Stability of the human respiratory control system. I. Analysis of a two-dimensional delay state-space model

Analysis of linear lung models based on state-space models

BACKGROUND AND OBJECTIVES

Linear parametric respiratory system models have been used in the model-based analysis of the respiratory system. Although there are studies exploring the physiological correctness and fitting accuracy of the models, they are not analysed in terms of interaction between parameters and dynamics of the model. In this study we propose to use state-space modelling to yield the time-varying nature of the system incorporated by the parameters.

METHODS

We tested controllability, observability and stability characteristics of the equation of motion, 2-comp. parallel, 2-comp. series, viscoelastic, 6-element and mead models while using the parameters given in the literature. In the sensitivity analysis we proposed to use dual Desensitized Linear Kalman Filter (DKF) and Extended Kalman Filter (EKF) method. In this method, state error covariance revealed the parameter sensitivities for each model.

RESULTS

Results showed that all models, except 2-comp. parallel and mead models, are both controllable and observable models. On the other hand all models, except mead model, are stable models. Regarding to the sensitivity analysis, dual DKF - EKF method estimated states of the models successfully with a low estimation error. Sensitivity analysis results showed that airway parameters have higher effects on the state estimation than the other parameters have.

CONCLUSION

We proved that state-space evaluation of the previously proposed parametric models of the respiratory system led us to quantitative and qualitative assessments of the respiratory models. Moreover parameter values found in the literature have different effects on the models.

Diagnosis, management and pathophysiology of central sleep apnea in children

Central sleep apnea (CSA) is thought to occur in about 1-5% of healthy children. CSA occurs more commonly in children with underlying disease and the presence of CSA may influence the course of their disease. CSA can be classified based on the presence or absence of hypercapnia as well as the underlying condition it is associated with. The management of CSA needs to be tailored to the patient and may include medication, non-invasive ventilation, and surgical intervention. Screening children at high risk will allow for earlier diagnosis and timely therapeutic interventions for this population. The review will highlight the pathophysiology, prevalence and diagnosis of CSA in children. An algorithm for the management of CSA in healthy children and children with underlying co-morbidities will be outlined.

Modulation-demodulation hypothesis of periodic breathing in human respiration

Periodic breathing (PB) is a diseased condition of the cardiorespiratory system, and mathematically it is modelled as an oscillation. Modeling approaches replicate periodic oscillation in the minute ventilation due to a higher than normal gain of the feedback signals from the chemoreceptors coupled with a longer than normal latency in feedback, and do not consider the waxing-waning pattern of the oronasal airflow. In this work, a noted regulation model is extended by integrating respiratory mechanics and respiratory central pattern generator (rCPG) model, using modulation-demodulation¹ hypothesis. This is a top-down modeling approach, and it is assumed that the sensory feedback signal from the chemoreceptors modulates the output of the rCPG model. It is also assumed that the brainstem network is responsible for the demodulation process. The respiratory mechanics is modeled as a multi-input multi-output (MIMO) system, where modulated and demodulated neural signals are applied as input and the minute ventilation and the oronasal airflow are specified as output. The minute ventilation signal drives the regulation model, completing the feedback loop. The proposed model is validated by comparing the model output with the clinical data. Using the modulation-demodulation hypothesis, a respiratory mechanics model is formulated in the form of a linear state-space model, which can be useful for providing assisted ventilation in clinical conditions.

The relationship between intermittent limit cycles and postural instability associated with Parkinson's disease

BACKGROUND

Many disease-specific factors such as muscular weakness, increased muscle stiffness, varying postural strategies, and changes in postural reflexes have been shown to lead to postural instability and fall risk in people with Parkinson's disease (PD). Recently, analytical techniques, inspired by the dynamical systems perspective on movement control and coordination, have been used to examine the mechanisms underlying the dynamics of postural declines and the emergence of postural instabilities in people with PD.

METHODS

A wavelet-based technique was used to identify limit cycle oscillations (LCOs) in the anterior-posterior (AP) postural sway of people with mild PD (n = 10) compared to age-matched controls (n = 10). Participants stood on a foam and on a rigid surface while completing a dual task (speaking).

RESULTS

There was no significant difference in the root mean square of center of pressure between groups. Three out of 10 participants with PD demonstrated LCOs on the foam surface, while none in the control group demonstrated LCOs. An inverted pendulum model of bipedal stance was used to demonstrate that LCOs occur due to disease-specific changes associated with PD: time-delay and neuromuscular feedback gain.

CONCLUSION

Overall, the LCO analysis and mathematical model appear to capture the subtle postural instabilities associated with mild PD. In addition, these findings provide insights into the mechanisms that lead to the emergence of unstable posture in patients with PD.

Using dominant eigenvalue analysis to predict formation of alternans in the heart

Ventricular fibrillation at the whole heart level is often preceded by the alternation of action potential duration (APD), i.e., alternans, at the cellular level. As proven in many experiments, traditional approaches based on the slope of the restitution curve have not been successful in predicting alternans formation. Recently, a technique has been theoretically developed based on dominant eigenvalue analysis to predict alternans formation in isolated cardiac myocytes. Here, we aimed to demonstrate that this technique can be applied to predict alternans formation at the whole heart level. Optical mapping was performed in Langendorff-perfused hearts from New Zealand white rabbits (n = 4), which were paced at decreasing basic cycle lengths to introduce APD alternans. In each heart, the basic cycle length corresponding to the local onset of alternans, B(onset), was determined and two regions of the heart were identified at B(onset): one region which exhibited alternans (1:1(alt)) and one which did not (1:1). Corresponding two-dimensional eigenvalue (λ) maps were generated using principal component analysis by analyzing action potentials after short perturbations from the steady state, and mean eigenvalues (λ[over ¯]) were calculated separately for the 1:1 and 1:1(alt) regions. We demonstrated that λ[over ¯] calculated at B(onset) was significantly different (p<0.05) between the two regions. Our results suggest that this dominant eigenvalue technique can be used to successfully predict the local alternans formation in the heart.

Integrative approaches for modeling regulation and function of the respiratory system

Mathematical models have been central to understanding the interaction between neural control and breathing. Models of the entire respiratory system-which comprises the lungs and the neural circuitry that controls their ventilation-have been derived using simplifying assumptions to compartmentalize each component of the system and to define the interactions between components. These full system models often rely-through necessity-on empirically derived relationships or parameters, in addition to physiological values. In parallel with the development of whole respiratory system models are mathematical models that focus on furthering a detailed understanding of the neural control network, or of the several functions that contribute to gas exchange within the lung. These models are biophysically based, and rely on physiological parameters. They include single-unit models for a breathing lung or neural circuit, through to spatially distributed models of ventilation and perfusion, or multicircuit models for neural control. The challenge is to bring together these more recent advances in models of neural control with models of lung function, into a full simulation for the respiratory system that builds upon the more detailed models but remains computationally tractable. This requires first understanding the mathematical models that have been developed for the respiratory system at different levels, and which could be used to study how physiological levels of O2 and CO2 in the blood are maintained.

Estimating eigenvalues of dynamical systems from time series with applications to predicting cardiac alternans

The stability of a dynamical system can be indicated by eigenvalues of its underlying mathematical model. However, eigenvalue analysis of a complicated system (e.g. the heart) may be extremely difficult because full models may be intractable or unavailable. We develop data-driven statistical techniques, which are independent of any underlying dynamical model, that use principal components and maximum-likelihood methods to estimate the dominant eigenvalues and their standard errors from the time series of one or a few measurable quantities, e.g. transmembrane voltages in cardiac experiments. The techniques are applied to predicting cardiac alternans that is characterized by an eigenvalue approaching −1. Cardiac alternans signals a vulnerability to ventricular fibrillation, the leading cause of death in the USA.

Computational models for the study of heart-lung interactions in mammals

The operation and regulation of the lungs and the heart are closely related. This is evident when examining the anatomy within the thorax cavity, in the brainstem and in the aortic and carotid arteries where chemoreceptors and baroreceptors, which provide feedback affecting the regulation of both organs, are concentrated. This is also evident in phenomena such as respiratory sinus arrhythmia where the heart rate increases during inspiration and decreases during expiration, in other types of synchronization between the heart and the lungs known as cardioventilatory coupling and in the association between heart failure and sleep apnea where breathing is interrupted periodically by periods of no-breathing. The full implication and physiological significance of the cardiorespiratory coupling under normal, pathological, or extreme physiological conditions are still unknown and are subject to ongoing investigation both experimentally and theoretically using mathematical models. This article reviews mathematical models that take heart-lung interactions into account. The main ideas behind low dimensional, phenomenological models for the study of the heart-lung synchronization and sleep apnea are described first. Higher dimensions, physiology-based models are described next. These models can vary widely in detail and scope and are characterized by the way the heart-lung interaction is taken into account: via gas exchange, via the central nervous system, via the mechanical interactions, and via time delays. The article emphasizes the need for the integration of the different sources of heart-lung coupling as well as the different mathematical approaches.

Time delay in physiological systems: analyzing and modeling its impact

This article examines the functional and clinical impact of time delays that arise in human physiological systems, especially control systems. An overview of the mathematical and physiological contexts for considering time delays will be illustrated, from the system level to cell level, by examining models that incorporate time delays. This examination will highlight how such delays in combination with other system structures and parameters influence system dynamics. Model analysis that reveals the influence of delays can also reveal related physiological effects which may have medical consequences and clinical applications.

Modeling alveolar volume changes during periodic breathing in heterogeneously ventilated lungs

A simplified model of periodic breathing, proposed by Whiteley et al. (Math. Med. Biol. 20:205-224, 2003), describes a non-uniform breathing pattern for a lung with an inhomogeneous gas distribution, such as that observed in some subjects suffering from respiratory disease. This model assumes a constant alveolar volume, and predicts incidence of irregular breathing caused by small, poorly ventilated regions of the lung. Presented here is an extension to this work which, by allowing variable lung volume, facilitates the investigation of pulmonary collapse in poorly ventilated compartments. A weakness of the original model is that a very small alveolar volume is required for periodic breathing to occur. The model presented within, which removes the assumption of constant compartment volume and allows alveolar volume to vary with time, predicts periodic breathing at higher, more realistic alveolar volumes. Furthermore, the predicted oscillations in ventilation match experimental data more closely. Thus the model that allows for alveolar collapse has improved upon these earlier results, and establishes a theoretical link between periodic breathing and atelectasis.

A computer model of the artificially ventilated human respiratory system in adult intensive care

A multi-technique approach to modelling artificially ventilated patients on the adult general intensive care unit (ICU) is proposed. Compartmental modelling techniques were used to describe the mechanical ventilator and the flexible hoses that connect it to the patient. 3D CFD techniques were used to model flow in the major airways and a Windkessel style balloon model was used to model the mechanical properties of the lungs. A multi-compartment model of the lung based on bifurcating tree structures representing the conducting airways and pulmonary circulation allowed lung disease to be modelled in terms of altered V/Q ratios within a lognormal distribution of values and it is from these that gas exchange was determined. A compartmental modelling tool, Bathfp, was used to integrate the different modelling techniques into a single model. The values of key parameters in the model could be obtained from measurements on patients in an ICU whilst a sensitivity analysis showed that the model was insensitive to the value of other parameters within it. Measured and modelled values for arterial blood gases and airflow parameters are compared for 46 ventilator settings obtained from 6 ventilator dependent patients. The results show correlation coefficients of 0.88 and 0.85 for the arterial partial pressures of the O(2) and CO(2), respectively (p<0.01) and of 0.99 and 0.96 for upper airway pressure and tidal volume, respectively (p<0.01). The difference between measured and modelled values was large in physiological terms, suggesting that some optimisation of the model is required.

Model-Based Analysis of Mechanisms Responsible for Sleep-Induced Carbon Dioxide Differences

This work describes a comprehensive mathematical model of the human respiratory control system which incorporates the central mechanisms for predicting sleep-induced changes in chemical regulation of ventilation. The model integrates four individual compartments for gas storage and exchange, namely alveolar air, pulmonary blood, tissue capillary blood, body tissues, and gas transport between them. An essential mechanism in the carbon dioxide transport is its dissociation into bicarbonate and acid, where a buffering mechanism through hemoglobin is used to prevent harmfully low pH levels. In the current model, we assume high oxygen levels and consider intracellular hydrogen ion concentration as the principal respiratory control variable. The resulting system of delayed differential equations is solved numerically. With an appropriate choice of key parameters, such as velocity of blood flow and gain of a non-linear controller function, the model provides steady-state results consistent with our experimental observations measured in subjects across sleep onset. Dynamic predictions from the model give new insights into the behaviour of the system in subjects with different buffering capacities and suggest novel hypotheses for future experimental and clinical studies.

A cardiovascular-respiratory control system model including state delay with application to congestive heart failure in humans

This paper considers a model of the human cardiovascular-respiratory control system with one and two transport delays in the state equations describing the respiratory system. The effectiveness of the control of the ventilation rate is influenced by such transport delays because blood gases must be transported a physical distance from the lungs to the sensory sites where these gases are measured. The short term cardiovascular control system does not involve such transport delays although delays do arise in other contexts such as the baroreflex loop (see [46]) for example. This baroreflex delay is not considered here. The interaction between heart rate, blood pressure, cardiac output, and blood vessel resistance is quite complex and given the limited knowledge available of this interaction, we will model the cardiovascular control mechanism via an optimal control derived from control theory. This control will be stabilizing and is a reasonable approach based on mathematical considerations as well as being further motivated by the observation that many physiologists cite optimization as a potential influence in the evolution of biological systems (see, e.g., Kenner [29] or Swan [62]). In this paper we adapt a model, previously considered (Timischl [63] and Timischl et al. [64]), to include the effects of one and two transport delays. We will first implement an optimal control for the combined cardiovascular-respiratory model with one state space delay. We will then consider the effects of a second delay in the state space by modeling the respiratory control via an empirical formula with delay while the the complex relationships in the cardiovascular control will still be modeled by optimal control. This second transport delay associated with the sensory system of the respiratory control plays an important role in respiratory stability. As an application of this model we will consider congestive heart failure where this transport delay is larger than normal and the transition from the quiet awake state to stage 4 (NREM) sleep. The model can be used to study the interaction between cardiovascular and respiratory function in various situations as well as to consider the influence of optimal function in physiological control system performance.