Impact of baseline coronary flow and its distribution on fractional flow reserve prediction

Model‐based prediction of fractional flow reserve (FFR) in the context of stable coronary artery disease (CAD) diagnosis requires a number of modelling assumptions. One of these assumptions is the definition of a baseline coronary flow, ie, total coronary flow at rest prior to the administration of drugs needed to perform invasive measurements. Here we explore the impact of several methods available in the literature to estimate and distribute baseline coronary flow on FFR predictions obtained with a reduced‐order model. We consider 63 patients with suspected stable CAD, for a total of 105 invasive FFR measurements. First, we improve a reduced‐order model with respect to previous results and validate its performance versus results obtained with a 3D model. Next, we assess the impact of a wide range of methods to impose and distribute baseline coronary flow on FFR prediction, which proved to have a significant impact on diagnostic performance. However, none of the proposed methods resulted in a significant improvement of prediction error standard deviation. Finally, we show that intrinsic uncertainties related to stenosis geometry and the effect of hyperemic inducing drugs have to be addressed in order to improve FFR prediction accuracy.


| INTRODUCTION
Fractional flow reserve (FFR) is an index to characterise the functional significance of coronary artery stenoses. 1,2 Although FFR is computed as the ratio between invasively measured post-stenotic and central aortic pressures, this index was originally derived to represent the ratio between the actual transtenotic flow over the hypothetical flow that would be observed in the absence of the stenosis under examination. 3 The theoretical derivation of FFR assumes constant peripheral resistance, which in turn is considered to be achieved under maximal vasodilation. 3 As a consequence, FFR is measured in hyperemic conditions, normally caused by the administration of a drug, such as adenosine, that selectively vasodilates the coronary peripheral vasculature. 4 A key advantage of FFR over more conventional methods, such as quantitative coronary angiography (QCA), is that, in addition to taking into account the geometry of a given lesion, it implies considering information about flow (under hyperemic conditions). In fact, after having been tested in large trials, [5][6][7] FFR is nowadays recommended to guide revascularization strategy in patients with stable coronary artery disease (CAD) without evidence of ischemia in non-invasive testing. 8 In practice, if a lesion is below a certain threshold, FFR ≤ FFR threshold , with FFR threshold =0.8, the recommendation is to intervene by performing percutaneous coronary intervention (PCI) or in some instances bypass surgery, while a negative outcome, ie, FFR>FFR threshold , results in treating the patient with optimal medical therapy. 8 Besides the proven validity of FFR as a a tool for functional assessment of stenosis severity, it remains an invasive procedure with associated risks. Moreover, in a study comprising almost 400 000 patients with suspected CAD from 663 US hospitals, almost two-thirds of the patients who underwent elective cardiac catheterisation proved to have non-obstructive CAD as determined by invasive angiography. 9 These considerations have motivated the search for non-invasive tests to reduce the number of invasive procedures. One of the most promising methods so far is coronary computed tomography angiography (CCTA). CCTA is a non-invasive anatomical imaging modality that allows to quantify the geometrical significance of a lesion and has a high diagnostic accuracy when compared with invasive coronary angiography-based diagnosis. A recent randomised trial suggested that CCTA improves patient outcome compared with standard care, 10 and guidelines currently suggest considering CCTA as a first-line test in all patients with suspected stable CAD. 11 CCTA has shown to be very selective in terms of correctly identifying CAD (FFR ≤ FFR threshold ), while its performance to exclude CAD (FFR>FFR threshold ) is not as satisfactory, resulting in many false positive recommendations, ie, many patients undergo an invasive procedure for FFR measurement that could have been avoided if a more selective non-invasive method would have been used. 12 In this scenario, CCTA-derived FFR has emerged as a possible response to the need for reducing false positive CCTA recommendations. Over the last decade, a significant number of methods for non-invasive computation of FFR based on CCTA have been proposed. 13 Such methods aim at predicting FFR for a given patient by using non-invasive information only and have already shown potential to be used as a screening tool on top of CCTA assessment. 14 These methods, based on reproducing the fluid mechanics in coronary vessels, share some common general steps that must be followed to deliver FFR predictions: (i) define the computational domain of coronary vessels; (ii) define a mathematical model for fluid mechanics valid in the domain defined in (i); (iii) define boundary conditions; (iv) solve the mathematical model; and (v) evaluate predicted FFR at desired locations. Although such steps can be found in any model-based FFR prediction method, the way in which each of these steps is performed varies greatly. In this paper, we address two aspects of this pipeline. First, we consider steps (ii) and (iv), working on the improvement and validation of a reduced-order model for the coronary circulation that allows for fast and accurate FFR prediction. Then, we focus on step (iii), investigating the impact of several methods for baseline coronary flow estimation and flow distribution proposed so far in the literature on FFR prediction.
Step (ii) of the general modelling strategy defined above requires making a choice on the mathematical model to be used for describing blood flow in coronary arteries. While in principle, many options are available, the most frequent choices found in the literature are 3D incompressible Navier-Stokes in rigid domains and 1D blood flow models in deformable vessels or fully lumped-parameter models. In general, 1D or lumped-parameter models are called reduced-order models. Several reduced-order models for FFR prediction have been proposed previously. [15][16][17][18][19] However, validation by comparison of predicted FFR with respect to results obtained by using more complex models was undertaken only in. [17][18][19] In Boileau et al, 18 a virtual population constructed from a single patient was used, while in Blanco et al, 17 a population of 20 patients was considered. Here, we modify the method proposed in Fossan et al 19 and validate it on a set of 63 patients (105 FFR measurements). For each patient, we perform simulations using a 3D model for the coronary vessels' domain and use those results as reference. To the best of our knowledge, such an extensive validation of the capacity of a reduced-order model to reproduce fluid mechanical aspects of model-based FFR prediction has not been performed so far.
The definition of boundary conditions for model-based FFR prediction is an unavoidable step. The sensitivity of predicted FFR to these boundary conditions was explored in previous studies [19][20][21] and was shown to be extremely relevant. In particular, Fossan et al 19 and Morris et al 21 showed that parameters that determine the hyperemic coronary flow have the highest influence on predicted FFR, for realistic ranges of other parameters. Motivated by this fact, we consider two aspects related to the definition of coronary flow, namely, baseline coronary flow and its distribution among the vessels of the network. In fact, virtually all methods for FFR prediction proposed so far require a baseline flow to be imposed and a criterion to distribute it among vessels in the network. Here, we have selected a set of representative methods for the definition of baseline coronary flow from published works, as well as three alternative methods to distribute such flow among coronary vessels, in order to assess the impact of these modelling choices on FFR prediction. Finally, we perform a sensitivity analysis where we compare the influence of baseline coronary flow to other parameters that are known to be important, namely, the stenosis geometry and the reduction in coronary peripheral resistance from baseline to hyperemic conditions. The rest of this paper is structured as follows. In Section 2, we describe the acquisition of patient-specific data (Section 2.1), a reduced-order model for FFR prediction (Section 2.2), the 3D modelling framework used for validation purposes (Section 2.3), the overall modelling strategy for FFR prediction (Section 2.4), several methods to determine baseline coronary flow (Section 2.5) and its distribution among coronary vessels (Section 2.6), and the method by which a sensitivity analysis of predicted FFR to relevant model parameters was performed (Section 2.7). Section 3 provides a summary of main characteristics of patients involved in the study (Section 3.1), results on the validation of the reducedorder model (Section 3.2), and results on the impact of explored modelling strategies on FFR prediction (Section 3.3). Finally, Section 4 includes a detailed analysis of reported results, as well as considerations on the study limitations and the steps to be taken to (a) improve the reduced-order model description to better capture the fluid mechanical aspects of the problem under consideration and (b) reduce uncertainty in FFR predictions. Patients were recruited as part of an ongoing clinical trial at St. Olavs hospital, Trondheim, Norway. 22 Patients included in this study had undergone CCTA because of chest pain and suspicion of stable CAD. Patients were enrolled with the findings of at least one coronary stenosis at CCTA examination and were further referred to invasive coronary angiography (ICA) with invasive FFR measurements. Exclusion criteria included nondiagnostic quality of the CCTA, previous percutaneous coronary intervention or bypass surgery, contraindications to adenosine, age (75 years or older), obesity (body mass index greater than 40), and hospitalisation due to unstable CAD after CCTA.

| Medical data acquisition
CCTA CCTA was performed using 2×128 detector row scanners (Siemens dual source Definition Flash) and 256 detector row CT scanners (Revolution CT, GE Healthcare, Waukesha, Wisconsin, US) with a standardised protocol. 23 Left ventricle mass (LVM) was quantified using a commercial software (Syngo.via, Siemens, Germany).

Ultrasound
Echocardiographic imaging was performed using a GE Vivid E95 scanner (GE Vingmed Ultrasound, Horten, Norway). Cardiac output (CO) was calculated on the basis of the cross-sectional area of the left ventricle outflow tract (measured immediately proximal to the points of insertion of the aortic leaflets) and velocity time integral derived from PW Doppler.
Fractional flow reserve FFR was measured using Verrata Plus (Philips Volcano, San Diego, USA) pressure wires according to standard practice. Intra-coronary nitroglycerine (0.2 mg) was given to all patients before advancing the pressure wire into the coronary arteries, and hyperemia was induced by continuous intravenous infusion of adenosine at a rate of at least 140μg/kg/ min. Pressure was measured over several cardiac cycles, and FFR measurements were taken during the nadir (lowest observed value). After measurement, the interventional cardiologist removed the pressure wire back to the equalisation point at the tip of the guiding catheter to ensure that there was no drift. Invasive pressure tracings were recorded and made available for further processing.

Clinical data
Standard non-invasive diastolic/systolic pressure measurements were performed on both arms as part of clinical routine before ICA using an automatic, digital blood pressure device, Welch Allyn ProBP 3400.

| Reduced-order model
Here, we briefly describe the methodology presented in Fossan et al 19 for the computation of FFR using a reduced-order model. FFR predictions obtained with the exact setting proposed in Fossan et al 19 will be denoted as FFR RO * , while predictions obtained using the improved version of the reduced-order model introduced in this section will be denoted as FFR RO * . This is valid throughout the rest of the paper unless otherwise specified.

| Vessels segmentation and computational domain meshing
Segmentation of coronary vessels is performed using the open-source software ITK-SNAP. 24 The output of ITK-SNAP is a labelled voxel volume identifying segmented vessels and a surface mesh of the segmented volume (in VTK format). Coronary arteries are segmented until their presence is not distinguishable from surrounding tissue. With this, the resulting average (±standard deviation) outlet radius of coronary arteries included in the computational domain is 0.9±0.23 mm. Surface mesh processing, addition of flow extensions, and 3D meshing are performed using the opensource library Vascular Modeling ToolKit (VMTK). 25, 26 The 3D volume meshes form the basis for both the reference 3D model and the reduced-order model. For the latter, centerlines are extracted from 3D domains using a centerline extraction algorithm available in VMTK.

| Domain definition
The resulting network of centerlines obtained by the processing steps briefly illustrated in Section 2.2.1 can be conceptually described as a directed graph G = (V, E), where • V are the vertices of the graph, which in this application can represent junctions/bifurcations, a root node, and terminal nodes of the network, hereafter called outlets. • E is a set of ordered pairs of vertices, in this case representing vessels.
Graph G will have M edges (or vessels) and N vertices. v root is the vertex at the root of the network, while v j out ; j ¼ 1; …; N out , are outlets and v j b ; j ¼ 0; …; N b , are vertices representing coupling points among vessels, see Figure 1. Vessel e j is described by a set of K j nodes produced by the centerline extraction algorithm cited in Section 2.2.1. Each node k j l ; l ¼ 1; …; jK j j, is marked as belonging to a bifurcation region (K j b ∈ K j ), belonging to a stenosis (K j s ∈ K j ), or belonging to a 1D domain (K j 1D ∈ K j ). The masking of such regions is explained in detail in Fossan et al. 19 Here, we modify the domain definition reported in Fossan et al 19 as follows: • the spacing between nodes k is reduced from 0.5mm to 0.125mm; • an additional criteria for masking stenotic regions on the basis of the gradient of the radius in the longitudinal direction is added. In Fossan et al, 19 a detected stenosis was marked until the estimated stenosis degree (SD) was below 12%. Here, we require that SD > 12% ∨ j dr dx j < 0:05. In practice, even though the estimated stenosis degree is below 12%, we continue to mark the region as a stenosis if the location represents a compression dr dx < 0:05 or an expansion dr dx > 0:05.

| Mathematical models
Let us consider a single vessel e j . In regions of e j labelled as 1D domain, ie, K j 1D ∈ K j , blood flow is modelled according to a 1D steady state blood flow model, ie, where A is the cross-sectional area of the vessel, Q is the blood flow rate, and P is the blood pressure. Moreover, ζ=4.31, 19 ρ=1.05g/cm 3 is the blood density, and μ=0.035 dyne/cm 2 s is the blood viscosity. The cross-sectional area A is assumed constant in time and equal to the area obtained from the segmentation of medical images. Then, the problem unknowns are pressure P and flow rate Q.
Regions masked as bifurcations and stenoses are not modelled using a continuous model. Nodes belonging to such regions are collapsed into a single point, and coupling conditions apply. See Figure 2 for a graphical illustration of this aspect. Coupling for both connection types (bifurcations and stenoses) is performed by enforcing the following relations at the coupling point ∑ where T is the number of vessels sharing a vertex with the bifurcation/stenosis. For stenoses, we have that T=2, while in the case of bifurcations, we have T ≥ 2. ΔP is an additional pressure loss, and λ is a coefficient that can assume values between zero and one. At bifurcations, ΔP is set to zero and λ is set to one, so that Equation (2a) describes continuity of total pressure. At coupling points representing stenoses, we set λ=0, and ΔP is computed as proposed in Seeley and Young, 27 namely where A 0 and A s refer to cross-sectional areas of the normal and stenotic segments, respectively. Similarly, D 0 and D s represent the normal and stenotic diameters. Furthermore, K v and K t are empirical coefficients, with 27 whereas L s is the length of the stenosis.

| Boundary conditions
In this work, we consider two alternative sets of outlet boundary conditions: prescribed flow rate at outlets or resistive elements coupled to outlets. See Section 2.4 for motivation on the two different setups introduced here.

Prescribed flow rates at outlets
Pressure is prescribed at v root , namely P root ¼P root , whereP root is the prescribed pressure. Then, we set Q j out ¼Q j ; j ¼ 1; …; N out , whereQ j ; j ¼ 1; …; N out , are the flows to be prescribed deriving from methods described in Sections 2.5 and 2.6. Defining flow rate at outlets implies that flow rate over the entire network is fixed. Then, the only remaining unknown is pressure along 1D domains, and pressure drops over stenotic regions. Such pressure is obtained integrating (1) for Q given along the 1D domains and evaluating coupling relations (2b) and (3) where appropriate. A reasonable strategy is to start at v root and traverse the entire tree, but other choices are possible.

Resistive elements coupled to outlets
As in the previous case, pressure is prescribed at v root , namely P root ¼P root . Then, we consider resistive elements coupled to outlets with resistances R j out ; j ¼ 1; …; N out . In this case, flow rate is unknown over the entire network, and a nonlinear algebraic system has to be solved to find the flow rates at outlets Q j out ; j ¼ 1; …; N out , which solve (1), (2b), (3), and P root −P root ¼ 0. More details on the numerical treatment of the modelling setup strategies presented here are given in Fossan et al. 19 2.3 | 3D modelling 3D simulations are used to validate the reduced-order model proposed in Fossan et al 19 and improved in this work. These simulations are performed considering segmented coronary trees as rigid domains with a prescribed pressure as inlet boundary condition and either prescribed flows (via prescribed parabolic velocity profile) or lumped-parameter models attached to each network outlet, according to the modelling pipeline described in Section 2.4. Furthermore, the flow is assumed laminar, and blood is modelled as an incompressible Newtonian fluid. The open-source library CBCFLOW, 28 based on FEniCS 29 is used to solve the resulting mathematical model. The incompressible Navier-Stokes equations are solved using the incremental pressure correction scheme, described in Simo and Armero. 30 Tetrahedral elements compose the computational mesh where the velocity field is approximated using piecewise-quadratic polynomials, while linear polynomials are used for pressure. The solver implementation follows very closely the one reported in Mortensen and Valen-Sendstad. 31 3D meshes were constructed using the open-source library VMTK. 25, 26 The meshing refinement level was determined by a meshing algorithm parameter called edge-length factor l f , which was set to l f =0.21 for all simulations. A mesh independence study showed that such discretisation provides meshindependent FFR predictions for a set of four patient-specific geometries. A full description of the underlying mathematical models and their numerical treatment is provided in Fossan et al 19 and references cited therein.

| Overall modelling strategy
FFR is normally measured at one or more points in the coronary tree. In the current computational context, we simulate a hyperemic state for the entire coronary tree. Therefore, we refer to a hyperemic state as the condition under which virtual FFR measurements are performed. Assuming that a network of coronary branches for a given patient with related parameters is available (see Section 2.2), we proceed as follows in order to obtain a hyperemic state: 1. define total baseline coronary flow by one of the methods described in Section 2.5: q; 2. distribute flows among the network's N out outlets according to one of the methods described in Section 2.6: Q out,l , with l=1,…,N out ; 3. perform a simulation with prescribed inlet pressure P root =P proximal and prescribed outlet flows defined in previous step to find P bln out;l ; with l ¼ 1; …; N out ; 4. compute resistances at outlets as R bln out;l ¼ where P bln out;l is the pressure at the l-th outlet resulting from the simulation performed in previous step and P v is a reference venous pressure, which is set to P v =5 mmHg throughout this work; 5. perform a simulation with prescribed inlet pressure P root =P proximal and prescribed resistances at outlets. Such resistances are computed as where TCRI is the so-called total coronary resistance index, ie, the factor by which peripheral coronary resistance drops from its value at baseline conditions to its value in hyperemia. In this work we use TCRI=4, unless otherwise stated. 4  Table 3 Branch split - Simulation results from point 5 of the previous list represent the hyperemic state from which FFR can be evaluated at any point of the network as where P j is the predicted pressure at the jth node of the network. We remind the reader about the fact that if predictions are obtained with the setting proposed in Fossan et al, 19 ie, without the modifications introduced in Section 2.2.2, then those predictions will be denoted as FFR RO * . Table 1 summarises necessary data and parameters for prediction of FFR with the reduced order model.

| Methods for baseline coronary flow estimation
In this work, we explore a number of methods to estimate baseline coronary flow, q. In selecting the methods to be used, we sought to include methods that use different types of data. In particular, we included methods that rely on CO estimation, as well as methods that use LVM and a method based on population studies. A description of such methods follows.

| CO-based methods for baseline coronary flow estimation
We consider two alternative methods to determine CO. The first method consists of simply using the CO derived from patient-specific ultrasound (US) measurements acquired as explained in Section 2.1.2. CO derived from US measurements will be called CO US hereafter. The second method uses a formula to estimate stroke volume from patient-specific parameters. 32 Such formula, originally proposed in de Simone et al, 33 is given as where W is weight in kilograms, Y is age in years, and HR is heart rate in beats per minute. Moreover, PP * is where PP is pulse pressure, PP=SBP−DBP, with SBP and DBP being systolic and diastolic blood pressure (in mmHg), respectively. Then, CO is computed as Once CO is estimated by either of the two methods, we compute baseline coronary flow as where γ=0.045 is the fraction of CO that flows into the coronary branches, 34 and X={US,DeSimone} are the two considered CO estimation methods.

| LVM-based methods for baseline coronary flow estimation
Alternatively to the use of CO to determine total baseline coronary flow q, several investigators have used LVM to perform such estimate. Here, we select two methods and propose modifications to them. In Kishi et al, 35 q is computed as where β=0.8 mL/min/g is a reference value for myocardial tissue perfusion, 36 and TMM stands for total myocardial mass, computed as It is worth noting that no other patient-specific information than LVM is used. A method that accounts for additional patient-specific information is the one used for FFR prediction in Sharma et al. 37 In this case, the observed association between the product SBP×HR is considered in the computation of baseline flow, which reads where TMM is computed with (12). Since, as it will be shown later in Section 3.1, Table 2, baseline coronary flows computed using (11) and (13) are very low compared with reference values, we propose to modify both methods. The reason for the underestimation of baseline coronary flow in these cases is related to the rather low TMM derived from the hypothesis stated in (12). In both modified methods, we use The constant value 2.4 was determined by considering that the ratio of reference TMM reported in Molina and DiMaio 38,39 over LVM for patients included in this study (2.39 (0.34) and 2.34 (0.42) for men and women, respectively). Moreover, in the case of q Sharma , the constant 0.08 was replaced by 0.14 in order to force average flow over the population to be in accordance with reference values. 40 See Table 2 for details. Modified baseline flow estimates are denoted as b q Kishi and b q Sharma for q Kishi and q Sharma , respectively.

| Population-based methods for baseline flow estimation
Here, we simply specify flows to left and right coronary branches as specified in Sakamoto et al. 40 Average flows are differentiated according to the dominance of the coronary vasculature. Table 2 summarises the different values for total coronary flow, as well as for left and right branches. Table 3 summarises the different methods used to determine coronary flow and how they are computed.

| Flow distribution
There is a variety of methods used in the literature to distribute baseline coronary flow among the coronary vessels. A major difference is given by the fact that some practitioners model a single branch (left or right), while others consider both branches simultaneously. In order to account for this aspect, we have applied the methods described below in two different settings, namely, applying the methods on single branches or on both coupled branches.

| Distal Murray
Murray's law 41 is the most commonly used method to distribute coronary flow; see previous studies 37, 42 for example. It expresses a proportionality between flow and vessel diameter with ξ=3 from theoretical considerations on minimum work. Distal Murray (DM) flow distribution implies that for a given network with N out outlets, flow is distributed among all outlets using (15) and outlets' diameters.

| Proximal Murray
In this case, one traverses a given network from its root, and flow is split at each bifurcation applying (15) among daughter vessels. The diameter used here was taken as the average vessel diameter, computed over vessel nodes that were not marked as belonging to stenoses or bifurcation areas. See Section 2.2 for an explanation and further references on how these nodes are marked. This method was used in previous studies, 35,43 in the context of FFR prediction.

| Transluminal Attenuation Gradient
Here, we implement the method proposed in Kishi et al, 35 where authors hypothesise an inverse relation between flow and transluminal attenuation gradient (TAG), ie, Q∝−1/TAG. The interesting feature of this approach is that it uses information on flow distribution properties of a given patient by directly looking into how contrast is advected in the coronary tree during CCTA acquisition. TAG is essentially the gradient of voxel intensities, quantified as Hounsfield Units (HU) in the CCTA image averaged along a given vessel. TAG is computed for each terminal point of the vessel network by computing HU along centerline nodes and fitting a first-order polynomial to the resulting data points (arc-length versus HU: HU(x)=HU 0 +TAG·x, with x the centerline arc-length). Intensities along the path are computed by averaging CCTA voxel intensities around each centerline node. Candidate voxels are those in a volume of 5×5×5 voxels centred at a given centerline node. In order to be included, a candidate voxel has to fall within the segmented vessel lumen and has to have a HU value that is below a threshold in order to exclude calcifications and artefacts. Such threshold is set equal to the average plus two standard deviations of HU values for all voxels contained in the segmented lumen volume. For a detailed definition of TAG and its computation in coronary trees; see Kishi et al 35 and references cited therein. TAG is computed for each patient considered in this study using CCTA images, segmented vessels lumen volume, and extracted centerlines.

| Branch flow split for single branch distribution
In the case in which a single coronary branch is considered for simulation, a further assumption is necessary since we depart from total baseline coronary flow, and a branch-specific flow is needed. Two strategies are followed. One strategy consists of splitting flow between left and right branches as observed in Sakamoto et al 40 and reported in Table 4. The second one regards simply imposing average coronary flows to left and right branches, specified in Table 2. Having defined the three flow distribution strategies that will be explored in this work, we can define a set of simulation settings that will be used for each of the baseline flow estimation methods described in Section 2.5 and summarised in Table 3. In fact, for each one of those baseline flows, we have Y Z flow splits, with Y={PM,DM,TAG} and Z={S,C}. S and C correspond to cases where flow distribution is applied to single and coupled branches, respectively. Table 5, together with Table 3, provides the reader with a complete overview of all modelling assumption combinations used in this study, so that the equations and parameters used for a specific simulation setup can be easily retrieved.

| Sensitivity analysis
We conduct uncertainty quantification and sensitivity analysis (UQ&SA) to rank the influence of uncertain input parameters on FFR prediction. In Fossan et al, 19 a wide range of parameters were analysed in terms of UQ&SA; however, the most influential parameters were those related to coronary flow, TCRI, and minimum stenotic radius. Other input parameters such as P proximal , Murray coefficient, and rheological parameters (ρ, μ) were less influential. On the basis of these considerations, we perform a sensitivity analysis on the uncertain input parameters q, TCRI, and SD, with the uncertainty in SD affecting the minimum radius. q was modelled as a (truncated at ±3 std) normal distribution with a mean and standard deviation deriving from the per patient mean and standard deviation obtained with four of the patient-specific baseline coronary flow estimates studied in this work, q US Guyton , q DeSimone Guyton ,q Kishi , andq Sharma . We model the hyperemic factor TCRI as a gamma distribution with shape parameter 3, scale factor 0.75, and shifted to 1. 19 For our dataset of 105 FFR measurements, clinically relevant stenoses were quantified in terms of SD using QCA and QCA CT , where QCA CT denotes QCA evaluated on the basis of segmented geometries. The standard deviation of QCA−QCA CT was 20% SD. To account for the uncertainty in stenosis geometry, we introduce a global parameter ΔSD, to be applied to all stenotic regions of a network such that the stenosis degree is given by SD=SD segmented +ΔSD, where SD segmented is the stenosis degree obtained from the original segmentation. Since we adopt a conservative approach where all detected stenoses are included in a global parameter, we model ΔSD as a normal distribution with a mean of 0 and with a standard deviation of 15%.    Sobol sensitivity indices, first-order (S i ) and total (S T,i ), are widely employed 44 and defined as Here, E Y jZ ¬i ½ represents the expected value of the output Y for a fixed value of the uncertain input, Z i , and V X ½ is the variance of some variable, X. Z is a vector that represents the uncertain input variables, Z ¼ q; TCRI; SD ½ ð Þ , and Z ¬i contains all elements of Z except Z i . SA results (which are obtained on a per patient basis) are summarised by averaging the first-order and total Sobol sensitivity indices over the population. Moreover, we compute weighted first-order sensitivity indices and total sensitivity indices where Si k and S k T;i are the sensitivities due to input i for FFR-prediction k, and V Y k ½ is the variance of FFR-prediction k. 19,45 Measures of uncertainty and sensitivity are estimated by the Monte Carlo method as described by Saltelli,46 and the accuracy of UQ&SA results are assessed by evaluating the standard deviation of the estimates from 10 bootstrapped samples 47 until the standard deviation is below 0.033 (ie, 99% confident that obtained value is within ±0.1.).

| Patient population and invasive measurements characteristics
A total of 63 patients that underwent invasive angiography and FFR measurements after clinical and CCTA examinations indicated stable CAD were recruited. From these patients, a total of 105 FFR measurements were collected. FFR measurements had a mean of 0.81 and a standard deviation of 0.14, with a per vessel positive FFR prevalence of 31.43% and a per patient positive FFR prevalence of 39.68% for a cut-off value of FFR ≤ 0.8. See Table 6 for a summary of population characteristics. Table 7 shows a comparison of LVM for the study population versus reference values taken from a large CT study for healthy young subjects. 48   Figure 3 shows scatter plots and Bland-Altman plots for a comparison of FFR predicted by 3D and reduced-order models, while a numerical characterisation of the comparison is given in Table 8. The modelling setup used to obtain such results was described in Section 2.4, while the particular model parameter definition used included single branch treatment, q US Guyton baseline coronary flow, and DM flow distribution. Results are shown for the domain definition setup described in Fossan et al 19 and the modified version proposed here. Figure 4 and Table 8 show results for FFR predictions obtained with current method and the one originally proposed in Fossan et al 19 for the case in which the hyperemic simulation is performed replacing resistive elements at outlets with the flows extracted from 3D simulations. Table 9 reports indexes on the agreement of predicted FFR by the reduced-order model with respect to results obtained using 3D simulations, as well as comparisons of reduced-order model predicted FFR and 3D model predicted FFR versus invasive measurements, for the three flow distribution strategies considered in this work. Table 10 shows resulting baseline coronary flows and perfusion indexes for strategies proposed in Section 2.5. Table 11 shows accuracy indexes for predicted FFR with respect to invasive FFR for all possible combinations of baseline flow estimation and flow distribution strategies. Only results using q Kishi and q Sharma are not reported, since baseline flows obtained with these two methods resulted in values well below physiological ranges with 151±32mL/ min and 93±27mL/min for q Kishi and q Sharma , respectively (see Table 10). Moreover, Table 12 shows results for coupled branches treatment and q US Guyton baseline flow on a per vessel basis, differentiating results among (major branches departing from) left anterior descending artery (LAD), left circumflex artery (LCX), and right coronary artery (RCA). Figure 5 shows results from the sensitivity analysis described in Section 2.7 in terms of the averaged and uncertainty weighted (see (17) and (18)

| Patient population and derived measurements
The study population is considered representative for patients with suspected CAD being referred for CCTA.
Since two of the methods to determine baseline coronary flow introduced in Section 2.5 depend on TMM, it is important to discuss results presented in Table 7. We observe that average LVM values for males and females are normally larger than the ones reported as reference values for healthy subjects in Fuchs et al. 48 The higher-than-reference LVM values observed for the population under study are supported by reported left ventricle hypertrophy in patients with stable treated chest angina 49 and by the fact that elevated LVM is a recognised marker for cardiovascular risk. 50 Moreover, TMM values obtained using relation (12) are well below reference values, while usage of relation (14) produces TMM estimates slightly above TMM reference values, in line with what is observed for LVM values. This apparent underestimation of TMM by relation (12) will have an impact in flow computations, as we will discuss later in Section 4.3.

| Validation of reduced-order model
Validation of the reduced-order model using a 3D model as reference is significantly more extensive than previously published results. In Fossan et al, 19 an earlier version of the reduced-order model described here was validated on a population of 13 patients and 23 FFR measurements for a single flow configuration. Moreover, in Blanco et al, 17 a population of 20 patients and 32 FFR measurements for a single flow configuration was considered. Here, we consider 63 patients, 105 FFR measurements, and three different flow configurations. Results depicted in Figure 3 show that the agreement between predicted FFR values by both models is satisfactory, as confirmed by results reported in Table 8. It is worth noting that the presented results allow to make a two-fold assessment of the reduced-order model results. On one hand, while using resistive elements at terminals, we can see how the entire modelling pipeline described in Section 2.4 is affected by using a simpler and computationally cheaper model with respect to using the 3D model. On the other hand, by imposing terminal flows from the results obtained using  the 3D model, we can focus on how well the reduced-order model reproduces pressure drops for fixed flows. As it can be clearly seen, the mismatch between results obtained using both models are larger in the case of prescribed flows at outlets of the reduced-order model. Consequently, this is the case where the benefits obtained by modifications to the original model proposed in Fossan et al 19 can be better observed. Furthermore, results reported in Table 9 show that predicted FFR deviations of the reduced-order model with respect to FFR predicted with the 3D model are small compared with errors in FFR prediction versus invasive measurements with bias (standard deviation) of −0.01 (0.03) and −0.05 (0.13) for FFR 3D versus FFR RO and FFR m versus FFR RO , respectively. Importantly, when comparing diagnostic accuracy indexes for reduced-order and 3D model predicted FFR versus invasive measurements, one can see that these are only marginally affected by the used model. These considerations allow us to conclude that results discussed in the following sections are mostly affected by the different modelling hypotheses used (boundary conditions) and not by errors in the description of the fluid mechanics introduced by the reduced-order model. Nevertheless, we note that the disagreement between reduced-order model results and 3D model results leaves room for further reduced-order model improvement, as discussed in Section 4.5.

| Predicted baseline coronary flow
When comparing FFR predictions by different model setups made in Section 3.3, we have excluded results obtained using baseline coronary flows q Sharma and q Kishi . The reason for this choice can be inferred from average baseline coronary flows reported in Table 10. In fact, predicted average flows for q Sharma and q Kishi , as well as average myocardium perfusion rate for q Sharma , are well below physiological values of 250 mL/min and 0.8 mL/g/min. 34 Moreover, it is remarkable how these quantities are in line with reference values for methods using relation (14) to compute TMM and where flow is not derived from perfusion rates, ie, q US Guyton , q DeSimone Guyton , q Sakamoto , andq Sharma .

| Effect of baseline coronary flow and its distribution on FFR prediction
Results presented in Table 11 clearly show that the flow distribution strategy for which our reduced-order model delivers better agreement with invasive measurements is the one based on TAG. In fact, linear fit coefficients and Pearson correlation coefficient are always better for TAG for fixed branch treatment and baseline flow estimation strategy. A closer look into how flow distribution methods effect FFR prediction on a vessel basis can be done from data shown in Table 12. Here, it becomes evident that TAG outperforms other flow distribution strategies for LAD and LCX but not for RCA. The overall better performance is due to the fact that lesions located in LAD and LCX (and their branches) account for almost 83 % of total lesion number, in combination with the fact that TAG systematically results in more blood flowing into the LAD with respect to flows obtained with DM and Proximal Murray (PM). For example, for coupled branches treatment and q US Guyton , average (standard deviation) flow in the LAD are 74 (26), 71 (26), and 89 (32)mL/min for DM, PM, and TAG flow distribution strategies, respectively. Such a pattern can be observed for all simulations reported in Table 11. The larger flow can compensate for underestimated stenosis severity or it can simply mean that TAG is better reflecting the flow distribution among vessels. This last observation is supported by the fact that TAG incorporates patient-specific information about flow distribution (via intensity gradients along vessels), while DM and PM are based on geometrical information alone.
The best performing method in terms of correlation to invasive measurements has the following setting: coupled branches treatment, q Sakamoto , and TAG flow distribution. On the other hand, the best method in terms of diagnostic accuracy has the following setting: coupled branches treatment, q US Guyton , and TAG flow distribution. Since this last method ranks second in terms of linear fitting coefficients, we will denote it as FFR OPT RO throughout the rest of this work. Diagnostic performance for different modelling choices can vary significantly. In particular, sensitivity among explored model setups varies between 48.48% and 78.79%. It is interesting to observe that maximum sensitivity (78.79%) is achieved for coupled branches treatment, TAG flow distribution, and all baseline coronary flow estimation methods, pointing out to the fact that flow distribution is more determinant than baseline coronary flow estimation.
Although errors (FFR RO −FFR m ) standard deviation varies for different model setups, such fluctuations are very small with respect to standard deviation absolute values, with larger variations among results obtained using different baseline flow definition methods rather than within single baseline flow definition methods. In other words, standard deviation changes seem to be mainly due to the use of different baseline flow estimation methods, rather than by flow distribution strategies. Another conclusion that can be made from standard deviation variation across used modelling setups is that none of the investigated methods seems to reduce error standard deviation significantly, although several of them imply using patient-specific information. This result is not surprising if one considers the origin of patientspecific information used by the different methods. In q US Guyton , we use ultrasound-derived CO, for which measurement errors of 690mL/min have been reported. Such errors propagate to baseline coronary flow, which, in this case, is computed as 4.5 % of CO and imply an error in flow of approximately 31 mL/min or 14 % of baseline coronary flow. The standard deviation of q US Guyton in our population is 42 mL/min and is thus comparable with the measurement error. Another method using CO estimates is q DeSimone Guyton , where the formula for determining stroke volume (7) was obtained by fitting experimental data, with a resulting Pearson correlation coefficient r=0.45, 33 denoting large uncertainty in predicted values. Moving forward among methods, we consider nowq Sharma . In this case, (13) is used. This relation involves a fitting of experimental data (on dogs) relating HR and SBP to oxygen consumption, with r=0.87, 51 combined then to another fitting of experimental data on the relation between coronary flow and oxygen consumption. 52 There are additional sources of errors for these patient-specific baseline flow estimation methods. The fraction of CO that flows into the coronaries, γ in (10), is assumed constant. Moreover, as already discussed, some methods make use of SBP and HR. These quantities are known to vary for the same patient depending on many factors. For our patient population, we observe that Pearson correlation coefficients for SBP and HR when comparing quantities acquired non-invasively and quantities measured during invasive FFR procedures are 0.09 and 0.29, for SBP and HR, respectively. All these considerations support the results shown here that evidence no benefit in terms of error standard deviation reduction by using patient-specific baseline flow estimation methods. However, this conclusion does not preclude that a better, currently nonreported baseline flow estimation method would result in improved FFR predictions.
Previously reported relative low impact of prescribed inlet pressure in FFR prediction frameworks 19,21 was confirmed in this study. Table 13 shows the completely neglectable impact of using different values for the prescribed inlet pressure, including pressure measured non-invasively, pressure acquired during invasive FFR measurement (the actual value to compute invasive FFR), and a fixed value of 100 mmHg, demonstrating that this parameter is of little importance when predicting FFR. Figure 6, left panel, shows a scatter plot of invasive FFR versus average predicted FFR (for all methods shown in Table 11), including maximum variation ranges and standard deviation. Average standard deviation is 0.048, with larger values for lower FFR values, as it can be seen in Figure 6, right panel. These results reinforce observations made while analysing data reported in Table 11 on the fact that baseline coronary flow estimation and flow distribution strategies    TCRI in this case is calculated on the basis of the baseline resistance of simulations corresponding to the best performing setting (coupled branches treatment, q US Guyton baseline flow, and TAG flow distribution) and hyperemic resistance corresponding to FFR best . Measurement locations which had errors (FFR RO,best −FFR m ) larger than two standard deviations where excluded and cases which resulted in TCRI values larger than 15 were excluded from this analyses. FFR, fractional flow reserve; TCRI, total coronary resistance index that the uncertainty in the stenosis degree has the largest influence on FFR-prediction uncertainty, with average firstorder and total sensitivity indices of 0.71 and 0.74, and uncertainty weighted sensitivity indices of 0.87 and 0.89, respectively. TCRI has a significant but much lower influence with average first-order and total sensitivity indices of 0.22 and 0.27 and uncertainty weighted sensitivity indices of 0.08 and 0.11, respectively. Baseline coronary flow has a minor influence in comparison, with average first order and total sensitivity indices of 0.04 and 0.07 and uncertainty weighted sensitivity indices of 0.03 and 0.04, respectively. The influence of TCRI and q is slightly increased when only cases in the critical region 0.7<FFR RO <0.9 are considered. The average standard deviation in predicted FFR is 0.12. However, we note that the uncertainty in how the flow is distributed is not considered and may be seen in relation to the fact that the average standard deviation resulting from the UQ&SA (0.12) is slightly lower than average predicted FFR measurement error standard deviation (0.15).
The above made observations allow us to conclude that other parameters than the ones studied here have to be addressed in order to improve FFR prediction accuracy. In particular, we consider now TCRI and make a postprocessing experiment by: • extracting best matching FFR predictions of all simulations shown in Table 11; • considering FFR OPT RO as the setting for reference baseline conditions; and • compute TCRI as the ratio of peripheral resistance (with respect to FFR measurement location) for best matching simulation versus peripheral resistance for reference baseline setting.
Selected best matching predictions versus measurements are shown in Figure 7, while Figure 8 shows a histogram of obtained TCRIs. Such results show that, for the given modelling framework, available segmentations and flows in physiological ranges, rather accurate FFR predictions can be obtained, if a highly variable TCRI is considered. Corresponding accuracy, sensitivity, and specificity are 94%, 85%, and 99% when all measurements are included and 97%, 90%, and 100% when outliers are excluded. Interestingly, TCRI distribution emerging from this analysis reflects the high variability observed in coronary flow reserve, which has shown to display a gamma distribution shape when directly measured on patients. 53

| Use of reduced-order models for the development of clinical decision support tools
The capacity of exploring a large number of modelling hypotheses, as well as parameter space via sensitivity analysis, is facilitated by the fact that we use an accurate reduced-order model of the problem under investigation. This fact points to one of the strengths of reduced-order models, which is that of being computationally cheaper and generally more robust than more complex models in terms of numerical issues such as convergence and stability of the overall algorithm. Advantages in terms of problem setup, preprocessing, and postprocessing over more complex models are also relevant. Moreover, errors introduced by reduced-order models can be compensated by the fact that one can explore a larger portion of possible solutions. However, such errors must be acceptable, which in the context of this work means that clinical decisions should not be affected by them.

| Limitations and future work
The present work has several limitations. The most relevant one is probably the fact that, although we explore certain modelling hypotheses, other are kept fixed and are inherent to our FFR prediction framework (segmentation method, modelling pipeline, etc). This fact is kept in mind throughout the text and only claims that we think are valid for any model-based FFR prediction framework are mentioned as such. The impact of patient-specific TCRI on FFR prediction diagnostic accuracy was only partially addressed. In fact, in Section 4.3.2 we show that a highly variable TCRI would have a significant impact in the diagnostic accuracy of our FFR prediction method. Ideally, we should assess how a patient-specific TCRI computation would perform in comparison to other methods, but to the best of our knowledge, no methodology to estimate TCRI non-invasively and without the use of drugs or additional imaging is currently available. Another component that can vary greatly among patients is total baseline coronary flow. In this case, we have considered quantification methods that depend on patient-specific information such as CO or LVM. However, one should ideally use a direct total baseline coronary flow measurement method. Such methods are being developed in the context of ultrasound imaging and magnetic resonance imaging and, once available, have the potential to provide highly valuable information for FFR prediction tools. See Fiorentini et al 54 and Schwitter et al 55 for recent developments in ultrasound and magnetic-resonance coronary flow estimation methods.
Another aspect to be considered is the fact that the reduced-order model is not perfectly reproducing 3D model results. Therefore, results are influenced not only by boundary conditions but also by errors in solving fluid mechanics within the computational domain representing epicardial vessels. This is common to any FFR prediction strategy based on reducedorder models. Here, we prefer to be aware of such errors and try to minimise them, rather than to embed them in the modelling framework and lose control on error causes when comparing FFR prediction and invasive measurements.
The prevalence of positive FFR values is not optimal, and one would desire a more homogeneous distribution of invasive FFR values to work with. However, we observe that this is an issue faced by virtually all published works on FFR prediction methods and is probably related to the intrinsic dynamics of how FFR is used in the clinic, with many measurements performed on lesions that reveal to be functionally nonsignificant.
Finally, the results from the UQ & SA depend on the assumed input uncertainties, and here we did not include uncertainties/variability due to differences in how baseline coronary flow is distributed. More importantly, the way in which the uncertainty of the stenosis geometry was modelled, via a global parameter effecting all stenotic regions, may have synthetically increased the influence of this parameter and is not necessarily entirely realistic. Nevertheless, results shown here should still be considered as valid for giving a clear indication to where attention should be focused in order to reduce uncertainty in FFR-prediction.

| Conclusions and future work
In this work, we have first improved and validated a reduced-order model for FFR prediction and then tested the impact of different methods proposed in the literature to estimate and distribute baseline coronary flow on FFR predictions. Both tasks were performed on a dataset regarding 63 patients with stable CAD and 105 invasive FFR measurements.
The proposed reduced-order model introduces errors with respect to the 3D model, which are significantly smaller than errors observed for predicted FFR versus invasive measurements. Moreover, diagnostic performance was only marginally affected by the use of the reduced-order model for cases in which 3D simulation results were available (63 patients, 105 invasive FFR measurements, 3 different setups for 3D simulations).
We have shown that methods for baseline coronary flow estimation and its distribution can affect the diagnostic performance of an FFR-prediction framework significantly. Moreover, the influence of methods for flow distribution is greater than that of methods for baseline flow estimation, and we find that TAG flow distribution outperforms flow distribution methods based on Murray's law for all tested setups. However, we observe that none of the investigated methods for baseline coronary flow estimation and distribution results in a significant reduction in error standard deviation for predicted FFR versus invasive FFR measurements. Finally, we see that for our modelling framework the diagnostic accuracy on a per-vessel basis can vary significantly.
In future work, we will regard the further improvement of our reduced-order model in terms of its capacity to reproduce the physics of 3D models in a more accurate manner. Moreover, we will look into better ways of characterising baseline coronary flow by means of, for example, non-invasive determination of baseline flow at selected vessels. Finally, we will consider how to better define a patient-specific TCRI, since we think that this factor plays a crucial role in representing patient-specific variability in coronary physiology relevant for FFR prediction.