Download 36. Mathematical Models of Cheyne-Stokes Breathing

Indian Journal of Biomechanics: Special Issue (NCBM 7-8 March 2009)
Mathematical Models of Cheyne-Stokes Breathing in understanding
Cardiovascular and Respiratory Disorders
Ruchi Singhal1, V. K. Katiyar2
1
.Department of Mathematics, MIT, Pune, India-411038.
2
.Department of Mathematics, I I T, Roorkee, India-247667.
Abstract
The interaction between the cardiovascular system and the respiratory system is
complicated and may be used for the rehabilitation of the carotid artery disease.
Periodic breathing is an unusual form of breathing with oscillations in minute
ventilations and with repetitive apneas or near apneas. Cheyne-Stokes like breathing
during sleep may be associated with obstructive sleep apnea (OSA) in some
individuals. Therefore, it was reasoned that stroke patients with periodic breathing in
sleep would be susceptible to OSA and for this purpose a mathematical model is
established. The basic material is the balance relationships for the lung-blood-tissue
gas transport and exchange system have been expressed in a number of dependent
time delays. Additional equations were written to define the chemical details of
transport and acid-base buffering, concentration equilibria, and blood flow behavior.
The result is simulated analytically and compared with the experimental observations.
This work may suggest that a proper directed respiration exercise can be used in the
treatment of carotid artery disease.
Keywords: Carotid Artery, Cheyne-Stokes breathing, Mathematical model, Time
delay, Periodic breathing.
1. Introduction
The term periodic breathing is used here to encompass different forms of breathing in
which minute ventilation has an obviously repetitive oscillatory pattern. The bestknown form of periodic breathing, Cheyne–Stokes respiration (CSR) is characterized
by a regularly symmetrical waxing and waning of breathing tidal volume and rate,
culminating in episodic apnoea. Periodic breathing does not always have the classic
architecture seen in CSR (Cherniack and Longobardo 2005, Grodins et. al. 1967,
Longobardo et. al. 2005).
Probably because of the linkages that exist among physiological systems and the
pervasive action of the respiratory rhythm on the activity of brain neurons, the swings
in ventilation are often accompanied by fluctuations of blood gases, heart rate, blood
pressure, EEG, sympathetic activity and the diameter of the pupil of the eye
(McNaughton 1998).
The mechanism of the interaction between respiration and cardiovascular systems
should be studied in order to understand the respiratory treating methods (Bai et. al.
1998). Periodic breathing has been reported in several diseases but is especially
associated with cerebrovascular and cardiovascular diseases. However, it also occurs
in normal asleep (Nopmaneejumruslers et. al. 2005, Thalhofer et. al. 2000). Therefore
this study is performed with the following aspects:
1) Establishing a proper model for the study of interaction of respiratory and
cardiovascular systems of carotid artery.
2) Simulation study on the effect of respiration mode which includes
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•
•
Simulation of regular respiration.
Simulation study of the effects of different synchronizing phase of step-leap
respiration mode.
In this paper, the formulation of an extended model and different results are
presented.
2. Model Description
In our model the material balance equations for CO2 and O2 for each of the three
compartments lung, brain and tissue are derived. In case of lung, the volumetric
fraction of gases add to one in both inspired and alveolar air and that the sum must be
equal to zero as its volume is constant, so we can express VE (volume exhaled) in
terms of VI (volume inhaled):
863
(1)
VE = VI +
Q Cv ( co2 ) − Ca ( co2 ) + Cv ( o2 ) − Ca ( o2 )
PB − 47
Where B is the barometric pressure, C denotes concentration of gas in artery (a) and
venous (v).
The alveolar-arterial and venous blood-brain concentration equilibria for CO2 and O2
is used (Grodins et. al. 1967). Using the law of conservation of mass, the dynamic
equations for the arterial stores of CO2 and O2 can be written as follows:
V .Pa (CO2 ) d
Vr
(2)
=
QC Cv (CO2 ) − Ca ( CO2 ) −
.Pa (CO2 ) + Va .Ca ( CO2 )
PB − 47 dt PB − 47
[(
(
(
)]
) (
)
)
V .Pa (O2 )
V .PI (O2 )
d
Vr
(3)
=
.Pa (O2 ) + Va .Ca (O2 )
PB − 47 PB − 47 dt PB − 47
where on L.H.S. the first term is the net rate at which the gases are brought into the
arterial store by the blood, second term is the rate at which gases leaves the arterial
store and the R.H.S. term is the rate of change of gases.
Similar equations can be written for the venous blood-brain CO2 and O2 stores,
d
QB Ca (CO2 ) − Cv , B (CO2 ) + VB (CO2 ) =
VB .CB (CO2 )
(4)
dt
d
QB Ca (O2 ) − Cv , B (CO2 ) − VB (CO2 ) =
VB .CB (CO2 )
(5)
dt
An exactly analogous set of equations with values for the tissue reservoir and tissue
venous blood substituted for those of the brain and brain venous blood is used to
define venous-tissue equilibria for CO2 and O2.
d
(6)
QT Ca (CO2 ) − Cv ,T ( CO2 ) + VT (CO2 ) =
VT .CT (CO2 )
dt
d
QT Ca (O2 ) − Cv ,T (O2 ) + VT ( O2 ) =
VT .CT (O2 )
(7)
dt
where QC , QB , QT ,V are cardiac output, blood flow in arteries, tissue venous blood
flow and alveolar ventilation respectively.
The total cardiac output equals the sum of the blood flow to arteries and tissue, i.e.,
(8)
QC = QB + QT
On the assumption that the air tension by the gases in ‘all other tissues’ is equal to that
of the venous blood with which each is perfused, then
PB = Pv , B and
PT = Pv ,T for each gas
(9)
QC Cv (O2 ) − Ca (O2 ) −
(
)
(
(
)
(
(
+
(
)
)
)
)
(
(
)
)
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The first concerns the calculation of current values for
Ca (CO2 ) , Cv , B (CO2 ) , Cv ,T ( CO2 ) , etc. to put into this system of equations. Special problems
are involved here because of the interdependence of CO2 content, oxyhemoglobin
content and because of the fact that the buffer equation cannot be solved explicitly for
the desired variable. We do this by a process of iterative guessing, for finding the
value of one variable and repeat the entire process until we get that value which
satisfies the relation between the variables. When regular heart rate is 72 beats per
second, there will be four heart cycles within one respiration cycle. The breathing for
the step-leap mode is performed as follows: first making three quick inspirations and
hold the breath and then, making three quick expirations in a sequence.
3. Results and Discussion
The equations of the model and the corresponding simulations have evolved through
several stages of increasing complexity. In our result an attempt was made to produce
Cheyne-Stokes respiration by subjecting the model to particular disturbances and
observing the ventilatory pattern that ensued. Using the constants used in previous
studies (Grodins et. al. 1967) and a normal controller curve, a hyperventilation signal
was applied to the model until the carbon dioxide tension was reduced from its resting
level of 40 mmHg to 14 mmHg. As shown in Fig. 1, Cheyne-Stokes respiration
appeared after a period of apnea lasting 105 sec, the period of the cycle was 30 sec.
The ventilation varied from cycle to cycle as did the arterial blood gas tensions. It
may be seen that the cyclic respiration gradually disappeared as the arterial saturation
and carbon dioxide tension increased. The Pa (CO2 ) was highest and the arterial oxygen
saturation the lowest during hyperapnea, the converse was true during apnea. These
results agree with those described experimentally (Bai et. al. 1998, Cherniack and
Longobardo 2005, Grodins et. al. 1967). Also shown in Fig. 1 is the effect of asphyxia
on the model. When the circulation was normal (6 sec) no periodic breathing
occurred. On the other hand, periodic breathing was induced in the model by asphyxia
when the circulation time was prolonged (30 sec).
Since long circulation times are associated clinically with reduced cardiac outputs, the
model was hyperventilated to a Pa (CO2 ) of 18 mmHg, the cardiac output used was 2.8
liters/min and the circulation time was 30 sec. The response obtained is shown in Fig.
2. It is similar to that obtained with a long circulation time alone (Fig.1). CheyneStokes breathing could not be produced by combining a normal circulation time with
a half-normal cardiac output. Cheyne-Stokes respiration appeared, nonetheless,
despite the higher level of oxygen saturation.
Fig. 3 shows the variation of average carotid artery blood volume with respiration for
both regular respiration mode and step-leap respiration mode. The result shown in
Fig. 3 indicates that the average carotid artery blood volume under the step-leap
respiration mode is much larger than that under the regular respiration. These results
suggest that with the step-leap respiration exercise, much more effects on the
cardiovascular circulation are generated.
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4. Conclusion
The results of this study indicate that the respiration mode has an effect on the
circulation system. By adapting different respiration modes, the effect can be
changed. The step-leap respiration mode can provide more assistance to the recovery
of the cardio-pulmonary disease. If one can control practicing the step-leap respiration
in such a mode that the inspirations are started at the beginning of diastole and the
expirations are started at the beginning of systole, the assistance of respiration will be
maximal within all the modes. These results have similarity with the clinical
experiments (Bai et. al. 1998, Grodins et. al. 1967). This work may suggest that a
proper directed respiration exercise can be used in the treatment of carotid artery
disease and may be used for the designing of a respiration rehabilitation device for
cardiovascular disease patients which can guide the user in practicing the proper stepleap respiration.
References
1. Bai J, Lu H, Zhang J, Zhao B, Zhou X. Optimization and mechanism of stepleap respiration exercise in treating of cor pulmonale. Computers in Biology
and Medicine, 1998; 28, 289-307.
2. Cherniack NS and Longobardo GS. Mathematical models of periodic breathing
and their usefulness in understanding cardiovascular and respiratory disorders.
Experimentally Physiology, 2005; 9(2), 295-305.
3. Grodins FS, Buell J and Bart AJ. Mathematical analysis and digital simulation
of the respiratory control system. Journal of Applied Physiology, 1967; 22(2),
260-276.
4. Longobardo G, Evangelisti CJ and Cherniack NS. Introduction of respiratory
pattern generators into models of respiratory control. Respir Physiol
Neurobiol, 2005; 50, 285–301.
5. McNaughton MT. Pathophysiology and treatment of Cheyne-Stokes
respiration. Thorax, 1998; 53, 514–518.
6. Nopmaneejumruslers C, Kaneko Y, Hajek V, Zivanovic V and Bradley TD.
Cheyne-Stokes respiration in stroke: relationship to hypocapnia and occult
cardiac dysfunction. Am J Respir Crit Care Med, 2005; 171, 1048–1052.
7. Thalhofer SA, Kiwus U and Dorow P. Influence of orthotopic heart
transplantation on breathing pattern disorders in patients with dilated
cardiomyopathy. Sleep Breath, 2000; 4, 121–126.
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Fig. 1 Production of Cheyne-stokes breathing based on a normal controller
curve. Periodic breathing began after a period of apnea and is shown to
be a function of the Paco2 level to which the subject is hyperventilated
(upper left and upper right), and the circulation delay (upper left and
lower left). Also shown (lower right), is the effect on asphyxia with a
normal circulation delay (no periodic breathing.
Fig 2 Cheyne-Stokes respiration in
a patient with low respiration and
step-leap respiration mode on the
average artery blood volume
Fig.3 Comparison of the effects of regular
cardiac output and a lung time of 30 sec.
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