Vol. 281, Issue 6, H2661-H2679, December 2001
A human cardiopulmonary system model applied to the analysis
of the Valsalva maneuver
K.
Lu1,
J. W.
Clark Jr.1,
F. H.
Ghorbel1,
D. L.
Ware2, and
A.
Bidani2
1 Dynamical Systems Group, Rice University, Houston 77005;
and 2 Department of Internal Medicine, University of Texas
Medical Branch, Galveston, Texas 77555
 |
ABSTRACT |
Previous models combining the
human cardiovascular and pulmonary systems have not addressed
their strong dynamic interaction. They are primarily cardiovascular or
pulmonary in their orientation and do not permit a full exploration of
how the combined cardiopulmonary system responds to large amplitude
forcing (e.g., by the Valsalva maneuver). To address this issue, we
developed a new model that represents the important components of the
cardiopulmonary system and their coupled interaction. Included in the
model are descriptions of atrial and ventricular mechanics,
hemodynamics of the systemic and pulmonic circulations, baroreflex
control of arterial pressure, airway and lung mechanics, and gas
transport at the alveolar-capillary membrane. Parameters of this
combined model were adjusted to fit nominal data, yielding accurate and
realistic pressure, volume, and flow waveforms. With the same set of
parameters, the nominal model predicted the hemodynamic responses to
the markedly increased intrathoracic (pleural) pressures during the
Valsalva maneuver. In summary, this model accurately represents the
cardiopulmonary system and can explain how the heart, lung, and
autonomic tone interact during the Valsalva maneuver. It is likely that
with further refinement it could describe various physiological states and help investigators to better understand the biophysics of cardiopulmonary disease.
cardiopulmonary modeling; ventricular interaction; closed-loop
hemodynamics; baroreflex control; airway mechanics; gas exchange
 |
INTRODUCTION |
THE DIAGNOSIS AND
TREATMENT of cardiopulmonary disease may be improved by using
mathematical models of the cardiovascular and pulmonary systems. With
this in mind, we developed a model of the cardiopulmonary system of the
normal human subject that not only represents the system accurately but
also predicts its response to a variety of commonly used diagnostic
procedures. To our knowledge, this is the first example of a truly
integrative model of the cardiopulmonary system.
Recently, our group (5, 25) developed a multicompartment
model of the canine circulation. We have now modified and extended this
cardiovascular model to encompass human heart mechanics, a circulatory
loop, baroreflex control of arterial pressure, airway mechanics, and
gas transport at the alveolar-capillary membrane.
Distributed circulatory models of the systemic and pulmonic
circulations have been developed (1, 3, 12, 37). However, the mechanics of the lung and airways were not detailed in any of
these, and the heart was modeled rather simply. The gas exchange at the
alveolar-capillary membrane (an obvious link between cardiovascular and
pulmonary system) was considered only in the model of Hardy et al.
(12). Of these models, baroreflex control of arterial pressure was included only in the work of Ursino et al.
(37).
Distributed airway mechanics models [e.g., Elad et al.
(7) and Lambert et al. (16)] can be too
complex for a combined cardiopulmonary model, making lumped lower-order
compartment models [such as that of Lutchen et al. (19)]
preferred. The lumped compartment model we (18) developed
describes ventilation, perfusion, mechanics, and gas transport over the
full range of normal lung volumes. A modified version of this model was
used in the current study.
Our heart model was based on our previous work in dogs (5,
25). The parameters of that model were adjusted to better fit
the flow, volume, and temporal relationships of the human cardiac
cycle. Similar adjustments were made in the systemic and pulmonic
component models of the canine circulatory loop (25). The
resulting model is of intermediate complexity and simulates pressure,
volume, and flow distribution of the human subject in the supine position.
To better simulate the cardiovascular response to perturbation, we
added nonlinear descriptions of the venous system and a description of
how the baroreflexes influence heart rate, myocardial contractility,
and vasomotor tone. We based our baroreceptor control model on the work
of Spickler et al. (35) and Wesseling et al. (38) and included descriptions of both parasympathetic
(vagal) and sympathetic pathways.
Our new lung model combines models previously developed by our group,
namely, an airway mechanics model [from Athanasiades et al.
(2)] and a gas exchange model [modified from Liu et al.
(18)]. It characterizes the nonlinear resistive-compliant properties of the airways and the nonlinear pressure-volume
characteristics of the lung. A distributed pulmonary circulatory model
containing 35 contiguous capillary segments characterizes gas exchange
at the alveolar-capillary membrane and yields good fits to expired O2 and CO2 data measured at the mouth.
This integrated cardiopulmonary model describes heart-lung interactions
and the timing of baroreflex changes in heart rate, myocardial
contractility, and vasomotor tone. Its parameters fit available
cardiovascular and pulmonary data obtained during tidal breathing and
can predict the responses to large-scale perturbations in pleural
pressure, such as those occurring in the forced vital capacity and
Valsalva maneuvers.
Glossary
Activation functions
| e(t) |
Time-varying activation function
|
| ea(t) |
Activation function of the atrium
|
| ev(t) |
Activation function of the ventricle
|
Airflows
CA |
Airflow from collapsible airways to alveolar region
|
DC |
Airflow from upper supported airway to collapsible airway
|
ED |
Airflow from environment to upper supported airway
|
Blood flows
Ao |
Aortic flow
|
PA |
Pulmonary arterial flow
|
Compliances
| CAo,P |
Aortic root compliance
|
| CAo,D |
Distal aortic compliance
|
| CPA |
Pulmonary artery compliance
|
| CPA,D |
Distal pulmonary artery compliance
|
| CPC |
Pulmonary capillary compliance
|
| CPV |
Pulmonary venous compliance
|
| CSA,D |
Distal systemic artery compliance
|
| CSC |
Systemic capillary compliance
|
Constants and scaling parameters
| a |
Time constant
|
| amin |
Dimensionless constant
|
| ax |
Normalized frequency offset
|
| Ai |
Parameter of activation function of the heart
|
| bmin |
Dimensionless constant
|
| bx |
Dimensionless constant
|
| Bi |
Parameter of activation function of the heart
|
| Ci |
Parameter of activation function of the heart
|
| CLT |
Lung tissue elastic constant
|
| D0 |
Volume parameter
|
| D1 |
Stressed pressure offset
|
| D2 |
Unstressed pressure offset
|
| h1 |
Constant
|
| h2 |
Constant
|
| h3 |
Constant
|
| h4 |
Constant
|
| h5 |
Constant
|
| h6 |
Constant
|
| K |
Gain
|
| K1 |
Stressed scaling pressure
|
| K2 |
Unstressed scaling pressure
|
| Ka |
Scaling parameter
|
| Kb |
Scaling parameter
|
| Kc |
Scaling parameter
|
| Kp1 |
Constant scaling parameter
|
| Kp2 |
Constant scaling parameter
|
| Kr |
Resistance scaling factor
|
| KR |
Resistance scaling factor
|
| Kv |
Scaling factor for pressure
|
|
Diastolic elastance coefficient
|
p |
Passive exponential constant
|
x |
Time constant
|
Gas diffusion and flux
C |
Gas species i blood concentration in the jth
capillary
|
| DLi |
Diffusion capacity for the ith gas species
|
| DL,CO2 |
Lung diffusion capacity of CO2
|
| DL,N2 |
Lung diffusion capacity of N2
|
| DL,O2 |
Lung diffusion capacity of O2
|
tot |
Total gas flux rate
|
Inertances
| LAo,D |
Distal aortic inertance
|
| LAo,P |
Aortic root inertance
|
| LPA |
Pulmonary arterial inertance
|
Neural control
| Fcon |
Normalized sympathetic efferent discharge frequency controlling
contractility
|
| FHr,S |
Normalized sympathetic controlling HR frequency
|
| FHr,V |
Normalized vagal controlling HR frequency
|
| Fsymp |
Sympathetic discharge frequency
|
| Fvagus |
Vagal discharge frequency
|
| Fvaso |
Normalized sympathetic efferent discharge frequency controlling
vasomotor tone
|
| Fx |
Discharge frequency
|
| x |
Generic output index representing heart rate, contractility, or
vasomotor tone
|
| N1 |
Baroreceptor firing frequency
|
| N2 |
Derivative of baroreceptor firing frequency
|
| Ncon |
Sympathetic discharge at central nervous system controlling
contractility
|
| NHr,S |
Sympathetic discharge at central nervous system controlling heart rate
|
| NHr,V |
Vagal discharge at central nervous system controlling heart rate
|
| N(s) |
Laplace transform of N(t)
|
| N(t) |
Baroreceptor discharge frequency
|
| Nvaso |
Sympathetic discharge at central nervous system controlling vasomotor
tone
|
| Nvaso(s) |
Laplace transform of Nvaso
|
| Nx,0 |
Base frequency
|
| Nx(t) |
Discharge frequency of neural pathways of the central nervous system
|
Physiology
| AoD |
Distal aorta
|
| AoP |
Proximal aorta
|
| BR |
Baroreceptor element
|
| CNS |
Central nervous system
|
| LA |
Left atrium
|
| LV |
Left ventricle
|
| LVF |
Left ventricular free wall
|
| PA |
Pulmonary arterioles
|
| PAD |
Distal pulmonary arterioles
|
| PAP |
Proximal pulmonary arterioles
|
| PC |
Pulmonary capillaries
|
| PCD |
Pericardium
|
| PV |
Pulmonary veins
|
| RA |
Right atrium
|
| RV |
Right ventricle
|
| RVF |
Right ventricular free wall
|
| SAD |
Distal systemic arterioles
|
| SAP |
Proximal systemic arterioles
|
| SC |
Systemic capillaries
|
| SPT |
Septum
|
| SV |
Systemic veins
|
| VC |
Vena cava
|
Pressures
| P0 |
Diastolic pressure magnitude
|
| Patm |
Atmospheric pressure
|
| Patm,i |
Partial pressure of gas species i in the atmosphere
|
| PAi |
Partial pressure of gas species i in the small airway
|
| PA |
Alveolar pressure
|
| PA,CO2 |
Alveolar CO2 partial pressure
|
| PA,O2 |
Alveolar O2 partial pressure
|
| PAo |
Aortic arch pressure
|
| Pb,CO2 |
CO2 partial pressure in the blood
|
| Pb,O2 |
O2 partial pressure in the blood
|
| PCi |
Partial pressure of gas species i in the middle airway
|
| PC |
Pressure in the lumen of the midairway segment
|
| PC,CO2 |
CO2 partial pressure in the collapsible airway
|
| PC,O2 |
O2 partial pressure in the collapsible airway
|
| PCW |
Recoil pressure of the chest wall
|
| PCO2 |
Partial pressure of CO2
|
| PDi |
Partial pressure of gas species i in the upper airway
|
| PD |
Pressure in the lung dead space
|
| PD,CO2 |
CO2 partial pressure in the lung dead space
|
| PD,O2 |
O2 partial pressure in the lung dead space
|
| PEL |
Lung elastic recoil pressure
|
| PES(V) |
End-systolic pressure
|
P |
Partial pressure of gas species i in the jth
capillary
|
| PLA |
Left atrial pressure
|
| PLV |
Left ventricular pressure
|
| Pmus |
Pressure of the respiratory muscles
|
| PO2 |
Partial pressure of O2
|
| PPL |
Pleural pressure
|
P |
Systemic arterial pressure in the active state
|
P |
Systemic arterial pressure in the passive state
|
| PSV |
Transmural pressure of systemic veins
|
| PTM |
Transmural pressure of collapsible midairway
|
| PVC |
Transmural pressure of the vena cava
|
Resistances
| R0 |
Offset parameter
|
| RAo,P |
Aortic root flow resistance
|
| RAo,D |
Distal aortic flow resistance
|
| RC |
Resistance of collapsible midairway
|
| RCOR |
Coronary flow resistance
|
| RCRB |
Cerebral flow resistance
|
| RLA |
Left atrial flow resistance
|
| RLT |
Lung tissue resistive constant
|
| RM |
Mitral valve flow resistance
|
| RPA |
Pulmonary arteriolar flow resistance
|
| RPA,D |
Distal pulmonary arterial flow resistance
|
| RPA,P |
Proximal pulmonary arterial flow resistance
|
| RPC |
Resistance of pulmonary capillaries
|
| RPC,0 |
Magnitude of pulmonary capillary resistance
|
| RPS |
Pulmonary shunt flow resistance
|
| RPV |
Pulmonary venous flow resistance
|
| RRA |
Right atrial flow resistance
|
| RS |
Small airways resistance
|
| RSA |
Resistance of systemic arteries
|
| RSA,D |
Systemic arteriolar flow resistance
|
| RSC |
Systemic capillary flow resistance
|
| RSV |
Systemic venous flow resistance
|
| RTAo |
Viscoelastic resistance of proximal aorta wall
|
| RTAo,D |
Viscoelastic resistance of distal aorta wall
|
| RTA |
Tricuspid valve flow resistance
|
| RTPA |
Pulmonary artery wall viscoelastic resistance
|
| Ruaw |
Upper supported airway resistance
|
| RVC |
Resistance of the vena cava
|
Variables and measurements
| EES |
End-systolic elastance
|
| EDPVR |
End-diastolic pressure-volume relationship
|
| ESPVR |
End-systolic pressure-volume relationship
|
| FRC |
Functional residual capacity
|
| FVC |
Forced vital capacity
|
| i |
Gas species (O2, CO2, or N2)
|
| j |
Number of a specific capillary in a series
|
| Nseg |
Number of capillary segments
|
| P-V |
Pressure-volume (relationship)
|
| s |
Laplace variable
|
| STPD |
Standard temperature, pressure, dry weight
|
| t |
Time
|
| TLC |
Total lung capacity
|
v |
Blood flow velocity in the jth capillary
|
| z |
Length coordinate of the pulmonary capillary
|
Volumes
| V0 |
Unstressed volume
|
| VA |
Alveolar volume
|
| VA,max |
Maximal alveolar volume
|
| VC |
Collapsible airway volume
|
| VCW |
Chest wall volume
|
| VD |
Systolic volume offset
|
| VED |
End-diastolic volume
|
| VES |
End-systolic volume
|
V |
Blood volume contained in the jth capillary
|
| VL |
Lung volume
|
| VLV |
Left ventricular volume
|
| Vmax |
Maximal volume
|
| Vmin |
Minimum volume
|
| VPC |
Blood volume of pulmonary capillaries
|
| VPC,max |
Maximal blood volume of pulmonary capillaries
|
| VSA |
Blood volume of systemic arteries
|
| VSA,0 |
Minimal volume of systemic arteries
|
| VSA,max |
Maximal lumen volume of systemic arteries
|
| VSV |
Luminal volume of systemic veins
|
| VVC |
Luminal volume of the vena cava
|
| VVE |
Viscoelastic volume
|
 |
MODEL DEVELOPMENT |
Ventricular Model
Our ventricular model is based on the work of Chung et al.
(5), wherein each ventricular compartment is characterized
by a time-varying elastance function (Tables
1-3).
The elastance function is developed by three curves, as
established in Ref. 5, namely, the end-systolic
P-V relationship (ESPVR), the end-diastolic P-V relationship (EDPVR),
and a time-varying activation function [e(t)].
The activation function e(t) consists of a
series of Gaussian curves and serves to produce a smooth
transition between the EDPVR and the ESPVR. A detailed description of
the ventricular model can be found in Ref. 5.
Circulatory Model
The general framework of our human circulatory loop model (Fig.
1 and Table
4) is similar to that of Olansen et
al. (25) with certain extensions and modifications. We
included 1) nonlinear P-V relationships to describe the
peripheral venous system, 2) a nonlinear collapsible
description of the P-V relationship for the vena cava, and
3) separate descriptions of baroreceptor-mediated control of
heart rate, myocardial contractility, and vasomotor tone.

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Fig. 1.
A
hydraulic equivalent representation of the closed-loop circulatory
model. For abbreviations, see Glossary.
|
|
Nonlinear P-V Characteristics of Systemic Veins
and the Vena Cava
Systemic veins.
The nonlinear P-V relationship of veins has been modeled previously by
Kresch (15) and by Snyder and Rideout (34).
As volume increases, the vessels stiffen. The resulting P-V curve can
be represented as follows
|
(1)
|
where PSV and VSV are the transmural
pressure and luminal volume of systemic veins,
Kv is a scaling factor (in mmHg), and Vmax is the maximal volume (in ml) of the lumped systemic
veins (Table 5).
Vena cava.
Under some conditions, the vena cava may collapse. For example, when
pleural pressure is greater than the luminal pressure of the vena cava,
total caval volume decreases substantially, and the resistance to flow
is increased. To account for this, we described the P-V relationship as
follows
|
(2)
|
|
(3)
|
where PVC and VVC denote the transmural
pressure and luminal volume of the vena cava, respectively,
V0 is the unstressed volume, and Vmin is the
minimum volume. We adjusted the parameters K1,
K2, D1, and
D2 to produce P-V curves similar to those used in the human venous model of Snyder and Rideout (34).
The resistance of the vena cava (RVC) is
a nonlinear function of its luminal blood volume (VVC)
according to the following equation
|
(4)
|
where KR is a scaling factor (in
mmHg · s · ml
1), Vmax denotes
the maximum volume, and R0 is an offset
parameter (in mmHg · s · ml
1) (Table 5).
Arterial Baroreflex Control
Our previous study (25) did not consider baroreflex
control of heart rate, myocardial contractility, and vasomotor tone. We
have now included lumped characterizations of the baroreceptors and
their reflex pathways in the present study, according to the general
structure used by Wesseling et al. (38).
Baroreceptors.
Figure 2 includes four functional blocks
that represent the baroreceptor, the central nervous system (CNS), the
efferent pathways, and the effector organ. The input to the
baroreceptor element (BR) is central arterial pressure [aortic arch
pressure (PAo)], and the output [N(t)] is the
instantaneous firing frequency of the BR. Following Spickler et al.
(35), we characterized the input-output relationship in
terms of the following transfer function
|
(5)
|
The corresponding differential equation is as follows
|
(6)
|
where K is the gain and a is a time
constant [35].

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Fig. 2.
Block diagram of baroreflex control of arterial pressure. A fast
vagal (dashed arrow) pathway and 3 slow sympathetic pathways are
included to control heart rate, myocardial contractility, and vasomotor
tone. The overall control scheme is based on the modeling concept of
Wesseling et al. (38). For abbreviations, see text.
|
|
Central nervous system.
The medullary cardiovascular control center is modeled in terms of four
noninteracting pathways, each characterized by filtering, gain, and a
delay as per the modeling concept of Wessling et al. (38).
One vagal (fast) and one sympathetic (slow) pathway each controls heart
rate, whereas two other sympathetic pathways control myocardial
contractility and vasomotor tone. The fast vagal pathway has a 0.2-s
delay, whereas each sympathetic pathway has a 3-s delay.
Efferent pathways.
We described each efferent pathway according to the following generic
equation in normalized form (Table 6)
|
(7)
|
The generic parameter x represents heart rate,
contractility, or vasomotor tone. The parameters
x and Nx,0 were fitted to the representative data. This equation provides a
sigmoidal input-output relationship (threshold and saturation) between
central neuron activity (output of central delay box) and the discharge
frequency of the particular motor neuron (6, 11, 22, 29,
35).
Because increases in BR firing frequency increase vagal discharge
frequency,
x in the vagal efferent pathway is negative, producing a monotonically increasing input-output
relationship for the linear part of the curve (Fig. 2). Sympathetic
pathways use positive
x values, because BR
and sympathetic discharge frequencies change in opposite directions.
Figure 2 shows that the discharge frequency (Fx)
of each efferent pathway inputs to the final block of the diagram,
which contains characterization of the input-output response of the
effector organ itself (the heart or vessel).
Effector organs.
Heart rate is controlled by vagal and sympathetic neural activity and
has been characterized by Sunagawa as a three-dimensional response
surface [36]. We developed the following equation to characterize the human heart rate response surface to vagal and sympathetic input (Table 7)
|
(8)
|
where HR (in beats/min) represents heart rate, FHr,V
and FHr,S are the normalized vagal and sympathetic
frequencies, and h1-h6 are constants.
This formula generates a normalized heart rate response surface
analogous to that of Sunagawa et al. (36).
In our study, the heart period (calculated as 60/HR, in s) is
explicitly determined by the vagal-sympathetic mechanism according to
Eq. 8, and the systolic period is mediated by the
sympathetic frequency (Fig. 3). The
diastolic filling time is the difference between the two and is thus
controlled indirectly.

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Fig. 3.
Model representation of the sympathetically regulated
activation function e(t). Four different levels
of contractility corresponding to different sympathetic efferent
frequencies (Fcon) are shown.
|
|
Greater sympathetic tone increases myocardial elastance and shortens
ventricular systole. Therefore, we modified the ventricular activation
function to describe the change in ventricular elastance [e(t)] as a function of sympathetic efferent
discharge frequency (Fcon) (see Fig. 3).
A rise in Fcon increases maximum elastance and
shortens the systolic period. The expression for the end-systolic P-V
relationship [PES(V)] becomes (notation from Ref.
25 and Table 1)
|
(9)
|
and the activation function
[ev(t)] becomes
|
(10)
|
where
|
(11)
|
|
(12)
|
Here, amin and
bmin are dimensionless constants representing
the minimum values of the functions a and b,
respectively, and Ka and
Kb are scaling parameters.
Arteries and arterioles are the major resistance vessels. When their
smooth muscle constricts, lumen diameter decreases, axial resistance to
flow increases, and the muscle wall stiffens. Therefore, a change in
vasomotor tone involves a change in both axial resistance and in wall
compliance. We transformed the passive and fully activated length-tension relationships previously described by Gore and Davis (10) into an equivalent P-V relationship for a
cylindrical vessel. Figure 4 shows the
passive and fully activated P-V curves used in our model, which are
represented as follows. Fully activated
|
(13)
|
and passive
|
(14)
|
where P
and P
represent
the arterial pressures in the fully activated and passive states,
respectively, VSA is the blood volume contained in systemic arteries, and VSA,0 (in ml) is the minimal volume. We
assume VSA
VSA,0 in Eqs. 13 and 14. Kc, Kp1,
and Kp2 (in mmHg) are constant scaling
parameters, D0 (in ml) is a volume parameter, and
p (in ml
1) is constant. During sympathetic
stimulation, the compliance of the vessel is characterized by Eq. 13; when the sympathetic tone is abolished, the compliance of
vessel wall is described by Eq. 14. The normalized
sympathetic efferent frequency (Fvaso) serves as a scaling
factor for the transition between these states
|
(15)
|
Axial resistance is also affected by sympathetic activity.
Resistance (RSA; in
mmHg · s · ml
1) and sympathetic efferent
frequency (Fvaso) are related by
|
(16)
|
The first term is regulated by the sympathetic frequency and the
second term is a function of lumen volume (VSA).
VSA,max is the maximal lumen volume and
Kr (in mmHg) is a pressure scaling constant.

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Fig. 4.
Active and passive P-V curves of systemic arteries.
PSA and VSA represent the pressure and volume
in the systemic arteries. Fvaso is the normalized
sympathetic discharge frequency controlling the vasomotor tone.
|
|
Airway/Lung Mechanics Model
The pulmonary portion of our cardiopulmonary model combines two
models previously developed. One focuses on airway/lung mechanics (2) and the other focuses on gas exchange
(18). Figure 5 shows an
equivalent pneumatic circuit model of the airways and lung of the
normal human. The lung mechanics model (2) includes nonlinear characterizations of airway resistance, airway and chest wall
compliance, and lung tissue viscoelasticity. This particular model
(2) has also been used in a related context to analyze the
"work of breathing" during clinical breathing maneuvers [see Athanasiades et al. (2) for details].

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Fig. 5.
Airway/lung mechanics model. A: components of airway
mechanics, pulmonary circulation, and gas exchange. B:
equivalent pneumatic circuit representation of airway/lung mechanics
and gas exchange [modified from Athanasiades et al.
(2)]. For abbreviations, see Glossary and
text.
|
|
In the supine human, the lungs and their airways are subject to the
same time-varying intrathoracic pleural pressure (PPL). Figure 5 indicates that this pressure is generated by the
respiratory muscles (Pmus) and the recoil pressure of the
chest wall (PCW). Measured PPL is also the
driving pressure for our airway mechanics model. The upper airway is
assumed rigid and is characterized by a nonlinear flow-dependent
resistor (Rohrer resistor). The midairways are assumed collapsible and
are characterized by a nonlinear volume-dependent resistance
[RC(VC)] and a nonlinear P-V
relationship [PTM(VC)], where VC
is the collapsible segment volume (Fig. 5). Pressure in the lumen of
the midairway segment of the model is denoted as PC, and
the transmural pressure across the wall is denoted as PTM.
PA is the alveolar pressure and PEL is the lung
elastic recoil pressure. Small airways resistance (RS) is characterized as a nonlinear function of
the alveolar volume (VA).
From an analysis of the pneumatic circuit according to Newton's first
law
|
(17)
|
|
(18)
|
|
(19)
|
The component air flows (in ml/s) in the airway system are
computed according to the equations below, which are derived from the
continuity equation applied to each node of the pneumatic network. The
resulting differential equations are as follows
|
(20)
|
|
(21)
|
|
(22)
|
As such, the rate of the volume changes in the airway may be
written as follows
|
(23)
|
|
(24)
|
where
tot denotes the total gas flux rate (in
ml/s) of all gaseous species across the alveolar-capillary membrane, as
given by Eq. 31.
Gas Exchange Model
Gas exchange between air and blood occurs across the
alveolar-capillary membrane. For modeling purposes, we assumed
1) inspired air is instantly warmed to body temperature and
saturated with water vapor, 2) gaseous content obeys the
ideal gas law, 3) blood is characterized as a uniform
homogeneous medium, and 4) reactions between the gaseous
species and blood are assumed to equilibrate instantaneously. The
empirical O2 and CO2 dissociation curves relate
the content of each species with their corresponding partial pressures
in blood. The diffusing capacity for the ith gaseous species
(DLi) characterizes its diffusion
across the alveolar-capillary membrane. O2 is taken up by
the blood, CO2 is removed, and N2 diffuses
either way depending on the direction of their instantaneous partial
pressure gradients.
The species conservation law is applied to inspiration and expiration.
Inspiration can be described as follows
|
(25)
|
|
(26)
|
|
(27)
|
and expiration can be described as follows
|
(28)
|
|
(29)
|
|
(30)
|
Here, PDi,
PCi, and
PAi are partial pressures of gas
species i (O2 or CO2) in the upper,
middle, and small airways, respectively; Patm,i
is the partial pressure of the gas species i in the
atmosphere; and VPC is the blood volume contained in
pulmonary capillary. N2 partial pressure in the airways was
obtained by subtracting the partial pressures of O2,
CO2, and H2O from the total airway pressure.
Nseg is the number of capillary segments. In the
gas exchange model, the lumped pulmonary capillary was divided into 35 segments, as in Liu et al. (18).
P
represents the partial pressure of gas species i in the
jth capillary segment, and
V
denotes the blood volume contained
in the jth capillary segment.
The total flux rate (
tot; in ml/s) of all gaseous
species across the alveolar membrane can be expressed as follows
|
(31)
|
Here, i = 1, 2, or 3 and represents the three
gaseous species (O2, CO2, and N2).
Species molar balance was employed to describe the dynamics of the
species blood concentration in each segment. The corresponding equation
for gas species i in the jth capillary segment is
given by
|
(32)
|
The formula of the lung diffusion capacity for each gaseous
species was taken from Liu et al. (18) (with a change in
units from ml
STPD · min
1 · mmH2O
1
to ml STPD · s
1 · mmHg
1).
These formulas are as follows
|
(33)
|
|
(34)
|
|
(35)
|
where VPC,max is the maximal blood volume in the
pulmonary capillaries.
Cardiopulmonary Interactions
Any combined cardiovascular and pulmonary model must account for
interactions that can occur between these systems. These interactions
take a variety of forms and frequently are quite subtle. In general, to
test for system interaction, a variable in one system is perturbed and
the effects on both systems are assessed. We accomplished this by using
only perturbations in pleural pressure (PPL). The following
sections provide simple examples of this coupled interaction.
How PPL mediates cardiac and vascular
mechanics.
PPL affects both intracardiac pressures and the pressures
within the large intrathoracic vessels, but alveolar pressure has the
greatest effect on pulmonary capillaries (18, 23).
Consequently, in our model, the capillary transmural pressure is
mediated by the alveolar pressure, whereas the pressures of the
pulmonary arteries and veins are changed by PPL.
How lung air volume changes lung perfusion.
The pulmonary capillary bed forms an extensive network of vessels,
which surround the alveolar region. During lung inflation, these
vessels are stretched and constricted by the expanding alveolar volume.
This increases capillary resistance and reduces blood flow, thus
facilitating gas exchange. The relationship we used to describe the
capillary resistance (RPC) changes with alveolar volume (VA) is as follows
|
(36)
|
Here, RPC,0 is a constant chosen to set
the magnitude of capillary resistance and VA,max represents
the maximum alveolar volume.
 |
COMPUTATIONAL ASPECTS |
To summarize, we modified and combined previous cardiac and
pulmonary models developed by our group to form a cardiopulmonary model
of the normal human (Tables
8-10).
The pulmonary models employed (2, 18) were originally
developed as human models and were verified using data obtained from
normal human subjects. However, the cardiovascular model used as a
basis for designing our human circulatory model (25) was
validated using data from the dog. To develop the human model, we first
scaled up our canine model to provide an initial model of the normal
human cardiovascular system. This has been done by others (see, e.g.,
Ref. 17). Because human and canine blood pressures and
blood velocities are similar, scaling factors are related closely to
the ratios of blood volume. (Blood volume is directly related to body
weight and body surface area.) In a second phase, we manually adjusted
the parameters of the initial human circulatory model to yield a
reasonable fit to typical human pressure data and hemodynamic indexes
available in the literature.
First, we determined that the cardiac output of a 70-kg human is ~2.5
times that of a 25-kg dog. Because the mean systemic arterial pressures
in the human and dog are similar, we calculated a set of human
cardiovascular parameters by decreasing all the resistive and inertial
parameters of the canine model by 2.5 and by similarly increasing the
compliant parameters. This scaling provided a reasonable initial
representation of the human cardiovascular system, although additional
adjustments were necessary for better regional representations of
typical hemodynamic waveforms.
The representations used for certain elements of the canine and human
circulatory models were different. Specifically, the linear
representations of venous compliance in the canine model were replaced
by nonlinear P-V relationships in the human model. Nonlinear active and
passive P-V curves were also incorporated to describe arterial compliance.
The structure of the human circulatory model also differs in that
several parallel circulation pathways were added. In the pulmonary
circulation, the average pulmonary shunt flow is 2% of the pulmonary
blood flow, whereas in the systemic circulation, the mean coronary and
cerebral flows are set to 6% and 14%, respectively, of the cardiac
output. The nominal distribution of blood volume in the pulmonic and
systemic circulations are set at a level of 8.8% and 84%,
respectively. The remaining 7.2% of the blood is contained in the
heart. These figures agree with the results shown in Ref.
24 (p. 30 and 124).
We approximated the first-order spatial derivative in Eq. 32
using a four-point biased quadratic interpolation formula
(31) and eliminated fictitious points at the entrance of
the capillary bed using constant inlet conditions (i.e., partial
pressures of 40 mmHg for O2 and 46 mmHg for
CO2).
The PPL data reported previously by Liu et al.
(18) were used to directly drive the pulmonary model.
Therefore, the respiratory frequency was determined directly from the
experimental data. The model begins at end expiration, when there is no
flow and air volume in the lung equals the functional residual capacity (FRC), which is set to the typical value of 2,200 ml.
The combined model has 77 nonlinear differential equations and 116 parameters associated with its component models. In all, 149 outputs
were generated simultaneously. Table 11
shows the distribution of the state variables and model parameters in
the combined cardiopulmonary model.
The model was programmed in C programming language and solved using the
variable step-size Runge-Kutta-Merson algorithm, with a maximum time
step size of 2 × 10
2 s and an error tolerance of
1 × 10
6. On average, it takes 20 min of CPU time on
a Pentium II 333-MHz machine to simulate 35 s of cardiopulmonary events.
Simulation Results
Hemodynamics.
Figure 6 compares a model-generated and
human systemic pressure waveform (Fig. 22.15 in Ref. 20).
The left ventricular end-systolic pressure is 125 mmHg and the systolic
duration is 0.3 s, or about one-third of the cardiac cycle (0.8 s). The aortic root pressure ranges from 80 to 120 mmHg. The dicrotic
notch can be clearly seen in both the simulation and the data.

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Fig. 6.
Model-predicted systemic pressure waveforms (A) compared
with the textbook figure [McClintic (20); B]
showing left ventricular pressure (PLV ), aortic root
pressure (PAo), and left atrial pressure
(PLA).
|
|
Figure 7 compares experimental data to
the model-generated left ventricular volume and aortic and pulmonary
arterial flow waveforms (Fig. 6-1 in Ref. 24). The left
ventricular volume ranged from 150 ml [end-diastolic volume
(VED)] to 70 ml [end-systolic volume (VES)],
giving a stroke volume of 80 ml and an ejection fraction of 80/150, or
0.533. The volume added as the result of atrial systole (
V) was 30 ml, ~20% of the VED. The peak aortic flow rate was 750 ml/s and the peak pulmonary arterial flow rate was 400-500 ml/s.
The aortic flow rate has a higher peak value and shorter time span
compared with the pulmonic flow rate because of the stronger
contractile force and higher afterload of the left ventricle. Numerical
integration of the aortic and pulmonic flow waveforms over one cycle
showed that the mean values of the area enclosed by the two waveforms
were the same, although during individual cardiac cycles they may be
different from each other due to variations of intrathoracic pressure.

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Fig. 7.
Model-predicted volumes and flows (left) compared with
reported experiment data [Mountcastle (24);
right]. A: left ventricular volume
(VLV); B: aortic flow ( Ao);
C: pulmonary arterial flow ( PA). For
abbreviations, see text.
|
|
Table 12 compares indexes of the
model-predicted and measured data. Our model predicts that at peak
inspiration, the stroke volume of the left heart decreases, whereas the
stroke volume of the right heart increases. At peak expiration, the
opposite occurs. These changes agreed well with measured data (4,
9, 13, 26, 28). Our model helps explain the mechanism underlying this physiological relationship.
Right and left ventricular volumes respond to PPL because
of both direct and series ventricular interaction. When PPL
are negative (e.g., with inspiration), an increase in venous return augments right ventricular filling and stroke volume. The increased right ventricular filling causes the septum to encroach upon the left
ventricle, because the pericardium limits total cardiac volume. As a
result, left ventricular stroke volume is reduced. Simulations that
ignore this ventricular interaction (rigid septum) underestimate the
percent fall in left ventricular stroke volume occurring during inspiration (2.5% vs. 5% with ventricular interaction). Without pericardial constraint, there is little respiratory variation in left
ventricular stroke volume (1%).
Expiration causes a volume shift from the pulmonary to the systemic
circulation. The blood pooling in the systemic vascular bed then
increases left ventricular afterload with the next inspiration. Simulations show that both the end-systolic transmural pressure and
volume of the left ventricle are highest at early inspiration, consistent with increased afterload (27, 30, 32). During inspiration, both decreased filling and increased afterload decrease left ventricular stroke volume. The same mechanism explains why the
left and right ventricles respond differently to elevated PPL during expiration.
Airway Mechanics and Gas Exchange
Figure 8 depicts the airway
pressures and lung volumes predicted by our model. During inspiration,
subatmospheric PPL is transmitted to the alveoli,
facilitating air flow into the lungs. As this occurs, lung elastic
recoil increases and the alveolar and atmospheric pressures equalize,
marking the end of inspiration and the start of expiration. During
expiration, the less negative PPL and the resulting changes
in lung elastic recoil cause positive alveolar pressure, pushing air
from the lungs. The inspiratory and expiratory changes in lung volume
are depicted in Fig. 8B. Here, total lung volume is the sum
of the air volumes contained in the alveoli, collapsible airways, and
dead space. The model predicts an average tidal volume of 500 ml and a
functional residual capacity (FRC) of 2.2 l, which agreed with
measured values.

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Fig. 8.
Model-predicted airway pressures and lung volume in
normal breathing. A: model-generated alveolar pressure
(PA). PPL, experimental data of the pleural
pressure. B: lung volume (VL). The lung volume
includes air volumes in the alveolar region, collapsible airways, and
dead space.
|
|
Inspiration fills the alveoli with O2-enriched air, whereas
expiration removes CO2. Figure
9 depicts the model-generated variation in airway gas composition in terms of changes in the partial pressures of O2 and CO2 (PO2 and
PCO2, respectively). Alveolar
PCO2 and PO2 are
relatively constant. Alveolar PO2 varies from
95 to 105 mmHg, and alveolar PCO2 varies from
35 to 40 mmHg. With inspiration, PO2 in
the upper airways (dead space) rises sharply, whereas
PCO2 drops sharply. However, not all inhaled
air enters the alveoli, and inhaled and residual air mix, making
variations in alveolar PO2 and
PCO2 much smaller than those in the dead space.
Expiration lowers the PO2 and raises the
PCO2 in the dead space, whereas gases
continuously diffuse across the alveolar-capillary membrane. Consequently, alveolar PO2 decreases and
alveolar PCO2 increases.

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Fig. 9.
Variations of airway gaseous partial pressure during normal
breathing in the simulation. A: PO2
variations in the dead space and alveolar regions. B:
PCO2 variations in the dead space and alveolar
regions.
|
|
Alveolar capillary gas exchange.
Figure 10 depicts the flux of
O2 and CO2 at the alveolar-capillary membrane,
which is modeled as 35 contiguous segments. For each segment, there is
a gaseous flux waveform that pulsates with capillary blood flow. Most
gaseous diffusion occurs at the initial capillary segments but later
diminishes when blood and alveolar gas content has equilibrated. Thus
the flux rates decrease exponentially from the first (entrance) to the
last segment (exit). We tested our model against the known changes that
occur during the FVC and Valsalva maneuvers.

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Fig. 10.
Three-dimensional representations of the regional
gaseous flux calculated at each of the 35 capillary segments.
A: O2 gas flux; B: CO2
gas flux.
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|
Forced Vital Capacity Maneuver
The FVC maneuver is a commonly used pulmonary function test. The
subject fully exhales and then inhales to total lung capacity (TLC)
without pausing. Immediately, the subject exhales as rapidly as
possible, until airflow is no longer detected at the mouth. We applied
the measured FVC PPL data reported in Ref. 18
to our model.
Figure 11 compares the model
predictions to data measured from a human. During the rapid inspiration
phase, lung volume increased to full capacity (Fig. 11A) and
PPL decreased (Fig. 11B). At the beginning of
forced expiration, PPL increases sharply, and lung volume
decreased until it reached residual volume. The predicted and measured
data correlated nicely.

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Fig. 11.
Model predictions of lung volume variations and flow at
the mouth (solid lines) compared with experimental data (dashed lines)
from Liu et al. (18). The dotted vertical lines show the
first expiration (e), inspiration (i), and second expiration (e*).
A: lung volume (VL) variations from residual
volume. B: flow at the mouth. C: measured
PPL data.
|
|
During the FVC maneuver, the expired PO2
decreased constantly and reached a minimum value of 118 mmHg. In
contrast, PCO2 increased steadily until its
maximum value of 38 mmHg. Figure 12
compares the predicted temporal profile of PO2
and PCO2 in expired air with data from Liu et
al. (18). Again, the model prediction agreed well with the
measured data.

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Fig. 12.
Model-predicted PO2 and
PCO2 (A) in the expired air at the
mouth (solid lines) during the forced expiration in the forced vital
capacity (FVC) maneuver compared with experimental recordings (dashed
lines) from Liu et al. (18).
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|
Hemodynamic changes are seldom recorded during the FVC maneuver, but
our model can predict them. Figure 13
shows the predicted change in left and right heart stroke volumes. As
in normal respiration, the left and right ventricles showed opposite
responses during inspiration and the early part of the forced
expiration. However, after a few beats into the prolonged second phase
of forced expiration, the stroke volumes of both ventricles decreased
quickly and then returned to baseline after an overshoot (Fig. 13).
Stroke volume decreases because elevated PPL decreases
venous return. The recovery and the overshoot may be caused by neural
factors (discussed below).

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Fig. 13.
Model-predicted percent changes in left (A)
and right ventricular stroke volumes (B) during the FVC
maneuver. C: PPL data from Liu et al.
(18). The dotted vertical lines show the first expiration
(e), inspiration (i), and second expiration (e*).
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|
Figure 14 demonstrates a similar
recovery and overshoot in the systemic arterial pressure waveform
(A) and the temporal variations in heart rate
(B), vagal discharge (C), and sympathetic
discharge (D). Heart rate increases slowly during the
maneuver. Vagal efferent activity slows, and a burst of sympathetic
activity occurs later. These findings are consistent with the faster
and slower activity of the vagal and sympathetic pathways,
respectively. The decreasing vagal and increasing sympathetic outputs
correlated with the observed increases in heart rate, myocardial
contractility, and vasomotor tone and explained the partial recovery of
arterial blood pressure that occurs during and shortly after the
maneuver.

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Fig. 14.
Model-predicted variables during the FVC maneuver.
A: systemic arterial pressure (PSA);
B: heart rate; C: vagal efferent discharge;
D: sympathetic efferent discharge; E:
PPL data from Liu et al. (18). The dotted
vertical lines show the first expiration (e), inspiration (i), and
second expiration (e*).
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Valsalva Maneuver
During the Valsalva maneuver, the subject forcefully exhales
against a closed glottis (or nose and mouth). The maneuver markedly elevates intrathoracic pressure and affects venous return, myocardial contractility, vasomotor tone, and baroreflex heart rate control. It is
a widely used test of baroreceptor reflexes (6).
The hemodynamic response to the Valsalva maneuver has four distinct
phases: phase 1 (an initial increase in arterial pressure), phase 2 (a rapid fall in arterial pressure, followed by a
partial recovery and tachycardia), phase 3 (a reduction in
arterial pressure upon the sudden termination of breath holding,
accompanied by a continued tachycardia), and phase 4 (an
overshoot in arterial pressure accompanied by a slowing of heart rate).
In our model, we simulated the Valsalva maneuver by elevating
PPL to a higher value for 15 s, starting at the end of
both inspiration and diastole. PPL of 10, 20, 30, and 40 mmHg were used in the simulation to represent different levels of
expiratory effort. Airflow in the airways was set to zero during the
maneuver to simulate the closed glottis.
Figure 15 compares the model-generated
changes in arterial pressure and heart rate when PPL is
40mmHg during the Valsalva maneuver with the experimental data from
Bannister (33). The predicted increase in arterial
pressure during phase 1 (~120% of baseline), recovery of
the arterial pressure during phase 2, and overshoot during
phase 4 (20% above baseline) all fitted well with the
measured data, as did the predicted heart rate changes. Heart rate
peaked at 110 beats/min and dropped to 62 beats/min after the maneuver. However, the predicted heart rate changes before and after the maneuver
were much smoother than the measured data. This may be because an
idealized square PPL waveform was used in the Valsalva maneuver simulation. In reality, PPL recordings show
fluctuations before and after the maneuver.

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Fig. 15.
Comparison of experimental data [Bannister
(33); A] with model-based predictions
(B) of the changes in PSA (top) and
heart rate (bottom) during the Valsalva maneuver. The arrows
in B, bottom, denote the beginning and end of the
maneuver. The four distinct phases of the Valsalva maneuver are
indicated by the vertical dashed lines and numbers.
|
|
Figure 16 shows heart rate, cardiac
output, and mean arterial pressure as a function of PPL
during the Valsalva maneuver. Experimental data from Korner et al.
(14) were superimposed on the plot. Both heart rate and
mean arterial pressure increased nearly linearly as PPL
increased. Because of reduced venous return, cardiac output declined
with the increase in PPL.

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Fig. 16.
Relationships between PPL and heart rate
(A), cardiac output (B), and mean arterial
pressure (C) during the Valsalva maneuver. [Data source:
Korner et al. (14).]
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Table 13 shows that a variety of
calculated hemodynamic indices evaluated from the model predictions
during the Valsalva maneuver agreed well with those obtained from
humans [Fox et al. (8)].
Baroreflex control during the Valsalva maneuver.
The Valsalva maneuver changes autonomic tone, central nervous system
activity, arterial blood pressure, and heart rate. Figure 17 illustrates how heart rate,
myocardial contractility, and vasomotor tone responded to baroreflex
control during each phase of the Valsalva maneuver.

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Fig. 17.
Model-generated arterial pressure waveforms during the
Valsalva maneuver under four baroreflex control conditions.
A: no baroreflex control present; B: only the
vasomotor tone control; C: vasomotor tone + myocardial
contractility control; D: all 3 control components
(vasomotor tone, myocardial contractility, and heart rate). The arrows
indicate the start and the end of the maneuver.
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Phase 1 elevation of arterial pressure occurs without
baroreflex control, suggesting that the elevation is due only to the mechanical forces of increased intrathoracic pressure (6), which compresses the heart chambers and augments output to the periphery.
Phase 1 lasts about one to two heartbeats. As venous return
is reduced by the elevated intrathoracic pressure, diastolic filling and stroke volume decrease. Therefore, in phase 2, blood
pressure decreases and heart rate increases, with baroreflexes helping maintain arterial pressure and cardiac output. Simulations demonstrate the importance of baroreflex control. When baroreflex control was
abolished (Fig. 17A), arterial pressure dropped during this phase. When the sympathetic vasomotor tone was added (Fig.
17B), the reduction in arterial pressure leveled off after
five to six beats, and arterial pressure stabilized. When myocardial
contractility was added (Fig. 17C), a slow and gradual
recovery of arterial pressure toward baseline occurred after six beats.
The delay corresponded to the late increase in sympathetic efferent
traffic (Fig. 18C), which
constricts arterial resistive vessels and augments myocardial contractility. When baroreflex control of heart rate was included (Fig.
17D), heart rate increased. These simulations demonstrate the importance of baroreflex control during phase 2 of the
Valsalva maneuver.

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Fig. 18.
Model-generated sympathetic and parasympathetic (vagal)
efferent bursts during the Valsalva maneuver. A:
PSA response. B: vagal discharge frequency
[showing both the spike representation (a) and the relative
changes of the frequency value (Fvagus; b)].
C: sympathetic discharge frequency [showing both the
spike representation (a) and the relative changes of the
frequency value (Fsymp; b)]. The arrows
indicate the start and the end of the maneuver.
|
|
During phase 3, the model predicts a fall in arterial
pressure without baroreflex input (Fig. 17A). The decrease
in intrathoracic pressure decreased intracardiac pressure and stroke
volume. Heart rates remained high due to the increased sympathetic tone
(Fig. 18C).
During phase 4, venous return became normal. However, the
delayed sympathetic response maintained the heightened myocardial contractility, tachycardia, and vasoconstriction. With no baroreflex control (Fig. 17A), arterial pressure slowly returned to
baseline. With vasoconstriction (Fig. 17B), there was a
small overshoot in arterial pressure even though arterial pressure
returned to normal. This overshoot was more prominent (120% of the
baseline level) when baroreflex control of myocardial contractility was
added (Fig. 17C). Adding heart rate control (Fig.
17D) augmented this overshoot to a lesser degree.
The variations of vagal and sympathetic discharge frequencies during
the maneuver are shown in Fig. 18. During phase 2, vagal discharges diminished and sympathetic discharges increased. The increase in sympathetic tone occurred after vagal tone decreased. Immediately after the release of the strain, vagal discharge frequency quickly returned toward control, whereas the elevated sympathetic tone
continued into the late part of phase 4.
Figure 19 summarizes how the individual
baroreflex pathways maintain arterial pressure. In phase 2,
vasoconstriction prevented arterial pressure from dropping at a
constant rate, whereas increased myocardial contractility and
tachycardia helped restore arterial pressure and cardiac output.
Continued increases in myocardial contractility and vasomotor control
contributed strongly to the overshoot of arterial pressure in
phase 4.

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Fig. 19.
Schematic representation of the contribution of the
individual baroreflex pathway to the PSA response during
the Valsalva maneuver. HR, heart rate; CON, myocardial contractility;
VASO, vasomotor tone.
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 |
DISCUSSION |
We presented a mathematical model of the human cardiopulmonary
system that combines several component models previously developed by
our group. Physiological data predicted by this combined model agreed
well with data taken from resting and normal subjects in the supine
position. The model was further validated by accurately predicting the
sudden and large physiological changes that occurred during the FVC
test and all four stages of the Valsalva maneuver.
Although the Valsalva maneuver is well understood, it remains a
relatively complicated physiological response, with mechanical and
autonomic components that are difficult to separate when used clinically. This limits what information might be obtained from a test
that otherwise is very helpful, easy to perform, and commonly used to
assess patients with a wide variety of cardiovascular disorders.
Because our cardiopulmonary system can mimic a combined response from
separate pulmonary, circulatory, and neural components, actual
(clinical) responses to the Valsalva maneuver might be analyzed in
terms of these components (see Fig. 18 and its accompanying discussion), and a specific physiological defect may be more easily identified. It is also possible that variations between or within groups may be more amenable to statistical analysis, because of the
purely quantitative nature of the model. Good statistical backing would
certainly improve the meaning and significance of future studies using
the Valsalva maneuver.
The utility of our cardiopulmonary model should not be limited to the
Valsalva maneuver, however. With further modification and extension, it
might help diagnose or analyze other normal or disordered physiological
responses, such as orthostatic hypotension, and could characterize
disease states such as atherosclerosis, valvular stenosis, and the
pulmonary effects of congestive heart failure and the adult respiratory
distress syndrome. It could also be helpful in assessing the prognosis
of patients with congestive heart failure and/or coronary artery
disease, because in these patients both autonomic and mechanical
dysfunction are major determinants of premature death.
Model Limitations
All models have limitations, and ours is no exception. The
following are a discussion of the limitations of our model:
We employed a circulatory model of intermediate complexity for
use in the larger cardiopulmonary model. It mimics the hemodynamics of
the circulation quite well. The objectives of the study are general,
however, and if questions such as flow in a particular circulation or
pulse wave propagation were asked of this model, its foundational
assumptions would be too crude to provide adequate predictions (e.g.,
wave propagation delay is approximated by a phase shift). In addition,
as a supine model, it cannot simulate the hemodynamic responses related
to changes in body position or gravitational forces, e.g., when
subjects stand up from the supine position or enter different
gravitational environments as in space flight. To address such
problems, additional bandwidth (structural changes) would have to be
provided in the form of the adoption of a more distributed model or
certain nonlinear elements would have to be included. This, however, is
the subject matter of another study.
Ventricular elastance is defined as the instantaneous transmural
pressure-to-ventricular volume ratio. At each point in time, it
represents a linear relationship between pressure and volume. Therefore, it provides only an approximation to the curvilinear Frank-Starling relationship (5). This approximation works
well around the operational point of the human heart, but with large increase in volume, it would overestimate the pressure developed by the
ventricle. To represent the Frank-Starling mechanism more faithfully,
the expression for elastance should be modified and made a function of
both end-diastolic volume (VED) and time, as in Ref.
10a.
The lung was characterized as a single compartment, and homogeneous
ventilation was assumed. This is unsuitable when the lungs have
regional disease. A multiple-compartment model would be required in
that case.
The neural control scheme currently employed in the
cardiopulmonary model includes only the baroreceptor-mediated control of heart rate, myocardial contractility, and vasomotor tone. It does
not contain an explicit description of the splanchnic circulation, which in humans has a venous bed richly innervated by adrenergic nerve
fibers. To develop quantitative descriptions of venoconstriction in
humans, more data are needed. Therefore, we neglected venoconstriction as an important baroreceptor-mediated effect in the Valsalva maneuver. Other important factors, such as the cardiopulmonary baroreceptors, hormonal effects, central and peripheral chemoreceptor-mediated ventilation, and autoregulation of special circulations, are not considered in the current model. These factors can also exert important
effects on the heart, circulatory hemodynamics, pulmonary mechanics,
and ventilatory control.
The model characterizes gas exchange only at the alveolar-capillary
membrane. However, gaseous partial pressures in pulmonary arterial
blood at the inlet to this membrane are not constant, because gas
exchange occurs at other tissue sites in the body. Modifying the model
to characterize this additional tissue gas exchange would affect the
gaseous content of the pulmonary arterial blood presented to the
alveolar membrane.
Finally, the model must be refined by comparing its predictions with
clinical data obtained prospectively.
 |
ACKNOWLEDGEMENTS |
The authors acknowledge the helpful comments of Drs. Dirar Khoury
and Sherif Nagueh of Baylor College of Medicine in the preparation of
the manuscript.
 |
FOOTNOTES |
This work was supported by the Bioengineering Center, University of
Texas Medical Branch, Galveston, TX.
Address for reprint requests and other correspondence: J. W. Clark, Dept. of Electrical and Computer Engineering, Rice Univ., 6100 Main St., Houston, TX 77005 (E-mail:
jwc{at}ece.rice.edu).
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 12 March 2001; accepted in final form 13 August 2001.
 |
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