Mette S. Olufsen

Numerical Simulation and Experimental Validation of Blood Flow in Arteries with Structured-Tree Outflow Conditions

AbstractBlood flow in the large systemic arteries is modeled using one-dimensional equations derived from the axisymmetric Navier–Stokes equations for flow in compliant and tapering vessels. The arterial tree is truncated after the first few generations of large arteries with the remaining small arteries and arterioles providing outflow boundary conditions for the large arteries. By modeling the small arteries and arterioles as a structured tree, a semi-analytical approach based on a linearized version of the governing equations can be used to derive an expression for the root impedance of the structured tree in the frequency domain. In the time domain, this provides the proper outflow boundary condition. The structured tree is a binary asymmetric tree in which the radii of the daughter vessels are scaled linearly with the radius of the parent vessel. Blood flow and pressure in the large vessels are computed as functions of time and axial distance within each of the arteries. Comparison between the simulations and magnetic resonance measurements in the ascending aorta and nine peripheral locations in one individual shows excellent agreement between the two.

New roles for the gamma rhythm: population tuning and preprocessing for the Beta rhythm.

Gamma (30–80 Hz) and beta (12–30 Hz) oscillations such as those displayed by in vitro hippocampal (CA1) slice preparations and by in vivo neocortical EEGs often occur successively, with a spontaneous transition between them. In the gamma rhythm, pyramidal cells fire together with the interneurons, while in the beta rhythm, pyramidal cells fire on a subset of cycles of the interneurons. It is shown that gamma and beta rhythms have different properties with respect to creation of cell assemblies. In the presence of heterogeneous inputs to the pyramidal cells, the gamma rhythm creates an assembly of firing pyramidal cells from cells whose drive exceeds a threshold. During the gamma to beta transition, a slow outward potassium current is activated, and as a result the cell assembly vanishes. The slow currents make each of the pyramidal cells fire with a beta rhythm, but the field potential of the network still displays a gamma rhythm. Hebbian changes of connections among the pyramidal cells give rise to a beta rhythm, and the cell assemblies are recovered with a temporal separation between cells firing in different cycles. We present experimental evidence showing that such a separation can occur in hippocampal slices.

ON DERIVING LUMPED MODELS FOR BLOOD FLOW AND PRESSURE IN THE SYSTEMIC ARTERIES

Windkessel and similar lumped models are often used to represent blood flow and pressure in systemic arteries. The windkessel model was originally developed by Stephen Hales (1733) and Otto Frank (1899) who used it to describe blood flow in the heart. In this paper we start with the onedimensional axisymmetric Navier-Stokes equations for time-dependent blood flow in a rigid vessel to derive lumped models relating flow and pressure. This is done through Laplace transform and its inversion via residue theory. Upon keeping contributions from one, two, or more residues, we derive lumped models of successively higher order. We focus on zeroth, first and second order models and relate them to electrical circuit analogs, in which current is equivalent to flow and voltage to pressure. By incorporating effects of compliance through addition of capacitors, windkessel and related lumped models are obtained. Our results show that given the radius of a blood vessel, it is possible to determine the order of the model that would be appropriate for analyzing the flow and pressure in that vessel. For instance, in small rigid vessels ( R < 0.2 cm) it is adequate to use Poiseuilles law to express the relation between flow and pressure, whereas for large vessels it might be necessary to incorporate spatial dependence by using a one-dimensional model accounting for axial variations.

Fractal network model for simulating abdominal and lower extremity blood flow during resting and exercise conditions

We present a one-dimensional (1D) fluid dynamic model that can predict blood flow and blood pressure during exercise using data collected at rest. To facilitate accurate prediction of blood flow, we developed an impedance boundary condition using morphologically derived structured trees. Our model was validated by computing blood flow through a model of large arteries extending from the thoracic aorta to the profunda arteries. The computed flow was compared against measured flow in the infrarenal (IR) aorta at rest and during exercise. Phase contrast-magnetic resonance imaging (PC-MRI) data was collected from 11 healthy volunteers at rest and during steady exercise. For each subject, an allometrically-scaled geometry of the large vessels was created. This geometry extends from the thoracic aorta to the femoral arteries and includes the celiac, superior mesenteric, renal, inferior mesenteric, internal iliac and profunda arteries. During rest, flow was simulated using measured supraceliac (SC) flow at the inlet and a uniform set of impedance boundary conditions at the 11 outlets. To simulate exercise, boundary conditions were modified. Inflow data collected during steady exercise was specified at the inlet and the outlet boundaries were adjusted as follows. The geometry of the structured trees used to compute impedance was scaled to simulate the effective change in the cross-sectional area of resistance vessels and capillaries due to exercise. The resulting computed flow through the IR aorta was compared to measured flow. This method produces good results with a mean difference between paired data to be 1.1 ± 7 cm3 s− 1 at rest and 4.0 ± 15 cm3 s− 1 at exercise. While future work will improve on these results, this method provides groundwork with which to predict the flow distributions in a network due to physiologic regulation.

BLOOD FLOW IN THE CIRCLE OF WILLIS: MODELING AND CALIBRATION.

A numerical model based on one-dimensional balance laws and ad hoc zero-dimensional boundary conditions is tested against experimental data. The study concentrates on the circle of Willis, a vital subnetwork of the cerebral vasculature. The main goal is to obtain efficient and reliable numerical tools with predictive capabilities. The flow is assumed to obey the Navier-Stokes equations, while the mechanical reactions of the arterial walls follow a viscoelastic model. Like many previous studies, a dimension reduction is performed through averaging. Unlike most previous work, the resulting model is both calibrated and validated against in vivo data, more precisely transcranial Doppler data of cerebral blood velocity. The network considered has three inflow vessels and six outflow vessels. Inflow conditions come from the data, while outflow conditions are modeled. Parameters in the outflow conditions are calibrated using a subset of the data through ensemble Kalman filtering techniques. The rest of the data is used for validation. The results demonstrate the viability of the proposed approach.

Analysis of Viscoelastic Wall Properties in Ovine Arteries

In this paper, we analyze how elastic and viscoelastic properties differ across seven locations along the large arteries in 11 sheep. We employ a two-parameter elastic model and a four-parameter Kelvin viscoelastic model to analyze experimental measurements of vessel diameter and blood pressure obtained in vitro at conditions mimicking in vivo dynamics. Elastic and viscoelastic wall properties were assessed via solutions to the associated inverse problem. We use sensitivity analysis to rank the model parameters from the most to the least sensitive, as well as to compute standard errors and confidence intervals. Results reveal that elastic properties in both models (including Youngs modulus and the viscoelastic relaxation parameters) vary across locations (smaller arteries are stiffer than larger arteries). We also show that for all locations, the inclusion of viscoelastic behavior is important to capture pressure-area dynamics.

Linear and Nonlinear Viscoelastic Modeling of Aorta and Carotid Pressure–Area Dynamics Under In Vivo and Ex Vivo Conditions

A better understanding of the biomechanical properties of the arterial wall provides important insight into arterial vascular biology under normal (healthy) and pathological conditions. This insight has potential to improve tracking of disease progression and to aid in vascular graft design and implementation. In this study, we use linear and nonlinear viscoelastic models to predict biomechanical properties of the thoracic descending aorta and the carotid artery under ex vivo and in vivo conditions in ovine and human arteries. Models analyzed include a four-parameter (linear) Kelvin viscoelastic model and two five-parameter nonlinear viscoelastic models (an arctangent and a sigmoid model) that relate changes in arterial blood pressure to the vessel cross-sectional area (via estimation of vessel strain). These models were developed using the framework of Quasilinear Viscoelasticity (QLV) theory and were validated using measurements from the thoracic descending aorta and the carotid artery obtained from human and ovine arteries. In vivo measurements were obtained from 10 ovine aortas and 10 human carotid arteries. Ex vivo measurements (from both locations) were made in 11 male Merino sheep. Biomechanical properties were obtained through constrained estimation of model parameters. To further investigate the parameter estimates, we computed standard errors and confidence intervals and we used analysis of variance to compare results within and between groups. Overall, our results indicate that optimal model selection depends on the artery type. Results showed that for the thoracic descending aorta (under both experimental conditions), the best predictions were obtained with the nonlinear sigmoid model, while under healthy physiological pressure loading the carotid arteries nonlinear stiffening with increasing pressure is negligible, and consequently, the linear (Kelvin) viscoelastic model better describes the pressure–area dynamics in this vessel. Results comparing biomechanical properties show that the Kelvin and sigmoid models were able to predict the zero-pressure vessel radius; that under ex vivo conditions vessels are more rigid, and comparatively, that the carotid artery is stiffer than the thoracic descending aorta; and that the viscoelastic gain and relaxation parameters do not differ significantly between vessels or experimental conditions. In conclusion, our study demonstrates that the proposed models can predict pressure–area dynamics and that model parameters can be extracted for further interpretation of biomechanical properties.

A practical approach to parameter estimation applied to model predicting heart rate regulation

Mathematical models have long been used for prediction of dynamics in biological systems. Recently, several efforts have been made to render these models patient specific. One way to do so is to employ techniques to estimate parameters that enable model based prediction of observed quantities. Knowledge of variation in parameters within and between groups of subjects have potential to provide insight into biological function. Often it is not possible to estimate all parameters in a given model, in particular if the model is complex and the data is sparse. However, it may be possible to estimate a subset of model parameters reducing the complexity of the problem. In this study, we compare three methods that allow identification of parameter subsets that can be estimated given a model and a set of data. These methods will be used to estimate patient specific parameters in a model predicting baroreceptor feedback regulation of heart rate during head-up tilt. The three methods include: structured analysis of the correlation matrix, analysis via singular value decomposition followed by QR factorization, and identification of the subspace closest to the one spanned by eigenvectors of the model Hessian. Results showed that all three methods facilitate identification of a parameter subset. The “best” subset was obtained using the structured correlation method, though this method was also the most computationally intensive. Subsets obtained using the other two methods were easier to compute, but analysis revealed that the final subsets contained correlated parameters. In conclusion, to avoid lengthy computations, these three methods may be combined for efficient identification of parameter subsets.

Impaired Cerebral Autoregulation Is Associated with Brain Atrophy and Worse Functional Status in Chronic Ischemic Stroke

Dynamic cerebral autoregulation (dCA) is impaired following stroke. However, the relationship between dCA, brain atrophy, and functional outcomes following stroke remains unclear. In this study, we aimed to determine whether impairment of dCA is associated with atrophy in specific regions or globally, thereby affecting daily functions in stroke patients. We performed a retrospective analysis of 33 subjects with chronic infarctions in the middle cerebral artery territory, and 109 age-matched non-stroke subjects. dCA was assessed via the phase relationship between arterial blood pressure and cerebral blood flow velocity. Brain tissue volumes were quantified from MRI. Functional status was assessed by gait speed, instrumental activities of daily living (IADL), modified Rankin Scale, and NIH Stroke Score. Compared to the non-stroke group, stroke subjects showed degraded dCA bilaterally, and showed gray matter atrophy in the frontal, parietal and temporal lobes ipsilateral to infarct. In stroke subjects, better dCA was associated with less temporal lobe gray matter atrophy on the infracted side ( = 0.029), faster gait speed ( = 0.018) and lower IADL score (0.002). Our results indicate that better dynamic cerebral perfusion regulation is associated with less atrophy and better long-term functional status in older adults with chronic ischemic infarctions.

A one-dimensional fluid dynamic model of the systemic arteries.

The systemic arteries can be modeled as a bifurcating tree of compliant tapering vessels while blood flow and pressure can be predicted by solving Navier-Stokes equations for each of the branches. If all branches are included the computational cost will become prohibitively large. Therefore, the tree must be truncated after a limited number of generations and a suitable outflow boundary condition must be applied. To this end we propose a structured tree in which the root impedance is calculated using a semi-analytical approach. In the structured tree the fluid dynamic equations are linearized giving a wave equation, which can be solved analytically for each vessel. This provides a dynamical boundary condition based on physiological principles which is computationally feasible. It exhibits the actual phase lag between flow and pressure as well as accommodating the wave propagation effects for the entire systemic arterial tree. Finally, the model has been compared with a standard and well established model, where outflow at the terminals are determined by attaching a Windkessel type boundary condition.