The residually stressed unloaded state of arteries: Membrane and thin cylinder approximations.

A solution is obtained for incompressible non-linearly elastic membranes that describes the bending of a cylindrical sector to form a perfect cylinder for a wide class of materials that includes isotropic materials and orthotropic materials reinforced by two families of mechanically equivalent fibres that are wound helically about the axial direction. Despite the relative simplicity of the physical problem, the solution of the corresponding boundary value problem for thick cylinders requires a numerical solution for even the simplest models. It is shown, however, that the thin shell solution provides an excellent approximation to the solution of the problem for cylindrical sectors whose thicknesses are an order of magnitude greater than that assumed for membranes. The approximate stress distribution in such thin shells is obtained. In these residually stressed cylinders, the radial stress is approximately zero but the hoop and axial stresses are finite. Estimates of the residual stresses in the unloaded state are obtained. A closed-form solution for the bending moment necessary to effect the deformation is also obtained.

Inflation of residually stressed Fung-type membrane models of arteries.

Elastic arteries are idealised as being incompressible orthotropic nonlinearly elastic cylinders. They are further idealised as being membranes in order to analyse the effect of the experimentally observed pre-stressing of arterial tissue on inflation. The pre-stress is modelled here using the opening-angle method. It is shown that there can be multiple unloaded states of arterial segments of Fung materials, suggesting the corresponding set of material parameters will not yield reliable predictions of arterial stress in three dimensions as there is no experimental evidence to support this non-uniqueness. It is also shown that the circumferential pre-strain has a simple magnifying scaling effect on the pressure-radial strain relation and on the axial force needed to maintain the membrane length during inflation; the effect of the axial pre-strain is more nuanced.

Segmental Variations in the Peel Characteristics of the Porcine Thoracic Aorta.

Aortic dissection occurs predominantly in the thoracic aorta and the mechanisms for the initiation and propagation of the tear in aortic dissection are not well understood. We study the tearing characteristics of the porcine thoracic aorta using a peeling test and we estimate the peeling energy per unit area in the ascending and the descending segments. The stretch and the peel force per unit width undergone by the peeled halves of a rectangular specimen are measured. We find that there can be significant variation in the stretch within the specimen and the stretch between the markers in the specimen varies with the dynamics of peeling. We found that in our experiment the stretch achieved in the peeled halves was such that it was in the range of the stretch at which the stress-stretch curve for the uniaxial experiment starts deviating from linearity. Higher peeling energy per unit area is required in the ascending aorta compared to the descending aorta. Longitudinal specimens required higher peeling energy per unit area when compared to the circumferential specimens.

On a possible methodology for identifying the initiation of damage of a class of polymeric materials.

In this paper, we provide a possible methodology for identifying the initiation of damage in a class of polymeric solids. Unlike most approaches to damage that introduce a damage parameter, which might be a scalar, vector or tensor, that depends on the stress or strain (that requires knowledge of an appropriate reference configuration in which the body was stress free and/or without any strain), we exploit knowledge of the fact that damage is invariably a consequence of the inhomogeneity of the body that makes the body locally 'weak' and the fact that the material properties of a body invariably depend on the density, among other variables that can be defined in the current configuration, of the body. This allows us to use density, for a class of polymeric materials, as a means to identify incipient damage in the body. The calculations that are carried out for the biaxial stretch of an inhomogeneous multi-network polymeric solid bears out the appropriateness of the thesis that the density of the body can be used to forecast the occurrence of damage, with the predictions of the theory agreeing well with experimental results. The study also suggests a meaningful damage criterion for the class of bodies being considered.

Implicit constitutive relations for nonlinear magnetoelastic bodies.

Implicit constitutive relations that characterize the response of elastic bodies have greatly enhanced the arsenal available at the disposal of the analyst working in the field of elasticity. This class of models were recently extended to describe electroelastic bodies by the present authors. In this paper, we extend the development of implicit constitutive relations to describe the behaviour of elastic bodies that respond to magnetic stimuli. The models that are developed provide a rational way to describe phenomena that have hitherto not been adequately described by the classical models that are in place. After developing implicit constitutive relations for magnetoelastic bodies undergoing large deformations, we consider the linearization of the models within the context of small displacement gradients. We then use the linearized model to describe experimentally observed phenomena which the classical linearized magnetoelastic models are incapable of doing. We also solve several boundary value problems within the context of the models that are developed: extension and shear of a slab, and radial inflation and extension of a cylinder.

Fidelity of the estimation of the deformation gradient from data deduced from the motion of markers placed on a body that is subject to an inhomogeneous deformation field.

Practically all experimental measurements related to the response of nonlinear bodies that are made within a purely mechanical context are concerned with inhomogeneous deformations, though, in many experiments, much effort is taken to engender homogeneous deformation fields. However, in experiments that are carried out in vivo, one cannot control the nature of the deformation. The quantity of interest is the deformation gradient and/or its invariants. The deformation gradient is estimated by tracking positions of a finite number of markers placed in the body. Any experimental data-reduction procedure based on tracking a finite number of markers will, for a general inhomogeneous deformation, introduce an error in the determination of the deformation gradient, even in the idealized case, when the positions of the markers are measured with no error. In our study, we are interested in a quantitative description of the difference between the true gradient and its estimate obtained by tracking the markers, that is, in the quantitative description of the induced error due to the data reduction. We derive a rigorous upper bound on the error, and we discuss what factors influence the error bound and the actual error itself. Finally, we illustrate the results by studying a practically interesting model problem. We show that different choices of the tracked markers can lead to substantially different estimates of the deformation gradient and its invariants. It is alarming that even qualitative features of the material under consideration, such as the incompressibility of the body, can be evaluated differently with different choices of the tracked markers. We also demonstrate that the derived error estimate can be used as a tool for choosing the appropriate marker set that leads to the deformation gradient estimate with the least guaranteed error.

A model for the formation, growth, and lysis of clots in quiescent plasma. A comparison between the effects of antithrombin III deficiency and protein C deficiency.

A mathematical model comprised of 23 reaction-diffusion equations is used to simulate the biochemical changes and transport of various reactants involved in coagulation and fibrinolysis in quiescent plasma. The growth and lysis of a thrombus, as portrayed by the model equations, is governed by boundary conditions that include the surface concentration of TF-VIIa, the generation of XIa by contact activation (in vitro), and the secretion of tPA due to endothelial activation. We apply the model to two clinically relevant hypercoagulable states, caused by deficiency of either antithrombin III or protein C. These predictions are compared with published experimental data which validate the utility of the developed model under the special case of static conditions. The incorporation of varying hemodynamic conditions in to the current fluid static model remains to be performed.

Towards an understanding of the mechanics underlying aortic dissection.

Acute aortic dissection and associated aortic catastrophes are among the most devastating forms of cardiovascular disease, with a remarkably high morbidity and mortality despite current medical and surgical treatment. The mechanics underlying aortic dissection are incompletely understood, and a further understanding of the relevant fluid and solid mechanics may yield not only a better appreciation of its pathogenesis, but also the development of improved diagnostic and therapeutic strategies. After illustrating some of the inadequacies with respect to the extant work on the mechanics of aortic dissection, we alternatively postulate that the clinical hemodynamic disturbances that render the aorta susceptible to the initiation of dissection are principally elevated maximum systolic and mean aortic blood pressure, whereas the hemodynamic disturbances that facilitate propagation of dissection are principally elevated pulse pressure and heart rate. Furthermore, abnormal aortic mechanical properties and/or geometry are requisite for dissection to occur. Specifically, we propose that the degree of anisotropy will directly influence the probability of future aortic dissection. Imaging of the aorta may provide information regarding aortic anisotropy and geometry, and in combination with a hemodynamic risk assessment, has the potential to be able to prospectively identify patients at high risk for future aortic dissection thereby facilitating prophylactic intervention. The aim of the paper is to identify the main mechanical issues that have a bearing on aortic dissection, and to suggest an appropriate mathematical model for describing the problem.

A theoretical model of enlarging intracranial fusiform aneurysms.

The mechanisms by which intracranial aneurysms develop, enlarge, and rupture are unknown, and it remains difficult to collect the longitudinal patient-based information needed to improve our understanding. We submit, therefore, that mathematical models hold promise by allowing us to propose and test competing hypotheses on potential mechanisms of aneurysmal enlargement and to compare predicted outcomes with limited clinical information--in this way, we may begin to narrow the possible mechanisms and thereby focus experimental studies. In this paper, we present a constrained mixture model of evolving thin-walled, fusiform aneurysms and compare multiple competing hypotheses with regard to the production, removal, and alignment of the collagen that provides the structural integrity of the wall. The results show that this type of approach has the capability to infer potential means by which lesions enlarge and whether such changes are likely to produce a stable or unstable process. Such information can better direct the requisite histopathological examinations, particularly on the need to quantify collagen orientations as a function of lesion geometry.

Heat-induced changes in the finite strain viscoelastic behavioir of a collaagenous tissue.

Supra-physiological temperatures are increasingly being used to treat many different soft need for injuries. To identify improved clinical treatments, however, there is a need for better information on the effect of the mechanics on the thermal damage process as well as the effect of the incurred damage on the subsequent mechanical properties. In this paper we report the first biaxial data on the stress relaxation behavior of a collagenous tissue before and after thermal damage. Based on a two-dimensional finite strain viscoelastic model, which incorporates an exponential elastic response, it is shown that the thermal damage can significantly decrease the characteristic time for stress relaxation and the stress residual.

A model for the formation and lysis of blood clots.

Both biochemical and mechanical factors have to be taken into account if a meaningful model for the formation, growth, and lysis of clots in flowing blood is to be developed. Most models that are currently in use neglect one or the other of these factors. We have previously reported a model [J Theoret Med 2003;5:183-218] that we believe is a step in this direction, incorporating many of the crucial biochemical and rheological factors that play a role in the formation, growth, and lysis of clots. While this model takes into account the extrinsic pathway of coagulation, it largely ignores the intrinsic pathway. Here, we discuss some of the general issues with respect to mathematical modeling of thrombus formation and lysis, as well as specific aspects of the model that we have developed.

A constrained mixture model for arterial adaptations to a sustained step change in blood flow.

A sustained change in blood flow results in an arterial adaptation that can be thought to consist of two general steps: an immediate vasoactive response that seeks to return the wall shear stress to its homeostatic value, and a long-term growth and remodeling process that seeks to restore the intramural stresses and, if needed, the wall shear stress toward their homeostatic values. Few papers present mathematical models of arterial growth and remodeling in general, and fewer yet address flow-induced changes. Of these, most prior models build upon the concept of "kinematic growth" proposed by Skalak in the early 1980s (Skalak R (1981) In: Proceedings of the IUTAM Symposium on finite elasticity. Martinus Nijhoff, The Hague, pp 347-355). Such approaches address important consequences of growth and remodeling, but not the fundamental means by which such changes occur. In this paper, therefore, we present a new approach for mathematically modeling arterial adaptations and, in particular, flow-induced alterations. The model is motivated by observations reported in the literature and is based on a locally homogenized, constrained mixture theory. Specifically, we develop a 3-D constitutive relation for stress in terms of the responses of the three primary load-bearing constituents and their time-varying mass fractions, with the latter accounting for the kinetics of the turnover of cells and extracellular matrix in changing, stressed configurations. Of particular importance is the concept that the natural configurations of the individual constituents can evolve separately and that this leads to changes in the overall material properties and empirically inferred residual stress field of the vessel. Potential applications are discussed, but there is a pressing need for new, theoretically motivated data to allow the prescription of specific functional forms of the requisite constitutive relations and the values of the associated material parameters.

Gas mixing for achieving suitable conditions for single point aerosol sampling in a straight tube: experimental and numerical results.

Experimental measurements of velocity and tracer gas concentration are taken in a straight tube to evaluate the effectiveness of mixing in achieving conditions as required by ANSI N13.1-1999 for single point extractive sampling from stacks and ducts of nuclear facilities. Mixing is evaluated for inlet turbulent intensities of 1.5%, 10%, and 20%, achieved by introducing various bi-plane grids, and for conditions generated by a commercial static gas mixer. The data obtained (at Reynolds number = 15,000) highlight the importance of inlet turbulence intensity in the process of turbulent dispersion of a dilute gas. The gas mixer does not introduce significant pressure losses and unlike bi-plane grids, the turbulence downstream of the mixer is not homogenous. A judicious choice of the release location that uses the large scale eddies and inhomogeneity of the turbulence ensures that the specified ANSI N13.1-1999 criteria are attained within 7 diameters downstream of the duct inlet. This is significantly more effective than a bi-plane grid where even with 20% inlet intensity the criteria are met only at 21 diameters downstream. The predictions of a proposed semi-empirical correlation match favorably with data. For example, at 18 diameters downstream with inlet intensities of 1.5% and 10%, the predicted coefficients of variation (COVs) of 150% and 65% are close to the actual values of 154% and 50%; where the COV of a set of measurements is the ratio of the standard deviation of the set to its mean value. The corresponding results obtained using commercially available software are 141% and 12%. Results from a particle-tracking model show good qualitative trends, but they should not be used to determine compliance with the requirements of the ANSI standard.

Lung function during moderate hypobaric hypoxia in normal subjects and patients with chronic obstructive pulmonary disease.

BACKGROUND: We sought to describe changes in spirometric variables and lung volume subdivisions in healthy subjects and patients with chronic obstructive pulmonary disease (COPD) during moderate acute hypobaric hypoxia as occurs during air travel. We further questioned whether changes in lung function may associate with reduced maximum ventilation or worsened arterial blood gases. METHODS: Ambulatory patients with COPD and healthy adults comprised the study populations (n = 27). We obtained baseline measurements of spirometry, lung volumes and arterial blood gases from each subject at sea level and repeated measurements during altitude exposure to 8000 ft (2438 m) above sea level in a man-rated hypobaric chamber. RESULTS: Six COPD patients and three healthy subjects had declines in FVC during altitude exposure greater than the 95% confidence interval (CI) for expected within day variability (p < 0.05). Average forced vital capacity (FVC) declined by 0.123 +/- 0.254 L (mean +/- SD; 95% CI = -0.255, -0.020; p < 0.05) for all subjects combined. The magnitude of decline in FVC did not differ between groups (p > 0.05) and correlated with increasing residual volume (r = -0.455; <0.05). Change in maximum voluntary ventilation (MVV) in the COPD patients equaled -1.244 +/- 4.797 L x min(-1) (95% CI = -3.71, 1.22; p = 0.301). Decline in maximum voluntary ventilation (MVV) in the COPD patients correlated with decreased FVC (r = 0.630) and increased RV (r = -0.546; p < 0.05). Changes in spirometric variables for patients and controls did not explain significant variability in the arterial blood gas variables PaO2, PaCO2 or pH at altitude. CONCLUSIONS: We observed a decline in forced vital capacity in some COPD patients and normal subjects greater than expected for within day variability. Spirometric changes correlated with changes in reduced maximum voluntary ventilation in the patients but not with changes in resting arterial blood gases.

A single integral finite strain viscoelastic model of ligaments and tendons.

A general continuum model for the nonlinear viscoelastic behavior of soft biological tissues was formulated. This single integral finite strain (SIFS) model describes finite deformation of a nonlinearly viscoelastic material within the context of a three-dimensional model. The specific form describing uniaxial extension was obtained, and the idea of conversion from one material to another (at a microscopic level) was then introduced to model the nonlinear behavior of ligaments and tendons. Conversion allowed different constitutive equations to be used for describing a single ligament or tendon at different strain levels. The model was applied to data from uniaxial extension of younger and older human patellar tendons and canine medial collateral ligaments. Model parameters were determined from curve-fitting stress-strain and stress-relaxation data and used to predict the time-dependent stress generated by cyclic extensions.

A mathematical model for shear-induced hemolysis.

The time-varying history of stress exposure within a rotary blood pump makes it difficult to arrive at a quantifiable design criterion for predicting cell traumatization. Constant stress experiments have revealed that there is a threshold stress level above which damage to blood cells occurs depending upon the time of exposure. The shear stress history experienced by cells within a rotary blood pump, however, is highly unsteady. In order to better predict cell trauma under these realistic conditions, a mathematical damage model based on a concept of "damage accumulation" has been developed. This model is evaluated within the context of red cell trauma. Experimental results support the hypothesis that the rate of damage accumulation increases nonlinearly with the stress level as well as the age of the cell.

Identification of elastic properties of homogeneous, orthotropic vascular segments in distension.

Characterization of the constitutive behavior of normal and pathological blood vessel segments could provide the clinician with a means to predict the onset and assess the severity of certain vascular maladies. Many of the constitutive models that have been proposed to date either fail to properly consider certain features of the anatomic structure and function of vascular tissue or are so mathematically complex that their utilization is intractable. We have developed a material identification technique that first required the adaptation and validation of a constitutive law describing the nonlinear, three-dimensional behavior of orthotropic, compressible, hyperelastic vascular segments. By coupling a nonlinear finite element program and experimental data with a robust nonlinear least-squares regression algorithm, a set of elastic parameters (moduli) is obtained. Regressions on data for a canine carotid artery and rabbit infrarenal aorta yielded coefficients of variation of 0.21 and 0.08, respectively. The estimated moduli demonstrated certain trends found by other investigators: both the canine carotid artery and rabbit aorta were found to be stiffer radially than circumferentially, and the former was found to be stiffer circumferentially than longitudinally. Using these material constants and measured arterial pressures, the stress distribution was computed for each specimen. The predicted radial stress was consistent with a transmural variation of approximately--p (applied luminal pressure) to approximately zero in both specimens, while the circumferential stresses ranged from 2.2p to 0.7p for the canine carotid, and from 6.4p to 3.7p for the rabbit aorta. The stress distributions qualitatively agreed with those reported in previous investigations, as well as with certain physiologic observations. Based on the results of our two sample cases, we believe that our technique could be beneficial to the assessment of the three-dimensional, anisotropic behavior of vascular tissue.

Hemodynamic effects of altitude exposure and oxygen administration in chronic obstructive pulmonary disease.

PURPOSE: Cardiovascular events are the leading cause of death during air travel. Because patients with chronic obstructive pulmonary disease (COPD) develop severe hypoxemia at altitude, we sought to determine whether changes in systemic hemodynamics may contribute to health risks during hypobaric hypoxia. PATIENTS AND METHODS: We recorded radial artery catheter blood pressure, cardiac frequency, and cardiac ectopy in 18 men (aged 68 +/- 6 years, mean +/- SD) with severe COPD (forced expiratory volume in 1 second 0.97 L +/- 0.32 L) at sea level, after 45 minutes of steady-state hypobaric hypoxia at 2,438 m in a hypobaric chamber, and after oxygen supplementation at 2,438 m. RESULTS: Mean arterial pressure (mm Hg), systolic blood pressure (SBP), diastolic blood pressure, and pulsus paradoxus during acute hypobaric exposure did not differ from baseline. During oxygen supplementation, SBP declined (p = 0.028). Decreases in pulsus paradoxus and pulse pressure were noted on oxygen (p < 0.05). We found no changes in cardiac frequency. Cardiac ectopy was uncommon; for one subject, ectopy increased with hypobaric hypoxia and decreased with oxygen administration. CONCLUSION: Vasopressor responses to hypoxia do not add to the risk of air travel in patients with severe COPD. Supplemental oxygen may cause beneficial hemodynamic changes in patients with COPD during acute hypobaric exposure.

Sensitivity and specificity of bronchial provocation testing. An evaluation of four techniques in exercise-induced bronchospasm.

The thresholds used to define a positive result for bronchial provocation challenges (BPC) are arbitrary. Requiring smaller decrements in expired flow to define a positive study would capture more cases of reactive airways (increased sensitivity) but would include some "normal" responses (decreased specificity). To examine the relationship between threshold definition and the ability to correctly classify subjects as either normal or as having airways hyperresponsiveness (AHR), four different BPC tests were administered on different days to 20 patients with a clinical diagnosis of exercise-induced bronchospasm (EIB) and 20 control subjects. The four BPC tests were indoor exercise on a cycle ergometer, methacholine inhalation challenge (MIC), eucapnic voluntary hyperventilation (EVH) with dry gas, and EVH with cold gas. Our results indicate that the thresholds which best separate the two groups are different for each of the four BPC techniques. For methacholine inhalation (MIC), a fall in FEV1 (d%FEV1) of 15 percent or greater at 188 cumulative breath units was 100 percent specific for AHR but had a sensitivity of only 55 percent. Eucapnic voluntary hyperventilation (EVH) with room temperature dry gas was 100 percent specific at a d%FEV1 of 11 percent, but, at that threshold, sensitivity was only 50 percent. EVH with cold air was 100 percent specific at a d%FEV1 of 12 percent but sensitivity was only 35 percent. The bicycle ergometer challenge was far too insensitive to be of value in evaluating AHR. Based on their respective receiver operating characteristic curves, the best separation of the two subject groups occurred at a d%FEV1 of 5 percent and 12 percent for the two EVH techniques and MIC, respectively. An individual's response to one test was highly correlated with the response to either of the other two (r = 0.66, p less than 0.001 for dry vs cold gas EVH; r = 0.56, p less than 0.001 for dry gas EVH vs methacholine; and r = 0.69, p less than 0.001 for cold gas EVH vs methacholine). Thus, MIC and EVH techniques are equally useful in defining AHR and each has its optimal threshold for a positive test result.

Exercise responses prior to pregnancy and in the postpartum state.

Increased women in the work force and requirements for maximal employee productivity have necessitated examination of the optimal time for parturients to resume normal activities. This prospective study was designed to determine whether prepregnancy measures of aerobic capacity are regained by 4-8 wk postpartum. Weight, percent body fat, recall energy expenditure, and exercise responses via a stage 1, graded cycle ergometer exercise test were determined in 11 subjects (mean age = 27.56 +/- 2.2) in a postabsorptive state prior to pregnancy and 4-8 wk postpartum. Subject characteristics were compared by the Student's t-test and differences across workloads and time by analysis of variance with repeated measures. Prepregnant weight (mean = 58.80 +/- 7.26 kg) was significantly less (P less than 0.05) than postpartum weight (mean = 62.81 +/- 9.12 kg), and prepregnant energy expenditure (1352 +/- 453 kJ) per day was significantly higher (P less than 0.05) than in the postpartum period (274 +/- 333 kJ). Maximal oxygen uptake was significantly higher (35.2 +/- 0.7 vs 30.5 +/- 2.0 ml.kg-1min-1) in the prepregnant as compared with the postpartum period. Further, heart rate at 125 and 150 W was significantly lower prepregnancy as compared with postpregnancy. Results support a detraining effect in the early postpartum period. Whether this detraining is an inevitable factor associated with pregnancy or whether exercising throughout pregnancy can ameliorate the decline in aerobic capacity postpartum is uncertain.