An integrative model of respiratory and cardiovascular control in sleep-disordered breathing

Limei Cheng; Olga Ivanova; Hsing-Hua Fan; Michael C K Khoo

doi:10.1016/j.resp.2010.06.001

. Author manuscript; available in PMC: 2011 Nov 30.

Published in final edited form as: Respir Physiol Neurobiol. 2010 Jun 11;174(1-2):4–28. doi: 10.1016/j.resp.2010.06.001

An integrative model of respiratory and cardiovascular control in sleep-disordered breathing

Limei Cheng ¹, Olga Ivanova ¹, Hsing-Hua Fan ¹, Michael C K Khoo ¹

PMCID: PMC2965826 NIHMSID: NIHMS213885 PMID: 20542148

Abstract

While many physiological control models exist in the literature, none thus far has focused on characterizing the interactions among the respiratory, cardiovascular and sleep-wake regulation systems that occur in sleep-disordered breathing. The model introduced in this study integrates the autonomic control of the cardiovascular system, chemoreflex and state-related control of respiration, including respiratory and upper airway mechanics, along with a model of circadian and sleep-wake regulation. The integrative model provides realistic predictions of the physiological responses under a variety of conditions including: the sleep-wake cycle, hypoxia-induced periodic breathing, Cheyne-Stokes respiration in chronic heart failure, and obstructive sleep apnoea (OSA). It can be used to investigate the effects of a variety of interventions, such as isocapnic and hypercapnic and/or hypoxic gas administration, the Valsalva and Mueller maneuvers, and the application of continuous positive airway pressure on OSA subjects. By being able to delineate the influences of the various interacting physiological mechanisms, the model is useful in providing a more lucid understanding of the complex dynamics that characterize state-cardiorespiratory control in the different forms of sleep-disordered breathing.

Keywords: physiological model simulation, autonomic pathways and reflexes, sleep regulation, obstructive sleep apnoea, Cheyne-Stokes respiration, cardiorespiratory interactions

1. Introduction

Many simulation models of the cardiovascular and respiratory systems have been published in the past several decades (Coleman and Randall, 1983; Dickinson, 1977; Grodins, 1959; Grodins et al. 1954; Khoo and Yamashiro, 1989; Madwed et al, 1989; Milhorn et al., 1965; Ursino and Magosso, 2003), starting with the highly simplified models proposed by Grodins et al. (1954). For instance, Grodins’ early electrical analog of the cardiovascular system (1959) was largely an “uncontrolled” model, consisted of only 2 linear resistors and 4 compliances, along with flow sources that implemented the Frank-Starling mechanism. Subsequent models became more realistic through the incorporation of progressively more functional components (Coleman and Randall, 1983; Dickinson, 1977; Ursino and Magosso, 2003). As well, many models were designed to solve specific issues in cardiorespiratory physiology. As such, these models focused primarily on the functional characterization of the pertinent components of these systems while capturing as realistically as necessary the detailed feature in these components. Some examples include the recent sophisticated models that provide very detailed mathematical representations of the structure and function of the heart (Miculloch et al., 1998) and lungs (Tawhai et al., 2004). While most models have addressed the cardiovascular and respiratory systems separately, only a subset have examined the interactions between these systems (Bai et al., 1997; Liang and Liu, 2006; Lu et al., 2001). While existing models are capable of characterizing the responses of the cardiovascular and respiratory systems to a wide range of interventions or conditions, very few have been developed for the simulation of the physiological response to sleep onset or the effect of circadian rhythms (Khoo et al., 1991; Stephenson, 2004). Certainly, none have addressed the integrated cardiovascular and respiratory responses that accompany normal or pathological sleep.

It has become widely recognized that the cardiovascular consequences of sleep-related breathing disorders, such as the obstructive sleep apnoea syndrome (OSAS) and Cheyne-Stokes respiration (CSR) can be quite profound (Bradley and Floras, 2003a, 2003b; Leung and Bradley, 2001). During both central and obstructive apnoea, absence of inspiratory airflow results in asphyxia, which subsequently stimulates breathing through the chemoreflexes and triggers a transient arousal from sleep. In OSAS, the arousal also leads to restoration of upper airway patency. These large fluctuations in chemoreflex and sleep state-related stimuli also produce concomitant changes in outflow in autonomic branches. There is accumulating evidence that chronic exposure to these repetitive swings in autonomic outflow and intermittent hypoxia can lead ultimately to the development of systemic hypertension, heart failure, myocardial infarction and stroke ( Leung and Bradley, 2001; Stephenson, 2004).

Present knowledge about the mechanisms that link sleep-disordered breathing to cardiovascular disease remains incomplete. The multifactorial dynamics and closed-loop aspects of the physiological events that underlie respiratory and state instability during sleep apnoea make it difficult to distinguish cause from effect under such conditions. By employing a mathematical modeling approach, it is possible to extricate the influences of the various interacting physiological mechanisms and thus attain a more lucid understanding of how they contribute to the complex dynamics that characterize cardiorespiratory control in the different forms of sleep-disordered breathing, such as OSAS and CSR. Subsequently, comparing the model predictions to the features extracted from empirical data provides insight into what the limitations of the current understanding are and what future experiments could be performed to address those limitations. This information can potentially be used to facilitate the identification of particular physiological defects, and the prediction or evaluation of specific outcomes for a given therapy.

In this paper, we introduce an integrative model which incorporates the essential features from existing submodels of cardiovascular, respiratory and sleep regulation and allows us to examine the interactions among these subsystems. The primary purpose for developing this integrative model is to provide the capability of performing virtual experiments and computer simulations of cardiorespiratory interactions under conditions of changing sleep-wake state as well as a variety of interventions, such as hypoxic gas administration, Valsalva and Mueller maneuvers, application of continuous positive airway pressure (CPAP), and Cheyne-Stokes respiration during sleep. The inclusion of a sleep regulation model allows simulations to be executed over time scales that span multiple sleep-wake cycles in months or even years.

The integrative model takes the form of a software package named “PNEUMA”. PNEUMA is implemented using SIMULINK^®/MATLAB^® (The Mathworks, Natick, MA), which provides a graphical environment that allows users to design the model components from built-in blocks or user defined functions, thus allowing the software to be executed across multiple computer platforms. The model code and supporting materials, including a user manual, are freely available to the research community. More information about the structure and implementation of the model can be downloaded from the USC Biomedical Simulations Resource website: http://bmsr.usc.edu/Software/PNEUMA/PNEUMA.html.

2. Methods

The model is developed by employing a hierarchical structure so that the degree of complexity at each level of organization is tailored appropriately to the investigation of the physiological processes at that level. This allows the overall model to be represented in a compact and efficient manner. At the highest level of organization, PNEUMA consists of four major interconnected “blocks”: (a) the first representing the respiratory subsystem; (b) the second representing the cardiovascular subsystem; (c) the third representing central neural control; and (d) the last representing the sleep mechanism. The respiratory subsystem includes upper airway dynamics, respiratory mechanics, gas exchange, and circulatory transport/mixing of blood gases. The cardiovascular subsystem contains the pulsating heart and hemodynamics of pulmonary and systemic circulations with its peripheral circulations, the automaticity of the sinoatrial (SA) node, the carotid baroreflex regulation, and the local blood flow control in the peripheral circulations. The central neural control system consists of the ventilatory controller, a variable respiratory rhythm generator, autonomic control, several key reflexes: chemoreflex, baroreflex and lung stretch reflex, and cardiorespiratory interactions that include respiratory sinus arrhythmia, pleural pressure modulation and central nervous system (CNS) response to arterial P_O2 and arterial P_CO2 in the cerebral blood flow that are necessary to model sympathetic and parasympathetic branches of autonomic control. The sleep mechanism counts for both circadian and ultradian rhythms. A schematic diagram of the model is displayed in Fig. 1.

Fig. 1 — Schematic diagram of the integrative model of cardiovascular and respiratory interactions with sleep-wake state control.

In this section, the major features that characterize these subsystems are highlighted, as well as key assumptions employed in the modeling process. All model equations are given in the Appendix. A glossary of the model variables and parameters, as well as their nominal values, is provided in Table 1. To facilitate the flow of logic in the presentation of the model below, we have found it useful to include the characterizations of the cardiovascular- and respiratory-related components of the central neural control block in the descriptions of the cardiovascular and respiratory subsystems, respectively.

Table 1.

PNEUMA simulation parameters and initial conditions

Parameter	Definition	Values	Units	Source
Cardiovascular System
Resistances
R_PA	Pulmonary arterial flow resistance	0.023	mmHg*s/mL	(Ursino, 1998)
R_PP	Pulmonary peripheral flow resistance	0.0894	mmHg*s/mL	(Ursino, 1998)
R_PV	Pulmonary venous flow resistance	0.0056	mmHg*s/mL	(Ursino, 1998)
R_SA	Systemic arterial flow resistance	0.06	mmHg*s/mL	(Ursino and Magosso, 2000)
R_SP	Splanchnic peripheral flow resistance	3.307	mmHg*s/mL	(Ursino and Magosso, 2000)
R_EP	Extra-splanchnic peripheral resistance	3.52	mmHg*s/mL	(Ursino and Magosso, 2000)
R_MPN	Skeletal muscle peripheral flow resistance	4.48	mmHg*s/mL	(Ursino and Magosso, 2000)
R_BPN	Cerebral peripheral flow resistance	6.57	mmHg*s/mL	(Ursino and Magosso, 2000)
R_HPN	Coronary peripheral flow resistance	19.71	mmHg*s/mL	(Ursino and Magosso, 2000)
R_SV	Splanchnic venous flow resistance	0.038	mmHg*s/mL	(Ursino and Magosso, 2000)
R_EV	Extra-splanchnic venous resistance	0.04	mmHg*s/mL	(Ursino and Magosso, 2000)
R_MV	Skeletal muscle venous flow resistance	0.05	mmHg*s/mL	(Ursino and Magosso, 2000)
R_BV	Cerebral venous flow resistance	0.075	mmHg*s/mL	(Ursino and Magosso, 2000)
R_HV	Coronary venous flow resistance	0.224	mmHg*s/mL	(Ursino and Magosso, 2000)
R_{VC_0}	Nominal vena cava flow resistance	0.025	mmHg*s/mL	Model
R_LA	Left atrial flow resistance	0.0025	mmHg*s/mL	(Ursino, 1998)
R_RA	Right atrial flow resistance	0.0025	mmHg*s/mL	(Ursino, 1998)
Compliances
C_PA	Pulmonary arterial compliances	0.76	mL/mmHg	(Ursino, 1998)
C_PP	Pulmonary peripheral compliances	5.8	mL/mmHg	(Ursino, 1998)
C_PV	Pulmonary venous compliances	25.37	mL/mmHg	(Ursino, 1998)
C_SA	Systemic arterial compliances	0.28	mL/mmHg	(Ursino and Magosso, 2000)
C_SP	Splanchnic peripheral compliances	2.05	mL/mmHg	(Ursino and Magosso, 2000)
C_EP	Extra-splanchnic peripheral compliances	0.668	mL/mmHg	(Ursino and Magosso, 2000)
C_MP	Skeletal muscle peripheral compliances	0.525	mL/mmHg	(Ursino and Magosso, 2000)
C_BP	Cerebral peripheral compliances	0.358	mL/mmHg	(Ursino and Magosso, 2000)
C_HP	Coronary peripheral compliances	0.119	mL/mmHg	(Ursino and Magosso, 2000)
C_SV	Systemic venous compliances	61.11	mL/mmHg	(Ursino and Magosso, 2000)
C_EV	Extra-splanchnic venous compliances	20	mL/mmHg	(Ursino and Magosso, 2000)
C_MV	Skeletal muscle venous compliances	15.71	mL/mmHg	(Ursino and Magosso, 2000)
C_BV	Cerebral venous compliances	10.71	mL/mmHg	(Ursino and Magosso, 2000)
C_HV	Coronary venous compliances	3.57	mL/mmHg	(Ursino and Magosso, 2000)
C_LA	Left atrial compliances	19.23	mL/mmHg	(Ursino, 1998)
C_RA	Right atrial compliances	31.25	mL/mmHg	(Ursino, 1998)
Inertances
L_PA	Pulmonary arterial inertance	0.00018	mmHg*s²/mL	(Ursino, 1998)
L_SA	Systemic arterial inertance	0.00022	mmHg*s²/mL	(Ursino, 1998)
Unstressed Volume
V_UPA	Pulmonary arterial unstressed volume	0	mL	(Ursino, 1998)
V_UPP	Pulmonary peripheral unstressed volume	123	mL	(Ursino, 1998)
V_UPV	Pulmonary venous unstressed volume	120	mL	(Ursino, 1998)
V_USA	Systemic arterial unstressed volume	0	mL	(Ursino and Magosso, 2000)
V_USP	Splanchnic peripheral unstressed volume	274.4	mL	(Ursino and Magosso, 2000)
V_UEP	Extra-splanchnic peripheral unstressed volume	134.64	mL	(Ursino and Magosso, 2000)
V_UMP	Skeletal muscle peripheral unstressed volume	105.8	mL	(Ursino and Magosso, 2000)
V_UBP	Cerebral peripheral unstressed volume	72.13	mL	(Ursino and Magosso, 2000)
V_UHP	Coronary peripheral unstressed volume	24	mL	(Ursino and Magosso, 2000)
V_USV	Splanchnic venous unstressed volume	1121	mL	(Ursino and Magosso, 2000)
V_UEV	Extra-splanchnic venous unstressed volume	550	mL	Ursino and Magosso, 2000)
V_UMV	Skeletal muscle venous unstressed volume	432.14	mL	(Ursino and Magosso, 2000)
V_UBV	Cerebral venous unstressed volume	294.64	mL	(Ursino and Magosso, 2000)
V_UHV	Coronary venous unstressed volume	98.21	mL	(Ursino and Magosso, 2000)
V_{VC_0}	Vena cava unstressed volume	130	mL	Model
V_ULA	Left atrial unstressed volume	25	mL	(Ursino, 1998)
V_URA	Right atrial unstressed volume	25	mL	(Ursino, 1998)
V_ULV	Left ventricular unstressed volume	16.77	mL	(Ursino, 1998)
V_URV	Right ventricular unstressed volume	40.88	mL	(Ursino, 1998)
Vena Cava
Kr_vc	Gain for vena cava flow resistance	0.001	mmHg*s/mL	(Lu et al., 2001)
Vvc_max	Maximum volume of vena cava	350	mL	(Lu et al., 2001)
Vvc_min	Minimum volume of vena cava	50	mL	(Lu et al., 2001)
D₁	Parameter for P–V curve of vena cava	0.3855	mmHg	Model
D₂	Parameter for P–V curve of vena cava	−5	mmHg	(Lu et al., 2001)
K_{1_}vc	Parameter for P–V curve of vena cava	0.15	mmHg	(Lu et al., 2001)
K_{2_}vc	Parameter for P–V curve of vena cava	0.4	mmHg	(Lu et al., 2001)
Respiratory System
Pleural Pressure and Alveolar Pressure
Rcw	Chest wall resistance	1.03	cmH₂O*s/L	(Schuessler et al., 1997)
R_LT	Lung transmural resistance	1.69	cmH₂O *s/L	(Schuessler et al., 1997)
Raw	Airway wall resistance	1.016	cmH₂O *s/L	Model
Ecw	Chest wall elastance	5	cmH₂O /L	(Schuessler et al., 1997)
E_LT	Lung transmural elastance	5	cmH₂O /L	(Schuessler et al., 1997)
k_1,aw	Constant for upper airway pressure	1.85	cmH₂O *s²/ L²	(Schuessler et al., 1997)
k_2,aw	Constant for upper airway pressure	0.43	cmH₂O *s²/ L²	(Schuessler et al., 1997)
Gas Exchange and Transport
Dead Space
Dead_(i),co2IC	Initial condition for i^th CO₂ dead space	39.562	L	(Khoo, 1990)
Dead_(i),co2IC	Initial condition for i^th CO₂ dead space	39.674	L	(Khoo, 1990)
Dead_(i),co2IC	Initial condition for i^th CO₂ dead space	39.813	L	(Khoo, 1990)
Dead_(i),co2IC	Initial condition for i^th CO₂ dead space	40.006	L	(Khoo, 1990)
Dead_(i),o2IC	Initial condition for i^th O₂ dead space	104.36	L	(Khoo, 1990)
Dead_(i),o2IC	Initial condition for i^th O₂ dead space	104.23	L	(Khoo, 1990)
Dead_(i),o2IC	Initial condition for i^th O₂ dead space	104.05	L	(Khoo, 1990)
Dead_(i),o2IC	Initial condition for i^th O₂ dead space	103.8	L	(Khoo, 1990)
V_d(i)	i^th dead space volume (i={1,..4}	0.03	L	(Khoo, 1990)
P_I,co2	Inspiratory CO₂ partial pressure	0	Torr	(Khoo, 1990)
P_I,co2	Inspiratory O₂ partial pressure	150	Torr	(Khoo, 1990)
V_t'	Respiratory flow	variable	L/sec	Model
V_t	Tidal Volume	variable	L	Model
P_dO2	Dead space O₂ partial pressure	variable	Torr	Model
P_dCO2	Dead space CO₂ partial pressure	variable	Torr	Model
Alveolar Gas Exchange
V_co2, V_Lco2	Lungs storage volume for CO₂	3	L	(Khoo, 1990)
V_o2, V_Lo2	Lungs storage volume for O₂	2.5	L	(Khoo, 1990)
P_Aco2IC	Initial condition for Partial CO₂ pressure	40.943	Torr	(Khoo, 1990)
P_Ao2IC	Initial condition for Partial O₂ pressure	102.52	Torr	(Khoo, 1990)
P_Ao2IC	Initial condition for Partial O₂ pressure	102.52	Torr	(Khoo, 1990)
P_ACO2	Alveolar CO₂ partial pressure	variable	Torr	Model
P_ACO2	Alveolar O₂ partial pressure	variable	Torr	Model
P_alv	Alveolar partial gas pressure	variable	Torr	Model
Q	Blood flow	variable	L/sec	Model
Cardiovascular Transport
tau_chemo	Peripheral chemoreceptors delay time constant	2	s	(Khoo, 1990)
T₁	Time constant for cardiovascular mixing	1	s	(Khoo, 1990)
T₂	Time constant for cardiovascular mixing	2	s	(Khoo, 1990)
T_a	Lung to chemoreceptor circulation delay	variable	s	(Lange et al., 1966)
LCTV₀	Lung to chemoreceptor transportation volume constant	0.588	liter	(Lange et al., 1966)
P_aO2firstIC	Initial condition for first order P_ao2 system	0.3557	Torr	(Khoo, 1990)
P_aO2secondIC	Initial condition for second order P_ao2 system	103.14	Torr	(Khoo, 1990)
P_aCO2firstIC	Initial condition for first order P_aco2 system	−0.2465	Torr	(Khoo, 1990)
P_aCO2secondIC	Initial condition for second order P_aco2 system	40.393	Torr	(Khoo, 1990)
P_{aO2_delay}IC	Initial condition for O₂ convection	103.12	Torr	(Khoo, 1990)
P_{aco2_delay}IC	Initial condition for CO₂ convection	40.445	Torr	(Khoo, 1990)
P_aCO2	CO₂ partial pressure	variable	Torr	Model
P_aO2	O₂ partial pressure	variable	Torr	Model
Cardiovascular Dissociation
C1	Maximum concentration of hemoglobin-bound oxygen	9	mL/mL	(Khoo, 1990)
C2	Maximum carbon dioxide concentration	87	mL/mL	(Khoo, 1990)
a1	Parameter in O₂ dissociation equation	0.3836	dimensionless	(Spencer et al., 1979)
a2	Parameter in CO₂ dissociation equation	1.819	dimensionless	(Spencer et al., 1979)
alpha1	Parameter in O₂ dissociation equation	0.02598	dimensionless	(Spencer et al., 1979)
alpha2	Parameter in CO₂ dissociation equation	0.05591	dimensionless	(Spencer et al., 1979)
K1	Parameter in O₂ dissociation equation	13	dimensionless	(Spencer et al., 1979)
K2	Parameter in CO₂ dissociation equation	194.4	dimensionless	(Spencer et al., 1979)
beta1	Parameter in O₂ dissociation equation	0.012275	dimensionless	(Spencer et al., 1979)
beta2	Parameter in CO₂ dissociation equation	0.03255	dimensionless	(Spencer et al., 1979)
S_{ao2_delay}IC	Initial Condition for Oxygen Saturation Delay	98.92	sec	(Spencer et al., 1979)
Brain Compartment
MR_bco2	Metabolic production rate for CO₂ in the brain tissue	0.0517	1/s STPD	(Khoo, 1990)
S_co2	Dissociation slope for CO₂ in the blood	0.0043	mL/(mL*Torr)	(Khoo, 1990)
S_bco2	Dissociation slope for CO₂ in the brain tissue	0.36	mL*100g⁻ ¹/Torr	(Khoo, 1990)
P_bco2IC	Initial condition for partial CO₂ pressure from the brain	48.538	Torr	(Khoo, 1990)
Body Tissues Compartment
V_tco2	Body tissue storage volume for CO₂	6	L	(Khoo, 1990)
V_to2	Body tissue storage volume for O₂	7.7	L	(Khoo, 1990)
MR_co2	Metabolic production rate for CO₂	0.0033	1/s STPD	(Khoo, 1990)
MR_o2	Metabolic consumption rate for O₂	0.0038	1/s STPD	(Khoo, 1990)
C_vco2IC	Initial condition for mixed venous CO₂ concentration	0.5247	mL/mL	(Khoo, 1990)
C_vo2IC	Initial condition for mixed venous O₂ concentration	0.1639	mL/mL	(Khoo, 1990)
Upper Airway Model
R_uaw	Upper airway wall resistance	1000000	cmH₂O*s/L	Model
A_0ua	Maximum area of opening in upper airway	1	a.u.	Model
K_ua	Proportionality coefficient between A_ua and Yua;	1	L/(s*cmH₂O)	Model
P_{crit_awake}	Critical upper airway pressure in wakefulness	−40	cmH₂O	Model
S_ua	Upper airway sensitivity to collapse	0.01	a.u.	Model
C_ua	Upper airway compliance	variable	L/cmH₂O	Model
P_ua	Upper airway pressure	variable	cmH₂O	Model
V̇_ua	Upper airway flow	variable	cmH₂O	Model
V̇	Total flow in airways	variable	cmH₂O	Model
Respiratory Muscle Activity
FlowIC	Initial air flow	0	L/s	Model
VC	Vital Capacity	5	L	Model
Pt_frcIC1	Initial condition for respiratory muscle reaction	0	spikes/s	Model
Pt_frcIC2	Initial condition for respiratory muscle reaction	0	spikes/s	Model
FlowIC	Initial condition for airflow	0	L/s	Model
VtIC	Initial condition for lung volume	0	L	Model
Central Neural Control
Carotid Baroreceptors
Pn	Center pressure for sigmoidal function	92	mmHg	(Ursino, 1998)
Kcs	Parameter for sigmoidal slope control	11.758	mmHG	(Ursino, 1998)
Pn_sleep	Parameter for sleep effects	0	mmHg	Model
Kcs_sleep	Parameter for sleep effect	0	mmHG	Model
fcs,min	Lower threshold for sigmoidal function	2.52	spikes/s	(Ursino, 1998)
fcs,max	Upper saturation for sigmoidal function	47.78	spikes/s	(Ursino, 1998)
τ_Z	Time constant for baroreflex	6.37	s	(Ursino, 1998)
τ_P	Time constant for baroreflex	2.076	s	(Ursino, 1998)
Ventilatory Response
I_c	Central apneic threshold	45	dimensionless	Model
I_pCO2	Peripheral apneic threshold for CO₂	38	dimensionless	Model
I_pO2	Peripheral apneic threshold for O₂	102.4	dimensionless	Model
Gc	Gain for central chemical drive	0.075	dimensionless	Model
Gp	Gain for peripheral chemical drive	0.0063	dimensionless	Model
S_wake	Factor of wakefulness to sleep	0.3	dimensionless	Model
Chemoreflex Control of Variable Respiratory Rhythm
F_b	Basal breathing frequency	12.5	Breath /min	(Duffin et al., 2000)
V_b	Basal ventilation	6.7	L/min	Model
T_D	Chemoreflex drive threshold	1539	mL	(Duffin et al., 2000)
T_P	Chemoreflex drive threshold	2879	mL	Model
S1_F	Scaling factor	0.00518	dimensionless	(Duffin et al., 2000)
S1_V	Scaling factor	0.024	dimensionless	(Duffin et al., 2000)
S2_F	Scaling factor	0.0105	dimensionless	(Duffin et al., 2000)
S2_V	Scaling factor	0.0367	dimensionless	(Duffin et al., 2000)
Chemoreflex
fchemo,max	Upper saturation for the sigmoidal function	12.3	spikes/s	(Magosso and Ursino, 2001)
fchemo,min	Lower saturation for the sigmoidal function	0.835	spikes/s	(Magosso and Ursino, 2001)
fchemo_contro l	Basal level for the chemoreflex	1.4	dimensionless	(Magosso and Ursino, 2001)
Kchemo	Slope control parameter for the sigmoidal function	29.27	mmHg	(Magosso and Ursino, 2001)
K_H	Constant value for the static response	3	dimensionless	(Magosso and Ursino, 2001)
τ_chemo	Time constant for the chemoreflex	2	s	(Magosso and Ursino, 2001)
Lung Stretch Receptors Reflex
Gls	Constant gain	23.29	spikes/sec/liter	(Magosso and Ursino, 2001)
τls	Time constant	2	sec	(Magosso and Ursino, 2001)
Offsets
X_sa	Saturation for the offset of α-sympathetic activity on peripheral resistance	6	Torr	(Magosso and Ursino, 2001)
θ_san	Nominal level of offset of α-sympathetic activity on peripheral resistance	13.2	spikes/sec	(Magosso and Ursino, 2001)
PO2n_sa	Central point for the sigmoidal function	30	Torr	(Magosso and Ursino, 2001)
kisc_sa	Parameter of α-sympathetic activity on peripheral resistance	2	dimensionless	(Magosso and Ursino, 2001)
X_sb	Saturation for the offset of β-sympathetic activity	21.2	Torr	Model
θ_sbn	Nominal level of offset of β-sympathetic activity	3.6	spikes/sec	(Magosso and Ursino, 2001)
PO2n_sb	Central point for the sigmoidal function	45	Torr	(Magosso and Ursino, 2001)
kisc_sb	Parameter of β-sympathetic activity	4	dimensionless	(Magosso and Ursino, 2001)
X_sp	Saturation for the offset of α-sympathetic activity on peripheral resistance	6	dimensionless	(Magosso and Ursino, 2001)
θ_spn	Nominal level of offset of α-sympathetic activity on peripheral resistance	13.2	spikes/sec	(Magosso and Ursino, 2001)
PO2n_sp	Central point for the sigmoidal function	30	Torr	(Magosso and Ursino, 2001)
kisc_sp	Parameter of α-sympathetic activity on unstressed volume of veins	2	dimensionless	(Magosso and Ursino, 2001)
τ_isc	Time constant for oxygen response	30	s	(Magosso and Ursino, 2001)
τ_cc	Time constant for carbon dioxide response	20	s	(Magosso and Ursino, 2001)
Autonomic Control
fcs,0	Center point for the sigmoidal function for parasympathetic	25	spikes/s	(Magosso and Ursino, 2001)
fpara,0	Lower saturation of the parasympathetic exponential decay function	3.2	spikes/s	(Magosso and Ursino, 2001)
fpara,∞	Upper limit of the parasympathetic exponential decay function	6.3	spikes/s	(Magosso and Ursino, 2001)
kp	Slope control parameter for the sigmoidal function	7.06	dimensionless	(Magosso and Ursino, 2001)
G_RSA,p	Central RSA gain for parasympathetic response	0.4	dimensionless	Model
Gchemo,p	Chemoreflex gain for parasympathetic response	0.03	dimensionless	Model
Glung, p	Lung stretch receptor reflex gain for parasympathetic response	0.24	dimensionless	(Magosso and Ursino, 2001)
f_s,0	Upper limit of the sympathetic exponential decay function	16.11	spikes/s	(Magosso and Ursino, 2001)
f_s,∞	Lower saturation of the sympathetic exponential decay function	2.1	spikes/s	(Magosso and Ursino, 2001)
Ks	Constant for the exponential function	0.07	s	(Magosso and Ursino, 2001)
G_RSA,bs	Central RSA gain for β-sympathetic response	0.4	dimensionless	Model
Gchemo,bs	Chemoreflex gain for β-sympathetic response	2.8	dimensionless	Model
Glung,bs	Lung stretch receptor reflex gain for β- sympathetic	0.24	dimensionless	Model
G_RSA,as	Central RSA gain for α-sympathetic response	0.4	dimensionless	Model
Gchemo,as	Chemoreflex gain for α-sympathetic response	4	dimensionless	Model
Glung,as	Lung stretch receptor reflex gain for α- sympathetic	0.34	dimensionless	(Magosso and Ursino, 2001)
β-Sympathetic Response
ftbsIC	β-sympathetic initial output after time delay	3.8576	spikes/s	(Ursino, 1998)
ftbs_min	Lower limit for the natural log function	2.66	spikes/s	(Ursino, 1998)
Gbs	β-sympathetic Gain varied with sleep drive	−0.13	dimensionless	(Ursino, 1998)
Gbs_sleep	β-sympathetic sleep gain factor	0.2	dimensionless	Model
τbs	β-sympathetic time constant	2	s	(Ursino, 1998)
Dbs	Delay for β-sympathetic time constant	2	s	(Ursino, 1998)
Parasympathetic Response
ftpIC	Para sympathetic initial output after time delay	4.2748	spikes/s	(Ursino, 1998)
Gpara	Parasympathetic Gain varied with sleep drive	0.09	dimensionless	(Ursino, 1998)
Gpara_sleep	Parasympathetic sleep gain factor	0.2	dimensionless	Model
τpara	Parasympathetic time constant	1.5	s	(Ursino, 1998)
Dbs	Delay for parasympathetic time constant	0.2	s	(Ursino, 1998)
Neuromuscular Drive
Inhale	Boolean variable for inhalation	1	dimensionless	Model
Sino-Atrial Node
HPbasal	Basal value for HP for denervated heart	0.58	s	(Ursino, 1998)
Maximum End-systolic Elastance
Glv	Elastance gain for left ventricle	0.475	mmHg /ml/v	(Ursino, 1998)
D_lv	Delay for elastance of left ventricle	2	s	(Ursino, 1998)
τ_lv	Time constant for elastance of left ventricle	8	s	(Ursino, 1998)
Emax0_lv	Basal level of maximum end-systolic elastance of left ventricle	2.392	mmHg /ml	(Ursino, 1998)
Grv	Elastance gain for right ventricle	0.282	mmHg /ml/v	(Ursino, 1998)
D_rv	Delay for elastance of right ventricle	2	s	(Ursino, 1998)
τ_rv	Time constant for elastance of right ventricle	8	s	(Ursino, 1998)
Emax0_rv	Basal level of maximum end-systolic elastance of right ventricle	1.412	mmHg /ml	(Ursino, 1998)
α-Sympathetic Control of Peripheral Resistance
fasIC	α-sympathetic initial output after time delay	34.793	spikes/s	(Ursino, 1998)
fas_min	Lower limit for the natural log function	2.66	spikes/s	(Ursino, 1998)
Gas_sleep	α-sympathetic Gain varied with sleep	0.3	dimensionless	Model
Gas_sp	α-sympathetic Gain for splanchnic peripheral resistance	0.695	dimensionless	(Ursino, 1998)
τas_sp	α-sympathetic time constant	2	s	Model
Das_sp	Delay α-sympathetic time constant	2	s	(Ursino, 1998)
Gas_ep	α-sympathetic Gain for extra-splanchnic peripheral resistance	1.94	dimensionless	(Ursino, 1998)
τas_ep	α-sympathetic time constant	2	s	(Ursino, 1998)
Das_ep	Delay α-sympathetic time constant	2	s	(Ursino, 1998)
Gas_mp	α-sympathetic Gain for skeletal muscle peripheral resistance	2.47	dimensionless	(Ursino, 1998)
τas_mp	α-sympathetic time constant	2	s	Model
Das_mp	Delay α-sympathetic time constant	2	s	(Ursino, 1998)
Vusv0	Basal level of unstressed volume of splanchnic venous circulation	1435.4	ml	(Ursino, 1998)
Gas_usv	α-sympathetic Gain for unstressed volume of splanchnic venous circulation	−265.4	ml/v	(Ursino, 1998)
τ_{as_usv}	α-sympathetic time constant	20	s	(Ursino, 1998)
D_{as_usv}	Delay α-sympathetic time constant	5	s	(Ursino, 1998)
Local Blood Flow Control of Peripheral Resistance
P_{aCO2_n}	Nominal arterial CO₂ partial pressure i	40	Torr	(Magosso and Ursino, 2001)
CvO2n_b	Nominal venous O₂ concentration in cerebral peripheral circulation	0.14	dimensionless	(Magosso and Ursino, 2001)
CvO2n_m	Nominal venous O₂ concentration in skeletal muscle peripheral circulation	0.155	dimensionless	(Magosso and Ursino, 2001)
CvO2n_h	Nominal venous O₂ concentration in coronary peripheral circulation	0.11	dimensionless	(Magosso and Ursino, 2001)
Tau_CO₂	Time constant for peripheral CO₂ response	20	s	(Magosso and Ursino, 2001)
Tau_O₂	Time constant for peripheral O₂ response	10	s	(Magosso and Ursino, 2001)
A	Parameter for flow regulation equation	20.9	dimensionless	(Magosso and Ursino, 2001)
B	Parameter for flow regulation equation	92.8	dimensionless	(Magosso and Ursino, 2001)
C	Parameter for flow regulation equation	10570	dimensionless	(Magosso and Ursino, 2001)
G_{O2_b}	Gain of local O2 response on cerebral vascular bed	10	dimensionless	(Magosso and Ursino, 2001)
G_{O2_h}	Gain of local O2 response on coronary vascular bed	35	dimensionless	(Magosso and Ursino, 2001)
G_{O2_m}	Gain of local O2 response on muscular vascular bed	30	dimensionless	(Magosso and Ursino, 2001)
Sleep Mechanism
A	Amplitude of the skewed sine function	20.9	dimensionless	Model
X_H	Bias of the skewed sine function for process CH	0.9	dimensionless	Model
X_L	Bias of the skewed sine function for process CL	0.15	dimensionless	Model
α_gc	Constant for sleep decaying	0.2/60	dimensionless	Model
α_rc	Rising rate of slow wave activity	0.4/60	dimensionless	Model
α_fc	Falling rate of slow wave activity	0.008/60	dimensionless	Model
SWAo	Initial value of sleep wake activity	0.007	dimensionless	Model

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2.1. Respiratory subsystem and its central neural control subsystem (see Appendix Eqs. A.1–A.40)

The respiratory subsystem allows simulation of both mechanical and gas exchange characteristics of breathing. The compartment representing respiratory mechanics receives input (total respiratory drive) from the central neural control subsystem and generates the airflow based on the curvilinear relationship between the respiratory muscles and the lung volume according to the Younes model (Riddle and Younes, 1981). Pleural and alveolar pressures are obtained by solving the lumped parameter equations that take into consideration the viscous and elastic properties of the chest wall and lung tissues (Schuessler et al., 1997). Compared to other published models, the advantages of our model are that: 1) it is able to generate pleural pressure by itself such that the model does not need any external input, and 2) it is able to to run by itself in closed-loop mode (Lu et al., 2001).

The gas exchange section consists of a tidal-breathing single-alveolar compartment lung model connected to a dead space that is comprised of 5 small serial compartments (Khoo, 1990). The nonlinear blood-gas dissociation curves, which also incorporate the Bohr-Haldane effects, are based on those published by Spencer et al.(1979). Mixing in the heart and vasculature is modeled after empirical transfer functions derived by Lange et al. (1966). Brain P_CO2, which affects the central chemoreflex response, is computed from the first-order differential equation proposed by Read and Leigh (1967). Gas exchange that occurs outside of the lungs and the brain is modeled as taking place in a single body tissues compartment (Khoo, 1990).

The ventilatory controller incorporates both central and peripheral chemoreceptor contributions. The central component is assumed to respond to only brain P_CO2, while the peripheral component is influenced by arterial P_CO2 and P_O2 and their multiplicative interaction (Khoo et al., 1991). The total ventilatory drive is assumed to be the sum of the central and peripheral chemoreflex components along with an additive state-related component which is modulated by a sleep-wake state index (SI). The output of the ventilator controller feeds into a series of control blocks that represents the neural processes responsible for short-term potentiation (STP) or the “ventilatory afterdischarge” (Dempsey et al., 2002). Our model of STP consists of an initial block that nonlinearly transforms the input from the ventilatory controller and sends the resulting output into a linear dynamic system composed of a static gain placed in parallel with a leaky integrator. The static gain is responsible for the direct effect of STP, converting part of the input instantaneously into a portion of the response. The leaky integrator contributes to the relatively slower afterdischarge, which decays with a time constant of approximately 40 s. A nonlinear process with threshold and saturation is placed at the output end of the STP model. Finally, the “filtered” total ventilatory drive is decomposed into tidal volume and respiratory rhythm (breath period) components using the model proposed by Duffin et al (2000). The continuously-varying breath period component is converted into stair-like respiration rate signal, so that breathing frequency is constant at each breath, using an integral pulse frequency modulation (IPFM) model, similar to what is used for generating heart period.

An important feature of the respiratory subsystem not derived from existing models is the component representing upper airway dynamics. Although sophisticated finite-element models of upper airway mechanics exist (Huang et al., 2005), for the purposes of this study, we opted for computational simplicity by employing a lumped-parameter model. Here, upper airway conductance Y_ua (the inverse of upper airway resistance) is assumed to be a linear function of the upper airway cross-sectional area, which in turn is dependent on the local transmural upper airway pressure (P_ua). When P_ua is less than ambient pressure but greater than the critical pressure (P_crit), Y_ua is a linear function of P_ua with a slope that depends on P_crit. When P_ua is equal to or greater than ambient pressure, the upper airway is fully open and thus upper airway conductance assumes its maximum value. When P_ua is equal to or less than the critical pressure P_crit, the upper airway collapses and the corresponding conductance becomes zero (or equivalently, upper airway resistance becomes infinite). P_crit itself is a function of sleep-wake state (SI): in wakefulness, P_crit takes on a negative value, but in sleep, P_crit becomes less negative. Thus, upper airway conductance is nonlinearly dependent on sleep-wake state, changing between higher values in wakefulness and lower values during sleep. The upper airway sensitivity parameter, S_ua, controls the dependence of P_crit on SI. In the simulated subject with OSAS, S_ua is assigned a value substantially higher than its default value in the control case.

2.2. Cardiovascular subsystem and its central neural control subsystem (Appendix Eqs. A.41–A.73)

The cardiovascular subsystem is capable of simulating the pulsatile nature of the heart and blood flow through the pulmonary and systemic circulations. Included in the model are descriptions of atria-ventricular mechanics, hemodynamics of the systemic and pulmonary circulations, SA node, change of total peripheral resistance and baroreflex. The models of the pulsatile heart, pulmonary circulation and systemic circulation, splanchnic, extrasplanchnic, muscular, coronary and cerebral peripheral circulations are based largely on the work of Ursino but have been modified in terms of different sleep-wake states (Ursino, 1998; Ursino and Magosso, 2000).

The SA node is modeled as a simple pacemaker that receives parasympathetic and beta-sympathetic inputs, but the gain of which is modulated by the sleep index and arousal index. An IPFM model is used to convert the continuous variable representing instantaneous heart rate (the driving signal) into a discrete period signal, analogous to the R-R interval. IPFM has been used in previous simulation studies for this purpose (Dempsey et al., 2002). The vagal effect on heart rate is modeled assuming first-order dynamics with the sleep index modulating the gain. Beta-sympathetic activity is assumed to affect heart rate along with ventricular contractility through a gain that is also modulated by the sleep index. The effect on ventricular contractility takes the form of a time-varying nonlinear elastance function that modulates systolic period and thus cardiac output. The diastolic filling time, the difference between the heart period and systolic period, is thus controlled indirectly. The activation of the right and left hearts is assumed to be fully synchronized and to occur simultaneously. Stroke volume can vary under the influence of venous return, heart period and contractility, and circulatory blood flows in pulmonary and systemic circulation systems are determined by the stroke volume and peripheral vascular tone.

Alpha-sympathetic activity controls the main part of the total peripheral resistance (TPR) change in terms of peripheral vascular tone, as characterized by the changes in peripheral resistances and unstressed volumes of blood vessels in splanchnic, extrasplanchnic and muscular peripheral circulations. Local blood flow in the peripheral vasculature is mediated by changes in the resistances of muscular, coronary and cerebral peripheral circulations, as determined by the local P_O2 and P_CO2 of the blood in these vascular beds. The changes in these local resistances contribute to the overall TPR change. The carotid sinus perceives the changes of arterial blood pressure and converts them into nerve activities for autonomic control, considered as baroreflex autonomic input.

The model of autonomic control assumes that the autonomic inputs that affect the cardiovascular system include the efferent neural outputs of the baroreceptors, chemoreceptors and pulmonary stretch receptors, and CNS response to arterial P_CO2 and P_O2. These inputs are integrated to produce autonomic outputs independently, including beta-sympathetic activity and parasympathetic efferent activity to the heart, and alpha-sympathetic activity to the peripheral vasculature. For sympathetic activities, the efferent firing rates are exponential functions of the weighted sum of the afferent activities and offsets. This section of the model allows the simulation of respiratory to cardiovascular effects such as respiratory sinus arrhythmia, pleural pressure modulation of pulmonary circulatory pressures and blood flow, and chemoreflex effects on peripheral vascular resistance. The cardiovascular changes in turn affect respiratory control through changes in gas exchange dynamics and circulatory delays resulting from changes in blood flow. The CNS response to arterial P_O2 and P_CO2 in the cerebral blood flow provides outputs of the offset terms for alpha-sympathetic activity corresponding to peripheral resistance change and vein constriction and offsets for beta-sympathetic activity with respect to heart rate modulation, based on the work of Magosso and Ursino (2001).

2.3. Sleep Mechanism (Appendix Eqs. A.74–A.83)

The sleep mechanism is a two process model from the Borbely group (Achermann and Borbely 2003; Borbely and Achermann 2000; Daan et al 1984). The total sleep propensity is determined by the combination of two processes: a sleep independent circadian process C and a sleep dependent slow-wave process S. The process C sets up a high and a low time-variant circadian threshold C_H and C_L to modulate the sleep indication, respectively. The circadian process C is described using a skewed sine function which provides the normal 16 hours of wake state and 8 hours of sleep period. The process S is oscillating between these two thresholds, and when it raises upon or beyond the high one, the sleep is triggered; when it falls on or below the low one, the sleep is terminated and awaked activity starts. The sleep wake activity (SWA) wave represents the stage of sleep and it is regulated by both process S and C and non-rapid eye movement (NREM) sleep and rapid eye movement (REM) sleep with a falling and a rising factors, respectively. By varying the interval between those two thresholds, the duration of sleep is manipulated and automatically different night by night. Because of this flexibility, not only can the model simulate regular sleep patterns, it can also evaluate abnormal cases such as the effects of sleep deprivation. The time course of process S is chosen based on experimental data on recordings of slow wave sleep. The time course for process S activity during awake is described from the interpolation of experimental data points which show an exponential increase with saturation.

The sleep mechanism model affects the respiratory and cardiovascular subsystems mainly through the sleep state index, SI, which provides an indication of whether the model is “awake” (SI=0) or “asleep” (SI=1). During sleep onset, SI is a scaled version of SWA as S_SWA, following the same time-course and attaining the value of unity (“asleep”) when SWA reaches its first and highest maximum value. The NREM and REM stages during sleep are simulated through a pulse train and are represented by changes in SWA with no activity in REM stage and an overall tendency to decay throughout the night during NREM sleep.

In the case of obstructive sleep apnoea, during the upper airway obstruction, arterial P_CO2 increases as arterial P_O2 and S_aO2 decrease, allowing the resulting chemical drive to progressively rise. When the total ventilatory drive exceeds the arousal threshold, arousal is triggered and the upper airway opens up allowing the resumption of airflow. The arousal threshold is assumed to be linearly related to S_SWA: the threshold is higher when S_SWA is higher and close to 1 during NREM sleep and it is lower when S_SWA is close to zero during REM sleep (see Eq. A.80). This assumption is based on empirical data showing that the threshold for arousal is lower , during REM stage than in NREM stage (Berry et al., 1998; Berry and Gleeson, 1997; Khoo et al., 1996). During arousal, the wakefulness stimulus reappears, but as ventilatory drive drops back below the threshold, the wakefulness stimulus disappears once again as the system reverts to its previous level of sleep propensity.

2.4. Software Implementation

PNEUMA is implemented using SIMULINK^®, which provides a powerful high level visually-oriented environment for modeling, simulating, and analyzing dynamic systems. The model contains 472 parameters and 80 states, and allows graphical access to 113 variables. It takes approximately 114 minutes to run a 48 hour simulation and about 60 hours to run a 10 week simulation on a PC with Intel Core 2 Duo E8500 CPU. PNEUMA is a hybrid model that contains both discrete and continuous modes of implementation, allowing it to simulate dynamic behavior over a time-scale that ranges from milliseconds to days (i.e. intra-heart beat events to circadian rhythms). An accompanying graphical user interface panel allows users to conveniently change the values of a large number of parameters or impose a variety of physiological conditions without having to modify the program directly. Advanced users can vary parameter values or make changes to the underlying models by directly modifying the graphical objects in the Simulink^® code.

2.5. Model parameters and Verification Issues

Wherever possible, the parameter values employed in the model are based on population values published in the literature, as indicated in Table 1. In some cases where the physiological values are unclear, the parameters have been tuned to ensure that the dynamic behavior of the model under the various conditions explored remains realistic. Model verification and validation are performed by comparing the simulation output under baseline conditions in normal breathing and sleep-disordered breathing and under different interventions to the general population-averaged cardiorespiratory data reported in the literature. Since the focus of this study is on the dynamic interactions among the various mechanisms at play in cardiorespiratory control, verification and validation are guided by employing a qualitative goodness of fit approach. This contrasts with smaller, more focused models in which the key parameters are estimated based on quantitative fits to experimental data obtained from individual subjects. In the present case, there is no single complete experimental dataset that the model can be validated against. Rather, we focus on comparing the model responses with the empirically derived responses that represent the “average subject” in each patient/subject population. The utility of this comprehensive, highly parameterized model is “proven” by testing the internal consistency of the simulated responses of a significant number of state variables over a range of perturbations and conditions.

3. Results: model simulations

3.1. Simulation of cardiorespiratory control during sleep onset in normals

Normal sleep onset is generally associated with a reduction in sympathetic activity and increase in parasympathetic tone, leading to a fall in heart rate and reductions in systolic and diastolic blood pressures, along with an accompanying decrease in ventilation (Mancia, 1980). Subsequently, there are ultradian fluctuations in autonomic activity controlling coronary artery tone (Kirby and Verrier, 1989; Verrier and Dickerson, 1991), systemic blood pressure (Dempsey et al., 2002) and heart rate (Baust and Bohnert, 1969; Mancia, 1980). The transition from wakefulness to sleep in normal breathing as predicted by the model is shown in Fig. 2. There is a reduction in total ventilatory drive due to the withdrawal of the wakefulness stimulus and a decrease in chemoreflex gain. Thus, ventilation drops, resulting in a small increase in P_aCO2 and slight decrease in S_aO2 which allows the respiratory system to compensate partially for the initial sleep state-related drop in ventilation. As such, both tidal volume and respiratory frequency first exhibit a small undershoot before settling at levels lower than during wakefulness. In the normal subject, the increase in parasympathetic tone and decrease in sympathetic tone results in the lower heart rate (higher heart period) and lower arterial blood pressure.

Fig. 2 — Simulation of wakefulness to sleep transition in a normal subject.

3.2. Valsalva maneuver

In the Valsalva maneuver, a subject performs a forced expiration against an occluded airway after a deep inspiration. The Valsalva maneuver is frequently employed as a test of autonomic cardiovascular function (Bannister, 1980; Fox et al., 1966). The maneuver markedly elevates intrathoracic pressure, thus affecting venous return, myocardial contractility, vasomotor tone and baroreflex heart rate control (Fox et al., 1966). The hemodynamic response to the Valsalva maneuver has four distinct phases (Bannister, 1980). At the beginning of forced expiration (phase 1), the contraction of thoracic cage compresses the lungs and causes a large rise in pleural pressure (P_pl). This intrathoracic pressure compresses the blood vessels within the chest. Aortic compression results in a transient increase in arterial blood pressure (ABP), which produces a brief bradycardia (heart rate decrease). Because the thoracic vena cava also becomes compressed, venous return to the heart is decreased, resulting decrease in cardiac output. This leads to an increased tachycardia (heart rate increase) for the 10 second of sustained forced expiration (phase 2) and a fall in ABP. But the decrease in ABP ceases after a few seconds if the peripheral sympathetic vasoconstriction response is normal. Once the occlusion is released (phase 3), the immediate drop in Ppl induces a decrease in ABP. The heart rate remains high due to the increased sympathetic tone. In phase 4, the delayed sympathetic response leads to an ABP overshoot and a compensatory bradycardia. Fig. 3B displays the results of a model simulation of the effects of the Valsalva maneuver on respiration, ABP and heart rate. The forced expiration against closed airway starts at t=84 s and ends at t=98 s. The simulated results for Phase 1 through 4 are as indicated on Fig.3B. Comparison with published experimental data from Bannister (1980) displayed in Fig.3A, shows good agreement. Although the results are not shown here, the model is able to also realistically simulate the effects of the Mueller maneuver – some of these effects involve the same mechanisms that dominate during obstructive apnoea (see Section 3.4).

Fig. 3 — Valsalva maneuver. A. Experimental data from Bannister *(4)*. B. Model results. The maneuver begins at time 80 s with a deep inspiration and ends at ~100 s with release of the breathhold (indicated by arrows).

3.3. Effects of changes in chemoreflex stimulation

The model assumes that hypercapnia stimulates both central and peripheral chemoreceptors while hypoxia affects only the peripheral chemoreceptors (Cooper et al., 2005; Spicuzz et al., 2005). Central depression by hypoxia has not been included in this model. Central chemoreceptor stimulation leads to increased ventilation as well as increased sympathetic outflow – the latter produces peripheral vasoconstriction, increased cardiac contractility and heart rate acceleration (Magosso and Ursino, 2001). Carotid chemoreceptor stimulation also produces peripheral vasoconstriction via the sympathetic pathways, but at the same time, there is an increased cardiac vagal activity leading to a slowing of heart rate (Ursino and Magosso, 2003). However, there is a significant degree of both competitive and reinforcing interaction among these different factors. For instance, during hypoxia, the increased ventilation acts via the lung stretch receptors to inhibit vagal outflow to the heart, while both the peripheral chemoreflex and the baroreflex act to increase cardiac vagal activity. Whether the net result is tachycardia or bradycardia depends on which of these competing effects dominate. The effect of hypoxia on sympathetic activity also consists of excitatory influence from the peripheral chemoreceptors balanced by the inhibitory influence of the lung stretch receptors (Somers et al., 1989a, 1989b). All these features have been incorporated into the model.

Fig. 4 illustrates the behavior of the model when the inhaled level of O₂ concentration is lowered from 21% to 10% to induce hypoxia, but the P_aCO2 level is dynamically maintained around its normal level of ~38.8 mmHg by increasing inspired P_CO2 accordingly. The simulation shows a transient period of ~1 minute immediately following the induction of hypoxia before P_aCO2 is restored to a stable level. The hypoxia simulation begins at t =800 s and lasts for 300 seconds. Accompanying the increased ventilatory drive and ventilation, are parallel increases in heart rate and ABP. ABP is now displayed in terms of beat-to-beat systolic (SBP) and diastolic (DBP) pressures. Note that respiratory sinus arrhythmia also increases because of the larger tidal volumes in the breathing pattern. These model predictions are consistent with reported experimental findings (Magosso and Ursino, 2001; Narkiewicz et al., 1999; Somers et al., 1989a, 1989b).

Fig. 5 summarizes the steady state responses of the major cardiorespiratory variables in the model to the various combinations of hypoxia with hypocapnia, normocapnia or hypercapnia, induced through simulated inhalation of 8.5% O₂ along with 0% (hypocapnia), 2.63% CO₂ (normocapnia) and 5.26% CO₂ (hypercapnia). These results are displayed as percent changes of each variable from its baseline control value (air inhalation). As might be expected, P_aCO2 and ventilation increase progressively from hypocapnic to isocapnic to hypercapnic hypoxia, while the reductions in mean S_aO2 and P_aO2 become progressively smaller. In hypocapnia hypoxia, total peripheral resistance (TPR) actually decreases slightly due to the local vasodilatory effect of hypoxia at the capillary level, but nevertheless, SBP and DBP increase because of the increases in heart rate (HR) and stroke volume. As P_aCO2 is increased with progressively higher levels of inhaled CO₂, TPR increases as sympathetic drive is increased, leading to increased blood pressure and lower cardiac output. These results from our simulations are consistent with what has been published in the literature (Koehler et al., 1980; Kontos et al., 1967; Magosso and Ursino, 2001).

3.4. Obstructive Sleep Apnoea

Obstructive apnoeas during sleep elicit a series of mechanical, hemodynamic, chemical and neural responses (Bradley and Floras, 2003a). The inspiratory efforts against the occluded pharynx lead to abrupt reductions in intrathoracic pressure. While sympathetic nervous system activity is normally reduced in sleep (Dempsey et al., 2002; Lanfranchi and Somers, 2001), during apnoea, the inhibitory vagal input from the pulmonary stretch receptors that restrains sympathetic activities during normal breathing is reduced This leads to a considerably higher level of central sympathetic outflow during obstructive apnoea. The consequent hypercapnic hypoxia further increases chemoreflex-induced sympathetic discharge, while the resulting vasoconstriction from increased alpha-sympathetic activity elevates peripheral resistance and blood pressure. However, arousal from sleep is triggered when the progressively increasing ventilatory effort resulting from the buildup in chemical drive during apnoea exceeds the arousal threshold – this results in a resumption of airflow through the stimulation of pharyngeal dilator muscle activity which restores airway patency. But the accompanying excitatory input from the chemoreflexes causes a further burst of sympathetic outflow along with a loss of vagal tone (Horner et al., 1995; Somers et al., 1995). As such, arousals are characterized by an intense surge in heart rate and increase in blood pressure (Lanfranchi and Somers, 2001; Leung and Bradley, 2001). The hyperventilatory breaths that accompany arousal rapidly restore blood gases towards normal levels and as the effects of the transient arousal wear off and the system falls quickly back to sleep, allowing the pharynx to be once again susceptible to collapse. In severe OSAS, these cycles of apnoea and arousal can recur several hundred times each night, exposing the cardiovascular system to high amplitude oscillations in heart rate and blood pressure.

To enable the model to simulate OSAS following sleep onset, the upper airway sensitivity parameter (S_ua) is increased to a value of least 0.38 from its control level of 0.01. Several obstructive apnoea episodes are displayed on Fig. 6. Each episode of obstructive apnoea leads to hypoxia and hypercapnia, which generate progressively larger respiratory efforts, as reflected in the augmented pleural pressure fluctuations. Eventually, these ventilatory efforts become large enough to trigger an arousal. The arousal brings with it a surge in sympathetic activity and withdrawal of parasympathetic activity, thus leading to significant increases in heart rate and blood pressure. As upper airway patency and airflow to the lungs are restored, there is a rapid resumption of sleep. Subsequent reductions in the state-related ventilatory drive lead to loss of upper airway patency again and a new cycle of obstructive apnoea followed by arousal begins. The model predicts periodicities on the order of ~52 s, which is consistent with what has been reported in the literature (Lanfranchi and Somers, 2001; Leung and Bradley, 2001).

Fig. 6 — Simulation of obstructive sleep apnoea.

Fig. 7 displays, on a more extended time-scale, the initiation and development of repetitive obstructive apnoea episodes following sleep onset, as predicted by the model. At ~2700 s (0.75 h) into the simulation, continuous positive airway pressure (CPAP) at a level of 15 cm H₂O is applied. Acute abolition of the obstructive apnoea episodes by CPAP prevents recurrent hypoxia, reducing the large swings in heart rate and blood pressure that accompany the alternating periods of upper airway obstruction and arousal. However, once CPAP treatment is terminated the obstructive apnoea episodes resume. Note the increase in mean SBP and DBP following the start of occurrence of the obstructive apnoea episodes, but this elevation of blood pressure is reversed once CPAP is administered. This is consistent with what is known about patients with OSAS being “non-dippers” as opposed to normal “dippers” (Lanfranchi and Somers, 2001; Leung and Bradley, 2001).

Fig. 8 shows the development of repetitive obstructive apnoea episodes in two simulated subjects with OSA: one with normal chemoreflex gain and the other with peripheral chemoreflex gain increased six fold. The case with normal chemoreflex gain (left panels) displays obstructive hypopnoeas that eventually become obstructive apnoeas. Cycle durations are on the order of 58 s. In the case with elevated peripheral gain (right panels), a combination of central apnoea and obstructive hypopnoea alternating with hyperpnoeas during arousal occurs. The periodicity of these oscillations is substantially shorter, being on the order of 37 s. The main purpose of these simulations is to demonstrate the capability of the model in generating OSA characteristics that range from purely obstructive apnoeas and hypopnoeas to mixed apnoeas with significant durations of central apnoea.

3.5. Hypoxia induced periodic breathing during sleep

Although periodic breathing (PB) can be induced in awake subjects through hypoxic gas administration or during altitude acclimatization, the tendency to develop PB is enhanced during sleep due to increased hypoxic, hypocapnic and hypercapnic sensitivities (Berssenbrugge et al., 1983). As displayed in Fig. 8, inducing hypoxia by reducing inhaled O₂ concentration from 21% to 10% in during sleep leads to PB, Induced periodic breathing does not occur immediately following hypoxic gas administration but starts to occur only after ~300 s of exposure to hypoxia. Accompanying arousals occur after ~800 s of hypoxic gas administration. Consistent with previous observations (Berssenbrugge et al., 1983; Khoo et al., 1996), full-fledged periodic breathing occurs following a duration over which initially small oscillations in ventilation become progressively amplified. After another 300 s, the model demonstrates stable periodic breathing oscillations. Fig. 8 shows an expanded view of this period of time. It should be noted that the periodicity of the ventilatory oscillation is on the order of 23 s. As well, arousals do not appear at every cycle of PB but only after several PB cycles. Each arousal, when it does appear, tends to occur when ventilatory drive is at its peak. The model also demonstrates hypoxia-induced PB in awake although the results are not shown here. And these results are in good agreement with the literature (Berssenbrugge et al., 1983; Khoo et al., 1991, 1996; Lorenzi-filho et al., 1999).

3.6. Cheyne-Stokes respiration in Chronic Heart Failure: Central Sleep Apnoea

Cheyne-Stokes respiration (CSR) is characterized by rhythmic oscillations of tidal volume, forming the periodic breathing pattern in which apnoeas and hypopnoeas alternate. CSR with central sleep apnoea (CSA) is common among patients with chronic heart failure (CHF) (Andreas et al., 1996; Bradley and Floras, 2003b). CSR-CSA and OSA have common pathophysiological features including episodic hypercapnia and arousals from sleep (Leung et al., 2005). However, unlike OSA, no exaggerated negative intrathoracic pressure is presented in CSR-CSA. The cumulative evidence indicates that Cheyne-Stokes respiration with central sleep apnoea is a part of the pathophysiological cycle involving the cardiovascular, pulmonary and autonomic nervous system that contributes to the progression of CHF (Bradley and Floras, 2003b; Ponikowski et al., 1999).

CSR-CSA provokes oscillations in heart rate and blood pressure similar to those seen during OSA: peaks occur during hyperpnoea and valleys occur during apnoea. However, increases in heart rate and blood pressure progress to peak levels more gradually, paralleling the less abrupt increases in ventilation. A necessary factor that contributes to the occurrence of CSR-CSA in patients with severe CHF is the reduced cardiac output which results in prolongation of the circulatory delay between the pulmonary capillaries and the central and peripheral chemoreceptors (Khoo et al., 1991; Leung et al., 2005). However, this factor appears to be insufficient to produce CSR-CSA, since not all patients with CHF exhibit CSR-CSA. As a recent modeling study suggests (Topor et al., 2007), the additional factor of elevated peripheral or central chemoreflex gain appears to be necessary for CSR-CSA to occur. This is supported by experimental studies that have found that CHF subjects who exhibit CSR-CSA also tend to have higher ventilatory responses to hypercapnia (Javaheri, 1999; Wilcox et al., 1998).

Fig. 9 displays a detailed view of several episodes of CSR-CSA simulated by the model. In this case, peripheral chemoreflex gain was increased six fold, the effective lung-to-chemoreceptor vascular volume was increased 1.5 times and cardiac output was decreased by 42% from normal levels by reducing the contractility of the left ventricle. Following sleep onset, the ventilatory drive starts to oscillate visibly and central apnoeas start to appear, creating the PB pattern associated with CSR. CSR-CSA can also be simulated with smaller reductions in cardiac output if peripheral chemoreflex gain is simultaneously increased more. This is consistent with the notion that PB is a manifestation of feedback instability in respiratory control: CSR-CSA can be produced by a combination of two or more of the following factors - increased system delay, increased controller gain, and increased plant gain (Khoo et al., 1991; Topor et al., 2007). The model predicts that the cycle duration of the PB pattern is approximately 60 s; this is consistent with the literature. It should be noted that not every hyperpnoeic phase of the CSR-CSA cycle is associated with an arousal.

Two commonly employed forms of treatment for CSR-CSA are the nocturnal administration of low concentrations of inhaled CO₂ or supplemental O₂. Fig. 10 displays simulations of the effects of CO₂ inhalation and O₂ supplementation on CSR-CSA during sleep. When the inhalation is changed from air to 3.5% CO₂, there is a rapid and complete disappearance of apnoea episodes and periodic breathing, followed by restoration of a stable sleep. In the case of administration of a hyperoxic mixture containing 30% O₂, the model also predicts the complete disappearance of apnoea episodes (slower than CO₂ administration) and marked attenuation of CSR. However, unlike the case with CO₂ administration, PB with hypopnoeas but without accompanying arousals still persists; these simulated characteristics are consistent with empirical data (Lorenzi-filho et al., 1999).

Fig. 10 — Simulation of Cheyne-Stokes respiration during sleep in a subject with chronic heart failure.

3.7. Simulation of overnight sleep – Normal Control vs. Obstructive Sleep Apnoea

In the Borbely model of sleep-wake regulation that we have employed, the sleep propensity index, “Process S”, diminishes as sleep progresses. This continues until S(t) hits the lower circadian threshold C_L of Process C, which represents the circadian rhythm, following which wakefulness appears and S(t) reverses course and starts to increase again (Panel B, Fig.11). In normal sleep, this model results in total sleep duration of 7.8 hours (Panel A, Fig.11). However, during OSA or CSR-CSA, the occurrence of repetitive arousals fragments sleep, prolonging the decay of S(t). As a consequence, our model predicts that the overall duration of sleep is about 8.5 hours when it is used to simulate an OSA subject (Panels C and D, Fig.11), and about 8.3 hours in a CHF subject with CSR-CSA (not displayed). These total sleep durations for sleep apnoeas are noticeably longer compared to the 7.8 hours predicted for normal sleep. However, the overnight administration of CPAP in OSA restores sleep duration back towards the normal duration at ~7.9 hours (Fig.11, panels E and F). These predictions imply that if the subject with OSA is awakened before sleep propensity is reduced to the levels that normally accompany the end of overnight sleep and the start of wakefulness, there will be a tendency for sleep propensity in the next 24 hours to have a higher average value than the previous 24 hours. Thus, the model predicts that sleep fragmentation resulting from repetitive arousal in OSA or CSR-CHF would lead to increased daytime sleepiness the following day.

Fig. 11 — Simulation of the effects of CO₂ and O₂ inhalation on a CHF subject with CSR-CSA.

4. Discussion

To date, much has been learnt about the mechanisms that underlie sleep-disordered breathing, as well as the physiological bases through which autonomic cardiovascular control is affected by the repetitive episodes of apnoea and arousal. Developing a quantitative dynamic model such as the one introduced here is useful for consolidating all this knowledge and providing a systematic means of predicting how the component mechanisms involved may interact under a variety of conditions. Model validation has been accomplished in two ways. First, the model has been shown to make reasonably realistic predictions of cardiovascular and respiratory control during wakefulness and sleep, consistent with a wide range of observed population-averaged physiological responses, under those conditions or interventions most pertinent to sleep disordered breathing. Secondly, we have demonstrated the internal consistency of the simulated responses over a wide range of perturbations.

By superseding and integrating its various component subsystems – respiratory, cardiovascular, sleep-wake and autonomic nervous systems – the present model is substantially more “self-contained” relative to previous models. For instance, the cardiorespiratory model by Lu et al (2001) requires the time-course of pleural pressure derived from experimental measurements as an input for the model to simulate the effects of the Valsalva maneuver; by contrast, in our model, the pleural pressure waveform is generated internally since the model also contains all the features pertinent to respiratory mechanics and control.

4.1. Incorporation of circadian and sleep effects

The most significant difference between the present and previous models of cardiorespiratory interactions in the literature is the inclusion of a circadian and wakefulness/sleep mechanism, even though a large database of knowledge concerning the effect of sleep on the cardiovascular and respiratory systems has been developed over the past few decades. With the implemented wake/sleep mechanism, this integrative model has been able to predict the dynamic changes that occur in the cardiovascular and the respiratory systems, as well as how they interact with each other, following transitions into and out of sleep. A more surprising outcome is that this feature allows the model to predict how arousals from sleep, triggered by cardiorespiratory events during obstructive or central apnoea, affect the total duration of the sleep. This implies that constraining the total duration of sleep to remain constant would lead to the accumulation of excess sleep propensity, analogous to what is commonly referred to as “sleep debt” in the experimental literature. In its current form, the model does not yet have the capability of directly simulating sleep debt, but this remains an important feature to add in future work.

4.2. Sensitivity Analyses

As mentioned earlier, the model consists of 472 parameters, a fraction of which correspond to physiological entities and therefore are assigned values (if available) that are appropriate for the subject group and condition under study. Since this model is largely an integration of smaller sub-models that have already been peer-reviewed and validated, many of the other parameters were assigned values employed in these previously published sub-models. Thus, we focused on identifying the key parameters that mediate the most significant cross-interactions across the modeled subsystems and delineating them from the non-sensitive parameters that could set as constants, in order to minimize the number of free parameters that have to be modified prior to running a simulation. Sensitivity analyses were conducted to determine the relative contributions of the key parameters that played the most significant roles in various conditions. These sensitivity analyses were also useful as consistency checks. The results of the most important of these analyses are as follows:

(a) Interaction between key parameters contributing to severity of obstructive sleep apnoea effects

Table 2 displays the model predictions obtained with various combinations of values for the sensitivity of upper airway collapsibility to state changes, S_ua, and the arousal threshold. In simulations of sleep in the normal subject, S_ua is set equal to 0.01, and during sleep, the critical closing pressure, P_crit, is −29.6 cm H₂O. Stable sleep is achieved and thus the arousal index (AI) is zero. The fluctuations in heart period are consistent with normal respiratory sinus arrhythmia, while the fluctuations in SBP and DBP are due primarily to the mechanical effects of respiration. When S_ua is increased to 0.3, so that during sleep, P_crit becomes much less negative (−3.5 cm H₂O), representing a substantially greater degree of collapsibility in the upper airway – but no apnoeas or arousals occur at this stage. However, when S_ua is increased further to 0.38, P_crit becomes even less negative (−2.8 cm H₂O), and now obstructive apnoeas terminated by arousals start to occur, albeit not frequently. Increasing S_ua a little more (to 4) leads to greater periodicity of obstructive apnoea and arousals (AI=18), and these are the course of substantially larger amplitudes of fluctuation in heart period, SBP and DBP. Increasing S_ua even further to 0.5 produces more frequent apnoea and arousals (AI = 46), but the amplitudes of the swings in heart period, SBP and DBP remain relatively constant. Increasing the arousal threshold delays the appearance of obstructive apnoea as S_ua is progressively increased. For the same S_ua (for instance, when S_ua=0.5), AI remains approximately the same as the case with the lower arousal threshold. However, the amplitudes of the fluctuations in heart period, SBP and DBP are substantially larger.

Table 2.

Sensitivity analysis of the effects of changing upper airway sensitivity and arousal threshold on respiratory, cardiovascular and state responses.

Sua		0.01(Control)	0.3	0.38	0.4	0.5

Pcrit during Sleep(cmH₂O)		−29.6	−3.5	−2.8	−2.7	−2.2
Arousal Threshold Upper Limit =0.75 (Control Value)	AI (per hour)	0	0	1	18	46

	ΔHP(msec)	50	52	277	285	282
	mean HP(msec)	796	790	781	768	761
	ΔSBP (mmHg)	6	4	48	52	48
	mean SBP(mmHg)	106	107	110	120	119
	ΔDBP (mmHg)	4	12	47	51	49
	mean DBP(mmHg)	66	67	69	77	77

Arousal Threshold Upper Limit =1.15	AI (per hour)	0	0	0	16	45

	ΔHP(msec)	50	52	56	302	307
	mean HP(msec)	796	790	783	790	778
	ΔSBP (mmHg)	6	4	32	66	65
	mean SBP(mmHg)	106	107	109	136	132
	ΔDBP (mmHg)	4	12	7	63	68
	mean DBP(mmHg)	66	67	68	87	85

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All results shown are based on 9-hour overnight sleep simulations. Sua: upper airway sensitivity to changes in sleep state index, AI: Arousal Index, Pcrit: critical closing pressure of the upper airway during sleep (cmH₂O), ΔHP: maximum amplitude of fluctuation in heart period (ms), ΔSDP: maximum swing in systolic blood pressure (mmHg), ΔDBP: maximum swing in diastolic blood pressure (mmHg).

(b) Interaction between key parameters contributing to CSR effects

Table 3 displays the results of the sensitivity analysis conducted to determine the contributions of the key parameters that influence the model during simulations of Cheyne-Stokes respiration with chronic heart failure. We have found, as might be expected, that the ability of the model to simulate CSR depends most sensitively on peripheral chemoreflex gain (G_p) and the circulation delay. However, circulation delay can be altered primarily by changing the basal value of maximum left ventricle elastance (pressure-volume ratio), Emaxlv, and/or the effective vascular volume between the lungs and carotid bodies. Reducing Emaxlv, and thus reduces cardiac contractility, increases circulation delay by decreasing cardiac output. Increasing the effective lung-to-carotid bodies vascular volume increases circulation delay for constant cardiac output. The increased vascular volume is most likely the consequence of cardiomegaly, which commonly occurs in patients with CHF.

Table 3.

Sensitivity Analysis of Key Parameters that contribute to Cheyne-Stokes Respiration in simulated subjects with Chronic Heart Failure.

Model Parameters^*					Simulation Results
% Increase in Gp^*	% Decrease in basal Emaxlv^*	% Increase in Vascular Volume ^*	% Reductio n in Cardiac Output	% Increase in Circula- tion Delay	CSR?	Central Apnoea?	AI (per h)	ΔHP (ms)	mean HP (ms)	ΔSBP (mmHg)	mean SBP (mmHg)	ΔDBP (mmHg)	mean DBP (mmHg)	Cycle dura- tion (s)
0	0	0	0	0	No	No	0	42	806	3	105	3	65	N/A
200	70	20	26.73	46.73	Yes	No	0	119	777	6	82	8	52	38.4
200	90	20	42.26	62.26	Yes	Yes	0	210	716	11	65	23	43	47.4
200	70	50	27.24	77.24	Yes	No	0	181	773	23	82	16	53	42.8
200	90	50	43.11	93.11	Yes	Yes	0	226	714	10	65	13	43	52.0
400	70	20	27.99	47.99	Yes	Yes	15	396	769	22	81	26	52	40.5
400	90	20	41.66	61.66	Yes	Yes	29	422	697	24	65	28	43	48.4
400	70	50	27.97	77.97	Yes	Yes	23	403	761	24	81	28	52	45.8
400	90	50	42.19	92.19	Yes	Yes	36	437	685	26	65	30	43	56.7
600	70	20	28.37	48.37	Yes	Yes	32	403	762	22	80	27	51	44.4
600	90	20	42.56	62.56	Yes	Yes	77	430	674	24	64	28	43	49.0
600	70	50	28.80	78.80	Yes	Yes	79	410	738	25	82	29	53	48.2
600	90	50	43.44	93.44	Yes	Yes	66	443	665	27	65	32	43	57.1

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All results displayed are based on 9-hour overnight sleep simulations. Gp: peripheral chemoreflex gain, AI: Arousal Index (per hour), ΔHP: maximum amplitude of fluctuation in heart period (ms), ΔSDP: maximum amplitude of fluctuation in systolic blood pressure (mmHg), ΔDBP: maximum amplitude of fluctuation in diastolic blood pressure (mmHg).

represents the model parameters.

As G_p is increased by 200% with concomitant increases in circulation delay, CSR appears but the periodic breathing pattern occurs with or without central apnoeas and without arousals. Periodic apnoeas and arousals appear when G_p is increased beyond 200%, leading to progressively higher AI with further increases in G_p. The amplitudes of the periodic fluctuations in heart period, SBP and DBP that are entrained by the ventilatory oscillations also increase. However, with increases in G_p above 400%, the increases in these fluctuation amplitudes tend to saturate. As G_p increases, there is relatively little change in cycle duration of these oscillations.

For the same level of G_p, a reduction in Emaxlv produces a corresponding decrease in cardiac output and increase in circulation delay, which in turn lead to a significant increase in oscillation cycle duration. The magnitudes of these changes are relatively unaffected by the magnitude of G_p. The accompanying periodic fluctuations in heart period, SBP and DBP also increase, but these increases tend to saturate as G_p is progressively raised. AI also increases as Emaxlv is progressively reduced. When G_p and Emaxlv are kept constant while the effective vascular volume between the lungs and carotid bodies is increased, circulation delay is increased correspondingly, whereas cardiac output remains unchanged. This leads to increased AI, as well as increases in the amplitudes of fluctuations in heart period, SBP and DBP. However, at much higher levels of G_p, increased circulation delays exert considerably smaller effects on the swings in heart period and blood pressure. The overall conclusion that one can draw from this sensitivity analysis is that G_p exerts a stronger influence on the strength of the ventilatory oscillations and the amplitude of the concomitant cardiovascular oscillations that are entrained, whereas the factors that influence circulation delay bear stronger effects on oscillation cycle duration. Although not shown here, we have found that increasing central chemoreflex gain, G_c, combined with at least 30% reduction in cardiac output, can also produce CSR-CSA. However, to achieve self-sustained instability, G_c has to be increased 8-fold compared to 3-fold increase in G_p. On the other hand, if G_c and G_p are both simultaneously increased and by the same proportion, the model remains stable. These results are compatible with the findings derived from another model (Topor et al., 2007).

(c) Interaction between baroreflex and chemoreflex inputs to the ANS

Another form of sensitivity analysis consists of blocking the effects of each of the governing variables and determining the corresponding outcomes on the overall control system. This kind of approach is in fact tantamount to the experimental technique of pharmacologically or surgically eliminating the influence of specific physiological variables in order to determine their contribution to the overall system response. The example shown in Fig. 13 illustrates the competing effects of the baroreflex and chemoreflex in regulating blood pressure. In the control case, the steady state model-predicted response in blood pressure to hypoxia, induced by lowering inhaled O₂ from 21% to 8.5%, is displayed. When both baroreflex and chemoreflex inputs to the autonomic nervous system are kept intact, hypoxia elevates blood pressure by ~30%. When the baroreflex input is blocked, blood pressure is increased by ~140% relative to levels resulting from normoxia. However, when chemoreflex input is blocked while baroreflex input is restored, blood pressure becomes lower than control levels (normoxia). These results demonstrate the following points: (i) stimulation of the peripheral chemoreceptors (by hypoxia) leads to sympathoexcitation which elevates blood pressure through autonomic control of the heart and peripheral vasculature; (ii) when chemoreflex inputs to the autonomic nervous system are blocked, the main effect of hypoxia on the peripheral vasculature is local vasodilation, leading to a drop in blood pressure; and (iii) the arterial baroreflex is substantially more powerful than the chemoreflex in regulating blood pressure.

Fig. 13 — Sensitivity analysis of the effects of hypoxia on blood pressure. Hypoxia is induced by lowering inhaled O₂ from 21% to 8.5%. Control: both baroreflex and chemoreflex inputs to the autonomic nervous system are left intact. “Block Baroreflex”: baroreflex inputs are blocked. “Block Chemoreflex”: chemoreflex inputs are blocked.

4.3. Model limitations

The sleep module provides the option of generating the NREM/REM ultradian rhythm. Changes in the depth of NREM sleep are modeled as a continuous process, which likely corresponds well with the underlying physiological processes. However, we have not determined how the modeled continuous states translate into the discrete NREM sleep stages that are employed in standard polysomnography. As well, we have been overly simplistic in modeling REM stage activity and have not distinguished tonic from phasic REM. Experimental studies have found that the threshold for arousal can substantially different between NREM and REM stages. There also appears to be a greater reduction in hypoxic ventilatory response in REM, as well as differences in many other physiological parameters, such as metabolic rate, autonomic tone, upper airway resistance and respiratory compliance. The time-courses of change in these parameters also remain unclear. These details have not been addressed in the current version of the model and further work is needed to overcome these limitations.

Other limitations include the fact that the current model is capable of simulating cardiorespiratory responses in sleep and resting wakefulness, but does not include other physiological states such as exercise, ventilatory loading or ventilation-perfusion abnormality. Although the present model does incorporate neural dynamics of the ventilatory controller in the form of short-term potentiation, other features of chemoreflex plasticity, such as hypoxic depression, or baroreflex plasticity, such as baroreflex resetting, have not been included. Presynaptic or postsynaptic gating of sensory input as a mechanism for controlling the balance of competing afferent inputs has also not been incorporated. As well, in the present model, the initiation of arousals is determined by setting ventilatory drive thresholds and the effects of arousal on respiratory and cardiovascular responses are assumed to be instantaneous. These assumptions are overly simplistic and future work could include more realistic dynamics based on empirical data.

Future improvements should provide the model with the capability of simulating different long-term disease progression scenarios (on time scales that range from days to years) with fine short-term detail (on a time scale of milliseconds). This would enable us to better understand the chronic effects of sleep-disordered breathing on autonomic function and daytime sleepiness.

Fig. 12 — Simulations with the model over a 24-hour circadian cycle – comparison among different cases. Panels A and B: normal sleep; Panels C and D: OSA; Panels E and F: OSA with CPAP treatment.

Acknowledgements

This work was supported by the USC Biomedical Simulations Resource through NIH Grant EB-001978.

Appendix : Model Equations

For a summary and glossary of all variables and parameters, please see Table 1.

Respiratory Gas Exchange and Transport

(a) Gas transport through the dead space(Khoo, 1990)

Inspiration

V_{d (1)} {\dot{P}}_{d {(1)}_{{CO}_{2}}} = \dot{V} [P_{I_{{CO}_{2}}} - P_{d {(1)}_{{CO}_{2}}}]

(A.1a)

\begin{array}{l} V_{d (i)} {\dot{P}}_{d {(i)}_{{CO}_{2}}} = \dot{V} [P_{d {(i - 1)}_{{CO}_{2}}} - P_{d {(i)}_{{CO}_{2}}}] & 2 \leq i \end{array} \leq 5

(A.1b)

V_{d (1)} {\dot{P}}_{d {(1)}_{O_{2}}} = \dot{V} [P_{I_{O_{2}}} - P_{d {(1)}_{O_{2}}}]

(A.2a)

\begin{array}{l} V_{d (i)} {\dot{P}}_{d {(i)}_{O_{2}}} = \dot{V} [P_{d {(i - 1)}_{O_{2}}} - P_{d {(i)}_{O_{2}}}] & 2 \leq i \leq 5 \end{array}

(A.2b)

Expiration

\begin{array}{l} V_{d (i)} {\dot{P}}_{d {(i)}_{{CO}_{2}}} = \dot{V} [P_{d {(i + 1)}_{{CO}_{2}}} - P_{d {(i)}_{{CO}_{2}}}] & 1 \leq i \leq 4 \end{array}

(A.3a)

V_{d (5)} {\dot{P}}_{d {(5)}_{{CO}_{2}}} = \dot{V} [P_{A_{{CO}_{2}}} - P_{d {(5)}_{{CO}_{2}}}]

(A.3b)

\begin{array}{l} V_{d (i)} {\dot{P}}_{d {(i)}_{O_{2}}} = \dot{V} [P_{d {(i + 1)}_{O_{2}}} - P_{d {(i)}_{O_{2}}}] & 1 \leq i \leq 4 \end{array}

(A.4a)

V_{d (5)} {\dot{P}}_{d {(5)}_{O_{2}}} = \dot{V} [P_{A_{O_{2}}} - P_{d {(5)}_{O_{2}}}]

(A.4b)

(b) Alveolar Gas Exchange (Khoo, 1990)

Inspiration

V_{{co}_{2}} {\dot{P}}_{{Aco}_{2}} = [863 \dot{Q} (C_{{vco}_{2}} - C_{{aco}_{2}}) + {\dot{V}}_{A} (P_{d (5) {co}_{2}} - P_{{Aco}_{2}})]

(A.5)

V_{o_{2}} {\dot{P}}_{{Ao}_{2}} = [863 \dot{Q} (C_{{vo}_{2}} - C_{{ao}_{2}}) + {\dot{V}}_{A} (P_{d (5) o_{2}} - P_{{Ao}_{2}})]

(A.6)

Expiration

V_{{co}_{2}} {\dot{P}}_{{Aco}_{2}} = [863 \dot{Q} (C_{{vco}_{2}} - C_{{aco}_{2}})]

(A.7)

V_{o_{2}} {\dot{P}}_{{Ao}_{2}} = [863 \dot{Q} (C_{{vo}_{2}} - C_{{ao}_{2}})]

(A.8)

(c) Cardiovascular Mixing (Heart and Vasculature) (Lange et al, 1966)

{\ddot{P}}_{{aCO}_{2}} = \frac{1}{(T 1^{*} T 2)} [P_{{ACO}_{2}} (t - T_{a}) - (T 1 + T 2) {\dot{P}}_{{aCO}_{2}} - P_{{aCO}_{2}}]

(A.9a)

{\ddot{P}}_{{ao}_{2}} = \frac{1}{(T 1^{*} T 2)} [P_{{Ao}_{2}} (t - T_{a}) - (T 1 + T 2) {\dot{P}}_{{ao}_{2}} - P_{{ao}_{2}}]

(A.9b)

T_{a} = LCTV / \dot{Q}

(A.10)

(d) Blood-Gas Dissociation Curves (Spencer et al, 1979)

CO₂

{Ca}_{{CO}_{2}} = \bar{C} a_{{CO}_{2}} \frac{F_{{CO}_{2}}^{1 / a_{2}}}{1 + F_{{CO}_{2}}^{1 / a_{2}}}

(A.11)

where F_{{co}_{2}} = \frac{P_{{Aco}_{2}} (1 + β_{2} P_{{Ao}_{2}})}{K_{2} (1 + α_{2} P_{{Ao}_{2}})}

(A.12)

O₂

{Ca}_{O_{2}} = \bar{C} a_{O_{2}} \frac{F_{O_{2}}^{1 / a_{1}}}{1 + F_{O_{2}}^{1 / a_{1}}}

(A.13)

where F_{O_{2}} = \frac{P_{A_{O_{2}}} (1 + β_{1} P_{A_{{CO}_{2}}})}{K_{1} (1 + α_{1} P_{A_{{CO}_{2}}})}

(A.14)

(e) Brain Compartment CO₂ exchange (Read and Leigh, 1967)

S_{{bCO}_{2}} {\dot{P}}_{{bCO}_{2}} = [{MR}_{{bCO}_{2}} + {\dot{Q}}_{b} S_{{CO}_{2}} (P_{{aCO}_{2}} - P_{{bCO}_{2}}) - h]

(A.15)

(f) Cerebral Blood Flow (Khoo, 1990)

Cerebral blood flow is calculated by solving the following quadratic equation:

{\dot{Q}}_{b}^{2} - [1 + 0.03 ({\dot{P}}_{{bco}_{2}} - 40)] {\dot{Q}}_{b 0} {\dot{Q}}_{b} + 0.03 ({MR}_{{bco}_{2}} - h) {\dot{Q}}_{b 0} / S_{{co}_{2}} = 0

(A.16)

(g) Gas Exchange in the Body Tissues Compartments (Khoo, 1990)

{Vt}_{{CO}_{2}} {\dot{C}}_{{VCO}_{2}} = [{MR}_{{CO}_{2}} + \dot{Q} (C_{{aCO}_{2}} - C_{{VCO}_{2}})]

(A.17)

{MR}_{CO 2} = {MR}_{CO 2}^{awake} (0.375 (1 - 0.4 SI) + 0.625)

(A.18)

{Vt}_{O_{2}} {\dot{C}}_{{VO}_{2}} = [- {MR}_{O_{2}} + \dot{Q} (C_{{aO}_{2}} - C_{{VO}_{2}})]

(A.19)

{MR}_{O 2} = {MR}_{O 2}^{awake} (0.375 (1 - 0.4 SI) + 0.625)

(A.20)

Ventilatory Controller (Khoo, 1990)

(a) For normal breathing,

D_{Total} = {\begin{array}{l} Y, {TH}_{L} \leq Y \leq & {TH}_{H} \\ 0, & OW \end{array}

(A.21)

Y = X / 2 + X / 2 \otimes [z_{p 0} \cdot e^{- z_{p 0} \cdot t} \cdot u (t)]

(A.22)

X = {\begin{array}{l} 2 / (1 + e^{- 5 D_{Total_O}}) - 1, & D_{Total_O} > 0 \\ 0, & D_{Total_O} \leq 0 \end{array}

(A.23)

(b) For sleep-disordered breathing,

D_{Total} = D_{Total_O}

(A.24)

where

D_{Total_O} = (1 - 0.4 SI) (D_{vent} - D_{state})

(A.25)

D_{state} = SI \cdot S_{wake}

(A.26)

D_{vent} = D_{c} + D_{p}

(A.27)

D_{C} = {\begin{array}{l} G_{C} (P_{{bCO}_{2}} - I_{C}), & P_{{bCO}_{2}} > I_{C} \\ 0, & Otherwise \end{array}

(A.28)

D_{P} = {\begin{array}{l} G_{P} (P_{{aCO}_{2}} - I_{{pCO}_{2}}) \cdot (I_{{pO}_{2}} - SAO 2), P_{{aCO}_{2}} > I_{{pCO}_{2}} & & I_{{pO}_{2}} > SAO 2 \\ 0, & Otherwise \end{array}

(A.29)

Respiratory Mechanics (Riddle and Younes, 1981)

P_{isom} = G_{neuromusc} D_{Total}

(A.30)

Y_{rs} = \frac{Y_{ua}}{1 + (R_{AW} + R_{LT} + R_{CW}) \cdot Y_{ua}}

(A.31)

Note: when Yua=0, then Yrs = 0.

\begin{array}{l} {\dot{V}}_{t} = \frac{\sqrt{{({0.25}^{P_{isom} (t) e^{- V_{t} / 0.28 VC}} \cdot Y_{rs} + b^{V_{t}} + V_{t} E_{rs} Y_{rs} - P_{ao} Y_{rs})}^{2} + 4 b^{V_{t}} (P_{isom} e^{- V_{t} / 0.28 VC} Y_{rs} - V_{t} E_{rs} Y_{rs} + P_{ao} Y_{rs})}}{2} \\ - \frac{{0.25}^{P_{isom} (t) e^{- V_{t} / 0.28 VC}} \cdot Y_{rs} + b^{v_{t}} + V_{t} E_{rs} Y_{rs} - P_{ao} Y_{rs}}{2} \end{array}

(A.32)

P_{mus} = \frac{P_{isom} e^{- V_{t} / 0.28 VC} (b^{V_{t}} - 0.25 {\dot{V}}_{t})}{({\dot{V}}_{t} + b^{V_{t}})}

(A.33)

P_{PL} = R_{CW} {\dot{V}}_{t} + E_{CW} V_{t} - P_{mus}

(A.34)

P_{alv} = R_{LT} {\dot{V}}_{t} + E_{LT} V_{t} + P_{PL}

(A.35)

Upper airway mechanics

(P_{ao} - P_{ua}) \cdot Y_{ua} = \dot{V} - {\dot{V}}_{ua}

(A.36)

where Y_{ua} = 1 / R_{ua}

(A.37)

P_{ua} = - \frac{P_{crit} (SI)}{b_{ua}} \int {\dot{V}}_{ua} dt - {\dot{V}}_{ua} \cdot R_{uaw}

(A.38)

Y_{ua} (SI) = k_{ua} \cdot A_{ua}, where A_{ua} = {\begin{array}{l} 0, \\ A_{0 ua} \cdot (1 - P_{ua} / P_{crit} (SI)), \\ A_{0 ua}, \end{array} \begin{matrix} P_{ua} \leq P_{crit} \\ P_{crit} < P_{ua} \leq 0 \\ P_{ua} > 0 \end{matrix}

(A.39)

P_{crit} (SI) = {\begin{array}{l} P_{crit_awake}, \\ P_{crit_awake} / (1 + S_{ua} \cdot {(SI)}^{2}) / (1 - Sleepawake), \\ P_{crit_awake} / (1 + S_{ua}), \end{array} \begin{array}{r} SI = 0 (awake) \\ 0 < SI < 1 \\ SI = 1 (sleep) \end{array},

(A.40)

where Sleepawake is 0 during sleep and 1 during wakefulness, S_ua is upper airway sensitivity and is directly related to P_crit.

Autonomic Inputs

(a) Carotid Baroreceptors (Ursino, 1998)

f_{cs} = \frac{[f_{cs, min} + f_{cs, max} exp (\frac{P_{ABP} - P_{n} + θ_{Pn}}{k_{cs} + θ_{kcs}})]}{[1 + exp (\frac{P_{ABP} - P_{n} + θ_{Pn}}{k_{cs} + θ_{kcs}})]}

(A.41a)

where P_{ABP} = \frac{τ_{z} \cdot s + 1}{τ_{p} \cdot s + 1} \cdot ABP

(A.41b)

θ_{Pn} = P_{n_sleep} \cdot (1 - AI) \cdot SI

(A.41c)

θ_{kcs} = K_{cs_sleep} \cdot (1 - AI) \cdot SI

(A.41d)

(b) Chemoreceptors (Magosso and Ursino, 2001)

φ_{chemo} ({Pa}_{O_{2}}, {Pa}_{{CO}_{2}}) = \frac{f_{chemo, max} + f_{chemo, min} exp (\frac{{Pa}_{O_{2}} - \bar{{Pa}_{O_{2}}}}{k_{chemo}})}{1 + exp (\frac{{Pa}_{O_{2}} - \bar{{Pa}_{O_{2}}}}{k_{chemo}})} • [K • ln (\frac{{Pa}_{{CO}_{2}}}{\bar{{Pa}_{CO 2}}}) + f_{chemo_control}]

(A.42)

where coefficient K is defined as follows:

K = {\begin{array}{l} K_{H} & {if Pa}_{O_{2}} > 80 \\ K_{H} - 1.2 (\frac{{Pa}_{O_{2}} - 80}{30}) & if 40 \leq {Pa}_{O_{2}} \leq 80 \\ K_{H} - 1.6 & {if Pa}_{O_{2}} < 40 \end{array}

(A.43)

\frac{{df}_{chemo}}{dt} = \frac{1}{τ_{chemo}} (- f_{chemo} + φ_{chemo})

(A.44)

(c) Lung Stretch Receptors (Magosso and Ursino, 2001)

φ_{lung} = G_{lung} V_{t}

(A.45)

\frac{{df}_{lung}}{dt} = \frac{1}{τ_{lung}} (- f_{lung} + φ_{lung})

(A.46)

(d) Offsets (Magosso and Ursino, 2001)

Offse t_{res,vein,heart} = θ_{san,spn,sbn} - θ_{O 2 sa, O 2 sp, O 2 sb} - θ_{CO 2 sa, CO 2 sp, CO 2 sb}

(A.47)

\frac{d θ_{O 2 sa, O 2 sp, O 2 sb}}{dt} = \frac{1}{τ_{isc}} (- θ_{O 2 sa, O 2 sp, O 2 sb} + W_{sa,sp,sb})

(A.48)

\frac{d θ_{CO 2 sa, CO 2 sp, CO 2 sb}}{dt} = \frac{1}{τ_{cc}} [- θ_{CO 2 sa, CO 2 sp, CO 2 sb} + {gcc}_{sa,sp,sb} \cdot (P_{aCO 2} - P_{aCO 2 n})]

(A.49)

W_{sa,sp,sb} = X_{sa,sp,sb} / (1 + exp ((P_{aO 2} - PO 2 n_{sa,sp,sb}) / {kisc}_{sa,sp,sb}))

(A.50)

Autonomic Control: Integration of autonomic inputs Magosso and Ursino, 2001)

(a) Alpha-Sympathetic Activity

\begin{array}{l} f_{tas_res,vein} = f_{s, \infty} + (f_{s, 0} - f_{s, \infty}) . \\ exp [k_{s} \cdot (- G_{baro,as} f_{cs} + G_{chemo,as} f_{chemo} - G_{lung,as} f_{lung} - G_{RSA, as} N_{t} - {Offset}_{res,vein})] \end{array}

(A.51)

(b) Beta-Sympathetic Activity

\begin{array}{l} f_{tbs} = f_{s, \infty} + (f_{s, 0} - f_{s, \infty}) . \\ exp [k_{s} \cdot (- G_{baro,bs} f_{cs} + G_{chemo,bs} f_{chemo} + G_{lung,bs} f_{lung} + G_{RSA,bs} N_{t} - {Offset}_{heart})] \end{array}

(A.52)

(c) Parasympathetic Activity

\begin{array}{l} f_{tp} = \frac{[f_{para, 0} + f_{para, \infty} \cdot exp (\frac{f_{cs} - f_{cs, 0}}{k_{p}})]}{[1 + exp (\frac{f_{cs} - f_{cs, 0}}{k_{p}})]} + G_{chemo,p} f_{chemo} + G_{lung,p} f_{lung} - G_{RSA,p} N_{t} - {Offset}_{para_n} \end{array}

(A.53)

{\dot{N}}_{t} = D_{total} \cdot {Rhythm}_{respiratory}

(A.54)

Autonomic Control of Sino-Atrial Node (Ursino, 1998)

HP = Δ {HP}_{bs} + Δ {HP}_{p} + {HP}_{basal}

(A.55)

(a) Beta-Sympathetic Control

σ_{bs} (t) = {\begin{array}{l} G_{bs} \cdot G_{bs} (SI) \cdot ln [f_{tbs} (t - D_{bs}) - f_{tbs min} + 1], & f_{tbs} \geq f_{tbs min} \\ 0, & f_{tbs} < f_{tbs min} \end{array}

(A.56)

G_{bs} (SI) = 1 - SI \cdot (1 - AI) \cdot G_{bs_sleep}

(A.57)

\frac{d}{dt} Δ {HP}_{bs} = \frac{1}{τ_{bs}} (- Δ {HP}_{bs} + σ_{bs})

(A.58)

(b) Parasympathetic Control

σ_{ps} (t) = \frac{G_{ps}}{G_{ps} (SI)} \cdot f_{tp} (t - D_{ps})

(A.59)

\frac{d}{dt} Δ {HP}_{ps} = \frac{1}{τ_{ps}} (- Δ {HP}_{ps} + σ_{ps})

(A.60)

G_{ps} (SI) = 1 - SI \cdot (1 - AI) \cdot G_{para_sleep}

(A.61)

Alpha-Sympathetic Control of Peripheral Resistance (Ursino, 1998)

Z_{j} = {\begin{array}{l} G_{j} \cdot G_{as} (SI) \cdot ln [f_{tas_i} (t - D_{j}) - f_{tas min} + 1], & f_{tas_i} \geq f_{tas min} \\ 0, & f_{tas_i} < f_{tas min} \end{array}

(A.62)

\frac{d Δ {TPR}_{j}}{dt} = \frac{1}{τ_{j}} (- Δ {TPR}_{j} + Z_{j})

(A.63)

{TPR}_{j} (t) = Δ {TPR}_{j} + {TPR}_{j 0}

(A.64)

G_{as} (SI) = 1 - SI \cdot (1 - AI) \cdot G_{as_sleep}

(A.65)

Local Blood Flow Control of Peripheral Resistance Magosso and Ursino, 2001)

G_{bp} = 1 / R_{bp} = G_{bpn} \cdot f_{chemo_b}

(A.66)

f_{chemo_b} = 1 + y_{O 2_b} + y_{CO 2_b}

(A.67)

\frac{{dy}_{O 2_b}}{dt} = \frac{1}{τ_{O 2}} [- y_{O 2_b} - \frac{G_{O 2_b}}{G_{a s} (SI)} (C_{vO 2_b} \cdot - C_{vO 2 n_b})]

(A.68)

\frac{{dy}_{CO 2_b}}{dt} = \frac{1}{τ_{CO 2}} (- y_{CO 2_b} + \frac{1}{G_{as} (SI)} \cdot \frac{A + \frac{B}{1 + {Ce}^{D {log}_{10} (P_{{aCO}_{2}_b})}}}{A + \frac{B}{1 + {Ce}^{D {log}_{10} (P_{{aCO}_{2} n_b})}}} - 1)

(A.69)

R_{hp,mp} = R_{hpn,mpn} \cdot f_{chemo_h, m}

(A.70)

f_{chemo_h, m} = (1 + y_{CO 2_h, m}) / (1 + y_{O 2_h . m})

(A.71)

\frac{{dy}_{O 2_h, m}}{dt} = \frac{1}{τ_{O 2}} [- y_{O 2_h, m} - \frac{G_{O 2_h, m}}{G_{as} (SI)} (C_{vO 2_h, m} - C_{vO 2 n_h, m})]

(A.72)

\frac{{dy}_{C O 2_h, m}}{dt} = \frac{1}{τ_{CO 2}} [- y_{CO 2_h, m} + G_{as} (SI) \cdot \frac{1 - e^{\frac{P_{aCO 2_h, m} - P_{aCO 2 n_h, m}}{k_{co 2_h, m}}}}{1 + e^{\frac{P_{aCO 2_h, m} - P_{aCO 2 n_h, m}}{k_{co 2_h, m}}}}]

(A.73)

Sleep Mechanism (Achermann and Borbely, 2003; Khoo, 1990)

Process C

\begin{array}{l} C_{H / L} = A [0.97 sin θ + 0.22 sin (2 θ) + 0.07 sin (3 θ) + 0.03 sin (4 θ) + 0.001 sin (5 θ)] + X \\ where A = 0.12, X = X_{H} = 0.9 for C_{H}, X = X_{L} = 0.15 for C_{L} . \end{array}

(A.74)

Process S

Awake State (S \geq C_{L}) : S (t) = 1 - [1 - S (t - Δ t)] e^{(- 0.055 Δ t / 3600)}

(A.75)

Sleep State (S \leq C_{H}) : \frac{dS}{dt} = - α_{gc} \cdot SWA

(A.76)

\frac{dSWA}{dt} = α_{rc} \cdot SWA (1 - \frac{SWA}{S}) - α_{fc} \cdot SWA \cdot REMT (t) + SWA \cdot n (t)

(A.77)

REMT (t) = REM + 0.2 \cdot AI

(A.78)

REM = {\begin{array}{r} 1 & REM \\ 0 & NREM \end{array}

(A.79)

AI = {\begin{matrix} 1, \\ 0, \end{matrix} \begin{matrix} D_{Total} > Thre \\ Otherwise \end{matrix}, where Thre = I_{vent} (0.7 + 0.3 S_{SWA})

(A.80)

S_{SWA} = 1.2 \cdot \frac{SWA}{S}

(A.81)

D_{state} = SI = {\begin{array}{l} S_{SWA}, \\ S_{SWAcombined}, \\ 0, \end{array} \begin{array}{r} sleep onset transition \\ sleep \\ awake \end{array}

(A.82)

S_{SWAcombined} = {\begin{array}{l} 1, & AI = 0 \\ 0, & AI = 1 \end{array}

(A.83)

Footnotes

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PERMALINK

An integrative model of respiratory and cardiovascular control in sleep-disordered breathing

Limei Cheng

Olga Ivanova

Hsing-Hua Fan

Michael C K Khoo

Abstract

1. Introduction

2. Methods

Fig. 1.

Table 1.

2.1. Respiratory subsystem and its central neural control subsystem (see Appendix Eqs. A.1–A.40)

2.2. Cardiovascular subsystem and its central neural control subsystem (Appendix Eqs. A.41–A.73)

2.3. Sleep Mechanism (Appendix Eqs. A.74–A.83)

2.4. Software Implementation

2.5. Model parameters and Verification Issues

3. Results: model simulations

3.1. Simulation of cardiorespiratory control during sleep onset in normals

Fig. 2.

3.2. Valsalva maneuver

Fig. 3.

3.3. Effects of changes in chemoreflex stimulation

Fig. 4.

Fig. 5.

3.4. Obstructive Sleep Apnoea

Fig. 6.

Fig. 7.

Fig. 8.

3.5. Hypoxia induced periodic breathing during sleep

3.6. Cheyne-Stokes respiration in Chronic Heart Failure: Central Sleep Apnoea

Fig. 9.

Fig. 10.

3.7. Simulation of overnight sleep – Normal Control vs. Obstructive Sleep Apnoea

Fig. 11.

4. Discussion

4.1. Incorporation of circadian and sleep effects

4.2. Sensitivity Analyses

(a) Interaction between key parameters contributing to severity of obstructive sleep apnoea effects

Table 2.

(b) Interaction between key parameters contributing to CSR effects

Table 3.

(c) Interaction between baroreflex and chemoreflex inputs to the ANS

Fig. 13.

4.3. Model limitations

Fig. 12.

Acknowledgements

Appendix : Model Equations

Respiratory Gas Exchange and Transport

(a) Gas transport through the dead space(Khoo, 1990)

Inspiration

Expiration

(b) Alveolar Gas Exchange (Khoo, 1990)

Inspiration

Expiration

(c) Cardiovascular Mixing (Heart and Vasculature) (Lange et al, 1966)

(d) Blood-Gas Dissociation Curves (Spencer et al, 1979)

CO2

O2

(e) Brain Compartment CO2 exchange (Read and Leigh, 1967)

(f) Cerebral Blood Flow (Khoo, 1990)

(g) Gas Exchange in the Body Tissues Compartments (Khoo, 1990)

Ventilatory Controller (Khoo, 1990)

Respiratory Mechanics (Riddle and Younes, 1981)

Upper airway mechanics

Autonomic Inputs

(a) Carotid Baroreceptors (Ursino, 1998)

(b) Chemoreceptors (Magosso and Ursino, 2001)

(c) Lung Stretch Receptors (Magosso and Ursino, 2001)

(d) Offsets (Magosso and Ursino, 2001)

Autonomic Control: Integration of autonomic inputs Magosso and Ursino, 2001)

(a) Alpha-Sympathetic Activity

(b) Beta-Sympathetic Activity

(c) Parasympathetic Activity

Autonomic Control of Sino-Atrial Node (Ursino, 1998)

(a) Beta-Sympathetic Control

(b) Parasympathetic Control

Alpha-Sympathetic Control of Peripheral Resistance (Ursino, 1998)

Local Blood Flow Control of Peripheral Resistance Magosso and Ursino, 2001)

Sleep Mechanism (Achermann and Borbely, 2003; Khoo, 1990)

Process C

CO₂

O₂

(e) Brain Compartment CO₂ exchange (Read and Leigh, 1967)