On the sensitivity of wall stresses in diseased arteries to variable material properties

Accurate estimates of stress in an atherosclerotic lesion require knowledge of the material properties of its components (e.g., normal wall, fibrous plaque, calcified regions, lipid pools) that can only be approximated. This leads to considerable uncertainty in these computational predictions. A study was conducted to test the sensitivity of predicted levels of stress and strain to the parameter values of plaque used in finite element analysis. Results show that the stresses within the arterial wall, fibrous plaque, calcified plaque, and lipid pool have low sensitivities for variation in the elastic modulus. Even a +/- 50% variation in elastic modulus leads to less than a 10% change in stress at the site of rupture. Sensitivity to variations in elastic modulus is comparable between isotropic nonlinear, isotropic nonlinear with residual strains, and transversely isotropic linear models. Therefore, stress analysis may be used with confidence that uncertainty in the material properties generates relatively small errors in the prediction of wall stresses. Either isotropic nonlinear or anisotropic linear models provide useful estimates, however the predictions in regions of stress concentration (e.g., the site of rupture) are somewhat more sensitive to the specific model used, increasing by up to 30% from the isotropic nonlinear to orthotropic model in the present example. Changes resulting from the introduction of residual stresses are much smaller.     

The effects of plaque morphology and material properties on peak cap stress in human coronary arteries

Heart attacks are often caused by rupture of caps of atherosclerotic plaques in coronary arteries. Cap rupture occurs when cap stress exceeds cap strength. We investigated the effects of plaque morphology and material properties on cap stress. Histological data from 77 coronary lesions were obtained and segmented. In these patient-specific cross sections, peak cap stresses were computed by using finite element analyses. The finite element analyses were 2D, assumed isotropic material behavior, and ignored residual stresses. To represent the wide spread in material properties, we applied soft and stiff material models for the intima. Measures of geometric plaque features for all lesions were determined and their relations to peak cap stress were examined using regression analyses. Patient-specific geometrical plaque features greatly influence peak cap stresses. Especially, local irregularities in lumen and necrotic core shape as well as a thin intima layer near the shoulder of the plaque induce local stress maxima. For stiff models, cap stress increased with decreasing cap thickness and increasing lumen radius (R = 0.79). For soft models, this relationship changed: increasing lumen radius and increasing lumen curvature were associated with increased cap stress (R = 0.66). The results of this study imply that not only accurate assessment of plaque geometry, but also of intima properties is essential for cap stress analyses in atherosclerotic plaques in human coronary arteries.     

Optical determination of anisotropic material properties of bovine articular cartilage in compression

The precise nature of the material symmetry of articular cartilage in compression remains to be elucidated. The primary objective of this study was to determine the equilibrium compressive Young's moduli and Poisson's ratios of bovine cartilage along multiple directions (parallel and perpendicular to the split line direction, and normal to the articular surface) by loading small cubic specimens (0.9 x 0.9 x 0.8 mm, n =15) in unconfined compression, with the expectation that the material symmetry of cartilage could be determined more accurately with the help of a more complete set of material properties. The second objective was to investigate how the tension-compression nonlinearity of cartilage might alter the interpretation of material symmetry. Optimized digital image correlation was used to accurately determine the resultant strain fields within the specimens under loading. Experimental results demonstrated that neither the Young's moduli nor the Poisson's ratios exhibit the same values when measured along the three loading directions. The main findings of this study are that the framework of linear orthotropic elasticity (as well as higher symmetries of linear elasticity) is not suitable to describe the equilibrium response of articular cartilage nor characterize its material symmetry; a framework which accounts for the distinctly different responses of cartilage in tension and compression is more suitable for describing the equilibrium response of cartilage; within this framework, cartilage exhibits no lower than orthotropic symmetry.     

A model for the nonlinear elastic response of large arteries

The nonlinear elastic response of large arteries subjected to finite deformations due to action of biaxial principal stresses, is described by simple constitutive equations. Generalized measures of strain and stress are introduced to account for material nonlinearity. This also ensures the existence of a strain energy density function. The orthotropic elastic response is described via quasi-linear relations between strains and stresses. One nonlinear parameter which defines the measures of strain and stress, and three elastic moduli are assumed to be constants. The lateral strain parameters (equivalent to Poisson's ratios in infinitesimal deformations) are deformation dependent. This dependence is defined by empirical relations developed via the incompressibility condition, and by the introduction of a fifth material parameter. The resulting constitutive model compares well with biaxial experimental data of canine carotid arteries.     

Identification of Bacillus subtilis genes for septum placement and shape determination.

The Bacillus subtilis divIVB1 mutation causes aberrant positioning of the septum during cell division, resulting in the formation of small, anucleate cells known as minicells. We report the cloning of the wild-type allele of divIVB1 and show that the mutation lies within a stretch of DNA containing two open reading frames whose predicted products are in part homologous to the products of the Escherichia coli minicell genes minC and minD. Just upstream of minC and minD, and in the same orientation, are three genes whose products are homologous to the products of the E. coli shape-determining genes mreB, mreC, and mreD. The B. subtilis mreB, mreC, and mreD genes are the site of a conditional mutation (rodB1) that causes the production of aberrantly shaped cells under restrictive conditions. Northern (RNA) hybridization experiments and disruption experiments based on the use of integrational plasmids indicate that the mre and min genes constitute a five-cistron operon. The possible involvement of min gene products in the switch from medial to polar placement of the septum during sporulation is discussed.     

Identification of heterogeneous elastic properties in stenosed arteries: a numerical plane strain study

Assessing the vulnerability of atherosclerotic plaques requires an accurate knowledge of the mechanical properties of the plaque constituents. It is possible to measure displacements in vivo inside a plaque using magnetic resonance imaging. An important issue is to solve the inverse problem that consists in estimating the elastic properties inside the plaque from measured displacements. This study focuses on the identifiability of elastic parameters, e.g. on the compromise between identification time and identification accuracy. An idealised plane strain finite element (FE) model is used. The effects of the FE mesh of the a priori assumptions about the constituents, of the measurement resolution and of the data noise are numerically investigated.     

Effects of collagen deposition on passive and active mechanical properties of large pulmonary arteries in hypoxic pulmonary hypertension

Proximal pulmonary artery (PA) stiffening is a strong predictor of mortality in pulmonary hypertension. Collagen accumulation is mainly responsible for PA stiffening in hypoxia-induced pulmonary hypertension (HPH) in mouse models. We hypothesized that collagen cross-linking and the type I isoform are the main determinants of large PA mechanical changes during HPH, which we tested by exposing mice that resist type I collagen degradation (Col1a1 $^\mathrm{R/R})$ and littermate controls (Col1a1 $^{+/+})$ to hypoxia for 10 days with or without $\beta $ -aminopropionitrile (BAPN) treatment to prevent cross-link formation. Static and dynamic mechanical tests were performed on isolated PAs with smooth muscle cells (SMC) in passive and active states. Percentages of type I and III collagen were quantified by histology; total collagen content and cross-linking were measured biochemically. In the SMC passive state, for both genotypes, hypoxia tended to increase PA stiffness and damping capacity, and BAPN treatment limited these increases. These changes were correlated with collagen cross-linking ( $p<0.05$ ). In the SMC active state, hypoxia increased PA dynamic stiffness and BAPN had no effect in Col1a1 $^{+/+}$ mice ( $p<0.05$ ). PA stiffness did not change in Col1a1 $^\mathrm{R/R}$ mice. Similarly, damping capacity did not change for either genotype. Type I collagen accumulated more in Col1a1 $^{+/+}$ mice, whereas type III collagen increased more in Col1a1 $^\mathrm{R/R}$ mice during HPH. In summary, PA passive mechanical properties (both static and dynamic) are related to collagen cross-linking. Type I collagen turnover is critical to large PA remodeling during HPH when collagen metabolism is not mutated and type III collagen may serve as a reserve.     

Identification and characterisation of regional variations in the material properties of ureter according to microstructure

There are few previous studies on the elastic properties of ureter and most have been limited to essentially one-dimensional deformation measurements. The object of this study was, therefore, to identify regional variations in the multiaxial behaviour of rabbit ureter, subjected to in vitro inflation/extension testing under a physiological range of intraluminal pressures and longitudinal forces. A microstructure-motivated material model (via two- and four-fibre families in turn for elastin and collagen) was implemented and its capacity to mathematically characterise the experimental data contrasted favourably with that of the well-established phenomenological models, but it was compromised by parameter covariance. Extensive optimisation studies confirmed that the reduced model without contribution to the elastin and circumferential-fibre (collagen) families characterised the data equally well without over-parameterisation. In view of the fitted parameters, the ureteral tissue was stiffer longitudinally, justified by the preferential alignment of collagen along that axis and the lower ureter was stiffer than the upper ureter, justified by the histological observation of a thickest lamina propria, i.e. of highest collagen content, there. The lower ureter was less anisotropic than the upper ureter, possessing a comparatively larger amount of diagonally arranged collagen fibres in tunica mucosa, while having the usual amounts of longitudinally arranged fibres in tunica adventitia and of circumferentially arranged fibres in tunica muscularis. The present data may be used as inputs to mathematical models of the ureter, assessing regional and intramural stress distributions, through which it is hoped that an improved appreciation of ureteral function may be attained in both health and disease.     

Measuring the microelastic properties of biological material.

We have used the atomic force microscope (AFM) to measure the local rigidity modulus at points on the surface of a section of hydrated cow tibia. These data are obtained either from contrast changes that occur as the contact force is altered, or from force versus distance curves obtained at fixed points. These two methods yield the same values for rigidity modulus (at a given point). At low resolution, the elastic morphology and topography mirror the features seen in optical and electron micrographs. At high resolution we see dramatic variations in elastic properties across distances as small as 50 nm.     

The fine structure of endothelium of large arteries

Endothelium of large arteries from several species was studied in thin sections with the electron microscope. Before sacrifice, some animals received an intravenous injection of colloidal thorium dioxide which was visualized in the sections. Surface replicas were prepared by carbon evaporation on either frozen-dried endothelium or on endothelium dried by sublimation of naphthalene with which the tissue had been impregnated. Cell boundaries, stained with silver, were observed in sections and also from the surface by stripping off the inner part of the endothelium. In addition to the usual cytoplasmic organelles, the endothelial cells showed certain characteristic features, namely, large invaginated pockets communicating with the arterial lumen, numerous much smaller vesicular structures immediately under the plasma membrane and apparently also communicating with the lumen, and inclusions, into which injected thorium particles were incorporated. Intercellular boundaries appeared as regular double membranes in thin sections, and they were outlined by a double row of silver granules after silver staining. No evidence was obtained of permeation of intracellular spaces by colloidal thorium.     

Spectrophotometric method for the determination of microquantities of lactic acid in biological material.

A simple spectrophotometric method for the determination of microquantities of lactic acid is described. Lactic acid was quantitatively oxidized to acetaldehyde by cerium(IV) under prescribed conditions. Oxidation was terminated by reducing cerium(IV) to cerium(III) with nitrite. The acetaldehyde was reacted with copper ions and oxalydihydrazide under alkaline conditions to form an intense blue complex which is measured spectrophotometrically. The developed citrate-phosphate-borate-sodium hydroxide alkalizing reagent enhanced color stability by removing cerium from solution and also ensured extremely low blank values. Beer's law was followed over the range 0 to 150 micrograms of lactate. The method was successfully applied to the routine determination of lactate in silage and blood.