What is needed to make low-density lipoprotein transport in human aorta computational models suitable to explore links to atherosclerosis? Impact of initial and inflow boundary conditions.

Personalized computational hemodynamics (CH) is a promising tool to clarify/predict the link between low density lipoproteins (LDL) transport in aorta, disturbed shear and atherogenesis. However, CH uses simplifying assumptions that represent sources of uncertainty. In particular, modelling blood-side to wall LDL transfer is challenged by the cumbersomeness of protocols needed to obtain reliable LDL concentration profile estimations. This paucity of data is limiting the establishment of rigorous CH protocols able to balance the trade-offs among the variety of in vivo data to be acquired, and the accuracy required by biological/clinical applications. In this study, we analyze the impact of LDL concentration initialization (initial conditions, ICs) and inflow boundary conditions (BCs) on CH models of LDL blood-to-wall transfer in aorta. Technically, in an image-based model of human aorta, two different inflow BCs are generated imposing subject-specific inflow 3D PC-MRI measured or idealized (flat) velocity profiles. For each simulated BC, four different ICs for LDL concentration are applied, imposing as IC the LDL distribution resulting from steady-state simulations with average conditions, or constant LDL concentration values. Based on CH results, we conclude that: (1) the imposition of realistic 3D velocity profiles as inflow BC reduces the uncertainty affecting the representation of LDL transfer; (2) different LDL concentration ICs lead to markedly different patterns of LDL transfer. Given that it is not possible to verify in vivo the proper LDL concentration initialization to be applied, we suggest to carefully set and unambiguously declare the imposed BCs and LDL concentration IC when modelling LDL transfer in aorta, in order to obtain reproducible and ultimately comparable results among different laboratories.

Uncertainty propagation of phase contrast-MRI derived inlet boundary conditions in computational hemodynamics models of thoracic aorta.

This study investigates the impact that uncertainty in phase contrast-MRI derived inlet boundary conditions has on patient-specific computational hemodynamics models of the healthy human thoracic aorta. By means of Monte Carlo simulations, we provide advice on where, when and how, it is important to account for this source of uncertainty. The study shows that the uncertainty propagates not only to the intravascular flow, but also to the shear stress distribution at the vessel wall. More specifically, the results show an increase in the uncertainty of the predicted output variables, with respect to the input uncertainty, more marked for blood pressure and wall shear stress. The methodological approach proposed here can be easily extended to study uncertainty propagation in both healthy and pathological computational hemodynamic models.

Automatic extraction of three-dimensional thoracic aorta geometric model from phase contrast MRI for morphometric and hemodynamic characterization.

PURPOSE: To propose and assess a new method that automatically extracts a three-dimensional (3D) geometric model of the thoracic aorta (TA) from 3D cine phase contrast MRI (PCMRI) acquisitions. METHODS: The proposed method is composed of two steps: segmentation of the TA and creation of the 3D geometric model. The segmentation algorithm, based on Level Set, was set and applied to healthy subjects acquired in three different modalities (with and without SENSE reduction factors). Accuracy was evaluated using standard quality indices. The 3D model is characterized by the vessel surface mesh and its centerline; the comparison of models obtained from the three different datasets was also carried out in terms of radius of curvature (RC) and average tortuosity (AT). RESULTS: In all datasets, the segmentation quality indices confirmed very good agreement between manual and automatic contours (average symmetric distance < 1.44 mm, DICE Similarity Coefficient > 0.88). The 3D models extracted from the three datasets were found to be comparable, with differences of less than 10% for RC and 11% for AT. CONCLUSION: Our method was found effective on PCMRI data to provide a 3D geometric model of the TA, to support morphometric and hemodynamic characterization of the aorta.

A rational approach to defining principal axes of multidirectional wall shear stress in realistic vascular geometries, with application to the study of the influence of helical flow on wall shear stress directionality in aorta.

The distribution of arterial lesions is attributed by the prevalent mechanistic theory to the proatherogenic role played by low and oscillatory wall shear stress (WSS). However, discrepancies observed when comparing WSS distribution with location of regions with lesion prevalence challenge this theory and have recently stimulated the idea that a role in endothelial mechanosensing is played by WSS multidirectionality, which could contribute to explain the observed discrepancies. Here an approach is presented for analyzing the multidirectional nature of WSS in complex vascular geometries. Using an essential geometric attribute of the vessel (its centerline), the local WSS vector is projected along an "axial" direction (aligned with the tangent to the vessel׳s centerline), and "secondary" direction (orthogonal to centerline׳s tangent), which is related to secondary flow. The WSS projection scheme is applied: (1) to a realistic computational hemodynamic model of human aorta, with the aim to come up with a plausibility checking regarding its consistency; and (2) to investigate if an aortic hemodynamics characterized by different amount and topology of helical flow (HF) could influence WSS directionality. The projection scheme confirmed its consistency and plausibility in realistic arterial geometries and allowed to get insight into the relationship between aortic intravascular fluid structures and WSS directionality. The findings of this study clearly show the potential of the projection scheme as quantitative tool for an in depth investigation of the WSS multidirectional nature. The proposed approach enriches the arsenal of tools available to study and exploit the role played by local hemodynamics in vascular disease.

Analysis of thoracic aorta hemodynamics using 3D particle tracking velocimetry and computational fluid dynamics.

Parallel to the massive use of image-based computational hemodynamics to study the complex flow establishing in the human aorta, the need for suitable experimental techniques and ad hoc cases for the validation and benchmarking of numerical codes has grown more and more. Here we present a study where the 3D pulsatile flow in an anatomically realistic phantom of human ascending aorta is investigated both experimentally and computationally. The experimental study uses 3D particle tracking velocimetry (PTV) to characterize the flow field in vitro, while finite volume method is applied to numerically solve the governing equations of motion in the same domain, under the same conditions. Our findings show that there is an excellent agreement between computational and measured flow fields during the forward flow phase, while the agreement is poorer during the reverse flow phase. In conclusion, here we demonstrate that 3D PTV is very suitable for a detailed study of complex unsteady flows as in aorta and for validating computational models of aortic hemodynamics. In a future step, it will be possible to take advantage from the ability of 3D PTV to evaluate velocity fluctuations and, for this reason, to gain further knowledge on the process of transition to turbulence occurring in the thoracic aorta.

Inflow boundary conditions for image-based computational hemodynamics: impact of idealized versus measured velocity profiles in the human aorta.

Here we analyse the influence of assumptions made on boundary conditions (BCs) extracted from phase-contrast magnetic resonance imaging (PC-MRI) in vivo measured flow data, applied on hemodynamic models of human aorta. This study aims at investigating if the imposition of BCs based on defective information, even when measured and specific-to-the-subject, might lead to misleading numerical representations of the aortic hemodynamics. In detail, we focus on the influence of assumptions regarding velocity profiles at the inlet section of the ascending aorta, incorporating phase flow data within the computational model. The obtained results are compared in terms of disturbed shear and helical bulk flow structures, when the same measured flow rate is prescribed as inlet BC in terms of 3D or 1D (axial) measured or idealized velocity profiles. Our findings clearly indicate that: (1) the imposition of PC-MRI measured axial velocity profiles as inflow BC may capture disturbed shear with sufficient accuracy, without the need to prescribe (and measure) realistic fully 3D velocity profiles; (2) attention should be put in setting idealized or PC-MRI measured axial velocity profiles at the inlet boundaries of aortic computational models when bulk flow features are investigated, because helical flow structures are markedly affected by the BC prescribed at the inflow. We conclude that the plausibility of the assumption of idealized velocity profiles as inlet BCs in personalized computational models can lead to misleading representations of the aortic hemodynamics both in terms of disturbed shear and bulk flow structures.

Synthetic dataset generation for the analysis and the evaluation of image-based hemodynamics of the human aorta.

Here, we consider the issue of generating a suitable controlled environment for the evaluation of phase contrast (PC) MRI measurements. The computational framework, tailored to build synthetic datasets, is based on a two-step approach, i.e., define and implement (1) an accurate CFD model and (2) an image generator able to mime the overall outcomes of a PC MRI acquisition starting from datasets retrieved by the computational model. About 20 different datasets were built by changing relevant image parameters (pixel size, slice thickness, time frames per cardiac cycle). Focusing our attention on the thoracic aorta, synthetic images were processed in order to: (1) verify to which extent the fluid dynamics into the aortic arch is influenced by the image parameters; (2) establish the effect of spatial and temporal interpolation. Our study demonstrates that the integral scale of the aortic bulk flow could be described satisfactorily even when using images which are nowadays acquirable with MRI scanners. However, attention must be paid to near-wall velocities that can be affected by large inaccuracy. In detail, in bulk flow regions error values are well bounded (below 5% for most of the analyzed resolutions), while errors greater than 100% are systematically present at the vessel's wall. Moreover, also the data interpolation process can be responsible for large inaccuracies in new data generation, due to the inherent complexity of the flow field in some connected regions.

On the importance of blood rheology for bulk flow in hemodynamic models of the carotid bifurcation.

Here we present a study on the impact of assumptions on image-based hemodynamic simulations of healthy carotid bifurcations. In particular, we evaluate to which extent assumptions on blood rheology influence bulk flow features, driven by the fact that few studies have provided adequate insights into the influence of assumptions to confidently model the 4D hemodynamics within the bifurcation. The final goal is to complement, integrate and extend with a quantitative characterization of the bulk flow the description currently adopted to classify altered hemodynamics, which is based on wall shear stress (WSS). Hemodynamic simulations of two image-based carotid bifurcation geometries were carried out assuming a reference Newtonian viscosity, two non-Newtonian rheology models and Newtonian viscosities based on characteristic shear rates. WSS-based and Lagrangian-based metrics for helical flow quantification and for vorticity dynamics quantification were calculated. Our findings suggest that the assumption of Newtonian rheology: (1) could be reasonable for bulk flow metrics (differences from non-Newtonian behavior are lower than 10%); (2) influences at different levels the WSS-based indicators, depending on the bifurcation model, even if in our study it is lower than the major source of uncertainty as recognized by the literature (i.e., uncertainty on geometry reconstruction).

Mechanistic insight into the physiological relevance of helical blood flow in the human aorta: an in vivo study.

The hemodynamics within the aorta of five healthy humans were investigated to gain insight into the complex helical flow patterns that arise from the existence of asymmetries in the aortic region. The adopted approach is aimed at (1) overcoming the relative paucity of quantitative data regarding helical blood flow dynamics in the human aorta and (2) identifying common characteristics in physiological aortic flow topology, in terms of its helical content. Four-dimensional phase-contrast magnetic resonance imaging (4D PC MRI) was combined with algorithms for the calculation of advanced fluid dynamics in this study. These algorithms allowed us to obtain a 4D representation of intra-aortic flow fields and to quantify the aortic helical flow. For our purposes, helicity was used as a measure of the alignment of the velocity and the vorticity. There were two key findings of our study: (1) intra-individual analysis revealed a statistically significant difference in the helical content at different phases of systole and (2) group analysis suggested that aortic helical blood flow dynamics is an emerging behavior that is common to normal individuals. Our results also suggest that helical flow might be caused by natural optimization of fluid transport processes in the cardiovascular system, aimed at obtaining efficient perfusion. The approach here applied to assess in vivo helical blood flow could be the starting point to elucidate the role played by helicity in the generation and decay of rotating flows in the thoracic aorta.

Outflow conditions for image-based hemodynamic models of the carotid bifurcation: implications for indicators of abnormal flow.

Computational fluid dynamics (CFD) models have become very effective tools for predicting the flow field within the carotid bifurcation, and for understanding the relationship between local hemodynamics, and the initiation and progression of vascular wall pathologies. As prescribing proper boundary conditions can affect the solutions of the equations governing blood flow, in this study, we investigated the influence to assumptions regarding the outflow boundary conditions in an image-based CFD model of human carotid bifurcation. Four simulations were conducted with identical geometry, inlet flow rate, and fluid parameters. In the first case, a physiological time-varying flow rate partition at branches along the cardiac cycle was obtained by coupling the 3D model of the carotid bifurcation at outlets with a lumped-parameter model of the downstream vascular network. Results from the coupled model were compared with those obtained by imposing three fixed flow rate divisions (50/50, 60/40, and 70/30) between the two branches of the isolated 3D model of the carotid bifurcation. Three hemodynamic wall parameters were considered as indicators of vascular wall dysfunction. Our findings underscore that the overall effect of the assumptions done in order to simulate blood flow within the carotid bifurcation is mainly in the hot-spot modulation of the hemodynamic descriptors of atherosusceptible areas, rather than in their distribution. In particular, the more physiological, time-varying flow rate division deriving from the coupled simulation has the effect of damping wall shear stress (WSS) oscillations (differences among the coupled and the three fixed flow partition models are up to 37.3% for the oscillating shear index). In conclusion, we recommend to adopt more realistic constraints, for example, by coupling models at different scales, as in this study, when the objective is the outcome prediction of alternate therapeutic interventions for individual patients, or to test hypotheses related to the role of local fluid dynamics and other biomechanical factors in vascular diseases.

Womersley number-based estimates of blood flow rate in Doppler analysis: in vivo validation by means of phase-contrast MRI.

A common clinical practice during single-point Doppler analysis is to measure the centerline maximum velocity and to recover the time-averaged flow rate by exploiting an assumption on the shape of velocity profile (a priori formula), either a parabolic or a flat one. In a previous study, we proposed a new formula valid for the peak instant linking the maximum velocity and the flow rate by including a well-established dimensionless fluid-dynamics parameter (the Womersley number), in order to account for the hemodynamics conditions (Womersley number-based formula). Several in silico tests confirmed the reliability of the new formula. Nevertheless, an in vivo confirmation is missing limiting the clinical applicability of the formula. An experimental in vivo protocol using cine phase-contrast MRI (2-D PCMRI) technique has been designed and applied to ten healthy young volunteers in three different arterial districts: the abdominal aorta, the common carotid artery, and the brachial artery. Each PCMRI dataset has been used twice: 1) to compute the value of the blood flow rate used as a gold standard and 2) to estimate the flow rate by measuring directly the maximum velocity and the diameter (i.e., emulating the intravascular Doppler data acquisition) and by applying to these data the a priori and the Womersley number-based formulae. All the in vivo results have confirmed that the Womersley number-based formula provides better estimates of the flow rate at the peak instant with respect to the a priori formula. More precisely, mean performances of the Womersley number-based formula are about three times better than the a priori results in the abdominal aorta, five times better in the common carotid artery, and two times better in the brachial artery.

Quantitative analysis of bulk flow in image-based hemodynamic models of the carotid bifurcation: the influence of outflow conditions as test case.

Although flow-driven mechanisms associated with vascular physiopathology also deal with four-dimensional phenomena such as species transport, the majority of the research on the subject focuses primarily on wall shear stress as indicator of disturbed flow. Indeed, the role that bulk flow plays in vascular physiopathology has not been thoroughly investigated, partly because of a lack of descriptors that would be able to reduce the intricacy of arterial hemodynamics. Here, an approach is proposed to investigate, in silico, the bulk flow within the carotid bifurcation. For this purpose, we coupled a three-dimensional model of carotid bifurcation with a lumped model of the downstream vasculature. For the sake of comparison, we also imposed three different fixed flow rate repartitions between the internal and external carotid arteries on the three-dimensional model. The bulk flow was characterized by applying a descriptor of helical motion, the helical flow index (HFI) to the model; the HFI has recently been shown to provide an accurate representation of complex flows. Moreover, a new metric is presented to investigate the vorticity dynamics within the bifurcation. Our results highlight the effectiveness of these metrics in the following contexts: (i) identifying and ranking emerging hemodynamic features and (ii) quantifying the influence of the outflow boundary conditions on the composition of the translational and rotational components of the fluid motion. The metrics applied herein allow for a more comprehensive analysis, which may lead to the development of an instrument to relate the bulk flow to vascular pathophysiological events that involve not only fluid-related forces, but also transport phenomena within blood.

Womersley number-based estimation of flow rate with Doppler ultrasound: sensitivity analysis and first clinical application.

In this paper we continue in investigating the approach we have proposed in a paper recently published, for a reliable estimate of (peak systolic) blood flow rate from velocity Doppler measurements. Basic features of this approach together with some in silico test cases were discussed in that work. Here, we provide more insights of this approach by performing a sensitivity analysis of the formulas relating blood flow rate to velocity. In particular we analyze how our estimates are affected by perturbation or errors in measurements in comparison with a standard method for catheter based estimates based on the assumption of a parabolic velocity profile. A first glance to in vivo clinical applications is given as well.

Bubble tracking through computational fluid dynamics in arterial line filters for cardiopulmonary bypass.

Gaseous embolism is still a concern in cardiopulmonary bypass, and the use of arterial line filters (ALFs) is widespread because of their recognized role in increasing safety. Currently, the methods used for the optimization/evaluation of ALF designs are based on a trial-and-error approach. In this work, we propose a method to objectively assess the air-handling capabilities of ALFs, using computational fluid dynamics (CFD) simulations to track the trajectory of large numbers of bubbles traversing the device under examination. We applied the CFD method to ALF prototypes, whose design featured the classical purge/screen configuration, to establish the relative roles of the bubble-trap and bubble-barrier deairing effects. Simulations were run at the maximum rated blood flows. Clusters of hundred bubbles in the micro- to macroembolic size range (10-1,000 microm diameter) were tracked. The results quantified the relative amount of bubbles whose fate is either to reach the purge line or to be intercepted by the filter screen. For microbubbles, the analysis detailed how the screen barrier is exploited, mapping the percent distribution of the intercepted bubbles along the screen surface. We conclude that the method proposed increases knowledge and awareness in designing an optimal exploitation of the filtration mechanisms involved in ALFs.

Blood damage safety of prosthetic heart valves. Shear-induced platelet activation and local flow dynamics: a fluid-structure interaction approach.

Thromboembolism and the attendant risk of cardioembolic stroke remains an impediment to the development of prosthetic cardiovascular devices. In particular, altered haemodynamics are implicated in the acute blood cell damage that leads to thromboembolic complications, with platelet activation being the underlying mechanism for cardioemboli formation in blood flow past mechanical heart valves (MHVs) and other blood re-circulating devices. In this work, a new modeling paradigm for evaluating the cardioembolic risk of MHVs is described. In silico fluid-structure interaction (FSI) approach is used for providing a realistic representation of the flow through a bileaflet MHV model, and a Lagrangian analysis is adopted for characterizing the mechanism of mechanically induced activation of platelets by means of a mathematical model for platelet activation state prediction. Additionally, the relationship between the thromboembolic potency of the device and the local flow dynamics is quantified by giving a measure of the role played by the local streamwise and spanwise vorticity components. Our methodology indicates that (i) mechanically induced activation of platelets when passing through the valve is dependent on the phase of the cardiac cycle, where the platelet rate of activation is lower at early systole than late systole; (ii) local spanwise vorticity has greater influence on the activation of platelets (R>or=0.94) than streamwise vorticity (R>or=0.78). In conclusion, an integrated Lagrangian description of key flow characteristics could provide a more complete and quantitative picture of blood flow through MHVs and its potential to activate platelets: the proposed "comprehensive scale" approach could represent an efficient and novel assessment tool for MHV performance and may possibly lead to improved valve designs.

In vivo quantification of helical blood flow in human aorta by time-resolved three-dimensional cine phase contrast magnetic resonance imaging.

The mechanics of blood flow in arteries plays a key role in the health of individuals. In this framework, the role played by the presence of helical flow in the human aorta is still not clear in its relation to physiology and pathology. We report here a method for quantifying helical flow in vivo employing time-resolved cine phase contrast magnetic resonance imaging to obtain the complete spatio-temporal description of the three-dimensional pulsatile blood flow patterns in aorta. The method is applied to data of one healthy volunteer. Particle traces were calculated from velocity data: to them we applied a Lagrangian-based method for helical flow quantification, the Helical Flow Index, which has been developed and evaluated in silico in order to reveal global organization of blood flow. Our results: (i) put in evidence that the systolic hemodynamics in aorta is characterized by an evolving helical flow (we quantified a 24% difference in terms of the content of helicity in the streaming blood, between mid and early systole); (ii) indicate that in the first part of the systole helicity is ascrivable mainly to the asymmetry of blood flow in the left ventricle, joined with the laterality of the aorta. In conclusion, this study shows that the quantification of helical blood flow in vivo is feasible, and it might allow detection of anomalies in the expected physiological development of helical flow in aorta and accordingly, could be used in a diagnostic/prognostic index for clinical practice.

Numerical simulation of the dynamics of a bileaflet prosthetic heart valve using a fluid-structure interaction approach.

The main purpose of this study is to reproduce in silico the dynamics of a bileaflet mechanical heart valve (MHV; St Jude Hemodynamic Plus, 27mm characteristic size) by means of a fully implicit fluid-structure interaction (FSI) method, and experimentally validate the results using an ultrafast cinematographic technique. The computational model was constructed to realistically reproduce the boundary condition (72 beats per minute (bpm), cardiac output 4.5l/min) and the geometry of the experimental setup, including the valve housing and the hinge configuration. The simulation was carried out coupling a commercial computational fluid dynamics (CFD) package based on finite-volume method with user-defined code for solving the structural domain, and exploiting the parallel performance of the whole numerical setup. Outputs are leaflets excursion from opening to closure and the fluid dynamics through the valve. Results put in evidence a favorable comparison between the computed and the experimental data: the model captures the main features of the leaflet motion during the systole. The use of parallel computing drastically limited the computational costs, showing a linear scaling on 16 processors (despite the massive use of user-defined subroutines to manage the FSI process). The favorable agreement obtained between in vitro and in silico results of the leaflet displacements confirms the consistency of the numerical method used, and candidates the application of FSI models to become a major tool to optimize the MHV design and eventually provides useful information to surgeons.

Doppler derived quantitative flow estimate in coronary artery bypass graft: a computational multiscale model for the evaluation of the current clinical procedure.

In order to investigate the reliability of the so called mean velocity/vessel area formula adopted in clinical practice for the estimation of the flow rate using an intravascular Doppler guide wire instrumentation, a multiscale computational model was used to give detailed predictions on flow profiles within Y-shaped coronary artery bypass graft (CABG) models. At this purpose three CABG models were built from clinical patient's data and used to evaluate and compare, in each model, the computed flow rate and the flow rate estimated according to the assumption of parabolic velocity profile. A consistent difference between the exact and the estimated value of the flow rate was found in every branch of all the graft models. In this study we showed that this discrepancy in the flow rate estimation is coherent to the theory of Womersley regarding spatial velocity profiles in unsteady flow conditions. In particular this work put in evidence that the error in flow rate estimation can be reduced by using the estimation formula recently proposed by Ponzini et al. [Ponzini R, Vergara C, Redaelli A, Veneziani A. Reliable CFD-based estimation of flow rate in haemodynamics measures. Ultrasound Med Biol 2006;32(10):1545-55], accounting for the unsteady nature of blood, applicable in the clinical practice without resorting to further measurements.

Helical flow as fluid dynamic signature for atherogenesis risk in aortocoronary bypass. A numeric study.

The main purpose of the study was to verify if helical flow, widely observed in several vessels, might be a signature of the blood dynamics of vein graft anastomosis. We investigated the existence of a relationship between helical flow structures and vascular wall indexes of atherogenesis in aortocoronary bypass models with different geometric features. In particular, we checked for the existence of a relationship between the degree of helical motion and the magnitude of oscillating shear stress in conventional hand-sewn proximal anastomosis. The study is based on the numerical evaluation of four bypass geometries that are attached to a simplified computer representation of the ascending aorta with different angulations relative to aortic outflow. The finite volume technique was used to simulate realistic graft fluid dynamics, including aortic compliance and proper aortic and graft flow rates. A quantitative method was applied to evaluate the level of helicity in the flow field associated with the four bypass models under investigation. A linear inverse relationship (R = -0.97) was found between the oscillating shear index and the helical flow index for the models under investigation. The results obtained support the hypothesis that an arrangement of the flow field in helical patterns may elicit damping in wall shear stress temporal gradients at the proximal graft. Accordingly, helical flow might play a significant role in preventing plaque deposition or in tuning the mechanotransduction pathways of cells. Therefore, results confirm that helical flow constitutes an important flow signature in vessels, and its strength as a fluid dynamic index (for instance in combination with magnetic resonance imaging flow visualization techniques) for risk stratification, in the activation of both mechanical and biological pathways leading to fibrointimal hyperplasia.

Reliable CFD-based estimation of flow rate in haemodynamics measures.

Physically useful measures in current clinical practice refer often to the blood flow rate, that is related to the mean velocity. However, the direct measurement of the latter is currently not possible using a Doppler velocimetry technique. Therefore, the usual approach to calculate the flow rate with this technique consists in measuring the maximum velocity and in estimating the mean velocity, making the hypothesis of parabolic profile that in realistic situations results in strongly inaccurate estimates. In this paper, we propose a different way for estimating the flow rate regarded as a function of maximum velocity and Womersley number. This relation is obtained by fixing a parametrised representation and by evaluating the parameters by means of a least-square approach working on the numerical results of CFD simulations (about 200). Numerical simulations are carried out by prescribing the flow rate, not the velocity profile. In this way, no bias is implicitly induced in prescribing boundary conditions. Validation tests based on numerical simulations show that the proposed relation improves the flow rate estimation.