It is extremely difficult to develop a proper numerical model of coronary vessels that are absolutely consistent with the clinical data. This is predominantly due to the complex geometry, vessel compliance, pulsatile flow, variable vascular resistance in the cycle, and non-Newtonian properties of blood. Other issues are the lack of detailed clinical hemodynamic data and the diverse physiologies of the circulatory system for each patient. Nevertheless, complex parameters can be analysed in a much broader scope due to the recent developments of diagnostic methods based on medical images, computer-based methods, and improved efficiency of numerical calculations. Thus, it is now possible to perform CFD analysis of the blood flow with variable flow conditions of parameterised models of complex circulatory systems in a short time.
Flow rate
During the simulation, the inlet flow changes for both constant and pulsatile flow. In the case of a constant flow, the flow rate increases linearly. In the case of a pulsed flow, the change in the flow rate was caused by a change in the amplitude of the pulse. It should also be noted that the volumetric flow rate (cardiac output) increases as a result of an increase in heart rate, but this does not affect the flow rate during one cardiac cycle. The stroke volume in our study is constant.
According to the Poisseuille’s formula, flow rate in the vessel is proportional to the fourth power of its radius. Therefore, very subtle changes in vascular diameter have a profound impact on blood flow and the autoregulation is more effective in vessels without atherosclerosis, with more flexible walls [
22]. Without taking into account autoregulation in numerical simulation, for the vessel without stenosis, the increase in the flow rate causes a linear increase in the pressure gradient.
However, the flow disturbance induced by stenosis causes a deviation from this law. The appearance of the vessel narrowing causes an additional pressure drop and the pressure gradient flow-rate relationship can be described by means of a quadratic polynomial [
6]. In the case of vessel with stenosis greater than 90%, an increase in the pressure gradient takes place much faster and therefore, this relationship is better described by means of higher order polynomial.
However, the increase of flow rate cause a decrease in pressure drop coefficient. For higher flow rate, the CDP becomes practically independent of the flow rate. The point at which this occurs depends on the degree of stenosis and is pronounced only at higher stenosis values, so it can be a parameter facilitating the determination of significant stenosis.
It should be noted that increase in flow rate or in amplitude of pulsatile flow rate results in increases in the AWSS values. For this reason, the region of low AWSS values is much more pronounced and the analysis of the places, where the atherosclerosis plaque progression may occur, is more precise. That is why, it seems that, the assessment of the risk of stenosis progression and their hemodynamic significance should be performed at high flow rates or high amplitudes of pulsation.
Heart rate
The unsteady flow phenomena play a major role in the blood flow through coronary arteries. In general, the pulsatile flow is characterized by complex spatial and temporal velocity distributions and reversed flow regions.
The increase in heart rate increases the number of cardiac cycles and thus increases of cardiac output [
22]. However, it should be noted, that practically the heart stroke volume is decreased at higher heart rate. Therefore heart rate remains the most practical indicator of the heart’s work and the functions of cardiovascular system during exercise stress test.
In our virtual cardiac test, an increase of the heart rate frequency causes an increase of the cardiac output, but it does not cause any increase in the stroke volume (Fig.
1). Therefore, the values of OSI and RRT are influenced by inertial forces resulting from the shortening of the cardiac cycle and the increasing in both blood acceleration and temporal fluctuation of the WSS.
Low-density cholesterol and blood pressure may contribute to the formation and development of atherosclerosis. Probably, heart rate (HR) also can affect the low values of WSS and high values of OSI [
22]. The highest values of OSI are observed for both high heart rate frequency and high amplitude of pulsatile flow rate in contrast to RRT. However, it should be noted that the value of OSI is more sensitive to the heart rate frequency and amplitude than RRT.
The spatial average OSI increase when increase in HR and reach the maximum value of 0.5 at 60% narrowing, practically independently of HR.
An increase in HR increases only the area of maximal OSI values in all branches of the coronary vessel. This parameter for high HR shows the places of a possible narrowing. However, it is difficult to determine the significance of the already existing narrowing only on the basis of OSI.
In contrast, RRT is independent of HR for low degree of narrowing. Above 60% of the degree of stenosis, there is a clear dependence on HR, which is very complex and difficult to clearly define.
There was no impact of HR on the surface- and time-averaged FFR
CT as ratio of average distal and proximal pressures, [
7,
23]. However, there are some limitations in clinical use of indices invasively determined. Variability of hemodynamic factors such as heart rate, blood pressure, cardiac contractility, vascular compliance and the measurement itself may affect the measurement result.
Additionally, it should be noted that the numerical determination FFRCT has also limitations.
The most important one is low image quality of CCTA and inaccurate fluid dynamics model for individual patients. In our study we have assumed of stroke volume independent of HR and lack of autoregulation.
An increased shear stress rate and turbulence at higher HR may contribute to a more frequent rupture of the atherosclerotic plaque. However, the mechanism of impact of HR on plague rupture is not yet fully understood [
24].
Stenosis degree
At a certain values of flow rate and stenosis degree, the flow is turbulent with distinct separation zones both upstream and downstream of stenosis. The transition from laminar to transient or turbulent flow causes an additional increase in the flow resistance, which is an unfavourable phenomenon that reduces the volumetric flow.
The significance of the stenosis may be assessed by comparison of the parameters directly related to the flow turbulence in the narrowed coronary vessel in a specific patient with those of modelled flow in a vessel without atherosclerosis.
Generally, an increase in degree of vessel narrowing increases in pressure drop and pressure drop coefficient. The surface average WSS is constant, independent from degree of stenosis.
This result is consistent with Murray’s law [
25], according to which the WSS is constant, proportional to the velocity gradient and independent of the vessel radius. However, the flow disturbance induced by stenosis causes a deviation from this law, as a result of which the average value of WSS clearly depends on the degree of narrowing and active radius of the vessel. Above a certain value of stenosis, depending on the flow rate, there is a significant increase of WSS values (Fig.
4b).
The OSI is equal to zero for laminar flow in one direction. The maximum OSI value is 0.5 for purely oscillatory flow, without a resultant flow in a specific direction. The significance of low WSS stresses and a high OSI has been confirmed in the arteriosclerotic process [
26]. However, the OSI does not provide a clear result if there are disturbances in the flow velocity. The index may be the same for both large velocity changes (strongly oscillatory flow) and small changes.
Therefore, the relative time of residence was introduced as a function of OSI and WSS averaged over time (AWSS). The analysis of this parameter also takes into account the impact of areas with low WSS index and high OSI [
9] in order to more complete determine areas at risk of atherosclerosis.
Increasing the degree of stenosis results in an increase in mean and maximum OSI and RRT values. Maximum OSI value of 0.5 is achieved in a particular value of stenosis degree (~ 60%).
The local wall areas of maximum values of OSI and RRT, also increases. The areas of low WSS and high OSI and RRT occur downstream to the stenosis, but a “creeping” effect of these areas was observed which the size and location strongly depend on the degree of stenosis and artery geometry.
An axially non-symmetrical stenosis arranged on the internal wall of a bend or near a bifurcation introduces additional disturbances to the flow in the plane perpendicular to the axis of the vessel. Poisseuille’s formula implies that a 10% reduction in the radius of a vessel increases its resistance by 52%. In practice, however, the resistance increase is much lower due to interaction of autoregulatory mechanisms and the complex arrangement of vessels with parallel and serial connections.
Studies [
7] are focused on the effect of fluctuations in HR and area stenosis on diagnostic indexes in a porcine model. They pointed out that, according to our study, FFR can be used for diagnose stenosis severity due to its independence from heart rate.
A clear impact on the FFRCT stenosis occurs at severe degrees of stenosis. In the low degree of stenosis in the range of 10 to 60%, time and surface averaged FFRCT changes by about 10% only. Therefore, so averaged FFRCT seems not sensitive enough for the assessment of hemodynamic significance of stenosis in the 10 to 60% range. Additionally, a decrease in FFRCT due to an increase in the degree of stenosis in one branch causes a slight increase in FFR in other branches, therefore the averaged FFR value at low values of constrictions is practically constant.
Even small changes in geometry can induce significant alterations of WSS [
9,
20,
26]. Many articles supports the low or oscillatory shear theory. According to Kleinstreuer [
27], areas with high time-averaged zero-dimensional WSS (> 1.5) as well as areas with low individual values (< 0.5) increase the probability of arteriosclerosis. Similarly, the probability of adhesion of pro-atherogenic participles and leukocytes to the endothelium may be higher in the event of a longer time of residence of blood near the arterial wall, a high OSI value [
10], AWSS value lower than 0.4 Pa [
20], and RRT above 10 Pa [
28].
Authors [
29] concluded that OSI and the relative residence time (RRT) could explain changes in plaque distribution as a result of changes in flow conditions and OSI-based risk factor was a better predictor than an AWSS-based one of subsequent stenosis.
Blood flow and disease localization was assessed at different stages in the disease process.
However, the conducted literature review revealed no reports concerning the impact of variable hemodynamic conditions (SV, CO, HR) on clinically indicated coronary index values. The current, most of the research present only the determination some hemodynamic parameters in a wide range of stenoses [
30].
Blood flow and disease localization was assessed at different stages in the disease process. However, the conducted literature review revealed no reports concerning the impact of variable hemodynamic conditions (SV, CO, HR) on clinically indicated coronary indices values. The current, only a part of research present the numerical determination some hemodynamic parameters in a wide range of stenoses [
30].
In clinical trials, measurements of coronary indices are made in anatomic and hemodynamic conditions specific for the patient, at the time of the medical examination. In turn, numerical simulations are based on patient-specific geometries using incomplete clinical data [
5]. Therefore, our virtual numerical tests is carried out in variable hemodynamic conditions and in a wide range of stenosis, with the possibility of a task of varying initial/boundary conditions and with the possibility of numerical reconstruction of the geometry. In the case of a normal coronary vessel, without a narrowing, it is possible to model of the stenosis in places where the coronary hemodynamic indices reach the critical values. Inversely, in the case of stenotic coronary artery, on the basis 3D surface approximation of coronary wall, we can reconstruct the normal vessel without stenosis. Comparison of the results obtained from these two models (with and without stenosis) may allow for more accurate assessment of the significance and progression of stenosis.
The curves of relationship between flow condition and hemodynamic indices, contains the inflection points and/or local extrema reflecting the nature of the flow changes due to changes of vessel geometry and hemodynamic conditions. These characteristic points can help to assess the relevance of existing stenosis as well as allow you to determine the probability of atherosclerosis progression and location of places particularly exposed to the growth of atherosclerotic plaques in normal vessels. The same degree of stenosis may have different clinical significance. The hemodynamic significance of the coronary stenosis depend not only with severity of the stenosis but also anatomy of the coronary artery. In addition, stenosis in one branch has also effects on the hemodynamic parameters in side branches. For this reason, to non-invasive assessment of hemodynamic significance of coronary stenosis the diagnostic indexes should be studied under variable hemodynamic conditions, taking into account the impact of heart rate, cardiac output and degree stenosis on extreme values of hemodynamic indices, both near stenosis (locally) and in side branches (globally).