In Vitro Analysis of Hemodynamics in the Ascending Thoracic Aorta: Sensitivity to the Experimental Setup

Abstract

We perform a stochastic sensitivity analysis of the experimental setup of a mock circulatory loop for in vitro hemodynamics analysis in the ascending thoracic aorta at a patient-specific level. The novelty of the work is that, for the first time, we provide a systematic sensitivity analysis of the effect of the inflow conditions, viz. the stroke volume, the cardiac cycle period, and the spatial distribution of the velocity in in-vitro experiments in a circulatory mock loop. We considered three different patient-specific geometries of the ascending thoracic aorta, viz. a healthy geometry, an aortic aneurysm, and a coarctation of the aorta. Three-dimensional-printed phantoms are inserted in a mock circulatory loop, and velocity and pressure measurements are carried out for the different setup conditions. The stochastic approach, performed using the generalized polynomial chaos, allows us to obtain continuous and accurate response surfaces in the parameter space, limiting the number of experiments. The main contributions of this work are that (i) the flow rate and pressure waveforms are mostly affected by the cardiac cycle period and the stroke volume, (ii) the impact of the spatial distribution of the inlet velocity profile is negligible, and (iii), from a practical viewpoint, this analysis confirms that in experiments it is also important to replicate the patient-specific inflow waveform, while the length of the pipe connecting the pump and the phantom of the aorta can be varied to comply with particular requirements as, for instance, those implied by the use of MRI in experiments.

1. Introduction

Congenital heart diseases are defined as deformities of the structure of the large blood vessels or of the heart. Ascending thoracic aortic aneurysms and coarctations are life-threatening diseases, since they can lead to major complications, such as aortic dissection and rupture. An aneurysm is a permanent dilatation of the aorta, conventionally presenting an increase in diameter greater than $50\%$ [1]. In the case of dissections originating in the ascending aorta, the mortality rate is between $15\%$ and $30\%$ [2], and even >94% for aortic ruptures [3]. Surgical repair is not safe either, as it implies a mortality risk of about $5\%$ [4]. On the other hand, the coarctation of the aorta is an obstructive congenital heart disease corresponding to approximately $8\%$ of all congenital heart diseases [5]. It appears as a narrowing of the cross-section of the aorta, which usually occurs in the thoracic area. The restriction compromises the functionality of the aorta, leading to an alteration of the blood flow and an increase in the pressure values in the upper part of the body [6].

Biomechanics and fluid dynamics studies of cardiovascular systems have assumed increasing importance in clinical terms, since they can give support in the analysis of the physical mechanisms ruling possible pathologies [7,8]. A huge amount of experimental and numerical work has been carried out on modeling blood flow in the cardiovascular network to analyze flow patterns for implants, medical drug delivery, and rupturing of vessels (see, e.g., [9,10,11,12,13]). Available tools for hemodynamic investigations at a patient-specific level include in silico analyses through computational fluid dynamics (CFD) simulations and in vitro experiments. In recent years, both approaches have been increasingly used for cardiovascular studies, thanks to procedures that integrate in vivo data (from magnetic resonance imaging (MRI), computer tomography (CT), and color Doppler ultrasound (CD-US)) in simulations and experiments (see, e.g., [14,15,16,17,18,19]).

CFD simulations, usually carried out by using finite-element and/or finite-volume approaches, allow the computation of the spatial and temporal distributions of the hemodynamic parameters with a level of accuracy not obtainable by any in vivo measurement. To obtain patient-specific predictions, the results of clinical exams are usually used as boundary conditions for the simulations (as in [16,17,20,21,22]). However, in silico analyses suffer from the presence of multiple sources of uncertainty that may reduce the accuracy of the results. Particular attention should be devoted to the segmentation of in vivo-acquired geometries and to the definition of inflow/outflow boundary conditions (see, e.g., [23,24,25,26,27,28,29,30,31]).

In vitro approaches may be synergically used with in silico ones to contribute to the validation of numerical simulations, since they avoid some of the sources of uncertainties that may lead to discrepancies between the in silico and in vivo results (see, e.g., [32,33]). Recent patient-specific experiments have been carried out using phantoms that are 3D-printed or produced using a lost casting technique from in vivo acquisitions (see, e.g., [34,35,36]). Among the different experimental apparatus, the sensorized mock circulatory loop (MCL) developed in [37,38] ensures perfect control and knowledge of the flow rate and pressure conditions and of the vessel geometry and wall properties. The proposed MCL has a versatile setup composed of a custom pulsatile left ventricular pump system, patient-specific 3D-printed phantoms, and chambers and pinch valves to emulate the effects of the downstream organs and vessels. It was demonstrated in [37] that the system was able to cover a wide range of aortic and mitral flows and to reproduce patient-specific cases both in terms of flow and pressure waveforms. Flow and pressure waveforms have been successfully validated with in vivo data from magnetic resonance analysis of patients in the preliminary results reported in [39,40]).

In the experimental framework, the assessment of a reliable and accurate setup for device testing represents a pivotal step in the reproduction of realistic hemodynamic conditions in a controlled environment (see, e.g., [41]). Thus, in this paper, we carried out a systematic analysis of the sensitivity of the experimental setup conditions to evaluate their impact on in vitro-measured quantities. We consider different healthy and disease geometries of the aorta, and we focus on the inflow conditions that have shown to be a crucial issue in both experiments and simulations (see, e.g., [21,22,30,31,42]). In particular, we consider two parameters characterizing the inlet flow-rate waveform, which are the cardiac-cycle period and the volume of blood introduced in the ascending aorta at each cycle. These two quantities have been shown to significantly influence the blood-flow repartition between the descending aorta and the supra-aortic branches and the magnitude of the wall shear stresses in the vessel in the numerical results in [31]). The third parameter is the spatial distribution of the inflow velocity. Indeed, the possible presence of cardiac valves upstream of the portion of the vessel reproduced in the 3D-printed experimental phantom may cause velocity patterns that are not uniform in space (see, e.g., [43,44,45]). Reproducing such a complex and time-dependent pattern is not possible in the MCL but, following what is also usually conducted in state-of-the-art numerical simulations, different idealized velocity profiles are obtained by changing the length of the connecting pipe between the pump and the phantom. We consider inflow velocity distribution ranging from the plug flow condition, i.e., uniform velocity distribution thanks to a very short connecting pipe ([17,23,24,46,47,48]), to a fully-developed parabolic distribution ([20,46,47,49,50,51,52]).

Continuous response surfaces in the setup parameter space of the measured quantities of interest are obtained by using a stochastic approach (uncertainty quantification (UQ). The setup parameters are defined as aleatory variables characterized by a known probability density function (PDF). Similarly to [25,31,53,54], the generalized polynomial chaos (gPC) expansion is used, because it converges much faster than Monte-Carlo-based methods (see, e.g., [55]). This approach allows to limit the number of experiments and, at the same time, ensures a good level of accuracy of the results.

The reference clinical dataset, the experimental setup, and the stochastic procedure are described in Section 2. The main flow features and the sensitivity analysis results are presented in Section 3 and Section 4. Final conclusions are given in Section 5.

2. Clinical Dataset, Experimental Setup, and Stochastic Sensitivity Analysis Procedure

2.1. Clinical Dataset

The healthy aorta dataset was obtained from clinical measurements in [17]. Morphological and functional patient-specific data were acquired from MRI analysis of a healthy 28-year-old male. The aorta geometry, shown in Figure 1a, was obtained through segmentation of the MRI data. It is characterized by the ascending aorta, AA, the descending aorta, DA, and the three supra-aortic branches (brachiocephalic artery, BCA, left common carotid artery, LCCA, and left subclavian artery, LSA). Figure 1b shows the waveform of the flow rate entering AA, which is acquired by processing the functional data obtained by means of phase-contrast MRI. The following acquisition parameters were set for PC-MRI: a flip angle of 10${}^{\circ }$, a repetition time of 5.32 ms, an echo time of 3 ms, a $2\times 2\times 2$ mm³ isotropic voxel, and a velocity encoding (VENC) interval of 250 cm/s.

The geometry of the healthy aorta was locally modified using a morphing procedure to generate two diseased geometries, viz. an aneurismatic aorta (Figure 2a) and a coarctated aorta (Figure 2b). The aneurysm develops in AA, and it is characterized by an increase in vessel diameter of about $60\%$. On the other hand, the coarctation is located in the descending branch with a decrease in the aortic diameter of up to $70\%$. The flow rate waveform from Figure 1b is also used as a reference for the diseased aortas.

2.2. Experimental Setup

The MCL is shown in Figure 3a and consists of a pump system, a reservoir, and resistive (pinch valves) and capacitive (rigid air-filled chambers) components for the set of the outlet pressure conditions at DA and at the supra-aortic branches [37]. To obtain the physiological pressure waveform, the three-element Windkessel model is used. The values of each lumped parameter are determined as in [15].

The aorta phantom is placed as in Figure 3b. It is realized with a stereolithographic 3D printing machine (Form 2, Formlabs). A semitransparent rigid resin (Clear Resin, Formlabs) was used. Cylindrical junctions are designed at the phantom’s inlet and outlets to link it with the circulatory mock loop. The inlet flow rate is set through the pump system, whereas the velocity profile is set by changing the length of the pipe connecting the pump and the phantom. A mixture of water (60%) and glycerol (40%) was chosen as the working fluid. The mixture has the same density and viscosity as blood ($\rho =$1.06 g/${\mathrm{cm}}^3$ and $\nu =$ 0.036 g/cm s).

The flow-rate waveforms are acquired by means of ultrasonic clamp-on flow sensors (Sonotec), whereas those of pressure through a strain gauge transducer (Truwave, Edwards). The flow field characterization is achieved through echography measurements (Doppler tracking methods), a technique that is widely used in the diagnosis of aortic diseases (see, e.g., [56]).

2.3. Stochastic Sensitivity Analysis Procedure

The stochastic sensitivity analysis to inlet boundary conditions is carried out using the gPC method in its nonintrusive form. In the gPC approach, the dependence between a stochastic variable of interest, Y, and the vector of independent uncertain parameters $\mathsf{\eta }$ is expressed by means of a polynomial expansion [57]. So, by employing term-base indexing:

$${\displaystyle Y\left(\psi \right)=\sum _{i=0}^{\infty }{a}_i{\Theta }_i\left(\mathsf{\eta }\left(\psi \right)\right)}$$

(1)

where $\psi$ is an aleatory event, ${\Theta }_i\left(\mathsf{\eta }\right)$ is the i-th gPC polynomial, and $a_i$ is the coefficient of the aforementioned variable within the expansion. To practically obtain the response surface, it is necessary to truncate the previous sum from Equation (1). This finite limit (N) is here determined as follows:

$$N=\prod _{j=1}^K(R_j+1)-1$$

(2)

where K is equal to the number of uncertainty variables of the problem, whereas $R_j$ is the highest polynomial order selected for the variable of index j. To calculate the coefficients $a_i$, the following mathematical expression is used:

$$a_i=\frac{〈Y,{\Theta }_i〉}{〈{\Theta }_i,{\Theta }_i〉}=\frac{1}{〈{\Theta }_i,{\Theta }_i〉}{\int }_{\mathrm{supp}\mathsf{\eta }}Y{\Theta }_{i}w\left(\mathsf{\eta }\right)\mathrm{d}\mathsf{\eta }$$

(3)

where $w\left(\mathsf{\eta }\right)$ is the weight function of ${\Theta }_i$. The Gaussian quadrature rule is herein used to calculate the coefficients $a_i$. The polynomial family ${\Theta }_i$ is a priori specified. Its optimal choice has a weight function in Equation (3) similar to the PDF assumed for the uncertain parameters.

Partial sensitivities are computed through the Sobol indices [58], which measure the direct influence of individual input variables or couples of them. The Sobol index $I_i$ is defined as the ratio between the variance only due to the i-th uncertain input parameter or to combinations of them, ${\sigma }_i^2$, and the total variance ${\sigma }^2$, as follows:

$$I_i=\frac{{\sigma }_i^2}{{\sigma }^2}$$

(4)

In the present work, the uncertain parameters of the experimental setup are the following: the cardiac-cycle period, T, the stroke volume, $SV$, and the length of the connecting pipe between the pump and the phantom, L. As conducted in [31], the PDFs for the parameters T and $SV$ are derived fitting clinical data from 23 patients with beta distributions:

$$g(y;\alpha ,\beta )=A{(1-y)}^{\alpha }{(1+y)}^{\beta }$$

(5)

where A is a constant introduced to normalize the integration of $g\left(y\right)$ over the interval $(-1,1)$. Exponents $\alpha$ and $\beta$ are the coefficients to be calibrated to match the input data. Considering as ranges of variation of the parameters $T\in [0.6,1.4]$ $\mathrm{s}$ and $SV\in [20,170]$ $\mathrm{m}\mathrm{L}$, we obtained $\alpha =5.5$ and $\beta =1.3$ for T and $\alpha =4.3$ and $\beta =2.6$ for $SV$. As for the third uncertain parameter, its range of variation is $L/D\in [0,70]$, the pipe diameter being $D=1$ cm. The lower limit is such that it provides the plug flow condition, and the upper limit is the length necessary for carrying out MRI measurements on the phantom if needed (the MCL is MRI-compatible, except for the pump that should be placed outside the MRI-dedicated room). We assume a uniform PDF distribution for $L/D$, obtained by using $\alpha =1$ and $\beta =1$ in Equation (5).

We choose the Jacobi polynomial family for the gPC expansion, which is the optimal one when dealing with beta PDF distributions. For all the uncertainty parameters, the expansion in Equation (1) is truncated to the third order. Thus, 4 quadrature points are needed for T, $SV$, and $L/D$, which are summarized in

Table 1. Quadrature points for the parameters T, $SV$, and $L/D$.

T [$\mathrm{s}$]	0.67	0.79	0.95	1.14
$\mathit{SV}$ [$\mathrm{m}\mathrm{L}$]	41.4	69.1	101.5	133.8
$\mathit{L}/\mathit{D}$ [−]	8	25	45	62

. Figure 4 shows the 16 inlet flow-rate waveforms for all the tensor–product combinations of T and $SV$. The flow-rate waveforms are obtained by properly scaling the one in Figure 1b. Moreover, the effects of $L/D$ on the distribution in space of the velocity at the inlet section of the phantom at the systolic peak are shown in Figure 5a,b for two combinations of T and SV, giving the lowest and the highest flow rates, respectively.

3. Main Flow Features

The description of the flow features in the three considered geometries refers to the echographic measurements shown in Figure 6, Figure 7 and Figure 8 for healthy, aneurismatic, and coarctated aortas, respectively. We carried out the measurements with a long-axis view, and the investigated portions of the aorta (A and B) are sketched for the sake of clarity. We considered the flow-rate waveform in Figure 1b and three temporal instants along the cardiac cycle: $t/T=0.156$ (systolic peak), $t/T=0.208$ (early diastole), and $t/T=0.260$ (diastole).

No significant flow separations or recirculations were found in the healthy aorta at the systolic peak, as shown in Figure 6b. The flow exhibits a regular behavior both along the ascending and descending aortic portions at the systolic peak and also in the early diastole (Figure 6c). Some irregularities are found during the diastole, typical of flow deceleration (Figure 6d). A significantly different flow pattern occurs in the ascending aorta (A view) when affected by the aneurysmatic pathology. The increase in the vessel cross-section leads to wide flow recirculations, both at the systolic peak and during the diastole (Figure 7b–d). Similar flow patterns are instead found for healthy and aneurismatic aortas in DA (B view). The coarctated aorta does not exhibit flow separations in the ascending aorta (A view in Figure 8) and in the descending aorta upstream of the striction induced by the coarctation (B view in Figure 8; the flow entering the coarctation can be clearly viewed on the right side of the figure). A similar flow pattern as for the healthy aorta is found upstream of the coarctation.

4. Results of the Sensitivity Analysis to Setup Parameters

4.1. Deterministic Results

The effects of the variation of $SV$, T, and $L/D$ are first analyzed in terms of flow-rate waveforms at the outlet sections (BCA, LCCA, LSA, and DA) and of the differential pressure waveforms, $\Delta p\left(t\right)$, evaluated between the ascending (${\Gamma }_{\mathrm{AA}}$) and the descending (${\Gamma }_{\mathrm{DA}}$) aorta as follows:

$$\Delta p\left(t\right)=p_{{\Gamma }_{\mathrm{AA}}}\left(t\right)-p_{{\Gamma }_{\mathrm{DA}}}\left(t\right)$$

(6)

Figure 9 shows the time behavior of the flow rates at the four outlet sections of the healthy aorta in the 64 in vitro measurements. It is clear that $SV$ and T affect the flow-rate magnitude, which increases by increasing $SV$ and/or reducing T, but slight differences are found among the shapes of the waveforms. The length L does not significantly influence the waveforms. The same trends also hold for the diseased geometries (Figure 10 and Figure 11). For the aneurismatic aorta, the flow repartition between DA and the supra-aortic branches and the related flow-rate waveforms are also similar to the ones of the heathy geometry (compare Figure 10 with Figure 9), and again, $SV$ and T are the main ones responsible for waveform variations. Despite the presence of the coarctation leading to a flow-rate reduction in the descending aorta for the 64 experiments presented in Figure 11, the main parameters influencing the waveform magnitude are still the same.

As for the differential pressure waveforms, it is evident from Figure 12 that this quantity is massively influenced by the presence of the coarctation. For a given triple (T, $SV$, and $L/D$), the pressure drop increase of about one order of magnitude in the case of coarctation compared with healthy and aneurysmatic geometries gives lower and similar results in terms of waveforms. This is due to the pressure drop occurring in the coactation region, and we recall that the guideline for coarctation treatment recommends surgical restoration for $\Delta p\ge 20\mathrm{mmHg}$ at rest [59]. For each geometry, stroke volume and cardiac cycle period significantly impact the magnitude of the waveforms, whereas the inlet velocity profiles, i.e., of the lengths $L/D$, have a negligible effect.

4.2. Stochastic Results

The PDF distributions and the partial sensitivities obtained from the stochastic analysis are shown in the top and bottom panels of Figure 13, Figure 14 and Figure 15 for the flow rates in the healthy, aneurismatic, and coarctated aortas, respectively. Significant variability of the results is present for all the outlet sections during the systolic part of the cardiac cycle, $t/T\in [0.1,0.35]$, whereas variability is lower during the diastole. During the systole, the main parameter affecting the stochastic variability of the flow-rate waveform is $SV$, followed by the period T. The parameter T becomes the most important in the diastolic phase for the flow-rate waveforms in DA and in BCA, followed by $SV$. However, it should be noted that Sobol indexed in diastole is computed over a total variability of the quantity of interest more than one order of magnitude lower than the one in systole. Finally, the parameter $L/D$ and interaction effect between the parameters always have a negligible impact on flow-rate variability.

As for the PDFs and Sobol indexes for the differential pressure waveforms shown in Figure 16, the stroke volume is still the main parameter affecting result variability in the systole, with a significant effect of T as well and negligible ones for $L/D$ and interactions. During the diastole, again characterized by a lower variability in terms of differential pressure, $SV$ is the main parameter when dealing with healthy and aneurismatic geometries, whereas the period T is the most effective for the diastolic phase in the coarctated aorta.

5. Conclusions

This study investigates the effects of the experimental setup on the main quantities of interest that can be measured in a mock circulatory loop though a stochastic approach. This represents a fundamental step towards the reproduction of realistic hemodynamic conditions in a controlled environment. We consider different healthy and disease geometries of the aorta, and we focus on the inflow conditions, provided in terms of flow-rate waveform and spatial distribution of the velocity at the inlet section. We only consider rigid walls in the present study, and we measure the flow-rate split among the outlets, as well as the differential pressure between the ascending and descending aorta.

From the flow-rate waveforms, we found that during the systolic phase, the parameter providing the highest variability of the measurements is the stroke volume, followed by T. During the diastole, T and $SV$ have a comparable importance, but for a decreased total variability. Analogous results are found for the differential pressure and for all three investigated geometries. Regarding the spatial velocity distribution, it always has a negligible impact. This is a good point, because at the state of the art, the adoption of more advanced techniques of image acquisition, such as 4D-flow magnetic resonance, is having a significant impact on experimental research [60] and the inspection of flow inside the mock circuit via MRI, planned as a future development of the research activity, will imply the use of different lengths of inlet pipe connecting the pump and the phantom of the aorta, and thus possibly different velocity profiles at the phantom inlet section. This will be allowed, on one hand, by the fact that the elements of the circuit, including the lumped parameter components, are managed to be MRI-compatible (without relying on hybrid hardware-in-the-loop solutions [61]) and, on the other, by the negligible impact of the spatial distribution of the velocity on the measured hemodynamic quantities in all the tested operating conditions. Moreover, similar sensitivity analyses could be repeated to investigate the effect of (i) wall compliance and (ii) outflow conditions. Compliant phantoms may be used to account for wall deformation together with FSI-CFD simulations ([62]). On the other hand, to investigate the effects of the outflow conditions, a more controllable system is needed that includes hybrid hardware-in-the-loop units ([61]) to emulate the Windkessel model at the outlet sections.