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A cardiovascular-respiratory control system modelincluding state delay with application to con-gestive heart failure in humans
Jerry Batzel Contact AuthorSpecial Research Center ”Optimization and Control,University of Graz,Heinrichstraße 22, A-8010 Graz, Austria.
Susanne Timischl-TeschlFachhochschule Technikum WienHoechstaedtplatz 5, 1200 WienAustria
Franz KappelSpecial Research Center ”Optimization and Control andInstitute of MathematicsUniversity of GrazHeinrichstraße 36, A-8010 Graz, Austria.
Email: jerry.batzel@uni-graz.at - Contact emailEmail: susanne.teschl@technikum-wien.atEmail: franz.kappel@uni-graz.atcontact FAX: +43 316 380 9795contact 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 systemmodel including state delay with applicationto congestive heart failure in humans
the date of receipt and acceptance should be inserted later – c© Springer-Verlag2004
Abstract. This paper considers a model of the human cardiovascular-respiratorycontrol system with one and two transport delays in the state equations describ-ing the respiratory system. The effectiveness of the control of the ventilation rateVA is influenced by such transport delays because blood gases must be trans-ported a physical distance from the lungs to the sensory sites where these gasesare measured. The short term cardiovascular control system does not involve suchtransport delays although delays do arise in other contexts such as the barore-flex loop (see [46]) for example. This baroreflex delay is not considered here. Theinteraction between heart rate, blood pressure, cardiac output, and blood vesselresistance is quite complex and given the limited knowledge available of this inter-action, we will model the cardiovascular control mechanism via an optimal controlderived from control theory. This control will be stabilizing and is a reasonableapproach based on mathematical considerations as well as being further motivatedby the observation that many physiologists cite optimization as a potential influ-ence in the evolution of biological systems (see, e.g., Kenner [30] or Swan [62]). Inthis paper we adapt a model, previously considered (Timischl [63] and Timischlet al. [64]), to include the effects of one and two transport delays. We will firstimplement an optimal control for the combined cardiovascular-respiratory modelwith one state space delay. We will then consider the effects of a second delay inthe state space by modeling the respiratory control via an empirical formula withdelay while the the complex relationships in the cardiovascular control will stillbe modeled by optimal control. This second transport delay associated with thesensory system of the respiratory control plays an important role in respiratorystability. As an application of this model we will consider congestive heart fail-ure where this transport delay is larger than normal and the transition from thequiet awake state to stage 4 (NREM) sleep. The model can be used to study the
Jerry J. Batzel: SFB ”Optimierung und Kontrolle”, Karl-Franzens-Universitat,Heinrichstraße 22, 8010 Graz, Austria
Franz Kappel: Mathematics Institute and SFB ”Optimierung und Kontrolle”,Karl-Franzens-Universitat, Heinrichstraße 36, 8010 Graz, Austria
Susanne Timischl-Teschl: Fachhochschule Technikum Wien, Austria
Supported by FWF (Austria) under grant F310 as a subproject of the SpecialResearch 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 situationsas well as to consider the influence of optimal function in physiological controlsystem performance.
1. Introduction
The cardiovascular system functions to maintain adequate blood flow tovarious regions of the body. This function depends upon the interaction ofa large number of factors including blood pressure, cross-section of arteries,cardiac output, and partial pressures of CO2 and O2 in the blood. Thereare global control mechanisms that act on the entire system to maintainappropriate blood flow and these mechanisms are supplemented by localmechanisms in each vascular region which act to shunt blood to those regionswhere demand is high and away from areas where demand is low. Theoverall control process which stabilizes the system is quite complicated andnot fully elucidated. Principles of optimal control theory will be applied todesign a control mechanism for this system. For further details about thecardiovascular system and control see, e.g., Rowell [55].
When breathing is not under voluntary control or subject to neurolog-ically induced changes, the human respiratory control system varies theventilation rate in response to the levels of carbon dioxide CO2 and oxy-gen O2 in the body (via partial pressures PaCO2
and PaO2). This chemical
control system depends upon information fed back from two sensory siteswhich monitor the blood gas levels (producing a negative feedback controlloop). These sensory sites at which the blood gas levels are measured are aphysical distance from the lungs (where blood gas levels are adjusted) andthus there are transport delays (which vary depending on blood flow) in thenegative feedback loop. Under normal conditions (even with delays in thefeedback control loop) the control system is sufficiently stable to maintainblood 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 cardiovascularsystems. Function of the respiratory system depends on blood flow throughthe lungs and tissues. The amount of oxygen O2 transported to the tissuesand carbon dioxide CO2 transported away from the tissues depends oncardiac output Q and blood flow F through the pulmonary and systemiccircuits. Q and F depend in turn upon heart rate H , stroke volume Vstr ,resistance in the vascular system R, and blood pressure P . Arterial bloodpressure Pas is controlled via the baroreceptor negative feedback loop whichhas important effects on H , Vstr, R, and hence Q. Systemic resistance whichimpacts blood pressure is also influenced by local metabolic control actingon the resistance of the blood vessels of various tissues. This local control isin turn influenced by local concentrations of CO2 and O2, thus illustratinganother important link between the two systems. The effect of concentrationof O2 on the resistance of the systemic blood vessel is included in this model.Furthermore, PaCO2
and PaO2can affect cardiac output and contractility as
Cardiovascular-respiratory control system 3
well (see, e.g., Richardson et al. [54]). Neither these blood gas effects norsynchronization of heart rate and ventilation are included in this model.
An optimal control approach will be used to model the complex inter-actions in the cardiovascular-respiratory control system. The cardiovascu-lar and respiratory controls are represented by a linear negative feedbackcontrol which minimizes a quadratic cost functional defining optimal per-formance. Reasons and motivation for incorporating an optimal control ap-proach is given in Section 3. This modeling approach was previously appliedby Kappel and Peer [24] and Timischl [63] to study transition from rest toexercise under a constant ergometric workload and the role of pulmonaryresistance during exercise.
The equations describing the state of the system are developed followingthe ideas of Khoo et al. [31], Grodins et al. [15], and Grodins [16,17] andKappel 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 congestiveheart 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 ofthe model is defined by equations (1) to (5) and is based on equations givenin Khoo et al. [31]. Two compartments, a lung compartment and a generaltissue compartment, are used to model the respiratory component of thesystem (see Figure 1).
The lung compartment equation (1) represents a mass balance equationfor CO2 and equation (2) similarly represents a mass balance equation forO2. The mass balance equations for CO2 and O2 in the tissue compartmentare given by Equations (3) and (4). Equation (5) tracks CO2 in the brainwhich 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 compart-ment. Transport delays appear in the mass balance equations as it takestime for tissue venous blood to reach the lungs and vice versa. The com-partment blood gas levels are adjusted by the ventilation rate VA which willbe further discussed in Section 3.
Note that the role of VA in the state equations for the lung compartment(1) and (2) is that of effective ventilation reflecting net ventilation after deadspace effects are removed.
Among the assumptions incorporated in the model we mention that themodel is an average flow model and thus ventilation represents minute ven-tilation and cardiovascular flow is non-pulsatile. Given the time scales andfocus of this study, these assumptions are reasonable. Other assumptionsare given in the appendix.
In passing we note that alveolar minute ventilation does not reflect mod-ulation 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 flowcan influence the distribution of blood and play a role in the baroreflexcontrol.
VACO2PaCO2
(t) = 863Fp(t)(CvCO2(t− τV )− CaCO2
(t)) (1)
+ VA(t)(PICO2− PaCO2
(t)),
VAO2PaO2
(t) = 863Fp(t)(CvO2(t− τV )− CaO2
(t)) (2)
+ VA(t)(PIO2− PaO2
(t)),
VTCO2CvCO2
(t) = MRCO2 + Fs(t)(CaCO2(t− τT )− CvCO2
(t)), (3)
VTO2CvO2
(t) = −MRO2 + Fs(t)(CaO2(t)− CvO2
(t− τT )), (4)
VBCO2CBCO2
(t) = MRBCO2+ FB(t)(CaCO2
(t− τB)− CBCO2(t)), (5)
casPas(t) = Ql(t)− Fs(t), (6)
cvsPvs(t) = Fs(t)−Qr(t), (7)
cvpPvp(t) = Fp(t)−Ql(t), (8)
Sl(t) = σl(t), (9)
Sr(t) = σr(t), (10)
σl(t) = −γlσl(t)− αlSl(t) + βlH(t), (11)
σr(t) = −γrσr(t)− αrSr(t) + βrH(t), (12)
H(t) = u1(t), (13)
VA(t) = u2(t). (14)
Table 1. Respiratory symbols
Symbol Meaning unit
Ca concentration of blood gas in arterial blood lSTPD · l−1
Cv concentration of blood gas in mixed venous blood lSTPD · l−1
MR metabolic production rate lSTPD ·min−1
Pa partial pressure of blood gas in arterial blood mmHgPv partial pressure of blood gas in mixed venous blood mmHgPI partial pressure of inspired gas mmHgB brain compartment -
u2 control function, u2 = VA lBTPS ·min−2
VA alveolar ventilation lBTPS ·min−1
VA time derivative of alveolar ventilation lBTPS ·min−2
VA effective gas storage volume of the lung compartment lBTPS
VT effective tissue gas storage volume lCO2,O2 carbon dioxide and oxygen respectively -
τ transport delay secIp, Ic cutoff thresholds mmHg
Cardiovascular-respiratory control system 5
The cardiovascular component of the model is based on the work ofGrodins and coworkers [15–17] and Kappel and coworkers [24,28,48] andis described by equations (6) to (12). This component includes two circuits(systemic and pulmonary) which are arranged in series, and two pumps (leftand right ventricle). See Figure 1. Each circuit subsumes the system of ar-teries 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, bloodflow and blood pressure are to be interpreted as mean values over the lengthof a pulse.
Mass balance equations for blood flowing through the systemic arteryand vein components are given by equations (6) and (7) respectively. Equa-tion (8) gives the mass balance equation for the pulmonary venous compo-nent. Under the assumption of a fixed blood volume V0, the equation forthe pulmonary arterial pressure can then be derived from the other cardio-vascular compartment pressures:
Pap(t) =1
cap(V0 − casPas(t)− cvsPvs(t)− cvpPvp(t)). (15)
Table 2. Cardiovascular symbols
Symbol Meaning Unit
α coefficient of S in the differential equation for σ min−2
Apesk Rs = ApeskCvO2mmHg ·min ·l−1
β coefficient of H in the differential equation for σ mmHg ·min−1
ca arterial compliance l ·mmHg−1
cv venous compliance l ·mmHg−1
F blood flow perfusing compartment l ·min−1
H heart rate min−1
γ coefficient of σ in the differential equation for σ min−1
Pas mean blood pressure in systemic arterial region mmHgPvs mean blood pressure in systemic venous region mmHgPap mean blood pressure in pulmonary arterial region mmHgPvp mean blood pressure in pulmonary venous region mmHgQ cardiac output l ·min−1
R resistance in the peripheral region of a circuit mmHg ·min ·l−1
S contractility of a ventricle mmHgσ derivative of S mmHg ·min−1
u1 control function, u1 = H min−2
Vstr stroke volume of a ventricle lV0 total blood volume ll,r left and right heart -p,s pulmonary and systemic circuits -
6 Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel
Lungs
PaO2
PaCO2
PaO2
PaCO2
PaCO2 PaO2
PvCO2PvO2
PvCO2
PvCO2
PvO2
PvO2
PvpPap
Pvs Pas
Qr Ql
Fs
Fp
Tissue
Controller
Auto-regulation
VA
H
PvO2
Rs
O2MR CO2MR
debitvalues
Fig. 1. Model block diagram
Blood flow F , which appears in equations (6) through (8) is related to bloodpressure via a form of Ohm’s law
Fs(t) =Pas(t)− Pvs(t)
Rs(t), (16)
Fp(t) =Pap(t)− Pvp(t)
Rp, (17)
where Pa is arterial blood pressure, Pv is venous pressure, and R is vascularresistance. Details can be found in [24,28,48].
As mentioned above, cardiac output Q is defined as the mean blood flowover 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 andr are used to distinguish between left and right ventricle. Subindices s andp represent systemic and pulmonary circuits respectively. A complex rela-tionship between stroke volume and blood pressure is given in Kappel andPeer [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 thearterial blood pressure opposing the ejection of blood, and c denotes thecompliance of the relaxed ventricle.
The Bowditch effect, which describes the observation that contractil-ity Sl (respectively Sr) increases if heart rate increases, is introduced viaEquations (9) through (12). This relation is essentially modeled via a secondorder differential equation. For details see Kappel and Peer [24].
Equations (13) and (14) define the variation of heart rate (H(t)) andvariation in ventilation (VA(t)) as mathematical control variables. The func-tions u1(t) and u2(t) will be derived using an optimality criterion which actsto minimize deviations in several quantities including these variations H(t)and VA(t) (see Section 3). Thus, limits are placed on the magnitude andvariation of the changes in the physiological controls H(t) and VA(t) whichreflects an assumption of minimal energy expense effort as an optimal con-trol criterion for the physiological control process.
Local metabolic autoregulation of systemic resistance is modeled usingthe assumption that systemic resistance Rs depends on venous oxygen con-centration CvO2
. Thus Rs is described by
Rs(t) = ApeskCvO2(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]. Theabove relationship was also used in Kappel and Peer [24]. Essentially, thisequation describes an important local constriction/relaxation mechanismacting on small vascular elements in response to local oxygen concentrationCvO2
(some tissues respond also to CvCO2). Global changes in Rs will be
discussed in Section 7. Delay in the control process of global resistance isnot analyzed in this paper.
Links between the respiratory and cardiovascular components can beseen in the equations. The respiratory mass balance equations include ex-pressions for the blood flows Fs and Fp. Levels of CvO2
which influencessystemic resistance via equation (20) is in turn affected by the respira-tory system. Heart rate H and ventilation rate VA influence both systemsthrough the control functions u1 and u2, while Pas, PaCO2
, and PaO2affect
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 designedto transfer the state of the organism from an initial perturbation (initialstate) to a final steady state in an optimal way that will be defined below.Given the complex and interrelated nature of the control systems discussedhere, the interaction of the various control effects will be represented as astabilizing control derived from optimal control theory. This approach todesigning a stabilizing control is motivated primarily by mathematical con-siderations and is reasonable given the lack of detailed information aboutparticular control interactions. Furthermore, this approach can provide in-formation on the nature and function of the controller as well as to helpidentify and study key controlling and controlled quantities.
As mentioned above, this approach is motivated primarily on mathe-matical grounds, but such a control derived from optimal control theory isfurther motivated by the view that optimal function likely plays a role inphysiological design. See, for example, Kenner [30] or Swan [62]. Minimizingstress on the system either by avoiding extreme actions or inefficient oper-ating states would represent such an optimal design criterion. The degree towhich physiological systems behave optimally is an open question of greatinterest.
In the model we focus on two physiological quantities which influencethe system: heart rate H and the ventilation rate VA. In the cardiovascularsystem, H is adjusted via the baroreceptor control loop and VA is adjustedby the respiratory control loop. These quantities are varied so that themean arterial blood pressure Pas and the partial pressures of carbon dioxidePaCO2
and oxygen PaO2in arterial blood are stabilized (along with the whole
system) to their steady state operating points when an initial perturbationoccurs. Parameters are chosen to define the steady state values which willbe the operating points (final steady state) as well as to derive the initialperturbed condition of the system.
The cost function we will use enforces the condition that the transi-tion from initial condition to final steady state is optimal in the sense thatPas, PaCO2
, and PaO2are 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 thefurther restriction that excessive heart rate and ventilation change are re-stricted (effort is efficient). In this way, the stabilizing feedback control canbe considered also as an optimizing feedback control.
In the mathematical setting for this problem, it is the variations in heartrate (H(t)) and ventilation (VA(t)) that represent the control functions u1(t)and u2(t). By including u1(t) and u2(t) in the cost functional, limits areplaced on the degree to which H and VA can be varied to stabilize thesystem, a reasonable physiological constraint which also reflects an efficiencyof 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) toanother (steady) state.
The control problem is then formulated as follows: We determine controlfunctions u1 and u2 that transfer the system from one state to another suchthat the cost functional
∫ ∞
0
(qas(Pas(t)− P feas )2 + qc(PaCO2
(t)− P feaCO2)2 (21)
+qo(PaO2(t)− P feaO2
)2 + q1u1(t)2 + q2u2(t)2)dt
is minimized under the restriction of the model equations:
x(t) = f(x(t), x(t − τT );W s) +B u(t), x0 = φ.
y(t) = Dx(t).(22)
where x(t) ∈ R14 is given by
x(t) = (PaCO2, PaO2
, CvCO2, CvO2
, CBCO2, Pas, Pvs, Pvp, Sl, Sr, σl, σr , H, VA)T .
The vector f represents the system equations, W s represents the vectorof associated weights in the cost functional, and y(t) is a vector whichrepresents the observation of controlled values. The delay τT ∈ R+ is afixed point delay and the initial condition is a function φ ∈ C where Cdenotes C([−τT , 0],R14). The positive scalar coefficients qas, qc, qo, q1, andq2 determine how much weight is associated to each term in the integrand.Superscript ”fe” refers to the final equilibrium or steady state to whichthe system is transfered by the control. We note that partial pressures andconcentrations are interchangeable according to the dissociation formulas.We use concentrations in some state equations to simplify the form of theequations. For further information related to applications of optimal controltheory 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 thequantities in the cost functional have values all set equal to one with theexception of qo, the weighting factor of PaO2
, which is set to 0.3. The moti-vation for a smaller weight for qo is that only large deviations in PaO2
act tosignificantly alter ventilation because, as can be seen from the adult oxyhe-moglobin saturation curve, there is a significant reserve of oxygen. Ventila-tory control response to PaO2
will be more pronounced only at lower levelsof PaO2
. In normal operating conditions a deviation of 1 mmHg in PaCO2
produces a larger percentage change in ventilation than does a proportionalmmHg deviation in PaO2
, thus suggesting that PaCO2is the primary focus of
control. These factors are also expressed in empirical relationships betweenPaO2
, PaCO2, and VA 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 deviationsin PaO2
in the cost functional. In regards to the other weights it appearsthat the system reacts in a more sensitive way to deviations in the othervariables than to deviations in PaO2
. Because we lack more information, wetake the same weight for all variables except PaO2
. In order to get moreinformation on these weights one would have to parameter estimation onthe basis of data obtained from appropriate tests which we plan to pursue.
In simulation studies [64] it was found that respiratory and cardiovascu-lar quantity cross-interaction through the cost functional is minimal. Con-sidering the respiratory component weights in isolation, simulations indicatethat the initial drops in VA and PaO2
at the transition to sleep (see Figures(2) thru (7)) are much more extreme when the PaO2
weight is small. Whenequal weights are given to PaO2
and PaCO2the initial undershoot of the final
steady states is much smaller. Simulations indicate that the undershoot issmall for qo greater than 0.3. The weights chosen for these simulations arereasonable given that there is certainly a range of responses for VA to givenPaO2
and PaCO2levels 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 systemfrom the initial condition ”resting awake” denoted by xa to the final steadystate ”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”. Thesestates are defined by parameter choices. We will consider two cases. In thefirst case we implement both the respiratory and cardiovascular controls asoptimal controls and derive formulations for both heart rate H and venti-lation rate VA which transfer the system from ”resting awake” xa to ”stage4 sleep” xs in an optimal way. Thus we do not consider explicitly the respi-ratory control sensory system and hence equation (5) is not required untilSection 10. For this case we have only the delays in transport between thelung and tissue compartments. Here τT is the transport delay from the lungsto the tissue compartment (see Grodins, [15] τT = 24 s). The transport delayfrom the tissue to the lung compartment τV is somewhat longer but due tothe relatively stable behavior of the venous side blood gases under normalconditions (situations where the state variables are in the physiologicallymeaningful range and without extreme variations) it is reasonable to con-sider τV = τT . Indeed, the dynamics of the system change almost not at all ifτv is varied, given that the venous side state variable variations are minimaland much damped compared to arterial side changes. This approximationof τv = τT is chosen to simplify computations.
In the second case we will incorporate VA into the state equations viaan empirical formula with delay relating VA 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 PaCO2and
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 two-
degree of freedom set of steady states. Thus it is necessary to choose steadystate values of two state variables as parameters when calculating the awakeand sleep steady states for the system. In general we choose values for PaCO2
and H . These quantities are chosen as the parameters for the equilibriabecause PaCO2
is tightly controlled independently of the special situationand 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 parameterchanges to be discussed in Section 7.
2. The control functions u1 and u2 which transfer system (22) from theinitial steady state ”awake”, xa, to the final steady state ”sleep”, xs, arefound as follows. We consider the linearized system around xs with initialcondition x(0) = xa, and the cost functional equation (21). The controlfunctions u1 and u2 are then computed such that the cost functionalis minimized subject to the linearized system. This is accomplished bysolving an associated algebraic matrix-Riccati equation which is used todefine the feedback gain matrix. In particular, u1 and u2 are given asfeedback control functions.
3. This control is used to stabilize the nonlinear system (22) defined byequations (1) to (14). The control will be suboptimal for the nonlinearsystem in the sense of Russell [56] and stabilizing.
6. Analytical considerations
We consider first the case where both heart rate H and the ventilationrate VA are modeled as optimal controls. We give the mathematical settingfor the system with one delay. In this case, since VA is defined by optimalcontrol we don’t need equation (5). We carry it along here for reference inthe 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
x(t) = f(x(t), x(t − τT );W s) +B u(t), x0 = φ
y(t) = Dx(t)(23)
where x(t) ∈ R14 is given by
x(t) = (PaCO2, PaO2
, CvCO2, CvO2
, CBCO2, Pas, Pvs, Pvp, Sl, Sr, σl, σr , H, VA)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. Theinitial 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” fromwhich the system will be transferred to the final steady state sleep xs bythe control. As controlled variables we have the observations
y(t) = Dx(t) = (PaCO2(t), 0.3PaO2
(t), Pas(t))T , (24)
where D ∈M3,14(R) is the weighting on the observations given by
D =
1 0 0 0 0 0 . . . 00 .3 0 0 0 0 . . . 00 0 0 0 0 1 . . . 0
(25)
The control u(t) ∈ R2 denotes the vector
u(t) = (u1(t), u2(t))T = (H, VA)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 byintroducing new vector variables ξ and η
ξ(t) = x(t) − xs,η(t) = y(t)− ys. (27)
Next, we approximate ξ linearly. To this aim we replace x in (23) by ξand make a Taylor expansion around xs for fixed time t. We are treatingx(t− τT ) as an independent variable for the expansion. This yields
x(t) = ξ(t) = f(xs + ξ(t), xs + ξ(t− τT );W s) +B u
= A1 ξ(t) +A2 ξ(t− τT )) +B u+ o(ξ).(28)
Here o(·) denotes the Landau symbol (h(x) = o(k(x)) :⇔ ‖h(x)‖/‖k(x)‖ −→0 as x −→ ∞). Note that the original state equations were already linearwith respect to the control u. The matrices Ai ∈M14,14(R), i = 1, 2 are theJacobians of f with respect to x(t), and x(t − τT ), respectively, evaluatedat x = xs,
A1 =∂f
∂x(t)(xs;W s),
A2 =∂f
∂x(t− τT )(xs;W s).
(29)
Analogously,
η(t) = (PaCO2(t)− P saCO2
, 0.3(PaO2(t)− P saO2
), Pas(t)− P sas)T
= 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,
ξ`(t) = A1ξ`(t) +A2 ξ`(t− τT ) +B u(t),
η`(t) = Dξ`(t),
ξ`(0) = xa − xs.(31)
This is a special case of the general linear hereditary control system
x(t) = Lxt +Bu(t), t ≥ 0,
y(t) = Dx(t).(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 abovecase Lxt = A1xt(0) + A2xt(−τT ) and h = τT . With this setting we applythe results on approximation of feedback control for delay systems usingLegendre polynomials found in Kappel and Propst [26] (see also [27]). In thisapproach the control is found for approximating systems defined on finitedimensional subspaces of Rn x L2 [−h, 0] utilizing Legendre polynomials. Forthis approach we use the first 5 Legendre polynomials. Thus, the calculatedcontrol will be an approximate control for the actual system. In the abovepaper it was shown that the control for the approximating system convergesto the control for the actual system as the approximating system convergesto the actual system in an appropriate sense.
7. Modeling Sleep
During sleep, as a result of physiological changes in the body (sometimesreferred to as the withdraw of the ”wakefulness drive”), the ventilatorycontrol system is less effective for a given level of blood gases. For example,lower muscle tone during quiet sleep affects the reaction of the respiratorymuscles to control signals. This reduction in responsiveness results in VAfalling as one transits from the ”awake” state through stage 1 to stage 4quiet or NREM sleep. The net effect is a decrease in PaO2
and an increasein 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 andblood pressure fall. Cardiac output is generally reduced though the degreeof 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, strokevolume are reduced during NREM-sleep. The reduction of sympathetic ner-vous system activity (see [60]) in the transition from quiet awake to NREMsleep would trigger these changes.
Given the reduction in sympathetic activity, a reduction in peripheralresistance 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 reducedstroke volume. It appears that counter influences such as an increase incardiac filling pressure (and end diastolic volume) resulting from being inthe supine position would tend to raise stroke volume. In the simulationshere, while not explicitly modeling the influence of position, we do includethe effect of the reduction in contractility caused by reduced sympatheticactivity. As a result a small drop in stroke volume is observed. However, themajor influence on Q is the drop in H .
The effect of reduced sympathetic activity on systemic resistance inNREM sleep is implemented as follows. In the model in the awake reststate, variation in Rs is only produced by the local control described above(with the base line global resistance fixed by the parameter Apesk). Thislocal control would not disappear during sleep but global influence on sys-temic resistance by reduced sympathetic activity should result in a modestreduction in systemic resistance. To model this change, note that Apeskwhich appears in relation (20), acts as a gain constant, relating oxygen con-centration and resistance. This constant will be reduced to model the globalreduction due to sympathetic effects. Further research will include a morecomplete picture of the two contributing and interacting controllers of Rs,one local, and one global. Based on steady state relations in the model, thesympathetic influence on contractility is modeled via a reduction in param-eters βl in equation (11) and βr in equation (12). In the simulations givenhere, Apesk is reduced by 5%, and in the contractility equations (Bowditcheffect), the parameters βl and βr are reduced by 10% in the NREM-steadystate (thus reducing contractility). Tables such as 3 and 4 list these valuechanges. In summary, the steady state ”sleep” is implemented by the fol-lowing parameter changes (recall PaCO2
and H are chosen as parameters forthe 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 Sec-tion 5.
The transition to stage 4 sleep is in reality not instantaneous but takessome minutes. We consider a transition time of 3-4 minutes. We include forthe dynamic simulation a time dependent decrease in the metabolic ratesover the transition time to stage 4 sleep and the same for the changes incontractility and systemic resistance. We still implement the control func-tions u1 and u2 calculated for a time-independent linear system thoughthese changes are time dependent. This further reduces the optimality butthe thereby obtained (suboptimal) control still stabilizes the system and isuseful for dynamic studies.
Cardiovascular-respiratory control system 15
The sleep dependent changes in contractility (βl and βr), the metabolicrates, and resistance (Apesk) are assumed to be mostly accomplished bystages one and two. This assumption is made for purposes of exploringthe dynamics of transition and because not much is available in the liter-ature about the actual time course of these parameter changes changes insleep transition. With this model it is possible to explore various parameterchange time courses.
8. One delay simulations
In the first simulation we consider the transition between xa and xs fora normal adult with slightly elevated Rs. We assume that heart rate Hfalls 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 nonlinearsystem and in this case, the control will be suboptimal but stabilizing. Allcalculations are performed using Mathematica 3.0 Tool boxes. The Mathe-matica package NDelayDSolve by Allen Hayes gives the numerical solutionof delay-differential equations.
Tables 3 and later parameter tables give the chosen parameters used formodeling given conditions such as the ”resting awake state”, ”NREM sleepstate” or ”congestive heart failure state”. Table 4 and later steady statevariable tables give the steady states computed from the model with thechosen parameters. In this case Tables 3 and 4 give values for resting awakeand stage 4 sleep with optimal control for a normal adult. Tables 18 to 20in the appendix give some comparison values from the literature.
Table 3. Optimal control parameters: normal adult sleep transition
Parameter Awake Sleep
Apesk 147.16 139.80βl 85.89 77.30βr 2.083 1.87H 75.0 68.0MRCO2 0.266 0.224MRO2 0.310 0.260PaCO2
40.0 44.0τT 24.0 24.0
Figures (2) thru (7) give the dynamics of the system produced by thecontrol 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 givenin Table 4. The qualitative changes in steady state values derived from themodel agree with observed behavior of the cardiovascular-respiratory con-trol 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 Awake Sleep
H 75.00 68.00Pas 101.7 87.04Pap 17.18 16.00Pvs 3.619 3.912Pvp 7.477 7.486PaCO2
40.0 44.0PaO2
103.38 98.92PvCO2
48.29 51.95PvO2
34.46 35.23Ql 4.938 4.335Qr 4.938 4.335Rs 19.86 19.18Sl 71.999 58.751Sr 5.488 4.478
VA 5.739 4.393Vstr,l 0.0659 0.0637Vstr,r 0.0659 0.0637
1 2 3 4 5minutes
38
40
42
44
46
mm Hg
PACO2
1 2 3 4 5minutes
85
90
95
100
105
110mm Hg
PAO2
Fig. 2. Optimal control: normal case
1 2 3 4 5minutes
46474849505152mm Hg
PVCO2
1 2 3 4 5minutes
34.234.434.634.8
3535.235.4
mm Hg
PVO2
Fig. 3. Optimal control: normal case
(see below) of commonly reported values for the physiological conditions weare considering. We note that there exists a variety of response combina-tions for various individuals requiring a parameter identification if specificdata is compared. The model predicts decreases in Pas and VA as experi-mentally observed in the sleep state (see, e.g., Krieger et al. [38], Phillipson
Cardiovascular-respiratory control system 17
1 2 3 4 5minutes
85
90
95
100
105mm Hg
PAS
1 2 3 4 5minutes
3.2
3.4
3.6
3.8
4mm Hg
PVS
Fig. 4. Optimal control: normal case
1 2 3 4 5minutes
15.5
16
16.5
17
17.5
18mm Hg
PAP
1 2 3 4 5minutes
7.4257.45
7.4757.5
7.5257.55
7.575
mm Hg
PVP
Fig. 5. Optimal control: normal case
1 2 3 4 5minutes
0.063
0.064
0.065
0.066
0.067
0.068liters
VSTR
1 2 3 4 5minutes
3.5
4
4.5
5
5.5litersper min
QL
Fig. 6. Optimal control: normal case
[50], Podszus [52], Somers et al. [60], and Mateika et al. [43]). Decreases inQ 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 PaCO2is con-
sistent with data provided in Koo et al. [37], Phillipson [50], and Shepard[59]. Further, the model reflects the drop in systemic resistance as well aspredicting an increase for Pvs. See Tables 18 and 20 in the appendix for asummary of state values derived from research literature for the awake andNREM sleep states.
Using the parameter values and steady states from Tables 3 and 4 wecalculate 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] forthe dynamic time course of various state transitions. The data provided inBurgess et al. [8] suggest a disproportionate drop in H during the initialphase of sleep onset. Model simulations also show that the controlH declines
18 Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel
1 2 3 4 5minutes
3.54
4.55
5.56
liters per minute
VA.
º
1 2 3 4 5minutes
62.565
67.570
72.575
77.580bpm
H
Fig. 7. Optimal control: normal case
with the largest part of the decline occurring during the initial stage of sleeponset consistent with the results in Burgess et al. [8]. In contrast, there isa continual smooth decline in Vstr,l and Q, more or less unaffected by sleepstage, The model can be used to explore various parameter effects on thesetime courses.
As can be seen by comparison with the simulations presented in [64]little dynamical difference is produced by introduction of delay into themass balance equations of the respiratory system. Indeed, even if the delayis increased by a factor of 15, no significant differences in dynamics appears.
The important delay introducing dynamic instability into the system isthe delay in the feedback control loop of the respiratory control system (seee.g. [2,3]). We will consider this in Section 10. An important physiologicalcondition which increases transport delay in the feedback control loop isfound in congestive heart failure.
9. Congestive heart condition and transport delay
Heart failure is a generic term covering a number of physiopathologies inheart performance which result in a decrease in general blood flow. Thiscondition is often referred to as congestive heart failure to focus on a mainconsequence, namely pulmonary or systemic edema.
Chronic heart failure is to be distinguished from ”heart attack” whichresults in blockage in coronary blood flow or blockage in ventricular or atrialflow. Heart failure can be categorized in a number of ways: forward versusbackward, left versus right, systolic versus diastolic, and low output versushigh output. These classifications are not uniformly consistently appliedbut they are useful in focusing on specific features of heart failure. Giventhe inherent connectedness of the circulation such divisions are to someextent artificial and indeed these distinctions can overlap. For example, inthe forward/backward division:
– forward failure focuses on reduced blood delivery, reduced ejection ofblood from the ventricles and insufficient Q for metabolic needs.
– backward failure focuses on reduced filling of the ventricles, reducedemptying of the venous system, or reduced Q unless high ventricularpressures exist.
Cardiovascular-respiratory control system 19
On the other hand, in the division systolic/diastolic:
– systolic failure focuses on insufficient systolic action, often impaired con-tractility, and consequently reduced ejection fraction and Q.
– diastolic failure refers to the impairment of ventricular filling withoutnecessarily an impairment of ejection fraction.
These classifications are subdivided into left and right heart categoriesand ”typical” clinical heart failure is due to impairment of left ventricularfunction and reflects systolic disfunction and forward failure. Causes of heartfailure 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 toprovide sufficient blood flow to meet the metabolic needs of the body. Inmost cases, this means that the heart exhibits a deterioration of the heart’spumping ability. Pumping impairment that is due to a reduction in con-tractility is a consequence of the heart muscle being damaged or weakenedin some way.
In chronic heart failure, there is a progressive deterioration in heartfunction over time which is the reason the condition is so serious. The con-dition becomes progressively more sever due to the compensatory mecha-nisms which try to maintain normal cardiovascular function. In a left heartfailure scenario, for example, if heart muscle is damaged so that contractil-ity is reduced, stroke volume and cardiac output will be decreased. Arterialblood pressure falls due to the impaired pumping efficiency of the heart. Thebaroreceptors, sensing reduced pressure, trigger compensatory sympatheticsystem activity and vasoconstriction. These responses can produce signifi-cant elevation of afterload, which can further reduce stroke volume. Overtime, the added stress to the heart results in damaging cardiac muscle com-pensatory changes (remodeling) which further weakens heart function. Thusthe deterioration in heart function is self-reinforcing. This form of heart fail-ure is referred to as chronic in contrast to acute heart failure which is theresult of heart damage occurring over a short time frame.
The kidneys may also respond to reduced cardiac function by induc-ing fluid retention to increase blood volume. This compensatory responseis triggered by the perceived reduction in circulating blood volume andacts to raise blood pressure. This fluid retention will increase preload orfilling pressure but the increased pressure and excess fluids can cause pul-monary or systemic fluid congestion and edema. In left ventricular failure,the reduced left ventricular function results in blood accumulating in thepulmonary venous system (raising pulmonary venous pressure) and can re-sult in significant pulmonary congestion and difficulty in breathing. Hence,the term ”congestive heart failure” is often used, though not every form ofheart failure exhibits this quality.
20 Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel
Given the interconnectedness of the circulatory system, progressive dete-rioration in one ventricle can lead to the other ventricle becoming impairedand the occurrence of simultaneous left and right heart failure. Diastolicand systolic failure can also occur together. In this paper we will refer toleft ventricular failure as that clinical condition of failure of forward blooddelivery due to reduced systolic function and reduced contractility of theheart tissue.
An example of stable pressure states for four types of clinical heartfailure are given in Table 5. For example, in clinical left heart failure (leftventricular 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 Pas Pvs Pap Pvp
LV failure no change ↑ moderately ↑ moderately ↑ significantlyRV failure no change ↑ significantly small change small changeLB failure no change no change no change ↑RB failure no change ↑ significantly no change no change
The impairment of the heart’s pumping action will be modeled by areduction in contractility and consequently the ejection fraction. Given theeffects of remodeling of the heart tissue due to the stress on the system, wecould also include a reduction in the ventricular compliance parameter inEq. (19) as a contributing factor to the reduction of stroke volume.
Tables 6 and 7 give the parameters and computed steady states for rest-ing 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. Giventhat there is little change in arterial blood pressure, this implies a similarreduction in ejection fraction consistent with clinical observations found inNiebauer et al. (1999) [44]. We also assume a small drop in right contrac-tility of 8% from normal. Due to the compensatory mechanism describedabove, 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 upperend of values reported in Javaheri (1999) [23]. Pulmonary resistance Rp isalso increased by 10% (see, e.g., Moraes et al. (2000) [12]). In this chroniccondition water retention and other mechanisms act to increase total bloodvolume 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 Parm-ley [47] as well as Chiariello and Perone-Filardi [10]. As a consequence ofthese changes Q decreases by 20% and hence transport delay is increasedby about 25% (we ignore the effects of the cerebral blood flow). The cardio-vascular steady state values for this case and further simulations presentedlater (see Section 11) can be compared with values presented in Tsuruta etal. (1994) [66]. In that paper, a model is developed and used to identify car-diovascular parameters relating to the four classes of severity of heart failureas defined by the New York Heart Association. The parameter estimationdepended on steady state values of H , Pas, Pap, Pvs, Pvp, V0, and cardiacoutput. The values for these state variables in the four classes depended oninterpolation from certain known values. Among the parameters which wereidentified were the vascular resistances.
Table 6. Optimal control parameters: chronic left ventricular heart failure sleeptransition
Parameter Awake Sleep
Apesk 198.7 188.7βl 25.8 23.19βr 1.67 1.50Rp 2.16 2.16V0 6.25 6.25H 86.02 78.02PaCO2
40.5 44.5τT 30.0 30.0
In contrast, Tables 8 and 9 give the parameters and computed steadystates for resting awake and stage 4 sleep for left heart failure where thereis no increase in blood volume V0 as might be the case when there is acuteheart failure. A small decrease in Sr is assumed. In this case, no change isassumed in Rp or PaCO2
. In this case, no increase in Pvs is seen but rathera drop occurs.
Figures (8) thru (13) give the dynamics of the control for the optimalcontrol case of acute left heart failure.
10. Modeling two delays in the state space
Previously we have modeled the transition to sleep considering ventilationas an optimal control. We are now going to use formula (33) which describesan empiric relation between VA and the blood gas partial pressures PaCO2
,
PBCO2, and PaO2
. Thus VA becomes incorporated into the state equationsand we can consider the transport delay in this control. We consider a singledelay for both the peripheral and central controls. This is reasonable as thetransport 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 sleeptransition
Steady State Awake Sleep
H 86.02 78.02Pas 98.4 83.72Pap 23.19 21.94Pvs 3.74 4.07Pvp 14.58 14.46PaCO2
40.5 44.5PaO2
102.8 98.35PvCO2
50.78 54.46PvO2
29.73 30.36Ql 3.98 3.46Qr 3.98 3.46Rs 23.79 23.02Sl 24.77 20.22Sr 5.04 4.11
VA 5.67 4.34Vstr,l 0.0463 0.0444Vstr,r 0.0463 0.0444
Table 8. Optimal control parameters: acute left heart failure sleep transition
Parameter Awake Sleep
Apesk 198.7 188.7βl 25.8 23.19βr 1.77 1.59Rp 1.965 1.965V0 5.0 5.0H 86.02 78.02PaCO2
40.0 44.5τT 32.0 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 thedependence of VA on PaCO2
, PaO2and PBCO2
is given by
VA(t) = Gpe−0.05PaO2
(t−τp)max(0, PaCO2
(t− τp)− Ip) (33)
+Gc max(0, PBCO2(t)− MRBCO2
KCO2FB− Ic).
The first term above describes the effect on ventilation of the blood gasesPaCO2
and PaO2as measured by peripheral sensors located in the carotid
artery. This will be referred to as the peripheral control. The second termdescribes the effect of the brain CO2 level (PBCO2
) and will be referred toas 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 Awake Sleep
H 86.02 78.02Pas 86.6 74.08Pap 19.24 18.30Pvs 2.77 3.02Pvp 11.97 11.96PaCO2
40.0 44.5PaO2
103.38 98.35PvCO2
51.06 55.17PvO2
28.12 28.83Ql 3.70 3.23Qr 3.70 3.23Rs 22.64 22.00Sl 24.77 20.22Sr 5.35 4.37
VA 5.739 4.34Vstr,l 0.0430 0.0414Vstr,r 0.0430 0.0414
1 2 3 4 5minutes
38
40
42
44
46
mm Hg
PACO2
1 2 3 4 5minutes
85
90
95
100
105
110mm Hg
PAO2
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 appearsin this equation. Note that Ip and Ic denote cutoff thresholds, so that therespective ventilation terms become zero when the quantities fall below thethresholds.
Ventilatory dead space effects are accounted for by defining the quantityVA = K · VE where VE is minute ventilation and K is a constant smallerthan one. In this way effective ventilation is reduced by a fixed dead spacepercent which corresponds to modeling change in ventilation as a changein rate of breathing. This is implemented here by a scale reduction in thecontrol gains Gc and Gp. See, e.g., Batzel and Tran [3].
The optimal control now only models the cardiovascular control. Therespiratory control is given by an empirical formula with delay.
24 Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel
1 2 3 4 5minutes
44464850525456mm Hg
PVCO2
1 2 3 4 5minutes
27.5
28
28.5
29
29.5mm Hg
PVO2
Fig. 9. Optimal control: acute left heart failure case
1 2 3 4 5minutes
65
70
75
80
85
90mm Hg
PAS
1 2 3 4 5minutes
2.22.42.62.8
33.23.4
mm Hg
PVS
Fig. 10. Optimal control: acute left heart failure case
1 2 3 4 5minutes
17
18
19
20
mm Hg
PAP
1 2 3 4 5minutes
11.4
11.6
11.8
12
12.2
12.4
mm Hg
PVP
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 fac-tor Gs and there is an increase in a shift term Kshift altering the operating
point of ventilation. The effective drive during sleep Vsleep is described by
Vsleep(t) = Gs(t)[max(0, Vawake(t)−Kshift(t))]. (34)
The time dependencies for Gs and Kshift reflect the smooth changein these parameters that occurs in the transition from ”awake” state to
Cardiovascular-respiratory control system 25
1 2 3 4 5minutes
0.038
0.04
0.042
0.044
0.046liters
VSTR
1 2 3 4 5minutes2.4
2.62.8
33.23.43.63.8
4liters per min
QL
Fig. 12. Optimal control: acute left heart failure case
1 2 3 4 5minutes
3.54
4.55
5.56
liters per minute
VA.
º
1 2 3 4 5minutes
72.575
77.580
82.585
87.590bpm
H
Fig. 13. Optimal control: acute left heart failure case
”stage 4 quiet sleep ”. Gs is set during the awake state at 1 and reducessmoothly to a minimum (normally 0.6) at stage 4 sleep. Kshift begins at 0and increases to a maximum (normally about 4 mmHg) by the beginning ofstage 1 sleep. These changes reflect the reduction in the normal ventilatoryresponse Vawake as a result of physiological changes during sleep. Once stage4 sleep is reached these values are constant. For these simulations we use abase line transit time to ”stage 4 sleep” to be three minutes. The changesin Gs and Kshift will be modeled by incorporating exponential functionswhich change smoothly through the various sleep stages between awake andstage 4 NREM sleep. The parameters in these expressions can be adjustedto simulate an essentially linear decrease over the entire transition fromawake to stage 4 sleep or bias the decrease to the early stages of sleep.
Similar decreases for the metabolic rates, sleep contractility, and sys-temic resistance reflecting the physiological changes during sleep transition(discussed above) are incorporated. The system is nonautonomous, howeverwe still implement the control functions u1 and u2 as calculated for a time-independent linear system around the final steady state ”stage 4 sleep”.This reduces the optimality of the control for the original nonlinear systembut the thereby obtained (suboptimal) control still stabilizes the system andis 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 func-tional
∫ ∞
0
(qas(Pas(t)− P sas)2 + q1u1(t)2
)dt (35)
is minimized under the restriction
x(t) = f(x(t), x(t − τp), x(t− τT );W s) +B u(t), x0 = φ.
y(t) = Dx(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 linearizedsystem now expanding the system using two delays.
x(t) = ξ(t) = f(xs + ξ(t), xs + ξ(t− τp), xs + ξ(t− τT );W s) +B u
= 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 respectto x(t), x(t − τp), and x(t− τT ) , respectively, evaluated at x = xs,
A1 =∂f
∂x(t)(xs;W s),
A2 =∂f
∂x(t− τp)(xs;W s),
A3 =∂f
∂x(t− τT )(xs;W s).
(38)
Analogously,
η(t) = (Pas(t)− P sas)T = Dξ(t). (39)
By neglecting terms of order o(ξ) we get linear approximations ξ`(t) andη`(t) for ξ(t) and η(t), respectively,
ξ`(t) = A1ξ`(t) +A2 ξ`(t− τp) +A3 ξ`(t− τT ) + B u(t),
η`(t) = Dξ`(t),
ξ`(0) = xr − xs.(40)
Again, we apply the results on approximation of feedback control for delaysystems using Legendre polynomials found in Kappel and Propst [26].
Cardiovascular-respiratory control system 27
Table 10. VA empirical control parameters: normal adult sleep transition
Parameter Awake Sleep
Gc 1.44 1.44Gp 30.24 30.24Gs 1.0 0.6Kshift 0 4.2IC 35.5 35.5IP 35.5 35.5H 75.02 68.02MRCO2 0.266 0.224MRBCO2
0.042 0.040MRO2 0.310 0.260Apesk 147.16 139.80βl 85.89 77.30βr 2.083 1.874τp 7.8 7.8τT 24.0 24.0S 4 transit - 3 min
Table 11. VA empirical control steady states: normal adult sleep transition
Steady State Awake Sleep
H 75.02 68.02Pas 101.77 87.14Pap 17.18 16.01Pvs 3.618 3.909Pvp 7.478 7.489PaCO2
39.16 42.67PaO2
104.37 100.47PvCO2
47.44 50.62PvO2
34.50 35.30PBCO2
47.23 50.34Ql 4.938 4.334Qr 4.938 4.334Rs 19.88 19.21Sl 72.02 58.77Sr 5.49 4.48VA 5.86 4.53Vstr,l 0.0658 0.0637Vstr,r 0.0658 0.0637
11. Simulations with two delays
Tables 10 and 11 give the parameters and computed steady states for restingawake and stage 4 sleep for a normal adult with borderline elevated Rsvalues and with moderate sleep transition profile. From this point on, weare using the empirical control for VA while maintaining the optimal controlfor the cardiovascular system. We will refer to this case as the VA empiricalcase. 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 durationclearly show the stabilizing influence of the control.
Figures (14) thru (20) give the dynamics of the control for the normal VAempirical control case. For this case the transition to stage 4 sleep is assumedto be 3 minutes. Furthermore, the shift Kshift is assumed to occur by stage1 (one fourth of the transition time) as in [35]. Gs reduces smoothly fromstage 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 Apeskare also assumed to be significantly reduced by stage 1. This assumption ismade for purposes of exploring the dynamics of transition and because notmuch is known about the actual time course of these parameter changes insleep transition.
1 2 3 4 5minutes
39
40
41
42
43
44mm Hg
PACO2
1 2 3 4 5minutes
7580859095
100105110mm Hg
PAO2
Fig. 14. VA empirical control: normal case
1 2 3 4 5minutes
46474849505152mm Hg
PVCO2
1 2 3 4 5minutes
34.234.434.634.8
3535.235.4
mm Hg
PVO2
Fig. 15. VA empirical control: normal case
Figures (21) thru (24) give the dynamics of the control for the normalVA empirical control case with a reduced sleep transition time. For thiscase the transition to stage 4 sleep is assumed to be 2 minutes. The shiftKshift is again assumed to occur by stage one (one fourth of the transitiontime) and the reductions in Gs, βl, βr and the metabolic rates are assumedto be reduced as in the previous case. The shift Kshift is increased to 5.2and the gain Gs at stage 4 is 0.4. The quicker transition time and largershift create a deeper drop in the ventilation rate with sleep onset than in
Cardiovascular-respiratory control system 29
1 2 3 4 5minutes
85
90
95
100
105mm Hg
PAS
1 2 3 4 5minutes
3.2
3.4
3.6
3.8
4mm Hg
PVS
Fig. 16. VA empirical control: normal case
1 2 3 4 5minutes
15.5
16
16.5
17
17.5
18mm Hg
PAP
1 2 3 4 5minutes
7.4257.45
7.4757.5
7.5257.55
7.575
mm Hg
PVP
Fig. 17. VA empirical control: normal case
1 2 3 4 5minutes
4.2
4.4
4.6
4.8
5liters per min
FS
1 2 3 4 5minutes
18
18.5
19
19.5
20
mm Hg min per lit
RS
Fig. 18. VA empirical control: normal case
the previous case. This behavior will be compared now with the congestiveheart condition case which includes an increased delay time.
Tables 12 and 13 give the parameters and computed steady states forresting 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. Forejection 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 by10% ([12]). Heart rate H is increased by 15%. The PvCO2
increase and PvO2
decrease are a consequence of the reduced cardiac output. In this chroniccondition water retention and other mechanisms act to increase total bloodvolume and we assume V0 is increased by 23%. The increase in V0 acts toraise Pvs. As a consequence of these changes Q decreases by 20% and hencetransport delay is increased by about 25%.
30 Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel
1 2 3 4 5minutes
0.063
0.064
0.065
0.066
0.067
0.068liters
VSTR
1 2 3 4 5minutes
3.5
4
4.5
5
5.5liters per min
QL
Fig. 19. VA empirical control: normal case
1 2 3 4 5minutes
3.54
4.55
5.56
liters per minute
VA.
º
1 2 3 4 5minutes
68
70
72
74
76bpm
H
Fig. 20. VA empirical control: normal case
1 2 3 4 5minutes
39
40
41
42
43
44mm Hg
PACO2
1 2 3 4 5minutes
7580859095
100105110mm Hg
PAO2
Fig. 21. VA empirical control: fast sleep case
Tables 14 and 15 give the parameters and computed steady states forresting awake and stage 4 sleep where the congestive heart condition involvessignificant 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 leftside) will eventually extend to deterioration of function on the other side[58]. We maintain the CO2 ventilation thresholds simulating an operatingpoint of PaCO2
in the middle range of values given in Javaheri (1999) [23].In general, PaCO2
levels in congestive heart patients are little changed fromlevels found in normal individuals even when there is reduced exchangeefficiency 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 isnow increased by 50%. For comparative state values in the case of severecongestive heart failure, see Bruschi et al. (1999) [7] and Bocchi et al. (2000)
Cardiovascular-respiratory control system 31
1 2 3 4 5minutes
46474849505152mm Hg
PVCO2
1 2 3 4 5minutes
34.234.434.634.8
3535.235.4
mm Hg
PVO2
Fig. 22. VA empirical control: fast sleep case
1 2 3 4 5minutes
15.5
16
16.5
17
17.5
18mm Hg
PAP
1 2 3 4 5minutes
7.4257.45
7.4757.5
7.5257.55
7.575
mm Hg
PVP
Fig. 23. VA empirical control: fast sleep case
1 2 3 4 5minutes
2
3
4
5
6liters per minute
VA.
º
1 2 3 4 5minutes
68
70
72
74
76bpm
H
Fig. 24. VA empirical control: fast sleep case
[6] for values of Q, Hanly et al. (1993) [21] for values of transport delay, andBocchi et al. (2000) [6] for comparative values of H , Rs, and Vstr. See alsoTable 22 in the appendix and Tsuruta et al. (1994) [66] and Hambrecht etal. (2000) [20] for comparative Rs values. Arterio-venous oxygen contentdifference 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 Niebaueret al. (1999) [44] for very severe heart failure cases.
It is well known that delays in feedback control can create instabilityin a control system. In congestive heart failure, the reduced cardiac outputinduces an increased transport delay which will reduce the efficiency of thecentral and peripheral controllers of ventilation. This reduced efficiency isdue to the increased time it takes for blood gases to be transported from thesite 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. VA empirical control parameters: left ventricular heart failure sleeptransition
Parameter Awake Sleep
Apesk 198.66 188.73βl 25.77 23.19βr 1.67 1.50Rp 2.16 2.16V0 6.25 6.25H 86.02 78.02IC 35.5 35.5IP 35.5 35.5Gs 1.0 0.6Kshift 0 4.2τp 9.75 9.75τT 30.0 30.0
Table 13. VA empirical control steady states: left ventricular heart failure sleeptransition
Steady State Awake Sleep
H 86.02 78.02Pas 98.49 83.81Pap 23.19 21.94Pvs 3.74 4.06Pvp 14.59 14.46PaCO2
39.16 42.67PaO2
104.37 100.47PvCO2
49.44 52.63PvO2
29.77 30.43PBCO2
47.23 50.34Ql 3.98 3.46Qr 3.98 3.46Rs 23.82 23.06Sl 24.77 20.22Sr 5.035 4.11
VA 5.86 4.53Vstr,l 0.046 0.044Vstr,r 0.046 0.044
where these levels are measured. One form of respiratory instability associ-ated with CHF is a form of periodic breathing (PB) known as Cheyne-Stokesrespiration (CSR). This form of involuntary respiration involves periods ofregular waxing and waning of tidal volume interspersed with central apnea(CA). Cheyne-Stokes respiration seems to be a complicating factor for CHFbut the actual mechanisms inducing CSR in congestive heart patients arestill under active investigation. The increased feedback delay due to reducedcardiac output, in conjunction with other factors may be sufficient to con-tribute to the onset, characteristics, or persistence of central sleep apnea,
Cardiovascular-respiratory control system 33
Table 14. VA empirical control parameters: left and right ventricular heart failuresleep transition
Parameter Awake Sleep
Apesk 250.17 237.7βl 12.88 11.60βr 1.46 1.31Rp 2.16 2.16V0 6.9 6.9H 92.02 80.02IC 35.5 35.5IP 35.5 35.5Gs 1.55 0.465Kshift 0 5.5τp 11.6 11.6τT 36.0 36.0S 4 transit - 2 min
Table 15. VA empirical control steady states: left and right ventricular heartfailure sleep transition
Steady State Awake Sleep
H 92.02 80.02Pas 90.82 72.78Pap 26.74 25.17Pvs 3.60 4.05Pvp 19.55 19.14PaCO2
37.95 44.44PaO2
105.77 98.41PvCO2
50.25 56.81PvO2
25.74 25.49PBCO2
46.02 52.12Ql 3.33 2.79Qr 3.33 2.79Rs 26.22 24.67Sl 13.25 10.37Sr 4.71 3.69VA 6.05 4.35Vstr,l 0.0362 0.0348Vstr,r 0.0362 0.0348
PB, or CSR associated with CHF. See, e.g., Hall et al. (1996) [19], Pinnaet al. (2000) [51] and Cherniack (1999) [9]. For analytical results see Batzeland Tran (2000) [3].
Figures (25) thru (30) give the congestive heart failure dynamics for theVA empirical control case simulating transition to sleep with fast transitionparameters and in this case we also assume an awake feedback gain which is50% higher than normal which in effect increases CO2 sensitivity. Increasesin 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 furtherincrease the shift term Kshift by 33% (to 5.5 mmHg). Fast sleep onset timescan occur in patients with fragmented sleep cycles as occurs when multiplesleep apneas induce arousal and sleep disturbance. See, e.g., Bennet et al.(1998) [4]. The change in sleep control parameters in this case simulates anincreased influence of the sleep state on control effectiveness.
These deviations in standard operating points drive ventilation to nearapnea and exhibit one mechanism for inducing Cheyne-Stokes respiration(CSR) and central sleep apnea. The oscillatory behavior is induced by theincreased delay as can be observed by comparing this case with the fasttransition case for a normal adult represented in Figures (21) thru (24).Javaheri (1999) [23] and Quaranta et al. (1997) [53] indicate that a num-ber of factors influencing control stability (such as higher CO2 sensitivityand circulation delay) may contribute to central sleep apnea and CSR incongestive heart patients. Lorenzi-Filho et al. (1999) [40] report that re-ductions in PaCO2
sensed at the peripheral chemoreceptors can also triggercentral apneas during Cheyne-Stokes respiration. It is clear that the inter-action of various respiratory factors can act in complex ways to influencethe production of CSR and apnea in congestive heart failure.
The larger reduction in sleep control gain Gs in this simulation actuallyacts to reduce the magnitude of the oscillatory cycles. On the other hand,simulations indicate and, in general, theory confirms that the higher con-trol gain (CO2 sensitivity) prolongs and exaggerates oscillatory behavior.Likewise, a longer time course in the reduction in control gain from stage1 to stage 4 sleep (thus maintaining higher gain for a longer time) wouldcontribute to unstable behavior. It is the degree and speed of the shiftKshift that is responsible for the initial steep drop in ventilation which cantrigger apnea and repetitive cycles similar to CSR and the increased delayreinforces and perpetuates the oscillatory behavior.
1 2 3 4 5minutes
37.540
42.545
47.550mm Hg
PACO2
1 2 3 4 5minutes
70
80
90
100
110mm Hg
PAO2
Fig. 25. VA empirical control: severe left and right ventricular failure sleep case
Cardiovascular-respiratory control system 35
1 2 3 4 5minutes
45
50
55
60
65
mm Hg
PVCO2
1 2 3 4 5minutes
24.5
25
25.5
26
26.5
27mm Hg
PVO2
Fig. 26. VA empirical control: severe left and right ventricular failure sleep case
1 2 3 4 5minutes
65707580859095mm Hg
PAS
1 2 3 4 5minutes
3.23.43.63.8
44.24.4
mm Hg
PVS
Fig. 27. VA empirical control: severe left and right ventricular failure sleep case
1 2 3 4 5minutes
24
25
26
27
28mm Hg
PAP
1 2 3 4 5minutes
18.518.75
1919.2519.5
19.7520mm Hg
PVP
Fig. 28. VA empirical control: severe left and right ventricular failure sleep case
12. Conclusion
In this paper we have considered a model of the cardiovascular-respiratorycontrol system with constant state equation delays. The model utilizes anoptimal control approach to represent the complex control features of thecardiovascular component in this system. The respiratory control is consid-ered both from an optimal control approach and from an empirical approachwhich introduces a respiratory feedback delay into the state equations. Themodel was applied to study the transition from the awake state to NREMsleep for normal individuals and for individuals suffering from congestiveheart problems. The model steady states are consistent with observationboth for the normal and congestive heart states. The dynamical simulationsshow that the transport delay between respiratory compartments does notcontribute to instability even at large delays. However, the transport delay
36 Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel
1 2 3 4 5minutes
0.034
0.035
0.036
0.037
0.038liters
VSTR
1 2 3 4 5minutes
2.6
2.8
3
3.2
3.4
3.6liters per min
QL
Fig. 29. VA empirical control: severe left and right ventricular failure sleep case
-1 1 2 3 4 5minutes
2
4
6
8liters per minute
VA.
º
1 2 3 4 5minutes
75
80
85
90
95
bpm
H
Fig. 30. VA empirical control: severe left and right ventricular failure sleep case
to the peripheral sensor is significant and can result in Cheyne-Stokes typerespiration for a severe congestive heart condition with certain respiratoryparameters during the transition to NREM sleep.
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Cardiovascular-respiratory control system 41
APPENDIX
We assume the following:
PAO2= PaO2
,PACO2
= PaCO2,
PBO2= PBvO2
,PBCO2
= PBvCO2,
PTO2= PTvO2
,PTCO2
= PTvCO2,
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 VA for the optimal case and control gains Gc andGp 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 incorpo-
rated 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 meanvalues 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 theS-shaped O2 dissociation curve which relates blood gas concentrations topartial pressures we will use the relation
CO2(t) = K1(1− e−K2PO2 (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 alinear dependence of CCO2 on PCO2 ,
CCO2(t) = KCO2PCO2(t) + kCO2 . (42)
A linear relationship was also used by Khoo et al. [31]. Other parameterand 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 Value Unit Source
Gc 1.440 * l/(min ·mmHg) [31]3.2 l/(min ·mmHg) [32]
Gp 30.240 * l/(min ·mmHg) [31]26.5 l/(min ·mmHg) [32]
Ic 35.5 * mmHg [31], [32]45.0 mmHg [35]
Ip 35.5 * mmHg [31], [32]38.0 mmHg [35]
K1 0.2 lSTPD/l [13]K2 0.046 mmHg−1 [13]kCO2 0.244 lSTPD/l [31]KCO2 0.0065 lSTPD/(l ·mmHg) [31]
0.0057 lSTPD/(l ·mmHg) [35]mB 1400 g [36], p. 745
MRBCO20.042 * lSTPD/min [32]0.031 lSTPD/(min ·kg brain tissue) [35]0.050 lSTPD/min [15]0.054 lSTPD/min [13]
MRCO2 0.21 lSTPD/min [35]0.235 lSTPD/min [32]0.200 lSTPD/min [33]0.26 * lSTPD/min [36] p. 239
MRO2 0.26 lSTPD/min [35]0.290 lSTPD/min [32]0.240 lSTPD/min [33]0.31 * lSTPD/min [36] p. 239
CO2 sensitivity 2.1 +/- 1.0 * lBTPS/(min ·mmHg) [23]
Cardiovascular-respiratory control system 43
Table 17. Parameter values (awake rest)
Parameter Value Unit Source
PICO20 mmHg [31], [15]
PIO2150 mmHg [31]
Patm 760 mmHg [31],[15]Rp 0.965 mmHg ·min /l [36] p. 233
1.4 mmHg ·min /l [36] p. 1441.95 * mmHg ·min /l [63]1.5-3 mmHg ·min /l [42]
RQ 0.88 - [15]0.81 - [31], [32]
0.84 * - [36] p. 239VAO2
2.5 * lBTPS [31]3.0 lBTPS [15]0.5 lBTPS [18], p. 1011
VACO23.2 * lBTPS [31]3.0 lBTPS [35], [15]
VTCO215 l [31], [35], [32]
VTO26 * l [31], [35], [32]1.55 l [18], p. 1011
VBCO20.9 * l [32]1.0 l [15]1.1 l [13]
VBO21.0 l [15]1.1 l [13]
VD 0.15 lBTPS [35],[36] p. 239
VD 2.4 lBTPS/min [36] p. 2392.28 lBTPS/min [31]
FB 0.5 l/(min ·kg brain tissue) [35], [36], p. 7450.75-0.8 * l/min [15], [13]
12-15% of Q l/min [55],p. 242
Table 18. Nominal steady state values (awake rest)
Quantity Value Unit Source
CaCO20.493 lSTPD/l [36], p. 253
CaO20.197 lSTPD/l [36], p. 253
CvCO20.535 lSTPD/l [36], p. 253
CvO20.147 lSTPD/l [36], p. 253
H 70 min−1 [36] p. 144Pap 12 mmHg [36] p. 144
15 mmHg [66] p. 410-22 mmHg [42] Chptr. 8
Pas 100 mmHg [36] p. 14493 mmHg [66] p. 4
Pvp 5 mmHg [36] p. 1448 mmHg [66] p. 4
Pvs 2-4 mmHg [36] p. 1445 mmHg [66] p. 4
44 Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel
Table 19. Nominal steady state values (awake rest)
Quantity Value Unit Source
PACO240 mmHg [18],p. 495, [36] p. 239
PAO2104 mmHg [18],p. 494100 mmHg [36] p. 239
PaCO240 mmHg [18],p. 495, [36], p. 253
PaO295 mmHg [18],p. 49490 mmHg [36], p. 253
PvCO245 mmHg [18],p. 49546 mmHg [36], p. 253
40-50 mmHg [42] Chptr. 8PvO2
40 mmHg [18],p. 494, [36], p. 25335-40 mmHg [42] Chptr. 8
Ql = Qr = Fp = Fs 6 l/min [31], [15]6.2 l/min [36] p. 2395 l/min [36] p. 144
4-7 l/min [42] Chptr. 8Rs 20. mmHg ·min /l [36] p. 144
11-18 mmHg ·min /l [42]
VA 4.038 lBTPS/min [13]5.6 lBTPS/min [36] p. 239
VE 8 lBTPS/min [36] p. 239Vstr,l 0.070 l [36] p. 144
Table 20. Nominal steady state values (NREM sleep)
Quantity % change Model Value Source
PACO2↑ 2-8 mmHg 44 mmHg [59]
PAO2↓ 3-11 mmHg 98.9 mmHg [59,37]
PvCO2↑ 6 % 51.9 mmHg estimate
PvO2↑ 1 % 35.2 mmHg estimate
VE ↓ 14-19 % 6 lBTPS/min [59,38]Pas ↓ 5-17 % 87.0 mmHg [43,59,60]H ↓ 10 % 68 mmHg [59,60]Q ↓ 0-10 % 4.3 l/min [59,57]VA ↓ 14-19 % 4.4 lBTPS/min [38]
MRO2 ↓ 15 % 0.26 lSTPD/min [35]MRCO2 ↓ 15 % 0.23 lSTPD/min [35]
sympathetic activity ↓ significantly - [60]Rs ↓ 5-10 % 19.2 mmHg ·min /l estimateSl ↓ 5-15 % 58.7 mmHg estimateSr ↓ 5-15 % 4.5 mmHg estimate
Cardiovascular-respiratory control system 45
Table 21. Miscellaneous parameters (awake and sleep unless otherwise noted)
Quantity Value Unit Source
V0 5.0 l [24]Apesk 177.47 mmHg ·min ·l−1 [24]αl 89.47 min−2 [24]αr 28.46 min−2 [24]βl 73.41 mmHg ·min−1 [24]βr 1.78 mmHg ·min−1 [24]γl 37.33 min−1 [24]γr 11.88 min−1 [24]cap 0.03557 l ·mmHg−1 [24]cas 0.01002 l ·mmHg−1 [24]cvp 0.1394 l ·mmHg−1 [24]cvs 0.643 l ·mmHg−1 [24]cl 0.01289 l ·mmHg−1 [24]cr 0.06077 l ·mmHg−1 [24]
Table 22. Estimated state variable values for congestive heart failure categoriesas presented in Tsuruta et al. [66]
Quantity Normal Stage A Stage B Stage C Stage D
H 70 85 85 85 85Q 5.6 5.0 4.4 3.8 3.2Vstr .08 .059 .052 .045 .038Pap 15.0 19.0 23.0 27.0 31.0Pas 93 .3 93.3 93.3 93.3 93.3Pvp 8.0 12.0 16.0 20.0 24.0Pvs 5.0 5.0 5.0 5.0 5.0Rs 15.45 17.30 19.66 22.76 27.03
Recommended