## Abstract

Umbilical artery Doppler velocimetry waveform notching has long been associated with umbilical cord abnormalities, such as distortion, torsion, and/or compression (i.e., constriction). The physical mechanism by which the notching occurs has not been elucidated. Flow velocity waveforms (FVWs) from two-dimensional pulsatile flows in a constricted channel approximating a compressed umbilical cord are analyzed, leading to a clear relationship between the notching and the constriction. Two flows with an asymmetric, semi-elliptical constriction are computed using a stabilized finite-element method. In one case, the constriction blocks 75% of the flow passage, and in the other the constriction blocks 85%. Channel width and prescribed flow rates at the channel inflow are consistent with typical cord diameters and flow rates reported in the literature. Computational results indicate that waveform notching is caused by flow separation induced by the constriction, giving rise to a vortex (core) wave and associated eddies. Notching in FVWs based on centerline velocity (centerline FVW) is directly related to the passage of an eddy over the point of measurement on the centerline. Notching in FVWs based on maximum cross-sectional velocity (envelope FVW) is directly related to acceleration and deceleration of the fluid along the vortex wave. Results show that notching in envelope FVW is not present in flows with less than a 75% constriction. Furthermore, notching disappears as the vortex wave is attenuated at distances downstream of the constriction. In the flows with 75 and 85% constriction, notching of the envelope FVW disappears at ∼3.8 and ∼4.3 cm (respectively) downstream of the constriction. These results are of significant medical importance, given that envelope FVW is typically measured by commercial Doppler systems.

- pulsatile channel flow
- flow velocity waveform, computational fluid dynamics
- umbilical cord abnormality

the umbilical cord is the structure that channels the circulation of blood between the fetus and the placenta during pregnancy. It consists of two arteries and a single vein embedded in Wharton's jelly. The cord may twist and turn, given the fact that, throughout pregnancy, the cord is floating in amniotic fluid within the amniotic cavity. Knots (called true knots) can result from complete looping of the umbilical cord through these movements.

The majority of studies suggest that true knot formation occurs in early pregnancy (7). The incidence of true knots in the umbilical cord is ∼1% of live-born infants and 5% of stillborns (6). Fox (4) reported that true knots are associated with a perinatal mortality rate of 8–11%. When true knots do occur, the mechanical compression of the umbilical vein and arteries could compromise the blood flow between the fetus and the placenta, depending on how tight the knot is.

Advances in ultrasound imaging and Doppler technology make it possible to observe patterns that are highly suggestive of true knots, alerting the obstetrician to follow the pregnancy more closely, looking for a possible compromise of blood flow to the fetus.

For example, Jakobi et al. (6) associated the presence of a true knot of the cord with a systolic notch in umbilical artery flow velocity waveforms (FVWs), which, when present, should “alert the obstetrician to the possibility of the presence of a true knot or other abnormalities in the cord.” FVW consists of the time history of velocity at any fixed location within the artery. Notching in FVW is defined as a period of deceleration during a larger period of overall or net acceleration. Conversely, notching is also defined as a period of acceleration during a larger period of overall deceleration.

Although umbilical artery Doppler velocimetry waveform notching has also been suspected of being associated with other umbilical cord abnormalities, such as cord entanglement in monoamniotic twins, velamentous cord insertion, and umbilical artery narrowing (11), the notching has been described more often in cases of a true knot.

The mechanism by which notching occurs in FVWs of the umbilical artery has not been elucidated. Plausible causes are as follows: distortion, torsion, and/or compression (i.e., constriction) (11). Our study consists of various numerical simulations of pulsatile blood flow through a constricted channel modeling or approximating flow in a compressed artery in the umbilical cord. Our goal is to establish a relationship between the constriction and the appearance of the notch in FVWs downstream of the constriction.

We consider pulsatile laminar flow in a two-dimensional channel with an asymmetric, semi-elliptical constriction, shown in Fig. 1. Similar flows have been simulated numerically by a number of researchers. For example, Tutty (13) studied pulsatile flow in a constricted channel to understand the oscillating wall shear stress and its relationship to the development of atherosclerosis in arterial vessels. Rosenfeld and Einav (12) studied the effect of constriction size on the dynamics of the resulting flow. The study of Tutty (13) was performed with either a sinusoidal or a nonsinusoidal (physiological) waveform boundary condition at the inflow. Li and Yamaguchi (9) extended the work of Tutty (13) to a wide range of nonsinusoidal pulsatile inflows to understand the effect of the prescribed inflow waveform on the flow dynamics. Further studies of two- and three-dimensional constricted flows can be found in the review by Berger and Jou (2). In revisiting pulsatile constricted flow, we focus on FVWs induced by the constriction. Our motivation is not the classical flow constriction due to atherosclerosis, but rather a constriction that may be due to the formation of a true knot in the umbilical cord.

## METHODS

#### Governing equations.

In our computations, the flow is governed by the two-dimensional, unsteady, incompressible Navier-Stokes equations, representing conservation of momentum (*Eq. 1*) and mass (*Eq. 2*), respectively:
^{3}, and viscosity is in the range 0.001 N·s·m^{−2} < μ < 0.008 N·s·m^{−2}, as referenced by Womersley (16). In our cases, we use μ = 0.004 N·s·m^{−2}. Solution of the Navier-Stokes equations yields the two-dimensional fluid velocity vector, (*u*, v), and the fluid static pressure, *p*, as functions of space, (*x*, *y*), and time, *t*. Note that *u* is the velocity component in the streamwise direction, and v is the velocity component in the spanwise direction.

In the case of pulsatile channel flow considered here, the flow may be characterized by the Strouhal number (St) and Reynolds number (Re). The Strouhal number is representative of the oscillation frequency characterizing the flow and, following Tutty (13), can be defined as
*L* is the unperturbed (nonconstricted) channel width, *Q*_{peak} is the peak volumetric flow rate per unit width of channel, resulting from the prescribed oscillating velocity profile at the channel inflow (to be defined further below), and *T* is the period of oscillation of the prescribed inflow. The Reynolds number reflects the importance of inertial forces relative to viscous forces and can be defined as

#### Problem formulation: channel domain, initial conditions, and boundary conditions.

The Navier-Stokes equations are solved numerically inside a constricted channel of unperturbed width *L* = 0.42 cm (Fig. 1), which is the maximum umbilical artery diameter during pregnancy reported by Weissman et al. (15). Typically, the length of the umbilical artery is on the order of 0.6 m [see Table 2 of van den Wijngaard et al. (14)]. In this study, we focus on the flow structures and FVW induced by the constriction. Thus we have taken a much shorter length of 0.1 m, but sufficiently long such that the outflow boundary condition does not impact the flow structures induced by the constriction, located 0.08915 m away from the outflow (i.e., the right-hand side end of the domain; see Fig. 1).

Initially, the flow is taken to be at rest; thus the initial condition consists of *u* = v = 0 at *t* = 0. Furthermore, *p* = 0 at *t* = 0 as well.

Next, we describe boundary conditions for the model with respect to the *x*-*y* coordinate system shown in Fig. 1. Note that the origin of this coordinate system is at the bottom left-hand corner of the domain.

At the inflow (at *x* = 0), velocity is prescribed at all times. At the outflow (at *x* = 0.1 m), velocity is left free. Instead, pressure is set as *p* = 0 for all times. Furthermore, no-slip boundary conditions are assigned to the velocity at the channel walls (i.e., *u* = v = 0 at channel walls for all times).

The inflow boundary condition at *x* = 0 is given by a pulsating parabolic velocity profile assigned to the *x*-component of the velocity:
*y*-component of the velocity at the inflow is set to zero [v(*x* = 0, *y*, *t*) = 0]. Note that the inflow *x*-velocity profile in *Eq. 5* possesses a parabolic shape, varying between 0 at the channel walls (i.e., at *y* = 0 and *y* = 0.42 cm) and centerline inflow velocity *U*_{CL}(*t*) at the centerline (i.e., at *y* = 0.21 cm). The inflow oscillating centerline velocity is
*t* = *0* [at which point *U*_{CL} = 0 up to *t* = *t** (at which point *U*_{CL} = *U*_{peak})]. For *t* > *t**, the inflow centerline velocity oscillates sinusoidally between *U*_{peak} and *U*_{min} with period *T*. Time *t** is arbitrary and simply serves to define an initial time period (between *t* = *0* and *t* = *t**) over which the flow spins up from rest, reaching an oscillatory state for *t* > *t** (*t** is set as 0.25 s). Note that flow rate *Q*_{peak} appearing in *Eqs. 3* and *4* is computed as *Q*_{peak} = *U*_{peak}*L*.

Velocities *U*_{peak} and *U*_{min} are chosen as 0.23743 m/s and 0.01965 m/s, respectively. These values correspond to the maximum fetal end-systolic velocity (*U*_{peak}) and the minimum fetal end-diastolic velocity (*U*_{min}) reported by Acharya et al. (1) in their Table 4.

Finally, the period, *T*, of the prescribed oscillating inflow velocity (appearing in *Eq. 6*) is set as *T* = 0.42254 s. This period corresponds to 142 beats/min, which is the mean fetal heart rate over the last 10 wk of normal gestation reported by Gabbe et al. (5).

For the values of *U*_{peak}, *T*, *L*, μ, and ρ chosen above, the Strouhal number is St = 0.042 and the Reynolds number is Re = 263, comparable to the values studied by Tutty (13).

An alternate inflow centerline velocity to that in *Eq. 6*, more representative of physiological conditions, can be defined following (9) as
*T* − 2, *A*_{0} = 0.055360393108630, *A*_{1} = 0.076532481162692, *A*_{2} = 0.063546459052423, *A*_{3} = 0.032363480900000, and *A*_{4} = 0.009640185800000.

The wave period is chosen as before, *T* = 0.42254 s. A plot of *Eqs. 6* and *7* is given in Fig. 2.

#### Numerical algorithm.

The Navier-Stokes equations, together with boundary and initial conditions previously described, are solved numerically using the streamline upwind/Petrov Galerkin finite-element method implemented in COMSOL (Burlington, MA). The equations are solved in the domain shown in Fig. 1, discretized by a mesh consisting of triangular finite elements. For example, the mesh in Fig. 1 consists of 8,912 elements and 4,709 mesh points. Note the finer mesh in the constriction area to resolve the larger velocity gradients expected in this region.

The Galerkin approximation of the streamline upwind/Petrov Galerkin finite-element method weak form of the Navier-Stokes equations is made through velocity and pressure given in terms of bilinear basis (shape) functions. Gaussian quadrature of weak form integrals leads to a nonlinear system of ordinary differential equations. Time integration of these ordinary differential equations via a variable time-stepping, fifth-order backward differential formula (ensuring minimum numerical damping) and subsequent linearization yield a linear system solved efficiently with the UMFPACK direct solver implemented in COMSOL.

## RESULTS

We present results from our simulations of pulsatile flow in a constricted channel. Two levels of constrictions are studied, 75 and 85%. In the upcoming subsections, we describe the flow structure, followed by a discussion of the associated FVWs and results of medical significance.

Figure 1 shows the domain with 75% constriction of the channel width. As mentioned earlier, the mesh for this case consists of 8,912 triangular elements and 4,709 nodes. The mesh for the 85% constriction case consists of 8,792 triangular elements and 4,661 nodes. Note that the flow with 85% constriction was also computed on a finer mesh with 18,064 triangular elements and 9,399 nodes, yielding results nearly indistinguishable from those of the coarser mesh. Here we only show results from the coarser mesh.

#### Flow structure.

Figure 3 shows the pulsatile flow structure over one cardiac cycle in the channel with 85% constriction with the sinusoidal inflow condition in *Eqs. 5* and *6*. The flow structure is shown in terms of streamlines and fluid speed. The structure is similar to that described by Tutty (13). The constriction induces flow separation characterized by a displaced core undulating between the channel walls. The vortex wave propagates downstream and is ultimately damped at distances downstream of the constriction. Eddies or recirculating zones form underneath the peak and above the trough of the core wave. These eddies are generated at the beginning of the cycle and wash away at the end. Unlike the displaced core wave, which propagates downstream, the eddies do not propagate as much. The center of each eddy moves slightly to the downstream end of the eddy, giving way to the formation of a corotating center in the location vacated by the original center. As many as three centers can coexist beneath the peak or above the trough of the core wave. As seen in Fig. 3, the flow accelerates in the peak and the trough regions of the wave and decelerates in the midregion between peaks and troughs. This behavior is attributed to conservation of mass and leads to notching in FVWs to be described in the upcoming subsection. Further description of the flow structure is given by Tutty (13) and Rosenfeld and Einav (12).

The previously described general flow structure is independent of the shape of the constriction. Simulations not shown were conducted with different shapes of the constriction (other than a semi-elliptical constriction), leading to a similar flow structure, in addition to notching of the FVW. This is in agreement with the results of Rosenfeld and Einav (12), who found that the shape of the constriction has a minor effect on the global features of the flow, and the constriction size is the most significant geometrical factor. We also performed a simulation with a symmetric constriction using the inflow condition in *Eqs. 5* and *6* and found a similar flow structure to the one previously described, which also led to notching in the FVW.

Finally, as noted by Rosenfeld and Einav (12), the previously described pulsatile flow characteristics in a constricted channel have been found to be vastly different than the flow characteristics in a constricted channel with steady (time-independent) inflow. In the latter case, the flow exhibits separation eddies or recirculation zones behind the constriction, and the flow becomes unsteady for Reynolds numbers 0(1,000). In the case of a pulsatile inflow boundary condition, such as is our case, a system of eddies is found at a much lower Reynolds number of 0(100).

#### FVWs.

Figures 4⇓⇓–7 show FVWs associated with the pulsatile flow structure previously described. The cases presented in these figures are characterized by sinusoidal inflow velocity given in *Eqs. 5* and *6*. FVWs are shown at various *x-*locations. Figures 4 and 5 correspond to the flow with 75%, whereas Figs. 6 and 7 correspond to the flow with 85% constriction. These figures also contain FVWs for unconstricted flow. Two types of FVWs are analyzed: one based on the centerline *x-*velocity (*u*_{cl}) at *y* = 0.21 cm at a particular *x-*location (Figs. 4 and 6), and a second based on the maximum *x-*velocity (*u*_{max}) in a particular cross section (Figs. 5 and 7). The latter type is referred to as the envelope FVW and is often measured by commercial Doppler systems (8). We will refer to the former type as the centerline FVW.

Centerline and envelope FVWs in Figs. 4⇑⇑–7 possess an abnormality in the form of a notch. Recall that notching in FVW is defined as a period of deceleration during a larger period of overall or net acceleration. Conversely, notching is also defined as a period of acceleration during a larger period of overall deceleration.

In the channel flow with 75% constriction, the notch is more apparent in terms of centerline FVW. In terms of envelope FVW, the notch is clearly evident in the flow with 85% constriction. Envelope FVW notching was not observed in flows with constriction levels <75%. Both types of FVWs exhibit much higher velocities for constricted flows compared with their unconstricted counterparts. This is due to the flow acceleration caused by the constriction. Note that, sufficiently downstream of the constriction, the notch disappears and the FVWs of the constricted flows nearly coincide with the FVWs of the unconstricted flow. This occurs as the vortex wave generated by the constriction is attenuated downstream. For example, notching in envelope FVW disappears at ∼3.8 and ∼4.3 cm downstream of the constriction in the flows with 75 and 85% constriction, respectively.

The mechanism by which the notch appears in the centerline and envelope FVWs is directly related to the core wave and associated eddies described earlier. To understand this mechanism, consider the notch that occurs in both types of FVWs in the flow with 85% constriction at *x* = 0.19 cm during the interval 0.55 s < *t* < 0.65 s (Figs. 6*C* and 7*C*).

Figure 8 shows streamlines superimposed with fluid speed at various times during the occurrence of this notch. As the core wave propagates toward the channel cross section at *x* = 0.19 cm (*t* = 0.576 and 0.587 s), both the centerline FVW and the envelope FVW show an increase in velocity. Once the wave reaches *x* = 0.19 cm (*t* = 0.595 and 0.606 s), the envelope FVW continues to rise. However, the centerline FVW begins to decrease as the crest of the wave brings high velocity to the region near the upper wall of the channel, while the midpoint of the cross section is engulfed by a low-velocity eddy formed beneath the wave crest. Eventually, the eddy moves downstream of the cross section at *x* = 0.19 cm, and the midpoint of the cross section is once again covered by the core wave bringing high-speed fluid (*t* = 0.646, 0.656, and 0.667 s). In contrast with the centerline FVW, the envelope FVW (which registers maximum *x-*velocity in the cross section and thus maximum velocity in the core wave) continues to increase as the core wave reaches the top wall region at *x* = 0.19 cm. The envelope FVW begins to decrease when the midregion of the wave (i.e., the wave region between the wave crest and wave trough) coincides with the cross section at *x* = 0.19 cm (*t* = 0.628 and 0.635 s). Once the trough of the wave reaches *x* = 0.19 cm (*t* = 0.646, 0.656, and 0.667 s), the envelope FVW rises again. In summary, as the core wave propagates past the cross section at *x* = 0.19 cm, the envelope FVW detects (or reflects) the low-speed fluid in the wave midregion between the crest and trough. That is, the fluid in the wave midregion possesses low speed compared with fluid in the crest and trough. As mentioned earlier, the reason for this is attributed to conservation of mass. Low-speed fluid circulates in the eddies beneath the wave crest and above the wave trough, thereby causing an increase in fluid speed along the wave crest and trough. Finally, we note that the notch in FVWs is more evident in the flow with 85% constriction than in the flow with 75% constriction. The reason for this is that the decrease in velocity along the core wave is sharper in the former case.

The results previously presented are based on the sinusoidal inflow condition in *Eqs. 5* and *6*. A physiological inflow given through *Eqs. 5* and *7* characterized by systole and diastole components has also been considered (see Fig. 2). As seen in Figs. 9 and 10, the simulation with 85% constriction and this physiological inflow also exhibits notching in FVWs. In the case of centerline velocity FVW, notching occurs during systole and diastole because each component induces an undulating core wave.

## DISCUSSION

Finite-element analysis of FVWs in pulsatile constricted channel flow has been performed with the purpose of establishing a relationship between notching in umbilical artery Doppler velocimetry waveforms and the constriction. Typically, notching in umbilical artery FVWs alerts obstetricians to the possibility of abnormalities in the umbilical cord, such as a constriction impairing the flow. The constriction may result from knotting of the umbilical cord. Computations highlighted in this paper demonstrate that notching directly results from periodic flow separation induced by the constriction, giving rise to a propagating vortex wave and associated eddies. Notching in centerline FVWs is directly related to the passage of an eddy over the point of measurement along the centerline, while notching in envelope FVWs is directly related to acceleration and deceleration of the fluid along the vortex wave.

The main goal of the present study has been to understand the fundamental physical mechanism by which notching occurs; thus modeling aspects related to flow geometry and boundary conditions have been idealized. For example, the present study is two-dimensional and has rigid walls. However, the main result obtained (i.e., the identification of the characteristic feature of the flow giving rise to notching in FVW) is general.

Berger and Jou (2) in their review of stenotic flows present an instantaneous snapshot of the flow in a symmetrically constricted three-dimensional channel with rigid walls. Similar to our two-dimensional flows, the three-dimensional flow exhibits a core jet behind the constriction, accompanied by a system of eddies. Furthermore, flow separation characterized by an undulating core jet has also been observed in the three-dimensional computation of a symmetrically constricted vessel with deformable walls of Figueroa et al. (3). In brief, a similar structure giving rise to FVW notching in our two-dimensional constricted flow is also present in three-dimensional constricted flow with deformable walls.

The effect of arterial compliance on FVWs in an umbilical artery has been studied via mathematical (analytic) modeling by Kleinner-Assaf et al. (8). Kleinnar-Assaf et al. obtained results for a range of Young's modulus (a measure of the stiffness of an isotropic elastic material) between 8 and 1 × 10^{5} Pa, which is the typical range for an umbilical artery. The flow was driven via the imposition of physiological pressures at the placental and fetal ends of the artery. An increase in stiffness led to a damping of the envelope FVW. For example, the maximum to minimum velocity ratio in FVW in a case with Young's modulus at 8 × 10^{5} Pa was ∼1.97, while a case with Young's modulus = 1 × 10^{5} Pa was characterized by a ratio of 8.33. These results indicate that perhaps stiffness would serve to damp notching in FVW. Thus FVW notching may be more pronounced in a model that takes into consideration arterial wall deformations compared with the notching in our rigid wall model.

The effect of three-dimensional curving or coiling on the flow structure in a constricted vessel has been discussed by Liu and Yamaguchi (9). Coiling of a constricted artery may lead to a relatively strong cross-flow component, inducing an altered undulating core wave, characterized by complex three-dimensional structure. Although coiling may have profound effects on the flow structure, Kleiner-Assaf et al. (8) note that in vivo images have revealed a fairly uncoiled cord with longitudinal curvatures much larger than its diameter during the first 5–6 mo [Nilsson (10)].

### Perspectives and Significance

We are currently conducting further studies involving realistic physiological inflow and outflow boundary conditions, vessel wall deformation, and three-dimensional geometry to understand their effects on FVW notching. Ultimately, we aim to develop a computational methodology that can provide guidance to obstetricians commonly monitoring pregnancies via umbilical cord arterial blood FVWs measured using Doppler ultrasound techniques. In the meantime, several useful conclusions of medical significance have been drawn from our current simplified simulations. For example, the magnitude of the notching is strongly dependent on the type of FVWs being measured and on the constriction level. Our studies indicate that notching in envelope FVW is not present in flows with ∼75% constrictions or less; however, this is not the case for centerline FVW. Furthermore, the magnitude of the notch depends largely on the location downstream of the constriction where the FVW is measured. Generally, notching disappears as the vortex wave is attenuated at distances downstream from the constriction. As highlighted earlier, in the flows with 75 and 85% constriction with sinusoidal centerline inflow velocity, envelope FVW notching disappears at ∼3.8 and ∼4.3 cm (respectively) downstream of the constriction. However, note that notching may not be present at locations slightly downstream or slightly upstream from locations in which notching is evident (e.g., see Fig. 7, *B*–*D*). This information may be of clinical importance, as notching in a knotted umbilical cord may not be detected with Doppler upstream of the knot and only beyond a certain downstream distance from the knot, which may affect the sensitivity of the test.

## DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

- Copyright © 2011 the American Physiological Society