Leif Rune Hellevik

Estimation of the pressure gradient across the fetal ductus venosus based on doppler velocimetry

In the fetus, the umbilical vein is directly linked to the inferior vena cava by the narrow ductus venosus. Thus, the ductus venosus blood velocity probably reflects the pressure gradient between the umbilical vein and the central venous system. In a longitudinal study that included 29 normal fetuses, pulsed Doppler velocimetry was carried out in the umbilical vein and the ductus venosus during the last half of the pregnancy. By applying the Bernoulli equation, we estimated the pressure gradient across the ductus venosus to vary between 0-3 mm Hg during the heart cycle; it remained within those ranges during gestational weeks 18-40. During fetal inspiratory movement, pressure gradients up to 22 mm Hg were estimated. The estimated ductus venosus pressure gradient seems to be within ranges compatible with known umbilical venous pressures, and may provide a new opportunity to understand central venous hemodynamics and respiratory force in the fetus once methodological limitations are controlled.

A benchmark study of numerical schemes for one-dimensional arterial blood flow modelling

Haemodynamical simulations using one-dimensional (1D) computational models exhibit many of the features of the systemic circulation under normal and diseased conditions. Recent interest in verifying 1D numerical schemes has led to the development of alternative experimental setups and the use of three-dimensional numerical models to acquire data not easily measured in vivo. In most studies to date, only one particular 1D scheme is tested. In this paper, we present a systematic comparison of six commonly used numerical schemes for 1D blood flow modelling: discontinuous Galerkin, locally conservative Galerkin, Galerkin least-squares finite element method, finite volume method, finite difference MacCormack method and a simplified trapezium rule method. Comparisons are made in a series of six benchmark test cases with an increasing degree of complexity. The accuracy of the numerical schemes is assessed by comparison with theoretical results, three-dimensional numerical data in compatible domains with distensible walls or experimental data in a network of silicone tubes. Results show a good agreement among all numerical schemes and their ability to capture the main features of pressure, flow and area waveforms in large arteries. All the information used in this study, including the input data for all benchmark cases, experimental data where available and numerical solutions for each scheme, is made publicly available online, providing a comprehensive reference data set to support the development of 1D models and numerical schemes.

Blood velocity profile in the ductus venosus inlet expressed by the mean/maximum velocity ratio

Mean blood velocity (Vmean) is needed for calculating blood flow and possibly the pressure gradient across the ductus venosus. Interference from low velocities from neighbouring vessels makes the direct Doppler measurement of Vmean unreliable. Therefore, it is suggested that Vmean can be derived more reliably from the maximum velocity (Vmax) once the velocity profile, expressed as the ratio Vmean/Vmax, is known. To determine this ratio, ultrasound was performed in 10 fetal sheep during acute experiments under general anaesthesia to ensure good recording control and optimal insonation. Based on 33 Doppler measurements at the ductus venosus inlet, the ratio Vmean/Vmax was determined to be 0.69 (SD +/- 0.07) regardless of Vmax, pulsatility index, vessel diameter, or angle of insonation. These results confirm the previous prediction based on a computational model that the velocity profile is partially blunted. The equation Vmean = 0.7Vmax is recommended for determining Vmean in the ductus venosus.

Venous pulsation in the fetal left portal branch: the effect of pulse and flow direction

To determine whether the waveform in the left portal branch is reciprocal to the waveform found in the ductus venosus and umbilical vein due to difference in pulse direction compared to flow.

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.

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.

Mechanical properties of the fetal ductus venosus and umbilical vein.

SummaryDuring 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 0mmHg, 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.

FSI simulation of asymmetric mitral valve dynamics during diastolic filling

In this article, we present a fluid–structure interaction algorithm accounting for the mutual interaction between two rigid bodies. The algorithm was used to perform a numerical simulation of mitral valve (MV) dynamics during diastolic filling. In numerical simulations of intraventricular flow and MV motion, the asymmetry of the leaflets is often neglected. In this study the MV was rendered as two rigid, asymmetric leaflets. The 2D simulations incorporated the dynamic interaction of blood flow and leaflet motion and an imposed subject-specific, transient left ventricular wall movement obtained from ultrasound recordings. By including the full Jacobian matrix in the algorithm, the speed of the simulation was enhanced by more than 20% compared to using a diagonal Jacobian matrix. Furthermore, our results indicate that important features of the flow field may not be predicted by the use of symmetric leaflets or in the absence of an adequate model for the left atrium.

A guide to uncertainty quantification and sensitivity analysis for cardiovascular applications

As we shift from population-based medicine towards a more precise patient-specific regime guided by predictions of verified and well-established cardiovascular models, an urgent question arises: how sensitive are the model predictions to errors and uncertainties in the model inputs? To make our models suitable for clinical decision-making, precise knowledge of prediction reliability is of paramount importance. Efficient and practical methods for uncertainty quantification (UQ) and sensitivity analysis (SA) are therefore essential. In this work, we explain the concepts of global UQ and global, variance-based SA along with two often-used methods that are applicable to any model without requiring model implementation changes: Monte Carlo (MC) and polynomial chaos (PC). Furthermore, we propose a guide for UQ and SA according to a six-step procedure and demonstrate it for two clinically relevant cardiovascular models: model-based estimation of the fractional flow reserve (FFR) and model-based estimation of the total arterial compliance (CT ). Both MC and PC produce identical results and may be used interchangeably to identify most significant model inputs with respect to uncertainty in model predictions of FFR and CT . However, PC is more cost-efficient as it requires an order of magnitude fewer model evaluations than MC. Additionally, we demonstrate that targeted reduction of uncertainty in the most significant model inputs reduces the uncertainty in the model predictions efficiently. In conclusion, this article offers a practical guide to UQ and SA to help move the clinical application of mathematical models forward. Copyright

Stochastic sensitivity analysis for timing and amplitude of pressure waves in the arterial system

In the field of computational hemodynamics, sensitivity quantification of pressure and flow wave dynamics has received little attention. This work presents a novel study of the sensitivity of pressure-wave timing and amplitude in the arterial system with respect to arterial stiffness. Arterial pressure and flow waves were simulated with a one-dimensional distributed wave propagation model for compliant arterial networks. Sensitivity analysis of this model was based on a generalized polynomial chaos expansion evaluated by a stochastic collocation method. First-order statistical sensitivity indices were formulated to assess the effect of arterial stiffening on timing and amplitude of the pressure wave and backward-propagating pressure wave in the ascending aorta, at the maximum pressure and inflection point in the systolic phase. Only the stiffness of aortic arteries was found to significantly influence timing and amplitude of the backward-propagating pressure wave, whereas other large arteries in the systemic tree showed marginal impact. Furthermore, the ascending aorta, aortic arch, thoracic aorta, and infrarenal abdominal aorta had the largest influence on amplitude, whereas only the thoracic aorta influenced timing. Our results showed that the non-intrusive polynomial chaos expansion is an efficient method to compute statistical sensitivity measures for wave propagation models. These sensitivities provide new knowledge in the relative importance of arterial stiffness at various locations in the arterial network. Moreover, they will significantly influence clinical data collection and effective composition of the arterial tree for in-silico clinical studies.