A MATHEMATICAL MODEL OF THE HUMAN RESPIRATORY CONTROL SYSTEM

Systems Engineering Approach to Modeling and Analysis of Chronic Obstructive Pulmonary Disease Part II: Extension for Variable Metabolic Rates

Recently, we developed a systems engineering model of the human cardiorespiratory system [Kurian et al. ACS Omega2023, 8 (23), 20524-20535. DOI: 10.1021/acsomega.3c00854] based on existing models of physiological processes and adapted it for chronic obstructive pulmonary disease (COPD)-an inflammatory lung disease with multiple manifestations and one of the leading causes of death in the world. This control engineering-based model is extended here to allow for variable metabolic rates established at different levels of physical activity. This required several changes to the original model: the model of the controller was enhanced to include the feedforward loop that is responsible for cardiorespiratory control under varying metabolic rates (activity level, characterized as metabolic equivalent of the task-Rm-and normalized to one at rest). In addition, a few refinements were made to the cardiorespiratory mechanics, primarily to introduce physiological processes that were not modeled earlier but became important at high metabolic rates. The extended model is verified by analyzing the impact of exercise (Rm > 1) on the cardiorespiratory system of healthy individuals. We further formally justify our previously proposed adaptation of the model for COPD patients through sensitivity analysis and refine the parameter tuning through the use of a parallel tempering stochastic global optimization method. The extended model successfully replicates experimentally observed abnormalities in COPD-the drop in arterial oxygen tension and dynamic hyperinflation under high metabolic rates-without being explicitly trained on any related data. It also supports the prospects of remote patient monitoring in COPD.

Systems Engineering Approach to Modeling and Analysis of Chronic Obstructive Pulmonary Disease

Chronic obstructive pulmonary disease (COPD) is a progressive lung disease characterized by airflow limitation. This study develops a systems engineering framework for representing important mechanistic details of COPD in a model of the cardiorespiratory system. In this model, we present the cardiorespiratory system as an integrated biological control system responsible for regulating breathing. Four engineering control system components are considered: sensor, controller, actuator, and the process itself. Knowledge of human anatomy and physiology is used to develop appropriate mechanistic mathematical models for each component. Following a systematic analysis of the computational model, we identify three physiological parameters associated with reproducing clinical manifestations of COPD: changes in the forced expiratory volume, lung volumes, and pulmonary hypertension. We quantify the changes in these parameters (airway resistance, lung elastance, and pulmonary resistance) as the ones that result in a systemic response that is diagnostic of COPD. A multivariate analysis of the simulation results reveals that the changes in airway resistance have a broad impact on the human cardiorespiratory system and that the pulmonary circuit is stressed beyond normal under hypoxic environments in most COPD patients.

Self-Regulating Adaptive Controller for Oxygen Support to Severe Respiratory Distress Patients and Human Respiratory System Modeling

Uncontrolled breathing is the most critical and challenging situation for a healthcare person to patients. It may be due to simple cough/cold/critical disease to severe respiratory infection of the patients and resulting directly impacts the lungs and damages the alveoli which leads to shortness of breath and also impairs the oxygen exchange. The prolonged respiratory failure in such patients may cause death. In this condition, supportive care of the patients by medicine and a controlled oxygen supply is only the emergency treatment. In this paper, as a part of emergency support, the intelligent set-point modulated fuzzy PI-based model reference adaptive controller (SFPIMRAC) is delineated to control the oxygen supply to uncomforted breathing or respiratory infected patients. The effectiveness of the model reference adaptive controller (MRAC) is enhanced by assimilating the worthiness of fuzzy-based tuning and set-point modulation strategies. Since then, different conventional and intelligent controllers have attempted to regulate the supply of oxygen to respiratory distress patients. To overcome the limitations of previous techniques, researchers created the set-point modulated fuzzy PI-based model reference adaptive controller, which can react instantly to changes in oxygen demand in patients. Nonlinear mathematical formulations of the respiratory system and the exchange of oxygen with time delay are modeled and simulated for study. The efficacy of the proposed SFPIMRAC is tested, with transport delay and set-point variations in the devised respiratory model.

Contribution of Cardiorespiratory Coupling to the Irregular Dynamics of the Human Cardiovascular System

Irregularity is an important aspect of the cardiovascular system dynamics. Numerical indices of irregularity, such as the largest Lyapunov exponent and the correlation dimension estimated from interbeat interval time series, are early markers of cardiovascular diseases. However, there is no consensus on the origin of irregularity in the cardiovascular system. A common hypothesis suggests the importance of nonlinear bidirectional coupling between the cardiovascular system and the respiratory system for irregularity. Experimental investigations of this theory are severely limited by the capabilities of modern medical equipment and the nonstationarity of real biological systems. Therefore, we studied this problem using a mathematical model of the coupled cardiovascular system and respiratory system. We estimated and compared the numerical indices of complexity for a model simulating the cardiovascular dynamics in healthy subjects and a model with blocked regulation of the respiratory frequency and amplitude, which disturbs the coupling between the studied systems.

MATHEMATICAL MODELS OF HUMAN RESPIRATORY AND BLOOD CIRCULATORY SYSTEMS

Aim. To analyze modern approaches to mathematical modeling of human respiratory and blood circulatory systems. Methods. Comprehensive review of scientific literature sources extracted from domestic and international resources databases. Results. Historical information and modern data concerning mathematical modeling of human functional respiratory and blood circulatory systems were summarized and analyzed in present ¬review; current trends in approaches to the construction of these models were revealed. Conclusions. Currently, two main approaches to the mathematical modeling of respiratory and blood circulatory systems exist. One of them is the construction of models of the mechanics of respiration and blood circulation. They are based on the models of mechanics of solid deformable body, thermomechanics, hydromechanics, and continuum mechanics. This approach uses complex mathematical apparatus, including Navier-Stokes equation, which makes it possible to obtain a number of theoretical results, but it is hardly usable for real problems solutions at present time. The second approach is based on the model of F. Grodins, who represented the process of breathing as a controlled dynamic system, described by ordinary differential equations, in which the process control is carried out according to the feedback principle. There is a significant number of modifications of this model, which made it possible to simulate various disturbing influences, such as physical activity, hypoxia and hyperemia, and to predict parameters characterizing functional respiratory system under these disturbing influences.

Modeling cerebral blood flow and ventilation instability due to CO2

A minimal model of cerebral blood flow and respiratory control was developed to describe hypocapnic and hypercapnic responses. Important nonlinear properties such as cerebral blood flow changes with arterial partial pressure of carbon dioxide ([Formula: see text]) and associated time-dependent circulatory time delays were included. It was also necessary to vary cerebral metabolic rate as a function of [Formula: see text]. The cerebral blood flow model was added to a previously developed respiratory control model to simulate central and peripheral controller dynamics for humans. Model validation was based on previously collected data. The variable time delay due to brain blood flow changes in hypercapnia was an important determinant of predicted instability due to nonlinear interaction in addition to linear loop gain considerations. Peripheral chemoreceptor gains above a critical level, but within normal limits, were necessary to produce instability. Instability was observed in recovery from hypercapnia and hypocapnia. The 20-s breath-hold test appears to be a simple test of brain blood flow-mediated instability in hypercapnia. Brain blood flow was predicted to play an important role with nonlinear properties. There is an important interaction predicted by the current model between central and peripheral control mechanisms related to instability in hypercapnia recovery. Posthyperventilation breathing pattern can also reveal instability tied to brain blood flow. Previous data collected in patients with chronic obstructive lung disease were closely fitted with the current model and instability predicted. Brain vascular volume was proposed as a potential cause of instability despite cerebral autoregulation promoting constant brain flow.NEW & NOTEWORTHY Prior models of brain blood flow and respiratory control have not focused on instability. Time varying time delay resulting from brain blood flow changes due to carbon dioxide (CO₂) and peripheral chemoreceptor gain were predicted to be important determinants of instability due to nonlinear interaction in addition to linear control loop gain. Time delay was assumed to be set by the ratio of brain arterial vascular volume and blood flow. This vascular volume was predicted to also significantly change with CO₂.

Drug-disease modeling in the pharmaceutical industry - where mechanistic systems pharmacology and statistical pharmacometrics meet

Modeling & simulation (M&S) methodologies are established quantitative tools, which have proven to be useful in supporting the research, development (R&D), regulatory approval, and marketing of novel therapeutics. Applications of M&S help design efficient studies and interpret their results in context of all available data and knowledge to enable effective decision-making during the R&D process. In this mini-review, we focus on two sets of modeling approaches: population-based models, which are well-established within the pharmaceutical industry today, and fall under the discipline of clinical pharmacometrics (PMX); and systems dynamics models, which encompass a range of models of (patho-)physiology amenable to pharmacological intervention, of signaling pathways in biology, and of substance distribution in the body (today known as physiologically-based pharmacokinetic models) - which today may be collectively referred to as quantitative systems pharmacology models (QSP). We next describe the convergence - or rather selected integration - of PMX and QSP approaches into 'middle-out' drug-disease models, which retain selected mechanistic aspects, while remaining parsimonious, fit-for-purpose, and able to address variability and the testing of covariates. We further propose development opportunities for drug-disease systems models, to increase their utility and applicability throughout the preclinical and clinical spectrum of pharmaceutical R&D.

Undiscovered Physiology of Transcript and Protein Networks

The past two decades have witnessed a rapid evolution in our ability to measure RNA and protein from biological systems. As a result, new principles have arisen regarding how information is processed in cells, how decisions are made, and the role of networks in biology. This essay examines this technological evolution, reviewing (and critiquing) the conceptual framework that has emerged to explain how RNA and protein networks control cellular function. We identify how future investigations into transcriptomes, proteomes, and other cellular networks will enable development of more robust, quantitative models of cellular behavior whilst also providing new avenues to use knowledge of biological networks to improve human health. © 2016 American Physiological Society. Compr Physiol 6:1851-1872, 2016.

Periodic ventilation: Consequences for the bodily CO2 stores and gas exchange efficiency

Using a mathematical model of CO2 transport, we investigated the underlying cause of why and to what extent periodic ventilation is less efficient for CO2 excretion/elimination compared to continuous/tidal ventilation leading to elevated CO2 stores unless mean alveolar minute ventilation () is elevated. The model predicts that the reduced efficiency of periodic ventilation is intrinsic to the sequential arrangement and differences in the relative storage capacities (product of size and CO2 capacitance coefficient) of the lungs, blood and tissues that leads to predominant blood and tissue storage during apnoeic periods. Consequently, overall CO2 transport becomes more prone to perfusion and diffusion limitation during periodic ventilation. At constant cardiac output (Q.) inefficiency will increase with the apnoeic duration (tap) concomitant with increasing blood and tissues CO2 storage and with the relative time spent apnoeic (tap/tcyc) due to increasing V.A/Q. mismatch. Conversely, temporal variation of Q. to better match V.A can reduce inefficiency radically. Thus such adjustment in blood flow is necessary for efficient CO2 elimination in periodic ventilation.

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.

Development of a patient simulator for teaching and evaluation of the basic cardio-pulmonary reanimation protocol

Providing appropriate cardio-pulmonary reanimation after cardio-pulmonary arrest is paramount for survival. An effective and low-cost approach to learn and practice the cardio-pulmonary reanimation is through a computerized life-size patient simulator. The present work describes the development of a patient simulator for the Centre of Education and Certification of Medical Aptitudes (CECAM) from the UNAM's Faculty of Medicine. This patient simulator has many new and innovative features, such real-time feedback to the medical student, which improves the whole teaching/learning experience.

Design and construction of a real simulator for calibrating lung servo-ventilators

This work shows the theoretical and practical development of a lung simulator for the calibration of Servoventilators of common use in health centers. It shows the development of a prototype device, Shown in the paper the formulation of a model to consider factors that exist in a human respiratory system in order to simulate normal and pathological conditions. Includes the calculation and construction of electronical and fluidic systems that were developed to set up an emulator that allows real lung adequate to connect with any type of servoventilator; as well as the results in terms of graphics of the required functions, highlighting the practical part that behaves like a real lung subsequently introduced into a torso anthropomorphic designed to better emulate real operating conditions of the lung embedded in a actual context closest to where the components behave as does the lung of a patient.

An integrative model of respiratory and cardiovascular control in sleep-disordered breathing

While many physiological control models exist in the literature, none thus far has focused on characterizing the interactions among the respiratory, cardiovascular and sleep-wake regulation systems that occur in sleep-disordered breathing. The model introduced in this study integrates the autonomic control of the cardiovascular system, chemoreflex and state-related control of respiration, including respiratory and upper airway mechanics, along with a model of circadian and sleep-wake regulation. The integrative model provides realistic predictions of the physiological responses under a variety of conditions including: the sleep-wake cycle, hypoxia-induced periodic breathing, Cheyne-Stokes respiration in chronic heart failure, and obstructive sleep apnoea (OSA). It can be used to investigate the effects of a variety of interventions, such as isocapnic and hypercapnic and/or hypoxic gas administration, the Valsalva and Mueller maneuvers, and the application of continuous positive airway pressure on OSA subjects. By being able to delineate the influences of the various interacting physiological mechanisms, the model is useful in providing a more lucid understanding of the complex dynamics that characterize state-cardiorespiratory control in the different forms of sleep-disordered breathing.

Delay equation analysis of human respiratory stability

A mathematical analysis of the stability in human respiration, based on the tau-decomposition method, is conducted on a simple, but realistic CO2 model of the respiratory system. This model incorporates a two-compartment representation (lungs and tissues) for the plant and a very general class of controller. By deriving an explicit stability criterion, the stability domain of the respiratory system can be characterized. We quantify the influence of four major parameters of respiratory instability, i.e. transport delay, lung volume, and equilibrium values of lung CO2 partial pressure and controller gain. We demonstrate the existence of a bifurcation point and periodic solutions, giving some characteristics of solutions near the bifurcation point.

Mathematical analysis and computer simulation of the respiratory system in the newborn infant

A mathematical model of neonatal respiratory control is proposed which can be used to stimulate the system under different physiological conditions. The model consists of a continuous plant and a discrete controller. Included in the plant are lungs, body tissue, brain tissue, a cerebrospinal fluid compartment, and central and peripheral receptors. The effect of shunt in the lungs is included in the model and the lung volume and the dead space are time varying. The controller utilizes outputs from peripheral and central receptors to adjust the depth and rate of breathing and the effects of prematurity of peripheral receptors are included in the system. Hering-Breuer type reflexes are embodied in the controller to accomplish respiratory synchronization. The model is examined and its simulation results under test conditions in hypoxia and hypercapnia are presented.

Adaptive control of inspired oxygen delivery to the neonate

Adaptive adjustment of inspired oxygen (FIO2), based on a desired percent arterial hemoglobin saturation (SO2) was achieved by on-line bedside control of the oxygen concentration delivered to the neonate. Fourteen infants with bronchopulmonary dysplasia (BW, 860 +/- 80 g; GA, 26 +/- 1 weeks; study age, 41 +/- 8 days) receiving oxygen-air mixtures by hood were studied. The desired range of SO2 from 92 to 96% with a target value of 95% was determined by pulse oximetry and maintained with adjustment of FIO2 using three modes: 1) standard neonatal intensive care protocol with oxygen delivery evaluated at 20 minutes intervals; 2) bedside manual control with FIO2 manipulation every 2 to 5 minutes; and 3) adaptive control with on-line adjustment of FIO2 according to a specifically designed adaptive program. Each study period was of 40 minute duration. SO2 values within a steady 94 to 96% range was achieved for 54% of the time with standard protocol, compared to 69% (P less than 0.01) with bedside manual control and 81% (P less than 0.01) with adaptive control. In addition, fluctuations in SO2 values and overshoots were less apparent with adaptive control of oxygen delivery. These data describe adaptive FIO2 control as an efficient alternative technique for achieving a stable desired range of oxygenation in neonates.

A model of the maturation of respiratory control in the newborn infant

A model of automatic neonatal respiratory control has been constructed as an aid in the investigation of a possible maturation in respiratory control loops during the newborn period. The primary objective was to provide a framework for investigating this hypothesis without the need for external stimuli or invasive measurements. Spontaneous sighs provide a physiological disturbance to the respiratory system by transiently altering the levels of the blood gases. The dynamic ventilatory response following such a disturbance was modeled. A change from a highly damped to less damped pattern was found when model parameter values were varied to mimic maturation in the neonatal period. A perturbation model analysis demonstrated the dynamic ventilatory response is most sensitive to factors affecting the gain of the peripheral chemoreflex loop. It is concluded that the model provides valuable insight into the hypothesis that the peripheral chemoreflex matures during the neonatal period and provides a viable method for testing this in the human infant.

Expiratory airflow patterns and gas exchange in the newborn infant: results of model simulations

A mathematical model simulating the newborn human infant's respiratory system was used to study the effects on gas exchange of varying expiratory airflow pattern and end expiratory lung volume (FRC). Inspiratory flow was modelled as a square wave and was constant for all simulations as were inspiratory and expiratory times. Expiratory airflow was also modelled as a square wave and was varied between 21 and 75 ml/sec with FRC held constant at either 30.2 or 21.2 ml/kg for each simulation. At a given FRC, expiratory airflow pattern had only a trivial effect on blood gases in the steady state. Comparing the extreme cases, fast expiration (75 ml/sec) at low FRC (21.1 ml/kg) with slow expiration (21 ml/sec) at high FRC (30.2 ml/kg), arterial PO2 was 3.8 mm Hg higher and arterial PCO2 1.0 mm Hg lower under the latter conditions. However, when short apneas were imposed, blood gases deteriorated less precipitously following the slow expiration at high FRC. We conclude that expiratory airflow retardation and the resultant elevation in end expiratory lung volume do not greatly enhance gas exchange in the healthy full term infant. However, mechanisms which slow expiratory airflow do provide a buffer for gas exchange during the short apneas often observed in infants.

A mathematical model of the human respiratory system

A model of the human respiratory system is proposed which has a satisfactory performance under different physiological conditions. The model comprises a continuous plant and a discrete controller which generates and updates the drive signal to the plant at the end of every breath to represent the Hering-Breuer reflex. Arterial and central medullary sensors are included. The lung volume, dead space volume, cardiac output and cerebral blood flow are time varying. The respiratory work is minimized. The model is examined and simulation results of its performance in hypercapnia, hypoxia, periodic breathing and moderate exercise are presented. The responses presented include the relatively fast transients of Cheyne-Stokes breathing and the slower transients associated with carbon dioxide inhalation.

Oscillation and chaos in physiological control systems

First-order nonlinear differential-delay equations describing physiological control systems are studied. The equations display a broad diversity of dynamical behavior including limit cycle oscillations, with a variety of wave forms, and apparently aperiodic or "chaotic" solutions. These results are discussed in relation to dynamical respiratory and hematopoietic diseases.

[Mathematical simulation of the respiratory system (author's transl)]

The respiratory system is described as a feedback control system. The controller consists of the peripheral chemoreceptors and the central chemosensitive structures, the respiratory centre in the medulla oblongata and the thorax-lung pump which they drive. The controlled system is comprised of three compartments (lung, brain and the remaining tissue) connected by the blood circulation. The controlled values are arterial pH and arterial O2 partial pressure and cerebral extracellular pH. Earlier models have been improved by: (1) the dead space description, (2) the thermodynamic formulation of the CO2 dissociation equation and the simple but accurate O2 dissociation equation of the blood, (3) the alteration of the CO2 dissociation equation for the brain and the remaining tissue to accommodate recent results, (4) the application of the one-receptor-theory of central chemosensitivity, (5) the pH dependence of brain circulation, (6) the bicarbonate exchange between blood and extracellular fluid of the brain and (7) the introduction of variable circulation times. Respiratory and metabolic disturbances of the respiratory system are analyzed. The mathematical formulation of the respiratory system is a differential difference equation system. In the steady state the experimental results are reproduced fairly well. A slight discrepancy is found in the simulation of metabolic acidosis. Apparently we have assumed the sensitivity of the peripheral chemoreceptors to be too large so that the respiratory response is not correctly predicted. In the numerical solution there is an overshoot in the on-transient and a damped oscillation in the off-transient of the alveolar CO2 partial pressure during respiratory acidosis. We have varied the parameters to make deviations small. The best agreement seems to result, if the central threshold is near the normal extracellular pH of the brain. A further deviation from experimental findings is that the cerebral CO2 and H+ concentration, the blood circulation of the brain, the alveolar O2 partial tension and the ventilation show a slight oscillation in the off-transient. Except for these discrepancies the experimental results, especially the stability of the extracellular pH of the brain, are reproduced fairly well. During hypoxia there are deviations form the experimental results if the central residual activity is constant and the central threshold deviates from the normal extracellular pH of the brain. But if the central residual activity is pH dependent and if the central threshold is equal to the normal extracellular pH of the brain, then the time course of VE and the other variables agree fairly well with experimental results. There is also a good correspondence between the theoretical and experimental data during hyperoxia. During metabolic acidosis the time constant of the bicarbonate exchange between blood and extracellular fluid of the brain is important. If a time constant of one minute is assumed, then the predicted and the experimental results correspond sufficiently well.

Stimulation of paraventricular neurosecretory cells by oxytocin applied iontophoretically

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A mathematical model of the controlled plant of the respiratory system

Ability to predict the dynamic response of oxygen, carbon dioxide tensions, and pH in blood and tissues to abrupt changes in ventilation is important in the mathematical modeling of the respiratory system. In this study, the controlled plant (the amount and distribution of O(2) and CO(2)) of the respiratory system is modeled. Although the body tissues are divided into a finite number of "compartments" (three tissue groups), in contrast to earlier models, the blood and tissue gas tensions within each compartment are considered to be continuously distributed in time and in one spatial coordinate. The mass conservation equations for oxygen and carbon dioxide involved in the blood-tissue gas exchange are described by a set of partial differential equations which take into account convection of O(2) and CO(2) caused by the flow of blood as well as diffusion due to local tension gradients. Nonlinear algebraic equations for the dissociation curves, which take into account the Haldane and Bohr effects in blood, are used to obtain the relationships between concentrations and partial pressures. Time-variable delays caused by the arterial and venous transport of the respiratory gases are also included. The model so constructed successfully reproduced actual O(2) and CO(2) tensions in arterial blood, and in muscle venous and mixed venous blood when ventilation was abruptly changed.