Department of Electronics, Computer Science and Systems,
University of Bologna, 40136 Bologna, Italy
The objective of this study was to
determine the impact of a total cavopulmonary connection on the main
hemodynamic quantities, both at rest and during exercise, when compared
with normal biventricular circulation. The analysis was performed by
means of a mathematical model of the cardiovascular system. The model
incorporates the main parameters of systemic and pulmonary circulation,
the pulsating heart, and the action of arterial and cardiopulmonary
baroreflex mechanisms. Furthermore, the effect of changes in
intrathoracic pressure on venous return is also incorporated. Finally,
the response to moderate dynamic exercise is simulated, including the
effect of a central command, local metabolic vasodilation, and the
"muscle pump" mechanism. Simulations of resting conditions indicate
that the action of baroreflex regulatory mechanisms alone can only partially compensate for the absence of the right heart. Cardiac output
and mean systemic arterial pressure at rest show a large decrease
compared with the normal subject. More acceptable hemodynamic quantity
values are obtained by combining the action of regulatory mechanisms
with a chronic change in parameters affecting mean filling pressure.
With such changes assumed, simulations of the response to moderate
exercise show that univentricular circulation exhibits a poor capacity
to increase cardiac output and to sustain aerobic metabolism,
especially when the oxygen consumption rate is increased above
1.2-1.3 l/min. The model ascribes the poor response to exercise in
these patients to the incapacity to sustain venous return caused by the
high resistance to venous return and/or to exhaustion of volume
compensation reserve.
 |
INTRODUCTION |
COMPLEX
FORMS of congenital heart disease may impose surgical
reconstructive procedures, creating new cardiovascular anatomy and
hemodynamics. The most striking examples are right heart-bypass operations (generally termed Fontan's operations) used in a variety of
congenital cardiac malformations such as tricuspid atresia, right
ventricle hypoplasia, and pulmonary atresia (6, 7, 10, 27,
35). In these patients, only the left side of the heart pumps
blood properly. One such operation consists in a total cavopulmonary
connection, whereby the systemic venous blood in the inferior and
superior vena cava is rerouted directly to the pulmonary arteries
without the benefit of the normal right ventricle. In this situation
the pulmonary and systemic circulation are in series with only one
pumping chamber.
Generally, patients who have undergone Fontan's procedure have a good
prognosis, although they have subnormal cardiac output (CO) at rest
(36, 42) while central venous pressure is significantly elevated (15, 22). Nevertheless, many studies report an
attenuated CO response to exercise in Fontan's subjects, even in
asymptomatic patients (2, 36, 42). The abnormal cardiac
response to exercise is attributed to cardiac factors, such as the
absence of right ventricle function, defective sinus node rhythm, and impaired left ventricular function. However, because of the
mechanical coupling between heart and peripheral circulation,
inadequate CO response to exercise might also depend on insufficient
peripheral vascular adjustments. Unfortunately, only a few studies have
investigated to what extent the exclusion of the pumping chamber
between the systemic and pulmonary side may affect the entire
circulation (15, 16, 34); in particular, the role of the
autonomic regulatory mechanisms is unclear, especially in the
compensatory response to exercise.
Accordingly, the present work was designed with two main
purposes: 1) to determine the influence of the main
hemodynamic factors in the maintenance of univentricular circulation
(UC) under resting conditions; and 2) to test the hypothesis
that during exercise, compensatory mechanisms are unable to maintain
venous return in the UC, thus resulting in insufficient CO and a severe
limitation of the maximal oxygen consumption rate. Hence, the abnormal
state may be concealed at rest but could appear under stressful
conditions, such as dynamic exercise, when the system, which has in
part exhausted its resources, is greatly stimulated.
Addressing this issue directly in the univentricular patients is
obviously limited by technical and ethical reasons. Moreover, in vivo
it is almost impossible to examine the influence of a single parameter
on the total circulation because of the complex interrelationships
existing among pressure, flow, resistance, and capacitance, further
complicated by the action of reflex regulatory mechanisms. Mathematical
cardiovascular system models and computer simulations may represent a
valid support for the analysis of this problem. A computer model, in
fact, allows the hemodynamic effects of individual parameter changes to
be investigated in rigorously quantitative terms.
To answer the abovementioned points, in this study we improved a
mathematical model of the cardiovascular system, including baroreflex
control under pulsating conditions (40). Hemodynamic data,
reproduced by this model with both left and right pumps, are considered
as the reference for functioning circulation. The model was then
modified through a direct connection of systemic and pulmonary
circulation bypassing the right heart. This modified model has been
used to study which compensatory mechanisms are effective in
maintaining UC both at rest and during exercise.
Glossary
| A |
Peak value of intramuscular pressure, mmHg
|
| AP |
Arterial pressure, mmHg
|
B |
Offset term to simulate a chronic alteration in the corresponding
effector response, spikes/s
|
| CO |
Cardiac output, ml/s
|
| Cj (j = ep, mp) |
Extrasplanchnic and skeletal muscle peripheral compliance,
respectively, ml/mmHg
|
| Cj (j = ev, sv, mv, v) |
Extrasplanchnic, splanchnic, muscle, and systemic thoracic venous
compliance, respectively, ml/mmHg
|
| Emax,lv |
Left ventricular end-systolic elastance (i.e., the slope of the
left ventricular end-systolic pressure-volume curve), mmHg/ml
|
| Emax,rv |
Right ventricular end-systolic elastance (i.e., the slope
of the right ventricular end-systolic pressure-volume curve), mmHg/ml
|
| fab |
Afferent activity from arterial baroreceptors, spikes/s
|
| fac |
Afferent activity from cardiopulmonary baroreceptors, spikes/s
|
| fes,j (j = p, v, h) |
Discharge frequency in efferent sympathetic fibers to arterioles,
veins, and the heart, respectively, spikes/s
|
| fes,min |
Minimum sympathetic stimulation, spikes/s
|
| fev |
Efferent vagal discharge frequency, spikes/s
|
| fmax,l |
Upper saturation of discharge frequency at cardiopulmonary
baroreceptors, spikes/s
|
| fes,cc, fev,cc |
Offset term in the efferent sympathetic responses and in the efferent
vagal response, respectively, reproducing the effect of motor central
command on cardiovascular control centers, spikes/s
|
| Fe |
Total extrasplanchnic flow (i.e., flow through the parallel
of Rep and Rmp), ml/s
|
| Fo,m |
Blood flow leaving leg muscle veins (i.e., blood flow
through Rmv), ml/s
|
| Gab,j (j = p, v, h) |
Constant gain linking afferent activity from arterial baroreceptors to
efferent sympathetic activity directed to peripheral arterioles, veins,
and heart, respectively, dimensionless
|
| Gab,vag |
Constant gain linking afferent activity from arterial
baroreceptors to vagal efferent activity, dimensionless
|
| Gac,j (j = p, v, h) |
constant gain linking afferent activity from cardiopulmonary
baroreceptors to efferent sympathetic activity directed to peripheral arterioles, veins, and heart, respectively, dimensionless
|
| Gac,vag |
Constant gain linking afferent activity from cardiopulmonary
baroreceptors to vagal efferent activity, dimensionless
|
| Gd |
Active muscle conductance, ml/(s · mmHg)
|
| Gl |
Gain at the central point of the cardiopulmonary,
baroreceptor response, spikes/ (s ·mmHg)
|
| G |
Strength of the corresponding effector mechanism, (effector
dimension) · s
|
| HR |
Heart rate, beats/min
|
| kl |
Parameter related to the central gain of cardiopulmonary
baroreceptor response, mmHg
|
| LBNP |
Lower body negative pressure, mmHg
|
| MAP |
Mean arterial pressure, mmHg
|
| NC |
Normal (biventricular) circulation
|
| P0 |
Constant parameter in the pressure-volume relationship of active muscle
veins, with the dimension of pressure, mmHg
|
| Pl |
Output variable of the low-pass filter, mmHg
|
| Pim |
Intramuscular pressure (i.e., extravascular pressure at active
muscle veins), mmHg
|
| Pla |
Pressure inside left atrium, mmHg
|
| Psa |
Systemic arterial pressure, mmHg
|
| Ppa |
Pressure inside pulmonary arteries, mmHg
|
| Ppv |
Pressure inside pulmonary veins, mmHg
|
| Pmcf |
Mean circulatory filling pressure, mmHg
|
| Pmv |
Pressure inside leg skeletal muscle veins, mmHg
|
| Pthor |
Intrathoracic pressure (i.e., extravascular pressure at vessels located
inside the thoracic cavity), mmHg
|
| Pthor,max |
Intrathoracic pressure at the end of expiration, mmHg
|
| Pthor,min |
Intrathoracic pressure at the end of inspiration, mmHg
|
| Ptn |
Pulmonary venous transmural pressure at the central point of the static
sigmoidal characteristic of cardiopulmonary baroreceptors, mmHg
|
| Pv |
Pressure inside systemic thoracic veins, mmHg
|
| Rd ( = 1/Gd) |
Resistance arranged in parallel to Rmp to
simulate muscle vasodilation during exercise,
mmHg · s · ml 1
|
| R'ep |
Total extrasplanchnic resistance (i.e., the parallel of
Rep and Rmp),
mmHg · s · ml1
|
| Rj (j = ev, mv, v) |
Extrasplanchnic, leg skeletal muscle, and thoracic venous resistance,
respectively, mmHg · s · ml1
|
| Rj (j = pa, pp, pv) |
Pulmonary artery, pulmonary peripheral, and pulmonary venous
resistance, respectively,
mmHg · s · ml1
|
| Rj (j = ep, sp, mp) |
Extrasplanchnic, splanchnic, and leg skeletal muscle
peripheral resistance, respectively,
mmHg · s · ml1
|
| s(t) |
Dimensionless variable ranging between 0 and 1, which represents
the fraction of the respiratory cycle
|
| T |
Heart period, s
|
| Tc |
Duration of the muscular contraction, s
|
| Te |
Expiration time, s
|
| Ti |
Inspiration time, s
|
| Tim |
Overall duration of the muscular contraction-relaxation cycle, s
|
| Tresp |
Respiratory period, s
|
| UC |
Univentricular circulation
|
 O2 |
Oxygen consumption rate, l/min
|
| O2 max |
Upper bound for oxygen consumption rate, l/min
|
| V'u,ev |
Total extrasplanchnic venous unstressed volume (i.e.,
Vu,ev + Vu,mv), ml
|
| Vu,lv |
x-Axis intercept of the left ventricular end-systolic
pres- sure-volume curve, ml
|
| Vu,j (j = ep, mp) |
Unstressed volume in extrasplanchnic and leg skeletal muscle
peripheral circulation, respectively, ml
|
| Vu,j (j = ev, sv, mv) |
Extrasplanchnic, splanchnic, and skeletal muscle venous unstressed
volume, respectively, ml
|
| Vmv |
Total blood volume in leg skeletal muscle, ml
|
| Vt |
Total amount of blood contained in cardiovascular system, ml
|
 l |
Time constant of the cardiopulmonary baroreceptor response, s
|
 0 |
Value of the effector in the absence of any sympathetic
drive, effector dimension
|
| |
Generic effector: peripheral resistances
(mmHg · s · ml1); venous unstressed
volumes (ml); heart contractility (mmHg/ml)
|
 |
State variable used to define s(t); i.e.,
s(t) = frac[(t)],
dimensionless
|
 (t) |
Activation function of skeletal muscle fibers, dimensionless
|
 (t) |
Dimensionless variable ranging between 0 and 1, which
represents the fraction of the muscular contraction-relaxation cycle
|
 (t) |
State variable used to define (t); i.e.,
(t)= frac [(t)], dimensionless
|
| |
QUALITATIVE MODEL DESCRIPTION |
Hemodynamics in a normal subject, at rest and in response to a
stress condition, were simulated using the mathematical model presented
by Ursino (40), to which a few improvements were made. The
improvements concern the following points. The first point concerns the
inclusion of the mechanical effects of respiration on the
cardiovascular system occurring through variations in intrathoracic pressure (see APPENDIX). Inclusion of the negative
intrathoracic pressure in the model is important because respiratory
factors may contribute to drive the blood flow into the lungs,
especially in patients with UC. The second point concerns the inclusion
of cardiopulmonary baroreceptors. The latter, in fact, may be
particularly important in cardiovascular regulation, especially under
the hemodynamic conditions (characterized by a reduced pulmonary venous
pressure) typical of UC. The third point is the subdivision of the
systemic circulation into three distinct parallel branches. These
represent the splanchnic circulation, circulation in the skeletal
muscle of legs, and circulation in the remaining extrasplanchnic
vascular beds. Separation of the vascular bed of the skeletal muscle of legs from the others is important to simulate dynamical exercise of the
lower limbs (see below). The final point is a description of the main
cardiovascular adjustments (both reflex and local) that occur during
moderate dynamic exercise.
The model for the univentricular patient has been developed starting
from the one valid for the normal subject, bypassing the right heart.
An accurate model description for a normal subject, including equations
and parameter assignment, may be found in a previous paper
(40). In the APPENDIX, only the mathematical equations describing the new aspects of the model are reported. All new
parameter values are listed in Table 1.
In the following, the main characteristics of the models are presented
in qualitative terms, stressing new features in particular.
The cardiovascular system.
The hydraulic analog of the circulatory system in the normal subject is
shown in Fig. 1. The vascular system
includes pulmonary and systemic circulation. The former consists of the
serial arrangement of three compartments mimicking the arterial,
peripheral, and venous pulmonary vascular beds (subscripts pa, pp, and
pv, respectively). Systemic circulation is described by means of eight
compartments. These include the large arteries (subscript sa), the
peripheral and venous circulation in the splanchnic (subscripts sp and
sv), leg skeletal muscle (subscripts mp and mv), and extrasplanchnic (subscripts ep and ev) vascular beds, and the systemic veins into the
thorax (subscript v). The latter compartment, which was not included in
the previous version, has been added to take into account the fact that
veins traverse cavities with a different pressure ambient
(25). Each compartment is akin to an elastic chamber that
exchanges flow with the downstream and upstream compartments through
hydraulic resistances. Blood volume in each compartment is expressed as
the sum of two contributions: the unstressed volume (defined as the
volume at null transmural pressure) and the stressed volume, which
accounts for the elastic deformation due to vessel compliance. A more
complex pressure-volume curve has been adopted for the veins in the
active muscle at negative transmural pressure to describe the so-called
"muscle pump" during exercise (Eq. A1 in the
APPENDIX). Inertial effects have only been included in the large artery compartments (inertances Lsa and
Lpa), where blood acceleration is significant.
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Fig. 1.
Hydraulic analog of the cardiovascular system in the normal
subject. P, pressures; R, hydraulic resistances; C,
compliances; L, inertances; F, flows; sa, systemic arteries;
sp and sv, splanchnic peripheral and splanchnic venous circulation; ep
and ev, extrasplanchnic peripheral and extrasplanchnic venous
circulation; mp and mv, peripheral and venous circulation in active
muscle compartment; v, systemic thoracic veins; ra, right atrium; rv,
right ventricle; pa, pulmonary arteries; pp and pv, pulmonary
peripheral and pulmonary venous circulation; la, left atrium; lv, left
ventricle. Gd, conductance used to simulate muscle
vasodilation during exercise. Dashed line delimits the portion of
cardiovascular system located inside the thoracic chamber;
Pthor, intrathoracic pressure; Pim,
intramuscular pressure; i.e., the extravascular pressure of active
muscle veins (surrounded by a dash-dot line).
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The right and left sides of the heart (see Fig. 1) embody a passive
atrium (described via a linear capacity) and a pulsating ventricle. The
contractile activity of the ventricle is simulated by a time-varying
elastance, reproducing the isometric pressure-volume curve in series
with a time-varying resistance, which mainly reflects the viscosity of
the ventricle. The shift from the end-diastolic to the end-systolic
pressure-volume curve is governed by a periodic excitation, mimicking
the sinus pacemaker.
To account for the respiratory effects on cardiovascular hemodynamics,
the extravascular pressure has been given a different value in the
portion located inside the thoracic cavity (i.e., heart, lungs, and
thoracic veins, surrounded in Figs. 1 and 2 by a dashed line) and in
the remaining vessels. The intrathoracic pressure changes periodically
as a consequence of respiration (Eqs. A3 and A4
in the APPENDIX). According to experimental data
(25), we assumed that intrathoracic pressure
(Pthor) in the model falls linearly during inspiration
(down to 
9 mmHg) and then rises linearly during expiration to recover
the steady value of the respiratory pause (approximately equal to 4
mmHg). Moreover, as will be described in Response to moderate
dynamic exercise, the pattern of intrathoracic pressure varies
(both as its duration and amplitude) during exercise. Furthermore, we assumed that the extravascular pressure outside the
thoracic chamber remains constant at the same value as atmospheric pressure, with the exception of extravascular pressure in the active
muscle, which varies rhythmically during dynamic exercise. These
assumptions allow respiratory fluctuations in the main hemodynamic quantities (arterial blood pressure, venous return, left and right stroke volume) to be reproduced fairly well (13, 19).
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Fig. 2.
Hydraulic analog of the cardiovascular system in a patient with
cavopulmonary connection. Pulmonary arteries have been coupled directly
to systemic veins. The meaning of the symbols is the same as in Fig.
1.
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To reproduce the right heart bypass (see Fig.
2), the pulmonary artery compartment has
been directly coupled to the systemic veins. This arrangement mimics
the conditions occurring after a total cavopulmonary operation. Under
this condition the left ventricle faces the serial arrangement of the
systemic and pulmonary resistances, i.e., the total vascular
resistance. The effect of different surgical procedures might be
simulated by suitably varying parameters Rv and
Cv in Fig. 2.
The baroreceptor control mechanisms.
The baroreflex model is the same in both circulatory models. In fact,
experiments of lower body negative pressure (LBNP) suggest that
univentricular patients have an intact baroreceptor response (15). The model distinguishes among the afferent pathways
from arterial and cardiopulmonary baroreceptors, the efferent
(sympathetic and vagal) activities, and the responses of several
distinct effectors (see block diagram in Fig.
3).
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Fig. 3.
Block diagram showing the baroreflex regulatory actions according
to the present model. Psa, arterial pressure;
Ppv Pthor, transmural pressure at
pulmonary veins; fab, afferent activity from
arterial baroreceptors; fac, afferent activity
from cardiopulmonary baroreceptors;
fes,j (j =
p, v, h), frequency discharge in efferent sympathetic activity to
arterioles, to veins, and to heart, respectively;
fev, discharge frequency in the vagus;
Rep, Rsp,
Rmp, extrasplanchnic, splanchnic, and active
muscle peripheral resistance; Vu,ev, Vu,sv,
Vu,mv, extrasplanchnic, splanchnic and active muscle
venous unstressed volume; Emax,RV and
Emax,LV, right ventricular and left ventricular
end-systolic elastance; T, heart period.
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Afferent information from arterial baroreceptors is described by
accounting for both static and rate-dependent gains in series with a
sigmoidal function. As has been shown in a previous work (40), this representation allows for a fairly good
replication of experimental results on baroreceptor stimulation.
The model of the cardiopulmonary baroreceptors, not included in our
previous studies, is based on the series arrangement of a low-pass
filter (which reproduces baroreceptor dynamics) and a sigmoidal
relationship with lower threshold and upper saturation (see Eqs.
A5 and A6 in the APPENDIX). Because these
receptors are mainly located in the left atria and pulmonary veins,
transmural pressure at pulmonary veins was used as their input quantity.
The efferent pathways include both sympathetic and parasympathetic
divisions. Vagal activity to the heart is a nonlinear function of
arterial baroreceptor activity and of cardiopulmonary activity (Eqs. A10 and A11 in the APPENDIX).
Distinct equations were used to describe sympathetic activity to
peripheral resistance, to the veins, and to the heart. According to
experimental results, the frequency of the sympathetic discharge is a
decreasing function of afferent activity, whereas the latter is
computed as the weighted sum of the activity from arterial and
cardiopulmonary baroreceptors (see Eqs. A8 and A9
in the APPENDIX). However, because the role of arterial and
cardiopulmonary baroreceptors is dissimilar in the control of
peripheral resistance, venous unstressed volume, and heart period,
different weights were used to compute the sympathetic activity
directed to the peripheral arterioles, to the venous circulation, and
to the heart.
The model includes four different effectors to fulfil the regulatory
actions. Three of them (heart contractility, peripheral systemic
resistance, and systemic venous unstressed volume) change in response
to sympathetic stimulation alone. In particular, an increase in the
frequency of the efferent sympathetic nerves causes an increase in
peripheral systemic resistances (splanchnic, muscular, and
extrasplanchnic) in end-systolic elastance, but a decrease in systemic
venous unstressed volumes (splanchnic, muscular, and extrasplanchnic).
The control of the heart period involves a balance between the
sympathetic and parasympathetic efferent activities: the heart period
decreases by rising the frequency of sympathetic fibers, whereas it
increases rising the frequency of spikes in the vagus. In the model we
assumed a simple linear interaction between the two heart period
control mechanisms because this choice can reproduce several
experimental data quite well (20).
Response to moderate dynamic exercise.
During exercise, the cardiovascular system is challenged to supply the
increased metabolic needs of working muscles while at the same time
maintaining the requirements of other essential organs. This is
achieved by an increased pulmonary ventilation, an increased CO, a
slight hypertension, and a redistribution of blood flow toward the
active muscles (5, 33). These cardiovascular adjustments
result from the superimposition between local vascular control
mechanisms and a reconfiguration of autonomic neural activity; in
particular, sympathetic activity to the heart and blood vessels is
increased while parasympathetic activity to the heart is decreased. Several experimental results support the idea that motor command from
the cerebral cortex, besides initiating movements, activates the
autonomic nervous system (either directly or through a shift in the
baroreceptor characteristic) (9, 18). Furthermore, it
seems likely that the central command plays a major role during mild
and moderate exercise, whereas during severe exercise afferent information from muscle receptors also affects cardiovascular response
(5, 33). In this study, only moderate exercise was considered, therefore, no other reflex mechanism was introduced. The
central motor command has been reproduced through offset terms in
sympathetic and vagal activities (see Eqs. A8 and A10 in the APPENDIX).
The increase in muscle blood flow during exercise has been ascribed to
two concurrent mechanisms: metabolic vasodilation and the effect of the
so-called "muscle pump." The first mechanism has been mimicked by
reducing the peripheral resistance in the leg skeletal muscle. To this
end, a parallel conductance in the skeletal muscle compartment of the
leg was inserted (see Figs. 1 and 2). Naturally, this conductance value
depends on exercise intensity. The effect of the active muscle pump has
been mimicked assuming that, during dynamic exercise, extravascular
pressure for the muscle veins (i.e., intramuscular pressure,
Pim, in Figs. 1 and 2) oscillates between 0 and a positive
level with a rhythmic pattern (Eqs. A13-A15 in the
APPENDIX) (31). Moreover, the pressure-volume characteristic of the muscle veins (hence, venous compliance) have been
given two different expressions, depending on whether transmural
pressure is positive or negative. At positive values of transmural
pressure the classic linear pressure-volume relationship was used,
i.e., constant compliance, where the x-axis intercept defines the unstressed volume. In contrast, at negative levels of
transmural pressure a nonlinear relationship of collapsing tubes taken
from Pedley (29) was adopted (Eq. A1). This
relationship implies that the veins become extremely elastic during
collapse, thus resulting in the expulsion of blood volume. Because of
the presence of valves (Eq. A2), this intermittent venous
squeezing during dynamic exercise favors venous return and CO.
To reproduce the increase in frequency and depth of breathing,
parameters characterizing the pattern of intrathoracic pressure have
been changed accordingly (11, 23, 37, 41). Dependence of
the central command and muscular vasodilation parameters (i.e., the
offset terms in sympathetic and vagal activities and peripheral conductance of the skeletal muscle vascular bed of the leg) on the
intensity of exercise has been given to simulate the response of a
normal subject to exercise (see RESULTS).
For the sake of simplicity, a possible involvement of the myogenic
response to sustain hypertension during exercise, as observed by Lash
and Shoukas (17), was not included explicitly in this work; hence, all experimental vasoconstriction is ascribed to sympathetic influences alone.
Simulated conditions.
The model has been used as follows. First, to assign a value to the
parameters describing cardiopulmonary baroreceptors, we preliminarily
simulated the response of the main hemodynamic quantities (MAP, CO, and
HR) in the NC to different levels of LBNP. Subsequently, hemodynamics
at rest have been compared in the NC and the UC using the basal
parameter values (40). Because the values of CO and MAP in
the UC patient were too low compared with the literature, a sensitivity
analysis was performed on the key cardiovascular parameters to
recognize which parameter changes may allow for the restoration of more
acceptable hemodynamics in the single pump circulation. Finally, the
response to moderate exercise was simulated, both in the NC and the UC
patient, taking suggestions from the sensitivity analysis into account.
| |
RESULTS |
Simulation of LBNP.
The values of the parameters that characterize the cardiopulmonary
baroreflex (i.e., parameters in the static sigmoidal relationship, Eq. A6, and the weighting factors in the expressions of
sympathetic and vagal activity, Eqs. A9 and A11
of the APPENDIX) have been given to reproduce MAP, CO, and
HR in normal humans at different levels of LBNP ranging between 10
and 50 mmHg. To simulate the effect of LBNP, it was assumed that any
10-mmHg depression around the legs causes a volume pooling as high as
about 100 ml (4). This is the same as assuming a lower
body venous compliance as high as 10 ml/mmHg, i.e., the compliance of
the skeletal muscle compartment of the legs (Cmv = 6.6 ml/mmHg) augmented by a small portion (3.4 ml/mmHg) of the
extrasplanchnic venous compliance.
Simulation results are shown in Fig. 4,
whereas parameter numerical values are reported in Table 1. The
agreement between model results and in vivo data (4, 24)
is acceptable.
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Fig. 4.
Percentage changes of mean systemic arterial pressure, heart rate,
and cardiac output (CO) simulated with the model in response to
different levels of lower body negative pressure (LBNP, continuous
line) and compared with in vivo data in normal volunteers (4,
24). In performing these simulations we assumed that venous
compliance in the lower body is 10 ml/mmHg.
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Comparison of normal and univentricular hemodynamics at rest.
The mean values of the main cardiovascular quantities over the cardiac
cycle, computed with the mathematical model of a normal subject at
rest, are listed in the first column of Table
2. These data are considered as the
reference conditions for a functioning biventricular circulation.
The second column shows the same quantities computed in the case of UC.
The absence of the right pump results in an unloading of the
baroreceptors (both cardiopulmonary and arterial), which respond by
increasing sympathetic activity and decreasing vagal activity. As a
consequence, splanchnic and extrasplanchnic resistances, heart rate and
cardiac contractility (Emax) are increased in
comparison with the reference case, whereas unstressed venous volumes
are reduced. These compensatory actions, however, are insufficient in
restoring the reference conditions (compare the first and second columns), i.e., the absence of the right heart can be only partially offset. In particular, MAP and pulmonary arterial pressure settle at a
value significantly lower than normal. Despite the significant rise in
HR, CO remains significantly low due to the peripheral vasoconstriction
and, above all, due to insufficient venous return (proportional to the
difference between pulmonary arterial pressure and left atrial
pressure). In this regard, it can be observed that a comparison of
venous return in NC and UC can be achieved by calculating the quantity
(Ppa Pla)/(Rpa + Rpp + Rpv), where the numerator represents the overall perfusion pressure of the pulmonary vascular bed and the denominator is the overall pulmonary resistance. Because the latter term is kept equal in NC and UC in the
present study, venous return differences can be ascribed to differences
in the numerator only.
The values of MAP and CO in the second column of Table 2 are
insufficient compared with the values that can be found in the clinical
literature. Clinical studies (15, 42) suggest that MAP in
UC at rest is normal or even slightly increased (mainly due to an
increase in diastolic pressure), whereas CO is about two-thirds normal.
To achieve hemodynamic values in UC in closer agreement with the
clinical literature, we assumed that univentricular hemodynamics is
maintained not only by baroreflex activation, but also by a chronic
alteration in some parameters. These alterations may reflect the action
of long-term regulation mechanisms, not considered explicitly in the
present model. To support this assumption, the UC was simulated by
individually modifying: 1) each of the major factors
governing mean filling pressure, i.e., venous unstressed volumes (Table
2, column 3), total blood volume (column 4), and venous compliances (column 5); 2) the left
ventricular contractility described by means of the end-systolic
pressure volume relationship (column 6); and 3)
the peripheral systemic resistances (column 7). However, the
end-systolic elastance, unstressed volumes, and peripheral resistances
in the model are not constant parameters but are actively controlled by
the sympathetic nerve fibers. Hence, to modify the latter quantities,
the static characteristics "parameter value versus sympathetic
activity" were shifted to the left. These changes mimic a chronic
alteration in the effector response. A more accurate description of the
parameter changes performed during the sensitivity analysis can be
found in the APPENDIX.
As expected, permanent variations in the quantities affecting mean
filling pressure (Table 2, columns 3-5) cause an
increase in MAP and especially in CO because of an increase in venous
return. In particular, noteworthy is the increase in pulmonary arterial pressure. However, the price to be paid is a significant increase in
systemic venous pressure that, in accordance with clinical data
(12), is significantly higher than the value observed in NC (14).
The simulated increase in the left ventricle contractility
(column 6 of Table 2) causes a mild benefit to CO compared
with the previous cases and only a partial restoration of the MAP
level. Moreover, it should be noted that greater increases in the
Emax parameter provide only negligible further improvements.
Finally, the increase of peripheral resistances in both the
extrasplanchnic (Rep), leg skeletal muscles
(Rmp), and splanchnic (Rsp) districts (see column 7 of
Table 2) causes only a small increase in MAP. Moreover, this result is
obtained through a dramatic fall in CO (down to 46 ml/s). The latter
result was well expected and agrees with clinical observations
(1).
Of course, the increase in end-systolic elastance has greater benefits
when it is paralleled by an increase in venous return. For this reason,
the effect of a simultaneous increase in total blood volume and
end-systolic characteristic was tested in the last column of Table 2 by
combining the parameter changes separately used in columns 4 and 6. Results show that an increase in cardiac contractility may be efficacious toward improving MAP and CO, provided
it is supported by an increase in venous return (compare results in
Table 2, columns 8 and 6). However, the latter
condition is unlikely in UC for two reasons. First, clinical and
theoretical studies suggest that cardiac contractility is depressed
rather than enhanced in patients with UC (1, 26). Second,
results in the last column of Table 2 show an increase in systolic
arterial pressure. The latter result is in disagreement with clinical
data (15).
In conclusion, the sensitivity analysis suggests that the achievement
of hemodynamic values in the UC patients in agreement with the clinical
literature can be ascribed to a chronic shift in a parameter affecting
mean filling pressure combined with the regulatory action of the
cardiopulmonary baroreceptors (which are triggered by the decrease in
pulmonary venous pressure). Arterial baroreceptors play a minor role
and are activated only if the previous compensations fail to maintain
normal systemic arterial pressure.
Response to moderate exercise.
Simulation of moderate exercise was performed by considering two levels
of oxygen consumption rate (O2, 1 and 2 l/min). The values of the parameters that simulate dynamic
exercise (i.e., the shift in the sympathetic and vagal activities,
fes,cc and fev,cc, see
Eqs. A8 and A10 in the APPENDIX, and
the increase in active muscle conductance, Gd) have been
tuned so that simulation results on the normal subject agree with
experimental data (28) (Fig. 5). Moderate exercise is accompanied by a
large increase in CO, an increase in HR, and a dramatic fall in
systemic peripheral resistance. The decrease in systemic peripheral
resistance is mostly due to muscle vasodilation and counters the
increase in systemic arterial pressure; as a consequence, arterial
pressure only rises modestly.
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Fig. 5.
Values of mean arterial pressure (MAP), CO, heart rate (HR), and
systemic vascular resistance (SVR) simulated with the model at two
different levels of dynamic exercise, corresponding to an overall
oxygen consumption rate as high as 1 (A) and 2 l/min
(B), respectively. All quantities are expressed as
percentages of the basal value occurring in a normal subject in resting
conditions. Experimental data (28) refer to normal
subjects.
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Figure 6 shows the results of four
different cardiovascular responses to moderate exercise (1 l/min)
simulated in the univentricular subject, presupposing the occurrence,
at rest, of no compensation and of one of the three compensations on
mean filling pressure analyzed in Table 2 (columns
3-5). The case analyzed in column 7 (i.e., a
change in peripheral resistances) was not examined because of its poor
effectiveness in maintaining basal CO. The increase in
Emax (either alone or with a simultaneous
increase in blood volume, columns 6 and 8) was
not simulated because univentricular patients seem to have depressed
rather than augmented contractility (1, 26). To facilitate
the comparison, the simulated cardiovascular adjustments in a normal
subject are repeated in each histogram. Moreover, all quantities are
normalized to the value occurring in the normal subject at rest
(assumed to be 100%).
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Fig. 6.
Comparison between model simulation response to moderate
exercise [O2 consumption
(O2) = 1 l/min] in the normal
subject (NC) and in the univentricular patient without any compensation
(UC) and for the three most efficient compensations presented in Table
2 [reduced venous unstressed volume: UC (Vuv); increased
total blood volume: UC (Vt); reduced venous compliance: UC
(Cv)]. All the quantities are expressed as percentages of
the basal value occurring in a normal subject in resting conditions.
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As clearly shown in Fig. 6, a decrease in venous unstressed volume
(Vuv) is the compensation that results in the worsened CO
response to moderate exercise. In fact, a chronic decrease in
Vuv implies that the venous system has already mobilized
part of its blood reserves, thus a smaller amount of blood is available to be displaced in response to physiological stress. The CO exhibits a
greater increase in the other two kinds of resting compensation. In
particular, we can observe that an elevated stiffness in the venous
wall (i.e., reduced compliance) allows CO to be sustained during
exercise better than an elevated total blood volume.
Figure 7A depicts CO as a
function of exercise intensity, quantified by the
O2. The shaded area represents the
normal value range experimentally obtained in a control group
(42). The symbols show our simulation results on normal
subject and on a UC patient with the three different kinds of resting
compensation. The model predictions for the normal subject fall into
the experimental range. Predictions obtained in the UC are also in
acceptable agreement with clinical data (42). In all three
cases examined, CO at rest falls into the normal range, although below
the median line. At a moderate exercise level (1 l/min) CO in the UC
still lies within the normal range. However, when
O2 exceeds 1.2-1.3 l/min, UC
patients become unable to significantly increase CO, mainly due to an
insufficient venous return. The model results quite well agree with
clinical data by Zellers et al. (42) (Fig. 7B).
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Fig. 7.
Static relationship between CO and
O2 rate. A: simulation
results. Circles, rest and exercise CO obtained with the model
simulating a normal subject. Triangles, rhombi, and squares, rest and
exercise CO obtained with the univentricular model simulating the three
resting compensations shown in Table 2 (Vuv,
Vt, and Cv, respectively). Dashed lines
represent the portions of the curves where O2 delivery to
the muscle does not warrant aerobic metabolism. Shaded area between
broken lines, normal range of values experimentally obtained on a
control group (from Ref. 42). B: redrawn
from data measured by Zellers et al. (42) in normal
subjects (shaded area) and in 18 Fontan patients.
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Starting from data in Fig. 7, it is possible to approximately evaluate
a threshold between aerobic and anaerobic metabolism during exercise.
During exercise, in fact, arteriovenous oxygen difference may increase
up to ~0.16 ml O2/ml blood (19). Assuming such an elevated oxygen extraction, an upper bound
O2 max for the oxygen consumption rate
(that is CO multiplied by arterovenous oxygen difference) can be
computed as O2 
0.16 · CO = CO/6.25 = O2 max. Accordingly,
the threshold between aerobic and anaerobic metabolism (i.e., the point
when O2 = O2 max) can be approximated in Fig. 7
by the following equation CO = 6.25 · O2. This is the equation
of a straight line passing through the origin and through the point of
coordinates (1, 6.25). As clearly shown in Fig. 7, at a moderate
exercise level (O2 = 1.0 l/min)
oxygen extraction to the active muscle in UC can still sustain the
O2, because oxygen delivery may surpass
the O2. By contrast, when
O2 exceeds 1.2-1.3 l/min, oxygen
extraction in UC patients becomes inadequate to the
O2 (dashed lines in Fig. 7), and the
patient must rely on anaerobic metabolism.
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DISCUSSION |
It is well documented that the recipients of Fontan circulation
exhibit quite a normal systemic arterial pressure, a moderate reduction
in CO and stroke volume, but an elevated venous pressure at rest. This
hemodynamic scenario allows for a basal functioning of the
cardiovascular system but becomes dangerously critical in response to
exercise compared with healthy subjects. This impairment is mainly
evident in the insufficient capacity to increase stroke volume and
hence CO (42). Although the previous aspects have been
well documented (2, 15, 36, 42), a clear understanding of
the phenomena involved and the role of individual hemodynamic factors
is still lacking.
To achieve a deeper understanding of hemodynamic adjustments in
patients with cavopulmonary connection, both at rest and during moderate exercise, a modified mathematical model of cardiovascular dynamics, integrated with the arterial and cardiopulmonary baroreflex system, was used in the present work.
Several computer models of the cardiovascular system in subjects with
univentricular heart have been proposed in recent years with different
purposes. The model developed by Pennati et al. (30) has
been used in particular to investigate local hemodynamics at the
reconstructive junction for the purpose of optimizing the surgical
procedure and reducing the postoperative risks for the patient. Other
mathematical models (16, 34) have been constructed to
analyze the effect of changes in some cardiovascular parameters on
systemic hemodynamics in this type of circulation. Nevertheless, none
of these models incorporates a detailed description of regulatory feedback mechanisms such as the arterial and cardiopulmonary
baroreflex. The latter modulate several cardiovascular parameters and
may thus contribute significantly to the responses observed in vivo.
Here the results obtained at rest will be discussed first.
Subsequently, new hypotheses on the response to moderate exercise will
be outlined below.
Hemodynamics of univentricular patients at rest.
The first simulation was performed by simply removing the right heart
from the model and computing the consequent values of hemodynamic
quantities; results suggest that a single heart in series with the
entire circulation cannot assure proper hemodynamics, even when
physiological baroreflex compensation and the respiration effect on
venous pulsatility are considered. The model revealed that the main
cause behind this deterioration in hemodynamics is the insufficient
venous return, which abates stroke volume, and thus CO and systemic
arterial pressure. Consequently, we performed a sensitivity analysis to
evaluate which chronic changes should take place to restore hemodynamic
data more similar to those clinically observed in univentricular
patients. This analysis demonstrated that the compensation able to
produce hemodynamic in close agreement with that occurring in
univentricular patients at rest is a moderate chronic alteration in a
parameter affecting the mean filling pressure, reinforced by the
response of cardiopulmonary baroreceptors. The latter cause a
significant increase in systemic resistance and a moderate increase in
heart rate.
The increase in mean filling pressure required to reproduce
hemodynamics in univentricular patients is probably the result of
additional long-term adjustments not considered in the present model.
This increase can be achieved by means of different alternative strategies: increasing total blood volume (perhaps by means of capillary fluid reabsorption and renal regulation), reducing venous unstressed volume [which might occur via a long-term increase in
venous vessel tone, perhaps through sympathetic influences (32,
38)], or reducing venous compliance (which might reflect alterations in venous wall mechanical properties).
Arterial baroreceptors seem to play a minor role in univentricular
patients at rest, because the combined action of cardiopulmonary baroreceptors and the long-term increase in mean filling pressure permit the restoration of a normal MAP level (15). Still,
a certain effect of arterial baroreceptors may ensue from the decrease in pressure pulsatility (see Table 2 and Ref. 15); in
fact, these receptors are sensitive not only to the instant arterial pressure level, but also to the rate of change of pressure waveform (40).
Results of the previous sensitivity analysis agree with the theoretical
and clinical findings by Kresh et al. (16). These authors,
with a simple computer model and animal experiments, reached the
conclusion that near-normal blood flow in the absence of the right
heart can be warranted only by an increase in stressed blood volume or
by a selective reduction in systemic venous compliance. More recently,
Macé et al. (21), through experiments in
anesthetized pigs, observed that mean circulatory filling pressure must
be elevated in Fontan versus biventricular circulation (21.8 ± 1.3 vs. 10.6 ± 0.8 mmHg) to achieve comparable values of systemic blood flow. The authors analyzed their data in terms of the Guytonian relationship among CO, mean circulatory filling pressure, and venous
return. These results agree with those shown in Table 2 of the present
work (columns 3-5) thinking that the compensations on
mean circulatory filling pressure accomplished during the sensitivity analysis (Pmcf in the range of 13.5-14.2 mmHg) allow
for only a partial restoration of the basal CO level.
This scenario is supported by the clinical data reported by Kelley et
al. (15). These authors observed that Fontan subjects have
a reduced venous capacitance (as measured by forearm venous congestion
or LBNP), increased forearm vascular resistance, increased HR, and
elevated resting plasma norepinephrine levels compared with control
subjects. Hence, the activity of the noradrenergic sympathetic nervous
system appeared elevated in these patients and venous tone appeared
higher than normal. Furthermore, data by Kelley et al.
(15) suggest that arterial pressure pulse amplitude is
reduced in Fontan versus biventricular subjects, especially due to a
reduction in systolic pressure, while diastolic pressure is moderately
elevated. As a consequence, MAP unchanged or mildly increased and
pressure pulsatility reduced. The latter observation agrees
with results in columns 3-5 of Table 2.
Response to moderate exercise.
According to the observations reported in clinical studies, the
simulated response to moderate exercise in UC is considerably reduced
compared with NC, especially when O2
increases above 1 l/min (2, 36, 42). In particular,
results of Fig. 6 suggest that univentricular patients have a greater
HR but reduced CO during moderate exercise compared with normal
subjects, hence, they have lower stroke volume. Furthermore, the
present model predicts that univentricular patients can have a maximum
O2 for aerobic metabolism as high as
~1.2-1.3 l/min, depending also on the type of compensation
adopted, whereas normal subjects can increase
O2 up to 2 l/min without reaching the
anaerobic threshold. The latter predictions agree with clinical data
quite well (8, 39, 42).
However, the interpretation provided by the model is quite different
from the one usually reported in the literature. Generally, a depressed
CO response to exercise is ascribed to heart factors, such as impaired
ventricular function, abnormal activity in the sinus node, or abnormal
conduction (2). Zellers et al. (42) hypothesized that a single systemic ventricle may frequently be compromised by prolonged hypoxemia and/or chronic volume overload. Vascular factors affecting preload have also received attention as a
potential cause of the abnormal stroke volume responsiveness. Shachar
et al. (36) measured an increased pressure gradient in the
conduit between the right atrium and the pulmonary arteries during
exercise in patients 4-25 mo after the operation. Hence, they
concluded that conduit obstruction may have contributed to a poor
cardiac response to exercise in these subjects. The present study
suggests a further possible mechanism for the reduced response to
exercise. Simulation results, reported in Figs. 5-7, have been obtained by considering normal characteristic for the left ventricle and normal pulmonary hemodynamic parameters. Hence, none of the hypotheses mentioned above has been incorporated in the model. In
contrast, results suggest that a subnormal stroke volume increase to
moderate exercise may be the consequence of an insufficient increase in
mean filling pressure to sustain venous return, which in turn results
in a poor stroke volume response and insufficient CO increase. To test
this hypothesis, it may be of value to measure the difference
Ppa Pla in UC, especially in conditions
of increased CO demand.
The insufficient capacity of univentricular patients to increase venous
return and stroke volume is the consequence of various concurrent
phenomena. First, univentricular patients exhibit an increase in the
resistance to venous return, which includes the series arrangement of
the systemic and pulmonary vascular beds. Hence, even if the systemic
resistance decreases dramatically during exercise, the resistance to
venous return is still quite elevated due to the series arrangement of
the pulmonary vascular bed. Of course, this result strictly depends on
the assumption that pulmonary resistance does not vary significantly
during exercise, i.e., pulmonary vessel recruitment is already maximal
at rest. Second, systemic resistance is elevated in univentricular
patients due to vasoconstriction of peripheral arterioles caused by the baroreflex control. This phenomenon contributes to a reduced CO (1). Finally, to simulate the hemodynamics of
univentricular patients at rest, we assumed a chronic shift in a
parameter affecting mean filling pressure. As a consequence of this
adjustment, the venous vascular bed may have exploited or attenuated
its blood volume compensation reserve; i.e., it may possess a smaller
capacity to further mobilize blood volume. Among the three permanent
alterations tested in Figs. 5-7 (i.e., the shift in the unstressed
volume characteristic, the increase in total blood volume, or the
decrease in venous compliance), a shift in the venous unstressed volume
causes the more precocious exhaustion of the compensatory reserve; as
in this case the increase in sympathetic activity during moderate exercise can only produce a minimal additional decrease in unstressed volume. By way of contrast, a reduction in venous compliance warrants a
more extended compensation. Indeed, if compliance is reduced, the
increase in stressed blood volume caused by sympathetic activation occurs against a more rigid venous compartment; this leads to a greater
pressure rise and improved venous return. Even in this more favorable
case, however, the capacity of univentricular patients to increase
venous return and CO appears lower during exercise compared with NC.
Of course, the previous explanation supported by model simulations does
not exclude the possibility that other phenomena, such as impairment in
left ventricle performance or obstruction in the cavopulmonary conduit,
may further contribute to the reduced exercise response. In this
regard, it can be observed that CO in UC during exercise is still a
little higher in the present simulations than in clinical data (compare
A and B of Fig. 7). One can expect that a
reduction in heart contractility [i.e., a reduction in
Emax and/or a right shift in the end-systolic
characteristic, as observed in some univentricular patients
(1)] may further abate the response to exercise, thus
providing even better agreement between model and clinical
observations. Reduced response to exercise is presumably a
multifactorial phenomenon caused by different factors acting together;
i.e., insufficient venous return, depressed heart contractility,
increased resistances, etc.
An important drawback to the present study concerns the limited
comparison between model predictions and real data in the literature.
Unfortunately, existing data to support model predictions are in very
short supply and concern only a few hemodynamic parameters. Performing
a deeper comparison would be of great value in future works to confirm
the validity of the main model assumptions or to discover aspects that
require improvement or modifications. This comparison, however, calls
for important hemodynamic quantities to be monitored in UC patients,
both at rest and during moderate exercise. Among the main model
assumptions, which need extensive validation, we can mention the
constancy of pulmonary resistance, the maintenance of an adequate
cardiac contractility, and the direct coupling between the systemic
veins and pulmonary arteries, which can mimic only certain types of
Fontan procedures.
In conclusion, the present study suggests that relatively normal
hemodynamics in patients with cavopulmonary connection at rest may be
sustained by a combination of increased sympathetic activity (possibly
caused by baroregulation), associated with a permanent change in a
parameter affecting mean filling pressure (venous unstressed volume
decrease, an increase in total blood volume, compliance decrease).
Moreover, the response to moderate exercise is depressed compared with
normal subjects, because the UC patients may fail to raise the venous
return to a level capable of supporting CO and the oxygen consumption
rate. This deficit can be ascribed to an elevated resistance to venous
return and/or to a reduced capacity to mobilize blood volume and
increase mean filling pressure. This phenomenon, together with other
possible mechanisms (such as a depressed cardiac contractility), can
explain the pathophysiology of UC.
The subdivision of extrasplanchnic circulation has been performed
assuming that the active muscle branch reproduces skeletal muscle in
the legs. This choice is justified because pedaling exercise on a cycle
ergometer is the test commonly used in the clinical practice to assess
cardiopulmonary performance. Individual parameters of the new two
branches m and e, in basal resting conditions, have been assigned
considering that normal blood flow entering the skeletal muscle in the
legs is ~13% of total CO (31) (thus blood flow in the
remaining extrasplanchnic vascular beds is ~57% of CO) and that the
ratio between analogous parameters in the two segments is the same as
between flows. Finally, the parallel arrangement of the two segments
provides the parameter values for the overall extrasplanchnic
circulation. Because strong intramuscular contractions during exercise
may cause veins to collapse (the so-called "muscle pump"), the
relationship between transmural pressure and blood volume in the active
muscle veins has been reproduced using the equations proposed by
(29) for collapsible tubes
Intrathoracic pressure in the model changes periodically according to
the following equations
Afferent information from arterial baroreceptors is described by using
the same equations as in Ref. 40. Hence, they are not
reported again for the sake of brevity.
We assumed that the afferent activity from the cardiopulmonary
receptors depends on transmural pressure at pulmonary veins through a
first-order, low-pass filter in series with a sigmoidal static
characteristic. Hence, the following equations have been used
TOP
ABSTRACT
INTRODUCTION
0.5 mmHg) in the thoracic veins. The value of Cv
has been given assuming that thoracic veins account for about one-third
of the total systemic venous compliance (3). Compared with
the previous model version (40), values of total extrasplanchnic and splanchnic venous compliance have been modified so
that total venous compliance provides the same value as used in the
previous study (i.e., 111.11 ml/mmHg) (see Table 1 for numeric values).
