Effects of arterial wall models and measurement uncertainties on cardiovascular model predictions.

We developed a methodology to assess and compare the prediction quality of cardiovascular models for patient-specific simulations calibrated with uncertainty-hampered measurements. The methodology was applied in a one-dimensional blood flow model to estimate the impact of measurement uncertainty in wall model parameters on the predictions of pressure and flow in an arterial network. We assessed the prediction quality of three wall models that have been widely used in one-dimensional blood flow simulations. A 37-artery network, previously used in one experimental and several simulation studies, was adapted to patient-specific conditions with a set of three clinically measurable inputs: carotid-femoral wave speed, mean arterial pressure and area in the brachial artery. We quantified the uncertainty of the predicted pressure and flow waves in eight locations in the network and assessed the sensitivity of the model prediction with respect to the measurements of wave speed, pressure and cross-sectional area. Furthermore, we developed novel time-averaged sensitivity indices to assess the contribution of model parameters to the uncertainty of time-varying quantities (e.g., pressure and flow). The results from our patient-specific network model demonstrated that our novel indices allowed for a more accurate sensitivity analysis of time-varying quantities compared to conventional Sobol sensitivity indices.

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.

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.

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.

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.

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.

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

OBJECTIVE: 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. METHODS: Ten fetuses (gestational age, 18-33 weeks), six with intrauterine growth restriction, three with non-immune hydrops and one with sacrococcygeal teratoma, were examined using ultrasound imaging and pulsed Doppler. Techniques were adjusted to record simultaneously the waveform from neighboring sections of the veins, relate wave components to each other and determine degree of pulsatility. The corresponding vessel diameters were determined. ANOVA with t-test or Wilcoxon signed rank test was used to compare paired measurements. RESULTS: Pulsation in the left portal branch was noted in all fetuses. The pulsatility index was higher than in the umbilical vein (P = 0.005) and the diameter smaller (P = 0.001). In the left portal branch the atrial contraction wave appeared as a velocity peak while there was a nadir during ventricular systole. Simultaneous recordings showed that the waveform was reciprocal to that found in the ductus venosus and umbilical vein. In three cases an augmented pulsatility represented a pendulation of blood in the left portal branch with time-averaged velocity near zero. CONCLUSIONS: The velocity waveform recorded in the left portal vein is an inverse image of that in the ductus venosus, proving that pulse wave and blood flow run in the same direction in the left portal vein. Low compliance (i.e. small diameter) is probably a main reason for the high incidence of pulsation in this vein. Time-averaged velocity near zero recorded in three fetuses indicates that this area acts also as a watershed.

Ultrasonographic velocimetry of the fetal ductus venosus.

In fetal lambs, the ductus venosus shunts well-oxygenated blood directly to the heart, a pattern expected to be found also in the human fetus. We aimed to describe the human ductus venosus in a longitudinal sonographic study of two-dimensional imaging, colour flow mapping, and pulsed doppler velocimetry every 3-4 weeks during the second half of pregnancy. The fetuses of 29 healthy women were studied. The ductus venosus and its blood flow were identified and recorded for later analysis that included maximum velocity tracing. In the 184 examinations analysed, the ductus venosus appeared as a narrow vessel projecting a high-velocity jet posteriorly to reach the foramen ovale. The mean peak velocity in the ductus venosus increased from 65 cm/s in week 18 to 75 cm/s at term. Low values of the time-averaged maximum velocity were found in 2 fetuses with cardiovascular abnormalities (1 supraventricular tachycardia, 1 congestive heart failure), as a result of reversed flow in the ductus venosus during atrial systole. The high peak velocity in the ductus venosus, which is comparable with arterial velocities, probably gives the blood sufficient momentum to reach the foramen ovale without extensive mixing with deoxygenated blood. Velocimetry of the ductus venosus carries new diagnostic possibilities.

Foramen ovale: an ultrasonographic study of its relation to the inferior vena cava, ductus venosus and hepatic veins.

According to the literature, oxygenated blood from the ductus venosus and hepatic veins may either enter the right atrium before flowing through the foramen ovale to the left atrium, or flow directly from the ductus venosus and the hepatic veins to the foramen ovale, bypassing the right atrium. To address this problem, 103 normal fetuses were examined by two-dimensional imaging, M-mode and color Doppler at an average gestational age of 27 weeks (range, 15-40 weeks). The position of the ventricular septum and foramen ovale, and the angle and flow direction of the inferior vena cava, ductus venosus and hepatic veins were recorded. Two pathways for blood were described: a left ductus venosus-foramen ovale pathway that delivers blood directly to the foramen ovale circumventing the right atrium, and a right inferior vena cava-right atrium pathway that delivers blood into the right atrium through the right portion of the proximal inferior vena cava at an angle of 13 degrees to the long axis of the spine. The left and medial hepatic veins enter the left ductus venosus-foramen ovale pathway, and the right hepatic vein enters the right inferior vena cava-right atrium pathway. This supports the hypothesis that oxygenated blood from the ductus venosus and left hepatic veins flows directly through the foramen ovale to the left atrium avoiding extensive mixture in the inferior vena cava and an intermediate entrance to the right atrium.

Early development of the hindbrain: a longitudinal ultrasound study from 7 to 12 weeks of gestation.

Twenty-nine healthy pregnant women were examined by transvaginal ultrasound to evaluate embryonic development in vivo between 7 and 12 weeks of gestation. The rhombencephalon with its fourth ventricle, the cerebellum and the choroid plexuses of the fourth ventricle were identified and measured. The cavity of the rhombencephalon, the future fourth ventricle, was always visible from 7 weeks, initially lying superiorly in the head of the embryo. The cerebellum and the choroid plexuses of the fourth ventricle became distinguishable during week 8. The volume of the rhombencephalic cavity was estimated. The shape and size of these rhombencephalic structures, their position in relation to each other and their relation to other brain structures changed specifically during the embryonic and early fetal period. This sonoembryological development corresponded to the descriptions in classical embryological literature.

Early development of the forebrain and midbrain: a longitudinal ultrasound study from 7 to 12 weeks of gestation.

The purpose of this longitudinal study was to describe embryonic development in vivo. Twenty-nine healthy pregnant women were examined five times with transvaginal ultrasound between 7 and 12 weeks of gestation. Brain structures such as the hemispheres, the choroid plexus of the lateral ventricles, the diencephalon, and the mesencephalon were identified and, if possible, measured. It was possible to identify the cavities of the hemispheres, the diencephalon and the mesencephalon during week 7. The choroid plexus of the lateral ventricles became visible during week 8. The growth of the length, width and height of the hemispheres and the choroid plexus of the lateral ventricles was curvilinear, that of the mesencephalon and diencephalon was linear except for the width of the diencephalon. The width of the diencephalon, the future third ventricle, was 1.1 mm during week 7. It decreased to 0.8 mm at 12 weeks. Apart from the rhombencephalon, the cavity of the diencephalon was the large dominating brain structure during embryonic development. In early fetal life the cerebral hemispheres took over this dominance. The study was in full agreement with descriptions in the embryological literature, both concerning the anatomical features and their chronological formation.

Ductus venosus blood velocity and the umbilical circulation in the seriously growth-retarded fetus.

Based on the assumption that the ductus venosus is regulator of oxygenated blood in the fetus, the present study investigated the blood flow velocity of the ductus venosus in relation to the umbilical circulation in the that seriously growth-retarded fetus. The study group of 38 fetuses (gestational week 17-39) had no chromosomal aberrations or structural malformations and had an ultrasonographic biometry of < 2.5th centile and birth weight of

Early development of the abdominal wall, stomach and heart from 7 to 12 weeks of gestation: a longitudinal ultrasound study.

The purpose of this ultrasound study was to describe longitudinally the normal embryonic development in vivo. Twenty-nine healthy pregnant women were examined five times each with transvaginal ultrasound between 7 and 12 weeks of gestation measured from the last menstrual period. Structures such as the midgut herniation into the umbilical cord, the stomach and the heart were recognized and measured. It was possible to identify the physiological midgut herniation during weeks 7-8. It was always present from 8.5 to 10.5 weeks. At 12 completed weeks, the gut was retracted into the abdominal cavity for all the fetuses. We visualized the stomach in nine embryos (31%) during week 8, in 22 embryos (76%) before 10 weeks, and in all fetuses before 11 weeks. The heart rate increased rapidly to a mean of 175 beats per minute (bpm) at the beginning of week 9. Thereafter it decreased slowly to a mean of 166 bpm at 12 weeks. The mean heart diameter was 22% of the crown-rump length at 7 weeks, 17% at 9 weeks and only 13% at 12 weeks.

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.

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.

Individualizing the aorto-radial pressure transfer function: feasibility of a model-based approach.

We fitted a three-segment transmission line model for the radial-carotid/aorta pressure transfer function (TFF) in 31 controls and 30 patients with coronary artery disease using noninvasively measured (tonometry) radial and carotid artery pressures (P(car)). Except for the distal reflection coefficient (0.85 +/- 0.21 in patients vs. 0.71 +/- 0.25 in controls; P < 0.05), model parameters were not different between patients or controls. Parameters were not related to blood pressure, age, or heart rate. We further assessed a point-to-point averaged TFF (TFF(avg)) as well as upper (TFF(max)) and lower (TFF(min)) enveloping TFF. Pulse pressure (PP) and augmentation index (AIx) were derived on original and reconstructed P(car) (P(car,r)). TFF(avg) yielded closest morphological agreement between P(car) and P(car,r) (root mean square = 4.3 +/- 2.3 mmHg), and TTF(avg) best predicted PP (41.5 +/- 11.8 vs. 41.1 +/- 10.0 mmHg measured) and AIx (-0.02 +/- 0.19 vs. 0.01 +/- 0.19). PP and AIx, calculated from P(car) or P(car,r), were higher in patients than in controls, irrespectively of the TFF used. We conclude that 1) averaged TFF yield significant discrepancies between reconstructed and measured pressure waveforms and subsequent derived AIx; and 2) different TFFs seem to preserve the information in the pressure wave that discriminates between controls and patients.

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.