Int J Biomed Comput, 31 (1992) 127-139 Elsevier Scientific Publishers Ireland Ltd.
MATHEMATICAL MODELING BLOOD CIRCULATION
A. GUETTOUCHEa, J.C. CHALLIERb, and A. AZANCOT-BENISIV
127
OF THE HUMAN
FETAL ARTERIAL
Y. ITOE, C. PAPAPANAYOTOU’,
Y. CHERRUAULTa
aVniversitP P.M Curie, Luboratoire MEDIMAT, 1.5, Rue de I’Ecole de Mkdecine, 75270 Paris Cedex 06, bVniversitP P.M. Curie, Laboratoire Biologic de la Reproduction, Brit A, 7, Quai St Bernard, 75252 Paris cedex OS and cService d’Exploration Fonctionnelles. Hbpital Robert DebrP. 48, boulevard Serurier, 75019 Paris (France) (Received Feburary 26th, 1992) (Revision received March 27th, 1992) (Accepted March 27th, 1992)
A mathematical model of the human fetal arterial circulation based on mass and momentum conservation for one-dimensional flow is presented. We simplified the fetal arterial vascular system from the heart to the placenta, defined 16 anatomical segments and studied the characteristics of the vascular system in relation to changes in morphology and hemodynamics. The two-step Lax-Wendroff finite difference scheme was used to solve the system of equations, after introducing the rheological constants, the diameter and length of the segments measured by two-dimensional imaging and the mean arterial velocity at the inlet segments obtained by pulsed Doppler. The model was validated by comparing the numerical results to our non-invasive ultrasound direct measurements and to previous published data. Key words: Mathematical model; Fetus; Blood circulation; Ultrasound
Introduction Research in the area of the fetal physiology has yielded physical and mathematical models of the fetal blood circulation in animals based on experimental data [1,2]. In the human fetus, the development of two-dimensional ultrasound imaging and pulsed Doppler techniques allows non-invasive measurements of vessel size and blood velocity in major arteries and veins [3-51. Clinicians rely on various parameters based on systolic and diastolic flow velocity waveforms for evaluating fetal and placental hemodynamics. There is a crucial need to develop tools in order to interpret local hemodynamic data obtained by non-invasive methods. A mathematical model can be an alternative to explore the overall fetal circulation. Lambert [6], appears to have been the first to apply mathematical methods to blood flow problems, considering non-viscous fluid and one-dimensional flow. In their contribution to blood flow modelling, Streeter et al. [7] presented a basis of mathematical modelling of pulsatile pressure and flow through distensible vessels. Correspondence ro: A. Guettouche, Universite P.M Curie, Laboratoire MEDIMAT, 15, Rue de 1’Ecole de Mkdecine, 75270 Paris Cedex 06, France.
0020-7101/92/$05.00 0 1992 Elsevier Scientific Publishers Ireland Ltd. Printed and Published in Ireland
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Many other authors have applied similar laws of mass and momentum conservation to hemodynamic modeling and differed by the choice of resistance terms, wall elasticity relationships and numerical methods of resolution [8-141. To allow the investigation and prediction of the overall human fetal-placental circulation, we developed a mathematical model based on mass and momentum conservation laws, utilizing ultrasound measurements as input. Our approach was to simplify the human fetal arterial system to 16 anatomical segments. We applied to every vascular segment the equations of mass and momentum conservation and the wall elasticity relationships (state equation) for one-dimensional flow. Such a model requires mean blood velocity at the inlet segments (ascending aorta and pulmonary trunk) which were obtained by pulsed Doppler; the diameter and length of the vascular segments measured by two-dimensional imaging. The quasilinear hyperbolic system of partial differential equations was numerically solved by the two-step Lax-Wendroff finite difference scheme [ 151. The model was validated by comparing the numerical results to our non-invasive ultrasound direct measurements and to previous published data. Anatomical and Mathematical Model Considering its anatomical complexity, the human fetal arterial system has been simplified (Fig. 1). The new configuration starts at the proximity of the cardiac valves and terminates at the placental level. The ascending aorta (segment 1) and the pulmonary trunk (segment 4) represent the inlet segments of the anatomical model. All segments refer to individual blood vessels except the vessels of the cephalic region (carotid, innominate and subclavian arteries) and the pulmonary arteries which were regrouped in segments 2 and 5, respectively. The upper part of the fetal body (head and upper members), lower members and organs as lung, kidney and placenta were represented by peripheral resistances (R2, R13 and R14; R5; R8 and R9; R15 and R16). Each vascular segment was mathematically modelled by a system of three equations: the mass and momentum conservation equations and the elasticity relationships which related the cross-sectional area to pressure. The basic assumptions of the model were: (i) the segment of blood vessel was modelled as a cylindrical and distensible tube of constant thickness; (ii) blood was considered as an incompressible, viscous and Newtonian fluid and its flow was supposed laminar; (iii) pressure and flow velocity were assumed uniform over arterial cross section. Using the above assumptions, the equations of mass and momentum conservation for a one-dimensional flow in each segment were:
(1) c$+v~+_-=_--
i
ap
P
ax
8?r/.~V P
A
(2)
where A, V and P are, respectively the cross-sectional area, the axial velocity and
Modeling fetal circulation
78
R
129
18
Fig. 1. Anatomical model of the human fetal arterial system. Segment number (1) ascending aorta; (2) innominate, carotid and subclavian arteries; (3) aortic isthmus; (4) pulmonary trunk; (5) pulmonary arteries; (6) ductus arteriosus; (7) thoracic aorta; (8) and (9) right and left renal arteries; (10) abdominal aorta; (I 1) and (12) right and left common iliac arteries; (13) and (14) right and left iliac arteries; (15) and (16) right and left umbilical arteries. -: Forwarded flow. The black labels (W) correspond to the peripheral resistance (R) of organs or regions.
the pressure, p stands for the density and p for the viscosity of blood. The unknown functions (A, V, P) depend on time (t) and on the axial coordinate (x). The right side of Eqn. (2) figures the Poiseuille’s formula for laminar steady flow representing only the effects of blood viscosity. It applies strictly to steady flow. For small values (C 10) of the Womersley frequency parameter a! = d(wp/p), where r is the vessel’s radius and w the angular frequency of the oscillatory motion [ 16,171. In fetal ascending aorta (r = 0.5 cm) and umbilical arteries (r = 0.23 cm), the values of the a! parameter were 4.03 and 1.85, respectively. Since arteries are tapered and distensible, the cross-sectional area A varies with
130
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pressure (P) and distance (x). The previous Eqns. (1) and (2) were completed by the wall elasticity relationship (state equation) given by: A = A(P,X) = A&l
- PDOIEb)
(3)
in which AO, Do, ho are, respectively, cross-sectional area, internal diameter, wall thickness under zero pressure of the vessels and E the Young’s modulus of elasticity. Numerical Solution The previous equations (1,2 and 3) constitute a hyperbolic quasilinear system. We considered as many set of equations as the number of segments. Boundary conditions at the proximal, internal and distal ends of the arterial segments and numerical solution are now examined. Proximal boundary conditions
The blood velocity waveforms were obtained by Doppler at the ascending aorta and pulmonary trunk inlet segments. The variation of blood velocity over a cardiac cycle can be described using Fourier’s approximation. N
V(O,t) =
c
aicos(ht)
+ bi sin&t)
0 I
t I
TO
(4)
i=O vd
To < t I
T
where w = 2rlT is the angular frequency, To and Tare the time of systolic ejection and that of the cardiac cycle, respectively, vd is the diastolic velacity. For a steady flow, we considered the mean blood velocity over a cardiac cycle as a proximal boundary condition at the inlet segment. The value of mean velocity obtained from the internal computer software of the pulse Doppler (Toshiba SSH160) introduced into the model, were 23 cm/s for ascending aorta and 21.4 cm/s for pulmonary trunk. Internal conditions (junctions and bifurcations)
At junctions and bifurcations of three or four vessels, we imposed the mass conservation and the pressure continuity as follows: in the case of a bifurcation:
Qi t&t) = QjCOJ>+ QdOvt) (5) Pj(&t)
= Pi(O,t) = Pk(O,t),
in the case of a junction:
Qi C&t) + Qj(LjJ> = QdOJ) Pi(Li,t)
= Pj(Lj,t) = Pk(O,t),
Modeling fetal circulation
131
where Li denotes the length of the segment number i, arriving at a bifurcation or a junction, Q(O,t) and Qi(Li,f) stand for the instantaneous volume flow rate at the inlet of segments i and at the outlet of segment j, respectively. Distal boundary conditions The effect of distal organs or regions was mimicked without considering their vascular structure by applying a resistance defined as a ratio of the driving pressure across the organ capillary bed to volume flow rate through this bed:
%rs= VW)
- f’JQ(LO (7)
with Q(L,t) = A(L,t) . V(L,t), where P(L,t) and Q(L,t) are the pressure and volume flow rate at the end of the peripheral arterial segment, P, the venous pressure considered constant (P, = 5.3 mmHg), L the length of this segment. The peripheral resistance at the end of segments 2, 5, 8, 9, 13, 14, 15 and 16, constitutes the distal boundary conditions. In default of values of arterial pressure in the human fetus, the resistances were assumed inversely proportional to the organ weight and symmetrically distributed in the kidneys, lower members and placenta (R8 = R9, R13 = R14 and R15 = R16). Thereafter, the resistance values were adjusted under steady flow conditions by the identification method of Vignes [18]. We minimized the following functional:
where Q&) and Q,&) are the calculated and measured flow rates respectively, and n the number of vascular segments (n = 16). Table I shows that the values of the peripheral resistances varied from 14 500 up to 142 000 dyne * s/cm5. Finite differences scheme With the above boundary conditions, we numerically solved the quasilinear hyper-
TABLE I PERIPHERAL
RESISTANCES OF THE MODEL
Regions R2 RS R8 R9 R13 R14 RlS R16
Resistance (dyne.s/cm5) upper part of fetal body lungs right kidney left kidney right lower member left lower member right placental part left placental part
21 ooo 38 000 142000 142000 45 500 45 500 14500 14500
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bolic partial differential equations (1, 2 and 3) by the two-step Lax-Wendroff finite differences scheme, [l&19-22], utilizing Vignes’ method for optimization of the resistance values (Fig. 2). In the first step we calculated the functions A, V and P at the point of coordinates ((j + l/2) Ax, (t + l/2) At) and in the second step we calculated these functions at (j Ax, (n + 1) At) as follows: step 1 (prediction):
(10) n+112
P j+li2=-
&, Do
l-
Ao -$I
(11) ::
>
step 2 (correction):
(13)
(14)
The schemes at proximal and distal boundaries were (15), (16) and (17), respectively: n+l A0
=A;-z
WI: - (AV)“o >
(1%
133
Modelingfetal circulation
Introduction 11, p,
p, E,
V(l),
of
ho,
constants:
Do(i),
V(4).
L(i).
Ax(i),
f
Q,(i)
i=l
to
At 18 1
1 Initialization Resistance Trr=10000,
of
parameters
coefficients: N=loO,
Vignes’ Optieization
R(Z)..
t = 0.001, 1
.
K=O
eathod
of
the
resistance
va I ues 1 Initial
conditions
of
finite
differences
aethod
i Lax-Uendroff t= tiAt
calculation
finite of
A(i),
Minimization
differences
soheee
V(i).
P(i).
of
J = ? i? n i Q i-1
Print A(i).
V(i),
P(i).
the (Q(i)
-
R,(i))’
results P(i). 1 End
Fig. 2. Numerical resolution diagram.
functional
Q(i).
R(i)
(L-U):
i=l
J
to 16
134
A. Guettouche et al.
II+1
Piv
= P,,+ RA;+’
pN+’
(17)
with:
where R is the resistance coefficient, At and Ax are the discretization steps of time and space respectively. The quantities of upper indice n were known and the quantities of upper indice n + 1 were to be calculated. At the junctions and bifurcations, we used the same method for discretizing Eqns. (5) and (6). The numerical calculation started with the following initial conditions: V(x, 0) = 0
P(x, 0) = P, A(x, 0) = A&l
(18) - P,DcjEh,,)
At = 0.0002 s and Axi = L/N, where Li and Ni are, respectively, the length and the number of the discretization points of the segment i (i.e. for ascending aorta i = 1, L, = 1.2 cm, N, = 6 and Axr = 0.2 cm). The model requires the rheological parameters (blood density p = 1.06 g/cm3, blood viscosity p = 0.052 dyn . s/cm2, Young’s modulus E = 6 lo6 dyn/cm2, wall thickness ho = 0.08 cm and venous pressure P, = 5.3 mmHg), the diameter and length of the segments measured by two-dimensional imaging (Table II). The length of each equivalent tube (segments 2, 5, 13 and 14) was fixed at 1 cm and its diameter was approximated from regional distribution of cardiac output. Results and Discussion The mean velocities obtained at the inlet segments by Doppler at 27.5 weeks of pregnancy (range 25-30) were introduced as input into the model. Table I shows the values of resistance identified by the Vignes’ method [18]. In this method, the quadratic difference between the flow rate values calculated by the model and those
Modeling fetal circulation
135
TABLE II ECHOGRAPHIC
MEASUREMENTS
OF VESSELS
Diameter and length at 27.5 weeks of pregnancy (range 25-30). R, right; L, left, Co, common.
(3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16)
Vessels
Diameter (cm)
Length (cm)
Ascending aorta Innominate artery Carotid artery Subclavian artery Aortic isthmus Pulmonary trunk Pulmonary arteries Ductus arteriosus Thoracic aorta R. renal artery L. renal artery Abdominal aorta R. co. iliac artery L. co. iliac artery R. iliac artery L. iliac artery R. umbilical artery L. umbilical artery
0.50
1.2
0.34
1.0
0.32 0.57 0.28 0.36 0.46 0.14 0.14 0.42 0.28 0.28 0.22 0.22 0.23 0.23
1.4 0.86 1.0 0.75 5.2 1.2 1.2 1.5 1.0 1.0 1.0 1.0 50.0 50.0
computed from velocity and vessel diameter measurements was minimized with a final accuracy of 7.1%. In the case of the placenta, the resistance (Rp = 7250 dyne . s/cm5) was calculated using the formula: l/Rp = l/R1 5 + i/R16 where R15 and R16 stand for placental resistances at right and left umbilical arteries. The value of placental resistance was comparable to the data observed by Huisseling et al. [23] in sheep between 100 and 130 days gestation: 0.093 mmHg/min per ml or 7438 dyne. s/cm5. These data show that the lowest resistance are located in placental and upper body circulations (i.e., 7250 and 21 000 dyne * s/cm5, respectively). Consequently, they receive the highest flow rate and changing their resistance could have major effects on systemic circulation. The highest resistance was noted in the kidney (142 000 dyne. s/cm5) in which the lowest flow rate can be expected. Pressure (dyne/cm2) and blood flow rate (cm3/s) units were converted to mmHg and to ml * mm/kg, respectively, using a fetal weight 1.25 kg at 27.5 weeks gestation [4]. The mean values of the various parameters, which were evaluated in the middle of each vascular segment are depicted in Table III. A good agreement was noticed between the velocities calculated by the model and those measured by Doppler, except in abdominal aorta. In Ductus arteriosus, the velocity calculated by the model 39.5 cm/s compares well with that measured by us (37.4 f 9.2 cm/s) and is within the values (24.8 f 3.9 at 20 weeks versus 50.1 f 14.5 cm/s at 38 weeks) reported by Mooren et al. 1241.In the abdominal aorta, the value computed by the model (35.2 cm/s) exceeded the mean velocity measured by Doppler (28.1 f 2.7 cm/s). This discrepancy
MODEL
(3) (4) (5) (6) (7) (8) (9) (IO) (11) (12) (13) (14) (15) (16)
(2)
(1)
Vessel
c
Ascending aorta Innominate artery Carotid artery Subclavian artery Aortic isthmus Pulmonary trunk Pulmonary arteries Ductus arteriosus Thoracic aorta R. renal artery L. renal artery Abdominal aorta R. co. iliac artery L. co. iliac artery R. iliac artery L. iliac artery R. umbilical artery L. umbilical artery 2.3 1.6 1.6 2.7 2.7 2.7
zt 5.7 f 5.7
f f f zt zt zt
?? 9.2
zt 2.1 f 2.2 85.5 283.9 81.1 202.9 288.5 20.5 20.5 247.5 123.7 123.7 64.2 64.2 59.4 59.4
22.7 21.4 37.4 34.3 25.3 25.3 28.1 39.7 39.7 28.6 28.6
21.2 21.4 26.4 39.5 33.9 27.3 27.3 35.2 40.3 40.3 34.2 34.2 28.9 28.9
232.4
C
146.9
23.0 zt 2.7
M
87.6 f 7.9 261.7 f 27.5 104.7 zt 45.6 273.6 zt 14.4 18.7 f 1.2 18.7 f 1.2 186.6 zt 18.1 117.3 f 8 117.3 zt 8 57.1 ?? 11.5 57.1 * 11.5
216.4 f 25.2
M
Flow rate (ml/mm per kg)
31.2
23.0
C
Velocity (ds)
C, calculated by the model; M, measured by ultrasound; R, right; L, left; Co, common.
RESULTS OF THE MATHEMATICAL
TABLE III
53.73 53.88 53.65 53.72 53.09 51.75 51.75 52.42 52.01 52.01 51.35 51.35 35.36 35.36
53.70
53.92
Pressure (mmHg) C
16.6 55.0 15.7 39.3 55.8 3.9 3.9 47.9 23.9 23.9 12.4 12.4 11.5 11.5
28.5
45.0
% Cardiac output C
z R
a
Modeling fetal circulation
137
might be ascribed either to the value of the diameter measured by echography or to the measurement of velocity recorded by Doppler, because of the difficulty to explore the aorta in this area. The model takes into account a diameter (0.42 cm) in the abdominal aorta which is analogous to the reference data (0.4 cm [4]). On the other hand, the angle between the transducer axis and the direction of flow reached at to 60” increasing the relative error on velocity by a factor of 2. The measured value of velocity (28.1 cm/s) and flow rate (186.6 f 18.1 ml/min per kg) in this vessel were straightened by the model, in using the mass conservation principle at kidney bifurcation, to 35.2 cm/s and 247.5 ml/min per kg, respectively. This allows 3.9% (20.5 ml/min per kg) of cardiac output to enter each kidney. A similar value of abdominal aorta flow rate (236.2 ml/mm per kg) was attained by difference between the values measured in thoracic aorta and renal arteries (273.6 and 37.4 ml/min per kg). In the other area, the values of calculated flow rates were comparable to our measurements. The fraction of cardiac output derived from the lungs, 8 1.1 ml/min per kg (15.7% of total cardiac output), is greater than in the fetal lamb (7%, Itskovitz [25]). The total cardiac output value estimated by the model (516.3 ml/min per kg) combined the flow rates of ascending aorta (232.4 ml/min per kg) and pulmonary trunk (283.9 rnl/min per kg). Reed et al. [26] reported similar values for mitral and tricuspid flow rates (232 f 25 and 307 f 30 ml/min per kg, respectively) over 26 to 30 weeks gestation. At the same period, Griffin [27] observed in the thoracic aorta a value of 246 f 30 ml/min per kg, slightly lower than our calculated (288.5 ml/mm/kg) and measured (273.6 f 14.4 ml/min per kg) data. The total placental umbilical flow rate (118.8 ml/min per kg) computed by the model is in agreement with our measured value (114.2 f 11.5 ml/min per kg) and with the results obtained by other authors: 115 f 36 [28], 122 & 21 [27] and 138.7 f 76 ml/min per kg [4]. Such a flow rate, which accounts for about 23% of cardiac output in human fetus, contrasts to the value of 40-45% (180-200 ml/mm per kg) derived from the sheep placenta [25]. In addition, the pressures were estimated by the model. To date, there are no reference on arterial pressure available in the human fetus. Our estimation by the model of the iliac arterial mean pressure (51.3 mmHg) is in the range of the observations reported in fetal lamb for the distal aortic pressure (femoral artery) by Reuss and Rudolph [29], Itskovitz et al. [30] and Rudolph et al. [31] which were 43 f 3, 46.1 f 5.6 and 48 f 4 mmHg, respectively. A good agreement was noted between the results of the model; the data collected by us and the reference data allowing a validation of the mathematical model. It further provides a rationale adjustment to measurements which do not conform to the basic principles of hemodynamic and proposes more appropriate values. This mathematical model of the overall fetal arterial circulation bears potential interesting issues. The model allowed an investigation of velocity in vascular segments difficult to examine routinely by non-invasive methods. Simulations of the different rheological parameters can be undertaken to explore adaptation or pathological evolution of the fetal arterial circulation. References I
Etekey GA, Darms DA, Manson WA and Assali NS: Analog computer simulation of the cardiovascular system of the fetal lamb. In San Diego Symposium for Biomed. Eng., 1963, pp. 172- 180.
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Rankin JHG and Phernetton TM: The construction and usefulness of physical models of the placenta, J Biomech, 12 (1979) 447-452. Reed KL, Anderson CF and Shenker L: Changes in intracardiac Doppler blood flow velocities in fetuses with absent umbilical artery diastolic flow, Am J Obsfet Gynecol, 157 (1987) 774-779. Lingman G and Marsal K: Fetal central blood circulation in the third trimester of normal pregnancy - a longitudinal study. I. Aortic and umbilical blood flow, Early Hum Dev, 13 (1986) 137- 150. Cope1 JA, Grannum PA, Green JJ, Belanger K; Hanna N, Jaffe CC, Hobbins JC and Kleinman CS: Fetal cardiac output in the isoimmunized pregnancy: A pulsed Doppler-echocardiographic study of patients undergoing intravascular intrauterine transfusion, Am J Obster Gynecol. 161 (1989) 361-365. Lambert JW: On the nonlinearities of fluid flow in nonrigid tubes, J Franklin Inst. 266 (1958) 83-102.
1 8 9
Streeter VL, Keitzer WF and Bohr DF: Pulsatile pressure and flow through distensible vessels, Circ Res, 13 (1963) 3-19. Anliker M, Rockwell RL and Ogden E: Nonlinear analysis of flow pulses and shock waves in arteries, Part I: Derivation and properties of mathematical model, ZAMP, 22 (1971) 217-246. Vander Werff TJ: Significant parameters in arterial pressure and velocity development, J Biomech. 7 (1974) 437-441.
10 Kufahl RH and Clark ME: A circle of Willis simulation using distensible vessels and pulsatile flow, J Biomech Eng, 107 (1985) 112-122.
11 Gerrard JH: Numerical analysis and linear theory of pulsatile flow in cylindrical deformable tubes: The testing of a numerical model for blood flow calculation, Med Biol Eng Cornput. 20 (1982) 49-57. 12 Smit CH: On the introduction of viscoelasticity into one-dimensional models of arterial blood flow. Acta Mech, 43 (1982) 15-26. 13 Saito GE and Vander Werff TJ: The importance of viscoelastiaity in arterial blood flow models, J Biomech, 8 (1975) 237-245.
14 Raines JK, Jaffrin MY and Shapiro AH: A computer simulation of arterial dynamics in the human leg, J Biomech, 7 (1974) 77-91. 15 Richtmyer RD and Morton KW: Diff erence Methods for Initial- Value Problems, 2 Edn, Interscience publishers, 1967. 16 Womersley JR: An elastic tube theory of pulse transmission and oscillatory flow in mammalian arteries. WADC tech. Report TR 56-614, Defence Documentation Center, 1957. 17 McDonald DA: Blood Flow in Arteries, Edward Arnold, Ltd., London, 1960. 18 Vignes J: Algorithmes numhiques. Analyse et Mise en Oeuvre, ED. Tech., 1980. 19 Smith GD: Numerical Solution of Partial Differential Equations: Finite Difference Merhods, Clarendon Press, 3 Edn., Oxford, 1985. 20 Euvrard D: Resolution numerique des Equations aux Derivees purtielles, Dlfferencesfinies, Elements finis. Ed. Masson, Paris, 1987. 21 Mitchell AR: Computational Methods in Partial Dij/erenrial Equations, John Wiley and Sons, 1969. 22 Lax PD: Hyperbolic Systems of Conservarions Laws the Mathematical Theory of Shock Waves, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1973. 23 Huisseling HV, Hasaart THM, Ruissen CJ, Muijsers GJJM and Jelte de Haan: Umbilical artery flow velocity waveforms during acute hypoxemia and the relationship with hemodynamic changes in the fetal lamb, Am J Obsre? Gynecol, 161 (1989) 1061-1604. 24 Mooren KDD, Barendregt LG and Wladimiroff JW: Flow velocity waveforms in the human fetal ductus arteriosus during the normal second half of pregnancy, Pediatr Res, 30, 5 (1991) 487-490. 25 Itskovitz J: Maternal-Fetal Hemodynamics (Eds: D Maulik and D McNellis), Perinatology Press, 1987, pp. 13-42. 26 Reed KL, Meijboom EJ, Sahn DJ, Scagnelli SA, Valdescruz LM and Shenker L: Cardiac Doppler flow velocities in human fetuses, Circulafion. 73, 41 (1986). 27 Griffin D, Cohen-Overbeek T and Campbell S: Fetal and utero-placental blood flow, Clin Obsret Gynaecol, 10 (1983) 565-602. 28 Eik-Nes SH, Marsal K, Brubakk AO, Kristofferson K and Ulstein M: Ultrasonic measurement of human fetal blood flow, J Biomed Eng, 4 (1982) 28-36.
Modeling fetal circulation 29 30 31
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Reuss ML and Rudolph AM: Distribution and recirculation of umbilical and systemic venous blood flow in fetal lambs during hypoxia, J Dev Physiol, 2 (1980) 71-84. Itskovitx J, LaGamma EF and Rudolph AM: Baroreflex control of the circulation in chronically instrumented fetal lambs, Circ Res, 52 (1983) 589-596. Rudolph CD, Meyers RL, Paulick RP and Rudolph AM: Effects of ductus venosus obstruction on liver and regional blood flows in the fetal lamb, Pediatr Res, 29 (1991) 347-352.
MATHEMATICAL MODELING BLOOD CIRCULATION
A. GUETTOUCHEa, J.C. CHALLIERb, and A. AZANCOT-BENISIV
127
OF THE HUMAN
FETAL ARTERIAL
Y. ITOE, C. PAPAPANAYOTOU’,
Y. CHERRUAULTa
aVniversitP P.M Curie, Luboratoire MEDIMAT, 1.5, Rue de I’Ecole de Mkdecine, 75270 Paris Cedex 06, bVniversitP P.M. Curie, Laboratoire Biologic de la Reproduction, Brit A, 7, Quai St Bernard, 75252 Paris cedex OS and cService d’Exploration Fonctionnelles. Hbpital Robert DebrP. 48, boulevard Serurier, 75019 Paris (France) (Received Feburary 26th, 1992) (Revision received March 27th, 1992) (Accepted March 27th, 1992)
A mathematical model of the human fetal arterial circulation based on mass and momentum conservation for one-dimensional flow is presented. We simplified the fetal arterial vascular system from the heart to the placenta, defined 16 anatomical segments and studied the characteristics of the vascular system in relation to changes in morphology and hemodynamics. The two-step Lax-Wendroff finite difference scheme was used to solve the system of equations, after introducing the rheological constants, the diameter and length of the segments measured by two-dimensional imaging and the mean arterial velocity at the inlet segments obtained by pulsed Doppler. The model was validated by comparing the numerical results to our non-invasive ultrasound direct measurements and to previous published data. Key words: Mathematical model; Fetus; Blood circulation; Ultrasound
Introduction Research in the area of the fetal physiology has yielded physical and mathematical models of the fetal blood circulation in animals based on experimental data [1,2]. In the human fetus, the development of two-dimensional ultrasound imaging and pulsed Doppler techniques allows non-invasive measurements of vessel size and blood velocity in major arteries and veins [3-51. Clinicians rely on various parameters based on systolic and diastolic flow velocity waveforms for evaluating fetal and placental hemodynamics. There is a crucial need to develop tools in order to interpret local hemodynamic data obtained by non-invasive methods. A mathematical model can be an alternative to explore the overall fetal circulation. Lambert [6], appears to have been the first to apply mathematical methods to blood flow problems, considering non-viscous fluid and one-dimensional flow. In their contribution to blood flow modelling, Streeter et al. [7] presented a basis of mathematical modelling of pulsatile pressure and flow through distensible vessels. Correspondence ro: A. Guettouche, Universite P.M Curie, Laboratoire MEDIMAT, 15, Rue de 1’Ecole de Mkdecine, 75270 Paris Cedex 06, France.
0020-7101/92/$05.00 0 1992 Elsevier Scientific Publishers Ireland Ltd. Printed and Published in Ireland
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Many other authors have applied similar laws of mass and momentum conservation to hemodynamic modeling and differed by the choice of resistance terms, wall elasticity relationships and numerical methods of resolution [8-141. To allow the investigation and prediction of the overall human fetal-placental circulation, we developed a mathematical model based on mass and momentum conservation laws, utilizing ultrasound measurements as input. Our approach was to simplify the human fetal arterial system to 16 anatomical segments. We applied to every vascular segment the equations of mass and momentum conservation and the wall elasticity relationships (state equation) for one-dimensional flow. Such a model requires mean blood velocity at the inlet segments (ascending aorta and pulmonary trunk) which were obtained by pulsed Doppler; the diameter and length of the vascular segments measured by two-dimensional imaging. The quasilinear hyperbolic system of partial differential equations was numerically solved by the two-step Lax-Wendroff finite difference scheme [ 151. The model was validated by comparing the numerical results to our non-invasive ultrasound direct measurements and to previous published data. Anatomical and Mathematical Model Considering its anatomical complexity, the human fetal arterial system has been simplified (Fig. 1). The new configuration starts at the proximity of the cardiac valves and terminates at the placental level. The ascending aorta (segment 1) and the pulmonary trunk (segment 4) represent the inlet segments of the anatomical model. All segments refer to individual blood vessels except the vessels of the cephalic region (carotid, innominate and subclavian arteries) and the pulmonary arteries which were regrouped in segments 2 and 5, respectively. The upper part of the fetal body (head and upper members), lower members and organs as lung, kidney and placenta were represented by peripheral resistances (R2, R13 and R14; R5; R8 and R9; R15 and R16). Each vascular segment was mathematically modelled by a system of three equations: the mass and momentum conservation equations and the elasticity relationships which related the cross-sectional area to pressure. The basic assumptions of the model were: (i) the segment of blood vessel was modelled as a cylindrical and distensible tube of constant thickness; (ii) blood was considered as an incompressible, viscous and Newtonian fluid and its flow was supposed laminar; (iii) pressure and flow velocity were assumed uniform over arterial cross section. Using the above assumptions, the equations of mass and momentum conservation for a one-dimensional flow in each segment were:
(1) c$+v~+_-=_--
i
ap
P
ax
8?r/.~V P
A
(2)
where A, V and P are, respectively the cross-sectional area, the axial velocity and
Modeling fetal circulation
78
R
129
18
Fig. 1. Anatomical model of the human fetal arterial system. Segment number (1) ascending aorta; (2) innominate, carotid and subclavian arteries; (3) aortic isthmus; (4) pulmonary trunk; (5) pulmonary arteries; (6) ductus arteriosus; (7) thoracic aorta; (8) and (9) right and left renal arteries; (10) abdominal aorta; (I 1) and (12) right and left common iliac arteries; (13) and (14) right and left iliac arteries; (15) and (16) right and left umbilical arteries. -: Forwarded flow. The black labels (W) correspond to the peripheral resistance (R) of organs or regions.
the pressure, p stands for the density and p for the viscosity of blood. The unknown functions (A, V, P) depend on time (t) and on the axial coordinate (x). The right side of Eqn. (2) figures the Poiseuille’s formula for laminar steady flow representing only the effects of blood viscosity. It applies strictly to steady flow. For small values (C 10) of the Womersley frequency parameter a! = d(wp/p), where r is the vessel’s radius and w the angular frequency of the oscillatory motion [ 16,171. In fetal ascending aorta (r = 0.5 cm) and umbilical arteries (r = 0.23 cm), the values of the a! parameter were 4.03 and 1.85, respectively. Since arteries are tapered and distensible, the cross-sectional area A varies with
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pressure (P) and distance (x). The previous Eqns. (1) and (2) were completed by the wall elasticity relationship (state equation) given by: A = A(P,X) = A&l
- PDOIEb)
(3)
in which AO, Do, ho are, respectively, cross-sectional area, internal diameter, wall thickness under zero pressure of the vessels and E the Young’s modulus of elasticity. Numerical Solution The previous equations (1,2 and 3) constitute a hyperbolic quasilinear system. We considered as many set of equations as the number of segments. Boundary conditions at the proximal, internal and distal ends of the arterial segments and numerical solution are now examined. Proximal boundary conditions
The blood velocity waveforms were obtained by Doppler at the ascending aorta and pulmonary trunk inlet segments. The variation of blood velocity over a cardiac cycle can be described using Fourier’s approximation. N
V(O,t) =
c
aicos(ht)
+ bi sin&t)
0 I
t I
TO
(4)
i=O vd
To < t I
T
where w = 2rlT is the angular frequency, To and Tare the time of systolic ejection and that of the cardiac cycle, respectively, vd is the diastolic velacity. For a steady flow, we considered the mean blood velocity over a cardiac cycle as a proximal boundary condition at the inlet segment. The value of mean velocity obtained from the internal computer software of the pulse Doppler (Toshiba SSH160) introduced into the model, were 23 cm/s for ascending aorta and 21.4 cm/s for pulmonary trunk. Internal conditions (junctions and bifurcations)
At junctions and bifurcations of three or four vessels, we imposed the mass conservation and the pressure continuity as follows: in the case of a bifurcation:
Qi t&t) = QjCOJ>+ QdOvt) (5) Pj(&t)
= Pi(O,t) = Pk(O,t),
in the case of a junction:
Qi C&t) + Qj(LjJ> = QdOJ) Pi(Li,t)
= Pj(Lj,t) = Pk(O,t),
Modeling fetal circulation
131
where Li denotes the length of the segment number i, arriving at a bifurcation or a junction, Q(O,t) and Qi(Li,f) stand for the instantaneous volume flow rate at the inlet of segments i and at the outlet of segment j, respectively. Distal boundary conditions The effect of distal organs or regions was mimicked without considering their vascular structure by applying a resistance defined as a ratio of the driving pressure across the organ capillary bed to volume flow rate through this bed:
%rs= VW)
- f’JQ(LO (7)
with Q(L,t) = A(L,t) . V(L,t), where P(L,t) and Q(L,t) are the pressure and volume flow rate at the end of the peripheral arterial segment, P, the venous pressure considered constant (P, = 5.3 mmHg), L the length of this segment. The peripheral resistance at the end of segments 2, 5, 8, 9, 13, 14, 15 and 16, constitutes the distal boundary conditions. In default of values of arterial pressure in the human fetus, the resistances were assumed inversely proportional to the organ weight and symmetrically distributed in the kidneys, lower members and placenta (R8 = R9, R13 = R14 and R15 = R16). Thereafter, the resistance values were adjusted under steady flow conditions by the identification method of Vignes [18]. We minimized the following functional:
where Q&) and Q,&) are the calculated and measured flow rates respectively, and n the number of vascular segments (n = 16). Table I shows that the values of the peripheral resistances varied from 14 500 up to 142 000 dyne * s/cm5. Finite differences scheme With the above boundary conditions, we numerically solved the quasilinear hyper-
TABLE I PERIPHERAL
RESISTANCES OF THE MODEL
Regions R2 RS R8 R9 R13 R14 RlS R16
Resistance (dyne.s/cm5) upper part of fetal body lungs right kidney left kidney right lower member left lower member right placental part left placental part
21 ooo 38 000 142000 142000 45 500 45 500 14500 14500
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bolic partial differential equations (1, 2 and 3) by the two-step Lax-Wendroff finite differences scheme, [l&19-22], utilizing Vignes’ method for optimization of the resistance values (Fig. 2). In the first step we calculated the functions A, V and P at the point of coordinates ((j + l/2) Ax, (t + l/2) At) and in the second step we calculated these functions at (j Ax, (n + 1) At) as follows: step 1 (prediction):
(10) n+112
P j+li2=-
&, Do
l-
Ao -$I
(11) ::
>
step 2 (correction):
(13)
(14)
The schemes at proximal and distal boundaries were (15), (16) and (17), respectively: n+l A0
=A;-z
WI: - (AV)“o >
(1%
133
Modelingfetal circulation
Introduction 11, p,
p, E,
V(l),
of
ho,
constants:
Do(i),
V(4).
L(i).
Ax(i),
f
Q,(i)
i=l
to
At 18 1
1 Initialization Resistance Trr=10000,
of
parameters
coefficients: N=loO,
Vignes’ Optieization
R(Z)..
t = 0.001, 1
.
K=O
eathod
of
the
resistance
va I ues 1 Initial
conditions
of
finite
differences
aethod
i Lax-Uendroff t= tiAt
calculation
finite of
A(i),
Minimization
differences
soheee
V(i).
P(i).
of
J = ? i? n i Q i-1
Print A(i).
V(i),
P(i).
the (Q(i)
-
R,(i))’
results P(i). 1 End
Fig. 2. Numerical resolution diagram.
functional
Q(i).
R(i)
(L-U):
i=l
J
to 16
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A. Guettouche et al.
II+1
Piv
= P,,+ RA;+’
pN+’
(17)
with:
where R is the resistance coefficient, At and Ax are the discretization steps of time and space respectively. The quantities of upper indice n were known and the quantities of upper indice n + 1 were to be calculated. At the junctions and bifurcations, we used the same method for discretizing Eqns. (5) and (6). The numerical calculation started with the following initial conditions: V(x, 0) = 0
P(x, 0) = P, A(x, 0) = A&l
(18) - P,DcjEh,,)
At = 0.0002 s and Axi = L/N, where Li and Ni are, respectively, the length and the number of the discretization points of the segment i (i.e. for ascending aorta i = 1, L, = 1.2 cm, N, = 6 and Axr = 0.2 cm). The model requires the rheological parameters (blood density p = 1.06 g/cm3, blood viscosity p = 0.052 dyn . s/cm2, Young’s modulus E = 6 lo6 dyn/cm2, wall thickness ho = 0.08 cm and venous pressure P, = 5.3 mmHg), the diameter and length of the segments measured by two-dimensional imaging (Table II). The length of each equivalent tube (segments 2, 5, 13 and 14) was fixed at 1 cm and its diameter was approximated from regional distribution of cardiac output. Results and Discussion The mean velocities obtained at the inlet segments by Doppler at 27.5 weeks of pregnancy (range 25-30) were introduced as input into the model. Table I shows the values of resistance identified by the Vignes’ method [18]. In this method, the quadratic difference between the flow rate values calculated by the model and those
Modeling fetal circulation
135
TABLE II ECHOGRAPHIC
MEASUREMENTS
OF VESSELS
Diameter and length at 27.5 weeks of pregnancy (range 25-30). R, right; L, left, Co, common.
(3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16)
Vessels
Diameter (cm)
Length (cm)
Ascending aorta Innominate artery Carotid artery Subclavian artery Aortic isthmus Pulmonary trunk Pulmonary arteries Ductus arteriosus Thoracic aorta R. renal artery L. renal artery Abdominal aorta R. co. iliac artery L. co. iliac artery R. iliac artery L. iliac artery R. umbilical artery L. umbilical artery
0.50
1.2
0.34
1.0
0.32 0.57 0.28 0.36 0.46 0.14 0.14 0.42 0.28 0.28 0.22 0.22 0.23 0.23
1.4 0.86 1.0 0.75 5.2 1.2 1.2 1.5 1.0 1.0 1.0 1.0 50.0 50.0
computed from velocity and vessel diameter measurements was minimized with a final accuracy of 7.1%. In the case of the placenta, the resistance (Rp = 7250 dyne . s/cm5) was calculated using the formula: l/Rp = l/R1 5 + i/R16 where R15 and R16 stand for placental resistances at right and left umbilical arteries. The value of placental resistance was comparable to the data observed by Huisseling et al. [23] in sheep between 100 and 130 days gestation: 0.093 mmHg/min per ml or 7438 dyne. s/cm5. These data show that the lowest resistance are located in placental and upper body circulations (i.e., 7250 and 21 000 dyne * s/cm5, respectively). Consequently, they receive the highest flow rate and changing their resistance could have major effects on systemic circulation. The highest resistance was noted in the kidney (142 000 dyne. s/cm5) in which the lowest flow rate can be expected. Pressure (dyne/cm2) and blood flow rate (cm3/s) units were converted to mmHg and to ml * mm/kg, respectively, using a fetal weight 1.25 kg at 27.5 weeks gestation [4]. The mean values of the various parameters, which were evaluated in the middle of each vascular segment are depicted in Table III. A good agreement was noticed between the velocities calculated by the model and those measured by Doppler, except in abdominal aorta. In Ductus arteriosus, the velocity calculated by the model 39.5 cm/s compares well with that measured by us (37.4 f 9.2 cm/s) and is within the values (24.8 f 3.9 at 20 weeks versus 50.1 f 14.5 cm/s at 38 weeks) reported by Mooren et al. 1241.In the abdominal aorta, the value computed by the model (35.2 cm/s) exceeded the mean velocity measured by Doppler (28.1 f 2.7 cm/s). This discrepancy
MODEL
(3) (4) (5) (6) (7) (8) (9) (IO) (11) (12) (13) (14) (15) (16)
(2)
(1)
Vessel
c
Ascending aorta Innominate artery Carotid artery Subclavian artery Aortic isthmus Pulmonary trunk Pulmonary arteries Ductus arteriosus Thoracic aorta R. renal artery L. renal artery Abdominal aorta R. co. iliac artery L. co. iliac artery R. iliac artery L. iliac artery R. umbilical artery L. umbilical artery 2.3 1.6 1.6 2.7 2.7 2.7
zt 5.7 f 5.7
f f f zt zt zt
?? 9.2
zt 2.1 f 2.2 85.5 283.9 81.1 202.9 288.5 20.5 20.5 247.5 123.7 123.7 64.2 64.2 59.4 59.4
22.7 21.4 37.4 34.3 25.3 25.3 28.1 39.7 39.7 28.6 28.6
21.2 21.4 26.4 39.5 33.9 27.3 27.3 35.2 40.3 40.3 34.2 34.2 28.9 28.9
232.4
C
146.9
23.0 zt 2.7
M
87.6 f 7.9 261.7 f 27.5 104.7 zt 45.6 273.6 zt 14.4 18.7 f 1.2 18.7 f 1.2 186.6 zt 18.1 117.3 f 8 117.3 zt 8 57.1 ?? 11.5 57.1 * 11.5
216.4 f 25.2
M
Flow rate (ml/mm per kg)
31.2
23.0
C
Velocity (ds)
C, calculated by the model; M, measured by ultrasound; R, right; L, left; Co, common.
RESULTS OF THE MATHEMATICAL
TABLE III
53.73 53.88 53.65 53.72 53.09 51.75 51.75 52.42 52.01 52.01 51.35 51.35 35.36 35.36
53.70
53.92
Pressure (mmHg) C
16.6 55.0 15.7 39.3 55.8 3.9 3.9 47.9 23.9 23.9 12.4 12.4 11.5 11.5
28.5
45.0
% Cardiac output C
z R
a
Modeling fetal circulation
137
might be ascribed either to the value of the diameter measured by echography or to the measurement of velocity recorded by Doppler, because of the difficulty to explore the aorta in this area. The model takes into account a diameter (0.42 cm) in the abdominal aorta which is analogous to the reference data (0.4 cm [4]). On the other hand, the angle between the transducer axis and the direction of flow reached at to 60” increasing the relative error on velocity by a factor of 2. The measured value of velocity (28.1 cm/s) and flow rate (186.6 f 18.1 ml/min per kg) in this vessel were straightened by the model, in using the mass conservation principle at kidney bifurcation, to 35.2 cm/s and 247.5 ml/min per kg, respectively. This allows 3.9% (20.5 ml/min per kg) of cardiac output to enter each kidney. A similar value of abdominal aorta flow rate (236.2 ml/mm per kg) was attained by difference between the values measured in thoracic aorta and renal arteries (273.6 and 37.4 ml/min per kg). In the other area, the values of calculated flow rates were comparable to our measurements. The fraction of cardiac output derived from the lungs, 8 1.1 ml/min per kg (15.7% of total cardiac output), is greater than in the fetal lamb (7%, Itskovitz [25]). The total cardiac output value estimated by the model (516.3 ml/min per kg) combined the flow rates of ascending aorta (232.4 ml/min per kg) and pulmonary trunk (283.9 rnl/min per kg). Reed et al. [26] reported similar values for mitral and tricuspid flow rates (232 f 25 and 307 f 30 ml/min per kg, respectively) over 26 to 30 weeks gestation. At the same period, Griffin [27] observed in the thoracic aorta a value of 246 f 30 ml/min per kg, slightly lower than our calculated (288.5 ml/mm/kg) and measured (273.6 f 14.4 ml/min per kg) data. The total placental umbilical flow rate (118.8 ml/min per kg) computed by the model is in agreement with our measured value (114.2 f 11.5 ml/min per kg) and with the results obtained by other authors: 115 f 36 [28], 122 & 21 [27] and 138.7 f 76 ml/min per kg [4]. Such a flow rate, which accounts for about 23% of cardiac output in human fetus, contrasts to the value of 40-45% (180-200 ml/mm per kg) derived from the sheep placenta [25]. In addition, the pressures were estimated by the model. To date, there are no reference on arterial pressure available in the human fetus. Our estimation by the model of the iliac arterial mean pressure (51.3 mmHg) is in the range of the observations reported in fetal lamb for the distal aortic pressure (femoral artery) by Reuss and Rudolph [29], Itskovitz et al. [30] and Rudolph et al. [31] which were 43 f 3, 46.1 f 5.6 and 48 f 4 mmHg, respectively. A good agreement was noted between the results of the model; the data collected by us and the reference data allowing a validation of the mathematical model. It further provides a rationale adjustment to measurements which do not conform to the basic principles of hemodynamic and proposes more appropriate values. This mathematical model of the overall fetal arterial circulation bears potential interesting issues. The model allowed an investigation of velocity in vascular segments difficult to examine routinely by non-invasive methods. Simulations of the different rheological parameters can be undertaken to explore adaptation or pathological evolution of the fetal arterial circulation. References I
Etekey GA, Darms DA, Manson WA and Assali NS: Analog computer simulation of the cardiovascular system of the fetal lamb. In San Diego Symposium for Biomed. Eng., 1963, pp. 172- 180.
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Rankin JHG and Phernetton TM: The construction and usefulness of physical models of the placenta, J Biomech, 12 (1979) 447-452. Reed KL, Anderson CF and Shenker L: Changes in intracardiac Doppler blood flow velocities in fetuses with absent umbilical artery diastolic flow, Am J Obsfet Gynecol, 157 (1987) 774-779. Lingman G and Marsal K: Fetal central blood circulation in the third trimester of normal pregnancy - a longitudinal study. I. Aortic and umbilical blood flow, Early Hum Dev, 13 (1986) 137- 150. Cope1 JA, Grannum PA, Green JJ, Belanger K; Hanna N, Jaffe CC, Hobbins JC and Kleinman CS: Fetal cardiac output in the isoimmunized pregnancy: A pulsed Doppler-echocardiographic study of patients undergoing intravascular intrauterine transfusion, Am J Obster Gynecol. 161 (1989) 361-365. Lambert JW: On the nonlinearities of fluid flow in nonrigid tubes, J Franklin Inst. 266 (1958) 83-102.
1 8 9
Streeter VL, Keitzer WF and Bohr DF: Pulsatile pressure and flow through distensible vessels, Circ Res, 13 (1963) 3-19. Anliker M, Rockwell RL and Ogden E: Nonlinear analysis of flow pulses and shock waves in arteries, Part I: Derivation and properties of mathematical model, ZAMP, 22 (1971) 217-246. Vander Werff TJ: Significant parameters in arterial pressure and velocity development, J Biomech. 7 (1974) 437-441.
10 Kufahl RH and Clark ME: A circle of Willis simulation using distensible vessels and pulsatile flow, J Biomech Eng, 107 (1985) 112-122.
11 Gerrard JH: Numerical analysis and linear theory of pulsatile flow in cylindrical deformable tubes: The testing of a numerical model for blood flow calculation, Med Biol Eng Cornput. 20 (1982) 49-57. 12 Smit CH: On the introduction of viscoelasticity into one-dimensional models of arterial blood flow. Acta Mech, 43 (1982) 15-26. 13 Saito GE and Vander Werff TJ: The importance of viscoelastiaity in arterial blood flow models, J Biomech, 8 (1975) 237-245.
14 Raines JK, Jaffrin MY and Shapiro AH: A computer simulation of arterial dynamics in the human leg, J Biomech, 7 (1974) 77-91. 15 Richtmyer RD and Morton KW: Diff erence Methods for Initial- Value Problems, 2 Edn, Interscience publishers, 1967. 16 Womersley JR: An elastic tube theory of pulse transmission and oscillatory flow in mammalian arteries. WADC tech. Report TR 56-614, Defence Documentation Center, 1957. 17 McDonald DA: Blood Flow in Arteries, Edward Arnold, Ltd., London, 1960. 18 Vignes J: Algorithmes numhiques. Analyse et Mise en Oeuvre, ED. Tech., 1980. 19 Smith GD: Numerical Solution of Partial Differential Equations: Finite Difference Merhods, Clarendon Press, 3 Edn., Oxford, 1985. 20 Euvrard D: Resolution numerique des Equations aux Derivees purtielles, Dlfferencesfinies, Elements finis. Ed. Masson, Paris, 1987. 21 Mitchell AR: Computational Methods in Partial Dij/erenrial Equations, John Wiley and Sons, 1969. 22 Lax PD: Hyperbolic Systems of Conservarions Laws the Mathematical Theory of Shock Waves, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1973. 23 Huisseling HV, Hasaart THM, Ruissen CJ, Muijsers GJJM and Jelte de Haan: Umbilical artery flow velocity waveforms during acute hypoxemia and the relationship with hemodynamic changes in the fetal lamb, Am J Obsre? Gynecol, 161 (1989) 1061-1604. 24 Mooren KDD, Barendregt LG and Wladimiroff JW: Flow velocity waveforms in the human fetal ductus arteriosus during the normal second half of pregnancy, Pediatr Res, 30, 5 (1991) 487-490. 25 Itskovitz J: Maternal-Fetal Hemodynamics (Eds: D Maulik and D McNellis), Perinatology Press, 1987, pp. 13-42. 26 Reed KL, Meijboom EJ, Sahn DJ, Scagnelli SA, Valdescruz LM and Shenker L: Cardiac Doppler flow velocities in human fetuses, Circulafion. 73, 41 (1986). 27 Griffin D, Cohen-Overbeek T and Campbell S: Fetal and utero-placental blood flow, Clin Obsret Gynaecol, 10 (1983) 565-602. 28 Eik-Nes SH, Marsal K, Brubakk AO, Kristofferson K and Ulstein M: Ultrasonic measurement of human fetal blood flow, J Biomed Eng, 4 (1982) 28-36.
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Reuss ML and Rudolph AM: Distribution and recirculation of umbilical and systemic venous blood flow in fetal lambs during hypoxia, J Dev Physiol, 2 (1980) 71-84. Itskovitx J, LaGamma EF and Rudolph AM: Baroreflex control of the circulation in chronically instrumented fetal lambs, Circ Res, 52 (1983) 589-596. Rudolph CD, Meyers RL, Paulick RP and Rudolph AM: Effects of ductus venosus obstruction on liver and regional blood flows in the fetal lamb, Pediatr Res, 29 (1991) 347-352.
