properties and not pressure/volume changes. This method is especially useful for determining how changes in bladder biomechanics are altered in pathologies where profound bladder wall remodeling occurs, such as diabetes and spinal cord injury.

     

A morphoelastic model for dermal wound closure

We develop a model of wound healing in the framework of finite elasticity, focussing our attention on the processes of growth and contraction in the dermal layer of the skin. The dermal tissue is treated as a hyperelastic cylinder that surrounds the wound and is subject to symmetric deformations. By considering the initial recoil that is observed upon the application of a circular wound, we estimate the degree of residual tension in the skin and build an evolution law for mechanosensitive growth of the dermal tissue. Contraction of the wound is governed by a phenomenological law in which radial pressure is prescribed at the wound edge. The model reproduces three main phases of the healing process. Initially, the wound recoils due to residual stress in the surrounding tissue; the wound then heals as a result of contraction and growth; and finally, healing slows as contraction and growth decrease. Over a longer time period, the surrounding tissue remodels, returning to the residually stressed state. We identify the steady state growth profile associated with this remodelled state. The model is then used to predict the outcome of rewounding experiments designed to quantify the amount of stress in the tissue, and also to simulate the application of pressure treatments.     

Strain-energy function and three-dimensional stress distribution in esophageal biomechanics

Knowledge of the transmural stress and stretch fields in esophageal wall is necessary to quantify growth and remodeling, and the response to mechanically based clinical interventions or traumatic injury, but there are currently conflicting reports on this issue and the mechanical properties of intact esophagus have not been rigorously addressed. This paper offers multiaxial data on rabbit esophagus, warranted for proper identification of the 3D mechanical properties. The Fung-type strain-energy function was adopted to model our data for esophagus, taken as a thick-walled (1 or 2-layer) tubular structure subjected to inflation and longitudinal extension. Accurate predictions of the pressure–radius–force data were obtained using the 1-layer model, covering a broad range of extensions; the calculated material parameters indicated that intact wall was equally stiff as mucosa–submucosa, but stiffer than muscle in both principal axes, and tissue was stiffer longitudinally, concurring our histological findings (Stavropoulou et al., Journal of Biomechanics. 42 (2009) 2654–2663). Employing the material parameters of individual layers, with reference to their zero-stress state, a reasonable fit was obtained to the data for intact wall, modeled as a 2-layer tissue. Different from the stress distributions presented hitherto in the esophagus literature, consideration of residual stresses led to less dramatic homogenization of stresses under loading. Comparison of the 1- and 2-layer models of esophagus demonstrated that heterogeneity induced a more uniform distribution of residual stresses in each layer, a discontinuity in circumferential and longitudinal stresses at the interface among layers, and a considerable rise of stresses in mucosa, with a reduction in muscle.     

Three-dimensional stress and strain distribution in a two-layer model of a coronary artery

For a right coronary artery, three-dimensional stress and strain distributions at a physiological intraluminal pressure and an axial extension ratio were computed on the basis of a two-layer elastic model. To validate the model, curves of external radius versus pressure and of axial force versus pressure were computed for three axial extension ratios. To analyze mechanical properties, stress-free configurations of media and adventitia, and the constitutive law of each layer in literature, were used. The present study showed that the peak circumferential stress and the peak axial stress appear in the media at the boundary between the media and adventitia. This result is due to the opening angle of the media being larger than π (rad) and the larger value of a material constant of the strain energy function for the media than for the adventitia. The circumferential stress and strain were discontinuous at the boundary. On the other hand, the radial stress was continuous at the boundary because of the boundary condition for stress. The circumferential stress and axial stress in the adventitia were almost uniformly distributed, and smaller than in the media. The residual stress and strain were also computed. The circumferential residual stress and strain were almost linearly distributed in each layer, although discontinuity appeared at the boundary between the two layers.     

Determination of human arterial wall parameters from clinical data

This study suggests a method to compute the material parameters for arteries in vivo from clinically registered pressure-radius signals. The artery is modelled as a hyperelastic, incompressible, thin-walled cylinder and the membrane stresses are computed using a strain energy. The material parameters are determined in a minimisation process by tuning the membrane stress to the stress obtained by enforcing global equilibrium. In addition to the mechanical model, the study also suggests a preconditioning of the pressure-radius signal. The preconditioning computes an average pressure-radius cycle from all consecutive cycles in the registration and removes, or reduces, undesirable disturbances. The effect is a robust parameter identification that gives a unique solution. The proposed method is tested on clinical data from three human abdominal aortas and the results show that the material parameters from the proposed method do not differ significantly (p < 0.01) from the corresponding parameters obtained by averaging the result from consecutive cycles.     

Patient-specific stress analyses in the ascending thoracic aorta using a finite-element implementation of the constrained mixture theory

It is now a rather common approach to perform patient-specific stress analyses of arterial walls using finite-element models reconstructed from gated medical images. However, this requires to compute for every Gauss point the deformation gradient between the current configuration and a stress-free reference configuration. It is technically difficult to define such a reference configuration, and there is actually no guarantee that a stress-free configuration is physically attainable due to the presence of internal stresses in unloaded soft tissues. An alternative framework was proposed by Bellini et al. (Ann Biomed Eng 42(3):488–502, 2014). It consists of computing the deformation gradients between the current configuration and a prestressed reference configuration. We present here the first finite-element results based on this concept using the Abaqus software. The reference configuration is set arbitrarily to the in vivo average geometry of the artery, which is obtained from gated medical images and is assumed to be mechanobiologically homeostatic. For every Gauss point, the stress is split additively into the contributions of each individual load-bearing constituent of the tissue, namely elastin, collagen, smooth muscle cells. Each constituent is assigned an independent prestretch in the reference configuration, named the deposition stretch. The outstanding advantage of the present approach is that it simultaneously computes the in situ stresses existing in the reference configuration and predicts the residual stresses that occur after removing the different loadings applied onto the artery (pressure and axial load). As a proof of concept, we applied it on an ideal thick-wall cylinder and showed that the obtained results were consistent with corresponding experimental and analytical results of the well-known literature. In addition, we developed a patient-specific model of a human ascending thoracic aneurysmal aorta and demonstrated the utility in predicting the wall stress distribution in vivo under the effects of physiological pressure. Finally, we simulated the whole process preceding traditional in vitro uniaxial tensile testing of arteries, including excision from the body, radial cutting, flattening and subsequent tensile loading, showing how this process may impact the final mechanical properties derived from these in vitro tests.     

Airway smooth muscle tone modulates mechanically induced cytoskeletal stiffening and remodeling.

The application of mechanical stresses to the airway smooth muscle (ASM) cell causes time-dependent cytoskeletal stiffening and remodeling (Deng L, Fairbank NJ, Fabry B, Smith PG, and Maksym GN. Am J Physiol Cell Physiol 287: C440-C448, 2004). We investigated here the extent to which these behaviors are modulated by the state of cell activation (tone). Localized mechanical stress was applied to the ASM cell in culture via oscillating beads (4.5 mum) that were tightly bound to the actin cytoskeleton (CSK). Tone was reduced from baseline level using a panel of relaxant agonists (10(-3) M dibutyryl cAMP, 10(-4) M forskolin, or 10(-6) M formoterol). To assess functional changes, we measured cell stiffness (G') using optical magnetic twisting cytometry, and to assess structural changes of the CSK we measured actin accumulation in the neighborhood of the bead. Applied mechanical stress caused a twofold increase in G' at 120 min. After cessation of applied stress, G' diminished only 24 +/- 6% (mean +/- SE) at 1 h, leaving substantial residual effects that were largely irreversible. However, applied stress-induced stiffening could be prevented by ablation of tone. Ablation of tone also inhibited the amount of actin accumulation induced by applied mechanical stress (P < 0.05). Thus the greater the contractile tone, the greater was applied stress-induced CSK stiffening and remodeling. As regards pathobiology of asthma, this suggests a maladaptive positive feedback in which tone potentiates ASM remodeling and stiffening that further increases stress and possibly leads to worsening airway function.     

Free body analysis,beam mechanics,and finite element modeling of the mandible of Alligator mississippiensis

The mechanical behavior of mammalian mandibles is well‐studied, but a comprehensive biomechanical analysis (incorporating detailed muscle architecture, accurate material properties, and three‐dimensional mechanical behavior) of an extant archosaur mandible has never been carried out. This makes it unclear how closely models of extant and extinct archosaur mandibles reflect reality and prevents comparisons of structure–function relationships in mammalian and archosaur mandibles. We tested hypotheses regarding the mechanical behavior of the mandible of Alligator mississippiensis by analyzing reaction forces and bending, shear, and torsional stress regimes in six models of varying complexity. Models included free body analysis using basic lever arm mechanics, 2D and 3D beam models, and three high‐resolution finite element models of the Alligator mandible, incorporating, respectively, isotropic bone without sutures, anisotropic bone with sutures, and anisotropic bone with sutures and contact between the mandible and the pterygoid flange. Compared with the beam models, the Alligator finite element models exhibited less spatial variability in dorsoventral bending and sagittal shear stress, as well as lower peak values for these stresses, suggesting that Alligator mandibular morphology is in part designed to reduce these stresses during biting. However, the Alligator models exhibited greater variability in the distribution of mediolateral and torsional stresses than the beam models. Incorporating anisotropic bone material properties and sutures into the model reduced dorsoventral and torsional stresses within the mandible, but led to elevated mediolateral stresses. These mediolateral stresses were mitigated by the addition of a pterygoid‐mandibular contact, suggesting important contributions from, and trade‐offs between, material properties and external constraints in Alligator mandible design. Our results suggest that beam modeling does not accurately represent the mechanical behavior of the Alligator mandible, including important performance metrics such as magnitude and orientation of reaction forces, and mediolateral bending and torsional stress distributions. J.Morphol. 2011. © 2011 Wiley‐Liss, Inc.     

Modelling of the mechanical and mass transport properties of auxetic molecular sieves: an idealised organic (polymeric honeycomb) host–guest system

Force field-based simulations have been employed to model the mechanical properties of a range of undeformed molecular polymeric honeycombs having conventional and re-entrant hexagon pores. The conventional and re-entrant hexagon honeycombs are predicted to display positive and negative in-plane Poisson's ratios, respectively, confirming previous simulations. The structure, and mechanical and mass transport properties of a layered re-entrant honeycomb ((2,8)-reflexyne) were studied in detail for a uniaxial load applied along the x ₂ direction. The mechanical properties are predicted to be stress- (strain-) dependent and the trends can be interpreted using analytical expressions from honeycomb theory. Transformation from negative to positive Poisson's ratio behaviour is predicted at an applied stress of σ₂ = 2 GPa. Simulations of the loading of C₆₀ and C₇₀ guest molecules into the deformed layered (2,8)-reflexyne host framework demonstrate the potential for tunable size selectivity within the host framework. The entrapment and release of guest molecules is attributed to changes in the size and shape of the pores in this host–guest system.     

Computational simulation of hemodynamic-driven growth and remodeling of embryonic atrioventricular valves

Embryonic heart valves develop under continuous and demanding hemodynamic loading. The particular contributions of fluid pressure and shear tractions in valve morphogenesis are difficult to decouple experimentally. To better understand how fluid loads could direct valve formation, we developed a computational model of avian embryonic atrioventricular (AV) valve (cushion) growth and remodeling using experimentally derived parameters for the blood flow and the cushion stiffness. Through an iterative scheme, we first solved the fluid loads on the axisymmetric AV canal and cushion model geometry. We then applied the fluid loads to the cushion and integrated the evolution equations to determine the growth and remodeling. After a set time of growth, we updated the fluid domain to reflect the change in cushion geometry and resolved for the fluid forces. The rate of growth and remodeling was assumed to be a function of the difference between the current stress and an isotropic homeostatic stress state. The magnitude of the homeostatic stress modulated the rate of volume addition during the evolution. We found that the pressure distribution on the AV cushion was sufficient to generate leaflet-like elongation in the direction of flow, through inducing tissue resorption on the inflow side of cushion and expansion on the outflow side. Conversely, shear tractions minimally altered tissue volume, but regulated the remodeling of tissue near the cushion surface, particular at the leading edge. Significant shear and circumferential residual stresses developed as the cushion evolved. This model offers insight into how natural and perturbed mechanical environments may direct AV valvulogenesis and provides an initial framework on which to incorporate more mechano-biological details.     

Functional adaptation of cancellous bone in human proximal femur predicted by trabecular surface remodeling simulation toward uniform stress state

Two-dimensional simulation of trabecular surface remodeling was conducted for a human proximal femur to investigate the structural change of cancellous bone toward a uniform stress state. Considering that a local mechanical stimulus plays an important role in cellular activities in bone remodeling, local stress nonuniformity was assumed to drive trabecular structural change to seek a uniform stress state. A large-scale pixel-based finite element model was used to simulate structural changes of individual trabeculae over the entire bone. As a result, the initial structure of trabeculae changed from isotropic to anisotropic due to trabecular microstructural changes caused by surface remodeling according to the mechanical environment in the proximal femur. Under a single-loading condition, it was shown that the apparent structural property evaluated by fabric ellipses corresponded to the apparent stress state in cancellous bone. As is observed in the actual bone, a distributed trabecular structure was obtained under a multiple-loading condition. Through these studies, it was concluded that trabecular surface remodeling toward a local uniform stress state at the trabecular level could naturally bring about functional adaptation phenomenon at the apparent tissue level. The proposed simulation model would be capable of providing insight into the hierarchical mechanism of trabecular surface remodeling at the microstructural level up to the apparent tissue level.     

Continuum modeling of forces in growing viscoelastic cytoskeletal networks

Mechanical properties of the living cell are important in cell movement, cell division, cancer development and cell signaling. There is considerable interest in measuring local mechanical properties of living materials and the living cytoskeleton using micromechanical techniques. However, living materials are constantly undergoing internal dynamics such as growth and remodeling. A modeling framework that combines mechanical deformations with cytoskeletal growth dynamics is necessary to describe cellular shape changes. The present paper develops a general finite deformation modeling approach that can treat the viscoelastic cytoskeleton. Given the growth dynamics in the cytoskeletal network and the relationship between deformation and stress, the shape of the network is computed in an incremental fashion. The growth dynamics of the cytoskeleton can be modeled as stress dependent. The result is a consistent treatment of overall cell deformation. The framework is applied to a growing 1-d bundle of actin filaments against an elastic cantilever, and a 2-d cell undergoing wave-like protrusion dynamics. In the latter example, mechanical forces on the cell adhesion are examined as a function of the protrusion dynamics.