Pressure-radius Curves Research Articles

Airway Bistability Is Modulated by Smooth Muscle Dynamics and Length-Tension Characteristics

Airway closure has important implications for lung disease, especially asthma; in particular, the prospect of bistability between open and closed (or effectively closed) airway states has been thought to play a prominent role in airway closure associated with the formation of clustered ventilation defects in asthma. However, many existing analyses of closure consider only static airway equilibria; here we construct, to our knowledge, a new model wherein airway narrowing and closure dynamics are modulated by coupling the airway to cross-bridge models of airway smooth muscle dynamics and force generation. Using this model, we show that important qualitative features of airway pressure-radius hysteresis loops are highly dependent on both airway smooth muscle dynamics, and the length-tension relationship. Furthermore, we show that two recent experimental results from intact bronchial segments are both expressions of the same phenomenon: that a monotonically increasing length-tension relationship, with sharply higher tension at longer lengths, is needed to drive the observed changes in low-compliance regions of the baseline pressure-radius curve. We also explore the potential implications of this finding for airway closure in coupled airway models.

Effects of the three-dimensional residual stresses on the mechanical properties of arterial walls

Effects of the three-dimensional residual stresses on the mechanical properties of arterial walls are analyzed in this paper, based on the model which considered the bending and stretching both in the circumferential and axial directions of the three distinct arterial layers. Moreover, different constitutive models are proposed to quantify the nonlinear mechanics of the three distinct layers and the important constituents, i.e. elastin, collagen fibers and smooth muscle cells (SMCs), are all taken into account. The stress distributions and pressure–radius curves of the arterial wall are given in details. Results demonstrate that the maximum values of the circumferential stress and the corresponding stress gradient in the media under the mean arterial pressure are reduced significantly as a consequence of the SMCs. The bending in the axial direction of the media and the opening angle of the intima have an obvious impact on the mechanical behaviors of arterial walls. This study may not only develop the understanding of effects of the three-dimensional residual stresses on the arterial wall response, but also can increase the accuracy of the analyses for patient-specific studies used for the treatments of arterial diseases.

Experimental characterization of the distribution of collagen fiber recruitment

The passive biomechanical behavior of blood vessels is generally modeled by a parallel arrangement of elastin and collagen, with collagen recruitment depending on vessel strain. We experimentally determined the collagen recruitment distribution using confocal microscopy. Digital images from sections of rabbit carotid artery under increasing circumferential stretch ratio were acquired. The straightness of the fibers was measured to compute the fraction of recruited fibers for each stretch ratio. The experimental distribution obtained was then used in the model proposed by Zulliger et al. This model is based on a strain energy function (SEF), which provides a constituent-based description of the wall mechanics. Using this model, a fit of the pressure–radius curve was then performed by using the experimental collagen recruitment distribution with the collagen elastic constant as a free fit parameter. The fit was good, but the value of the collagen elastic constant obtained was below values reported in the literature. A second fit of the pressure–radius curve was also performed, whereby the collagen distribution was left free to adapt while the collagen elastic constant was set to a physiological value. The differences between the experimental collagen engagement distribution and the distribution obtained when the collagen elastic constant was fixed were analyzed. Good qualitative agreement was found between the experimental distribution of collagen recruitment and the model. Some experimental limitations and modeling approximations remain. Nevertheless, experimental proof of progressive collagen recruitment was established, which validates the basic hypothesis of the model.

A Structural Model of the Venous Wall Considering Elastin Anisotropy

The three-dimensional biomechanical behavior of the vascular wall is best described by means of strain energy functions. Significant effort has been devoted lately in the development of structure-based models of the vascular wall, which account for the individual contribution of each major structural component (elastin, collagen, and vascular smooth muscle). However, none of the currently proposed structural models succeeded in simultaneously and accurately describing both the pressure-radius and pressure-longitudinal force curves. We have hypothesized that shortcomings of the current models are, in part, due to unaccounted anisotropic properties of elastin. We extended our previously developed biomechanical model to account for elastin anisotropy. The experimental data were obtained from inflation-extension tests on facial veins of five young white New Zealand rabbits. Tests have been carried out under a fully relaxed state of smooth muscle cells for longitudinal stretch ratios ranging from 100% to 130% of the in vivo length. The experimental data (pressure-radius, pressure-force, and zero-stress-state geometries) provided a complete biaxial mechanical characterization of rabbit facial vein and served as the basis for validating the applicability and accuracy of the new biomechanical model of the venous wall. When only the pressure-radius curves were fitted, both the anisotropic and the isotropic models gave excellent results. However, when both pressure-radius and pressure-force curves are simultaneously fitted, the model with isotropic elastin shows an average weighted residual sum of squares of 8.94 and 23.9 in the outer radius and axial force, respectively, as compared to averages of 6.07 and 4.00, when anisotropic elastin is considered. Both the Alkaike information criterion and Schwartz criterion show that the model with the anisotropic elastin is more successful in predicting the data for a wide range of longitudinal stretch ratios. We conclude that anisotropic description of elastin is required for a full 3D characterization of the biomechanics of the venous wall.

A strain energy function for arteries accounting for wall composition and structure

Identification of a Strain Energy Function (SEF) is used when describing the complex mechanical properties of soft biological tissues such as the arterial wall. Classic SEFs, such as the one proposed by Chuong and Fung (J. Biomech. Eng. 105(3) (1983) 268), have been mostly phenomenological and neglect the particularities of the wall structure. A more structural model was proposed by Holzapfel et al. (J. Elasticity 61 (2000) 1–48.) when they included the characteristic angle at which the collagen fibers are helically wrapped, resulting in an excellent SEF for applications such as finite element modeling. We have expanded upon the idea of structural SEFs by including not only the wavy nature of the collagen but also the fraction of both elastin and collagen contained in the media, which can be determined by histology. The waviness of the collagen is assumed to be distributed log-logistically. In order to evaluate this novel SEF, we have used it to fit experimental data from inflation–extension tests performed on rat carotids. We have compared the results of the fit to the SEFs of Choung and Fung and Holzapfel et al. The novel SEF is found to behave similarly to that of Holzapfel et al., both succeed in describing the typical S-shaped pressure-radius curves with comparable quality of fit. The parameters of the novel SEF obtained from the fitting, bearing the physical meaning of the elastic modulus of collagen, the elastic modulus of elastin, the collagen waviness, and the collagen fiber angle, were compared to experimental data and discussed.

A constitutive formulation of arterial mechanics including vascular smooth muscle tone.

A pseudo-strain energy function (pseudo-SEF) describing the biomechanical properties of large conduit arteries under the influence of vascular smooth muscle (VSM) tone is proposed. In contrast to previous models that include the effects of smooth muscle contraction through generation of an active stress, in this study we consider the vascular muscle as a structural element whose contribution to load bearing is modulated by the contraction. This novel pseudo-SEF models not only arterial mechanics at maximal VSM contraction but also the myogenic contraction of the VSM in response to local increases in stretch. The proposed pseudo-SEF was verified with experimentally obtained pressure-radius curves and zero-stress state configurations from rat carotid arteries displaying distinct differences in VSM tone: arteries from normotensive rats displaying minimal VSM tone and arteries from hypertensive rats exhibiting significant VSM tone. The pressure-radius curves were measured in three different VSM states: fully relaxed, maximally contracted, and normal VSM tone. The model fitted the experimental data very well (r2 > 0.99) in both the normo- and hypertensive groups for all three states of VSM activation. The pseudo-SEF was used to illustrate the localized reduction of circumferential stress in the arterial wall due to normal VSM tone, suggesting that the proposed pseudo-SEF can be of general utility for describing stress distribution not only under passive VSM conditions, as most SEFs proposed so far, but also under physiological and pathological conditions with varying levels of VSM tone.

A strain energy function for arteries accounting for wall composition and structure

Identification of a Strain Energy Function (SEF) is used when describing the complex mechanical properties of soft biological tissues such as the arterial wall. Classic SEFs, such as the one proposed by Chuong and Fung (J. Biomech. Eng. 105(3) (1983) 268), have been mostly phenomenological and neglect the particularities of the wall structure. A more structural model was proposed by Holzapfel et al. (J. Elasticity 61 (2000) 1–48.) when they included the characteristic angle at which the collagen fibers are helically wrapped, resulting in an excellent SEF for applications such as finite element modeling. We have expanded upon the idea of structural SEFs by including not only the wavy nature of the collagen but also the fraction of both elastin and collagen contained in the media, which can be determined by histology. The waviness of the collagen is assumed to be distributed log-logistically. In order to evaluate this novel SEF, we have used it to fit experimental data from inflation–extension tests performed on rat carotids. We have compared the results of the fit to the SEFs of Choung and Fung and Holzapfel et al. The novel SEF is found to behave similarly to that of Holzapfel et al., both succeed in describing the typical S-shaped pressure-radius curves with comparable quality of fit. The parameters of the novel SEF obtained from the fitting, bearing the physical meaning of the elastic modulus of collagen, the elastic modulus of elastin, the collagen waviness, and the collagen fiber angle, were compared to experimental data and discussed.

Assessment of Mechanical Homogeneity of the Arterial Wall by an Artery-Inversion Test

The aim of this study was to apply a method to check, by means of a simple mechanical test, the homogeneity in the elastic properties of an artery. The method is based on in vitro measurements of the pressure–radius relationships of the artery in the normal configuration and when the artery is turned inside out. The mechanical properties of the arterial wall are described by a strain energy function (SEF). The wall is assumed to be homogeneous and the SEF is determined using data from the pressure–radius curves obtained in vitro or in vivo. Fresh pig carotids were mounted on an inflation–extension apparatus allowing the measurement of diameter by means of an ultrasonic device, and the internal pressure by means of a catheter. First, the SEF is determined using the pressure–radius relationship. Geometrical characteristics of the zero stress state (ZSS) are measured on a longitudinally cut ring of the artery. This state is used as reference state for the measurements of the deformations. The artery is then turned inside out to create a different initial strain distribution in the wall. The measured pressure–radius curve is then compared to the one predicted using the SEF obtained by the control test. The two curves should be identical for a homogeneous arterial wall and different if the wall is nonhomogeneous. The comparison between the pressure–radius relationship of the inverted artery and the theoretical one for a pig carotid showed that the pig carotid is elastically homogeneous.

The influence of aging and aortic stiffness on permanent dilation and breaking stress of the thoracic descending aorta.

To assess the influence of aging and aortic stiffness on the extent of irreversible deformation and breaking stress of the human thoracic aorta. From 14 human heart valve donors without aortic disease (mean age 35 years, range 8-59 years), 14 intact segments of the thoracic descending aorta were studied within 48 h after cardiac arrest. In an experimental setup, the segments were submitted to increasing hydrostatic pressure loads, both statically and dynamically, while radius and wall thickness were monitored echocardiographically. Pressure-radius curves were constructed. Radius and wall thickness were determined at a pressure of 100 mmHg. Radius at elastin resting length and collagen recruitment pressure (Pcol, mmHg) were derived from the pressure-radius relationship and stress-strain curves were constructed to yield Young's moduli of elastin and collagen. Distensibility (D, mmHg-1) was determined while loading the segment with a sinusoidal pressure wave of 120/50 mmHg at both 0.5 and 1 Hz. Subsequently increasing static pressure loads of 400, 800, 1200 and 1600 mmHg were applied. After each pressure load, the increase in aortic radius at a pressure of 100 mmHg (Rinc) was determined. The experiment continued until rupture occurred and breaking stress (sigma break, N m-2) was calculated, donor age and aortic stiffness were correlated with Rinc and sigma break of the aortic segments. Mean breaking stress of the 14 segments was 2.7 x 10(6) N m-2. Breaking stress was negatively correlated with age (r2 = 0.66) and positively with D (r2 = 0.44) and with Pcol (r2 = 0.18). Seven segments survived a pressure load of 800 mmHg, in these vessels, the extent of irreversible dilation was positively correlated with age (r2 = 0.42) and negatively with D (r2 = 0.40) and Pcol (r2 = 0.40). Permanent deformation and rupture of the human thoracic aorta following pressure overload are influenced by age, distensibility and collagen recruitment pressure.

Hemodynamic and mechanical performance of arterial grafts assessed by numerical simulation: a design oriented study.

The hemodynamic and mechanical characteristics of an end-to-end implanted prosthesis for a small artery were theoretically investigated. The changes in the main physical and geometrical properties of the prosthesis were simulated by means of a numerical model of arterial hemodynamics. Variation in the pressure-radius curve due to changes in lumen size, wall thickness, elasticity, and tapering were considered. The effects of such changes on pressure, flow, and wall stresses during the cardiac cycle were evaluated. To avoid superimposing the effects, only 1 of the graft properties was varied with respect to a reference condition in each simulation. A prosthesis with a reduced lumen size (20%) was subjected to higher shear stress, which was dangerously doubled. Wall thickening (200%) primarily determined decreased circumferential stress (300%) and increased wall shear stress (48%) because it caused a reduction of the graft lumen size. The time averages of flow and pressure over the cardiac cycle were not significantly influenced by the simulated changes, being imposed primarily by the proximal and distal circulations.

Changes in the distensibility of the intrapulmonary arteries in the normal newborn and growing pig

The static elastic properties and structure of the intra-pulmonary arteries have been studied in pigs from fetal life to 4 months old, by radiographic and histological methods. During this period the average radius of the vessels examined varied from approximately equal to 0.9 mm at birth to 3.3 mm at the age of 4 months. The form of the pressure radius curves was similar to that observed in the systemic circulation and was characteristic of the gradual recruitment of collagen fibres with increasing pressure. In vessels near the hilum, wall thickness decreased from approximately equal to 180 microns to 100 microns during the first 2 weeks and thereafter increased to approximately equal to 250 microns. Throughout the lung and in all age groups there was a linear relationship between the logarithm of pressure strain elastic modulus (used as a measure of functional stiffness) and pressure. In contrast to the systemic arteries there was little change in functional stiffness towards the periphery, although the incremental elastic modulus (structural stiffness) did increase at sites further removed from the heart. During the first two weeks of life there was little change in functional stiffness (Ep approximately equal to 1.1 X 10(4) Nm2) although by 4 months Ep had increased by a factor of three. During this period despite fluctuations with age, an overall increase in structural stiffness was also observed (EINC approximately equal to 1 X 10(5) at 4 to 6 h to 2 X 10(5) Nm-2 at 4 months). The observations suggest that the changes in elasticity observed in early life are a consequence of the adaptation of the circulation to the lower pulmonary arterial pressure of extra-uterine life.

Effect of smooth muscle activity on the static and dynamic elastic properties of the rabbit carotid artery.

The variation of radius with pressure was measured in vitro in the carotid artery of ten rabbits. Experiments were performed without prior conditioning of the vessel, over an inflation-deflation cycle at pressures in the range 0 to 24 kPa during treatment with, and in the absence of noradrenaline (referred to as active and passive conditions). The effect of a step change in pressure (2.7 kPa) on the radius of the vessel was investigated on a further three specimens. Under passive conditions, the variation of the static and the real part of the dynamic incremental elastic modulus with pressure and stress was similar during both inflation and deflation. Under active conditions a large degree of pressure-radius hysteresis was observed. At physiological pressures smooth muscle activity was associated with a decrease in elastic modulus when compared to passive values at the same pressure or stress. At a stress above 2 x 10(5) N.m-2 yield of the constricted vessels was observed and on further inflation and subsequent deflation the pressure radius curve was identical to that obtained under passive conditions. We suggest that conditioning a vessel, by means of repeated inflation to a high pressure and deflation to zero pressure, before measuring its elastic properties may give misleading results when applied to the vessels of living animals.

Active Changes in Tone in the Canine Vena Cava

Active changes in tone of the superior vena cava were studied in dogs several days after recovery from surgery. The diameter was measured continuously by sonar technique and distending pressure by Silastic cannulas. An occlusion cuff at the junction of the superior vena cava and atrium generated a passive range of pressure-radius data in the control state. Interventions included catecholamine infusion, fright, and exercise. In most instances, active constriction with norepinephrine could be shown by a pressure-radius curve that was steeper and displaced toward the pressure axis. The constricted curves were frequently, but not invariably, sigmoid, whereas control occlusion curves were arcuate. Isoproterenol also produced active constriction of the superior vena cava, as did fright and moderate exercise.