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1 Cardiothoracic Unit, Great Ormond Street Hospital for Children, WC1N 3JH London, United Kingdom; 2 Bioengineering and 3 Energy Engineering Departments and Laboratory of Biological Structure Mechanics, Politecnico di Milano, 20133 Milan, Italy; and 4 Pediatric Cardiovascular Surgery, The University of Michigan Health System, Ann Arbor, Michigan 48109
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ABSTRACT |
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 TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
REFERENCES
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Hypoplastic left heart syndrome is the most common lethal cardiac malformation of the newborn. Its treatment, apart from heart transplantation, is the Norwood operation. The initial procedure for this staged repair consists of reconstructing a circulation where a single outlet from the heart provides systemic perfusion and an interpositioning shunt contributes blood flow to the lungs. To better understand this unique physiology, a computational model of the Norwood circulation was constructed on the basis of compartmental analysis. Influences of shunt diameter, systemic and pulmonary vascular resistance, and heart rate on the cardiovascular dynamics and oxygenation were studied. Simulations showed that 1) larger shunts diverted an increased proportion of cardiac output to the lungs, away from systemic perfusion, resulting in poorer O2 delivery, 2) systemic vascular resistance exerted more effect on hemodynamics than pulmonary vascular resistance, 3) systemic arterial oxygenation was minimally influenced by heart rate changes, 4) there was a better correlation between venous O2 saturation and O2 delivery than between arterial O2 saturation and O2 delivery, and 5) a pulmonary-to-systemic blood flow ratio of 1 resulted in optimal O2 delivery in all physiological states and shunt sizes.
congenital heart disease; computer model; hypoplastic left heart syndrome
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INTRODUCTION |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
REFERENCES
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HYPOPLASTIC LEFT HEART
SYNDROME is a lethal condition characterized by a hypoplasia of
the left ventricle, the mitral valve, the aortic valve, and the
ascending aorta. Preoperative survival relies on the patency of the
ductus arteriosus to supply the systemic circulation. The initial
operation (Norwood stage I, Fig. 1)
consists in using the pulmonary valve and the main pulmonary artery as the systemic outflow and interposing a shunt between the innominate artery and the right pulmonary artery as a source of pulmonary blood
flow (5, 11). Despite improved prognosis with the
introduction and refinement of the palliative Norwood operation,
immediate postoperative mortality remains at 20-30%, even in
high-volume centers (4). During the second stage of the
operation, the pulmonary flow from the shunt is replaced by a superior
vena cava-to-pulmonary arterial anastomosis, and the final stage
consists in connecting the inferior vena cava to the pulmonary artery
(total cavopulmonary connection).
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After the first-stage reconstruction, the systemic and pulmonary circulations are in parallel, and the right ventricle acts as the sole hydraulic power source. The distribution of the cardiac output between the systemic and the pulmonary circulations is governed in part by the interposition shunt. Various pharmacological and therapeutic interventions in the early postoperative period are aimed at maintaining this delicate balance between the two circulations (3, 11).
We have developed a lumped-parameter model of the Norwood circulation to study the cardiovascular effects of changes in geometry of the interposition shunt, systemic and pulmonary vascular resistance, and heart rate.
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MATERIALS AND METHODS |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
REFERENCES
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Mathematical Model
A lumped-parameter model of the Norwood circulation was built following the methodology previously used to model the fetal (2, 12) and neonatal circulation (14, 15). The model is made of three subsystems: the hypoplastic heart, the systemic circulation, and the pulmonary circulation (Fig. 2).
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Heart. Models for both atria [left (LA) and right (RA)] and the single ventricle (SV) are mathematically similar; differences are reflected by defining appropriate values of the various parameters within each model.
Pressure within any cardiac chamber (Pcc) varies throughout the cardiac cycle because of changes of volume within the chamber (Vcc) and contractile activity of the sarcomeres. Active (systolic) and passive (diastolic) properties of the myocardium account for the total chamber pressure (Pcc). This can be expressed mathematically as (27)
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![P<SUB>cc,active</SUB>(<IT>t</IT>)<IT>=E</IT><SUB>cc</SUB>(V<SUB>cc</SUB>, <IT>t</IT>)<IT> · </IT>[V<SUB>cc</SUB>(<IT>t</IT>)<IT>−</IT>V<SUB>u,cc</SUB>]<IT>+R</IT><SUB>wall,cc</SUB><IT> · </IT><FR><NU>dV<SUB>cc</SUB>(<IT>t</IT>)</NU><DE>d<IT>t</IT></DE></FR>](/content/vol280/issue5/fulltext/H2076/img002.gif)
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T) and
temporal lapse (Fig. 2). The expression of
E*cc(Vcc) depends on the
pressure-volume relationship during systole of the cardiac chamber. For
both atria, we assumed a linear pressure-volume function so that
E*RA(VRA) and
E*LA(VLA) are constant. For
the single ventricle, a second-order polynomial function is adopted
where the elastance decreases linearly with increasing volume
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![P<SUB>cc,passive</SUB>(<IT>t</IT>)<IT>=</IT>P<SUB>0,cc</SUB><IT> · </IT>{<IT>e</IT><SUP><IT>K</IT><SUB><IT>E,</IT>cc</SUB>[V<SUB>cc</SUB>(<IT>t</IT>)<IT>−</IT>V<SUB>cc,u</SUB>]</SUP><IT>−1</IT>}](/content/vol280/issue5/fulltext/H2076/img005.gif)
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P(t)]
and the volume flow rate [
(t)] across them
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Vascular system. The vascular circulation was divided into two subsystems connected by the systemic-to-pulmonary shunt. The systemic subsystem consists of four compartments: descending aorta, systemic arterial bed, systemic venous bed, and systemic large veins. The pulmonary subsystem consists of proximal pulmonary arteries, pulmonary arterial bed, and pulmonary venous bed.
Each compartment (depicted as a block in Fig. 2) in the model was assumed to have a constant compliance (C), where C =
V(t)/P(t), where P(t) is the
instantaneous local pressure and V(t) is the instantaneous
compartmental blood volume. Law of mass conservation was applied to
each compartment and, with adoption of the above definition for C, can
be expressed as
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in(t) and
out(t) indicate the respective sum of
the instantaneous volumetric flow rates at the inlet and the outlet of
the compartment, respectively.
The momentum conservation law for each interconnection (depicted as
lines in Fig. 2) between two compartments can be expressed as
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P(t) is the instantaneous pressure
difference applied to the line ends, (t) is the
instantaneous volumetric flow rate, R is the purely viscous
resistance, and L is the inertance. Inertial effects were
taken into consideration in the large arteries only.
For the interposition shunt, local fluid dynamics are important. Hence,
a more comprehensive expression for the momentum conservation law is
used
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(t)2 accounts for
the nonlinear effects of convective energy loss due to flow
separations, eddies, vortexes, and turbulence associated with flow.
Previously, we showed that inertial effects in the shunt are negligible
(9). Inasmuch as shunt parameters R and
K can be expressed also as functions of the shunt diameter
(D) (9, 29), the previous equation can be
rewritten as
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1 · mm4 and
k2 of 960 and 5,200 mmHg · (l · min)2 · mm4
(9).
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Clinical Data
Cardiac catheterization and angiographic data of 28 Norwood patients were available for this study. Some of these data were used as input parameters; other data were used for validation of the model (Table 1). Twenty patients received a 3.5-mm shunt; a 4-mm shunt was used in the remaining eight patients. Table 1 summarizes the anatomic and hemodynamic data.
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Parameter Values
The parameter values of the shunt have been described above; those of the single ventricle (passive and active pressure relationships) were obtained from previous studies of univentricular circulation (14, 15, 20, 23). No data were available on the pressure-volume relationship of the atria or on the flow resistance across an atrioventricular valve or an ASD in neonatal univentricular heart. Those parameters were extracted from a published model of the fetal heart at full term (17) after appropriate rescaling (13). They are listed in Table 2. Heart rate (HR) and duration of the cardiac cycle (Tc = 1/HR) were derived from our clinical data set. Duration of ventricular systole (Ts,SV = 0.16 + 0.3Tc) increases linearly with duration of the cycle (1), and duration (Ts,LA = Ts,RA = 0.3Tc) and time advance (T = 0.02Tc) of
atrial systole were calculated as fractions of
Tc (24).
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The parameter values of the remaining compartments of the circulation
were extrapolated from our previous data of hydraulic lumped-parameter
models of the fetal and neonatal circulation (12, 14, 15).
They were scaled down (13) for an infant with a body
surface area of 0.33 m2 (average surface area of our
clinical subjects). Total pulmonary and systemic vascular resistances
(PVR and SVR, respectively) were obtained from clinical data, and their
values were distributed among the model compartments in analogy with
the models of the human circulation in the literature (8, 24,
28). Table 3 lists these values.
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O2 Calculations
The O2 transport model had the following unknown variables: O2 contents (ml/dl blood) of the systemic arteries (Cart), of the systemic veins (Cven), and of the pulmonary veins (CPV). Because of the systemic-to-pulmonary shunt, O2 content in the pulmonary arteries was equal to Cart. The first two governing equations were
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P and S are
pulmonary and systemic mean blood volumetric flow rates, respectively,
CO is cardiac output (CO = P + S), and C
O2 is the
whole body O2 consumption. Volumetric flow rates were
obtained from the lumped-parameter model of the newborn circulation.
A third equation was derived from the lung O2 exchange
model, based on the work by Hill et al. (7). This
represented effects of the capillaries in the pulmonary bed as a single
unit with volume (Vtot) of 3.5 ml. Hence, variation of the
partial pressure of O2 (PO2) with
distance along the capillary bed was equivalent to the change of
PO2 in an element of blood moving with time
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1 · mmHg1),
PAO2 is the alveolar
PO2 (100 mmHg), 
is the O2
solubility in blood (0.03 × 103
ml · ml1 · mmHg1), and
O2Cap was the maximal O2 carrying capacity [in
ml/dl blood; O2Cap = 1.34 · Hb, where Hb is
the Hb content (in g/dl)]. Slope was the slope of the oxyhemoglobin
saturation (Sat) curve, described as a function of
PO2
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4 and 2.7, respectively, being
PO2 expressed in mmHg (22).
O2 content was proportional to Sat by the factor
O2Cap.
PO2 and pulmonary venous O2
saturation were obtained by integrating the above differential equation
throughout the transit time interval, calculated as the ratio of
Vtot to p. The integration was carried
out under the condition that
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O2 is the known
O2 uptake in the lung, which is assumed to be equal to
CO2.
Systemic O2 delivery (in ml/min) and O2 balance
were then calculated as
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O2 = 159.64 ml · min1 · m2.
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RESULTS |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
REFERENCES
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The mathematical model generated flow and pressure temporal tracings, as well as mean values of saturations and O2 delivery.
Validity of the Model
The mathematical model was validated by comparing the results of simulations with a part of the averaged (Table 1) and individual clinical data. The clinical parameters averaged from 28 patients, used as input of the model, were as follows: HR = 125.6 beats/min, indexed PVR (PVRa) = 2.3 mmHg · m2 · l1 · min,
indexed SVR (SVRa) = 21.6 mmHg · m2 · l1 · min,
and D = 3.5 mm. With those input parameters, the
simulation showed a CO of 2.22 l/min with a cardiac index (CI) of 6.73 l · min1 · m2,
P/S of 1.14, pulmonary flow of
1.18 l/min, and mean systemic and pulmonary arterial pressures of 69 and 12 mmHg, respectively. Calculated O2 delivery was 546.9 ml · min1 · m2, and
arterial and venous O2 saturations were 79% and 55.9%, respectively.
Ventricular pressure-volume loops, temporal relationships of
ventricular, aortic, and pulmonary arterial pressures, and flow in
shunt and aorta are depicted on Fig. 4.
The end-diastolic and systolic volumes were 34.9 and 16.9 cm3, respectively, with an ejection fraction of 51.6%.
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There was a good correlation between the simulation and the averaged clinical data (Table 1) as well as published data (20, 23).
If one takes individual cases, however, whereas good correlations were
obtained in some, there were discrepancies among others. Table
4 shows the ranges of predicted values
for various parameters and the percent differences from the observed
data. For these simulations, the measured pulmonary venous saturation
for each patient was used.
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Effects of Shunt Size
With the use of D as the independent variable, shunt hemodynamics were simulated for diameters of 3, 3.5, 4, 4.5, and 5 mm. Figure 5A demonstrates the result of the simulations where CI andP/S are reported as functions of
shunt diameter. Enlarging the shunt diameter was associated with
increases in CI and P/S, resulting in higher pulmonary flow. Augmentation in CI led to increased
arterial O2 saturation (Fig. 5B), but not venous
O2 saturation. On the other hand, O2 delivery
(Fig. 5C) slightly increased when shunt diameter increased
from 3 to 3.5 mm and sharply decreased for larger shunts as more blood
is diverted to the pulmonary circulation at the expense of the systemic
output. Effects of shunt size on mean pulmonary arterial pressure were
nearly linear, inasmuch as varying the shunt size from 3 to 5 mm
resulted in a pressure increase from 9 to 18 mmHg. Conversely, mean
systemic arterial pressure decreased from 81 to 59 mmHg. As expected,
the pressure gradient along the shunt decreased as shunt diameter was
enlarged, ranging from 72 to 41 mmHg for shunt diameter of 3-5 mm.
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Effects of Vascular Resistances
Figure 6 shows CI,P/S, arterial and venous
saturations, and O2 delivery as functions of
PVRa (0.5-20
mmHg · m2 · l1 · min)
for shunt diameter of 3.5 mm and SVRa of 21.6 mmHg · m2 · l1 · min.
CI slightly decreased (from 7.0 to 5.9 l · min1 · m2), while
P/S diminished from 1.23 to 0.64 (Fig. 6A). Arterial and venous saturations showed a
similar decline (Fig. 6B), but O2 delivery
reached a maximum at PVRa of ~4-5
mmHg · m2 · l1 · min
(Fig. 6C). Mean systemic arterial pressure remained nearly constant (73-79 mmHg), while mean pulmonary arterial pressure showed abrupt elevation with higher PVRa (from 5.5 to 48 mmHg).
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Figure 7 depicts the effects of
SVRa (4-64
mmHg · m2 · l1 · min)
when PVRa was kept constant at 2.3 mmHg · m2 · l1 · min.
Small increments in SVRa resulted in a large drop in CI and
near-linear increases in P/S as
flow became preferentially pulmonary (Fig. 7A). Arterial
O2 saturation (Fig. 7B) increased steeply to
reach an asymptote near 80% when SVRa reached 20 mmHg · m2 · l1 · min.
Venous O2 saturation behaved differently, however. At low SVRa, venous O2 saturation increased until
a peak of 62% was reached at 8 mmHg · m2 · l1 · min;
then a progressive decrease was observed. O2 delivery to
the body decreased as SVRa increased (Fig.
7C). Mean pulmonary arterial pressure did not show
significant variations (10-12 mmHg), while mean systemic arterial
pressure increased from 39 to 82 mmHg.
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Effects of HR
Figure 8 illustrates the effects of increasing HR on CI,P/S,
arterial and venous O2 saturations, and O2
delivery. CI increased with HR, while
P/S remained relatively
unchanged (Fig. 8A). Arterial and venous O2
saturations increased with respect to HR (Fig. 8B). An
increase in HR from 60 to 180 beats/min produced an increase in
O2 delivery from 380 to 650 ml
O2 · min1 · m2
(Fig. 8C).
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Elevated pressure gradients along the shunt were also observed with higher HR, ranging from 45 to 70 mmHg.
Simulation of Early Postoperative Period
In the early postoperative period, PVRa and HR are often increased by inotropic drugs. To evaluate this condition, simulations with PVRa twice the baseline value (4.6 mmHg · m2 · l1 · min)
and HR of 150 beats/min were performed with various shunt diameters
(3-5 mm). CI and
P/S were lower than
the baseline values (PVRa = 2.3 mmHg · m2 · l1 · min
and HR = 120 beats/min at rest). This difference was
accentuated for larger shunts (Fig. 9,
A and B). O2 balance remained almost unchanged (Fig. 9C).
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Simulation of Exercise
Dynamic exercise is characterized by a marked increase in O2 consumption, reduction in SVRa, and elevation in HR (5, 6). We simulated this condition by increasing HR to 180 beats/min, reducing SVRa by one-half (10.8 mmHg · m2 · l1 · min),
and assuming a CO2 that is twice that at rest.
As expected, CI increased and P/S
decreased (Fig. 9, A and B). Mean pulmonary
arterial pressure increased from 9 to 17 mmHg for respective shunt
diameters of 3 and 5 mm. O2 balance as a function of shunt
diameter is shown in Fig. 9C. Whereas a 3.5-mm shunt
provided a balance >3.5 at rest, O2 balance during exercise was always lower, despite use of larger shunts. Nevertheless, O2 balance during exercise remained >1.5 and increased
with larger shunts. It is possible that a combination of highly
intensive exercise (i.e., higher CO2)
and small shunts (3 or 3.5 mm) could lead to O2 deficit.
Figure 9 demonstrates that when O2 balance was plotted
against P/S, the maximum level
was reached at P/S ~ 1 for
all three simulated physiological conditions. Similarly, O2
delivery was at a maximum at
P/S = 1 in all ranges of
PVRa (Fig. 6) and shunt sizes (Fig. 5) simulated in our study.
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DISCUSSION |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
REFERENCES
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This study confirms the clinical experience that the shunt is crucial in the regulation of pulmonary and systemic blood flow and, consequently, of tissue oxygenation.
Shunts of 3.5 mm diameter are most commonly used in neonates. Simulations of the early postoperative state confirm that a 3.5-mm shunt allows for a flow distribution between the systemic and the pulmonary circulations that provides a maximum level of O2 delivery. In the simulation of exercise, however, a 3.5-mm shunt did not provide enough O2 to achieve an optimal O2 balance. Whereas larger shunts would meet these increased O2 requirements during exercise, they would result in a reduction of O2 delivery at rest because of excessive pulmonary flow and reduced systemic flow.
Inasmuch as demand on shunt performance varies with changes in physiological conditions, one limitation of the Norwood operation is the inability of the shunt to satisfy all these conditions. Similarly, the nature of the shunt does not provide for adjustments for growth.
The optimal O2 delivery is achieved when balanced pulmonary
and systemic perfusion is established, namely, when
P/S ~ 1. This is true
regardless of shunt size, physiological state, or absolute values of
PVR. In other words, when pulmonary and systemic perfusions are equal,
the Norwood circulation is optimized.
In practice, despite the fixed nature of the shunt, when O2 requirements change, manipulation of the PVR and SVR is used to maintain the optimal flow ratio. This was confirmed by the introduction in the lumped-parameter model of changes in SVR and PVR similar to those produced by therapeutic manipulation, such as changes in inspired PO2, positive end-expiratory pressure, nitric oxide inhalation, and supplemental CO2 (10, 16-18).
The importance of monitoring venous O2 saturation as an indicator of tissue O2 delivery in the postoperative period has been previously reported (19) and confirmed by our simulations. In all situations, there was a better correlation between venous O2 saturations and O2 delivery than between arterial saturations and O2 delivery. However, the measurement of mixed venous saturation in clinical practice remains difficult (21).
Limitations of the Mathematical Model
CO2 was kept constant when the
effect of HR was simulated. As far as the exercise physiology is
concerned, the calculations of O2 delivery and saturations
were based on an O2 consumption that was twice the baseline
value. The effect of respiration on cardiopulmonary performance was not
examined. However, the Frank-Starling mechanism was implemented. Other
active regulatory mechanisms were not considered in the present model.
The discrepancies between the calculated and clinical values are most
likely related to the deficiencies of the latter. They result from a
wide range of anatomic variations, such as anastomotic strictures,
kinking, and intimal thickening, of the shunt. These will result in an
overestimation of P/S, which was
our most common error. In addition, flow calculations in clinical
practice are based on the measurement of mixed venous saturations
(i.e., a mixture of inferior and superior vena cava blood in the right atrium). This value is not available in the Norwood patients, who have
an obligatory left-to-right shunt at atrial level (all the pulmonary
venous blood mixes with the systemic venous blood in the right atrium
to reach the single ventricle). Minor changes in O2
saturations can lead to major differences in
P/S and, thus, the deficiencies
of some of these clinical data.
Finally, the mathematical model presented was not compared with specific patient cases to include a comprehensive validation of all possible clinical scenarios. In addition to the impractical aspects of manipulating physiological parameters in extremely ill children, the ethical implications of such maneuvers in the clinical setting prevented us from acquiring the necessary data. As a consequence, the model in its present form may be useful for some, but not all, patients to whom it is applied. Application of this model is further limited by the inability to predict a priori in which patients the model does not provide good correlation. However, this modeling methodology represents an important new paradigm in the analytic and numerical applications in clinical cardiopulmonary physiology. As such, it remains useful in the understanding and prediction of hemodynamic changes in these patients after their surgical reconstruction.
Conclusions
With the use of the lumped-parameter model of the Norwood circulation, the following points were demonstrated or confirmed.With increasing shunt size a greater proportion of the CO is directed into the lungs, and this can lead to systemic hypoperfusion and poor O2 delivery.
Changes in SVR have a more dramatic influence on hemodynamics than changes in PVR.
Levels of oxygenation were mostly unchanged by HR. However, extremely low HRs resulted in lower O2 saturations.
There is a better correlation of mixed venous O2 saturation with systemic O2 delivery than of arterial saturation with O2 delivery.
P/S = 1 results in optimal
O2 balance and O2 delivery in all physiological
states and shunt sizes.
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ACKNOWLEDGEMENTS |
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This work was supported by the British Heart Foundation.
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FOOTNOTES |
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Address for reprint requests and other correspondence: F. Migliavacca, Dipartimento di Bioingegneria, Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133 Milan, Italy (E-mail: Migliavacca{at}biomed.polimi.it).
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 13 September 1999; accepted in final form 29 November 2000.
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REFERENCES |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
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