A human cardiopulmonary system model appliedto the analysis of the Valsalva maneuver
K. LU,1 J. W. CLARK, JR.,1 F. H. GHORBEL,1 D. L. WARE,2 AND A. BIDANI2
1Dynamical Systems Group, Rice University, Houston 77005; and 2Department of InternalMedicine, University of Texas Medical Branch, Galveston, Texas 77555Received 12 March 2001; accepted in final form 13 August 2001
Lu, K., J. W. Clark, Jr., F. H. Ghorbel, D. L. Ware,and A. Bidani. A human cardiopulmonary system modelapplied to the analysis of the Valsalva maneuver. Am JPhysiol Heart Circ Physiol 281: H2661–H2679, 2001.—Pre-vious models combining the human cardiovascular and pulmo-nary systems have not addressed their strong dynamic inter-action. They are primarily cardiovascular or pulmonary intheir orientation and do not permit a full exploration of howthe combined cardiopulmonary system responds to large am-plitude forcing (e.g., by the Valsalva maneuver). To addressthis issue, we developed a new model that represents theimportant components of the cardiopulmonary system andtheir coupled interaction. Included in the model are descrip-tions of atrial and ventricular mechanics, hemodynamics ofthe systemic and pulmonic circulations, baroreflex control ofarterial pressure, airway and lung mechanics, and gas trans-port at the alveolar-capillary membrane. Parameters of thiscombined model were adjusted to fit nominal data, yieldingaccurate and realistic pressure, volume, and flow waveforms.With the same set of parameters, the nominal model pre-dicted the hemodynamic responses to the markedly increasedintrathoracic (pleural) pressures during the Valsalva maneu-ver. In summary, this model accurately represents the car-diopulmonary 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 describevarious physiological states and help investigators to betterunderstand the biophysics of cardiopulmonary disease.
THE DIAGNOSIS AND TREATMENT of cardiopulmonary dis-ease may be improved by using mathematical modelsof the cardiovascular and pulmonary systems. Withthis in mind, we developed a model of the cardiopul-monary system of the normal human subject that notonly represents the system accurately but also predictsits response to a variety of commonly used diagnosticprocedures. To our knowledge, this is the first exampleof a truly integrative model of the cardiopulmonarysystem.
Recently, our group (5, 25) developed a multicom-partment model of the canine circulation. We have nowmodified and extended this cardiovascular model to
encompass human heart mechanics, a circulatory loop,baroreflex control of arterial pressure, airway mechan-ics, and gas transport at the alveolar-capillary mem-brane.
Distributed circulatory models of the systemic andpulmonic circulations have been developed (1, 3, 12,37). However, the mechanics of the lung and airwayswere not detailed in any of these, and the heart wasmodeled rather simply. The gas exchange at the alve-olar-capillary membrane (an obvious link between car-diovascular and pulmonary system) was consideredonly in the model of Hardy et al. (12). Of these models,baroreflex control of arterial pressure was includedonly in the work of Ursino et al. (37).
Distributed airway mechanics models [e.g., Elad etal. (7) and Lambert et al. (16)] can be too complex for acombined cardiopulmonary model, making lumpedlower-order compartment models [such as that ofLutchen et al. (19)] preferred. The lumped compart-ment model we (18) developed describes ventilation,perfusion, mechanics, and gas transport over the fullrange of normal lung volumes. A modified version ofthis model was used in the current study.
Our heart model was based on our previous work indogs (5, 25). The parameters of that model were ad-justed to better fit the flow, volume, and temporalrelationships of the human cardiac cycle. Similar ad-justments were made in the systemic and pulmoniccomponent models of the canine circulatory loop (25).The resulting model is of intermediate complexity andsimulates pressure, volume, and flow distribution ofthe human subject in the supine position.
To better simulate the cardiovascular response toperturbation, we added nonlinear descriptions of thevenous system and a description of how the barore-flexes influence heart rate, myocardial contractility,and vasomotor tone. We based our baroreceptor controlmodel on the work of Spickler et al. (35) and Wesselinget al. (38) and included descriptions of both parasym-pathetic (vagal) and sympathetic pathways.
Our new lung model combines models previouslydeveloped by our group, namely, an airway mechanicsmodel [from Athanasiades et al. (2)] and a gas ex-
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: [email protected]).
The costs of publication of this article were defrayed in part by thepayment of page charges. The article must therefore be herebymarked ‘‘advertisement’’ in accordance with 18 U.S.C. Section 1734solely to indicate this fact.
Am J Physiol Heart Circ Physiol281: H2661–H2679, 2001.
change model [modified from Liu et al. (18)]. It char-acterizes the nonlinear resistive-compliant propertiesof the airways and the nonlinear pressure-volumecharacteristics of the lung. A distributed pulmonarycirculatory model containing 35 contiguous capillarysegments characterizes gas exchange at the alveolar-capillary membrane and yields good fits to expired O2and CO2 data measured at the mouth.
This integrated cardiopulmonary model describesheart-lung interactions and the timing of baroreflexchanges in heart rate, myocardial contractility, andvasomotor tone. Its parameters fit available cardiovas-cular and pulmonary data obtained during tidalbreathing and can predict the responses to large-scaleperturbations in pleural pressure, such as those occur-ring in the forced vital capacity and Valsalva maneu-vers.
Glossary
Activation functions
e(t) Time-varying activation functionea(t) Activation function of the atriumev(t) Activation function of the ventricle
Airflows
Q̇CA Airflow from collapsible airways to al-veolar region
Q̇DC Airflow from upper supported airway tocollapsible airway
Q̇ED Airflow from environment to upper sup-ported airway
VED End-diastolic volumeVES End-systolic volumeVbi
(j) Blood volume contained in the jth cap-illary
VL Lung volumeVLV Left ventricular volume
Vmax Maximal volumeVmin Minimum volumeVPC Blood volume of pulmonary capillaries
VPC,max Maximal blood volume of pulmonarycapillaries
VSA Blood volume of systemic arteriesVSA,0 Minimal volume of systemic arteries
VSA,max Maximal lumen volume of systemic ar-teries
VSV Luminal volume of systemic veinsVVC Luminal volume of the vena cavaVVE 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 atime-varying elastance function (Tables 1–3). The elastancefunction 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 activa-tion function [e(t)]. The activation function e(t) consists of aseries of Gaussian curves and serves to produce a smooth
transition between the EDPVR and the ESPVR. A detaileddescription of the ventricular model can be found in Ref. 5.
Circulatory Model
The general framework of our human circulatory loopmodel (Fig. 1 and Table 4) is similar to that of Olansen et al.(25) with certain extensions and modifications. We included1) nonlinear P-V relationships to describe the peripheralvenous system, 2) a nonlinear collapsible description of theP-V relationship for the vena cava, and 3) separate descrip-tions of baroreceptor-mediated control of heart rate, myocar-dial contractility, and vasomotor tone.
Nonlinear P-V Characteristics of Systemic Veins and theVena Cava
Systemic veins. The nonlinear P-V relationship of veinshas been modeled previously by Kresch (15) and by Snyderand Rideout (34). As volume increases, the vessels stiffen.The resulting P-V curve can be represented as follows
PSV 5 2Kv 3 log SVmax
VSV2 0.99D (1)
where PSV and VSV are the transmural pressure and luminalvolume of systemic veins, Kv is a scaling factor (in mmHg),and Vmax is the maximal volume (in ml) of the lumpedsystemic veins (Table 5).
Vena cava. Under some conditions, the vena cava maycollapse. For example, when pleural pressure is greater thanthe luminal pressure of the vena cava, total caval volumedecreases substantially, and the resistance to flow is in-creased. To account for this, we described the P-V relation-ship as follows
if VVC $ V0, then PVC 5 D1 1 K1 3 ~VVC 2 V0! (2)
if VVC , V0, then PVC 5 D2 1 K2 3 e(VVC/Vmin) (3)
where PVC and VVC denote the transmural pressure andluminal volume of the vena cava, respectively, V0 is theunstressed volume, and Vmin is the minimum volume. Weadjusted the parameters K1, K2, D1, and D2 to produce P-Vcurves similar to those used in the human venous model ofSnyder and Rideout (34).
Table 1. Parameter values of the ventricular model
Parameters LVF RVF PCD
EES, mmHg/ml 4.3 0.6 N/AP0, mmHg 1.7 0.67 0.5V0, ml 25 25 200VD, ml 40 40 N/Al, ml21 0.015 0.015 0.005
For abbreviations, see Glossary and text.
Table 2. Parameter values of the atrial model
Parameters LA RA
EES, mmHg/ml 0.2 0.2P0, mmHg 0.5 0.5V0, ml 20 20VD, ml 20 20l, ml21 0.025 0.025
The resistance of the vena cava (RVC) is a nonlinear func-tion of its luminal blood volume (VVC) according to the fol-lowing equation
RVC 5 KR 3 SVmax
VVCD2
1 R0 (4)
where KR is a scaling factor (in mmHg zs zml21), Vmax denotesthe maximum volume, and R0 is an offset parameter (inmmHg zs zml21) (Table 5).
Arterial Baroreflex Control
Our previous study (25) did not consider baroreflex controlof heart rate, myocardial contractility, and vasomotor tone.We have now included lumped characterizations of thebaroreceptors 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 blocksthat represent the baroreceptor, the central nervous system(CNS), the efferent pathways, and the effector organ. Theinput to the baroreceptor element (BR) is central arterialpressure [aortic arch pressure (PAo)], and the output [N(t)] isthe instantaneous firing frequency of the BR. FollowingSpickler et al. (35), we characterized the input-output rela-tionship in terms of the following transfer function
N~s!
PAo~s!5
K 3 ~1 1 0.036s!
~1 1 0.0018s!~1 1 as!where a , 0.0018 (5)
Fig. 1. A hydraulic equivalent representation of theclosed-loop circulatory model. For abbreviations, seeGlossary.
Table 3. Parameter values for the activation function
Parameters
ev(t)
ea(t)i 5 1 i 5 2 i 5 3 i 5 4
Ai 0.3 0.35 0.5 0.55 0.9Bi, s2 0.045 0.035 0.037 0.036 0.018Ci, s 0.175 0.23 0.275 0.3 0.025
Parameter Ai is dimensionless. For abbreviations, see Glossaryand text.
Table 4. Nominal parameter values in the systemicand pulmonic circulations
The corresponding differential equation is as follows
0.0018ad2N~t!
dt2 1 ~0.0018 1 a!dN~t!
dt1 N~t!
5 KFPAo~t! 1 0.036KdPAo~t!
dt G (6)
where K is the gain and a is a time constant [35].Central nervous system. The medullary cardiovascular con-
trol center is modeled in terms of four noninteracting path-ways, each characterized by filtering, gain, and a delay as perthe 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 myocardialcontractility and vasomotor tone. The fast vagal pathway hasa 0.2-s delay, whereas each sympathetic pathway has a 3-sdelay.
Efferent pathways. We described each efferent pathwayaccording to the following generic equation in normalizedform (Table 6)
Fx~t! 5 ax 1bx
etx@Nx~t! 2 Nx,0# 1 1.0(7)
The generic parameter x represents heart rate, contractility,or vasomotor tone. The parameters tx and Nx,0 were fitted tothe representative data. This equation provides a sigmoidalinput-output relationship (threshold and saturation) be-tween 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 vagaldischarge frequency, tx in the vagal efferent pathway isnegative, producing a monotonically increasing input-outputrelationship for the linear part of the curve (Fig. 2). Sympa-thetic pathways use positive tx values, because BR and sym-pathetic discharge frequencies change in opposite directions.Figure 2 shows that the discharge frequency (Fx) of eachefferent pathway inputs to the final block of the diagram,which contains characterization of the input-output responseof the effector organ itself (the heart or vessel).
Effector organs. Heart rate is controlled by vagal andsympathetic neural activity and has been characterized bySunagawa as a three-dimensional response surface [36]. Wedeveloped the following equation to characterize the humanheart rate response surface to vagal and sympathetic input(Table 7)
HR 5 h1 1 h2 3 FHr,S 2 h3 3 FHr,S2 2 h4 3 FHr,V
1 h5 3 FHr,V2 2 h6 3 FHr,V 3 FHr,S
(8)
where HR (in beats/min) represents heart rate, FHr,V andFHr,S are the normalized vagal and sympathetic frequencies,and h1–h6 are constants. This formula generates a normal-ized heart rate response surface analogous to that of Su-nagawa et al. (36).
Table 5. Parameter values for nonlinear P-V curves ofsystemic veins and the vena cava
Parameters Values
Systemic Veins
Kv, mmHg 40Vmax, ml 3,500
Vena Cava
D1, mmHg 0.0D2, mmHg 25.0K1, mmHg 0.15K2, mmHg 0.4KR, mmHg zs zml21 0.001R0, mmHg zs zml21 0.025V0, ml 130Vmax, ml 350Vmin, ml 50
For abbreviations, see Glossary and text.
Fig. 2. Block diagram of baroreflex control of arterialpressure. A fast vagal (dashed arrow) pathway and 3slow sympathetic pathways are included to controlheart rate, myocardial contractility, and vasomotortone. The overall control scheme is based on the mod-eling concept of Wesseling et al. (38). For abbreviations,see text.
In our study, the heart period (calculated as 60/HR, in s) isexplicitly determined by the vagal-sympathetic mechanism ac-cording to Eq. 8, and the systolic period is mediated by thesympathetic frequency (Fig. 3). The diastolic filling time is thedifference between the two and is thus controlled indirectly.
Greater sympathetic tone increases myocardial elastanceand shortens ventricular systole. Therefore, we modified theventricular activation function to describe the change inventricular elastance [e(t)] as a function of sympathetic effer-ent discharge frequency (Fcon) (see Fig. 3).
A rise in Fcon increases maximum elastance and shortens thesystolic period. The expression for the end-systolic P-V relation-ship [PES(V)] becomes (notation from Ref. 25 and Table 1)
PES(V) 5 a~Fcon! 3 EES 3 ~V 2 VD) (9)
and the activation function [ev(t)] becomes
ev~t, Fcon! ; (i 5 1
n
Aie 212Sb~Fcon! 3 t 2 Ci
BiD2
(10)
where
a~Fcon! 5 amin 1 Ka 3 Fcon (11)
b~Fcon! 5 bmin 1 Kb 3 Fcon (12)
Here, amin and bmin are dimensionless constants represent-ing the minimum values of the functions a and b, respec-tively, and Ka and Kb are scaling parameters.
Arteries and arterioles are the major resistance vessels.When their smooth muscle constricts, lumen diameter de-creases, axial resistance to flow increases, and the musclewall stiffens. Therefore, a change in vasomotor tone involvesa change in both axial resistance and in wall compliance. Wetransformed the passive and fully activated length-tensionrelationships previously described by Gore and Davis (10) intoan equivalent P-V relationship for a cylindrical vessel. Figure 4shows the passive and fully activated P-V curves used in ourmodel, which are represented as follows. Fully activated
p represent the arterial pressures in thefully activated and passive states, respectively, VSA is theblood volume contained in systemic arteries, and VSA,0 (inml) is the minimal volume. We assume VSA $ VSA,0 in Eqs. 13and 14. Kc, Kp1, and Kp2 (in mmHg) are constant scalingparameters, D0 (in ml) is a volume parameter, and tp (inml21) is constant. During sympathetic stimulation, the com-pliance of the vessel is characterized by Eq. 13; when the
sympathetic tone is abolished, the compliance of vessel wallis described by Eq. 14. The normalized sympathetic efferentfrequency (Fvaso) serves as a scaling factor for the transitionbetween these states
PSA(VSA) 5 Fvaso 3 PSAa (VSA) 1 ~1 2 Fvaso! 3 PSA
p (VSA) (15)
Axial resistance is also affected by sympathetic activity.Resistance (RSA; in mmHg zs zml21) and sympathetic efferentfrequency (Fvaso) are related by
RSA 5 Kr 3 e4 3 Fvaso 1 Kr 3 SVSA,max
VSAD2
(16)
The first term is regulated by the sympathetic frequency andthe second term is a function of lumen volume (VSA). VSA,max
Fig. 3. Model representation of the sympathetically regulated acti-vation function e(t). Four different levels of contractility correspond-ing to different sympathetic efferent frequencies (Fcon) are shown.
Table 6. Parameter values for the baroreflex pathway
is the maximal lumen volume and Kr (in mmHg) is a pressurescaling constant.
Airway/Lung Mechanics Model
The pulmonary portion of our cardiopulmonary model com-bines two models previously developed. One focuses on air-way/lung mechanics (2) and the other focuses on gas ex-change (18). Figure 5 shows an equivalent pneumatic circuitmodel of the airways and lung of the normal human. Thelung mechanics model (2) includes nonlinear characteriza-tions of airway resistance, airway and chest wall compliance,and lung tissue viscoelasticity. This particular model (2) hasalso been used in a related context to analyze the “work ofbreathing” during clinical breathing maneuvers [see Atha-nasiades et al. (2) for details].
In the supine human, the lungs and their airways aresubject to the same time-varying intrathoracic pleural pres-sure (PPL). Figure 5 indicates that this pressure is generated
by the respiratory muscles (Pmus) and the recoil pressure of thechest wall (PCW). Measured PPL is also the driving pressure forour airway mechanics model. The upper airway is assumedrigid and is characterized by a nonlinear flow-dependent resis-tor (Rohrer resistor). The midairways are assumed collapsibleand are characterized by a nonlinear volume-dependent resis-tance [RC(VC)] and a nonlinear P-V relationship [PTM(VC)],where VC is the collapsible segment volume (Fig. 5). Pressure inthe lumen of the midairway segment of the model is denoted asPC, and the transmural pressure across the wall is denoted asPTM. PA is the alveolar pressure and PEL is the lung elasticrecoil pressure. Small airways resistance (RS) is characterizedas a nonlinear function of the alveolar volume (VA).
From an analysis of the pneumatic circuit according toNewton’s first law
PA 5 PEL 1 RLTV̇A 1 PPL (17)
PC 5 PTM 1 PPL (18)
PPL 5 PCW 1 Pmus (19)
The component air flows (in ml/s) in the airway system arecomputed according to the equations below, which are de-rived from the continuity equation applied to each node of thepneumatic network. The resulting differential equations areas follows
Q̇CA 5PC 2 PA
RS(20)
Q̇DC 5PD 2 PC
RC(21)
Q̇CA 5Patm 2 PD
Ruaw5 Q̇ED (22)
As such, the rate of the volume changes in the airway maybe written as follows
V̇C 5 Q̇DC 2 Q̇CA (23)
V̇A 5 Q̇CA 2 Ftot (24)
where Ftot denotes the total gas flux rate (in ml/s) of allgaseous species across the alveolar-capillary membrane, asgiven by Eq. 31.
Fig. 4. Active and passive P-V curves of systemic arteries. PSA andVSA represent the pressure and volume in the systemic arteries. Fvaso
is the normalized sympathetic discharge frequency controlling thevasomotor tone.
Fig. 5. Airway/lung mechanics model. A:components of airway mechanics, pulmo-nary circulation, and gas exchange. B: equiv-alent pneumatic circuit representation ofairway/lung mechanics and gas exchange[modified from Athanasiades et al. (2)]. Forabbreviations, see Glossary and text.
Gas exchange between air and blood occurs across thealveolar-capillary membrane. For modeling purposes, we as-sumed 1) inspired air is instantly warmed to body tempera-ture and saturated with water vapor, 2) gaseous contentobeys the ideal gas law, 3) blood is characterized as a uniformhomogeneous medium, and 4) reactions between the gaseousspecies and blood are assumed to equilibrate instanta-neously. The empirical O2 and CO2 dissociation curves relatethe content of each species with their corresponding partialpressures in blood. The diffusing capacity for the ith gaseousspecies (DLi) characterizes its diffusion across the alveolar-capillary membrane. O2 is taken up by the blood, CO2 isremoved, and N2 diffuses either way depending on the direc-tion of their instantaneous partial pressure gradients.
The species conservation law is applied to inspiration andexpiration. Inspiration can be described as follows
dPDi
dt5
1VD
~Q̇EDPatm,i 2 Q̇DCPDi! (25)
dPCi
dt5
1VC
SQ̇DCPDi2 Q̇CAPCi
2 PCi
dVC
dt D (26)
dPAi
dt5
1VA 5Q̇CAPCi
2 PAi
dVA
dt2 (
j 5 1
Nseg DLi@PAi
2 Pbi
~j!#DVPCj
VPC 6 (27)
and expiration can be described as follows
dPDi
dt5
1VD
~Q̇EDPDi2 Q̇DCPCi
! (28)
dPCi
dt5
1VC
SQ̇DCPCi2 Q̇CAPAi
2 PCi
dVC
dt D (29)
dPAi
dt5
1VA 5Q̇CAPAi
2 PAi
dVA
dt2 (
j 5 1
Nseg DLi@PAi
2 Pbi
~j!#DVPCj
VPC 6 (30)
Here, PDi, PCi, and PAi are partial pressures of gas species i(O2 or CO2) in the upper, middle, and small airways, respec-tively; Patm,i is the partial pressure of the gas species i in theatmosphere; and VPC is the blood volume contained in pul-monary capillary. N2 partial pressure in the airways wasobtained by subtracting the partial pressures of O2, CO2, andH2O from the total airway pressure. Nseg is the number ofcapillary segments. In the gas exchange model, the lumpedpulmonary capillary was divided into 35 segments, as in Liuet al. (18). Pbi
( j ) represents the partial pressure of gas speciesi in the jth capillary segment, and DVPC
(j) denotes the bloodvolume contained in the jth capillary segment.
The total flux rate (Ftot; in ml/s) of all gaseous speciesacross the alveolar membrane can be expressed as follows
Ftot 5 (i 5 1
3
(j 5 1
Nseg DLi@PAi
2 Pbi
~j!#DVPCj
VPC(31)
Here, i 5 1, 2, or 3 and represents the three gaseous species(O2, CO2, and N2).
Species molar balance was employed to describe the dy-namics of the species blood concentration in each segment.The corresponding equation for gas species i in the jth cap-illary segment is given by
]Cbi
~j!
]t5 2
]vZb
~j!Cbi
~j!
]z1
DLi@PAi
2 Pbi
~j!#
VPC(32)
The formula of the lung diffusion capacity for each gaseousspecies was taken from Liu et al. (18) (with a change in unitsfrom ml STPD zmin21 zmmH2O21 to ml STPD zs21 zmmHg21).These formulas are as follows
where VPC,max is the maximal blood volume in the pulmonarycapillaries.
Cardiopulmonary Interactions
Any combined cardiovascular and pulmonary model mustaccount for interactions that can occur between these sys-tems. These interactions take a variety of forms and fre-quently are quite subtle. In general, to test for system inter-action, a variable in one system is perturbed and the effectson both systems are assessed. We accomplished this by usingonly perturbations in pleural pressure (PPL). The followingsections provide simple examples of this coupled interaction.
How PPL mediates cardiac and vascular mechanics. PPL
affects both intracardiac pressures and the pressures withinthe large intrathoracic vessels, but alveolar pressure has thegreatest effect on pulmonary capillaries (18, 23). Conse-quently, in our model, the capillary transmural pressure ismediated by the alveolar pressure, whereas the pressures ofthe pulmonary arteries and veins are changed by PPL.
How lung air volume changes lung perfusion. The pulmo-nary capillary bed forms an extensive network of vessels,which surround the alveolar region. During lung inflation,these vessels are stretched and constricted by the expandingalveolar volume. This increases capillary resistance and re-duces blood flow, thus facilitating gas exchange. The relation-ship we used to describe the capillary resistance (RPC)changes with alveolar volume (VA) is as follows
RPC(VA) 5 RPC,0S VA
VA,maxD2
(36)
Here, RPC,0 is a constant chosen to set the magnitude ofcapillary resistance and VA,max represents the maximumalveolar volume.
COMPUTATIONAL ASPECTS
To summarize, we modified and combined previouscardiac and pulmonary models developed by our groupto form a cardiopulmonary model of the normal human(Tables 8–10). The pulmonary models employed (2, 18)were originally developed as human models and wereverified using data obtained from normal human sub-jects. However, the cardiovascular model used as abasis for designing our human circulatory model (25)was validated using data from the dog. To develop thehuman model, we first scaled up our canine model to
provide an initial model of the normal human cardio-vascular system. This has been done by others (see,e.g., Ref. 17). Because human and canine blood pres-sures and blood velocities are similar, scaling factorsare related closely to the ratios of blood volume. (Bloodvolume is directly related to body weight and bodysurface area.) In a second phase, we manually adjustedthe parameters of the initial human circulatory modelto yield a reasonable fit to typical human pressure dataand hemodynamic indexes available in the literature.
First, we determined that the cardiac output of a70-kg human is ;2.5 times that of a 25-kg dog. Becausethe mean systemic arterial pressures in the humanand dog are similar, we calculated a set of humancardiovascular parameters by decreasing all the resis-
tive and inertial parameters of the canine model by 2.5and by similarly increasing the compliant parameters.This scaling provided a reasonable initial representa-tion of the human cardiovascular system, althoughadditional adjustments were necessary for better re-gional representations of typical hemodynamic wave-forms.
The representations used for certain elements of thecanine and human circulatory models were different.Specifically, the linear representations of venous com-pliance in the canine model were replaced by nonlinearP-V relationships in the human model. Nonlinear ac-tive and passive P-V curves were also incorporated todescribe arterial compliance.
The structure of the human circulatory model alsodiffers in that several parallel circulation pathwayswere added. In the pulmonary circulation, the averagepulmonary shunt flow is 2% of the pulmonary bloodflow, whereas in the systemic circulation, the meancoronary and cerebral flows are set to 6% and 14%,respectively, of the cardiac output. The nominal distri-bution of blood volume in the pulmonic and systemiccirculations are set at a level of 8.8% and 84%, respec-tively. The remaining 7.2% of the blood is contained inthe heart. These figures agree with the results shownin Ref. 24 (p. 30 and 124).
We approximated the first-order spatial derivative inEq. 32 using a four-point biased quadratic interpola-tion formula (31) and eliminated fictitious points at theentrance of the capillary bed using constant inlet con-ditions (i.e., partial pressures of 40 mmHg for O2 and46 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 determineddirectly from the experimental data. The model beginsat end expiration, when there is no flow and air volumein 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 differentialequations and 116 parameters associated with its com-ponent models. In all, 149 outputs were generatedsimultaneously. Table 11 shows the distribution of thestate variables and model parameters in the combinedcardiopulmonary model.
The model was programmed in C programming lan-guage and solved using the variable step-size Runge-Kutta-Merson algorithm, with a maximum time stepsize of 2 3 1022 s and an error tolerance of 1 3 1026.On average, it takes 20 min of CPU time on a Pentium
Table 8. Initial conditions used in the cardiovascularmodel
II 333-MHz machine to simulate 35 s of cardiopulmo-nary events.
Simulation Results
Hemodynamics. Figure 6 compares a model-gener-ated and human systemic pressure waveform (Fig.22.15 in Ref. 20). The left ventricular end-systolic pres-sure is 125 mmHg and the systolic duration is 0.3 s, orabout one-third of the cardiac cycle (0.8 s). The aorticroot pressure ranges from 80 to 120 mmHg. The di-crotic notch can be clearly seen in both the simulationand the data.
Figure 7 compares experimental data to the model-generated left ventricular volume and aortic and pul-monary 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 ejectionfraction of 80/150, or 0.533. The volume added as theresult of atrial systole (DV) was 30 ml, ;20% of theVED. The peak aortic flow rate was 750 ml/s and thepeak pulmonary arterial flow rate was 400–500 ml/s.The aortic flow rate has a higher peak value andshorter time span compared with the pulmonic flowrate because of the stronger contractile force andhigher afterload of the left ventricle. Numerical inte-gration of the aortic and pulmonic flow waveforms overone cycle showed that the mean values of the areaenclosed by the two waveforms were the same, al-though during individual cardiac cycles they may bedifferent from each other due to variations of intratho-racic pressure.
Table 12 compares indexes of the model-predictedand measured data. Our model predicts that at peakinspiration, the stroke volume of the left heart de-creases, whereas the stroke volume of the right heartincreases. At peak expiration, the opposite occurs.These changes agreed well with measured data (4, 9,13, 26, 28). Our model helps explain the mechanismunderlying this physiological relationship.
Right and left ventricular volumes respond to PPLbecause of both direct and series ventricular interac-tion. When PPL are negative (e.g., with inspiration), anincrease in venous return augments right ventricularfilling and stroke volume. The increased right ventric-ular filling causes the septum to encroach upon the leftventricle, because the pericardium limits total cardiacvolume. As a result, left ventricular stroke volume isreduced. Simulations that ignore this ventricular in-teraction (rigid septum) underestimate the percent fallin left ventricular stroke volume occurring during in-spiration (2.5% vs. 5% with ventricular interaction).Without pericardial constraint, there is little respira-tory variation in left ventricular stroke volume (1%).
Expiration causes a volume shift from the pulmo-nary to the systemic circulation. The blood pooling inthe systemic vascular bed then increases left ventric-ular afterload with the next inspiration. Simulationsshow that both the end-systolic transmural pressureand volume of the left ventricle are highest at earlyinspiration, consistent with increased afterload (27, 30,32). During inspiration, both decreased filling and in-creased afterload decrease left ventricular stroke vol-ume. The same mechanism explains why the left andright ventricles respond differently to elevated PPLduring expiration.
Airway Mechanics and Gas Exchange
Figure 8 depicts the airway pressures and lung vol-umes predicted by our model. During inspiration, sub-atmospheric PPL is transmitted to the alveoli, facilitat-ing air flow into the lungs. As this occurs, lung elasticrecoil increases and the alveolar and atmospheric pres-sures equalize, marking the end of inspiration and thestart of expiration. During expiration, the less negativePPL and the resulting changes in lung elastic recoilcause positive alveolar pressure, pushing air from the
Table 11. Distribution of state variables, parameters,and outputs in the combined humancardiopulmonary model
Number ofState
VariablesNumber ofParameters
Number ofOutputs
Cardiovascular system 32 52 76Airway mechanics 4 10 7Gas exchange 36 24 50Baroreflex control 5 30 16Total 77 116 149
Fig. 6. Model-predicted systemic pressurewaveforms (A) compared with the text-book figure [McClintic (20); B] showingleft ventricular pressure (PLV ), aortic rootpressure (PAo), and left atrial pressure(PLA).
lungs. The inspiratory and expiratory changes in lungvolume are depicted in Fig. 8B. Here, total lung volumeis the sum of the air volumes contained in the alveoli,collapsible airways, and dead space. The model pre-dicts an average tidal volume of 500 ml and a func-tional residual capacity (FRC) of 2.2 l, which agreedwith measured values.
Inspiration fills the alveoli with O2-enriched air,whereas expiration removes CO2. Figure 9 depicts themodel-generated variation in airway gas compositionin terms of changes in the partial pressures of O2 andCO2 (PO2 and PCO2, respectively). Alveolar PCO2 andPO2 are relatively constant. Alveolar PO2 varies from 95to 105 mmHg, and alveolar PCO2 varies from 35 to40 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 inalveolar PO2 and PCO2 much smaller than those in thedead space. Expiration lowers the PO2 and raises thePCO2 in the dead space, whereas gases continuouslydiffuse across the alveolar-capillary membrane. Conse-
Fig. 7. Model-predicted volumes and flows(left) compared with reported experimentdata [Mountcastle (24); right]. A: left ven-tricular volume (VLV); B: aortic flow (Q̇Ao);C: pulmonary arterial flow (Q̇PA). For ab-breviations, see text.
Fig. 8. Model-predicted airway pressures and lung volume in normalbreathing. A: model-generated alveolar pressure (PA). PPL, experi-mental data of the pleural pressure. B: lung volume (VL). The lungvolume includes air volumes in the alveolar region, collapsible air-ways, and dead space.
Table 12. Comparison of hemodynamic indexesdrawn from model-predicted and measured data
quently, alveolar PO2 decreases and alveolar PCO2 in-creases.
Alveolar capillary gas exchange. Figure 10 depictsthe flux of O2 and CO2 at the alveolar-capillary mem-brane, which is modeled as 35 contiguous segments.For each segment, there is a gaseous flux waveformthat pulsates with capillary blood flow. Most gaseousdiffusion occurs at the initial capillary segments but
later diminishes when blood and alveolar gas contenthas equilibrated. Thus the flux rates decrease exponen-tially from the first (entrance) to the last segment(exit). We tested our model against the known changesthat occur during the FVC and Valsalva maneuvers.
Forced Vital Capacity Maneuver
The FVC maneuver is a commonly used pulmonaryfunction test. The subject fully exhales and then in-hales to total lung capacity (TLC) without pausing.Immediately, the subject exhales as rapidly as possi-ble, until airflow is no longer detected at the mouth. Weapplied the measured FVC PPL data reported in Ref. 18to our model.
Figure 11 compares the model predictions to datameasured from a human. During the rapid inspirationphase, lung volume increased to full capacity (Fig. 11A)and PPL decreased (Fig. 11B). At the beginning offorced expiration, PPL increases sharply, and lung vol-ume decreased until it reached residual volume. Thepredicted and measured data correlated nicely.
During the FVC maneuver, the expired PO2 de-creased constantly and reached a minimum value of118 mmHg. In contrast, PCO2 increased steadily untilits maximum value of 38 mmHg. Figure 12 comparesthe predicted temporal profile of PO2 and PCO2 in ex-pired air with data from Liu et al. (18). Again, themodel prediction agreed well with the measured data.
Hemodynamic changes are seldom recorded duringthe FVC maneuver, but our model can predict them.Figure 13 shows the predicted change in left and rightheart stroke volumes. As in normal respiration, the left
Fig. 9. Variations of airway gaseous partialpressure during normal breathing in the sim-ulation. A: PO2 variations in the dead spaceand alveolar regions. B: PCO2 variations inthe dead space and alveolar regions.
Fig. 10. Three-dimensional representations of the regional gaseousflux calculated at each of the 35 capillary segments. A: O2 gas flux; B:CO2 gas flux.
and right ventricles showed opposite responses duringinspiration and the early part of the forced expiration.However, after a few beats into the prolonged secondphase of forced expiration, the stroke volumes of bothventricles decreased quickly and then returned to base-line after an overshoot (Fig. 13). Stroke volume de-creases because elevated PPL decreases venous return.The recovery and the overshoot may be caused byneural factors (discussed below).
Figure 14 demonstrates a similar recovery and over-shoot in the systemic arterial pressure waveform (A)and the temporal variations in heart rate (B), vagaldischarge (C), and sympathetic discharge (D). Heartrate increases slowly during the maneuver. Vagal ef-ferent activity slows, and a burst of sympathetic activ-ity occurs later. These findings are consistent with thefaster and slower activity of the vagal and sympatheticpathways, respectively. The decreasing vagal and in-creasing sympathetic outputs correlated with the ob-
served increases in heart rate, myocardial contractil-ity, and vasomotor tone and explained the partialrecovery of arterial blood pressure that occurs duringand shortly after the maneuver.
Valsalva Maneuver
During the Valsalva maneuver, the subject forcefullyexhales against a closed glottis (or nose and mouth).The maneuver markedly elevates intrathoracic pres-sure and affects venous return, myocardial contractil-ity, vasomotor tone, and baroreflex heart rate control.It is a widely used test of baroreceptor reflexes (6).
The hemodynamic response to the Valsalva maneu-ver has four distinct phases: phase 1 (an initial in-crease in arterial pressure), phase 2 (a rapid fall inarterial pressure, followed by a partial recovery andtachycardia), phase 3 (a reduction in arterial pressureupon the sudden termination of breath holding, accom-panied by a continued tachycardia), and phase 4 (an
Fig. 11. Model predictions of lung volume variations and flow at themouth (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: measuredPPL data.
Fig. 12. Model-predicted PO2 and PCO2 (A)in the expired air at the mouth (solid lines)during the forced expiration in the forcedvital capacity (FVC) maneuver comparedwith experimental recordings (dashedlines) from Liu et al. (18).
Fig. 13. Model-predicted percent changes in left (A) and right ven-tricular stroke volumes (B) during the FVC maneuver. C: PPL datafrom Liu et al. (18). The dotted vertical lines show the first expiration(e), inspiration (i), and second expiration (e*).
overshoot in arterial pressure accompanied by a slow-ing of heart rate).
In our model, we simulated the Valsalva maneuverby elevating PPL to a higher value for 15 s, starting atthe end of both inspiration and diastole. PPL of 10, 20,30, and 40 mmHg were used in the simulation torepresent different levels of expiratory effort. Airflowin the airways was set to zero during the maneuver tosimulate the closed glottis.
Figure 15 compares the model-generated changes inarterial pressure and heart rate when PPL is 40mmHgduring the Valsalva maneuver with the experimentaldata from Bannister (33). The predicted increase inarterial pressure during phase 1 (;120% of baseline),recovery of the arterial pressure during phase 2, andovershoot during phase 4 (20% above baseline) all fit-ted well with the measured data, as did the predictedheart 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 andafter the maneuver were much smoother than themeasured data. This may be because an idealizedsquare PPL waveform was used in the Valsalva maneu-ver simulation. In reality, PPL recordings show fluctu-ations before and after the maneuver.
Figure 16 shows heart rate, cardiac output, andmean arterial pressure as a function of PPL during theValsalva maneuver. Experimental data from Korner etal. (14) were superimposed on the plot. Both heart rateand mean arterial pressure increased nearly linearlyas PPL increased. Because of reduced venous return,cardiac output declined with the increase in PPL.
Table 13 shows that a variety of calculated hemody-namic indices evaluated from the model predictionsduring the Valsalva maneuver agreed well with thoseobtained from humans [Fox et al. (8)].
Baroreflex control during the Valsalva maneuver.The Valsalva maneuver changes autonomic tone, cen-tral nervous system activity, arterial blood pressure,and heart rate. Figure 17 illustrates how heart rate,myocardial contractility, and vasomotor tone re-sponded to baroreflex control during each phase of theValsalva maneuver.
Phase 1 elevation of arterial pressure occurs withoutbaroreflex control, suggesting that the elevation is dueonly to the mechanical forces of increased intrathoracicpressure (6), which compresses the heart chambersand augments output to the periphery.
Phase 1 lasts about one to two heartbeats. As venousreturn is reduced by the elevated intrathoracic pres-sure, diastolic filling and stroke volume decrease.Therefore, in phase 2, blood pressure decreases andheart rate increases, with baroreflexes helping main-tain arterial pressure and cardiac output. Simulationsdemonstrate the importance of baroreflex control.
Fig. 14. Model-predicted variables during the FVC maneuver. A:systemic arterial pressure (PSA); B: heart rate; C: vagal efferentdischarge; D: sympathetic efferent discharge; E: PPL data from Liu etal. (18). The dotted vertical lines show the first expiration (e),inspiration (i), and second expiration (e*).
Fig. 15. Comparison of experimental data [Bannister (33); A] withmodel-based predictions (B) of the changes in PSA (top) and heartrate (bottom) during the Valsalva maneuver. The arrows in B,bottom, denote the beginning and end of the maneuver. The fourdistinct phases of the Valsalva maneuver are indicated by the ver-tical dashed lines and numbers.
When baroreflex control was abolished (Fig. 17A), ar-terial pressure dropped during this phase. When thesympathetic vasomotor tone was added (Fig. 17B), thereduction in arterial pressure leveled off after five tosix beats, and arterial pressure stabilized. When myo-cardial contractility was added (Fig. 17C), a slow andgradual recovery of arterial pressure toward baselineoccurred after six beats. The delay corresponded to thelate increase in sympathetic efferent traffic (Fig. 18C),which constricts arterial resistive vessels and aug-ments myocardial contractility. When baroreflex con-trol of heart rate was included (Fig. 17D), heart rateincreased. These simulations demonstrate the impor-tance of baroreflex control during phase 2 of the Val-salva maneuver.
During phase 3, the model predicts a fall in arterialpressure without baroreflex input (Fig. 17A). The de-crease in intrathoracic pressure decreased intracardiacpressure and stroke volume. Heart rates remainedhigh due to the increased sympathetic tone (Fig. 18C).
During phase 4, venous return became normal. How-ever, the delayed sympathetic response maintained theheightened myocardial contractility, tachycardia, andvasoconstriction. With no baroreflex control (Fig. 17A),
arterial pressure slowly returned to baseline. Withvasoconstriction (Fig. 17B), there was a small over-shoot in arterial pressure even though arterial pres-sure returned to normal. This overshoot was moreprominent (120% of the baseline level) when baroreflexcontrol of myocardial contractility was added (Fig.17C). Adding heart rate control (Fig. 17D) augmentedthis overshoot to a lesser degree.
The variations of vagal and sympathetic dischargefrequencies during the maneuver are shown in Fig. 18.During phase 2, vagal discharges diminished and sym-pathetic discharges increased. The increase in sympa-thetic tone occurred after vagal tone decreased. Imme-diately after the release of the strain, vagal dischargefrequency quickly returned toward control, whereasthe elevated sympathetic tone continued into the latepart of phase 4.
Figure 19 summarizes how the individual baroreflexpathways maintain arterial pressure. In phase 2, va-soconstriction prevented arterial pressure from drop-ping at a constant rate, whereas increased myocardialcontractility and tachycardia helped restore arterialpressure and cardiac output. Continued increases inmyocardial contractility and vasomotor control contrib-
Fig. 16. Relationships between PPL and heart rate (A), cardiac output (B), and mean arterial pressure (C) duringthe Valsalva maneuver. [Data source: Korner et al. (14).]
uted strongly to the overshoot of arterial pressure inphase 4.
DISCUSSION
We presented a mathematical model of the humancardiopulmonary system that combines several compo-nent models previously developed by our group. Phys-iological data predicted by this combined model agreedwell with data taken from resting and normal subjectsin the supine position. The model was further validatedby accurately predicting the sudden and large physio-
Fig. 17. Model-generated arterial pressure waveforms during theValsalva maneuver under four baroreflex control conditions. A: nobaroreflex control present; B: only the vasomotor tone control; C:vasomotor tone 1 myocardial contractility control; D: all 3 controlcomponents (vasomotor tone, myocardial contractility, and heartrate). The arrows indicate the start and the end of the maneuver.
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 representa-tion (a) and the relative changes of the frequency value (Fsymp; b)]. Thearrows indicate the start and the end of the maneuver.
Fig. 19. Schematic representation of the contribution of the individ-ual baroreflex pathway to the PSA response during the Valsalvamaneuver. HR, heart rate; CON, myocardial contractility; VASO,vasomotor tone.
Table 13. Summary of model-predicted changes inhemodynamic indexes during the Valsalva manuever
logical changes that occurred during the FVC test andall four stages of the Valsalva maneuver.
Although the Valsalva maneuver is well understood,it remains a relatively complicated physiological re-sponse, with mechanical and autonomic componentsthat are difficult to separate when used clinically. Thislimits what information might be obtained from a testthat otherwise is very helpful, easy to perform, andcommonly used to assess patients with a wide varietyof cardiovascular disorders. Because our cardiopulmo-nary system can mimic a combined response fromseparate pulmonary, circulatory, and neural compo-nents, actual (clinical) responses to the Valsalva ma-neuver might be analyzed in terms of these compo-nents (see Fig. 18 and its accompanying discussion),and a specific physiological defect may be more easilyidentified. It is also possible that variations between orwithin groups may be more amenable to statisticalanalysis, because of the purely quantitative nature ofthe model. Good statistical backing would certainlyimprove the meaning and significance of future studiesusing the Valsalva maneuver.
The utility of our cardiopulmonary model should notbe limited to the Valsalva maneuver, however. Withfurther modification and extension, it might help diag-nose or analyze other normal or disordered physiolog-ical responses, such as orthostatic hypotension, andcould characterize disease states such as atherosclero-sis, valvular stenosis, and the pulmonary effects ofcongestive heart failure and the adult respiratory dis-tress syndrome. It could also be helpful in assessingthe prognosis of patients with congestive heart failureand/or coronary artery disease, because in these pa-tients both autonomic and mechanical dysfunction aremajor determinants of premature death.
Model Limitations
All models have limitations, and ours is no exception.The following are a discussion of the limitations of ourmodel:
We employed a circulatory model of intermediatecomplexity for use in the larger cardiopulmonarymodel. It mimics the hemodynamics of the circula-tion quite well. The objectives of the study are gen-eral, however, and if questions such as flow in aparticular circulation or pulse wave propagationwere asked of this model, its foundational assump-tions would be too crude to provide adequate predic-tions (e.g., wave propagation delay is approximatedby a phase shift). In addition, as a supine model, itcannot simulate the hemodynamic responses relatedto changes in body position or gravitational forces,e.g., when subjects stand up from the supine positionor enter different gravitational environments as inspace flight. To address such problems, additionalbandwidth (structural changes) would have to beprovided in the form of the adoption of a more dis-tributed model or certain nonlinear elements wouldhave to be included. This, however, is the subjectmatter of another study.
Ventricular elastance is defined as the instanta-neous transmural pressure-to-ventricular volume ra-tio. At each point in time, it represents a linear rela-tionship between pressure and volume. Therefore, itprovides only an approximation to the curvilinearFrank-Starling relationship (5). This approximationworks well around the operational point of the humanheart, but with large increase in volume, it wouldoverestimate the pressure developed by the ventricle.To represent the Frank-Starling mechanism morefaithfully, the expression for elastance should be mod-ified and made a function of both end-diastolic volume(VED) and time, as in Ref. 10a.
The lung was characterized as a single compart-ment, and homogeneous ventilation was assumed. Thisis unsuitable when the lungs have regional disease. Amultiple-compartment model would be required in thatcase.
The neural control scheme currently employed inthe cardiopulmonary model includes only the barore-ceptor-mediated control of heart rate, myocardialcontractility, and vasomotor tone. It does not containan explicit description of the splanchnic circulation,which in humans has a venous bed richly innervatedby adrenergic nerve fibers. To develop quantitativedescriptions of venoconstriction in humans, moredata are needed. Therefore, we neglected venocon-striction as an important baroreceptor-mediated ef-fect in the Valsalva maneuver. Other important fac-tors, such as the cardiopulmonary baroreceptors,hormonal effects, central and peripheral chemore-ceptor-mediated ventilation, and autoregulation ofspecial circulations, are not considered in the cur-rent model. These factors can also exert importanteffects on the heart, circulatory hemodynamics, pul-monary mechanics, and ventilatory control.
The model characterizes gas exchange only at thealveolar-capillary membrane. However, gaseous par-tial pressures in pulmonary arterial blood at the inletto this membrane are not constant, because gas ex-change occurs at other tissue sites in the body. Modi-fying the model to characterize this additional tissuegas exchange would affect the gaseous content of thepulmonary arterial blood presented to the alveolarmembrane.
Finally, the model must be refined by comparing itspredictions with clinical data obtained prospectively.
The authors acknowledge the helpful comments of Drs. DirarKhoury and Sherif Nagueh of Baylor College of Medicine in thepreparation of the manuscript.
This work was supported by the Bioengineering Center, Univer-sity of Texas Medical Branch, Galveston, TX.
REFERENCES
1. Amoore JN and Santamore WP. Model studies of the contri-bution of ventricular interdependence to the transient changesin ventricular function with respiratory efforts. Cardiovasc Res23: 683–694, 1989.
2. Athanasiades A, Ghorbel F, Clark JW, Niranjan SC,Olansen JB, Zwichenberger JB, and Bidani A. Energy anal-ysis of a nonlinear model of the normal human lung. J Biol Sys8: 115–139, 2000.
3. Beyar R and Goldstein Y. Model studies of the effects of thethoracic pressure on the circulation. Ann Biomed Eng 15: 373–383, 1987.
4. Brinker JA, Weiss JL, Lappe DL, Rabson JL, Summer WR,Permutt S, and Weisfeldt ML. Leftward septal displacementduring right ventricular loading in man. Circulation 61: 626–633, 1980.
5. Chung D, Niranjan S, Clark JW, Bidani A, Johnston WE,Zwischenberger JB, and Traber DL. A dynamic model ofventricular interaction and pericardial influence. Am J PhysiolHeart Circ Physiol 272: H2942–H2962, 1997.
6. Eckberg DL and Sleight P. Human Baroreflexes in Health andDisease. Oxford, UK: Clarendon, 1992.
7. Elad D and Kamm R. Parametric evaluation of forced expira-tion using a numerical model. J Biol Mech Eng 111: 192–199,1989.
8. Fox IJ, Crowley WP, Grace JB, and Wood EH. Effect of theValsalva maneuver on blood flow in the thoracic aorta in man.J Appl Physiol 21: 1553–1560, 1966.
9. Gabe IT, Gault JH, Ross J Jr, Mason DT, Mills CJ, Schill-ingford JP, and Braunwald E. Measurement of instanta-neous blood flow velocity and pressure in conscious man with acatheter-tip velocity probe. Circulation 40: 603–614, 1969.
10. Gore RW and Davis MJ. Mechanics of smooth muscle inisolated single microvessels. Ann Biomed Eng 12: 511–520,1984.
10a. Greene ME, Clark JW Jr, Mohr DN, and Bourland HM. Amathematical model of left-ventricular function. Med Biol EngComput 1973, 126–134.
11. Hainsworth R. The physiological approach to cardiovascularreflexes. Clin Sci 91, Suppl: 43–49, 1996.
12. Hardy HH, Collins RE, and Calvert RE. A digital computermodel of the human circulatory model. Med Biol Eng Comput 20:550–564, 1982.
13. Karam M, Wise RA, Natarajan TK, Permutt S, and WagnerNH. Mechanism of decreased left ventricular stroke volumeduring inspiration in man. Circulation 69: 866–873, 1984.
14. Korner PI, Tonkin AM, and Uther JB. Reflex and mechanicalcirculatory effects of graded Valsalva maneuvers in normal man.J Appl Physiol 40: 434–440, 1976.
15. Kresch E. Design of a non-linear electrical model for veins (PhDthesis). Philadelphia, PA: University of Pennsylvania, 1968.
16. Lambert RK, Wilson TA, Hyatt RE, and Rodarte JR. Acomputational model for expiratory flows. J Appl Physiol 52:44–56, 1982.
17. Li JK-J. Comparative Cardiovascular Dynamics of Mammals.Boca Raton, FL: CRC, 1996.
18. Liu CH, Niranjan SC, Clark JW, San KY, ZwischenbergerJB, and Bidani A. Airway mechanics, gas exchange, and bloodflow in a nonlinear model of the normal human lung. J ApplPhysiol 84: 1447–1469, 1998.
19. Lutchen KR, Suki B, Zhang Q, Petak F, Daroczy B, andHantos Z. Airway and tissue mechanics during physiologicalbreathing and bronchoconstriction in dogs. J Appl Physiol 77:373–385, 1994.
20. McClintic JR. Basic Anatomy and Physiology of the HumanBody (2nd ed.). New York: Wiley, 1975.
22. Melchior FM, Srinivasan RS, and Charles JB. Mathemati-cal modeling of human cardiopulmonary system for simulation oforthostatic response. Am J Physiol Heart Circ Physiol 262:H1920–H1933, 1992.
23. Milnor WR. Pulmonary hemodynamics. In: CardiovascularFluid Dynamics, edited by Bergel DH. New York: Academic,1972, vol. 2, p. 299–340.
24. Mountcastle VB, ed. Medical Physiology. St. Louis, MO:Mosby, 1968, vol. 1.
25. Olansen JB, Clark JW, Khoury D, Ghorbel FH, and BidaniA. A closed-loop model of the canine cardiovascular system thatincludes ventricular interaction. Comput Biomed Res 33: 260–295, 2000.
26. Olsen CO, Tyson GS, Maier GW, Davis JW, and Rankin JS.Dimished stroke volume during inspiration: a reverse thoracicpump. Circulation 72: 668–679, 1985.
27. Parsons GH and Green JF. Mechanisms of pulses paradoxusin upper airway obstruction. J Appl Physiol 45: 598–603, 1978.
28. Ruskin J, Bache RJ, Rembert JC, and Greenfield J. Pres-sure-flow studies in man: effect of respiration on left ventricularstroke volume. Circulation 48: 79–85, 1973.
29. Sato T, Kawada T, Inagaki M, Shishido T, Yakaki H, Sugi-machi M, and Sunagawa K. New analytic framework forunderstanding sympathetic baroreflex control of arterial pres-sure. Am J Physiol Heart Circ Physiol 276: H2251–H2261, 1999.
30. Scharf SM, Brown R, Saunders N, and Greens LH. Effect ofnormal and loaded spontaneous inspiration on cardiovascularfunction. J Appl Physiol 47: 582–590, 1979.
32. Shuler RH, Ensor C, Ginning RE, Moss WG, and JohnsonV. The differential effects of respiration on the left and rightventricles. Am J Physiol 137: 620–627, 1942.
33. Bannister R. Arterial Blood pressure and Hypertension, editedby Sleight P. Oxford, UK: Oxford University Press, 117–121,1980.
34. Snyder MF and Rideout VC. Computer simulation studies ofthe venous circulation. IEEE Trans Biomed Eng 16: 325–334,1969.
35. Spickler JW, Kezdi P, and Geller E. Transfer characteristicsof the carotid sinus pressure control system. In: Baroreceptorsand Hypertension, edited by Kezdi P. Dayton, OH: Pergamon,1965, p. 31–40.
36. Sunagawa K, Kawada T, and Nakahara T. Dynamic nonlin-ear vago-sympathetic interaction regulating heart rate. HeartVessels 13: 157–174, 1998.
37. Ursino M and Magosso E. Acute cardiovascular response toisocapnic hypoxia. I. A mathematical model. Am J Physiol HeartCirc Physiol 279: H149–H165, 2000.
38. Wesseling KH and Settels JJ. Circulatory model of baro- andcardio-pulmonary reflexes. In: Blood Pressure and Heart RateVariability. Amersterdam, The Netherlands: IOS, 1993, p.56–67.
