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

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

TOP ABSTRACT INTRODUCTION MODEL DEVELOPMENT COMPUTATIONAL ASPECTS DISCUSSION REFERENCES

Ventricular Model

Circulatory Model

View larger version (25K): [in this window] [in a new window]   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 P_SV and V_SV are the transmural pressure and luminal volume of systemic veins, K_v is a scaling factor (in mmHg), and V_max 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 P_VC and V_VC denote the transmural pressure and luminal volume of the vena cava, respectively, V₀ is the unstressed volume, and V_min 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).

(4)

1

1

Arterial Baroreflex Control

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 (P_Ao)], 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].

View larger version (37K): [in this window] [in a new window]   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 N_x,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).

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, F_Hr,V and F_Hr,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).

View larger version (16K): [in this window] [in a new window]   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.

(9)

(10)

(11)

(12)

(13)

(14)

1

(15)

1

(16)

View larger version (13K): [in this window] [in a new window]   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

View larger version (22K): [in this window] [in a new window]   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 (P_PL). Figure 5 indicates that this pressure is generated by the respiratory muscles (P_mus) and the recoil pressure of the chest wall (P_CW). Measured P_PL 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 [R_C(V_C)] and a nonlinear P-V relationship [P_TM(V_C)], where V_C is the collapsible segment volume (Fig. 5). Pressure in the lumen of the midairway segment of the model is denoted as P_C, and the transmural pressure across the wall is denoted as P_TM. P_A is the alveolar pressure and P_EL is the lung elastic recoil pressure. Small airways resistance (R_S) is characterized as a nonlinear function of the alveolar volume (V_A).

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

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, P_Di, P_Ci, and P_Ai are partial pressures of gas species i (O₂ or CO₂) in the upper, middle, and small airways, respectively; P_atm,i is the partial pressure of the gas species i in the atmosphere; and V_PC 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. N_seg 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 · min¹ · mmH₂O¹ to ml STPD · s¹ · mmHg¹). These formulas are as follows

(33)

(34)

(35)

where V_PC,max is the maximal blood volume in the pulmonary capillaries.

Cardiopulmonary Interactions

How P_PL mediates cardiac and vascular mechanics. P_PL 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 P_PL.

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 (R_PC) changes with alveolar volume (V_A) is as follows

(36)

Here, R_PC,0 is a constant chosen to set the magnitude of capillary resistance and V_A,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.

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 P_PL 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.