PBFT optimization algorithm based on community contributions

Pengpeng Wang; Xu Wang; Yumin Shen; Jinlong Wang; Xiaoyun Xiong; Pengpeng Wang; Xu Wang; Yumin Shen; Jinlong Wang; Xiaoyun Xiong

doi:10.3934/mbe.2023447

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 6: 10200-10222. doi: 10.3934/mbe.2023447

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Research article Special Issues

PBFT optimization algorithm based on community contributions

School of Information and Control Engineering, Qingdao University of Technology, Qingdao 266520, China

Academic Editor: Jerry Chun-Wei Lin

Received: 29 December 2022 Revised: 05 March 2023 Accepted: 10 March 2023 Published: 30 March 2023

Community governance is the basic unit of social governance, and it is also an important direction for building a social governance pattern of co-construction, co-governance and sharing. Previous studies have solved the problems of data security, information traceability and participant enthusiasm in the process of community digital governance by building a community governance system based on blockchain technology and incentive mechanisms. The application of blockchain technology can solve the problems of low data security, difficulty in sharing and tracing and low enthusiasm on the part of multiple subjects regarding participation in community governance. The process of community governance involves the cooperation of multiple government departments and multiple social subjects. Under the blockchain architecture, the number of alliance chain nodes will reach 1000 with the expansion of community governance. The existing consensus algorithms for coalition chains are difficult to meet the high concurrent processing requirements under such large-scale nodes. An optimization algorithm has improved the consensus performance to a certain extent, but the existing systems still cannot meet the data needs of the community and are not suitable for community governance scenarios. Since the community governance process only involves the participation of relevant departments in users, all nodes in the network are not required to participate in the consensus under the blockchain architecture. Therefore, a practical Byzantine fault tolerance (PBFT) optimization algorithm based on community contribution (CSPBFT) is proposed here. First, consensus nodes are set according to different roles of participants in community activities, and participants are given different consensus permissions. Second, the consensus process is divided into different stages, and the amount of data processed by each consensus step is reduced. Finally, a two-level consensus network is designed to perform different consensus tasks, and reduce unnecessary communication between nodes to reduce the communication complexity of consensus among nodes. Compared with the PBFT algorithm, CSPBFT reduces the communication complexity from O(N2) to O(N2/C3). Finally, the simulation results show that, through rights management, network level setting and consensus phase division, when the number of nodes in the CSPBFT network is 100–400, the consensus throughput can reach 2000 TPS. When the node in the network is 1000, the instantaneous concurrency is guaranteed to be above 1000 TPS, which can meet the concurrent needs of the community governance scenario.

Keywords:

Citation: Pengpeng Wang, Xu Wang, Yumin Shen, Jinlong Wang, Xiaoyun Xiong. PBFT optimization algorithm based on community contributions[J]. Mathematical Biosciences and Engineering, 2023, 20(6): 10200-10222. doi: 10.3934/mbe.2023447

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Abstract

1. Introduction

Coronary artery disease (CAD) develops when coronary arteries become damaged or diseased because of the dominant prevalence of stenosis. Relying only on anatomical information when doing noninvasive or invasive coronary angiography has proven to be insufficient to detect hemodynamically significant epicardial stenosis ^[1]. Invasive fractional flow reserve (FFR) has been accepted as the gold standard because of its high accuracy for the assessment of myocardial ischemia ^[2]. FFR determines the likelihood that the stenosis is impeding oxygen delivery to the heart muscle, and it is defined as the pressure after (distal to) a stenosis relative to the pressure before (proximal to) the stenosis. However, the application rate of gold standard invasive FFR in clinical practice is low, mainly due to its high measurement cost and potential risks during catheter insertion ^[3].

Clinical vascular stenosis measurement by invasive FFR ^[4] can be effectively solved by calculating computed tomography (CT) angiography-derived noninvasive fractional flow reserve (FFR_CT) with the help of computational fluid dynamics (CFD) solvers ^[5,6,7]. Noninvasive FFR_CT provides incremental diagnostic value over coronary CT angiography (CCTA) alone by adding functional information ^[8]. Over the past decade, a series of randomized trials have established noninvasive CCTA as a safe and potentially cost-effective strategy in the workup of patients with acute chest pain and low-intermediate risk of acute coronary syndrome who present to the emergency department ^[9].

FFR_CT has been widely used and studied by clinicians and researchers in recent years. The diagnostic performance of FFR_CT has been investigated experimentally by some multicenter studies, including DISCOVER-FLOW (Diagnosis of Ischemia-Causing Stenoses Obtained Via Noninvasive Fractional Flow Reserve) ^[10,11], DeFACTO (Determination of Fractional Flow Reserve by Anatomic Computed Tomographic Angiography) ^{[12,13,14,15]} and NXT (Analysis of Coronary Blood Flow Using CT Angiography: Next Steps) ^{[16,17,18,19]}, comprising a total of 609 patients and 1050 vessels. The trial in 2014 showed that the accuracy of FFR_CT was 81% ^[17], which needs to be improved to satisfy clinical requirements. Recently, a total of 338 consecutive patients with 422 vessels from 9 Chinese medical centers undergoing CCTA and invasive FFR were retrospectively analyzed. Their FFR_CT values were obtained as a novel on-site computational fluid dynamics-based FFR_CT (uCT-FFR). A new model of FFR_CT relying on boundary conditions derived from structural deformation of the coronary lumen and aorta, with a transluminal attenuation gradient and assumptions regarding microvascular resistance, was developed and applied to assess the performance characteristics of uCT-FFR and CCTA in detecting lesion-specific ischemia in all lesions, intermediate lesions (luminal stenosis 30 to 70%), and "gray zone" lesions (FFR 0.75 to 0.80). The accuracy of this novel approach reached 91% ^[20]. A prospective, multicenter, cross-sectional study, Dan-NICAD 2, including approximately 2000 patients with low to intermediate pretest probability of CAD and without previous history of CAD, is an ongoing study ^[21] that is expected to contribute to the improvement of diagnostic strategies for patients suspected of CAD in 3 different steps: risk stratification prior to coronary CTA, diagnostic strategy after coronary CTA, and invasive wireless quantitative flow ratio (QFR) analysis as an alternative to invasive FFR.

The insufficient accuracy of FFR_CT is the major obstacle to progress in its clinical application. The main aspects of modeling that affect FFR_CT estimation are the accuracy of the model reconstruction, the choice of inlet and outlet boundary conditions and the estimation of the flow distribution in the coronary network ^[22], where the boundary condition directly determines the blood flow pattern. Boundary condition settings can be simplified by calculating the total coronary blood flow ^[5]. The calculation of total coronary blood flow could be completed based on the scaling relationship between myocardial mass and coronary blood flow ^[23]. However, myocardial mass is not the only factor affecting coronary blood flow, and there are individual differences in the correlation between myocardial mass and total coronary blood flow. Heart rate and aortic peak systolic pressure were input into the mathematical model of coronary blood flow of Arthurs et al ^[24]. This model can provide patient-specific estimates of coronary blood flow changes between rest and exercise situations. Duanmu et al. considered the ventricular contraction function of patients and simulated patient-specific coronary blood flow changes under intramyocardial stress changes at the terminus of the coronary artery ^[25]. Duanmu et al. incorporated the head loss at the junctions of inlets to could provide patient-specific estimates of coronary blood flow changes under cross-sectional area changes at the inlets of the coronary arteries ^[26]. In addition to heart rate and aortic peak systole pressure, ventricular contraction function and head loss at the junctions of inlets were also considered input parameters, but they were not easy to detect.

In the studies of Taylor et al and Choy et al. ^[5,23], the myocardial mass satisfied the power law, the diameter and length of branch arteries having exponents of 3/8 and 3/4, respectively. Huo et al established the scaling law on length and volume relationship with an exponent of 7/9 based on experiments with rats of different ages ^[27]. These mathematical models provide approaches to calculate the distribution of flow rates in coronary artery branches. However, the models by Choy 2008 and Huo 2009 are based on animal experimental data, so they might not be applicable to human beings.

Based on this situation, this study proposed a FFR_CT calculation method with personalized boundary conditions, which can realize the non-invasive simulation of coronary blood flow distribution and assist the clinical treatment of coronary stenosis.

2. Methods

2.1. Geometry model of coronary artery

The exclusion criteria for this study were as follows: 1) myocardial damage or recent myocardial infarction (within one month); 2) left ventricular dysfunction; 3) significant 3-vessel disease; and 4) poor-quality CT images. All patients underwent CCTA and coronary FFR. Intravenous adenosine (140 μg/kg/min) was administered to the inlet coronary artery to induce maximum hyperemia. Based on the maximum hyperemia value, we obtained the pressure waveform of the aortic pressure (P_a) and distal arterial pressure (P_d) using a pressure guide wire. We further calculated FFR as the ratio of the mean pressure at a cross-section 3 cm downstream of the stenosis (P_d) to the mean pressure at the coronary entry region (P_a) during maximal hyperemia ^[5,28]. Sixteen patients with steady coronary artery disease were included in our study. Among them, 20 vessels with coronary stenosis were included, 14 (88%) with right coronary artery dominance. Informed consent was waived due to the retrospective nature of the study.

Medical image segmentation and 3D reconstruction of the geometric model of coronary and aortic roots were accomplished by using the commercial tool MIMICS (Materialise, Leuven, Belgium). Vessels with diameters larger than 1 mm were saved and reconstructed. The centerline of each vessel was identified, and the diameter of each vessel was calculated by manually selecting each vessel in MIMICS. Then, the tool Freeform (Artec 3d, Luxembourg) was used to remove the noise in the 3D model, and the software GEOMAGIC (Geomagic, Research Triangle Park, North Carolina) was used to repair and further optimize the model surface. GEOMAGIC software was used to further optimize the model surface and calculate the volume of the 3D model. Then, a branch of the coronary artery was cut in the 3D model, and the volume of the remaining 3D model was calculated using GEOMAGIC. The volume of a branch of the coronary artery was the volume change before and after cutting the 3D model. Repeating the above operation on each vessel of the 3D model, the volume of each vessel of the coronary artery was calculated for further research. SOLIDWORKS (Dassault Systemes, Waltham, Massachusetts) was used to cut the entrance port into a plat section. Finally, the geometric model was imported to ANASYS CFX (ANSYS Corporation, Canonsburg, Pennsylvania) for hemodynamics simulation.

2.2. Mathematic model of coronary blood flow rate

Myocardium is perfused by coronary arteries. Some coronary vessels buried deeply in the myocardium will be compressed during cardiac systole, which will affect coronary blood flow. Blood flow will reach a climax at early diastole and then decelerate slowly ^[29]. In general, coronary blood flow during left ventricular systole is only 20–30% of that during diastole and will decrease when myocardial contraction strengthens.

Some independent physiological parameters, including myocardial mass, diastolic blood pressure and heart rate, were selected to calculate coronary blood flow.

Myocardial mass. The scaling laws between coronary blood flow (Q_cor) and the myocardial mass (M) can be described by Eq (1) ^[23]:

${Q}_{cor}\propto {M}^{0.75}$

(1)

In a broad sense, the myocardium includes the right ventricular myocardium, atrial myocardium and other myocardial tissue in addition to the left ventricular myocardium. The left ventricular myocardial mass (M_L) accounts for approximately 85% of the total myocardial mass ^[30,31]. Therefore, the scaling law can be revised as Eq (2):

${Q}_{cor}\propto {\left({M}_{L}/0.85\right)}^{0.75}$

(2)

Diastolic blood pressure. Coronary perfusion is mainly affected by diastolic blood pressure since coronary blood flow perfusion is mainly accomplished in diastole. According to Poiseuille's law, Eq (3) is satisfied:

${Q}_{cor} = ∆P\left(\pi {r}^{4}/8\eta L\right)$

(3)

where the perfusion pressure △P, which is the pressure drop along the artery with a length of L, can be calculated from Eq (4),

$∆P = {P}_{D}-{P}_{Ra}$

(4)

where P_D is the aortic diastolic blood pressure and P_Ra is the right atrial pressure (-2–10 mmHg). Studies have shown that aortic diastolic blood pressure can be used alone to calculate the coronary blood flow rate, and diastolic blood pressure is proportional to the coronary blood flow rate ^[32,33,34]. For the sake of simplification, the perfusion pressure was set as the aortic diastolic blood pressure by ignoring the right atrial pressure in this study. This let the relationship between coronary blood flow rate and diastolic blood pressure be described by Eq (5):

${Q}_{cor}\propto {P}_{D}$

(5)

Heart rate. Effective perfusion time is another factor affecting coronary blood flow. The diastole of a cardiac cycle is the time for coronary blood perfusion ^[29]. Diastolic duration (T) becomes relatively compressed with increasing heart rate (HR).

According to the correlation between heart rate and diastolic duration (Figure 1) ^[35], the perfusion time per minute can be described as a function of heart rate as in Eq (6):

${T}_{m} = T*HR = HR[-0.568\mathrm{ln}\left(HR\right)+3.246]$

(6)

Figure 1. Correlation between left ventricular diastole and heart rate during a cardiac cycle.

DownLoad: Full-Size Img PowerPoint

As the perfusion rate is assumed to be constant, the blood flow of the coronary artery is proportional to the perfusion time. Then, the relationship between coronary blood flow rate and heart rate can be described by Eq (7):

${Q}_{cor}\propto HR[-0.568\mathrm{ln}\left(HR\right)+3.246]$

(7)

Combining Eqs (2), (5) and (7), the mathematical model of coronary blood flow rate during maximal hyperemia with adenosine can be established as Eq (8):

${Q}_{cor} = kHR[-0.568\mathrm{ln}\left(HR\right)+3.246]{P}_{D}{\left({M}_{L}/0.85\right)}^{0.75}$

(8)

where k is a constant coefficient that needs to be determined according to clinical data.

Due to the lack of clinical data, coronary blood flow was derived from the simulation in this study. Twenty-nine randomly selected patients with mild stenosis and FFR < 0.90 were adopted to perform the simulation and calculate the value of k that made the simulation result close to the real physiological value. The value of k for this study was the average of those from the 29 patients.

FFR_CT was the ratio of the mean pressure at a cross-section 3 cm downstream of the stenosis (P_d) to the mean pressure at the coronary entry region (P_a) ^[5]. Figure 2 shows the technical flow of coronary FFR_CT. First, a three-dimensional model of the coronary artery was reconstructed from CCTA. Blood was modeled as a Newtonian fluid, and the blood density and dynamic viscosity were constants with values of 1050 kg/m³ and 0.0035 Pa·s, respectively ^[28]. The blood flow was steady, and the material properties of the arterial walls were rigid.

Figure 2. Technical flow chart of calculating coronary FFR_CT. (a) 3D reconstruction of the coronary artery model based on CCTA; (b) meshing of fluid and boundary layer; (c) individualized settings of the boundary conditions; (d) CFD simulation calculation and postprocessing.

DownLoad: Full-Size Img PowerPoint

The mesh of the geometries was generated by using nonstructural tetrahedron elements. To improve the accuracy of the boundary layer calculation, the boundary layer near the vessel wall was set to three layers. After testing the target residual and convergence step dependence, the target residual was determined to be 1e-4. The cardiac output at the root of the aorta was assigned as the inlet boundary condition. The mean value of aortic diastolic pressure at the outlet of the aorta was assigned as the outlet boundary condition. Based on the mean value of the coronary blood flow rate of the patient, the coronary blood flow rate was initialized as the product of myocardial mass and myocardial blood flow during hyperemia, at 3.37 mL/min/g ^[36]. The blood flow rate in a vessel was proportional to the 3rd power of the vessel diameter according to Poiseuille's solution and Murray's law ^[37]. The flow rate at each terminal branch was identified as the outlet boundary condition. Simulations were carried out by using ANSYS Workbench. Steady flow simulation was employed to significantly reduce the computational cost. The computational time of FFR_CT was only 10 minutes, which was 24-fold less than with Taylor's method ^[5].

The coronary blood flow rate was initialized as the product of myocardial mass and myocardial blood flow during hyperemia, at 3.37 mL/min/g ^[36], which is the mean value in humans, and then it was updated according to the following steps until the simulation result of FFR_CT matched the invasive FFR value.

(a) Perform simulation with the initial value of flow rate, and calculate the coronary FFR_CT from the simulation results.

(b) If the simulation result of FFR_CT was larger than the invasive value of FFR, increase the coronary blood flow rate; otherwise, decrease it.

(c) Adjust the flow rate by 50, 20, 10, and 5 mL/min until the simulation result of FFR_CT matched the invasive value of FFR.

(d) Take the matched flow rate as the reference coronary blood flow rate.

(e) Substitute the reference coronary blood flow rate into Eq (6) to solve for the constant k. The mean value of k (k = 0.003) obtained from the 29 patients was taken as the value of k in the mathematical model, and thus Eq (8) could be written as Eq (9):

${Q}_{cor} = 0.003HR[-0.568\mathrm{ln}\left(HR\right)+3.246]{P}_{D}{\left({M}_{L}/0.85\right)}^{0.75}$

(9)

Then, Eq (9) was applied to perform the simulation for the 20 vessels with coronary stenosis in our 16 patients.

2.3. Branch flow distribution approaches for coronary artery

Several morphological (diameter, length, and volume) and functional (flow) parameters of the coronary arterial tree related to myocardial mass were quantified in in vitro animal experiments by Choy et al ^[23]. The results showed that arterial volume was linearly related to regional myocardial mass ^[23], whereas the sum of coronary arterial branch lengths, vessel diameters, and volumetric flow showed a 3/4, 3/8, and 3/4 power-law relationship, respectively. These findings were consistent with the earlier experimental results of Seiler et al ^[38,39]. Based on the physiological observation that longer coronary arteries have more daughter branches feeding a larger mass of cardiac muscle, Lee et al presented a vessel length-based method calculating the coronary flow distribution over coronary major arteries ^[40].

Given that the volume of a vessel contains its diameter and the length, we proposed a vessel volume-based method to calculate the flow distribution over coronary major arteries. Since the arterial volume is linearly related to regional myocardial mass ^[23], the regional myocardial mass of the left anterior descending (LAD) artery, the left circumflex (LCX) artery, and the right coronary artery (RCA) can be described by Eq (10):

${M}_{LAD} = \alpha {V}_{vessel-LAD}; \; {M}_{LCX} = \alpha {V}_{vessel-LCX}; \;{M}_{RCA} = \alpha {V}_{vessel-RCA}$

(10)

where M_LAD, M_LCX and M_RCA represent the regional myocardial mass of the LAD, LCX and RCA, respectively; V_vessel-LAD, V_vessel-LCX and V_vessel-RCA represent the vessel volume of the LAD, LCX and RCA, respectively; and α is a constant scale coefficient.

The volumetric flow shows a 0.75 power-law relationship with myocardial mass, so Eq (11) can be obtained:

${Q}_{LAD} = \beta {{M}_{LAD}}^{0.75} ; \; {Q}_{LCX} = \beta {{M}_{LCX}}^{0.75} ; \; {Q}_{RCA} = \beta {{M}_{RCA}}^{0.75}$

(11)

where Q_LAD, Q_LCX and Q_RCA represent the volumetric flow of the LAD, LCX and RCA respectively, and β is a constant scale coefficient.

Equation (12) can be deduced from the above two equations. Namely, the volumetric flow shows a 0.75 power-law relationship with the vessel volume. Although such a relationship was not presented in the paper by Choy 2008 ^[23], it follows from the results presented therein, but it has had no direct experimental validation:

${Q}_{LAD} = {\alpha }^{0.75}\beta {{V}_{vessel-LAD}}^{0.75} ; \; {Q}_{LCX} = {\alpha }^{0.75}\beta {{V}_{vessel-LCX}}^{0.75} ; \; {Q}_{RCA} = {\alpha }^{0.75}\beta {{V}_{vessel-RCA}}^{0.75}$

(12)

The total coronary blood flow rate is the sum of the blood flow rate over the LAD, LCX and RCA, as shown in Eq (13):

${Q}_{cor} = {Q}_{LAD}+{Q}_{LCX}+{Q}_{RCA}$

(13)

The flow distributions of the LAD, LCX and RCA is respectively are described in Eq (14):

${Q}_{LAD} = {Q}_{cor}\left[{{V}_{vessel-LAD}}^{0.75}/({{V}_{vessel-LAD}}^{0.75}+{{V}_{vessel-LCX}}^{0.75}+{{V}_{vessel-RCA}}^{0.75})\right]$

${Q}_{LCX} = {Q}_{cor}\left[{{V}_{vessel-LCX}}^{0.75}/({{V}_{vessel-LAD}}^{0.75}+{{V}_{vessel-LCX}}^{0.75}+{{V}_{vessel-RCA}}^{0.75})\right]$

(14)

${Q}_{RCA} = {Q}_{cor}\left[{{V}_{vessel-RCA}}^{0.75}/({{V}_{vessel-LAD}}^{0.75}+{{V}_{vessel-LCX}}^{0.75}+{{V}_{vessel-RCA}}^{0.75})\right]$

Ideally, the blood flow rate in a vessel is proportional to the 3rd power of the vessel diameter according to Poiseuille's solution and Murray's law ^[36], as in Eq (15):

$Q = \left(\mathrm{\pi }{d}^{3}/4\right)\sqrt{\lambda /\mu }$

(15)

where Q is the flow rate through a blood vessel, d is its diameter, μ is the fluid viscosity, and λ is a constant. This equation represents the energy consumed by the metabolism per unit volume.

If d_LAD, d_LCX_, and d_RCA are respectively assigned as the ostium diameter of each branch of the LAD, LCX and RCA (the ostium diameter of each branch can be estimated according to the cross-sectional area of the branch), the flow rate of each branch can be inferred from Eqs (15) and (16) can be obtained:

$\begin{array}{c}{Q}_{LAD-j} = {Q}_{LAD}\left({{d}_{LAD-j}}^{3}/{\sum }_{i = 1}^{n}{{d}_{LAD-i}}^{3}\right)\\ {Q}_{LCX-j} = {Q}_{LCX}\left({{d}_{LCX-j}}^{3}/{\sum }_{i = 1}^{m}{{d}_{LCX-i}}^{3}\right)\\ {Q}_{RCA-j} = {Q}_{RCA}\left({{d}_{RCA-j}}^{3}/{\sum }_{i = 1}^{s}{{d}_{RCA-i}}^{3}\right)\end{array}$

(16)

where i and j are, respectively, the number of terminals and the identity of any branch of the LAD, LCX or RCA; and n, m and s are the total number of terminal branches of the LAD, LCX and RCA, respectively.

The coronary artery model reconstructed from CCTA does not completely satisfy Murray's law, so Eq (16) needs to be modified as follows. The flow distribution of each branch was calculated in the direction of blood flow according to the classification of vascular branches, as shown in Figure 3, where di and dij (i, j = 1, 2) represent the diameter of each daughter branch of the coronary artery, and Rij (i, j = 1, 2) represents the ratio of the flow rate at branch ij to that at the major root. This flow distribution approach was defined as the diameter-flow approach. The diameter-flow relationship showed that the blood flow rate in a vessel was proportional to the 3rd power of the vessel diameter.

Figure 3. Schematic diagram of the diversion ratio calculation (R represents the diversion ratio).

DownLoad: Full-Size Img PowerPoint

Similarly, we can assume that the flow rate is proportional to the volume of the corresponding coronary branches: Q∝V. As shown in Figure 4, Q₀ = Q₁+Q₂; Q₁ = Q₁₁+Q₁₂; Q₂ = Q₂₁+Q₂₂; Q₁:Q₂ = V₁^0.75: V₂^0.75; Q₁₁:Q₁₂ = V₁₁^0.75: V₁₂^0.75; Q₂₁:Q₂₂ = V₂₁^0.75: V₂₂^0.75… This flow distribution approach was defined as the volume-flow approach. The volume-flow relationship showed that the blood flow rate in a vessel was proportional to the 0.75 power of the vessel volume.

Figure 4. Diagram of flow distribution in branch vessels.

DownLoad: Full-Size Img PowerPoint

Based on the mean values of aortic diastolic pressure, myocardial mass and heart rate of the patient, we calculated a patient-specific coronary blood flow rate. Based on the volume or diameter from CCTA, we achieved patient-specific distribution of the blood flow rate. Based on the coronary blood flow rate and the distribution rule, we achieved a patient-specific coronary blood flow rate at each terminal branch. Combining the mean value of aortic diastolic pressure as the inlet boundary condition and the coronary blood flow rate at each terminal branch as the outlet boundary condition, we obtained the patient-specific boundary conditions of the fluid dynamics analysis.

2.4. Application of the calculation of FFR_CT of coronary stenosis

Based on the two approaches to the patient-specific boundary conditions of the fluid dynamics analysis, the distribution of pressure in a vessel was identified to calculate the FFR_CT value.

As measures of the performance of the two approaches in calculating FFR_CT, we chose the relative error and the diagnostic accuracy at the threshold of FFR ≤ 0.80 through comparison with the clinical FFR values of all 20 vessels with coronary stenosis of the 16 patients. To evaluate the consistency of the two approaches with the clinically measured FFR, we used the consistency with the 95% CI of the Bland-Altman method and the metrics of sensitivity, specificity, positive predictive value (PPV) and negative predictive value (NPV).

3. Results

Based on the information of 29 patients, including the mean values of aortic diastolic pressure, myocardial mass and heart rate, we obtained a mean value of k of 0.003. The distribution of the parameters is shown in Table 1.

Table 1. The distribution of the parameters to identify the coronary blood flow rate.

Parameter details	Range
The mean value of aortic diastolic pressure	51–101 mmHg
Myocardial mass	98–186 g
Heart rate	57–96 bpm
Cardiac output	2.75–6.9 L/min
The coronary blood flow rate	2.3–5.19 ml/min

| Show Table

DownLoad: CSV

Based on the structure of the coronary artery from CCTA, we used the diameter-flow approach and volume-flow approach to calculate the corresponding FFR_CT values. The distribution of the parameters of the two approaches is shown in Table 2, and they were within the range of numbers in previous coronary morphology studies ^[23,28]. Based on Eqs (10) and (11), the regional myocardial masses of the LAD, LCX and RCA were replaced by the corresponding blood vessel volumes. The blood vessel volumes of the LAD, LCX and RCA are shown in Table 2.

Table 2. The distribution of the parameters of 16 patients.

Parameter details	Range
The mean value of aortic diastolic pressure	70–98 mmHg
Myocardial mass	96–175 g
Heart rate	54–84 bpm
Cardiac output	2.1–6.3 L/min
V_vessel-LAD	502.46–2755.81 mm³
V_vessel-LCX	432.42–1855.67 mm³
V_vessel-RCA	316.6–924.18 mm³
d₁	3.12–3.88 mm
d₁₁	2.78–2.89 mm
d₁₂	2.64–2.96 mm
d₂	2.11–2.44 mm
d₂₁	1.13–1.88 mm
d₂₂	1.02–1.66 mm
V₁	76.42–118.82 mm³
V₁₁	151.7–163.9 mm³
V₁₂	158.7–199.5 mm³
V₂	139.79–186.94 mm³
V₂₁	60.14–166.47 mm³
V₂₂	57.17–151.42 mm³

| Show Table

DownLoad: CSV

3.1. Rationality

Due to the lack of actual values of clinical coronary blood flow, the ratio of total coronary blood flow to cardiac output (Per_Q) and myocardial blood flow (Q_myo) were selected as the evaluation indices to assess the rationality of the model of coronary blood flow. The mean values of Per_Q and Q_myo in the 20 vessels with coronary stenosis of the 16 patients, under the condition of hyperemia, were 16.97% and 4.07 mL/min/g, respectively, which were in accordance with the rule of medical statistics ^[36].

The coronary flow distribution to coronary major branches for 20 vessels with coronary stenosis was calculated (Table 3), and they were almost identical to those based on clinical measurements ^[41].

Table 3. Average fraction of coronary flow distribution for LAD, LCX, and RCA.

	LAD (%)	LCX (%)	RCA (%)
Calculated data	32.9	20.6	46.5
Clinical data (Sakamoto 2013)	31.1	26.7	41.9

| Show Table

DownLoad: CSV

3.2. Accuracy

The diagnostic performance of FFR_CT was estimated by the reference of clinical coronary FFR, with FFR ≤ 0.80 as the threshold.

The relative errors between the FFR_CT based on CFD simulation and the invasively measured FFR were obtained. These relative errors were 10.47% and 11.76%, respectively, under the volume-flow relationship and diameter-flow relationship. These differences may have arisen both from the neglect of branches with diameters less than 1 mm and from the subjective operation of the 3D model reconstruction and smoothing. Table 4 shows the results of P_a and P_d and the flow rates of the diameter-flow and volume-flow approaches.

Table 4. The pressure and flow rates of different vessels.

Vessel number		The P_d of diameter approach (mmHg)	The P_d of volume approach (mmHg)	P_a (mmHg)	The flow rate of diameter approach (ml/min)	The flow rate of volume approach (ml/min)
1	82		83	86	0.158	0.286
2	34		72	85	0.121	0.237
3	56		83	101	0.445	0.704
4	86		79	101	0.305	0.424
5	55		67	70	0.170	0.367
6	84		84	94	0.699	1.06
7	81		76	90	1.284	1.502
8	74		76	82	0.081	0.249
9	74		78	82	0.937	0.949
10	47		49	51	0.062	0.155
11	64		60	92	0.049	0.142
12	90		80	92	0.054	0.126
13	64		67	83	0.125	0.315
14	63		74	80	0.424	0.668
15	67		78	80	0.224	0.375
16	74		69	98	0.343	0.485
17	75		80	85	0.358	0.529
18	83		82	85	0.157	0.252
19	67		72	82	0.293	2.89
20	76		77	87	0.184	0.22

| Show Table

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In Table 5, the numbers 2 and 6 represent that 2 vessels were calculated as having FFR_CT ≤ 0.8 and 6 vessels were FFR_CT > 0.8 among the vessels with a clinically measured FFR ≤ 0.8 in the 20 vessels of the 16 patients. The numbers 1 and 11 indicate that 1 vessel had a calculated FFR_CT ≤ 0.8 and 11 vessels were FFR_CT > 0.8 among those with a clinically measured FFR > 0.8.

Table 5. Diagnostic accuracy of FFF_CT based on the volume-flow relationship.

FFR_CT	FFR
FFR_CT	≤ 0.80	> 0.80
≤ 0.80	2	1
> 0.80	6	11

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In Table 6, the numbers 6 and 2 represent that 6 vessels were calculated as having FFR_CT ≤ 0.8 and 2 vessels were FFR_CT > 0.8 among vessels with a clinically measured FFR ≤ 0.8 among the 20 vessels. The numbers 1 and 11 indicate that 1 vessel had a calculated FFR_CT ≤ 0.8 and 11 vessels were FFR_CT > 0.8 among those with a clinically measured FFR > 0.8.

Table 6. Diagnostic accuracy of FFF_CT based on the diameter-flow relationship.

FFR_CT	FFR
FFR_CT	≤ 0.80	> 0.80
≤ 0.80	6	1
> 0.80	2	11

| Show Table

DownLoad: CSV

The diagnostic accuracies of FFR_CT obtained from the volume-flow approach and diameter-flow approach were 65% and 85%, respectively, as illustrated in Tables 5 and 6.

3.3. Consistency

The Bland-Altman method was used to evaluate the consistency of FFR_CT and FFR. The 95% confidence interval between FFR_CT and FFR was approximately [-0.2, +0.2], and most of the data fell within the interval, which indicated that FFR_CT and FFR had good consistency (Figure 5). The consistency was greater than 80%.

Figure 5. Bland-Altman diagrams. (a) Volume-flow approach; (b) diameter-flow approach.

DownLoad: Full-Size Img PowerPoint

The PPV, NPV, sensitivity and specificity under the volume-flow approach were 25%, 91.67%, 66.67% and 64.7%, respectively, and were 75%, 91.67%, 85.71% and 84.62% under the diameter-flow approach.

4. Discussion

This study presented a mathematical model for estimating the coronary blood flow rate and two approaches for setting the patient-specific flow boundary conditions in calculating coronary FFR_CT. The present study validated the diagnostic performance of FFR_CT compared with invasive FFR.

Under the condition of hyperemia, the flow rate in the coronary artery reaches its peak value, which makes up approximately 15% of the cardiac output ^[36]. The calculated values of the flow rate in the coronary artery for the 16 patients were similar to the mean values of the human body ^[36].

With regard to 3D geometric model reconstruction, no more details were investigated in the present study. Because the 3D geometrical model of the coronary arterial tree derived from CT images was used, the spatial resolution in the small-diameter vessels of CCTA could result in deviation between reconstructed geometries and the actual anatomy, adding bias to the stenosis and FFR_CT computations ^[42].

The error of diagnostic accuracy under the volume-flow approach was larger than that under the diameter-flow approach. Compared with clinical FFR, the diagnostic accuracy of noninvasive FFR_CT, which is currently widely used in clinics, was 84.3% ^[10,16,17]. However, this study showed that the diagnostic accuracy of CCTA was only 64%, which meant that CCTA would only be a strategy for preliminary screening of coronary artery stenosis ^{[11,12,13,14]}. This means CCTA will have difficulty meeting the requirements of clinical use. The large shortcoming may result from the fact that the volume-flow approach is more sensitive to the gray zone of insecurity (with FFR ≤ 0.80 as the threshold). There is no absolute cutoff point at which the FFR becomes abnormal; rather, there is a smooth transition, with a large gray zone of insecurity. In clinical trials, a cutoff point of 0.75 to 0.80 has been used; higher values indicate nonsignificant stenosis, whereas lower values indicate a significant lesion. Kato et al evaluated the optimal cutoff value of FFR_CT by using fluidstructure interactions ^[43]. They concluded that it was reasonable to support the optimal cutoff value of 0.80 when the disease prevalence is more than 30% if the goal is minimizing 1-year cardiac events. However, underestimation should be adjusted to reduce cardiac events, especially when disease prevalence is low.

There are some limitations to this method. First, individual differences in coronary microcirculation resistance were not considered. Coronary microcirculation resistance, namely, after-loading, which is affected by the activity of the myocardium, directly affects the perfusion of the coronary artery ^[44]. In this study, we assumed that the patients all had normal coronary microcirculation and had the same response to adenosine and other vasodilator drugs. However, the actual situation is that there is some difference in the coronary microcirculation between different individuals or different regions of individuals, and most of them have different degrees of microcirculatory disturbance ^[45,46]. Second, the model can only be used in the simulation of coronary arteries during hyperemia. Whether the model can be applied to the resting state still needs to be investigated in depth. Third, the estimated parameter k was based on 29 randomly selected patients. The value of k might change if the set of patients used for estimation has different characteristics. Estimation from a large sample size and precise stratification would be more representative. Finally, the simulations in this study were all in the steady state, corresponding to the average condition of hyperemia, so the characteristics of pulsatile blood flow were not reflected.

5. Conclusions

A mathematical method to set patient-specific flow boundary conditions was proposed for the sake of exploring the effect of new strategies to specify hyperemic coronary flow and flow distribution parameters in FFR_CT simulations. As shown by comparing CCTA assessment, coronary FFR_CT and invasive FFR, FFR_CT has good consistency with FFR under the proposed patient-specific flow boundary conditions. This study offers a new way to calculate FFR_CT and supports the clinical application of FFR_CT.

In summary, this work employed coronary blood flow distribution methods based on the volume-flow relationship and diameter-flow relationship to set the boundary conditions in the numerical simulation of FFR_CT and compare their application feasibility. The results demonstrated good consistency between the noninvasive value of FFR_CT and the invasive value of FFR, even though there were large errors between the diagnostic accuracies of these two methods. The mathematical model for estimating the coronary blood flow rate and the coronary branch flow distribution method can be applied to calculate the value of clinical noninvasive FFR_CT. Further in-depth research with large cohorts is needed to modify, verify and clarify the application of the proposed models in the calculation of FFR_CT, and an in-depth comparison of the proposed methodology to other methodologies in the literature is needed.

Acknowledgement

This work was supported by National Natural Science Foundation of China (11772015, 11832003).

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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