Ultrasonic estimation of flow velocity and pressure in blood vessels

Abstract

The invention relates to a method for determining a velocity of a fluid and / or a pressure distribution in a vessel, the method comprising the steps of: - Acquiring a series of ultrasonic signal data comprising information on tracers comprised by a fluid flow in a vessel, - Three-dimensionally localizing and tracking at least some of the tracers in the series of the ultrasonic signal data, such that for each tracked tracer flow data is obtained, the flow data comprising information on a plurality of locations and associated velocities at different time points of the tracer, - From the locations of the tracers, determining, particularly segmenting a flow volume delimiting a volume within which the fluid flow takes place in the vessel, - Estimating the velocity and / or the pressure for at least one time point of the fluid in the vessel with a physics-informed machine learning method that is trained with at least some of the flow data and with location data, wherein the location data comprises a plurality of locations within the flow volume and associated time points.

Description

[0001] Ultrasonic estimation of flow velocity and pressure in blood vessels

[0002] Specification

[0003] The invention relates to a method for determining a velocity of a fluid and / or a pressure in a vessel, a computer program for executing the method on a computer and a measurement system configured to execute the method.

[0004] Attempts to ultrasonic super-resolution vascular imaging have been seen in the past decade. This is achieved by accumulating centroids of a sparse contrast agent (e.g. microbubbles) signals over hundreds or thousands of ultrasound images. The agent is comprised by the fluid, e.g. the blood in form of tracers. The localized centroids of the tracers are used to represent the locations of the tracers confined within the vessel lumen. With this localization-based technique, the spatial resolution of ultrasound imaging can be improved by a factor of 10. However, many difficulties exist when applying this technique for imaging stenosed large vessels, such as the coronary arteries. First, the accumulated location map is sparse, and it takes several minutes to get a dense reconstruction of centroids which allows a conclusion on a structure. Second, the blood velocity in the vessel can exceed 1m / s, for example in case the vessel is stenosed. This velocity is so high that - due to the too low temporal resolution of the methods known in the art other than ultrasound imaging - the related fluid dynamics cannot be captured in detail. Third, no pressure information is available from super-resolution ultrasonic imaging technics in the art.

[0005] On the other hand, machine learning has become an increasing popular tool in imaging applications. Specifically, physics-informed neural networks, PINNs, allow encoding of the governing equations that describe the physics behind the observed problem that is to be solved by the neural network.

[0006] Neural networks, however, require extensive training data, wherein acquiring training data in a patient-specific way, i.e. from a single patient, is in most applications unfeasible, particularly, as it takes too long or these data are simply not there to this extend required.

[0007] Therefore, the application of neural networks is limited in patient-specific applications. The invention sets out to solve the drawbacks of super-resolution ultrasonic imaging as laid out above and to provide a method and a system configured to non-invasively estimate in a patient-specific way both the velocity and / or a pressure information in vessels in a matter of seconds.

[0008] Compared with MRI and CT, the unprecedented high frame rate capability of ultrasound imaging (kHz) is critical and eventually allows tracking the high-speed tracers in vessels. The method has the direct application of quantifying vessel stenosis.

[0009] An object of the present invention is to provide a cost-efficient, non-invasive imaging method that allows for generating patient-specific rapid high-resolution information on a fluid flow in a vessel.

[0010] Applications of the invention are to limited to living patients, but can be extended to other field of technology, such as fluid flow imaging in general.

[0011] The object is achieved by the method for determining a velocity of a fluid and / or a pressure, particularly a pressure distribution in a vessel having the features of claim 1.

[0012] Advantageous embodiments are described in the dependent claims.

[0013] According to a first aspect of the invention, a method, particularly a computer- implemented method for determining a velocity of a fluid and / or a pressure, particularly a pressure distribution in a vessel comprises at least the steps of:

[0014] Acquiring a series of ultrasonic signal data comprising information on tracers comprised by a fluid flow in a vessel,

[0015] Three-dimensionally localizing and tracking at least some of the tracers in the series of the ultrasonic signal data, such that for each tracked tracer flow data is obtained, the flow data comprising information on a plurality of locations and associated velocities at different time points of the tracer,

[0016] From the locations of the tracers, determining, particularly segmenting a flow volume delimiting a volume within which the fluid flow takes place in the vessel

[0017] Estimating the velocity and / or the pressure for at least one time point of the fluid in the vessel with a physics-informed machine learning method, particularly with a physics-informed deep learning method, more particularly with a physics-informed neural network, that is trained with at least some of the flow data and location data, wherein the location data comprises a plurality of locations within the flow volume and associated time points.

[0018] The invention allows for high-resolution ultrasonic 3D-imaging of vessels that may not be resolvable with conventional ultrasonic measurement methods. But not only is the spatial resolution limit overcome by the method according to the invention, but the method allows to determine with high accuracy a fluid flow in the vessel in terms of velocity along three dimensions and a pressure in the vessel. The velocity and the pressure are furthermore determined and predicted in a time-resolved fashion at sampling rate that is seemingly unprecedented in X-ray-imaging, MRI, CT, or optical imaging.

[0019] Therefore, the method provides a very cost efficient, high-resolution imaging modality that may be applied to existing ultrasonic imaging devices. The method does not require e.g. radioactive markers, or expensive CT-equipment

[0020] The method may be particularly advantageous when imaging coronary artery stenosis. Therefore, the vessel may be a coronary artery.

[0021] The tracers may be any kind tracers that provide contrast in the ultrasonic signal data for localizing the tracers.

[0022] The tracers may be or comprise microbubbles injected into the vessel. Microbubbles are particularly advantageous when imaging blood vessels of a living patient, as for example, the use of microbubbles is clinically approved for cardiac ultrasonic imaging.

[0023] It is noted that the vessel may not be visible or recorded in the ultrasonic signal data, but only the tracers.

[0024] Each or at least some of the tracers may be recorded in the series of ultrasonic signal data at two or more time points. This allows for tracking the tracers.

[0025] The acquisition of the series of ultrasonic signal data may be facilitated directly from an ultrasonic measurement system or in form of a file obtained from a computer or a non-transitory storage medium. Training of the method requires only minimal ultrasonic acquisition time - in the order of seconds only. The acquisition time in essence corresponds to a recording length of the ultrasonic signal data.

[0026] Particularly, acquisition times are within the range of 0.25 s to 10 s, more particularly within 0.5 s and 5 s.

[0027] According to the method at least some or all of the tracers are localized and tracked in three spatial dimensions in the series of the ultrasonic signal data. From the localizations of the same tracers in the series it is possible to determine an associated three-dimensional velocity of the tracer for at least some time points at which the tracer is localized. Flow data is obtained and designed to comprise the information on time points, determined location form the localization, and the associated velocity.

[0028] Thus, it is possible to obtain for each tracked tracer a 3D-trajectory in the flow volume reflecting velocity and location over time.

[0029] According to another embodiment of the invention, at least some of the trajectories are graphically displayed. This allows to gain a better understanding on the flow volume and a velocity of the tracers.

[0030] For localizing the tracers, the tracers need to fulfil a sparsity criterion, i.e. a signal of the tracer should be spatially separable from the signal of other tracers at the time of localization of the tracer.

[0031] For example, in the ultrasonic signal data if rendered as a series of ultrasonic images, the tracers point spread functions in the images should be spatially resolvable from each other. The term “spatially resolvable” in turn depends inter alia on the specifics of the ultrasonic imaging system generating the ultrasonic signal data, the signal quality and the processing of said signals. The skilled person is aware of several similar definitions for resolution in ultrasonic signal data. A basic requirement is that the locations of each tracer in the signal data may be determined by means of a centroid method determining a center of the signal originating from the tracer. If two tracers are too close to each other to be spatially resolved, the localization may be discarded or omitted.

[0032] The flow volume particularly comprises information on a shape and a size of the flow volume delimiting a volume within which the fluid flow takes place in the vessel. The flow volume, particularly the shape and the size of the flow volume may be determined by determining an envelope from all tracer localizations of the flow data.

[0033] One advantage of localizing the tracers is that a dramatically improved spatial resolution is obtained for the locations as well as for the determination of the associated velocity of each tracer. This advantage is then further reflected in the very precise determination of the flow volume, which may resemble the actual vessel volume. As a consequence, the velocity and pressure for vessels that are considered to be too small for imaging using ultrasonic imaging systems becomes possible. As ultrasonic imaging systems intrinsically provide high recording rates, the combination of the method with such an ultrasonic imaging system provides a cost-efficient solution for high resolution, high-speed imaging of vessels.

[0034] Based on the very precise flow data on location, velocity, the method estimates the velocity and / or the pressure for at least one time point of the fluid in the vessel with a physics-informed machine learning method.

[0035] The estimation of the velocity and / or the pressure is typically made for locations and time points other than the ones in the flow data. This allows to estimate velocities and pressures in the flow volume at any desired sampling density, allowing to generate time- and spatially-resolved velocity distributions and / or time- and spatially resolved pressure distributions for the flow volume.

[0036] According to another embodiment of the invention, the physics-informed machine learning method is or comprises a physics-informed deep learning method.

[0037] According to another embodiment of the invention, the physics-informed machine learning method is or comprises a physics-informed neural network.

[0038] The physics-informed machine learning method is trained with at least some of the flow data and location data.

[0039] Training with the flow data allows for specifically adapting the method to any vessel geometry and fluid flow, further, the flow data provide instances of a “ground truth”, wherein the location data serve for generating sufficient training data for the physics- informed machine learning method. The location data can be automatically generated and are not required to stem from localizations of tracers. This allows for generation of as many locations for training as desired and deemed necessary to obtain robust estimations for the velocity and pressure of the fluid flow in the flow volume.

[0040] Particularly, the number of tracked tracers is at least one or two orders of magnitude smaller than a number of locations generated for the location data.

[0041] This approach allows for example to record a series of ultrasonic signal data governing a recording time of about one second at a sampling rate of 1kHz, and to train from this amount of data the physics-informed machine learning method sufficiently well to generate robust and accurate estimates of the velocity and / or the pressure.

[0042] The ultrasonic signal data may comprise beamformed ultrasonic data, more particularly ultrasonic image data.

[0043] The term “physics-informed” particularly relates to a machine learning method that is trained by a loss function that at least in part comprises equations or functions that govern a physics problem or a physical context, such as equations governing the physics of flow dynamics of a fluid in a vessel.

[0044] This way, the machine learning method tends to find estimates that are close or identical to the analytical or numerical solution of the governing equations or functions.

[0045] According to another embodiment of the invention, the fluid is a liquid, particularly an aqueous liquid.

[0046] In the current specification the term “ultrasonic” and “ultrasound” are used synonymously.

[0047] According to another embodiment of the invention, training of the physics-informed machine learning method comprises the step of providing the flow data of at least some tracked tracers as input to the physics-informed machine learning method, wherein the physics- informed machine learning method determines, particularly outputs for each location and time point an estimated velocity, wherein a flow data loss function determines a flow data loss value indicative of a deviation of the velocities of the provided flow data from velocities estimated by the physics-informed machine learning method. The flow data loss function takes into account the flow data and thus provides a reference to the physics-informed machine learning method with respect to a ground truth, i.e. the flow data, that have to be met for valid estimates.

[0048] The deviation of the velocities of the provided flow data from velocities estimated by the physics-informed machine learning method may be for example a mean square deviation, particularly a root mean square deviation, of the estimated velocities from the velocities form the flow data.

[0049] The flow data loss function however, does not account for any physics law underlying the fluid flow in the vessel, but only references estimates from the physics-informed machine learning method to essentially experimental data, i.e. the flow data.

[0050] According to another embodiment of the invention, training of the physics-informed machine-learning method comprises providing the location data as input to the physics-informed machine learning method, wherein the physics-informed machine learning method determines, particularly outputs for each provided location and time point an estimated velocity and an estimated pressure of the fluid at the location and the time point, wherein the estimated velocities and the estimated pressures are provided to a physics loss function comprising, the Navier-Stokes equations, particularly a left hand side of the Navier-Stokes equations resolved to zero, wherein the physics loss function determines a physics loss value indicative of a deviation of the velocities and the pressures estimated by the physics-informed machine-learning method from velocities and pressures that would solve the Navier-Stokes equations.

[0051] This embodiment allows for creation of the physics-informed machine learning method configured to estimate physically meaningful estimates of the fluid flow in the vessel.

[0052] The Navier-Stokes equations are differential equations that describe fluid flows and the dynamic of these.

[0053] The Navier-Stokes equations in the context of the current specification may also govern the continuity equation for fluids.

[0054] While the Navier-Stokes equations equate a left-hand side of terms with a right-hand side of terms, it is also possible to resolve these equations to zero, such that without loss of generality, on the left-hand side of the equations all terms of the Navier- Stokes equations are placed and equated to zero.

[0055] This resolved form allows to create a physics loss function comprising the right-hand side of the equations, wherein the physics loss function if a solution is provided to the physics loss function would yield a physics loss value of zero, wherein in any other case, the physics loss value would be different to zero, indicating that the proposed solution is not exact and / or wrong. Thus, the deviation determined by the physics loss value may be assigned to the output of the left-hand side of the Navier-Stokes equations resolved to zero.

[0056] Training of the physics-informed machine learning method therefore biases the method toward physical solutions for the velocities and / or the pressure.

[0057] Particularly, the physics loss function is provided only with the location data, i.e. no additional information on velocities or a ground truth is provided, which allows as many training episodes - e.g. by providing more and more location data - as desired until the estimates of the velocity and / or the pressure are robust and physically meaningful, e.g. when a gradient descend training estimator indicates a minimum.

[0058] Information regarding the flow volume is in essence encoded in the locations of the location data, as no location outside the flow volume is provided during training.

[0059] According to another embodiment of the invention, training comprises a first stage, a second stage, and a third stage, wherein in the first stage, the data loss value is determined, in the second stage the physics loss value is determined, and in the third stage a composite loss value is determined from a composite loss function comprising the data loss function and the physics loss function, particularly wherein the composite loss function determines a composite loss value, particularly wherein the composite loss function comprises a sum, particularly a weighted sum, of the data loss function and the physics loss function, wherein the composite loss function is minimized during training episodes, such that the composite loss value is minimal, e.g. by adjusting weights of neurons comprised by a neural network of the machine learning method.

[0060] It is noted that the composite loss value may be calculated from a sum of the magnitudes of the physics loss value and the data loss value, i.e. the values are taken as positive regardless of their sign.

[0061] The composite loss function allows to guide the physics loss function toward meaningful solutions of the Navier-Stokes equations, as the data loss function and its corresponding data loss value penalizes solutions that minimize the physics loss function but that are not connected to the flow data as determined from the ultrasonic signal data. In turn, the physics loss function and its corresponding physics loss value penalizes solutions that minimize the data loss function but comprise velocities that that are non-physical and thus artifacts. That is, the composite loss function contributes and enables the training of the physics-informed machine learning method based on comparably few flow data from the ultrasonic signal data. The flow data may be used solely on the data loss function, during the first stage, wherein the location data may be exclusively used for the second stage. Obviously, providing the location of the flow data during the second stage does not harm the training.

[0062] Particularly, these three stages are executed within each episode of the training.

[0063] According to another embodiment of the invention, the Navier-Stokes equations correspond to Navier-Stokes equations adjusted for incompressible fluids, conservation of momentum, and conservation of mass.

[0064] The latter may be reflected by the continuity equation comprised by the Navier- Stokes equations.

[0065] This embodiment for example allows modeling of most water-based liquids and is thus particularly useful when applying the method to flow estimations in a blood vessel.

[0066] According to another embodiment of the invention, the physics loss function comprises physics residual functions, fa, f2,f3, fa, based on the terms of Navier- Stokes equations, particularly wherein the physics residual functions, fa, f2,f3, fa, comprises all terms of the Navier-Stokes equations resolved to zero: wherein denotes a partial differential operator, t denotes time, p denotes a particularly constant fluid density, p denotes a viscosity of the fluid, u, v, w are velocity components of the velocity estimated by the physics-informed machine learning method along three dimensions x , y , z , wherein p is the pressure as estimated by the physics-informed machine learning method, wherein the physics loss value Lphysicsand the physics loss function is determined according to: Wherein 7Vphysicsdenotes the number of locations for which the physics loss function has been evaluated.

[0067] The Navier-Stokes equations may be provided either in dimensional or in non- dimensional expressions.

[0068] The physics residual equations are defined and used to enforce physics constraints within the flow volume and time points.

[0069] Particularly, the physics residual functions , f2,f3, relate to the Navier-Stokes equations for conservation of momentum, wherein f4, relates to the continuity equation for conservation of mass.

[0070] According to another embodiment of the invention, the data loss function and the data loss value Ldatais determined according to: wherein u, v, w are velocity components for location x^y^Zj at time point t, of the velocity estimated by the physics-informed machine learning method and wherein “reference . “reference . Reference are the corresponding velocity components of the velocity of the provided flow data, wherein Ndata, denotes the number of locations and time points for which the data loss function has been evaluated during an episode.

[0071] According to another embodiment of the invention, the composite loss function and its corresponding composite loss value is minimized by a gradient descent method.

[0072] The composite loss value ^composite might be defined as ^composite = “ |idata l + b |Lphysics b wherein a and b are positive numbers for weighting the contributions of the physics loss value and the data loss value.

[0073] According to another embodiment of the invention, for the tracking of the tracers, a localization method for the tracers, particularly the spatially resolvable tracers is applied to the ultrasonic signal data, wherein the localization method determines a centroid of a point spread function of the tracer in the ultrasonic signal data, particularly such that the location of each tracer is determined with a spatial resolution that is greater than a spatial resolution of the ultrasonic signal data.

[0074] For this purpose, the ultrasonic signal data may be transformed to a series of images in which an information on an ultrasonic contrast is encoded in the pixel values. Each tracer will give rise to a point spread function, wherein the localization method is configured to determine a centroid of the point spread function.

[0075] This allows determination of the location of the tracer beyond the spatial resolution of the ultrasonic measurement system and thus a particularly precise and accurate estimation of the flow volume and the flow data.

[0076] According to another embodiment of the invention, the vessel has a diameter in the range from 6 to 15 wavelength of the sound waves used for recording the ultrasonic signal data, particularly at an imaging depth of approximately 234 wavelengths.

[0077] This for example corresponds to a diameter in the range of 1.5 mm (@ 6 MHz) up to 3.8 mm at an imaging depth of about 60 mm.

[0078] However, the skilled person is aware of achievable spatial resolutions with respect to imaging depth and wavelength. Therefore, these numbers are provided solely for illustrative purposes.

[0079] According to another embodiment of the invention, the vessel is a blood vessel in a living body, particularly wherein the blood vessel is a coronary artery.

[0080] The method may therefore be used as an imaging method on the clinical context, which in turn allows to diagnose various blood vessel pathologies fast and reliable using existing ultrasonic measurement systems upgraded with the capabilities provided by method according to the invention. Particularly with regard to living bodies, a short measurement time is advantageous for example in terms patient cooperation.

[0081] The method may be used for determining a stenosis of an artery, particularly a stenosis of a coronary artery.

[0082] According to another embodiment of the invention, the tracers are so sparse in the fluid that at least some of the tracers are spatially separated from other tracers so that most of time and at most time points their associated point spread functions don’t overlap, particularly wherein the ultrasonic signal data is processed in time steps.

[0083] According to another embodiment of the invention, the method is executed during an ultrasonic imaging examination of a patient, particularly a blood vessel of the patient, wherein the training of the physics-informed machine learning method is executed for each patient at least once, particularly during examination.

[0084] This embodiment details the applicability of the method in the context of clinical examinations of patient, such as humans. Particularly, the method is suited to replace or at least complement any existing method for diagnosing stenosis of an artery, such as invasive and non-invasive X-ray-based methods, such as Fractional flow reserve methods (FFR, CFR), or CT coronary angiography (CTCA), which are elaborate, and require radioactive markers.

[0085] The method may be applied in tumor imaging and / or cardiology.

[0086] According to another embodiment of the invention, for determining the flow volume from the locations of the tracers, a boundary is determined that delimits the flow volume, wherein the boundary is parametrized by means of finite elements, such as triangles.

[0087] According to another embodiment of the invention, a time-resolved velocity and / or a time-resolved pressure distribution for at least a portion of the flow volume, particularly for the complete flow volume is estimated by the trained physics-informed machine learning method.

[0088] Gaining knowledge of the velocity distribution and or the pressure distribution allows for visualization of the fluid flow in the flow volume, which in many applications is crucial. The distribution may comprise sampling of the velocity and / or pressure at a regular grid in the flow volume, such that a distribution may be obtained.

[0089] Sampling density is essentially only limited by the location data that may be used to query the trained physics-informed machine learning method, i.e. how many location and time points are submitted to the trained physics-informed machine learning method.

[0090] For example, according to another embodiment of the invention, the distribution(s) are generated over the course of several cardiac cycles.

[0091] Additionally, the prediction method can be applied to any segment of the series of ultrasonic signal data, allowing for the possibility of utilizing multiple GPUs in parallel to handle different parts of the ultrasonic signal data.

[0092] According to another embodiment of the invention, the velocity and / or the pressure estimated by the trained physics-informed machine learning method is displayed, particularly wherein the estimated velocity and / or the estimated pressure is displayed in combination with the flow volume, more particularly wherein the estimated velocity and / or the estimated pressure is displayed such that the velocity and / or the pressure is associated to a three-dimensional position in the flow volume.

[0093] This embodiment allows for visualization of the estimated velocity and / or the pressure in a suitable and easy conceivable format. According to another embodiment of the invention, a sequence of velocities and / or pressures as estimated by the trained physics-informed machine learning method is generated, particularly in form of a movie, such that a temporal evolution of the velocity and / or pressure, particularly a temporal evolution of the velocity distribution and / or the estimated pressure distribution is generated, wherein said sequence(s) is / are displayed, particularly in form of a movie.

[0094] The time points may be selected according to a user input or in a predefined fashion. The time points may be selected to lie in the future or in the past with regard to the time points associated to the flow data.

[0095] In order to generate a sequence of the velocity and / or the pressure distribution a plurality of locations with associated time points may be submitted to the physics- informed machine learning method.

[0096] According to a second aspect of the invention, a computer program comprises computer program code, that when executed on a computer causes the computer to execute the method according to one of the preceding claims.

[0097] The computer program may be in form of a computer program product, e.g. stored on a non-transitory storage medium. For the purpose of executing the computer program, the code may be executed by one or more processors of the computer.

[0098] According to a third aspect of the invention, an ultrasonic measurement system comprises at least the following components:

[0099] An ultrasonic transducer head system configured to generate and record the ultrasonic signal data;

[0100] A computer connected to the ultrasonic transducer head system and configured to receive the ultrasonic signal data. The ultrasonic measurement system is characterized in that the computer is configured to execute the method according to the first aspect or the computer program according to the second aspect of the invention.

[0101] For the purpose of communication with any necessary devices or systems for executing the method, such as for example with the transducer head system or a display for displaying the velocity and / or the pressure, the computer may comprise interfaces configured to communicate with the respective device or system, e.g. via cable or wirelessly.

[0102] Definitions, embodiments and / or features provided in the context of the first aspect of the invention are analogously applicable to the invention according to the second and third aspect and vice versa. Particularly, exemplary embodiments are described below in conjunction with the Figures. The Figures are appended to the claims and are accompanied by text explaining individual features of the shown embodiments and aspects of the present invention. Each individual feature shown in the Figures and / or mentioned in said text of the Figures may be incorporated (also in an isolated fashion) into a claim relating to the device according to the present invention.

[0103] In Fig. 1 a simulated vessel 1 with a fluid flowing through the vessel 1 with a pulsating pressure is depicted. The fluid, which, in this exemplary simulation, is an incompressible liquid with the viscosity and density of blood, comprises tracers in form of simulated microbubbles that give rise to a contrast during recording with an ultrasonic measurement system.

[0104] The arrows point to the direction of flow. The vessel 1 may be a blood vessel.

[0105] The simulation was performed with the following parameters: a flow vessel phantom with a 60% stenosis 2 was simulated in Ansys Fluent. The vessel 1 had a diameter of 3.8mm, and a length of 18mm, matching the size of the left main coronary artery of a human. Unsteady flow simulations were run with a peak velocity of about 1m / s. The density of the fluid was 1060 kg / m3, and the viscosity was 3.5 mPas. Flow velocity and pressure data were saved every 1 ms.

[0106] Approximately twelve random scatters representing contrast agents, i.e. tracers were generated at the 1st time step, their locations were then updated every 1 ms based on the closest velocity obtained from the above Ansys Fluent simulation. Tracer replenishment was performed from the inlet 3 when the number of scatters within the vessel was less than twelve.

[0107] For ultrasound simulation, the software package KWave was employed. Each pulseecho simulation consisted of the transmission of a diverging wave with a virtual source placed behind the center of a 32x32-element matrix transducer at -27 mm, and the reception of the radio frequency (RF) data with all 1024 elements. The excitation signal was a 2-cycle sinusoid at 6 MHz, and the sampling frequency was 120 MHz. The pulse repetition frequency (frame rate) was 1 kHz, and the tracers generated in the previous step were imaged every 1 ms. This data may be considered as exemplary ultrasonic signal data 100. The recorded RF data were beamformed with a 3D delay-and-sum beamformer (cf Fig. 2A). This way, a series of images could be obtained from the ultrasonic signal data that were used for localizing the tracers in each time step. The centroids of the tracers were localized by using a Gaussian kernel, matching the ultrasound point spread function (PSF) 101- however also other kernel shapes for centroiding may be used. Separation thresholds were set to reject localizations, if they were too close with each other, potentially representing overlapping PSFs. The localized centroids were then linked between every two frames by minimizing the total displacement. This is depicted for a single time step in Fig. 2B. It is noted that the time step may be used synonymously with the term time point in the context of the current specification.

[0108] Fig. 3: From the series of tracked tracers it is possible to determine trajectories 107, which comprise information of the time-resolved locations as determined by the localizations 103 and associated velocities 104 for the tracers. The trajectories 107 that last for at least 10 ms, i.e. 10 images, were preserved, cf. Fig. 3A. The trajectories 108 were separately filtered with a Kalman Filter employing a second order motion model (displacement, velocity and acceleration), cf. Fig. 3B. The displacement (Ad) for a specific tracer between every two adjacent frames was used to render the corresponding velocity components in 3D (u, v, w) at that specific time (f) - Ad / At, and At is the time step of 1ms in this case.

[0109] This results for each time step, i.e. time point of the simulation in flow data 100 comprising localizations 103 and associated tracer velocities 104 is shown in Fig. 2B.

[0110] Fig. 4A depicts determined locations of the tracers from a different perspective. Locations 104 of the tracers are depicted as dots. Using specific algorithms, the flow volume 105 of the vessel 1 was determined based on the locations 104 of the tracers with the following steps:

[0111] Accumulation of locations 104 of the tracers are used to create a three-dimensional boundary surface (enclosing all localizations), which represents the flow volume 105, cf Fig. 4A.

[0112] In particular the ‘boundary’ Matlab function has been used followed by triangulation (‘trisurf’ Matlab function) to obtain the flow volume 105 in parametrized form, cf. Fig. 4B. In Fig. 5 the density of location data 106 as used for training the physics-informed machine learning method is depicted. The density of locations 109 in the location data 106 is much higher, typically several orders of magnitude, then the locations 106 as determined by from the flow data 102. This allows for a dense sampling of the flow volume 105. For example, the flow data 102 may comprise 5 to 10 locations 103 per time point, wherein the location data 106 comprises about 3,000 to 10,000 locations per time step.

[0113] Construction of the neural network, training and prediction

[0114] The physics-informed machine learning method used in the example is a physics- informed neural network. Said network used spatial coordinates x, y, z and t as the inputs and outputs three-dimensional velocity components u, v, w and / or pressure p at said time. In one implementation, there were 10 hidden layers in between the input and output layers, with 120 neurons at each hidden layer. A composite loss function was composed of one data loss function and one physics loss function, and the later was used to constrain the solution space and make the predictions physically meaningful.

[0115] More specifically, the data loss function has as inputs the centroids of sparse tracers, i.e. the locations 103 of the tracers as determined from the ultrasonic signal data 100, the corresponding data loss value was calculated as the mean square error between the predicted velocity 114 components from the physics-informed neural network and the ones 104 obtained from the flow data 102.

[0116] The physics loss function, comprised 3D incompressible and dimensionless Navier- Stokes equations. The input for the physics loss function are different from those adopted for data loss, as physically plausible predictions need to be valid over a broader flow region. Furthermore, there is a too little number of locations 103 in the flow data 102 to train the physics-informed neural network over the complete flow volume 105. As no vessel boundary is in this implementation (the same situation in practical ultrasound scanning, where vessel segmentation with sparse tracers signals is not possible after removing tissue signal with filters - e.g. singular value decomposition filter), the 3D flow volume 105 was identified by closing the area around the sparse tracers 103. Denser spatial locations 109 (on regular grids) were used to calculate the physics driven loss. These denser spatial locations 109 are comprised in the location data 106. In one implementation, the location data 106 comprises about 3000 locations 109 fixed for each time step (1ms). For comparison, there are only about 7-10 localized tracer locations 103 per time step (1ms) available from the flow data 102.

[0117] These two loss terms, the physics loss function and the data loss function, are weighted during the training. The parameters of the physics-informed neural network are updated and optimized by using the Adam optimizer, starting with a learning rate of 1e-3, and decreasing by 25% after every 1000 epochs (also referred to as episodes in the context of the current specification). In this implementation, a total 20000 epochs were used for training.

[0118] In Fig. 6 the result for an exemplary embodiment of the method is shown with regard to a pressure 115 in the vessel 1 as simulated.

[0119] In this embodiment the method determines a sequence of images depicting the pressure 115 in the flow volume 105 over time. The gray-value coding for the pressure 115 is given at the colorbar right to the respective image taken from the sequence of images. From the sequence, that comprises 1 ,000 images, each for a subsequent time step of 1 ms, two instances -at 0.024s and at 0.126s - have been selected and are shown in Fig. 6A and B respectively. Thus, the flow dynamics in the vessel 1 have been sampled with a lms sampling rate.

[0120] The spatial sampling of the flow volume 105 is so high that barely single locations 109 in form of dots are recognizable. In Col 1 of Fig. 6A and B the pressure as obtained from the ground truth from the simulation is depicted, wherein in Col 2 of Fig. 6A and B the pressure 115 as estimated by the trained physics-informed neural network is depicted after performing the method according to the invention on the simulated data.

[0121] As the simulation comprised a pulsed flow profile, the pressure distributions vary between Fig. 6A and Fig. 6B. In comparison with the ground truth, the estimated pressure 115 is very similar and shows only minor deviations from the ground truth.

[0122] In Fig. 6 it can be seen how the pressure drop at the stenosis 2 takes place, and how the pressure increases upstream the stenosis 2, while remaining relatively constant downstream the stenosis 2. The method predicts the pressure 115 in the stenosed vessel 1 with striking accuracy, while having been trained on flow data 102 obtained with one second at an acquisition rate of only 1 kHz yielding approximately 7,000 to 10,000 locations in total. Nonetheless due to the inventive training procedure, it is possible to train the network so well that prediction over the entire flow volume 105 remaining accurate at high spatial and temporal resolution. The spatial resolution in particular is beyond the spatial resolution of the ultrasonic signal data 100.

[0123] In Fig. 7 from the same simulation and with the same physics-informed network the velocity 114 is depicted for the same two instances in time. The velocity components along x, y and z (u, v, w) is depicted separately.

[0124] In Fig. 7A the velocity component u along the x-axis is depicted. In Fig. 7B the velocity component v along the y-axis is depicted. In Fig. 7C the velocity component w along the z-axis is depicted.

[0125] Gray-scale coding is according to the color bars associated to the respective plot. The spatial and temporal sampling for the velocity 114 corresponds to the spatial and temporal sampling of the pressure 115 as depicted in Fig. 6.

[0126] In Col 1 of Fig. 7 the respective velocity component is depicted according to the ground truth obtained from the simulation at the first instance t= 0.024s.

[0127] In Col 2 of Fig. 7 the respective velocity component is depicted as estimated by the method according to the invention executed on the simulation data for the first instance t= 0.024s.

[0128] In Col 3 of Fig. 7 the respective velocity component is depicted according to the ground truth obtained from the simulation at the second instance t= 0. 126s.

[0129] In Col 4 of Fig. 7 the respective velocity component is depicted as estimated by the method according to the invention executed on the simulation data for the second instance t= 0. 126s.

[0130] As in Fig. 6, a striking agreement between ground truth and estimated velocity 114 is observable. The pulsating velocity 114 along the x-axis is clearly visible in Fig. 7A Col 2 and Col 4, wherein along y and z the velocity 114 is comparably low at all time.

[0131] The method according to the invention allows for non-invasive, rapid, accurate and cost-efficient estimation of fluid flows in a vessel 1 based on ultrasonic signal data 100 obtained from the fluid flow in the vessel 1. List of reference numerals

[0132] 1 vessel

[0133] 2 stenosis

[0134] 3 inlet 4 outlet

[0135] 100 ultrasonic signal data

[0136] 101 PSF of tracer

[0137] 102 flow data

[0138] 103 determined location of the tracer 104 determined associated velocity of the tracer

[0139] 105 flow volume

[0140] 106 location data

[0141] 107 trajectories

[0142] 108 filtered trajectories 109 locations of location data

[0143] 114 predicted flow velocity

[0144] 115 predicted pressure

[0145] *****

Claims

Claims1. A method for determining a velocity of a fluid and / or a pressure in a vessel (1), the method comprising the steps of:- Acquiring a series of ultrasonic signal data (100) comprising information (101) on tracers comprised by a fluid flow in a vessel, Three-dimensionally localizing and tracking at least some of the tracers in the series of the ultrasonic signal data (100), such that for each tracked tracer flow data (102) is obtained, the flow data (102) comprising information on a plurality of locations (103) and associated velocities (104) at different time points of the tracer, From the locations (103) of the tracers, determining a flow volume (105) delimiting a volume within which the fluid flow takes place in the vessel (1), Estimating the velocity (114) and / or the pressure (115) for at least one time point of the fluid in the vessel with a physics-informed machine learning method that is trained with at least some of the flow data (102) and with location data (106), wherein the location data (106) comprises a plurality of locations (109) within the flow volume (105) and associated time points.

2. The method according to claim 1, wherein training of the physics-informed machine learning method comprises providing the flow data (102) of at least some tracked tracers as input to the physics-informed machine learning method, wherein the physics-informed machine learning method determines for each location (103) and time point an estimated velocity (114), wherein a flow data loss function determines a flow data loss value indicative of a deviation of the velocities (104) of the provided flow data (102) from velocities (114) estimated by the physics-informed machine learning method.

3. The method according to claim 1 or 2, wherein training of the physics-informed machine-learning method comprises providing the location data (106) as input to the physics-informed machine learning method, wherein the physics- informed machine learning method determines for each location (109) and time point an estimated velocity (114) and an estimated pressure (115) of the fluid at the location (109) and the time point, wherein the estimated velocities (114) and the estimated pressures (115) are provided to a physics loss function comprising the Navier-Stokes equations, wherein the physics loss functiondetermines a physics loss value indicative of a deviation of the velocities (114) and the pressures (115) estimated by the physics-informed machine-learning method from velocities and pressures that would solve the Navier-Stokes equations.

4. The method according 2 and 3, wherein training comprises a first stage, a second stage, and a third stage, wherein in the first stage, the data loss value is determined, in the second stage the physics loss value is determined, and in the third stage a composite loss value is determined from a composite loss function comprising the data loss function and the physics loss function, wherein the composite loss function is minimized during training episodes.

5. The method according to one of the claims 3 and 4, wherein the Navier-Stokes equations correspond to Navier-Stokes equations adjusted for incompressible fluids, conservation of momentum, and conservation of mass.

6. The method according to one of the claims 3 to 5, wherein the physics loss function comprises physics residual functions, , fa, fa, fa, based on the terms of Navier-Stokes equations:wherein denotes a partial differential operator, t denotes time, p denotes a constant fluid density, p denotes a viscosity of the fluid, u, v, w are velocity components of the velocity (114) estimated by the physics-informed machine learning method along three dimensions x, y, z, wherein p is the pressure (115) as estimated by the physics-informed machine learning method, wherein the physics loss value, Lphysics, and the physics loss function is determined according to:Wherein / Vphysicsdenotes the number of locations for which the physics loss function has been evaluated.

7. The method according to one of the claims 2 to 6, wherein the data loss function and the data loss value, Ldata, is determined according to:wherein u, v, w are velocity components for location Xj, y}, Zj at time point ty of the velocity (114) estimated by the physics-informed machine learning method and wherein ureference . Reference . wreference are the corresponding velocity components of the velocity (104) of the provided flow data (102), wherein / Vclata, denotes the number of locations (109) and time points for which the data loss function has been evaluated.

8. The method according to one of the preceding claims, wherein for the tracking of the tracers, a localization method for the tracers is applied to the ultrasonic signal data (100), wherein the localization method determines a centroid of a point spread function (101) of the tracer in the ultrasonic signal data (100).

9. The method according to one of the preceding claims, wherein the vessel (1) is a blood vessel in a living body.

10. The method according to one of the preceding claims, wherein the method is executed during an ultrasonic imaging examination of a patient, wherein the training of the physics- informed machine learning method is executed for each patient at least once.

11. The method according to one of the preceding claims, wherein for determining the flow volume (105) from the locations (103) of the tracers, a boundary is determined that delimits the flow volume, wherein the boundary is parametrized by means of finite elements, such as triangles.

12. The method according to one of the preceding claims, wherein a time-resolved velocity and / or a pressure distribution for at least a portion of the flow volume(105) flow volume is estimated by the trained physics-informed machine learning method.

13. The method according to one of the preceding claims, wherein a sequence of velocities (114) and / or pressures (115) as estimated by the trained physics- informed machine learning method is generated such that a temporal evolution of the velocity (114) and / or the pressure (115) is generated, wherein said sequence(s) is / are displayed.

14. A computer program comprising computer program code, that when executed on a computer causes the computer to execute the method according to one of the preceding claims.

15. An ultrasonic measurement system comprising at least the following components:- An ultrasonic transducer head system configured to generate and record the ultrasonic signal data (100);- A computer connected to the ultrasonic transducer head system and configured to receive the ultrasonic signal data (100), characterized in that the computer is configured to execute the method according to one of the claims 1 to 13 or the computer program according to claim 14.*****

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