INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES APPENDIX A APPENDIX B

OBSTETRIC DOPPLER ultrasonography is commonly used for evaluation of the flow velocity waveform (FVW) from fetal and maternal blood vessels (16). These waveforms are used to calculate Doppler indexes (DI) such as the systolic-to-diastolic ratio (S/D), the pulsatility index (PI), and the resistance index (RI) (see APPENDIX A). Because the fetus is completely dependent on the supply of oxygen and nutrients from the placenta, noninvasive examination of blood flow through the umbilical circulation is very important for assessment of fetal well-being. Because the DI provide empirical information on peripheral resistance to umbilical circulation, they are widely used for clinical diagnosis of high-risk pregnancies such as pregnancy-induced hypertension and of complications as seen in fetal intrauterine growth restriction (28, 29).

Direct noninvasive measurements of fetal blood flow through the umbilical arteries were conducted with modified ultrasonographic probes that combine the Doppler and B-mode techniques and allow for simultaneous measurement of local blood velocity and vessel diameter (7-9, 11, 12, 24, 30). These methods are not routinely used clinically because the arteries are subjected to pulsatile flow that continuously changes their diameter, rendering the measured data inaccurate and nonreproducible and thereby limited in diagnostic value (13).

The existing computational studies for analysis of the relationship between DI and fetal blood flow rate are based on lumped-parameter models of electrical elements for simulations of the umbilical and placental arteries (25, 26). These models are analogous electrical circuits of resistors and capacitors that represent the umbilical arteries, the first radial branches, and the placental villous tree. They were used to examine the effect of different physiological variables on the PI. Similar electric networks were also used to study the correlation between normal and pathological placental blood flow and the DI (14, 27). Recently, hemodynamic models were developed to study uteroplacental and fetomaternal blood flow for comparison with Doppler velocity waveforms (18, 23). However, the correlation between the measured values of DI and umbilical blood flow as well as their dependence on anatomic and physiological parameters (e.g., pressure gradients or arterial geometry and wall properties) have not been satisfactorily studied with a distributed physical model.

The rapidly expanding utilization of ultrasonography in prenatal care for clinical diagnosis of complications in pregnancy has led to the need for more in-depth and sophisticated analysis of the interpretation of Doppler measurements. In this work, we used a mathematical model for pulsatile flow of a viscous fluid in an inflated tube to study the dependence of DI on anatomic and physiological parameters and their correlation with umbilical blood flow.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES APPENDIX A APPENDIX B

Description of model. The umbilical cord and the arteries along its axis are very coiled at parturition. However, in vivo images taken during gestation with optic fibers (19) revealed a fairly uncoiled cord with longitudinal curvatures much larger than its diameter during the first 5-6 mo. During the last 2-3 mo of gestation the umbilical cord became coiled as a result of increased fetal movements. Accordingly, it is reasonable to assume that a first-order physical model of umbilical blood flow will adequately simulate the umbilical artery between the umbilicus and the placenta by a finite elastic cylinder with thin walls (Fig. 1). The tube is able to undergo local deformations in both longitudinal (z) and circumferential () directions.

View larger version (18K): [in this window] [in a new window]   Fig. 1.   Umbilical blood flow model. Pu, pressure at umbilicus insert; Pp, pressure at placental insert; r, radial direction; z, longitudinal direction; L, length.

Steady flow in an elastic tube. The steady component of the flow through an elastic tube is assumed to be laminar (parabolic distribution) and to have a local resistance as in Poiseuille's flow (10). The radius of the elastic tube R₀(z) changes with the tube mean internal pressure P₀(z), which varies along the z axis. Accordingly, the pressure-flow relationship can be described in a way similar to laminar flow by

(1)

where Q_s is the steady volumetric flow rate and µ is the fluid viscosity.

(2)

(3)

(4)

(5)

(6)

(7)

Pulsatile flow in an elastic tube. The unsteady component of the pulsatile flow is assumed to be induced by propagation of small waves in a pressurized elastic tube. The mathematical approach is based on the classic model for the fluid-structure interaction problem, which describes the dynamic equilibrium between the fluid and the thin tube wall (4, 31, 32). The dynamic equilibrium is expressed by the hydrodynamic equations (Navier-Stokes) for the incompressible fluid flow and the equations of motion for an elastic tube, which are coupled together by the boundary conditions (BC) at the fluid-wall interface. The motion of the liquid is described in a fixed laboratory coordinate system, and the dynamic equilibrium of a tube element in its deformed state is expressed in a Lagrangian (material) coordinate system, which is attached to the surface of the tube (Fig. 2).

View larger version (18K): [in this window] [in a new window]   Fig. 2.   Mechanics of arterial wall. A: axisymmetric wall deformation. B: element of tube wall under biaxial loading. R0, elastic tube radius; , , , fixed system coordinates; , , , Lagrangian coordinates; T, internal stress.

(8)

(9)

(10)

(11)

(12)

Umbilical blood flow. The complete description of blood flow in the umbilical artery is obtained by adding the unsteady component (Eq. 9) to the steady component (Eq. 7). Thus the longitudinal velocity in the umbilical artery as a function of axial and radial location is given by

(13)

The time-dependent umbilical blood flow rate at a given point z between the umbilicus and the placenta is

(14)

The umbilical cord normally consists of two umbilical arteries. Accordingly, the mean umbilical flow rate at a given point z in one period is

(15)

where T is the time of one heartbeat.

Computational technique. The longitudinal blood velocity (W) at any location within the umbilical artery was computed from Eq. 13 for a given set of independent system parameters and an assumed pressure waveform within the tube. Following Womersley (31, 32), we assumed a harmonic wave for the oscillatory component of the pressure (see Eq. B15), which is given in Eq. 10 for the first harmonic that represents the heart rate. A more realistic blood pressure wave is composed of k harmonics and can be described by the following Fourier series

(16)

where A_n is a complex number and is given by an amplitude (M_n) and phase shift (_n)

(17)

In the absence of data from direct measurement of the pressure waveform in umbilical arteries of humans or experimental animals, we used here the first six harmonics listed in Table 1, which were derived from pressure measurements in the pulmonary artery of a dog (5). This choice may be justified by the fact that the pulmonary arterial pressure of the fetus is the same as its systemic arterial pressure (20), and the systolic and diastolic values of this waveform are similar to those of umbilical blood pressure (70 and 45 mmHg, respectively).

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES APPENDIX A APPENDIX B

The model directly provided the axial velocity at any given time and location within the tube for a specified anatomic and physiological variable of umbilical blood flow, and this allowed the derivation of FVW at selected measurement locations. Figure 3 depicts envelope FVW for a normal sample and shows the reference parameters together with the variations for extreme values (Table 2) of the system parameters. The DI for the reference set of parameters at the umbilicus end (z = 0) were PI = 1.00, RI = 0.62, and S/D = 2.62, and the mean umbilical blood flow (Q_mean) was 92 ml/min.

View larger version (29K): [in this window] [in a new window]   Fig. 3.   Variation of flow velocity waveforms (FVW) at fetal site (z = 0) with anatomic and physiological parameters. Thick curve is FVW for reference values (Table 2) that represent normal umbilical blood flow. W, velocity; t, time; d, diameter; E, Young's modulus; h, wall thickness; R0, tube radius; , blood density; µ, blood viscosity; Pmax, peak input pressure; P0,P, mean arterial pressure at placental insert; P0,U, mean arterial pressure at umbilical insert.

The present model enabled the analysis of the dependence of DI on the location of Doppler ultrasound measurements. Accordingly, the DI were evaluated from FVW computed for the set of reference parameters at 10 different radial locations (r* = r/R₀ = 0-0.9) on the fetal umbilicus. Similarly, the DI were also computed at 11 equally spaced locations (z* = z/L = 0-1.0) along the umbilical artery (from umbilicus to placenta). The results are shown in Fig. 4. The DI are almost independent of radial location for most of the cross section, whereas pulsatility decreased (up to 50%) toward the placenta.

View larger version (15K): [in this window] [in a new window]   Fig. 4.   Spatial variation of Doppler indexes (DI). Left: radial variation at z = 0. Right: axial variation for indexes extracted from envelope FVW. , pulsatility index (PI); , resistance index (RI); , systolic-to-diastolic ratio (S/D); r* and z*, radial and axial locations within tube, respectively.

The role of arterial anatomy in the determination of DI was examined by computing FVW for a range of arterial diameters (d = 1.0-4.4 mm), lengths (L = 20-150 cm), wall thicknesses (h/R₀ = 0.05-0.25), and elastic moduli (E = 1-8 × 10⁵ Pa). The computed DI for arterial diameter and length are shown in Fig. 5 as are the corresponding Q_mean. The vertical lines (indicated by "Ref") represent the results for the reference values given in Table 2. Generally, the DI decreased as the artery diameter increased or its length decreased. Q_mean increased with increases in the arterial diameter and decreased as the arterial length increased. In an artery with increased wall thickness the DI decreased, but Q_mean was almost unaffected (Fig. 6). The role of increasing Young's modulus (E) of the artery is mechanically equivalent to that of thickening its wall thickness; thus the variability of DI and Q_mean is similar to that of wall thickness (Fig. 6).

View larger version (24K): [in this window] [in a new window]   Fig. 5.   Variability of DI and mean umbilical blood flow rate (Qmean) with d and L of umbilical artery. , PI; , RI; , S/D; Ref, results for reference values given in Table 2.

View larger version (23K): [in this window] [in a new window]   Fig. 6.   Variability of DI and Qmean with E and h/R0 of umbilical artery. , PI; , RI; , S/D.

The contribution of blood properties to the pattern of DI and the corresponding blood flow rate was investigated by varying blood density ( = 500-2,150 kg/m³) and viscosity (µ = 0.01-0.08 Poise). The results are depicted in Fig. 7 and show that DI decreased as blood density increased or viscosity decreased. The mean blood flow rate was independent of blood density but decreased as blood viscosity increased.

View larger version (24K): [in this window] [in a new window]   Fig. 7.   Variability of DI and Qmean with  and µ. , PI; , RI; , S/D.

The influence of physiological parameters was studied by varying blood pressure and FHR. Figure 8 shows that FHR had no effect on either DI or the mean blood flow rate. To examine the contribution of peak input pressure (steady  unsteady), we varied the reference value of P_max by multiplying the fundamental harmonic of the unsteady pressure to yield a range of values around the reference value (P_max = 52-96 mmHg) with negligible changes in the mean arterial pressure (55-54 mmHg). The resulting DI increased with P_max, whereas Q_mean was almost unaffected (Fig. 8).

View larger version (24K): [in this window] [in a new window]   Fig. 8.   Variability of DI and Qmean with fetal heart rate (FHR) and Pmax (umbilicus). , PI; , RI; , S/D; bpm, beats/min.

Variation of the mean arterial pressure at the placental insert (P_0,P) in the range of 5-45 mmHg with maintenance of the mean arterial pressure at the umbilicus insert (P_0,U) constant at 50 mmHg is shown in Fig. 9. As downstream pressure increased, Q_mean decreased and the DI increased. Variation of P_0,U in the range of 15-75 mmHg while P_0,P was held constant at 10 mmHg resulted in peak pressures of 25-85 mmHg. The corresponding DI decreased as P_0,U increased, whereas Q_mean increased (Fig. 9). Another simulation was conducted by changing P_0,U and P_0,P in such a way that the ratio P_0,P/P_0,U = 0.5 was kept constant. This yielded results that are similar to the simulations with P_0,P held constant at 10 mmHg (Fig. 9).

View larger version (23K): [in this window] [in a new window]   Fig. 9.   Variability of DI and Qmean with mean driving pressures P0,U, P0,P, and both P0,U and P0,P while their ratio is held constant. , PI; , RI; , S/D.

The large volume of computed values of DI and Q_mean for a range of selected anatomic and physiological parameters (Table 2) allowed investigation of the correlation between the blood flow rate and the DI. For this purpose we plotted Q_mean versus each of the DI for six system parameters that were found to affect the measured DI (Fig. 10). The diameter of the umbilical artery was found to be the most influential parameter, whereas peak umbilicus pressure had the least effect.

View larger version (29K): [in this window] [in a new window]   Fig. 10.   Relationships between Qmean and DI for different anatomic and physiological parameters. Extreme values of each parameter are marked.

The FVW is strongly dependent on the spectral content of the input pressure wave. Attempts were made to simulate an abnormal FVW with a postsystolic notch, a condition that is known to be representative of high-resistance circulation and to have good correlation to poor obstetrical outcome (22). For this purpose, the second and fourth harmonics were multiplied by 2 and the third harmonic by 3. The resulting pressure wave and FVW at three locations along the artery in comparison with the corresponding waves in normal conditions are shown in Fig. 11. It can be seen that the postsystolic notch depth becomes smaller at locations closer to the placenta. The DI for the dicrotic notch condition measured at z = 0 (umbilicus) were PI = 1.34, RI = 0.8, and S/D = 4.88, values that are very close to those in normal conditions.

View larger version (30K): [in this window] [in a new window]   Fig. 11.   Normal versus pathological (postsystolic notch) FVW for simulated Doppler measurements at 3 sites along umbilical artery.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES APPENDIX A APPENDIX B

A mathematical model of blood flow in the umbilical artery was used to examine the variability of DI with respect to anatomic and physiological parameters that dictate fetal blood flow. Knowing the blood flow field at any axial and radial location of the umbilical artery provided the FVW and enabled a detailed investigation of the spatial variability of DI. In addition, a direct correlation between the estimated DI and umbilical blood flow rate could be obtained from the model.

Measurement site. Medical ultrasound machines including Doppler instruments usually measure the envelope FVW for calculations of DI. In this study, we investigated the variability of DI that were derived at different radial locations (Fig. 4) and from the envelope FVW. The difference between DI calculated at different radial locations (but away from the arterial wall) are relatively small and may be of the order of magnitude of the system resolution. The DI calculated from the envelope FVW were almost identical to those measured at the FVW from the centerline. Hence, the exact radial site of measurement does not affect the accuracy of the obtained DI. On the other hand, the longitudinal site of Doppler measurements greatly affects the FVW and, as a result, the DI as well (Fig. 4). Accordingly, it is useful to conduct measurements at a defined location to ensure reproducibility and provide data that can be used for comparison. Because only the umbilicus (the fetal side) and the placental insert of the umbilical artery can be identified with certainty, it is now largely recommended to measure DI of the umbilical artery on the placental site (1).

Anatomic factors. The diameter and the length of the umbilical artery have a strong effect on the estimated DI. The results showed that either long umbilical arteries or small diameters yielded increased values of DI (Fig. 5), which can be explained by the increased resistance to blood flow as one would expect in a steady laminar flow according to Poiseuille law. The prediction of DI for small-diameter arteries can be related to pathologies and abnormalities such as arterial occlusions or the presence of a single artery. Pernoll (21) reported that 14% of infants with a single umbilical artery will die perinatally and >50% of them will have structural defects. The decrease of DI as the gestational age increases toward the end of pregnancy (2) may be explained by the increase of the umbilical artery diameter.

Blood properties. Variation in blood viscosity induces changes in the expected DI similar to those induced by changing arterial length because both have a similar contribution to the resistance to blood flow (the denominator of Poiseuille's law for laminar flow). Increasing blood viscosity increases the DI (Fig. 7); however, extreme changes in blood viscosity occur very rarely. For example, in abnormal conditions such as marked anemia or Rh isoimmunization, blood viscosity may decrease and, as a result, smaller values of DI are to be expected. On the other hand, the density of blood has very little effect on blood flow rate but induces changes in the expected DI (Fig. 7) like those for different diameters of the artery (Fig. 5) because it appears in the expressions for the pulsatile component of blood velocity (Eqs. 8 and 9).

Blood pressure. The influence of blood pressure was examined from different aspects because the umbilical arterial pressure was assumed to be composed of an oscillating component superimposed on a constant pressure level that decreases along the artery toward the placenta. Examination of the effect of maximal blood pressure at the fetal end of the umbilical artery (umbilicus) was conducted by variation of the fundamental harmonic of the unsteady component with negligible changes in mean blood pressure and blood flow rate (Fig. 8). Figure 8 also shows that FHR does not affect blood flow rate and the DI. A similar result was also obtained in other models that used a sinusoidal function for the pressure wave rather than a realistic signal (26). Nevertheless, DI measured from the umbilical artery tend to decrease slightly with FHR over the normal clinical range (26).

Abnormal waveforms. In clinical practice, the shape of the FVW is also examined for cases with the absence of end-diastolic flow or the presence of a postsystolic notch. In this study, we simulated irregular FVW by changing the spectral content of the input pressure wave. In particular, we were interested in simulations of a postsystolic notch on the FVW, which usually reflects a high distal resistance to umbilical blood flow even in cases with normal DI. For this purpose, we conducted simulations of cases in which we changed the magnitude of some harmonics (either separately or in combinations) and found that increasing the ratio between the second harmonic and the first harmonic generates a postsystolic notch on the FVW. The values of DI from the FVW with the notch were similar to normal values except for S/D. Moreover, we conducted simulations with this abnormal FVW and varied the diameter of the umbilical artery (as shown for normal FVW in Fig. 5) and obtained the same variability of the DI as with normal FVW.