Quantitative General Theory for Periodic Breathing inChronic Heart Failure and its Clinical Implications

Darrel P. Francis, MA, MRCP; Keith Willson, MSc, MIPEM; L. Ceri Davies, BSc, MRCP;Andrew J.S. Coats, DM, FRCP; Massimo Piepoli, MD, PhD

Background—In patients with chronic heart failure (CHF), periodic breathing (PB) predicts poor prognosis. Clinicalstudies have identified numerous risk factors for PB (which also includes Cheyne-Stokes respiration). Computersimulations have shown that oscillations can arise from delayed negative feedback. However, no simple general theoryquantitatively explains PB and its mechanisms of treatment using widely-understood clinical concepts. Therefore, weintroduce a new approach to the quantitative analysis of the dynamic physiology governing cardiorespiratory stabilityin CHF.

Methods and Results—An algebraic formula was derived (presented as a simple 2D plot), enabling prediction from easilyacquired clinical data to determine whether respiration will be unstable. Clinical validation was performed in 20 patientswith CHF (10 with PB and 10 without) and 10 healthy normal subjects. Measurements, including chemoreflexsensitivity (S) and delay (d), alveolar volume (VL), and end-tidal CO2 fraction (C# ), were applied to the stability formula.The breathing pattern was correctly predicted in 28 of the 30 subjects. The principal combined parameter (C# S)3(d/VL)was higher in patients with PB (14.263.0) than in those without PB (3.160.5; P50.0005) or in normal controls(2.460.5;P50.0003). This was because of differences in both chemoreflex sensitivity (17496235 versus 6206103 and5266104 L/min per atm CO2; P50.0001 andP,0.0001, respectively) and chemoreflex delay (0.5360.06 vs 0.4060.06and 0.3060.04 min;P5NS andP50.02).

Conclusion—This analytical approach identifies the physiological abnormalities that are important in the genesis of PB andexplicitly defines the region of predicted instability. The clinical data identify chemoreflex gain and delay time (ratherthan hyperventilation or hypocapnia) as causes of PB.(Circulation. 2000;102:2214–2221.)

Key Words: heart failuren ventilationn physiology

I n congestive heart failure (CHF), periodic breathing (PB)in the daytime1,2 or during sleep3 results in elevated

mortality, which may result from the numerous repetitivefluctuations in blood gases, blood pressure, and heart rate.4

These repeated insults may induce chronic sympathetic over-activation, impede exercise capacity,5 and precipitate ventric-ular arrhythmias.6

It has long been speculated that PB may arise frompathological feedback in ventilatory control.7–10Clinical stud-ies have identified several possibilities in patients withPB,11,12 particularly hyperventilation or hypocapnia.6,13–16

Prolonged circulation delay8,17 and increased chemoreceptorsensitivity18 have also been implicated. The role of circulationdelay is controversial, partly because of animal work19 thathad to prolong it to a biologically implausible 2 to 5 minutesto engender PB. Aside from the clinical approach, there are 2conceptually different mathematical approaches.

Computer simulations, including an ingenious analog elec-trical circuit7 and numerical iterative models in digital com-puters,9,20,21 have shown oscillations arising with certain

configurations of system physiology. The drawback is that animmediate overview of system behavior across a variety ofphysiological and pathological states is not gained becausecomputerized resimulation is required for each proposedstate. As the number of variables increases, the impact ofchanges in starting conditions becomes increasingly time-consuming to tabulate and difficult to conceptualize.

The alternative approach is to solve analytically the dy-namics of respiratory control in PB. The attraction of thisapproach is that it should make obvious the range of physi-ological states that result in PB and predict and explain themechanisms of effective treatments, while removing the needfor computer recalculation for each possible combination ofclinical variables. Models have been developed using thefrequency domain22 or by seeking critical values of circula-tion delay,23 but their complexity has limited widespreadappreciation. An important step towards a directly clinicallyapplicable model was taken by Mackey and Glass,24 but closeexamination reveals that the stability criterion proposed iscorrect only when ventilation and cardiac output are zero.

Received February 14, 2000; revision received June 6, 2000; accepted June 14, 2000.From Royal Brompton Hospital, (D.P.F., K.W., L.C.D., A.J.S.C.) and the National Heart and Lung Institute (D.P.F., L.C.D., A.J.S.C., M.P.), London,

UK, and Piacenza Hospital, Italy (M.P.).Correspondence to D.P. Francis, Heart Failure Unit, Royal Brompton Hospital, Sydney St, London SW36NP, UK. E-mail d.francis@cheerful.com© 2000 American Heart Association, Inc.

Circulation is available at http://www.circulationaha.org

2214

The lack of an analytical unified general theory of PB thatis both mathematically explicit and comprehensible to clini-cians has hindered a deeper understanding of the physiolog-ical processes; this understanding would enable rationalinterventions to be planned. Currently, the underlying delay-differential equations are difficult to solve analytically usingthe techniques that have hitherto been applied.

We aimed to solve these fundamental equations governingcardiorespiratory stability by applying a new technique thatwould give a rigorous quantitative basis for understanding thepathophysiology of PB in CHF. This would give rise not onlyto testable predictions of cardiorespiratory stability fromprimary clinical data, but also provide a framework forunderstanding the effect on stability of changes in physiolog-ical variables, such as those from therapy.

Methods

Analytical ModelVentilation is regulated by delayed negative feedback. The chemo-sensors are located in the aorta, the carotids, and the brainstem and,therefore, they receive information on blood gas status some timeafter a change occurs in the lung; a further neurological delay mayoccur before ventilation changes. The purpose of the control systemis to regulate blood gases and damp away any small randomdisturbances. However, the delay before response provides thepotential for a vicious circle of increasing stimuli and responses (ofalternating signs) to develop; this is PB. The analytical approachseeks to identify explicitly the conditions that cause the oscillationsto grow.

What determines whether PB will occur is control system behaviornear the steady state. Physiological fluctuations in this region can bedescribed with linear mathematics. Carbon dioxide (CO2) andoxygen oscillate during PB in antiphase.25 Therefore, we representthe state of the blood gases by a single variable. It is useful todevelop a variable c, which represents the displacement of thealveolar gas stores (nominally CO2 fraction) away from its meanvalue, C# (Table 1). Likewise v represents the displacement ofalveolar ventilation from its mean value V˙

A. Because of the timedelay, the value of v at time t (vt) depends on the value of c at sometime d previously (ct2d). Near the steady state, therefore, vt5S3ct2d,where S represents chemoreflex gain (additional ventilation per unitincrease in c). Fluxes of CO2 into and out of the lung arise frommetabolism, ventilation, and exchange with blood stores. Metabolicproduction of CO2 by the body is expressed as V˙

A C# . The rate of CO2removal from the lung by ventilation is expressed as (V˙

A1v)(C# 1c).Oscillations in arterial CO2 necessitate a net transfer of CO2 from thelung into extrapulmonary stores (in comparison with the steady state) ata rate ofbQ̇c, whereb indicates marginal solubility of CO2 in blood and

Q̇ , cardiac output, assuming that pulmonary venous CO2 is stable. Therate of increase of lung CO2 stores is VL(dc/dt), where VL indicatesalveolar volume and dc/dt, the rate of change of alveolar CO2 fraction.Thus, we obtain the following equation.

(1) VL

dc

dt5V̇AC# 2@~V̇A1v!( #C1c!]2bQ̇c

System stability depends on behavior with small displacements c andv, during which the product term vc is negligible. This results in thefollowing equation.

(2) VL

dc

dt1c~V̇A1bQ̇!1vC# 50

After a small disturbance, the system may relax to its steady statewithout oscillating or it may show oscillations that either decay or grow.All these behaviors can be described by defining a complex number,r5g1jw, where g and w are the real and imaginary parts, respectively,and addressing the exponential ert5egt1jvt5egt3ejvt5egt(cosvt1jsinvt).The (cosvt1jsinvt) factor represents oscillation with period 2p/v, andegt represents the changing amplitude of the periodic waveform. If g.0,any disturbance will lead to oscillations that grow until their sizebecomes limited by nonlinearities in the system; if g,0, the oscillationsdecay (ie, breathing will stabilize after any transient disturbance).

To solve equation 2, we make 3 substitutions. First, we replace cby ert. Second, dc/dt5rert5rc. Third, v5(S3ct2d), and the value of cat a timed previously (ct2d) is e2rdc, so v5S3e2rd3c. This givesequation 3.

(3) VLr1~V̇A1bQ̇!1SC# e2rd50

Because this equation is the ultimate determinant of the stability ofrespiratory control, a general solution for r would give a means ofdirectly predicting PB from primary clinical data. Previous workusing conventional trigonometric and logarithmic functions has beenunsuccessful. In the Results section, we introduce a novel approachthat allows a full solution.

Clinical Measurement of Physiological ParametersWe performed clinical measurements of the physiological parametersV̇A, C# , S, d, and VA on 20 patients with CHF (10 who exhibited PBduring quiet rest in the daytime and 10 who did not) whose mean agewas 61612 years and on 10 age-matched normal controls with a meanage of 58615 years. Clinical characteristics of the patients are shown inTable 2. The presence of PB was determined from respiration record-ings, which were obtained by respiratory impedance plethysmographywith the subjects awake and semirecumbent on a couch by one blinded

TABLE 1. Variables

C# Steady-state alveolar CO2 fraction

c Displacement of alveolar CO2 fraction

V̇A Steady-state alveolar ventilation

v Displacement of alveolar ventilation

t Time

d Chemoreflex delay

S Chemoreflex sensitivity

b CO2 solubility

Q̇ Cardiac output

VL Alveolar volume

Any internally consistent units are valid.

TABLE 2. Patients’ Clinical Characteristics

Patients WithPeriodic Breathing

Patients WithStable Breathing

Age, y 57.864.2 66.968.3

Peak oxygen uptake,mL z kg21 z min21

16.161.9 17.861.0

LVEF, % 19.462.2 26.864.2

Cause, n

Ischaemic 2 3

Nonischaemic 8 7

Treatment, n

Diuretic 10 10

ACE inhibitor 6 7

All blocker 3 3

Digoxin 2 1

b-Blocker 1 2

Values are mean6SEM or No. of patients.

Francis et al Quantitative Theory for Periodic Breathing 2215

investigator (L.C.D.). Informed consent and ethical approval wereobtained.

Subjects sat and breathed through a calibrated heated pneumota-chograph into a metabolic cart that used a mass spectrometer(Innovision) to measure ventilation, CO2, and O2. Subjects under-went 2 tests. In the first, mean alveolar ventilation (V˙

A) and meanend-tidal CO2 fraction (C# ) were measured. They then rebreathedfrom a 6-L bag initially containing 100% oxygen until theend-tidal PCO2 reached 10 kPa or the test became uncomfortable.The gain (S) of the chemoreflex system was measured as the ratioof the rate of rise of alveolar ventilation to the rate of rise ofend-tidal PCO2 in L/min per atmosphere CO2. Effective lungvolume (VL) for ventilatory exchange was determined by fitting amonoexponential curve to the pattern of increase in end-tidal PO2

when the inspired gas was switched to oxygen.In the second test, a series of stimuli of 7% inspired CO2 were

applied over 12 to 15 minutes, and the resulting ventilation andend-tidal PCO2 sequences were resampled at 1 Hz. Chemoreflex timedelay (d) was determined through cross-correlation analysis as thelag giving the maximal correlation between them. We used values of0.05 L z L21 z kPa2126 (and atmospheric pressure of 100 kPa) forbCO2

and 0.06 Lz kg21 z min21 for resting Q̇.

Statistical AnalysisComparisons between the 3 groups are made with ANOVA.Within-subject paired comparisons are made with the pairedt test.P,0.05 is considered significant. Data are presented asmean6SE.

ResultsGeneral Solution of Analytical ModelEquation 3, on which hinges the stability of cardiorespiratorycontrol in CHF, can be solved by introducing Lambert’s Wfunction27 (defined by its inverse, W21:z3zez), as detailed inthe Appendix. The solution is as follows.

(4) r51

dFWS2

#CSd

VLe@~V̇A#

1bQ̇!d]/V LD2(V̇A1Q̇b)d

VLG

Although this may at first appear to be a complicatedexpression, it is constructed from 2 combination variables,C# S(d/VL) and [(V̇A1bQ̇)(g/VL)]. Inserting values for the 6physiological variables immediately yields an explicit de-scription of the response to any disturbance in respiration.

This response is described by the function ert, which isequal to (egt3ejvt), because r contains real and imaginary parts(r5g1jv). The rate of growth or decay of oscillations is eg,which depends on the underlying physiological variables inthe manner shown in Figure 1. The period of oscillation 2p/valso varies (as shown in Figure 2); however, for most patientswith PB, this period is between 2 and 2.5 times the chemore-flex delay.

Thus, if a patient has (V˙A1bQ̇)(d/VL)54.5, C# S(d/

VL)510.0, andd50.45 minutes, Figure 1 shows that the

Figure 1. Effect of (V̇A1bQ̇)(d/VL) and C# S(d/VL) on the growth factor of ventilatory oscillations. Growth factors .1 cause breathing tobecome periodic; values ,1 give steady breathing.

2216 Circulation October 31, 2000

response to a small disturbance is an oscillation that grows bya factor of 1.75 every 0.45 minutes, and Figure 2 shows thatthe period of oscillation will be 1.1 min. The underlyingmathematics, which do not to be performed once the 2 plotsare available, are done by equation 4, such thatr5(0.5612.7j)/d, so the growth factor is e0.56 (ie, 1.75 per0.45 minutes) and the period is (2p/2.7)30.45 (ie, 1.1 min).

Generality of SolutionThe solutions plotted in Figures 1 and 2 are completelygeneral: their axes are dimensionless, because all unitscancel. The intercept of the stability plot on the C# S(d/VL) axis(which at first may appear to be arbitrary) isp/2, and it, thewhole boundary of stability, and indeed the whole shape ofthe 3D surfaces are independent of all physical constants. Theplots will not require recalculation for changes in any variableand are thus applicable to a wide range of situations. Copiesare available from the authors.

Clinical ResultsThe observed values of C# S(d/VL) and (V̇A1bQ̇)(d/VL) areplotted in Figure 3. The boundary between oscillatory andsteady breathing, predicted by equation 4, is also shown. PBwas correctly predicted from primary clinical measurementsin 10 of 10 patients with PB, and steady breathing wascorrectly predicted in 9 of the remaining 10 patients and in 9of the 10 normal controls.

Pathophysiological Components Contributingto PBThe principal combined parameter C# S(d/VL) was muchhigher in patients with PB (14.263.0) than in patients withoutPB (3.160.5; P50.0005) or in normal controls (2.460.5;P50.0003), as shown in Figure 4. Of its 4 contributory

Figure 2. Effect of (V̇A1bQ̇)(d/VL) and C# S(d/VL) onperiod of oscillations.

Figure 3. Observed values of (V̇A1bQ̇)(d/VL) and C# S(d/VL) inpatients with PB (h), patients without PB (‚) and normal con-trols (E). Oblique line indicates the boundary, predicted byequation 5, between oscillatory and steady breathing.

Francis et al Quantitative Theory for Periodic Breathing 2217

factors, the most important were chemoreflex sensitivity(17496235 versus 6206103 and 5266104 L/min per atmCO2 in patients with PB versus those without PB and normalcontrols; P50.0001 andP,0.0001, respectively) and che-moreflex delay (0.5360.06 versus 0.4060.06 and 0.3060.04min; P5NS andP50.02, respectively). No significant dif-ference existed between group means for C# (4.960.1%versus 4.860.1% and 4.860.2%;P.0.05 for both compari-sons) or for VL (3.560.3 versus 3.560.2 and 3.260.2;P.0.05 for both comparisons). The second combined param-eter, (V̇A1bQ̇)(d/VL), did not differ between the groups(5.060.9 versus 3.460.5 and 3.160.5; P.0.05 for bothcomparisons).

Cycle Time of PBFor the 10 patients with PB, the observed cycle time averaged1.260.2 min. The prediction from our model, using theirphysiological parameters, was for a cycle time of 1.260.1min. the predicted time, using the 4d formula of Mackey andGlass,24 was 2.160.3 min. The difference between observedand predicted cycle times are plotted against their means forour model (Figure 5A) and the model of Mackey and Glass24

(Figure 5B). With our model, the prediction error averaged

0.0 min, with a SD of 0.4 min (ie, no significant bias;P50.8).The Mackey and Glass24 model showed a significant bias(prediction error averaging 0.9 min;P50.001) and a widerprediction error SD of 0.6 min.

DiscussionThis new and explicit criterion for stability enables a directconsideration of the physiological factors that contribute to

Figure 4. Comparison between normal controls, patients without PB, and patients with PB of (A) C# S(d/VL); (B) (V̇A1bQ̇)(d/VL); (C) S; (D)d; (E) C# ; and (F) VL.

Figure 5. Comparison between predicted and observed cycletimes of periodic breathing, using (A) our analytical model and(B) the formula of Mackey and Glass.24 Each plot shows the dif-ference between the predicted and observed values plottedagainst the mean of both.

2218 Circulation October 31, 2000

PB and a comparison of their relative importance. The resultsare general: the stability diagram (Figure 1) is independent ofall physical constants, and it does not require recalculation forchanges in physiological variables. Moreover, the clinicaldata show good validity in predicting both instability andperiod.

This now illuminates clinical controversies regardingpathogenesis and may help in the rational development oftherapy. A first step is to simplify Figure 1 to focus on theprincipal question of whether breathing is doomed to beperiodic by showing only the dividing line between stabil-ity and instability and by representing the effects ofchanges in physiological variables by arrows. Figure 6Ashows why instability is favored by increased chemoreflexslope (or lag) and by decreased lung volume (or cardiacoutput). Figure 6B shows how treatment that increasescardiac output, lung volume, ventilation or inspired CO2 ordecreases lag time or effective chemoreflex slope canstabilize ventilation.

Hyperventilation and Hypocapnia: Red Herrings?Chronic hyperventilation and/or hypocapnia are often consid-ered important in the pathogenesis of PB for 2 reasons.6,13–16

First, CHF patients with PB have a lower mean arterial PCO2

and a higher mean ventilation. Second, the application ofinspired CO2 stabilizes ventilation.

There are also 3 reasons to question the importance of thisin causation. First, respiratory control is undoubtedly lessstable during sleep14 when mean ventilation is lower; indeed,even normal subjects frequently develop PB while asleep.14

Second, exercise, which raises mean ventilation, reliablyattenuates PB.28 Third, and perhaps most importantly, ouranalysis shows that increased ventilation and decreased PCO2

each favor stability.How can these observations be reconciled? The answer lies

in an important physiological property infrequently measuredin clinical studies: chemoreflex gain.

Chemoreflex GainThe mean value of the product of alveolar ventilation andalveolar CO2 fraction must match the rate of the metabolicproduction of CO2. Thus, for any given metabolic rate, thepossible steady-state values of ventilation and alveolar CO2

fraction form a hyperbola (Figure 7). The position of therespiratory system depends on where the chemoreflex con-troller response crosses the hyperbola. An increased chemore-flex slope causes it to meet the hyperbola higher.

Clinically, this means that a higher ventilation shouldalways be suspected of concealing a large increase in che-moreflex gain. Elevated hypercapnic gain is associated withrespiratory instability in patients with CHF.29,30 In onestudy,31 the hypercapnic gain was 124% greater in those withcentral sleep apnea than in those with obstructive sleep apnea,whereas the PCO2 values were only 16% lower (and, byimplication, the alveolar ventilation'16% higher). Thestability chart shows that these abnormalities in PCO2 andventilation both favor stability but are readily overpoweredby an underlying large enhancement of chemoreflex gain.

Lung VolumeIf the association between hyperventilation-hypocapnia andPB seen in observational studies can be explained by anunderlying difference in chemoreflex slope, what mechanismcan be offered for the therapeutic trials of continuous positiveairway pressure, which have found stabilized breathing,increased PCO2, and decreased mean ventilation?32 We sug-gest that continuous positive airway pressure invokes another

Figure 6. Effects of physiological changes on stability: (A) clini-cal abnormalities favoring instability; (B) treatment maneuversfavoring stability. Changes in d or VL affect both coordinatesproportionately, in the direction of the dotted lines.

Francis et al Quantitative Theory for Periodic Breathing 2219

potent confounding effect that is also rarely measured, anincrease in mean lung volume.33 This change may be moreimportant than the changes in PCO2 and ventilation, whichmay be of the order of 10% when compared with placebo.32

Indeed, stabilizing ventilation by increasing lung volumemight even contribute to the treatment of obstructive sleepapnea.

The Importance of Time DelayThe importance of delay in the ventilatory response isdisputed. The strongest countervailing argument has beenGuyton’s dog study,19 which revealed that purely alteringcirculation delay requires substantial prolongation, to 2 to 5minutes, before PB arises. This can now be explained by thestability diagram (Figure 6), which shows that pure prolon-gation of delay is inefficient in destabilizing breathing. Alarge increment would be required because the direction isdiagonal. However, on a background of increased chemore-flex sensitivity (vertical displacement), a small prolongationcan push the state into instability. In support of this explana-tion are studies showing that breathing can be stabilized34,35

by treatments that shorten delay time, such as valve opera-tions, transplantation, milrinone, or theophylline.

The explicit solution of the delay equation also predicts acycle time proportional to delay time (just over twice thedelay in most cases). This prediction is supported by theclinical observations in this study and those of others.8 It ismore appropriate than Mackey and Glass’ prediction24 of aratio of 4. The clinically observed correlation between lung-to-ear circulation time and the period of PB14 is thusexplained.

Independence of Initiating DisturbanceNo particular pattern of disturbance is necessary to initiatePB. Any disturbance, however small, will become amplifiedinto PB if the scale factor is.1 and damped away if the scalefactor is,1. The period of oscillation of the system is alsoindependent of the initial stimulus and depends only onsystem physiology. It is analogous to the case of the “feed-back” phenomena heard when a microphone is accidentallybrought too close to a loudspeaker to which it is connected.The pitch of the resulting unwanted note is independent of the

initiating sounds. Moreover, attempting to remain silentcannot prevent the squeak. Ultimately, system behaviordepends only on the properties of the system. Likewise, ifbreathing control is in distinctly unstable territory (Figure 1),breathing is doomed to become periodic, with stereotypedperiod.

Study LimitationsThis analytical study and its clinical validation data areprincipally directed at the critical determinants of cardiore-spiratory stability in PB resulting from CHF. We used asingle variable representing blood gas variation. Respiratorystability is determined by system behavior near the steadystate, where most patients with CHF have high oxygensaturations.36 Here, hypoxic chemoreflex responses are muchsmaller than hypercapnic responses for the same change inpartial pressure. Thus, even though swings in PCO2 during theonset phase of PB may be smaller than those of PO2,respiratory stability depends on the CO2 chemoreflex.

There are three aspects, however, in which hypoxia mayplay a more prominent role. First, any baseline hypoxemiamay increase hypercapnic chemoreflex gain. Second, onceoscillations become established and episodic desaturationsoccur, the hypoxic chemoreflex response is steeper and couldcontribute to determining the size to which oscillations grow(which our model does not study). Third, if the PB is not dueto CHF but a different cause, such as altitude hypoxia,deoxygenation may play a pre-eminent role. The model couldbe re-expressed in terms of hypoxic rather than hypercapnicstimuli: this would affect the measurement of the variablesequivalent to C# and S but would not affect the generalpredictions of the model.

Conclusion and Clinical ImplicationsThe 6 principal physiological factors that favor PB are a steepchemoreflex slope, long lag to chemoreflex response, lowventilation, low cardiac output, high alveolar-atmosphericCO2 difference, and small lung volume. Of these, chemore-flex enhancement and prolonged lag to ventilatory responsemay be the most important factors in CHF. Hyperventilationand hypocapnia, long considered prime factors, may beepiphenomena of increased chemoreflex slope. The beneficialeffects of therapies can now be categorized as follows:oxygen reduces effective chemoreflex gain (by removing anyenhancement of the hypercapnic chemoreflex gain and atten-uating any independent hypoxic chemoreflex component);continuous positive airway pressure increases mean lungvolume; and inotropes and corrective surgery reduce circula-tion delay and increase cardiac output. Inspired CO2 increasesventilation (favoring stability) and also causes the alveolarCO2 to resettle at a slightly higher level. Mathematically, it isthe difference between alveolar and inspired CO2 that isrepresented by C# . When this is reduced, it favors stability.Nevertheless, in each case, smaller secondary effects on otherparameters cannot be excluded.

In PB, basic elements that are essentially smooth showspontaneous pattern formation because of time-delayed neg-ative feedback. This chemoresponse time delay can bemeasured using simple clinical equipment as easily as can

Figure 7. Increase in chemoreflex slope causes a small increasein ventilation and a small fall in PCO2. Position of respiratory sys-tem (P1) depends on where chemoreflex controller response(Slope1) crosses the hyperbola. An increased chemoreflex slope(Slope2) causes it to meet hyperbola higher (P2).

2220 Circulation October 31, 2000

chemoreflex gain. An explicitly quantitative framework interms of clinical concepts is now available with which toconsider the mechanisms of therapies at an intellectual level.We hope to stimulate colleagues to consider, measure, anddiscuss these factors (which exert mathematically indepen-dent effects on breathing stability) whenever studying PB orits treatments.

AppendixThere are only 2 independent unknowns (g andv, with r5g1jv),and the equation lies in the complex plane (and is thus equivalent to2 standard real equations). A general solution might therefore beexpected to be readily obtainable. Unfortunately, not only dostandard steps fail to reach a general solution, but the generalsolution is intrinsically so strange that it cannot even be described byconventional mathematical functions. It requires Lambert’s transcen-dental W function.27 This is the mapping W:z ez3z. For example,because 3e3'60.26, W(60.26)'3.

Multiplying equation 3 by the following

(5)d

VLerd1$@~V̇A

#1bQ̇!d#/VL%

and rearranging, we obtain equation 6.

(6) S rd1~V̇A1bQ̇)d

VLD erd{[(V̇ A

#1bQ)̇d]/V L}52

#CSd

VLe@~V̇A#

1bQ̇!d#/VL

The W function can be applied to both sides as follows.

(7) rd1~V̇A1bQ̇!d

VL5WS2

#CSd

VLe@~V̇A#

1bQ̇!d#/VLDFrom this equation, the general solution is as follows.

(8) r51

dWS2

#CSd

VLe@(V̇A#

1bQ̇!d]/V LD2~V̇A#1bQ̇!

VL

This explicitly yields the predicted growth rate and oscillatory periodfrom primary clinical data. Once Figures 1 and 2 are constructed,however, no further W computation is necessary because they arecompletely general.

AcknowledgmentsD.P. Francis is supported by the British Heart Foundation, L.C.Davies by the Robert Luff Foundation, M. Piepoli by the WellcomeTrust, and A.J.S. Coats by the Viscount Royston Trust.

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