A Mathematical Model of Human Respiration at Altitude

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.

Prediction of exhaled carbon dioxide concentration using a computer-simulated person that included alveolar gas exchange

Accurate prediction of inhaled CO₂ concentration and alveolar gas exchange efficiency would improve the prediction of CO₂ concentrations around the human body, which is essential for advanced ventilation design in buildings. We therefore, developed a computer-simulated person (CSP) that included a computational fluid dynamics approach. The CSP simulates metabolic heat production at the skin surface and carbon dioxide (CO₂ ) gas exchange at the alveoli during the transient breathing cycle. This makes it possible to predict the CO₂ distribution around the human body. The numerical model of the CO₂ gas exchange mechanism includes both the upper and lower airways and makes it possible to calculate the alveolar CO₂ partial pressure; this improves the prediction accuracy. We used the CSP to predict emission rates of metabolically generated CO₂ exhaled by a person and assumed that the tidal volume will be unconsciously reduced as a result of exposure to poor indoor air quality. A reduction in tidal volume resulted in a decrease in CO₂ emission rates of the same magnitude as was observed in our published experimental data. We also observed that the predicted inhaled CO₂ concentration depended on the flow pattern around the human body, as would be expected.

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.

Ventilatory oscillations at exercise in hypoxia: A mathematical model

We evaluated the mechanisms responsible for the instability of ventilation control system under simultaneous metabolic (exercise) and environmental (hypoxia) stresses, promoting the genesis of periodic breathing. A model following the main concepts of ventilatory control has been tested, including cardiovascular and respiratory parameters, characteristics of peripheral and central chemoreceptors, at mild exercise in hypoxia (FIO₂=0.145). Interaction between O₂ and CO₂ sensing was introduced following three different modalities. A sensitivity and multivariate regression analyses closely matched with physiological data for magnitude and period of oscillations. Low FIO₂ and long circulatory delay from lungs to peripheral chemoreceptors (DeltaTp) lengthen the period of oscillations, while high peripheral and central chemoresponses to O₂ and CO₂, low FIO₂ and high DeltaTp increased their magnitude. Peripheral and central O₂/CO₂ interactions highlight the role of CO₂ on peripheral gain to O₂ and the contribution of peripheral afferences on central gain to CO₂. Our model supports the key role of peripheral chemoreceptors in the genesis of ventilatory oscillations. Differences in the dynamics of central and peripheral components might be determinant for the system stability.

A Comparative Data-Based Modeling Study on Respiratory CO2 Gas Exchange during Mechanical Ventilation

The goal of this study is to derive a minimally complex but credible model of respiratory CO2 gas exchange that may be used in systematic design and pilot testing of closed-loop end-tidal CO2 controllers in mechanical ventilation. We first derived a candidate model that captures the essential mechanisms involved in the respiratory CO2 gas exchange process. Then, we simplified the candidate model to derive two lower-order candidate models. We compared these candidate models for predictive capability and reliability using experimental data collected from 25 pediatric subjects undergoing dynamically varying mechanical ventilation during surgical procedures. A two-compartment model equipped with transport delay to account for CO2 delivery between the lungs and the tissues showed modest but statistically significant improvement in predictive capability over the same model without transport delay. Aggregating the lungs and the tissues into a single compartment further degraded the predictive fidelity of the model. In addition, the model equipped with transport delay demonstrated superior reliability to the one without transport delay. Further, the respiratory parameters derived from the model equipped with transport delay, but not the one without transport delay, were physiologically plausible. The results suggest that gas transport between the lungs and the tissues must be taken into account to accurately reproduce the respiratory CO2 gas exchange process under conditions of wide-ranging and dynamically varying mechanical ventilation conditions.

Comprehensive diagnosis of whole-body acid-base and fluid-electrolyte disorders using a mathematical model and whole-body base excess

A mathematical model of whole-body acid-base and fluid-electrolyte balance was used to provide information leading to the diagnosis and fluid-therapy treatment in patients with complex acid-base disorders. Given a set of measured laboratory-chemistry values for a patient, a model of their unique, whole-body chemistry was created. This model predicted deficits or excesses in the masses of Na(+), K(+), Cl(-) and H2O as well as the plasma concentration of unknown or unmeasured species, such as ketoacids, in diabetes mellitus. The model further characterized the acid-base disorder by determining the patient's whole-body base excess and quantitatively partitioning it into ten components, each contributing to the overall disorder. The results of this study showed the importance of a complete set of laboratory measurements to obtain sufficient accuracy of the quantitative diagnosis; having only a minimal set, just pH and PCO2, led to a large scatter in the predicted results. A computer module was created that would allow a clinician to achieve this diagnosis at the bedside. This new diagnostic approach should prove to be valuable in the treatment of the critically ill.

Transient respiratory response to hypercapnia: analysis via a cardiopulmonary simulation model

In recent years, our group has developed a comprehensive cardiopulmonary (CP) model that comprises the heart, systemic and pulmonary circulations, lung mechanics and gas exchange, tissue metabolism, and cardiovascular and respiratory control mechanisms. In this paper, we analyze the response of the model to hypercapnic conditions and hence focus on the chemoreflex control mechanism. Particularly, we have enhanced the peripheral chemoreceptor model in order to better reflect respiratory control physiology. Using the CO(2) fraction in the inspired air as input to the CP model, we were able to analyze the transient response of the system to CO(2) step input at different levels, in terms of alveolar gas partial pressures, tidal volume, minute ventilation and respiratory frequency. Model predictions were tested against experimental data from human subjects [1]. Results show good agreement for all the variables under study during the transient phases and low root mean square errors at steady state. This indicates the potential for the model to be used as a valid tool for clinical practice and medical research, providing a complementary way to experience-based clinical decisions.

The role of the central chemoreceptors: a modeling perspective

After introducing the respiratory control system, a previously developed model of the respiratory chemoreflexes, based on rebreathing test data, is briefly described. This model is used to gain insights into the respiratory chemoreflex characteristics of a selection of individuals, and so discover the role of their central chemoreceptors. The chemoreflex model characteristics for each individual were estimated by adjusting the model parameters so that its predictions fit their rebreathing test results. To gain a steady state description of the control of breathing at rest the chemoreflex model is combined with a model of the cerebrovascular reactivity and converted from P(CO)₂ to [H(+)] chemoreceptor inputs. This description is used to illustrate how acid-base and cerebrovascular reactivity factors affect the environment of the central chemoreceptors and determine their role in breathing control. Finally, a dynamic model incorporating the chemoreflex model, acid-base and cerebrovascular reactivity is used to show the role of the central chemoreceptors in stabilizing breathing during sleep at altitude.

Disequilibrium between alveolar and end-pulmonary-capillary O2 tension in altitude hypoxia and respiratory disease: an update of a mathematical model of human respiration at altitude

We have previously formulated and validated a mathematical model specifically designed to describe human respiratory behavior at altitude. In that model, we assumed equality of alveolar and end-pulmonary-capillary oxygen tensions. However, this equality may not hold true during rapid and prolonged changes to high altitudes producing severe hypoxia as can occur in aircraft cabin decompressions and in some respiratory diseases. We currently investigate this possibility by modifying our previous model to include the dynamics of oxygen exchange across the pulmonary capillary. The updated model was validated against limited experimental data on ventilation and gas tensions in various altitude-decompression scenarios. The updated model predicts that during rapid and sustained decompressions to high altitudes the disequilibrium of gas tensions between alveolar gas and capillary blood could be 10 Torr, or larger. Neglecting this effect underestimates the severity of a decompression and its potential to produce unconsciousness and subsequent brain damage. In light of these results, we also examined the effect of this disequilibrium on the diminished oxygen diffusion capacity that can occur in some respiratory diseases. We found that decreases in diffusion capacity which would have minimal effects at sea level produced significant disequilibrium of gas tensions and a large fall in hemoglobin oxygen saturation at a cabin altitude of 4000-8000 ft. As demonstrated, this new model could serve as an important tool to examine the important physiological consequences of decompression scenarios in aircraft and the pathophysiological situations in which the equilibrium of gas tensions along the pulmonary capillary are particularly critical.