Credibility assessment of patient-specific modeling in transcatheter aortic valve implantation. I. A population-based validation of patient-specific modeling

Abstract

Transcatheter aortic valve implantation (TAVI) has become a standardized treatment for aortic valve stenosis, supported by computational modeling to enhance procedural planning. However, the credibility of TAVI simulations requires rigorous validation following regulatory standards. This study aims to perform a population-based validation of the structural and hemodynamic simulation performance of the SAPIEN 3 (S3) Ultra device by comparing computational predictions with clinical post-TAVI data, following the ASME V&V 40 framework. A patient-specific structural model followed by fluid–structure interaction was developed to simulate S3 deployment and then assess post-TAVI hemodynamics in 20 patients. Structural parameters (device diameters) and hemodynamic indices (effective orifice area, EOA, and transmural pressure gradient, TPG) were extracted. Validation was performed using empirical cumulative distribution function (ECDF) analysis, with an acceptance threshold of 5% for model credibility. EOA and TPG predictions showed reasonable agreement with echocardiographic data (errors within 10%). ECDF-based comparison demonstrated a high level of accuracy for device diameters (≤5% area metric), whereas hemodynamic parameters exhibited slightly greater discrepancies, potentially due to clinical measurement variability. This study establishes a robust computational validation framework for patient-specific TAVI modeling, ensuring regulatory compliance and clinical applicability. These findings highlight the potential of in silico trials to support TAVI planning and decision-making. This study is complemented by a second part dedicated to uncertainty quantification and sensitivity analysis.

I. INTRODUCTION

Since the first-in-human implantation in 2002, transcatheter aortic valve implantation (TAVI) has advanced from a complicated technology to a safe and standardized therapy for the treatment of aortic valve stenosis in the elderly.^1–3 This achievement results not only from improvements in device design and delivery systems, as well as clinician confidence in the percutaneous treatment, but also from clinical and fundamental research efforts.

Using patient-specific geometries and physiological boundary conditions, computational modeling can replicate the structural and hemodynamic processes associated with TAVI, supporting clinicians in patient interpretation and treatment. Recent works on TAVI include fluid–structure interaction (FSI) using either arbitrary Lagrangian–Eulerian⁴ or smoothed particle hydrodynamics (SPH)⁵ methods for self- and balloon-expandable transcatheter heart valves. These in silico methods are categorized as software as a medical device from a regulatory perspective and must undergo a validation process to evaluate the prediction accuracy of the model. This process is known as credibility assessment to quantify the in silico methodology through evidence and is a risk-based framework that defines credibility requirements for in silico models. To provide a standard for the credibility assessment in computational models, the American Society of Mechanical Engineering (ASME) has developed a generalized framework for assessing model credibility known as “V&V40—Assessing Credibility of Computational Modeling through Verification and Validation: Application to Medical Devices.”⁶ The ASME V&V 40 standard asserts that model credibility can be achieved using verification, validation, and uncertainty quantification to demonstrate the model accuracy and reliability. Verification determines if a computational model matches the mathematical model and solution. Validation evaluates how well the model matches in vitro or in vivo data for the intended purpose. However, validation studies in the context of TAVI simulations were based on comparisons with postoperative angiographies,⁷ without considering quantitative evaluations of the three-dimensional device's deformed shape. Recently, a comprehensive validation using experimental crimping data for collecting radial stent-frame forces was carried out by Grossi et al. in accordance with ASME V&V 40.⁸

In light of the lacks in the model credibility of TAVI simulations, this work seeks to perform a population-based validation of the structural and hemodynamic model performance of the SAPIEN 3 (S3) Ultra device (Edwards Lifesciences, Irvine, USA) in comparison to quantitative post-TAVI clinical data. The validation process was organized in alignment with ASME V&V 40 standards, beginning with the specification of the question of interest (QoI), the context of use (CoU), and associated model risks. The structural and hemodynamic parameters of the S3 device were obtained using both a previously established finite-element model for S3 deployment⁹ and an FSI post-TAVI simulation based on the SPH approach here described. The SPH-based approach allowed us to characterize the interaction between the TAVI components required for the FSI problem using the same solver for both structural and hemodynamic analysis, despite the fact that SPH has a number of limitations, such as low spatial discretization near boundaries. To assess the rigor of output comparison, probability distributions of the structural and hemodynamic parameters were calculated for both clinical and in silico data, and the agreement within an acceptance criterion was evaluated to determine the model credibility. Specifically, the probability density functions were derived from scalar values of chosen structural and hemodynamic parameters collected from ten patients for each device size. We empathize that this work represents the first part of a two-part study. In part 1, we focus on the validation of a patient-specific computational model for TAVI, assessing the structural and hemodynamic model predictive capabilities against clinical data. Part 2 of the study, published separately,¹⁰ is dedicated to uncertainty quantification and sensitivity analysis, investigating how variations in model parameters—such as material properties, boundary conditions, and procedural factors—affect simulation outcomes. Together, these two studies provide a comprehensive credibility assessment of the TAVI modeling framework in accordance with the ASME V&V 40 standard.

II. RESULTS

The simulated deformed shape of the S3 device stent frame was qualitatively compared to that derived from post-TAVI computed tomography (CT) imaging, as illustrated for a representative patient case in Fig. 1. A good agreement was qualitatively seen between the simulation and the actual clinical imaging regarding device implantation depth and conformability to the aortic root. Figure 2 illustrates the final configuration of the S3 device, together with the fabric skirt and the mapped valve leaflets for the 23 and 26 mm S3 devices. Varying degrees of expansion among patients according to the amount of calcifications and aortic root morphology were observed.

FIG. 1.

(a) Frontal and (b) axial view of the simulated S3 stent frame (red) against the actual segmented device (blue) for a representative patient case.

FIG. 2.

Representative final configuration resulting from TAVI simulation for different patient cases with either (a) and (b) the 23 mm S3 or (c) and (d) the 26 mm S3.

The blood flow velocity and S3 valve kinematics demonstrated consistent patterns aligned with expected physiological behavior during the simulated cardiac cycle. Figure 3 illustrates the flow velocity field at different phases of the cardiac cycle for a representative patient with the 26 mm S3 device. The S3 leaflets progressively open during the acceleration phase, developing a robust central flow jet in the systolic phase [Fig. 3(b)]. A realistic valve closure coaptation occurs during diastole, showing minimal retrograde flow and faithfully sealing the left ventricular chamber [Fig. 3(e)]. Figure 4 similarly illustrates the systolic flow velocity jet for four patient cases and different valve sizes. The peak velocity for the whole patient study group was 2.44 ± 0.45 m/s, in contrast to the echocardiographic assessment of 2.50 ± 0.41 m/s. The SPH-related effective orifice area (EOA) was 1.47 ± 0.45 cm², in contrast to echocardiographic values of 1.41 ± 0.38 cm², while the transvalvular pressure gradient (TPG) was 29.96 ± 8.30 mm Hg, opposed to clinical assessments of 27.35 ± 7.51 mm Hg.

FIG. 3.

Flow velocities for a representative patient case at (a) opening, (b) flow acceleration, (c) peak flow, (d) deceleration, and (e) early-diastolic phases.

FIG. 4.

Flow velocity distribution for four representative patient cases at systolic phase with (a) and (d) size of 23 mm and (b) and (c) size of 26 mm.

The empirical cumulative distribution function (ECDF) curves calculated for each patient for both in silico predictions and actual in vivo measurements were obtained for the validation analysis [Figs. 5(a)–5(e)]. For each quantity of interest, the area metric enabled a quantitative evaluation of the level of agreement for the acceptance criterion of 5%. The area metric values for the device diameter were 1.50%, 5.10%, and 3.10% for the inflow, mid-flow, and outflow cross-sectional levels, respectively. The evaluation of fluid-related validation measures resulted in an area metric of 9.60% for the TPG and 6.30% for the EOA. Figure 5(f) illustrates an assessment of the distribution of relative errors between computational and clinical data for the investigated quantities of interest. Both EOA and TPG demonstrated greater discrepancies from clinical measures than from device diameter estimations.

FIG. 5.

ECDF plots for both the clinical measurements and the predictions of (a) TPG, (b) EOA, (c) inflow diameter, (d) mid-flow diameter, (e) outflow diameter, and (f) ECDF of relative error for all quantities of interest.

III. DISCUSSION

The framework established in this study is based on rigorous verification and validation processes to assure the credibility of the patient-specific model for structural and hemodynamic evaluations of TAVI-related medical devices. The model credibility for clinical and regulatory decision-making was established by specifying the QoI and CoU. We initially outlined the validation process to identify the quantity of interest and the comparator, subsequently proposing an analytical comparison based on a predetermined acceptance criterion to evaluate the degree of rigor and the related model risk. The novelty of the suggested methodology is contingent not only upon the robust development of the patient-specific TAVI model but also on the probabilistic evaluation to determine the agreement between in silico and in vivo data.

While other research groups aimed to validate patient-specific TAVI models,^11,12 this study is the first one to carry out a probabilistic evaluation of the entire patient cohort for the balloon-expandable S3 device. Bosi et al.⁷ performed a qualitative validation of the self-expandable CoreValve device using angiographic imaging on 28 patients. Recently, Grossi et al.⁸ completed a validation of the Evolut R and Acurate Neo2 devices against in vitro and in vivo data utilizing the ASME V&V 40 framework to establish model credibility. They validated the model by analyzing the average percentage differences of the relative errors between the predicted deformed device shapes and those seen in postoperative angiographies. This study presented a more rigorous and comprehensive validation from various perspectives. The implementation of post-TAVI CT imaging as a standard procedure in TAVI permitted accurate diameter measurements due to CT's superior resolution compared to angiography. Second, the fluid-dynamic behavior associated with the S3 was evaluated against echocardiographic data for an in-depth assessment of the device performance using standard metrics for cardiac valve prosthesis design. The rigor of the agreement was evaluated by probabilistic distributions of each validation parameter to enhance the knowledge of the real variability and distribution of the in silico prediction. We used the ECDF of the relative error to show that the estimates of the inflow and outflow device diameters had very little variation across all patients, with almost 80% of the simulated data points having relative errors that were less than 5% of the clinical data. The variability was slightly higher at the mid-level of the S3 device compared to the inflow and outflow levels, possibly due to the distinct calcification patterns specific to the aortic root models. The area metric was also used to quantify the degree to which the distribution ECDF shapes fit, as opposed to differences in a single value. A stringent level of rigor (area metric <5% acceptance level) was achieved for the device diameter, indicating the strong accuracy of the structural TAVI model in driving decisions on S3 deployment. A significant difference was noted between computational and clinical data for the flow-based parameters. We hypothesize that disparities may arise not only from the FSI model's ability to simulate post-TAVI hemodynamics but also from possible errors in clinical assessment. The EOA and TPG parameters were obtained by echocardiography through an indirect estimation utilizing measured velocities and the Bernoulli equation. In contrast, the FSI analysis allowed the direct computation of flow-related parameters from the valve kinematics and the pressure field. Echocardiographic assessments exhibit greater sensitivity to intra-observer variability and instrument accuracy. However, the EOA and TPG area metrics were still smaller than the 10% discrepancy between the clinical and computational data. Indeed, the acceptance requirement of 5% here proposed is a stringent value in consideration of the fact that the ASME V&V 40 does not specify a specific threshold for validation activities.⁶ It should also be mentioned that the area metric has the advantage of allowing for the detection of varying degrees of variance in both clinical and numerical results while maintaining an identical mean value. However, this metric loses its sensitivity to variance differences if the clinical and simulation do not intersect.

Establishing trust in the predicted accuracy of computational models designed for clinical or regulatory decisions is essential. Aldieri et al.¹³ carried out a thorough implementation to assess the reliability of a computer model for predicting femur fracture risk. Curreli et al.¹⁴ utilized ASME V&V 40 to evaluate the credibility of an agent-based model employed in drug development. In the cardiovascular field, Galappaththige et al.¹⁵ expanded the ASME V&V 40 framework to encompass patient-specific cardiac models. In a similar way, Santiago et al.¹⁶ utilized the standard for flow modeling following LVAD implantation, whereas Ramella et al.¹⁷ introduced a structural model of the TEVAR approach, which was successfully tested and validated against experimental data. Nonetheless, the implementation of the comprehensive credibility plan via verification and validation operations remains poorly examined. Our principal contribution is the generation of a realistic structural and fluid-dynamic model, which was validated at each component level. Although the SPH technique does not facilitate the estimate of shear forces, the proposed FSI model achieved full two-way coupling among anatomical components, the device, and blood, thereby surpassing the rigid part assumptions employed in prior investigations. We stress that both structural and hemodynamic boundary conditions were customized to each patient's physiological state. It should be noted that the ASME indicates an uncertainty quantification study to complete the validation process. Uncertainty quantification seeks to uncover potential model limitations either from inherent variability or insufficient knowledge. While this study addressed the model validation, a comprehensive investigation of how errors in material characteristics, implantation depth, and flow boundary conditions affect the simulation outcome is given in the second part of the study as described by Scuoppo et al.¹⁰ In order to determine how applicable the chosen validation activities are within the context of use, the ASME also recommends an applicability assessment to complete the credibility assessment. This was, however, not carried out in this study.

IV. CONCLUSIONS

This work proposed a rigorous and comprehensive methodology for the population-based validation of patient-specific structural and hemodynamic models of TAVI in relation to clinical evidence. The patient-specific TAVI model was validated for each component to ensure robustness in accordance with the ASME V&V 40 framework. The validation method included a risk-informed credibility framework for assessing the necessary levels of model validation, indicating that the model had sufficient credibility for the specified context of use. Probabilistic analysis confirmed the accomplishment of a 5% acceptability threshold for predictions regarding the S3 deployment configuration for the entire study group. The fluid-based quantities of interest (i.e., EOA and TPG) did not satisfy the acceptance criterion relative to the comparator; however, the error distribution remained within the 10% threshold deemed acceptable despite possible sources of error in clinical estimations. This study offered insights into the systematic and rigorous assessment of the credibility of computational modeling for TAVI and exemplified how population-based evidence for the credibility of virtual populations should be generated through statistical comparisons to ensure that predictions align with comparative clinical trials.

V. METHODS

A. Patient study group

A prospective study was carried out to enroll patients with severe aortic valve stenosis treated by TAVI using both 23- and 26-mm devices. Patients with the 29 mm S3 device were excluded from our study due to its infrequent use in our clinical institution (incidence of ≤10). Prospective enrollment was necessary, as structural validation of the in silico model involved comparison with post-TAVI contrast-enhanced computed tomography (CT) imaging of the implanted bioprosthesis. Following ethical approval and the signing of informed consent, each patient had post-TAVI CT imaging to assess the S3 diameter at three cross-sectional levels—inflow, mid-flow, and outflow. After TAVI, transesophageal echocardiography was performed on each patient to assess the effective orifice area (EOA) and transvalvular pressure gradient (TPG) for model validation purposes. Blood pressure, heart rate, and echocardiographic flow velocity were collected and utilized as input for the boundary conditions of FSI models. Specifically, blood pressure was measured using a cuff prior to the TAVI procedure, along with the patient's heart rate. Echocardiographic flow velocities were obtained upon in-hospital admission and utilized in the fluid-dynamic model due to the absence of direct catheterization measurements of TPG across the diseased valve.

The planned enrollment comprised 100 TAVI patients; however, the current in silico validation was limited to ten patients with the 23-mm S3 device and ten patients with the 26-mm S3 device due to considerable computing demand in both structural and hemodynamic analyses. Nevertheless, the 20 patients included for validation were derived by stratified sampling to provide a test sample that reflects the same variability as the broader cohort for age, blood pressure, and heart rate parameters. Table I outlines the range of clinical and demographic characteristics of the patient study group.

TABLE I.

Clinical demographic, device size, and post-procedural echocardiography data. M/F = male/female; Psys = systolic pressure; Pdias = diastolic pressure; HR = heart rate; TPG = transvalvular pressure gradient; EOA = effective orifice area; TAV = transcatheter aortic valve; SD = standard deviation.

	Age (years)	Sex	Psys (mm Hg)	Pdias (mm Hg)	HR (bpm)	Flow velocity (m/s)	EOA (cm2)	TPG (mm Hg)	S3 size (mm)
#1	79	F	126.0	60.0	61.0	2.40	1.62	23.0	26.00
#2	85	M	118.0	58.0	88.0	2.75	1.53	30.0	23.00
#3	81	F	119.0	85.0	67.0	2.42	1.70	23.0	26.00
#4	64	M	165.0	72.0	67.0	2.68	1.23	29.0	23.00
#5	82	F	105.0	55.0	65.0	2.10	1.60	18.0	26.00
#6	77	M	142.0	75.0	69.0	2.67	1.40	28.0	23.00
#7	83	M	147.0	37.0	70.0	2.66	1.26	28.0	23.00
#8	79	M	140.0	64.0	66.0	1.95	1.54	24.0	23.00
#9	82	M	135.0	75.0	78.0	2.24	1.75	20.0	23.00
#10	78	F	119.0	58.0	77.0	2.97	2.30	35.0	26.00
#11	80	M	100.0	50.0	65.0	2.74	0.62	30.0	23.00
#12	75	F	128.0	47.0	62.0	2.00	1.28	24.0	26.00
#13	84	F	123.0	78.0	68.0	1.85	1.62	14.0	26.00
#14	82	M	130.0	66.0	60.0	2.09	1.10	18.0	23.00
#15	78	F	120.0	83.0	69.0	2.80	0.80	32.0	26.00
#16	83	M	132.0	63.0	69.0	2.42	1.75	24.0	23.00
#17	69	F	95.0	52.0	74.0	2.70	1.00	29.0	26.00
#18	84	F	137.0	66.0	61.0	3.35	1.12	45.0	26.00
#19	78	F	111.0	44.0	72.0	3.14	1.50	39.0	26.00
#20	82	F	104.0	61.0	75.0	2.10	1.45	34.0	23.00
Mean	79.25		124.8	62.45	69.15	2.5	1.41	27.35
SD	5.16		17.22	12.97	6.82	0.41	0.38	7.51

B. Validation for ASME V&V 40

The credibility assessment of the computation model of TAVI was performed in accordance with the ASME V&V 40 standard, beginning with the definition of a scientific question of interest (QoI) and the context of use (CoU). The QoI establishes the specific prediction or simulation objective of the model, whereas the CoU states the circumstances for the model use and the related risks. The subsequent QoI and CoU have been formulated:

• QoI—Can a patient-specific TAVI model of S3 deployment at a given position replicate the stent diameters, EOA, and TPG measured after the intervention with <5% difference?

• CoU—Given the significant variability in calcification patterns, annulus sizes, and patient demographics, the utilization of patient-specific modeling for mimic device deployment and post-TAVI patient hemodynamics enables the prediction of the aortic valve's structural and fluid-dynamic behavior.

Following the delineation of the CoU and QoI, the ASME V&V 40 standard recommends establishing a model risk that considers the concepts of decision consequence and model influence. Decision consequence pertains to the severity of potential negative outcomes resulting from the incorrect decision based on the model, whilst model influence indicates the model contribution to the final decision. This implies that models with elevated risks have more rigorous credibility assessments. The standard also includes the specification of credibility goals, which define the acceptable levels of accuracy in the model predictions and are established on the intended application and the associated risk.

We assumed that the decision consequence was deemed significant since the in silico model output may yield erroneous predictions, potentially causing serious harm to patients when applied in clinical contexts. Likewise, the model influence was deemed significant, as the model is designed to assess the performance of TAVI devices of varying sizes and hemodynamic conditions on a certain subject to guide regulatory decisions. Thus, the model was designated a risk rating of 5 on a 1–5 scale, suggesting the necessity for high credibility goals. This indicates that the comparison of the quantity of interest between the comparator (i.e., the clinical data) and the computational model is predicated on differences of less than 5%. For validation, the structural and fluid-related quantities of interest associated with the QoI were (a) inflow, mid-flow, and outflow diameters of the implanted device; (b) the EOA at peak systole; and (c) the TPG across the S3 valve leaflets.

C. Patient-specific TAVI model

Based on the established QoI and CoU, a computational model was developed to serve as a cardiac tool for estimating S3-related structural and hemodynamic model performance. The patient-specific TAVI model enables the calculation of the device's deformed configuration using a previously developed structural finite-element simulation⁹ and the assessment of both EOA and TPG using FSI analysis.

1. Structural simulation

Early-diastolic CT scans were utilized to develop the patient-specific TAVI model for each patient. The geometries comprised the aorta wall, calcific plaques, and native aortic leaflets using a segmentation approach described in our previous studies.^9,18 The anatomies of the artery and calcifications were segmented using semi-automatic and fully automatic thresholding procedures, respectively. Stenotic aortic valve leaflets were generated using a parametric model based on anatomical landmarks as described by Catalano et al.⁹ A specific study was conducted to evaluate the accuracy of the semi-automatic segmentation process in comparison to the ISO:50 thresholding method, with the aim of enhancing the reliability of the current patient-specific approach. Results on the accuracy of patient-specific geometries are described by Scuoppo et al.¹⁸

Following segmentation, the patient-specific TAVI model for S3 deployment was developed in accordance with verification activities required by ASME V&V 40. The verification analysis sought to assess discretization error (i.e., element size and formulation), numerical solver error (e.g., penalty, scale factor, viscous pressure, etc.), and numerical code verification (i.e., benchmarking against established solutions) as previously described by our group for the present TAVI model.¹⁹ The main goal of verification activities is to maintain numerical errors at an acceptable level to ensure that the rigor of the patient-specific TAVI model does not undermine the quality of the validation process. A neo-Hookean material model was employed for each component of the aortic root. For the calcification, the material parameter was derived from existing research data from Bosi et al.²⁰ For the aortic wall and native valve leaflets, the material parameters were determined by an inverse approach that minimizes the difference in aortic wall strain and peak systolic valve area between the simulation and the pre-TAVI imaging data.⁹ Thus, material parameters of the aortic wall and native valve leaflets were tuned for each patient model.

In the S3 model, the stent frame was generated by integrating surface elements to describe the external device surface with beam elements to account for the device skeleton. This ensured precise modeling of the device's mechanical behavior during crimping and deployment as described previously.⁹ To account for hardening and rate-dependent material properties for the plasticity behavior, the S3 cobalt-chromium stent frame was modeled using a combination of isotropic elasticity and Johnson–Cook plasticity. The device valve leaflets were modeled using a second-order Ogden formulation, whilst the skirt exhibited neo-Hookean material behavior. Tie constraints were implemented to replicate the suturing between the stent framework and the fabric skirt. The balloon geometry was derived from literature data²¹ and was modeled with the neo-Hookean material model. After assembling the S3 and the delivery system, the implantation depth measured in the post-TAVI CT scan was adopted for determining the medical device position in each patient-specific TAVI model. Details on the development and validation of the S3 model are described elsewhere.²²

The TAVI procedure was modeled using the Abaqus/Explicit solver via three separate simulation steps using restart analysis to consider the final state after each simulation. In the S3 delivery systems configuration [Fig. 6(a)], the stent frame of the device was crimped to achieve a nominal diameter of 6.7 mm [Fig. 6(b)]. Following the elastic recoil achieved by detaching the stent frame from the crimper [Fig. 6(c)], the deployment is modeled by inflating the balloon with nominal fluid volume [Figs. 6(d)–6(f)]. To prevent numerical complications arising from significant crimping-induced distortion, the device leaflets and skirt were mapped on the deformed S3 stent frame at the end of the deployment simulation [Fig. 6(g)]. This was achieved by tying the device leaflets and skirt models onto the stent-frame surface and applying nodal displacements from the first simulation step through a shrink-fit method [Fig. 6(h)].⁹

FIG. 6.

Stages of TAVI simulation including (a) undeformed configuration, (b) crimping of the device and opening of stenotic leaflets, (c) S3 elastic recoil, (d)–(f) S3 implantation from balloon inflation to deflation, (g) device skirt and leaflets mapping on deformed stent frame, and (h) frontal and axial views of the final configuration.

2. FSI simulation

Following the deployment of the S3 device, post-TAVI hemodynamics were assessed using the SPH methodology implemented in the Abaqus solver. The fluid domain incorporates the post-TAVI structural model with a predefined field to account for the resultant stress field from the structural simulation, along with two fluid reservoirs attached to each end of the aorta to simulate left ventricular and arterial compliance [Figs. 7(a) and 7(b)]. A parametric approach was employed to design the reservoir geometry, with the particle volume being double that of the left ventricular stroke volume. The aortic wall boundary was populated with SPH particles, except for the area surrounding the S3 device, to prevent initial penetration of particles into adjacent anatomical structures. Thus, an initial infilling simulation step was conducted to populate the entire fluid domain prior to the cardiac simulation. The interaction between all anatomical structures and the components of the medical device was simulated using contact pairs with the general contact method in Abaqus/Explicit.

FIG. 7.

(a) Geometric domain of SPH model showing the left ventricular and aortic compliance chambers, (b) particle mesh of the fluid domain, (c) pressure waveforms, and (d) flow rate boundary conditions applied for the two compliance chambers.

The Hugoniot's equation of state was utilized to represent blood, which was considered incompressible and Newtonian, with a density of 1060 kg/m³, a wave speed of 7.5 × 10⁴ mm/s, a viscosity of 3.0 × 10⁻⁹ MPa s and a bulk viscosity damping coefficient of 0.006.²³ Boundary conditions were implemented differently for each patient based on the clinical data presented in Table I. Specifically, cuff blood pressure and heart rate were used as boundary conditions to calibrate the duration and pressure drop of a pressure waveform [Fig. 7(c)]. In a similar way, the heart rate and systolic flow jet of S3 from echocardiography were utilized to determine the duration and peak of a velocity boundary condition imposed on the left ventricular reservoir [Fig. 7(d)]. During cardiac flow simulation, forward flow is governed during systole by the velocity boundary condition generated by the left ventricular reservoir. In diastole, the velocity boundary condition is removed, and the flow is moved by the pressure gradient between the left ventricular and aortic pressure waveforms. The aorta reservoir functions as a fluid collector during systole and contracts during diastole or relaxes in systole to mimic in vivo vascular compliance.

Concerning the numerical verification, a discretization error analysis was assessed to examine the impact of varying particle sizes on the quantity of interest. Spatial discretization convergence was obtained for a particle size of 0.9 mm with errors on the three quantities of interest ≤5%. The impact of mass scaling and viscous pressure on the dynamic response of the SPH model was assessed in relation to numerical solver error. For numerical code verification, a representative benchmark problem was identified for the patient-specific FSI model. Specifically, the approach consisted of comparing the parabolic SPH-related velocity profile at the left ventricular outflow tract level with the parabolic velocity profile of a pipe under laminar flow. The appendix summarizes the results of verification activities.

3. Validation

The validation for the ASME V&V 40 standard seeks to assess the degree to which model outputs align with real-world evidence, including the evaluation of prediction error and related uncertainty in the patient-specific TAVI model. This study of the work outlines the validation process, which involves comparing model predictions with clinical post-TAVI imaging and echocardiographic data (i.e., the comparator) to assess the model's capability for realistic simulation of the physical phenomenon. The description of the uncertainty quantification is described in the second part of this investigation,¹⁰ where the model uncertainties due to material properties, boundary conditions, and TAVI-related procedural parameters were investigated in four patients by quasi-Monte Carlo analysis and Pareto plots.

For each quantity of interest, a one-to-one comparison was performed between model predictions and clinical measurements using empirical cumulative distribution functions (ECDFs). For the finite sample ( $n$) of data, the ECDF curves can be computed as

$$F_n\left(x\right)=\frac{1}{n}\sum _{i=1}^{n}1(X_i\le 1).$$ (1)

The ECDF shows the proportion of observed data points $\left(X_i\right)$ less than or equal to a given value. The ECDFs of the relative error of each validation metric were also computed to interpret the distribution of the difference between predictions and actual clinical data. For quantitative assessment, the area under the model and comparator ECDF curves was computed (referred to as the area metric), and the percentage difference was determined to evaluate the model accuracy.

APPENDIX: VERIFICATION ANALYSIS

A grid convergence analysis was carried out for particle sizes of 0.9, 1.15, 1.4, and 1.7 mm, using 0.9 mm as the reference size. The calculation of REs did not exhibit a monotonic increase with size, except for EOA, which had a positive correlation with values of 4.6%, 7.44%, and 13.79% for diameters of 1.15, 1.4, and 1.7 mm, respectively.

For numerical solver errors, the influence of solver configurations on the response of the S3 device in a representative FSI model was evaluated. The assessment included the analysis of mass scaling and viscous pressure on maximum principal stress of the aortic wall. Target time increments of 0.5 × 10⁻⁶, 1.0 × 10⁻⁶ (as reference), and 2.5 × 10⁻⁶ s, together with viscous pressures of 1.4 × 10⁻⁵, 1.0 × 10⁻⁶ (as reference), and 9.0 × 10⁻⁶ MPa were analyzed. Mass scaling exhibited minimal impact on EOA and TPG, with negligible variations across the configurations. Velocity and TPG exhibited considerable variation among analyses so that a target time increment value of 2.5 × 10⁻⁶ was adopted to achieve an error of 3.3%. The viscous pressure exhibited a reference value of 1.4 × 10⁻⁵ MPa, with negligible variances across configurations. For numerical code verification, the parabolic velocity profile at the left ventricular outflow tract level was compared to the parabolic velocity profile of a pipe under laminar flow, as delineated by Poiseuille's Law.¹³ Considering the elliptical configuration of the left ventricular outflow tract (L = 50 mm), a minor radius of 6 mm was employed for the theoretical computation of blood velocity. A pressure gradient of 4 mm Hg was assumed to compute the theoretical velocity profile. Solver setting and mesh size were defined according to results from discretization and numerical solver error analyses. After simulation a relative error of 9% between the numerical and analytical estimations was achieved.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Ethics Approval

This study was approved by the IRCCS ISMETT Ethics Committee (Approval No. IRRB04/04). All participants provided written informed consent prior to enrollment in the study.

Author Contributions

Chiara Catalano: Data curation (equal); Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal). Roberta Scuoppo: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal). Tahir Turgut: Conceptualization (equal); Formal analysis (equal). Vincent Bouwman: Conceptualization (equal); Software (equal). Nils Götzen: Conceptualization (equal); Funding acquisition (equal); Supervision (equal). Stefano Cannata: Formal analysis (equal); Supervision (equal). Giovanni Gentile: Formal analysis (equal); Supervision (equal). Caterina Gandolfo: Conceptualization (equal); Supervision (equal). Salvatore Pasta: Conceptualization (equal); Funding acquisition (equal); Resources (equal); Writing – original draft (equal); Writing – review & editing (equal).

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.