On Intlet Pressure Boundary Conditions for CT-Based Computation of FFRSS: Clinical Measurement of Aortic Pressure

Background and Objectives: With the development of medical imaging and computational uid dynamics (CFD), a fast calculation method of steady-state fractional ow reserve (FFRss) based on CTA images has been applied to predict myocardial ischemia. Whilst this is a reliable non-invasive method of calculating FFR, assumptions involved in the analysis still need to be investigated for further improvement in predictive accuracy. In this study, we analyzed the inuence of inlet and outlet boundary conditions on FFR SS. Methods: A clinical trial was carried out in Peking University People's Hospital. We enrolled 15 patients with coronary heart disease. All patients underwent coronary CTA examination, aortic pressure measurement and FFR catheter surgery. In order to better reect the relationship between different entrance boundary conditions and FFRss, we used the invasive measurement of FFR as the standard, and calculated FFRss for 15 patients with coronary heart disease with different degrees of stenosis. The boundary conditions are divided into two groups: (1) pressure calculated based on physiological formula; (2) clinically measured aortic pressure. Based on the boundary conditions calculated by the physiological formula, we further studied the changes of FFRss under different coronary vasodilation responses (12%-48% baseline). Results: The research results show that although the pressure difference between the two pressure boundary conditions is 15mmHg, the FFRss calculation result does not change signicantly. With the change of the vasodilation state, the microcirculation resistance of the exit boundary condition gradually increased, and the calculated FFR value increased. Conclusions: We found that changes in the microcirculation resistance of coronary stenoses have a huge impact on FFRss. The changes in FFRss values caused by boundary conditions are due to the overestimation of the vasodilation response. Therefore, individualizing the hyperemia state of the stenotic vessel microcirculation resistance value is an effective method to improve the calculation of FFR.

myocardium when the same coronary artery has no stenosis. Its de nition can be simpli ed as FFR = Pd/Pa, where Pa and Pd are respectively the maximum coronary hyperemia state (need to be achieved by injecting coronary microcirculation dilatation drugs such as adenosine to achieve [9] ) the average pressure of the aortic root, The average pressure of the coronary artery distal to the stenotic lesion. FFR > 0. 8 indicates that the lesion does not cause signi cant myocardial ischemia and no intervention is required. Fame-1 and Fame-2 experiments proved that FFR is an effective diagnostic index to determine the suitability of stent therapy [10] .
With the development of medical imaging and computational uid dynamics (CFD), non-invasive computation of FFR has been developed. Based on coronary CT angiography (CTA), the method of computational uid dynamics is used to simulate the hemodynamics in the coronary arteries, to obtain the pressure distribution in the state of coronary congestion, and then to calculate the ratio of the distal end of coronary artery stenosis to the aortic pressure. That is, the non-invasive numerical calculation of FFR (Noninvasive Fractional Flow Reserve Derived From Coronary CT Angiography, FFR CT ). FFR CT technology provides a new non-invasive method for the functional detection of coronary artery stenosis [11][12][13][14][15][16][17][18][19][20] . In order to further develop this technology, it is mainly to simplify it to facilitate faster calculation of FFR, and many simpli ed calculation methods such as QFR, CAFFR, VFFR and FFRSS (based on steady calculation) used in this research have been derived. However, any simpli ed FFR calculation algorithm is inseparable from the assumptions of the model. Sensitivity analysis of boundary conditions and input parameters is essential for calculating FFR.
In the measurement of clinical FFR, doctors usually need to use vasodilators (such as adenosine) to cause the expansion of blood vessels. Adenosine activates A2A receptors and causes coronary arteries to dilate, increasing the myocardial ow of healthy people by 3.5 to 4 times [21] . In the non-invasive numerical calculation of FFR, a special challenge is to simulate the maximum congestion state of blood vessels and simulate the effect of adenosine on reducing the peripheral resistance of coronary microcirculation [22] .
Studies have shown that the total coronary artery resistance drops to 0.24 times the resting value of venous blood when the blood vessels are in the state of maximum congestion. At this time, 140 mg/kg/min of vascular adenosine is clinically administered. In coronary microvascular ischemic disease, the vasodilation response is reduced, and in some extreme cases, adenosine does not cause changes in resting blood vessels [21] . It is an assumption to simulate the maximum congestion state in the FFR model based on CT calculation, which changes the microcirculation resistance to 0.24 times of the original [14] .
However, because the expansion ability of microvessels varies from individual to region, the resistance of microcirculation also has a certain in uence on the ow rate of coronary arteries. Microvascular resistance is used as the boundary condition of the outlet in the calculation model. In the coronary circulation, the in uence of calculating FFR has not been thoroughly studied. Therefore, in this study, we have also evaluated the in uence of microcirculation resistance on the calculation model.
In this study, we used different entry boundary conditions to perform CFD calculation and analysis on the coronary artery model of personalized patients, and explored the impact on FFR. The in uence of simulated different hyperemia states on FFR was analyzed.

Results
The display of FFR calculated according to the aortic pressure waveform obtained in clinical trials, the pressure calculated based on physiological parameters and the two boundary conditions is shown in Table 2.  Table 2 gives a quantitative comparison of the FFR values of the two boundary conditions. It can be seen that there is a good correlation with the clinical FFR, and the FFR calculated by the two boundary conditions is basically the same, which shows that the boundary conditions are calculated based on the physiological formula FFR has good accuracy. We also listed the FFR cloud image of each patient to observe the pressure changes of each patient's stenotic vessels under different boundary conditions. As shown in Fig. 6.
As different patients grow older, the state of vasodilation of blood vessels will change. In calculating FFR, we simulated three different sets of changes in the maximum congestion state (under the boundary conditions calculated based on the physiological formula), respectively calculated the calculated FFR of 15 patients in these three cases, and compared with the clinical FFR. Table 3 lists the in uence of microcirculation resistance and ow rate changes on the FFR value under different conditions.

Discussion
With the development of medical images and uid dynamics, the physiological calculations of coronary arteries have developed rapidly. The evaluation of myocardial ischemia based on the coronary artery function of CT images has become the mainstream of the development of coronary physiology. Any CTbased calculation model is inseparable from the discussion of boundary conditions. Taylor et al. discussed the effects of coronary imaging quality, segmentation uncertainty, blood viscosity, etc. on FFR [23] , and did not conduct a detailed study on the assumption of the maximum congestion state in the calculation model.
In this study, we separately discussed the sensitivity of the in ow boundary condition pressure to FFR in the calculation model and the sensitivity of the out ow boundary condition microcirculation resistance to FFR. The results show that the boundary conditions of the two pressures have almost the same effect on FFR, but the slight change in microcirculation resistance has a greater impact on FFR. Because the change of microcirculation resistance affects the ow in the stenotic blood vessel. The ow rate is an important factor affecting FFR. The higher the ow rate with the lower the FFR.
The effect of pressure on FFR is mainly due to the undetectable boundary conditions of active pressure. Therefore, we have previously adopted the assumption of aortic pressure and the physiological calculation formula based on pressure as the boundary condition of the entrance. We conducted clinical trials for verify the difference between the boundary conditions and the actual clinical boundary conditions. The results show that the FFR values calculated by the pressure boundary conditions of the two inlets have a high correlation. It can be seen from the results that in patients 9, 10, the aortic pressure calculated by the two methods has exceeded 15mmHg, but the calculated FFR value is the same.
For patients with mild stenosis (such as patients 5, 15), the pressure drop at both ends of the stenosis is also relatively small, and the difference is very small. Therefore, even if the maximum hyperemia is reached, the two different inlet pressure boundary conditions have no change in the pressure drop at both ends of the stenosis. The maximum congestion state has a great in uence on the FFR calculation of severe stenosis (such as patients 10 and 13). Because the stenosis resistance generated in the maximum congestion state is greater, and the pressure drop at both ends of the stenosis is relatively large, so the calculation is larger difference.
The change of congestion state is more obvious for patients with severe stenosis. According to Poisson's law , when the stenosis remains unchanged, the increase in ow will increase the pressure drop, and the increase in resistance will also increase the pressure drop, thereby reducing FFR value. This is a simple principle of uid mechanics. When the ow velocity in a narrow place decreases, it will lead to a high FFR. If there are any diseases in the coronary microcirculation vessels, it may lead to an increase in FFR, resulting in a decrease in blood ow. In clinical surgery, an increase in FFR can lead to false positives, leading to misdiagnosis of the patient's condition.
In patients with stable angina, 65% of women and 32% of men have mild stenosis [30] , which means that there is no stenosis but myocardial ischemia has occurred. Some of these patients suffer from coronary microvascular dysfunction, which is re ected in coronary heart disease, which leads to a mismatch P = Q R between coronary stenosis and myocardial ischemia [30] . The current CT-based FFR technology cannot best serve this part of patients, because the technology assumes the use of the greatest possible vasodilation function [14] , For patients with impaired vasodilation response, the simulation will overestimate the blood ow through the stenosis, which will result in a lower value than the clinical real FFR, which will lead to a false diagnosis of myocardial ischemia.
As the patient's age status changes, the diastolic state of stenotic vessels is also affected to a certain extent. For example, patient 7, as the diastolic state changes, the FFR value no longer changes. In patient 13, FFR changed signi cantly with the change of vasodilation state. Daniel et al. through the post-mortem analysis of ADVIESE ii con rmed that with age, the maximum congestion state gradually decreases, and the FFR value gradually decreases [31] . Coronary microcirculation resistance has a greater impact on FFR, mainly because the change of coronary microcirculation resistance is due to the change of vascular ow.
The value of microcirculation resistance quanti es the magnitude of this change.
In this study, we con rmed that different coronary microcirculation resistances have a huge impact on the calculation of FFR. Therefore, for patients who do not have coronary artery stenosis but have myocardial ischemia, it should be noted that the FFR value calculation based on CT should take into account that the maximum hyperemia change of coronary microvascular resistance may no longer be 0.24 this phenomenon. One way to solve this problem is to be able to quantify the changes in coronary microvessels in different patients, and the changes in the maximum congestion state of microcirculation resistance. In future work, we will establish the relationship between the diastolic state of microcirculation resistance and age to solve this problem. Another method is to use resting diagnostic indicators, which do not depend on the congestion state of blood vessels, such as IFR, RFR, etc. to diagnose myocardial ischemia.
The quality of the CTA image may affect the reconstruction of the 3D model, and the accuracy of the reconstruction of the coronary artery model affects the accuracy of calculating the FFR to a certain extent.
On the other hand, the method of calculating FFR based on steady state mentioned in this study is a more reliable non-invasive method of calculating FFR. The model is improved based on the geometric multiscale model [14] . This study explored the boundary conditions of the entrance and exit of the model, but did not discuss other physiological parameters in the model too much, which should be considered in the future work. What this model gives is the boundary condition of the inlet pressure. In other assumptions, the boundary condition based on the inlet ow is also considered in our future work.

Conclusions
In this study, we separately discussed the sensitivity of the entrance boundary conditions and exit boundary conditions of the non-invasive calculation FFR model based on CT images to FFR. We found that the pressure change of the inlet pressure boundary condition has no obvious effect on the results of FFRCT. The main reason for the change of FFRCT is the change of the microcirculation resistance at the end of the blood vessel. Therefore, in future work, personalizing the patient's microcirculation resistance is an effective method to improve the calculation of FFR based on CT.

Enrolled patients
In cooperation with Peking University People's Hospital, we enrolled 16 blood vessels in 15 patients with stable angina pectoris for obtain true pressure aortic pressure waveform. All patients underwent 256-slice CT scan, transcatheter FFR operation, and acquired the patient's aortic pressure waveform and clinical physiological parameters during the operation. The institutional review boards of the participating centers approved the study protocol, and each patient signed an informed consent. The Biomechanics Laboratory of Beijing University of Technology analyzed the anonymized data independently. The participants' data are shown in Table 1.

Three-dimensional reconstruction of coronary image
The resolution of the coronary artery CTA images of the enrolled patients was 512*512. The interval between adjacent CTA image slices was 1mm, and the pixel quality of each slice was 0.5 mm*0.5 mm. Nitroglycerin was administered before CT and FFR acquisition. We used a 3-D model reconstruction software to reconstruct the epicardial coronary artery model. According to the coronary artery lumen segmentation [23] , the segmented coronary arteries' minimum diameter was 1mm, and coronary arteries greater than 1 mm in diameter were included. The mesh of the coronary model is divided by ANSYS software, using a tetrahedral mesh with 6 layers of prismatic elements on the boundary, and the number of meshes for a single model is 200,000. The grid sensitivity analysis of the model is carried out, and the calculation effect is the best when the grid is around 200,000.

Model for calculating FFR based on CT: FFR SS
A fast calculation method of steady-state blood ow reserve based on CTA images (FFRss) [24] has been applied to the non-invasive diagnosis of myocardial ischemia. Taylor et al pioneered a non-invasive calculation of FFR based on CT [14] , and we proposed a steady-state method for calculating FFR based on the calculation model. This method is a simpli ed version of the geometric multi-scale FFR calculation model. The speci c calculation process is as follows: Construct a three-dimensional model of the coronary artery based on the patient's coronary artery CTA image. The average pressure of the aorta is used as the boundary condition of the inlet, and the microcirculation resistance calculated by the coronary ow is used as the boundary condition of the outlet. We can quickly calculate the ow rate and pressure distribution in patients with coronary congestion. The calculation structure is shown in Fig. 1.
The boundary conditions of the 3D coronary artery model are very important for numerical calculation of FFRss, and the accuracy of FFRss calculation strongly depends on the boundary conditions. The inlet boundary condition of the FFRss calculation model is based on the assumption of a physiological situation. The aortic pressure is calculated by the physiological formula of the human body. The coronary artery outlet of the model is connected with a resistance that simulates the coronary microcirculation resistance. Therefore, it is necessary to analyze the in uence of the boundary conditions of the entrance and exit of the calculation model on FFRss to improve the accuracy of the calculation of the FFR model to predict myocardial ischemia.
In the solution of the FFRss calculation model, we assume that the blood vessel wall is rigid and simulate the blood ow as an incompressible viscous Newtonian uid. The density is usually selected as 105 0kg/m3, and the dynamic viscosity is 0.0035 Pa·s. Use commercial software solver ANSYS-CFX to solve Navier-Stokes and continuity equations to obtain the distribution of coronary artery stenosis vessel pressure and ow.

Entry boundary condition acquisition
In the previous FFRss boundary conditions, the boundary conditions of the patient's 3D coronary entrance were set to aortic pressure (Pa), but the aortic pressure needs to be measured clinically, so we contracted based on the patient's physiological parameters provided in the literature [25] Pressure (SBP), diastolic blood pressure (DBP) and heart rate (HR) to obtain: In addition, we have obtained the real aortic pressure waveform clinically for verify the in uence of the aortic pressure based on the physiological formula on the calculation of FFR. During the FFR operation, the professional clinician obtained the true aortic pressure waveform of the patient for several cycles through the pressure guide wire, as shown in Fig. 2.
We use the obtained two pressures as the boundary conditions for FFRss calculation to calculate FFR, and compare and analyze the results of the two boundary conditions calculations.

Simulation of microcirculation resistance at the coronary arteries
We equate the resistance of the microcirculation downstream of the coronary artery to the resistance, as the boundary condition of the coronary artery exit of the calculation model, determine the resistance according to the structure of the coronary artery [27] . According to the allometric scaling law, there is a power-law relationship between ow velocity and vessel diameter [28] , and the ow velocity of each coronary artery branch can be calculated according to : Qm is the total ow of coronary artery branches of the m sublevel, and Qmn is the ow of the branch n in the m sublevel, and n is the total number of branches of the m sublevel.
According to the ow of each branch of the coronary artery, based on the basic principles of electricity. Therefore, the microcirculation resistance of each coronary artery branch outlet can be obtained. The calculation formula is shown in (2)(3). The schematic diagram of the calculation of coronary microcirculation resistance is shown in Fig. 3.
Rn_res represents the microcirculation afterload resistance value of the coronary artery branch n in the static state, Pv is the venous pressure of the venous blood vessel, Pa is the pressure of the coronary artery outlet, and Q is the ow of the coronary artery branch at this time.
The simulation method based on CT calculation of FFR for the simulation of the maximum vascular congestion state adopts the assumption that when the blood vessel reaches the maximum diastolic state, the microcirculation resistance becomes 0.24 times of the resting state [26] , Rn_hyp represents the congestion state, and the coronary artery branch n Load resistance value after microcirculation. As shown in (2)(3)(4): In the calculation model, we use the resistance boundary condition to simulate the change of microcirculation resistance. Therefore, the coronary afterload resistance in the hyperemic state is 24% of the static afterload resistance, but the afterload resistance of different patients may be 18% -32% [26] .
In order to be able to test whether the microcirculation resistance becomes 0.24 times that of the resting state during hyperemia, and how sensitive the FFR is to this, we simulated the microcirculation resistance,

Declarations
Ethics approval and consent to participate This study passed the inspection by the medical ethics committee of People's Hospital, Peking University. All participants have signed an informed consent.

Consent for publication
All patients of this study have agreed to publish. This study passed the inspection by the medical ethics committee of People's Hospital, Peking University. All participants have signed an informed consent.

Availability of data and materials
The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.

Competing interests
The authors declare that there is no con ict of interests of this article.
Funding and the downstream microcirculation resistance of the diseased branch was adjusted to be based on the basic condition of the coronary artery tree structure model differs in the reduction level (24%, 48%, 72%, 96%).

The calculation of FFR
In the clinical FFR operation, professional clinicians measured the aortic pressure of the enrolled patients, and obtained the pressure pd at the distal end of coronary artery stenosis 3cm [29] of 15 patients through the pressure guide wire. According to the simpli ed the calculation . In the FFR simulation calculation model, when the steady-state calculation model reaches convergence, the pressure at the distal end of the stenosis is extracted, and the CT-based FFR value is obtained using the formula. As is shown in  Figure 1 FFRss model structure diagram. The coronary artery is a three-dimensional model, the microcirculation part of the coronary artery is an 0D circuit model, and 3D and 0D are coupled for calculation.

Figures
Page 17/19 Figure 2 Schematic diagram of the clinically measured aortic pressure waveform (a single cycle).