Johnny T. Ottesen

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.

The minimal model of the hypothalamic–pituitary–adrenal axis

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.

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.

Mathematical modeling of the hypothalamic–pituitary–adrenal gland (HPA) axis, including hippocampal mechanisms

This paper presents a mathematical model of the HPA axis. The HPA axis consists of the hypothalamus, the pituitary and the adrenal glands in which the three hormones CRH, ACTH and cortisol interact through receptor dynamics. Furthermore, it has been suggested that receptors in the hippocampus have an influence on the axis. A model is presented with three coupled, non-linear differential equations, with the hormones CRH, ACTH and cortisol as variables. The model includes the known features of the HPA axis, and includes the effects from the hippocampus through its impact on CRH in the hypothalamus. The model is investigated both analytically and numerically for oscillating solutions, related to the ultradian rhythm seen in data, and for multiple fixed points related to hypercortisolemic and hypocortisolemic depression. The existence of an attracting trapping region guarantees that solution curves stay non-negative and bounded, which can be interpreted as a mathematical formulation of homeostasis. No oscillating solutions are present when using physiologically reasonable parameter values. This indicates that the ultradian rhythm originate from different mechanisms. Using physiologically reasonable parameters, the system has a unique fixed point, and the system is globally stable. Therefore, solutions converge to the fixed point for all initial conditions. This is in agreement with cortisol levels returning to normal, after periods of mild stress, in healthy individuals. Perturbing parameters lead to a bifurcation, where two additional fixed points emerge. Thus, the system changes from having a unique stable fixed point into having three fixed points. Of the three fixed points, two are stable and one is unstable. Further investigations show that solutions converge to one of the two stable fixed points depending on the initial conditions. This could explain why healthy people becoming depressed usually fall into one of two groups: a hypercortisolemic depressive group or a hypocortisolemic depressive group.

Novel characteristics of valveless pumping

This study investigates the occurrence of valveless pumping in a fluid-filled system consisting of two open tanks connected by an elastic tube. We show that directional flow can be achieved by introducing a periodic pinching applied at an asymmetrical location along the tube, and that the flow direction depends on the pumping frequency. We propose a relation between wave propagation velocity, tube length, and resonance frequencies associated with shifts in the pumping direction using numerical simulations. The eigenfrequencies of the system are estimated from the linearized system, and we show that these eigenfrequencies constitute the resonance frequencies and the horizontal slope frequencies of the system; “horizontal slope frequency” being a new concept. A simple model is suggested, explaining the effect of the gravity driven part of the oscillation observed in response to the tank and tube diameter changes. Results are partly compared with experimental findings.

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 the Afferent Dynamics of the Baroreflex Control System

In this study we develop a modeling framework for predicting baroreceptor firing rate as a function of blood pressure. We test models within this framework both quantitatively and qualitatively using data from rats. The models describe three components: arterial wall deformation, stimulation of mechanoreceptors located in the BR nerve-endings, and modulation of the action potential frequency. The three sub-systems are modeled individually following well-established biological principles. The first submodel, predicting arterial wall deformation, uses blood pressure as an input and outputs circumferential strain. The mechanoreceptor stimulation model, uses circumferential strain as an input, predicting receptor deformation as an output. Finally, the neural model takes receptor deformation as an input predicting the BR firing rate as an output. Our results show that nonlinear dependence of firing rate on pressure can be accounted for by taking into account the nonlinear elastic properties of the artery wall. This was observed when testing the models using multiple experiments with a single set of parameters. We find that to model the response to a square pressure stimulus, giving rise to post-excitatory depression, it is necessary to include an integrate-and-fire model, which allows the firing rate to cease when the stimulus falls below a given threshold. We show that our modeling framework in combination with sensitivity analysis and parameter estimation can be used to test and compare models. Finally, we demonstrate that our preferred model can exhibit all known dynamics and that it is advantageous to combine qualitative and quantitative analysis methods.

Patient-specific modelling of head-up tilt

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.

Patient-specific modeling of the neuroendocrine HPA-axis and its relation to depression: Ultradian and circadian oscillations.

In the Western world approximately 10% of the population experience severe depression at least once in their lifetime and many more experience a mild form of depression. Depression has been associated with malfunctions in the hypothalamus-pituitary-adrenal (HPA) axis. We suggest a novel mechanistic non-linear model capable of showing both circadian as well as ultradian oscillations of the hormone concentrations related to the HPA-axis. The fast ultradian rhythm is assumed to originate from the hippocampus whereas the slower circadian rhythm is assumed to be caused by the circadian clock. The model is able to describe the oscillatory patterns in hormone concentration data from 29 patients and healthy controls. Using non-linear mixed effects modeling with statistical hypothesis testing, three of the model parameters are identified to be related to depression. These parameters represent underlying physiological mechanisms controlling the average levels as well as the ultradian frequencies and amplitudes of the hormones ACTH and cortisol. The results are promising since they point toward an exact etiology for depression. As a consequence new biomarkers and pharmaceutical targets may be identified.

The Mathematical Microscope – Making the Inaccessible Accessible

In this chapter we introduce a new term, the “mathematical microscope”, as a method of using mathematics in accessing information about reality when this information is otherwise inaccessible. Furthermore, we discuss how models and experiments are related: none of which are important without the other. In the sciences and medicine, a link that is often missing in the chain of a system can be made visible with the aid of the mathematical microscope. The mathematical microscope serves not only as a lens to clarify a blurred picture but more important as a tool to unveil profound truths. In reality, models are most often used in a detective-like manner to investigate the consequences of different hypothesis. Thus, models can help clarify connections and relations. Consequently, models also help to reveal mechanisms and to develop theories. Case studies are presented and the role of mathematical modelling is discussed for type 1 and type 2 diabetes, depression, cardiovascular diseases and the interactions between the combinations of these, the so-called grey triangle in the metabolic syndrome.