Simulation of human breathing gas exchange for the ventilation regulation study

Ermolaev Evgenii Sergeevich ORCID: 0000-0001-9325-703X PhD in Biology Researcher; SSC RF — IMBP RAS 76 A Khoroshevskoe highway, Moscow, 123007, Russia 1861894@mail.ru D'yachenko Aleksandr Ivanovich ORCID: 0000-0002-5272-222X Doctor of Technical Science Head of laboratory, IMBP RAS 76A Khoroshevskoe shosse, Moscow, 123007, Russia alexander-dyachenko@yandex.ru

Shulagin Yurii Alekseevich PhD in Biology Leading researcher, IMBP RAS 76A Khoroshevskoe shosse, Moscow, 123007, Russia shulagin-yury@yandex.ru

1. INTRODUCTION

Central and peripheral chemoreflex play a key role in the regulation of pulmonary ventilation in response to changes in the partial pressure of CO ₂ and O ₂ in the inhaled air.

Recently, data have been actively published related to the effect of chronic intermittent hypoxia on respiratory regulation, changes that occur during sleep apnea. In addition, the effects of extreme conditions, such as during emergency work in mines or deep-sea diving, during space flights, as well as the effects of microgravity, can change the reaction of the respiratory system to CO ₂ and O ₂ and other gases. Respiratory regulation research is of particular practical use for the development of personal respiratory protection equipment used in various conditions unsuitable for normal human activity, such as fires, mine collapses and other disasters.

To understand the effect of various effects of extreme conditions on respiratory regulation, it is necessary to study the ventilation response to hypoxia and hypercapnia both at rest and in conditions of long-term and short-term stay in such conditions.

A typical example of such changes in the regulation of respiration in extreme conditions may be the effect on the respiratory system of such physiological effects of short-term space flights as the redistribution of body fluids. The effects of weightlessness can also be modeled in terrestrial experiments with water or dry immersion, as well as in experiments with changes in body position ^{[1, 2]}.

For example, studies of the ventilation response to hypoxia and hypercapnia in various body positions have shown that in the horizontal position of the body there is an increase in intra-food pressure in response to an increase in P _ET CO ₂. At the same time, despite the fact that the ventilation response to hypoxia decreased without hypercapnic stimulus, the intra-food pressure in response to the hypoxic stimulus remained unchanged ^[3].

There are many different methods of studying the ventilation reaction, each of which has its advantages and disadvantages. In the traditional method of recurrent respiration, the subject breathes into a small bag so that the partial pressure of gases in the bag-lung-blood system is quickly equalized ^[4]. We investigated the ventilation reaction to modified gas mixtures in various body positions, in an anti-orthostatic body position ^[5] and in conditions of "dry" immersion ^[6].

At the same time, using mathematical modeling, it is possible to estimate the change in partial gas pressures in isolated compartments of the body in response to various external influences. Modeling makes it possible to reduce the number of real experiments in extreme conditions or when testing personal protective equipment ^[7], as well as to estimate the time of effective human work in such conditions. The use of such simulation models to predict the behavior of physiological parameters based on numerical solutions of differential equations can be useful for studying the respiratory control system ^[8] or for designing various breathing apparatus ^[9].

The first mathematical models of the "respiratory chemostat" were described back in 1945 by J.S. Gray. 20 years later, F.S.Grodins and G. James presented one- and two-compartment models for quantifying the reaction of pulmonary ventilation in response to inhalation of carbon dioxide in stationary conditions with arterial hypoxemia and metabolic disorders in the acid-base balance ^[10]. Unlike the J.S. Gray model, the F.S. Grodins model made it possible to study the ventilation reaction not only at rest, but also during stationary physical exertion. At the same time, the models did not take into account the transport properties of blood during the interaction of hypoxic and hypercapnic influences, as well as all the time delays of chemoreflexions.

The mathematical model of E.Magosso and M.Ursino ^[8], which we adopted as a model of respiratory chemoregulation for our updated four-compartment model, has a convenient form for studying these phenomena. For example, in 2001, E.Magosso and M.Ursino studied the effects of changes in the content of CO ₂ and O ₂ in arterial blood in the cardiorespiratory system using a mathematical model and obtained interesting results that are useful for the analysis and rational interpretation of data on the physiology of human respiration ^[8]. The model describes afferent chemoreceptor pathways, efferent sympathetic activity, and CNS response using differential equations.

The L.M. Ellwein compartmental mathematical model ^[11] makes it possible to predict the response of the cardiorespiratory system to hypercapnia in patients with congestive heart failure. The model calculates blood pressure, blood flow, and gas concentrations (CO ₂ and O ₂) in the brain, lung, and tissue compartments in response to the stepwise effects of hypercapnia.

However, in these models, the authors did not investigate the dynamics of gas exchange and blood circulation during recurrent respiration.

A group of scientists J.Duffin and A.Mohan have developed a piecewise linear mathematical model of the central and peripheral chemoreflex ^[12]. The model made it possible to evaluate the parameters of the respiratory chemoreflex based on experimental data on the ventilation reaction to changes in the gas composition of alveolar air. The mathematical model of J. Duffin ^[12] identifies 3 independent components of the ventilation response – the basal component, peripheral and central stimuli, which are described by simple equations. Thus, there are some thresholds of partial pressure of _CO2 in arterial blood and brain tissues, at which these components are triggered, each of these reactions has a linear character. However, such a model is not convenient for studying the interaction of baroreflex and chemoreflex in various conditions of body position.

Mathematical models are used to interpret physiological data on the acute reaction of the cardiorespiratory system under extreme conditions, as well as to clarify contradictory results in order to formulate a unified theoretical basis for the physiology of the cardiorespiratory system. Using experimental data on the regulation of respiration at rest, during hyperventilation and during recurrent respiration, mathematical modeling makes it possible to predict the reaction of the cardiorespiratory system in response to acute exposures, which is important for estimating the estimated time of effective human work in such conditions.

The purpose of this work was to compare the results of mathematical modeling of ventilation reactions to hypercapnia obtained by various methods, to compare them with the results of D. Reed's experiments.

2. METHODS

The mathematical model describes the gas exchange in the human cardiorespiratory system and the external space of a given volume. The model is based on general physical concepts of mass transfer and transfer of matter in living systems, represented as systems with concentrated parameters.

Fig. 1 is a block diagram of a four-compartment mathematical model of human gas exchange during breathing at rest, hyperventilation and during recurrent respiration, in which arterial blood belongs to the "lung" compartment, and venous blood belongs to the "brain" and "other tissues" compartments. Individual compartments are highlighted in bold lines.

The model is based on a previously presented system of equations describing the ventilation response to hypoxia and hypercapnia during recurrent respiration ^[13] and describes the general principle of gas exchange between compartments not only in the special case of exposure by the method of recurrent respiration, but also under any other conditions, including at rest and hyperventilation.

2.1. MODEL COMPARTMENTS AND BASIC EQUATIONS

In our previous studies, 2 reservoirs were isolated in the human body – the "lung" compartment and the "tissue" compartment ^[7]. In this study, a third reservoir has been identified – the "brain" compartment. The fourth reservoir of the model, the respiratory circuit (or external compartment), is located outside the human body, and describes the external environment — the atmosphere or any other limited space from which a person inhales and exhales air, including devices designed to study the regulation of breathing or respiratory protection with appropriate parameters of temperature, pressure, humidity and the content of inhaled and exhaled gases.

Previously, similar models were used to analyze stationary and transient processes in the cardiorespiratory system in response to various stationary levels, as well as stepwise input disturbances of the content of CO ₂ and O ₂ ^[14]. The purpose of such studies was to obtain basic equations describing the regulation of respiration, but not to study the reaction of respiration under altered conditions, for example, as with recurrent respiration.

The mass balance equation for each compartment and gas component is written as follows:

where j = T, L, B and S represent the "tissue", "lung", "brain" and "external" compartments, respectively; i = 1, 2, 3 represent O ₂, CO ₂, and a mixture of other gases (nitrogen (N ₂) and others) that are not involved in gas exchange and chemical reactions, respectively; t is time; M _ji is the amount of gas i in compartment j; J _ji is the amount of gas i passing into compartment j from any other compartments, or the release and consumption of gas i in compartment j.

Thus, the mathematical model is described by a system of the following mass balance equations:

The "fabric" compartment:

The "pulmonary" compartment:

The "brain" compartment:

where C _Tvi, C _Bvi and C _ai are the content of gas i in the venous blood of the "tissue" and "brain" compartments and arterial blood flowing from the "pulmonary" compartment, respectively; F _ai and F _Ii are the fractional concentrations of gas i in exhaled and inhaled alveolar gas (the ratio of gas molecules i to the total the number of molecules of the gas mixture); Q _T is the volumetric velocity of blood flow between the "tissue" and "pulmonary" compartment, and Q _B is the volumetric velocity of blood flow between the "cerebral" and "pulmonary" compartment; V_'AI and V_'AE are inspiratory and expiratory alveolar ventilation, taking into account the V _DC of the physiological dead space (see equation (18)), which is not involved in gas exchange, respectively; J _B i is the release or consumption of gas i in the "brain" compartment; J _T i is the release or consumption of gas i in the "tissue" compartment; t ₁ and t ₃ are the time of blood transfer from the lungs to the tissues, namely, to the "tissue" and "brain" compartments, respectively; t ₂ and t ₄ are, respectively, the time of blood transfer from tissues to lungs. The assumption that the consumption of O2 and the release of CO 2 in the tissues of the "pulmonary" compartment is significantly less than in the "tissue" and "brain" compartments allows us to exclude the corresponding parameters from equation (3).

In a particular case, all other gas components, with the exception of O ₂ and CO ₂, can be excluded from metabolic and chemical reactions, which means J ₃=0. This special case occurs during normal air breathing, when changes in the nitrogen content in the body can be neglected. Changes in the nitrogen content (and in general the "third gas") cannot be neglected in hyperbaria and hypobaria, i.e. when modeling underwater and high-altitude situations, when switching to oxygen respiration.

The equations describing the relationship between the partial pressure of gases and their fractional content have the form:

where P _B is the total pressure in the "external" compartment;P is the pressure of the dry gas mixture in the inhaled and alveolar gas, excluding water vapor; P _k i is the partial pressure O ₂, C O ₂, and N ₂ in the inhaled and exhaled (alveolar) gas; P _H20 = 47 mmHg. the pressure of saturated water vapor at a body temperature of 37 ° C.

The diffusion of gases between the pulmonary blood and the alveoli is described by the following equation:

where D _L i are the diffusion coefficients for the i-th gas, the values of which are presented in Table 1 and selected from the literature ^[15-17].

The fractional content of gases in the inhaled and exhaled alveolar gas mixture can be determined by means of equations 2-7 and an equation describing gas exchange in the "external" compartment:

The "external" compartment may include devices for absorbing and releasing gases, for example, a chemical absorber CO ₂ or a regenerative cartridge producing O ₂ (for example, a "self-rescuer"), the operation of which is described by the parameter R _i, in the special case R _i = 0.

The external environment represented by the 4th compartment of the mathematical model can be limited by the volume of the device to which a person is connected, or by the volume of a closed hermetic object, or represented by a sufficiently large volume, conditionally being an atmosphere with appropriate parameters of temperature, pressure, humidity and gas content.

The simulation of breathing maneuvers begins with a calm exhalation. In a particular case, when modeling recurrent respiration, the total volume of the system consisting of the alveolar space, connecting tubes, mask and the working volume of the container for recurrent respiration ranges from 5 to 40 liters. For example, when breathing from the atmosphere, the volume of the "external" compartment is significantly larger than the volume of the alveolar space and a volume of 10,000 liters is selected in the calculations.

The system of four compartments is a closed system, while the dynamics of gases in the "external" compartment obeys the law of an ideal gas.

2.2. SPECIAL EQUATIONS

The amount of gases in physically dissolved or bound form in the "lung" and "tissue" compartment can be described by the following equations ^[18]:

The amount of gases in physically dissolved or bound form in the "brain" compartment is described by the following equation:

where V _LT, V _BT and V _CT are the volumes of lung, brain and other human tissues, respectively; α _L i solubility of gas i in lung tissue or tissues of the "pulmonary" compartment; β _T i and β _B i solubility of gas i in tissues, including brain tissue or in the tissue component "tissue" and "brain" compartments, respectively; V _Lbl, V _Cbl and V _Bbl blood volume in the "pulmonary", "tissue" and "brain" compartments, respectively; α _bl i solubility of gas i in the blood of the "pulmonary" compartment; β _Tbl i and β _Bbl i solubility of gas i in blood of the "tissue" and "brain" compartments, respectively; N _i relative affinity for hemoglobin O ₂ and CO ₂; S ^Hb _a i, S ^Hb _Tv i and S ^Hb _Bv i saturation of hemoglobin O ₂ and CO ₂ in arterial and venous blood circulating between compartments.

The transport properties of blood are described by Hill's equations, taking into account the effects of Bohr and Haldane ^[19]. It should be noted that variants of describing the transport properties of blood according to G.R.Kelman and A.R.Douglas have also been studied ^{[20, 21]}. Since the calculated dynamics of gas exchange in all variants of the description of transport properties were close, and the calculation procedure is simpler for J.Spencer, in this paper we present only the results obtained using formulas according to J.Spencer.

The respiratory cycle is defined as a change in the oral respiratory flow according to the trigonometric law:

where ω is the cyclic respiratory rate; and T _BC is the duration of the respiratory cycle, including inhalation and exhalation.

This means that the instantaneous value of the alveolar volume V _alv can be expressed in terms of a certain integral of equation (13):

where V _0alv is the volume of gas in the alveolar space at the initial moment of time with a calm exhalation;

Earlier, A. Ben-Tal presented a hierarchical classification of mathematical models ^[22]. According to this classification, our model refers to models with gas exchange, gas transmission properties of blood and periodic respiration. However, the models presented in the work of A. Ben-Tal do not describe the feedback of the dependence of ventilation on the level of O2 and CO 2 in arterial blood and are not applicable to the study of the ventilation reaction during recurrent respiration.

Full ventilation V' in the model is regulated by peripheral and central chemoreflex, V_'p and V_'c, respectively. Chemoreflexions and dependences of ventilation, respiratory volume and frequency on changes in partial pressure of CO ₂ and O ₂ in arterial blood are described by the equations presented in the works of E.Magosso and M.Ursino ^[8]. In particular, total lung ventilation is calculated as follows:

In this case, the respiratory volume changes with ventilation and is calculated from the total ventilation and respiratory rate

Alveolar ventilation is calculated as follows, taking into account the V _DC of the physiological dead space:

where V_'AI and V_'AE are inspiratory and expiratory alveolar ventilation.

The relationship between the dynamic pressure in the alveolar space and the velocity of the respiratory flow is described by the F.Rohrer equation ^[23]:

where ΔP _alv is the relative pressure difference in the alveolar space and the atmosphere; V' is the rate of respiratory flow, m ₁ and m ₂ are coefficients characterizing the viscous and turbulent components of respiratory resistance that occur during breathing.

Hypercapnia and hypoxia also stimulate the cardiovascular system. The change in blood flow velocity in response to hypoxic and hypercapnic stimuli is described by experimentally obtained equations presented in 2006 in the work of H. Zhou and his co-authors ^[24]:

where Q _j is the blood flow between the "pulmonary" and "tissue" compartment or between the "pulmonary" and "cerebral" compartment, Q jrest is the _blood flow at rest, ΔQ _ji is the increase in volumetric blood flow rate in response to hypoxic or hypercapnic stimuli.

2.3. SIMULATION PARAMETERS

The initial amounts of O ₂ and CO ₂ in the "lung", "tissue" and "brain" compartments are calculated using equations 10, 11 and 12 according to the mass ratios presented in ^[25]. The initial values of partial pressure O2 and CO 2 in arterial and venous blood are shown in Table 1.

Table 1

Adjusted parameters for the average person

(the initial data are taken from the literature).

The consumption of O2 _{(Jo2) and the release of CO 2} (Jco ₂) in the "brain" and "tissue" compartments are 60 ml/min and 56 ml/min for brain tissues and 154 ml/min and 110.4 ml/min for other tissues, respectively ^[26-28]. Table 1 and Table 2 show the main parameters of the three compartments of the model adopted for the "average person".

Table 2

The initial data of the model characterizing the pulmonary ventilation of the "average person" at rest.

The parameters characterizing the ventilation reaction are taken from the work of E.Magosso and M.Ursino and adjusted to match the average values of the ventilation response of the subjects from D.Reed's experiments ^[29].

The total volume of gases in the biotechnical system, including respiratory volumes at full inhalation, with a fully deflated mixing chamber and the volume of tubes entering the device for return respiration is from 5 to 40 liters ^[29]. The initial concentration of oxygen in the device is 20.93%, carbon dioxide is 0.03%, the temperature is 27 °C, and the pressure in the system is 760 mmHg, taking into account gas vapor.

According to D. Reed's experimental data, on average, the return breathing maneuver lasted 4 minutes for return breathing with a small bag and about 10 minutes with a large bag, which means that the time of simulation of the return breathing maneuver using a mathematical model was chosen accordingly.

2.4. NUMERICAL SIMULATION PARAMETERS AND SENSITIVITY OF MODEL PARAMETERS

All calculations were performed in the MATLAB R2017b software package (2017, Mathworks, USA) using an automatic solver (DDES procedure) with an initial step Δt = 0.01 sec for the numerical solution of differential equations.

The mathematical model is a closed system of 10 ordinary differential equations with delays and 15 algebraic equations.

The sensitivity analysis of the model was carried out to simulate calm breathing, whereas the partial pressure parameters in each compartment are balanced and do not have an upward or downward trend. The change in the average partial pressure value was estimated from the moment the equilibrium between the compartments was established, which occurred for all parameters after 20 thousand simulation samples. A change in the control parameters by 20% up or down led to a change in the equilibrium value of the partial pressure in all compartments and is shown in Table 3.

Table 3

Analysis of the sensitivity of the behavior of a mathematical model to parameters, where J _B O ₂, J _B CO ₂, J _T O ₂, J _T CO ₂, Q _T, Q _B are control parameters, and P _a O ₂, P _a CO ₂, P _v O ₂, P _v CO ₂, P _Bv O ₂, P _Bv CO ₂ – controlled parameters.

Table 3 considered the sensitivity of the controlled parameters to the control parameters under conditions of recurrent respiration at times of the order of tens of minutes. It was also found that when the parameters m _1, m ₂, D _L i varied, the controlled parameters almost did not change.

In addition, the sensitivity of equations (10), (11) and (12) to "capacitive" parameters determining the amount of gases in physically dissolved or bound form in each of the compartments was analyzed. The change in the value of the total amount of gas in the compartment was estimated in response to a 20% change in the parameter up or down. Note that if several control parameters are changed simultaneously, the effect may not be additive. The results are shown in Table 4.

Table 4

Analysis of the sensitivity of the amounts of gases in the compartments to the volumes and solubilities of gases in the compartments, where V _alv, V _Lbl, V _Cbl, V _Bbl, V _LT, V _CT, V _BT, α _bl, β _bl, α _T, β _T are the control parameters, and M _L o ₂, M _T about ₂, M _B about ₂, M _L about ₂, M _T about ₂, M _B about ₂ controlled parameters.

The transport properties of blood, described by Hill's equations taking into account the Bohr and Haldane effects, have a significant impact on the output parameters of the model. In particular, when the threshold values of Po ₂ describing hypoxic effects are reached, it slows down the growth of the curve describing the increasing hypercapnia during recurrent respiration of Rco ₂. The effects of blood transport properties are shown in Figures 2 and 3 on the ranges of Rho ₂ and Rco ₂ studied by us, characteristic of hypoxia and hypercapnia during recurrent respiration.

Fig. 2 – Effects of blood transport properties. The figure shows the dependence of the carbon dioxide voltage in the blood at the appropriate levels of hypercapnia (Rco ₂ equal to 20, 30, 40, 50, 70 mmHg) and a gradual change in the oxygen voltage in the blood. It can be seen that when the oxygen voltage drops below 80 mmHg, there is a sharp change in the behavior of the curve describing changes in the voltage of carbon dioxide in the blood.

Fig. 3 - Effects of blood transport properties. The figure shows the dependence of oxygen concentration in the blood at the appropriate levels of oxygen partial pressure (Po ₂ equal to 60, 80, 100, 150, 200 mmHg) and a gradual increase in hypercapnia (as with recurrent respiration). It can be seen that the growth of hypercapnia, combined with hypoxia, is accompanied by the effect of "flooding" the curve describing the change in oxygen tension in the blood, for example, during recurrent respiration.

Thus, when modeling the combined effects of hypoxia and hypercapnia during recurrent respiration, the effects of mutual influence of oxygen and carbon dioxide contents may be observed when thresholds describing hypoxia or hypercapnia are reached.

In addition, Figure 4 shows the sensitivity of ventilation parameters in response to the duration of numerical simulation of recurrent respiration, demonstrating the inertial nature of the mathematical model of chemosensitivity described by the equations presented in the works of E.Magosso and M.Ursino, and used in this work. Sensitivity was assessed when the equations describing gas exchange between compartments were switched off without hypoxic effects (in the Ro ₂ and Rco ₂ ranges, at which the peripheral chemoreflex is switched off). The simulation was carried out with a linear increase in hypercapnia from 40 to 70 mmHg, but with a different rate of increase in hypercapnia – 30 mmHg for 3, 5 and 8 minutes.

Fig. 4 – The effect of the growth rate of Pcr ₂ on the growth of ventilation in the model of E.Magosso and M.Ursino [equations 16-18, Appendix, E.Magosso et al. 2001]. Along the abscissa axis, the partial pressure of carbon dioxide (mmHg), along the ventilation ordinate axis (l/min). A simulation of recurrent respiration to the level of 70 mmHg is presented in the absence of hypoxic exposure for 3, 5 and 8 minutes. It can be seen that for longer modeling approaches, with all other identical parameters (such as the physiological parameters characterizing chemoreflex and the absence of the influence of equations describing gas exchange), ventilation in response to increasing hypercapnia is more pronounced, thus manifesting the inertial properties of the central chemoreflex.

2.5. SIMULATION OF RECURRENT RESPIRATION BY THE METHOD OF D. REED

The results of the return breathing simulation were compared with the results of D. Reed's return breathing experiments described in his works. D. Reed conducted the following experiments in his research ^[29].

The standard method of return breathing according to D. Reed: a bag with a volume of 4 to 6 liters for return breathing is filled with a gas mixture with 7% carbon dioxide, 50% oxygen and 43% nitrogen. The subjects were previously trained in the technique, and prepared for the experiment for 30 minutes sitting in comfortable conditions, and breathing a hyperoxic gas mixture. The initial partial pressure of inhaled oxygen in the bag before return breathing was about 350 mmHg. Return breathing began after maximum exhalation and lasted for 4 minutes.

The period between tests in a series of experiments with recurrent breathing was tried to be maintained in equal intervals and ranged from 6 to 11 minutes.

P _ET co ₂ was recorded in the range from 35 mmHg to 85 mmHg with a measurement accuracy of 0.5 mmHg, while P _ET o ₂ was recorded in the range from 0 mmHg to 400 mmHg with a measurement accuracy of 4 mmHg. Full ventilation was calculated in 4-minute sections of time.

Stationary methods. The curves were obtained as a result of 7 experiments on 3 subjects. The results were obtained by two methods: a) 30 minutes of breathing in a hyperoxic gas mixture with 3% and 7% carbon dioxide with an interval of calm breathing in room air for 30 minutes; b) 20 minutes of breathing in gas mixtures with 3%, 5% and 7% carbon dioxide without intervals of calm breathing, All gas mixtures contained from 35 to 40% oxygen.

Fig. 5 – Comparison on the "Rso ₂-ventilation" plane of ventilation reactions obtained as a result of modeling of recurrent respiration and stationary methods described in the works of D. Reed. On the abscissa axis, the partial pressure of carbon dioxide (mmHg), on the ordinate axis, ventilation (l/min). A simulation of recurrent respiration to the level of 65 mmHg is presented in the absence of hypoxic exposure and with preliminary hyperoxic preparation (blue curve (x) on the graph). The simulation of the ventilation reaction obtained by stationary methods according to the following schemes is also presented: a) 30 minutes of breathing in a gas mixture with 3% and 7% carbon dioxide with an interval of calm breathing of 30 minutes (red curve (o) on the graph); b) 20 minutes of breathing in gas mixtures with 3%, 5% and 7% carbon dioxide without intervals of calm breathing (green curve (a) on the chart). The simulation results are close to the results of D. Reed's experiments ^[29].

In the experiments of D.Reed ^[29], the average values of the ventilation reaction were 2.07±0.69 and 2.03±0.41 l/min/mmHg, ventilation at 55 mmHg - 25.43±5.11 and 44.00±7.86 l/min, and the apnea point - 42.14±2.10 and 33.20±0.95 mmHg for the method of recurrent respiration and stationary method, respectively. Thus, both in D. Reed's experiment and in our model calculations, the values of the ventilation reaction differed slightly between the methods of stationary and recurrent respiration, whereas the position of the ventilation reaction curve on the Rco ₂-ventilation plane obtained by the stationary method was significantly higher. This difference is manifested in the fact that the value of the "apnea point" parameter, i.e. the value of Rco ₂ at zero ventilation, obtained by linear approximation of the curve of dependence of ventilation on Rco ₂, is less for the stationary method compared with the method of recurrent respiration.

Additional validation of the model can be performed based on a comparison of experimental results with bags of different volumes and corresponding theoretical calculations.

In D.Reed's experiments and tests of recurrent respiration with different bag volumes, different rates of growth of P _a co ₂ were observed, with recurrent respiration with a bag of 40 liters, the rate of increase of P _a co ₂ decreased by 2-3 times compared with recurrent respiration with a bag of 5 liters ^[29].

Fig. 6 – Comparison of ventilation reactions obtained as a result of modeling return respiration for a small (5 l) and a large bag (40 l) described in the works of D. Reed. On the abscissa axis, the partial pressure of carbon dioxide (mmHg), on the ordinate axis, ventilation (l/min). A simulation of recurrent respiration to the level of 65 mmHg is presented in the absence of hypoxic exposure and with preliminary hyperoxic preparation [Read, 1967].

According to D.Reed ^[29], we found the average values of the ventilation reaction of 2.10±0.53 and 3.04±1.82 l/min/mmHg, ventilation at 55 mmHg - 27.43±5.94 and 34.07±7.08 l/min, ventilation at 65 mmHg - 48.44±8.38 and 64.41±23.79 l/min, and the apnea point is 41.26±4.88 and 42.42±3.18 mmHg. for the method of recurrent respiration with small and large bags, respectively, while the rate of CO2 increase is 6.12±0.57 and 2.00±0.15 mmHg./min.

It can be seen that the simulation results are close to the results of D. Reed's experiments. Also, as in D. Reed's experiments, the ventilation reaction obtained by using a smaller volume is lower than when using a large bag.

A comparison of the calculated and experimental data is shown in Table 5, indicating the spread of experimental data.

Table 5

Comparison of ventilation reactions obtained in simulation (mod.) and real experiments by D.Reed (ex.). A series of experiments with return breathing by the standard method (RB) with a small bag, stationary methods (SS), and return breathing by the standard method with a large bag (LB) were compared.

A comparison of the results of numerical modeling with D. Reed's experimental data showed the adequacy of the mathematical model.

The proposed mathematical model makes it possible to investigate the ventilation response to hypoxia and combined hypoxia and hypercapnia, gas exchange and regulatory results of arbitrary lung ventilation. Additionally, changes in the frequency and depth of breathing characteristics, effects of changes in blood flow, pulmonary volumes, and redistribution of liquid media as a result of gravitational influences can be included in the model.

However, the model has certain assumptions and limitations. The choice of the parameters of the compartments and the initial values of the modeled variables was made on the basis of various literature sources, therefore, it is impossible to expect full compliance of the simulation results and experimental data. In particular, the transport properties of blood (to which the behavior of the model is quite sensitive) may differ from those specified in the algorithms used. In addition, experimental data are obtained with the participation of various groups of subjects, the full set of necessary parameters of which remains unknown.

Validation of the mathematical model was limited by experimental data obtained in D. Reed's experiments.

In the future, the model is expected to be validated using a wider set of experiments on the same sample of subjects, in different body positions and under the influence of microgravity effects. In addition, validation of the mathematical model on experimental data obtained in experiments with hypoxia is required. It is also necessary to study the transport properties of the subjects' blood as closely as possible for adequate numerical simulation of experiments.

3. CONCLUSION

The mathematical model describes the dynamics of the content of _CO2, _O2 and other gases in 3 compartments of the biological system and the external environment, represented by the 4th "external" compartment of the mathematical model. The external environment can be limited by the volume of the device to which a person is connected, or by the volume of a closed hermetic object, or represented by a sufficiently large volume, conditionally being an atmosphere with appropriate parameters of temperature, pressure, humidity and gas content.

As part of the study, a stationary method for studying ventilation regulation and recurrent respiration was modeled. The sensitivity of the model parameters was analyzed as part of the study. The ventilation response to hypercapnia changes qualitatively in accordance with experimental data.

The authors declare the absence of a conflict of interest.

The work is supported by the program of fundamental research of the SSC RF – IMBP RAS (topic FMFR-2024-0038).