Application of artificial intelligence in mathematical modeling of coronary blood flow

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Introduction

Cardiovascular disease (CVD) is the leading cause of death and disability worldwide. In 2021 alone, CVD accounted for more than 20 million deaths, representing approximately one-third of all deaths worldwide [1].

The age-standardized mortality rate from cardiovascular disease varies from country to country depending on the overall prevalence of cardiovascular disease in the region, which is determined by the presence of risk factors in the population, such as obesity, physical inactivity, hyperlipidemia, hypertension, diabetes mellitus, chronic kidney disease, and environmental risks [2].

According to Rosstat, in 2022, 831,557 people died from CVD (43.8 % of the total mortality rate). This category includes coronary artery disease (CAD), cerebrovascular diseases, and acute cerebrovascular accidents. According to data from the Perm Territorial Office of the Federal State Statistics Service, mortality in the Perm Territory due to diseases of the circulatory system in 2022 amounted to 16,525 people. In total, 35,018 people died from various causes in the Perm Territory. Consequently, mortality due to diseases of the circulatory system accounts for about 47 % of the total mortality rate.

An important factor influencing the mortality rate from cardiovascular diseases is the diagnostic and therapeutic strategies used to treat CAD. Close attention to this problem and the allocation of resources to this area over the past 25 years have led to a decrease in the mortality rate from CVD in countries with a high socio-demographic index [3; 4]. Accurate diagnosis is the first step in choosing the appropriate treatment method [5].

The diagnostic tools used to diagnose CAD are divided into presumptive and definitive.

The former includes primary diagnostic criteria such as clinical symptoms, electrocardiogram (ECG) changes, and biochemical markers that can be used to quickly suspect acute myocardial infarction (AMI). Precise criteria are more important for choosing a therapeutic approach, including invasive coronary angiography (CAG), computed tomography (CT) of the heart, positron emission tomography (PET), hybrid PET/CT, and conventional single photon emission computed tomography (SPECT) [6; 7].

The most accurate diagnostic indicator for determining the need for revascularization in hemodynamically significant coronary artery lesions is the fractional reserve of coronary blood flow (CFR), which is defined as the difference in pressure in the coronary artery between the proximal and distal segments in the presence of significant stenotic lesions under conditions of pharmacological vasodilation [8]. CFR is traditionally calculated using a guidewire catheter during invasive coronary angiography, but it can subsequently be assessed using invasive coronary angiography with fractional flow reserve (FFR) determination without the need for a guidewire catheter [9]. Given the invasive nature of FFR, new methods have been developed to assess the discrepancy between the anatomical and functional status of significant coronary lesions, such as a non-invasive method for assessing coronary blood flow reserve using CT (CT FFR) [10].

Traditional Invasive Coronary Angiography and Percutaneous Coronary Intervention

Selective CAG was first used by Souns in 1958. Since then, CAG has revolutionized our understanding of the pathophysiology and treatment of heart disease and has become one of the major achievements in cardiology. It is used as the standard and accepted method for diagnosing CAD [3; 11]. Changes were made to the original CAG technique based on growing numbers of observations by doctors, including the acquisition of digital images, a reduction in the volume of contrast medium and the number of catheters used, and the development of quantitative measurements, which made it possible to obtain more accurate images, reduce the number of complications, increase the percentage of successful operations, and shorten the duration of the procedure [11; 12]. This procedure is one of the most common invasive procedures in the world. More than 634,000 examinations were performed in Russia in 2023, resulting in 324,000 percutaneous coronary interventions (PCI) and more than 29,000 aortocoronary interventions.

CAG is performed using an X-ray angiography system, which includes an X-ray source and an image detector mounted on a movable C-arm. The C-arm for imaging can rotate around the object located on the moving patient table. The figure shows the position of the C-arm in neutral (vertical) and anterior-posterior (AP) orientation when the X-ray source is located under the patient table. X-ray imaging is performed by a trained medical professional who uses a catheter to inject a contrast agent into the arteries and positions the equipment to obtain optimal two-dimensional projections of the vascular system. The C-arm can rotate around the isocenter using two angles of rotation: primary (α) and secondary (β). The angle α varies from +180° (left anterior oblique, LAO) to -180° (right anterior oblique, RAO), and the angle β varies from +90° (cranial, CRA) to -90° (caudal, CAU). It is possible to change the distance between the patient and the X-ray source (SOD) and the distance between the source and the detector (SID). All these parameters, together with the images obtained, are saved in a DICOM file, which allows data to be systematically recorded during X-ray imaging.

 

Fig. Parameters of the X-ray angiography system

 

Another important advantage of CAG is the possibility of simultaneous PCI, which has a high success rate (over 95%) [13]. PCI has also shown greater effectiveness in treating patients with chronic total occlusion compared to drug therapy [14; 15]. Early ICD implantation also reduced overall and cardiac mortality in patients with acute coronary disease [16]. A comparison of PCI with coronary artery bypass grafting (CABG) showed that PCI (using drug-eluting stents) is as safe and effective as CABG in patients with left main coronary artery disease with low surgical risk and a lower frequency of repeat revascularization required after CABG [17; 18]. However, CABG is considered the method of choice in patients with multivessel coronary artery disease, with higher survival rates, lower rates of major cardiovascular events (MCE), and repeat revascularization than with PCI [19; 20]. In addition, the success of revascularization in patients using PCI depends on several factors, such as the type and severity of CAD, as well as the characteristics of the technique, instruments, and drugs used during and after PCI [21].

Patient-specific computational models of blood flow in coronary arteries have become fundamental as a clinical technology in recent years, contributing to improved diagnosis and treatment of cardiovascular disease. These models allow clinicians to more accurately assess hemodynamic parameters and the significance of coronary stenosis, which facilitates informed decisions about the need for interventions such as stenting. Evidence of their clinical effectiveness is supported by a significant number of studies demonstrating a high degree of agreement with invasive measurement methods such as catheterization.

Such models have advantages in terms of safety, as they do not require invasive procedures, which reduces risks for patients. The effectiveness and economic feasibility of introducing these technologies into clinical practice have been confirmed by a number of studies, which have shown a reduction in treatment costs and improved clinical outcomes. Thus, the use of patient-specific computational models of blood flow in the coronary arteries is a promising direction in cardiology, contributing to more accurate and safer diagnosis and treatment of cardiovascular diseases.

The process of patient-specific modeling of coronary blood flow based on angiographic data can be divided into several key stages.

The first stage involves segmenting the coronary arteries on angiographic images. Various methods are used for this purpose, such as intensity thresholding, the active contour method, and modern algorithms based on machine learning. Combining these methods significantly improves segmentation accuracy and minimizes the impact of noise and artifacts.

It is critically important to note that correct segmentation serves as the basis for subsequent modeling stages, since any errors at this stage can lead to data distortion and, consequently, negatively affect the accuracy of hemodynamic parameter calculations [22].

The next important step is to reconstruct the three-dimensional geometry of the coronary tree. Three-dimensional geometry plays a key role in creating a realistic blood flow model, as it allows for complex anatomical and topographical features, including bends, bifurcations, and stenoses, to be taken into account.

Creating such a three-dimensional model is greatly simplified when using three-dimensional medical imaging data, such as computed tomography angiography (CTA). However, for two-dimensional X-ray angiography, it is necessary to apply methods for reconstructing three-dimensional geometry from a series of images.

There are various approaches to reconstructing coronary arteries from angiographic images based on the principles of projective geometry and stereoscopic vision. These methods can be divided into two categories: “bottom-up” and “top-down.” Bottom-up methods are based on epipolar constraints, which assign each point on the first projection a corresponding point on the second projection. Once such a pair of points has been established, the position of the three-dimensional point can be uniquely reconstructed [23].

Errors in angiographic angle measurements can complicate the process of matching points on projections. Using an uncalibrated angiograph can distort the resulting geometry.

In some studies, the device was calibrated using phantom objects. In studies where uncalibrated geometry was used, an additional calibration step is performed before reconstruction [23].

However, matching points on projections can be difficult due to errors in measuring angiographic angles. Calibrating the system using phantoms can help minimize distortions. A number of studies have used additional calibration steps to improve the accuracy of reconstruction based on uncalibrated geometry [22].

“Top-down” methods for coronary artery reconstruction use a three-dimensional model of the coronary tree, whose projections are adapted to the image of the vessels on two-dimensional X-ray images.

The deformable three-dimensional model develops under the influence of external energy, which is calculated from the discrepancy between the projection of the deformable model and the 2D X-ray image of the coronary artery, and internal energy, which is determined by the smoothness and topology of the deformable model itself. The main difficulty in reconstructing the geometry of coronary vessels based on active contours is the design of external and internal energy conditions [22].

The projection methods described require the identification of relevant anatomical features in images. Such actions are usually performed manually, and their automation is a complex task [22; 23]. The accuracy of the reconstruction will also depend on the accuracy of the angiographic angle measurements. However, the main problem is intersecting branches and overlaps, which can lead to uncertainty in the resulting geometry. Movements of the coronary arteries caused by cardiac and respiratory contractions also create difficulties in establishing correspondences between segmented two-dimensional images [22].

In recent years, the use of neural networks for coronary artery reconstruction has become a hot topic in medical imaging. In particular, work [27] represents an important contribution to this field. The study describes a multi-stage approach to training a neural network for 3D coronary tree reconstruction. Data for training the neural network is generated using a synthetic vessel generator. The generator uses a reference coronary tree geometry obtained from computed tomography. The generator works by creating synthetic coronary tree geometry from the reference geometry by introducing vessel geometry. To simulate pathological changes, stenoses with varying degrees of narrowing are randomly introduced. Unlike projection methods, this approach does not require precise shooting angles or equipment calibration.

However, one of the main limitations of these studies is the lack of consideration of vessel movement between images, which is often observed in single-plane X-ray systems. This movement can significantly affect the accuracy and reliability of vascular geometry reconstruction, especially when interpreting temporal changes associated with physiological processes. In work [28], an approach using rigid transformation is proposed to simulate real non-simultaneous projections in synthetic vessel geometry. This method allows for the creation of a more accurate model of the coronary tree, taking into account changes in vessel geometry caused by heart and respiratory movements.

Although the approach using a neural network to reconstruct three-dimensional vessel geometry from angiographic images still requires further optimization to achieve high accuracy, it retains the potential for fully automated reconstruction of coronary arteries without the need for manual routine operations, which is a significant advantage over classical projection methods.

Invasive Assessment of Fractional Blood Flow Reserve

Assessment of coronary blood flow is of great importance during PCI. Initially, intracoronary pressure measurements were used to determine pressure gradients and evaluate the results of angioplasty. The advent of intracoronary pressure measurement catheters and fractional flow reserve (FFR) assessment made it possible to determine the effect of epicardial stenosis on myocardial perfusion [26]. In clinical trials, FFR measurement has optimized patient selection for revascularization and improved clinical outcomes at a lower economic cost [28–30].

Despite compelling evidence and recommendations, the use of pressure-based coronary flow assessment remains low in many regions of the world. The clinical application of coronary flow data in the US and Europe ranges from 10 to 20 % [30]. Factors limiting the use of these data include the high cost and limited effectiveness of modern pressure transducers, the need for a hypertensive stimulus, the increased duration of the procedure, and the presumed unnecessary nature of physiological studies [31].

Methods for quantitative assessment of coronary artery stenosis severity using angiography have undergone changes. Historically, a reduction in lumen diameter of more than 50 % compared to a healthy reference diameter has been used to define significant stenosis requiring treatment. However, inter-expert agreement in the assessment of coronary artery stenosis severity is low, ranging from 40 % to 70 % [32]. Furthermore, only half of intermediate stenoses are associated with ischemia, and PCI does not improve outcomes in such cases. The use of angiography to assess blood flow preceded quantitative coronary angiography (QCA) and FFR. In 1970, Rutishauser et al. first measured absolute coronary blood flow in humans using cine densitometry [33]. In 1984, Vogel and his colleagues used densitometry to determine the average transit time of contrast medium at rest and during hypertension to determine coronary blood flow reserve (CFR). Three decades later, thanks in part to advances in the field, including three-dimensional (3D) quantitative computed tomography and computational fluid dynamics (CFD), a surrogate measure of flow reserve (FFR) obtained through angiography was proposed. In 2014, two publications by Papafaklis and Tu described an approach to virtual FRV (?) assessment using only conventional angiography and without the use of a pressure catheter [34–36]. This milestone marked the emergence of an alternative to the use of a pressure catheter for FRV assessment. Over the past decade, there has been a significant increase in the number of systems capable of assessing FRV directly from angiography results.

Noninvasive Assessment of Fractional Blood Flow Reserve

The use of three-dimensional modeling of coronary blood flow improves visualization and understanding of coronary artery pathology, which is important for clinical decision-making. Hemodynamic calculations can not only assist in planning interventions such as angioplasty or stenting, but also contribute to a more accurate assessment of patient risks [37].

The integration of computational modeling with clinical data enhances our understanding of the complex dynamics occurring within the coronary arteries, which has a positive impact on improving clinical assessment metrics. Among the key hemodynamic indicators, several play an important role in analyzing the condition of coronary reflections.

Fractional flow reserve (FFR) is one of the main parameters for assessing coronary artery hemodynamics. FFR is calculated by comparing blood pressure before and after stenosis. Using this ratio, doctors can determine how much the narrowing impedes blood flow to the heart. FFR can provide valuable information about whether coronary stenosis requires intervention, such as angioplasty or stenting, or whether it can be treated with medication [38].

Coronary flow reserve (CFR) is an important indicator calculated as the ratio of coronary blood flow during maximum vasodilation to blood flow at rest [39]. It takes into account vascular resistance and the characteristics of coronary artery dilation, which helps in assessing their functionality. By comparing the CFR of a stenosed coronary artery with a reference segment, such as a non-stenosed segment or a segment with minimal disease, the relative CFR can be calculated. It is particularly useful in assessing functional consequences by comparing blood flow through a specific stenosis with blood flow in another, less affected artery.

Results and Discussion

Modern medicine places high demands on both the qualifications of doctors and the methods used. The total amount of information about diseases increases every year, and one person is unable to accurately assess the importance of the available material for medical practice, which is where mathematics comes to the rescue, helping to structure the material. The choice of mathematical models for describing and researching medical objects depends on the individual knowledge of the specialist and the specifics of the tasks being solved. Numerical methods are not only the most accurate, but also allow for the correct construction of computational algorithms, which is important in the diagnosis of diseases.

In the early stages of developing mathematical models that qualitatively determine the relationship between blood flow and pressure, researchers had to resort to significant simplifications of the Navier–Stokes equations relating to incompressible flows. In general, mathematical models of blood circulation can be classified into three categories: models with concentrated parameters (0D models), models of arterial hemodynamics (1D), and three-dimensional numerical models.

Three-dimensional (3D) models use numerical methods to solve Navier–Stokes equations. This approach allows for accurate representation of the geometry of the vascular system, simulation of three-dimensional pulsatile flow (including turbulence components), and implementation of complex models of blood behavior and vessel properties.

One-dimensional (1D) models are created by averaging the Navier–Stokes equations across the cross-section of the vessel. This leads to the neglect of non-axial velocity components, the assumption of an axial velocity profile in vessels, and the condition of constant pressure across the cross-section. However, these models have limitations near bifurcations, side branches, or affected areas, especially for complex lesions in the area of branches and bifurcations.

The 0D hemodynamics model is a simplified approach to describing blood circulation. This method is used to analyze the dynamics of pressure and blood flow in the vascular system, ignoring the spatial characteristics of the system. The flow in such a model is assumed to be constant, axisymmetric, and unidirectional, where vessel segments are considered as circular cylinders, which leads to significant simplifications in the analysis of fluid resistance. This often creates a high level of inaccuracy in modeling blood flow in affected coronary arteries, where stable unidirectional flow, axisymmetry, or circular vessel cross-sections are absent.

Hybrid models aim to optimize computation time while ensuring accurate simulation results by combining different approaches, including parameterized models and one-dimensional models, as well as combinations of one-dimensional and three-dimensional models.

The ultimate goal of the proposed methodology is to create a digital twin, i.e., a virtual copy of a specific patient's coronary artery affected by atherosclerosis, which would allow for a more thorough analysis of the patient's clinical condition and analysis of possible and most optimal treatment, as well as suggest further adaptation of the developed copy over time during subsequent examinations.

The only data required as input for the developed software are X-ray angiographic images of the patient in question. Subsequently, the software will enable the physician to perform virtual stenting manually (by marking the desired location for the stent and selecting the stent diameter) or perform automated optimized virtual stenting, which will provide information about the most suitable location and diameter of the stent to be installed, as well as analyze several options for stent placement and expansion diameter to determine the most optimal option. During the optimization process, not only FFR should be taken into account as the sole parameter, as is usually the case, but also hemodynamic data in the narrowed and dilated sections of the artery (using WSS (wall shear stress) analysis)). Therefore, a computational approach may also be useful, as it allows for computer simulation of stent implantation. Methods for implanting stents into the coronary arteries of a specific patient have been described in the literature [40–43]. The main advantage of this type of computer modeling is that it allows analysis of the impact of stenting on vFFR recalculation after intervention and prediction of hemodynamic effects and treatment benefits in a specific coronary artery, enabling more thorough treatment planning. It would be useful for doctors to have a “virtual stenting” tool that would allow them to study the effects of several alternative intervention strategies in “simulation” mode (using computer modeling) and obtain an assessment of the best and most effective stent placement before performing clinical treatment in vivo.

Conclusions

The practical application of mathematical models will enable the reconstruction of 3D images of coronary arteries and blood flow modeling, which will optimize CAD treatment. This will allow for effective planning of endovascular interventions based on patient data in the absence of the patient. Further study of this issue promises enormous prospects for the development of mathematical modeling of coronary blood flow and effective decision-making during interventional procedures, which will potentially reduce morbidity and mortality from cardiovascular diseases.

About the authors

A. A. Porodikov

S.G. Sukhanov Federal Center for Cardiovascular Surgery

Email: faridun.azimov.98@list.ru
ORCID iD: 0000-0003-3624-3226

PhD (Medicine), Cardiovascular Surgeon

Russian Federation, Perm

A. N. Biyanov

S.G. Sukhanov Federal Center for Cardiovascular Surgery

Email: faridun.azimov.98@list.ru
ORCID iD: 0000-0002-9314-3558

PhD (Medicine), Pediatric Cardiologist

Russian Federation, Perm

V. B. Arutyunyan

S.G. Sukhanov Federal Center for Cardiovascular Surgery

Email: faridun.azimov.98@list.ru
ORCID iD: 0000-0002-1730-9050

PhD (Medicine), Cardiovascular Surgeon

Russian Federation, Perm

F. F. Azimov

S.G. Sukhanov Federal Center for Cardiovascular Surgery

Author for correspondence.
Email: faridun.azimov.98@list.ru
ORCID iD: 0009-0006-3286-6951

Medical Intern

Russian Federation, Perm

M. A. Barulina

Perm State National Research University

Email: faridun.azimov.98@list.ru
ORCID iD: 0000-0003-3867-648X

DSc (Physics and Mathematics), Director of the Institute of Physics and Mathematics

Russian Federation, Perm

Ya. N. Ivanov

Perm State National Research University

Email: faridun.azimov.98@list.ru
ORCID iD: 0000-0003-3974-9011

Master of Physics and Mathematics Institute

Russian Federation, Perm

References

Supplementary files