Download A cardiovascular-respiratory control system

A cardiovascular-respiratory control system model
including state delay with application to congestive heart failure in humans
Jerry Batzel Contact Author
Special Research Center ”Optimization and Control,
University of Graz,
Heinrichstraße 22, A-8010 Graz, Austria.
Susanne Timischl-Teschl
Fachhochschule Technikum Wien
Hoechstaedtplatz 5, 1200 Wien
Austria
Franz Kappel
Special Research Center ”Optimization and Control and
Institute of Mathematics
University of Graz
Heinrichstraße 36, A-8010 Graz, Austria.
Email: [email protected] - Contact email
Email: [email protected]
Email: [email protected]
contact FAX: +43 316 380 9795
contact phone: +43 316 380 8552
Journal of Mathematical Biology manuscript No.
(will be inserted by the editor)
Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel
A cardiovascular-respiratory control system
model including state delay with application
to congestive heart failure in humans
c Springer-Verlag
the date of receipt and acceptance should be inserted later – 2004
Abstract. This paper considers a model of the human cardiovascular-respiratory
control system with one and two transport delays in the state equations describing the respiratory system. The effectiveness of the control of the ventilation rate
V̇A is influenced by such transport delays because blood gases must be transported a physical distance from the lungs to the sensory sites where these gases
are measured. The short term cardiovascular control system does not involve such
transport delays although delays do arise in other contexts such as the baroreflex loop (see [46]) for example. This baroreflex delay is not considered here. The
interaction between heart rate, blood pressure, cardiac output, and blood vessel
resistance is quite complex and given the limited knowledge available of this interaction, we will model the cardiovascular control mechanism via an optimal control
derived from control theory. This control will be stabilizing and is a reasonable
approach based on mathematical considerations as well as being further motivated
by the observation that many physiologists cite optimization as a potential influence in the evolution of biological systems (see, e.g., Kenner [30] or Swan [62]). In
this paper we adapt a model, previously considered (Timischl [63] and Timischl
et al. [64]), to include the effects of one and two transport delays. We will first
implement an optimal control for the combined cardiovascular-respiratory model
with one state space delay. We will then consider the effects of a second delay in
the state space by modeling the respiratory control via an empirical formula with
delay while the the complex relationships in the cardiovascular control will still
be modeled by optimal control. This second transport delay associated with the
sensory system of the respiratory control plays an important role in respiratory
stability. As an application of this model we will consider congestive heart failure where this transport delay is larger than normal and the transition from the
quiet awake state to stage 4 (NREM) sleep. The model can be used to study the
Jerry J. Batzel: SFB ”Optimierung und Kontrolle”, Karl-Franzens-Universität,
Heinrichstraße 22, 8010 Graz, Austria
Franz Kappel: Mathematics Institute and SFB ”Optimierung und Kontrolle”,
Karl-Franzens-Universität, Heinrichstraße 36, 8010 Graz, Austria
Susanne Timischl-Teschl: Fachhochschule Technikum Wien, Austria
Supported by FWF (Austria) under grant F310 as a subproject of the Special
Research Center F003 ”Optimization and Control”
Version August 23, 2004.
Key words: Respiratory system, Cardiovascular System, Optimal control, Delay
Mathematics Subject Classification (2000): 92C30, 49J15
2
Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel
interaction between cardiovascular and respiratory function in various situations
as well as to consider the influence of optimal function in physiological control
system performance.
1. Introduction
The cardiovascular system functions to maintain adequate blood flow to
various regions of the body. This function depends upon the interaction of
a large number of factors including blood pressure, cross-section of arteries,
cardiac output, and partial pressures of CO2 and O2 in the blood. There
are global control mechanisms that act on the entire system to maintain
appropriate blood flow and these mechanisms are supplemented by local
mechanisms in each vascular region which act to shunt blood to those regions
where demand is high and away from areas where demand is low. The
overall control process which stabilizes the system is quite complicated and
not fully elucidated. Principles of optimal control theory will be applied to
design a control mechanism for this system. For further details about the
cardiovascular system and control see, e.g., Rowell [55].
When breathing is not under voluntary control or subject to neurologically induced changes, the human respiratory control system varies the
ventilation rate in response to the levels of carbon dioxide CO2 and oxygen O2 in the body (via partial pressures PaCO2 and PaO2 ). This chemical
control system depends upon information fed back from two sensory sites
which monitor the blood gas levels (producing a negative feedback control
loop). These sensory sites at which the blood gas levels are measured are a
physical distance from the lungs (where blood gas levels are adjusted) and
thus there are transport delays (which vary depending on blood flow) in the
negative feedback loop. Under normal conditions (even with delays in the
feedback control loop) the control system is sufficiently stable to maintain
blood levels of these gases within very narrow limits. See, e.g., [11] or [14]
for more information on this system.
There are a number of links between the respiratory and cardiovascular
systems. Function of the respiratory system depends on blood flow through
the lungs and tissues. The amount of oxygen O2 transported to the tissues
and carbon dioxide CO2 transported away from the tissues depends on
cardiac output Q and blood flow F through the pulmonary and systemic
circuits. Q and F depend in turn upon heart rate H, stroke volume Vstr ,
resistance in the vascular system R, and blood pressure P . Arterial blood
pressure Pas is controlled via the baroreceptor negative feedback loop which
has important effects on H, Vstr , R, and hence Q. Systemic resistance which
impacts blood pressure is also influenced by local metabolic control acting
on the resistance of the blood vessels of various tissues. This local control is
in turn influenced by local concentrations of CO2 and O2 , thus illustrating
another important link between the two systems. The effect of concentration
of O2 on the resistance of the systemic blood vessel is included in this model.
Furthermore, PaCO2 and PaO2 can affect cardiac output and contractility as
Cardiovascular-respiratory control system
3
well (see, e.g., Richardson et al. [54]). Neither these blood gas effects nor
synchronization of heart rate and ventilation are included in this model.
An optimal control approach will be used to model the complex interactions in the cardiovascular-respiratory control system. The cardiovascular and respiratory controls are represented by a linear negative feedback
control which minimizes a quadratic cost functional defining optimal performance. Reasons and motivation for incorporating an optimal control approach is given in Section 3. This modeling approach was previously applied
by Kappel and Peer [24] and Timischl [63] to study transition from rest to
exercise under a constant ergometric workload and the role of pulmonary
resistance during exercise.
The equations describing the state of the system are developed following
the ideas of Khoo et al. [31], Grodins et al. [15], and Grodins [16,17] and
Kappel and coworkers [24,28,48].
The model can also be used to study difficult to measure parameters
(such as pulmonary resistance) in other conditions as well such as congestive
heart failure.
2. Model equations with delay
The general model equations including delays are given in equations (1) to
(14). Symbols are defined in Tables 1 and 2. The respiratory component of
the model is defined by equations (1) to (5) and is based on equations given
in Khoo et al. [31]. Two compartments, a lung compartment and a general
tissue compartment, are used to model the respiratory component of the
system (see Figure 1).
The lung compartment equation (1) represents a mass balance equation
for CO2 and equation (2) similarly represents a mass balance equation for
O2 . The mass balance equations for CO2 and O2 in the tissue compartment
are given by Equations (3) and (4). Equation (5) tracks CO2 in the brain
which is needed as input to the central respiratory sensor (see Section 10).
We note that the brain is considered as part of the general tissue compartment. Transport delays appear in the mass balance equations as it takes
time for tissue venous blood to reach the lungs and vice versa. The compartment blood gas levels are adjusted by the ventilation rate V̇A which will
be further discussed in Section 3.
Note that the role of V̇A in the state equations for the lung compartment
(1) and (2) is that of effective ventilation reflecting net ventilation after dead
space effects are removed.
Among the assumptions incorporated in the model we mention that the
model is an average flow model and thus ventilation represents minute ventilation and cardiovascular flow is non-pulsatile. Given the time scales and
focus of this study, these assumptions are reasonable. Other assumptions
are given in the appendix.
In passing we note that alveolar minute ventilation does not reflect modulation of ventilation by the rate or depth of breathing which can influence
4
Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel
stability (see Batzel and Tran 2000 [1–3]) and that pulsatility in blood flow
can influence the distribution of blood and play a role in the baroreflex
control.
VACO2 ṖaCO2 (t) = 863Fp (t)(CvCO2 (t − τV ) − CaCO2 (t))
(1)
VAO2 ṖaO2 (t) = 863Fp (t)(CvO2 (t − τV ) − CaO2 (t))
(2)
+ V̇A (t)(PICO2 − PaCO2 (t)),
+ V̇A (t)(PIO2 − PaO2 (t)),
VTCO2 ĊvCO2 (t) = MRCO2 + Fs (t)(CaCO2 (t − τT ) − CvCO2 (t)),
VTO2 ĊvO2 (t) = −MRO2 + Fs (t)(CaO2 (t) − CvO2 (t − τT )),
(4)
VBCO2 ĊBCO2 (t) = MRBCO2 + FB (t)(CaCO2 (t − τB ) − CBCO2 (t)),
cas Ṗas (t) = Ql (t) − Fs (t),
cvs Ṗvs (t) = Fs (t) − Qr (t),
cvp Ṗvp (t) = Fp (t) − Ql (t),
(3)
(5)
(6)
(7)
(8)
Ṡl (t) = σl (t),
Ṡr (t) = σr (t),
(9)
(10)
σ̇l (t) = −γl σl (t) − αl Sl (t) + βl H(t),
σ̇r (t) = −γr σr (t) − αr Sr (t) + βr H(t),
(11)
(12)
Ḣ(t) = u1 (t),
(13)
V̈A (t) = u2 (t).
(14)
Table 1. Respiratory symbols
Symbol
Ca
Cv
MR
Pa
Pv
PI
B
u2
V̇A
V̈A
VA
VT
CO2 ,O2
τ
Ip , I c
Meaning
concentration of blood gas in arterial blood
concentration of blood gas in mixed venous blood
metabolic production rate
partial pressure of blood gas in arterial blood
partial pressure of blood gas in mixed venous blood
partial pressure of inspired gas
brain compartment
control function, u2 = V̈A
alveolar ventilation
time derivative of alveolar ventilation
effective gas storage volume of the lung compartment
effective tissue gas storage volume
carbon dioxide and oxygen respectively
transport delay
cutoff thresholds
unit
lSTPD · l−1
lSTPD · l−1
lSTPD · min−1
mmHg
mmHg
mmHg
lBTPS · min−2
lBTPS · min−1
lBTPS · min−2
lBTPS
l
sec
mmHg
Cardiovascular-respiratory control system
5
The cardiovascular component of the model is based on the work of
Grodins and coworkers [15–17] and Kappel and coworkers [24,28,48] and
is described by equations (6) to (12). This component includes two circuits
(systemic and pulmonary) which are arranged in series, and two pumps (left
and right ventricle). See Figure 1. Each circuit subsumes the system of arteries and veins, arterioles, and capillary networks under three components:
a single elastic artery, a single elastic vein, and a single resistance vessel.
Blood flow is assumed to be unidirectional and non-pulsatile. Thus, blood
flow and blood pressure are to be interpreted as mean values over the length
of a pulse.
Mass balance equations for blood flowing through the systemic artery
and vein components are given by equations (6) and (7) respectively. Equation (8) gives the mass balance equation for the pulmonary venous component. Under the assumption of a fixed blood volume V0 , the equation for
the pulmonary arterial pressure can then be derived from the other cardiovascular compartment pressures:
Pap (t) =
1
(V0 − cas Pas (t) − cvs Pvs (t) − cvp Pvp (t)).
cap
(15)
Table 2. Cardiovascular symbols
Symbol
α
Apesk
β
ca
cv
F
H
γ
Pas
Pvs
Pap
Pvp
Q
R
S
σ
u1
Vstr
V0
l,r
p,s
Meaning
coefficient of S in the differential equation for σ
Rs = Apesk CvO2
coefficient of H in the differential equation for σ
arterial compliance
venous compliance
blood flow perfusing compartment
heart rate
coefficient of σ in the differential equation for σ
mean blood pressure in systemic arterial region
mean blood pressure in systemic venous region
mean blood pressure in pulmonary arterial region
mean blood pressure in pulmonary venous region
cardiac output
resistance in the peripheral region of a circuit
contractility of a ventricle
derivative of S
control function, u1 = Ḣ
stroke volume of a ventricle
total blood volume
left and right heart
pulmonary and systemic circuits
Unit
min−2
mmHg · min ·l−1
mmHg · min−1
l · mmHg−1
l · mmHg−1
l · min−1
min−1
min−1
mmHg
mmHg
mmHg
mmHg
l · min−1
mmHg · min ·l −1
mmHg
mmHg · min−1
min−2
l
l
-
6
Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel
Lungs
PaCO 2 PaO2
VA
Fp
Pap
PvCO 2
PvO2
Pvp
PaCO 2
PaO2
Ql
Qr
Pvs
PvCO 2
PvO2
H
debit
values
Pas
PaCO 2
PaO2
Fs
Tissue
PvO2 PvCO 2
Controller
Rs
Autoregulation
PvO2
MRO2 MRCO2
Fig. 1. Model block diagram
Blood flow F , which appears in equations (6) through (8) is related to blood
pressure via a form of Ohm’s law
Pas (t) − Pvs (t)
,
Rs (t)
Pap (t) − Pvp (t)
,
Fp (t) =
Rp
Fs (t) =
(16)
(17)
where Pa is arterial blood pressure, Pv is venous pressure, and R is vascular
resistance. Details can be found in [24,28,48].
As mentioned above, cardiac output Q is defined as the mean blood flow
over the length of a pulse,
Q(t) = H(t)Vstr (t),
(18)
Cardiovascular-respiratory control system
7
where H is the heart rate and Vstr is the stroke volume. Subindices l and
r are used to distinguish between left and right ventricle. Subindices s and
p represent systemic and pulmonary circuits respectively. A complex relationship between stroke volume and blood pressure is given in Kappel and
Peer [24] which reflects the Frank-Starling law and the basic relation
Vstr (t) = S(t)
cPv (t)
.
Pa (t)
(19)
Here S denotes the contractility, Pv is the venous filling pressure, Pa is the
arterial blood pressure opposing the ejection of blood, and c denotes the
compliance of the relaxed ventricle.
The Bowditch effect, which describes the observation that contractility Sl (respectively Sr ) increases if heart rate increases, is introduced via
Equations (9) through (12). This relation is essentially modeled via a second
order differential equation. For details see Kappel and Peer [24].
Equations (13) and (14) define the variation of heart rate (Ḣ(t)) and
variation in ventilation (V̈A (t)) as mathematical control variables. The functions u1 (t) and u2 (t) will be derived using an optimality criterion which acts
to minimize deviations in several quantities including these variations Ḣ(t)
and V̈A (t) (see Section 3). Thus, limits are placed on the magnitude and
variation of the changes in the physiological controls H(t) and V̇A (t) which
reflects an assumption of minimal energy expense effort as an optimal control criterion for the physiological control process.
Local metabolic autoregulation of systemic resistance is modeled using
the assumption that systemic resistance Rs depends on venous oxygen concentration CvO2 . Thus Rs is described by
Rs (t) = Apesk CvO2 (t),
(20)
where Apesk is a parameter. This relationship was introduced by Peskin
[49] and is based on work on autoregulation by Huntsman et al. [22]. The
above relationship was also used in Kappel and Peer [24]. Essentially, this
equation describes an important local constriction/relaxation mechanism
acting on small vascular elements in response to local oxygen concentration
CvO2 (some tissues respond also to CvCO2 ). Global changes in Rs will be
discussed in Section 7. Delay in the control process of global resistance is
not analyzed in this paper.
Links between the respiratory and cardiovascular components can be
seen in the equations. The respiratory mass balance equations include expressions for the blood flows Fs and Fp . Levels of CvO2 which influences
systemic resistance via equation (20) is in turn affected by the respiratory system. Heart rate H and ventilation rate V̇A influence both systems
through the control functions u1 and u2 , while Pas , PaCO2 , and PaO2 affect
the dynamical behavior through the cost functional.
8
Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel
3. Control of the system
The control for the cardiovascular and respiratory system will be designed
to transfer the state of the organism from an initial perturbation (initial
state) to a final steady state in an optimal way that will be defined below.
Given the complex and interrelated nature of the control systems discussed
here, the interaction of the various control effects will be represented as a
stabilizing control derived from optimal control theory. This approach to
designing a stabilizing control is motivated primarily by mathematical considerations and is reasonable given the lack of detailed information about
particular control interactions. Furthermore, this approach can provide information on the nature and function of the controller as well as to help
identify and study key controlling and controlled quantities.
As mentioned above, this approach is motivated primarily on mathematical grounds, but such a control derived from optimal control theory is
further motivated by the view that optimal function likely plays a role in
physiological design. See, for example, Kenner [30] or Swan [62]. Minimizing
stress on the system either by avoiding extreme actions or inefficient operating states would represent such an optimal design criterion. The degree to
which physiological systems behave optimally is an open question of great
interest.
In the model we focus on two physiological quantities which influence
the system: heart rate H and the ventilation rate V̇A . In the cardiovascular
system, H is adjusted via the baroreceptor control loop and V̇A is adjusted
by the respiratory control loop. These quantities are varied so that the
mean arterial blood pressure Pas and the partial pressures of carbon dioxide
PaCO2 and oxygen PaO2 in arterial blood are stabilized (along with the whole
system) to their steady state operating points when an initial perturbation
occurs. Parameters are chosen to define the steady state values which will
be the operating points (final steady state) as well as to derive the initial
perturbed condition of the system.
The cost function we will use enforces the condition that the transition from initial condition to final steady state is optimal in the sense that
Pas , PaCO2 , and PaO2 are stabilized such that the cumulative deviations of
these quantities from their final steady state values are as small as possible,
while the presence of u1 (t) and u2 (t) in the cost functional implements the
further restriction that excessive heart rate and ventilation change are restricted (effort is efficient). In this way, the stabilizing feedback control can
be considered also as an optimizing feedback control.
In the mathematical setting for this problem, it is the variations in heart
rate (Ḣ(t)) and ventilation (V̈A (t)) that represent the control functions u1 (t)
and u2 (t). By including u1 (t) and u2 (t) in the cost functional, limits are
placed on the degree to which H and V̇A can be varied to stabilize the
system, a reasonable physiological constraint which also reflects an efficiency
of effort. The calculated control acts in the optimal way as defined by the
Cardiovascular-respiratory control system
9
cost functional to transfer the system from one state (initial condition) to
another (steady) state.
The control problem is then formulated as follows: We determine control
functions u1 and u2 that transfer the system from one state to another such
that the cost functional
Z ∞
fe 2
e
qas (Pas (t) − Pas
) + qc (PaCO2 (t) − PafCO
)2
(21)
2
0
+qo (PaO2 (t) − PafOe2 )2 + q1 u1 (t)2 + q2 u2 (t)2 dt
is minimized under the restriction of the model equations:
ẋ(t) = f (x(t), x(t − τT ); W s ) + B u(t),
y(t) = Dx(t).
x0 = φ.
(22)
where x(t) ∈ R14 is given by
x(t) = (PaCO2 , PaO2 , CvCO2 , CvO2 , CBCO2 , Pas , Pvs , Pvp , Sl , Sr , σl , σr , H, V̇A )T .
The vector f represents the system equations, W s represents the vector
of associated weights in the cost functional, and y(t) is a vector which
represents the observation of controlled values. The delay τT ∈ R+ is a
fixed point delay and the initial condition is a function φ ∈ C where C
denotes C([−τT , 0], R14 ). The positive scalar coefficients qas , qc , qo , q1 , and
q2 determine how much weight is associated to each term in the integrand.
Superscript ”f e” refers to the final equilibrium or steady state to which
the system is transfered by the control. We note that partial pressures and
concentrations are interchangeable according to the dissociation formulas.
We use concentrations in some state equations to simplify the form of the
equations. For further information related to applications of optimal control
theory in biomedicine see, e.g., Swan [62] or Noordergraaf and Melbin [45],
and for general reference on mathematical control theory see Russell [56].
4. Effects of the weights
In the simulations presented in this paper the weights associated with the
quantities in the cost functional have values all set equal to one with the
exception of qo , the weighting factor of PaO2 , which is set to 0.3. The motivation for a smaller weight for qo is that only large deviations in PaO2 act to
significantly alter ventilation because, as can be seen from the adult oxyhemoglobin saturation curve, there is a significant reserve of oxygen. Ventilatory control response to PaO2 will be more pronounced only at lower levels
of PaO2 . In normal operating conditions a deviation of 1 mmHg in PaCO2
produces a larger percentage change in ventilation than does a proportional
mmHg deviation in PaO2 , thus suggesting that PaCO2 is the primary focus of
control. These factors are also expressed in empirical relationships between
PaO2 , PaCO2 , and V̇A given by Wasserman et al. [67] or Khoo et al. [31] (see
10
Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel
Section 10). For these reasons, a smaller weight is associated with deviations
in PaO2 in the cost functional. In regards to the other weights it appears
that the system reacts in a more sensitive way to deviations in the other
variables than to deviations in PaO2 . Because we lack more information, we
take the same weight for all variables except PaO2 . In order to get more
information on these weights one would have to parameter estimation on
the basis of data obtained from appropriate tests which we plan to pursue.
In simulation studies [64] it was found that respiratory and cardiovascular quantity cross-interaction through the cost functional is minimal. Considering the respiratory component weights in isolation, simulations indicate
that the initial drops in V̇A and PaO2 at the transition to sleep (see Figures
(2) thru (7)) are much more extreme when the PaO2 weight is small. When
equal weights are given to PaO2 and PaCO2 the initial undershoot of the final
steady states is much smaller. Simulations indicate that the undershoot is
small for qo greater than 0.3. The weights chosen for these simulations are
reasonable given that there is certainly a range of responses for V̇A to given
PaO2 and PaCO2 levels and of heart rate H to arterial blood pressure Pas
levels. Parameter identification could set these values for individual cases.
5. Simulation Method
In the simulations we will consider the control which transfers the system
from the initial condition ”resting awake” denoted by xa to the final steady
state ”stage 4 quiet sleep” denoted by xs . We view the ”resting awake state”
as an initial perturbation of the steady state ”stage 4 quiet sleep”. These
states are defined by parameter choices. We will consider two cases. In the
first case we implement both the respiratory and cardiovascular controls as
optimal controls and derive formulations for both heart rate H and ventilation rate V̇A which transfer the system from ”resting awake” xa to ”stage
4 sleep” xs in an optimal way. Thus we do not consider explicitly the respiratory control sensory system and hence equation (5) is not required until
Section 10. For this case we have only the delays in transport between the
lung and tissue compartments. Here τT is the transport delay from the lungs
to the tissue compartment (see Grodins, [15] τT = 24 s). The transport delay
from the tissue to the lung compartment τV is somewhat longer but due to
the relatively stable behavior of the venous side blood gases under normal
conditions (situations where the state variables are in the physiologically
meaningful range and without extreme variations) it is reasonable to consider τV = τT . Indeed, the dynamics of the system change almost not at all if
τv is varied, given that the venous side state variable variations are minimal
and much damped compared to arterial side changes. This approximation
of τv = τT is chosen to simplify computations.
In the second case we will incorporate V̇A into the state equations via
an empirical formula with delay relating V̇A to levels of PaCO2 , PaO2 , and
PBCO2 . These delays are in reality state dependent and nonconstant since
they depend upon blood flow Fs (t) which in turn is affected by cardiac
Cardiovascular-respiratory control system
11
output Q(t), systemic resistance, and indeed the blood gases PaCO2 and
PaO2 . However the decrease in cardiac output during the transition to stage
4 sleep is about 10% and we will assume the delays are constant.
The equilibrium equations for the system (1) to (14) determine a twodegree of freedom set of steady states. Thus it is necessary to choose steady
state values of two state variables as parameters when calculating the awake
and sleep steady states for the system. In general we choose values for PaCO2
and H. These quantities are chosen as the parameters for the equilibria
because PaCO2 is tightly controlled independently of the special situation
and H is easily and reliably measured.
In summary, the transition from the ”resting awake” steady state to
”stage 4 (NREM) sleep” is simulated by carrying out the following steps:
1. Compute the steady states ”resting awake” xa and ”stage 4 sleep” xs .
The steady states ”awake ” and ”sleep” are defined by a set of parameter
changes to be discussed in Section 7.
2. The control functions u1 and u2 which transfer system (22) from the
initial steady state ”awake”, xa , to the final steady state ”sleep”, xs , are
found as follows. We consider the linearized system around xs with initial
condition x(0) = xa , and the cost functional equation (21). The control
functions u1 and u2 are then computed such that the cost functional
is minimized subject to the linearized system. This is accomplished by
solving an associated algebraic matrix-Riccati equation which is used to
define the feedback gain matrix. In particular, u1 and u2 are given as
feedback control functions.
3. This control is used to stabilize the nonlinear system (22) defined by
equations (1) to (14). The control will be suboptimal for the nonlinear
system in the sense of Russell [56] and stabilizing.
6. Analytical considerations
We consider first the case where both heart rate H and the ventilation
rate V̇A are modeled as optimal controls. We give the mathematical setting
for the system with one delay. In this case, since V̇A is defined by optimal
control we don’t need equation (5). We carry it along here for reference in
the two delay case. The nonlinear system described by Eq. (1) to Eq. (14)
with one constant point delay is represented by the vector system (23) as
ẋ(t) = f (x(t), x(t − τT ); W s ) + B u(t),
y(t) = D x(t)
x0 = φ
(23)
where x(t) ∈ R14 is given by
x(t) = (PaCO2 , PaO2 , CvCO2 , CvO2 , CBCO2 , Pas , Pvs , Pvp , Sl , Sr , σl , σr , H, V̇A )T .
The vector W s represents the associated weights for the sleep steady state
(in general, we use the same weights for ”awake” and ”sleep” states. The
initial condition function φ ∈ C will be chosen as a constant function,
12
Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel
φ = xa , where xa is equal to the initial steady state vector ”awake” from
which the system will be transferred to the final steady state sleep xs by
the control. As controlled variables we have the observations
y(t) = D x(t) = (PaCO2 (t), 0.3PaO2 (t), Pas (t))T ,
where D ∈ M3,14 (R) is the weighting on the observations given by


1 0 0 0 0 0 ... 0
D =  0 .3 0 0 0 0 . . . 0 
0 0 0 0 0 1 ... 0
(24)
(25)
The control u(t) ∈ R2 denotes the vector
u(t) = (u1 (t), u2 (t))T = (Ḣ, V̈A )T
where B ∈ M2,14 (R). Explicitly, we compute a feedback control
u(t) = −Fm x(t)
(26)
where Fm is the feedback gain matrix.
We want to stabilize system (23) around the stage 4 sleep equilibrium
xs . Therefore our first step is a shift in the origin of the state space by
introducing new vector variables ξ and η
ξ(t) = x(t) − xs ,
η(t) = y(t) − y s .
(27)
Next, we approximate ξ˙ linearly. To this aim we replace x in (23) by ξ
and make a Taylor expansion around xs for fixed time t. We are treating
x(t − τT ) as an independent variable for the expansion. This yields
˙ = f (xs + ξ(t), xs + ξ(t − τT ); W s ) + B u
ẋ(t) = ξ(t)
= A1 ξ(t) + A2 ξ(t − τT )) + B u + o(ξ).
(28)
Here o(·) denotes the Landau symbol (h(x) = o(k(x)) :⇔ kh(x)k/kk(x)k −→
0 as x −→ ∞). Note that the original state equations were already linear
with respect to the control u. The matrices Ai ∈ M14,14 (R), i = 1, 2 are the
Jacobians of f with respect to x(t), and x(t − τT ), respectively, evaluated
at x = xs ,
∂f
(xs ; W s ),
∂x(t)
∂f
A2 =
(xs ; W s ).
∂x(t − τT )
A1 =
(29)
Analogously,
s T
η(t) = (PaCO2 (t) − PasCO2 , 0.3(PaO2 (t) − PasO2 ), Pas (t) − Pas
)
= D ξ(t).
(30)
Cardiovascular-respiratory control system
13
By neglecting terms of order o(ξ) we derive linear approximations ξ` (t)
and η` (t) for ξ(t) and η(t), respectively,
ξ̇` (t) = A1 ξ` (t) + A2 ξ` (t − τT ) + B u(t),
η` (t) = Dξ` (t),
a
(31)
s
ξ` (0) = x − x .
This is a special case of the general linear hereditary control system
ẋ(t) = Lxt + Bu(t),
y(t) = Dx(t).
t ≥ 0,
(32)
Here xt (s) = x(t + s), −h ≤ s ≤ 0, h > 0, where x(t) ∈ R14 , u(t) ∈
R2 , and y(t) ∈ R3 . Also B ∈ M2,14 (R) and D ∈ M3,14 (R). In the above
case Lxt = A1 xt (0) + A2 xt (−τT ) and h = τT . With this setting we apply
the results on approximation of feedback control for delay systems using
Legendre polynomials found in Kappel and Propst [26] (see also [27]). In this
approach the control is found for approximating systems defined on finite
dimensional subspaces of Rn x L2 [−h, 0] utilizing Legendre polynomials. For
this approach we use the first 5 Legendre polynomials. Thus, the calculated
control will be an approximate control for the actual system. In the above
paper it was shown that the control for the approximating system converges
to the control for the actual system as the approximating system converges
to the actual system in an appropriate sense.
7. Modeling Sleep
During sleep, as a result of physiological changes in the body (sometimes
referred to as the withdraw of the ”wakefulness drive”), the ventilatory
control system is less effective for a given level of blood gases. For example,
lower muscle tone during quiet sleep affects the reaction of the respiratory
muscles to control signals. This reduction in responsiveness results in V̇A
falling as one transits from the ”awake” state through stage 1 to stage 4
quiet or NREM sleep. The net effect is a decrease in PaO2 and an increase
in PaCO2 (see Shepard [59]) even though metabolic rates also fall. See, e.g.,
Krieger et al. [38], Batzel and Tran [1], or Khoo et al. [35] for further details.
In sleep, general sympathetic activity is reduced and heart rate and
blood pressure fall. Cardiac output is generally reduced though the degree
of reduction varies with situation and individual. See, e.g., Somers et al.
[60], Mancia [41], Podszus [52] and Shepard [59].
Research suggests (cf., eg., Mancia [41], Podszus [52], Bevier et al. [5],
and Somers et al. [60]), that peripheral resistance, as well as, perhaps, stroke
volume are reduced during NREM-sleep. The reduction of sympathetic nervous system activity (see [60]) in the transition from quiet awake to NREM
sleep would trigger these changes.
Given the reduction in sympathetic activity, a reduction in peripheral
resistance is a reasonable consequence and, in general, such a reduction in
14
Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel
sympathetic activity should also impact contractility, resulting in a reduced
stroke volume. It appears that counter influences such as an increase in
cardiac filling pressure (and end diastolic volume) resulting from being in
the supine position would tend to raise stroke volume. In the simulations
here, while not explicitly modeling the influence of position, we do include
the effect of the reduction in contractility caused by reduced sympathetic
activity. As a result a small drop in stroke volume is observed. However, the
major influence on Q is the drop in H.
The effect of reduced sympathetic activity on systemic resistance in
NREM sleep is implemented as follows. In the model in the awake rest
state, variation in Rs is only produced by the local control described above
(with the base line global resistance fixed by the parameter Apesk ). This
local control would not disappear during sleep but global influence on systemic resistance by reduced sympathetic activity should result in a modest
reduction in systemic resistance. To model this change, note that Apesk
which appears in relation (20), acts as a gain constant, relating oxygen concentration and resistance. This constant will be reduced to model the global
reduction due to sympathetic effects. Further research will include a more
complete picture of the two contributing and interacting controllers of Rs ,
one local, and one global. Based on steady state relations in the model, the
sympathetic influence on contractility is modeled via a reduction in parameters βl in equation (11) and βr in equation (12). In the simulations given
here, Apesk is reduced by 5%, and in the contractility equations (Bowditch
effect), the parameters βl and βr are reduced by 10% in the NREM-steady
state (thus reducing contractility). Tables such as 3 and 4 list these value
changes. In summary, the steady state ”sleep” is implemented by the following parameter changes (recall PaCO2 and H are chosen as parameters for
the system):
–
–
–
–
–
lower heart rate H,
higher PaCO2 concentration in arterial blood,
lower O2 demand (MRO2 ) and lower CO2 production (MRCO2 ),
decrease in Rs by reducing Apesk ,
decrease in contractility by reducing βl and βr .
Once the parameters are chosen, we implement the steps outlined in Section 5.
The transition to stage 4 sleep is in reality not instantaneous but takes
some minutes. We consider a transition time of 3-4 minutes. We include for
the dynamic simulation a time dependent decrease in the metabolic rates
over the transition time to stage 4 sleep and the same for the changes in
contractility and systemic resistance. We still implement the control functions u1 and u2 calculated for a time-independent linear system though
these changes are time dependent. This further reduces the optimality but
the thereby obtained (suboptimal) control still stabilizes the system and is
useful for dynamic studies.
Cardiovascular-respiratory control system
15
The sleep dependent changes in contractility (βl and βr ), the metabolic
rates, and resistance (Apesk ) are assumed to be mostly accomplished by
stages one and two. This assumption is made for purposes of exploring
the dynamics of transition and because not much is available in the literature about the actual time course of these parameter changes changes in
sleep transition. With this model it is possible to explore various parameter
change time courses.
8. One delay simulations
In the first simulation we consider the transition between xa and xs for
a normal adult with slightly elevated Rs . We assume that heart rate H
falls from 75 to 68 bpm and that PaCO2 rises from 40 mmHg to 44 mmHg.
We will apply the calculated control for the linear system to the nonlinear
system and in this case, the control will be suboptimal but stabilizing. All
calculations are performed using Mathematica 3.0 Tool boxes. The Mathematica package NDelayDSolve by Allen Hayes gives the numerical solution
of delay-differential equations.
Tables 3 and later parameter tables give the chosen parameters used for
modeling given conditions such as the ”resting awake state”, ”NREM sleep
state” or ”congestive heart failure state”. Table 4 and later steady state
variable tables give the steady states computed from the model with the
chosen parameters. In this case Tables 3 and 4 give values for resting awake
and stage 4 sleep with optimal control for a normal adult. Tables 18 to 20
in the appendix give some comparison values from the literature.
Table 3. Optimal control parameters: normal adult sleep transition
Parameter
Apesk
βl
βr
H
MRCO2
MRO2
PaCO2
τT
Awake
147.16
85.89
2.083
75.0
0.266
0.310
40.0
24.0
Sleep
139.80
77.30
1.87
68.0
0.224
0.260
44.0
24.0
Figures (2) thru (7) give the dynamics of the system produced by the
control for the optimal control case.
Using the above parameter assumptions, the steady state values for
”resting awake” and ”stage 4 sleep” are calculated from the model and given
in Table 4. The qualitative changes in steady state values derived from the
model agree with observed behavior of the cardiovascular-respiratory control system. Quantitatively, the simulated values fall within cited ranges
16
Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel
Table 4. Optimal control steady states: normal adult sleep transition
Steady State
H
Pas
Pap
Pvs
Pvp
PaCO2
P aO 2
PvCO2
P vO 2
Ql
Qr
Rs
Sl
Sr
V̇A
Vstr,l
Vstr,r
Awake
75.00
101.7
17.18
3.619
7.477
40.0
103.38
48.29
34.46
4.938
4.938
19.86
71.999
5.488
5.739
0.0659
0.0659
Sleep
68.00
87.04
16.00
3.912
7.486
44.0
98.92
51.95
35.23
4.335
4.335
19.18
58.751
4.478
4.393
0.0637
0.0637
mm Hg
110
mm Hg
46
44
105
PACO2
PAO2
100
42
95
40
90
38
85
1
2
3
4
5
minutes
1
2
3
4
5
minutes
Fig. 2. Optimal control: normal case
mm Hg
52
51
50
49
48
47
46
mm Hg
35.4
35.2
35
34.8
34.6
34.4
34.2
PVCO2
1
2
3
4
5
minutes
PVO2
1
2
3
4
5
minutes
Fig. 3. Optimal control: normal case
(see below) of commonly reported values for the physiological conditions we
are considering. We note that there exists a variety of response combinations for various individuals requiring a parameter identification if specific
data is compared. The model predicts decreases in Pas and V̇A as experimentally observed in the sleep state (see, e.g., Krieger et al. [38], Phillipson
Cardiovascular-respiratory control system
17
mm Hg
105
mm Hg
4
100
3.8
95
3.6
PAS
90
PVS
3.4
85
3.2
1
2
3
4
5
minutes
1
2
3
4
5
minutes
Fig. 4. Optimal control: normal case
mm Hg
18
mm Hg
7.575
7.55
7.525
7.5
7.475
7.45
7.425
17.5
17
16.5
PAP
16
15.5
1
2
3
4
5
minutes
PVP
1
2
3
4
5
minutes
Fig. 5. Optimal control: normal case
litersper min
5.5
liters
0.068
0.067
0.066
5
VSTR
0.065
4.5
0.064
4
0.063
1
2
3
4
5
minutes
QL
3.5
1
2
3
4
5
minutes
Fig. 6. Optimal control: normal case
[50], Podszus [52], Somers et al. [60], and Mateika et al. [43]). Decreases in
Q and stroke volume as reported in Shepard [59] or Schneider et al. [57]
are indicated by the model. The drop in PaO2 and increase in PaCO2 is consistent with data provided in Koo et al. [37], Phillipson [50], and Shepard
[59]. Further, the model reflects the drop in systemic resistance as well as
predicting an increase for Pvs . See Tables 18 and 20 in the appendix for a
summary of state values derived from research literature for the awake and
NREM sleep states.
Using the parameter values and steady states from Tables 3 and 4 we
calculate the controls u1 and u2 which transfer the system from xa to xs .
Reference data can be found in Burgess et al. [8] and Bevier et al. [5] for
the dynamic time course of various state transitions. The data provided in
Burgess et al. [8] suggest a disproportionate drop in H during the initial
phase of sleep onset. Model simulations also show that the control H declines
18
Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel
liters per minute
6
5.5
5
4.5
4
3.5
º
.
VA
1
2
3
4
5
minutes
bpm
80
77.5
75
72.5
70
67.5
65
62.5
H
1
2
3
4
5
minutes
Fig. 7. Optimal control: normal case
with the largest part of the decline occurring during the initial stage of sleep
onset consistent with the results in Burgess et al. [8]. In contrast, there is
a continual smooth decline in Vstr,l and Q, more or less unaffected by sleep
stage, The model can be used to explore various parameter effects on these
time courses.
As can be seen by comparison with the simulations presented in [64]
little dynamical difference is produced by introduction of delay into the
mass balance equations of the respiratory system. Indeed, even if the delay
is increased by a factor of 15, no significant differences in dynamics appears.
The important delay introducing dynamic instability into the system is
the delay in the feedback control loop of the respiratory control system (see
e.g. [2,3]). We will consider this in Section 10. An important physiological
condition which increases transport delay in the feedback control loop is
found in congestive heart failure.
9. Congestive heart condition and transport delay
Heart failure is a generic term covering a number of physiopathologies in
heart performance which result in a decrease in general blood flow. This
condition is often referred to as congestive heart failure to focus on a main
consequence, namely pulmonary or systemic edema.
Chronic heart failure is to be distinguished from ”heart attack” which
results in blockage in coronary blood flow or blockage in ventricular or atrial
flow. Heart failure can be categorized in a number of ways: forward versus
backward, left versus right, systolic versus diastolic, and low output versus
high output. These classifications are not uniformly consistently applied
but they are useful in focusing on specific features of heart failure. Given
the inherent connectedness of the circulation such divisions are to some
extent artificial and indeed these distinctions can overlap. For example, in
the forward/backward division:
– forward failure focuses on reduced blood delivery, reduced ejection of
blood from the ventricles and insufficient Q for metabolic needs.
– backward failure focuses on reduced filling of the ventricles, reduced
emptying of the venous system, or reduced Q unless high ventricular
pressures exist.
Cardiovascular-respiratory control system
19
On the other hand, in the division systolic/diastolic:
– systolic failure focuses on insufficient systolic action, often impaired contractility, and consequently reduced ejection fraction and Q.
– diastolic failure refers to the impairment of ventricular filling without
necessarily an impairment of ejection fraction.
These classifications are subdivided into left and right heart categories
and ”typical” clinical heart failure is due to impairment of left ventricular
function and reflects systolic disfunction and forward failure. Causes of heart
failure include any condition which reduces heart performance such as:
– myocardial damage which weakens the myocardial muscle,
– insufficient coronary blood flow,
– reduced myocardial contractility.
In general, heart failure implies the consequence that the heart fails to
provide sufficient blood flow to meet the metabolic needs of the body. In
most cases, this means that the heart exhibits a deterioration of the heart’s
pumping ability. Pumping impairment that is due to a reduction in contractility is a consequence of the heart muscle being damaged or weakened
in some way.
In chronic heart failure, there is a progressive deterioration in heart
function over time which is the reason the condition is so serious. The condition becomes progressively more sever due to the compensatory mechanisms which try to maintain normal cardiovascular function. In a left heart
failure scenario, for example, if heart muscle is damaged so that contractility is reduced, stroke volume and cardiac output will be decreased. Arterial
blood pressure falls due to the impaired pumping efficiency of the heart. The
baroreceptors, sensing reduced pressure, trigger compensatory sympathetic
system activity and vasoconstriction. These responses can produce significant elevation of afterload, which can further reduce stroke volume. Over
time, the added stress to the heart results in damaging cardiac muscle compensatory changes (remodeling) which further weakens heart function. Thus
the deterioration in heart function is self-reinforcing. This form of heart failure is referred to as chronic in contrast to acute heart failure which is the
result of heart damage occurring over a short time frame.
The kidneys may also respond to reduced cardiac function by inducing fluid retention to increase blood volume. This compensatory response
is triggered by the perceived reduction in circulating blood volume and
acts to raise blood pressure. This fluid retention will increase preload or
filling pressure but the increased pressure and excess fluids can cause pulmonary or systemic fluid congestion and edema. In left ventricular failure,
the reduced left ventricular function results in blood accumulating in the
pulmonary venous system (raising pulmonary venous pressure) and can result in significant pulmonary congestion and difficulty in breathing. Hence,
the term ”congestive heart failure” is often used, though not every form of
heart failure exhibits this quality.
20
Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel
Given the interconnectedness of the circulatory system, progressive deterioration in one ventricle can lead to the other ventricle becoming impaired
and the occurrence of simultaneous left and right heart failure. Diastolic
and systolic failure can also occur together. In this paper we will refer to
left ventricular failure as that clinical condition of failure of forward blood
delivery due to reduced systolic function and reduced contractility of the
heart tissue.
An example of stable pressure states for four types of clinical heart
failure are given in Table 5. For example, in clinical left heart failure (left
ventricular failure):
–
–
–
–
–
cardiac output is reduced 20 to 50%;
significant elevation in pulmonary venous pressure occurs;
modest elevation in pulmonary arterial pressure is observed;
modest elevation in systemic venous pressure occurs;
no significant change in systemic arterial pressure is found.
Table 5. Congestive heart failure pressure changes at left/right ventricular
(LV/RV) failure and left/right backward (LB/RB) failure.
Condition
LV failure
RV failure
LB failure
RB failure
Pas
Pvs
Pap
Pvp
no
no
no
no
↑ moderately
↑ significantly
no change
↑ significantly
↑ moderately
small change
no change
no change
↑ significantly
small change
↑
no change
change
change
change
change
The impairment of the heart’s pumping action will be modeled by a
reduction in contractility and consequently the ejection fraction. Given the
effects of remodeling of the heart tissue due to the stress on the system, we
could also include a reduction in the ventricular compliance parameter in
Eq. (19) as a contributing factor to the reduction of stroke volume.
Tables 6 and 7 give the parameters and computed steady states for resting awake and stage 4 sleep for serious chronic left ventricular heart failure.
In this model, the left contractility Sl is reduced by 65% from normal. Given
that there is little change in arterial blood pressure, this implies a similar
reduction in ejection fraction consistent with clinical observations found in
Niebauer et al. (1999) [44]. We also assume a small drop in right contractility of 8% from normal. Due to the compensatory mechanism described
above, the systemic resistance Rs parameter, Apesk , is increased by 35%.
Heart rate H is increased by 15%. We set PaCO2 to 40.5 mmHg, at the upper
end of values reported in Javaheri (1999) [23]. Pulmonary resistance Rp is
also increased by 10% (see, e.g., Moraes et al. (2000) [12]). In this chronic
condition water retention and other mechanisms act to increase total blood
volume and we assume V0 is increased by 25%. The increase in V0 acts to
Cardiovascular-respiratory control system
21
raise Pvs . These assumptions are consistent with the observations in Parmley [47] as well as Chiariello and Perone-Filardi [10]. As a consequence of
these changes Q decreases by 20% and hence transport delay is increased
by about 25% (we ignore the effects of the cerebral blood flow). The cardiovascular steady state values for this case and further simulations presented
later (see Section 11) can be compared with values presented in Tsuruta et
al. (1994) [66]. In that paper, a model is developed and used to identify cardiovascular parameters relating to the four classes of severity of heart failure
as defined by the New York Heart Association. The parameter estimation
depended on steady state values of H, Pas , Pap , Pvs , Pvp , V0 , and cardiac
output. The values for these state variables in the four classes depended on
interpolation from certain known values. Among the parameters which were
identified were the vascular resistances.
Table 6. Optimal control parameters: chronic left ventricular heart failure sleep
transition
Parameter
Apesk
βl
βr
Rp
V0
H
PaCO2
τT
Awake
198.7
25.8
1.67
2.16
6.25
86.02
40.5
30.0
Sleep
188.7
23.19
1.50
2.16
6.25
78.02
44.5
30.0
In contrast, Tables 8 and 9 give the parameters and computed steady
states for resting awake and stage 4 sleep for left heart failure where there
is no increase in blood volume V0 as might be the case when there is acute
heart failure. A small decrease in Sr is assumed. In this case, no change is
assumed in Rp or PaCO2 . In this case, no increase in Pvs is seen but rather
a drop occurs.
Figures (8) thru (13) give the dynamics of the control for the optimal
control case of acute left heart failure.
10. Modeling two delays in the state space
Previously we have modeled the transition to sleep considering ventilation
as an optimal control. We are now going to use formula (33) which describes
an empiric relation between V̇A and the blood gas partial pressures PaCO2 ,
PBCO2 , and PaO2 . Thus V̇A becomes incorporated into the state equations
and we can consider the transport delay in this control. We consider a single
delay for both the peripheral and central controls. This is reasonable as the
transport delay in the central controller is only about 15% more than the
22
Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel
Table 7. Optimal control steady states: chronic left ventricular heart failure sleep
transition
Steady State
H
Pas
Pap
Pvs
Pvp
PaCO2
P aO 2
PvCO2
P vO 2
Ql
Qr
Rs
Sl
Sr
V̇A
Vstr,l
Vstr,r
Awake
86.02
98.4
23.19
3.74
14.58
40.5
102.8
50.78
29.73
3.98
3.98
23.79
24.77
5.04
5.67
0.0463
0.0463
Sleep
78.02
83.72
21.94
4.07
14.46
44.5
98.35
54.46
30.36
3.46
3.46
23.02
20.22
4.11
4.34
0.0444
0.0444
Table 8. Optimal control parameters: acute left heart failure sleep transition
Parameter
Apesk
βl
βr
Rp
V0
H
PaCO2
τT
Awake
198.7
25.8
1.77
1.965
5.0
86.02
40.0
32.0
Sleep
188.7
23.19
1.59
1.965
5.0
78.02
44.5
32.0
peripheral controller delay (Khoo, [31], τp = 6s and τc = 7s). Furthermore,
it is the peripheral control which is responsible for instability in the control
(Khoo et al. [31], Batzel and Tran, [2,3]). A relationship describing the
dependence of V̇A on PaCO2 , PaO2 and PBCO2 is given by
V̇A (t) = Gp e
−0.05PaO (t−τp )
max(0, PaCO2 (t − τp ) − Ip )
MRBCO2
+Gc max(0, PBCO2 (t) −
− Ic ).
KCO2 FB
2
(33)
The first term above describes the effect on ventilation of the blood gases
PaCO2 and PaO2 as measured by peripheral sensors located in the carotid
artery. This will be referred to as the peripheral control. The second term
describes the effect of the brain CO2 level (PBCO2 ) and will be referred to
as the central control. This formula taken from Khoo et al. [31] is based on
Cardiovascular-respiratory control system
23
Table 9. Optimal control steady states: acute left heart failure sleep transition
Steady State
H
Pas
Pap
Pvs
Pvp
PaCO2
P aO 2
PvCO2
P vO 2
Ql
Qr
Rs
Sl
Sr
V̇A
Vstr,l
Vstr,r
Awake
86.02
86.6
19.24
2.77
11.97
40.0
103.38
51.06
28.12
3.70
3.70
22.64
24.77
5.35
5.739
0.0430
0.0430
Sleep
78.02
74.08
18.30
3.02
11.96
44.5
98.35
55.17
28.83
3.23
3.23
22.00
20.22
4.37
4.34
0.0414
0.0414
mm Hg
110
mm Hg
46
105
PACO2
44
PAO2
100
42
95
40
90
38
85
1
2
3
4
5
minutes
1
2
3
4
5
minutes
Fig. 8. Optimal control: acute left heart failure case
experimental observations such as presented in the Handbook of Physiology
[14]. A transport delay τp between the lungs and peripheral control appears
in this equation. Note that Ip and Ic denote cutoff thresholds, so that the
respective ventilation terms become zero when the quantities fall below the
thresholds.
Ventilatory dead space effects are accounted for by defining the quantity
V̇A = K · V̇E where V̇E is minute ventilation and K is a constant smaller
than one. In this way effective ventilation is reduced by a fixed dead space
percent which corresponds to modeling change in ventilation as a change
in rate of breathing. This is implemented here by a scale reduction in the
control gains Gc and Gp . See, e.g., Batzel and Tran [3].
The optimal control now only models the cardiovascular control. The
respiratory control is given by an empirical formula with delay.
24
Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel
mm Hg
56
54
52
50
48
46
44
mm Hg
29.5
PVCO2
29
PVO2
28.5
28
27.5
1
2
3
4
5
minutes
1
2
3
4
5
minutes
Fig. 9. Optimal control: acute left heart failure case
mm Hg
90
mm Hg
85
3.4
3.2
3
2.8
2.6
2.4
2.2
PAS
80
75
70
65
1
2
3
4
5
minutes
PVS
1
2
3
4
5
minutes
Fig. 10. Optimal control: acute left heart failure case
mm Hg
mm Hg
12.4
20
12.2
PAP
19
PVP
12
18
11.8
17
11.6
1
2
3
4
5
minutes
11.4
1
2
3
4
5
minutes
Fig. 11. Optimal control: acute left heart failure case
A model describing the transition to sleep was given by Khoo et al. [35].
During sleep, ventilatory drive is diminished by reducing the sleep gain factor Gs and there is an increase in a shift term Kshif t altering the operating
point of ventilation. The effective drive during sleep V̇sleep is described by
V̇sleep (t) = Gs (t)[max(0, V̇awake (t) − Kshif t (t))].
(34)
The time dependencies for Gs and Kshif t reflect the smooth change
in these parameters that occurs in the transition from ”awake” state to
Cardiovascular-respiratory control system
liters
0.046
0.044
VSTR
0.042
0.04
0.038
1
2
3
4
5
minutes
25
liters per min
4
3.8
3.6
3.4
3.2
3
2.8
2.6
2.4
1
QL
2
3
4
5
minutes
Fig. 12. Optimal control: acute left heart failure case
liters per minute
6
5.5
5
4.5
4
3.5
.
VA
º
1
2
3
4
5
minutes
bpm
90
87.5
85
82.5
80
77.5
75
72.5
H
1
2
3
4
5
minutes
Fig. 13. Optimal control: acute left heart failure case
”stage 4 quiet sleep ”. Gs is set during the awake state at 1 and reduces
smoothly to a minimum (normally 0.6) at stage 4 sleep. Kshif t begins at 0
and increases to a maximum (normally about 4 mmHg) by the beginning of
stage 1 sleep. These changes reflect the reduction in the normal ventilatory
response V̇awake as a result of physiological changes during sleep. Once stage
4 sleep is reached these values are constant. For these simulations we use a
base line transit time to ”stage 4 sleep” to be three minutes. The changes
in Gs and Kshif t will be modeled by incorporating exponential functions
which change smoothly through the various sleep stages between awake and
stage 4 NREM sleep. The parameters in these expressions can be adjusted
to simulate an essentially linear decrease over the entire transition from
awake to stage 4 sleep or bias the decrease to the early stages of sleep.
Similar decreases for the metabolic rates, sleep contractility, and systemic resistance reflecting the physiological changes during sleep transition
(discussed above) are incorporated. The system is nonautonomous, however
we still implement the control functions u1 and u2 as calculated for a timeindependent linear system around the final steady state ”stage 4 sleep”.
This reduces the optimality of the control for the original nonlinear system
but the thereby obtained (suboptimal) control still stabilizes the system and
is useful for dynamic studies.
26
Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel
Thus we now determine a control function u1 such that the cost functional
Z
∞
0
s 2
qas (Pas (t) − Pas
) + q1 u1 (t)2 dt
(35)
is minimized under the restriction
ẋ(t) = f (x(t), x(t − τp ), x(t − τT ); W s ) + B u(t),
x0 = φ.
y(t) = D x(t)
(36)
Clearly τp < τT and, again, φ ∈ C where C denotes C([−τT , 0], R14 ).
In an analogous fashion with the one delay case we form the linearized
system now expanding the system using two delays.
˙ = f (xs + ξ(t), xs + ξ(t − τp ), xs + ξ(t − τT ); W s ) + B u
ẋ(t) = ξ(t)
= A1 ξ(t) + A2 ξ(t − τp ) + A3 ξ(t − τT ) + B u + o(ξ).
(37)
The matrices Ai ∈ M14,14 (R), i = 1, 2, 3 are the Jacobians of f with respect
to x(t), x(t − τp ), and x(t − τT ) , respectively, evaluated at x = xs ,
∂f
(xs ; W s ),
∂x(t)
∂f
A2 =
(xs ; W s ),
∂x(t − τp )
∂f
(xs ; W s ).
A3 =
∂x(t − τT )
A1 =
(38)
Analogously,
s T
η(t) = (Pas (t) − Pas
) = D ξ(t).
(39)
By neglecting terms of order o(ξ) we get linear approximations ξ` (t) and
η` (t) for ξ(t) and η(t), respectively,
ξ̇` (t) = A1 ξ` (t) + A2 ξ` (t − τp ) + A3 ξ` (t − τT ) + B u(t),
η` (t) = Dξ` (t),
r
(40)
s
ξ` (0) = x − x .
Again, we apply the results on approximation of feedback control for delay
systems using Legendre polynomials found in Kappel and Propst [26].
Cardiovascular-respiratory control system
27
Table 10. V̇A empirical control parameters: normal adult sleep transition
Parameter
Gc
Gp
Gs
Kshif t
IC
IP
H
MRCO2
MRBCO2
MRO2
Apesk
βl
βr
τp
τT
S 4 transit
Awake
1.44
30.24
1.0
0
35.5
35.5
75.02
0.266
0.042
0.310
147.16
85.89
2.083
7.8
24.0
-
Sleep
1.44
30.24
0.6
4.2
35.5
35.5
68.02
0.224
0.040
0.260
139.80
77.30
1.874
7.8
24.0
3 min
Table 11. V̇A empirical control steady states: normal adult sleep transition
Steady State
H
Pas
Pap
Pvs
Pvp
PaCO2
P aO 2
PvCO2
P vO 2
PBCO2
Ql
Qr
Rs
Sl
Sr
V̇A
Vstr,l
Vstr,r
Awake
75.02
101.77
17.18
3.618
7.478
39.16
104.37
47.44
34.50
47.23
4.938
4.938
19.88
72.02
5.49
5.86
0.0658
0.0658
Sleep
68.02
87.14
16.01
3.909
7.489
42.67
100.47
50.62
35.30
50.34
4.334
4.334
19.21
58.77
4.48
4.53
0.0637
0.0637
11. Simulations with two delays
Tables 10 and 11 give the parameters and computed steady states for resting
awake and stage 4 sleep for a normal adult with borderline elevated Rs
values and with moderate sleep transition profile. From this point on, we
are using the empirical control for V̇A while maintaining the optimal control
for the cardiovascular system. We will refer to this case as the V̇A empirical
case. In all figures, we exhibit simulations for the first few minutes to focus
28
Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel
on the early transition dynamics in detail. Simulations of longer duration
clearly show the stabilizing influence of the control.
Figures (14) thru (20) give the dynamics of the control for the normal V̇A
empirical control case. For this case the transition to stage 4 sleep is assumed
to be 3 minutes. Furthermore, the shift Kshif t is assumed to occur by stage
1 (one fourth of the transition time) as in [35]. Gs reduces smoothly from
stage 1 to stage 4 with most of the change occurring in the first two stages.
As in the optimal case the reductions in βl , βr , metabolic rates, and Apesk
are also assumed to be significantly reduced by stage 1. This assumption is
made for purposes of exploring the dynamics of transition and because not
much is known about the actual time course of these parameter changes in
sleep transition.
mm Hg
44
mm Hg
110
105
100
95
90
85
80
75
43
42
41
PACO2
40
39
1
2
3
4
5
minutes
PAO2
1
2
3
4
5
minutes
Fig. 14. V̇A empirical control: normal case
mm Hg
52
mm Hg
51
35.4
35.2
35
34.8
34.6
34.4
34.2
PVCO2
50
49
48
47
46
1
2
3
4
5
minutes
PVO2
1
2
3
4
5
minutes
Fig. 15. V̇A empirical control: normal case
Figures (21) thru (24) give the dynamics of the control for the normal
V̇A empirical control case with a reduced sleep transition time. For this
case the transition to stage 4 sleep is assumed to be 2 minutes. The shift
Kshif t is again assumed to occur by stage one (one fourth of the transition
time) and the reductions in Gs , βl , βr and the metabolic rates are assumed
to be reduced as in the previous case. The shift Kshif t is increased to 5.2
and the gain Gs at stage 4 is 0.4. The quicker transition time and larger
shift create a deeper drop in the ventilation rate with sleep onset than in
Cardiovascular-respiratory control system
29
mm Hg
105
mm Hg
4
100
3.8
95
3.6
PAS
90
PVS
3.4
85
3.2
1
2
3
4
5
minutes
1
2
3
4
5
minutes
Fig. 16. V̇A empirical control: normal case
mm Hg
18
mm Hg
7.575
7.55
7.525
7.5
7.475
7.45
7.425
17.5
17
16.5
PAP
16
15.5
1
2
3
4
5
PVP
minutes
1
2
3
4
5
minutes
Fig. 17. V̇A empirical control: normal case
liters per min
5
mm Hg min per lit
20
4.8
4.6
RS
19.5
FS
19
4.4
18.5
4.2
18
1
2
3
4
5
minutes
1
2
3
4
5
minutes
Fig. 18. V̇A empirical control: normal case
the previous case. This behavior will be compared now with the congestive
heart condition case which includes an increased delay time.
Tables 12 and 13 give the parameters and computed steady states for
resting awake and stage 4 sleep for the left ventricular heart failure case.
In this model, the left contractility Sl is reduced by 65% from normal. For
ejection fraction values in heart failure see Niebauer et al. (1999) [44].
We also assume a small drop in right contractility of 8% from normal.
Systemic resistance Rs is increased by 35% and pulmonary resistance Rp by
10% ([12]). Heart rate H is increased by 15%. The PvCO2 increase and PvO2
decrease are a consequence of the reduced cardiac output. In this chronic
condition water retention and other mechanisms act to increase total blood
volume and we assume V0 is increased by 23%. The increase in V0 acts to
raise Pvs . As a consequence of these changes Q decreases by 20% and hence
transport delay is increased by about 25%.
30
Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel
liters
0.068
liters per min
5.5
0.067
0.066
0.065
4.5
0.064
4
0.063
1
2
3
4
5
QL
5
VSTR
minutes
3.5
1
2
3
4
5
4
5
minutes
Fig. 19. V̇A empirical control: normal case
bpm
76
liters per minute
6
5.5
5
4.5
4
3.5
1
º
74
.
VA
H
72
70
68
2
3
4
5
minutes
1
2
3
minutes
Fig. 20. V̇A empirical control: normal case
mm Hg
44
mm Hg
110
105
100
95
90
85
80
75
43
42
41
PACO2
40
39
1
2
3
4
5
minutes
PAO2
1
2
3
4
5
minutes
Fig. 21. V̇A empirical control: fast sleep case
Tables 14 and 15 give the parameters and computed steady states for
resting awake and stage 4 sleep where the congestive heart condition involves
significant reduction in contractility of both the left and right ventricle.
Often it is the case that deterioration on one side of the heart (here the left
side) will eventually extend to deterioration of function on the other side
[58]. We maintain the CO2 ventilation thresholds simulating an operating
point of PaCO2 in the middle range of values given in Javaheri (1999) [23].
In general, PaCO2 levels in congestive heart patients are little changed from
levels found in normal individuals even when there is reduced exchange
efficiency in the lungs due to congestion. See, e.g., Sullivan et al. (1988)
[61].
As a consequence of the reduced cardiac output the transport delay is
now increased by 50%. For comparative state values in the case of severe
congestive heart failure, see Bruschi et al. (1999) [7] and Bocchi et al. (2000)
Cardiovascular-respiratory control system
mm Hg
52
31
mm Hg
51
35.4
35.2
35
34.8
34.6
34.4
34.2
PVCO2
50
49
48
47
46
1
2
3
4
5
minutes
PVO2
1
2
3
4
5
minutes
Fig. 22. V̇A empirical control: fast sleep case
mm Hg
18
mm Hg
7.575
7.55
7.525
7.5
7.475
7.45
7.425
17.5
17
16.5
PAP
16
15.5
1
2
3
4
5
PVP
minutes
1
2
3
4
5
4
5
minutes
Fig. 23. V̇A empirical control: fast sleep case
bpm
76
liters per minute
6
74
º
.
VA
5
H
72
4
70
3
68
2
1
2
3
4
5
minutes
1
2
3
minutes
Fig. 24. V̇A empirical control: fast sleep case
[6] for values of Q, Hanly et al. (1993) [21] for values of transport delay, and
Bocchi et al. (2000) [6] for comparative values of H, Rs , and Vstr . See also
Table 22 in the appendix and Tsuruta et al. (1994) [66] and Hambrecht et
al. (2000) [20] for comparative Rs values. Arterio-venous oxygen content
difference for the severe CHF case is consistent with Kugler et al. (1982)
[39]. The very low contractility implies (given the small change in pressure)
an ejection fraction consistent with clinical observations found in Niebauer
et al. (1999) [44] for very severe heart failure cases.
It is well known that delays in feedback control can create instability
in a control system. In congestive heart failure, the reduced cardiac output
induces an increased transport delay which will reduce the efficiency of the
central and peripheral controllers of ventilation. This reduced efficiency is
due to the increased time it takes for blood gases to be transported from the
site where these blood gas levels are adjusted (the lungs) to the sensory sites
32
Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel
Table 12. V̇A empirical control parameters: left ventricular heart failure sleep
transition
Parameter
Apesk
βl
βr
Rp
V0
H
IC
IP
Gs
Kshif t
τp
τT
Awake
198.66
25.77
1.67
2.16
6.25
86.02
35.5
35.5
1.0
0
9.75
30.0
Sleep
188.73
23.19
1.50
2.16
6.25
78.02
35.5
35.5
0.6
4.2
9.75
30.0
Table 13. V̇A empirical control steady states: left ventricular heart failure sleep
transition
Steady State
H
Pas
Pap
Pvs
Pvp
PaCO2
P aO 2
PvCO2
P vO 2
PBCO2
Ql
Qr
Rs
Sl
Sr
V̇A
Vstr,l
Vstr,r
Awake
86.02
98.49
23.19
3.74
14.59
39.16
104.37
49.44
29.77
47.23
3.98
3.98
23.82
24.77
5.035
5.86
0.046
0.046
Sleep
78.02
83.81
21.94
4.06
14.46
42.67
100.47
52.63
30.43
50.34
3.46
3.46
23.06
20.22
4.11
4.53
0.044
0.044
where these levels are measured. One form of respiratory instability associated with CHF is a form of periodic breathing (PB) known as Cheyne-Stokes
respiration (CSR). This form of involuntary respiration involves periods of
regular waxing and waning of tidal volume interspersed with central apnea
(CA). Cheyne-Stokes respiration seems to be a complicating factor for CHF
but the actual mechanisms inducing CSR in congestive heart patients are
still under active investigation. The increased feedback delay due to reduced
cardiac output, in conjunction with other factors may be sufficient to contribute to the onset, characteristics, or persistence of central sleep apnea,
Cardiovascular-respiratory control system
33
Table 14. V̇A empirical control parameters: left and right ventricular heart failure
sleep transition
Parameter
Apesk
βl
βr
Rp
V0
H
IC
IP
Gs
Kshif t
τp
τT
S 4 transit
Awake
250.17
12.88
1.46
2.16
6.9
92.02
35.5
35.5
1.55
0
11.6
36.0
-
Sleep
237.7
11.60
1.31
2.16
6.9
80.02
35.5
35.5
0.465
5.5
11.6
36.0
2 min
Table 15. V̇A empirical control steady states: left and right ventricular heart
failure sleep transition
Steady State
H
Pas
Pap
Pvs
Pvp
PaCO2
P aO 2
PvCO2
P vO 2
PBCO2
Ql
Qr
Rs
Sl
Sr
V̇A
Vstr,l
Vstr,r
Awake
92.02
90.82
26.74
3.60
19.55
37.95
105.77
50.25
25.74
46.02
3.33
3.33
26.22
13.25
4.71
6.05
0.0362
0.0362
Sleep
80.02
72.78
25.17
4.05
19.14
44.44
98.41
56.81
25.49
52.12
2.79
2.79
24.67
10.37
3.69
4.35
0.0348
0.0348
PB, or CSR associated with CHF. See, e.g., Hall et al. (1996) [19], Pinna
et al. (2000) [51] and Cherniack (1999) [9]. For analytical results see Batzel
and Tran (2000) [3].
Figures (25) thru (30) give the congestive heart failure dynamics for the
V̇A empirical control case simulating transition to sleep with fast transition
parameters and in this case we also assume an awake feedback gain which is
50% higher than normal which in effect increases CO2 sensitivity. Increases
in CO2 sensitivity have been reported in cases of central sleep apnea in
34
Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel
heart failure. See, e.g., Topor et al. (2001) [65] and Javaheri (1999) [23].
We simulate a quick transition to sleep with sleep gain Gs reduced by 70%
in stage 4 sleep as compared to the normal reduction of 40%. We further
increase the shift term Kshif t by 33% (to 5.5 mmHg). Fast sleep onset times
can occur in patients with fragmented sleep cycles as occurs when multiple
sleep apneas induce arousal and sleep disturbance. See, e.g., Bennet et al.
(1998) [4]. The change in sleep control parameters in this case simulates an
increased influence of the sleep state on control effectiveness.
These deviations in standard operating points drive ventilation to near
apnea and exhibit one mechanism for inducing Cheyne-Stokes respiration
(CSR) and central sleep apnea. The oscillatory behavior is induced by the
increased delay as can be observed by comparing this case with the fast
transition case for a normal adult represented in Figures (21) thru (24).
Javaheri (1999) [23] and Quaranta et al. (1997) [53] indicate that a number of factors influencing control stability (such as higher CO2 sensitivity
and circulation delay) may contribute to central sleep apnea and CSR in
congestive heart patients. Lorenzi-Filho et al. (1999) [40] report that reductions in PaCO2 sensed at the peripheral chemoreceptors can also trigger
central apneas during Cheyne-Stokes respiration. It is clear that the interaction of various respiratory factors can act in complex ways to influence
the production of CSR and apnea in congestive heart failure.
The larger reduction in sleep control gain Gs in this simulation actually
acts to reduce the magnitude of the oscillatory cycles. On the other hand,
simulations indicate and, in general, theory confirms that the higher control gain (CO2 sensitivity) prolongs and exaggerates oscillatory behavior.
Likewise, a longer time course in the reduction in control gain from stage
1 to stage 4 sleep (thus maintaining higher gain for a longer time) would
contribute to unstable behavior. It is the degree and speed of the shift
Kshif t that is responsible for the initial steep drop in ventilation which can
trigger apnea and repetitive cycles similar to CSR and the increased delay
reinforces and perpetuates the oscillatory behavior.
mm Hg
110
mm Hg
50
PACO2
47.5
100
45
PAO2
90
42.5
80
40
37.5
70
1
2
3
4
5
minutes
1
2
3
4
5
minutes
Fig. 25. V̇A empirical control: severe left and right ventricular failure sleep case
Cardiovascular-respiratory control system
mm Hg
35
mm Hg
27
65
26.5
PVCO2
60
26
55
PVO2
25.5
50
25
45
24.5
1
2
3
4
5
minutes
1
2
3
4
5
minutes
Fig. 26. V̇A empirical control: severe left and right ventricular failure sleep case
mm Hg
95
mm Hg
90
4.4
4.2
4
3.8
3.6
3.4
3.2
PAS
85
80
75
70
65
1
2
3
4
5
minutes
PVS
1
2
3
4
5
minutes
Fig. 27. V̇A empirical control: severe left and right ventricular failure sleep case
mm Hg
28
mm Hg
20
19.75
27
PAP
26
PVP
19.5
19.25
19
25
18.75
24
18.5
1
2
3
4
5
minutes
1
2
3
4
5
minutes
Fig. 28. V̇A empirical control: severe left and right ventricular failure sleep case
12. Conclusion
In this paper we have considered a model of the cardiovascular-respiratory
control system with constant state equation delays. The model utilizes an
optimal control approach to represent the complex control features of the
cardiovascular component in this system. The respiratory control is considered both from an optimal control approach and from an empirical approach
which introduces a respiratory feedback delay into the state equations. The
model was applied to study the transition from the awake state to NREM
sleep for normal individuals and for individuals suffering from congestive
heart problems. The model steady states are consistent with observation
both for the normal and congestive heart states. The dynamical simulations
show that the transport delay between respiratory compartments does not
contribute to instability even at large delays. However, the transport delay
36
Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel
liters
0.038
liters per min
3.6
3.4
0.037
0.036
3.2
VSTR
QL
3
0.035
2.8
0.034
1
2
3
4
5
minutes
2.6
1
2
3
4
5
minutes
Fig. 29. V̇A empirical control: severe left and right ventricular failure sleep case
bpm
liters per minute
8
95
6
.
VA
4
90
85
º
2
-1
H
80
75
1
2
3
4
5
minutes
1
2
3
4
5
minutes
Fig. 30. V̇A empirical control: severe left and right ventricular failure sleep case
to the peripheral sensor is significant and can result in Cheyne-Stokes type
respiration for a severe congestive heart condition with certain respiratory
parameters during the transition to NREM sleep.
References
1. J. J. Batzel and H. T. Tran, Modeling Instability in the Control System for
Human Respiration: applications to infant non-REM sleep, Appl. Math. Comput. 110, pp. 1-51, 2000.
2. J. J. Batzel and H. T. Tran, Stability of the human respiratory control system.
Part1: Analysis of a two dimensional delay state-space model, J. Math. Biol.
41(1), pp. 45-79, 2000.
3. J. J. Batzel and H. T. Tran, Stability of the human respiratory control system.
Part2: Analysis of a three dimensional delay state-space model, J. Math. Biol.
41(1), pp. 80-102, 2000.
4. L. S. Bennett, B. A. Langford, J. R. Stradling, and R. J. Davies, Sleep fragmentation indices as predictors of daytime sleepiness and nCPAP response in
obstructive sleep apnea, Am. J. Resp. Crit. Care Med. 158(3), pp. 778-786,
1998.
5. W. C. Bevier, D. E. Bunnell, and S. M. Horvath, Cardiovascular function
during sleep of active older adults and the effects of exercise, Exp. Gerontol.
22(5), pp. 329-37, 1987.
6. E. A. Bocchi, A.V. Vilella de Moraes, A. Esteves-Filho, F. Bacal, J. O. Auler,
M. J. Carmona, G. Bellotti, and A. F. Ramires, L-arginine reduces heart rate
and improves hemodynamics in severe congestive heart failure, Clin. Cardiol.
23(3), pp. 205-210, 2000.
7. C. Bruschi, F. Fanfulla, E. Traversi, V. Patruno, G. Callegari, L. Tavazzi, and
C. Rampulla, Identification of chronic heart failure patients at risk of CheyneStokes respiration, Monaldi Arch. Chest Dis. 54(4), pp. 319-324, 1999.
Cardiovascular-respiratory control system
37
8. H. J. Burgess, J. Kleiman, and J. Trinder, Cardiac activity during sleep onset
, Psychophysiology Vol. 36(3), pp. 298-306, 1999.
9. N.S. Cherniack, Apnea and periodic breathing during sleep, New Engl. J. Med.
341(13), pp. 985-987, 1999.
10. M. Chiariello and P. Perrone-Filardi, Pathophysiology of heart failure,
Miner Electrolyte Metab. 25(1-2), pp. 6-10, 1999.
11. R. G. Crystal, J.B. West, et al., eds., The Lung: Scientific foundations,
Lippincott-Raven Press, Philadelphia, 1997.
12. D. L. Moraes, W. S. Colucci, and M. M. Givertz, Secondary pulmonary hypertension in chronic heart failure: the role of the endothelium in pathophysiology
and management., Circulation 102(14), pp. 1718-1723, 2000.
13. W. F. Fincham and F. T. Tehrani, A mathematical model of the human respiratory system, J. Biomed. Eng. 5(2), pp. 125-133, 1983.
14. A. P. Fishman, N. S. Cherniack, J. G. Widdicombe, and S. R. Geiger,
eds. Handbook of Physiology: The Respiratory System, Volume II, Control
of Breathing, Part 2 Am. Phys. Soc., Bethesda, Maryland, 1986.
15. F. S. Grodins, J. Buell, and A. J. Bart, Mathematical analysis and digital
simulation of the respiratory control system, J. Appl. Physiol. 22(2), pp. 260276, 1967.
16. F. S. Grodins, Integrative cardiovascular physiology: a mathematical model
synthesis of cardiac and blood vessel hemodynamics, Quart. Rev. Biol. 34(2),
pp. 93-116, 1959.
17. F. S. Grodins, Control Theory and Biological Systems, Columbia University
Press, New York, 1963.
18. A. C. Guyton, Textbook of Medical Physiology, 7th ed., W. B. Saunders Company, Philadelphia, 1986.
19. M. J. Hall, A. Xie, R. Rutherford, S. Ando, J..S. Floras, and T.D. Bradley,
Cycle length of periodic breathing in patients with and without heart failure,
Am. J. Respir. Crit. Care. Med. 154 (2 pt. 1), pp. 376-381, 1996.
20. R. Hambrecht, S. Gielen, A. Linke, E. Fiehn, J. Yu, C. Walther, N. Schoene,
and G. Schuler, Effects of exercise training on left ventricular function and
peripheral resistance in patients with chronic heart failure: A randomized trial,
Jama 283(23), pp. 3095-3101, 2000.
21. P. Hanly, N. Zuberi, and R. Gray, Pathogenesis of Cheyne-Stokes respiration in patients with congestive heart failure: relationship to arterial P CO 2 .,
Chest 104(4), pp. 1079-1084, 1993.
22. L. L. Huntsman, A. Noordergraaf, and E. O. Attinger, Metabolic autoregulation of blood flow in skeletal muscle: a model, in J. Baan, A. Noordergraaf, and
J. Raines (eds.), Cardiovascular System Dynamics, pp. 400-414, MIT Press,
Cambridge, 1978.
23. S. Javaheri, A mechanism of central sleep apnea in patients with heart failure,
New Engl. J. Med. 341(13), pp. 949-54, 1999.
24. F. Kappel and R. O. Peer, A mathematical model for fundamental regulation
processes in the cardiovascular system, J. Math. Biol. 31(6), pp. 611-631, 1993.
25. F. Kappel and R. O. Peer, Implementation of a cardiovascular model and algorithms for parameter identification, SFB Optimierung und Kontrolle, KarlFranzens-Universität Graz, technical report Nr. 26, 1995.
26. F. Kappel and G. Propst, Approximation of feedback control for delay systems
using Legendre polynomials, Conf. Sem. Mat. Univ. Bari. Nr. 201, 1984.
27. F. Kappel and D. Salamon, Spline approximation for retarded systems and
the Riccati equation, Siam J.Control Optim. 25(4), pp. 1082-1117, 1987.
28. F. Kappel, S. Lafer, and R. O. Peer, A model for the cardiovascular system
under an ergometric workload, Surv. Math. Ind. 7, pp. 239-250, 1997.
29. A. M. Katz, Physiology of the heart: 2nd edition, Raven Press, N.Y., 1992.
38
Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel
30. T. Kenner, Physical and mathematical modeling in cardiovascular systems,
in: Clinical and Research Applications of Engineering Principles edited by
N. H. C. Hwang, E. R. Gross, and D. J. Patel, University Park Press, Baltimore, pp. 41-109, 1979.
31. M. C. K. Khoo, R. E. Kronauer, K. P. Strohl, and A. S. Slutsky, Factors inducing periodic breathing in humans: a general model, J. Appl. Physiol. 53(3),
pp. 644-659, 1982.
32. M. C. K. Khoo, A model of respiratory variability during non-REM sleep,
in: Respiratory Control: A Modeling Perspective, edited by G. D. Swanson,
F. S. Grodins, and R. L. Hughson, Plenum Press, New York, 1989.
33. M. C. K. Khoo, Modeling the effect of sleep-state on respiratory stability,
in: Modeling and Parameter Estimation in Respiratory Control, pp. 193-204,
edited by M. C. K. Khoo, Plenum Press, New York, 1989.
34. M. C. K. Khoo and S. M. Yamashiro, Models of control of breathing, in:
Respiratory Physiology: an analytical approach edited by H. A. Chang and
M. Paiva, pp. 799-829, Marcel Dekker, New York, 1989.
35. M. C. K. Khoo, A. Gottschalk, and A. I. Pack, Sleep-induced periodic breathing
and apnea: a theoretical study, J. Appl. Physiol. 70(5), pp. 2014-2024, 1991.
36. R. Klinke and S. Silbernagl (eds.), Lehrbuch der Physiologie, Georg Thieme
Verlag, Stuttgart, 1994.
37. K. W. Koo, D.S. Sax, and G. L. Snider, Arterial blood gases and pH during
sleep in chronic obstructive pulmonary disease, Am. J. Med. 58(5), pp. 663670, 1975.
38. J. Krieger, N. Maglasiu, E. Sforza, and D. Kurtz Breathing during sleep in
normal middle age subjects, Sleep 13(2), pp. 143-154, 1990.
39. J. Kugler, C. Maskin, W. H. Frishman, E. H. Sonnenblick, and T. H. LeJemtel,
Regional and systemic metabolic effects of angiotensin-converting enzyme inhibition during exercise in patients with severe heart failure, Circulation 66(6),
pp. 1256-61, 1982.
40. G. Lorenzi-Filho, F. Rankin, I. Bies, and T. Douglas Bradley, Effects of inhaled
carbon dioxide and oxygen on Cheyne-Stokes respiration in patients with heart
failure, Am. J. Respir. Crit. Care Med. 159 (5 pt.1), pp. 1490-98, 1999.
41. G. Mancia, Autonomic modulation of the cardiovascular system during sleep,
New Engl. J. Med. 328(5), pp. 347-49, 1993.
42. L. Martin, Pulmonary physiology in clinical practice : the essentials for patient
care and evaluation, Mosby Press, St. Louis, 1987.
43. J. H. Mateika, S. Mateika, A. S. Slutsky, and V. Hoffstein, The effect
of snoring on mean arterial blood pressure during non-REM sleep, Am.
Rev. Respir. Dis. 145(1), pp. 141-146, 1992.
44. J. Niebauer, A. L. Clark, S. D. Anker, and A. J. S. Coats, Three year mortality in heart failure patients with very low left ventricular ejection fractions,
Int. J. Cardiol. 70(3), pp. 245-247, 1999.
45. A. Noordergraaf and J. Melbin, Introducing the pump equation, in: Cardiovascular System Dynamics: Models and Measurements, edited by T. Kenner,
R. Busse, and H. Hinghofer-Szalkay, pp. 19-35, Plenum Press, New York, 1982.
46. J.T. Ottesen, Modeling of the baroreflex-feedback mechanism with time-delay,
J. Math. Biol., 36(1), pp. 41-63, 1997.
47. W.W. Parmley, Pathophysiology of congestive heart failure, Am-J-Cardiol.
56(2), pp. 7A-11A, 1985.
48. R.
O.
Peer,
Mathematische
Modellierung
von
grundlegenden
Regelungsvorgängen im Herzkreislauf-System, Technical Report, KarlFranzens-Universität und Technische Universität Graz, 1989.
49. C. S. Peskin, Lectures on mathematical aspects of physiology, in: Mathematical Aspects of Physiology edited by F. C. Hoppensteadt, Lect. Appl. Math. 19,
pp. 1-107, Am. Math. Soc, Providence, Rhode Island, 1981.
Cardiovascular-respiratory control system
39
50. E. A. Phillipson and G. Bowes, Control of breathing during sleep, in: Handbook of Physiology, Section 3: The Respiratory System, Volume II, Control of
Breathing, Part 2, edited by A. P. Fishman, N. S. Cherniack, J. G. Widdicombe, and S. R. Geiger, pp. 649-690, Am. Phys. Soc., Bethesda, Maryland,
1986.
51. G. D. Pinna, R. Maestri, A. Mortara, M. T. La Rovere, F. Fanfulla, and
P. Sleight, Periodic breathing in heart failure patients: testing the hypothesis
of instability of the chemoreflex loop, J. Appl. Physiol. 89(6) , pp. 2147-2157,
2000.
52. T. Podszus, Kreislauf und Schlaf, in: Kompendium Schlafmedizin für Ausbildung, Klinik und Praxis, Deutsche Gesllschaft für Schlafforschung und
Schlafmedizin, edited by H. Schulz, Landsberg, Lech : Ecomed-Verl.-Ges.,
1997.
53. A. J. Quaranta, G. E. D’Alonso, and S. L. Krachman, Cheyne-Stokes respiration during sleep in congestive heart failure , Chest 111(2), pp. 467-473,
1997.
54. D. W. Richardson, A. J. Wasserman, and J. L. Patterson Jr., General and regional circulatory responses to change in blood pH and carbon dioxide tension,
J. Clin. Invest. 40, pp. 31-43, 1961.
55. L. B. Rowell, Human Cardiovascular Control, Oxford University Press, 1993.
56. D. L. Russell, Mathematics of finite-dimensional control systems, Marcel
Dekker, New York, 1979.
57. H. Schneider, C.D. Schaub, K.A. Andreoni, A.R. Schwartz, P.L. Smith,
J.L. Robotham, and C.P. O’Donnell, Systemic and pulmonary hemodynamic
responses to normal and obstructed breathing during sleep., J. Appl. Physiol. 83(5), pp. 1671-1680, 1997.
58. F. J. Schoen, The heart-a pathophysiological perspective, Ch. 13, in Robbins
Pathologic Basis of Disease, Sixth Edition, edited by R. S. Cotran et al.,
W. B. Saunders, Philadelphia, 1999.
59. J.W. Shepard Jr., Gas exchange and hemodynamics during sleep,
Med. Clin. North Am. 69(6), pp. 1243-64, 1985.
60. V. K. Somers, M.E. Dyken, A. L. Mark, and F. M. Abboud, Sympatheticnerve activity during Sleep in normal subjects, New Engl. J. Med. 328(5),
pp. 303-307, 1993.
61. M. J. Sullivan, M.B. Higginbotham, and F. R. Cobb, Increased exercise ventilation in patients with chronic heart failure: intact ventilatory control despite
hemodynamic and pulmonary abnormalities, Circulation 77(3), pp. 552-559,
1988.
62. G. W. Swan, Applications of optimal control theory in biomedicine, Marcel
Dekker, Inc., New York 1984.
63. S. Timischl, A Global Model of the Cardiovascular and Respiratory System,
PhD thesis, Karl-Franzens-Universität Graz, 1998.
64. S. Timischl, J. J. Batzel, and F. Kappel, Modeling the human cardiovascularrespiratory control system: an optimal control application to the transition to
non-REM sleep Spezialforschungsbereich F-003 Technical Report 190, KarlFranzens-Universität, 2000.
65. Z. L. Topor, L. Johannson, J. Kasprzyk, and J. E. Remmers, Dynamic ventilatory response to CO2 in congestive heart failure patients with and without
central sleep apnea, J. Appl. Physiol. 91(1), pp. 408-416, 2001.
66. H. Tsuruta, T. Sato, M. Shirataka and N. Ikeda, Mathematical model of the
cardiovascular mechanics for diagnostic analysis and treatment of heart failure: part 1 model description and theoretical analysis, Med. Biol. Eng. Comp.
32(1), pp. 3-11, 1994.
67. K. Wasserman, B. J. Whipp, and R. Casaburi, Respiratory control during exercise, in: Handbook of Physiology, Section 3: The Respiratory System, Volume
40
Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel
II, Control of Breathing, Part 2, edited by A. P. Fishman, N. S. Cherniack,
J. G. Widdicombe, and S. R. Geiger, Am. Phys. Soc., Bethesda, Maryland,
1986.
Cardiovascular-respiratory control system
41
APPENDIX
We assume the following:
P AO 2 = P aO 2 ,
PACO2 = PaCO2 ,
PBO2 = PBvO2 ,
PBCO2 = PBvCO2 ,
P TO 2 = P T v O 2 ,
PTCO2 = PT vCO2 ,
where v = mixed venous blood, T is tissue compartment.
Further we assume:
– The alveoli and pulmonary capillaries are single well-mixed spaces;
– constant temperature, pressure and humidity are maintained in the gas
compartment;
– gas exchange is by diffusion; ventilatory dead space is incorporated via
the optimal control V̇A for the optimal case and control gains Gc and
Gp for the empirical case (see text);
– the delay in the respiratory controller signal to effector muscles is zero;
– delay in the baroreceptor signal to the controller and from controller to
effector muscles is zero;
– metabolic rates and other parameters are constant in a given state;
– pH effects on dissociation laws and other factors are ignored or incorporated into parameters;
– acid/base buffering, material transfer across the blood brain barrier, and
tissue buffering effects are ignored;
– no inter-cardiac shunting occurs;
– intrathoracic pressure is ignored for this average flow model;
– unidirectional non-pulsatile blood flow through the heart is assumed;
hence, blood flow and blood pressure have to be interpreted as mean
values over the length of a pulse;
– fixed blood volume V0 is assumed.
The parameters for α, β, γ, as well as the compliances cas , cap , cvs ,
cvp , cl , and cr are chosen as in the paper by Kappel and Peer [24]. For the
S-shaped O2 dissociation curve which relates blood gas concentrations to
partial pressures we will use the relation
CO2 (t) = K1 (1 − e−K2 PO2 (t) )2 .
(41)
This relation was also used by Fincham and Tehrani [13]. Khoo et al. [31]
assumes a piecewise linear relationship.
42
Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel
For CO2 , considering the narrow working range of PCO2 we assume a
linear dependence of CCO2 on PCO2 ,
CCO2 (t) = KCO2 PCO2 (t) + kCO2 .
(42)
A linear relationship was also used by Khoo et al. [31]. Other parameter
and steady state values from the literature are given in the following tables.
Parameters normally used for simulations are marked with an asterisk.
Table 16. Parameter values (awake rest)
Parameter
Gc
Gp
Ic
Ip
K1
K2
kCO2
KCO2
mB
MRBCO2
MRCO2
MRO2
CO2 sensitivity
Value
1.440 *
3.2
30.240 *
26.5
35.5 *
45.0
35.5 *
38.0
0.2
0.046
0.244
0.0065
0.0057
1400
0.042 *
0.031
0.050
0.054
0.21
0.235
0.200
0.26 *
0.26
0.290
0.240
0.31 *
2.1 +/- 1.0 *
Unit
l/(min ·mmHg)
l/(min ·mmHg)
l/(min ·mmHg)
l/(min ·mmHg)
mmHg
mmHg
mmHg
mmHg
lSTPD /l
mmHg−1
lSTPD /l
lSTPD /(l · mmHg)
lSTPD /(l · mmHg)
g
lSTPD / min
lSTPD /(min ·kg brain tissue)
lSTPD / min
lSTPD / min
lSTPD / min
lSTPD / min
lSTPD / min
lSTPD / min
lSTPD / min
lSTPD / min
lSTPD / min
lSTPD / min
lBTPS /(min ·mmHg)
Source
[31]
[32]
[31]
[32]
[31], [32]
[35]
[31], [32]
[35]
[13]
[13]
[31]
[31]
[35]
[36], p. 745
[32]
[35]
[15]
[13]
[35]
[32]
[33]
[36] p. 239
[35]
[32]
[33]
[36] p. 239
[23]
Cardiovascular-respiratory control system
43
Table 17. Parameter values (awake rest)
Parameter
PICO2
P IO 2
Patm
Rp
RQ
VAO 2
VACO2
VTCO2
VT O 2
VBCO2
VBO 2
VD
V̇D
FB
Value
0
150
760
0.965
1.4
1.95 *
1.5-3
0.88
0.81
0.84 *
2.5 *
3.0
0.5
3.2 *
3.0
15
6*
1.55
0.9 *
1.0
1.1
1.0
1.1
0.15
2.4
2.28
0.5
0.75-0.8 *
12-15% of Q
Unit
mmHg
mmHg
mmHg
mmHg · min /l
mmHg · min /l
mmHg · min /l
mmHg · min /l
lBTPS
lBTPS
lBTPS
lBTPS
lBTPS
l
l
l
l
l
l
l
l
lBTPS
lBTPS / min
lBTPS / min
l/(min ·kg brain tissue)
l/ min
l/ min
Source
[31], [15]
[31]
[31],[15]
[36] p. 233
[36] p. 144
[63]
[42]
[15]
[31], [32]
[36] p. 239
[31]
[15]
[18], p. 1011
[31]
[35], [15]
[31], [35], [32]
[31], [35], [32]
[18], p. 1011
[32]
[15]
[13]
[15]
[13]
[35],[36] p. 239
[36] p. 239
[31]
[35], [36], p. 745
[15], [13]
[55],p. 242
Table 18. Nominal steady state values (awake rest)
Quantity
CaCO2
Ca O 2
CvCO2
Cv O 2
H
Pap
Pas
Pvp
Pvs
Value
0.493
0.197
0.535
0.147
70
12
15
10-22
100
93
5
8
2-4
5
Unit
lSTPD /l
lSTPD /l
lSTPD /l
lSTPD /l
min−1
mmHg
mmHg
mmHg
mmHg
mmHg
mmHg
mmHg
mmHg
mmHg
Source
[36], p. 253
[36], p. 253
[36], p. 253
[36], p. 253
[36] p. 144
[36] p. 144
[66] p. 4
[42] Chptr. 8
[36] p. 144
[66] p. 4
[36] p. 144
[66] p. 4
[36] p. 144
[66] p. 4
44
Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel
Table 19. Nominal steady state values (awake rest)
Quantity
PACO2
P AO 2
PaCO2
P aO 2
PvCO2
P vO 2
Ql = Q r = F p = F s
Rs
V̇A
V̇E
Vstr,l
Value
40
104
100
40
95
90
45
46
40-50
40
35-40
6
6.2
5
4-7
20.
11-18
4.038
5.6
8
0.070
Unit
mmHg
mmHg
mmHg
mmHg
mmHg
mmHg
mmHg
mmHg
mmHg
mmHg
mmHg
l/ min
l/ min
l/ min
l/ min
mmHg · min /l
mmHg · min /l
lBTPS / min
lBTPS / min
lBTPS / min
l
Source
[18],p. 495, [36] p. 239
[18],p. 494
[36] p. 239
[18],p. 495, [36], p. 253
[18],p. 494
[36], p. 253
[18],p. 495
[36], p. 253
[42] Chptr. 8
[18],p. 494, [36], p. 253
[42] Chptr. 8
[31], [15]
[36] p. 239
[36] p. 144
[42] Chptr. 8
[36] p. 144
[42]
[13]
[36] p. 239
[36] p. 239
[36] p. 144
Table 20. Nominal steady state values (NREM sleep)
Quantity
PACO2
P AO 2
PvCO2
P vO 2
V̇E
Pas
H
Q
V̇A
MRO2
MRCO2
sympathetic activity
Rs
Sl
Sr
% change
↑ 2-8 mmHg
↓ 3-11 mmHg
↑6%
↑1%
↓ 14-19 %
↓ 5-17 %
↓ 10 %
↓ 0-10 %
↓ 14-19 %
↓ 15 %
↓ 15 %
↓ significantly
↓ 5-10 %
↓ 5-15 %
↓ 5-15 %
Model Value
44 mmHg
98.9 mmHg
51.9 mmHg
35.2 mmHg
6 lBTPS / min
87.0 mmHg
68 mmHg
4.3 l/ min
4.4 lBTPS / min
0.26 lSTPD / min
0.23 lSTPD / min
19.2 mmHg · min /l
58.7 mmHg
4.5 mmHg
Source
[59]
[59, 37]
estimate
estimate
[59, 38]
[43, 59, 60]
[59, 60]
[59, 57]
[38]
[35]
[35]
[60]
estimate
estimate
estimate
Cardiovascular-respiratory control system
45
Table 21. Miscellaneous parameters (awake and sleep unless otherwise noted)
Quantity
V0
Apesk
αl
αr
βl
βr
γl
γr
cap
cas
cvp
cvs
cl
cr
Value
5.0
177.47
89.47
28.46
73.41
1.78
37.33
11.88
0.03557
0.01002
0.1394
0.643
0.01289
0.06077
Unit
l
mmHg · min ·l−1
min−2
min−2
mmHg · min−1
mmHg · min−1
min−1
min−1
l · mmHg−1
l · mmHg−1
l · mmHg−1
l · mmHg−1
l · mmHg−1
l · mmHg−1
Source
[24]
[24]
[24]
[24]
[24]
[24]
[24]
[24]
[24]
[24]
[24]
[24]
[24]
[24]
Table 22. Estimated state variable values for congestive heart failure categories
as presented in Tsuruta et al. [66]
Quantity
H
Q
Vstr
Pap
Pas
Pvp
Pvs
Rs
Normal
70
5.6
.08
15.0
93 .3
8.0
5.0
15.45
Stage A
85
5.0
.059
19.0
93.3
12.0
5.0
17.30
Stage B
85
4.4
.052
23.0
93.3
16.0
5.0
19.66
Stage C
85
3.8
.045
27.0
93.3
20.0
5.0
22.76
Stage D
85
3.2
.038
31.0
93.3
24.0
5.0
27.03