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### Summary

Apnoea due to airway obstruction is an ever present concern in anaesthesia and critical care practice and results in rapid development of hypoxaemia that is not always remediable by manual bag-mask ventilation. As it is often difficult or impossible to study experimentally (although some historical animal data exist), it is useful to model the kinetics of hypoxaemia following airway obstruction. Despite being a complex event, the consequences of airway obstruction can be predicted with reasonable fidelity using mathematical and computer modelling. Over the last 15 years, a number of high fidelity mathematical and computer models have been developed, that have thrown light on this important event.

Broadly speaking, a ‘model’ is a set of rules (equations) that is grounded in some immutable physical laws of nature, bounded by a set of underlying assumptions. So, if we drop a ball from a certain height, we can predict its velocity and position at any point after release, because we can construct a simple model of its descent based on Newton’s Second Law. Our assumptions might include the fact that the resistance of the medium through which the ball falls has a negligible effect. If we wanted more precision, we could certainly take account of air resistance, but if we try to include an exhaustive array of influencing factors, we will sacrifice simplicity of the model for enhanced precision. So often, in modelling, the balance between the beauty of parsimony and the necessity of complexity has to be carefully considered [1–4].

Models can be divided into ‘forward’ and ‘inverse’ [5]. The simpler forward models create a mathematical model based on physical laws, such as in the falling ball example given above, but in physiology, the model might be based on the conservation of mass, Fick’s law of diffusion, the gas laws etc. Inputting different values for model parameters (parameters are constants in an equation that link functions to variables) yields an output. This is the kind of model that has been developed to describe the development of hypoxaemia during obstructed apnoea that will be discussed later.

Inverse models are more complex because they do not usually involve the inputting of values for chosen parameters. Instead, they consist of a set of equations that are used to determine the properties of the model that are most consistent with the actual measured data. So, in our simple ball example, we might actually measure the velocity and position of the ball during its descent and use these data to determine which gravitational law, from a selection of guesses (assuming we did not know or believe Newton’s suggestion) is most likely to fit these data. A well-known example of this kind of inverse modelling in our discipline is the Multiple Inert Gas Elimination Technique (MIGET) to estimate ventilation/perfusion (V/Q) distributions in the lung. Here, measurements of arterial and venous partial pressures of various infused dissolved inert gases are made, and a computer determines which possible distribution or permutation of V/Q ratios (from many thousands of possibilities) is most consistent with the actual measurements [6]. Another example is pulse contour analysis for the determination of cardiac output [7].

### Modelling obstructed apnoea

Apnoea is the cessation of pulmonary (alveolar) ventilation. Superficially, it might seem that the rate of arterial desaturation during obstructed apnoea is simply determined by the imbalance between oxygen uptake at the lung and consumption in the tissues. Given that the rate of oxygen consumption in the tissues is largely constant under a given set of conditions, the critical factor would appear to be the size of the reservoir of oxygen in the lungs and blood. However, such a model would be too simplistic because the events of apnoea are classically in non-steady state (and, as it turns out, non-linear). In other words, the uptake of oxygen from the alveoli to the pulmonary capillary blood is not constant, but diminishes with time. The shape/position of the oxyhaemoglobin dissociation curve also changes as the pH and partial pressure of carbon dioxide (Pco2), change during the apnoea (the Bohr shift). Progressive systemic hypoxaemia might have a further effect on pulmonary vascular resistance, and if this were inhomogeneous (i.e. unevenly distributed across the lung), this might alter the distribution of pulmonary blood flow, thereby altering the degree of shunt or V/Q matching in the lungs in different ways. We can now see why this seemingly simple forward-modelling task might be actually quite complex.

According to Sands et al. [4], the first theoretical analysis of the factors influencing apnoeic desaturation was that of Farmery and Roe [3]. This analysis was constructed from first principles using discrete equations, solved symbolically, and so this now seems rather quaint compared with more contemporary analyses that invariably use computational numerical techniques or simulation tools, such as Simulink or Matlab (both of Mathworks, Cambridge, UK), that can be more powerful, yet are more opaque.

The basis of all these analyses lies in a physical model with stated assumptions. Most physical models are actually quite simple and consist of a single alveolar compartment, a pulmonary capillary blood flow and a shunt blood flow, both perfusing a single metabolic ‘tissue’ compartment, as shown in Fig. 1.

The alveolar compartment is the most important reservoir of oxygen, although as we will discover later, under normal circumstances it is not the largest. In the steady state during normal ventilation, the net rate of oxygen flowing into the alveolar compartment from the trachea equals the rate of removal of oxygen by the pulmonary blood. During obstructed apnoea, there is no influx from the trachea, yet oxygen continues to be removed from the alveolar compartment by the pulmonary blood, and so the alveolar Po2 (PAo2) and arterial Po2 (Pao2) begin to fall. Some models assume that alveolar volume remains constant during apnoea [3], whereas, in fact, the lung volume will diminish somewhat, because in early apnoea, oxygen uptake from the lung is greater than CO2 delivery to it. This property is incorporated into other models [2], although the differences in model behaviours are not great.

#### Modelling the events of early apnoea

From Fick’s law, we can determine that this oxygen flux (or uptake) is given by:

• (1)

where and are the instantaneous oxygen uptake and arterial (actually strictly pulmonary venous) oxygen content at the level of the alveolus, and these are functions of time (t), and and are the pulmonary blood flow and mixed venous (or pulmonary arterial) oxygen content, which are treated as constants, (i.e. time invariant). Note the distinction between and tissue oxygen consumption ().

Figure 2a shows that after a few seconds, the removal of oxygen from the alveolus will result in a fall in alveolar partial pressure and this in turn will result in a fall in arterial saturation (i.e. pulmonary venous saturation, Spv). From Equation 1, we can see that this reduced value will result in a smaller value of oxygen uptake, . If we were considering an inert gas, then this process would be classically exponential, i.e. that longer the process proceeds, the slower the gas uptake becomes. However, things are a bit more complicated for oxygen. The shape of the oxyhaemoglobin dissociation curve means that the fall in alveolar PO2 produces a non-linear fall in , so what might have been a simple exponential decline in oxygen uptake is now a compound of an exponential and a sigmoidal function.

The consequence of this non-linearity in oxygen dissociation is that, in a pre-oxygenated patient, will remain constant and maximal as long as the alveolar partial pressure is greater than ∼15 kPa. This will result in a period of constancy of oxygen uptake before the ultimate decline. The factors that determine the rate of fall of arterial saturation in this period are predominantly the alveolar volume and the initial FAo2. Interestingly, although cardiac output appears as a term in Equation 1, its value does not independently influence the uptake of oxygen, because it is mathematically coupled with the other terms in the equation (namely Ca and Cv). We can illustrate this by considering the immediate pre-apnoeic state, wherein, for a given metabolic rate and for a maximal arterial oxygen saturation, the mixed venous oxygen content is fixed by, and coupled to, the cardiac output. If we were to increase the cardiac output, the tissues will extract less oxygen per unit volume of blood passing through them, and so the mixed venous content will be elevated too. So, if in equation 1, has a high value, too will be elevated such that the product will be little changed (or more likely, somewhat reduced) compared with a lower cardiac output state. Consequently, the steady-state cardiac output has little influence in this early stage of apnoea.

#### Modelling the events of late stage apnoea

Now, imagine that oxygen has been taken up from the alveolus to the pulmonary capillary blood (albeit at a diminishing rate) such the arterial saturation is now almost equal to the mixed venous saturation. Now all three lung compartments (mixed venous blood, alveolar gas and arterial blood) are almost in equilibrium. Both and will continue to fall (together as one) as the tissues continue to extract oxygen from the blood, but there will be little difference between the arterial and mixed venous saturations across the lung and hence, little uptake of oxygen from the lung even though it is still far from empty of oxygen. The rate of fall of and from this point on are determined primarily by the size of the oxygen store within the blood, the lung being now much less relevant, and so the predominant factors here are the total blood volume and the haemoglobin concentration.

Some readers may have spotted an apparent (but readily explicable) inconsistency. Normally, there is always a fixed oxygen saturation or oxygen content difference between arterial and venous blood at the level of the tissues. However, we have described above a situation in which there is little or no arterial-venous content difference at the level of the lung. How can this be, given that the circulation is continuous between tissue and lung? The picture becomes clear when we consider that both and are functions of both time and space. This is depicted in Fig. 2b. In this example, at a time point 100 s after the start of apnoea, the Spv at the level of the alveolus has fallen to about 60%. However, the contemporaneous saturation in the peripheral artery (Sa) is around 70% because there is a transport lag between these sites, and the blood leaving the lungs with a saturation of 60% has not yet arrived. At the level of the tissues there remains a fixed peripheral arterial-venous saturation difference determined by the metabolic rate, cardiac output and haemoglobin concentration. This is shown by the vertical line connecting Sa and Sv. The contemporaneous mixed venous or pulmonary arterial saturation (Spa) is higher than the tissue venous saturation because of the transport lag between these sites. As a result of the combined arterial and venous transit lags, the transpulmonary saturation difference (ΔS) is indeed much smaller than the fixed peripheral arterio-venous saturation difference. The small transpulmonary saturation difference means that oxygen uptake from the lung is also small, whereas the tissues continue to consume oxygen at a fixed rate, thus depleting the remaining reservoir of oxygen; i.e. the blood.

#### The events of intermediate stage apnoea

We have noted how in early stage apnoea, the reservoir of oxygen within the lung buffers the rate of arterial desaturation, and so the lung volume and initial alveolar oxygen fraction are important determinants of the rate of desaturation. In contrast, in late stage apnoea, the lung is less relevant because oxygen uptake is small, and it is the reservoir of oxygen within the blood store that buffers the rate of desaturation. Not surprisingly, the haemoglobin concentration and blood volume are key determinants of the rate of desaturation in this phase. What connects these two phases? The transition between the early and late phases is determined by the circulatory transit time, which is inversely related to cardiac output. Examining Fig. 2b, it is clear that if the circulatory transit time were short, then the transpulmonary saturation difference (and hence the oxygen uptake from the lung) would be preserved at a value close to the peripheral saturation difference, and the lung reservoir would be depleted more quickly.

### Using model examples in practice

We can refine the model described by considering acid–base status. During the course of an apnoea, the blood becomes progressively acidaemic which produces a right shift (Bohr shift) in the oxyhaemoglobin dissociation curve. Given that we can estimate the CO2 production rate and the CO2 buffering capacity of the body, we can therefore estimate the rate of rise of Paco2. Assuming that this produces a pure uncompensated respiratory acidosis, we can also estimate the resultant pH change, and so, armed with estimates of Paco2 and pH at any instant, we can estimate the shift in the P50 value (i.e. the Pao2 value at 50% saturation of haemoglobin) as time progresses [8]. This progressive right shifting oxyhaemoglobin dissociation curve has the effect of accelerating the rate of desaturation.

Figures 3 and 4 show some examples of what are the key influences on the rate of desaturation in clinical practice. Figure 3 shows the effects of different lung volumes (Fig. 3a) and blood volumes (Fig. 3b) in models of the preterm infant [4]. This model shows extreme desaturation, as the patients have high metabolic rates compared with their lung volumes and total haemoglobin content. Note in Fig. 3a how the effect of lung volume on desaturation is limited to the early phase, and how the late phase desaturation rates are unaffected. Conversely, note in Fig. 3b how the effect of blood volume is limited to the late phase, and how the early phase desaturation rates are unaffected.

Figure 4 shows the more modest desaturation rates in adult subjects with varying metabolic rates and shunt fractions. Whilst we have neglected the effect of pulmonary shunt in our discussion so far, we can simulate this effect by adding a shunt flow that is a fixed fraction of the cardiac output as depicted in Fig. 1. Not surprisingly, this produces a greater level of desaturation at any time, but the rate of desaturation is not affected by a fixed shunt. However, it is highly likely that the shunt fraction is not fixed, but progressively worsens during a ‘real-life’ obstructed apnoea because of the effects of reduced functional residual capacity, increased abdominal pressure, reduced chest wall recoil and complex effects on pulmonary blood flow distributions during hypoxia and anaesthesia. These are not easy to model without a lot of guesswork, but one can envisage that during the course of an apnoea, patients may move from one iso-shunt line progressively down to the shunt line beneath, and so accelerating the rate of apparent desaturation.

The metabolic rate markedly influences the rate of desaturation as shown in Fig. 4b. Febrile, anxious and shivering patients would therefore experience more rapid apnoeic desaturation.

### Relevance to the ‘can’t intubate can’t ventilate’ situation

Both the American Society of Anesthesiologists and the Difficult Airway Society difficult airway algorithms recommend that if attempts to intubate the patient are unsuccessful, the patient should be allowed to awaken, by which it is assumed that the patient will return to an unparalysed state capable of resuming spontaneous ventilation and maintaining their own airway patency [9, 10]. Benumof used a modelling approach to determine what the arterial saturation would be by the time 1 mg.kg−1 suxamethonium had worn off and muscle twitch amplitudes had returned to between 10% and 90% of normal [9]. These data are reproduced in Fig. 5. The patients under study were pre-oxygenated. The normal 70-kg patient was able to maintain arterial saturation above critical levels for about 8 min, which is just within the recovery window for the return of neuromuscular function. Data are also presented for the ‘unwell’ 70-kg patient. In this example, the assumptions are that alveolar volume is reduced by 20%, metabolic rate increased by 20%, the cardiac output reduced by 20%, haemoglobin concentration reduced by 30% and there is a 10% shunt. Data from an obese patient (with reduced functional residual capacity and increase in oxygen consumption) are also displayed. For both the unwell and the obese patients, critical desaturation is predicted to occur before functional neuromuscular recovery has even begun, despite pre-oxygenation. Even in the healthy patient, full neuromuscular recovery is not predicted to occur before critical desaturation supervenes.

### Wider physiological airway modelling

This review has only focussed on modelling the development of hypoxaemia following apnoea. An extensive review of all aspects of airway modelling is outside the scope of this article, but ‘the airway’ and ‘gas exchange’ are rich grounds for biomathematical modellers. Most clinicians and physiologists model the conducting airways and alveolus as a single ‘balloon on a straw’ because the reality of the branching geometry of the conducting airways, and its relationship with the blood–gas interface, is highly complex. The classical texts of Weibel and Paiva [18, 19] are an introduction to modelling an anatomically realistic branching structure whose cross-sectional area grows rapidly with successive airway generations (the so called ‘trumpet model’). Their partial differential equations describe gas concentrations as functions of both space and time, and this leads to the conclusion that there is in fact no discrete boundary between convective gas transport in the upper airway and diffusive gas transport in the alveoli. Consequently, we learn from these models that there is no discrete boundary between what we used to consider to be alveolar and upper airway dead space compartments [5]. The relevance of this kind of modelling is that we can apply it to real clinical situations to get answers or predictions that match reality. For example, according to classical teaching, for a constant alveolar ventilation rate, the arterial and end-tidal Pco2 should not be influenced by the specific ventilatory pattern (flow-control, volume control, inspired:expired time ratio, etc.) that is delivered to patients’ lungs during active ventilation. However, in reality, we know that ventilatory pattern does indeed influence arterial and end-tidal Pco2, and this can be predicted using the complex ‘trumpet’ model, but not using simpler models [18, 19]. A challenge for the future is to develop personal, patient-specific models that will allow individualised prediction and direction of therapy.

The Human Physiome Project (http://www.physiome.org) is an integrated multicentre programme that began in 1997 with the aim of creating integrated models of functional behaviour of human physiology, starting with microscopic modelling of the outputs of genes and compiling these into behaviours at the level of the cell, tissue, organ and ultimately the entire organism [21]. A major part of this project is the Lung Physiome that is a supercomputer-driven, anatomically based model of asymmetrically branching conducting airways and acini, that is informed by knowledge of behaviours at the subcellular and cellular levels [20]. This may seem far away from being immediately clinically applicable, but it is widely believed that such modelling will play an important role in the development of new therapies (that can be ‘tested’ in these ‘in silico’ models) and in furthering our understanding of the mechanisms and treatments of conditions such as obstructive airways disease and acute respiratory distress syndrome.

### Conclusions

The kinetics of desaturation following apnoea is relatively complex. Understanding apnoea and the genesis of hypoxaemia as a result of it is highly relevant to several issues in airway management. This more fundamental understanding has potential to take ‘airway management’ beyond simple considerations of which device to use to support the airway. There are a number of key physiological factors that operate in apnoea, and these interact in non-linear and often counterintuitive ways such that it is not always possible to determine the effect of any single factor in isolation. Modelling allows us to examine these complex events and their interactions, so that we can better understand the consequences of our clinical actions or inactions.

### Competing interests

No external funding or competing interests declared.

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