An assessment of ductus venosus tapering and wave transmission from the fetal heart.

Pressure and flow pulsations in the fetal heart propagate through the precordial vein and the ductus venosus (DV) but are normally not transmitted into the umbilical vein (UV). Pulsations in the umbilical vein do occur, however, in early pregnancy and in pathological conditions. Such transmission into the umbilical vein is not well understood. In particular, the effect of the impedance changes in the DV due to its tapered geometry is not known. This paper presents a mathematical model that we developed to study the transmission of pulsations, originating in the fetal heart, through the DV to the umbilical vein. In our model, the tapered geometry of the DV was found to be of minor importance and the only effective reflection site in the DV appears to be at the DV inlet. Differences between the DV inlet and outlet flow were also found to be minor for medium to large umbilical vein-DV diameter ratios. Finally, the results of a previously proposed lumped model were found to agree well with the present model of the DV-umbilical vein bifurcation.

A mathematical model of umbilical venous pulsation.

Pulsations in the fetal heart propagate through the precordial vein and the ductus venosus but are normally not transmitted into the umbilical vein. Pulsations in the umbilical vein do occur, however, in early pregnancy and in pathological conditions. Such transmission into the umbilical vein is poorly understood. In this paper we hypothesize that the mechanical properties and the dimensions of the vessels do influence the umbilical venous pulsations, in addition to the magnitude of the pressure and flow waves generated in the fetal atria. To support this hypothesis we established a mathematical model of the umbilical vein/ductus venosus bifurcation. The umbilical vein was modeled as a compliant reservoir and the umbilical vein pressure was assumed to be equal to the stagnation pressure at the ductus venosus inlet. We calculated the index of pulsation of the umbilical vein pressure ((max-min)/mean), the reflection and transmission factors at the ductus venosus inlet, numerically and with estimates. Typical dimensions in the physiological range for the human fetus were used, while stiffness parameters were taken from fetal sheep. We found that wave transmission and reflection in the umbilical vein ductus venosus bifurcation depend on the impedance ratio between the umbilical vein and the ductus venosus, as well as the ratio of the mean velocity and the pulse wave velocity in the ductus venosus. Accordingly, the pulsations initiated by the fetal heart are transmitted upstream and may arrive in the umbilical vein with amplitudes depending on the impedance ratio and the ratio between the mean velocity and the pulse wave velocity in the ductus venosus.

Mechanism of pulmonary venous pressure and flow waves.

The pulmonary venous systolic flow wave has been attributed both to left heart phenomena, such as left atrial relaxation and descent of the mitral annulus, and to propagation of the pulmonary artery pressure pulse through the pulmonary bed from the right ventricle. In this study we hypothesized that all waves in the pulmonary veins originate in the left heart, and that the gross wave features observed in measurements can be explained simply by wave propagation and reflection. A mathematical model of the pulmonary vein was developed; the pulmonary vein was modeled as a lossless transmission line and the pulmonary bed by a three-element lumped parameter model accounting for viscous losses, compliance, and inertia. We assumed that all pulsations originate in the left atrium (LA), the pressure in the pulmonary bed being constant. The model was validated using pulmonary vein pressure and flow recorded 1 cm proximal to the junction of the vein with the left atrium during aortocoronary bypass surgery. For a pressure drop of 6 mmHg across the pulmonary bed, we found a transit time from the left atrium to the pulmonary bed of tau approximately 150ms, a compliance of the pulmonary bed of C approximately 0.4 ml/mmHg, and an inertance of the pulmonary bed of 1.1 mmHgs2/ml. The pulse wave velocity of the pulmonary vein was estimated to be c approximately 1m/s. Waves, however, travel both towards the left atrium and towards the pulmonary bed. Waves traveling towards the left atrium are attributed to the reflections caused by the mismatch of impedance of line (pulmonary vein) and load (pulmonary bed). Wave intensity analysis was used to identify a period in systole of net wave propagation towards the left atrium for both measurements and model. The linear separation technique was used to split the pressure into one component traveling from the left atrium to the pulmonary bed and a reflected component propagating from the pulmonary bed to the left atrium. The peak of the reflected pressure wave corresponded well with the positive peak in wave intensity in systole. We conclude that the gross features of the pressure and flow waves in the pulmonary vein can be explained in the following manner: the waves originate in the LA and travel towards the pulmonary bed, where reflections give rise to waves traveling back to the LA. Although the gross features of the measured pressure were captured well by the model predicted pressure, there was still some discrepancy between the two. Thus, other factors initiating or influencing waves traveling towards the LA cannot be excluded.

Equations for estimating muscle fiber stress in the left ventricular wall.

Left ventricular muscle fiber stress is an important parameter in cardiac energetics. Hence, we developed equations for estimating regional fiber stresses in rotationally symmetric chambers, and equatorial and apical fiber stresses in prolate spheroidal chambers. The myocardium was modeled as a soft incompressible material embedding muscle fibers that support forces only in their longitudinal direction. A thin layer of muscle fibers then contributes with a pressure increment determined by the fiber stress and curvature. The fiber curvature depends on the orientation of the fibers, which varies continuously across the wall. However, by assuming rotational symmetry about the long axis of the ventricle and including a longitudinal force balance, we obtained equations where fiber stress is completely determined by the principal curvatures of the middle wall surface, wall thickness, and cavity pressure. The equations were validated against idealized prolate spheroidal chambers, whose wall thicknesses are such that the fiber stress is uniform from the equator to the apex. Because the apex is free to rotate, the resultant moment about the long axis of the LV must be zero. By using this constraint together with our fiber-stress equations, we were able to estimate a muscle fiber orientation distribution across the wall that was in qualitative agreement with published measurements.

Mechanical properties of the fetal ductus venosus and umbilical vein.

During fetal circulatory compromise, velocity pulsations in the precordial veins increase and are commonly transmitted through the ductus venosus into the umbilical vein, indicating a serious prognosis. The nature of the pulsations and their transmission into the periphery, specifically the umbilical vein, is poorly understood. We present information on the mechanical properties of fetal veins as a basis for describing the pulse wave propagation. Five fetal sheep livers with connecting veins (gestational age 0.8-0.9) were studied in vitro. The transmural pressure, obtained with a fluid-filled catheter, was reduced stepwise from 10.3 to 0 mmHg, and the diameter determined by ultrasonography. Each data set was fitted to an exponential function to determine the stiffness parameter and the area at a standard pressure, which we proposed to be 5 mmHg for the fetal venous circulation. The stiffness parameter was 6.2+/-1.8 at the ductus venosus outlet, 3.4+/-1.3 at the ductus venosus inlet, and 4.0+/-1.0 in the umbilical vein. Correspondingly, values for compliance and pulse wave velocity for the three venous sections were established for a physiological pressure range. The estimated pulse wave velocity of 1-3m/s is comparable with values estimated for veins in adults. The mechanical properties of fetal veins are comparable with those described for veins later in life. The stiffness parameter represents the elastic properties at all pressure levels and conveniently permits inference of compliance and pulse wave velocity.

Simulation of pressure drop and energy dissipation for blood flow in a human fetal bifurcation.

The pressure drop from the umbilical vein to the heart plays a vital part in human fetal circulation. The bulk of the pressure drop is believed to take place at the inlet of the ductus venosus, a short narrow branch of the umbilical vein. In this study a generalized Bernoulli formulation was deduced to estimate this pressure drop. The model contains an energy dissipation term and flow-scaled velocities and pressures. The flow-scaled variables are related to their corresponding spatial mean velocities and pressures by certain shape factors. Further, based on physiological measurements, we established a simplified, rigid-walled, three-dimensional computational model of the umbilical vein and ductus venosus bifurcation for stationary flow conditions. Simulations were carried out for Reynolds numbers and umbilical vein curvature ratios in their respective physiological ranges. The shape factors in the Bernoulli formulation were then estimated for our computational models. They showed no significant Reynolds number or curvature ratio dependency. Further, the energy dissipation in our models was estimated to constitute 24 to 31 percent of the pressure drop, depending on the Reynolds number and the curvature ratio. The energy dissipation should therefore be taken into account in pressure drop estimates.