Mathematical model of fetal circulation and oxygen
delivery
F. J. Huikeshoven, I. D. Hope, G. G. Power, R. D. Gilbert and L. D. Longo
Am J Physiol Regulatory Integrative Comp Physiol 249:192-202, 1985.
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Mathematical model of fetal circulation
and oxygen delivery
FRANS
J. HUIKESHOVEN,
INEZ D. HOPE,
GORDON
G. POWER,
RAYMOND
D. GILBERT,
AND LAWRENCE
D. LONG0
Division of Perinatal Biology, Department
of Physiology, School of Medicine,
Loma Linda University,
Loma Linda, California 92350
means of assessing the relative importance of these parameters.
This report presents a new model of the fetal circulation as a transport system of oxygen, based primarily on
laboratory data from chronically
catheterized fetal
lambs. Although many results from such preparations
have been published during the last decade, values for
the oxygen consumption of many fetal organs have only
recently become available (4, 9, 11, 22, 23, 35). Thus a
model of the fetal circulation as a transport system of
oxygen based on experimental data was not possible until
recently. Previous mathematical models of the fetal circulation either did not include specific information on
the oxygen consumption of several key organs (5) or, of
necessity, were based on estimates of oxygen consumption of these organ systems (1). Other models of the fetal
circulation did not address the role of oxygen transport
(X),34). The model described here is thus unique in that
it incorporates recent experimental data on the oxygen
consumption of the liver (4), intestines (9), brain (23),
and myocardium (11) in a comprehensive analysis. Our
aim is to understand a physiological system as the sum
of all of its parts, and in general we do not use advanced
pure or numerical mathematics.
oxygen consumption; congenital
occlusion; placental blood flow
The fetal circulation is divided into 16 vascular compartments as shown in Fig. 1. Both fetal and maternal
placental compartments are included. Each fetal compartment (i) has a fixed compliance (Ci) and unstressed
volume (Vu;). The blood pressure within each compartment is determined as
RAYMOND
l
l
heart disease; umbilical cord
HAS BEEN A STEADY APPEARANCE
of valuable
reports of data on the circulation of the fetus and metabolic properties of many of its important organ systems.
In a physiological system as complicated as the fetal
circulation, no one result can be viewed in isolation but
must be compatible with other individual results. The
construction of a mathematical model of the fetal circulation as a transport system provides a means of assembling and integrating individual results into a comprehensive coherent description. Such a model, when implemented on a digital computer, allows the investigator to
efficiently evaluate the compatibility of individual results
and to study the effects of changes in isolated parameters
on total system performance. The model also serves to
identify insufficient and nonexistent data and affords a
THERE
R192
MODEL
Pi = (Vi - VUi)/Ci
(1)
where Pi is pressure (in mmHg) and Vi is compartmental
blood volume (in ml). Lines connecting the compartments in Fig. 1 indicate the possibility of blood flow
between
compartments. Flow from compartment i to j
.
i--*J,
in
ml/min),
with the exception of the flow between
(Q
the superior and inferior venae cavae and the output of
the heart, is linearly related to the pressure difference
between compartments i andj
*
Qi-j = (Pi - Pj)/Ri+
(2)
where R+
0363-6119/85
(in mmHg min ml-‘) is resistance to blood
$1.50 Copyright
l
l
0 1985 the American
Physiological
Society
Downloaded from ajpregu.physiology.org on January 17, 2007
FRANS J., INEZ D. HOPE,
GORDON
G.
D. GILBERT, AND LAWRENCE D. LONGO.
Mathematical
model of fetal circulation and oxygen delivery. Am,
J. Physiol. 249 (Regulatory
Integrative
Comp. Physiol. 18):
R192-R202,
1985.--o
better understand
the fetal circulation
and its regulation
we constructed
a dynamic model of fetal
circulation
as a transport system. The fetal vascular system is
divided into 16 compartments
which incorporate
the peculiarities of the fetal circulation
that produce a difference in oxygen
concentration
in blood supplying the upper and lower body.
Recently published data is used to provide a firm experimental
base for the mode.1. The model is used to examine how the
results on parts of the fetal cardiovascular
system and fetal
oxygen consumption
are compatible
and form a coherent description. We also studied the effects of disturbances
from the
normal steady state produced by changesin patterns of and
resistances to blood flow. A maternal placental blood flow of
<ZOO ml. min-’ *kg fetal wt-’ produces a steady-state value of
oxygen tension in the fetal ascending aorta of <17 mmHg,
which is incompatible
with normal oxygen delivery. A minimal
value of umbilical flow providing an adequate oxygen supply to
the fetal body is 87 ml min-’ kg fetal wt-l. Due to the geometry
of the fetal circulation,
the highest normal oxygen tension in
the fetal ascending aorta is -25 mmHg, only 8 mmHg above
the lowest normal tension of 17 mmHg. Dynamic studies using
the model demonstrate
differences in response of fetal arterial
oxygen tension to temporal cord occlusion and temporal decrease in maternal placental flow.
HUIKESHOVEN,
POWER,
MODEL OF FETAL OXYGEN DELIVERY
I(
R193
the product of &i-j and Co2i. Each compartment
has a
fixed oxygen consumption
specified in milliliters
per
minute. The oxygen saturation
for each compartment
( [ HbO&) is given as
\
4 L--AA
<
UB
[HbO&
= Co,JM
(4)
where M is maximal oxygen capacity of the red blood
cells, assuming that I g hemoglobin can bind 1.34 ml Oz.
From the oxygen saturation
we compute an oxygen tension (POzi, in mmHg) for each compartment
using the
oxyhemoglobin
dissociation
relationship
\
[HbOZl
log PO*i =
(5)
where k, and k2 are constants.
The effects of CO* and
pH are not considered.
Oxygen transfer
between the
maternal
and fetal placental compartments
(Qo,,) is
linearly related to the difference in oxygen tension between maternal (POT,) and fetal (Po,~) blood
V
.
i
(100 - [HbO&)
IN
, ,
f\
-
A
,
FPL
<
Parameter
*
1. Schematic
of model. AA, ascending
aorta; BR, brain; DA,
descending
aorta; FPL, fetal-side
placenta;
HE, liver; IN, intestines;
IVC, inferior
vena cava; LA, left atrium;
LB, lower body; MPL, maternal-side placenta;
MY, myocardium;
PA, pulmonary
artery;
PV, pulmonary
bed; RA, right atrium;
SVC, superior
vena cava; UB, upper
body; UV, umbilical
vein(s).
flow between the two compartments.
Blood flow from
the right heart (QJ is determined by the cardiac function
curve for the right heart and the’afterload,
as described
Pod;mf
_
_
_
“1
Values
The model values are scaled for a 3-kg fetal lamb.
Tables 1 and 2 give values for all parameters used in the
model. Many of these values, especially those for resistante, were derived using the steady-state
condition they
produce. The steady-state
values for blood flow, pressure,
total volume, and oxygen tension are given in Fig. 2.
TABLE I. Parameter values used in model
of circulation of 3-kg fetus
.----_
---Compartment
bY
Q = Qrl x [( 150 - P)/lOO]
-
l
htmrr*nrmr
FIG.
=m(Pyzm
where Rmf (in mmHg . mm ml-‘) is effective resistance
to O2 transfer from the maternal to the fetal side. Maternal blood flowing into the placenta has a fixed oxygen
concentration.
Blood leaving the placenta has the oxygen
concentration
of the compartment.
/&
/‘L
uv
1
Qopp
Compliance,
ml/mmHg
-
(3)
where &,I is given by the right atria1 pressure and the
cardiac function curve as designated in Fig. 1 (ml vs.
mmHg) and P is pulmonary arterial pressure (in mmHg).
The output from the left side of the heart is similarly
determined.
Flow is only allowed between the superior
and inferior caval veins if the pressure difference between
them exceeds a threshold value (P,), in which case the
flow is a linear function of the pressure difference minus
P,. The maternal component of the placenta is modeled
with a constant blood volume so that flow in and out are
equal.
Oxygen transport, delivery, and consumption
are modeled by assigning a total quantity of oxygen (Ozi, in ml)
to each compartment.
Dividing 02i by blood volume Vi
gives the oxygen concentration
(CO,;) in each compartment. Blood flow (Qi+j) determines the shift in oxygen
from compartment
i to compartment
j, which is equal to
AA
BR
DA
FPL
HE
IN
WC
LA
LB
MY
PA
PV
RA
svc
UB
uv
0.07
0.6
0.14
1.4
3.0
0.25
0.2
1.0
1.4
0.25
0.25
0.25
1.0
0.1
0.9
0.3
Unstressed
ml
11.5
9.0
23J4
32.0
1.0
0.25
39.0
6.0
26.0
1.25
4.0
1.25
6.0
19.5
16.0
10.5
Vol,
O2 Consumption,
ml/min
0
2.5
0
15*0
5.0
1.2
0
0
6.0
2.2
0
1.0
0
0
5.5
0
Fetal blood: total volume,
335.5 ml; 02 capacity,
0.15 ml On/ml; Fzl,
1.32; kp, 0.32. Maternal
placental
blood: volume,
20 ml; inflow,
1,500
ml/min;
outflow,
1,500 ml/min;
inflowing
02 concentration,
0.11 ml
OJml;
02 capacity,
0.13 ml Og/ml; kl, 1.62; 122, 0.22. Effective
resistance
to transplacental
02 diffusion,
0.4 mmHg . min - ml-‘. Abbreviations
as
in Fig. 1.
Downloaded from ajpregu.physiology.org on January 17, 2007
l
;,;
R194
HUIKESHOVEN
TABLE 2. Resistance
compartments
AA-BR
AA-DA
AA-MY
AA-UB
BR-IVC
BR-SVC
DA-FPL
DA-HE
DA-IN
1.
0.6
0.0067
0.6
0.125
0.7
0.08
0.053
1.36
0.577
to blood
flow between
DA-LB
0.144
FPL-UV
HE-NC
0,009 1
0.0086
0.116
0.00357
0,00204
0.0208
0.15
0.17
0.004
IN-HE
WC-LA
IVC-RA
LB-IVC
LB-SVC
MY-RA
PA-DA
Values
expressed.
in mmHg
* Threshold
for flow between
PA-PV
PV-LA
SVC-LA
SVC-RA
UB-IVC
UB-SVC
UV-HE
uv-IVC
svc4vc*
0.412
0.122
0.04
0.0029
0.15
0.017
0.027
0.0355
0.002
min.ml-‘.
Abbreviations
as in
SVC and IVC, 0.25 mmHg.
z IBR
Fig.
I
ET
AL.
from data obtained using thermodilution
techniques (16).
Well documented and consistent data are available for
blood flow to the brain (10, 13, 33), myocardium (11, 13,
33), upper body (2, 13, 33), intestines (4, 9, 13, 33), and
placenta (13,24,33). The liver has three sources of blood
supply: arterial supply, which has been well estimated
using the microsphere technique (4,13,33); portal venous
flow, which can be estimated from the intestinal blood
supply; and umbilical supply, which has been estimated
from experiments
involving injection of microspheres
into the umbilical vein itself (4, 33). Blood flow to the
lower body is also given in several reports (10, 13, 33).
The sum of all individual flows agrees well with the total
measured cardiac output.
No data are available for flow from the upper body
and brain to the inferior vena cava or from the lower
body to the superior vena cava. We estimated these flows
to be 10% of the total venous return of these compartments and then used the model to evaluate their impor(see
RESULTS).
Blood Pressures
I I
I
(563 .
I 58
V
IN
P
if\
--
5.
23.9
\
14.9
V
4.0
PO2 17.6
1
f"
Arterial pressures at different locations in the body
are given by Anderson et al. (2). We assumed a normal
mean arterial pressure of -50 mmHg. Experimental
values for venous and atria1 pressure range from 1.8 to 5
mmHg or more (2, 13, 14). Because the atria1 pressures
largely determine cardiac output, the selection of these
values is important
with respect to the choice of the
cardiac function curves. Our cardiac function curves,
discussed in detail elsewhere (ZO), are mainly based on
those reported by Gilbert (l3,14), and therefore we chose
atria1 pressure values to be compatible with these results.
The afterload was estimated from data on acute experiments in the newborn lamb (8). Data for pressures in the
organ compartments
are unavailable; thus we estimated
these values based on a mean systemic pressure of 13.8
mmHg (13). Blood flows and pressures determined the
resistances, using the relationship given in Eq. 2.
Volumes
.
I Po247.7I-
FIG. 2. Steady-state
values for pressures,
volumes,
and flows used in model. P, blood pressure in mmHg;
POT, oxygen tension in mmHg. Flows are in ml/min.
in Fig. 1.
oxygen tensions,
V, volume in ml;
Abbreviations
as
Compartmental
blood volumes are given in Fig. 2. The
available data on volumes are limited. Brace (3) discussed
several measurements and reported values of total volume. By using data on local circulation
times obtained
in the acute preparation with angiograms (31) and dye
(30), we roughly estimated the compartmental
values.
The sum of these estimates agrees with the reported total
volume (3).
The unstressed volumes are determined from the total
volume, pressure, and compliance using Eq. 1.
Blood Flows
The combined cardiac output (i.e., outputs of right and
left ventricles minus Dulmonary blood flow) ranges from
400 to 610 ml’min-’
gkg fetal wt-’ (2, 10, 13, 33). The
right ventricular contribution
is 50-70% of the total.
Pulmonary flow is only 6-7% of the combined ventricular
output (2). Most blood from the superior vena cava flows
to the right atrium, with only -10% going to the left
atrium, as estimated from acute experiments (34) and
Compliances
We estimated arterial and atria1 compliances from
pulse pressures. Values for hepatic and total fetal compliance have been reported (15), and a value for placental
compliance can be estimated (24). We estimated compliance for compartments
with small total volumes, i.e.,
lungs, myocardium, and intestines, from the total volume
and pressure in these compartments,
assuming that the
Downloaded from ajpregu.physiology.org on January 17, 2007
tance
MODEL
OF FETAL
OXYGEN
R195
DELIVERY
unstressed volume is also small. The contribution of the
sum of the compliances for these three compartments
represents only 7% of the total body compliance. The
division of the total body compliance after subtracting
these three values plus the compliances of the liver,
placenta, and umbilical vein was estimated based on the
anatomic structure, weight, and blood volume of each
remaining compartment.
Oxygen Consumption
Maternal Placental Compartment
Uterine blood flow is reported to range from 1,000 to
2,000 ml/min (32, 35). We selected a midrange value of
1,500 ml/min to represent the norm. Maternal placental
blood volume in the exchange area was estimated to be
20 ml, a value that may be high for sheep (19) and is
probablv low for humans. The amount of maternal blood
Summary of Data Used
There are sufficient hemodynamic data available on
regional blood flows for organ blood volumes to be estimated from circulation times. Although data are also
available on central blood pressures, values for local
pressures and compliances are scant, and several had to
be estimated. Data are available for oxygen consumption
for each compartment, but for several organs the data
are from only one study that has not yet been confirmed.
The patterns of oxygen flow and values for oxygen saturation in the compartments depend more heavily on the
regional blood flows, compartmental volume, and oxygen
consumption than on compartmental pressure and compliance. Thus there is a reasonable experimental base for
the model, especially as it is used to study fetal oxygen
transport.
Computations were performed on a Data General
Eclipse computer using a FORTRAN program. A listing
of the program is available on request.
RESULTS
Steady-State Peformance
Figure 2 gives the steady-state values for vascular
pressure, volume, oxygen tension, and blood flow. First,
we compare the PO* values in the compartments with
animal experimental data from reports on compartmental oxygen consumption. Because some reports give only
oxygen concentration or [HbOz], Table 3 gives the model’s steady-state values for oxygen tension, concentration, and saturation for more ready comparison.
In the study reporting oxygen consumption of the liver
(4), control values for the mean oxygen concentration in
the femoral artery were only 5.3-6.9 ml/100 ml, with Pop
20-21 mmHg. Thus the fetal hemoglobin concentrations
in these experiments must have been relatively low, -9
g/100 ml. Due to the difference in fetal hemoglobin
concentration in these experiments and the model, oxygen concentration levels in the model are -2 ml/f00 ml
higher than the published data. Therefore the mean
values reported for oxygen concentration in the umbilical
vein (9.0-10.2 ml/100 ml) and liver (6.9-8.5) agree well
with the model’s predictions. The value for oxygen con-
Downloaded from ajpregu.physiology.org on January 17, 2007
We assume that no oxygen consumption occurs in
large vessels or the heart except in the myocardial compartment. Values for oxygen consumption in the brain
have been reported by Jones et al. (23), and a value of
2.5 ml/min was selected. By using the data of Fisher et
al. (11, 12), we estimated the total myocardial oxygen
consumption to be 2.2 ml/min. The possibility of substantial errors exists in the values reported for oxygen
consumpti .on in the liver (4) because several computations are made using measu.red flow values that may
include substantial errors due to inadequate mixing of
microspheres. We estimated the liver consumption to be
5 ml/min and used the model to evaluate this value.
Oxygen consumption in the intestines is reported by
Edelstone and Holzman (9) to be only 1.2 ml/min. No
recent data are available for oxygen consumption by the
lungs, but using data from acute experiments (6), we
estimated the pulmonary oxygen consumption to be 1
ml/min, -4% of the total fetal oxygen consumption. If
we assume that the fetal hindlimb represents the upper
and lower body, we can estimate their oxygen consumption from data of Iwamoto and Rudolph (22) on oxygen
consumption in the hindlimb and from data of Morriss
et al. (28) on arteriovenous oxygen concentration differences in the hindlimb. This gives values of 4.0 (22) and
5.9 (28) for the upper body and 4.7 (22) and 7.0 ml/min
(28) for the lower body. However, the upper and lower
body compartments both contain small organs, such as
the thyroid and kidney, with higher oxygen consumption
(22). Therefore we selected a value of 5.5 ml/min for the
oxygen consumption of the upper body and 6 ml/min for
the lower body, both slightly larger than the means. The
total fetal oxygen consumption obtained by summing all
compartmental values used in this model is 23.4 ml/min,
a value in good agreement with recently published data
(26)
Okygen consumption of the placenta and uterus combined is reported by Wilkening and Meschia (35) to be
-16 ml/min. By using this value we estimated the placental oxygen consumption to be 15 ml/min. Because
this value is high compared with acute data (26), we
evaluated it using the model.
determines, in part, the total oxygen reserve for the fetus
and thus is important in dynamic studies. By using a
maternal hemoglobin level of 10 g/100 ml, the total
oxygen capacity of maternal blood was determined to be
13 ml 02/100 ml blood, and we assumed that maternal
arterial blood is 85% saturated. Fetal oxygen capacity
was chosen to be 15 ml O&O0 ml blood, based on a fetal
hemoglobin level of 11 g/100 ml. We estimated values
for the dissociation curve constants k, and k2 for fetal
and maternal blood to be consistent with the global value
for resistance to oxygen diffusion (35). This value for
diffusion incorporates factors for uneven distribution,
shunting, and other influences of the spatial arrangements of the exchange vessels and thus is not directly
comparable with diffusion data at the capillary level (19).
RI96
HUIKESHOVEN
ET
AL.
3. Compartmental steady-state values for oxygen concentration, saturation, and tension
under various conditions
----..--- -----TABLE
Normal
Compartment
Abbreviations
8.3
4.5
7.6
12.0
9.0
5.5
8.9
8.3
5.4
4.5
7.5
6.4
7.5
6.1
6.5
12.0
% Saturation
Tension,
mmHg
55
30
51
80
60
37
59
55
36
30
50
43
50
41
43
80
22.3
15.9
21.1
32.4
23.9
17.6
23.5
22.3
17.3
16.0
20.9
19.0
20.9
18.5
19.2
32.4
Concn,
ml 02/100
ml
9.5
5.7
7.6
12.0
9.0
5.5
7.5
9.5
5.4
5.8
7.3
6.2
7.3
7,2
7.8
12.0
% Saturation
63
38
51
80
60
37
50
63
36
38
49
41
49
48
52
80
Streaming
Blood
Tension,
mmHg
24.9
17.9
21.1
32.4
23.9
17.6
21.0
24.9
17.3
17.9
20.5
18.6
20.5
20.4
21.4
32.4
Transposition
of Great
Arteries
Ventricular
Septal
Defect
Concn,
ml O,/lOO
ml
6.9
3.2
7.6
12.0
9.0
5.5
8.8
8.2
5.3
3.2
8.2
7.1
6.9
4.9
5.2
12.0
%Saturation
Tension,
mmHg
46
21
51
80
60
37
58
55
36
22
55
47
46
32
34
80
19.8
13.7
21.0
32.4
23.9
17.6
23,3
22.3
17.3
13.8
22.3
20.2
19.8
16.5
17.0
32.4
Concn,
ml O,/lOO
ml
7.6
3.8
7.6
12.0
9.1
5.6
8.8
7.6
5.4
3.9
7.6
6.5
7.6
5.5
5.9
12.0
%Saturation
Tension,
mmHg
51
25
51
80
60
37
59
51
36
26
51
43
51
37
39
80
21.1
14.8
21.1
32.4
23.9
17.6
23.5
21.1
17*3
14.9
21.1
19.2
21.1
17.5
18.1
32.4
as in Fig. 1.
centration in the portal vein (3.8-4.3 ml/100 ml) is estimated values of oxygen consumption, we altered the
comparable with the model’s value for the intestines (5.5 individual oxygen consumption of each organ, &5O% one
ml/100 ml), and the reported value for the inferior vena at a time, and examined changes in all compartmental
cava (3.0-4.7 ml/100 ml) measured distal to the inflow
[HbOP] values. As expected, the maximal [HbOp] change
of the umbilical blood may be compared with the model’s was seen in those organs in which the oxygen consumpvalue for the lower body compartment (5.4 ml/100 ml). tion was altered. These changes were most pronounced
Therefore, after standardizing fetal hemoglobin concen- in the brain (d5%)
and myocardium (+14%), indicating
trations, the experimental values agree well with the that small errors in measured [HbOn] in these organs
model’s predicted steady state.
produce moderate errors in the computed oxygen conExperimental values of oxygen concentration (arterial
sumption of these organs. At the same time, [HbOz] in
7.8 and venous 5.8 ml/100 ml) and tension (arterial 20 the other compartments changed <3%, showing that
and venous 16 mmHg) supplying and leaving the intes- such errors only slightly affect overall results. Increasing
tines (9) agree well with the model’s predictions. Re- or decreasing lower body oxygen consumption produces
ported values for the oxygen concentration of the inflow
a tl4% change in [HbOz] of that region as well as
and outflow of the free left ventricular wall of the myo- moderate changes in almost all other compartments.
cardium (11, 12) cannot be compared with predicted
Disturbances produced by changing the upper body oxvalues because all myocardial tissue is treated as a unit
ygen consumption were more pronounced in lower body
in the model. For the lower body, a mean inflow oxygen compartments than in those of the upper body. Changes
concentration of 7.3 and outflow of 5.1 ml/100 ml is of k50% in the hepatic and intestinal oxygen consumpreported (28); both these values are close to the model’s tion produce a change of only *8% in the [Hb02] of the
predictions. Reported cerebral inflow and outflow oxygen organ itself, indicating that errors in experimentally
concentrations (23) also agree with the model’s values. measured oxygen consumption of the liver and intestines
In summary, previously published experimental results may be substantial; however, the influence of such errors
agree well with the model’s predictions of oxygen tension on the overall system is small. With a 50% decrease in
and saturation. We conclude that the reports on oxygen placental oxygen consumption,
all compartmental
consumption are in good agreement with each other and [Hb02] values increase 6%. Similarly, a 50% increase of
are comparable with values reported for total oxygen placental oxygen consumption decreases all compartconsumption (26).
mental [HbOz] values by 6%.
A description of the oxygen concentration at several
points in the fetal circulation based on a series of acute
Flow Alterations
experiments was given 30 yr ago by Dawes et al. (7). In
comparing these experimental results with those preDisturbance of normal resistance values. To analyze
dicted by this model, one finds that the most striking
the stability of the system, we changed each resistance
differences are the smaller values for the experimental
one at a time by doubling or halving it. A change in
[Hb02] in the upper and lower body compartments (25
resistance produces an overall change proportional to the
and 26.5%, respectively), probably due to flow restricrelative size of the flow through that resistance. With
tions induced by the acute experiment leading to differthe exception of a twofold increase in the resistance to
ences in oxygen consumption.
the upper body, changes in other steady-state parameters
To measure the sensitivitv of the svstem to errors in were small. A 50% decrease in the resistance between
4
Downloaded from ajpregu.physiology.org on January 17, 2007
AA
BR
DA
FPL
HE
IN
IVC
LA
LB
MY
PA
PV
RA
svc
UB
uv
Concn,
ml O,/lOO
ml
Complete
Preferential
of Ductus Venosus
Case
MODEL
OF
FETAL
OXYGEN
R197
DELIVERY
=r
(:%3)
ISt60
PAA
(MMHG)
r
t-4
800
a
r
600 -
6DV
(ML/MIN)
O/
0
8
QI
42
Q3
FIG. 3. Relationship
between
resistance
in ductus venosus
(RDV)
and ductus venosus blood flow (QDV),
fetal placental
blood flow (QFP),
combined
cardiac output (CO; i.e., sum of left and right ventricle
output
minus lung flow), mean blood pressure
in ascending
aorta (PAA),
and
oxygen tension in ascending
aorta (POT). Arrow,
control
value.
no major disturbance of the system. Thus errors in the
estimation of these flow values produce no significant
errors in the overall performance of the model.
Ductus uenosusflow. Figure 3 shows the relationships
between the ductus venosus resistance and its flow, umbilical flow, cardiac output, and mean blood pressure and
O2 tension in the ascending aorta. A 300% increase in
the resistance decreased ductus flow 65%, from 280 to
97 ml/min. It also lowered umbilical flow 14%, from 540
to 464 ml/min, and decreased combined cardiac output
7%, thereby resulting in a decrease in ascending aortic
oxygen tension from 22.3 to 21.2 mmHg. A 75% decrease
in the resistance resulted in an 88% increase in ductus
flow to 526 ml/min, an 18% increase in umbilical flow to
636 ml/min, and a 9% increase in combined cardiac
output, resulting in an increase of oxygen tension to 23.5
mmHg (+1.2 mmHg). Therefore although the ductus
venosus influences umbilical flow substantially, it does
not significantly influence cardiac output, central oxygen
tensions, or central blood pressures.
Edelstone and Rudolph (10) have suggested that ductus venosus flow m,ay influence the oxyg& tension in the
ascending a.orta as a result of preferential streaming. To
analyze the potenti al contribution in such a circumstance, we modeled maximal preferential streaming by
assuming a direct inflow of all umbilical blood through
the ductus venosus into the left atrium. The resultant
steady-state values are given in Table 3. Ascending aortic
[HbOz] increased only 8%, causing POT to rise from 22.3
to 24.9 mmHg.
Ductus arteriosus flow. Figure 4 shows the relationship
between the resistance in the ductus arteriosus and its
flow, combined cardiac output, blood pressures in the
descending aorta and pulmonary artery, and oxygen tension in the ascending aorta. These relationships were
studied earlier using a previous model (F. J. Huikeshoven
and H. W. Jongsma, unpu blished data) and will not be
discussed here. Additional information prov ided by the
present model is the predicted slight fall in ascending
aortic PO* due to a decrease in combined cardiac output
and umbilical flow when resistance in the ductus arteriosus is increased. In animal studies the administration
of indomethacin increased the ductus resistance 800%
(25). In the model a similar increase produced a fall in
PO:! in the ascending aorta from 22.3 to 21.7 mmHg, on
the same order of magnitude as that reported (25).
Changes in flow patterns as seen in congenital heart
diseases.Heymann and Rudolph (18) have estimated the
[HbOZ] values expected in several arteries of fetuses with
certain congenital heart malformations (18). Our model
provides a tool to analyze such problems in some detail.
For example, we examined transposition of the great
arteries and ventricular septal defects. Table 3 gives the
steady-state values for Paz and [HbO*] in the case of
transposition. We modeled this by assuming that the
entire right cardiac output enters the ascending aorta
while the- left cardiac output enters the pulmonary artery.
Ascending aortic oxygen tension fell to 19.8 mmHg (-2.5
mmHg), below that of the normal Paz in the descending
aorta. The descending aortic Pop did not increase, contrary to expectation (1 8)
Complete modeling of ventricular septal defects is
Downloaded from ajpregu.physiology.org on January 17, 2007
the ascending aorta and the upper body produced a 60%
increase (from 316 to 507 ml/min) of flow to the upper
body (excluding the myocardium and brain) and a small
decrease in flow to all other organs. This caused the
upper body flow, including the brain and the myocardium, to almost equal the left cardiac output, resulting
in a sharp decrease in flow from the ascending to the
descending aorta, typically to ~1 ml/min. Due to the
increased upper body venous return, the pressure difference between the superior and the inferior vena cava
exceeded the set threshold of 0.25 mmHg, and a small
flow (-55 ml/min) occurred between the venae cavae,
thus increasing mixing in the heart. The effect on the
arterial POT was Cl mmHg.
Flow between the lower body and the superior vena
cava represents flow from that part of the body deriving
its blood supply from the descending aorta with a venous
drainage to the superior vena cava. Doubling its resistance reduced this flow from 32 to 17 ml/min, but there
was no observable effect on overall system performance.
A 50% decrease in the resistance increased flow from 32
to 58 ml/min with a 0.3-mmHg rise in the Po2 in the
ascending aorta and inferior vena cava, due to a smaller
flow of less-oxygenated blood from the lower body to the
inferior vena cava. Flow from the upper body to the
inferior vena cava represents flow from that part of the
body deriving its blood supply from the ascending aorta
with a venous drainage to the inferior vena cava. As in
the case above, doubling or halving the resistance caused
R198
HUIKESHOVEN
more complicated.
of the blood in the
sure equilibration.
curves also results.
Such defects produce partial mixing
ventricles and interventricular
presAlteration
of the ventricular
function
We have only modeled the effect of
ET
AL.
SO
LOS
0x
I
T
0
1000
6MP
8
2000
3000
(ML/MIN)
FIG. 6. Relationship
between
maternal
placental
blood flow (QMP)
and oxygen tension in ascending
aorta (POT) and lowest oxygen saturation (LOS). Arrow,
control value.
I400
r
iDA 8oo ‘i,
(ML/MlN)600
RDA
(MMHG.ML-‘*MM)
FIG. 4. Relationship
between resistance
in ductus arteriosus
(RDA)
and ductus arteriosus
blood flow @DA),
combined
cardiac output (CO;
i.e., sum of left and right ventricle
output minus lung flow), mean blood
pressure in descending
aorta (PDA) and pulmonary
artery
(PPA), and
oxygen tension in ascending
aorta (POT). Arrow,
control
value,
COMBINED RESISTANCE
UV-HE
(% OF NORMAL)
uv-IVC
i)FP
(MUMIN)
FIG. 5. A: relationship
between
resistance
(76 normal)
to outflow
from umbilical
vein(s)
(UV) to both liver (HE) and inferior
vena cava
(WC) and fetal placental
blood flow (QFP), oxygen tension in ascending
aorta (Pop), lowest oxygen saturation
(LOS), and mean blood pressure
in ascending
aorta (PAA).
B: relationship
between
QFP and Paz and
LOS. Arrows,
control values.
complete mixing of the blood so as to produce equal Paz
in both ventricles. Table 3 presents these results for
steady-state values. Changes subsequent to complete
mixing are less pronounced than those changes seen in
the case of transposition, with a minimal rise in pulmonary [HbOz].
Changes in umbilical blood flow. We studied the role of
fetal placental flow by changing the resistances to the
outflow of umbilical venous blood, thus simulating partial cord occlusion. Figure 5A shows the relationship
between umbilical venous outflow resistance and umbilical flow, mean blood pressure and Paz in the ascending
aorta, and lowest O2 saturation, i.e., cerebral venous. The
lowest value of [Hb02] in the vascular compartments is
an indication of the available compartmental oxygen
reserve. Its value is nil when the compartmental oxygen
uptake is 100%. Figure 5B shows the relationship between umbilical blood flow and oxygen delivery to the
fetus. Changes in umbilical flow due to partial occlusion
of the umbilical vein(s) only slightly influence fetal oxygenation, provided that umbilical flow does not fall
below 400 ml/min. The figure also shows that normal
uncontrolled oxygen delivery is impossible when the
umbilical flow falls below 260 ml/min (87 ml. min-’ . kg
fetal wt-l). This prediction agrees with the reported
observation that fetal oxygen consumption is maintained
with umbilical blood flow reduced to about 50% of control
(21). The disagreement between the model’s slight fall in
mean arterial blood pressure with increasing umbilical
venous resistance and the experimentally observed unchanged blood pressure (21) suggests that reflex mechanisms may be activated in the intact fetus.
Changes in maternal placental flow. Figure 6 shows the
influence of maternal uteroplacental flow on fetal oxygenation. Changes in maternal placental flow in the
range of l,OOO-3,000 ml/min influence fetal ascending
aortic oxygen values only slightly. However, when the
maternal placental blood flow falls below 1,000 ml/min,
fetal ascending aortic Po2 is substantially decreased.
Thus changes in fetal oxygenation caused by temporary
decreases in maternal placental blood flow are strongly
dependent on the steady-state value of that flow before
the change. In addition, normal uncontrolled steady-
Downloaded from ajpregu.physiology.org on January 17, 2007
40L
MODEL
OF
FETAL
OXYGEN
I
25-
R199
DELIVERY
I
state fetal oxygenation
cannot be maintained when the
ascending aortic Paz falls below 17 mmHg, as occurs
when the maternal placental flow falls below 600 ml/min
(ZOO ml wmin-1 kg fetal wt-l). This value agrees with the
observation
that fetal oxygen consumption
decreases
sharply when the uterine flow falls below 500-600 ml/
min (35).
PO2 (MMHG)
IS-
l
60-
PAA
(MMHG) 4
40-
600 -
Dynamic
i]DAPL,,, _
(ML&IN)
I
0
I
I
I
I
1
2
I
3
TIME (MtN)
7. Predicted
response to 30-s increase in outflow
resistance
of
umbilical
vein(s).
Plot of fetal blood flow to placenta (@APL), mean
blood pressure
in ascending
aorta (PM),
and oxygen
tension
in asceding aorta (PO& Arrows, beginning and end of increased resistance.
FIG.
n_
IS 2000 6MP
(MLhIN)
Of
C
.-L
1
I
1
I
I
I
2sr
A
I
I
1
1
1
2
I
3
TIME (MN)
8. Predicted
response
in oxygen
tension
in ascending
aorta
(Pop) to 30-s d ecrease in maternal
placental
blood flow (QMP).
Temporal reduction
of maternal
placental
blood flow from normal
value of
1,500 to 750 ml/min
(A), from 1,500 to 375 ml/min
(B), and from
steady-state
value of 750 reduced to 375 ml/min
(C).
FIG.
6Mpz:/rr
(MLJHIN)
TIME (MN)
9, Predicted
responses
in oxygen
tension
(POT) to 30-s decrease in maternal
placental
blood
ing different
maternal
placental
volumes
(V).
FIG.
The model provides a tool to study dynamic responses
as well as normal and abnormal steady-state
conditions.
Although many animal experiments
give only steadystate results, they contribute
to an understanding
of
dynamic changes. We studied the consequences of a 30s decrease in maternal placental blood flow, as can occur
during a uterine contraction, and a 30-s decrease in fetal
placental flow due to a partial occlusion of the umbilical
vein(s). Figure 7 shows the predicted response to a partial
occlusion of the umbilical vein(s). Both fetal ascending
and descending aortic Po2 values decrease suddenly with
a rapid recovery after normalization of flow. Figure 8
shows the predicted response to decreased maternal
blood flow. A 50% reduction in maternal placental blood
flow produces only a small late decrease in fetal arterial
Paz when maternal blood flow is in its normal range
(Fig. 8A ). A 75% reduction in maternal flow (Fig. 8B)
induces a fall in fetal Po2 comparable with that caused
by 50% restriction of umbilical blood flow (Fig. 7). The
onset of the change, however, is delayed -10 s, and the
fall is sustained for -45 s after normalization of flow. If
the maternal placental flow starts with a steady-state
value of 750 ml/min, a further 50% reduction produces
a substantial decrease in fetal oxygen levels, and recovery
takes -90 s (Fig. SC). Because the dynamic response of
the model depends partly on the choice of blood volumes,
Fig. 9 shows the effect different initial maternal placental
blood volumes produce in response to flow reduction.
The differences in responses are minimal when the volume changes are between 4 and 100 ml.
in ascending
aorta
flow (QMP)
assum-
DISCUSSION
Every model is a simplication of the in vivo physiological situation. We chose to model the fetal circulation as
a set of compliant vascular compartments with all resistance to blood flow contained between compartments.
In actuality, no such sharp divisions exist; resistance to
flow occurs within as well as between compartments. An
almost infinite set of compliances and resistances would
produce a more physiological model, but computations
using such a model would be excessive. Moreover, dividing the circulation into a larger number of compartments
is of little value when no data are available for the smaller
segments into which it is divided. The most important
reason for selecting the construction of this model is that
most available data are from laboratory experiments in
which computations are based on the assumption that
the circulation consists of discrete compliant compartments with all resistances contained between them. This
approach is thus the most consistent with our goal to
utilize and combine available results. Other models have
Downloaded from ajpregu.physiology.org on January 17, 2007
PO2
(MM&)
Responses
R200
ET
AL.
20.5% of the spheres distributed
to these organs when
injected into the inferior vena cava (10). Both values are
closer to the usual distribution
of the right atria1 blood
(0% to the upper body in our model) than the usual
distribution
of the left atria1 blood (78% to the upper
body in our model), suggesting that the major part of
ductus venosus blood and inferior caval blood in these
animal experiments
was entering the right atrium. Moreover, in the same experiments
8.8% of the spheres injected into the ductus venosus entered the lungs, even
more than the 7.2% of the spheres injected into the
inferior vena cava (10). Both values compare well with
the 10-l 1% of the right atria1 blood normally entering
the lungs. Despite the scant evidence for preferential
streaming,
we computed its possible effects on oxygen
tensions. We calculated that total preferential streaming
of the ductus venosus flow to the left atrium would
increase the POT in the ascending aorta from 22.3 to 24.9
mmHg. Because both this value and the resultant difference in Paz of 3.8 mmHg between ascending and descending aortas are higher than the reported values, there
is probably
at most a small amount of preferential
streaming.
Computations
of the possible effects on fetal blood
gases in cases of congenital heart malformations
are
difficult. Heymann and Rudolph (18) suggest that certain
congenital
heart malformations
cause oxyhemoglobin
saturations
in several arteries to change substantially.
For instance, in the case of transposition
of the great
arteries, they predict an increase in pulmonary
arterial
[HbO,j from 55 to 70% and descending aortic [HbO,]
from 60 to 65%. Their results are based on computations
of flows in the heart and great vessels which assume that
flow patterns in the malformed heart are straightforward
deviations from the normal flow patterns. Such an assumption may be an oversimplification,
but it is incorrect
to assume that only the central part of the circulation is
affected by such changes in [HbOz]. During steady-state
conditions the normal placental oxygen uptake remains
unchanged, and with a normal umbilical flow the umbilical arteriovenous
[HbO,] difference should be the same
in normal fetuses and those with malformations.
This
fact was not considered
in the computations
reported
(18). Our model predicts less dramatic changes; in particular,
the maximal
increase in pulmonary
arterial
[HbOz] is only 5045%.
Calculations
of the steady-state
values after changes
in fetal and maternal placental flows indicate that the
normal range of both these flows provides a wide margin
of safety. In the normal range, moderate flow disturbances only slightly affect the fetal oxygen levels. In
contrast,
as shown in Fig. 6, fetal oxygenation
may
become compromised
if the Paz in the ascending aorta
falls below 17 mmHg. Under physiological
conditions,
reflex mechanisms
will come into play to redistribute
oxygen (29), but the long-term adequacy of these adjustments and their impact on fetal growth remains uncertain (12).
The instances of abnormal flow patterns examined in
this study indicate that the steady-state
value of PO:! in
the ascending aorta varies onlv a few millimeters
of
Downloaded from ajpregu.physiology.org on January 17, 2007
shown that such a construction
is valuable even when
only a small number of vascular compartments
are included. For instance the model of Guyton et al. (17) of
the adult circulation
contains only five compartments,
and the model of the fetal circulation of Huikeshoven
et
al. (ZU), which does not consider oxygen transport,
contains only six peripheral
compartments.
The present
model contains 16 vascular compartments,
comparable
with the model of Allen et al. (1) (25 compartments)
and
Cameron et al. (5) (17 compartments).
An important shortcoming
of any model consisting of
a limited number of compartments
is the unavoidable
assumption
of complete mixing within each compartment, This oversimplification
of the normal transport
and consumption
of oxygen is somewhat
minimized in
this model because it contains 16 compartments
and all
of the central pathways
consist of at least three compartments.
We modeled the oxygen consumption
in each organ
system by extracting a given amount of oxygen from the
total oxygen in the compartmental
blood, a simplification
consistent with the available experimental
data (4,9, 11,
22, 23). Oxygen diffusion in the placenta from maternal
to fetal blood is simplified by the assumption
that both
the fetal and maternal sides of the placenta are pools of
blood with O2 exchange depending only on the difference
in oxygen pressure between them. The possibilities
of
uneven distribution
and shunting of placental blood flow
are not considered.
In a sense, this model is only a framework
of a description of the fetal circulation as a transport
system without
feedback mechanisms such as the autoregulation
of blood
flow, baroreceptor
reflexes, chemoreceptor
reflexes, or
endocrine control. It does, however, provide the necessary framework
for the incorporation
of many of these
features, especially autoregulation
and the chemoreceptor reflexes. Although these shortcomings
presently limit
its application
for short-term
dynamic studies, during
long-term changes many of these feedback systems have
already adapted to the new steady state, rendering the
model valuable for steady-state
studies under many different conditions. Normal control values for oxygen tension demonstrate
that individual reports on specific organ oxygen consumptions
are compatible with each other
and collectively provide a coherent description
of fetal
oxygen delivery. This validates the model as an accurate
description of the fetal oxygen delivery system and allows
us to study the effects of certain abnormalities.
The model is useful in exploring the consequences
of
numerous alterations
in the fetal circulation.
For instance, the ductus venosus is obliterated early in fetal
life in some species, but its absence in the human fetus
is associated with severe pathology (27). We calculated
that marked changes in its resistance
and blood flow
have little impact on fetal oxygenation.
It has been
suggested that the ductus venosus plays an important
role in preferential
streaming of umbilical blood to the
upper body (10). However, in the reported animal experiments 26.8% of the microspheres
injected into the ductus venosus were distributed
to the upper body structures, excluding the lungs. onlv slightlv more than the
HUIKESHOVEN
MODEL
OF
FETAL
OXYGEN
R201
DELIVERY
ence in the time interval between the onset of the disturbance and the onset of the fall in PO* distinguishes
the two conditions. For a better understanding
of these
phenomena the model needs to be extended to include
the cardiovascular
reflex mechanisms.
One may speculate, however,
that decreased umbilical
and maternal
placental flows mediate changes in fetal heart rate primarily through a fall in PO2 and that the difference in
time delay of the two respon ses results solely from the
architect&e of the fetal vascular system.
We thank
Sheila
Whitson
for typing
the manuscript
and Tom
Bazemore
for preparing
the illustrations.
This work was supported
in part by the Netherlands
Organization
for the Advancement
of Pure Research
(ZWO)
and by National
Institute of Child Health
and Human
Development
Grants
HD-16827
and
HD-13949.
Present
address
of F. J. Huikeshoven:
Dept. of Obstetrics
and
Gynecology,
Sint Franciscus
Gasthuis,
PO Box 10900, Rotterdam
3004
BA, The Netherlands.
Address for reprint
requests: G. G, Power, Div.
of Perinatal
Biology,
Dept. of Physiolom,
School of Medicine,
Loma
Linda Univ., Loma Linda, CA 92350.
Received
24 May
1984; accepted
in final
form
25 February
1985.
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mercu rY u nder ex treme conditions,
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Under dynamic conditions, the Po2 in the aorta may
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