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C 2004) Cardiovascular Engineering: An International Journal, Vol. 4, No. 1, March 2004 (°
An Optimal Control Approach to Modeling the Cardiovascular-Respiratory System: An Application to Orthostatic Stress MARTIN FINK,∗,‡ JERRY J. BATZEL,∗ and FRANZ KAPPEL∗,†
PHYSIOLOGY
This paper introduces a model designed to study the cardiovascular-respiratory system and its control features. It has been previously applied to study several physiological situations and in this paper it will be applied to the simulation of the phenomenon of orthostatic stress. Orthostatic stress refers to stress placed on the cardiovascular system when the body is in the upright position as compared to the supine position. The model consists of cardiovascular and respiratory components and includes cardiovascular autoregulation, ventilation control, and the baroreflex loop. Instead of an explicit formula for calculating the control response from the pressures and blood gases, we use an optimal control. Steady state and dynamic model simulation is compared with experimental data we have collected using head up tilt (HUT) experiments. The simulations fit the measured data well and represent reasonable physiological values. This work also examines an important issue related to orthostatic stress experiments. The head up tilt experiment (where gravity creates extra pressure stress on the lower body) is to be distinguished from the lower body negative pressure (LBNP) experiment where the lower body is subject to reduced exterior air pressure. Both tests create blood volume shifts to the lower body but the two physiological conditions are not entirely equivalent. We discuss some of these issues and examine several aspects of implementation of orthostatically induced pressures.
General Cardiovascular and Respiratory System Function Cardiovascular Subsystem The cardiovascular system (CVS) can be divided into two main circuits: the pulmonary and the systemic circuit. These circuits can be subdivided into four main vascular circuit components: pulmonary arterial, pulmonary venous, systemic arterial, and systemic venous. The pulmonary system circuit connects the heart with the lung compartment, while the systemic system connects the heart to the tissue compartments where metabolism takes place. The lungs and the tissue compartments act as resistances in the flow of blood. An intricate control system acts to vary blood flow between tissues and lungs so as to maintain appropriate levels of nutrients, carbon dioxide CO2 , and oxygen O2 . Figure 1 provides a block diagram representation of this system. State variables for the CVS consist mainly of vascular pressures Pas , Pvs , Pap , and Pvp where Pas represents arterial systemic pressure and the other quantities are similary defined. Respiratory state variables are partial pressures for CO2 , and oxygen O2 in the lungs tissues. A main function of the CVS control system is to maintain arterial blood pressure Pas at appropriate levels by varying cardiac output Q (via variation in heart rate H and heart contractility) and vascular systemic resistance and venous capacitance. The steady state for various physiological conditions such as supine or standing is maintained by negative feedback loops which depend on sensing arterial and venous blood pressure at several sites.
Key words: orthostatic stress; cardiovascular system; baroreceptor loop; HUT; LBNP.
∗ SFB
“Optimierung und Kontrolle,” Karl-Franzens-Universit¨at, Graz, Austria. † Department of Mathematics, Karl-Franzens-Universit¨at, Graz, Austria. ‡ To whom correspondence should be a ddressed at SFB “Optimierung und Kontrolle,” Karl-Franzens-Universit¨at, Heinrichstraße 22, 8010 Graz, Austria. E-mail:
[email protected]
27 C 2004 Plenum Publishing Corporation 1567-8822/04/0300-0027/0 °
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Fink, Batzel, and Kappel
Figure 1. Orthostatic model block diagram.
These steady states are most likely optimal for the system, although the nature of this optimality is still debated. The short term control of Pas depends on global control mechanisms mediated by the sympathetic and parasympathetic nervous systems which modulate heart rate, contractility, systemic arterial resistance, and systemic venous capacitance. There are also local control mechanisms for controlling vasodilation and resistance to insure that enough blood is delivered to organs when needed and which depend on local O2 and CO2 concentrations. Longer term control mechanisms involve control of blood volume via the kidney and hormonal controls which have a variety of influences. Respiratory Subsystem The respiratory system acts to exchange O2 , which is needed by the various tissues for metabolism, for CO2 which is produced by metabolic activity. Efficient ex-
change of these gases in the lungs depends on the ventilation rate V˙ A which is controlled by a negative feedback loop depending on sensors for CO2 and O2 located in the carotid artery and also sensors for brain tissue CO2 . A significant delay occurs in this feedback loop which can effect stability when there are significant deviations in these blood gases. This delay is caused by the time needed for the blood to transport these gases from the lungs to the sensory sites (transport delay) and hence this delay depends on Q. There are certain conditions where hypoxia can impact response to orthostatic stress. See e.g., Rickards and Newman (2002). Blood gases can influence the local control of vascular resistance which is also influenced by the baroreflex. Metabolic effects can influence longer term control of blood pressure. Thus it is of value to include a respiratory component to the model so that such respiratory and metabolic effects can be examined although they are not considered in this report. In orthostatic stress simulated by the tilt table experiment there is usually little
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deviation in blood gases and hence the transport delays are ignored. As mentioned above, the respiratory system in turn influences CVS resistance via oxygen cocentration (brain and heart tissue also respond to CO2 concentration) and there are also other mechanisms which act to match ventilation and cardiac output as well as synchronize respiratory and heart rate frequencies. Orthostatic Stress Orthostatic stress refers to the stress induced on the cardiovascular system due to gravitational effects produced by the upright position as compared to the supine position. The main gravitational effect is the pooling of blood in the compliant venous compartment of the lower body. This induces a fall in blood pressure which must be counteracted by the baroreflex which senses this drop. Heart rate and contractility are increased to raise the blood pressure and sympathetic activity responds by decreasing the capacitance in the venous compartment and also influences systemic resistance. The net result is that, while there is a shift of about 500 mL of blood to the venous compartment, mean arterial blood pressure remains unchanged while heart rate H is increased. The overall short term response depends on a combination of physiological reactions which may vary greatly between individuals. In some individuals, blood pressure is maintained by a large increase in contractility, small increase in H , and decrease in venous capacitance while in other individuals other combinations of change in H , contractility, and venous capacitance can occur. See Rowell (1993) for further discussion. Figure 2 illustrates the tilt table test which is used to study the effect of orthostatic stress. This effect takes on medical significance in those individuals who have inadequate transient response to orthostatic stress (such
Figure 2. Orthostatic stress diagram.
29
as the elderly) and in astronauts who exhibit orthostatic intolerance upon return to normal gravity. To see the impact of the gravitational force on blood pressure, note that transmural pressure increases in the feet from 98 mmHg in supine position to 198 mmHg in upright position. The difference of 100 mmHg equals a blood column of 130 cm (see the Textbook of Physiology (Patton et al., 1989) for further details). The result is an increase in venous blood volume in upright position of about 500 mL as compared to the supine position. Among other orthostatic changes which occur between the supine and standing positions, there is general agreement that: – – – – –
venous capacitance decreases in upright position; Q decreases in upright position; H is generally higher in upright position; central venous pressure is higher in supine position; stroke volume Vstr is higher in the supine position.
Control Loops Control loops are shown in the block diagram depicted as Fig. 1 for the respiratory and cardiovascular systems and represent the short term control response to perturbations in the steady state of the system. System control depends on sensory mechanisms which monitor critical state variables and respond to changes in these variables via negative feedback loops. The two main short term system loops represented in the diagram and which will be implemented in the model consist of the baroreflex loop which stabilizes blood pressure and the respiratory control loop which controls levels of CO2 (and indirectly pH) and O2 . The baroreflex depends on arterial blood pressure sensors in the carotid and aortic bodies. These sensors send signals to the central control in the brain which responds via the sympathetic and parasympathetic systems to vary H, contractility, systemic resistance Rs , and venous capacitance all of which influence arterial blood pressure Pas . There are also cardiopulmonary sensors (low pressure sensors) which respond to pressure changes in the right atrium and pulmonary arteries. Another important control loop consists of the local autoregulation of tissue vascular resistance which responds to concentrations of O2 and CO2 . This response reflects the changing needs of the tissues due to metabolic activity. The effect of O2 concentration on vasodilation in muscle tissue is important for overall systemic resistance. The respiratory peripheral chemosensors located in the carotid bodies respond to CO2 and O2 while the central
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chemosensors in the brain respond to CO2 . These sensors send information to the central control processor in the medulla which varies the ventilatory rate V˙ A . Implementation of the control processes which are represented in Fig. 1 will be discussed in the following section. ORTHOSTATIC STRESS MODEL In this section we present the model equations which will be used to study orthostatic stress. The equations are adapted from earlier work of Kappel and Peer (1993) and Timischl (1998). This was a nonpulsatile model simulating mean behaviour. To see whether orthostatic effects could be captured by this modeling approach, the structure of the model was minimally adapted to incorporate orthostatic mechanisms. The model is augmented by two lower systemic vascular compartments where gravitational effects are introduced. Using this division, it is possible to distinguish between the upper and lower body and describe the influence of orthostatic stress and gravitational pressure on the lower compartment. The model consists of a set of nonlinear ordinary differential equations, auxiliary equations, and an optimal control mechanism. Model equations refer to states represented in block diagram (Fig. 1). The Basic ODEs Descriptions of blood flow depend on mass balance equations which are set up for the inflow and outflow of blood from six vascular compartments. Cardiac output denoted by Q and intercompartmental blood flows denoted by F determine these mass balances. The lung and tissue compartments act as resistances to the flows F. State equations describe the compartmental volume changes according to the flows for the arterial systemic, venous systemic, and venous pulmonary circuits. The respiratory equations are developed in a similar way. Mass balance equations are set up for the inflow and outflow of CO2 and O2 from the lung and tissue compartments as transported by the pulmonary blood flow Fp and systemic tissue blood flow Fs . Ventilation V˙ A exchanges these gases in the lungs and metabolic rates MR create the opposite exchange in the tissue compartment. The basic model differential equations representing mass balance equations for blood flow are given in Eqs. (1)–(5) while the mass balance for blood gases are given by Eqs. (6)–(9). Equations (10) through (13) represent a model for contractility as influenced by H
Fink, Batzel, and Kappel
(Bowditch effect) while Eqs. (14) through (16) represent the control. casUp P˙ asUp = Q l − Fa − FsUp
(1)
casLo P˙ asLo = Fa − FsLo
(2)
cvsLo P˙ vsLo = FsLo − Fv − c˙ vsLo PvsLo
(3)
cvsUp P˙ vsUp = Fv − Q r + FsUp
(4)
(5) cvp P˙ vp = Fp − Q l ¡ ¢ VACO2 P˙ aCO2 = 863Fp CvCO2 − CaCO2 ¡ ¢ + V˙ A PICO2 − PaCO2 (6) ¡ ¢ ¡ ¢ VAO2 P˙ aO2 = 863Fp CvO2 − CaO2 + V˙ A PIO2 − PaO2 VTCO2 C˙ vCO2 VTO2 C˙ vO2
¡ ¢ = MRCO2 + Fs CaCO2 − CvCO2 ¡ ¢ = −MRO2 + Fs CaO2 − CvO2
(7) (8) (9)
S˙ l = σl
(10)
S˙ r = σr
(11)
σ˙ l = −γl σl − αl Sl + βl H
(12)
σ˙ r = −γr σr − αr Sr + βr H
(13)
˙ = u1 H
(14)
V¨ A = u 2
(15)
c˙ vsLo = u 3 .
(16)
Tables 1 and 2 contain all symbol definitions. Control issues will be discussed further below in Section (The Optimal Control). “Lo” refers to lower compartment and “Up” refers to upper compartment. Cardiovascular Auxiliary Equations The following cardiovascular auxiliary equations are incorporated in the model: Pap (t) =
1 ¡ V0 − casUp PasUp (t) − casLo PasLo (t) cap − cvsUp PvsUp (t) − cvsLo (t)PvsLo (t) ¢ − cvp Pvp (t) ,
(17)
Q l (t) = H
cl Pvp (t) f (Sl (t), PasUp (t))(1 − kl ) , (18) PasUp (t)(1 − kl ) + f (Sl (t), PasUp (t))kl
Q r (t) = H
cr PvsU p (t) f (Sr (t), Pap (t))(1 − kr ) , Pap (t)(1 − kr ) + f (Sr (t), Pap (t))kr
(19)
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Table 1. Respiratory Symbols Symbol
Meaning
Unit
MRCO2 MRO2 CaCO2 CaO2 PaCO2 PaO2 PvCO2 PvCO2 V˙ A VACO2 VAO2 VTCO2 VTO2 K, k
metabolic CO2 production rate metabolic O2 consumption rate concentration of CO2 in arterial blood concentration of O2 in arterial blood partial pressure of CO2 in arterial blood partial pressure of O2 in arterial blood partial pressure of CO2 in mixed venous blood partial pressure of O2 in mixed venous blood alveolar ventilation effective CO2 storage volume of the lung compartment effective O2 storage volume of the lung compartment effective tissue storage volume for CO2 effective tissue storage volume for O2 dissociation constants relating concen to partial pressure
lSTPD · min−1 lSTPD · min−1 ml/ml ml/ml mmHg mmHg mmHg mmHg lBTPS · min−1 lBTPS lBTPS l l
where kl , kr , f , and td are defined as kl = exp(−td /(Rl cl )) and kr = exp(−td /(Rr cr )),
(20)
f (s, p) = 0.5(s + p) − 0.5(( p − s)2 + 0.01)1/2 , µ ¶1/2 60 60 . −κ td = H H
(21) (22)
Table 2. Cardiovascular Symbols Symbol α Apesk β frac cas cap cvs cvp Fp Fs H γ Pgrav Pas Pap Pvs Pvp Q Rp Rs S cl,r Rl,r σ u Vstr V0 l, r
Meaning
Unit
coefficient of S in the differential equation for σ Rs = Apesk CvO2 coefficient of H in the differential equation for σ upper compartment fraction of basic total prone systemic volume capacitance of the arterial part of the systemic circuit capacitance of the arterial part of the pulmonary circuit capacitance of the venous part of the systemic circuit capacitance of the venous part of the pulmonary circuit blood flow perfusing the lung compartment blood flow perfusing the tissue compartment heart rate coefficient of σ in the differential equation for σ hydrostatic pressure induced by gravity mean blood pressure in arterial region: systemic circuit mean blood pressure in arterial region: pulmonary circuit mean blood pressure in venous region: systemic circuit mean blood pressure in venous region: pulmonary circuit cardiac output resistance in the peripheral region of the pulmonary circuit peripheral resistance in the systemic circuit contractility of the ventricle compliance of the respective relaxed ventricle total viscous resistance of the respective ventricle derivative of S control function stroke volume of the ventricle total blood volume left, right heart circuit respectively
min−2 mmHg · min · l−1 mmHg · min−1 l l · mmHg−1 l · mmHg−1 l · mmHg−1 l · mmHg−1 l · min−1 l · min−1 min−1 min−1 mmHg mmHg mmHg mmHg mmHg l · min−1 mmHg · min · 1−1 mmHg · min · 1−1 mmHg l · mmHg−1 mmHg · min · 1−1 mmHg · min−1 l l
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The auxiliary cardiovascular equation, Eq. (17) describes pulmonary arterial pressure in terms of the other cardiovascular pressures. Equations (18) and (19) describe relations for Q in terms of stroke volume, preload, and afterload (see Kappel and Peer (1993) for details). Equation (21) implements a maximum condition so that stroke volume does not exceed the filling volume. Equation (22) describes the time of diastole td in terms of H . Respiratory Auxiliary Equations Auxiliary equations for the respiratory system consist of dissociation equations (Eqs. (23)–(26)) relating partial pressures and concentrations of blood gases. We will model alveolar ventilation V˙ A (related to minute ventilation and dead space ventilation by Eq. (27)). using an optimal control approach (See Section (The Optimal Control)). ¡ −K P (t) ¢2 , CaO2 (t) = K 1 1 − e 2 aO2 ¡ ¢ −K P (t) 2 , CvO2 (t) = K 1 1 − e 2 vO2
(23) (24)
CaCO2 (t) = K CO2 PaCO2 (t) + kCO2 ,
(25)
CvCO2 (t) = K CO2 PvCO2 (t) + kCO2 ,
(26)
V˙ A = V˙ E − V˙ D .
(27)
The Flows The intercompartment blood flows are defined by Fv = max(0; PvsLo − PvsUp − Pgrav )/Rv ,
(28)
Fa = (PasUp − PasLo + Pgrav )/Ra ,
(29)
FsUp = (PasUp − PvsUp )/Rs ,
(30)
FsLo = (PasLo − PvsLo )/Rs ,
(31)
Fp = (Pap − Pvp )/Rp .
(32)
where the max-function implements the action of the venous valves to avoid backflow. Pgrav represents the hydrostatic pressure induced on the lower compartments (further information on this implementation is given in Section (Derivation of Orthastatic Effects)). These auxiliary equations relating pressure, flow, and resistance are essentially the same as Ohm’s law. Also Rs = Apesk CvO2 ,
(33)
Fs = FsUp + FsLo , Pgrav = cρgh sin(Tilting Angle).
(34) (35)
where ρ = 1.05 mg/mm3 = 1050 kg/m3 , c = 1/ 133.322 mmHg/Pa, and h = 0.6 m are the physical parameters for hydrostatic pressure. Equation (33) describes the effect of oxygen partial pressure on systemic resistance. Mechanisms which act to match ventilation and cardiac output as well as synchronize respiratory and heart rate frequencies are ignored. Equation (34) describes the flow through the tissue compartment. Equation (35) describes the dependency of the gravitation effect on the degree of tilt. The Optimal Control The control functions u 1 , u 2 , and u 3 represent the variations in heart rate H , ventilation rate V˙ A , and lower venous capacitance cvsLo . The control of lower venous capacitance includes the baroreflex influence and may also reflect the nonlinearity in the pressure-volume relation. There is disagreement over the degree to which the change in capacitance is active or passive. Baroflex control changes of systemic resistance and contractility are modeled via parameter changes in these simulations. We model the control action as an optimal control via a cost functional. The transition from an initial (steady state) disturbance to the final steady state is optimal in the sense that controlled values Pas , Pvs , PaCO2 , and PaO2 are stabilized such that deviations from their final steady state values are as small as possible. Furthermore, H, V˙ A , and cvsLo are prevented from changing too fast or too extensively. The stationary equations for the system (Eqs. (1)– (16)) determine a three degrees of freedom set of steady states. Therefore we need to choose the steady state values of three state variables as parameters. We choose values for steady states PaCO2 , H , and cvsLo . The state PaCO2 is chosen because its value is rather standard at 40 mmHg. State H is chosen because it is easily measured. State cvsLo is chosen by doing a parameter estimation from the data. We could also have chosen PasUp as the third parameter calculated from systolic and diastolic pressure. Mathematically, this can be formulated in the following way. To transfer the system to the final or target steady state, we determine control functions such that the cost functional Z ∞ ¡ ¢2 ¡ ¢2 f f qa PasUp − PasUp + qv PvsUp − PvsUp 0
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¡ ¢2 ¡ ¢2 + qCO2 PaCO2 − PafCO + qO2 PaO2 − PafO 2
still stabilizes the system and is useful for dynamic studies.
2
+ q1 (u 1 )2 + q2 (u 2 )2 + q3 (u 3 )2 dt
33
(36) MODELING HISTORY
is minimized under the restriction of the model equations x˙ (t) = g(x(t)) + B(x(t))u(t), x(0) = x . i
(37)
The values qi are weights defining the degree of influence of quantities on the cost. The superscript “f” refers to the target or final steady state while superscript “i” refers to the initial condition or state. In modeling the transition from the (initial) “resting supine” steady state x s to the (final) “head up tilt” state x h the following steps are carried out: – Compute the steady states “resting supine,” x s , and “head up tilt,” x h . Here the steady state “head up tilt” is defined by parameter changes: • Pgrav increased based on the degree of tilt and Eq. (35), • higher heart rate H , • lowered capacitance, higher resistance, and increased contractility. – The control functions u 1 , u 2 , and u 3 which transfer the system (Eqs. (1)–(16)) from the initial steady state “supine,” x s , to the final steady state “head up tilt,” x h , are found as follows. We consider the linearized system around state x h with initial condition x(0) = x s , and the cost functional as given in Eq. (36). The states x h and x s are defined by parameter choices defining these states. We then compute the control functions u 1 , u 2 , and u 3 such that the cost functional is minimized subject to the linearized system. This is accomplished by solving an associated algebraic matrixRiccati equation, which is used to define the feedback gain matrix. – This control is used to stabilize the nonliner system Eqs. (1)–(16). This control will be suboptimal in the sense of Russell (1979) and still stabilizing. We may implement step changes in Pgrav , systemic resistance, and contractility, or introduce smooth changes over a short tilt time. In the later case, though the system is now nonautonomous, we still implement the control functions u 1 , u 2 , and u 3 calculated for a time-independent linear system around the final steady state “head up tilt.” This further reduces the optimality but the thereby obtained (suboptimal) control
Several articles have been published, which describe models including orthostatic stress—either head up tilt (HUT) or lower body negative pressure (LBNP). See for instance Boyers et al. (1972), Croston and Fitzjerrell (1974), Leonard et al. (1979), White et al. (1983), Sud et al. (1993), Karam et al. (1994), Melchior et al. (1994), Burton (2000), Peterson et al. (2002), Walsh et al. (2001), Heldt et al. (2002), and Hao et al. (2003). LBNP models modify the impact internal blood pressure has on the volume V of a compartment by varying the transmural pressure Ptrans and in some cases changing the capacitance c as well: V = f (c, Ptrans ).
(38)
HUT models most often change the blood flow q into, out of, or through the compartment according to the tilting angle 2: q = q 0 + G sin 2.
(39)
DERIVATION OF ORTHOSTATIC EFFECTS There are different ways to view the action of the gravitational force on the system of blood vessels, which are discussed in this section. In supine position the gravitational force is perpendicular to the body and thus doesn’t have significant influence on the cardiovascular system and in particular the flows. When standing the direction of gravitation is parallel to the body and thus it acts on the blood cells in the direction of the lower arteries and veins, thus the weight of the upper blood cells presses on the lower cells. Hydrostatic theory shows, that the pressure occurring at a certain level is independent of the form of the vessels above, it only depends on the height of the column above, because all other forces are compensated by the vessel walls. In our model we split the arterial and venous systemic compartments into an upper and a lower part, to distinguish vessels and tissue at about heart level and at a much lower level. The pressures at heart level are given by PasUp and PvsUp where PasUp is the pressure we measure when visiting the doctor. These pressures act on the lower compartments and influence the flow into or out of them. As mentioned before, in tilt there is an additional
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pressure (here called Pgrav ), depending on the tilting angle, which comes from the weight of blood in the upper compartments acting on the lower ones. In general the flow between two compartments is defined by Ohm’s law F=
Fv =
There are three different modeling approaches to discuss:
1 max(0; (PvsLo − Pgrav ) − PvsUp ), Rv P˜ asLo − P˜ vsLo = Rs =
FsLo
Pin − Pout . R
1 max(0; P˜ vsLo − P˜ vsUp ) Rv
=
(PasLo − Pgrav ) − (PvsLo − Pgrav ) , Rs
Pressure-Flow Considerations
which are equivalent to the flows given above.
Focusing for a moment only on the upper and lower compartments on the arterial side of the systemic circuit, the pressure-flow relationship would be given by
Volume Considerations
Fa =
(PasUp + Pgrav ) − PasLo . Ra
Fv =
1 max(0; PvsLo − (PvsUp + Pgrav )). Rv
FsLo
The volume of a compartment is related to the overall transmural pressure of the compartment (including hydrostatic pressure). Simplified, this function can be assumed to be linear, VasLo = casLo ( P˜ asLo + Pgrav ).
(47)
Differentiating Eq. (47) and combining it with Eq. (46) we get d (casLo ( P˜ asLo + Pgrav )) = Fa − FsLo dt PasUp − P˜ asLo P˜ asLo − P˜ vsLo = − Ra Rs
(48)
and similarly
Starting from the view that the kinetic energy and thus the flow is independent of hydrostatic effects (see Berne and Levy (1997) and Rowell (1993)) we develop the following relations. Since the flow is given by just the dynamical pressures P˜ and noting that PasLo = P˜ asLo + Pgrav and PasUp = P˜ asUp , we derive that P˜ asUp − P˜ asLo PasUp − (PasLo − Pgrav ) = Ra Ra
(46)
(42)
Energy Considerations
Fa =
d VasLo = Fa − FsLo . dt
(41)
Between the lower compartments we have the “normal” flow PasLo − PvsLo = . Rs
(45)
The change of a compartmental volume equals the difference between inflow and outflow, i.e.,
(40)
The pressures in this approach reflect the actual pressures inside the compartments, thus PasLo is a combination of the dynamic pressure ( P˜ asUp ) and the hydrostatic or tilting pressure (Pgrav ). That is PasLo = P˜ asLo + Pgrav , but in the upper compartment PasUp = P˜ asUp . Note that substituting this expression for PasLo into Eq. (40) the hydrostatic and gravitational effects cancel. Similarly on the venous side (including the maximum function representing the venous valves) we have
(44)
(43)
d (cvsLo ( P˜ vsLo + Pgrav )) dt P˜ vsLo − PvsUp P˜ asLo − P˜ vsLo − . = Rs Rv
(49)
The hydrostatic pressure increases the volume of the lower compartments, which indirectly influences the flows and the dynamical pressures of the system, which all happens simultaneously. To get the actual pressures (including the hydrostatic pressure) inside the compartments we use PasLo = P˜ asLo + Pgrav , and plug in P˜ asLo = PasLo − Pgrav . We proceed
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PasLo − PvsLo d (cvsLo PvsLo ) = dt Rs
similarly for P˜ vsLo = PvsLo − Pgrav . We find that PasUp − (PasLo − Pgrav ) d (casLo PasLo ) = dt Ra −
−
(PasLo − Pgrav ) − (PvsLo − Pgrave ) , Rs
(PasLo − Pgrav ) − (PvsLo − Pgrav ) d (cvsLo PvsLo ) = dt Rs −
(PvsLo − Pgrav ) − PvsUp , Rv
which is again equivalent to the two approaches from before and our model equations:
(PvsLo − Pgrav ) − PvsUp , (51) Rv
Note: Equations (48) and (49) can also be derived from the viewpoint of lower body negative pressure with Pgrav = −Pbias (see Heldt et al. (2002)). A closer comparison between this LBNP implementation and the HUT implementation discussed above shows that the flows are the same, but the internal compartment pressures will differ in the lower compartments.
RESULTS AND ONGOING WORK
PasUp − (PasLo − Pgrav ) d (casLo PasLo ) = dt Ra PasLo − PvsLo − , Rs
35
(50)
Figure 3 shows a set of data measured at a HUT from 0◦ to 70◦ in 10 sec with FinometerTM from a 25-year-old
Figure 3. Measurement of 70◦ tilt in 10 sec. The upper panel depicts systolic, mean and diastolic arterial pressure. The lower panel shows heart rate and tilt angle.
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Figure 4. Model fit of the experimental data.
male human. This data is used for comparison with simulations of the full model proposed in this paper. In Fig. 4 we show the results of a parameter fit, which was done by first adjusting some model parameters to fit the steady state and afterwards adjusting the weights in the cost functional to fit the dynamics (parameters and steady state values in the Tables 3 and 4). Fitting Steady States and Dynamics In the experimental data used for this paper the variable Tilting Angle changed from 0 to 70 degrees in 10 sec, heart rate went up from 59 to 90.6, and upper arterial systemic pressure PasUp decreased by 5% (see Fig. 4). At steady state supine, the volumes of the upper and lower systemic compartments were fixed by estimating the proportions for upper and lower arterial and venous capacitances. The quantities cvsLo , Apesk , and βl were estimated from steady state data for supine and tilted positions using
Table 3. Parameter Values Parameter
Value
Parameter
Value
V frac∗ casUp casLo ∗ cvsUp
5.0 0.3 0.01002∗ frac 0.01002∗ (1-frac) 0.70∗ frac
cl cr K1 K2 kCO2
0.01289 0.06077 0.2 0.05 0.244
∗ cvsLo cap cvp
See Table 4 0.03557 0.1394
0.0065 0.0 150.0
A∗pesk MRO2 MRCO2 αl αr
K CO2 PICO2 PIO2
See Table 4 0.290 0.244 89.47 28.46
Rl Rr Ra Rv VCO2
11.35 4.158 0.1 0.1 3.200
βl∗ βr γl γr
See Table 4 2.08 37.33 11.88
VO2 VTCO2 VTO2
2.500 15.000 6.000
Note. Parameters were either taken from Timischl (1998) or fitted∗ . Ra and Rv were chosen as small as reasonable.
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Table 4. Steady States State variable
Supine
Tilting angle
0
Supine
Tilt
Sl
84.96
87.02
PasUp PasLo PvsLo
82.35 78.53 Sr 82.12 121.88 σl 4.89 45.45 σr
4.32 0.00 0.00
6.63 0.00 0.00
PvsUp Pvp PaCO2
4.66 6.16 40.00
59.00 5.26 0.368
90.65 5.26 0.056
PaO2 PvCO2 PvO2
Tilt 70
State variable
1.69 H 4.00 V˙ A 40.00 cvsLo
102.46 102.46 Rs 48.14 48.08 Apesk 34.39 32.39 βl Q
16.79 124.60 128.83 4.61
18.22 141.65 85.89 4.21
a minimization scheme. The induced volume shift of blood predicted by the model into the venous leg compartment was 800 mL. To fit the dynamics we adjust the weights in the cost function for the optimal control. There are two different
Table 5. Nonzero Weights for Control State variable
Weights X
Control variable
Weights U
0.015 6.49 1.07 0.006
u1 u2 u3
0.0002 0.0105 8.3500
PasUp PvsUp PaCO2 PaO2
weight-vectors: Weights U and Weights X . Weights U puts weights on changes of the control, Weights X puts weights on deviations from the final steady state. The fit (Fig. 4) was done by estimating the weights for the control. We used all three control weights and weights for PasUp and H , PaCO2 and PaO2 . For this data set, the weights necessary to fit the data indicate a strong influence of the cardio-pulmonary control loop. For the respiratory
Figure 5. Complete model output for the fitted data.
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subsystem, due to the minimal dynamics, the weights qualitatively reflect the physiology, while the actual values are more determined by the numerical scheme. The next steps which are anticipated in terms of model development are as follows. – Analytical studies; 1. The analytical results on steady state analysis will be further developed; 2. Analytical studies on the comparison between LBNP and HUT modeling will be done to clarify their relationship. – Physiological modeling: 1. Further experimental data will be collected and analyzed as well as parameter identification performed with test subjects; 2. There are physiological differences between fast and slow tilt, as well as between small and large tilt. In small tilt low pressure sensors play a more prominent role and implementation of this control loop effect needs to be studied. Fast tilt involves buffering by the lung blood volume and this needs to be studied as well; 3. Implementation of sympathetic effect on contractility and systemic resistance should be explored as a control, currently implemented as parameter changes. 4. The baroreceptors in the carotid artery are situated above the heart and sense a slightly different pressure from the aortic sensors (which measure PasUp ). The effects of this variation in pressure are not modeled. ACKNOWLEDGMENTS Supported by FWF (Austria) under grant F310 as a subproject of the Special Research Center F003 “Optimization and Control.” Finometer purchased with Austrian BM:BWK infrastructure grant UGP4 to the Department of Physiology, IBMS, University of Graz. REFERENCES Berne R, and Levy M. Cardiovascular Physiology, 7th ed., St. Louis, MO: Mosby, 1997.
Fink, Batzel, and Kappel
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