Modelling pulse wave propagation in the rabbit systemic circulation to assess the effects of altered nitric oxide synthesis

Pulse wave propagation in the mature rabbit systemic circulation was simulated using the one-dimensional equations of blood flow in compliant vessels. A corrosion cast of the rabbit circulation was manufactured to obtain arterial lengths and diameters. Pulse wave speeds and inflow and outflow boundary conditions were derived from in vivo data. Numerical results captured the main features of in vivo pressure and velocity pulse waveforms in the aorta, brachiocephalic artery and central ear artery. This model was used to elucidate haemodynamic mechanisms underlying changes in peripheral pulse waveforms observed in vivo after administering drugs that alter nitric oxide synthesis in the endothelial cells lining blood vessels. According to our model, these changes can be explained by single or combined alterations of blood viscosity, peripheral resistance and compliance, and the elasticity of conduit arteries.     

Surgical Technique for the Implantation of Tissue Engineered Vascular Grafts and Subsequent In Vivo Monitoring

The development of Tissue Engineered Vessels (TEVs) is advanced by the ability to routinely and effectively implant TEVs (4-5 mm in diameter) into a large animal model. A step by-step protocol for inter-positional placement of the TEV and real-time digital assessment of the TEV and native carotid arteries is described here. In vivo monitoring is made possible by the implantation of flow probes, catheters and ultrasonic crystals (capable of recording dynamic diameter changes of implanted TEVs and native carotid arteries) at the time of surgery. Once implanted, researchers can calculate arterial blood flow patterns, invasive blood pressure and artery diameter yielding parameters such as pulse wave velocity, augmentation index, pulse pressures and compliance. Data acquisition is accomplished using a single computer program for analysis throughout the duration of the experiment. Such invaluable data provides insight into TEV matrix remodeling, its resemblance to native/sham controls and overall TEV performance in vivo.     

血管内超声评价川崎病晚期患儿颈动脉功能、颈动脉内膜-中膜厚度和组织特征的临床研究

摘要 目的：探究血管内超声评价川崎病晚期患儿颈动脉功能、颈动脉内膜-中膜厚度和组织特征的临床价值。方法：选取81例川崎病晚期患儿作为本研究的川崎病组,同时选择同一时期的81例健康体检者作为本研究的对照组。所有受试者均通过血管内超声检查,对比分析两组一般临床资料、颈动脉功能、颈动脉内膜-中膜厚度、组织特征、颈动脉血流动力学和血管内皮功能。结果：川崎病组和对照组儿童的一般临床资料比较,差异均无统计学意义（P>0.05）。川崎病组患儿颈动脉血管僵硬度、脉搏波传导速度、颈动脉内膜-中膜厚度、血管的压力-应变弹性系数、硬度指数、颈动脉阻力指数和颈动脉僵硬度均显著高于对照组,而血管顺应性、血管径向应变率、动脉最大剪切率、颈动脉最小剪切率、颈动脉搏动指数和血管内皮依赖性舒张功能则显著小于对照组（P<0.05）。结论：血管内超声可有效评价川崎病晚期患儿颈动脉功能、颈动脉内膜-中膜厚度和组织特征,能够为后续疾病的治疗及预后提供重要参考,值得推广使用。     

Effects of Blood Pressure and Sex on the Change of Wave Reflection: Evidence from Gaussian Fitting Method for Radial Artery Pressure Waveform

An early return of the reflected component in the arterial pulse has been recognized as an important indicator of cardiovascular risk. This study aimed to determine the effects of blood pressure and sex factor on the change of wave reflection using Gaussian fitting method. One hundred and ninety subjects were enrolled. They were classified into four blood pressure categories based on the systolic blood pressures (i.e., ≤110, 111–120, 121–130 and ≥131 mmHg). Each blood pressure category was also stratified for sex factor. Electrocardiogram (ECG) and radial artery pressure waveforms (RAPW) signals were recorded for each subject. Ten consecutive pulse episodes from the RAPW signal were extracted and normalized. Each normalized pulse episode was fitted by three Gaussian functions. Both the peak position and peak height of the first and second Gaussian functions, as well as the peak position interval and peak height ratio, were used as the evaluation indices of wave reflection. Two-way ANOVA results showed that with the increased blood pressure, the peak position of the second Gaussian significantly shorten (P<0.01), the peak height of the first Gaussian significantly decreased (P<0.01) and the peak height of the second Gaussian significantly increased (P<0.01), inducing the significantly decreased peak position interval and significantly increased peak height ratio (both P<0.01). Sex factor had no significant effect on all evaluation indices (all P>0.05). Moreover, the interaction between sex and blood pressure factors also had no significant effect on all evaluation indices (all P>0.05). These results showed that blood pressure has significant effect on the change of wave reflection when using the recently developed Gaussian fitting method, whereas sex has no significant effect. The results also suggested that the Gaussian fitting method could be used as a new approach for assessing the arterial wave reflection.     

Wave Propagation in a Viscous Fluid Contained in an Orthotropic Elastic Tube

The problem of pressure wave propagation through a viscous fluid contained in an orthotropic elastic tube is considered in connection with arterial blood flow. Solutions to the fluid flow and elasticity equations are obtained for the presence of a reflected wave. Numerical results are presented for both isotropic and orthotropic elastic tubes. In particular, the pressure pulse, flow rate, axial fluid velocity, and wall displacements are plotted vs. time at various stations along the ascending aorta of man. The results indicate an increase in the peak value of the pressure pulse and a decrease in the flow rate as the pulse propagates away from the heart. Finally, the velocity of wave propagation depends mainly on the tangential modulus of elasticity of the arterial wall, and anisotropy of the wall accounts in part for the reduction of longitudinal movements and an increase in the hydraulic resistance.     

Pulse Wave Velocity Testing in the Baltimore Longitudinal Study of Aging

Carotid-femoral pulse wave velocity is considered the gold standard for measurements of central arterial stiffness obtained through noninvasive methods¹. Subjects are placed in the supine position and allowed to rest quietly for at least 10 min prior to the start of the exam. The proper cuff size is selected and a blood pressure is obtained using an oscillometric device. Once a resting blood pressure has been obtained, pressure waveforms are acquired from the right femoral and right common carotid arteries. The system then automatically calculates the pulse transit time between these two sites (using the carotid artery as a surrogate for the descending aorta). Body surface measurements are used to determine the distance traveled by the pulse wave between the two sampling sites. This distance is then divided by the pulse transit time resulting in the pulse wave velocity. The measurements are performed in triplicate and the average is used for analysis.     

Digital Photoplethysmography for Assessment of Arterial Stiffness: Repeatability and Comparison with Applanation Tonometry

Introduction

Arterial stiffness is an independent risk factor for cardiovascular morbidity and can be assessed by applanation tonometry by measuring pulse wave velocity (PWV) and augmentation index (AIX) by pressure pulse wave analysis (PWA). As an inexpensive and operator independent alternative, photoelectric plethysmography (PPG) has been introduced with analysis of the digital volume pulse wave (DPA) and its second derivatives of wave reflections.

Objective

The objective was to investigate the repeatability of arterial stiffness parameters measured by digital pulse wave analysis (DPA) and the associations to applanation tonometry parameters.

Methods and Results

112 pregnant and non-pregnant individuals of different ages and genders were examined with SphygmoCor arterial wall tonometry and Meridian DPA finger photoplethysmography. Coefficients of repeatability, Bland-Altman plots, intraclass correlation coefficients and correlations to heart rate (HR) and body height were calculated for DPA variables, and the DPA variables were compared to tonometry variables left ventricular ejection time (LVET), PWV and AIX. No DPA variable showed any systematic measurement error or excellent repeatability, but dicrotic index (DI), dicrotic dilatation index (DDI), cardiac ejection elasticity index (EEI), aging index (AI) and second derivatives of the crude pulse wave curve, b/a and e/a, showed good repeatability. Overall, the correlations to AIX were better than to PWV, with correlations coefficients >0.70 for EEI, AI and b/a. Considering the level of repeatability and the correlations to tonometry, the overall best DPA parameters were EEI, AI and b/a. The two pansystolic time parameters, ejection time compensated (ETc) by DPA and LVET by tonometry, showed a significant but weak correlation.

Conclusion

For estimation of the LV function, ETc, EEI and b/a are suitable, for large artery stiffness EEI, and for small arteries DI and DDI. The only global parameter, AI, showed a high repeatability and the overall best correlations with AIX and PWV.     

Spectral and Coherent Analysis of the Relationship between Pulse Wave Forms in the Carotid Artery and Heart Rate

The purpose of this study was the assessment of the parameters of pulse waves in the carotid artery and their relationships with baroreflex-controlled heart rate variability. The early systolic peak (ESP) is closely connected with the filling of the carotid artery, whereas the late systolic peak (LSP) shows the amplitude of the wave reflected from the periphery and affects arterial pressure oscillations activating baroreceptors and changing the heart rate (HR). A cardiac rhythmogram and pulse pressure waves in the carotid artery were continuously recorded for 5 min. ESP and LSP amplitudes (in ml/l) were evaluated in each wave. Patients with coronary artery disease and healthy subjects were examined. Spectrum analysis and assessment of the coherent function between RR intervals, the ESP, and the LSP in the middle frequency band were performed. In healthy subjects, the amplitude of the ESP was higher than that of the LSP, whereas, in patients, the LSP amplitude surpassed that of the ESP. The LSP/ESP ratio was significantly higher in patients, indicating that artery dilation depended more on the reflected wave than on the primary wave. The study revealed that age had the most pronounced effect on the LSP/ESP ratio. Healthy subjects showed the highest age-dependent coherence between the RR interval and the ESP amplitude, whereas in patients this coherence was significantly lower. Measurement of the ESP and LSP amplitudes and their ratio allowed evaluation of vasomotor artery tone. Age and disease produced the maximum effects on the artery tone and decreased the correlation between the changes in the artery tone and the HR in the baroreflex range of frequency regulation.     

Parameters describing the pulse wave

Pulse wave analysis permits non-invasive assessment of arterial elasticity indices. The contour varies in different parts of the circulation. It depends on physiological or pathophysiological conditions of the organism. The pathological events like arteriosclerosis or diabetes have a primary effect to the artery elasticity. Hypertension or some heart diseases also influence the pulse wave velocity and resulted in earlier wave reflections. There are several methods of pulse wave measurements based on different principles and depending on the type of measured pulse wave. The evaluation parameters can be assessed from the time domain, derivations, velocity or frequency domain. The main aim of this review article is to offer a recent overview of pulse wave measurement parameters and main results obtained. The principles of pulse wave measurement and current experience in clinical practice are shortly discussed too.     

The influence of vessel wall elasticity and peripheral resistance on the carotid artery flow wave form: a CFD model compared to in vivo ultrasound measurements

The Doppler flow wave form and its derived measures such as the pulsatility index provide clinically important tools for the investigation of arterial disease. The typical shape of Doppler flow wave forms is physiologically known to be largely determined by both peripheral resistance and elastic properties of the arterial wall. In the present study we systematically investigate the influence of both vessel wall elasticity and peripheral resistance on the flow wave form obtained from a CFD-simulation of blood flow in the carotid bifurcation. Numerical results are compared to in vivo ultrasound measurements. The in vivo measurement provides a realistic geometry, local elasticities and an input flow wave form for the numerical experiment. Numerical and experimental results are compared at three different sites in the carotid branches. Peripheral resistance has a profoundly decreasing effect on velocities in the external carotid artery. If elasticity is taken into account, the computed peak systolic velocities are considerably lower and a more realistic smoothing of the flow wave form is found. Together, the results indicate that only if both vessel wall elasticity and positive peripheral resistance are taken into account, experimentally obtained Doppler flow wave forms can be reproduced numerically.     

Racial differences in central blood pressure and vascular function in young men

Young African-American men have altered macrovascular and microvascular function. In this cross-sectional study, we tested the hypothesis that vascular dysfunction in young African-American men would contribute to greater central blood pressure (BP) compared with young white men. Fifty-five young (23 yr), healthy men (25 African-American and 30 white) underwent measures of vascular structure and function, including carotid artery intima-media thickness (IMT) and carotid artery beta-stiffness via ultrasonography, aortic pulse wave velocity, aortic augmentation index (AIx), and wave reflection travel time (Tr) via radial artery tonometery and a generalized transfer function, and microvascular vasodilatory capacity of forearm resistance arteries with strain-gauge plethysmography. African-American men had similar brachial systolic BP (SBP) but greater aortic SBP (P<0.05) and carotid SBP (P<0.05). African-American men also had greater carotid IMT, greater carotid beta-stiffness, greater aortic stiffness and AIx, reduced aortic Tr and reduced peak hyperemic, and total hyperemic forearm blood flow compared with white men (P<0.05). In conclusion, young African-American men have greater central BP, despite comparable brachial BP, compared with young white men. Diffuse macrovascular and microvascular dysfunction manifesting as carotid hypertrophy, increased stiffness of central elastic arteries, heightened resistance artery constriction/blunted resistance artery dilation, and greater arterial wave reflection are present at a young age in apparently healthy African-American men, and conventional brachial BP measurement does not reflect this vascular burden.     

Noninvasive pressure pulse waveform analysis of flow-mediated vasodilation evoked by post-occlusive reactive hyperemia maneuver

Post-occlusive reactive hyperemia (PORH) assesses flow-mediated vasodilation at microvascular level due to bioactivity of endothelial-derived factors. Ordinary augmentation index that quantifies endothelial response is based on an ensemble-averaged waveform that limits its short-time application. This study proposes a mathematical model and two corresponding indices to evaluate arterial pressure response after blood flow restoration. Radial pressure pulse waveforms were acquired by a 12 bits acquisition board at a sampling rate of 1.0 kHz using a piezoelectric transducer. Signals were stored during 30 s at baseline condition and 60 s after 5-min occlusion using an arm-cuff placed over the brachial artery. In both conditions, the pressure pulse waveform presents systolic and diastolic phases with progressive and regressive pulse waveforms, respectively. Changes in pulse wave morphology were also observed and comprised attenuation of the pulse pressure amplitude (markedly first and second systolic peaks). This characteristic of the pulse pressure was described by the time-domain summation of two pairs of Gaussian-like waveforms (representing independent progressive and regressive components) with parameters related to amplitude, time lag, and duration for each component. A steepest descent optimization routine was used to fit the model parameters to experimental data of normotensive and subjects with hypertension. The optimized parameters were used to calculate two indices, RIx_1,2 (second-to-first systolic peak ratio) and RIx_1,3 (first diastolic-to-first systolic ratio). The observed responses between groups suggest that RIx_1,2 is related to an endothelial response to the ischemic process and could be used as a clinical tool to assess endothelial function in hypertension.     

A Non-invasive Method for Determining Biomechanical Properties of the Internal Carotid Artery

ObjectivesMechanical properties of the carotid artery play an important role in the progression of arterial disease such as atherosclerosis. An early change in the mechanical properties of the arteries can be introduced as a novel risk factor for cardiovascular events. The aim of this study is to estimate, in vivo, the elastic biomechanical properties of the internal carotid wall (ICA), from the noninvasive determination of the local arterial wave speed (c).Material and methodsTo achieve this objective, c was determined from a mathematical and physical model developed in our previous work that uses the measurement of the instantaneous blood velocity at two sites by contrast magnetic resonance (PCMR), the study having been conducted on 20 healthy, young and old subjects. The determination of Young's modulus (E) requiring the measurement of the arterial radius (R) and the wall thickness (h), we first estimated the arterial compliance (C). Then from a segmentation of the PCMR image, we evaluated R and thus the elastance given by the product Eh. Finally, in front of the difficulty of measuring h, E was estimated from a statistical study on h.ResultsOur method is sensitive to a variation of the parietal elasticity as it is the case with the age. A statistical test showed that there is a very significant difference between younger and older subjects in terms of speed wave, elastance, compliance, and Young's modulus (p<0.001). Furthermore, these results, in agreement with the reference values reported in the literature, are very promising for detecting a pathological change in parietal elasticity, as is the case in atherosclerosis.ConclusionThus, the in vivo application of this technique shows its potential for clinical evaluation of arterial stiffness ICA as it is fully quantitative, non-invasive and can be performed in real time.     

In vivo repeatability of the pulse wave inverse problem in human carotid arteries

Accurate arterial stiffness measurement would improve diagnosis and monitoring for many diseases. Atherosclerotic plaques and aneurysms are expected to involve focal changes in vessel wall properties; therefore, a method to image the stiffness variation would be a valuable clinical tool. The pulse wave inverse problem (PWIP) fits unknown parameters from a computational model of arterial pulse wave propagation to ultrasound-based measurements of vessel wall displacements by minimizing the difference between the model and measured displacements. The PWIP has been validated in phantoms, and this study presents the first in vivo demonstration. The common carotid arteries of five healthy volunteers were imaged five times in a single session with repositioning of the probe and subject between each scan. The 1D finite difference computational model used in the PWIP spanned from the start of the transducer to the carotid bifurcation, where a resistance outlet boundary condition was applied to approximately model the downstream reflection of the pulse wave. Unknown parameters that were estimated by the PWIP included a 10-segment linear piecewise compliance distribution and 16 discrete cosine transformation coefficients for each of the inlet boundary conditions. Input data was selected to include pulse waves resulting from the primary pulse and dicrotic notch. The recovered compliance maps indicate that the compliance increases close to the bifurcation, and the variability of the average pulse wave velocity estimated through the PWIP is on the order of 11%, which is similar to that of the conventional processing technique which tracks the wavefront arrival time (13%).     

Surface artery volume pulse wave measurement - verification of a new method

The aim of this paper is to prove the possible reproducibility of measurement with a new developed device for artery elasticity monitoring and determining the standard of major pulse wave parameters. As a measurement sensor, a conic probe with thin convex membrane was used. This technique allows setting an arbitrary pressure to a measured surface artery. We measured pulse waves on the radial arteries of 108 individuals. We expected similar features in arterial wall elasticity. We concentrated primarily on the amount of subcutaneous fat. For the measured waves we evaluated five following pulse wave parameters: relative crest time, elasticity index, dicrotic wave attenuation, dicrotic wave time and interwave distance. There were no significant differences in measured pulse wave parameters among the tested groups of subjects.     

Determination of delay time in individual transfer function for central aortic pressure reconstruction

In previous research,time-delay(Δt)was a more important parameter than the reflection coefficient in the individual transfer function of central aortic pressure reconstruction.TheΔt can be obtained by electrocardiography(ECG)or phonocardiography(PCG).Because the pre-ejection period remains an uncertain factor,the present study used ECG and PCG to define the delay time and analyzed the accuracy of the reconstruction results.TheΔtpre is the actual delay time derived from the aorta to the carotid pressure wave,ΔtPCG is the time delay between the aortic valve component of the second heart sound and the dicrotic incisura of the carotid pressure wave,andΔtECG represents the delay from the interval of the ECG R-peak to the foot of the carotid pressure wave.Compared with the measured aortic pressure,the reconstruction result obtained byΔt=ΔtPCG slightly differed from the best result estimated byΔt=Δtpre.However,the differences between the result obtained byΔt=ΔtECG and the best result were significant in terms of the diastolic blood pressure,and pulse pressure,and especially in terms of the augmentation index and root-mean-square-error.Thus,theΔt should be determined by PCG for central aortic pressure reconstruction in practice.     

Pulmonary vascular impedance and wave reflections in the hypoxic calf.

The alterations in pulsatile hemodynamics that occur during hypoxic pulmonary vasoconstriction have not been well characterized. Changes in oscillatory hemodynamics, however, may affect right ventricular-pulmonary vascular coupling and the dissipation of energy within the lung vasculature. To better define hypoxic pulsatile hemodynamics, we measured main pulmonary artery proximal and distal micromanometric pressures and ultrasonic flow in four open-chest calves during progressive hypoxia. Main pulmonary artery impedance and pressure transmission spectra were calculated using spectral analysis methods. Measured pressure and flow signals were separated in the time domain into forward and backward components. Hypoxia increased pulmonary blood pressure and resistance and produced multiple modifications in the impedance and pressure transmission spectra that indicated increased wave reflections and elasticity. The impedance and apparent phase velocity first-harmonic values were increased in amplitude, and the pressure transmission modulus plot showed an increased peak value. In addition, the impedance modulus plot demonstrated a rightward shift and increased oscillation in the mid- to high-frequency range. The time domain analysis also confirmed increased wave reflections and elasticity. Hypoxia produced large backward-traveling (reflected) pressure and flow waves. The initial portions of these waves arrived at the heart during systole, producing characteristic changes in the measured pressure and flow waveforms. With prolonged hypoxia, main pulmonary artery pulse wave velocity increased by 30%. Thus, hypoxia is associated with complex alterations in pulmonary artery elasticity and wave reflections that act to increase the oscillatory afterload of the right ventricle.     

Relationship of Adiposity to Subclinical Atherosclerosis in Obese Patients with Type 2 Diabetes

Objective: There is an increased prevalence of macrovascular disease in type 2 diabetes. The pathogenesis has been related to metabolic risk factors, insulin resistance, and obesity. One of the strongest predictors is the presence of subclinical atherosclerosis. This study was designed to examine the relationship between obesity and regional patterns of adiposity, insulin resistance, and five independent measures of subclinical atherosclerosis. Research Methods and Procedures: Fifty‐two overweight and obese men and women with type 2 diabetes of relatively short known duration were examined. Measures of subclinical vascular disease were assessment of arterial stiffness by pulse wave velocity, ultrasound measurement of the carotid artery intimal‐medial thickness and plaque index, and measurement of the extent of coronary and aortic calcification using electron beam computed tomography. Insulin resistance was measured using the hyperinsulinemic euglycemic clamp. Body composition was measured using DXA and computed tomography. Results: Adiposity was a strong determinant of pulse wave velocity. Carotid intimal‐medial thickness was correlated with age, low‐density lipoprotein‐cholesterol, and hyperglycemia, but not with adiposity. Hyperglycemia and plasma activator inhibitor‐1 were significant correlates of the carotid artery plaque index. Coronary calcium scores were significantly correlated with age and interleukin‐6 and significantly and negatively correlated to insulin sensitivity index. Discussion: These findings suggest that obesity may play an important role in the early phase of subclinical macrovascular disease related to vessel stiffness, whereas hyperglycemia and insulin resistance in conjunction with other risk factors have important roles in progression from vessel stiffness to atheroma formation in type 2 diabetes.     

Limitations of Augmentation Index in the Assessment of Wave Reflection in Normotensive Healthy Individuals

Objectives

Augmentation index (AIx) is widely used as a measure of wave reflection. We compared the relationship between AIx and age, height and sex with ‘gold standard’ measures of wave reflection derived from measurements of pressure and flow to establish how well AIx measures wave reflection.

Materials and Methods

Measurements of carotid pressure and flow velocity were made in the carotid artery of 65 healthy normotensive individuals (age 21–78 yr; 43 male) and pulse wave analysis, wave intensity analysis and wave separation was performed; waveforms were classified into type A, B or C. AIx, the time of the first shoulder (T_s), wave reflection index (WRI) and the ratio of backward to forward pressure (P_b/P_f) were calculated.

Results

AIx did not correlate with log WRI or P_b/P_f. When AIx was restricted to positive values AIx and log WRI were positively correlated (r = 0.33; p = 0.04). In contrast log WRI and P_b/P_f were closely correlated (r = 0.66; p<0.001). There was no correlation between the T_s and the timing of P_b or the reflected wave identified by wave intensity analysis. Wave intensity analysis showed that the morphology of type C waveforms (negative AIx) was principally due to a forward travelling (re-reflected) decompression wave in mid-systole. AIx correlated positively with age, inversely with height and was higher in women. In contrast log WRI and P_b/P_f showed negative associations with age, were unrelated to height and did not differ significantly by gender.

Conclusions

AIx has serious limitations as a measure of wave reflection. Negative AIx values derived from Type C waves should not be used as estimates of wave reflection magnitude.     

Arterial wall remodeling under sustained axial twisting in rats

Blood vessels often experience torsion along their axes and it is essential to understand their biological responses and wall remodeling under torsion. To this end, a rat model was developed to investigate the arterial wall remodeling under sustained axial twisting in vivo. Rat carotid arteries were twisted at 180° along the longitudinal axis through a surgical procedure and maintained for different durations up to 4 weeks. The wall remodeling in these twisted arteries was examined using histology, immunohistochemistry and fluorescent microscopy. Our data showed that arteries remodeled under twisting in a time-dependent manner during the 4 weeks post-surgery. Cell proliferation, MMP-2 and MMP-9 expressions, medial wall thickness and lumen diameter increased while collagen to elastin ratio decreased. The size and number of internal elastic lamina fenestrae increased with elongated shapes, while the endothelial cells elongated and aligned towards the blood flow direction gradually. These results demonstrated that sustained axial twisting results in artery remodeling in vivo. The rat carotid artery twisting model is an effective in vivo model for studying arterial wall remodeling under long-term torsion. These results enrich our understanding of vascular biology and arterial wall remodeling under mechanical stresses.