An Optimal Control Approach to Modeling the Cardiovascular-Respiratory System: An Application to Orthostatic Stress

Development and validation of a mathematical model to simulate human cardiovascular and respiratory responses to battlefield trauma

Mathematical models of human cardiovascular and respiratory systems provide a viable alternative to generate synthetic data to train artificial intelligence (AI) clinical decision-support systems and assess closed-loop control technologies, for military medical applications. However, existing models are either complex, standalone systems that lack the interface to other applications or fail to capture the essential features of the physiological responses to the major causes of battlefield trauma (i.e., hemorrhage and airway compromise). To address these limitations, we developed the cardio-respiratory (CR) model by expanding and integrating two previously published models of the cardiovascular and respiratory systems. We compared the vital signs predicted by the CR model with those from three models, using experimental data from 27 subjects in five studies, involving hemorrhage, fluid resuscitation, and respiratory perturbations. Overall, the CR model yielded relatively small root mean square errors (RMSEs) for mean arterial pressure (MAP; 20.88 mm Hg), end-tidal CO₂ (ETCO₂ ; 3.50 mm Hg), O₂ saturation (SpO₂ ; 3.40%), and arterial O₂ pressure (PaO₂ ; 10.06 mm Hg), but a relatively large RMSE for heart rate (HR; 70.23 beats/min). In addition, the RMSEs for the CR model were 3% to 10% smaller than the three other models for HR, 11% to 15% for ETCO₂ , 0% to 33% for SpO₂ , and 10% to 64% for PaO₂ , while they were similar for MAP. In conclusion, the CR model balances simplicity and accuracy, while qualitatively and quantitatively capturing human physiological responses to battlefield trauma, supporting its use to train and assess emerging AI and control systems.

A mathematical model of cardiovascular dynamics for the diagnosis and prognosis of hemorrhagic shock

A variety of mathematical models of the cardiovascular system have been suggested over several years in order to describe the time-course of a series of physiological variables (i.e. heart rate, cardiac output, arterial pressure) relevant for the compensation mechanisms to perturbations, such as severe haemorrhage. The current study provides a simple but realistic mathematical description of cardiovascular dynamics that may be useful in the assessment and prognosis of hemorrhagic shock. The present work proposes a first version of a differential-algebraic equations model, the model dynamical ODE model for haemorrhage (dODEg). The model consists of 10 differential and 14 algebraic equations, incorporating 61 model parameters. This model is capable of replicating the changes in heart rate, mean arterial pressure and cardiac output after the onset of bleeding observed in four experimental animal preparations and fits well to the experimental data. By predicting the time-course of the physiological response after haemorrhage, the dODEg model presented here may be of significant value for the quantitative assessment of conventional or novel therapeutic regimens. The model may be applied to the prediction of survivability and to the determination of the urgency of evacuation towards definitive surgical treatment in the operational setting.

Seven Mathematical Models of Hemorrhagic Shock

Although mathematical modelling of pressure-flow dynamics in the cardiocirculatory system has a lengthy history, readily finding the appropriate model for the experimental situation at hand is often a challenge in and of itself. An ideal model would be relatively easy to use and reliable, besides being ethically acceptable. Furthermore, it would address the pathogenic features of the cardiovascular disease that one seeks to investigate. No universally valid model has been identified, even though a host of models have been developed. The object of this review is to describe several of the most relevant mathematical models of the cardiovascular system: the physiological features of circulatory dynamics are explained, and their mathematical formulations are compared. The focus is on the whole-body scale mathematical models that portray the subject's responses to hypovolemic shock. The models contained in this review differ from one another, both in the mathematical methodology adopted and in the physiological or pathological aspects described. Each model, in fact, mimics different aspects of cardiocirculatory physiology and pathophysiology to varying degrees: some of these models are geared to better understand the mechanisms of vascular hemodynamics, whereas others focus more on disease states so as to develop therapeutic standards of care or to test novel approaches. We will elucidate key issues involved in the modeling of cardiovascular system and its control by reviewing seven of these models developed to address these specific purposes.

Cardiovascular dynamics during head-up tilt assessed via pulsatile and non-pulsatile models

This study develops non-pulsatile and pulsatile models for the prediction of blood flow and pressure during head-up tilt. This test is used to diagnose potential pathologies within the autonomic control system, which acts to keep the cardiovascular system at homeostasis. We show that mathematical modeling can be used to predict changes in cardiac contractility, vascular resistance, and arterial compliance, quantities that cannot be measured but are useful to assess the system's state. These quantities are predicted as time-varying parameters modeled using piecewise linear splines. Having models with various levels of complexity formulated with a common set of parameters, allows us to combine long-term non-pulsatile simulations with pulsatile simulations on a shorter time-scale. We illustrate results for a representative subject tilted head-up from a supine position to a [Formula: see text] angle. The tilt is maintained for 5 min before the subject is tilted back down. Results show that if volume data is available for all vascular compartments three parameters can be identified, cardiovascular resistance, vascular compliance, and ventricular contractility, whereas if model predictions are made against arterial pressure and cardiac output data alone, only two parameters can be estimated either resistance and contractility or resistance and compliance.

An optimal control approach for blood pressure regulation during head-up tilt

This paper presents an optimal control approach to modeling effects of cardiovascular regulation during head-up tilt (HUT). Many patients who suffer from dizziness or light-headedness are administered a head-up tilt test to explore potential deficits within the autonomic control system, which maintains the cardiovascular system at homeostasis. This system is complex and difficult to study in vivo, and thus we propose to use mathematical modeling to achieve a better understanding of cardiovascular regulation during HUT. In particular, we show the feasibility of using optimal control theory to compute physiological control variables, vascular resistance and cardiac contractility, quantities that cannot be measured directly, but which are useful to assess the state of the cardiovascular system. A non-pulsatile lumped parameter model together with pseudo- and clinical data are utilized in the optimal control problem formulation. Results show that the optimal control approach can predict time-varying quantities regulated by the cardiovascular control system. Our results compare favorable to our previous study using a piecewise linear spline approach, less a priori knowledge is needed, and results were obtained at a significantly lower computational cost.

A model of the effects of fluid variation due to body position on Cheyne-Stokes respiration

Cheyne-Stokes respiration is a distinct breathing pattern consisting of periods of hyperpnea followed by apnoeas, with unknown aetiology. One in two patients with congestive heart failure suffers from this condition. Researchers hypothesize that key factors in CSR are the fluid shift from the standing to supine position and the differences between genders. A mathematical model of the cardiorespiratory system was constructed using parameter values from real data. Hopf bifurcation analysis was used to determine regions of steady versus oscillatory breathing patterns. In the model, Cheyne-Stokes respiration is more likely to occur while in the supine position and males are more likely to develop Cheyne-Stokes than females. These findings, which are in agreement with clinical experience, suggest that both gender and fluid shift contribute to the pathogenesis of Cheyne-Stokes respiration and that physical quantities such as blood volumes and neural feedback may be sufficient to explain the observations of CSR.

Patterns of cardiovascular control during repeated tests of orthostatic loading

To investigate patterns of cardiovascular control, a protocol of head up tilt (HUT) followed by lower body negative pressure (LBNP), which represents a significant cardiovascular control challenge, was employed. Linear regression of beat-to-beat heart rate (HR) and mean blood pressure (MBP) data collected over repeated tests was used to analyze control response during the LBNP phase of the combined HUT + LBNP protocol. Four runs for each of 10 healthy young males reaching presyncope were analyzed. Subjects were classified into 2 groups based on the consistency of MBP regulation in response to central hypovolemia induced by LBNP. The consistent group tended to exhibit consistent HR slope (rate of change of HR over time as calculated by linear regression) whereas subjects in the inconsistent group could not be easily classified. Subjects with consistent MBP maintenance exhibited patterns suggesting a consistency of response in cardiovascular control whereas subjects less successful in maintaining MBP exhibited less clearly defined patterns over four runs.

Cerebral autoregulation: from models to clinical applications

Short-term regulation of cerebral blood flow (CBF) is controlled by myogenic, metabolic and neurogenic mechanisms, which maintain flow within narrow limits, despite large changes in arterial blood pressure (ABP). Static cerebral autoregulation (CA) represents the steady-state relationship between CBF and ABP, characterized by a plateau of nearly constant CBF for ABP changes in the interval 60-150 mmHg. The transient response of the CBF-ABP relationship is usually referred to as dynamic CA and can be observed during spontaneous fluctuations in ABP or from sudden changes in ABP induced by thigh cuff deflation, changes in posture and other manoeuvres. Modelling the dynamic ABP-CBFV relationship is an essential step to gain better insight into the physiology of CA and to obtain clinically relevant information from model parameters. This paper reviews the literature on the application of CA models to different clinical conditions. Although mathematical models have been proposed and should be pursued, most studies have adopted linear input-output ('black-box') models, despite the inherently non-linear nature of CA. The most common of these have been transfer function analysis (TFA) and a second-order differential equation model, which have been the main focus of the review. An index of CA (ARI), and frequency-domain parameters derived from TFA, have been shown to be sensitive to pathophysiological changes in patients with carotid artery disease, stroke, severe head injury, subarachnoid haemorrhage and other conditions. Non-linear dynamic models have also been proposed, but more work is required to establish their superiority and applicability in the clinical environment. Of particular importance is the development of multivariate models that can cope with time-varying parameters, and protocols to validate the reproducibility and ranges of normality of dynamic CA parameters extracted from these models.

Aspects of control of the cardiovascular-respiratory system during orthostatic stress induced by lower body negative pressure

This paper considers a model developed to study the cardiovascular control system response to orthostatic stress as induced by two variations of lower body negative pressure (LBNP) experiments. This modeling approach has been previously applied to study control responses to transition from rest to aerobic exercise, to transition to non-REM sleep and to orthostatic stress as produced by the head up tilt (HUT) experiment. LBNP induces a blood volume shift because negative pressure changes the volume loading characteristics of the compartment which is subject to the negative pressure. This volume shift induces a fall in blood pressure which must be counteracted by a complicated control response involving a variety of mechanisms of the cardiovascular control system. There are a number of medical issues connected to these questions such as orthostatic intolerance in the elderly resulting in dizziness or fainting during the transition from sitting to standing. The model presented here is used to study the interaction of changes in systemic resistance, unstressed venous volume, venous compliance, heart rate, and contractility in the control of orthostatic stress. The overall short term response depends on a combination of these physiological reactions which may vary from individual to individual. There remain open questions about which factors have greater importance. The model simulations are compared to experimental data collected for LBNP exerted from the hips to feet and from ribs to feet.