A Model Analysis of Arterial Oxygen Desaturation during Apnea in Preterm Infants

Rapid arterial O2 desaturation during apnea in the preterm infant has obvious clinical implications but to date no adequate explanation for why it exists. Understanding the factors influencing the rate of arterial O2 desaturation during apnea () is complicated by the non-linear O2 dissociation curve, falling pulmonary O2 uptake, and by the fact that O2 desaturation is biphasic, exhibiting a rapid phase (stage 1) followed by a slower phase when severe desaturation develops (stage 2). Using a mathematical model incorporating pulmonary uptake dynamics, we found that elevated metabolic O2 consumption accelerates throughout the entire desaturation process. By contrast, the remaining factors have a restricted temporal influence: low pre-apneic alveolar causes an early onset of desaturation, but thereafter has little impact; reduced lung volume, hemoglobin content or cardiac output, accelerates during stage 1, and finally, total blood O2 capacity (blood volume and hemoglobin content) alone determines during stage 2. Preterm infants with elevated metabolic rate, respiratory depression, low lung volume, impaired cardiac reserve, anemia, or hypovolemia, are at risk for rapid and profound apneic hypoxemia. Our insights provide a basic physiological framework that may guide clinical interpretation and design of interventions for preventing sudden apneic hypoxemia.


Introduction
Apnea and its accompanying arterial O 2 desaturation are common clinical complications in preterm infants, occurring in more than 50% of very low birth weight infants [1]. In preterm infants, apnea causes a reduction in heart rate [2] and cerebral perfusion [3], often requires mechanical ventilation, and is associated with neurodevelopmental impairment [4]. Apnearelated hypoxemia is of major concern in light of evidence that repetitive hypoxia in newborn animals results in irreversiblyaltered carotid body function [5], raising the possibility of impaired ventilatory control, and causes neurocognitive and behavioural deficits [6]. Respiratory arrest and hypoxemia are also strongly implicated in sudden infant death syndrome (SIDS) [7,8] where the speed at which hypoxemia develops is considered to be particularly dangerous.
In preterm infants, the rate of arterial O 2 desaturation ( _ S Sa O2 ) can be highly variable and rapid, with average rates as high as 4.3% s 21 during isolated apneas [9]. An earlier framework to describe _ S Sa O2 proposed that metabolic O 2 consumption relative to alveolar volume determines the speed at which alveolar P O2 falls [10]; it was envisaged that _ S Sa O2 is then a function of falling P O2 and the slope of the oxy-hemoglobin dissociation curve. However, such a model assumes that the rate of alveolar depletion of O 2 , denoted pulmonary O 2 uptake ( _ V Vp O2 ), is equal to tissue O 2 consumption during apnea (see Methods -Theory). Previous studies in adults have shown that _ V Vp O2 falls from metabolic consumption during apnea [11], and our previous modeling studies in lambs showed that the difference between _ V Vp O2 and metabolic O 2 consumption has a crucial role in determining _ S Sa O2 during recurrent apneas [12]. We found that apneic changes in _ V Vp O2 cause desaturation to occur in 2 stages. During stage 1, lung O 2 stores are depleted, and _ V Vp O2 falls below metabolic consumption. During stage 2, _ V Vp O2 is close to zero, and tissue O 2 needs are provided by depletion of blood O 2 stores.
To date, no complete theoretical analysis of the factors influencing desaturation during apnea has been published. The only available study [13] has a number of critical limitations. First, the model incorporated a constraint of a fixed difference between Sa O2 and mixed-venous saturation; thus dynamic changes in _ V Vp O2 could not occur and their influence on _ S Sa O2 could not be examined. Second, no assessment was made of the impact of cardiorespiratory factors on the two stages of O 2 desaturation. Third, in focusing on adults, the study did not examine profound desaturation to levels well below 60% as can often occur in preterm infants [9,14].
Accordingly, the aim of the current study was to quantify the importance of cardiorespiratory factors relevant to _ S Sa O2 during apnea, with particular reference to the preterm infant. Using a model that permits variation of _ V Vp O2 during apnea, we examine a number of factors known to influence _ S Sa O2 , such as lung volume [15], metabolic O 2 consumption [16] and pre-apneic arterial oxygenation [17] as well as factors that are particularly pertinent for the developing newborn, including anemia, hypovolemia, reduced O 2 affinity, and chronically and acutely reduced cardiac output. We use the results to develop a conceptual framework for the interpretation of mechanisms underlying rapid _ S Sa O2 during apnea.

Overview of the two-compartment model for gas exchange
To determine the independent influence of clinically relevant cardiorespiratory factors on _ S Sa O2 during a single isolated apnea, we used a two-compartment lung-body mathematical model which incorporated realistic blood O 2 stores and gas exchange dynamics (Figure 1), as described in Methods -Mathematical model (a full list of symbols is provided in Table 1). We used published parameters for healthy preterm infants born at ,30 wk gestational age (Table 2); the values represent measurements taken at approximately term equivalent age when surprisingly rapid desaturation has been observed [9]. We also derive analytic solutions for _ S Sa O2 to quantify the importance of cardiorespiratory factors on _ S Sa O2 to obtain a detailed view of the arterial O 2 desaturation process, as described in Methods -Theory.

Pulmonary gas exchange dynamics during apnea
To examine changes in O 2 /CO 2 exchange during apnea, a single apnea was imposed on the model. During apnea, changes in alveolar O 2 and CO 2 stores are not constant ( Figure 2); importantly, alveolar P O2 (PAO 2 ) did not continue to fall at its initial rate as governed by metabolic O 2 consumption ( _ V V O2 ), but instead the rate of fall in PAO 2 was reduced as it approached mixed venous P O2 (P v v O2 ), an observation also reflected in the falling _ V Vp O2 . As a result, two distinct phases for O 2 depletion can be seen, which we refer to as stage 1 and stage 2 [12]. During stage 1, PAO 2 fell rapidly and _ V Vp O2 decreased and became dissociated from _ V V O2 ; during stage 2, with _ V Vp O2 greatly reduced, both PAO 2 and P v v O2 fell together at a reduced rate. The two distinct phases were also observed for alveolar and arterial P CO2 (PACO 2 , Pa CO2 ) although stage 1 for CO 2 was substantially shorter than that for O 2 . Such an effect results from the earlier fall in pulmonary CO 2 uptake ( _ V Vp CO2 ) relative to the fall in _ V Vp O2 (Figure 2A) and is reflected in the reduction in respiratory exchange ratio ( _ V Vp CO2 = _ V Vp O2 ) ( Figure 2B). Consequently, a more rapid fall in PAO 2 was observed compared with the rise in PACO 2 (see Methods -Derivation of equations), such that PAO 2 fell by 100 mmHg in the time PACO 2 rose by just 14 mmHg ( Figure 2C).

Time-course of _ S Sa O 2 during apnea
The time-course of _ S Sa O2 is complex ( Figure 3), a consequence of the nonlinear O 2 -dissociation curve in combination with the fall in _ V Vp O2 . At apnea onset, Sa O2 started to fall with a rate equivalent to that predicted by Equation 12, where _ S Sa O2~0 :5% s {1 (Figure 3). During apnea, changes in the slope of the O 2 -dissociation curve (bHbO 2 ) and _ V Vp O2 dominated the time-course of desaturation as hypoxemia progressed. As Sa O2 started to fall after apnea onset, bHbO 2 increased with little change in _ V Vp O2 , resulting in a proportional increase in _ S Sa O2 . However, as arterial hypoxemia developed, there was a concurrent decline in _ V Vp O2 . As _ S Sa O2 is directly proportional to the product bHbO 2 | _ V Vp O2 (Equation 11) it follows that during apnea, the peak _ S Sa O2 of 3.5% s 21 occurred when bHbO 2 | _ V Vp O2 reached a maximum. This occurred when neither _ V Vp O2 nor bHb O2 was at its maximum (both ,50% of peak). Finally, with _ V Vp O2 greatly reduced during stage 2, _ S Sa O2 remained at a constant level ( _ S Sa O2~1 :7% s -1 ), close to that predicted by Equation 13 (1.8% s 21 ).

Factors influencing _ S Sa O 2
The following parameters were individually varied from their 'normal' values to quantify their influence on _ S Sa O2 : resting PAO 2 , Figure 1. Model schematic representing O 2 uptake, transport and consumption. O 2 stores are represented by the alveolar, arterial, and venous compartments. Two dynamically-independent levels of O 2 uptake are denoted: pulmonary O 2 uptake ( _ V Vp O2 ) and metabolic consumption ( _ V V O2 ). R-L shunt is also included. T a is the arterial transit time. Symbols are described in Table 1

Author Summary
When breathing stops, the flow of O 2 into and the flow of CO 2 out of the body cease. Such an event, termed an apnea, can be especially dangerous in preterm infants in whom it can lead to a rapid decline in arterial O 2 saturation, reaching rates of 3-8% per second, rapidly reducing O 2 to a level that could lead to neurological damage. Despite extensive experimental research, we have a poor mechanistic understanding of the causes of rapidly developing hypoxemia. We describe a new mathematical model that allows examination of the importance of the major cardiorespiratory factors that are likely to influence the speed at which arterial hypoxemia worsens during apnea. We found that high metabolic rate as well as reduced pre-apneic ventilation, lung volume, cardiac output, hemoglobin content, blood O 2 affinity, and blood volume accelerate the development of hypoxemia during apnea. Importantly, the cardiorespiratory factors that contribute to rapid hypoxemia are all pertinent to the preterm infant during early postnatal development. Thus the newborn is highly susceptible to rapid and severe desaturation, potentially explaining the propensity of preterm infants, particularly those with apnea, to neurological impairment.
To quantify _ S Sa O2 we used 3 different measures. First, since apnea is considered clinically significant if it lasts for .10 s and is accompanied by bradycardia or O 2 desaturation [18], we calculated the average rate of fall in Sa O2 between apnea onset and 10 s later ( _ S Sa 10 s O2 ); such a measure describes the immediacy of onset of desaturation and is analogous to the practical measurement of average _ S Sa O2 used in many clinical studies [9,15,19,20]. Second, we determined the peak instantaneous _ S Sa O2 during apnea Blood volume (Qb) 80 ml kg 21 [57] P 50 is the partial pressure at 50% saturation. VL is taken from data on functional residual capacity. For all simulations unless otherwise stated: respiratory exchange ratio (RER) was assumed to be 0.8; shunt fraction (Fs) was adjusted to 8.7% to achieve a resting Pa O2 of 72 mmHg as is typical for normal healthy infants [58]; resting alveolar ventilation ( _ V VI~168 ml min -1 kg -1 under normal conditions) was set to achieve resting PAO 2~1 00 mmHg. doi:10.1371/journal.pcbi.1000588.t002 ( _ S Sa peak O2 ), the value during the linear portion of arterial desaturation [10,21] which we find is not confounded by resting Sa O2 . Third, we report a measure of Sa O2 during stage 2 apnea ( _ S Sa stage 2

O2
). To quantify the sensitivity of _ S Sa O2 to changes in each cardiorespiratory factor, we defined the term impact ratio as the ratio of proportional increase in _ S Sa O2 to a small increase from the normal value of each factor. For example, an impact ratio of 1 indicates a one-to-one increase in _ S Sa O2 with an increase in the factor, and a negative ratio indicates an inverse relationship. The impact of each cardiorespiratory factor on _ S Sa 10 s O2 , _ S Sa peak O2 , and _ S Sa stage 2 O2 is summarised in Table 3.
Resting PAO 2 . Changes in PAO 2 , achieved via reduced resting ventilation or increasing inspired O 2 (FIO 2 ), had a substantial effect on the onset of desaturation. Reduced pre-apneic PAO 2 dramatically increased _ S Sa 10 s O2 ( Figure 4A), but had little effect on _ S Sa peak O2 or _ S Sa stage 2

O2
. In contrast, increasing pre-apneic PAO 2 with the application of supplemental O 2 achieved the opposite, essentially right-shifting or delaying the arterial desaturation curve, where one second of delay can be achieved by an increase in PIO 2 (DPIO 2 ) of ,7 mmHg, or DFIO 2 of ,1% (see Methods -Derivation of equations). These results occurred despite only a minor influence being visible on resting Sa O2 . For example, a reduction of PAO 2 from 100 to 60 mmHg caused a 6% reduction in resting Sa O2 but at the same time led to a more than 2-fold elevation in _ S Sa 10 s O2 ( Figure 4B). Additionally, a severe reduction in PAO 2 , to below 70 mmHg, was required to elevate _ S Sa peak O2 . Lung volume (VL) and blood volume (Qb). _ S Sa 10 s O2 and _ S Sa peak O2 were inversely related to VL during stage 1 ( Figure 5A, B), but changes in VL had no influence on _ S Sa stage 2

O2
. In direct contrast, reduced Qb strongly increased _ S Sa stage 2

O2
, but had no effect on stage   Impact ratio is defined as the ratio of proportional increase in _ S Sa O2 to the proportional increase in each factor, based on small changes around normal values. An impact ratio of 1 indicates a one-to-one increase in _ S Sa O2 >with an increase in the factor, and a negative ratio indicates an inverse relationship. CC = cardiac compensated, nCC = cardiac uncompensated.
; we refer to this procedure as 'cardiac compensation'. Under this condition, elevated _ V V O2 caused a directly proportional increase in Sa O2 throughout stages 1 and 2 ( Figure 6A, B). Without cardiac compensation, the effect of increased _ V V O2 on Sa O2 during stage 1 was magnified, as shown by the further increase in _ S Sa peak O2 ( Figure 6A, B). Hemoglobin content (Hb) and oxygen affinity (P 50 ). Reduced hemoglobin content (Hb) increased _ S Sa peak O2 and _ S Sa stage 2 O2 but had little effect on _ S Sa 10 s O2 ( Figure 7A, B). The increase in _ S Sa peak O2 occurred with an increase in the peak of the product bHbO 2 | _ V Vp O2 as _ V Vp O2 was higher at each level of Sa O2 . The simulation was repeated with cardiac compensation for the reduction in hemoglobin content, where D _ Q QT(%)~1=DHb(%), to maintain constant resting S v v O2 . Following such compensation, no changes in _ S Sa peak O2 or _ S Sa 10 s O2 were observed but reduced Hb continued to increase _ S Sa stage 2

O2
. In examining the influence of P 50 , P 90 was adjusted in equal proportion on the basis of published data [22]. Increased P 50 increased the immediate _ S Sa O2 , increased _ S Sa 10 s O2 , decreased _ S Sa peak O2 and had no effect on _ S Sa stage 2 O2 ( Figure 7C, D).
Cardiac output ( _ Q QT). Reduced resting _ Q QT increased _ S Sa peak O2 , but had little impact on _ S Sa 10 s O2 or _ S Sa stage 2 O2 ( Figure 8A, B). As with Hb, the increase in _ S Sa peak O2 with reduced resting _ Q QT occurred with an increase in the peak of the product bHbO 2 | _ V Vp O2 . To differentiate between the influence on _ S Sa O2 of an acute reduction in cardiac output, i.e. when bradycardia accompanies apnea, rather than a chronic reduction, we reduced cardiac output in a step-wise manner from the baseline value at the time of apnea onset. In constrast to the effect of reduced resting _ Q QT, a transient ( Figure 8C, D). Resting R-L shunt fraction (Fs). Increased Fs reduced resting Sa O2 and S v v O2 but had no effect on _ S Sa 10 s O2 , _ S Sa peak O2 , or _ S Sa stage 2 O2 ( Figure 9A, B).

Discussion
Our model analysis of the rate of arterial O 2 desaturation during apnea demonstrates that pre-apneic ventilation, lung volume, cardiac output, hemoglobin content and blood volume exert unique effects on _ S Sa O2 throughout the time-course of desaturation, while metabolic O 2 consumption is uniformly influential throughout the process. Our analysis reveals that lung volume and the slope of the O 2 -dissociation curve are important early in the process, during what we refer to as stage 1 [12], but not stage 2. For the first time, our study reveals that reduced cardiac output and hemoglobin content, and as a consequence resting mixed-venous saturation, substantially accelerate peak _ S Sa O2 . Finally, low blood volume and hemoglobin content, and therefore a low total blood O 2 capacity, increase the speed of desaturation, but only in stage 2. In addition to infants with elevated metabolic needs and low lung volume, those with anemia, cardiac dysfunction, or hypovolemia, which are common complications of prematurity, are at heightened risk of rapid and profound arterial desaturation during apnea.

Methodological considerations
To evaluate the independent effects of cardiorespiratory factors on _ S Sa O2 we used a two-compartment model, incorporating both alveolar and blood gas stores. The inclusion of a realistic blood store was crucial to reveal that changes in _ V Vp O2 occur as a consequence of arterial and mixed-venous saturation falling ; the influence on _ S Sa peak O2 is small in the normal range but becomes stronger at low PAO 2 . n = 'normal' 'values; S1, stage 1 slope; S2, stage 2 slope. doi:10.1371/journal.pcbi.1000588.g004 asynchronously during apnea (Figure 3). Our approach allowed us to extend the previous framework based on the assumption of constant _ V Vp O2 [23], which prevented the recognition that a steep O 2 -dissociation curve and low lung volume do not accelerate _ S Sa O2 beyond stage 1. Furthermore, the varying _ V Vp O2 permitted recognition that cardiac output, hemoglobin content, and blood volume have a major influence on _ S Sa O2 . In the current study, the typical value of _ S Sa O2 found using our model was 3.5% s 21 whereas Poets and Southall [9] using beat-bybeat oximetry in preterm infants reported a mean value for _ S Sa O2~4 :3% s -1 during isolated apneas. Reasons for our lower value may lie with our simplifying assumptions. Notably, we assumed a homogenous lung compartment and complete gas mixing and as such, the model incorporated neither limitation of alveolar-capillary diffusion nor an uneven ventilation-perfusion distribution, two factors that could cause an increase in _ S Sa O2 . In addition, we assumed a constant lung volume during apnea, equal to published values of functional residual capacity, whereas it is known that lung volume can fall during apnea [15,24]; based on our data, a fall in lung volume to 15.5 ml min 21 kg 21 immediately after apnea onset would achieve _ S Sa O2 of 4.3% s 21 ( Figure 5B). A final assumption implicit in our model is that all O 2 transfer to the blood occurs via the pulmonary circulation. However, in very preterm infants there is evidence of percutaneous respiration in the first few days of life in both room air and with supplemental O 2 [25]. With whole body exposure of 90% O 2 to the newborn skin, it has been calculated that _ V Vp O2 can be reduced by 8-10% [26], likely via an increased resting mixed-venous saturation; our study demonstrates that such an effect would decrease _ S Sa O2 during apnea.

Pulmonary gas exchange dynamics during apnea
Our study is consistent with previous observations that _ V Vp O2 and _ V Vp CO2 rapidly decline during apnea from their steady-state values [11], with _ V Vp CO2 falling faster than _ V Vp O2 . The relatively low blood capacitance for O 2 compared with that for CO 2 results in the resting alveolar-mixed-venous partial pressure difference being ,12-fold greater for O 2 than for CO 2 . Consequently, when apnea begins ,12 times more O 2 than CO 2 must diffuse across the lung to obliterate the alveolar-mixed-venous partial pressure difference. The slower fall in _ V Vp O2 vs. _ V Vp CO2 provides for a faster depletion of alveolar O 2 vs. CO 2 stores; such an effect results in complete desaturation of arterial blood in the time Pa CO2 rises by just 14 mmHg. These findings lead us to conclude that short-term O 2 homeostasis is more unstable than CO 2 homeostasis and thus that the danger of isolated apneas in infants is likely to be mediated via hypoxemia rather than hypercapnia.

Factors influencing _ S Sa O 2
Our study provides for the first time a comprehensive analysis of the factors that determine arterial desaturation during apnea in preterm infants. We show that resting oxygenation in the form of alveolar P O2 has the greatest influence on desaturation at apnea onset. When apnea begins at an increasingly lower alveolar P O2 , _ S Sa O2 more quickly reaches its maximum because P O2 rapidly arrives at the steepest part of the O 2 -dissociation curve. This effect explains the inverse relationship between mean _ S Sa O2 and preapneic Sa O2 during apnea [17], but as we show the peak slope itself is negligibly affected by reduced resting P O2 within the normal range.
We demonstrate that _ S Sa O2 is inversely related to lung volume during stage 1 of apnea as a result of the greater reduction in alveolar P O2 in poorly inflated lungs per unit of O 2 transferred into the pulmonary capillaries. This analysis is consistent with the inverse correlation between _ S Sa O2 and lung volume [15], with the view that active upper airway closure maintains lung volume and slows _ S Sa O2 [27,28], and with our recent report that the application of continuous positive airway pressure effectively slows _ S Sa O2 in lambs [29]. However, once stage 2 begins, the blood becomes the principal source of O 2 and thus the only store which influences _ S Sa O2 . A novel finding from our study is that reduced resting mixedvenous saturation, caused by either a reduced cardiac output or reduced hemoglobin content, strongly elevates peak _ S Sa O2 , independent of metabolic O 2 consumption. We show that reduced resting mixed-venous saturation accelerates _ S Sa O2 via an increase in the peak value of bHbO 2 | _ V Vp O2 ; in other words, low mixed-venous saturation provides for a greater pulmonary O 2 uptake even in the presence of a developing arterial hypoxemia, and thereby increases _ S Sa O2 . A role for hemoglobin in determining _ S Sa O2 is consistent with the finding that elevated hemoglobin content in adults slows _ S Sa O2 during apnea [21]. In contrast, blood transfusion to raise hemoglobin content in anemic preterm infants, a common clinical therapy, has little or no impact on the severity of apneic desaturation [30]. Our proposed explanation for the lack of benefit of raising hemoglobin content via transfusion is that it also reduces heart rate [30] and cardiac output. Thus, in the newborn, the rise in mixed-venous saturation expected after transfusion is counteracted by a tendency for mixed-venous saturation to fall as a result of reduced cardiac output. An investigation that failed to find an effect of cardiac output on _ S Sa O2 [23] did not account for our  finding that pre-apneic and transient changes in cardiac output have opposing influence on _ S Sa O2 . Importantly, we find that a transient fall in cardiac output, characteristic of bradycardia during apnea in preterm infants [2], conserves alveolar O 2 via reduced _ V Vp O2 and thus reduces _ S Sa O2 (see Equations 10 and 11). Consistent with this finding, apneic bradycardia prevents a rapid fall in Sa O2 in adults [21].
We found that each of the factors examined exerts a unique and therefore recognisable influence on the time course of the desaturation process ( Figure 10). Low alveolar P O2 can be recognised by a left-shift of the desaturation trajectory so that desaturation begins sooner following the onset of apnea. A steep desaturation slope in the early phase of stage 1 points to a low ratio of lung volume to metabolic O 2 consumption. In the late phase of stage 1, when desaturation proceeds in a linear fashion, a low resting mixed-venous saturation accelerates _ S Sa O2 and leaves the fingerprint of a low inflection point in arterial O 2 desaturation; low resting mixed-venous saturation reflects low cardiac output or hemoglobin content with respect to O 2 consumption. Lastly rapid _ S Sa O2 during stage 2 signifies a low total blood O 2 capacity with respect to O 2 consumption which would point to either low blood volume or anemia. The presence of a constant R-L shunt, while having no influence on _ S Sa O2 , causes a parallel downwards shift in the desaturation trajectory. The unique impact of different factors on the desaturation curve may be used to guide preventive clinical intervention.

Clinical significance
We show theoretically that the lower lung volume [31] and higher metabolic O 2 consumption [32] of preterm compared to term infants predisposes to a rapid onset and progression of desaturation during apnea. Two reports offer support for this view. First, rapid desaturation occurs in infants with low functional residual capacity [15], a finding that may help to explain the more frequent O 2 desaturation events during active sleep [33] when functional residual capacity is reduced. Second, frequent desaturation is characteristic of preterm infants with bronchopulmonary dysplasia (BPD) [34] whose O 2 consumption is 25% greater [35], and functional residual capacity is 25% less [36], than in preterm infants without BPD; Equations 11 and 12 predict that such differences increase both immediate and peak _ S Sa O2 by ,70%. In addition, hypoventilation and reduced resting PAO 2 in infants with BPD, as inferred from elevated PACO 2 [37], further increase desaturation at apnea onset. Our finding that each rise of 1% in inspired O 2 provides ,1 s of delay (right-shift) in the onset of apneic desaturation (Equation 15) may guide the titration of supplemental O 2 for the prevention of apneic hypoxemia while minimising the well known side-effects of long-term exposure to hyperoxia.
Our study has implications for the management of infants in clinical care. Metabolic O 2 consumption can be elevated after feeding [38], with reduced ambient temperature [39], and via the adminstration of methylxanthines [40]. Despite the success of methylxanthines in reducing the frequency of apnea and bradycardia, such treatment has surprisingly little impact on hypoxemic episodes [41]; we suggest that the elevated O 2 consumption and the absence of bradycardia are likely to increase _ S Sa O2 during those apneas that persist despite treatment. The severity of hypoxemic episodes is reduced by switching preterm infants from supine to prone [42], which may increase functional residual capacity [43] and improve diaphragm function, increase tidal volume and increase resting alveolar P O2 [44]. Our finding that low cardiac output leads to increased _ S Sa O2 during apnea leads to the suggestion that judicious adjustment of inotropic support in infants with cardiac abnormalities could improve resting mixedvenous saturation and reduce apneic hypoxemia.
Hypoxemic events become less frequent between infancy and childhood, despite an unchanged apnea frequency [28], perhaps as a result of a fall in O 2 consumption per body weight. However, before this occurs, infants experience a period of susceptibility to rapid desaturation during apnea as a result of a fall in hemoglobin content and O 2 affinity [22] and a rise in O 2 consumption [45]. The implications for SIDS are obvious in that these changes coincide with the peak incidence for SIDS at 2-3 months [46]. SIDS also occurs disproportionately in preterm infants [46], who manifest severe anemia [22] and greater O 2 consumption. Infants resuscitated from apparent life threatening events have been found to have lower hemoglobin content [47], pointing to a potential role for rapid _ S Sa O2 in the progression of such events. It is possible that the rapid development of apneic hypoxemia initiates prolonged hypoxic cardiorespiratory depression that in turn leads to SIDS.

Conclusion
We have provided a mathematical framework for quantifying the relative importance of key cardiorespiratory factors on the rate of arterial O 2 desaturation during apnea, with particular relevance to preterm infants. For the first time we have demonstrated that each of the factors examined has a signature influence on the trajectory of desaturation, providing quantitative insight into the causes of rapidly developing hypoxemia during apnea.

Mathematical model
Lung compartment. For the lung, a single homogeneous compartment is assumed based on the model of Grodins et al [48]. Each equation describing changes in the alveolar partial pressure of each gas (G) is based on the conservation of mass (specifically, the pressure-volume product) and is expressed in terms of inspired and expired alveolar ventilation and transfer of gases into the pulmonary capillary: where _ P PAG represents the rate of change of alveolar P O2 , P CO2 , and P N2 ; PIG represents the inspired alveolar partial pressure of each gas G; P 0 is atmospheric pressure converted from STP to BTP (863 mmHg); _ V Vp G represents _ V Vp O2 and { _ V Vp CO2 , pulmonary gas uptake (STPD) for O 2 and CO 2 ( _ V Vp N2 was neglected in this study for simplicity); _ V VI and _ V VE are inspired and expired alveolar ventilation (BTPS). Accounting for the difference in _ V VE and _ V VI due to a net pulmonary gas uptake into the pulmonary blood, yields: where PB = barometric pressure (760 mmHg); P vap = water vapour pressure (47 mmHg); _ V Vp total is the net pulmonary gas uptake, _ V Vp total~_ V Vp O2 { _ V Vp CO2 . Since purely obstructive apneas are relatively rare in preterm infants [49], an unobstructed airway was chosen as the standard model in this study. In the current study it was assumed that lung volume did not fall during apnea, as in active sleep [24], when apneic desaturation events are most common [33]. With lung volume constant, conservation of mass requires that passive airflow into the unobstructed airway must occur in response to a net pulmonary gas uptake into the pulmonary blood [11]. To account for this effect, we can write: Pilot simulations predicted that the volume of gas inflow during apnea is unlikely to exceed physiological deadspace. Thus, during apnea PI G is taken as PAG of the last exhaled breath prior to apnea onset. For the current study we assumed diffusion equilibrium at the pulmonary capillaries, such that PA G~P c 0 G . Gas uptake is determined from the Fick equation; specifically, pulmonary blood flow ( _ Q Qp), and the difference between end capillary (Cc 0 G ) and mixed venous (C v v G ) content: Utilising equations for R-L shunt, arterial content of each gas G is determined from its end capillary (Cc 0 G ) and mixed venous (C v v G ) content, and pulmonary shunt fraction (Fs): Fs defines the ratio of pulmonary blood flow to cardiac output, Body compartment. Assuming that the P O2 of the venous blood is equilibrated with the tissue P O2 , the mass-balance equations are: where Ca G (t{T a ) represents the gas content of O 2 and CO 2 in the arterioles; T a is arterial transit time; , the metabolic consumption of O 2 and production of CO 2 ; Qv G represents Qv O2 and Qv CO2 the combined venous/ tissue volumes for O 2 and CO 2 .
Blood O 2 stores were partitioned by assigning blood volume (Qb) to arterial (25%) and venous (75%) compartments [50] and they were modelled assuming an entirely unmixed arterial compartment, and an entirely mixed and homogenous venous compartment. The arterial transit time (T a ) is constrained by the arterial volume (Qa) by the relationship T a~Q a= _ Q QT. The body compartment volume Qv O2 is taken as the venous volume. Qv CO2 , the effective venous/tissue volume for CO 2 is taken as the same value for Qv O2 , based on published data (see Methods -Derivation of equations). Physiologically this represents no additional contribution of a specific tissue reservoir for CO 2 within the time frame of apnea.
To characterise the O 2 -dissociation curve we used a modified form of the equation of Severinghaus [51]. We re-expressed the equation with respect to the partial pressure at 50% (P 50 ) and at 90% (P 90 ) saturation: where k 1~( 9P 50 3 -P 90 3 )=(P 90 -9P 50 ) and k 2~P50 3 zk 1 P 50 . Values for P 50 (24.0 mmHg) and P 90 (53.6 mmHg), were obtained from the data of Delivoria-Papadopoulos [22] for a 9-10 wk-old preterm infant. O 2 content (C O2 , ml ml 21 ) includes that bound to hemoglobin (Hb, g ml 21 ) and that dissolved in plasma: The relationship between CO 2 content (C CO2 ) and P CO2 was assumed linear: where bb CO2~0 :0048 ml ml -1 mmHg -1 and k CO2~0 :364 ml ml -1 as adapted for STPD from Grodins et al. [52]. Simulations were performed using software written in MA-TLAB (The Mathworks; Natick, MA). Theory A general equation. In an earlier study we developed a general relationship that describes the factors influencing the magnitude of _ S Sa O2 at any instant in time during apnea [12]:  [11,12]. However, such an assumption is valid prior to any substantial fall in Sa O2 , and as therefore useful to explicitly describe _ S Sa O2 immediately upon apnea onset ( _ S Sa onset O2 ): increases dramatically with reduced resting PAO 2 ( Figure 11). Although no simple expression could be written to describe _ S Sa O2 explicitly for stage 1, we derived an expression for _ S Sa O2 during stage 2 (see Methods -Derivation of equations), given by:

Derivation of equations
Here we derive the explicit equations used within the current study to encapsulate key relationships pertaining to gas exchange and arterial desaturation during apnea. . Estimation of effective blood volume for CO 2 . Using the same methodology as described above, the ratio of _ C C v v O2 to _ C C v v CO2 during stage 2 can be used to estimate the ratio of Qb CO2 to Qb O2 . _ C C v v O2 and _ C C v v CO2 can be found using: where Qb O2 and Qb CO2 are the effective blood volumes for O 2 / Figure 11. Relationship between the slope of the oxyhemoglobin dissociation curve and alveolar P O2 O2 . Note that reduced alveolar P O2 (PAO 2 ) causes a substantial increase in the slope of the oxy-hemoglobin dissociation curve (bHbO 2 ; see inset) and in _ S Sa O2 at apnea onset ( _ S Sa onset O2 ; based on Equation 12). doi:10.1371/journal.pcbi.1000588.g011 CO 2 ; bb CO2 is the capacitance coefficient for CO 2 . Neglecting pulmonary gas exchange, combining Equation 17 for O 2 and CO 2 gives: Equation 18 permitted the calculation of Qv CO2 =Qv O2 based on published data [53; their Figure 3] where during apnea the rate of rise in C v v CO2 is very close to the rate of fall in the product of C v v O2 and the respiratory exchange ratio (RER); using { _ C C v v O2 = _ C C v v CO2~1 :29 from their data, and assuming resting RER = 0.8, we find that Qb CO2 =Qb O2~1 :03 or approximately 1. Thus Qv CO2 =Qv O2 is assumed to be 1.
Stage 1 hypercapnia. Here we develop a relationship to describe the time-course of alveolar/arterial hypercapnia during stage 1 for CO 2 . Using Equation 1 for CO 2 , taking _ V VI, _ V VE~0, gives the relationship _ P PACO 2~P0 _ V Vp CO2 =VL. Substituting the steady-state Fick equation, _ V Vp CO2~b b CO2 _ Q QT(Pa CO2 -P v v CO2 ), assuming alveolar-arterial equilibrium (Pa CO2~P ACO 2 ), using _ V Vp CO2~_ V V CO2 under resting conditions, assuming that P v v CO2 is constant, and solving for PACO 2 yields: Calculating the rate of rise in PACO 2 ( _ P PACO 2 ) by taking the derivative gives: Equations 19 and 20 describe the slowing of _ P PACO 2 from the initial rate _ P PACO 2~P0 _ V Vp CO2 =VL as PACO 2 rises towards P v v CO2 . Specifically, the time constant t~VL=(bb CO2 P 0 _ Q QT) demonstrates that high bb CO2 causes a rapid slowing of _ V Vp CO2 and hence of _ P PACO 2 as the arterial value approaches venous value. Indeed, fitting an exponential curve to the PACO 2 trace (Figure 2) during the first 5 s of apnea yielded a rapid time constant of 1.26 s, a value close to that predicted by VL=(bb CO2 P 0 _ Q QT). The corollary is that the low value of bb O2 prevents the slowing of _ V Vp O2 as desaturation progresses, giving rise to a rapid PAO 2 decline and thus rapid arterial desaturation. Likewise, further reducing bb O2 by lowering hemoglobin content potentiates such effect.
Impact of supplemental O 2 . The delay (right-shift) in arterial desaturation during apnea with increasing supplemental O 2 (DPI O2 ) can be predicted explicitly. Using Equation 1 for O 2 under the conditions of apnea, and assuming DPIO 2~D PAO 2 , the delay (Dt) in arterial desaturation is given by: Author Contributions