Patient-specific modeling of cardiovascular and respiratory dynamics during hypercapnia

This study develops a lumped cardiovascular-respiratory system-level model that incorporates patient-specific data to predict cardiorespiratory response to hypercapnia (increased CO(2) partial pressure) for a patient with congestive heart failure (CHF). In particular, the study focuses on predicting cerebral CO(2) reactivity, which can be defined as the ability of vessels in the cerebral vasculature to expand or contract in response CO(2) induced challenges. It is difficult to characterize cerebral CO(2) reactivity directly from measurements, since no methods exist to dynamically measure vasomotion of vessels in the cerebral vasculature. In this study we show how mathematical modeling can be combined with available data to predict cerebral CO(2) reactivity via dynamic predictions of cerebral vascular resistance, which can be directly related to vasomotion of vessels in the cerebral vasculature. To this end we have developed a coupled cardiovascular and respiratory model that predicts blood pressure, flow, and concentration of gasses (CO(2) and O(2)) in the systemic, cerebral, and pulmonary arteries and veins. Cerebral vascular resistance is incorporated via a model parameter separating cerebral arteries and veins. The model was adapted to a specific patient using parameter estimation combined with sensitivity analysis and subset selection. These techniques allowed estimation of cerebral vascular resistance along with other cardiovascular and respiratory parameters. Parameter estimation was carried out during eucapnia (breathing room air), first for the cardiovascular model and then for the respiratory model. Then, hypercapnia was introduced by increasing inspired CO(2) partial pressure. During eucapnia, seven cardiovascular parameters and four respiratory parameters was be identified and estimated, including cerebral and systemic resistance. During the transition from eucapnia to hypercapnia, the model predicted a drop in cerebral vascular resistance consistent with cerebral vasodilation.

Functionality of the baroreceptor nerves in heart rate regulation

Two models describing the afferent baroreceptor firing are analyzed, a basic model predicting firing using a single nonlinear differential equation, and an extended model, coupling K nonlinear responses. Both models respond to the the rate (derivative) and the rate history of the carotid sinus arterial pressure. As a result both the rate and the relative level of the carotid sinus arterial pressure is sensed. Simulations with these models show that responses to step changes in pressure follow from the rate sensitivity as observed in experimental studies. Adaptation and asymmetric responses are a consequence of the memory encapsulated by the models, and the nonlinearity gives rise to sigmoidal response curves. The nonlinear afferent baroreceptor models are coupled with an effector model, and the coupled model has been used to predict baroreceptor feedback regulation of heart rate during postural change from sitting to standing and during head-up tilt. The efferent model couples the afferent nerve paths to the sympathetic and parasympathetic outflow, and subsequently predicts the build up of an action potential at the sinus knot of the heart. In this paper, we analyze the nonlinear afferent model and show that the coupled model is able to predict heart rate regulation using blood pressure data as an input.

Linear and nonlinear viscoelastic modeling of ovine aortic biomechanical properties under in vivo and ex vivo conditions

This study uses linear and nonlinear viscoelastic models to describe the dynamic distention of the aorta induced by time-varying arterial blood pressure. We employ an inverse mathematical modeling approach on a four-parameter (linear) Kelvin viscoelastic model and two five-parameter nonlinear viscoelastic models (arctangent and sigmoid) to infer vascular biomechanical properties under in vivo and ex vivo experimental conditions in ten and eleven male Merino sheep, respectively. We used the Akaike Information Criterion (AIC) as a goodness-of-fit measure. Results show that under both experimental conditions, the nonlinear models generally outperform the linear Kelvin model, as judged by the AIC. Furthermore, the sigmoid nonlinear viscoelastic model consistently achieves the lowest AIC and also matches the zero-stress vessel radii measured ex vivo. Based on these observations, we conclude that the sigmoid nonlinear viscoelastic model best describes the biomechanical properties of ovine large arteries under both experimental conditions considered in this study.

Modeling heart rate regulation--part II: parameter identification and analysis

In part I of this study we introduced a 17-parameter model that can predict heart rate regulation during postural change from sitting to standing. In this subsequent study, we focus on the 17 model parameters needed to adequately represent the observed heart rate response. In part I and in previous work (Olufsen et al. 2006), we estimated the 17 model parameters by minimizing the least squares error between computed and measured values of the heart rate using the Nelder-Mead method (a simplex algorithm). In this study, we compare the Nelder-Mead optimization method to two sampling methods: the implicit filtering method and a genetic algorithm. We show that these off-the-shelf optimization methods can work in conjunction with the heart rate model and provide reasonable parameter estimates with little algorithm tuning. In addition, we make use of the thousands of points sampled by the optimizers in the course of the minimization to perform an overall analysis of the model itself. Our findings show that the resulting least-squares problem has multiple local minima and that the non-linear-least squares error can vary over two orders of magnitude due to the complex interaction between the model parameters, even when provided with reasonable bound constraints.

Stochastic and deterministic models for agricultural production networks

An approach to modeling the impact of disturbances in an agricultural production network is presented. A stochastic model and its approximate deterministic model for averages over sample paths of the stochastic system are developed. Simulations, sensitivity and generalized sensitivity analyses are given. Finally, it is shown how diseases may be introduced into the network and corresponding simulations are discussed.

Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions

Blood 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.

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.

Structured tree outflow condition for blood flow in larger systemic arteries

A central problem in modeling blood flow and pressure in the larger systemic arteries is determining a physiologically based boundary condition so that the arterial tree can be truncated after a few generations. We have used a structured tree attached to the terminal branches of the truncated arterial tree in which the root impedance is estimated using a semianalytical approach based on a linearization of the viscous axisymmetric Navier-Stokes equations. This provides a dynamic boundary condition that maintains the phase lag between blood flow and pressure as well as the high-frequency oscillations present in the impedance spectra. Furthermore, it accommodates the wave propagation effects for the entire systemic arterial tree. The result is a model that is physiologically adequate as well as computationally feasible. For validation, we have compared the structured tree model with a pure resistance and a windkessel model as well as with measured data.