(Circulation. 2000;102:2214.)
© 2000 American Heart Association, Inc.
Clinical Investigation and Reports |
Quantitative General Theory for Periodic Breathing in Chronic Heart Failure and its Clinical Implications
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{at}cheerful.com
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Abstract |
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Background—In patients with chronic heart failure (CHF), periodic breathing (PB) predicts poor prognosis. Clinical studies have identified numerous risk factors for PB (which also includes Cheyne-Stokes respiration). Computer simulations have shown that oscillations can arise from delayed negative feedback. However, no simple general theory quantitatively explains PB and its mechanisms of treatment using widely-understood clinical concepts. Therefore, we introduce a new approach to the quantitative analysis of the dynamic physiology governing cardiorespiratory stability in CHF.
Methods and Results—An algebraic formula was derived
(presented as a simple 2D plot), enabling prediction from
easily acquired clinical data to determine whether respiration will be
unstable. Clinical validation was performed in 20 patients with CHF (10
with PB and 10 without) and 10 healthy normal subjects. Measurements,
including chemoreflex sensitivity (S) and delay (
), alveolar volume
(VL), and end-tidal CO2 fraction (
),
were applied to the stability formula. The breathing pattern was
correctly predicted in 28 of the 30 subjects. The principal combined
parameter (S)x(/VL) was higher in
patients with PB (14.2±3.0) than in those without PB (3.1±0.5;
P=0.0005) or in normal controls (2.4±0.5;
P=0.0003). This was because of differences in both
chemoreflex sensitivity (1749±235 versus 620±103 and 526±104 L/min
per atm CO2; P=0.0001 and P<0.0001,
respectively) and chemoreflex delay (0.53±0.06 vs 0.40±0.06 and
0.30±0.04 min; P=NS and P=0.02).
Conclusion—This analytical approach identifies the physiological abnormalities that are important in the genesis of PB and explicitly defines the region of predicted instability. The clinical data identify chemoreflex gain and delay time (rather than hyperventilation or hypocapnia) as causes of PB.
Key Words: heart failure • ventilation • physiology
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Introduction |
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TopAbstractIntroductionMethodsResultsDiscussionAppendix 1References |
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In congestive heart failure (CHF), periodic breathing (PB) in the daytime1 2 or during sleep3 results in elevated mortality, which may result from the numerous repetitive fluctuations in blood gases, blood pressure, and heart rate.4 These repeated insults may induce chronic sympathetic overactivation, impede exercise capacity,5 and precipitate ventricular arrhythmias.6
It has long been speculated that PB may arise from pathological feedback in ventilatory control.7 8 9 10 Clinical studies have identified several possibilities in patients with PB,11 12 particularly hyperventilation or hypocapnia.6 13 14 15 16 Prolonged circulation delay8 17 and increased chemoreceptor sensitivity18 have also been implicated. The role of circulation delay is controversial, partly because of animal work19 that had to prolong it to a biologically implausible 2 to 5 minutes to engender PB. Aside from the clinical approach, there are 2 conceptually different mathematical approaches.
Computer simulations, including an ingenious analog electrical circuit7 and numerical iterative models in digital computers,9 20 21 have shown oscillations arising with certain configurations of system physiology. The drawback is that an immediate overview of system behavior across a variety of physiological and pathological states is not gained because computerized resimulation is required for each proposed state. As the number of variables increases, the impact of changes in starting conditions becomes increasingly time-consuming to tabulate and difficult to conceptualize.
The alternative approach is to solve analytically the dynamics of respiratory control in PB. The attraction of this approach is that it should make obvious the range of physiological states that result in PB and predict and explain the mechanisms of effective treatments, while removing the need for computer recalculation for each possible combination of clinical variables. Models have been developed using the frequency domain22 or by seeking critical values of circulation delay,23 but their complexity has limited widespread appreciation. An important step towards a directly clinically applicable model was taken by Mackey and Glass,24 but close examination reveals that the stability criterion proposed is correct only when ventilation and cardiac output are zero.
The lack of an analytical unified general theory of PB that is both mathematically explicit and comprehensible to clinicians has hindered a deeper understanding of the physiological processes; this understanding would enable rational interventions to be planned. Currently, the underlying delay-differential equations are difficult to solve analytically using the techniques that have hitherto been applied.
We aimed to solve these fundamental equations governing cardiorespiratory stability by applying a new technique that would give a rigorous quantitative basis for understanding the pathophysiology of PB in CHF. This would give rise not only to testable predictions of cardiorespiratory stability from primary clinical data, but also provide a framework for understanding the effect on stability of changes in physiological variables, such as those from therapy.
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Methods |
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TopAbstractIntroductionMethodsResultsDiscussionAppendix 1References |
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Analytical Model
Ventilation is regulated by delayed negative feedback. The chemosensors are located in the aorta, the carotids, and the brainstem and, therefore, they receive information on blood gas status some time after a change occurs in the lung; a further neurological delay may occur before ventilation changes. The purpose of the control system is to regulate blood gases and damp away any small random disturbances. However, the delay before response provides the potential for a vicious circle of increasing stimuli and responses (of alternating signs) to develop; this is PB. The analytical approach seeks to identify explicitly the conditions that cause the oscillations to grow.
What determines whether PB will occur is control system behavior near
the steady state. Physiological fluctuations in
this region can be described with linear mathematics. Carbon dioxide
(CO2) and oxygen oscillate during PB in
antiphase.25 Therefore, we represent the state of
the blood gases by a single variable. It is useful to develop a
variable c, which represents the displacement of the
alveolar gas stores (nominally CO2 fraction) away from its
mean value, (Table 1
). Likewise v
represents the displacement of alveolar ventilation from its
mean value 
. Because of the time
delay, the value of v at time t (vt) depends on the value
of c at some time previously (ct-). Near the steady
state, therefore, vt=Sxct-, where S
represents chemoreflex gain (additional ventilation per unit
increase in c). Fluxes of CO2 into and out of the lung
arise from metabolism, ventilation, and exchange with blood
stores. Metabolic production of CO2 by
the body is expressed as .
The rate of CO2 removal from the lung by ventilation is
expressed as (+v)(+c).
Oscillations in arterial CO2
necessitate a net transfer of CO2 from the lung into
extrapulmonary stores (in comparison with the steady state) at
a rate of ß
c, where ß indicates
marginal solubility of CO2 in blood and
, cardiac output, assuming that
pulmonary venous CO2 is stable. The rate of
increase of lung CO2 stores is VL(dc/dt), where
VL indicates alveolar volume and dc/dt, the rate of change
of alveolar CO2 fraction. Thus, we obtain the following
equation.
 | (1) |
 | (2) |
t=egtxejt=egt(cost+jsint).
The (cost+jsint) factor represents
oscillation with period 2
/, and egt
represents the changing amplitude of the periodic waveform. If
g>0, any disturbance will lead to oscillations
that grow until their size becomes limited by nonlinearities in the
system; if g<0, the oscillations decay (ie, breathing will
stabilize after any transient disturbance).
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To solve equation 2
, we make 3 substitutions. First, we replace c by
ert. Second, dc/dt=rert=rc. Third,
v=(Sxct-), and the value of c at a time previously
(ct-) is e-rc, so
v=Sxe-rxc. This gives equation 3.
 | (3) |
Clinical Measurement of Physiological Parameters
We performed clinical measurements of the
physiological parameters
, , S, , and VA
on 20 patients with CHF (10 who exhibited PB during quiet rest in the
daytime and 10 who did not) whose mean age was 61±12 years and on 10
age-matched normal controls with a mean age of 58±15 years. Clinical
characteristics of the patients are shown in Table 2. The presence of PB was determined from
respiration recordings, which were obtained by respiratory
impedance plethysmography with the subjects awake and semirecumbent on
a couch by one blinded investigator (L.C.D.). Informed consent and
ethical approval were obtained.
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Subjects sat and breathed through a calibrated heated pneumotachograph
into a metabolic cart that used a mass spectrometer
(Innovision) to measure ventilation, CO2, and
O2. Subjects underwent 2 tests. In the first, mean alveolar
ventilation () and mean end-tidal
CO2 fraction () were measured. They then rebreathed
from a 6-L bag initially containing 100% oxygen until the end-tidal
PCO2 reached 10 kPa or the test became
uncomfortable. The gain (S) of the chemoreflex system was measured as
the ratio of the rate of rise of alveolar ventilation to the rate of
rise of end-tidal PCO2 in L/min per atmosphere
CO2. Effective lung volume (VL) for ventilatory
exchange was determined by fitting a monoexponential
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 and
end-tidal PCO2 sequences were resampled at 1
Hz. Chemoreflex time delay () was determined through
cross-correlation analysis as the lag giving the maximal
correlation between them. We used values of 0.05 L ·
L-1 · kPa-126 (and
atmospheric pressure of 100 kPa) for ßCO2 and
0.06 L · kg-1 · min-1 for
resting 
.
Statistical Analysis
Comparisons between the 3 groups are made with ANOVA.
Within-subject paired comparisons are made with the paired t
test. P<0.05 is considered significant. Data are
presented as mean±SE.
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Results |
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General Solution of Analytical Model
Equation 3
, on which hinges the stability of cardiorespiratory
control in CHF, can be solved by introducing Lambert’s W
function27 (defined by its inverse,
W-1:z
zez), as detailed in the Appendix. The
solution is as follows.
 | (4) |
S(/VL) and
[(+ß)(g/VL)].
Inserting values for the 6 physiological
variables immediately yields an explicit description of the
response to any disturbance in respiration.
This response is described by the function ert, which is
equal to (egtxejt), because r contains real
and imaginary parts (r=g+j). The rate of growth or decay of
oscillations is eg, which depends on the
underlying physiological variables in the
manner shown in Figure 1. The period of
oscillation 2/ also varies (as shown in Figure 2); however, for most patients with PB,
this period is between 2 and 2.5 times the chemoreflex delay.
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Thus, if a patient has
(+ß)(/VL)=4.5,
S(/VL)=10.0, and =0.45 minutes, Figure 1 shows
that the response to a small disturbance is an
oscillation that grows by a factor of 1.75 every 0.45
minutes, and Figure 2 shows that the period of oscillation
will be 1.1 min. The underlying mathematics, which do not to be
performed once the 2 plots are available, are done by equation 4, such
that r=(0.56+2.7j)/, so the growth factor is e0.56 (ie,
1.75 per 0.45 minutes) and the period is (2/2.7)x0.45 (ie, 1.1
min).
Generality of Solution
The solutions plotted in Figures 1 and 2 are completely general:
their axes are dimensionless, because all units cancel. The intercept
of the stability plot on the S(/VL) axis (which
at first may appear to be arbitrary) is /2, and it, the whole
boundary of stability, and indeed the whole shape of the 3D surfaces
are independent of all physical constants. The plots will not require
recalculation for changes in any variable and are thus applicable
to a wide range of situations. Copies are available from the
authors.
Clinical Results
The observed values of S(/VL) and
(+ß)(/VL)
are plotted in Figure 3. The boundary
between oscillatory and steady breathing, predicted by equation 4, is
also shown. PB was correctly predicted from primary clinical
measurements in 10 of 10 patients with PB, and steady breathing was
correctly predicted in 9 of the remaining 10 patients and in 9 of the
10 normal controls.
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), patients
without PB (
) and normal controls (
). Oblique line indicates the
boundary, predicted by equation 5
Pathophysiological Components Contributing
to PB
The principal combined parameter
S(/VL) was much higher in patients with PB
(14.2±3.0) than in patients without PB (3.1±0.5; P=0.0005)
or in normal controls (2.4±0.5; P=0.0003), as shown in
Figure 4. Of its 4 contributory factors,
the most important were chemoreflex sensitivity (1749±235 versus
620±103 and 526±104 L/min per atm CO2 in patients with PB
versus those without PB and normal controls; P=0.0001 and
P<0.0001, respectively) and chemoreflex delay (0.53±0.06
versus 0.40±0.06 and 0.30±0.04 min; P=NS and
P=0.02, respectively). No significant difference existed
between group means for (4.9±0.1% versus 4.8±0.1% and
4.8±0.2%; P>0.05 for both comparisons) or for
VL (3.5±0.3 versus 3.5±0.2 and 3.2±0.2;
P>0.05 for both comparisons). The second combined
parameter,
(+ß)(/VL),
did not differ between the groups (5.0±0.9 versus 3.4±0.5 and
3.1±0.5; P>0.05 for both comparisons).
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Cycle Time of PB
For the 10 patients with PB, the observed cycle time averaged
1.2±0.2 min. The prediction from our model, using their
physiological parameters, was for a
cycle time of 1.2±0.1 min. the predicted time, using the 4 formula
of Mackey and Glass,24 was 2.1±0.3 min. The difference
between observed and predicted cycle times are plotted against their
means for our 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; P=0.8). The Mackey and
Glass24 model showed a significant bias (prediction error
averaging 0.9 min; P=0.001) and a wider prediction error SD
of 0.6 min.
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Discussion |
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TopAbstractIntroductionMethodsResultsDiscussionAppendix 1References |
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This new and explicit criterion for stability enables a direct consideration of the physiological factors that contribute to PB and a comparison of their relative importance. The results are general: the stability diagram (Figure 1
) is
independent of all physical constants, and it does not require
recalculation for changes in physiological
variables. Moreover, the clinical data show good validity in
predicting both instability and period.
This now illuminates clinical controversies regarding
pathogenesis and may help in the rational development of therapy. A
first step is to simplify Figure 1 to focus on the principal question
of whether breathing is doomed to be periodic by showing only the
dividing line between stability and instability and by
representing the effects of changes in
physiological variables by arrows. Figure 6A shows why instability is favored by
increased chemoreflex slope (or lag) and by decreased lung volume (or
cardiac output). Figure 6B shows how treatment that increases cardiac
output, lung volume, ventilation or inspired CO2 or
decreases lag time or effective chemoreflex slope can stabilize
ventilation.
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Hyperventilation and Hypocapnia: Red Herrings?
Chronic hyperventilation and/or hypocapnia are often
considered important in the pathogenesis of PB for 2
reasons.6 13 14 15 16 First, CHF patients with PB have a lower
mean arterial PCO2 and a higher
mean ventilation. Second, the application of inspired CO2
stabilizes ventilation.
There are also 3 reasons to question the importance of this in causation. First, respiratory control is undoubtedly less stable during sleep14 when mean ventilation is lower; indeed, even normal subjects frequently develop PB while asleep.14 Second, exercise, which raises mean ventilation, reliably attenuates PB.28 Third, and perhaps most importantly, our analysis 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 measured in clinical studies: chemoreflex gain.
Chemoreflex Gain
The mean value of the product of alveolar ventilation and
alveolar CO2 fraction must match the rate of the
metabolic production of CO2. Thus, for
any given metabolic rate, the possible steady-state values
of ventilation and alveolar CO2 fraction form a hyperbola
(Figure 7). The position of the
respiratory system depends on where the chemoreflex controller response
crosses the hyperbola. An increased chemoreflex slope causes it to meet
the hyperbola higher.
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Clinically, this means that a higher ventilation should always be
suspected of concealing a large increase in chemoreflex gain. Elevated
hypercapnic gain is associated with respiratory instability in patients
with CHF.29 30 In one study,31 the
hypercapnic gain was 124% greater in those with central sleep apnea
than in those with obstructive sleep apnea, whereas the
PCO2 values were only 16% lower (and, by
implication, the alveolar ventilation 
16% higher). The stability
chart shows that these abnormalities in PCO2
and ventilation both favor stability but are readily overpowered by an
underlying large enhancement of chemoreflex gain.
Lung Volume
If the association between hyperventilation-hypocapnia
and PB seen in observational studies can be explained by an underlying
difference in chemoreflex slope, what mechanism can be offered for the
therapeutic trials of continuous positive airway pressure, which have
found stabilized breathing, increased PCO2, and
decreased mean ventilation?32 We suggest that continuous
positive airway pressure invokes another potent confounding effect that
is also rarely measured, an increase in mean lung
volume.33 This change may be more important than the
changes in PCO2 and ventilation, which may be
of the order of 10% when compared with placebo.32 Indeed,
stabilizing ventilation by increasing lung volume might even contribute
to the treatment of obstructive sleep apnea.
The Importance of Time Delay
The importance of delay in the ventilatory response is disputed.
The strongest countervailing argument has been Guyton’s dog
study,19 which revealed that purely altering circulation
delay requires substantial prolongation, to 2 to 5 minutes, before PB
arises. This can now be explained by the stability diagram (Figure 6),
which shows that pure prolongation of delay is inefficient in
destabilizing breathing. A large increment would be required because
the direction is diagonal. However, on a background of increased
chemoreflex sensitivity (vertical displacement), a small prolongation
can push the state into instability. In support of this explanation are
studies showing that breathing can be stabilized34 35 by
treatments that shorten delay time, such as valve operations,
transplantation, milrinone, or theophylline.
The explicit solution of the delay equation also predicts a cycle time proportional to delay time (just over twice the delay in most cases). This prediction is supported by the clinical observations in this study and those of others.8 It is more appropriate than Mackey and Glass’ prediction24 of a ratio of 4. The clinically observed correlation between lung-to-ear circulation time and the period of PB14 is thus explained.
Independence of Initiating Disturbance
No particular pattern of disturbance is necessary to
initiate PB. Any disturbance, however small, will become
amplified into PB if the scale factor is >1 and damped away if
the scale factor is <1. The period of oscillation of the
system is also independent of the initial stimulus and depends only on
system physiology. It is analogous to the case of the "feedback"
phenomena heard when a microphone is accidentally brought 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 silent cannot prevent the squeak. Ultimately,
system behavior depends only on the properties of the system. Likewise,
if breathing control is in distinctly unstable territory (Figure 1),
breathing is doomed to become periodic, with stereotyped period.
Study Limitations
This analytical study and its clinical validation data are
principally directed at the critical determinants of cardiorespiratory
stability in PB resulting from CHF. We used a single variable
representing blood gas variation. Respiratory stability is
determined by system behavior near the steady state, where most
patients with CHF have high oxygen saturations.36 Here,
hypoxic chemoreflex responses are much smaller than hypercapnic
responses for the same change in partial pressure. Thus, even though
swings in PCO2 during the onset 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 may play a
more prominent role. First, any baseline hypoxemia may increase
hypercapnic chemoreflex gain. Second, once oscillations
become established and episodic desaturations occur, the hypoxic
chemoreflex response is steeper and could contribute to determining the
size to which oscillations grow (which our model does not
study). Third, if the PB is not due to CHF but a different cause, such
as altitude hypoxia, deoxygenation may play a
pre-eminent role. The model could be re-expressed in terms of hypoxic
rather than hypercapnic stimuli: this would affect the measurement of
the variables equivalent to and S but would not affect the
general predictions of the model.
Conclusion and Clinical Implications
The 6 principal physiological factors that
favor PB are a steep chemoreflex slope, long lag to chemoreflex
response, low ventilation, low cardiac output, high
alveolar-atmospheric CO2 difference, and small lung volume.
Of these, chemoreflex enhancement and prolonged lag to ventilatory
response may be the most important factors in CHF. Hyperventilation and
hypocapnia, long considered prime factors, may be
epiphenomena of increased chemoreflex slope. The beneficial effects of
therapies can now be categorized as follows: oxygen reduces effective
chemoreflex gain (by removing any enhancement of the hypercapnic
chemoreflex gain and attenuating any independent hypoxic chemoreflex
component); continuous positive airway pressure increases mean lung
volume; and inotropes and corrective surgery reduce circulation delay
and increase cardiac output. Inspired CO2 increases
ventilation (favoring stability) and also causes the alveolar
CO2 to resettle at a slightly higher level. Mathematically,
it is the difference between alveolar and inspired CO2 that
is represented by . When this is reduced, it favors
stability. Nevertheless, in each case, smaller secondary effects on
other parameters cannot be excluded.
In PB, basic elements that are essentially smooth show spontaneous pattern formation because of time-delayed negative feedback. This chemoresponse time delay can be measured using simple clinical equipment as easily as can chemoreflex gain. An explicitly quantitative framework in terms of clinical concepts is now available with which to consider the mechanisms of therapies at an intellectual level. We hope to stimulate colleagues to consider, measure, and discuss these factors (which exert mathematically independent effects on breathing stability) whenever studying PB or its treatments.
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Acknowledgments |
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D.P. Francis is supported by the British Heart Foundation, L.C. Davies by the Robert Luff Foundation, M. Piepoli by the Wellcome Trust, and A.J.S. Coats by the Viscount Royston Trust.
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Appendix 1 |
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TopAbstractIntroductionMethodsResultsDiscussionAppendix 1References |
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There are only 2 independent unknowns (g and
, with
r=g+j), and the equation lies in the complex plane (and is thus
equivalent to 2 standard real equations). A general solution might
therefore be expected to be readily obtainable. Unfortunately, not only
do standard steps fail to reach a general solution, but the general
solution is intrinsically so strange that it cannot even be described
by conventional mathematical functions. It requires Lambert’s
transcendental W function.27 This is the mapping W:z
ezz. For example, because 3e360.26,
W(60.26)3.
Multiplying equation 3 by the following
 | (5) |
.
 | (6) |
 | (7) |
 | (8) |
and 2 are
constructed, however, no further W computation is necessary because
they are completely general. Received February 14, 2000; revision received June 6, 2000; accepted June 14, 2000.
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TopAbstractIntroductionMethodsResultsDiscussionAppendix 1References |
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