Short-term cardiovascular responses to postural change from sitting to standing involve complex interactions between the autonomic nervous system, which regulates blood pressure, and cerebral autoregulation, which maintains cerebral perfusion. We present a mathematical model that can predict dynamic changes in beat-to-beat arterial blood pressure and middle cerebral artery blood flow velocity during postural change from sitting to standing. Our cardiovascular model utilizes 11 compartments to describe blood pressure, blood flow, compliance, and resistance in the heart and systemic circulation. To include dynamics due to the pulsatile nature of blood pressure and blood flow, resistances in the large systemic arteries are modeled using nonlinear functions of pressure. A physiologically based submodel is used to describe effects of gravity on venous blood pooling during postural change. Two types of control mechanisms are included: 1) autonomic regulation mediated by sympathetic and parasympathetic responses, which affect heart rate, cardiac contractility, resistance, and compliance, and 2) autoregulation mediated by responses to local changes in myogenic tone, metabolic demand, and CO(2) concentration, which affect cerebrovascular resistance. Finally, we formulate an inverse least-squares problem to estimate parameters and demonstrate that our mathematical model is in agreement with physiological data from a young subject during postural change from sitting to standing.

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This paper concerns ODE modeling of the hypothalamic-pituitary- adrenal axis (HPA axis) using an analytical and numerical approach, combined with biological knowledge regarding physiological mechanisms and parameters. The three hormones, CRH, ACTH, and cortisol, which interact in the HPA axis are modeled as a system of three coupled, nonlinear differential equations. Experimental data shows the circadian as well as the ultradian rhythm. This paper focuses on the ultradian rhythm. The ultradian rhythm can mathematically be explained by oscillating solutions. Oscillating solutions to an ODE emerges from an unstable fixed point with complex eigenvalues with a positive real parts and a non-zero imaginary parts. The first part of the paper describes the general considerations to be obeyed for a mathematical model of the HPA axis. In this paper we only include the most widely accepted mechanisms that influence the dynamics of the HPA axis, i.e. a negative feedback from cortisol on CRH and ACTH. Therefore we term our model the minimal model. The minimal model, encompasses a wide class of different realizations, obeying only a few physiologically reasonable demands. The results include the existence of a trapping region guaranteeing that concentrations do not become negative or tend to infinity. Furthermore, this treatment guarantees the existence of a unique fixed point. A change in local stability of the fixed point, from stable to unstable, implies a Hopf bifurcation; thereby, oscillating solutions may emerge from the model. Sufficient criteria for local stability of the fixed point, and an easily applicable sufficient criteria guaranteeing global stability of the fixed point, is formulated. If the latter is fulfilled, ultradian rhythm is an impossible outcome of the minimal model and all realizations thereof. The second part of the paper concerns a specific realization of the minimal model in which feedback functions are built explicitly using receptor dynamics. Using physiologically reasonable parameter values, along with the results of the general case, it is demonstrated that un-physiological values of the parameters are needed in order to achieve local instability of the fixed point. Small changes in physiologically relevant parameters cause the system to be globally stable using the analytical criteria. All simulations show a globally stable fixed point, ruling out periodic solutions even when an investigation of the 'worst case parameters' is performed.

An elastic rubber tube is connected with a stiffer rubber tube forming two halves of a torus and filled with water. Compressing one of the rubber tubes symmetrically and periodic at a point of asymmetry creates a remarkable unidirectional mean flow in the system. The size and the direction of the mean flow depend on the frequency of compression, the elasticity of the tubes, the compression ratio, and the type of compression with respect to time in a complicated manner. The system is modelled using a one-dimensional theory derived by averaging the Navier-Stokes equations ignoring higher order terms in a certain small quantity. The one-dimensional model is analysed partly analytically and partly numerically. A series of experiments on a physical realisation of the system are described. The theoretical findings and experimental results are compared; They show a remarkable agreement between the experiments and the predictions of the model. Frequencies at which the mean flow change direction are predicted numerically as well as analytically and the two results are compared.

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.

Short-term cardiovascular responses to head-up tilt (HUT) involve complex cardiovascular regulation in order to maintain blood pressure at homoeostatic levels. This manuscript presents a patient-specific model that uses heart rate as an input to fit the dynamic changes in arterial blood pressure data during HUT. The model contains five compartments representing arteries and veins in the upper and lower body of the systemic circulation, as well as the left ventricle facilitating pumping of the heart. A physiologically based submodel describes gravitational pooling of the blood into the lower extremities during HUT, and a cardiovascular regulation model adjusts cardiac contractility and vascular resistance to the blood pressure changes. Nominal parameter values are computed from patient-specific data and literature estimates. The model is rendered patient specific via the use of parameter estimation techniques. This process involves sensitivity analysis, prediction of a subset of identifiable parameters, and non-linear optimization. The approach proposed here was applied to the analysis of aortic and carotid HUT data from five healthy young subjects. Results showed that it is possible to identify a subset of model parameters that can be estimated allowing the model to fit changes in arterial blood pressure observed at the level of the carotid bifurcation. Moreover, the model estimates physiologically reasonable values for arterial and venous blood pressures, blood volumes and cardiac output for which data are not available.

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During orthostatic stress, arterial and cardiopulmonary baroreflexes play a key role in maintaining arterial pressure by regulating heart rate. This study presents a mathematical model that can predict the dynamics of heart rate regulation in response to postural change from sitting to standing. The model uses blood pressure measured in the finger as an input to model heart rate dynamics in response to changes in baroreceptor nerve firing rate, sympathetic and parasympathetic responses, vestibulo-sympathetic reflex, and concentrations of norepinephrine and acetylcholine. We formulate an inverse least squares problem for parameter estimation and successfully demonstrate that our mathematical model can accurately predict heart rate dynamics observed in data obtained from healthy young, healthy elderly, and hypertensive elderly subjects. One of our key findings indicates that, to successfully validate our model against clinical data, it is necessary to include the vestibulo-sympathetic reflex. Furthermore, our model reveals that the transfer between the nerve firing and blood pressure is nonlinear and follows a hysteresis curve. In healthy young people, the hysteresis loop is wide, whereas, in healthy and hypertensive elderly people, the hysteresis loop shifts to higher blood pressure values, and its area is diminished. Finally, for hypertensive elderly people, the hysteresis loop is generally not closed, indicating that, during postural change from sitting to standing, baroreflex modulation does not return to steady state during the first minute of standing.