Method for calculating pressure drop between cross sections of artery

Abstract

The present invention is a computer-implemented method for calculating a pressure drop between the cross-sections of the aorta, coronary, carotid, renal or peripheral arteries. The method includes steady-state computational fluid dynamics (CFD) simulation and separates the pressure drop into components: steady-state pressure loss and transient pressure loss. The steady-state pressure loss is represented as a linear combination of pressures (zero flow, viscous resistance and local resistance), and the transient pressure loss is calculated according to a second law of motion. In an embodiment, the steady-state pressure loss includes similar invariants (Euler number, Lagrange number, and Reynolds number). The present invention provides for rapid computer simulation assessment of human heart conditions, including fractional flow reserve, and is positively assessed using clinical trials.

Description

Field of the Invention

[0001] The present invention belongs to the field of medical diagnosis. The present invention is a computer-implemented method for in silico evaluation of human heart conditions. Background of the Invention

[0003] Although, in essence, every fluid flow is a transient flow, this aspect is actually rarely considered. Most computational problems associated with the use of fluid mechanics utilize steady flow methods, which provide sufficient information about the evolution of steady flows. Over several decades of research, only a small fraction of practical problems have required analysis of the overall complexity of flow evolution over time. These practical problems include hydraulic shock (water hammer, fluid hammer), flow measurement, and control devices used in said measurement, which include devices that use fluids as a medium for transmitting forces (e.g., an anti-lock braking system (ABS)), tides, and hydraulic waves. These practical problems exist in many technical fields, including aerospace and automotive. Two hydraulic engineers - Samuel Grace (e.g., in Grace SF (1928) Oscillatory motion of viscous liquid in a long straight tube, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 5(31):933-939) and Johan also studied the overall complexity of evolution over time at a more fundamental level. In 1949, an article was published ( JC (1949) Resistance and inertia of the flow of liquids in a tube or open channel, Flow Turbulence and Combustion, 1(1):169-197), in which he studied several types of motion of laminar flow and the general slow motion in several cases selected by the author. According to the article, the relationship between the total flow rate and the pressure drop can be described by the impedance of the channel or tube. Interestingly, in the early 20th century, cardiology was another application field where, due to the development of catheterization, it was necessary to consider the overall complexity of flow evolution over time.

[0004] The history of catheterization (including cardiac catheterization) is much longer and can be traced back four hundred years. William Harvey in 1628 and Stephen Hales 100 years later made significant progress in describing blood circulation and measuring arterial pressure respectively. The 19th century was the golden age of the development of cardiovascular physiology, and the most prominent contributions came from Claude Bernard, and Carl Ludwig. Human cardiac catheterization began in the 20th century, and Werner Forssman was the first to successfully perform cardiac catheterization on a human. Werner Forssman performed this procedure on himself in 1929. In the late 1940s, André Cournand and Dickinson Richards introduced diagnostic cardiac catheterization, and in the 1960s Mason Sones described selective coronary angiography. In the 1970s, Andreas Gruentzig initiated transcatheter interventional therapy, which led to the expansion and refinement of the main techniques. Today, the Sones technique is rarely used, and percutaneous coronary intervention and coronary angiography mainly consist of methods including percutaneous radial artery and percutaneous femoral artery (Bourassa MG (2005), The history of cardiac catheterization, The Canadian Journal of Cardiology, 21(12): 1011 - 1014).

[0005] The dissemination of knowledge about the benefits and risks associated with cardiac catheterization for the assessment of insufficiency occurred in the late 1960s and 1970s. This attracted the attention of hydraulic engineers to these techniques. In 1973, the results of two pioneering works by Donald Young and Frank Tsai were published.

[0006] The starting point for establishing the unsteady (transient) flow characteristics of arterial stenosis is the results of the theoretical analysis described in the article:

[0007]

[0008] where: Δp is the pressure drop, L is the distance, η is the fluid viscosity coefficient, D is the pipe diameter, ρ is the fluid density, v is the mean instantaneous velocity, and dv / dt is the rate of change of the mean velocity. The flow characteristics of a fluid have a well - defined meaning known to those skilled in the art and may include any one or all of velocity, pressure change, Reynolds number, and other parameters.

[0009] The first part on the right - hand side of Equation (1) above describes the pressure drop due to viscous effects in a steady flow, and the second part on the right - hand side is about the additional pressure required to accelerate the fluid. It is pointed out in the overview of this article that the value of the high - frequency inertia (H) is less than the value of the low - frequency inertia (L), and for laminar flow, the difference between them is: 20% in a wide open channel or slit, 38% in a rectangular tube, and 33% in a circular tube, and for turbulent flow, the difference between them is: 0.25% - 1.5% in a rough or smooth open channel, 0.5% - 3.5% in a slit or rough open channel, 0.5% - 3% in a smooth circular tube, and 1% - 8% in a rough circular tube ( (1949)).

[0010] Young and Tsai published another article in 1973 (Young DF, Tsai FY (1973) Flow characteristics in models of arterial stenosis. II. Unsteady flow, Journal of Biomechanics, 6(5):547 - 559), in which an approximate equation for the transient pressure drop across a stenosis was derived and experimentally verified. Experiments were conducted on tubes, and the researchers studied the effects of stenosis and instability in steady flow by using asymmetric and axisymmetric models. These two researchers also used hot - films to study turbulence and its development. Young and Tsai demonstrated in this article that: (i) for severely constricted models, steady flow is slightly more stable than oscillatory flow, (ii) for mildly constricted models, steady flow is less stable than the corresponding oscillatory flow, and (iii) three (3) dimensionless parameters can characterize the unsteady flow through a stenosis. These two researchers referred to the equations of Fry et al. (1956, 1959) and Daily et al. (1956). The study by Young and Tsai resulted in a pressure drop of the following form:

[0011]

[0012] where: K v is the viscous coefficient, η is the viscosity coefficient, D is the tube diameter, v is the cross - sectional mean instantaneous velocity in an unobstructed tube, K tis the turbulence coefficient, A0 is the cross-sectional area of the pipe without obstacles, and A1 is the minimum cross-sectional area of the constriction. is the fluid density, K u is the inertia coefficient, L is the distance between the pressure taps, dv / dt is the rate of change of the average velocity, and |v|v is the vector length value multiplied by v. The instantaneous average velocity of a steady flow can be written as So now the above equation can be written in dimensionless form:

[0013]

[0014] where:

[0015]

[0016] where: is the fluid density, v p is the peak value of the cross-sectional average velocity (v = v s + v n ), v s is the steady component of the cross-sectional average velocity, v n is the amplitude of the periodic component of the cross-sectional average velocity, K v is the viscosity coefficient, Re p is the peak Reynolds number is the velocity ratio (v / v p ), v is the instantaneous cross-sectional average velocity in the unobstructed pipe, K t is the turbulence coefficient, A0 is the cross-sectional area of the pipe without obstacles, and A1 is the minimum cross-sectional area of the constriction, K u is the inertia coefficient, L is the distance between the pressure taps, τ is the characteristic time, df / dt is the function varying with time, I v is the viscosity effect index, I t is the turbulence effect index, I u is the inertia effect index. is the vector length value multiplied by

[0017] In 1975, Young et al. (Young DF, Cholvin NR, Roth AC (1975) Pressure drop across artificially induced stenoses in the femoral arteries of dogs, Circulation Research, 36(6):735-743) described the in vivo pressure losses measured in artificially constricted arteries and tested the use of equation (2) for predicting pressure drop. The description disclosed experiments conducted on 13 dogs, which included inserting implants into their femoral arteries. Flow and pressure drop across the stenoses (severity 52.3%-92.2%) were measured, and the average pressure loss was between 2 mmHg and 30 mmHg, with peaks between 9 mmHg and 53 mmHg. The article also mentioned that Young et al. were able to predict pressure losses within about 20% of the experimental values.

[0018] The development of catheterization combined with vascular imaging (especially angiography) quickly provided a new context for the results of Young and Tsai. However, in the 1970s, there was no knowledge of which parameter (index, metric, etc.) should be used as the standard for functional assessment.

[0019] A 1977 scientific article by Young et al. (Young DF et al. (1977) Hemodynamics of arterial stenoses at elevated flow rates, Circulation Research, 41(1):99-107) focused directly on critical stenosis, which is the level of stenosis beyond which blood flow decreases. The article described experiments conducted on 12 large hybrid dogs, in which stenoses in the carotid and femoral arteries were induced in the range of 55.7%-91.0% reduction. In this experiment, the researchers used hollow cylindrical plugs and measured the pressure drop and instantaneous flow velocity, with average velocities between 3.9 cm / s and 88.88 cm / s. A useful metric for describing the impact of stenosis is the ratio of the maximum flow velocity in the stenosed vessel to the maximum flow velocity without stenosis By integrating equation (2) over a flow cycle, Young et al. proposed the following equation:

[0020]

[0021] where: K v is the viscous coefficient, η is the viscosity coefficient, D is the pipe diameter, K tis the turbulence coefficient, A0 is the cross-sectional area of the tube without obstacles, A1 is the minimum cross-sectional area of the stenosis, is the fluid density, is the mean instantaneous velocity averaged over time within the period. Young et al. noted that under large pressure losses from the distal end to the stenosis and under vasodilated conditions, the blood flow through a specific artery could be increased significantly by 4 to 5 times (in Young et al. (1977)).

[0022] Itzchak et al. (Itzchak Y et al. (1977) Determination of the pressure drop across an arterial stenosis utilizing angiocinedensitometry, The Yale Journal of Biology and Medicine, 50(4):375 - 381) examined 33 healthy patients and 23 patients with intermittent claudication and atherosclerotic obliterans. The severity of intermittent claudication and atherosclerotic obliterans was at the same level. The purpose of this medical examination was to present a method for calculating the pressure gradient across the stenosis by using flow data obtained from angiocinedensitometry. The researchers calculated the pressure loss across the stenotic lesion and made the calculation in the left external iliac artery. The Poiseuille flow formula was used to calculate the pressure gradient. The total pressure drop from the stenosis inlet to any point in the venous system can be expressed as:

[0023]

[0024] where: V1 is the flow velocity through the segment, η is the viscosity, L1 is the length of the stenosis, r1 is the average radius along the stenotic lesion, V2 is the average flow velocity through the periphery, L2 is the theoretical length of the venous system, and r2 is the theoretical average diameter of the periphery.

[0025] In summary, the researchers wrote that in healthy iliac arteries, the pressure gradient varied from 0.017 mmHg to 0.083 mmHg, with an average of 0.039 ± 0.018 mmHg, and the resistance to flow along the artery was 0.076 ± 0.082 mmHg·min / L. In atherosclerotic arteries, the pressure gradient varied between 0.0856 mmHg and 3.552 mmHg, with an average of 0.982 ± 0.918 mmHg, and the resistance across the stenosis was between 0.26 mmHg·min / L and 9.6 mmHg·min / L, with an average of 2.45 ± 2.4 mmHg·min / L (in Itzchak et al. (1977)).

[0026] Gould conducted an experiment on 10 long-term instrumented and non-sedated dogs to study pressure-flow characteristics (Gould KL (1978) Pressure-flow characteristics of coronary stenoses in unsedated dogs at rest and during coronary vasodilation, Circulation Research, 43(2):242-253). The experiment was conducted during rest and involved pharmacological coronary vasodilation in 100 cases of left circumflex artery stenosis. A Doppler flow transducer was placed around the dissected free proximal circumflex artery to induce constriction. A circumferential balloon constrictor was sutured 2 mm distally. A small tapered catheter was placed in the distal main circumflex artery, before the main branches. Catheters similar to the small tapered catheter were placed in the pulmonary artery, aortic root, and left atrium. According to this experiment, 50% or more of the pressure loss caused by arterial lumen constriction is related to the flow velocity and can be described in a simplified form as:

[0027] ΔP = FV + GV 2 , (7)

[0028] where: ΔP is the pressure drop in mmHg, F is the pressure loss coefficient due to viscous friction, V is the coronary artery flow velocity in cm / s, and G is the pressure loss coefficient due to flow separation or local turbulence downstream of the stenosis. At the same time, Gould demonstrated that the first and second parts on the right side of equation (7) describe 65% and 35% of the total pressure drop at rest, respectively, and 33% and 67% of the peak coronary artery flow during the hyperemic state, respectively (in Gould (1978)).

[0029] Mates et al. (Mates RE et al. (1978) Fluid dynamics of coronary artery stenosis, Circulation research, 42(1):152-162) studied the flow through an isolated stenotic segment in a large coronary artery. In an experiment, detailed velocity and pressure measurements were made using a large-scale coronary circulation model. A servo valve was used to simulate instantaneous peripheral resistance and pulsatile aortic pressure. The researchers examined many asymmetric and axisymmetric stenoses and ultimately used 4 geometric shapes of stenoses in the experiment. The pressure-flow relationship in the model can be expressed as:

[0030]

[0031] Where: L is the length of the stenosis, Q is the average flow rate, R is the peripheral bed resistance, η is the fluid viscosity, d0 is the diameter of the tube, d is the diameter of the canine coronary artery, d1 is the lowest equivalent, is the fluid density, C d is the orifice flow coefficient. In this paper, it is recognized that the selected model is very simple compared to the complex coronary circulation model. The selected model includes a straight rigid tube with a narrowing element inserted in series, and these elements have the resistance of the peripheral bed. It is also noted that the pressure recovery downstream of the constricted throat is reduced (or in some cases there is no recovery at all). The pressure loss is not determined by the geometry of the stenosis, but by its minimum area, and at normal heart rates, the flow is quasi-steady (in Mates et al. (1978)).

[0032] McMahon et al. (McMahon MM et al. (1979) Quantitative coronary angiography: measurement of the “critical” stenosis in patients with unstable angina and single-vessel disease without collaterals, Circulation, 60(1):106 - 113) examined 15 patients divided into 2 groups. The first group included 10 patients with normal left ventricular (LV) angiography and single coronary artery disease, without collateralization, ischemic ST-T changes, but no infarction and new refractory rest angina. The second group included 5 patients and was almost similar to the first group, but these 5 patients had subendocardial infarction (SEI). The researchers referred to the geometry obtained from the quantitative angiography of the patients, and they studied the proximal coronary artery critical stenosis. McMahon et al. proposed that the pressure drop across a moderately severe narrowing could theoretically be evaluated according to the following relationship:

[0033]

[0034] Where: k1 and k2 are constants based on assumptions of blood viscosity, blood density, lesion shape and length, and flow rate, Q is the flow rate, d min is the minimum diameter of the stenosis. The following results were obtained for the first and second groups respectively: the minimum stenosis diameters were 0.88 ± 0.14 mm and 0.64 ± 0.08 mm, the diameter reduction was 72% and 78%, and the cross-sectional area reduction was 92% and 95%. The differences observed between the results of the first and second groups stem from the significant differences in hemodynamics (in McMahon et al. (1979)).

[0035] In the extensive and in - depth work written by Bessems (Bessems DD (2007) On the propagation of pressure and flow waves through the patient specific arterial system, PhD thesis, TU Eindhoven), in Chapter 4, the author wrote an overview of the study on: the pressure loss across a stenosis as a function of the local flow characteristics and geometry of the stenosis described using the Womersley parameter and the Reynolds number. Bessems used the study by Young and Tsai in 1973 in this research. The author performed calculations based on an axisymmetric model of the stenosis geometry. The final equation describing the relationship between the flow characteristics and the pressure drop caused by the stenosis is expressed as:

[0036]

[0037] And:

[0038]

[0039]

[0040] Where: K v is the viscosity coefficient, α is the Womersley parameter, R s is the friction coefficient, q is the flow rate, is the blood density, K t is the turbulence coefficient, A0 is the cross - sectional area of the lumen blood vessel proximal to the stenosis, A s is the cross - sectional area of the lumen blood vessel at the neck of the stenosis, K u is the instability coefficient, L u is the coefficient of equation (18), t is the time, K c is the offset, a0 is the radius of the lumen blood vessel proximal to the stenosis, a is the stenosis geometry, L s is the stenosis length, R0 represents the friction coefficient of the steady flow per unit length in a straight tube, z is the axial position. The author compared the pressure drop from the numerical simulation with the pressure drop based on the equation in the study by Young and Tsai, and compared the pressure drop from the final equation with the numerical results - the differences were up to 80% and 10% respectively (in Bessems (2007)). Interestingly, the transient component of the pressure loss was mentioned again in the study by Bessems (2007).

[0041] A similar concept is shown in Liang et al. (Liang F et al. (2009) Multi-scale modelling of the human cardiovascular system with applications to aortic valvular and arterial stenoses, Medical & Biological Engineering & Computing, 47(7):743-755), and this concept is created on the basis of multi-scale modelling including a computational model of the entire cardiovascular system. The 55 largest arteries constituting the arterial tree are modelled in one dimension, and their geometry is described based on data published by Stergiopulos et al. and Olufsen in their works. Assuming that the pressure-flow relationship on a normal heart valve is consistent with Bernoulli's law and considering the contributions of blood inertia and viscous resistance, the pressure drop equation can be expressed as:

[0042]

[0043] Where: R cv is the viscosity coefficient, Q is the flow through the stenosis, B cv is the flow separation coefficient, L cv is the inertia term coefficient, is the time derivative of Q.

[0044] However, when it comes to severe aortic valve (AV) stenosis, the energy loss associated with the sudden expansion of the flow from the stenosis neck to the ascending aorta dominates the pressure loss across the valve. The researchers adopted the model proposed by Garcia et al.:

[0045]

[0046] Where: is the blood density, Q av is the flow across the valve, EOA is the effective orifice area, A ao is the cross-sectional area of the ascending aorta, and t is time. In summary, the researchers mentioned that stenosis location is a key factor in the overall hemodynamic effects of arterial constriction in the arterial system. Equation (15) shows a similarity to the previous evaluation model of the pressure drop across the stenosis under instantaneous conditions, but it only focuses on the local evaluation of the pressure drop and it cannot be directly applied to the entire vessel length (in Liang et al. (2009)).

[0047] Huo et al. provided an attempt at the direct diagnostic use of the cross - stenosis pressure loss assessment model initiated by Donald Young and Frank Tsai (Huo Y et al. (2012) A validated predictive model of coronary fractional flow reserve, Journal of the Royal Society, Interface, 9(71):1325 - 1338). These researchers derived a pressure drop and myocardial FFR model using the physics of coronary artery constriction. To validate the model, the researchers conducted in - vivo (in the arteries of 8 pigs, using an occluder cuff to induce stenosis) and in - vitro (in isolated arteries with asymmetric and symmetric tubes, where an inflatable occluder cuff was used to induce stenosis) experiments. The proposed analytical model used the general Bernoulli equation:

[0048] ΔP = ΔP 对流 +ΔP 收缩 +ΔP 扩散 +ΔP 扩大 , (16)

[0049] where: ΔP 对流 is the energy loss as a result of flow convection, ΔP 收缩 is the energy loss due to the sudden narrowing of the cross - sectional area (CSA) from the normal proximal vessel to the stenosis, ΔP 扩散 is the energy loss due to flow diffusion, ΔP 扩大 is the energy loss due to the rapid expansion of the cross - sectional area (CSA) from the stenosis to the normal distal vessel. The total pressure loss across the stenosis can be written in two ways. The first is (for (L s / Re s )·D s , when α≥0.05):

[0050]

[0051] The second is (for (L s / Re s )·D s , when α<0.05):

[0052]

[0053] The following symbols are used in the equations: is the blood density, Q is the flow rate, CSA s is the stenosis cross - sectional area, α is the radius of the inviscid core (dimensionless), For the velocity distribution of a blunt body at a narrow exit, the pressure loss caused by a sudden expansion of the CSA, L s is the narrow length, L e is the inlet length for α = 0.05, η is the dynamic viscosity, x is the position, For the parabolic velocity distribution at the narrow exit, the pressure loss caused by a sudden expansion of the CSA, Re s is the Reynolds number at the narrow inlet, D s is the inlet diameter of the stenosis. Huo et al. confirmed that the model proposed by the researchers was consistent with the experimental measurement results and concluded that their coronary stenosis analysis model could predict the FFR based on congestive coronary blood flow and stenosis size without any empirical parameters (in Huo et al. (2012)).

[0054] In an article written by Itu et al. (Itu L et al. (2013) Non-invasive hemodynamic assessment of aortic coarctation: validation with in vivo measurements, Annals of Biomedical Engineering, 41(4):669-681), an assessment based on CFD combined with an innovative non-invasive model personalized method for non-invasive hemodynamic assessment of pre-operative and post-operative patients with aortic coarctation (CoA) was proposed. The researchers proposed the total pressure drop for the coarctation:

[0055]

[0056] and:

[0057]

[0058] where: K v is the viscosity coefficient, α is the Womersley number, R vc is the viscous resistance, q is the flow rate, ρ is the fluid density, K t is the turbulence coefficient, A0 is the normal cross-sectional area, A c is the minimum cross-sectional area of the coarctation, K u is the inertia coefficient, L u is the length of the coarctation, t is the time, K c is the continuity coefficient, is the average flow rate, L c is the length of the coarctation, μ is the dynamic viscosity of the fluid.

[0059] In the preliminary results, the absolute error obtained by Itu et al. was less than 2 mmHg. To calculate the centerline of the artery and the different radius measurements, a 3D geometric model of vascular tree segmentation was used. To obtain the lumen of the vascular tree segmentation, the researchers used 3D contrast-enhanced MR angiography (MRA) (in Itu et al. (2013)).

[0060] A nice and practical summary of these studies, including Bessems (2007) and Itu L et al. (2013), can be seen in the patent specifications obtained by Sharma et al. ((2013) Non-invasive functional assessment of coronary artery stenosis including simulation of hyperemia by changing resting microvascular resistance, US Patent No. 10373700B2) and Sharma et al. ((2013) Non-invasive functional assessment of coronary artery stenosis including simulation of hyperemia by changing resting microvascular resistance, US Patent No. 10354744B2).

[0061] In clinical practice, the functional assessment of circulatory insufficiency follows two alternative strategies to achieve: (1) the magnitude of the pressure change caused by flow assessment, or (2) the change in blood flow supply under certain pressure conditions. Since the deterioration of circulatory pathological symptoms usually occurs under stress conditions rather than at rest, the assessment is therefore carried out under stress conditions (usually achieved by pharmacological means). According to the current standards (Lawton JS et al. (2022) 2021 ACC / AHA / SCAI Guideline for Coronary Artery Revascularization: Executive Summary: A Report of the American College of Cardiology / American Heart Association Joint Committee on Clinical Practice Guidelines, Circulation, 145(3): e4 - e17; Knuuti J et al. (2020) 2019 ESC Guidelines for the diagnosis and management of chronic coronary syndromes, European heart journal, 41(3): 407 - 477), the recommended strategy is the first strategy based on the magnitude of pressure change.

[0062] What is recommended is to perform the assessment using a coronary reserve ratio including FFR, iFR, or equivalent values (for example, the ESC guidelines also mention FFR - CT obtained by computer simulation). This ratio can be calculated as the ratio of the average of the distal pressure and the proximal pressure: FFR = P d / P a , so FFR = 1 - ΔP / P a . This is because the distal pressure (P d ) at a given cross - section can be expressed as the difference between the proximal pressure (for example, the pressure (P a ) at the coronary artery inlet cross - section) and the pressure loss (ΔP) along the flow path between the two cross - sections. The capital letter P represents the average pressure value, and the lowercase letter p represents its instantaneous value (so P=(1 / T)·∫ T p dt, where T is the average time interval). These methods are related to flow assessment, so in case of insufficient blood supply, even if being used, currently they are not the gold standard for diagnosis.

[0063] Under in - vivo conditions, the FFR measurement procedure is based on two pressure measurement results (P a and P d)And both use a pressure wire placed within the artery. Although the procedure is not time-consuming, its implementation can be complex and the procedure can be harmful to the patient. Placing the pressure wire in the artery is dangerous. The danger is not introduced by computer simulation methods, including computational fluid dynamics (CFD), which is a very important and promising alternative to in vivo methods. There is no doubt that computational fluid dynamics (although difficult but successful) is gaining the trust of more and more clinicians in diagnostic use (Cook CM et al. (2017) Diagnostic accuracy of computed tomography-derived fractional flow reserve: A systematic review, JAMA cardiology, 2(7):803-810; Driessen RS et al. (2019) Comparison of coronary computed tomography angiography, fractional flow reserve, and perfusion imaging for ischemia diagnosis, Journal of the American College of Cardiology, 73(2):161-173; Torii R, Yacoub MH (2021) CT-based fractional flow reserve: development and expanded application, Global cardiology science & practice, E202120; Pontone G et al. (2022) Clinical applications of cardiac computed tomography: a consensus paper of the European Association of Cardiovascular Imaging - part II, European heart journal, Cardiovascular Imaging, 23(4):e136-e161). In particular, coronary computed tomography angiography-derived fractional flow reserve has gained followers.

[0064] Users of computer simulation diagnostic tools expect to achieve the highest accuracy, sensitivity, specificity, etc. Naturally, this involves efforts to achieve the most precise geometric mapping and also to use computational grids with extremely high mesh densities. Even though this one-way strategy allows for an increase in accuracy, it does not take into account the practical aspects of the feasibility and usability of this type of tool in diagnosis. This is one of the objectives of the present invention. In fact, increasing the computational mesh density is always associated with an exponential increase in the computational workload. Additionally, the numerical CFD algorithm stability criteria, particularly the Courant-Friedrichs-Lewy (CFL) convergence condition (Courant R, Friedrichs K, Lewy H (1928) the partial differential equations of mathematical physics, Mathematische Annalen, 100, 32-74; de Moura CA, Kubrusly CS (2013) The Courant-Friedrichs-Lewy (CFL) condition: 80 years after its discovery, Birkhauser Boston) pose a requirement:

[0065]

[0066] where: v is the velocity, Δh is the mesh density factor, and Δt is the time step. The existence of these criteria leads to a paradoxical situation where Δh → 0 and Δt → 0 and computational instability.

[0067] In such a paradoxical situation, not only does the computational workload for one time step Δt increase exponentially with the increase in the number of grid nodes, but also the number of time steps (Δt) increases sharply at the same time. It can be roughly estimated that several thousand or even millions of time steps are required for a coronary artery flow simulation for one cardiac cycle. From the perspective of a clinician, a coronary artery flow simulation for one cardiac cycle takes a large amount of time to complete, while a standard in-vivo operation is usually completed within 10 minutes - 15 minutes to one hour. Additionally, high computational power on-demand availability is required to maintain the required availability of computer simulation diagnostic services. Providers of these diagnostic services today ensure the availability of computer diagnostic services using cloud computing technologies such as Microsoft Azure or Amazon Web Services. It is envisioned that once the computer simulation diagnostic services become routine worldwide, without improving the performance of computer diagnostic simulations, there will not be sufficient available resources to ensure the required on-demand availability. Another factor is that high computational power requires a large amount of electrical energy. This large amount of electrical energy has a negative impact on the environment (carbon emissions).

[0068] There is a need for an accurate computer simulation method for the diagnostic assessment of the hemodynamic conditions of the human heart that is available on-demand and does not require a large amount of computational effort to implement.

[0069] Brief Description of the Invention

[0070] The following brief description is for a better understanding of the principles and benefits of the present invention claimed in the appended claims. Therefore, it is not meant to be limiting in any sense.

[0071] The present invention is a computer-implemented method for calculating the pressure drop (Δp) between a first cross-section and a second cross-section of an artery. In an embodiment, the artery can be a coronary artery, a carotid artery, a renal artery, a peripheral artery, or the aorta. According to the present invention, the second cross-section is downstream of the first cross-section, and the two cross-sections are separated by a channel of length l. As those skilled in the art can understand, the length of the channel is the distance between the two cross-sections. Depending on the context, the present disclosure uses both distance and length to describe aspects of the present invention. The channel is part of the artery and, if unobstructed, allows blood (or any other fluid) to flow from the first cross-section to the second cross-section. The term "downstream" refers to the direction of blood flow in a healthy human. For example, a point in the artery downstream of the artery inlet is a point where blood flows from the inlet.

[0072] The method includes a steady-state computational fluid dynamics (CFD) simulation, but allows for the calculation of the pressure drop (Δp), and more importantly, the transient pressure drop. This is achieved by separating the pressure drop into a steady-state pressure loss component and a transient pressure loss component.

[0073] The present invention employs a method different from known methods in auxiliary diagnostic methods based on transient flow CFD simulations (such as vFFR, CT-FFR, FFR-CT, etc.). By avoiding the use of transient CFD simulations to directly calculate transient flow, the present invention formulates its simulations in a completely different manner from those methods. Instead, the present invention implements a steady-state CFD simulation to obtain transient flow. This allows for a simulation speed that is hundreds or even thousands of times faster while maintaining the accuracy level required for diagnosis. This allows for faster diagnosis and treatment. In certain embodiments intended for the practical implementation of the present invention, the present invention focuses on pressure loss assessment.

[0074] In one aspect of the present invention, the steady-state pressure loss component is formulated such that it allows for a reliable and rapid diagnosis.

[0075] The distal pressure p d is a function of the current flow rate q and is described as the proximal pressure p a minus the transient pressure drop Δp along the flow path l between the distal and proximal positions, i.e.:

[0076] p d = p a - Δp. (24)

[0077] The measured instantaneous pressure values obtained are averaged to calculate the FFR index. The value obtained from the calculation of the FFR index is instantaneous. For the calculation, patient-specific variables can be used to derive the distal pressure value, proximal pressure value, and current flow rate value in the cross-section, as described in the following literature: Kosior A, Mirota K, Tarnawski W (2019) Patient-specific modelling of hemodynamic parameters in coronary arteries (WO 2020 / 048642 A1, or in EP 3820357 B1). The distal pressure value can also be obtained from non-invasive measurements.

[0078] According to the present invention, the pressure drop (Δp) can be expressed as:

[0079] Δp = Δp s + Δp t , (25)

[0080] where Δp s is the steady-state pressure loss component, and Δp tis the transient pressure loss component. The various embodiments described in the detailed description illustrate how to calculate the steady-state pressure loss component and the transient pressure loss component. Unless otherwise stated, all embodiments and elements of the described embodiments can be combined together to form new embodiments within the scope of the present disclosure.

[0081] The main applications of the present invention are in the assessment of pressure drop across a stenosis and are associated with such assessment; in obtaining a diagnostic index, particularly a fractional flow reserve (FFR), through computer simulation. The diagnostic index is, for example, related to diagnosing and treating medical conditions related to the human heart.

[0082] In an embodiment, only the steady-state pressure loss component is used to calculate the pressure drop:

[0083] Δp = Δp s 。 (26)

[0084] The present invention provides various methods for calculating the steady-state pressure loss component. According to an embodiment, the steady-state pressure loss component is expressed as:

[0085] Δp s = c0 + c1·q + c2·|q|·q, (27)

[0086] where c0, c1, and c2 are empirical coefficients that can be calculated using steady-state computational fluid dynamics (CFD) simulations. This embodiment allows, for example, the calculation of FFR using steady-state CFD simulations. The benefit of this embodiment is the computational speed required to obtain the FFR index, and this is related to diagnosis and treatment.

[0087] In another embodiment, the steady-state pressure loss component is expressed as:

[0088] Δp s = Δp0 + Δp1 + Δp2, (28)

[0089] where Δp0 is the zero-flow pressure, Δp1 is the viscous resistance of the straight portion of the channel or the portion of the channel where the ratio of the radius of curvature (R) to the radius (r) of the portion is equal to r / R < 10, and Δp2 is the local resistance of the portion of the channel having a stenosis. The ratio of the radius of curvature (R) to the radius (r) of the portion being equal to r / R < 10 ensures that the channel or its portion is relatively straight. In an embodiment, r / R can be less than 9, 8, 7, or 6. A stenosis in the channel of an artery has a defined meaning known to those skilled in the art. A stenosis can be understood as a narrowing or blockage of an artery.

[0090] In an embodiment of the present invention, the viscous resistance (Δp1) is equal to:

[0091]

[0092] where f is the hemodynamic friction coefficient, and ρ is the density of blood. <d>is the "equivalent" inner diameter of the channel. <d>Not measured, but calculated for a channel model having a circular cross-section and a length equal to the channel length for which it was modeled. <d>It is calculated on the assumption that the pressure loss of laminar flow is the same for the channel and the channel model. <v>is the "equivalent" flow velocity and is also calculated for the channel model, assuming that the pressure loss for laminar flow is the same for the channel and the channel model. The local resistance (Δp2) is proportional to Calculating the pressure drop using Δp1 and Δp2 allows the results obtained in clinical trials to be reliably reproduced, without the need for large computational power.

[0093] "Equivalent" inner diameter and "equivalent" flow velocity are terms used in the field of the present invention and have a defined meaning therein. This meaning should not be confused with equivalents used in the law.

[0094] In embodiments, the pressure drop (Δp) includes a transient pressure loss component and has the following form:

[0095] Δp = Δp s + Δp t . (30)

[0096] In these embodiments, the transient pressure loss component (Δp t ) can be calculated according to Newton's second law of motion. More specifically, the transient pressure loss component (Δp t ) can be expressed as:

[0097]

[0098] where t is time and k t is an empirical exponent defined in the range of 1 to 1.2. In embodiments, k t can be equal to 1.6, 1.7, 1.8, 1.85, 1.9, 1.95, 1.96, 1.97 or 1.98. <a0>is a calculated rather than measured "equivalent" surface area. Using <d>to calculate <a0>, for example <a0> ＝(π / 4) <d> 4 。In a similar manner, those skilled in the art can use <a0>Calculation <v>. It includes the calculation of the steady-state pressure loss component and the transient pressure loss component (Δp t ), and the pressure drop Δp is the above transient pressure drop. The transient pressure drop is reconstructed based on the proposed relationship describing the flow characteristics, where the results of the steady-state series of CFD simulations are used to select empirical parameters. This aspect of the present invention allows the elimination of complex and difficult transient CFD simulations, thus saving the time required for calculations and also avoiding the need to use high computing capabilities. This allows the present invention to be available on demand and reduces any environmental impact of the computer simulation diagnostic method.

[0099] In an embodiment, the pressure drop Δp is defined between two cross-sections of the coronary artery. In a more specific embodiment, the pressure drop Δp is defined between the coronary artery inlet and a point in the coronary artery downstream of the coronary artery inlet.

[0100] In a more specific embodiment, the two cross-sections and / or the channel between the two cross-sections have the same or similar topology. A similarity index is used to define the similar topology, which takes into account the change in the inner diameter between the two cross-sections and / or the change in the inner diameter on the artery between the two cross-sections, the branches at the two cross-sections and / or the branches on the artery between the two cross-sections, or all of the above. In an embodiment of the present invention, the similarity index can have the following values: 50%, 55%, 60%, 65%, 70%, 75%, 80%, 85%, 90%, 95% or 99%. The similarity index can be defined as a range created using any of the above values. The similarity index can be defined as higher than 50%, higher than 55%, etc.

[0101] Various details and other embodiments of the present invention will be presented in the following description. Brief Description of the Drawings

[0103] Now, the present invention will be described in more detail with reference to the drawings, in which: Figure 1 The results of coronary computed tomography angiography (CCTA) and the visualization of the features of the present invention for obtaining the complete instantaneous pressure drop in the coronary artery aorta are presented, Figure 2 including schematic diagrams of embodiments of the present invention, Figure 3 An example of finding the "equivalent" diameter of a patient according to the present invention is presented, Figure 4 The calculation of the steady-state pressure loss including the Lagrangian number and the Reynolds number according to the present invention is presented, Figure 5 A comparison of the measurement results of the flow reserve fraction for 1960 in-vivo human cardiac cycles based on the approximate calculation of the steady-state pressure loss component is shown, and Figure 6 a comparison of the transient pressure drop from medical data and the transient pressure drop reconstructed by using the present invention is included.

[0104] Detailed Description

[0105] In an embodiment of the present invention, the relationship establishing the pressure drop Δp between two cross-sections of an artery is equal to:

[0106] Δp = Δp s + Δp t = c0 + c1·q + c2·|q|·q + Δp t , (32)

[0107] where c0, c1, and c2 are empirical coefficients, and q is the flow rate. The values of the empirical coefficients can be calculated using the results of a series of steady-state CFD simulations, enabling the calculation of the pressure loss Δp using steady-state simulations s . The values of the empirical coefficients can be calculated using the results of clinical trials or a combination of clinical trials and steady-state CFD simulations. This yields the steady-state component Δp of the pressure loss s . In an embodiment, only Δp s can be calculated (without Δp t ), which allows for obtaining, for example, FFR. In a more preferred embodiment, q is in the range of 1 ml / s to 15 ml / s. This ensures a higher level of accuracy of the results. Even more preferably, q is equal to 1.5 ml / s, 3.5 ml / s, or 5 mL / s. Alternatively, the most accurate results can be obtained by following these steps: (1) approximating the typical q0 value of a given artery using, for example, Milnor WR (1982) Normal hemodynamic state, in Hemodynamics, Baltimore, Williams & Wilkins, (2) establishing a range from q0 / 5 to 3q0, and (3) performing the calculations within this range

[0108] The second component Δp t describes the inertial effects that may occur under unsteady flow conditions. This yields Δp t the transient component. This component is not calculated using the results of steady-state CFD simulations but is calculated according to Newton's second law of motion:

[0109]

[0110] Thus, as the entropy of the force generated by the acceleration / deceleration dv / dt = (1 / A)·(dq / dt) of the mass between two cross-sections (e.g., between the proximal and distal cross-sections), where l is the distance and A is the surface area of the artery cross-section. k is the empirical exponent (usually k t is the empirical exponent (usually k t = 1), and in a preferred embodiment, k t = 1.2, which means that the inertial force is increased by 20% relative to the theoretical value. In an embodiment, the empirical exponent may have different values and includes, for example, 0.7, 0.8, 0.9, 1.1, 1.15, 1.16, 1.17, 1.18, 1.19, 1.21, 1.22, 1.23, 1.24, 1.25, 1.3 or 1.35.

[0111] For the present invention, the way to determine the transient component Δp t is fixed and given in the above equation (33), and the steady-state component Δp s can be determined using any one of the many possibilities contemplated by the present invention.

[0112] As can be seen from equation (33), this embodiment does not utilize transient CFD calculations but only steady-state CFD calculations to obtain the transient pressure drop. This provides an improvement in the speed of calculating the pressure drop Δp by several orders of magnitude.

[0113] In an embodiment of the present invention, the pressure loss Δp s under steady state is represented by the superposition of the sum of three physical components:

[0114] Δp s = Δp0 + Δp1 + Δp2. (34)

[0115] The component Δp0 is specific to coronary circulation simulation and it corresponds to the zero-flow pressure (Hoffman JIE (1990) Pressure-flow relationships of the coronary arteries, in: Kajiya F, Klassen GA, Spaan JAE, Hoffman JIE (eds) Coronary Circulation, Springer-Verlag Tokyo, 109-125). Generally, its value is so small that its contribution to the total value of Δp (and also ΔP) is negligible. For healthy individuals, this contribution is approximately 14 ± 7 mmHg, and in percutaneous coronary interventions for ST-segment elevation myocardial infarction only, this contribution can reach and even exceed 42 mmHg (Patel N et al. (2015) The zero-flow pressure measured immediately after a primary percutaneous coronary intervention for a ST-segment elevation myocardial infarction provides the most useful invasive index for predicting the extent of myocardial infarction after 6 months as showed in An OxAMI Study (Oxford Acute Myocardial Infarction), JACC, Cardiovascular interventions, 8(11):1410-1421). In one embodiment, Δp0 is a constant value.

[0116] The two remaining components, Δp1 and Δp2, vary significantly with flow rate. Δp1 and Δp2 are the viscous resistance and local resistance components (located in the region of artery narrowing or blockage, such as at a stenosis) of the straight section (or section with relatively small curvature - the ratio of the curvature radius (R) to the cross-sectional radius (r) of the blood vessel is r / R < 10, in view of Freidoonimehr et al. (2021) Effect of artery curvature on the coronary fractional flow reserve, Physics of Fluids, 33, 031906), respectively. Regardless of the method used (experimental or simulation), those skilled in the art recognize that in embodiments including Δp1 and Δp2, the values of Δp1 and Δp2 are most relevant to the accuracy of the functional assessment of pressure loss (clinical trials show up to 98%). For example, at a stenosis.

[0117] In one embodiment of the present invention, the viscous resistance component Δp1 is approximated according to Henry Darcy's law and thus approximated in the following form (Bird RB, Stewart WE, Lightfoot EN (2007) Transport phenomena, John Wiley & Sons; White FM (1999) Fluid Mechanics, Boston Mass: WCB / McGraw-Hill):

[0118]

[0119] where: f is the hemodynamic friction coefficient, l is the distance from the first cross-section (where the proximal pressure p a is measured, for example, from the inlet) to the second cross-section (where the distal pressure p d is to be calculated), is the density of the flowing medium (here blood), and <d>The "equivalent" inner diameter of a coronary artery channel calculated for a coronary artery channel model, the coronary artery channel model having a circular cross-section and a length equal to the length of the real coronary artery channel for which it is modeled, and simultaneously assuming that the pressure loss for laminar flow is the same for the real coronary artery channel and the coronary artery channel model. <v>Is the "equivalent" flow velocity calculated for a coronary artery channel model that has a circular cross-section and the same length as the length of the real coronary artery channel for which it is modeled, and simultaneously assumes that the pressure loss for laminar flow is the same for the real coronary artery channel and the coronary artery channel model. Those skilled in the art will recognize that this method gives for <v>and <d>reliable values because, in the case of tomographic scans, the resolution of the tomographic scan is too low to provide accurate values of the flow velocity and inner diameter of the artery (0.5 mm resolution, where d can be at the level of 4 mm). This embodiment has the same benefits as the embodiment including equation (34), which includes avoiding transient CFD simulations.

[0120] Figure 1 is shown using on the CCTA results <d>Visualization of the embodiments. In Figure 1 (A), the aorta and coronary artery tree obtained by CCTA are present, and in Figure 1 (B), the right coronary artery view and its curved planar reconstruction (CPR) are present. In Figure 1 (C), Figure 1 (D), and Figure 1 (E), the "equivalent" diameters are present respectively <d>The reconstruction principle is used to obtain the invariant of the transient pressure drop according to the present invention, as well as the relationship between the steady-state pressure drop and the transient pressure drop.

[0121] In Figure 3 In one embodiment shown, the following equation (53) is used for calculation <d>value, and use <d>Calculation <v>Value. Figure 3 An example of finding the "equivalent" diameter of a patient using the Poiseuille number is shown. The patient is a 67-year-old male (height: 176 cm, weight: 92 kg, BMI: 29.7, systolic and diastolic blood pressures are 145 mmHg and 85 mmHg respectively, heart rate: 72 / min). In Figure 3 it, the dotted curve depicts the variation of the Poiseuille number with respect to the "equivalent" diameter, and the solid curve represents the "equivalent" diameter when the Poiseuille number is equal to 32 (laminar flow). This is one of the methods for finding the "equivalent" diameter for a patient.

[0122] In one embodiment, <d> 、 <v>It can be the median of the inner diameter of the channel and the flow rate through the channel. In different embodiments, <v>And <d>The value is selected such that, for example, a mass flow rate, volume, momentum, or energy parameter corresponds to a parameter obtained from a flow velocity and an inner diameter obtained from measurement results (e.g., from tomographic measurement results).

[0123] In one embodiment of the present invention, (35) is transformed into a general dimensionless form:

[0124]

[0125] Considering that generally, the left - hand expression defines a similarity invariant in the form of an Euler number We finally have:

[0126]

[0127] This allows performing calculations using linear equations instead of quadratic equations, which provides a significant improvement in terms of calculation speed. As mentioned before, this also allows reducing the need for high computing power.

[0128] In another embodiment, under a wide range of physiological conditions, the blood flow in the straight section of the coronary artery (thus except for the region with plaque location) has the characteristics of laminar flow. This is because the Reynolds number (where η represents the viscosity coefficient) is rarely greater than a few hundred. Therefore, assuming a channel with a circular cross - section, the hydrodynamic friction coefficient is determined according to the Hagen - Poiseuille's law:

[0129]

[0130] where Re is a similarity invariant in the form of a Reynolds number:

[0131]

[0132] In a preferred embodiment, we can also assume that the Euler number of the loss component in the straight section is:

[0133]

[0134] In an embodiment, B is equal to 32.

[0135] In an embodiment, the third component of the pressure loss (Δp2) is a local loss and has a value proportional to the dynamic pressure:

[0136]

[0137] In a preferred embodiment, the Euler number is expressed as:

[0138]

[0139] This means that the Euler number is proportional to a certain constant. If one follows the approximation for the cross narrow Young-TSai loss, Bessems (2007), Itu (2012) and other researchers mentioned previously, they relate this specific constant to a general form ( <a0> / A s -1) 2 Combine the components of, where <a0>and A s are the cross-sectional areas of the vascular lumen based on the "equivalent" diameters at the narrow proximal and narrow neck, respectively. Considering the throttling effect of atherosclerotic plaques on flow described in the medical literature, we obtained (Carnicelli AP et al. (2013) Cross-sectional area for the calculation of carotid artery stenosis on computed tomographic angiography, Journal of vascular surgery, 58(3):659-665):

[0140]

[0141] where: ΔA represents the change in the cross-sectional area of the lumen due to stenosis, and S represents the degree of stenosis, which can be transposed to:[[]]

[0142]

[0143] Therefore, the dimensionless characteristics of the resistance across the stenosis are given as follows:[[]]

[0144]

[0145] where, k s is an empirical parameter that can be calculated using steady-state CFD simulations.[[]]

[0146] According to the present invention, the pressure loss under transient or steady-state conditions (involving Δp s or ΔP s , where Δp s is the instantaneous value of the pressure loss, and ΔP s is the intermediate value of the pressure loss) can be calculated using similarity flow invariants. This takes into account that in practice the above two pressure loss components are added together, resulting in Eu = Eu1 + Eu2 (see the above equations (40) and (45)), and thus:[[]]

[0147]

[0148] Therefore, by introducing the Lagrangian number, we obtain:[[]]

[0149]

[0150] In this way, we obtain a preferred embodiment for describing the flow characteristics of Δp1 + Δp2:[[]]

[0151]

[0152] Equivalently, the following equation is obtained:

[0153]

[0154] where K s = k s ·(S / (1 - S)) 2 is an empirical parameter describing the dissipation level across the stenosis. The benefit of this embodiment is the computational speed derived from using a linear equation rather than, for example, a quadratic equation. This is not at the expense of the accuracy of the results obtained. In the embodiment, the cross-section is cylindrical and B = 32.

[0155] Based on the Lagrangian number and the Reynolds number, the "equivalent" diameter <d>and K s As Figure 4 shown in the calculation. Use the medical data of the patient for Figure 3 to perform each calculation.

[0156] In another embodiment of the present invention, a system of similarity invariants provides a general relationship describing the accuracy of the steady-state component Δp s with a constant Δp0. Given the need to reconstruct the full transient of the variation of the distal pressure p d as a function of time, it must be supplemented with the Δp t component. To avoid doubt, "full transient" refers to a flow that has no reasonably observable trend in its temporal variation. The values of the similarity invariants (La, Re, Eu) are calculated according to their generally known definitions, and the pressure loss Δp s and the flow rate q, empirical parameters (such as B or K s ) can be obtained using steady-state CFD simulations (therefore, for example, only two calculations are required for B and K s which ensures a faster result). Diameter <d>can be selected to have a value equivalent to the value obtained from the segmentation of the geometric model (so the surface area is A = π <d> 2 / 4), and the dynamic viscosity coefficient η can be selected according to the patient-specific rheological properties of the blood (Baskurt OK (2007) Handbook of hemorheology and hemodynamics, IOS Press). Those skilled in the art know that various known models for non-Newtonian fluids can be used to calculate the viscosity, and for Newtonian fluids, a constant viscosity value can be used. In contrast, typical state-of-the-art methods require thousands of time-consuming CFD simulations.

[0157] In an embodiment of the present invention, the diameter <d>does not directly reflect the diameter in terms of the segmentation result, but it is chosen to be "equivalent" to the diameter obtained from the segmentation result in terms of the energy dispersion level. In this sense, the improved concept of the present invention abandons the idealized methods for evaluating the energy loss across stenosis proposed by the existing models ( JC (1949), Young & Tsai (1973), Young & Tsai (1975), Young & Tsai (1977) and recently). It would be possible to directly measure the diameter of the blood vessel from the geometry only in the case of large-diameter arteries (as in Young & Tsai (1977)) or artificial models of in vitro or computer-simulated geometries (even more so for stenosis). This limitation stems from the too low resolution of the experimental geometry, which only allows for sufficient accuracy for large-diameter arteries or for artificially simplified arteries (e.g., arteries represented as simple tubes).

[0158] In a preferred embodiment of the present invention, the "equivalent" diameter <d>is considered to be the diameter of a circular channel for which the energy loss under steady-state laminar flow conditions would be equal to the energy loss generated under the same flow conditions in the actual geometry of the artery. In this case, the hydrodynamic friction coefficient f and the Reynolds (Re) and Poiseuille (Po) similarity numbers satisfy the equation:

[0159]

[0160] At the same time, as previously stated regarding the pressure loss on straight or low-curvature sections, Therefore, the friction coefficient is equal to:

[0161]

[0162] Note We obtain the following form for the Poiseuille number:

[0163]

[0164] Or by replacing the "equivalent" velocity with the flow rate <v>:

[0165]

[0166] Among them, the ratio Δp / q = R0 represents the resistance, and in this embodiment, the resistance is a constant value.

[0167] Therefore, in a preferred embodiment of the present invention, the "equivalent" diameter <d>Values that are considered to satisfy the system of equations given below:

[0168]

[0169] This embodiment provides a further improvement in the calculation speed without compromising its accuracy.

[0170] Embodiments of the present invention include five steps: (1) "equivalent" diameter <d>The selection of (2) approximation of the viscous section of the flow characteristics, (3) approximation of the local pressure loss section of the flow characteristics, (4) calculation of the steady-state pressure drop, and (5) calculation of the transient (inertial) component of the pressure drop. This is illustrated in Figure 3 as exemplified in, where <d>Equal to approximately 2.064 nm. For Figure 3 the embodiment of <d>Calculation <v>, which is clear to those skilled in the art. Repeat the whole process for all locations and branches where steady-state CFD simulation results are available.

[0171] Step 1 is a preparatory step for reconstructing flow characteristics, which involves selecting scales for describing invariant flows by using similarity invariants. Of crucial importance here is the characteristic linear dimension scale, and thus of crucial importance is the equivalent diameter <d>, because of the velocity scale <v>It depends on it. In a more preferred embodiment, the scale is <d>。In another embodiment, the scale is <v>According to the present invention, the equivalent diameter <d>The value is calculated as described above to obtain consistency in the diffusion phenomenon between two cross-sections in the artery. In one example, the two cross-sections are a proximal cross-section and a distal cross-section.

[0172] Step 2 involves selecting from the descriptions of the flow characteristics in the coronary artery segments described herein, which are related to the viscous energy dissipation on straight segments and those with small curvatures. In one embodiment, this includes using the flow characteristics defined in equations (32), (34), or (54).

[0173] Under real conditions (physiological conditions for humans), the pressure loss is generally the sum of the linear pressure loss and the local pressure loss. This is of great significance in the case of flow in the coronary artery with high geometric variability. In fact, it may be very difficult to separate the linear (strictly speaking, viscous) component from the local component. In a preferred embodiment of the present invention, instead of separating the pressure loss into the sum of the linear pressure loss component and the local pressure loss component, the value of the flow velocity q and thus the velocity scale <v>and the Reynolds number is chosen so that the second component of the flow characteristics is negligibly small, so that B·(l / <d>) >> K s ·Re. Formally, under the idealized conditions of the embodiment, the value of parameter B is equal to 32. In other embodiments, the value of B will be different. In one embodiment, B is considered here as a constant at the fixed distal cross-section (at a distance l from the inlet). In this case, if K s ≠32 under real conditions, then K s = const and only this part of the flow characteristics is considered in the step of determining this parameter.

[0174] Step 3 relates to the description of the second component of the local losses of the flow characteristics. Its aim is to determine the empirical parameter K s value. Although there may be many stenotic parts on the coronary artery, which is known physiologically, the main ones are the stenoses formed by atherosclerotic plaques. If there is no functionally significant stenosis on a given branch, then in terms of the physiological flow conditions, this component will have no significant effect. In the embodiment, the value of parameter K s can be determined as the slope coefficient of a simple regression in the La = f(Re) system, whose intercept is equal to B·l / <d>。

[0175] Step 4 involves using a steady-state CFD simulation to calculate the pressure loss Δp s value. Since La = Eu·Re, the pressure loss Δp s can be calculated as This embodiment is applicable to the calculation of computer-simulated FFR. This application stems from the fact that by step 4, average values are obtained, and these values are used for the calculation of computer-simulated FFR.

[0176] Figure 5 summarizes the results of the clinical trial. The clinical trial provided medical data of patients. The patients were 30 females (8 with myocardial ischemia), with an average heart rate of 68.30 ± 5.06 / min, average systolic and diastolic blood pressures of 131.83 ± 11.03 mmHg and 79.47 ± 9.93 mmHg respectively; and 25 males (17 with myocardial ischemia), with an average heart rate of 68.72 ± 6.02 / min, average systolic and diastolic blood pressures of 136.72 ± 13.46 mmHg and 82.56 ± 8.09 mmHg respectively. The in-vivo FFR ranged from 0.61 to 0.92. This medical data was used to compare the experimental results with those obtained using the present invention.

[0177] The medical data included 1960 human cardiac cycles, in which in-vivo pressure measurements were performed using a pressure wire. For each of these cardiac cycles, the pressure loss (steady state) was calculated by following steps 1 to 4, and then the distal pressure was approximated by following (step 5). This allows for the calculation of FFR (computer-simulated) using only Δp s (the steady-state pressure loss component in steps 1 - 4) and in combination with Δp s and Δp t (the transient pressure loss component, steps 1 - 5).

[0178] Determine the correlation between in-vivo FFR and computer-simulated FFR. The correlation coefficient using only the linear component of Δp s (Δp1) was 80.3%. The correlation coefficient using the local resistance component (Δp2) was 93.8%, although the deviations were large and diagnostically unacceptable (+0.063 and +0.0848 respectively). However, when the pressure loss and FFR index were calculated based only on the two components (Δp1 + Δp2) obtained after step 4, the correlation increased to 98.7%, and the median difference was -0.009. The FFR values obtained for Δp1 + Δp2 were thus diagnostically acceptable and had an error comparable to or even smaller than that expected for in-vivo FFR.

[0179] Figure 6 presents a description of the use of the same as that used for Figure 5 Example of curves of in-vivo and computer-simulated pressures obtained from the same medical data. The curve obtained by computer simulation reproduces the curve obtained in-vivo. The relationship between the pressure drop value obtained by computer simulation and the flow rate reproduces the value obtained in-vivo.

[0180] Step 5 is used to reconstruct the pressure loss as a function of the flow rate, including its variation over time (Δp = f(q(t))). In this step, the steady-state pressure value Δp obtained in step 4 is used s , and the transient pressure loss component Δp is used t . As previously mentioned, Δp t is calculated according to the second principle of kinetics In a preferred embodiment, the empirical coefficient is equal to k t = 1.2.

[0181] The results of the computer-simulated FFR calculation obtained in step 5 are evaluated using the results obtained from clinical trials. Reference Figure 5 . The degree of correlation is 99.9%, and the median error is only 1.69813·10 -5 .

[0182] In Figure 2 an example of implementing steps 1 - 5 is given. In the first step, the "equivalent" diameter of the arterial branch flow channel is calculated <d>. During steps 2 and 3, an approximation of the linear resistance Δp1 and the local resistance Δp2 is performed. In the fourth step, the steady-state component of the pressure drop (Δp s ) is calculated, which allows entry into the last step, namely the calculation of the transient component of the pressure drop (Δp t ). These five steps allow the evaluation of the pressure drop in the coronary artery, including its transient component.

[0183] At any step, the density and the dynamic viscosity coefficient (η) can be calculated as described below. Basically, and η are patient-specific parameters. For example, the density of blood can be approximated by the Hawksley formula , where are the densities of plasma and red blood cells, respectively, and HCT is the hematocrit number (De Gruttola S et al. (2005) Computational simulation of anon - newtonian model of the blood separation process, Artif Organs, 29(12):949 - 59). There are also various applicable blood rheology models here, for example, the Carreau model, the Cross model, the Yasuda model, and many others (Yilmaz F, Gundogdu MY (2008) A critical review on blood flow in large arteries; relevance to blood rheology, viscosity models, and physiologic conditions, Korea - Australia Rheology Journal, 20(4):197 - 211). Although factors of individual variability can be introduced in this way, in the preferred embodiment, these parameters are chosen as constants. In this preferred embodiment, the accuracy of the calculation is not significantly affected.< / d> < / d> < / d> < / v> < / d> < / v> < / d> < / v> < / d> < / v> < / d> < / d> < / d> < / d> < / v> < / d> < / d> < / d> < / d> < / d> < / d> < / v> < / v> < / d> < / v> < / d> < / d> < / d> < / d> < / d> < / v> < / v> < / d> < / v> < / d> < / a0> < / d> < / v> < / d> < / d> < / d>

Claims

1. A computer-implemented method for calculating a pressure drop (Δp) between a first cross-section and a second cross-section of an artery, wherein, The second cross-section is downstream of the first cross-section, and the two cross-sections are separated by an arterial passage having a length (l), and wherein the pressure drop (Δp) is expressed as Δp = Δp s , where Δp s is a steady-state pressure loss component, wherein the method includes using computational fluid dynamics (CFD) simulation to calculate the steady-state pressure loss component (Δp s ), and characterized in that the computational fluid dynamics (CFD) simulation is a steady-state computational fluid dynamics (CFD) simulation, and wherein, Δp s = c0 + c1·q + c2·|q|·q, where c0, c1, c2 are empirical coefficients, and q is the flow rate, or wherein, Δp s is expressed as Δp s = Δp0 + Δp1 + Δp2, where Δp0 is the zero-flow pressure, Δp1 is the viscous resistance of the straight portion of the passage or the portion of the passage where the ratio of the radius of curvature (R) to the radius (r) of the portion is equal to r / R < 10, Δp2 is the local resistance of the portion of the passage having a stenosis, and wherein, the viscous resistance (Δp1) is equal to: where f is the hemodynamic friction coefficient, is the density of the blood, and wherein, <d>is the "equivalent" inner diameter of the channel calculated for the model of the channel, the model of the channel having a circular cross-section and a length equal to the length of the channel for which the model is created, and simultaneously assuming that the pressure loss for laminar flow is the same for the channel and for the model of the channel, and <v>The "equivalent" flow velocity is calculated for the model of the channel assuming that the pressure loss in laminar flow is the same for the channel and for the model of the channel, and wherein the local resistance (Δp2) is proportional to is proportional.< / v> < / d> 2. The computer-implemented method according to claim 1, wherein, The viscous resistance (Δp1) is calculated using the following formula: wherein, the Euler number is equal to 3. The computer-implemented method according to claim 2, wherein, The Euler number is expressed as: And Re is expressed as: where η is the dynamic viscosity coefficient.

4. The computer-implemented method according to any one of claims 1-3, wherein, The local resistance (Δp2) is calculated using the following formula: where k s is an empirical parameter, and: Where ΔA refers to the change in the cross-sectional area of the channel due to stenosis, and S refers to the degree of stenosis, <a0>is the "equivalent" diameter using the narrow proximal end <d>Calculated "equivalent" cross-sectional area, and A s is the cross-sectional area at the narrow neck.< / d> 5. The computer-implemented method according to any one of claims 1-4, wherein The sum of the viscous resistance and the local resistance (Δp1 + Δp2) is calculated using the Lagrangian number: where k s is an empirical parameter, and: where ΔA is the change in the cross-sectional area of the channel due to stenosis, and S is the degree of stenosis, <a0>is the "equivalent" diameter using the narrow proximal end <d>Calculated "equivalent" cross-sectional area, and A s is the cross-sectional area at the narrow neck.< / d> 6. The computer-implemented method according to any one of claims 1-5, wherein, "equivalent" diameter ( <d>) satisfies the following system of equations: < / d> where η is the dynamic viscosity coefficient.

7. The computer-implemented method according to any one of claims 1-6, wherein, The pressure drop (Δp) is defined as Δp = Δp s + Δp t , where Δp t is the transient pressure loss component, and Δp t is expressed as: where t is time, k t is an empirical exponent defined in the range of 1 to 1.2, and <a0>is used <d>Calculated "equivalent" surface area, and the method includes calculating the transient pressure loss component (Δp t ).< / d> 8. The computer-implemented method according to any one of claims 1-7, wherein, The artery is a coronary artery, carotid artery, renal artery, peripheral artery, or aorta.

9. The computer-implemented method according to any one of claims 1-8, wherein, q is in the range of 1 ml / s to 15 ml / s, or q is equal to 1.5 ml / s, 3.5 ml / s, or 5 ml / s, or where q is in the range of q0 / 5 to 3q0, where q0 is a typical value of the flow rate of the artery.

10. The computer-implemented method according to any one of claims 1-9, wherein, The two cross-sections or the arterial passage between the two cross-sections have the same or similar topological structures, where a similarity index is used to define the similar topological structures, the similarity index taking into account the change in the inner diameter between the two cross-sections or the change in the inner diameter on the passage between the two cross-sections, the branches at the two cross-sections or the branches on the passage between the two cross-sections, or all of the above, and where the similarity index has the following values: at least 50%, 55%, 60%, 65%, 70%, 75%, 80%, 85%, 90%, 95%, or 99%.

11. The computer-implemented method according to any one of claims 1-10, wherein, r / R is lower than 9, 8, 7, or 6.

12. The computer-implemented method according to any one of claims 1-11, wherein, The method further includes calculating the fractional flow reserve (FFR).

13. A data processing system comprising means for performing the method according to any one of claims 1 - 12.

14. A computer program product comprising instructions which, when executed by a computer, cause the computer to perform the method according to any one of claims 1 to 12.

15. A computer-readable storage medium comprising instructions which, when executed by a computer, cause the computer to perform the method according to any one of claims 1 to 12.

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