A mathematical model to serve as a clinical tool for assessing obstructive sleep apnea severity

Abstract

Obstructive sleep apnea (OSA) is a sleep disorder caused by periodic airway obstructions and has been associated with numerous health consequences, which are thought to result from tissue hypoxia. However, challenges in the direct measurement of tissue-level oxygenation make it difficult to analyze the hypoxia exposure pattern in patients. Furthermore, current clinical practice relies on the apnea-hypopnea index (AHI) and pulse oximetry to assess OSA severity, both of which have limitations. To overcome this, we developed a clinically deployable mathematical model, which outputs tissue-level oxygenation. The model incorporates spatial pulmonary oxygen uptake, considers dissolved oxygen, and can use time-dependent patient inputs. It was applied to explore a series of breathing patterns that are clinically differentiated. Supporting previous studies, the result of this analysis indicated that the AHI is an unreliable indicator of hypoxia burden. As a proof of principle, polysomnography data from two patients was analyzed with this model. The model showed greater sensitivity to breathing in comparison with pulse oximetry and provided systemic venous oxygenation, which is absent from clinical measurements. In addition, the dissolved oxygen output was used to calculate hypoxia burden scores for each patient and compared to the clinical assessment, highlighting the importance of event length and cumulative impact of obstructions. Furthermore, an intra-patient statistical analysis was used to underscore the significance of closely occurring obstructive events and to highlight the utility of the model for quantitative data processing. Looking ahead, our model can be used with polysomnography data to predict hypoxic burden on the tissues and help guide patient treatment decisions.

Keywords: breathing; desaturation; hypopnea; hypoxemia; hypoxia; mass transfer; oxygenation.

FIGURE 1

Results for normal subject. (A) Input simulated normal breathing pattern with V _End = 2.3 L, V _T = 0.5 L, V _D = 0.15 L, and b _r = 12 breaths/min. (B) Alveolar oxygen partial pressure. (C) Systemic arterial and venous hemoglobin oxygen saturation. (D) Dissolved oxygen concentration in systemic arteries and veins.

FIGURE 2

Analysis of simulated case of severe OSA. (A) Input breathing pattern with four 40-s apneas over a 3.3-min period, resulting in an AHI of 72. Hyperventilation is characterized by V _T = 1.0 L and b _r = 24 breaths/min. (B) Systemic arterial and venous hemoglobin oxygen saturation. (C) Dissolved oxygen concentration in systemic arteries and veins. The grayed portions indicate periods of ceased breathing (B,C). (D) Alveolar and pulmonary capillary oxygen partial pressures for normal condition (t = 360 s) and at the end of the breathing pattern (t = 560 s).

FIGURE 3

Analysis of reoxygenation following different ventilatory responses. (A) Simulation 1: Mild OSA breathing pattern (AHI = 12) with normal breathing following 20-s apnea. (B) Simulation 2: Mild OSA breathing pattern (AHI = 12) with 20-s period of hyperventilation (b _r = 24 breaths/min) following 20-s apnea, before resumption of normal breathing. (C) Systemic arterial and venous hemoglobin oxygen saturation for both simulations. (D) Dissolved oxygen concentration in systemic arteries and veins for both simulations.

FIGURE 4

Analysis of AHI scoring criteria for apneas using moderate OSA breathing patterns. (A) Simulation 3: Input breathing pattern with four 20-s apneas over a 10-min period (AHI = 24) and no unscored obstructive events. (B) Simulation 4: Input breathing pattern with four 20-s apneas over a 10-min period (AHI = 24) and six unscored obstructive events, each with a 5-s duration. (C) Systemic arterial and venous hemoglobin oxygen saturation for both simulations. (D) Dissolved oxygen concentration in systemic arteries and veins for both simulations.

FIGURE 5

Analysis of AHI scoring criteria for hypopneas. (A) Simulation 5: Input moderate OSA breathing pattern with four 10-s apneas over a 10-min period (AHI = 24). (B) Simulation 6: Input breathing pattern with four 25-s unscored hypopneas over a 10-min period (AHI = 0), each with a 50% reduction in airflow. (C) Systemic arterial and venous hemoglobin oxygen saturation for both simulations. (D) Dissolved oxygen concentration in systemic arteries and veins for both simulations.

FIGURE 6

Analysis of apnea frequency and duration. (A) Simulation 7: Input moderate OSA breathing pattern with four 30-s apneas over a 10-min period (AHI = 24). (B) Simulation 8: Input severe OSA breathing pattern with eight 15-s apneas over a 10-min period (AHI = 48). (C) Simulation 9: Input breathing pattern with four 15-s apneas over a 10-min period (AHI = 24). (D) Systemic arterial and venous hemoglobin oxygen saturation for Simulations 7 and 8. (E) Dissolved oxygen concentration in systemic arteries and veins for Simulations 7 and 8. (F) Systemic arterial and venous hemoglobin oxygen saturation for Simulations 7 and 9. (G) Dissolved oxygen concentration in systemic arteries and veins for Simulations 7 and 9.

FIGURE 7

Analysis of single apnea/hypopnea duration and AHI. (A) Comparison of some simulations for percent decrease in saturation. (B) Comparison of some simulations for percent decrease in concentration. (C) Comparison of some simulations for percent decrease in oxygen mass transfer. (D) Percent of normal saturation for simulations with variable single apnea duration and constant AHI. (E) Percent of normal concentration for simulations with variable single apnea duration and constant AHI.

FIGURE 8

OSA Patient 1 analysis over portion of sleep study. (A) Recorded heart rate and cardiac output. (B) Lung volume obtained from conversion of recorded nasal pressure. (C) Model output hemoglobin oxygen saturation in systemic arteries and veins, along with recorded pulse oximeter data $(S_{p,O_2})$ . The dashed line and arrow are used to indicate a point where the model predicts a lower value than $S_{p,O_2}$ following the RERA event. (D) Model output dissolved oxygen concentration in systemic arteries and veins. The grayed portions indicate respiratory events, as labeled in (B). Breaks in the y-axis are shown for (C,D).

FIGURE 9

OSA Patient 2 analysis over portion of sleep study. (A) Recorded heart rate and cardiac output. (B) Lung volume obtained from conversion of recorded nasal pressure. (C) Model output hemoglobin oxygen saturation in systemic arteries and veins, along with recorded pulse oximeter data $(S_{p,O_2})$ . (D) Model output dissolved oxygen concentration in systemic arteries and veins. The dashed lines in (D) are were used to represent the range of dissolved arterial oxygen during wakefulness. The grayed portions indicate respiratory events, as labeled in (B).