Analytical solution of phosphate kinetics for hemodialysis.

Chronic kidney diseases imply an ongoing need to remove toxins, with hemodialysis as the preferred treatment modality. We derive analytical expressions for phosphate clearance during dialysis, the single pass (SP) model corresponding to a standard clinical hemodialysis and the multi pass (MP) model, where dialysate is recycled and therefore makes a smaller clinical setting possible such as a transportable dialysis suitcase. For both cases we show that the convective contribution to the dialysate is negligible for the phosphate kinetics and derive simpler expressions. The SP and MP models are calibrated to clinical data of ten patients showing consistency between the models and provide estimates of the kinetic parameters. Immediately after dialysis a rebound effect is observed. We derive a simple formula describing this effect which is valid both posterior to SP or MP dialysis. The analytical formulas provide explanations to observations of previous clinical studies.

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.

Modeling ventricular contraction with heart rate changes.

Recently, a mathematical model of the pumping heart has been proposed describing the heart as a pressure source depending on time, volume and flow. The underlying concept is based on a new two-step paradigm that allows separation between isovolumic (non-ejecting) and ejecting heart properties. The first step describes the ventricular pressure in the isovolumic ventricle. In the following step, the isovolumic description is extended with the ejection effect in order to embrace the pumping heart during actual blood ejection. The description of the isovolumic heart properties plays a crucial role in this paradigm. However, only a single isovolumic model has previously been used restricting the heart rate to 1 Hz. In this paper, a family of models describing the isovolumic contracting ventricle are critically examined. A characterization of what constitutes an optimal model is given and used as a criteria for choosing the optimal model in this family. Moreover, and this is indeed a point, the proposed model in this study is valid for arbitrary heart rates and based on experimental data. The model exhibits all major features of the ejecting heart, including how ventricular pressure and flow vary in time for various heart rates and how stroke volume and cardiac output vary with heart rate. The modeling strategy presented embraces the same steps and demarcations as those suitable for clinical examination whereby new experiments are suggested.

Valveless pumping in a fluid-filled closed elastic tube-system: one-dimensional theory with experimental validation.

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.

Describing the pumping heart as a pressure source.

The pumping heart is described by a new mathematical approach which considers the heart as a pressure source depending on time, volume and flow. This new approach allows a separation between isovolumic (non-ejecting) and ejecting heart properties. The computed results cover most of the features of the human ventricle during normal and altered vascular conditions. It is shown that the time-varying elastance concept is disqualified as an independent description of the heart, it follows from isovolumic heart properties and an ejection effect which consists of positive and negative effects of ventricular blood ejection.

General compartmental models of the cardiovascular system.

A systematic discussion on linear as well as non-linear compartmental models of the cardiovascular system and its various feedback mechanism is given. Most of the results are independent of explicit functional expressions. The topological structure of the model is essential for the response to a local change in peripheral resistance. Inclusion of Parallel paths versus serial paths gives qualitatively different response. Global asymptotic stability follows from the general theory.

Modelling of the baroreflex-feedback mechanism with time-delay.

The cardiovascular system is considered. A direct modelling of the non-linear baroreflex-feedback mechanism, including time-delay, is developed based on physiological theory and empirical facts. The feedback model is then evaluated on an expanded, but simple, non-pulsatile Windkessel model of the cardiovascular system. The stability of the entire model is analyzed and the effect of the value of the time-delay is investigated and discussed. The time-delay may cause oscillations. A finite number of stability switches may occur dependent on the value of the time-delay. The location of these stability switches turns out to be sensitive to the value of the parameters in the model. We suggest a simple experiment to determine whether or not the time-delay is responsible for the 10 second Mayer waves. Data from an ergometer bicycle test is used for evaluation of the model.