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Am J Physiol Heart Circ Physiol 281: H2661-H2679, 2001;
0363-6135/01 $5.00
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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
TOP
ABSTRACT
INTRODUCTION
MODEL DEVELOPMENT
COMPUTATIONAL ASPECTS
DISCUSSION
REFERENCES

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
TOP
ABSTRACT
INTRODUCTION
MODEL DEVELOPMENT
COMPUTATIONAL ASPECTS
DISCUSSION
REFERENCES

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
 QCA   Airflow from collapsible airways to alveolar region
 QDC   Airflow from upper supported airway to collapsible airway
 QED   Airflow from environment to upper supported airway

Blood flows
 QAo   Aortic flow
 QPA   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
 lambda    Diastolic elastance coefficient
 tau p   Passive exponential constant
 tau x   Time constant

Gas diffusion and flux
C<UP><SUB>b<SUB><IT>i</IT></SUB></SUB><SUP>(<IT>j</IT>)</SUP></UP>   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
 Phi 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<UP><SUB>b<SUB><IT>i</IT></SUB></SUB><SUP>(<IT>j</IT>)</SUP></UP>   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<UP><SUB>SA</SUB><SUP>a</SUP></UP>   Systemic arterial pressure in the active state
P<UP><SUB>SA</SUB><SUP>p</SUP></UP>   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<UP><SUB>Z<SUB>b</SUB></SUB><SUP>(<IT>j</IT>)</SUP></UP>   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<UP><SUB>b<SUB><IT>i</IT></SUB></SUB><SUP>(<IT>j</IT>)</SUP></UP>   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
TOP
ABSTRACT
INTRODUCTION
MODEL DEVELOPMENT
COMPUTATIONAL ASPECTS
DISCUSSION
REFERENCES

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.

                              
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Table 1.   Parameter values of the ventricular model

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.


                              
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Table 2.   Parameter values of the atrial model


                              
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Table 3.   Parameter values for the activation function


                              
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Table 4.   Nominal parameter values in the systemic and pulmonic circulations

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
P<SUB>SV</SUB><IT>=</IT>−<IT>K</IT><SUB>v</SUB><IT>×</IT>log <FENCE><FR><NU>V<SUB>max</SUB></NU><DE>V<SUB>SV</SUB></DE></FR><IT>−</IT>0.99</FENCE> (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).

                              
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Table 5.   Parameter values for nonlinear P-V curves of systemic veins and the vena cava

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
if V<SUB>VC</SUB><IT>≥</IT>V<SUB>0</SUB>, then P<SUB>VC</SUB><IT>=D</IT><SUB>1</SUB><IT>+K</IT><SUB>1</SUB><IT>×</IT>(V<SUB>VC</SUB><IT>−</IT>V<SUB>0</SUB>) (2)

if V<SUB>VC</SUB><IT><</IT>V<SUB>0</SUB>, then P<SUB>VC</SUB><IT>=D</IT><SUB>2</SUB><IT>+K</IT><SUB>2</SUB><IT>×e</IT><SUP>(V<SUB>VC</SUB>/V<SUB>min</SUB>)</SUP> (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
R<SUB>VC</SUB><IT>=K<SUB>R</SUB>×</IT><FENCE><FR><NU>V<SUB>max</SUB></NU><DE>V<SUB>VC</SUB></DE></FR></FENCE><SUP>2</SUP><IT>+R</IT><SUB>0</SUB> (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
<FR><NU>N(<IT>s</IT>)</NU><DE>P<SUB>Ao</SUB>(<IT>s</IT>)</DE></FR><IT>=</IT><FR><NU><IT>K×</IT>(1<IT>+</IT>0.036<IT>s</IT>)</NU><DE>(1<IT>+</IT>0.0018<IT>s</IT>)(1<IT>+as</IT>)</DE></FR> where<IT> a<</IT>0.0018 (5)
The corresponding differential equation is as follows
0.0018a <FR><NU>d<SUP>2</SUP>N(<IT>t</IT>)</NU><DE>d<IT>t</IT><SUP>2</SUP></DE></FR><IT>+</IT>(0.0018<IT>+a</IT>) <FR><NU>dN(<IT>t</IT>)</NU><DE>d<IT>t</IT></DE></FR><IT>+</IT>N(<IT>t</IT>) (6)

<IT>=K</IT><FENCE>P<SUB>Ao</SUB>(<IT>t</IT>)<IT>+</IT>0.036<IT>K </IT><FR><NU>dP<SUB>Ao</SUB>(<IT>t</IT>)</NU><DE>d<IT>t</IT></DE></FR></FENCE>
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)
F<SUB><IT>x</IT></SUB>(<IT>t</IT>)<IT>=a<SUB>x</SUB>+</IT><FR><NU><IT>b<SUB>x</SUB></IT></NU><DE><IT>e</IT><SUP><IT>&tgr;<SUB>x</SUB></IT>[N<SUB><IT>x</IT></SUB>(<IT>t</IT>)<IT>−</IT>N<SUB><IT>x,</IT>0</SUB>]</SUP><IT>+</IT>1.0</DE></FR> (7)
The generic parameter x represents heart rate, contractility, or vasomotor tone. The parameters tau 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).

                              
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Table 6.   Parameter values for the baroreflex pathway

Because increases in BR firing frequency increase vagal discharge frequency, tau 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 tau 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)
HR<IT>=h</IT><SUB>1</SUB><IT>+h</IT><SUB>2</SUB><IT>×</IT>F<SUB>Hr,S</SUB><IT>−h</IT><SUB>3</SUB><IT>×</IT>F<SUP>2</SUP><SUB>Hr,S</SUB><IT>−h</IT><SUB>4</SUB><IT>×</IT>F<SUB>Hr,V</SUB> (8)

<IT>+h</IT><SUB>5</SUB><IT>×</IT>F<SUP>2</SUP><SUB>Hr,V</SUB><IT>−h</IT><SUB>6</SUB><IT>×</IT>F<SUB>Hr,V</SUB><IT>×</IT>F<SUB>Hr,S</SUB>
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).

                              
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Table 7.   Nominal parameter values for the effector organs in the baroreflex model

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)
P<SUB>ES</SUB>(V)<IT>=a</IT>(F<SUB>con</SUB>)<IT>×</IT>E<SUB>ES</SUB><IT>×</IT>(<IT>V−</IT>V<SUB>D</SUB>) (9)
and the activation function [ev(t)] becomes
e<SUB>v</SUB>(<IT>t, </IT>F<SUB>con</SUB>)<IT>≡</IT><LIM><OP>∑</OP><LL><IT>i=</IT>1</LL><UL><IT>n</IT></UL></LIM><IT> A<SUB>i</SUB>e−</IT><FR><NU>1</NU><DE>2</DE></FR><SUP> <FENCE><FR><NU><IT>b</IT>(F<SUB>con</SUB>)<IT>×t−C<SUB>i</SUB></IT></NU><DE><IT>B<SUB>i</SUB></IT></DE></FR></FENCE><SUP>2</SUP></SUP> (10)
where
a(F<SUB>con</SUB>)<IT>=a</IT><SUB>min</SUB><IT>+K<SUB>a</SUB>×</IT>F<SUB>con</SUB> (11)

b(F<SUB>con</SUB>)<IT>=b</IT><SUB>min</SUB><IT>+K<SUB>b</SUB>×</IT>F<SUB>con</SUB> (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
P<SUP>a</SUP><SUB>SA</SUB>(V<SUB>SA</SUB>)<IT>=K</IT><SUB>c</SUB><IT>×</IT>log <FENCE><FR><NU>V<SUB>SA</SUB><IT>−</IT>V<SUB>SA,0</SUB></NU><DE><IT>D</IT><SUB>0</SUB></DE></FR><IT>+</IT>1</FENCE> (13)
and passive
P<SUP>p</SUP><SUB>SA</SUB>(V<SUB>SA</SUB>)<IT>=K</IT><SUB>p1</SUB><IT>×e</IT><SUP><IT>&tgr;</IT><SUB>p</SUB><IT>×</IT>(V<SUB>SA</SUB><IT>−</IT>V<SUB>SA<IT>,</IT>0</SUB>)</SUP><IT>+K</IT><SUB>p2</SUB><IT>×</IT>(V<SUB>SA</SUB><IT>−</IT>V<SUB>SA<IT>,</IT>0</SUB>)<SUP>2</SUP> (14)
where P<UP><SUB>SA</SUB><SUP>a</SUP></UP> and P<UP><SUB>SA</SUB><SUP>p</SUP></UP> 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 tau 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
P<SUB>SA</SUB>(V<SUB>SA</SUB>)<IT>=</IT>F<SUB>vaso</SUB><IT>×</IT>P<SUP>a</SUP><SUB>SA</SUB>(V<SUB>SA</SUB>)<IT>+</IT>(1<IT>−</IT>F<SUB>vaso</SUB>)<IT>×</IT>P<SUP>p</SUP><SUB>SA</SUB>(V<SUB>SA</SUB>) (15)
Axial resistance is also affected by sympathetic activity. Resistance (RSA; in mmHg · s · ml-1) and sympathetic efferent frequency (Fvaso) are related by
R<SUB>SA</SUB><IT>=K<SUB>r</SUB>×e</IT><SUP>4<IT>×</IT>F<SUB>vaso</SUB></SUP><IT>+K<SUB>r</SUB>×</IT><FENCE><FR><NU>V<SUB>SA,max</SUB></NU><DE>V<SUB>SA</SUB></DE></FR></FENCE><SUP>2</SUP> (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
P<SUB>A</SUB><IT>=</IT>P<SUB>EL</SUB><IT>+R</IT><SUB>LT</SUB><A><AC>V</AC><AC>˙</AC></A><SUB>A</SUB><IT>+</IT>P<SUB>PL</SUB> (17)

P<SUB>C</SUB><IT>=</IT>P<SUB>TM</SUB><IT>+</IT>P<SUB>PL</SUB> (18)

P<SUB>PL</SUB><IT>=</IT>P<SUB>CW</SUB><IT>+</IT>P<SUB>mus</SUB> (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
<A><AC>Q</AC><AC>˙</AC></A><SUB>CA</SUB><IT>=</IT><FR><NU>P<SUB>C</SUB><IT>−</IT>P<SUB>A</SUB></NU><DE><IT>R</IT><SUB>S</SUB></DE></FR> (20)

<A><AC>Q</AC><AC>˙</AC></A><SUB>DC</SUB><IT>=</IT><FR><NU>P<SUB>D</SUB><IT>−</IT>P<SUB>C</SUB></NU><DE><IT>R</IT><SUB>C</SUB></DE></FR> (21)

<A><AC>Q</AC><AC>˙</AC></A><SUB>CA</SUB><IT>=</IT><FR><NU>P<SUB>atm</SUB><IT>−</IT>P<SUB>D</SUB></NU><DE><IT>R</IT><SUB>uaw</SUB></DE></FR><IT>=</IT><A><AC>Q</AC><AC>˙</AC></A><SUB>ED</SUB> (22)
As such, the rate of the volume changes in the airway may be written as follows
<A><AC>V</AC><AC>˙</AC></A><SUB>C</SUB><IT>=</IT><A><AC>Q</AC><AC>˙</AC></A><SUB>DC</SUB><IT>−</IT><A><AC>Q</AC><AC>˙</AC></A><SUB>CA</SUB> (23)

<A><AC>V</AC><AC>˙</AC></A><SUB>A</SUB><IT>=</IT><A><AC>Q</AC><AC>˙</AC></A><SUB>CA</SUB><IT>−&PHgr;</IT><SUB>tot</SUB> (24)
where Phi 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
<FR><NU>dP<SUB><IT>D<SUB>i</SUB></IT></SUB></NU><DE>d<IT>t</IT></DE></FR><IT>=</IT><FR><NU>1</NU><DE>V<SUB>D</SUB></DE></FR> (<A><AC>Q</AC><AC>˙</AC></A><SUB>ED</SUB>P<SUB>atm,<IT>i</IT></SUB><IT>−</IT><A><AC>Q</AC><AC>˙</AC></A><SUB>DC</SUB>P<SUB><IT>D<SUB>i</SUB></IT></SUB>) (25)

<FR><NU>dP<SUB><IT>C<SUB>i</SUB></IT></SUB></NU><DE>d<IT>t</IT></DE></FR><IT>=</IT><FR><NU>1</NU><DE>V<SUB>C</SUB></DE></FR> <FENCE><A><AC>Q</AC><AC>˙</AC></A><SUB>DC</SUB>P<SUB><IT>D<SUB>i</SUB></IT></SUB><IT>−</IT><A><AC>Q</AC><AC>˙</AC></A><SUB>CA</SUB>P<SUB><IT>C<SUB>i</SUB></IT></SUB><IT>−</IT>P<SUB><IT>C<SUB>i</SUB></IT></SUB> <FR><NU>dV<SUB>C</SUB></NU><DE>d<IT>t</IT></DE></FR></FENCE> (26)

<FR><NU>dP<SUB><IT>A<SUB>i</SUB></IT></SUB></NU><DE>d<IT>t</IT></DE></FR><IT>=</IT><FR><NU>1</NU><DE>V<SUB>A</SUB></DE></FR> <FENCE><A><AC>Q</AC><AC>˙</AC></A><SUB>CA</SUB>P<SUB><IT>C<SUB>i</SUB></IT></SUB><IT>−</IT>P<SUB><IT>A<SUB>i</SUB></IT></SUB> <FR><NU>dV<SUB>A</SUB></NU><DE>d<IT>t</IT></DE></FR><IT>−</IT><LIM><OP>∑</OP><LL><IT>j=</IT>1</LL><UL><IT>N</IT><SUB>seg</SUB></UL></LIM> <FR><NU>D<SUB>L<SUB><IT>i</IT></SUB></SUB>[P<SUB><IT>A<SUB>i</SUB></IT></SUB><IT>−</IT>P<SUP>(<IT>j</IT>)</SUP><SUB>b<SUB><IT>i</IT></SUB></SUB>]<IT>&Dgr;</IT>V<SUP><IT>j</IT></SUP><SUB>PC</SUB></NU><DE>V<SUB>PC</SUB></DE></FR></FENCE> (27)
and expiration can be described as follows
<FR><NU>dP<SUB><IT>D<SUB>i</SUB></IT></SUB></NU><DE>d<IT>t</IT></DE></FR><IT>=</IT><FR><NU>1</NU><DE>V<SUB>D</SUB></DE></FR> (<A><AC>Q</AC><AC>˙</AC></A><SUB>ED</SUB>P<SUB><IT>D<SUB>i</SUB></IT></SUB><IT>−</IT><A><AC>Q</AC><AC>˙</AC></A><SUB>DC</SUB>P<SUB><IT>C<SUB>i</SUB></IT></SUB>) (28)

<FR><NU>dP<SUB><IT>C<SUB>i</SUB></IT></SUB></NU><DE>d<IT>t</IT></DE></FR><IT>=</IT><FR><NU>1</NU><DE>V<SUB>C</SUB></DE></FR> <FENCE><A><AC>Q</AC><AC>˙</AC></A><SUB>DC</SUB>P<SUB><IT>C<SUB>i</SUB></IT></SUB><IT>−</IT><A><AC>Q</AC><AC>˙</AC></A><SUB>CA</SUB>P<SUB><IT>A<SUB>i</SUB></IT></SUB><IT>−</IT>P<SUB><IT>C<SUB>i</SUB></IT></SUB> <FR><NU>dV<SUB>C</SUB></NU><DE>d<IT>t</IT></DE></FR></FENCE> (29)

<FR><NU>dP<SUB><IT>A<SUB>i</SUB></IT></SUB></NU><DE>d<IT>t</IT></DE></FR><IT>=</IT><FR><NU>1</NU><DE>V<SUB>A</SUB></DE></FR> <FENCE><A><AC>Q</AC><AC>˙</AC></A><SUB>CA</SUB>P<SUB><IT>A<SUB>i</SUB></IT></SUB><IT>−</IT>P<SUB><IT>A<SUB>i</SUB></IT></SUB> <FR><NU>dV<SUB>A</SUB></NU><DE>d<IT>t</IT></DE></FR><IT>−</IT><LIM><OP>∑</OP><LL><IT>j=</IT>1</LL><UL><IT>N</IT><SUB>seg</SUB></UL></LIM> <FR><NU>D<SUB>L<SUB><IT>i</IT></SUB></SUB>[P<SUB><IT>A<SUB>i</SUB></IT></SUB><IT>−</IT>P<SUP>(<IT>j</IT>)</SUP><SUB>b<SUB><IT>i</IT></SUB></SUB>]<IT>&Dgr;</IT>V<SUP><IT>j</IT></SUP><SUB>PC</SUB></NU><DE>V<SUB>PC</SUB></DE></FR></FENCE> (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<UP><SUB><IT>b</IT><SUB><IT>i</IT></SUB></SUB><SUP>(<IT>j</IT>)</SUP></UP> represents the partial pressure of gas species i in the jth capillary segment, and Delta V<UP><SUB>PC</SUB><SUP>(<IT>j</IT>)</SUP></UP> denotes the blood volume contained in the jth capillary segment.

The total flux rate (Phi tot; in ml/s) of all gaseous species across the alveolar membrane can be expressed as follows
&PHgr;<SUB>tot</SUB><IT>=</IT><LIM><OP>∑</OP><LL><IT>i=</IT>1</LL><UL>3</UL></LIM> <LIM><OP>∑</OP><LL><IT>j=</IT>1</LL><UL><IT>N</IT><SUB>seg</SUB></UL></LIM> <FR><NU>D<SUB>L<SUB><IT>i</IT></SUB></SUB>[P<SUB><IT>A<SUB>i</SUB></IT></SUB><IT>−</IT>P<SUP>(<IT>j</IT>)</SUP><SUB>b<SUB><IT>i</IT></SUB></SUB>]<IT>&Dgr;</IT>V<SUP><IT>j</IT></SUP><SUB>PC</SUB></NU><DE>V<SUB>PC</SUB></DE></FR> (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
<FR><NU>∂C<SUP>(<IT>j</IT>)</SUP><SUB>b<SUB><IT>i</IT></SUB></SUB></NU><DE><IT>∂t</IT></DE></FR><IT>=</IT>−<FR><NU><IT>∂v</IT><SUP>(<IT>j</IT>)</SUP><SUB>Z<SUB>b</SUB></SUB>C<SUP>(<IT>j</IT>)</SUP><SUB>b<SUB><IT>i</IT></SUB></SUB></NU><DE><IT>∂z</IT></DE></FR><IT>+</IT><FR><NU>D<SUB>L<SUB><IT>i</IT></SUB></SUB>[P<SUB><IT>A<SUB>i</SUB></IT></SUB><IT>−</IT>P<SUP>(<IT>j</IT>)</SUP><SUB>b<SUB><IT>i</IT></SUB></SUB>]</NU><DE>V<SUB>PC</SUB></DE></FR> (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
D<SUB>L,O<SUB>2</SUB></SUB><IT>=</IT><RAD><RCD><FR><NU>V<SUB>PC</SUB></NU><DE>V<SUB>PC,max</SUB></DE></FR></RCD></RAD> (33)

<IT>×</IT>(0.397<IT>+</IT>0.0085 P<SC>o</SC><SUB>2</SUB><IT>−</IT>0.00013 P<SC>o</SC><SUP>2</SUP><SUB>2</SUB><IT>+</IT>5.1<IT>×</IT>10<SUP>−7</SUP> P<SC>o</SC><SUP>2</SUP><SUB>2</SUB>)

D<SUB>L,CO<SUB>2</SUB></SUB><IT>=</IT><RAD><RCD><FR><NU>V<SUB>PC</SUB></NU><DE>V<SUB>PC,max</SUB></DE></FR></RCD></RAD><IT>×</IT>16.67 (34)

D<SUB>L,N<SUB>2</SUB></SUB><IT>=</IT><RAD><RCD><FR><NU>V<SUB>PC</SUB></NU><DE>V<SUB>PC,max</SUB></DE></FR></RCD></RAD><IT>×</IT>0.25 (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
R<SUB>PC</SUB>(V<SUB>A</SUB>)<IT>=R</IT><SUB>PC,0</SUB><FENCE><FR><NU>V<SUB>A</SUB></NU><DE>V<SUB>A,max</SUB></DE></FR></FENCE><SUP>2</SUP> (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
TOP
ABSTRACT
INTRODUCTION
MODEL DEVELOPMENT
COMPUTATIONAL ASPECTS
DISCUSSION
REFERENCES

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.

                              
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Table 8.   Initial conditions used in the cardiovascular model

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.

                              
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