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.
When breathing stops, the flow of O2 into and the flow of CO2 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 O2 saturation, reaching rates of 3–8% per second, rapidly reducing O2 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 O2 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.
Citation: Sands SA, Edwards BA, Kelly VJ, Davidson MR, Wilkinson MH, et al. (2009) A Model Analysis of Arterial Oxygen Desaturation during Apnea in Preterm Infants. PLoS Comput Biol 5(12): e1000588. doi:10.1371/journal.pcbi.1000588
Editor: Kim Prisk, University of California, San Diego, United States of America
Received: July 28, 2009; Accepted: October 30, 2009; Published: December 4, 2009
Copyright: © 2009 Sands et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: We received no funding for this work.
Competing interests: The authors have declared that no competing interests exist.
Apnea and its accompanying arterial O2 desaturation are common clinical complications in preterm infants, occurring in more than 50% of very low birth weight infants . In preterm infants, apnea causes a reduction in heart rate  and cerebral perfusion , often requires mechanical ventilation, and is associated with neurodevelopmental impairment . Apnea-related hypoxemia is of major concern in light of evidence that repetitive hypoxia in newborn animals results in irreversibly-altered carotid body function , raising the possibility of impaired ventilatory control, and causes neurocognitive and behavioural deficits . Respiratory arrest and hypoxemia are also strongly implicated in sudden infant death syndrome (SIDS) , where the speed at which hypoxemia develops is considered to be particularly dangerous.
In preterm infants, the rate of arterial O2 desaturation () can be highly variable and rapid, with average rates as high as 4.3% s−1 during isolated apneas . An earlier framework to describe proposed that metabolic O2 consumption relative to alveolar volume determines the speed at which alveolar falls ; it was envisaged that is then a function of falling and the slope of the oxy-hemoglobin dissociation curve. However, such a model assumes that the rate of alveolar depletion of O2, denoted pulmonary O2 uptake (), is equal to tissue O2 consumption during apnea (see Methods – Theory). Previous studies in adults have shown that falls from metabolic consumption during apnea , and our previous modeling studies in lambs showed that the difference between and metabolic O2 consumption has a crucial role in determining during recurrent apneas . We found that apneic changes in cause desaturation to occur in 2 stages. During stage 1, lung O2 stores are depleted, and falls below metabolic consumption. During stage 2, is close to zero, and tissue O2 needs are provided by depletion of blood O2 stores.
To date, no complete theoretical analysis of the factors influencing desaturation during apnea has been published. The only available study  has a number of critical limitations. First, the model incorporated a constraint of a fixed difference between and mixed-venous saturation; thus dynamic changes in could not occur and their influence on could not be examined. Second, no assessment was made of the impact of cardiorespiratory factors on the two stages of O2 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 ,.
Accordingly, the aim of the current study was to quantify the importance of cardiorespiratory factors relevant to during apnea, with particular reference to the preterm infant. Using a model that permits variation of during apnea, we examine a number of factors known to influence , such as lung volume , metabolic O2 consumption  and pre-apneic arterial oxygenation  as well as factors that are particularly pertinent for the developing newborn, including anemia, hypovolemia, reduced O2 affinity, and chronically and acutely reduced cardiac output. We use the results to develop a conceptual framework for the interpretation of mechanisms underlying rapid during apnea.
Overview of the two-compartment model for gas exchange
To determine the independent influence of clinically relevant cardiorespiratory factors on during a single isolated apnea, we used a two-compartment lung-body mathematical model which incorporated realistic blood O2 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 . We also derive analytic solutions for to quantify the importance of cardiorespiratory factors on to obtain a detailed view of the arterial O2 desaturation process, as described in Methods – Theory.
Figure 1. Model schematic representing O2 uptake, transport and consumption.
O2 stores are represented by the alveolar, arterial, and venous compartments. Two dynamically-independent levels of O2 uptake are denoted: pulmonary O2 uptake () and metabolic consumption (). R-L shunt is also included. Ta is the arterial transit time. Symbols are described in Table 1.doi:10.1371/journal.pcbi.1000588.g001
Table 1. Mathematical symbols.doi:10.1371/journal.pcbi.1000588.t001
Table 2. Typical parameters for the preterm infant at term equivalent age.doi:10.1371/journal.pcbi.1000588.t002
Pulmonary gas exchange dynamics during apnea
To examine changes in O2/CO2 exchange during apnea, a single apnea was imposed on the model. During apnea, changes in alveolar O2 and CO2 stores are not constant (Figure 2); importantly, alveolar () did not continue to fall at its initial rate as governed by metabolic O2 consumption (), but instead the rate of fall in was reduced as it approached mixed venous (), an observation also reflected in the falling . As a result, two distinct phases for O2 depletion can be seen, which we refer to as stage 1 and stage 2 . During stage 1, fell rapidly and decreased and became dissociated from ; during stage 2, with greatly reduced, both and fell together at a reduced rate. The two distinct phases were also observed for alveolar and arterial (, ) although stage 1 for CO2 was substantially shorter than that for O2. Such an effect results from the earlier fall in pulmonary CO2 uptake () relative to the fall in (Figure 2A) and is reflected in the reduction in respiratory exchange ratio () (Figure 2B). Consequently, a more rapid fall in was observed compared with the rise in (see Methods – Derivation of equations), such that fell by 100 mmHg in the time rose by just 14 mmHg (Figure 2C).
Figure 2. Pulmonary gas exchange during apnea.
(A) Rate of pulmonary O2/CO2 exchange. and fall from resting levels during apnea. (B) Net alveolar-capillary gas uptake () and respiratory exchange ratio ()during apnea. (C) Changes in alveolar, arterial and mixed venous during apnea. Contrast the time-course in and as they fall/rise towards . (*) represents the fall in if was assumed equal to . S1 = stage 1; S2 = stage 2.doi:10.1371/journal.pcbi.1000588.g002
Time-course of during apnea
The time-course of is complex (Figure 3), a consequence of the nonlinear O2-dissociation curve in combination with the fall in . At apnea onset, started to fall with a rate equivalent to that predicted by Equation 12, where (Figure 3). During apnea, changes in the slope of the O2-dissociation curve () and dominated the time-course of desaturation as hypoxemia progressed. As started to fall after apnea onset, increased with little change in , resulting in a proportional increase in . However, as arterial hypoxemia developed, there was a concurrent decline in . As is directly proportional to the product (Equation 11) it follows that during apnea, the peak of 3.5% s−1 occurred when reached a maximum. This occurred when neither nor was at its maximum (both ~50% of peak). Finally, with greatly reduced during stage 2, remained at a constant level (), close to that predicted by Equation 13 (1.8% s−1).
Figure 3. The time course of during apnea.
Panel (A) shows the increase in the slope of the oxy-hemoglobin dissocation curve at the level of alveolar (), and the fall in pulmonary oxygen uptake () that occurs during apnea. Panel (B) shows that changes in the product explain the time course of the instantaneous slope of arterial O2 desaturation () during apnea. Note that the peak occurs when is substantially less than its resting value. Note also that the rate of fall of mixed-venous saturation () and become equal and constant after 20 s.doi:10.1371/journal.pcbi.1000588.g003
The following parameters were individually varied from their ‘normal’ values to quantify their influence on : resting , lung volume (), metabolic O2 consumption (), blood hemoglobin content (Hb), cardiac output (), R-L shunt fraction (Fs), and the at 50% saturation (P50). All other parameters were kept constant to remove confounding effects, unless specified otherwise.
To quantify we used 3 different measures. First, since apnea is considered clinically significant if it lasts for >10 s and is accompanied by bradycardia or O2 desaturation , we calculated the average rate of fall in between apnea onset and 10 s later (); such a measure describes the immediacy of onset of desaturation and is analogous to the practical measurement of average used in many clinical studies ,,,. Second, we determined the peak instantaneous during apnea (), the value during the linear portion of arterial desaturation , which we find is not confounded by resting . Third, we report a measure of during stage 2 apnea (). To quantify the sensitivity of to changes in each cardiorespiratory factor, we defined the term impact ratio as the ratio of proportional increase in 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 with an increase in the factor, and a negative ratio indicates an inverse relationship. The impact of each cardiorespiratory factor on , , and is summarised in Table 3.
Table 3. Impact ratios describing the effect of cardiorespiratory factors on .doi:10.1371/journal.pcbi.1000588.t003
Changes in , achieved via reduced resting ventilation or increasing inspired O2 (), had a substantial effect on the onset of desaturation. Reduced pre-apneic dramatically increased (Figure 4A), but had little effect on or . In contrast, increasing pre-apneic with the application of supplemental O2 achieved the opposite, essentially right-shifting or delaying the arterial desaturation curve, where one second of delay can be achieved by an increase in () of ~7 mmHg, or of ~1% (see Methods – Derivation of equations). These results occurred despite only a minor influence being visible on resting . For example, a reduction of from 100 to 60 mmHg caused a 6% reduction in resting but at the same time led to a more than 2-fold elevation in (Figure 4B). Additionally, a severe reduction in , to below 70 mmHg, was required to elevate .
Figure 4. Impact of pre-apneic alveolar (ventilation, supplemental O2) on .
(A) Effect of three levels of alveolar (), (i) 100 mmHg, (ii) 80 mmHg and (iii) 60 mmHg, on arterial () and mixed venous () O2 desaturation during apnea. Note that arterial O2 desaturation is substantially right-shifted with increased . (B) Sensitivity of to changes in pre-apneic (). Note that reduced has a major impact on but little impact on ; the influence on is small in the normal range but becomes stronger at low . n = ‘normal’ 'values; S1, stage 1 slope; S2, stage 2 slope.doi:10.1371/journal.pcbi.1000588.g004
Lung volume (VL) and blood volume (Qb).
and were inversely related to VL during stage 1 (Figure 5A, B), but changes in VL had no influence on . In direct contrast, reduced Qb strongly increased , but had no effect on stage 1 desaturation as reflected in no change in or (Figure 5C, D).
Figure 5. Impact of lung volume (VL) and blood volume (Qb) on .
(A) Effect of three levels of VL, (i) 30, (ii) 20 and (iii) 10 ml kg−1, on arterial () and mixed venous () O2 desaturation during apnea. (B) Sensitivity of to changes in VL. Note that reduced VL has a strong impact on and but no impact on . (C) Effect of three levels of Qb, (iv) 120, (v) 80 and (iv) 40 ml kg−1, on and during apnea. (D) Sensitivity of to changes in Qb. Note that reduced Qb has little impact on or but has a large impact on . n = ‘normal’ values; S1, stage 1 slope; S2, stage 2 slope.doi:10.1371/journal.pcbi.1000588.g005
Metabolic O2 consumption ().
To examine the impact of changing on , independent of resting , was adjusted to maintain resting constant, where ; we refer to this procedure as ‘cardiac compensation’. Under this condition, elevated caused a directly proportional increase in throughout stages 1 and 2 (Figure 6A, B). Without cardiac compensation, the effect of increased on during stage 1 was magnified, as shown by the further increase in (Figure 6A, B).
Figure 6. Impact of metabolic O2 consumption () on .
Panel (A) shows the effect of doubling on arterial () and mixed venous () O2 during apnea; (i) 10 ml min−1kg−1, (ii) 20 ml min−1kg−1 with cardiac compensation (CC), and (iii) 20 ml min−1kg−1 with no CC (nCC). Note that with CC, increased , from (i) to (ii), elevated uniformly at all levels of during both stages 1 and 2; note that the level of at the inflection point (shown by short black lines) is unchanged. With nCC (iii), increased caused a reduced resting and lower inflection, and greater during stage 1, compared to (ii). (B) Sensitivity of to changes in . Note that with increased : a uniform increase in occurred with CC, and a more-than-proportional increase was seen with nCC; is elevated in both cases, but more so with nCC; a uniform increase in is shown regardless of CC. n = ‘normal’ values; S1, stage 1 slope; S2, stage 2 slope.doi:10.1371/journal.pcbi.1000588.g006
Hemoglobin content (Hb) and oxygen affinity (P50).
Reduced hemoglobin content (Hb) increased and but had little effect on (Figure 7A, B). The increase in occurred with an increase in the peak of the product as was higher at each level of . The simulation was repeated with cardiac compensation for the reduction in hemoglobin content, where , to maintain constant resting . Following such compensation, no changes in or were observed but reduced Hb continued to increase . In examining the influence of P50, P90 was adjusted in equal proportion on the basis of published data . Increased P50 increased the immediate , increased , decreased and had no effect on (Figure 7C, D).
Figure 7. Impact of hemoglobin content (Hb) and O2 affinity (P50) on .
(A) Effect of three levels of Hb, (i) 12 g dl−1, (ii) 8 g dl−1 and (iii) 4 g dl−1, on arterial () and mixed venous () O2 desaturation during apnea. Note the fall in at the inflection point (shown by short black lines). Note also that the reduced Hb has little impact on desaturation above . (B) Sensitivity of to changes in Hb. (C) Effect of three levels of P50, (iv) 18 mmHg, (v) 24 mmHg, and (vi) 36 mmHg, on . (D) Sensitivity of to changes in P50. n = ‘normal’ values; S1, stage 1 slope; S2, stage 2 slope.doi:10.1371/journal.pcbi.1000588.g007
Cardiac output ().
Reduced resting increased , but had little impact on or (Figure 8A, B). As with Hb, the increase in with reduced resting occurred with an increase in the peak of the product . To differentiate between the influence on 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 , a transient reduction in decreased , but had a negligible impact on or (Figure 8C, D).
Figure 8. Impact of cardiac output () on .
(A) Effect of three levels of resting , (i) 375 ml min−1kg−1, (ii) 250 ml min−1kg−1, and (iii) 125 ml min−1kg−1, on arterial () and mixed venous () O2 during apnea. Note that reduced elevates , associated with a reduction in resting and reduction in at the stage 1–2 transition or inflection point (shown by short black lines). (B) Sensitivity of to changes in . Note the strong influence of on , but negligible effect on and . (C) Simulations in (A) repeated for a step change in at apnea onset by (iv) +125 ml min−1kg−1 (e.g. tachycardia), (v) 0 ml min−1kg−1, and (vi) −125 ml min−1kg−1 (e.g. bradycardia), following resting . Note that the transient effect of is opposite to the resting effect of on arterial desaturation during apnea. (D) Sensitivity of to acute changes in during apnea. Note the strong influence of a step-change in on , but negligible effect on and . n = ‘normal’ values.doi:10.1371/journal.pcbi.1000588.g008
Resting R-L shunt fraction (Fs).
Increased Fs reduced resting and but had no effect on , , or (Figure 9A, B).
Figure 9. Impact of R-L shunt (Fs) on .
(A) Effect of three levels of Fs, (i) 0%, (ii) 15%, and (iii) 30%, on arterial () and mixed venous () O2 during apnea. Note that resting R-L shunt fraction has a negligible impact on during apnea. (B) Sensitivity of to changes in Fs. n = ‘normal’ values; S1, stage 1 slope; S2, stage 2 slope.doi:10.1371/journal.pcbi.1000588.g009
Our model analysis of the rate of arterial O2 desaturation during apnea demonstrates that pre-apneic ventilation, lung volume, cardiac output, hemoglobin content and blood volume exert unique effects on throughout the time-course of desaturation, while metabolic O2 consumption is uniformly influential throughout the process. Our analysis reveals that lung volume and the slope of the O2-dissociation curve are important early in the process, during what we refer to as stage 1 , 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 . Finally, low blood volume and hemoglobin content, and therefore a low total blood O2 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.
To evaluate the independent effects of cardiorespiratory factors on 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 occur as a consequence of arterial and mixed-venous saturation falling asynchronously during apnea (Figure 3). Our approach allowed us to extend the previous framework based on the assumption of constant , which prevented the recognition that a steep O2-dissociation curve and low lung volume do not accelerate beyond stage 1. Furthermore, the varying permitted recognition that cardiac output, hemoglobin content, and blood volume have a major influence on .
In the current study, the typical value of found using our model was 3.5% s−1 whereas Poets and Southall  using beat-by-beat oximetry in preterm infants reported a mean value for 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 . 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 ,; based on our data, a fall in lung volume to 15.5 ml min−1kg−1 immediately after apnea onset would achieve of 4.3% s−1 (Figure 5B).
A final assumption implicit in our model is that all O2 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 O2 . With whole body exposure of 90% O2 to the newborn skin, it has been calculated that can be reduced by 8–10% , likely via an increased resting mixed-venous saturation; our study demonstrates that such an effect would decrease during apnea.
Pulmonary gas exchange dynamics during apnea
Our study is consistent with previous observations that and rapidly decline during apnea from their steady-state values , with falling faster than . The relatively low blood capacitance for O2 compared with that for CO2 results in the resting alveolar–mixed-venous partial pressure difference being ~12-fold greater for O2 than for CO2. Consequently, when apnea begins ~12 times more O2 than CO2 must diffuse across the lung to obliterate the alveolar–mixed-venous partial pressure difference. The slower fall in vs. provides for a faster depletion of alveolar O2 vs. CO2 stores; such an effect results in complete desaturation of arterial blood in the time rises by just 14 mmHg. These findings lead us to conclude that short-term O2 homeostasis is more unstable than CO2 homeostasis and thus that the danger of isolated apneas in infants is likely to be mediated via hypoxemia rather than hypercapnia.
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 has the greatest influence on desaturation at apnea onset. When apnea begins at an increasingly lower alveolar , more quickly reaches its maximum because rapidly arrives at the steepest part of the O2-dissociation curve. This effect explains the inverse relationship between mean and pre-apneic during apnea , but as we show the peak slope itself is negligibly affected by reduced resting within the normal range.
We demonstrate that is inversely related to lung volume during stage 1 of apnea as a result of the greater reduction in alveolar in poorly inflated lungs per unit of O2 transferred into the pulmonary capillaries. This analysis is consistent with the inverse correlation between and lung volume , with the view that active upper airway closure maintains lung volume and slows ,, and with our recent report that the application of continuous positive airway pressure effectively slows in lambs . However, once stage 2 begins, the blood becomes the principal source of O2 and thus the only store which influences .
A novel finding from our study is that reduced resting mixed-venous saturation, caused by either a reduced cardiac output or reduced hemoglobin content, strongly elevates peak , independent of metabolic O2 consumption. We show that reduced resting mixed-venous saturation accelerates via an increase in the peak value of ; in other words, low mixed-venous saturation provides for a greater pulmonary O2 uptake even in the presence of a developing arterial hypoxemia, and thereby increases . A role for hemoglobin in determining is consistent with the finding that elevated hemoglobin content in adults slows during apnea . 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 . Our proposed explanation for the lack of benefit of raising hemoglobin content via transfusion is that it also reduces heart rate  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  did not account for our finding that pre-apneic and transient changes in cardiac output have opposing influence on . Importantly, we find that a transient fall in cardiac output, characteristic of bradycardia during apnea in preterm infants , conserves alveolar O2 via reduced and thus reduces (see Equations 10 and 11). Consistent with this finding, apneic bradycardia prevents a rapid fall in in adults .
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 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 O2 consumption. In the late phase of stage 1, when desaturation proceeds in a linear fashion, a low resting mixed-venous saturation accelerates and leaves the fingerprint of a low inflection point in arterial O2 desaturation; low resting mixed-venous saturation reflects low cardiac output or hemoglobin content with respect to O2 consumption. Lastly rapid during stage 2 signifies a low total blood O2 capacity with respect to O2 consumption which would point to either low blood volume or anemia. The presence of a constant R-L shunt, while having no influence on , 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.
Figure 10. Conceptual framework depicting the temporal sequence of influence of the key cardiorespiratory factors on .
Note the regions of influence of lung volume (), cardiac output () and blood volume (), each with respect to metabolic O2 consumption (). Hemoglobin content (Hb) influences the latter phase of stage 1 as well as stage 2. The impact of reduced is limited to stage 1, and blood volume to stage 2. Reduced causes a leftward shift in the desaturation trajectory. Note that the point of inflection at the transition between stages reveals the resting .doi:10.1371/journal.pcbi.1000588.g010
We show theoretically that the lower lung volume  and higher metabolic O2 consumption  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 , a finding that may help to explain the more frequent O2 desaturation events during active sleep  when functional residual capacity is reduced. Second, frequent desaturation is characteristic of preterm infants with bronchopulmonary dysplasia (BPD)  whose O2 consumption is 25% greater , and functional residual capacity is 25% less , than in preterm infants without BPD; Equations 11 and 12 predict that such differences increase both immediate and peak by ~70%. In addition, hypoventilation and reduced resting in infants with BPD, as inferred from elevated , further increase desaturation at apnea onset. Our finding that each rise of 1% in inspired O2 provides ~1 s of delay (right-shift) in the onset of apneic desaturation (Equation 15) may guide the titration of supplemental O2 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 O2 consumption can be elevated after feeding , with reduced ambient temperature , and via the adminstration of methylxanthines . Despite the success of methylxanthines in reducing the frequency of apnea and bradycardia, such treatment has surprisingly little impact on hypoxemic episodes ; we suggest that the elevated O2 consumption and the absence of bradycardia are likely to increase during those apneas that persist despite treatment. The severity of hypoxemic episodes is reduced by switching preterm infants from supine to prone , which may increase functional residual capacity  and improve diaphragm function, increase tidal volume and increase resting alveolar . Our finding that low cardiac output leads to increased during apnea leads to the suggestion that judicious adjustment of inotropic support in infants with cardiac abnormalities could improve resting mixed-venous saturation and reduce apneic hypoxemia.
Hypoxemic events become less frequent between infancy and childhood, despite an unchanged apnea frequency , perhaps as a result of a fall in O2 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 O2 affinity  and a rise in O2 consumption . The implications for SIDS are obvious in that these changes coincide with the peak incidence for SIDS at 2–3 months . SIDS also occurs disproportionately in preterm infants , who manifest severe anemia  and greater O2 consumption. Infants resuscitated from apparent life threatening events have been found to have lower hemoglobin content , pointing to a potential role for rapid 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.
We have provided a mathematical framework for quantifying the relative importance of key cardiorespiratory factors on the rate of arterial O2 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.
For the lung, a single homogeneous compartment is assumed based on the model of Grodins et al . 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:(1)
where represents the rate of change of alveolar , , and ; represents the inspired alveolar partial pressure of each gas G; P0 is atmospheric pressure converted from STP to BTP (863 mmHg); represents and , pulmonary gas uptake (STPD) for O2 and CO2 ( was neglected in this study for simplicity); and are inspired and expired alveolar ventilation (BTPS). Accounting for the difference in and due to a net pulmonary gas uptake into the pulmonary blood, yields:(2)
where = barometric pressure (760 mmHg); = water vapour pressure (47 mmHg); is the net pulmonary gas uptake, .
Since purely obstructive apneas are relatively rare in preterm infants , 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 , when apneic desaturation events are most common . 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 . To account for this effect, we can write:(3)
Pilot simulations predicted that the volume of gas inflow during apnea is unlikely to exceed physiological deadspace. Thus, during apnea is taken as of the last exhaled breath prior to apnea onset.
For the current study we assumed diffusion equilibrium at the pulmonary capillaries, such that . Gas uptake is determined from the Fick equation; specifically, pulmonary blood flow (), and the difference between end capillary () and mixed venous () content:(4)
Utilising equations for R-L shunt, arterial content of each gas G is determined from its end capillary () and mixed venous () content, and pulmonary shunt fraction ():(5)
Fs defines the ratio of pulmonary blood flow to cardiac output, such that Fs = .
Assuming that the of the venous blood is equilibrated with the tissue , the mass-balance equations are:(6)
where represents the gas content of O2 and CO2 in the arterioles; Ta is arterial transit time; represents and , the metabolic consumption of O2 and production of CO2; represents and the combined venous/tissue volumes for O2 and CO2.
Blood O2 stores were partitioned by assigning blood volume (Qb) to arterial (25%) and venous (75%) compartments  and they were modelled assuming an entirely unmixed arterial compartment, and an entirely mixed and homogenous venous compartment. The arterial transit time (Ta) is constrained by the arterial volume (Qa) by the relationship . The body compartment volume is taken as the venous volume. , the effective venous/tissue volume for CO2 is taken as the same value for QvO2, based on published data (see Methods – Derivation of equations). Physiologically this represents no additional contribution of a specific tissue reservoir for CO2 within the time frame of apnea.
To characterise the O2-dissociation curve we used a modified form of the equation of Severinghaus . We re-expressed the equation with respect to the partial pressure at 50% (P50) and at 90% (P90) saturation:(7)
where and . Values for P50 (24.0 mmHg) and P90 (53.6 mmHg), were obtained from the data of Delivoria-Papadopoulos  for a 9–10 wk-old preterm infant. O2 content (, ml ml−1) includes that bound to hemoglobin (Hb, g ml−1) and that dissolved in plasma:(8)
The relationship between CO2 content () and was assumed linear:(9)
where and as adapted for STPD from Grodins et al. .
Simulations were performed using software written in MATLAB (The Mathworks; Natick, MA).
A general equation.
In an earlier study we developed a general relationship that describes the factors influencing the magnitude of at any instant in time during apnea :(10)
where is the capacitance co-efficient of blood for O2. To specifically demonstrate the role of gas exchange, it is more useful to represent in terms of . Using Equation 1 for O2 under conditions of apnea (,), assuming , and using , reveals:(11)
where (% mmHg−1) is defined as the slope of the O2-dissociation curve, specifically regarding end-capillary with respect to . It is clear from Equation 11 that is directly proportional to the product , which both vary substantially during apneic arterial desaturation. Although Equations 10 and 11 are useful conceptually, values for () or throughout apnea are unknown, and thus is not simple to predict explicitly.
The original framework to understand factors influencing was based on the assumption that , which does not hold true during apnea ,. However, such an assumption is valid prior to any substantial fall in , and as therefore useful to explicitly describe immediately upon apnea onset ():(12)
Notably, Equation 12 demonstrates that for any level of and , is intimately related to . Consequently, increases dramatically with reduced resting (Figure 11).
Figure 11. Relationship between the slope of the oxy-hemoglobin dissociation curve and alveolar .
Note that reduced alveolar () causes a substantial increase in the slope of the oxy-hemoglobin dissociation curve (; see inset) and in at apnea onset (; based on Equation 12).doi:10.1371/journal.pcbi.1000588.g011
Although no simple expression could be written to describe explicitly for stage 1, we derived an expression for during stage 2 (see Methods – Derivation of equations), given by:(13)
Since the total blood O2 capacity () is much greater than , is determined principally by () with negligible influence coming from lung volume () and the slope of the O2-dissociation curve (), as well as . Using the values for the preterm infant in Table 2 and maximum , Equation 13 predicts that . The little remaining during stage 2 can be found by combining Equations 11 and 13:(14)
Equation 14 predicts that of its resting value during stage 2. Notably, is increased by reducing Hb and Qb; the greater and thus a greater rate of alveolar O2 depletion with reduced blood O2 capacity () will increase .
How can we reconcile that Equation 13 shows that no longer influences during stage 2, when the general equation (Equation 11) implies that reduced will accelerate throughout apneic desaturation? Equation 14 reveals that during stage 2, elevated also acts to increase ; thus nearly entirely offsetting the direct influence on . The same applies for reduced , which acts to elevate and therefore no longer accelerates during stage 2.
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.
Stage 2 arterial O2 desaturation.
This section details the derivation of an explicit equation to predict the rate of both arterial and venous desaturation during the severe desaturation of stage 2, a phase where is substantially reduced below and both and fall at the same rate. Ignoring dissolved plasma O2, consideration of Equation 1 for O2 and assuming yields:(15)
By substituting the following relationships into Equation 15: ; ; Qb = Qa+Qv; ; it can be shown that is directly proportional to the difference between and , where:(16)
Combining Equations 11 and 16 yields Equation 13.
Estimation of effective blood volume for CO2.
Using the same methodology as described above, the ratio of to during stage 2 can be used to estimate the ratio of to . and can be found using:(17)
where and are the effective blood volumes for O2/CO2; is the capacitance coefficient for CO2. Neglecting pulmonary gas exchange, combining Equation 17 for O2 and CO2 gives:(18)
Equation 18 permitted the calculation of based on published data [53; their Figure 3] where during apnea the rate of rise in is very close to the rate of fall in the product of and the respiratory exchange ratio (RER); using from their data, and assuming resting RER = 0.8, we find that or approximately 1. Thus 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 CO2. Using Equation 1 for CO2, taking ,, gives the relationship . Substituting the steady-state Fick equation, , assuming alveolar-arterial equilibrium (), using under resting conditions, assuming that is constant, and solving for yields:(19)
Calculating the rate of rise in () by taking the derivative gives:(20)
Equations 19 and 20 describe the slowing of from the initial rate as rises towards . Specifically, the time constant demonstrates that high causes a rapid slowing of and hence of as the arterial value approaches venous value. Indeed, fitting an exponential curve to the 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 . The corollary is that the low value of prevents the slowing of as desaturation progresses, giving rise to a rapid decline and thus rapid arterial desaturation. Likewise, further reducing by lowering hemoglobin content potentiates such effect.
Impact of supplemental O2.
The delay (right-shift) in arterial desaturation during apnea with increasing supplemental O2 () can be predicted explicitly. Using Equation 1 for O2 under the conditions of apnea, and assuming , the delay () in arterial desaturation is given by:(21)
Conceived and designed the experiments: SAS PJB. Performed the experiments: SAS. Analyzed the data: SAS BAE MRD MHW PJB. Wrote the paper: SAS BAE VJK MRD MHW PJB. Designed and implemented the model: SAS VJK MRD MHW.
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