Adaptive Inert Gas Exchange Model for Improved Hypobaric Decompression Sickness Risk Estimation

METHODS

To illustrate our hypothesis, experimental data were compared with model predictions to identify and interpret the observed discrepancies. All current inert gas exchange models have one thing in common: they all use predetermined and fixed tissue half-value times. Here, we use a different approach and have modeled the underlying physiological parameters, i.e., the prevailing half-times are merely a result of the anthropometric, cardiovascular, and pulmonary parameters. Using nominal (fixed) physiological data, our adaptive biophysical gas exchange model emulates a model with fixed half-value times, similar to a conventional Bühlmann-like gas exchange model. Thus, we are able to assess the performance of both a fixed-parameter and an adaptive-parameter gas exchange model and compare their predictions with experimental data.

Materials

The basic model structure is presented in Fig. 1. The “human body” is divided into six tissue compartments with nominal parameters based on physiological data from Bühlmann.⁴ The tissue compartments A, B, C, D, E, and F represent the brain and spinal cord, the central circulation, the skin, joints and bones, skeletal musculature, and fatty tissue, respectively. The lung function is described by one single alveolar compartment of 4.1 L, neglecting the intrapulmonary shunt and the alveolar dead space.⁵ It is assumed that the compartments were well-mixed, i.e., no heterogeneity within a compartment. Also, although arterial and venous concentrations are transported by finite blood flows, any transport delays between the lungs and the tissues are neglected. The nitrogen (N₂) gas exchange flow and volume are derived by tracking the inhaled and exhaled N₂ concentrations, together with the alveolar ventilation.

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The transportation of N₂ is modeled by a set of ordinary differential equations based on the conservation of mass principle. The rate at which the amount of inert gas (volume × concentration) in a compartment changes per unit of time equals the rate of mass transfer into the compartment less the rate of removal from the compartment. The model includes a diffusional mass transfer between the pulmonary capillary bed and the alveoli, as well as between tissue capillaries and tissue cells. However, there is strong evidence that in normal healthy humans, the alveolar diffusional flux of inert gases is fast compared to the other dynamics within the system, leading to a fast equilibration between end-capillary blood and the alveolar gas, even under moderate subanaerobic exercise conditions.⁵ Also, the diffusion process between capillary and cell was assumed to be rapid, hence all diffusional fluxes are equalled to zero, leading to a perfusion-limited inert gas exchange for the complete system. Based on this instantaneous diffusion equilibrium, the N₂ concentration in arterial blood leaving the lungs is proportional to the N₂ concentration within the alveoli and vice versa, i.e., C_a = λ_B:A C_A, with λ_B:A denoting the N₂ blood-to-gas partition coefficient. Similarly, a venous equilibrium exists at the interface between the outgoing capillary blood and the tissue, i.e., C_V = λ_B:T C_T, with λ_B:T being the relevant N₂ blood-to-tissue partition coefficient. Another consequence of the diffusion equilibria is that compartment volumes, V_A (alveolar volume) and V_AB (alveolar blood volume), are combined into a single lung compartment, with an effective volume equal to V_A+V_AB λ_B:A. In the same way, each of the six tissue compartment volumes, V_T,n and V_TB,n, are combined into a tissue compartment with an effective volume of V_T,n+V_TB,n λ_B:T. Based on these simplifications, and assuming steady volumes (dV/dt = 0), the time derivates of the compartmental inert gas concentration, C_A and C_T,n, are mathematically described by a set of ordinary differential equations:$$(V_A+V_{AB}{\lambda }_{B:A})\frac{d{C}_A}{dt}=\dot{Q_c}{C}_V-\dot{Q_c}{C}_a+\dot{V_A}{C}_I-\dot{V_A}{C}_A$$(1)$$(V_{T,n}+V_{TB,n}{\lambda }_{B:T})\frac{d{C}_{T,n}}{dt}={\dot{Q}}_{c,n}{C}_{V,n}-{\dot{Q}}_{c,n}{C}_a$$(2)

Equation 1 describes the mass balance for the alveolar compartment, while Equation 2 represents the mass transfer for a tissue compartment and is applied separately to each of the six tissue compartments, with n ranging from 1–6. The inhaled N₂ concentration is denoted by C_I. The exhaled N₂ concentration is considered equal to the alveolar N₂ concentration, C_A. The mixed venous N₂ concentration, C_V, is given by:$$C_V={\displaystyle \sum }_{n=1}^6({\dot{Q}}_{c,n}{C}_{V,n})/\dot{Q_c}$$(3)

The concentrations are expressed in mol · l⁻¹. Using the molar volume, these values are converted to ml · l⁻¹, ppm, and fraction (%). Note that for a perfusion-limited gas exchange, the half-time of each compartment is derived from the physiological parameters according to:$$T_{\mathit{\text{Half}},n}=\ln \left(2\right)\frac{V_{T,n}{H}_{T,n}}{\dot{Q_{c,n}}{H}_B}$$(4)

The nominal volume parameters for the tissue compartments A, B, C, D, E, and F are set to 1.85, 2.64, 4.36, 13.07, 28.30, and 13.07 l, respectively. The alveolar ventilation is set to 6.8 l · min⁻¹ and the cardiac output to 5.67 l · min⁻¹. The nominal fractional blood flow to the tissue compartments A, B, C, D, E, and F are 0.85, 2.58, 0.32, 0.36, 1.20, and 0.36 l · min⁻¹, respectively. The solubility coefficients for N₂, expressed in ml · l⁻¹ (standard temperature and pressure dry) at a pressure of 1.0 bar at 37°C, for blood and all tissues (except fat) and for fat were set to 12.83 and 66.124, respectively.⁴ The resulting N₂ blood-to-gas partition coefficient, the tissue-to-blood partition coefficients for fat, and those for all other tissues were set to 0.0162, 0.194, and 1.0, respectively. The adaptive biophysical gas exchange model equations and parameters were programmed in Python.

The theoretical bubble growth was calculated using the TBDM model, which is extensively described elsewhere.^2,6,7 In brief, it calculates the bubble growth for a single (ever-present) nucleus of fixed initial size in each of 10 parallel compartments with fixed half-times between 5–480 min. The model calculates the inert gas tension in each tissue compartment, according to a perfusion-limited exponential model, which is then used as one of the inputs to calculate the change of the bubble radius. The Bubble Growth Index (BGI) is the maximum theoretical bubble size attained during the decompression profile in any of the compartments, relative to an initial preformed micronucleus of 3 µm. The peak BGI value is typically used as the primary measure of decompression stress and as input for a logistic-regression-based DCS probability model. The TBDM model equations and parameters were implemented in Python.

Procedure

Model predictions were compared against the data from a recent experimental study performed by the Swedish Aerospace Physiology Centre.⁸ This study evaluated the decompression stress, in terms of venous gas emboli (VGE) and N₂ exchange, during simulated high-altitude flights in a hypobaric chamber using different flight profiles (see Fig. 2 top panels). Eight men were investigated during three conditions. Condition A was a continuous exposure to 24,000 ft (7315 m) while breathing a normoxic gas mixture (Po₂ = 21 kPa, corresponding to an O₂ fraction of 52.5%). Conditions B and C had the same total exposure time at altitude, but with an intermittent recompression to 20,000 ft (6096 m) and 900 ft (274 m), respectively, simulating excursions to lower altitude, such as for airplane refueling. Four-chamber cardiac images were obtained using an ultrasound system and the level of VGE was assessed using the Eftedal-Brubakk 5-degree scale. N₂ gas exchange was measured using a modified closed-circuit rebreather. The N₂ washout and uptake, expressed in ml · min⁻¹, was provided as 20-min average values, averaged both over time and the eight subjects. Additional data from this study were basic biometrical data from the subjects, as well as cardiac output throughout the experiment.

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RESULTS

Considering the inert gas exchange, Fig. 2 presents the measured 20-min average N₂ volume flow values for the three conditions, compared to the model predictions using the nominal model parameters. The measured values, during exposures to 24,000 ft (7315 m) and intermittent lower altitudes, are shown using gray boxes. The solid line presents the continuous N₂ volume flow prediction, while the hatched boxes show the derived 20-min average prediction values.

During the experiment, three distinctly different gas exchange patterns were observed. Condition A exhibited a gradually (exponential) decreasing elimination rate, while alternating washout and uptake of N₂ was observed for Condition C. Condition B showed N₂ washout during every exposure to 24,000 ft (7315 m) and during the first two periods at 20,000 ft (6096 m), but a slight uptake was seen during the last period at 20,000 ft (6096 m). Although these general gas exchange patterns were indeed well predicted by the model, the biophysical gas exchange model using the nominal (fixed) parameters was not able to predict some important features. Two important discrepancies were observed. First, the measured washout for Condition A was lower than the simulated N₂ washout rate. Secondly, for Condition B, the model did not predict a N₂ uptake during the last period at 20,000 ft (6096 m), and it underestimated the N₂ washout during the last exposure to 24,000 ft (7315 m).

Using the adaptive physiological model, total N₂ flow can be decomposed into the flow components of the different model compartments. Fig. 3 (top panel) illustrates this decomposition for Condition A. During the first 30 min, the majority of the washed out N₂ is seen as coming from the muscle tissue compartment, while the fat compartment is the leading compartment afterwards. The washout flow is dependent on the cardiac output, especially during the first part of the washout, which is dominated by well perfused tissues. Adapting the model by introducing a 30% reduction in the cardiac output (as was measured during the exposures), resulted in a decreased estimated N₂ outwash flow, with a 20-min average value much closer to the observed outwash (Fig. 3, lower panel).

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Fig. 4 (top panel) focuses on the N₂ flow decomposition for the last intermittent recompression of Condition B (between 110–130 min). During this last period at 20,000 ft (6096 m), N₂ is washing out from the “slow” fat compartment, while there is N₂ uptake for the other “faster” compartments, especially the muscle compartment. The proportion of the flow from each compartment determines the total flow. For example, as illustrated in Fig. 4 (lower panel), two different people can have a different total flow pattern during this period at 20,000 ft (6096 m), depending on their body characteristics, e.g., the proportion of fat or muscle tissue in the body. Persons 1 and 2 represent a thin, lean individual with an increased perfusion and a larger, heavyset individual with decreased perfusion, respectively.

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To evaluate bubble growth predictions, we considered the prevalence of VGE that was measured every 5 min during the experiments.⁸ Fig. 5 (left panel), shows the median observed (Eftedal-Brubakk) bubble score at rest and after knee bends for the three conditions, with a score of 0 indicating no observable bubbles and a score of 3 indicating at least 1 bubble per cardiac cycle. The VGE scores during a (total of) 80-min exposure to 24,000 ft (7315 m) are substantially reduced by intermittent 20-min excursions to 900 ft, but are (slightly) increased by 20-min excursions to 20,000 ft (6096 m), especially during the last period at 20,000 ft (6096 m).⁸ The BGI predictions using the TBDM model are presented in Fig. 5 (middle panel). The prediction of the bubble growth pattern corresponded well with the observed VGE for Conditions A and B, but not for Condition C. The model predicted a maximum BGI of a little over 10.0 for Condition C, similar to the maximum BGI predicted for Condition A, while in reality, a distinctive lower VGE score was observed during the experiment for Condition C.

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Weighing factors can be introduced based on the N₂ flow components. Each tissue compartment has a different contribution to the total N₂ volume flow; see, e.g., Fig. 3 (top panel). Considering only the muscle and the fat compartment (compartments E and F, respectively), a different N₂ flow pattern was observed between Conditions B and C (Fig. 6). After the initial outwash, only a marginal participation of muscle tissue was observed in the inert gas flow for Condition B, while fatty tissues are releasing N₂ during the complete altitude exposure. In contrast, the overall N₂ release for fatty tissue was significantly decreased during Condition C due to the intermittent recompressions, while the overall (peak) N₂ release for muscle tissue was significantly increased. The N₂ gas exchange in Condition B was dominated by the fat compartment, while it was governed by the muscle tissues for Condition C. Considering these dominating compartments as a weighing factor, the resulting bubble growth prediction changes significantly. Fig. 7 shows the calculated bubble growth for each of the 10 compartments from the TBDM model for Conditions B (top panel) and C (lower panel). Traditionally, the BGI is defined as the maximum value, as a function of time, in any of the compartments. However, considering a weighing factor based on the dominating flow compartment and selecting only the fat compartment for Condition B, a gradual increase was observed in bubble growth during the periods at 24,000 ft (7315 m) and 20,000 ft (6096 m). In Condition C, including a weighing factor and selecting only the muscle compartment, a very modest bubble growth is predicted, followed by almost no bubble growth at all.

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DISCUSSION

Currently, the output of most inert gas exchange models in decompression models is limited to a prediction of the inert gas tension for several compartments with fixed and predetermined half-value times. It is possible, in theory, to convert this information to an estimate of the exhaled or absorbed N₂ volume and the associated N₂ flow. However, this requires additional information regarding the volume of each compartment. This information is not readily available for common gas exchange models like the ZH-L16, as it considers only theoretical compartments with no direct link to physiological parameters. The general approach is then to divide the total body volume, e.g., 75 l, equally over all the tissue compartments of the model. Instead, we choose to construct a physiological model with a more limited number of compartments, which are clearly relatable to anthropometric and cardiovascular parameters. This idea in itself is not new. Indeed, in the past, Jones,⁹ Mapleson,¹⁰ and Flook¹¹ investigated the use of such models. In the current paper, we added an alveolar compartment, thereby providing a direct link with measurable N₂ concentrations in the exhaled air and ventilation parameters. Also, the model is well suited to take into account interpersonal differences in anthropometric, cardiovascular, and pulmonary characteristics.

In general, discarding any possible measurement errors (bias), the N₂ outwash as predicted by the physiological model overestimated the measured 20-min average values. Ånell et al. reported a decrease of the cardiac output, most likely due to the subjects lying down during the experimental exposures.⁸ Accordingly, decreasing the cardiac output in our model by 30% results in a predicted N₂ outwash that was much closer to the measurements.

Focusing on the last exposure at 20,000 ft (6096 m) in Condition B, Ånell et al. noticed a significant interindividual variation in the N₂ exchange.⁸ Two subjects continued to have an N₂ washout, albeit very slowly, and five subjects showed an N₂ uptake. A traditional gas exchange model using fixed half-times would not be able to predict this actual N₂ gas exchange variation for different subjects. Our model, however, confirms that different subjects can indeed have a different N₂ gas exchange, depending on their anthropometric parameters (especially the proportion of the lean and fatty tissues) and the time point during the exposure.

One subject had a very high washout rate of 19.4 m ⋅ min⁻¹ on the last exposure at 20,000 ft (6096 m) in Condition B, which is even higher than the average N₂ washout rate during the first 20 min after the initial altitude exposure. The standard inert gas tension models do not predict this very high washout rate, and neither does our adapted physiological model, at the moment. It is hypothesized that this very high N₂ washout is due to bubble formation, as a high VGE score was observed for this subject.⁸ Although it has been previously suggested that N₂ bubbles in tissues can actually slow N₂ elimination from these tissues due to a reduced tissue–blood N₂ concentration gradient,¹² intravascular bubble formation has also been reported to potentially increase the blood N₂ transport capacity.¹³ This increased gas transport via bubbles in the venous blood should be further investigated and incorporated accordingly in the model. We strongly believe that the N₂ gas flow is a critical, but unexplored, parameter in DCS (modeling), which could be used as an additional decompression stress marker.

Successful saturation diving decompression profiles are based on limiting the (maximum) flow of inert gas.¹⁴ Assuming that “saturation diving decompression” is a good analog for altitude decompression, we suggest using N₂ flow as a weighing factor for modeling altitude decompressions. Traditionally, in gas exchange and bubble growth models, all compartments are assumed equally important. However, it is clear that some body compartments have a more pronounced impact on the inert gas exchange than others. Using the original TBDM model, and assuming that all 10 compartments are equally important, an incorrect evaluation is made of the decompression stress in different altitude exposure conditions. A clear difference in the N₂ flow patterns can be observed between the different altitude exposures. For Condition B, the N₂ gas exchange is governed by fatty tissue, while skeletal musculature dominates the N₂ flow in Condition C. Selecting only the tissues with corresponding half-times from the (10-compartment) TBDM leads to completely different conclusions regarding the decompression stress, with predicted bubble growth pattern now being much closer to the measured VGE scores (Fig. 5, right panel).

A similar discrepancy is noted when analyzing the repeated altitude exposures data described by Pilmanis et al.¹⁶ Although an exposure with four 30-min altitude exposures without ground interval resulted in a significant lower DCS incidence (7 cases; 22%) compared to a single exposure of 120 min (19 cases; 59%), the bubble growth and BGI values predicted by the TBDM model are similar for both exposures. This shows that, here as well, the TBDM model fails to discern between DCS risk differences in these exposures. It cannot be excluded at this point that other factors, e.g., the use of a fixed constant value for the metabolic gas tension for all tissue compartments, also contribute to the incorrect evaluation of the decompression stress using the original TBDM model. It is accepted that, while presence of VGE is an imperfect surrogate for DCS in decompression trials, the absence of bubbles is strongly correlated with a very low risk of DCS.¹⁵ Accurate predictions of low VGE scores or bubble growth would allow for correct selection of certain flight profiles. A significant difference in VGE grade probably indicates a difference in the DCS risk.

The design of new preoxygenation and hypobaric decompression strategies is often based on DCS probability models, like the Altitude Decompression Sickness Risk Assessment Computer model or the TBDM-derived probability model, where logistic regression quantitively relates the TBDM Bubble Growth Index to a DCS risk based on existing altitude DCS data.⁷ Our analysis shows that the predicted inert gas pattern and predicted bubble growth do not always correspond to the observed values during actual experiments, questioning the output of the current theoretical bubble growth models as a reliable DCS predictor. Therefore, using these predicted decompression stresses in probability models to design new hypobaric decompression procedures, across a wide range of space exploration and high-altitude decompression scenarios, warrants caution.

In conclusion, the traditional “Haldanean” type of gas exchange models do not provide an accurate view on the actual inert gas flows during hypobaric exposures. Hence, important information is missing, not only on the individual gas exchange, but also in correctly interpreting the subsequent bubble growth predictions and DCS risk estimation. The advantage of an adaptive physiological gas exchange model is that it can incorporate different anthropometric parameters, as well as account for physiological changes during decompression exposures. This improves N₂ gas exchange estimation significantly during different operational conditions. We therefore strongly recommend incorporating physiological parameters in any model to be used for the estimation of DCS risk and the design of mitigation procedures. This work clearly demonstrates that there is still a need to better integrate physiological factors when designing and building (probabilistic) decompression models and hypobaric risk estimation tools.