Anne M. Robertson

A theoretical and non-destructive experimental approach for direct inclusion of measured collagen orientation and recruitment into mechanical models of the artery wall

Gradual collagen recruitment has been hypothesized as the underlying mechanism for the mechanical stiffening with increasing stress in arteries. In this work, we investigated this hypothesis in eight rabbit carotid arteries by directly measuring the distribution of collagen recruitment stretch under increasing circumferential loading using a custom uniaxial (UA) extension device combined with a multi-photon microscope (MPM). This approach allowed simultaneous mechanical testing and imaging of collagen fibers without traditional destructive fixation methods. Fiber recruitment was quantified from 3D rendered MPM images, and fiber orientation was measured in projected stacks of images. Collagen recruitment was observed to initiate at a finite strain, corresponding to a sharp increase in the measured mechanical stiffness, confirming the previous hypothesis and motivating the development of a new constitutive model to capture this response. Previous constitutive equations for the arterial wall have modeled the collagen contribution with either abrupt recruitment at zero strain, abrupt recruitment at finite strain or as gradual recruitment beginning at infinitesimal strain. Based on our experimental data, a new combined constitutive model was presented in which fiber recruitment begins at a finite strain with activation stretch represented by a probability distribution function. By directly including this recruitment data, the collagen contribution was modeled using a simple Neo-Hookean equation. As a result, only two phenomenological material constants were required from the fit to the stress stretch data. Three other models for the arterial wall were then compared with these results. The approach taken here was successful in combining stress-strain analysis with simultaneous microstructural imaging of collagen recruitment and orientation, providing a new approach by which underlying fiber architecture may be quantified and included in constitutive equations.

Flow of Oldroyd-B fluids in curved pipes of circular and annular cross-section

Perturbation methods are used to study steady, fully developed flow of Oldroyd-B fluids through curved pipes for both pipes of circular and annular cross-section. The perturbation parameter is the curvature ratio, given by the cross-sectional radius of the pipe divided by the constant radius of the pipe centerline. We compare results for creeping and non-creeping flows for both viscoelastic and Newtonian fluids. In pipes of circular cross-section, the velocity field for creeping flows of Oldroyd-B fluids is qualitatively similar to that found for non-creeping flows of Newtonian fluids. Namely, in addition to the primary flow, there is a secondary motion consisting of counter-rotating vortices. In curved annular pipes, two pairs of counter-rotating vortices are generated by either inertial or elastic effects. In this geometry, the differences between creeping flow of viscoelastic fluids and non-creeping flow of Newtonian fluids are dramatically accentuated at small values of inner to outer pipe radius, riro. For Newtonian fluids, as riro approaches zero, the magnitude and size of the vortices adjacent to the inner cylinder shrink to zero. However, for creeping flow of Oldroyd-B fluids, the inner vortex pair is comparable to the outer vortex pair in both size and strength, even for values of riro as small as 0.01. For pipes of circular cross-section, the effect of elasticity on the drag is considered and earlier predictions by Bowen et al. for the upper convected Maxwell fluid are extended to the Oldroyd-B fluid for non-zero Reynolds number.

Flow of second order fluids in curved pipes

Abstract Beginning with the work of Dean in 1927, regular perturbation methods have been used to study flows of incompressible Newtonian, generalized Newtonian and viscoelastic fluids in curved pipes of circular cross section. In these studies, the perturbation parameter is the curvature ratio: the cross sectional radius of the pipe divided by the radius of curvature of the pipe centerline. Closed form perturbation solutions for the equations of motion for second order fluids have previously been obtained by other authors for the special case when the second normal stress coefficient is zero, H.G. Sharma, A. Prakash, Ind. J. Pure Appl. Math 8 (1977) 546–557; P.J. Bowen, Ph.D. Thesis, University of Wales, (1990); P.J. Bowen, A.R. Davies, K. Walters, J. Non-Newtonian Fluid Mech. 38 (1991) 113–126. Here, we obtain closed form solutions for the perturbation equations even when the second normal stress difference is non-zero. We show that for a countable number of combinations of non-dimensional parameters a perturbation solution exists but is not unique. For other combinations of parameters a perturbation solution does not even exist. This latter result implies that for these parameter values there does not exist a steady, fully developed solution for flow in curved pipes which is a perturbation of the straight pipe solution, regardless of the magnitude of the curvature ratio. We emphasize that these singular points do not arise when the second normal stress coefficient is zero. A solution to the perturbation equations exists and is unique for values of the material constants which correspond to real polymeric fluids. For these values of the material constants, the secondary motion at zero Reynolds number is qualitatively similar to that arising in Newtonian fluids due to inertial effects.

The numerical design of a parallel plate flow chamber for investigation of endothelial cell response to shear stress

Parallel plate chambers are frequently used to examine the response of biological cells to a constant wall shear stress. However, the stress can vary more than 80% across the chamber due to end effects. Earlier estimates of the magnitude of this inhomogeneity used boundary layer theory and experiments. Here, the full equations for steady, three-dimensional flow in a novel parallel plate device were solved numerically and used to identify an active test region where the shear stress is within 5% of a constant value. Endothelial cells can be confined to this region to assure a nearly uniform shear stress exposure.

Rheological models for blood

Rheology is the science of the deformation and flow of materials. It deals with the theoretical concepts of kinematics, conservation laws and constitutive relations, describing the interrelation between force, deformation and flow. The experimental determination of the rheological behaviour of materials is called rheometry. The object of haemorheology is the application of rheology to the study of flow properties of blood and its formed elements, and the coupling of blood and the blood vessels in living organisms. This field involves the investigation of the macroscopic behaviour of blood determined in rheometric experiments, its microscopic properties in vitro and in vivo and studies of the interactions among blood cellular components and between these components and the endothelial cells that line blood vessels.

AN AUTOMATED HIGHWAY SYSTEM LINK LAYER CONTROLLER FOR TRAFFIC FLOW STABILIZATION

Controls for the link layer in the Automated Highway System (AHS) hierarchy proposed in the Partners for Advanced Transit and Highways (PATH) are developed. The link layer is modeled using vehicle conservation flow models. Desired traffic conditions on the highway are given by a pair of density and velocity profiles which are assumed to be consistent with the demand on, and the capability of, the highway system. The link layer control laws presented in this paper then stabilize the actual traffic condition to the desired values. Control laws are derived for three highway topologies: a single lane highway, a highway with multiple discrete lanes and a two-dimensional highway with an arbitrary flow pattern. The control laws obtained for each of the topologies is distributed and are suited for implementation in the lower levels of the AHS control hierarchy. Simulation results are also presented.

A Parametric Model for Studies of Flow in Arterial Bifurcations

Regional differences in hemodynamic loads on arterial walls have been associated with localized vascular disease such as atherosclerosis and cerebral aneurysms. Due to their intrinsic geometric relevance, three-dimensional (3D) reconstructions of arterial segments are frequently used in hemodynamic studies of these diseases. However, it is not possible to use them to systematically vary geometric features for parametric studies. Idealized vascular models are inherently suited for parametric studies, but are limited by their tendency to oversimplify the vessel geometry. In this work, a hierarchy of three parametric bifurcation models is introduced. The models are relatively simple, yet capture all geometric features identified as common to cerebral bifurcations in the complex transition from parent to daughter branches. While these models were initially designed for parametric studies, we also evaluate the possibility of using them for 3D reconstruction of cerebral arteries, with the future goal of improving reconstruction of poor quality clinical data. The lumen surface and vessel hemodynamics are compared between two reconstructed cerebral bifurcations and matched parametric models. Good agreement is found. The average and maximum geometric differences are less than 3.1 and 10%, respectively for all three parametric models. The maximum difference in wall shear stress is less than 8% for the most complex parametric model.

A DIRECTOR THEORY APPROACH FOR MODELING BLOOD FLOW IN THE ARTERIAL SYSTEM: AN ALTERNATIVE TO CLASSICAL 1D MODELS

It remains computationally infeasible to model the full three-dimensional (3D) equations for blood flow in large sections of the circulatory system. As a result, one-dimensional (1D) and lumped parameter models play an important role in studies of the arterial system. A variety of 1D models are used, distinguished by the closure approximations employed. In this paper, we introduce a nine-director theory for flow in axisymmetric bodies as an alternative to the 1D models. Advantages of the director theory include (i) the theory makes use of all components of linear momentum; (ii) the flow is not assumed to be uni-directional; (iii) the theory is hierarchical; (iv) there is no need for closure approximations; and (v) wall shear stress enters directly as a dependent variable. In order to simplify the equations for mathematical analysis, for this work, attention is focused on cases where it is appropriate to model the flow as quasi-steady and the wall motion does not have a significant impact on bulk flow parameters. This work lays the foundation for future applications of the theory to unsteady flows in flexible walled vessels. For the geometries considered here, the nine-director theory has the same advantage as 1D models in providing a relatively simple relation between flow rate and average pressure drop. Conditions for existence, uniqueness and local stability of steady solutions are determined for both the 1D and nine-director equations. The predictive capability of classical 1D models found in the recent literature and a nine-director model7,15 are carefully evaluated through comparison with analytical and computational solutions to the axisymmetric, steady Navier–Stokes equations in geometries relevant to blood flow. For these benchmark problems over the range of Reynolds numbers considered, the nine-director theory is found to provide better results than the classical 1D models. A novel approach for parameter identification is in the 1D model is given and shown to substantially improve its predictive capability in these test cases.

Diversity in the Strength and Structure of Unruptured Cerebral Aneurysms

Intracranial aneurysms are pathological enlargements of brain arteries that are believed to arise from progressive wall degeneration and remodeling. Earlier work using classical histological approaches identified variability in cerebral aneurysm mural content, ranging from layered walls with intact endothelium and aligned smooth muscle cells, to thin, hypocellular walls. Here, we take advantage of recent advances in multiphoton microscopy, to provide novel results for collagen fiber architecture in 15 human aneurysm domes without staining or fixation as well as in 12 control cerebral arteries. For all aneurysm samples, the elastic lamina was absent and the abluminal collagen fibers had similar diameters to control arteries. In contrast, the collagen fibers on the luminal side showed great variability in both diameter and architecture ranging from dense fiber layers to sparse fiber constructs suggestive of ineffective remodeling efforts. The mechanical integrity of eight aneurysm samples was assessed using uniaxial experiments, revealing two sub-classes (i) vulnerable unruptured aneurysms (low failure stress and failure pressure), and (ii) strong unruptured aneurysms (high failure stress and failure pressure). These results suggest a need to refine the end-point of risk assessment studies that currently do not distinguish risk levels among unruptured aneurysms. We propose that a measure of wall integrity that identifies this vulnerable wall subpopulation will be useful for interpreting future biological and structural data.

Flow Conditions in the Intracranial Aneurysm Lumen Are Associated with Inflammation and Degenerative Changes of the Aneurysm Wall

BACKGROUND AND PURPOSE: Saccular intracranial aneurysm is a common disease that may cause devastating intracranial hemorrhage. Hemodynamics, wall remodeling, and wall inflammation have been associated with saccular intracranial aneurysm rupture. We investigated how saccular intracranial aneurysm hemodynamics is associated with wall remodeling and inflammation of the saccular intracranial aneurysm wall. MATERIALS AND METHODS: Tissue samples resected during a saccular intracranial aneurysm operation (11 unruptured, 9 ruptured) were studied with histology and immunohistochemistry. Patient-specific computational models of hemodynamics were created from preoperative CT angiographies. RESULTS: More stable and less complex flows were associated with thick, hyperplastic saccular intracranial aneurysm walls, while slower flows with more diffuse inflow were associated with degenerated and decellularized saccular intracranial aneurysm walls. Wall degeneration (P = .041) and rupture were associated with increased inflammation (CD45+, P = .031). High wall shear stress (P = .018), higher vorticity (P = .046), higher viscous dissipation (P = .046), and high shear rate (P = .046) were associated with increased inflammation. Inflammation was also associated with lack of an intact endothelium (P = .034) and the presence of organized luminal thrombosis (P = .018), though overall organized thrombosis was associated with low minimum wall shear stress (P = .034) and not with the flow conditions associated with inflammation. CONCLUSIONS: Flow conditions in the saccular intracranial aneurysm are associated with wall remodeling. Inflammation, which is associated with degenerative wall remodeling and rupture, is related to high flow activity, including elevated wall shear stress. Endothelial injury may be a mechanism by which flow induces inflammation in the saccular intracranial aneurysm wall. Hemodynamic simulations might prove useful in identifying saccular intracranial aneurysms at risk of developing inflammation, a potential biomarker for rupture.