Chapter 7Experimental Studies of Respiration and Apnea

Eugene N. Bruce

Abstract The use of physiologically-based computational models of chemoreflexcontrol of ventilation has provided general insights into the roles of specificmechanisms in the genesis of periodic breathing and apneas. Our early studiesutilized formal mathematical approaches to simplify complex models of this typeso that their behaviors could more easily be predicted from various combinations ofphysiological and environmental parameters. Because it is difficult to apply suchmodels to individual patients, we subsequently pursued a “black-box” approachin which the objective was to characterize the dynamic properties of the systemfor individual subjects, then relate these properties to physiological and environ-mental parameters. By stimulating ventilation through pseudorandom variationsin inspired CO2 (or O2) level, we estimated input–output models, both open-loop(i.e., from end-tidal PCO2 to ventilation) and closed-loop (i.e., from inspired CO2

to ventilation). We found that the dynamic properties of the resulting models differbetween normal subjects and both sleep apnea patients and heart failure patients.We also demonstrated in normal subjects that the closed-loop model does notchange between wakefulness and quiet sleep, even though the gain of the open-loop (or controller) model decreases. To explore the mechanistic basis for thesefindings using a detailed, physiologically-based, chemoreflex model, we enhancedthe typical model of this type by improving the representation of O2 transportand distribution beyond the usual, single lumped-compartment, approach. In ournew model, brain and muscle tissue each comprise two subcompartments withintercompartmental diffusion and arterio-venous shunting, as well as O2 bindingto myoglobin in muscle. We use this model to predict changes in brain tissue PO2

during sleep apnea. Chapter 8 provides another approach to respiratory controlsystem modeling while Chap. 6 discusses the role of transport delay in respiratorycontrol.

E.N. Bruce (�)Center for Biomedical Engineering, University of Kentucky, Lexington, KY, USAe-mail: Eugene.Bruce@uky.edu

J.J. Batzel et al. (eds.), Mathematical Modeling and Validation in Physiology,Lecture Notes in Mathematics 2064, DOI 10.1007/978-3-642-32882-4 7,© Springer-Verlag Berlin Heidelberg 2013

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7.1 Introduction

The use of physiologically-based computational models of chemoreflex controlof breathing has provided general insights into the roles of specific mechanismsinvolved in the feedback control of ventilation in sleep apnea and periodic breathing(see also Chap. 8). These models are based on the principle of conservation ofmass, usually applied to oxygen and carbon dioxide. Typically the models includelumped, uniform compartments representing the lungs, arterial blood, various largetissues (e.g., skeletal muscles, brain), and venous blood. In addition, these modelsusually incorporate equations to predict the chemoreflex control of ventilation ( PV )and cardiac output (Q) as functions of partial pressures of CO2 and O2 in brainor arterial blood compartments . Periodic breathing often is viewed as a form ofoscillation, and the models are examined for conditions which alter the stability ofthe system. On this basis, loop gain and time delays have received considerableemphasis [2, 8, 10, 20]. Because ventilatory control is subjected to frequent randomdisturbances, the potential contributions of transient “instabilities” (e.g., dampedoscillations leading to stochastic resonances) have also been noted [14].

To model sleep apnea, it is necessary to distinguish between obstructive andcentral apneas and to include mechanisms for both. Thus, the usual approach isto represent the chemoreflex control of patency of the upper airway as a separatefeedback pathway, with actual ventilation being dependent on both the chemoreflex-specified (or desired) ventilation and the degree of airway patency [13]. Obstructiveapnea occurs when the airway patency falls to zero while desired ventilation is stillnonzero.

In the cases of both periodic breathing and sleep apnea, these modeling studieshave motivated general concepts which are difficult to validate when applied toindividual patients. The major difficulty is the large number of parameters involvedin such models. It is not possible to measure all of them in one patient, so themodel might include a few which are measured and many which are typical ofnormal, healthy subjects. The quantitative differences between the model and datafrom individual patients are often too large to permit conclusions about specificmechanisms.

Other studies utilized formal mathematical approaches to simplify complexmodels of this type so that their behaviors could more easily be predicted forvarious combinations of physiological and environmental parameters [4, 9]. Theseapproaches improved the ability to anticipate the general consequences of alteringmodel parameters (such as changing the inspired gas) on the occurrence of periodicbreathing or apnea; however, even when the model was simplified to one having,for example, six parameters, it was not possible to measure these parameter valuesin individual patients. Furthermore, these parameters of reduced models representbroad conceptual relationships between variables rather than basic physiologicalproperties and the design of experimental protocols to estimate them in humansubjects is challenging.

7 Experimental Studies of Respiration and Apnea 123

FICO2

~ ~VE

Fig. 7.1 General scheme for estimating a “black-box” model for control of ventilation: Applyvariations in inspired CO2 (FI QCO2) and measure resulting ventilatory fluctuations ( QVE). The inter-nal mechanisms are not detected (such as the role of arterial carbon dioxide partial pressure PaCO2

7.2 “Black-box” Models of Chemoreflex Controlof Ventilation

The black-box approach identifies properties of the ventilatory control system inindividual subjects by applying a known stimulus and measuring the ventilatoryresponse. This method had been utilized for many years by investigators whoapplied a stepwise change in the composition of the inspired gas. The method wasmost effective when used with a protocol for forcing a stepwise change in end-tidalgas pressure (rather than inspired gas pressure) developed by Swanson [19] andSlessarov [18] and refined by Poulin and Robbins [17]. However, given the well-known problems with estimating parameters based on persistently exciting stimuli[12, Chap. 3], we utilized a pseudorandom stimulus for the same purpose [15].Figure 7.1 indicates the application of this method for determining the ventilatoryresponse to CO2 stimulation. The light gray lines represent feedback control ofventilation by CO2 in its most abstract form, and the heavy black lines indicate thatthis entire system comprises the contents of the “black box”. A computer monitorsthe subject’s breathing in real time and varies the inspired CO2 concentrationaccording to a pseudorandom binary sequence (indicated by the tilde) [15]. Thevariations in ventilation (also indicated by a tilde) are related to the stimulus usinga Box-Jenkins form of model [12], as shown in Fig. 7.2. The general input–outputrelation for this model is:

y.t/ D B.q/

F.q/u.t � k/ C C.q/

D.q/e.t/; (7.1)

where q is the unit delay operator, t has units of discrete time, and e.t/ isuncorrelated random noise. B.q/; C.q/; D.q/ and F.q/ are polynomials in q whosecoefficients are to be estimated. From continuous measurements of airflow andairway CO2 level (Fig. 7.3) one derives ventilation (eV E .t/) and end-tidal PCO2

(eP CO2 ) for each breath. (One may convert the discrete time interval “breath number”

124 E.N. Bruce

~VE

~PICO2

Delay Transfer Function

u(t)input

y(t)output

e(t)white noise

Noise Model

C(q)

D(q)

B(q)

F(q)qk +

Fig. 7.2 Box-Jenkins model structure used to estimate parameters of the black-box model ofventilatory control. PI QCO2 denotes applied variations of inspired CO2 pressure; QVE denotesvariations of ventilation from its pre-stimulus mean level

Fig. 7.3 Examples of data signals during pseudo-random CO2 stimulation (awake human subject)

into sampled “continuous time” data, if desired.) The polynomial coefficients areestimated using an iterative, least squares method [12]. Polynomial orders arechosen on the basis of an AIC measure, and the same orders are used for data whichare to be compared (e.g., before and after an intervention). In order to comparethe models in different situations, we evaluate the unit-pulse response of the model(i.e., the response to a single pulse of u.t//.

We applied this approach to the question of why periodic breathing occursmore frequently during sleep than during wakefulness. The usual explanationshave assumed that the loop gain of the ventilatory controller increases in sleep,leading to instability. Our analysis of closed-loop ventilatory response to a pulseof CO2 in hyperoxia found no increase in oscillatory behavior during NREM sleepin young subjects, on average [16], although some individual subjects showed aslight increase (Fig. 7.4). Also, there may be a decrease in closed-loop stabilitywhen peripheral chemoreceptors are not depressed by hyperoxia (Chap. 1). Thus,the mechanisms by which ventilation becomes unstable near sleep onset probablyinvolve fluctuations in state of arousal rather than state-dependent changes incontroller properties [13]. To the contrary, elderly subjects typically exhibited achange from a non-oscillatory unit-pulse response awake to a damped oscillationin sleep (Fig. 7.4). Thus, we conclude that the ventilatory control system is lessstable in NREM sleep in the elderly. In other comparisons we found that, comparedto normal subjects, the ventilatory response is more unstable in COPD patients [11]and congestive heart failure patients [unpublished observations].

7 Experimental Studies of Respiration and Apnea 125

0

0.03

0.06

0

0.1

0.2awakeasleep

awakeasleep

YOUNG SUBJECTELDERLY SUBJECT

0 100 200 300 0 100 200 300Time (sec) Time (sec)

Ven

tilat

ion

(L/m

in)

Fig. 7.4 Calculated unit-pulse responses of black-box models of ventilatory control for an elderlysubject and a young subject awake (solid) and during NREM sleep (dashed)

7.3 A Model for Predicting Brain Tissue Oxygen Tensions

It is often assumed that the most deleterious aspect of sleep apnea is the impaireddelivery of oxygen to the brain [1, 6]. The mechanisms which contribute to centraland obstructive apneas, and the physiological responses which ensue and affect gastransport, are numerous and highly complex. Even models of ventilatory controlbased on simple principles can become very complicated [5]. Because of thedifficulty of producing central and obstructive apneas in a model of ventilatorycontrol in a manner that would mimic an actual patient, we proposed to developa model of oxygen delivery to brain tissue that would be driven by a patient’sactual ventilation (Fig. 7.5). Because the typical single, lumped, compartmentrepresentation of skeletal muscle tissue is inadequate for evaluating the range of PO2

values that occur in a tissue, recently we developed a model of skeletal muscle tissuehaving two tissue and three vascular subcompartments [3]. Subsequently, we alsodeveloped a similar model for brain tissue; however, because gray matter and whitematter have such different levels of metabolism and perfusion, they were studiedseparately. The basic structure of the brain tissue model is shown in Fig. 7.6 and themodel equations are presented in the appendix.

To evaluate the model, its resting values were compared to values from theliterature (Table 7.1) for white matter (WM), gray matter (GM), and whole brain(GCW). Of particular importance, the predicted oxygen tensions in the two braintissue subcompartments lie in the lower half and upper half of the ranges of tissuePO2 found in the literature. Therefore, it was concluded that the model should be

126 E.N. Bruce

Fig. 7.5 Schematic of a model for predicting tissue oxygen levels (PO2 ) in the brain using asubject’s measured alveolar ventilation ( PVA). Q, QCBF, QM, QNM denote cardiac output, andblood flows to brain (CBF), muscle (M), and nonmuscle (NM) compartments respectively. QSF

denotes pulmonary shunt blood flow. Vart, Vren represent volumes of arterial and mixed venousblood compartments respectively. FI QCO2 and FI QO2 denote fractions of CO2 and O2 in inspiredgas respectively. MRbO2, MRnO2, and MRnmO2 denote metabolic rates in the brain tissue, muscletissue and non-muscle tissue, respectively

useful for predicting brain tissue oxygen tensions. Steady states were simulated forvarious constant ventilation levels to determine the effect of reducing arterial PO2 byhypoventilation (Fig. 7.7) during Waking and NREM sleep. The lower values of bothbrain and muscle PO2 are relatively resistant to arterial hypoxia for PaO2 > 45 Torr;however, in the model muscle PO2 drop in NREM sleep whereas brain tissue PO2

are preserved. Therefore, the decrease in ventilation accompanying NREM sleepalters brain PO2 through its effect on PaO2 , but there is no additional effect from theother factors that differ between waking and sleep. Subsequently we examined theeffect on brain PO2 of an apnea lasting one minute and of a 2 min hypoventilationfollowed by transient hyperventilation (Fig. 7.8). Apnea produces a rapid fall inbrain PO2 in both tissue subcompartments. PO2 during apnea falls below 10 Torr, alevel which may reflect even lower tensions and impairment to aerobic metabolismat some locations in the tissue. There is a tendency for the oxygen tensions of thetwo subcompartments to equalize.

Finally, we extracted the rib cage and abdominal signals from a polysomnogramof an elderly female subject (Fig. 7.9). Relative gain of these signals was determined

7 Experimental Studies of Respiration and Apnea 127

Diffusion

Gradient-driven flux

QCBF

PaO2HbO2

Subcompartment 1

Subcompartment 2

Ptb1 O2

Ptb2 O2

Mb1O2

Mb2 O2

Pbv O2HbO 2,bv

Pbv1 O2P bv2 O2

Pb1O2

Pb2O2

P b3 O2

Fig. 7.6 Structure of model of brain tissue, comprising two tissue sub-compartments and threevascular sub-compartments. PbkO2 denotes oxygen partial pressure in vascular sub-compartmentk; PbvkO2 denotes oxygen partial pressure at the distal (venous) end of vascular sub-compartment k.PbvO2 and HbO2;bv denote oxygen partial pressure and oxyhemoglobin content in venous outflowfrom the brain tissue vascular bed, respectively. PtbkO2 and MbkO2 denote oxygen partial pressureand metabolic rate in tissue sub-compartment k, respectively. QCBF denotes brain blood flow

Table 7.1 Measured (from various studies in the literature) and model predicted values of oxygenrelated variables in gray (GM) and white (WM) matter and for the whole brain (G C M). CBFdenotes cerebral blood flow and MRO2 denotes metabolic rates. PsagO2 and HbO2;sag denoteoxygen partial pressure and oxyhemoglobin saturation measured in sagittal sinus respectively. PtO2

denotes various brain tissue partial pressures of oxygen. PcapO2 and PvenO2 denote oxygen partialpressure measured near brain capillaries and venules, respectively. PaO2 represents arterial partialpressure of oxygen. For other symbols, see text in appendix

Parameter Units Measured Model

MRO2 (GM) ml O2/min/100 mg 5.7, 5.9 5.0CBF (GM) ml/min/100 mg 61, 65.3, 69, 62.0, 66.5 75MRO2 (WM) ml O2/min/100 mg 1.8, 1.43CBF (WM) ml/min/100 mg 19.0, 21.4, 22.2MRO2 (G C W) ml O2/min/100 mg 3.65, 3.2CBF (G C W) ml/min/100 mg 55, 53.5Pt O2 (GM) Torr 5–15, 42, 7–42 (mean 23) 18, 32PcapO2 (GM) Torr 37.0Pt O2 (G C W) Torr 10–40, 27–47, 12–48PvenO2 (G C W) Torr 37.9–40.9PvenO2 (GM) Torr 31.5PsagO2 (G C W) Torr 43.5, 44–60PsagO2 (GM) Torr y 31.6% HbO2;sag (G C W) Torr 68, 71.5% HbO2;sag (GM) Torr 63.4

128 E.N. Bruce

Brain Muscle

0

5

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35

40

0 20 40 60 80 100 1200

5

10

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0 20 40 60 80 100 120

PaO2 (Torr) PaO2 (Torr)

PO2 (Torr)PO2 (Torr)

Subcompartment 1Subcompartment 2

KEY Wake NREM

Fig. 7.7 Model predicted steady-state PO2 values in brain and skeletal muscle tissues awakeand in NREM sleep at various arterial oxygen partial pressures .PaO2 / levels resulting fromhypoventilation

0 5 10 15 20 255

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100

%H

bO2

VA.

(LPM)

Time (min)

Bra

in T

issu

e P

O2

(T

orr)

Fig. 7.8 Model predicted responses to an apnea lasting one minute and a subsequent 2 minhypopnea followed by a transient hyperventilation

by their excursions during an obstructive apnea, and amplitude scaling was eval-uated by matching the HbO2 predicted by the model when it was driven by thissubject’s ventilation signal during a period of uniform breathing. The predicted PO2

in the two brain tissue compartments are also shown in Fig. 7.9. During the first

7 Experimental Studies of Respiration and Apnea 129

0

10

20

30

alv

eola

r v

entil

atio

n(L

PM

)

80

85

90

95%

HbO

2B

rain

tis

sue

PO

2 (T

orr)

T ime (min)

Fig. 7.9 Model predicted brain tissue PO2 in response to two episodes of periodic breathing in anelderly female subject. Ventilation (Vdota) PVA was calculated from an actual polysomnogram asexplained in the text

episode of periodic breathing brain PO2 actually rise because of the hyperventilatoryperiods, whereas during the second episode these PO2 fall progressively but stillremain well above 10 Torr. The model predicts that brain tissue PO2 will be highlycorrelated with % HbO2.

It would be unwise to extrapolate from one subject but this simulation raisesthe question whether profound swings in ventilation during sleep will affectbrain tissue PO2 as profoundly. On the other hand, the predicted fluctuations in% HbO2 are more attenuated than the measured changes in oxygen saturation inthe polysomnogram, suggesting that predictions from the model about brain PO2

might be underestimating the actual changes.

7.4 Conclusions

Although integrative models of chemoreflex control of ventilation predict thatperiodic breathing is related to factors which alter stability of feedback loops thatcontrol ventilatory demand and upper airway patency, it is not possible to test theseconcepts directly in individual patients. Indirect approaches using black-box modelsapplied to individual subjects generally support these concepts; however, these latterapproaches do not support the idea that ventilatory control is less stable in sleep thanduring wakefulness. Thus, the mechanisms by which ventilation becomes unstable

130 E.N. Bruce

near sleep onset probably involve fluctuations in state rather than state-dependentchanges in physiological properties [7, 13]. The deleterious effects of longstandingsleep-disordered breathing include cognitive dysfunction [1, 6] which may be dueto episodic severe brain hypoxia. We propose to utilize a model of oxygen deliveryto the brain which is driven by the patient’s own ventilation in order to estimate thedegree of brain hypoxia experienced by the patient. Such a model will need to be“tuned” to each patient so that its dynamic behaviors represent the patient ratherthan merely mimicking average population behaviors.

Appendix

Conservation of mass is applied in each subcompartment for oxygen in thesame manner as described in previous Chaps. 5 and 6, the difference being thatoxygen can diffuse between the two tissue subcompartments via a flux driven bythe concentration gradient. Thus, the typical mass balance equation for a tissuesubcompartment has the form

dCvmk;g.t/

dtD Fluxmk;g.t/

VmkC Dmk;g.Cmk;g.t/ � Cmk0 ;g.t//

Dx;

whereFluxmk;g.t/ D ŒDbmk;g.Ptck;g.t/ � Pmk;g.t//� � MRmk;g

and g refers to either O2 or CO2, k represents subcompartment k (1 or 2), Vmk is thevolume of that subcompartment, Cvmk;g and Pmk;g are the concentration and partialpressure of gas g in subcompartment k; Dmk;g is the intratissue diffusion coefficientfor gas g. Dbmk;g is the blood-tissue conductance for gas g: Cmk;g is the dissolvedgas concentration, and MRmk;g is the metabolic rate of consumption .MR > 0/ orproduction .MR < 0/ of gas g in subcompartment k.

Ptck;g is the effective partial pressure of gas g in the vascular compartment ofsubcompartment k driving the flux of g from the blood to the tissue. For O2, thiseffective pressure is found by first determining the average HbO2 concentrationin the vascular compartment, then evaluating the corresponding pressure from theoxygen dissociation curve using an iterative method. The within-tissue diffusioncoefficient for O2 is set to 10 times the standard diffusivity [3]. Dbmk;O2 is theproduct of PSm, solubility, and tissue mass for both compartments. PSm is thepermeability-surface area for muscle, which was set to 37.5. For exchange of O2

between subcompartment 1 and the venular blood compartment, all blood-tissueconductances were set to 7.5 % of their values for subcompartment 1.

In vascular compartments O2 exists in dissolved and combined forms so that incompartment i , the total O2 concentration is CiO2 D HbO2 C SO2Pi O2. Otherphysiological factors are taken into account: .i/ control of Q by changes in arterialPO2 ; .ii/ control of brain blood flow .QCBF/ by changes in arterial PO2 and PCO2

(the latter estimated from changes in ventilation by assuming that metabolic rate isunchanged); .iii/ changes in muscle blood flow due to changes in arterial PO2 ; .iv/

7 Experimental Studies of Respiration and Apnea 131

effect of sleep state on brain and muscle tissue metabolic rates and perfusion. Thefirst two factors are discussed by Topor [20]; the third is discussed in [3], and theeffects of sleep state are represented as follows: whole body metabolic rate decreasesby 10 % in NREM sleep; both Q and QCBF are determined as functions of arterialpartial pressure of oxygen and carbon dioxide PaO2 and PaCO2 respectively, and bothdecrease by 10 % in NREM (relative to awake values). Changes in either blood flowoccur with a time constant of 2 min.

Acknowledgements This work was supported by grants OH008651 from NIOSH and NS050289from NIH.

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