How to calculate pressure drop across a cross section of an artery

Abstract

The present invention is a computer-implemented method for calculating pressure drop across cross sections of the aorta, coronary arteries, carotid arteries, renal arteries, or peripheral arteries. The method involves steady-state computational fluid dynamics (CFD) simulation and separates the pressure drop into two components: steady-state pressure loss and transient pressure loss. The former is expressed as a linear combination of pressures (zero flow, viscous resistance, and local resistance), while the latter is calculated according to the second law of motion. In embodiments, the steady-state pressure loss includes similarity invariants (Euler number, Lagrange number, and Reynolds number). The present invention enables rapid simulation assessment of human cardiac status (including fractional flow reserve) and has been positively evaluated in clinical trials.

Description

[Technical Field]

[0001] The present invention is in the field of medical diagnostics.The present invention is a computer-implemented method for assessing a person's cardiac condition on a computer. [Background technology]

[0002] Essentially, all fluid flows are transient, yet in practice this aspect is rarely considered. Most computational problems related to the use of fluid mechanics utilize a steady-flow approach, which provides ample information about the evolution of steady-state flows. Only a few practical problems over decades of research require the full complexity of time-dependent flow analysis. These practical problems include hydraulic shocks (water hammer, fluid hammer), flow measurement, and control devices that use fluids as a force transmission medium (e.g., antilock braking systems (ABS)), tides, and hydraulic waves. These practical problems exist in many technical fields, including aerospace and automotive. The full complexity of time-dependent evolution has also been studied at a more fundamental level by two hydraulic engineers, Samuel Grace (e.g., 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 Johann Schönfeld. Schonfeld published a paper in 1949 (Schonfeld 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, including laminar flow and general slow motion, in several cases selected by the author. According to the paper, the relationship between total flow rate and pressure drop can be explained by the impedance of the tube or pipe. Interestingly, in the early 20th century, with the development of catheter insertion technology, cardiology became another application field that required consideration of all the complex factors of how blood flow changes over time.

[0003] The evolution of catheterization techniques (including cardiac catheterization) and the history of cardiovascular physiology dates back much further, dating back 400 years. William Harvey in 1628 and Stephen Hales 100 years later made major advances in describing blood circulation and measuring arterial pressure, respectively. The 19th century was a golden age in the development of our understanding of cardiovascular physiology. Claude Bernard, Étienne Jules, and Carl Ludwig are among the most notable contributors. Human cardiac catheterization began in the 20th century, with Werner Forssmann becoming the first to successfully catheterize a human heart. Werner Forssmann performed the procedure himself in 1929. In the late 1940s, André Cournand and Dickinson Richards introduced diagnostic cardiac catheterization, and in the 1960s, Mason Thornes reported selective coronary angiography. In the 1970s, Andreas Gruenzig pioneered catheter-based interventions, and key techniques have expanded and improved. Today, the Soanes technique is rarely used, and percutaneous coronary interventions and coronary angiography primarily consist of approaches involving the percutaneous radial and percutaneous femoral arteries (Bourassa MG (2005), History of cardiac catheterization, The Canadian Journal of Cardiology, 21(12):1011-1014).

[0004] In the second half of the 20th century and into the 1970s, growing knowledge of the risks as well as the benefits of cardiac catheterization for dysfunction assessment attracted the attention of hydraulic engineers. In 1973, two groundbreaking studies were published by Donald Young and Frank Tsai.

[0005] The starting point for constructing the unsteady (transient) flow characteristics of arterial stenosis was the result of the theoretical analysis described in Schoenfeld's paper. TIFF2025540303000002.tif17133where Δp is the pressure drop, L is the distance, η is the liquid viscosity, D is the pipe diameter, ρ is the liquid density, v is the mean velocity, and dv / dt is the rate of change of the mean velocity. Fluid flow characteristics are well-established concepts well known to experts and include velocity, pressure change, Reynolds number, and other parameters.

[0006] The first half of the right-hand side of the above equation (1) represents the pressure drop due to viscous effects in steady flow, and the second half represents the additional pressure required to accelerate the liquid. The abstract of this paper states that the value of high-frequency inertial resistance (H) is smaller than the value of low-frequency inertial resistance (L), and that in the case of laminar flow, the difference is 20% for wide open channels or gaps, 38% for square pipes, and 33% for circular pipes. In the case of turbulent flow, the difference is 0.25-1.5% for rough or smooth open channels, 0.5-3.5% for gaps or rough open channels, 0.5-3% for smooth circular pipes, and 1-8% for rough circular pipes (Schonfeld (1949)). In another paper published in 1973, Young and Tsai (Young DF, Tsai FY (1973) Flow properties in models of arterial stenoses. II. Unsteady flow, Journal of Biomechanics, 6(5):547-559) developed and experimentally verified an approximate equation for the transient pressure drop across a stenosis. Experiments were performed in tubes, and the researchers investigated the effects of stenosis and unsteadiness on steady flow using asymmetric and axisymmetric models. The two researchers also investigated turbulence and its development using hot films. In this paper, Young and Tsai proved that (i) in models with severe stenosis, steady flow is slightly more stable than oscillatory flow, (ii) in models with mild stenosis, steady flow is less stable than the corresponding oscillatory flow, and (iii) three dimensionless parameters can characterize unsteady flow through a stenosis. These researchers referred to the equations of Fry et al. (1956, 1959) and Daily et al. (1956). In this study by Young and Tsai, the pressure drop is expressed as: TIFF2025540303000003.tif16134 where K v is the viscosity coefficient, η is the viscosity coefficient, D is the pipe diameter, v is the cross-sectional average instantaneous velocity in an obstacle-free pipe, K t is the turbulence coefficient, A0 is the cross-sectional area of ​​the pipe without obstacles, A1 is the minimum cross-sectional area with a constriction, ρ is the density of the fluid, K uは is the inertia coefficient, L is the distance between the pressure taps, dv / dt is the rate of change of the mean velocity, and |v|v is the vector length multiplied by v. The instantaneous mean velocity of a steady flow is v=v s +v n Since it can be expressed as f(t￣), the above equation becomes: TIFF2025540303000004.tif16134 and It can be made dimensionless like TIFF2025540303000005.tif16133, where ρ is the liquid density, v p is the peak value of the cross-sectional average velocity (v = v s +v n ), v s is the stationary 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 (ρv p D / η), v is the speed ratio (v / v p ), v is the cross-sectional average instantaneous velocity in an obstacle-free tube, K t is the turbulence coefficient, A0 is the cross-sectional area of ​​the tube without obstacles, A1 is the minimum cross-sectional area with a constriction, K u is the inertia coefficient, L is the distance between the pressure taps, τ is the characteristic time, df / dt is the change in function with time, I v is the viscous effect index, I t is the turbulence effect index, I u is the inertial effect exponent. |v￣|v￣ is the length of the vector multiplied by v￣.

[0007] 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 in vivo measurements of pressure drop across artificially stenotic arteries and the use of Equation (2) to predict pressure drop. Thirteen dogs were examined in this study, in which implants were inserted into the femoral arteries. Flow and pressure drop were measured across the stenotic area (severity range: 52.3-92.2%). Mean pressure drop ranged from 2-30 mmHg, with peak pressure drop ranging from 9-53 mmHg. The paper also noted that Young et al. were able to predict pressure drop within approximately 20% of experimental values. Developments in catheterization techniques combined with vascular imaging (particularly angiography) rapidly provided new context for Young and Tsai's findings. However, in the 70s of the 20th century, there is no knowledge as to which parameters (indicators, indicators, etc.) should be used as criteria for functional assessment.

[0008] A 1977 scientific paper by Young et al. (Young DF et al. (1977) Hemodynamics of arterial stenoses at elevated flow rates, Circulation Research, 41(1):99-107) focuses directly on critical stenosis, the level of stenosis at which blood flow is reduced. The paper describes experiments in 12 large mongrel dogs in which stenosis rates in the carotid and femoral arteries were reduced by 55.7 to 91.0%. In these experiments, the researchers used hollow cylindrical plugs and measured pressure drops and instantaneous flow rates. The average velocities ranged from 3.9 to 88.88 cm / s. The maximum flow rate (Q ms ) and the maximum flow rate without stenosis (Q mo ) ratio. Young et al. integrated equation (2) for one cycle of blood flow to obtain the following equation: TIFF2025540303000006.tif15134 is proposed, where Kv is the viscosity coefficient, η is the viscosity coefficient, D is the pipe diameter, K t where is the turbulence coefficient, A0 is the cross-sectional area of ​​the unobstructed vessel, A1 is the minimum cross-sectional area at the stenosis, ρ is the density of the fluid, and v is the time-averaged instantaneous flow velocity in the cycle. Young et al. have noted that when there is a large pressure loss distal to the stenosis and the blood flow through a particular artery is in a state of vasodilation, it can increase dramatically by 4-5 times (Young et al. (1977)).

[0009] Itzchak et al. (1977) Determination of the pressure drop across an arterial stenosis using angiocinedensitometry, The Yale Journal of Biology and Medicine, 50(4):375-381) conducted a study on 33 healthy patients and 23 patients with intermittent claudication and arteriosclerosis obliterans. The severity of intermittent claudication and arteriosclerosis obliterans is comparable. The purpose of this medical study was to demonstrate a method for calculating the pressure gradient across a stenosis using flow data obtained from angiocin concentration measurements. The researchers calculated the pressure drop across the stenosis, and the calculation was performed on the left external iliac artery. The Poiseuille flow equation was used to calculate the pressure gradient. The total pressure drop from the entrance to the stenosis to any point in the venous system can be expressed as follows: TIFF2025540303000007.tif17134where 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.

[0010] The researchers concluded that in healthy iliac arteries, the pressure gradient ranged from 0.017 to 0.083 mmHg, averaging 0.039 ± 0.018 mmHg, and the resistance to blood flow along the artery was 0.076 ± 0.082 mmHg min / L. In atherosclerotic arteries, the pressure gradient ranged from 0.0856 to 3.552 mmHg, averaging 0.982 ± 0.918 mmHg, and the resistance at the stenosis ranged from 0.26 to 9.6 mmHg min / L, averaging 2.45 ± 2.4 mmHg min / L (Itzchak et al., 1977).

[0011] Gould conducted an experiment in 10 chronically instrumented, unsedated dogs to examine pressure-flow characteristics (Gould KL (1978) Pressure-flow characters of coronary stenoses in unsedated dogs at rest and during coronary vasodilation, Circulation Research, 43(2):242-253). The experiment was performed at rest, and 100 left circumflex artery stenoses were subjected to pharmacological coronary dilation. A Doppler flowmeter was attached around the proximal portion of the dissected free circumflex artery to induce stenosis. A circumferential balloon deflater was sewn 2 mm distally. A small tapered catheter was placed in the distal main circumflex artery, anterior to the main branch. Catheters similar to the small tapered catheter were inserted into the pulmonary artery, aortic root, and left atrium. According to this experiment, a pressure loss of 50% or more caused by arterial luminal stenosis is related to flow velocity, which can be simply described as follows: TIFF2025540303000008.tif12134where ΔP is the pressure drop (mmHg), F is the pressure loss coefficient due to viscous friction, V is the coronary flow velocity (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 proved that the first and second halves of the right-hand side of equation (7) represent 65% and 35% of the total pressure loss at rest, respectively, and 33% and 67% of the total pressure loss at maximum coronary flow in a hyperemic state (Gould (1978)).

[0012] Mates et al. (1978) Fluid dynamics of coronary artery stenosis, Circulation research, 42(1):152-162) studied flow through isolated stenoses in large coronary arteries. In their experiments, detailed velocity and pressure measurements were performed using a large-scale coronary circulation model. Servovalves were used to simulate instantaneous peripheral resistance and pulsatile aortic pressure. The researchers analyzed a number of asymmetric and axisymmetric stenoses, and ultimately used four stenosis geometries for their experiments. The relationship between pressure and flow in the model was expressed as follows: TIFF2025540303000009.tif32134where L is the length of the stenosis, Q is the mean 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 minimum equivalent volume, ρ is the density of the fluid, and C d is the orifice discharge coefficient.

[0013] The paper acknowledges that the model chosen is very simple compared to more complex coronary circulation models. The chosen model consists of stenotic elements inserted in series into a straight rigid tube, and these elements have a distal bed resistance. It also points out that pressure recovery downstream of the stenosis is reduced (and in some cases even absent). Pressure loss is determined by the minimum area of ​​the stenosis, not its shape, and flow is quasi-steady at normal heart rates (Mates et al. (1978)). McMahon 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 two groups. The first group included 10 patients with normal left ventricular (LV) angiography, single-vessel coronary artery disease without collaterals, ischemic ST-T changes but no myocardial infarction, and new-onset refractory rest angina. The second group included five patients, similar to the first group, except these five patients presented with subendocardial infarction (SEI). The researchers studied the critical stenosis of the proximal coronary arteries based on the morphology obtained from the patients' quantitative angiography. McMahon et al. proposed that the pressure drop across a moderately severe stenosis can be theoretically estimated based on the following relationship: TIFF2025540303000010.tif33134 where k1 and k2 are constants based on blood viscosity, blood density, lesion shape and length, and flow rate assumptions; Q is the flow rate; d min is the minimum diameter of the stenotic area. The following results were obtained in Groups 1 and 2, respectively: minimum stenotic diameters of 0.88 ± 0.14 mm and 0.64 ± 0.08 mm, diameter reductions of 72% and 78%, and cross-sectional area reductions of 92% and 95%. The difference observed between the results of Groups 1 and 2 is due to the significant difference in hemodynamics (McMahon et al. (1979)).

[0014] Chapter 4 of Bessems's major work (Bessems DD (2007) On the propagation of pressure and flow waves through the patient-specific arterial system, PhD thesis, TU Eindhoven) provides an overview of his research into the relationship between pressure loss in a stenosis, described using the Womersley parameter and the Reynolds number, and the local flow characteristics and geometry of the stenosis. In this research, Bessems uses the results of Young and Tsai's research in 1973. The author calculates the geometry of the stenosis based on a uniaxially symmetric model. The final equation that expresses the relationship between flow characteristics and pressure loss through a stenosis is as follows: TIFF2025540303000011.tif11133 and TIFF2025540303000012.tif40133, where K v is the viscosity coefficient, α is the Womersley parameter, R s is the friction coefficient, q is the flow rate, ρ is the blood density, K t - turbulence coefficient, A0 is the cross-sectional area of ​​the lumen vessel proximal to the stenosis, A s is the cross-sectional area of ​​the lumen vessel at the neck of the stenosis, K u is the unsteady coefficient, L u is the coefficient of equation (18), t is time, K c is the offset, a0 is the radius of the lumen vessel proximal to the stenosis, a is the shape of the stenosis, L s is the length of the constriction, R0 is the friction coefficient for steady flow per unit length in the straight pipe, and z is the axial position. The authors compared the pressure loss from the numerical simulation with the pressure loss based on the equation in Young and Tsai's study, and the pressure loss from the final equation to the numerical result. The differences were up to 80% and 10%, respectively (Bessems (2007)). Interestingly, the Bessems (2007) study also mentioned a transient component in the pressure loss.

[0015] A similar concept was also presented by Liang et al. (Liang F et al. (2009) Multi-scale modeling of the human cardiovascular system with applications to aortic valvular and arterial stenoses, Medical & Biological Engineering & Computing, 47(7):743-755), which was based on multi-scale modeling including a computational model of the entire cardiovascular system. The 55 major arteries forming the arterial tree were modeled in one dimension, and their geometry was described based on data published in the studies of Stergiopulos et al. and Olufsen et al. If the relationship between pressure and flow in a normal heart valve is consistent with Bernoulli's law and the contributions of blood inertia and viscous resistance are taken into account, the pressure drop equation can be expressed as follows: TIFF2025540303000013.tif12133 where R cv is the viscosity coefficient, Q is the flow rate through the constriction, B cv is the flow separation factor, L cv is the coefficient of the inertia term, and Q is the time derivative of Q.

[0016] However, in severe aortic stenosis (AV), the energy loss associated with the sudden expansion of blood flow from the venous contraction to the ascending aorta is the primary cause of valvular pressure loss. TIFF2025540303000014.tif14133where ρ is blood density, Q av is the flow rate through the valve, EOA is the effective valve area, A ao is the cross-sectional area of ​​the ascending aorta, and t is time. In conclusion, the researchers stated that identifying the stenosis site is an important factor in the overall hemodynamic impact of arterial stenosis in the arterial system. Equation (15) shows similarity to a previous model for assessing pressure drop across a stenosis under transient conditions, but it only addresses the local assessment of pressure drop and cannot be directly applied to the entire vessel length (Liang et al. (2009)).

[0017] An attempt to directly utilize the stenotic pressure loss assessment model initiated by Donald Young and Frank Tsai for diagnosis was presented by Huo et al. (Huo Y et al. (2012) A validated predictive model of coronary fractional flow reserve, Journal of the Royal Society, Interface, 9(71):1325-1338).

[0018] These researchers used the physics of coronary artery stenosis to derive a pressure loss and myocardial FFR model. To validate this model, the researchers performed in vivo (stenosis induced using an occlusion cuff in eight pig arteries) and ex vivo (stenosis induced using an inflatable occlusion cuff in isolated arteries with asymmetric and symmetric tubing). The proposed analytical model uses the general Bernoulli equation: TIFF2025540303000015.tif9134 where ΔP convective is the energy loss due to flow convection, ΔP constriction is the energy loss due to the sudden narrowing of the cross-sectional area (CSA) from the normal proximal vessel to the stenotic site, and ΔP diffusive is the energy loss due to the diffusion of the flow, ΔP expansion is the energy loss due to the sudden expansion of the cross-sectional area (CSA) from the stenotic segment to the normal distal vessel. The total pressure loss at the stenotic segment can be expressed in two ways: first, (L s / Re s )·D s When α≧0.05, TIFF2025540303000016.tif13133And then (L s / Re s )·D s For α<0.05, TIFF2025540303000017.tif12133. In the formula, the following symbols are used: ρ is blood density, Q is flow rate, CSA s is the constriction cross-sectional area, α is the radius of the inviscid core (dimensionless), TIFF2025540303000018.tif2367 shows the pressure loss due to the sudden expansion of the CSA relative to the blunt flow velocity profile at the constriction exit, L s is the length of the stenosis, L e is the inlet length when α = 0.05, η is the kinematic viscosity, x is the position, TIFF2025540303000019.tif2465 shows the pressure loss due to the sudden expansion of the CSA for the parabolic flow velocity profile at the constriction exit, Re s is the Reynolds number at the entrance of the constriction, D s is the stenosis entrance diameter. Huo et al. confirmed that their proposed model was consistent with experimental results and also concluded that their coronary artery stenosis analysis model can predict FFR based on hyperemic coronary blood flow and stenosis dimensions without empirical parameters (Huo et al. (2012)).

[0019] In a paper by Itu et al. (Itu L et al. (2013) Non-invasive hemodynamic assessment of coarctation of the aorta: validation by in vivo measurements, Annals of Biomedical Engineering, 41(4):669-681), a method combining CFD-based assessment and an innovative non-invasive model personalization approach was proposed for non-invasive hemodynamic assessment of patients with coarctation of the aorta (CoA) before and after surgery. The researchers performed a comprehensive pressure drop analysis for coarctation of the aorta, TIFF2025540303000020.tif17133 and TIFF2025540303000021.tif36132, where K v is the viscosity coefficient, α is the Womersley number, R v c is the viscous resistance, q is the flow rate, ρ is the liquid density, K t is the turbulence coefficient, A0 is the normal cross-sectional area, Ac is the minimum cross-sectional area of ​​the contraction, K u is the inertia coefficient, L u is the contraction length, t is time, K c is the continuity coefficient, q is the mean flow rate, L c is the contraction length, and μ is the dynamic viscosity of the fluid.

[0020] In preliminary results, Itu et al. obtained an absolute error of less than 2 mmHg. A 3D geometric model that segmented the vascular tree was used to calculate arterial centerline and various radius measurements. To obtain the lumen of the vascular tree segments, the researchers used 3D contrast-enhanced MR angiograms (MRA) (Itu et al., 2013).

[0021] Beautiful and practical summaries of these studies, including those by Bessems (2007) and Itu L et al. (2013), are given in the patent specifications issued by Sharma et al. (2013): "Non-invasive functional assessment of coronary artery stenosis including simulation of hyperemia by changes in resting microvascular resistance" (US Patent No. 10373700B2) and Sharma et al. (2013) (2013) "Non-invasive functional assessment of coronary artery stenosis including simulation of hyperemia by changes in resting microvascular resistance," US Patent No. 10354744B2.

[0022] In clinical practice, functional assessment of circulatory dysfunction is based on two alternative strategies: (1) the magnitude of pressure changes assessed by blood flow assessment or (2) changes in blood flow volume under specific pressure conditions. Because worsening of cardiovascular symptoms often occurs under stress rather than at rest, assessment is typically performed under stress conditions (usually achieved through pharmacological measures). According to current standards (Lawton JS et al. (2022) 2021 ACC / AHA / SCAI Coronary Revascularization Guidelines: Summary: Report of the American College of Cardiology / American Heart Association Joint Clinical Practice Guidelines Committee, 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 changes.

[0023] It is recommended to use FFR, iFR, or equivalent coronary reserve ratio for the assessment (for example, the ESC guidelines also mention simulated FFR-CT). This ratio can be calculated as the ratio of the mean distal pressure to the mean proximal pressure: FFR = P d / P a , i.e., FFR=1-ΔP / P a This is the distal pressure (P d ) is the proximal pressure (e.g., coronary artery ostium (P a ) cross section) and the pressure loss (ΔP) in the flow path between both cross sections. The capital letter P represents the average pressure value, and the lower case letter p represents its instantaneous value (i.e., TIFF2025540303000022.tif20103 where T is the averaging time interval). Perfusion assessment, a method related to inadequate blood supply, is not currently the gold standard for diagnosis, even though it is used.

[0024] The in vivo FFR measurement procedure involves two pressure measurements (P a and P d), both of which are performed using a pressure wire placed inside an artery. This procedure is not time-consuming, but can be complicated to perform and may be harmful to the patient. Placing a pressure wire inside an artery is risky. This risk does not occur with simulation methods (including CFD (Computational Fluid Dynamics)), making simulation methods a very important and promising alternative to in vivo methods. Undoubtedly, computational fluid dynamics has been a challenging but successful approach, increasing clinicians' confidence in diagnostics (Cookcm et al. (2017) Diagnostic accuracy of computed tomography fractional flow reserve: a systematic review, JAMA cardiology, 2(7):803-810; Driessen RS et al. (2019) Comparison of coronary computed tomography, fractional flow reserve, and perfusion imaging in the diagnosis of ischemia, Journal of the American College of Cardiology, 73(2):161-173; Torii R, Yacoub MH (2021) CT-based fractional flow reserve: developments and expanded applications, Global cardiology science & practice, E202120; Pontone G et al. (2022) Clinical applications of cardiac computed tomography: a European Society of Cardiology consensus paper). Imaging Part II, European heart journal, Cardiovascular Imaging, 23(4):e136-e161). In particular, fractional flow reserve obtained from coronary CT angiography is gaining support.

[0025] Users of simulation diagnostic tools expect to achieve the highest accuracy, sensitivity, specificity, etc. Naturally, this involves using computational meshes with very high mesh densities, aiming to map the shape most accurately. Even if this one-way strategy can achieve improved accuracy, it does not take into account the practical aspects of the feasibility and usefulness of this type of tool in diagnosis. This is one of the aims of the present invention. In fact, an increase in computational mesh density is always accompanied by an exponential increase in the amount of computation. Furthermore, stability criteria for numerical CFD algorithms, especially the Courant-Friedrichs-Lewy (CFL) convergence condition (Courant R, Friedrichs K, Lewy H (1928) Uber die Partiellen Differenzengleichungen der mathematischen Physik, Mathematische Annalen, 100, 32-74; de Moura CA, Kubrusly CS (2013) Courant-Friedrichs-Lewy (CFL) condition: 80 years since its discovery, Birkhauser Boston) impose requirements on TIFF2025540303000023.tif15133Here, v is the velocity, Δh is the mesh density coefficient, and Δt is the time step width. These conditions result in a contradiction: Δh → 0, Δt → 0, making the calculation unstable.

[0026] In this paradoxical situation, not only does the computational complexity per time step Δt increase exponentially with the increase in the number of mesh nodes, but the number of time steps (Δt) also increases dramatically. It can be roughly estimated that simulating coronary artery blood flow over one cardiac cycle requires thousands or even millions of time steps. From a clinician's perspective, simulating coronary artery blood flow over one cardiac cycle requires an enormous amount of time, whereas a standard in-vivo examination typically takes 10–15 minutes to an hour. Furthermore, maintaining the necessary availability of simulation diagnostic services requires on-demand access to high-capacity computing power. Today, companies providing these diagnostic services use cloud computing technologies such as Microsoft Azure and Amazon Web Services to ensure the availability of their simulation diagnostic services. However, as simulation diagnostic services become routinely used on a global scale, it is expected that there will be insufficient resources to ensure the necessary on-demand availability without improving the performance of in silico simulations. Another factor is the large amount of electrical energy required for high-capacity computing. This large amount of electrical energy has a negative impact on the environment (carbon footprint). There is a need for an accurate simulation method for diagnostic assessment of human cardiac hemodynamic status that is available on demand and does not require extensive computational effort to implement. Summary of the Invention

[0027] The following brief description is intended to provide a better understanding of the principles and advantages of the invention as set forth in the appended claims, and is not intended to be limiting in any way. 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 embodiments, the artery can be a coronary artery, a carotid artery, a renal artery, a peripheral artery, or an 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 tube of length l. As will be understood by those skilled in the art, the length of the tube is the distance between the two cross-sections. Depending on the context, this disclosure uses both distance and length to describe aspects of the present invention. The tube is a portion of the artery that, when unoccluded, allows blood (or other fluids) 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 downstream from an artery inlet refers to the direction of blood flow from the inlet.

[0028] This method involves a steady-state computational fluid dynamics (CFD) simulation, but it is able to calculate not only the pressure drop (Δp) but also the transient pressure drop by separating the pressure drop into steady-state and transient pressure drop components.

[0029] The present invention takes a different approach from diagnostic support methods based on transient flow CFD simulation (e.g., vFFR, CT-FFR, FFR-CT). The present invention formulates the simulation in a completely different way by avoiding the use of transient CFD simulation to directly calculate transient flow. Instead, the present invention performs steady-state CFD simulation to arrive at the transient flow. This allows for simulation speeds hundreds or even thousands of times faster while maintaining the accuracy required for diagnosis. This allows for more rapid diagnosis and treatment. In a specific practical embodiment of the present invention, the present invention focuses on the evaluation of pressure loss.

[0030] In one aspect of the present invention, the steady-state pressure loss component is formulated to allow for reliable and rapid diagnosis.

[0031] Distal pressure p dis a function of the current flow q and the reduced proximal pressure p due to the transient pressure drop Δp across the flow path l between the distal and proximal locations. a is expressed as The FFR index is calculated by averaging the instantaneous pressure measurements. The FFR index is calculated as an instantaneous value. In the calculation, the distal pressure, proximal pressure, and current values ​​at the cross section can be derived using patient-specific variables. For details, see Kosior A, Mirota K, Tarnawski W (2019) Patient-specific modeling of hemodynamic parameters in coronary arteries (WO 2020 / 048642 A1 or EP 3820357 B1). Distal pressure values ​​can also be obtained from noninvasive measurements.

[0032] According to the present invention, the pressure drop (Δp) is expressed as: TIFF2025540303000025.tif12133where Δp s is the steady pressure loss component, and Δp t is the transient pressure loss component. Various embodiments described in the detailed description show how to calculate the steady-state and transient pressure loss components. Unless otherwise stated, all embodiments and elements of the embodiments can be combined to form new embodiments within the scope of this disclosure.

[0033] A primary application of the present invention is the assessment of transstenotic pressure loss and, in connection with this assessment, the calculation, through computer simulation, of diagnostic indices, in particular fractional flow reserve (FFR), which are relevant, for example, for the diagnosis and treatment of pathologies related to the human heart.

[0034] In one embodiment, the pressure loss is calculated using only the steady-state pressure loss component. TIFF2025540303000026.tif11134

[0035] The present invention provides various methods for calculating the steady-state pressure drop component. In one embodiment, the steady-state pressure drop component is expressed as: TIFF2025540303000027.tif11133 where c0, c1, and c2 are empirical coefficients that can be calculated using steady-state computational fluid dynamics (CFD) simulations. In this embodiment, the FFR can be calculated, for example, using steady-state CFD simulations. An advantage of this embodiment is the high computational speed required to obtain diagnostically and therapeutically relevant FFR metrics.

[0036] In another embodiment, the steady-state pressure drop component can be expressed as: TIFF2025540303000028.tif12133 where Δp0 is the zero flow pressure, Δp1 is the viscous resistance of a straight section of the tube or a section of the tube where the ratio of the radius of curvature (R) to the section radius (r) is r / R < 10, and Δp2 is the local resistance of a section of the tube with a stenosis. Having a ratio of the radius of curvature (R) to the section radius (r) of r / R < 10 ensures that the tube or section is relatively straight. In embodiments, r / R can be less than 9, 8, 7, or 6. A stenosis in an arterial vessel has an established meaning known to those skilled in the art. A stenosis can be understood as a narrowing or blockage of the artery.

[0037] In one embodiment of the present invention, the viscous drag (Δp1) is TIFF2025540303000029.tif15132, where f is the hemodynamic friction coefficient and ρ is the density of blood. 〈d〉 is the "equivalent" inner diameter of the tube. 〈d〉 is not measured but is calculated for a model tube with a circular cross section of the same length as the tube for which the model is created. 〈d〉 is calculated assuming that the laminar pressure drop is the same for the tube and the model tube. 〈v〉 is the "equivalent" flow velocity and is calculated for the model tube assuming that the laminar pressure drop is the same for the tube and the model tube. Local resistance (Δp2) is calculated as ρ〈v〉 2 / 2. Calculating the pressure drop using Δp1 and Δp2 reliably reproduces the results obtained in clinical trials without requiring significant computing power.

[0038] It should be noted that in the field of the present invention, "equivalent" internal diameter and "equivalent" flow rate are established terms and have different meanings from "equivalent" in law. In an embodiment, the pressure drop (Δp) includes a transient pressure loss component. TIFF2025540303000030.tif11133

[0039] In these embodiments, the transient pressure loss component (Δp t ) can be calculated according to Isaac Newton's second law of motion. More specifically, the transient pressure loss component (Δp t ) is expressed as follows: TIFF2025540303000031.tif12133 where t is time and k t is an empirical index defined in the range of 1 to 1.2. In an embodiment, k t is equal to 1.6, 1.7, 1.8, 1.85, 1.9, 1.95, 1.96, 1.97, or 1.98. <a0>is the "equivalent" surface area, which is calculated rather than measured. <a0>For example, <d>Using <a0> =(π / 4) <d> 4 Similarly, a person skilled in the art can calculate <a0>Using <v>The steady pressure loss component and the transient pressure loss component (Δp t The pressure drop Δp, which includes the calculation of Δp, is the transient pressure drop described above. The transient pressure drop is reconstructed based on a proposed relationship describing the flow characteristics, where empirical parameters are selected using the results of a series of steady-state CFD simulations. This aspect of the invention eliminates the need for complex and difficult transient CFD simulations, saving time and avoiding the use of high computer power. This makes the invention available on demand and reduces the environmental impact of simulation diagnostics.

[0040] 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 entrance of the coronary artery and a point within the coronary artery downstream from the entrance of the coronary artery.

[0041] In more specific embodiments, the two cross sections and / or the vessels between the two cross sections have the same or similar topology. Similar topology is defined using a similarity index that takes into account changes in internal diameter between the two cross sections, and / or crossing of arteries between the two cross sections, branches at the two cross sections, and / or crossing of arteries between the two cross sections, or all of these. In embodiments of the invention, the similarity index can have a value of 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 50% or greater, 55% or greater, etc. Various details and other embodiments of the invention are described in the following description. [Brief explanation of the drawings]

[0042] The invention will now be described in more detail with reference to the accompanying drawings. [Figure 1] FIG. 1 shows the results of a coronary computed tomography angiography (CCTA) and a visualization of the features of the present invention used to obtain a complete transient pressure drop in the coronary arteries. [Figure 2] FIG. 2 contains a diagram of one embodiment of the present invention. [Figure 3] FIG. 3 shows an example of determining the "equivalent" diameter of a patient according to the present invention. [Figure 4] FIG. 4 shows the calculation of steady state pressure drop including Lagrange and Reynolds numbers according to the present invention. [Figure 5] Figure 5 shows a comparison of fractional flow reserve measurements for 1960 in vivo human cardiac cycles, calculated based on approximations of the steady-state pressure loss component. [Figure 6] FIG. 6 includes a comparison of transient pressure drops from medical data and those reconstructed using the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0043] In one embodiment of the present invention, the established relationship representing the pressure drop Δp between two cross sections of an artery is given by the following equation: where c0, c1, and c2 are empirical coefficients and q is the flow rate. The empirical coefficient values ​​can be calculated using the results of a series of steady-state CFD simulations, which allows the pressure drop Δp to be calculated using the steady-state simulations. s The value of the empirical coefficient can be calculated using the results of clinical trials or a combination of clinical trials and steady-state CFD simulations. This allows the pressure drop Δp s is a steady component. In the embodiment, Δp s Only (Δp t (without q), which can be used to obtain, for example, FFR. In a more preferred embodiment, q is in the range of 1 ml / s to 15 ml / s. This ensures more accurate results. Even more preferably, q is 1.5 ml / s, 3.5 ml / s, or 5 ml / s. Alternatively, the most accurate results can be obtained by the following procedure: (1) approximating a typical q value for a particular artery using, for example, Milnor WR (1982) Normal hemodynamic state, in Hemodynamics, Baltimore, Williams & Wilkins; (2) establishing a range from q / 5 to 3q; and (3) performing calculations within this range.

[0044] Δp t represents the inertial effects that may occur under unsteady flow conditions. t is the transient component. This component is calculated according to Isaac Newton's second law of motion, not the result of a steady-state CFD simulation. TIFF2025540303000033.tif14134

[0045] Therefore, the quotient of the forces resulting from the acceleration / deceleration of a mass ρlA between two cross sections (for example, between the proximal and distal cross sections) is dv / dt = (1 / A)·(dq / dt), where l is the distance and A is the cross-sectional area of ​​the artery. t is an empirical index (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 embodiments, this empirical index can have different values, including, 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.

[0046] Transient component Δp in the present invention t The method for determining is fixed and is shown in the above equation (33). On the other hand, the steady-state component Δp s can be determined using any of a number of methods contemplated by the present invention.

[0047] As can be seen from equation (33), in this embodiment, transient CFD calculations are not used to calculate the transient pressure loss, but only steady-state CFD calculations are used, which improves the calculation speed of the pressure loss Δp by several orders of magnitude.

[0048] In one embodiment of the present invention, the pressure loss Δp in the steady state s is expressed as a superposition of the sum of three physical components. The Δp0 component is specific to coronary circulation simulations and corresponds to zero flow pressure (Hoffman JIE (1990) Pressure-flow relationship of the coronary arteries, in: Kajiya F, Klassen GA, Spaan JAE, Hoffman JIE (eds) Coronary Circulation, Springer-Verlag Tokyo, 109-125). Generally, this value is very small, so its contribution to the total Δp (and ΔP) value is negligible. In healthy subjects, the contribution is approximately 14±7 mmHg, but can reach or even exceed 42 mmHg in coronary interventions for ST-segment elevation myocardial infarction alone (Patel N et al. (2015) Zero-flow pressure measured immediately after primary percutaneous coronary intervention for ST-segment elevation myocardial infarction has been shown to be the most effective invasive index for predicting the progression of myocardial infarction 6 months later in "An OxAMI Study (Oxford Acute Myocardial Infarction)", JACC, Cardiovascular interventions, 8(11):1410-1421). In one embodiment, Δp is a constant value.

[0049] The remaining two components, Δp1 and Δp2, vary significantly with flow. Δp1 and Δp2 are the viscous resistance in straight sections (or sections with relatively low curvature, i.e., sections where the ratio of the radius of curvature (R) to the cross-sectional radius of the vessel (r) is r / R<10, from the perspective of Freidoonimehr et al. (2021) Effect of artery curvature on the coronary fractional flow reserve, Physics of Fluids, 33, 031906) and the local resistance (regions of the artery that are narrowed or occluded, e.g., at stenoses). Regardless of the methodology used (experimental or simulation), those skilled in the art will recognize that in embodiments that include Δp1 and Δp2, the values ​​of Δp1 and Δp2 are most relevant to the accuracy of the functional assessment of pressure loss (up to 98%, as shown in clinical trials, e.g., at stenoses).

[0050] In one embodiment of the present invention, the viscous drag component Δp1 is approximated according to Henry-Darcy's law and expressed in the form (Bird RB, Stewart WE, Lightfoot EN (2007) Transport phenomenon, John Wiley & Sons; White FM (1999) Fluid Mechanics, Boston Mass: WCB / McGraw-Hill), TIFF2025540303000035.tif13133where f is the hemodynamic friction coefficient, l is the first cross section (proximal pressure p a From the location where the distal pressure p d where ρ is the distance to the coronary artery (where ρ is calculated), ρ is the density of the flow medium (here, blood), and 〈d〉 is the "equivalent" inner diameter of the coronary artery, calculated for a model coronary artery with a circular cross section and the same length as the actual coronary artery for which the model was created. It is assumed that the laminar flow pressure drop is the same for the actual coronary artery and the model coronary artery. 〈v〉 is the "equivalent" flow velocity calculated for a model coronary artery with a circular cross section and the same length as the actual coronary artery for which the model was created, and it is assumed that the laminar flow pressure drop is the same for the actual coronary artery and the model coronary artery. Those skilled in the art will appreciate that this approach yields reliable values ​​for 〈v〉 and 〈d〉, since the resolution of tomography measurements is too low to provide accurate values ​​of arterial flow velocity and inner diameter (at a resolution of 0.5 mm, d may be on the order of 4 mm). This embodiment has the same advantages as the embodiment involving Equation (34), including avoiding transient CFD simulation.

[0051] A visualization of an embodiment using 〈d〉 for CCTA results is shown in Figure 1. Figure 1(A) shows the aortic and coronary artery tree obtained by CCTA, and Figure 1(B) shows an image of the right coronary artery and its curved planar reconstruction (CPR). Figures 1(C), (D), and (E) show, respectively, the reconstruction principle of the "equivalent" diameter 〈d〉, the invariant used to obtain the transient pressure drop according to the present invention, and the relationship between steady-state and transient pressure drop.

[0052] In one embodiment, shown in Figure 3, the value of <d> is calculated using Equation (53) below, and the value of <v> is calculated using <d>. Figure 3 shows an example of using the Poiseuille number to determine the "equivalent" diameter of a patient. The patient is a 67-year-old male (height 176 cm, weight 92 kg, BMI 29.7, systolic and diastolic blood pressures 145 mmHg and 85 mmHg, respectively, and heart rate 72 beats / min). The dotted line in Figure 3 shows the relationship between the Poiseuille number and the "equivalent" diameter, and the solid line shows the "equivalent" diameter when the Poiseuille number is 32 (laminar flow). This is one way to determine the "equivalent" diameter of a patient. In one embodiment, <d> and <v> can be intermediate values ​​of the inner diameter of the pipe and the flow velocity through the pipe. In a different embodiment, the values ​​of <v> and <d> are selected such that, for example, mass flow rate, volume, momentum, or energy parameters match the flow velocity and inner diameter parameters obtained from measurements (e.g., tomography measurements). In one embodiment of the present invention, (35) is converted to a general dimensionless form: TIFF2025540303000036.tif13134In general, the left-hand side of the equation is the Euler number Considering that we define similarity invariants in the form TIFF2025540303000037.tif20118, we end up with The result is TIFF2025540303000038.tif12134.

[0053] This allows calculations to be performed using linear equations rather than quadratic equations, greatly increasing the speed of calculations and, as mentioned above, reducing the need for high computing power. In yet another embodiment, under various physiological conditions, blood flow in straight coronary arteries (excluding areas where plaque is present) exhibits laminar characteristics. This is because the Reynolds number, Re = 〈v〉 · 〈d〉 · ρ / η, where η is the viscosity coefficient, rarely exceeds several hundred. As a result, the fluid friction coefficient in a tube of circular cross section is assumed to be determined according to the Hagen-Poiseuille law: TIFF2025540303000039.tif16134Here, Re is a similarity invariant of the form of the Reynolds number as follows: TIFF2025540303000040.tif13134

[0054] In a preferred embodiment, the Euler number of the loss component in the straight section may be assumed to be: TIFF2025540303000041.tif13134 In one embodiment, B is equal to 32.

[0055] In one embodiment, the third component of the pressure loss (Δp2) is a local loss and has a value proportional to the dynamic pressure. TIFF2025540303000042.tif17134 In the preferred embodiment, Euler's number is expressed as: This means that the Euler number is proportional to a certain constant. Following the approximation of the Young-Tsai loss over a constriction, Bessems (2007), Itu (2012), and others mentioned above define this constant as (in general form Combine with the component with TIFF2025540303000044.tif25122, where: <a0>and A s are the luminal cross-sectional area of ​​the vessel based on the "equivalent" diameter at the proximal and neck of the stenosis, respectively. Considering that the flow-restricting effect of atherosclerotic plaque, along with the level of stenosis, has been described in the medical literature, we come to the following conclusion (Carnicelli AP et al. (2013) Cross-sectional Area for Calculation of Carotid Artery Stenosis in Computed Tomography Angiography, Journal of Vascular Surgery, 58(3):659-665). TIFF2025540303000045.tif14134Here, ΔA means the change in the cross-sectional area of ​​the lumen due to stenosis, and S means the degree of stenosis, which can be replaced as follows: TIFF2025540303000046.tif14133As a result, the dimensionless characteristic of the constriction resistance is given as follows: TIFF2025540303000047.tif16133 where k s is an empirical parameter that can be calculated using steady-state CFD simulation.

[0056] According to the present invention, the pressure loss (Δp s or ΔP s is related to Δp s is the instantaneous value of pressure loss, ΔP s is the intermediate value of the pressure loss) can be calculated using similarity flow invariants. This takes into account that both pressure loss components are actually added together, resulting in Eu = Eu1 + Eu2 (see equations (40) and (45) above), TIFF2025540303000048.tif12132, and introducing the Lagrange number, we obtain the following equation. TIFF2025540303000049.tif12134

[0057] In this way, a preferred embodiment of the flow characteristic describing Δp1+Δp2 is obtained. Similarly, the following formula is obtained: TIFF2025540303000051.tif12132 where K s =k s (S / (1-S)) 2 is an empirical parameter that represents the level of transmission loss through the stenosis. An advantage of this embodiment is that the calculation speed is improved by using a linear equation instead of, for example, a quadratic equation, without reducing the accuracy of the results obtained. In one embodiment, the cross section is cylindrical and B=32.

[0058] Based on the Lagrange and Reynolds numbers, the "equivalent" diameter 〈d〉 and K s is calculated as shown in Figure 4. All calculations are performed using the patient's medical data used in Figure 3.

[0059] In another embodiment of the present invention, the homothetic invariant system is s provides a universal relationship that expresses the distal pressure p d If we need to reconstruct the complete transient of the change of Δp as a function of time, t components must be added. For the avoidance of doubt, "fully transient" refers to a flow that has no reasonably observable trends in time. The values ​​of the similarity invariants (La, Re, Eu) are calculated according to commonly known definitions and the pressure drop Δp s and flow rate q, B or K s Empirical parameters such as B and K can be obtained using steady-state CFD simulations (so that, for example, s , only two calculations are required, resulting in a faster result. The diameter 〈d〉 is obtained from the segmentation of the geometric model (hence the surface area A = π〈d〉). 2 / 4). Meanwhile, the dynamic viscosity coefficient η can be chosen according to the patient's specific hemorheological properties (Baskurt OK (2007) Handbook of hemorheology and hemodynamics, IOS Press). Those skilled in the art know that viscosity can be calculated using various known models used for non-Newtonian fluids, and that a constant viscosity can be used for Newtonian fluids. For comparison, typical state-of-the-art approaches require thousands of time-consuming CFD simulations.

[0060] In one embodiment of the present invention, the diameter 〈d〉 does not directly reflect the diameter in the segmentation result, but is selected 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 approach to evaluating energy loss through a stenosis proposed by existing models (Schonfeld JC (1949), Young & Tsai (1973), Young & Tsai (1975), Young & Tsai (1977), and recent models). Only in the case of large-diameter arteries (as in Young & Tsai (1977)) or artificial shape models (even stenosis models), is it possible to measure the vessel diameter directly from the shape in vitro or in silico. This limitation is due to the low resolution of experimental shapes, and sufficient accuracy can only be achieved for large-diameter arteries or artificially simplified arteries (e.g., arteries represented as simple tubes).

[0061] In a preferred embodiment of the present invention, the "equivalent" diameter 〈d〉 is considered to be the diameter of a circular tube whose energy loss under steady laminar flow conditions is equal to the energy loss occurring in the actual geometry of an artery under the same flow conditions. Under these circumstances, the fluid friction factor f and the scaling factors for the Reynolds number (Re) and the Poiseuille number (Po) satisfy the following equation: TIFF2025540303000052.tif14133

[0062] At the same time, as mentioned above about the pressure loss in the straight section or low curvature section, TIFF2025540303000053.tif21132. Therefore, the friction coefficient is TIFF2025540303000054.tif14133TIFF2025540303000055.tif2191, the Poiseuille number is expressed by the following formula: TIFF2025540303000056.tif14131 Or by replacing the "equivalent" velocity 〈v〉 with the flow rate TIFF2025540303000057.tif14132 Here, the ratio Δp / q=R0 represents the resistance, which is a constant value in this embodiment. Thus, in a preferred embodiment of the present invention, the "equivalent" diameter <d> is considered to be that value that satisfies the system of equations set out below: TIFF2025540303000058.tif19134

[0063] This embodiment further improves the calculation speed without compromising the calculation accuracy. One embodiment of the present invention consists of five steps: (1) selecting an "equivalent" diameter <d>, (2) approximating the viscous segment of the flow characteristic, (3) approximating the local pressure loss segment of the flow characteristic, (4) calculating the steady-state pressure drop, and (5) calculating the transient (inertial) component of the pressure drop. This is illustrated in Figure 3, where <d> is approximately 2.064 nm. In the embodiment of Figure 3, <d> can be used to calculate <v>, as will be apparent to those skilled in the art. This entire procedure is repeated for all positions and branches for which steady-state CFD simulation results are available.

[0064] Step 1, a preliminary step for the reconstruction of flow characteristics, involves selecting a scale for describing the invariant flow using similarity invariants. The key here is the characteristic linear dimension scale, and therefore the equivalent diameter 〈d〉, since the equivalent diameter 〈d〉 depends on the velocity scale 〈v〉. In a more preferred embodiment, the scale is 〈d〉. In another embodiment, the scale is 〈v〉. According to the present invention, the value of the equivalent diameter 〈d〉 is calculated as described above to achieve consistency of the dispersion phenomenon between two cross sections within the artery. In one example, the two cross sections are the proximal and distal cross sections.

[0065] In step 2, select a flow characteristic from those described herein that is related to viscous energy dissipation in straight and shallow curvature sections of the coronary artery, which in one embodiment includes the flow characteristics defined in equations (32), (34), or (54).

[0066] In practical conditions (physiological conditions in humans), the pressure loss is usually the sum of the linear pressure loss and the local pressure loss. This is of great significance in the flow in coronary arteries, where there are large geometric variations. In fact, it can be very difficult to separate the linear (strictly viscous) component from the sum of the local pressure loss. In preferred embodiments of the present invention, rather than separating the pressure loss into the sum of the linear pressure loss and the local pressure loss, the value of the flow rate q, and therefore the velocity scale 〈v〉 and the Reynolds number, are chosen so that the second component of the flow characteristic is negligibly small, and therefore TIFF2025540303000059.tif18112. Formally, under ideal conditions of the embodiment, the value of parameter B is equal to 32. In other embodiments, the value of B varies. In one embodiment, B is treated as a constant at a fixed distal cross-sectional area (l away from the inlet). In this case, K s If ≠ 32, then K s = const, and only this part of the flow characteristics is considered in the process of determining this parameter.

[0067] Step 3 refers to the second element of the flow characteristic, which describes the local losses. Its purpose is to obtain the empirical parameter K s The goal is to determine the value of K. It is known physiologically that there can be many stenotic elements in a coronary artery, but the main one is the stenosis formed by atherosclerotic plaque. If there is no functionally significant stenosis in a particular branch, this element does not play a significant role in terms of physiological flow conditions. In one embodiment, K s The value of the parameter can be determined as the slope coefficient of a simple regression line for the La = f(Re) system, the intercept of which is equal to B·l / 〈d〉.

[0068] Step 4 is to calculate the pressure loss Δp using a steady CFD simulation. s Since La = Eu Re, the pressure loss Δp s is Δp s =ρ〈v〉 2 This embodiment is applicable to the calculation of the simulated FFR. This application comes from the fact that the average value obtained by step 4 is used to calculate the simulated FFR.

[0069] Figure 5 summarizes the results of the clinical trial. Patient medical data were provided in the clinical trial. Thirty female patients (eight of whom had myocardial ischemia) had a mean heart rate of 68.30 ± 5.06 / min, and mean systolic and diastolic blood pressures of 131.83 ± 11.03 mmHg and 79.47 ± 9.93 mmHg, respectively. Twenty-five male patients (17 of whom had myocardial ischemia) had a mean heart rate of 68.72 ± 6.02 / min, and mean 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. The medical data included 1,960 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 according to steps 1 to 4, and then the distal pressure was approximated (step 5). This resulted in the steady-state pressure loss component of steps 1 to 4, Δp s Only the transient pressure loss component Δp s and Δp t The FFR (simulation) could be calculated using the combination of

[0070] A correlation between in vivo FFR and simulated FFR was established. Δp s When only the linear component (Δp1) of the Δp1-Δp2 analysis was used, the correlation coefficient was 80.3%. When the local resistance component (Δp2) was used, the correlation coefficient was 93.8%, but 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 in step 4, the correlation increased to 98.7%, with a median difference of -0.009. Thus, the FFR values ​​obtained with Δp1 + Δp2 are diagnostically acceptable, with errors comparable to or smaller than those expected for in vivo FFR.

[0071] Figure 6 shows examples of in vivo and simulated pressure curves obtained using the same medical data as Figure 5. The simulated curves reproduce the curves obtained in vivo. The simulated pressure drop versus flow values ​​reproduce the values ​​obtained in vivo. Step 5 is to reconstruct the pressure drop as a function of flow rate, which includes the time course of the flow rate. TIFF2025540303000060.tif2084 is also included. In this step, the steady-state pressure value Δp obtained in step 4 is s and the transient pressure loss component Δp t As mentioned above, Δp t is the second principle of dynamics In the preferred embodiment, the empirical coefficient k t is 1.2.

[0072] The simulated FFR calculation results obtained in step 5 were evaluated with the results obtained in the clinical trial. See Figure 5. The correlation was 99.9% with a median error of only 1.69813 × 10 -5 is.

[0073] An embodiment implementing steps 1-5 is shown in Figure 2. The first step calculates the "equivalent" diameter 〈d〉 of the arterial branch or flow channel. Steps 2 and 3 calculate approximations of the linear resistance Δp1 and the local resistance Δp2. Step 4 calculates the steady-state component of the pressure drop (Δp s ) is calculated. This allows us to calculate the transient component of the pressure drop (Δp t These five steps allow the pressure drop in the coronary arteries to be evaluated, including the transient component.

[0074] In either step, the density (ρ) and dynamic viscosity (η) can be calculated as follows: Basically, ρ and η are patient-specific parameters. For example, the density of blood is calculated using Hawksley's equation: ρ = ρ pl (1-HCT)+HCT·ρ RBC It can be approximated as follows: where ρ pl , ρ RBC are the densities of plasma and red blood cells, respectively, and HCT is the hematocrit value (De Gruttola S et al. (2005) Computational Simulation of a Non-Newtonian Model of the Blood Separation Process, Artif Organs, 29(12):949-59). A wide variety of blood rheology models can be applied here. For example, there are the Carreau, Cross, and Yasuda models, as well as many others (Yilmaz F, Gundogdu MY (2008) A Critical Review on Blood Flow in Large Arteries; Relevance to Blood Rheology, Viscosity Models, and Physiological Conditions, Korea-Australia Rheology Journal, 20(4):197-211). Although it is possible to introduce factors of individual differences in this way, in a preferred embodiment, these parameters are selected as constants. In this preferred embodiment, this does not significantly affect the calculation accuracy. < / v> < / d> < / a0> < / d>

Claims

1. 1. A computer-implemented method for calculating a pressure drop (Δp) between a first cross-section and a second cross-section of an artery, comprising: The second cross section is downstream of the first cross section, and the two cross sections are separated by an arterial tube having a length (l), and the pressure drop (Δp) is Δp = Δp ｓ and Δp ｓ is a steady pressure loss component, and the method uses computational fluid dynamics (CFD) simulation to calculate the steady pressure loss component (Δp ｓ ), wherein the computational fluid dynamics (CFD) simulation is a steady-state computational fluid dynamics (CFD) simulation; where c ０ , c １ , c ２ is an experience coefficient, q is the flow rate, Or, Δp ｓ is Δp ｓ = Δp ０ +Δp １ +Δp ２ and Δp ０ is the zero flow pressure, Δp １ is the viscous resistance of the straight section of the pipe or the section of the pipe where the radius of curvature (R) and the radius of the section (r) are r / R<10, Δp ２ is the local resistance of the tube part with the constriction, and the viscous resistance (Δp １ ) is expressed as where f is the hemodynamic friction coefficient, ρ is the density of blood, <d> is the "equivalent" inner diameter of a model tube with a circular cross section and a length identical to that of the tube for which the model was created, calculated by assuming that the laminar pressure loss is the same for the tube and the model tube, <v> is the "equivalent" flow velocity for the model tube, calculated by assuming that the laminar pressure loss is the same for the tube and the model tube, and the local resistance (Δp ２ ) is ρ(〈v〉 ２ / 2).

2. 2. The computer-implemented method of claim 1, wherein the viscous resistance (Δp １ )teeth, It is calculated using Here, the Euler number is Eu １ = Δp １ / (ρ・〈v〉 ２ ) is equivalent to the method.

3. 3. The computer-implemented method of claim 2, wherein the Euler number is: is expressed as Re is expressed as follows: where η is the dynamic viscosity coefficient.

4. 4. The computer-implemented method according to claim 1, wherein the local resistance (Δp ２ )teeth, It is calculated using Here, k ｓ is an empirical parameter, and where ΔA means the change in the cross-sectional area of ​​the tube due to constriction, S means the degree of constriction, and <A ０ > is the "equivalent" cross-sectional area calculated using the proximal "equivalent" diameter <d> of the stenosis, and A ｓ is the cross-sectional area of ​​the stenosis at the neck; method.

5. 5. The computer-implemented method according to claim 1, wherein the sum of the viscous resistance and the local resistance (Δp １ +Δp ２ ) is the Lagrange number is calculated using Here, k ｓ is an empirical parameter, and where ΔA means the change in the cross-sectional area of ​​the tube due to constriction, S means the degree of constriction, and <A ０ > is the "equivalent" cross-sectional area calculated using the proximal "equivalent" diameter <d> of the stenosis, and A ｓ is the cross-sectional area of ​​the stenosis at the neck, method.

6. 6. The computer-implemented method of claim 1, wherein the "equivalent" diameter (<d>) is solved by the following simultaneous equations: Fulfilling where η is the dynamic viscosity coefficient. method.

7. 7. The computer-implemented method of claim 1, wherein the pressure drop (Δp) is Δp=Δp ｓ +Δp ｔ is defined as Δp ｔ is the transient pressure loss component, and Δp ｔ teeth is expressed as where t is time, k ｔ is an empirical index defined in the range of 1 to 1.2, <A ０ > is the "equivalent" surface area calculated using <d>, and this method includes the transient pressure drop component (Δp ｔ ) method.

8. 8. The computer-implemented method of claim 1, wherein the artery is a coronary artery, a carotid artery, a renal artery, a peripheral artery, or an aorta.

9. 9. The computer-implemented method of claim 1, wherein q is in the range of 1 ml / sec to 15 ml / sec, or q is equal to 1.5 ml / sec, 3.5 ml / sec, or 5 ml / sec, or q is equal to q ０ / 5-3q ０ where q ０ is a typical value of the arterial flow rate.

10. 10. The computer-implemented method of claim 1, wherein the two cross sections or the arterial ducts between the two cross sections have the same or similar topology, where similar topology is defined using a similarity index that takes into account changes in internal diameter between the two cross sections or across the duct between the two cross sections, branches at the two cross sections or across the duct between the two cross sections, or all of these, and the similarity index has a value of at least 50%, 55%, 60%, 65%, 70%, 75%, 80%, 85%, 90%, 95%, or 99%. method.

11. 11. The computer-implemented method of claim 1, wherein r / R is less than 9, 8, 7, or 6.

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

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

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

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

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