Artery buckling analysis using a four-fiber wall model

Artery bent buckling has been suggested as a possible mechanism that leads to artery tortuosity, which is associated with aging, hypertension, atherosclerosis, and other pathological conditions. It is necessary to understand the relationship between microscopic wall structural changes and macroscopic artery buckling behavior. To this end, the objectives of this study were to develop arterial buckling equations using a microstructure-based 4-fiber reinforced wall model, and to simulate the effects of vessel wall microstructural changes on artery buckling. Our results showed that the critical pressure increased nonlinearly with the axial stretch ratio, and the 4-fiber model predicted higher critical buckling pressures than what the Fung model predicted. The buckling equation using the 4-fiber model captured the experimentally observed reduction of critical pressure induced by elastin degradation and collagen fiber orientation changes in the arterial wall. These results improve our understanding of arterial stability and its relationship to microscopic wall remodeling, and the model provides a useful tool for further studies.     

Mechanical instability of normal and aneurysmal arteries

Tortuous arteries associated with aneurysms have been observed in aged patients with atherosclerosis and hypertension. However, the underlying mechanism is poorly understood. The objective of this study was to determine the effect of aneurysms on arterial buckling instability and the effect of buckling on aneurysm wall stress. We investigated the mechanical buckling and post-buckling behavior of normal and aneurysmal carotid arteries and aorta’s using computational simulations and experimental measurements to elucidate the interrelationship between artery buckling and aneurysms. Buckling tests were done in porcine carotid arteries with small aneurysms created using elastase treatment. Parametric studies were done for model aneurysms with orthotropic nonlinear elastic walls using finite element simulations. Our results demonstrated that arteries buckled at a critical buckling pressure and the post-buckling deflection increased nonlinearly with increasing pressure. The presence of an aneurysm can reduce the critical buckling pressure of arteries, although the effect depends on the aneurysm’s dimensions. Buckled aneurysms demonstrated a higher peak wall stress compared to unbuckled aneurysms under the same lumen pressure. We conclude that aneurysmal arteries are vulnerable to mechanical buckling and mechanical buckling could lead to high stresses in the aneurysm wall. Buckling could be a possible mechanism for the development of tortuous aneurysmal arteries such as in the Loeys–Dietz syndrome.     

A biomechanical model for fibril recruitment: Evaluation in tendons and arteries

Simulations of soft tissue mechanobiological behaviour are increasingly important for clinical prediction of aneurysm, tendinopathy and other disorders. Mechanical behaviour at low stretches is governed by fibril straightening, transitioning into load-bearing at recruitment stretch, resulting in a tissue stiffening effect. Previous investigations have suggested theoretical relationships between stress-stretch measurements and recruitment probability density function (PDF) but not derived these rigorously nor evaluated these experimentally. Other work has proposed image-based methods for measurement of recruitment but made use of arbitrary fibril critical straightness parameters. The aim of this work was to provide a sound theoretical basis for estimating recruitment PDF from stress-stretch measurements and to evaluate this relationship using image-based methods, clearly motivating the choice of fibril critical straightness parameter in rat tail tendon and porcine artery. Rigorous derivation showed that the recruitment PDF may be estimated from the second stretch derivative of the first Piola-Kirchoff tissue stress. Image-based fibril recruitment identified the fibril straightness parameter that maximised Pearson correlation coefficients (PCC) with estimated PDFs. Using these critical straightness parameters the new method for estimating recruitment PDF showed a PCC with image-based measures of 0.915 and 0.933 for tendons and arteries respectively. This method may be used for accurate estimation of fibril recruitment PDF in mechanobiological simulation where fibril-level mechanical parameters are important for predicting cell behaviour.     

A molecular dynamics investigation of buckling behaviour of hydrogenated graphene

Molecular dynamics simulations have been performed to characterise the stability behaviour of graphene nanoribbons having different hydrogen coverage, subject to a uniaxial compressive load. The temperature is set at a very low value to circumvent the contribution of thermal agitations. The results show that hydrogen coverage promotes to a rapid drop in the strain of buckling onset due to the effects of easy rotation of newly unsupported sp³ bonds. Furthermore, we have also found a critical value of the hydrogen adsorption above which the declining trend in the stability behaviour of hydrogenated graphene nanoribbons is reversed.     

Evaluation of blood flow velocity waveform in common carotid artery using multi-branched arterial segment model of human arteries

Arteriosclerosis is considered to be a major cause of cardiovascular diseases, which account for approximately 30% of the causes of death in the world. We have recently demonstrated a strong correlation between arteriosclerosis (arterial elasticity) and two characteristics: maximum systolic velocity (S1) and systolic second peak velocity (S2) of the common carotid artery flow velocity waveform (CCFVW). The CCFVW can be measured by using a small portable measuring device. However, there is currently no theoretical evidence supporting the causes of the relation between CCFVW and arterial elasticity, or the origin of the CCFVW characteristics. In this study, the arterial blood flow was simulated using a one-dimensional systemic arterial segments model of human artery in order to conduct a qualitative evaluation of the relationship between arterial elasticity and the characteristics of CCFVW. The simulation was carried out based on the discretized segments with the physical properties of a viscoelastic tube (the cross-sectional area at the proximal and terminal ends, the length, and the compliance per unit area of the tube (C_S)). The findings obtained through this study revealed that the simulated CCFVW had shape similar characteristics to that of the measured CCFVW. Moreover, when the compliance C_S of the model was decreased, the first peak of the simulated-CCFVW decreased and the second peak increased. Further, by separating the anterograde pulse wave and the reflected pulse wave, which form the CCFVW, we found that the decrease in the first peak of the simulated CCFVW was due to the arrival of a reflected pulse wave from the head after the common carotid artery toward the arrival of a anterograde pulse wave ejected directly from the heart and that the increase in the second peak resulted from the arrival of the peak of the reflected pulse wave from the thoracic aorta. These results establish that the CCFVW characteristics contribute to the assessment of arterial elasticity.     

A biomechanical model of Peyronie's disease

Peyronie's disease is a pathological condition of the penis which is characterized by localized ossification of the tunica albuginea. A common symptom of the chronic stage is penile deformity during erection, which is frequently associated with pain and erectile dysfunction. A two-dimensional biomechanical model of the penis was applied to study the development of Peyronie’s disease by simulating the mechanical stress distribution which would result from the interaction of the ossified tunical tissue with other penile soft tissues. The model was solved by using commercial finite element software for a characteristic erectile pressure. The results demonstrate that Peyronie’s plaques may induce intensified stresses around the penile nerves and blood vessels, up to double those in the normal penis. These elevated stresses may cause a painful sensation of neural origin or ischemia in regions of compressed vascular tissue. Severe penile deformities have been shown to develop if Peyronie’s plaques develop only around one of the corpora cavernosa due to the non-homogeneous resistance of the tunica to expansion during erection. The present model can be clinically applied as an aid in the planning process of reconstructive surgery or insertion of a prosthesis.     

An analysis of age-and size-dependent flowering: A critical-production model

Three critical phenomena of flowering can be recognized in nature: critical age, critical initial size and critical switchover size. In order to understand the ecological significance of these flowering phenomena from the viewpoint of matter production, a simple model of flowering phenomean (critical-production model) was studied, assuming that plant flowering is controlled by the productive capacity of the plant (critical-production model, under the condition that plants forecast environmental conditions affecting matter production. Thus, we concluded that all three critical phenomena are various manifestation of the same critical-production principle. Then, using the model simulation, the reliability of each critical phenomenon for securing a given critical production was investigated in relation to the photosynthetic productivity of the habitat. The main predictions obtained from the simulation were as follows: 1) Only annual and biennial plants will show critical age phenomena, and most biennials will be facultative. 2) Among perennials, the critical initial size phenomenon will appear in low productive habitats, whereas in high productive habitats the critical switchover size phenomenon will be observed.     

A geometry-based model for non-invasive estimation of pressure gradients over iliac artery stenoses

The aim of this study was to develop and verify a model that provides an accurate estimation of the trans-lesion hyperemic pressure gradient in iliac artery stenoses in seconds by only using patient-specific geometric properties obtained from 3-dimensional rotational angiography (3DRA).Twenty-one patients with symptomatic peripheral arterial disease (PAD), iliac artery stenoses and an ultrasound based peak systolic velocity ratio between 2.5 and 5.0 underwent 3DRA and intra-arterial pressure measurements under hyperemic conditions. For each lesion, geometric properties were extracted from the 3DRA images using quantitative vascular analysis software. Hyperemic blood flow was estimated based on stenosis geometry using an empirical relation. The geometrical properties and hyperemic flow were used to estimate the pressure gradient by means of the geometry-based model. The predicted pressure gradients were compared with in vivo measured intra-arterial pressure measurements performed under hyperemic conditions.The developed geometry-based model showed good agreement with the measured hyperemic pressure gradients resulting in a concordance correlation coefficient of 0.86. The mean bias ± 2SD between the geometry-based model and in vivo measurements was comparable to results found by evaluating the actual computational fluid dynamics model (−1.0 ± 14.7 mmHg vs −0.9 ± 12.7 mmHg).The developed model estimates the trans-lesional pressure gradient in seconds without the need for an additional computational fluid dynamics software package. The results justify further study to assess the potential use of a geometry-based model approach to estimate pressure gradient on non-invasive CTA or MRA, thereby reducing the need for diagnostic angiography in patients suffering from PAD.     

A theoretical model for fibroblast-controlled growth of saccular cerebral aneurysms

A new theoretical model for the growth of saccular cerebral aneurysms is proposed by extending the recent constitutive framework of Kroon and Holzapfel [2007a. A model for saccular cerebral aneurysm growth by collagen fibre remodelling. J. Theor. Biol. 247, 775-787]. The continuous turnover of collagen is taken to be the driving mechanism in aneurysmal growth. The collagen production rate depends on the magnitude of the cyclic deformation of fibroblasts, caused by the pulsating blood pressure during the cardiac cycle. The volume density of fibroblasts in the aneurysmal tissue is taken to be constant throughout the growth process. The growth model is assessed by considering the inflation of an axisymmetric membranous piece of aneurysmal tissue, with material characteristics representative of a cerebral aneurysm. The diastolic and systolic states of the aneurysm are computed, together with its load-free state. It turns out that the value of collagen pre-stretch, that determines growth speed and stability of the aneurysm, is of pivotal importance. The model is able to predict aneurysms with typical berry-like shapes observed clinically, and the predicted wall stresses correlate well with the experimentally obtained ultimate stresses of this type of tissue. The model predicts that aneurysms should fail when reaching a size of about 1.2-3.6 mm, which is smaller than what has been clinically observed. With some refinements, the model may, however, be used to predict future growth of diagnosed aneurysms.     

Diffusion-controlled reactions of enzymes. An approximate analytic solution of Chou's model

Based on Chou's model, a criterion was derived, by which one can judge whether or not the physical picture of the critical spherical shell described by Chou for an enzyme-substrate fast reaction system can emerge. Furthermore, for those reaction systems with such a physical picture, an approximate analytical solution was presented, which can be easily handled to calculate the upper limit of the diffusion-controlled reaction and the corresponding concentration distribution of substrate molecules on the surface of the major protein outside the active site. The results thus obtained are in good agreement with those computed by Chou et al. through the approach of numerical solution. Furthermore, the physical significance of the criterion and its relation to the critical spherical shell are substantiated during the process of derivation, which is very helpful for gaining an insight into this kind of biomolecular system with surprisingly high reaction rates.     

A multiscale biomechanical model of platelets: Correlating with in-vitro results

Using dissipative particle dynamics (DPD) combined with coarse grained molecular dynamics (CGMD) approaches, we developed a multiscale deformable platelet model to accurately describe the molecular-scale intra-platelet constituents and biomechanical properties of platelets in blood flow. Our model includes the platelet bilayer membrane, cytoplasm and an elaborate elastic cytoskeleton. Correlating numerical simulations with published in-vitro experiments, we validated the biorheology of the cytoplasm, the elastic response of membrane to external stresses, and the stiffness of the cytoskeleton actin filaments, resulting in an accurate representation of the molecular-level biomechanical microstructures of platelets. This enabled us to study the mechanotransduction process of the hemodynamic stresses acting onto the platelet membrane and transmitted to these intracellular constituents. The platelets constituents continuously deform in response to the flow induced stresses. To the best of our knowledge, this is the first molecular-scale platelet model that can be used to accurately predict platelets activation mechanism leading to thrombus formation in prosthetic cardiovascular devices and in vascular disease processes. This model can be further employed to study the effects of novel therapeutic approaches of modulating platelet properties to enhance their shear resistance via mechanotransduction pathways.     

A unit-cell-based three-dimensional molecular mechanics analysis for buckling load,effective elasticity and Poisson's ratio determination of the nanosheets

By using a three-dimensional (3D) space-frame-like model, a molecular mechanics (MM) approach is proposed for determination of the buckling loads, effective Young's modulus and Poisson's ratio of the nanosheets, using a proper unit cell. The governing equations are derived based on the 3D kinematics of deformations and the principle of minimum total potential energy. The unit-cell-based results are employed for the space-frame-like finite element model of the nanosheet. The nonlinear MM equations are solved by representing bonds of the boron nitride nanosheet (BNNS) by beam elements to extract the local characteristics. These properties are employed in modelling of the nanosheet, as a space-frame-like finite element structure. The force field constants are chosen according to the Morse, AMBER, UFF and DREIDING models to determine the buckling strength, and effective Poisson's ratio and in-plane rigidity of the whole graphene and BNNSs. Silicon Carbide nanosheets are analysed based on different force constants. These results are concordant with the results available in the literature. The comparisons reveal that the DREIDING force field usually gives the most accurate predictions.     

A bidomain threshold model of propagating calcium waves

We present a bidomain fire-diffuse-fire model that facilitates mathematical analysis of propagating waves of elevated intracellular calcium (Ca²⁺) in living cells. Modeling Ca²⁺ release as a threshold process allows the explicit construction of traveling wave solutions to probe the dependence of Ca²⁺ wave speed on physiologically important parameters such as the threshold for Ca²⁺ release from the endoplasmic reticulum (ER) to the cytosol, the rate of Ca²⁺ resequestration from the cytosol to the ER, and the total [Ca²⁺] (cytosolic plus ER). Interestingly, linear stability analysis of the bidomain fire-diffuse-fire model predicts the onset of dynamic wave instabilities leading to the emergence of Ca²⁺ waves that propagate in a back-and-forth manner. Numerical simulations are used to confirm the presence of these so-called ‘tango waves’ and the dependence of Ca²⁺ wave speed on the total [Ca²⁺].      

Stability properties of pulse vaccination strategy in SEIR epidemic model

The problem of the applicability of the pulse vaccination strategy (PVS) for the stable eradication of some relevant general class of infectious diseases is analyzed in terms of study of local asymptotic stability (LAS) and global asymptotic stability (GAS) of the periodic eradication solution for the SEIR epidemic model in which is included the PVS. Demographic variations due or not to diseased-related fatalities are also considered. Due to the non-triviality of the Floquet's matrix associate to the studied model, the LAS is studied numerically and in this way it is found a simple approximate (but analytical) sufficient criterion which is an extension of the LAS constraint for the stability of the trivial equilibrium in SEIR model without vaccination. The numerical simulations also seem to suggest that the PVS is slightly more efficient than the continuous vaccination strategy. Analytically, the GAS of the eradication solutions is studied and it is demonstrated that the above criteria for the LAS guarantee also the GAS.     

A model of herd grazing as a travelling wave,chemotaxis and stability

Possible constitutive models are examined for the formation of a herd, under the assumption that a herd forms a travelling wave while grazing. Under quite general conditions, it is found that the only possibility for a travelling wave is a balance between food seeking and natural dispersion, such as in chemotaxis. The stability of the travelling wave previously conjectured, is shown both for one- and two-dimensional perturbations.