FR3157793A1 - COMPUTER-IMPLEMENTED METHOD FOR ESTIMATING THE FRACTION OF CORONARY RESERVE FLOW, SYSTEM - Google Patents

Abstract

COMPUTER-IMPLEMENTED METHOD FOR ESTIMATING THE CORONARY RESERVE FLOW FRACTION, SYSTEM Method for estimating the coronary reserve flow fraction (FFR) of a vessel comprising: Acquisition (ACQ1) of a first image from an imaging system; Extraction (EXT1) of a first data set (ENS1) defining anatomical descriptors of a first vessel; Acquisition (ACQ2) of a second data set (ENS2) comprising at least the heart rate, the systolic pressure, and the myocardial mass; Generation (GEN1) of an output (S1) defining a prediction of a coronary flow reserve fraction (FFR) quantity by means of executing a first machine learning model (MLA1) to produce an output (S1), said first machine learning model (MLA1) implementing a parameterizable loss function including at least a first factor modeling at least one parameterized physical constraint optimized during the training of said model (MLA1), said parameterized physical constraint resulting in particular from a numerical model of hemodynamic equations (MOD1). Figure for abstract: Fig.5

Description

COMPUTER-IMPLEMENTED METHOD FOR ESTIMATING THE FRACTION OF CORONARY RESERVE FLOW, SYSTEM Field of invention

The field of the invention relates to methods and systems for predicting a value of the coronary reserve called FFR for the characterization of coronary flow. The field of the invention relates more particularly to non-invasive methods implementing software means for characterizing coronary flow. The invention relates to means providing decision support for the placement of a stent, the invasive deepening of a pathology or on the contrary to rule out a risk of infarction for example.

State of the art

Currently, the characterization of coronary flow in a patient, particularly to consider whether or not to place a stent in the case of the presence of stenoses impacting coronary reserve, involves numerous steps to be implemented and generally requires an invasive intervention in the patient.

One of the most important constants in the evaluation and characterization of coronary flow is the coronary flow reserve, called FFR and meaning "Fractional Flow Reserve" in the English literature. Generally, the measurement of FFR with the required precision is carried out from an invasive coronary angiography by inserting a catheter with a pressure sensor. Indeed, the FFR can be expressed as the ratio between two pressures measured upstream and downstream of a vessel, considering an upstream pressure measured at the ostium in order to characterize the coronary flow of the vessel. Certain FFR values may lead to patient monitoring, one or more in-depth examinations, or even an intervention for the placement of a stent.

Some solutions are based solely on an approach implementing a hemodynamic model of the vessels from the Navier Stokes equations. This is for example the case of the solutions described in application US2012/0041318 published on February 16, 2012. A problem with this approach is that such a system is complex to solve because it is a system of differential equations and the latter is solved in 3D. Such a system can involve significant computational resources. Finally, such a model does not take into account a customization of the coefficients of the system of differential equations, in particular the resistance coefficients, the compliance coefficients, and other boundary conditions.

Other solutions based solely on a machine learning approach do not allow optimal learning of a model to be trained. Indeed, such a model requires training on thousands, even millions of data points due to the large number of numerical parameters measured in each patient corresponding to all the specifications of each patient and to all the flow configurations of blood flows within each vessel. This is particularly the case of the application published under number FR3096498 and filed on 05/23/2019, with a model trained on 200 patients. However, the configurations can be very diverse depending on the type of vessel, the elasticity of the vessels, their length, the number of stenoses or other singularities present in the vessel, the flow rate, the age of the patient, etc.

Finally, a major constraint is that data collection during training requires measuring variables invasively and at different points in each vessel. Therefore, to date, there is no training of a machine learning model based on in vivo training data that would allow the construction of a reliable model.

There is therefore a challenge in proposing a reliable, robust and non-invasive alternative to existing solutions.

• Acquisition d’au moins une première image d’un système d’imagerie ;
• Extraction d’un premier ensemble de données définissant des descripteurs anatomiques d’un premier vaisseau de ladite première image ;
• Acquisition d’un second ensemble de données, dites données patient, comportant au moins la fréquence cardiaque et la masse myocardique ;
• Génération d’une entrée comportant le premier ensemble de données, le second ensemble de données ;
• Génération d’une sortie définissant une prédiction d’une quantité de fraction de flux de réserve coronarienne au moyen de l’exécution d’un premier modèle d’apprentissage machine pour produire une sortie, ledit premier modèle d’apprentissage machine mettant en œuvre une fonction de perte paramétrable incluant au moins un premier facteur modélisant au moins une contrainte physique paramétrée optimisée lors de l’entrainement dudit modèle, ladite contrainte physique paramétrée résultant notamment d’un premier modèle numérique d’équations hémodynamiques.

According to one aspect, the invention relates to a method for estimating the coronary reserve flow fraction of a vessel comprising:

• Acquisition of at least a first image from an imaging system;
• Extracting a first data set defining anatomical descriptors of a first vessel of said first image;
• Acquisition of a second set of data, called patient data, comprising at least the heart rate and the myocardial mass;
• Generating an input comprising the first data set, the second data set;
• Generating an output defining a prediction of a coronary reserve flow fraction quantity by executing a first machine learning model to produce an output, said first machine learning model implementing a parameterizable loss function including at least a first factor modeling at least one parameterized physical constraint optimized during the training of said model, said parameterized physical constraint resulting in particular from a first numerical model of hemodynamic equations.

The method advantageously makes it possible to estimate the pressure at each point of the vessel and therefore makes it possible to estimate the fraction of the coronary reserve flow at each point of the vessel. One advantage is to have an estimate of the fraction of the coronary reserve flow that is precise and faithful to the specific case of the individual considered without using an invasive measurement. One advantage of using a cost function integrating a physical constraint is to make the learning of the machine learning model more efficient, in particular by using heterogeneous training data, i.e. from different sources.

According to one embodiment, the extraction of the first set of data is obtained by means of a first segmentation of a representation of a set of vessels acquired by the imaging system, said extraction resulting from a selection of at least one first vessel displayed on a display, the selection of a point of the vessel.

One advantage is to make the most of the images acquired by an imaging system, particularly to extract anatomical data.

According to one embodiment, the anatomical descriptors are extracted from a representation of the first vessel, at least one first descriptor comprising data characteristic of the radius or diameter of the section in each position of the first vessel and at least one second descriptor comprising the positioning within the length of the vessel of at least one narrowing of the diameter defining the presence of at least one stenosis.

One advantage is to use personalized patient data to train the machine learning model and leverage the trained machine learning model.

According to one embodiment, the second descriptor comprises data characteristic of the inlet section of at least one portion defining the narrowing of the first vessel and data characteristic of the outlet section of at least one portion defining the narrowing of said first vessel.

• Un troisième descripteur correspondant à la courbure locale du vaisseau et/ou ;
• Un quatrième descripteur correspondant à la longueur d’une sténose et/ou ;
• Un cinquième descripteur correspondant à une dilatation locale du vaisseau (anévrisme).

According to one embodiment, the extracted anatomical descriptors also include:

• A third descriptor corresponding to the local curvature of the vessel and/or;
• A fourth descriptor corresponding to the length of a stenosis and/or;
• A fifth descriptor corresponding to a local dilation of the vessel (aneurysm).

One advantage is that it allows certain parts of the vessel to be identified in order to characterize them and obtain a better estimate of the pressures thanks to the trained machine learning model.

According to one embodiment, the method comprises estimating the rate of incoming blood flow at rest from at least the heart rate, the systolic pressure and the myocardial mass, the second set of data comprising the value of said rate of incoming blood flow.

An advantage is having measurable input data for each patient, the consideration of which allows a better estimation of the outlet pressure at each point of the first vessel.

According to one embodiment, the estimation of the flow rate of the incoming hyperemic flow at rest from at least the heart rate, the systolic pressure and the myocardial mass, the second set of data comprising the value of said flow rate of the incoming hyperemic flow.

An advantage is having measurable input data for each patient, the consideration of which allows a better estimation of the outlet pressure at each point of the first vessel.

• Un Age de l’individu ;
• Un genre de l’individu ;
• Un indicateur de diabète de l’individu ;
• Un indicateur d’hypertension de l’individu.

According to one embodiment, the second set of patient data notably comprises:

• An Age of the Individual;
• A kind of individual;
• An indicator of the individual's diabetes;
• An indicator of individual hypertension.

One advantage is having personalized input data, especially from data that can be collected easily and allows for improved prediction of output pressure based on the patient profile.

According to one embodiment, the first machine learning model is a DeepONet network whose loss function incorporates a second factor defining a loss function of a differential mathematical operator.

One advantage is to exploit a learning model without having to model and simulate a system of differential equations when using the data to estimate pressures within a first vessel.

According to one embodiment, the first factor is modeled during the training of the machine learning model by a parametric function whose parameters are dependent on the solutions of the equations of the numerical model of hemodynamic equations characterizing the hemodynamic flow in the first vessel, said parametric function being implemented by a continuous non-linear operator, the resolution of said parametric function making it possible to generate value limits outside of which the cost function is penalized.

One advantage is to obtain a more efficient function whose learning has made it possible to remove inconsistent values, for example those violating physical laws.

• Un modèle de conservation de la masse ;
• Un modèle de conservation de la quantité de mouvement ;
• Une équation d’état de la pression.

According to one embodiment, the equations characterizing the hemodynamic flow in the first vessel to optimize the cost function and modeled by the first hemodynamic numerical model which comprises a system of equations comprising:

• A model of conservation of mass;
• A model of conservation of momentum;
• An equation of state for pressure.

An advantage is to take into account several constraints characterized by different physics equations and making it possible to limit the training domain.

According to one embodiment, the method comprises a training step carried out from a third set of training data characterizing portions of pipes each modeling an irregular geometry characteristic of at least one predefined section profile, the training of the machine learning model being supervised so that the values characterizing the flow of a fluid in each portion are measured on a test bench comprising at least one portion of test pipe and sensors making it possible to measure test data at different points of said test bench, including in particular the flow rate, the pressure, the viscosity, the resistance of the fluid, the Reynolds number and the density of the fluid.

An advantage of being able to generate a large amount of training data taking into account many anatomical geometries without requiring testing a large variety of vessel geometries from different patients.

According to one embodiment, the method comprises a characterization of the different portions of the test bench, each portion being characterized by a radius of artery to be modeled, a length of artery to be modeled, an elasticity or a compliance and a type of portion.

According to one embodiment, the method comprises a pump, a clock and a means for recording a pumping frequency so as to inject a fluid into the test bench reproducing a periodic activity relating to the cardiac cycle characterizing the different portions of the test bench.

An advantage is that it allows for the control of a wide variety of in vitro model parameters and the simulation of different virtual patients representing a diversity of patients.

• d’au moins une résistance de sortie modélisation la force qui s’oppose à l’écoulement du sang dans la portion du premier vaisseau ;
• d’au moins une impédance ;
• d’au moins une valeur de viscosité du fluide ;
• d’au moins une valeur de la pression à l’équilibre.

According to one embodiment, the third data set also comprises a measurement and a calculation for at least one point of each predefined pipeline portion:

• of at least one output resistance modeling the force which opposes the flow of blood in the portion of the first vessel;
• of at least one impedance;
• of at least one fluid viscosity value;
• of at least one value of the equilibrium pressure.

One advantage is that it allows the calibration of a test bench with characteristics similar to real vessel anatomical features.

According to one embodiment, the method comprises a training step carried out from a second set of training data characterizing the geometry of portions of vessels obtained from an imaging system, the training of the machine learning model being supervised so that the values characterizing the flow of a fluid in the first vessel are estimated by a second model of hemodynamic equations called a digital solver, to define test and validation data for the machine learning model.

One advantage is that it has a great possibility of training from medical data such as real patient images from which the values and flow characteristics can be estimated to train the machine learning model of the invention.

• un premier type de segment définissant des portions sensiblement régulières correspondant à des portions de vaisseaux présentant une section diminuant progressivement d’une extrémité proximale vers une extrémité distale ;
• un second type de segment définissant au moins une portion irrégulière, ladite au moins une portion irrégulière correspondant à une portion comportant au moins un rétrécissement du diamètre supérieur à un seuil prédéfini,

According to one embodiment, a second segmentation of the first vessel automatically generates a plurality of segment types including:

• a first type of segment defining substantially regular portions corresponding to portions of vessels having a section gradually decreasing from a proximal end towards a distal end;
• a second type of segment defining at least one irregular portion, said at least one irregular portion corresponding to a portion comprising at least one narrowing of the diameter greater than a predefined threshold,

each of the sets of segments having a system of equations modeling the hemodynamic flow in said portion.

• Un troisième type segment caractérisé par la présence d’une courbure locale du vaisseau supérieure à un seuil prédéfini et/ou ;
• Un quatrième type segment caractérisé par une dilatation locale du vaisseau (anévrisme).
• Un avantage est de modéliser l’écoulement au plus proche de la réalité de l’écoulement dans chaque portion tenant compte de l’anatomie du vaisseau considéré.

According to one embodiment, the second segmentation of the first vessel automatically generates a plurality of segment types including:

• A third segment type characterized by the presence of a local curvature of the vessel greater than a predefined threshold and/or;
• A fourth type segment characterized by local dilation of the vessel (aneurysm).
• An advantage is to model the flow as close as possible to the reality of the flow in each portion taking into account the anatomy of the vessel considered.

According to one embodiment, the numerical solver comprises a system of equations characterizing the physics of the flow of a hemodynamic flow from the incompressible Navier Stokes equations applied to at least one portion of a single-dimensional vessel.

One advantage is to simplify the equations and allow easier training by reducing the dimension of the solver.

• Un modèle de conservation de la masse et/ou ;
• Un modèle de conservation de la quantité de mouvement et/ou ;
• Une équation d’état de la pression et/ou ;
• Un premier modèle d’un delta de pression et/ou ;
• Un second modèle d’un delta de pression établi à partir des mesures in-vitro.

According to one embodiment, the system of equations comprises a second hemodynamic digital model comprising for each portion of vessel at least one model among which:

• A model of conservation of mass and/or;
• A model of conservation of momentum and/or;
• An equation of state for pressure and/or;
• A first model of a pressure delta and/or;
• A second model of a pressure delta established from in vitro measurements.

According to one embodiment, the method comprises a calculation of the blood flow characteristics in each portion of the first vessel at each point of the axis of the vessel from the second hemodynamic model applied locally to each of the portions.

According to one embodiment, the second hemodynamic digital model makes it possible to estimate for each portion of vessel of the second type a pressure delta from a pressure delta model comprising a viscous resistance coefficient and a turbulent resistance coefficient.

An advantage is having a semi-empirical model that can be refined for each anatomical profile considered.

According to one embodiment, the second hemodynamic digital model makes it possible to estimate for each portion of vessel of the second type, of the third type, and/or of the fourth type a pressure delta from a pressure delta model comprising a viscous resistance coefficient and a turbulent resistance coefficient, said coefficients being estimated from tests carried out on an in vitro test bench of the invention, said measurements carried out on the test bench making it possible to estimate flow characteristics of a fluid.

An advantage is to use in vitro measurements to obtain a model of equations as close as possible to the anatomy of the portion of vessel considered.

According to one embodiment, the first machine learning model is trained by considering the values estimated by the second model integrated over the entire first vessel.

One advantage is that it simplifies the use of the machine learning model by exploiting only a limited number of input data.

According to one embodiment, a first set of training data comprises geometry data extracted from a patient vessel imaging system, patient data comprising at least the heart rate, the systolic pressure and the mass of the myocardium and test and validation data from pressure measurements at different points of the portion of the vessel considered.

• Acquisition d’un premier ensemble de données d’entrainement à partir de :
  • Acquisition d’une première image d’un système d’imagerie ;
  • Extraction d’un premier ensemble de données définissant des descripteurs anatomiques d’un premier vaisseau de ladite première image ;
  • Acquisition d’un troisième ensemble de données, dites données patient, comportant au moins la fréquence cardiaque et la masse myocardique ;
  • Lecture de données caractérisant des points de mesures de pression d’un premier vaisseau définissant des données de test et de validation d’un modèle d’apprentissage machine ;
• Exécution du modèle d’apprentissage machine et apprentissage par la mise en œuvre d’une fonction de coût minimisant l’erreur entre une donnée produite par le modèle et les données de tests et de validation ;
• Acquisition d’un second ensemble de données d’entrainement à partir de :
  • Acquisition d’une première image d’un système d’imagerie ;
  • Extraction d’un premier ensemble de données définissant des descripteurs anatomiques d’un premier vaisseau de ladite première image ;
  • Résolution d’un solveur modélisant un modèle d’équations hémodynamiques à partir des équations incompressibles de Navier Stokes en une dimension, les données estimées par le solveur définissant des données de test et de validation d’un modèle d’apprentissage machine ;
• Exécution du modèle d’apprentissage machine et apprentissage par la mise en œuvre d’une fonction de coût minimisant l’erreur entre une donnée produite par le modèle et les données de tests et de validation ;
• Acquisition d’un troisième ensemble de données d’entrainement à partir de :
  • Modélisation d’un ensemble de géométries définissant des canalisations adaptées à acheminer un fluide dont la viscosité est sensiblement celle du sang et adaptées à être agencées au sein d’un banc de test ;
  • Exécution d’un ensemble de tests d’écoulement d’un fluide avec une variété de géométries différentes au sein du banc de test ;
  • Mesure de données caractérisant des points de mesures de pression d’une première portion de canal définissant des données de test et de validation d’un modèle d’apprentissage machine ;
• Exécution du modèle d’apprentissage machine et apprentissage par la mise en œuvre d’une fonction de coût minimisant l’erreur entre une donnée produite par le modèle et les données de tests et de validation.

According to another aspect the invention relates to a method of training a machine learning algorithm to produce a trained network for calculating the pressure of a fluid at a plurality of positions within a first vessel as a function of time within the cardiac cycle, said training method comprising:

• Acquisition of a first set of training data from:
  • Acquisition of a first image from an imaging system;
  • Extracting a first data set defining anatomical descriptors of a first vessel of said first image;
  • Acquisition of a third set of data, called patient data, comprising at least heart rate and myocardial mass;
  • Reading data characterizing pressure measurement points of a first vessel defining test and validation data for a machine learning model;
• Execution of the machine learning model and learning by implementing a cost function minimizing the error between data produced by the model and the test and validation data;
• Acquisition of a second set of training data from:
  • Acquisition of a first image from an imaging system;
  • Extracting a first data set defining anatomical descriptors of a first vessel of said first image;
  • Resolution of a solver modeling a hemodynamic equation model from the incompressible Navier Stokes equations in one dimension, the data estimated by the solver defining test and validation data for a machine learning model;
• Execution of the machine learning model and learning by implementing a cost function minimizing the error between data produced by the model and the test and validation data;
• Acquisition of a third training data set from:
  • Modeling of a set of geometries defining pipes suitable for conveying a fluid whose viscosity is substantially that of blood and suitable for being arranged within a test bench;
  • Performing a set of fluid flow tests with a variety of different geometries within the test bench;
  • Measurement of data characterizing pressure measurement points of a first portion of channel defining test and validation data for a machine learning model;
• Execution of the machine learning model and learning by implementing a cost function minimizing the error between data produced by the model and the test and validation data.

One advantage is having a wide variety of training data to train the model on a wide variety of cases. Another advantage is that different branches or training data can be used to configure another branch. Thus, in vitro data can be used to configure the training branch based on the CFD solver. Similarly, data acquired by imaging real patients can be used to configure the CFD solver.

Brief description of the figures

• FIG. 1: une représentation des différents systèmes permettant d’acquérir et traiter les données et les images de vaisseaux d’un patient pour prédire la FFR d’au moins un premier vaisseau ;
• FIG. 2: une représentation du résultat d’une étape de segmentation d’une artère en 3D comportant plusieurs vaisseaux avec la création de lignes centrales pour chaque vaisseau artériel ;
• FIG. 3: une représentation d’un vaisseau en particulier dont on souhaite estimer la FFR à partir d’une segmentation selon un mode de réalisation du procédé de l’invention ;
• FIG. 4: une représentation d’un vaisseau redressé selon un axe Ox permettant de mettre en œuvre une étape de segmentation d’un mode de réalisation du procédé de l’invention ;
• FIG. 5: un mode de réalisation des principales étapes du procédé d’estimation de la FFR ;
• FIG. 6: un mode de réalisation des principales étapes du procédé d’entrainement d’un modèle d’apprentissage machine selon l’invention ;
• FIG. 7: un exemple de réalisation d’une architecture à une sortie d’un réseau de type DeepONet comportant la mise en œuvre d’une fonction de coût paramétrée selon des contraintes physiques ;
• FIG. 7: un exemple de réalisation d’une architecture à plusieurs sorties d’un réseau de type DeepONet comportant la mise en œuvre d’une fonction de coût paramétrée selon des contraintes physiques ;
• FIG. 7: un exemple de réalisation d’une architecture de laFIG. 7mise en œuvre selon les procédés de l’invention ;
• FIG. 8: un exemple de banc de test permettant de réaliser des mesures pour définir un ensemble de données d’entrainement comportant des données pour tester et valider le modèle d’apprentissage machine ;
• FIG. 9: un exemple de modèle de géométrie pouvant être utilisée sur le banc de test de laFIG. 8;
• FIG. 10: d’autres exemples de modèle de géométrie d’une portion sténosée pouvant être utilisée sur le banc de test de laFIG. 8,
• FIG. 11: un exemple de seconde segmentation adaptées mis en œuvre pour découper différents vaisseaux selon des types de segments.

Other characteristics and advantages of the invention will emerge on reading the detailed description which follows, with reference to the appended figures, which illustrate:

• FIG. 1 : a representation of the different systems for acquiring and processing data and images of a patient's vessels to predict the FFR of at least one first vessel;
• FIG. 2 : a representation of the result of a 3D artery segmentation step comprising several vessels with the creation of central lines for each arterial vessel;
• FIG. 3 : a representation of a particular vessel for which the FFR is to be estimated from a segmentation according to an embodiment of the method of the invention;
• FIG. 4 : a representation of a vessel straightened along an Ox axis making it possible to implement a segmentation step of an embodiment of the method of the invention;
• FIG. 5 : an embodiment of the main steps of the FFR estimation process;
• FIG. 6 : an embodiment of the main steps of the method for training a machine learning model according to the invention;
• FIG. 7 : an example of the implementation of an architecture at one output of a DeepONet type network comprising the implementation of a cost function parameterized according to physical constraints;
• FIG. 7 : an example of the implementation of a multi-output architecture of a DeepONet type network including the implementation of a cost function parameterized according to physical constraints;
• FIG. 7 : an example of the realization of an architecture of the FIG. 7 implementation according to the methods of the invention;
• FIG. 8 : an example of a test bench allowing measurements to be carried out to define a training data set containing data to test and validate the machine learning model;
• FIG. 9 : an example of a geometry model that can be used on the test bench of the FIG. 8 ;
• FIG. 10 : other examples of geometry model of a stenosed portion that can be used on the test bench of the FIG. 8 ,
• FIG. 11 : an example of second adapted segmentation implemented to cut different vessels according to segment types.

Definitions

In the present invention, the term "FFR" refers to a ratio between an inlet pressure of a vessel and an outlet pressure.

We call “Pa”: the inlet pressure of a vessel.

We call “Pd”: the outlet pressure of a vessel.

We call “Pdi”: the pressure along the axis of a vessel.

ON calls "ST": a stenosis in the general case and "ST _i " when a particular stenosis is designated is described in an example.

A “DON” network is called a DeepONet network, which in English means “Deep Operator Network”.

A machine learning model trained to generate outputs from input data is called "MLA ₁ ".

The data needed to train the machine learning model used according to different data sources are called "ENS ₁ '", "ENS ₁ ''", "ENS ₁ '''".

ENS ₁ ' refers to training data used and measured in vivo.

ENS ₁ '' refers to training data obtained from a model modeling hemodynamic flows MOD ₂ .

ENS ₁ ''' refers to training data used and measured in vitro.

The data acquired by the machine learning model when operating the trained model is called "ENS ₁ ". Note that the data in the ENS ₁ set may include additional data relative to one of the ENS ₁ ' or ENS ₁ ' sets, in particular "patient data".

We call "ENS₂ » “Patient data” corresponding to measured physiological data, demographic data, history, etc., this data defines part of the input data of the MLA machine learning model₁trained.

We call "ENS ₂ '" the "patient data" to train the MLA ₁ machine learning model corresponding to the measured physiological data, demographic data, and history.

It is possible in some embodiments that the ENS ₂ data is used for training the MLA ₁ model, accordingly, data from the ENS ₂ set may be included in the ENS ₂ ' set.

The term "CFD" refers to a solver of Navier Stokes differential equations, which in English stands for "computational Fluid Dynamics" meaning numerical analysis of fluid mechanics.

A one-dimensional Navier Stokes differential equation solver is called "1-D CFD".

We call L _opé (θ): the term or factor of a cost function modeling the loss to be minimized of an operator model of a network integrating in-vivo and in-vitro physiological training data. It is also noted L _operator (θ).

L _phy (θ) is the term or factor of a cost function modeling the loss to be minimized in a system of physical equations of a network integrating physical constraints. It is also noted L _physics (θ).

A "first segmentation" is called SEG ₁ , the segmentation allowing to acquire images from a 3D imaging system and to select a vessel or an artery of interest and to calculate the interior, exterior volumes and to calculate the central lines of the vessels or arteries.

A "second segmentation" is called SEG ₂ , the segmentation allowing a vessel to be cut into several longitudinal portions in order to use a MOD ₂ hemodynamic equation model to model the blood flow in each of the portions.

The term "HIS" refers to the hospital information system, which stands for "Hospital Information System" in English. It can also refer to the electronic patient health records management system.

The name "CFS" is given to the cardiac analysis system, the acronym of which stands for "Coronary Flow System" in English.

Resistance is generally denoted R or R ₂ in equations. Resistance refers to the force that opposes the flow of blood through the blood vessel. It is influenced in particular by the diameter and length of the vessel, the viscosity of the blood, the type of flow: laminar or turbulent, and the elasticity of the vessel walls.

Impedance is denoted Zc in general and R ₁ in equations. Impedance allows for consideration of the dynamic aspects of flow and vessel, particularly as the vessel contracts and expands over time within the cardiac cycle. Impedance can also capture the influence of pressure waves and the frequency dependence of heart rate.

Compliance is denoted by C, and can be expressed as the inverse of elasticity. Elasticity can also be called elastance. It corresponds to the ability of a tissue to return to its initial state after undergoing deformation. Compliance refers to the ability of a vessel to dilate and increase in volume in response to an increase in internal pressure.

The values P ₀ , A ₀ correspond to values measured or estimated or calculated at equilibrium. It corresponds in the cardiac cycle to the instant in which the fluid, that is to say the blood, is least subject to dynamics. This corresponds for example to a local minimum of certain values in dynamics, in particular the speed or the flow rate.

The method applies equally to vessels and arteries. Thus, in the present invention, when the vessel 10 is designated, the embodiments described also apply in the same way to an artery.

The invention relates on the one hand to a method for predicting the pressure and a value of the FFR of a vessel of the myocardium of an individual from a trained machine learning model and a method for training said machine learning model making it possible to predict the FFR of a vessel of the myocardium of an individual. The invention can relate to any type of vessel other than those of the myocardium for which an expression of the flow rate or an incoming flow of blood is available as input. In the latter case, the “in vivo” training branch is then adapted according to the type of vessel considered and therefore the associated trained machine learning model.

Method for predicting FFR

The process for predicting FFR involves several steps.

• Acquisition d’une image

There FIG. 5 represents an embodiment of the steps of the method for estimating the FFR.

• Image acquisition

According to one embodiment, a first step comprises an acquisition of at least one image of a set of at least one vessel from an imaging system. According to one embodiment, a plurality of images is acquired so as to reconstruct a complete 3D image of a region of interest. The imaging system may be a scanner such as a coronary CT scanner, an MRI or an ultrasound imaging device. These images are called DICOM images and designate in English terminology “Digital Imaging and Communications in Medicine”. Such a format allows interoperability of the image with different systems. In addition, different data can be associated and recorded with the image, in particular data relating to the patient. Furthermore, the format allows the exploitation of the imaging data so as to carry out operations and display certain parameters.

This step is rated ACQ ₁ on the FIG. 5 The corresponding step during the training phase is noted ACQ ₁ on the FIG. 6 .

According to one embodiment, a coronary CT scanner is implemented to collect images of an individual. Such an imaging system is based on a computed tomography medical imaging system. It is also called a coronary CT angiography and noted in the literature CTCA designating "Coronary Computed Tomography Angiography" in English terminology. The image produced is obtained using an X-ray device. The data collected is then processed by a computer to create cross-sectional images of the heart. These images can be compiled to form a 3D view of the coronary arteries as shown in FIG. 2 The method of the invention makes it possible to recover clear images of the heart and arteries.

An advantage of this imaging system is that CTCA is non-invasive, fast, and offers high-resolution images, allowing detailed evaluation of the coronary arteries.

There FIG. 1 represents different components of a non-invasive imaging system such as a CTCA.

There FIG. 1 represents a first block denoted HIS allowing the collection of data measured on a patient by measuring devices such as imaging data like an MRI image or an image from a Coroscanner. This component is also capable of receiving reports or analyses produced by the CFS component.

A second block, called PACS, is used to archive, record, and transmit acquired medical data. The PACS component includes a data server.

A third block marked VIEW allows you to view the acquired images, create the first segmentation and use metadata and data from the first segmentation SEG ₁ .

A fourth block denoted CFS comprises a calculation unit allowing the use of the first segmentation SEG ₁ as input and the carrying out of the second segmentation SEG ₂ of said images if necessary during the training of the machine learning model MLA ₁ or during the exploitation of the CFD model MOD ₂ .

According to one embodiment, this calculation unit is also used to implement the CFD solver and/or the MLA ₁ model for the calculation of the FFR.

According to one embodiment, this calculation unit is configured to generate a report of the “report” type making it possible to restore the main outputs of the MLA ₁ model, the acquisition data and the data calculated from the data produced by the MLA ₁ model. This report can be displayed on a computer display so that an operator or a doctor can consult it.

There FIG. 2 represents a segmented 3D view of the aorta 11 and the coronary vessels (eg LAD, LCX, RCA). A vessel 10 is represented among other vessels. It is subsequently called the “first vessel” and is noted 10. The method of the invention makes it possible to select a vessel for which the FFR is to be estimated, such as the first vessel 10, from the method of the invention.

There FIG. 3 represents the first vessel 10 whose image was acquired using an imaging system such as a coronary CT scanner and which was selected from a selection operation carried out on the display.

According to another embodiment, the method automatically triggers the method of the invention on all of the acquired vessels or a pre-selection of a subset of identified vessels.

In the remainder of the description, the method applies to the analysis of a vessel 10 selected from a display.

There FIG. 3 in this case represents a 2D sectional view of a vessel 10.

In the case of the FIG. 3 , the first vessel 10 comprises different stenosed portions ST ₁ , ST ₂ or non-stenosed 6 which can be selected using a selector or a method of automatic detection of diameter variations.

Representation of images

The method of the invention comprises processing the images of the first vessel 10 so as to enable its display from a CFS display. To this end, an image recognition algorithm is implemented in order to isolate certain parts of interest and segment the image.

The method of the invention makes it possible to generate a two-dimensional representation of a vessel. The represented vessel is advantageously selected beforehand from a selector. An operator can select an artery, an opening vessel or a portion of a vessel from the operating system of the acquired images. For this purpose, an image processing algorithm can be implemented so as to automatically crop the images and extract the pixels of at least one vessel or artery within an image to display a representation thereof.

There FIG. 2 illustrates the result of the first segmentation with the creation of central lines for each arterial vessel. This step is implemented at the VIEW step of the FIG. 1 .

A first SEG ₁ segmentation makes it possible to collect images from the imaging system and to select the pixels of interest of a vessel or more generally of an organ.

A segmentation algorithm makes it possible to analyze the spectral densities of the acquired image, the contours of shapes and contrasts of the latter according to the imaging system used in order to exploit the portions of interest of an acquired image.

This first SEG ₁ segmentation allows to select a vessel for example by selecting a group of pixels of interest of an area to be kept. When analyzing a shape defining a vessel, a proximal reference point and a distal reference point are automatically calculated in order to select a longitudinal portion of a vessel.

According to one embodiment, a proximal reference point is defined with a parent artery. The distal reference point may correspond to a point in an area in which an anatomical reference can be taken, or a given diameter of a point located below a predefined diameter. This predefined diameter corresponds for example to a value of 1.5 mm. This limit corresponds to the current resolution limits of coronary CT scanners. However, if this limit were to decrease or if another more efficient imaging system were to be used, the invention could be applied to a vessel up to a distal end having a radius less than 1.5 mm in order to adjust a new distal reference.

The first segmentation SEG ₁ allows to delimit the external surface of the vessel and the internal surface of the vessel and to calculate an average central line of the vessel.

According to one embodiment, the method of the invention comprises implementing a rectification algorithm making it possible to generate a display of the vessel in a 2D representation along an Ox axis which can define, for example, the center line of the vessel.

The first segmentation SEG ₁ includes the possibility of defining a marker along the vessel on the 2D representation or the 3D representation in order to visualize section planes of the vessel. For this purpose, a selector allows a user to select a section plane from a marker.

According to one embodiment, the first segmentation SEG ₁ makes it possible to delimit portions of all or part of a vessel or an artery to select said portion and mark said portion in order to annotate the representation. One advantage is to make it possible to annotate portions of metadata that can be used during the training of the MLA ₁ machine learning model or during the exploitation of the CFD MOD ₂ model called CFD.

Representation of the selected vessel

There FIG. 4 illustrates a representation of a vessel 10 straightened along an Ox axis. The figure represents different parts of the vessel 10 and different descriptors of the vessel 10.

An advantage of this representation is that the diameter or internal radius of a portion of vessel or artery comprising at least one stenosis can also be visualized along the same longitudinal axis and therefore according to a common reference of all the diameters of said vessel or artery.

Within the FIG. 4 plaques marked 4 are represented, such as calcium plaques in this example which form on the inner wall of a vessel 10. The stenoses are detected using imaging; they correspond to local reductions in diameters or radii of the first vessel 10.

Preprocessing and extraction of anatomical features

Anatomical data is used to define part of the input data of the MLA ₁ machine learning model and is also used for training the MLA ₁ model.

According to one embodiment, the input data of the trained model comprises a first set of data ENS ₁ characterizing the anatomy of at least one vessel from an acquired image of the individual. The method makes it possible to select a first vessel 10 from an acquired image and to automatically extract from a processing of the image data characterizing each portion 6, ST of the first vessel 10.

The extraction step is noted EXT ₁ on the FIG. 5 illustrating the implementation of the method for estimating the FFR. The corresponding step during training is also noted EXT ₁ on the FIG. 6 .

The inner diameter is preferentially extracted over the entire length of the first vessel 10 using analysis and processing of the acquired images.

- Curvature

According to one embodiment, the method comprises a step of calculating the curvature of the vessel at different points of the vessel. According to one embodiment, a function C _b (x) defines the function making it possible to determine the curvature as a function of the position x on the longitudinal axis Ox of the representation of the straightened vessel of the FIG. 4 or the central axis of the unstraightened vessel.

The curvature of the vessel can therefore be recorded during this operation so as to be re-used subsequently as data to model certain portions of the vessel, for example during the training phase using a MOD ₂ hemodynamic model. The method makes it possible to annotate the portions having a curvature greater than a given threshold, for example in the second segmentation step SEG ₂ .

This operation is carried out in particular during the second segmentation SEG _2. Indeed, when the curvature is greater than a given threshold on a portion of the vessel, the hemodynamic flow in these portions can be modeled by one or more dedicated equation(s).

According to one embodiment, the curvature of certain portions of the vessel may be used as input to the MLA ₁ machine learning model.

The method of the invention makes it possible to index along a vessel 10 considered all of the radii or internal diameters along said vessel 10. According to an improved mode, the external radius can be measured and indexed along the vessel 10 considered. One advantage is to consider the thickness of the wall of a vessel 10. However, the invention can be implemented without consideration of the thickness of the vessel or artery considered.

During the image processing operation, a function of the radius r(x) can be generated by taking all the rays of the vessel or artery under consideration from an origin position x ₀ to a terminal position x _t . The sampling resolution of the function can be set so as to discretize the set of values on a predefined scale.

According to one embodiment, all of the curvatures of the vessel 10 are also indexed along the straightened vessel along the longitudinal axis Ox. Thus, the method of the invention makes it possible to collect, by automatic analysis of the image, all of the values of radii or internal diameters and all of the values of curvatures of the vessel 10 considered. These values are recorded in a memory and each of the values is associated with a position of the vessel 10 along the longitudinal axis of straightening.

- Singularity

The method of the invention comprises a step of detecting a set of singularities along the vessel 10 considered.

According to a first example, a first type of singularity is a plaque 4. According to a second example, a second type of singularity is a junction zone or a bifurcation zone. According to a third example, a third type of singularity is an aneurysm. Other types of singularities can be noted.

In the following description, the case where 4 plaques lead to the formation of one or more ST stenoses will be considered as an example.

- Stenosis

There FIG. 4 represents an example case in which two stenoses ST ₁ and ST ₂ are identified through image processing. Automatic detection of stenoses can be configured by automatic analysis of the variation of the diameters or internal radii of the vessel over the entire portion analyzed. It is noted that the FIG. 4 represents plates 4 not managing stenoses and others resulting in the formation of ST ₁ and ST ₂ stenoses. The method of the invention makes it possible to identify and annotate the presence of stenoses on an image of a vessel.

In one embodiment, direct identification of stenoses is not provided and autoencoders are used. Autoencoders are a type of neural network used in machine learning for unsupervised learning. They consist of an encoder and a decoder, and their objective is to learn a latent representation of the input data. The latent representation is provided with additional necessary properties and can be used, for example, to reduce the dimensionality of the data or to eliminate noise. The invention includes other implementations of autoencoders for 2D and 3D input data.

The presence of an ST stenosis is the consequence of a local reduction in the radius or the internal diameter of the first vessel 10. Each ST stenosis can be detected and identified using an image analysis algorithm. Different existing algorithms can be applied within the framework of the present invention, such as the autoencoders previously mentioned. The detection of stenoses allows their counting along the first vessel 10. The vessel portions comprising an ST stenosis are indexed and the presence of said ST stenosis can be characterized by a set of positions x _s along the Ox axis. The stenoses of the FIG. 4 are noted ST ₁ and ST ₂ .

Each ST stenosis can be characterized according to different criteria. A first criterion corresponds to its dimensions. The dimensions of an ST stenosis include at least the length along the Ox axis and the thickness along the axis perpendicular to the Ox axis of the ST stenosis. The occlusion diameter or the minimum radius in the stenotic portion can be extracted.

• Le diamètre minimal de la sténose en mm et/ou ;
• La surface minimale de la sténose en mm² selon un plan de coupe du vaisseau et/ou ;
• Le volume total de de la sténose en mm3 et/ou ;
• Le degré de sténose coronarienne maximum exprimé en pourcentage de diamètre du vaisseau et/ou ;
• Le degré de sténose coronarienne maximum exprimé en pourcentage de surface du vaisseau selon un plan de coupe du vaisseau et/ou ;
• La courbure de la zone correspondant à la sténose.

According to other embodiments, the following parameters may be used to characterize a stenosis:

• The minimum diameter of the stenosis in mm and/or;
• The minimum area of the stenosis in mm² according to a section plane of the vessel and/or;
• The total volume of the stenosis in mm3 and/or;
• The maximum degree of coronary stenosis expressed as a percentage of vessel diameter and/or;
• The maximum degree of coronary stenosis expressed as a percentage of vessel surface area according to a sectional plane of the vessel and/or;
• The curvature of the area corresponding to the stenosis.

These morphological parameters are for example calculated from the identified singularity such as a plate-type singularity.

According to one embodiment, the diameter or the internal radius of the vessel 10 at each position of a stenosed portion ST ₁ , ST ₂ can also be calculated. This latter data can supplement, for example, data characterizing the thickness of the stenosis in the method of the invention.

In the case of the FIG. 4 , two stenoses ST ₁ , ST ₂ were identified along vessel 10. A first stenosis has a length L _ST1 the second stenosis has a length L _ST2 . The radius or diameter in these portions can be noted.

According to one embodiment, the distance between two ST stenoses₁and ST₂ can also be taken into account by a measurement on the image. The measurement can advantageously be carried out automatically from an image processing algorithm. This data can be used for training and/or exploitation of the trained MLA model₁.

According to one embodiment, the vessel can be divided into several segments in order to calculate the inlet pressure and the outlet pressure of each segment, in particular during the second segmentation to train the model.

We consider the case in the following description where a vessel is considered entirely from an entry point, generally a junction zone with an artery and a distal end defined by a geometric or anatomical marker.

There FIG. 4 is used to describe a vessel used during the operating phase of the MLA ₁ model and also to describe a vessel used during the training phase, for example from a MOD ₂ hemodynamic model for which one equation can be associated with non-stenosed portions and another equation can be associated with stenosed portions.

The pressure curve allows you to visualize the evolution of pressure along the vessel. This curve, although represented on the FIG. 4 cannot be obtained directly from imaging. The method of the invention makes it possible to overcome this problem since the objective of the invention is to predict the pressure values along the vessel 10 from a trained MLA ₁ machine learning model.

When the pressure points are obtained, it is possible to consider two points at the entrance and exit of the vessel 10 to calculate the FFR.

The method of the invention makes it possible to estimate, as shown in the FIG. 4 , the pressure p(x) at any point of the vessel. In this case the FFR is then calculated in a second step from the predicted values of the pressure Pd and Pa.

According to one embodiment, the predicted value is a value of the FFR which is a ratio between an inlet pressure Pa of a vessel and a pressure at a considered point of the vessel, for example the outlet pressure Pd. We have the following relationship which is verified at any point x of the vessel: FFR(x) = P(x) / Pa.

Pressure estimation can also be done using the second CFD model called MOD ₂ in order to train the MLA ₁ model with the values estimated by the solver.

This representation illustrates the pressure drops after the presence of a stenosis.

During the training step using the training data, a pressure curve can be obtained along the Ox axis of the vessel in which a catheter equipped with a pressure probe navigates. The invention can use such data made available and recorded in a memory.

The input to the MLA ₁ machine learning model corresponds to the entire vessel or artery considered such as the one represented in FIG. 4 . The artery or vessel is a topological unit, going from the inlet to the outlet. In the case where there are several branches in the artery, for example an LDA and LCX branch or an RCA branch, the MLA ₁ model considers each of them independently. An exchange of data such as, the extractions of anatomical characteristics, the flow q or the pressure p at the junction zones allows to take into account the different junction constraints.

• Données d’entrée du procédé pour prédire la FFR

According to one embodiment, the length of the vessel 10 from a characteristic entry point to a characteristic exit point may also define an input data of the MLA machine learning model _1. The characteristic points may correspond to bifurcation points, to a point of the vessel corresponding to a narrowing of the diameter greater than a threshold, etc.

• Process input data for predicting FFR

The anatomical data defined previously defines a part of the input data, denoted ENS ₁ , of the machine learning model MLA ₁ .

According to one embodiment, the input data of the trained model MLA ₁ comprises a second set of data ENS ₂ characterizing the individual.

Input flow rate

According to one embodiment, the resting input flow rate Q _rest is calculated from the myocardial mass Mc which is data acquired at the input of the system. An algebraic model makes it possible to express the resting input flow rate Q _rest from the myocardial mass Mc.

According to one embodiment, the input data of the MLA ₁ model is the mass of the myocardium Mc or the mass supplied by one of the vessels, or the mass of a part of the myocardium such as the mass of a left ventricle, M _{C_LV.}

According to another embodiment, the input data of the MLA ₁ model is a flow rate calculated from the mass of the myocardium Mc or from the mass of a part of the myocardium.

In the latter case, the method comprises a calculation of the resting flow rate in the first vessel 10 whose myocardial mass Mc is known.

Depending on the vessel selection, the blood fraction of the artery supplying said vessel is considered.

The method of the invention makes it possible to deduce from the selection of the first vessel 10 considered the upstream flow rate supplying said first vessel 10. There are three main arteries supplying the heart and therefore all the vessels. The 3 arteries are noted LAD, LCX and RCA.

The first artery, LAD, is the artery called the "Left Anterior Descending artery" in English terminology and in French: "The anterior interventricular artery". This artery supplies the anterior part of the heart, particularly the left ventricle.

The second artery LCX refers to the artery called "Left Circumflex" in English terminology, or in French the left circumflex artery. It supplies blood to the lateral wall of the heart, mainly to the left ventricle.

The third RCA artery is the "Right Coronary Artery." It primarily supplies blood to the right ventricle and part of the left ventricle.

These arteries supply oxygenated blood to different parts of the myocardium. Blood flow in these arteries can be calculated based on known proportions of blood ejection in each artery.

A relationship allows the resting flow rate in a given vessel to be deduced.

According to one embodiment, the following relationship can be written:

Q _rest ^{= 2.5 * (M C 0.75 )} _[ ^ml /min],

According to another embodiment, if only the mass of the left ventricle M _{C_LV} is known, this relationship can be written:

Q_{rest_LV}= 12*((0.7*( f_c* P_system)/1000)-0.4) * M_{C_LV} / 100 [ml/min],

where f_cis the heart rate and P_system is the systolic pressure. This relationship is only valid if the flow rate meets the individual's oxygen demand.

In the case of characterizing a hyperemic flow, a hyperemic flow coefficient k _hyp can be calculated according to an empirical formula.

k _hyp = 4.5 – 0.018 * ds + 0.00056 * ds ² – 0.0000085 * ds ³

with ds: The maximum degree of coronary stenosis expressed as a percentage of vessel diameter.

The following relationship can be written: Q _hyp = Q _rest * k _hyp

If only the mass of the left ventricle is known, the relationship can be written as Q _hyp = Q _{rest_LV} * k _hyp

According to one embodiment, the model includes as input only the mass of the myocardium Mc. According to another embodiment, the model includes as input the mass of the myocardium and the resting flow rate as input. It is understood that the two inputs Q _rest and Mc are strongly correlated and therefore that only one value can be considered as input to the MLA ₁ model. However, the model can take into account both inputs.

Patient data

These data are noted ENS ₂ on the FIG. 5 and ENS ₂ ' on the FIG. 6 for “in vivo” training data. According to one embodiment, the individual’s heart rate is input to the MLA ₁ model.

According to one embodiment, additional data is taken into account with the value of the individual's systolic pressure.

According to one embodiment, the age of the individual is an input to the MLA ₁ model. According to one embodiment, the gender of the individual is also an input to the MLA ₁ model.

According to one embodiment, the weight and height of the individual are inputs to the MLA ₁ model.

According to one embodiment, background data are inputs of the MLA ₁ model such as the presence of diabetes, hypertension, the existence and count of previous percutaneous coronary intervention(s), the existence and count of myocardial infarction, the quantification of tobacco consumption.

Contextual data Standardization

The method for estimating the FFR of the invention comprises a step aimed at normalizing NM the data forming the anatomical descriptors extracted from the segments/portions of the first vessel 10 and at normalizing the patient data. This step is noted NML on the FIG. 5 and on the FIG. 6 . We therefore designate the NML normalization step in training or in exploitation of the MLA ₁ machine learning model. The normalization steps are by construction designed to generate normalized outputs whatever the training data. Thus, only the input data can possibly change at the entry of the NML blocks _.

The input data can have different formats depending on the data collected. The NML normalization block therefore allows potentially different transformations to be applied to the measured values.

According to one embodiment, the NML normalization comprises a step of scaling the values according to a predefined scale. The NML normalization also comprises a transformation of the data according to a predefined unit system.

According to another embodiment, the NML normalization step comprises an upgrade of the data precision and its format. The NML normalization makes it possible to generate the data format allowing the input vector to be processed in the same way as the input vector was used for training according to the different training branches.

According to one embodiment, a data imputation technique is applied to the input data to fill in missing values.

The input to the MLA ₁ machine learning model is denoted by E ₁ on the FIG. 5 .

• Sortie du modèle MLA

The corresponding entries on the FIG. 6 during the training phases are noted ENS ₁ ', ENS ₁ '', ENS ₁ ''', ENS ₂ '. The test and validation data E ₂ , E ₃ , E ₄ allow the model to be trained by correcting the error of a cost function Lf.

• MLA Model Release

The method of the invention implements a machine learning model MLA ₁ to produce output data denoted S ₁ on the FIG. 5 . In particular, the output data includes the predicted values of pressure along the vessel, flow rate along the vessel, and values of the vessel sections 10.

The predicted pressures are used to infer the FFR function along the axis.

The corresponding outputs during training are noted respectively S ₂ , S ₃ and S ₄ according to the training branches of the FIG. 6 .

Among the data produced by the MLA model₁, he is possible to exploit the inlet pressure Pa considered at the cutting plane defining the inlet of the vessel and the pressure Pd considered at the cutting plane defining the outlet of the vessel.

From the pressures produced by the MLA ₁ model, the method of the invention comprises a step of calculating the FFR, namely the Pd/Pa ratio. This step is represented in FIG. 5 illustrating the process for estimating FFR or outlet pressures. This step is also shown in FIG. 6 representing the training process.

According to one embodiment, the flow rate Q is predicted by the MLA ₁ model at any point x of the vessel and throughout the cardiac cycle, including as a function of time t.

Finally, the values of the input and output sections A can be generated and the evolution of this section A(t) during the cardiac cycle can also be calculated by the MLA ₁ model. Finally, the model can be trained to produce the section A ₀ at equilibrium.

According to one embodiment, the method of the invention comprises a step of processing the data generated by the MLA ₁ model to produce new values of parameters of interest.

According to an embodiment corresponding to that of the FIG. 7 , the values of interest are the equilibrium pressure P ₀ , the resistance R and the compliance C and the impedance Zc at the outlet of the portion considered, i.e. calculated at the outlet cutting plane. In the FIG. 4 , this output cutting plane is the pc ₆ plane. On the FIG. 11 , this cutting plane can be defined at point(s) OB ₁ , OB ₂ , OB ₃ or OB ₄ .

Model

The invention implements a model for predicting the pressure and other associated characteristics of a vessel, taking into account various geometric and physical constraints. The model is based on scientific machine learning SciML, a research field that uses deep learning to solve complex physical problems. An example of a SciML model is the physics-based neural network known as a “PINN” network. Such a network is particularly suitable for learning physical laws, particularly nonlinear ones, for example, approximated by differential equations. One advantage of such a network is that it allows working on physical processes in stenosed vessels. However, PINN can only provide solutions for defined vascular parameters, such as the geometry of the vessel or the position of the stenosis. The inclusion of each new parameter value requires a separate simulation, involving a complete retraining of the PINN. To overcome these limitations, one embodiment uses a physics-based DeepONet network, also known as a “PIDeepONet”. Such a network is referred to as an MLA model.₁ in this application. The PIDeepONet network can be used to learn various linear or nonlinear operators and impose physical constraints.

DeepONet Network

According to one embodiment, the method of the invention implements an MLA ₁ machine learning algorithm based on a DeepONet network, denoted DON. Such a network is designed to learn mathematical operators. In this context, such an operator is a function called "mapping" in English terminology. It transforms one function into another. The DeepONet network learns this transformation directly from ENS ₁ training data.

In a DON network, the loss function Lf is designed to capture the error between the network's predicted outputs and the actual or expected outputs for a given set of input functions. The general form of the loss function in a DON is typically a function of the error between the network's prediction and the true output for various input samples ENS1', ENS2', ENS1'', ENS ₁ '''.

In the case of a network trained to learn a mathematical operator, the loss function may include terms that assess how accurately the derivatives are predicted.

In order to represent the mathematical operator from the data, the DON network allows us to consider two components generally noted "trunk" and "branch" in Anglo-Saxon terminology. The "trunk" component or branch is noted Bt and the "branch" component or branch is noted Bb in the rest of the description.

There FIG. 7 represents an architecture of a network called in English terminology “Physics informed DeepONet”. This example of DON type architecture is advantageously executed in the method of the invention.

A physics-informed DeepONet network allows output functions to be consistent with physical constraints by minimizing a loss taking into account underlying physical laws.

This architecture includes inputs {u(x _i )} _i and {y _k (i)} _k defining the inputs respectively of two branches Bb and Bt of the network.

There FIG. 7 represents an example of a one-output architecture.

According to one embodiment, the methods of the invention implement a multi-output network architecture. FIG. 7 represents an example of a multi-output architecture. In this example, the outputs of branch Bb of the network are divided into n groups and the outputs of branch Bt of the network are equally divided into n groups. Then the ^kth outputs of each group are processed together and produce the ^kth solution.

There FIG. 7 represents an example of an n-output architecture used in the context of the invention so as to generate different outputs here represented by the pressure P, the flow rate Q, the section A and the section A ₀ at equilibrium. In this figure, the secondary outputs are represented. The secondary outputs are the data calculated in a second step from the data produced by the MLA ₁ network. They are therefore obtained indirectly. The FIG. 7 represent in particular the FFR obtained along the axis of the vessel, the equilibrium pressure P ₀ , the impedance Zc, the compliance C and the resistance R.

The model can be expressed as a function G(u)(y) taking as model inputs the two inputs {u(x _i )} _i and {y _k } _k to generate a prediction.
[Math 1]

Where i = 1, … , K

Where 0 = p ₀ < p ₁ < … < p _n

The inputs {u(x _i )} _{i ɛ [1 ; N]} represent N distinct input functions. And for each y _k (i) taken on an interval K ε [1 ; P], there are P values taken in the domain of the function G(u _(i) ). The generated output is denoted G(u _(i) ,y(i)).

In the case of the present invention, the values y _k (i) are the input data received and processed as input to the MLA ₁ model for each input sample. A sample comprises all the input data of an individual such as his age, heart rate f _c , myocardial mass Mc, etc. A sample can also correspond in the case of training the model to a sample of a case modeled in vitro or simulated from a CFD model to which data from fictitious/virtual patients have been assigned.

u(x _i ) is a function returning the parameters used to describe the different anatomies of vessels or portions of vessels defining the anatomical input data for ENS ₁ ' training. u(xi) is a function used to process the ENS ₁ ' training data in order to train the MLA ₁ model.

u(x _i ) corresponds to the input functions, which describe the geometric/anatomical characteristics of the vessel in the sensor points x. In one embodiment, u(x) is defined in the domain [0, 1] and depends on the distance to the stenosis. For such functions, 0 corresponds to 100% stenosis and 1 means no stenosis. In another embodiment, the u(x) functions are determined by an internal representation generated by the autoencoder network. In this case, the construction of u(x) is an unsupervised learning process. The Bb branch of the DON processes the operator input. This input is usually a function or a set of values that represent a function.

Branch Bb learns a representation of the input function. To do this, it can take as input samples of this function, for example, values of the function at different points, and produce a higher-dimensional representation that captures some characteristics of this function.

This Bb branch allows us to understand the structure and properties of the input function, allowing the network to respond correctly to variations in this input data.

The Bt branch is designed to process the points {y _k } _k of the domain for which the operator's output is to be evaluated. The Bt branch produces a representation of these points. This representation is then used to predict the operator's output at these specific points.

The Bt branch allows the DON network to understand how the operator acts on different parts of its domain, which helps improve the predictions of the DON model outputs.

The outputs of branches Bb and Bt are combined to produce the final output of the network. This combination can be done in several ways. For example, the combination is achieved by a scalar product.

By separately processing the operator input via the Bb branch and the output points via the Bt branch, the DON network can efficiently learn the relationship between the operator input and output. This separation allows for greater flexibility and accuracy in learning complex operators.

In an example implementation, the DON network only includes one branch Bb. Such a network is called an “unstacked DeepONet” in English terminology. It has the advantage of requiring less matrix computation.

In an example implementation of a DON network, a network architecture is configured such that the outputs of branches Bb and Bt are combined via a linear combination, such as a dot product, to produce the final output of the DON network.

According to one embodiment, the DON network takes into account physical constraints. The network is called “physics-informed DeepONet” in English terminology. Such a network makes it possible to take into account physical constraints in the configuration of the loss function, making it possible to penalize predicted values outside a predefined range of values.

Loss functions

According to an example, the loss function L _f (θ) of the DON network is configured to minimize a first term denoted L _opé (θ) which corresponds to the loss function of the operator and a second term denoted L _phy (θ) which corresponds to the loss function of the physical constraints.

According to one embodiment, when training the MLA ₁ model, the loss function Lf(θ) is configured to optimize an error by taking into account a factor making it possible to learn the operator L _opé (θ) optimizing the error of a function u(x) modeling the geometry of the vessels and one making it possible to learn physical constraints L _phy (θ) optimizing the error of a system of equations defining bounds and a compatible domain of the physics induced by the Navier Stokes equations.

The step of calculating and optimizing the error calculated by the loss function is represented in FIG. 6 when training the MLA ₁ model.

The loss function can be written: L _f (θ) = L _opé (θ) + L _phy (θ)

With, for example, the following expression of the two operators:
, [Math 2]
, [Math 3]

Where {y_k(i)}_{k=1 to Q}refers to a set of randomly sampled points in the domain G(u_(i)).

Where N is a linear or nonlinear differential operator that gives differential equations the following form: N(u, G) = 0

The physical constraints taken into account in equation [3] by the operator L _phy (θ) can take the form of the system of differential equations resulting from the incompressible Navier Stokes equations.

This system of equations takes the form of a numerical model of hemodynamic equations called MOD _1. It can be expressed as follows:
[Math 4]

In another embodiment, another system of hemodynamic equations may be used to define the MOD ₁ model or other equations may be integrated into the MOD ₁ model of equations [4]. These hemodynamic equations may be modeled in 1D, 2D, or 3D.

Other equations of state than that of pressure can also be used.

According to one embodiment, the empirical equation for calculating Young's modulus is used as described for CFD.

In one example, the Navier Stokes equations are represented in a dimensionless manner and are integrated over the cross-sectional area of the vessel or portion of vessel under consideration.

The parameters and variables are defined as follows:
[Math 5]

We consider x as a coordinate along the Ox axis of vessel straightening. t is the time parameter, p is the pressure, q is the flow rate, A is the section, δ is the boundary of the layer thickness and ν is viscosity.

In these equations, a new variable Re, denoting the Reynolds number, is introduced. The values of the variables in a 1D one-dimensional model are considered to be their known maximum values.

Furthermore, the method of the invention makes it possible to consider as another target variable. For additional variables , , , , additional constraints have been defined, namely: , , , .

Thus, if the following intervals are violated, the cost function will penalize learning:

= [0 ; 1]

= [0 ; 1]

= [7 ^e -06, 1]

= [7 ^e -06, 1]

• Entrainement du modèle

According to different embodiments, methods of optimization and minimization of each equation [4] of the MOD ₁ model described can be implemented to solve the system according to a residue or an error to be minimized.

• Model training

The invention also relates to a method for training the MLA ₁ model. An advantage of the training method of the invention is to consider different sets of test and validation data E ₂ , E ₃ and E ₄ coming from different training configurations to train the MLA ₁ model.

There FIG. 6 represents the MLA ₁ model during these different training phases noted B ₁ , B ₂ , B ₃ allowing three network inputs to be obtained to train it Three inputs are represented on the FIG. 6 by blocks B ₁ , B ₂ , B ₃ allowing a better understanding of these different training phases. The test and validation data of each of these branches E ₂ , E ₃ and E ₄ are represented at the output of the 3 data normalization blocks noted NML.

An advantage of training using different configurations is to obtain better predictions in the exploitation phase of the MLA ₁ model.

Indeed, the first input aims to train the MLA ₁ model with data tested in vivo, i.e. with patients for whom pressure measurements are made from a catheter. The second input aims to train the MLA ₁ model with a CFD solver allowing to solve a system of incompressible differential equations of Navier Stockes. The third branch aims to train the MLA ₁ model with data measured on a test bench, the measurements are said to be in vitro.

Such 3-phase training makes it possible to obtain an exhaustive panel of test data allowing to obtain data representative of the real case, taking into account the physics equations and finally making it possible to model a large number of different stenosis geometries.

According to one embodiment, the method of the invention may comprise training based solely on a single input {B ₁ }, {B ₂ } or {B ₃ }, or on two inputs {B ₁ , B ₂ }, {B ₁ , B ₃ }, {B ₂ , B ₃ } out of three. For example, according to one embodiment, the method of the invention only comprises the training data {ENS ₁ '', E ₃ } and {ENS ₁ ''', E ₄ }. In this case the model is trained solely with data from the CFD solver and in vitro data.

When the training method includes the first input B ₁ in vivo tests can be conducted in order to obtain the data E ₂ . In another case, the data E ₂ exists and is stored in a memory or a database. In the latter case, the method of the invention receives the data from a memory to train the model with the test data E ₂ which have already been collected.

Such training comprising data from several inputs B ₁ , B ₂ and B ₃ as represented in FIG. 6 makes it possible to obtain better predictions of the inlet pressure p _a of a first vessel 10 and better predictions of the pressure on the axis up to the outlet of the vessel including the outlet pressure p _d of the first vessel 10. The FFR deduced by the network thus trained is then more reliable depending on the specific case encountered.

Input B2: CFD solver training data

A second entry marked “cfd” on the FIG. 6 allows training the MLA ₁ machine learning model. According to one embodiment, a set of training data corresponding to the test and validation data E ₃ is obtained by solving a so-called “1D CFD” model.

• Seconde et troisième Segmentations SEG2, SEG3 pour le solveur CFD

The 1D CFD equation system is for example applied to data acquired from a vessel geometry obtained from an imaging system. FIG. 6 represents this example case in which the anatomical data of vessel 10 from the first segmentation SEG ₁ of the first input B ₁ are used to provide anatomical data to the 1D CFD solver of input B ₂ . Arrow 19 represents the acquisition of this data.

• Second and third Segmentations SEG2, SEG3 for the CFD solver

The method of the invention comprises a second segmentation SEG ₂ of the first vessel 10 into different segments or portions. This step is denoted SEG ₂ on the FIG. 6 The second segmentation step SEG ₂ is used in particular during the training phase jointly with the MOD ₂ modeling model of the hemodynamic flow also called CFD.

A third segmentation SEG ₃ can also be used during the training phase with in vitro test data. This training phase corresponds to the third branch of the FIG. 6 called “in vitro”.

The third segmentation relates to a pipeline modeled on a test bench comprising portions within which a flow is injected and parameters characterizing the flow are measured. These data can then be used to train the MLA ₁ model. The third segmentation aims to characterize portions of pipelines having topological properties representing the different singularities of a vessel.

To this end, the training method of the invention makes it possible to automatically generate sectional plans of the vessel in order to consider each segment as a portion of vessel whose blood flow properties can be calculated using the MOD ₂ hemodynamic model. One advantage is to calculate test values to train the machine learning model.

During the second segmentation SEG ₂ used to associate the Navier Stokes flow equations with characteristic portions requiring a given model, characteristic cutting planes can be defined. These cutting planes can be automatically generated using an image processing algorithm.

There FIG. 4 also represents different cutting planes generated in particular during the second segmentation SEG _2. These cutting planes are noted pc ₁ , pc ₂ , pc ₃ , pc ₄ pc ₅ , pc ₆ on the FIG. 4 . They are generated to delimit the different types of portions in the second segmentation.

There FIG. 4 represents different portions including a type of non-stenosed portion 6 and a type of stenosed portion ST ₁ , ST ₂ .

In the example of the FIG. 4 , the delimitation planes pc ₁ and pc ₂ delimit a first non-stenosed portion 6, the planes pc ₂ and pc ₃ delimit a first stenosed portion ST ₁ , the planes pc ₃ and pc ₄ delimit a second non-stenosed portion 6 and the section planes pc ₄ and pc ₅ delimit a second stenosed portion ST ₂ and finally the section planes pc ₃ and pc ₄ delimit a third non-stenosed portion 6.

The section planes are shown on the FIG. 4 to illustrate that after the cutting plane Pc ₃ or PC ₅ delimiting the end of a stenosed portion, a drop in pressure can be a consequence.

During training, the use of the MOD ₂ hemodynamic model in the cfd branch of the FIG. 6 allows the calculation of characteristic values of blood flow. The boundary conditions between the different portions considered allow the equations to be used in a data processing loop to integrate the results along the entire length of the vessel.

According to one embodiment, when using the MOD ₂ hemodynamic model, called CFD, each segment considered corresponds to a physical type of the part of the vessel considered, namely a non-stenosed portion, a stenosed portion, a curved portion, a portion comprising a bifurcation. According to this embodiment, this rule makes it possible to define that two neighboring segments are always of different type.

There FIG. 11 represents an example of a second segmentation SEG ₂ used to generate training data from a MOD ₂ own hemodynamic equation model. In this model different equations can be applied to different types of portions having a own hemodynamic equation model.

There FIG. 11 represents a case of modeling several vessels with different bifurcations AB ₁ , AB ₂ and AB ₃ representing areas of junction between vessels.

An inlet of the blood flow is represented by the element IB ₁ and the different outlets of the blood flow are denoted OB ₁ , OB ₂ , OB ₃ and OB ₄ .

A first type of segments sg ₁ , sg ₃ , sg ₄ , sg ₆ , sg ₈ , sg ₉ , sg ₁₀ , sg ₁₁ , sg ₁₃ , sg ₁₄ , sg ₁₆ , sg ₁₇ , sg ₁₉ , sg ₂₁ , corresponds to the portions of the vessel in which the flow can be modeled by equations of the CFD solver, called 1D Navier Stokes equations noted [1] and [2], allowing the flow to be modeled and the pressures within the vessel(s) to be calculated by considering initial hypotheses on the flow of the flow.

According to one embodiment, the method of the invention is carried out for each branch from the entry point IB ₁ to the different exit points OB ₁ , OB ₂ , OB ₃ and OB ₄ . The method of the invention makes it possible to obtain the pressure P(x, t), the flow rate Q(x, t) and the section A(x, t), that is to say at any point along each vessel and at each instant of the cardiac cycle.

According to one embodiment, a mesh is defined to model and solve the system. The number of cells in the mesh along the vessel axis and the time step are two control parameters. Other control parameters can be defined.

A second type of sg segments₅, sg₇, sg_12,sg₂₀corresponds to the portions of the vessel presenting a stenosis, that is to say a local reduction in the diameter of the vessel greater than a threshold. The portions presenting a stenosis require the use of a dedicated model of flow circulation different from the model used for circulation in a non-stenotic portion. In this case, an equation [21] described below can be used to model the blood flow and to calculate the inlet and outlet pressures of each stenotic portion by considering initial hypotheses on the flow of the flow.

A third type of segments sg ₂ , sg ₁₈ corresponds to vessel portions with a curvature greater than a threshold and which can affect the flow circulation with respect to a circulation in an uncurved portion. In this case, an additional equation can be used to model the blood flow by considering initial assumptions on the flow flow.

A fourth type of sg ₁₅ segments corresponds to portions of vessel with aneurysm, i.e. portions with a local bulge in the vessel diameter greater than a threshold. A local bulge can be detected due to a local increase in the inner and/or outer diameter of the vessel greater than a threshold. The threshold can be expressed as an absolute value or as a proportion of the average diameter of the vessel or the diameter in an adjacent portion. The threshold can also be characterized by the local variation in diameter. Portions with a local bulge can be better modeled by requiring the use of a dedicated flow circulation model that is different from the model used for circulation in a portion of the first type for example. In this case, a semi-empirical equation similar to the equation [21] described below dedicated can be used to model the blood flow by considering initial hypotheses

Thus, the second segmentation operation is carried out automatically by first labeling the different portions of the vessel according to specific geometric characteristics.

• Conditions de continuité aux limites des portions segmentées

In order to delimit the cutting planes during this second segmentation SEG ₂ , the training method of the invention makes it possible to detect variations in diameters or radii of the first vessel. Thus, after a reduction in the diameter greater than a predefined threshold, for example expressed as a percentage of the average radius, the maximum radius or the local average radius before the reduction, a cutting plane can be generated. And identically, after an increase in the diameter greater than a predefined threshold, for example expressed as a percentage of the average radius, the maximum radius or the local average radius after the increase, a cutting plane can be generated.

• Continuity conditions at the boundaries of segmented portions

The second SEG segmentation₂allows a continuity condition to be defined between two successive segmented portions, in particular with regard to the values estimated by the different MOD models₂applied depending on the type of segments. In particular, the pressure or flow rate at the outlet of a first portion is equal to the pressure or flow rate at the inlet of a second portion directly following the first portion. And conversely, the pressure or flow rate at the inlet of a second portion is equal to the pressure or flow rate at the outlet of a first portion directly preceding the second portion. This is also true for other parameters characterizing the flow or the boundary conditions at the junctions between segments.

Thus, it is possible to train the MLA ₁ model by the estimated values of the second MOD ₂ hemodynamic flow model within the training method of the invention by a step-by-step approach.

For this purpose, a feedback loop can be used to integrate the values calculated by applying the second hemodynamic model MOD ₂ to a plurality of segments linked together with the continuity conditions.

For stenosed portions, a mathematical method can be implemented to manage boundary conditions. For example, a so-called "ghost cells" method can be implemented. These ghost cells are added to the mesh cells.

According to one embodiment, the feedback loop makes it possible to test which type of segment is processed in the loop for a given position before repeating the calculations for an incremented position. If the position considered in the feedback loop is a stenotic position, the method of the invention makes it possible to implement a ghost cell.

The principle is to define a physical grid of space allowing calculations to be carried out in which each cell represents a volume of fluid. Ghost cells allow boundary conditions to be defined on pressure, flow rate, etc. in the ghost cells according to the boundary conditions of the problem. For example, the fluid velocity can be set to 0 to simulate the presence of a vessel wall. The conditions can be of different natures such as Dirichet conditions for fixed values of the conditions or Neumann conditions for gradient type values or more complex conditions such as sliding or periodicity conditions. For example, the condition on the time variable t can be induced from the heart rate constraint. The integration of the Navier Stokes conservation equations is carried out on each cell and also on the ghost cells.

When integrating the equations on each branch from the entry point IB ₁ to the different exit points, stenosed or not, the boundary conditions given by the resolution of the Windkessel equations [17] at the output, that is to say in the case of the FIG. 11 , at points OB ₁ , OB ₂ , OB ₃ or OB ₄ , allow the equations to be solved in close succession.

There FIG. 6 also represents the example case in which the anatomical data of the vessel 10 are obtained from the geometric data of the in vitro measurements, i.e. the geometric data of the training inputs B ₃ . Arrow 18 represents the acquisition of these data coming from the PROD ₂ block.

The different geometry segmentations created in vitro can be used as many different scenarios for MOD ₂ , or CFD, modeling. One advantage is that they produce a large amount of training data to cover a large number of scenarios. For example, cases in which several stenosed portions follow one another in the same vessel.

Furthermore, these geometry data from the 3rd SEG ₃ segmentation allow the validation of CFD models. Thus, the choice of equations, in particular equation [21] for the stenosed portions or equivalent semi-empirical equations, can be adjusted to model certain vessel portions. For example, the coefficients kv and k _T of equation [21] can be adjusted according to the different geometries of stenosed portions or other coefficients can be calculated and adjusted for other semi-empirical equations.

In another case, the data is obtained from ship data already stored in a memory. The source of the data is noted PROD ₁ on the FIG. 6 . The geometric data are denoted E ₁ ''. According to an example case, the geometric data come from the same images as the data made in-vivo from the entry B ₁ of the FIG. 6 One interest is to allow comparison of CFD predictions with in-vivo measured data.

The 1D CFD model is obtained by applying the Navier-Stokes equations for each segmented portion or group of portions of the first vessel 10.
, [Math 6]
, [Math 7]

Or :
, [Math 8]
, [Math 9]
, [Math 10]

We consider x as a coordinate along the Ox axis of vessel straightening. t is the time parameter, p is the pressure, q is the flow rate, A is the section, δ is the thickness of the boundary of the layer wall and ν is viscosity.

Equations (1) and (2) can be derived with an assumption of uniform pressure within a section of the vessel.

The speed profile is denoted here by u and can be expressed as follows:
[Math 11]

According to one embodiment, additional terms, such as gravity, may be included in the equations or different velocity profiles.

According to one embodiment, an equation of state that associates the pressure and the surface area of the vessel section can also be added to the system of equations:
, [Math 12]

Or
, [Math 13]
, [Math 14]

With the following definitions of the constants or parameters taken into consideration:
, [Math 15]

p ₀ and r ₀ are the pressure and radius of the vessel at equilibrium

E is Young's modulus.

h is the thickness of the vessel.

According to one embodiment, the experimental data give the following relationship allowing these latter parameters to be linked:
, [Math 16]

Or

k ₁ = 2∙10 ⁷ g/(s²∙cm)

k ₂ = 22.53 cm-1

k ₃ = 8.65∙10 ⁵ g/(s²∙cm)

Relation (4) may, in some embodiments, incorporate elasticity effects such as viscoelasticity without affecting the structure of the CFD solver.

According to one embodiment, the thickness of the wall of a vessel is a constant of the system. The method of the invention takes into account a default value. According to one embodiment, the thickness of the vessel can be modified according to parameters such as parameters of the individual profile: such as their age or gender and parameters extracted from the imaging. According to one embodiment, the thickness of the vessel considered is taken into account in the elasticity model of the vessels.

The boundary conditions allow us to solve equations [6] to [10]. The boundary conditions are given at the entrance and exit of the vessel considered.

Boundary conditions can be obtained directly or deduced indirectly from experimental measurements.

According to one embodiment among the values which are used for the boundary conditions, we find the resistance R₁ or the characteristic impedance Zc, the resistance R₂and compliance C.

The outlet conditions can be obtained from the Windkessel equation. The latter is written as a function of the pressure p and flow q parameters:
, [Math 17]

Or :
, [Math 18]

• l’impédance Zc, notée R₁dans l’équation [17];
• l’égalité suivante : R = R₁+ R_{2 .}

This equation is true for:

• the impedance Zc, noted R ₁ in equation [17];
• the following equality: R = R ₁ + R _{2 .}

Different ways can be used to determine the coefficients of resistance R or R ₂ and compliance C and impedance R ₁ or Zc,

According to one embodiment, values of constants R ₁ or Zc, R ₂ , and C can be estimated from the in-vitro data using the pressure wave propagation speed, noted PWV and designating in English terminology “pressure wave velocity”. It is calculated using two distant pressure sensors considering a known distance.

The impedance R ₁ and the compliance C can be obtained for example from the following equations:
, [Math 19]
, [Math 20]

According to another example, values of constants R ₁ or Zc, R ₂ and C can be considered according to the values published in the literature.

However, these methods do not allow for adjusting and adapting values to each individual.

Thus, training the machine learning model allows learning the learning function from the in-vivo measured data and the CFD model.

The invention makes it possible in a second step, when the machine learning model has been trained, to deduce these values from the application of the MLA ₁ machine learning model with operating data from the ENS ₁ data set.

Equations [6] to [15] with the boundary conditions given by the Windkessel model [17] allow to solve the 1D CFD solver for vessel branches not having stenosis or more generally not being obstructed.

For portions with a stenosis, bifurcation, or aneurysm, a semi-empirical equation can be used and solved.

For example, in the case of stenosis, according to one embodiment, a semi-empirical model expressing a pressure delta, available in the literature, can be used:
, [Math 21]

where K _ν and K _t are the viscous and turbulent resistance coefficients, and the subscripts s ₀ and s ₁ refer to the values immediately before the stenosis and inside the stenosis, respectively.

Such a model is called the first pressure delta model.

In this example, the cross-section is considered constant along the entire length of the stenosis. It is specified here that the notations are similar to those of the output of the MLA ₁ model noted S ₁ , however, they do not represent the same variables or parameters.

This expression is linked to a geometry as a whole and cannot be resolved for a point x of the Ox axis and a given instant t.

The precise expressions of the coefficients K _ν and K _t can be obtained experimentally or the data can be obtained from the scientific literature.

Equation [21] can be modified for each profile of the FIG. 10 according to experimental tests. To this end, the test bench of the FIG. 8 can be used to reproduce a near-vessel flow environment under given flow conditions.

For this, each portion of vessel can be characterized by its type: simple portion, curved portion, portion with a narrowing, or even a portion with a dilation.

In each of these portions, a pressure delta can be measured by an experimental method using sensors.

The measurements made can be used to define the viscous resistance coefficients kv and the turbulent resistance coefficients _kT for each identified portion. One advantage is to use a test bench 20 allowing the estimation of the flow characteristics of a fluid to subsequently estimate the coefficients kv and kT.

A method for estimating these coefficients can be based on the implementation of a machine learning model making it possible to learn the coefficients from a plurality of measurements for which the values characterizing the flow of the fluid are known due to the test bench.

Equations [6] to [15] can be solved numerically for example from the MacCormack method considering a given order of the system and the boundary conditions. The 1D CFD model can be applied to cases of different complexity, from a single unobstructed vessel to a network of branched vessels with multiple singularities such as stenosis, bifurcation, aneurysm, etc. In order to translate the geometry obtained by the sensors into a form suitable for numerical simulation, the pre-processing software identifies and divides the complete geometry into a set of "segments" and creates a topological diagram describing their interconnection as shown in FIG. 11 .

Each segment represents a section of the vessel corresponding to one of the predefined types, for example, an unobstructed straight vessel, an unobstructed curved vessel, a vessel with stenosis, a bifurcation, etc. Each segment is also characterized by a set of variables such as its length, diameter, type, neighboring segments with which it interacts, etc.

If the system of the vessel(s) being modeled includes one or more bifurcation points, the entire computational area is divided into "branches", each branch consisting of a continuous set of segments between the segments containing the entrance or bifurcation and the exit boundaries. An example of a bifurcation is shown in FIG. 11 at the junction points AB ₁ , AB ₂ , AB ₃ .

The values of the characteristic parameters of the stenotic portions are solved from equation [21].

Appropriate equations such as equation [21] are used in "special" segments e.g. a segment with stenosis, etc. The 1D CFD model is designed to be easily adaptable to the introduction of various types of "special" segments, requiring specialized treatment, as well as to modifications of the equations governing the baseline, which is accomplished by a set of switches activating different formulations of equations [6] to [15] e.g., various treatments of elasticity [12] and [15], velocity profile [11] or boundary conditions [17].

The resolution of the system makes it possible to provide the data to test the MLA ₁ model during training and to validate the latter to obtain a prediction with a sufficient confidence score.

A data normalization step can be carried out, noted NML on the FIG. 6 This data normalization phase has two objectives: on the one hand, to normalize the data from different training sources and on the other hand to train the model with data having the same format, the same ranges of values as the data processed during the exploitation of the trained MLA ₁ model.

The normalization step in training from simulations of the second MOD ₂ model, called CFD modeling, is also used to improve stability by ensuring that all primary variables have comparable magnitudes.

Entry B3: Training with in vitro measurements

A third entry B ₃ noted “In vitro” on the FIG. 6 allows to train the MLA ₁ machine learning model. According to this method, a test bench 20 is defined so as to master a flow geometry and measurements at different characteristic points. Such a bench has the advantage of allowing to carry out thousands of experiments reflecting different configurations without requiring testing on patients. In order to train the model, measurements can be carried out in different ways. One method aims to model sections of pipes and to measure the fluidic and mechanical properties of the flow within these sections, for example by measuring the pressure from a pressure sensor, the flow rate from a sensor or a Doppler-type ultrasound, the viscosity from a viscometer, the elasticity for example from a tensile test.

• Le rayon de l'artère imprimer en 3D, exprimé en cm
• la longueur du segment, exprimé en cm
• la vitesse maximale noté Up, exprimé en cm/s
• la pulsation noté f ou f_cexprimée en Hz/60
• le débit Q, exprimé en ml/min
• la viscosité, exprimée en Pa*s
• la densité exprimée en kg/m³
• le nombre de Reynolds noté Re
• la pression d'équilibre notée P₀et exprimée en mmHg
• l’élasticité noté E, et exprimé en Mpa ou la compliance C qui est relié à la valeur de l’élasticité
• le vecteur de gravité

Many parameters can be adjusted, such as resistance R, impedance Zc, compliance C and for example the following parameters:

• The radius of the 3D printed artery, expressed in cm
• the length of the segment, expressed in cm
• the maximum speed noted Up, expressed in cm/s
• the pulsation noted f or f _c expressed in Hz/60
• the flow rate Q, expressed in ml/min
• viscosity, expressed in Pa*s
• the density expressed in kg/m ³
• the Reynolds number noted Re
• the equilibrium pressure noted P ₀ and expressed in mmHg
• elasticity noted E, and expressed in Mpa or compliance C which is linked to the elasticity value
• the gravity vector

The parameters can be adjusted for example by the choice of the properties of the materials used, the choice of the thickness of the parts used, the method of manufacturing the part such as 3D printing or casting the part, and their configuration or arrangement between them or within the bench.

An example of a test bench 20 modeling the flow of fluid within different portions is shown in FIG. 8 . Such a representation is illustrated as an example, other examples of test benches can be produced within the framework of the present invention. An advantage of this test bench is that it allows any type of segmentation to be represented. This test bench 20 can correspond to a third segmentation SEG ₃ which can, for example, imitate or not concrete cases of vessels in vivo.

The test bench 20 may comprise an inlet 26, a main flow channel 25, a test section 23, a return flow channel 27, a pump 21, pressure sensors 24.

During such tests, the pressure sensors 24 make it possible to record inlet and outlet pressures for portions to be tested having different geometries.

Section 30 represents a curved section that can be tested under the same conditions as a straight section. Section 32 is used to model a bifurcation. Section 33 is used to test a narrowing of the canal and is intended to model a stenosis.

There FIG. 9 represents different portions of bifurcation 32 with parameters characterizing the angle alpha between the bifurcated portions which and characterizing the diameters d ₁ and d ₂ of said bifurcated portions.

There FIG. 10 represents different geometries 33 modeling various ST stenoses by different channel narrowings and different topologies. These geometries can be used to generate many test bench configurations 20 and thus adjust the coefficients kv and kT of equation [21] according to the topology of the singularities describing a stenosis likely to be present in a vessel.

Narrowing profiles are noted such that the P profile defines a narrowing along straight lines. An S20 profile defines a narrowing along curved lines. An _S1 profile defines a narrowing along curved lines with an asymmetry of variation of the narrowing between the downstream portion of the stenosis and the upstream portion of the stenosis. The S30 profile has an inverse asymmetry with respect to the S10 profile. The PU profile has a narrowing with straight lines with an asymmetry within the section. The SU profile has a variation of diameter within the asymmetric stenosed portion between the upper and lower part of the vessel. The PU2 profile also has an asymmetry and an induced orientation of the flow along an axis not collinear with that of the vessel. It is possible to use two profiles of the FIG. 10 to simulate a case in which two or more stenoses are present as in the case of the FIG. 4 .

The geometric data thus modeled can be used to produce numerous scenarios. The source of this data is noted PROD ₂ on the FIG. 6 . The PROD ₂ source corresponds to the production of the input anatomical descriptors allowing the definition of the ENS ₁ input data of the MLA ₁ model.

As in the CFD framework, the PROD ₂ component allows the production of geometric data and data from all in-vitro measurements Q, P, A from the sensors placed on the test bench 20.

According to one embodiment, when training the MLA ₁ machine learning model, pressure and flow rate measurements at different points in the test channel may be measured using appropriate sensors, including pressure sensors. These values are used to train the model.

• le débit Q(x,t) par exemple en tout point à partir d’une échographie type Doppler
• la pression P(t) en différents points, par exemple 5 à 10 points, et
• A(t) en différents points, par exemple 5 à 10 points.

During training, according to one embodiment, it is possible to measure the values of the following parameters at several points in the channel:

• the flow rate Q(x,t) for example at any point from a Doppler type ultrasound
• the pressure P(t) at different points, for example 5 to 10 points, and
• A(t) at different points, for example 5 to 10 points.

In addition, the equilibrium pressure P ₀ , the viscosity, as well as the resistance R, the compliance C and the impedance Zc at the output of the branch considered, can also be measured.

It is therefore possible to test the values predicted by the MLA ₁ learning model using the measured values.

The measured data define the E ₄ data represented in the FIG. 6 . The E ₄ data allows to test the model during supervised training and to validate the MLA ₁ machine learning model to obtain a confidence score of the desired predictions. In this regard, the E ₄ test data can be fragmented into different sets so as to define an optimized training phase.

Geometric data and measured data are noted ENS ₁ ''' and E ₄ on the FIG. 6 . The geometric data are defined to be normalized identically to the training data E ₁ ' or E ₁ '' or the operating dataset E ₁ of the MLA ₁ model.

This normalization step is denoted NML. This normalization step allows in particular the processing of multi-fidelity training data coming from different systems.

One approach to combining diverse data sources is to consider them in the context of so-called “multi-fidelity” methods. Multi-fidelity methods, denoted as MF, represent a significant advance in the field of computational science, especially for complex models that require significant computational resources. These methods combine high-fidelity data that are accurate but expensive to obtain, for example, through in-vitro, in-vivo tests, and low-fidelity data that are less accurate but less expensive, such as the CFD solver, to improve the prediction accuracy.

The multi-fidelity approach is particularly advantageous in optimization and uncertainty quantification applications because it reduces the need for high-fidelity model evaluations, such as in vitro measurements. Multi-fidelity models can use Gaussian process regression or nonlinear autoregressive network architectures to correlate different fidelity levels.

Entry B1: Training with in vivo measurements

A first entry B₁and noted “In-vivo” on theFIG. 6allows training the MLA machine learning model₁ from patient data. The notation "In vivo" does not necessarily refer to manipulations carried out on a living being, but refers to the fact that the data comes from living beings, more specifically from humans. ENS patient data₂’ used are recorded in a memory before use.

The method comprises a training phase of the MLA ₁ machine learning model using in-vivo measurements. The measurements can be made from a catheter introduced into an artery or vessel of a patient. The catheter advantageously has at least one pressure sensor. The pressure readings are taken at different points identified on the acquired and straightened image of the vessel. During these measurements, the measurements are preferably taken at different times in the cardiac cycle and with different heart rates. In addition, the in-vivo flow can be measured by different methods, including Doppler ultrasound, thermodilution, etc.

In one embodiment, demographic data, patient history data, and functional data such as systolic/diastolic pressure, cardiac MRI measurements, and cardiac ultrasound may be measured. Other data may also be collected and used in various embodiments.

Laboratory measurements, including hematocrit and protein, can be used, for example, to estimate viscosity.

Alternatively, the training method of the invention does not include the measurement step, but only a step of reading data already acquired and recorded in a memory.

The collected dataset can then be used to learn an MLA ₁ machine learning model. In this scenario, the geometry of the vessel is acquired by an imaging system. The vessel of interest is segmented and the portions without stenosis and with stenoses are identified as described previously in the FFR estimation process.

The training data, for the in vivo case, in this case includes the test data E ₂ represented in FIG. 6 . According to one embodiment, the training data comprise on the one hand the ENS ₁ ' data extracted from the anatomical descriptors and the ENS ₂ ' data called patient data as well as the measured data E ₂ making it possible to validate the predictions made by the MLA ₁ model.

The training data can be split into different groups for testing to define supervised training and to validate the model to obtain a sufficient confidence score.

List of equations [Math n]

[1]: function model of a DeepONet or “Physic informed DeepONet” type machine learning model;

[2]: equation of a loss function specific to the modeling of the loss reduction linked to the mathematical operator;

[3]: equation of a loss function specific to the modeling of the loss reduction linked to the mathematical operator;

[4]; Navier Stokes equations defining the MOD ₁ model to constrain the loss function of the MLA ₁ learning model;

[5]; standardized expressions of the different variables/parameters of the equations [4];

[6]: Navier Stokes mass conservation equation defining the second MOD ₂ model also called CFD model or CFD solver;

[7]: Navier Stokes motion conservation equation defining the second MOD ₂ model also called CFD model or CFD solver;

[8]: equation allowing the estimated values to be normalized, in particular the section, position and thickness of the wall;

[9]: equation allowing the estimated values to be normalized, in particular for time and pressure, and in particular in equations [61] and [72];

[10]: expression of the Reynods number;

[11]: equation describing the velocity profile;

[12]: pressure equation of state;

[13]: normalized expression of the radius of the vessel at equilibrium;

[14] normalized expression of Young’s modulus notably used in equation [7]

[15]: Equation of the section of the vessel at equilibrium;

[16]: Empirical equation for calculating Young's modulus;

[17]: Windkessel equation;

[18]: normalized expression of the resistance and impedance coefficients of equation [17];

[19]: expression of impedance from a measurement of the propagation speed of a pressure wave;

[20]: expression of compliance from a measurement of the propagation speed of a pressure wave;

[21]: semi-empirical equation modeling the flow within a stenosed portion.

Claims (26)

Method for estimating the coronary flow reserve fraction (FFR) of a vessel comprising:

• Acquisition (ACQ ₁ ) of at least one first image (IM ₁ ) from an imaging system;
• Extraction (EXT ₁ ) of a first data set (ENS ₁ ) defining anatomical descriptors of a first vessel (10) of said first image (IM ₁ );
• Acquisition (ACQ ₂ ) of a second set of data (ENS ₂ ), called patient data, comprising at least the heart rate (f _c ) and the myocardial mass (Mc);
• Generation of an input (E ₁ ) comprising the first set (ENS ₁ ) of data, the second set of data (ENS ₂ );
• Generation (GEN ₁ ) of an output (S ₁ ) defining a prediction of a quantity of coronary flow reserve fraction (FFR) by means of the execution of a first machine learning model (MLA ₁ ) to produce an output (S ₁ ), said first machine learning model (MLA ₁ ) implementing a parameterizable loss function (L _f ) including at least a first factor (L _phy ) modeling at least one parameterized physical constraint optimized during the training of said model (MLA ₁ ), said parameterized physical constraint resulting in particular from a first numerical model of hemodynamic equations (MOD ₁ ).

Method according to claim 1 characterized in that the extraction of the first set of data (ENS ₁ ) is obtained thanks to a first segmentation (SEG1) of a representation of a set of vessels acquired by the imaging system, said extraction resulting from a selection of at least one first vessel (10) displayed on a display, the selection of a point of the vessel. Method according to any one of claims 1 to 2, characterized in that the anatomical descriptors are extracted from a representation (2) of the first vessel (10), at least one first descriptor comprising data characteristic of the radius or diameter of the section in each position of the first vessel (10) and at least one second descriptor comprising the positioning within the length of the vessel (10) of at least one narrowing of the diameter defining the presence of at least one stenosis (ST, ST ₁ , ST ₂ ). Method according to claim 3 characterized in that the second descriptor comprises data characteristic of the inlet section of at least one portion defining the narrowing of the first vessel (10) and data characteristic of the outlet section of at least one portion defining the narrowing of said first vessel (10). Method according to any one of claims 3 to 4, characterized in that the extracted anatomical descriptors also include:

• A third descriptor corresponding to the local curvature of the vessel and/or;
• A fourth descriptor corresponding to the length of a stenosis (ST, ST ₁ , ST ₂ ) and/or;
• A fifth descriptor corresponding to a local dilation of the vessel (aneurysm).

Method according to claim 1 characterized in that it comprises the estimation of the flow rate of the incoming blood flow (Q _rest ) at rest from at least the heart rate (f _c ), the systolic pressure (P _syst ) and the myocardial mass (Mc), the second set of data (ENS ₂ ) comprising the value of said flow rate of the incoming blood flow (Q _rest ). Method according to claim 1 characterized in that it comprises the estimation of the flow rate of the incoming hyperemic flow at rest from at least the heart rate (f _c ), the systolic pressure (P _syst ) and the myocardial mass (Mc), the second set of data (ENS ₂ ) comprising the value of said flow rate of the incoming hyperemic flow. Method according to claim 1 characterized in that the second set (ENS₂) of patient data includes in particular:

• An Age of the Individual;
• A kind of individual;
• An indicator of the individual's diabetes;
• An indicator of individual hypertension.

Method according to any one of claims 1 to 8, characterized in that the first machine learning model (MLA ₁ ) is a DeepONet network whose loss function (Lf) integrates a second factor (L _opé ) defining a loss function of a differential mathematical operator. Method according to any one of claims 1 to 9, characterized in that the first factor (L _phy ) is modeled during the training of the machine learning model (MLA ₁ ) by a parametric function whose parameters are dependent on the solutions of the equations of the numerical model of hemodynamic equations (MOD ₁ ) characterizing the hemodynamic flow in the first vessel (10), said parametric function being implemented by a continuous non-linear operator, the resolution of said parametric function making it possible to generate value limits outside of which the cost function (L _phy ) is penalized. Method according to claim 10 characterized in that the equations characterizing the hemodynamic flow in the first vessel (10) to optimize the cost function (L_phy) and modeled by the first numerical hemodynamic model (MOD)₁) which includes a system of equations comprising:

• A model of conservation of mass;
• A model of conservation of momentum;
• An equation of state for pressure.

Method according to one of the preceding claims, characterized in that it comprises a training step (B ₃ ) carried out from a third set of training data (ENS ₁ ''', E ₄ ) characterizing portions of pipes each modeling an irregular geometry characteristic of at least one predefined section profile, the training of the machine learning model (MLA ₁ ) being supervised so that the values characterizing the flow of a fluid in each portion are measured on a test bench (20) comprising at least one portion of test pipe and sensors making it possible to measure test data (E ₄ ) at different points of said test bench (20) comprising in particular the flow rate, the pressure, the viscosity, the resistance of the fluid, the Reynolds number and the density of the fluid. Method according to claim 12 characterized in that it comprises a characterization of the different portions of the test bench (20), each portion being characterized by a radius of artery to be modeled, a length of artery to be modeled, an elasticity or a compliance (C) and a type of portion. Method according to claim 13 characterized in that it comprises a pump, a clock and a means of recording a pumping frequency so as to inject a fluid into the test bench (20) reproducing a periodic activity relating to the cardiac cycle characterizing the different portions of the test bench (20). Method according to any one of claims 13 to 14 characterized in that the third set of data (ENS₁’’’) also includes a measurement and calculation for at least one point of each predefined pipeline section:

• of at least one output resistance (R, R ₂ ) modeling the force which opposes the flow of blood in the portion of the first vessel (10);
• of at least one impedance (Zc, R ₁ );
• of at least one fluid viscosity value;
• of at least one value of the pressure (P ₀ ) at equilibrium.

Method according to any one of claims 1 to 15 characterized in that it comprises a training step (B ₂ ) carried out from a second set of training data (ENS ₁ '', E ₃ ) characterizing the geometry (ENS ₁ '') of portions of vessels obtained from an imaging system, the training of the machine learning model (MLA ₁ ) being supervised so that the values characterizing the flow of a fluid in the first vessel (10) are estimated by a second model of hemodynamic equations (MOD ₂ ) called digital solver (CFD ₁ ), to define test and validation data (E ₃ ) of the machine learning model (MLA ₁ ). Method according to claim 16 characterized in that a second segmentation (SEG₂) of the first vessel (10) automatically generates a plurality of segment types (6, ST₁, ST₂) of which:

• a first type of segment (6) defining substantially regular portions corresponding to portions of vessels having a section gradually decreasing from a proximal end towards a distal end;
• a second type of segment (ST, ST ₁ , ST ₂ , sg ₅ , sg ₇ , sg ₁₂ , sg ₂₀ ) defining at least one irregular portion, said at least one irregular portion corresponding to a portion comprising at least one narrowing of the diameter greater than a predefined threshold,

each of the sets of segments having a system of equations modeling the hemodynamic flow in said portion. Method according to any one of claims 16 to 17 characterized in that the second segmentation (SEG₂) of the first vessel (10) automatically generates a plurality of segment types (6, ST₁, ST₂) of which :

• a third segment type (sg18) characterized by the presence of a local curvature of the vessel greater than a predefined threshold and/or;
• A fourth segment type (sg15) characterized by local dilation of the vessel (aneurysm).

Method according to any one of claims 16 to 17, characterized in that the digital solver (CFD ₁ ) comprises a system of equations characterizing the physics of the flow of a hemodynamic flow from the incompressible Navier Stokes equations applied to at least one portion of a vessel of a single dimension. Method according to claim 19 characterized in that the system of equations comprises a second numerical hemodynamic model (MOD₂) comprising for each portion of vessel at least one model among which:

• A model of conservation of mass and/or;
• A model of conservation of momentum and/or;
• An equation of state for pressure and/or;
• A first model of a pressure delta and/or;
• A second model of a pressure delta established from in vitro measurements.

Method according to any one of claims 16 to 20, characterized in that it comprises a calculation of the blood flow characteristics in each portion of the first vessel (10) at each point of the axis of the vessel from the second hemodynamic model (MOD ₂ ) applied locally to each of the portions. Method according to one of claims 16 to 21, characterized in that the second hemodynamic digital model (MOD ₂ ) makes it possible to estimate for each portion of vessel of the second type (ST, ST ₁ , ST ₂ ) a pressure delta from a pressure delta model comprising a viscous resistance coefficient (kv) and a turbulent resistance coefficient (k _T ). Method according to one of claims 16 to 21, characterized in that the second hemodynamic digital model (MOD ₂ ) makes it possible to estimate for each portion of vessel of the second type (ST, ST ₁ , ST ₂ ), of the third type (sg18), and/or of the fourth type (sg15) a pressure delta from a pressure delta model comprising a viscous resistance coefficient (kv) and a turbulent resistance coefficient (k _T ), said coefficients being estimated from tests carried out on an in vitro test bench (20) of one of claims 12 to 15, said measurements carried out on the test bench (20) making it possible to estimate flow characteristics of a fluid. Method according to any one of claims 16 to 21, characterized in that the first machine learning model (MLA ₁ ) is trained by considering the values estimated by the second model (MOD ₂ ) integrated over the entire first vessel (10). Method according to one of the preceding claims, characterized in that a first set of training data (ENS ₁ ', ENS ₂ ', E ₂ ) comprises geometry data (ENS ₁ ') extracted from a patient vessel imaging system, patient data (ENS ₂ ') comprising at least the heart rate (f _c ), the systolic pressure (P _syst ) and the myocardial mass (Mc) and test and validation data (E ₂ ) from pressure measurements at different points of the portion of vessel considered. Method of training a machine learning algorithm to produce a trained network (MLA₁) making it possible to calculate the pressure of a fluid at a plurality of positions within a first vessel (10) as a function of time within the cardiac cycle, said training method comprising:

• Acquisition of a first set of training data (ENS ₁ ', E ₂ ) from:
  • Acquisition (ACQ ₁ ) of a first image (IM ₁ ) from an imaging system;
  • Extraction (EXT ₁ ) of a first data set (ENS ₁ ') defining anatomical descriptors of a first vessel (10) of said first image (IM ₁ );
  • Acquisition of a third set of data (ENS ₂ '), called patient data, comprising at least the heart rate (f _c ) and the myocardial mass (Mc);
  • Reading data (E ₂ ) characterizing pressure measurement points of a first vessel (10) defining test and validation data for a machine learning model (MLA ₁ );
• Execution of the machine learning model (MLA ₁ ) and learning by implementing a cost function minimizing the error between data produced (S ₂ ) by the model (MLA ₁ ) and the test and validation data (E ₂ );
• Acquisition of a second set of training data (ENS ₁ '', E ₃ ) from:
  • Acquisition (ACQ ₁ ) of a first image (IM ₁ ) from an imaging system;
  • Extraction (EXT ₁ ) of a first data set (ENS ₁ '') defining anatomical descriptors of a first vessel (10) of said first image (IM ₁ );
  • Resolution of a solver (CFD ₁ ) modeling a model (MOD ₂ ) of hemodynamic equations from the incompressible Navier Stokes equations in one dimension, the data estimated (E ₃ ) by the solver (CFD ₁ ) defining test and validation data for a machine learning model (MLA ₁ );
• Execution of the machine learning model (MLA ₁ ) and learning by implementing a cost function minimizing the error between data produced (S ₃ ) by the model (MLA ₁ ) and the test and validation data (E ₃ );
• Acquisition of a third set of training data (ENS ₁ ''', E ₄ ) from:
  • Modeling of a set of geometries defining pipes suitable for conveying a fluid whose viscosity is substantially that of blood and suitable for being arranged within a test bench (20);
  • Performing a set of fluid flow tests with a variety of different geometries within the test bench (20);
  • Data measurement (E ₄ ) characterizing pressure measurement points of a first portion of channel defining test and validation data for a machine learning model (MLA ₁ );
• Execution of the machine learning model (MLA ₁ ) and learning by implementing a cost function minimizing the error between data produced (S ₄ ) by the model (MLA ₁ ) and the test and validation data (E ₄ ).