Is our breathing optimal? Solving a piecewise linear model with constraints.

This paper is motivated by a question related to the control of amplitude and frequency of breathing. We present a simplified mathematical model, consisting of two piecewise linear ordinary differential equations, that could represent gas exchange in the lungs. We then define and solve an optimal control problem with unknown durations of inhalation and exhalation, subject to several constraints. The durations are divided such that one of the state variables is strictly increasing during the first phase and decreasing during the second phase. The optimal control problem can be solved analytically. One analytical solution is found when the forcing is a given sinusoidal function with unknown period and amplitude. Other analytical solutions are found when the forcing function, the period and the duration of the first phase are unknown but the amplitude is given. Our results show that different cost functions can produce different optimal forcing functions. We also show that the shape of these functions does not affect the average levels of oxygen in the lungs-the average level of oxygen is only dependent on the amplitude and period of breathing in the model we present.

Cardiorespiratory interactions in humans and animals: rhythms for life.

The cardiorespiratory system exhibits oscillations from a range of sources. One of the most studied oscillations is heart rate variability, which is thought to be beneficial and can serve as an index of a healthy cardiovascular system. Heart rate variability is dampened in many diseases including depression, autoimmune diseases, hypertension, and heart failure. Thus, understanding the interactions that lead to heart rate variability, and its physiological role, could help with prevention, diagnosis, and treatment of cardiovascular diseases. In this review, we consider three types of cardiorespiratory interactions: respiratory sinus arrhythmia (variability in heart rate at the frequency of breathing), cardioventilatory coupling (synchronization between the heart beat and the onset of inspiration), and respiratory stroke volume synchronization (the constant phase difference between the right and the left stroke volumes over one respiratory cycle). While the exact physiological role of these oscillations continues to be debated, the redundancies in the mechanisms responsible for its generation and its strong evolutionary conservation point to the importance of cardiorespiratory interactions. The putative mechanisms driving cardiorespiratory oscillations as well as the physiological significance of these oscillations will be reviewed. We suggest that cardiorespiratory interactions have the capacity to both dampen the variability in systemic blood flow as well as improve the efficiency of work done by the heart while maintaining physiological levels of arterial CO2. Given that reduction in variability is a prognostic indicator of disease, we argue that restoration of this variability via pharmaceutical or device-based approaches may be beneficial in prolonging life.

Robust Unidirectional Airflow through Avian Lungs: New Insights from a Piecewise Linear Mathematical Model.

Avian lungs are remarkably different from mammalian lungs in that air flows unidirectionally through rigid tubes in which gas exchange occurs. Experimental observations have been able to determine the pattern of gas flow in the respiratory system, but understanding how the flow pattern is generated and determining the factors contributing to the observed dynamics remains elusive. It has been hypothesized that the unidirectional flow is due to aerodynamic valving during inspiration and expiration, resulting from the anatomical structure and the fluid dynamics involved, however, theoretical studies to back up this hypothesis are lacking. We have constructed a novel mathematical model of the airflow in the avian respiratory system that can produce unidirectional flow which is robust to changes in model parameters, breathing frequency and breathing amplitude. The model consists of two piecewise linear ordinary differential equations with lumped parameters and discontinuous, flow-dependent resistances that mimic the experimental observations. Using dynamical systems techniques and numerical analysis, we show that unidirectional flow can be produced by either effective inspiratory or effective expiratory valving, but that both inspiratory and expiratory valving are required to produce the high efficiencies of flows observed in avian lungs. We further show that the efficacy of the inspiratory and expiratory valving depends on airsac compliances and airflow resistances that may not be located in the immediate area of the valving. Our model provides additional novel insights; for example, we show that physiologically realistic resistance values lead to efficiencies that are close to maximum, and that when the relative lumped compliances of the caudal and cranial airsacs vary, it affects the timing of the airflow across the gas exchange area. These and other insights obtained by our study significantly enhance our understanding of the operation of the avian respiratory system.

Forgotten Circulation: Reduced Mesenteric Venous Capacitance in Hypertensive Rats Is Improved by Decreasing Sympathetic Activity.

BACKGROUND: The mesenteric venous reservoir plays a vital role in mediating blood volume and pressure changes and is richly innervated by sympathetic nerves; however, the precise nature of venous sympathetic regulation and its role during hypertension remains unclear. We hypothesized that sympathetic drive to mesenteric veins in spontaneously hypertensive (SH) rats is raised, increasing mean circulatory filling pressure (MCFP), and impairing mesenteric capacitance. METHODS: Arterial pressure, central venous pressure, mesenteric arterial, and venous blood flow were measured simultaneously in conscious male Wistar and SH rats. MCFP was assessed using an intraatrial balloon. Hemodynamic responses to volume changes (±20%) were measured before and after ganglionic blockade and carotid body denervation. Sympathetic venoconstrictor activity was measured in situ. RESULTS: MCFP in vivo (10.8±1.6 versus 8.0±2.1 mm Hg; P=0.0005) and sympathetic venoconstrictor drive in situ (18±1 versus 10±2 µV; P<0.0001) were higher in SH rats; MCFP decreased in SH rats after hexamethonium and carotid body denervation (7.6±1.4; P<0.0001 and 8.5±1.0 mm Hg; P=0.0045). During volume changes, arterial pressure remained stable. With blood loss, net efflux of blood from the mesenteric bed was measured in both strains. However, during volume infusion, we observed net influx in Wistar (+2.3±2.6 mL/min) but efflux in SH rats (-1.0±1.0 mL/min; P=0.0032); this counterintuitive efflux was abolished by hexamethonium and carotid body denervation (+0.3±1.7 and 0.5±1.6 mL/min, respectively). CONCLUSIONS: In SH rats, excessive sympathetic venoconstriction elevates MCFP and reduces capacitance, impairing volume buffering by mesenteric veins. We propose selective targeting of mesenteric veins through sympathetic drive reduction as a novel therapeutic opportunity for hypertension.

Prolonged remission of advanced bronchoalveolar adenocarcinoma in a dog treated with autologous, tumour-derived chaperone-rich cell lysate (CRCL) vaccine.

PURPOSE: This paper presents the treatment of a 12-year-old female spayed Great Dane who presented with vestibular signs (ataxia, nystagmus, hind end collapse). Thoracic radiographs revealed a discrete pulmonary nodule in the right cranial lung lobe. Ultrasound-guided fine needle aspirate detected primary bronchoalveolar adenocarcinoma, verified via computed tomography, with a second smaller nodule discovered in the right cranial lung lobe. MATERIALS AND METHODS: A lateral thoracotomy with right cranial lung lobectomy was performed. Histopathological analysis of the nodules and an excised lymph node identified grade III bronchoalveolar adenocarcinoma with vascular infiltration and lymph node metastasis - a grim diagnosis with a reported median survival time of 6-27 days. A 10-g sample of the tumour was processed into a chaperone-rich cell lysate (CRCL) vaccine, which was administered weekly to the patient. Imiquimod - a Toll-like receptor 7 (TLR7) agonist - was applied topically for the first 12 treatments to stimulate local Langerhans cells. A single injection of bacillus Calmette-Guerin (BCG) was administered for additional immune stimulation at week 30 of treatment. RESULTS: The dog remained stable and in otherwise good health until diffuse relapse occurred 44 weeks after the initial treatment; following gastrointestinal bleeding, the dog was euthanised 50+ weeks post diagnosis. CONCLUSION: To the authors' knowledge, this is the first report of significantly prolonged survival following a diagnosis of grade III/stage III bronchoalveolar adenocarcinoma in a canine patient. This case report suggests that CRCL vaccine combined with topical imiquimod is a safe, effective treatment for canine tumours.

Mathematical modelling of atrial and ventricular pressure-volume dynamics and their change with heart rate.

This paper presents mathematical models that can simulate the cardiovascular system of a healthy sheep under normal resting conditions in which the heart rate changes significantly. The models include several new modelling features that are introduced progressively. The contraction of the cardiac chambers is modelled using a time-dependent muscle force with constant elasticity instead of time dependent elasticity. A new hypothesis about the mechanical contraction of the atria generates realistic pressure volume (PV) loops. The inter-ventricular interaction is modelled as well. Additionally, hysteresis is incorporated in the aortic valve to produce an end-systolic reverse (negative) flow. Most of the model parameter values are based on previous literature data while time periods of delay, atrial and ventricular contraction are derived using experimental data from 14 sheep. We provide new relationships between contraction time and delay as a function of heart period. The effects of different aspects of our modelling on the mean cardiac output, stroke volume, ejection time, ejection fraction and PV loops are studied. Model outputs are compared with published experimental results where possible, and are within a wide range of physiological observations.

The logic behind neural control of breathing pattern.

The respiratory rhythm generator is spectacular in its ability to support a wide range of activities and adapt to changing environmental conditions, yet its operating mechanisms remain elusive. We show how selective control of inspiration and expiration times can be achieved in a new representation of the neural system (called a Boolean network). The new framework enables us to predict the behavior of neural networks based on properties of neurons, not their values. Hence, it reveals the logic behind the neural mechanisms that control the breathing pattern. Our network mimics many features seen in the respiratory network such as the transition from a 3-phase to 2-phase to 1-phase rhythm, providing novel insights and new testable predictions.

A model for control of breathing in mammals: coupling neural dynamics to peripheral gas exchange and transport.

A new model for aspects of the control of respiration in mammals has been developed. The model integrates a reduced representation of the brainstem respiratory neural controller together with peripheral gas exchange and transport mechanisms. The neural controller consists of two components. One component represents the inspiratory oscillator in the pre-Bötzinger complex (pre-BötC) incorporating biophysical mechanisms for rhythm generation. The other component represents the ventral respiratory group (VRG), which is driven by the pre-BötC for generation of inspiratory (pre)motor output. The neural model was coupled to simplified models of the lungs incorporating oxygen and carbon dioxide transport. The simplified representation of the brainstem neural circuitry has regulation of both frequency and amplitude of respiration and is done in response to partial pressures of oxygen and carbon dioxide in the blood using proportional (P) and proportional plus integral (PI) controllers. We have studied the coupled system under open and closed loop control. We show that two breathing regimes can exist in the model. In one regime an increase in the inspiratory frequency is accompanied by an increase in amplitude. In the second regime an increase in frequency is accompanied by a decrease in amplitude. The dynamic response of the model to changes in the concentration of inspired O2 or inspired CO2 was compared qualitatively with experimental data reported in the physiological literature. We show that the dynamic response with a PI-controller fits the experimental data better but suggests that when high levels of CO2 are inspired the respiratory system cannot reach steady state. Our model also predicts that there could be two possible mechanisms for apnea appearance when 100% O2 is inspired following a period of 5% inspired O2. This paper represents a novel attempt to link neural control and gas transport mechanisms, highlights important issues in amplitude and frequency control and sets the stage for more complete neurophysiological control models.

Central regulation of heart rate and the appearance of respiratory sinus arrhythmia: new insights from mathematical modeling.

A minimal model for the neural control of heart rate (HR) has been developed with the aim of better understanding respiratory sinus arrhythmia (RSA)--a modulation of HR at the frequency of breathing. This model consists of two differential equations and is integrated into a previously-published model of gas exchange. The heart period is assumed to be affected primarily by the parasympathetic signal, with the sympathetic signal taken as a parameter in the model. We include the baroreflex, mechanical stretch-receptor feedback from the lungs, and central modulation of the cardiac vagal tone by the respiratory drive. Our model mimics a range of experimental observations and provides several new insights. Most notably, the model mimics the growth in the amplitude of RSA with decreasing respiratory frequency up to 7 breaths per minute (for humans). Our model then mimics the decrease in the amplitude of RSA at frequencies below 7 breaths per minute and predicts that this decrease is due to the baroreflex (we show this both numerically and analytically with a linear baroreflex). Another new prediction of the model is that the gating of the baroreflex leads to the dependency of RSA on mean vagal tone. The new model was also used to test two previously-suggested hypotheses regarding the physiological function of RSA and supports the hypothesis that RSA minimizes the work done by the heart while maintaining physiological levels of arterial CO2. These and other new insights the model provides extend our understanding of the integrative nature of vagal control of the heart.

A closed-loop model of the respiratory system: focus on hypercapnia and active expiration.

Breathing is a vital process providing the exchange of gases between the lungs and atmosphere. During quiet breathing, pumping air from the lungs is mostly performed by contraction of the diaphragm during inspiration, and muscle contraction during expiration does not play a significant role in ventilation. In contrast, during intense exercise or severe hypercapnia forced or active expiration occurs in which the abdominal "expiratory" muscles become actively involved in breathing. The mechanisms of this transition remain unknown. To study these mechanisms, we developed a computational model of the closed-loop respiratory system that describes the brainstem respiratory network controlling the pulmonary subsystem representing lung biomechanics and gas (O2 and CO2) exchange and transport. The lung subsystem provides two types of feedback to the neural subsystem: a mechanical one from pulmonary stretch receptors and a chemical one from central chemoreceptors. The neural component of the model simulates the respiratory network that includes several interacting respiratory neuron types within the Bötzinger and pre-Bötzinger complexes, as well as the retrotrapezoid nucleus/parafacial respiratory group (RTN/pFRG) representing the central chemoreception module targeted by chemical feedback. The RTN/pFRG compartment contains an independent neural generator that is activated at an increased CO2 level and controls the abdominal motor output. The lung volume is controlled by two pumps, a major one driven by the diaphragm and an additional one activated by abdominal muscles and involved in active expiration. The model represents the first attempt to model the transition from quiet breathing to breathing with active expiration. The model suggests that the closed-loop respiratory control system switches to active expiration via a quantal acceleration of expiratory activity, when increases in breathing rate and phrenic amplitude no longer provide sufficient ventilation. The model can be used for simulation of closed-loop control of breathing under different conditions including respiratory disorders.

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.

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.

Optimal nutritional intake for fetal growth.

The regular nutritional intake of an expectant mother clearly affects the weight development of the fetus. Assuming the growth of the fetus follows a deterministic growth law, like a logistic equation, albeit dependent on the nutritional intake, the ideal solution is usually determined by the birth-weight being pre-assigned, for example, as a percentage of the mother's average weight. This problem can then be specified as an optimal control problem with the daily intake as the control, which appears in a Michaelis-Menten relationship, for which there are well-developed procedures to follow. The best solution is determined by requiring minimum total intake under which the preassigned birth weight is reached. The algorithm has been generalized to the case where the fetal weight depends in a detailed way on the cumulative intake, suitably discounted according to the history. The optimality system is derived and then solved numerically using an iterative method for the specific values of parameter. The procedure is generic and can be adapted to any growth law and any parameterisation obtained by the detailed physiology.

Control of breathing: two types of delays studied in an integrated model of the respiratory system.

We use a recently developed mathematical model that integrates a reduced representation of the brainstem respiratory neural controller together with peripheral gas exchange and transport to study numerically the dynamic response of the respiratory system to several physiological stimuli. We compare between the system responses with two major sources of delay: circulatory transport vs. neural feedback dynamics, and we show that the dynamics of the neural feedback processes dictates the dynamic response to hypoxia and hypercapnia. The source of the circulatory delay (blood velocity vs. distance from the lungs to chemoreceptors) was found to be important. Our model predicts that periodic breathing is associated with the ventilatory "afterdischarge" (slow recovery of ventilation) after a brief perturbation of CO(2). We also predict that there could be two possible mechanisms for the appearance of periodic breathing and that circulatory delay is not a necessary condition for this to occur in certain cases.

Simplified models for gas exchange in the human lungs.

This paper presents a hierarchy of models with increasing complexity for gas exchange in the human lungs. The models span from a single compartment, inflexible lung to a single compartment, flexible lung with pulmonary gas exchange. It is shown how the models are related to well-known models in the literature. A long-term purpose of this work is to study nonlinear phenomena seen in the cardio-respiratory system (for example, synchronization between ventilation rate and heart rate, and Cheyne-Stokes respiration). The models developed in this paper can be regarded as the controlled system (plant) and provide a mathematical framework to link between "molecular-level", and "systems-level" models. It is shown how changes in molecular level affect the alveolar partial pressure. Two assumptions that have previously been made are re-examined: (1) the hidden assumption that the air flow through the mouth is equal to the rate of volume change in the lungs, and, (2) the assumption that the process of oxygen binding to hemoglobin is near equilibrium. Conditions under which these assumptions are valid are studied. All the parameters in the models, except two, are physiologically realistic. Numerical results are consistent with published experimental observations.