Devices and systems for determining pressure differentials in pipelines

Abstract

Apparatus and systems for determining pressure differentials in a pipe are provided. Embodiments of the invention provide a method for determining pressure differentials in a pipe caused by fluid flow within the pipe, the method comprising the steps of: obtaining time-related three-dimensional fluid velocity data at multiple points along the pipe; processing the time-related three-dimensional fluid velocity data to determine: i) the flow velocity (Q) of the fluid through the pipe; ii) the kinetic energy (K) of the fluid flow through the pipe; iii) the advection energy rate (A) of the fluid flow through the pipe; and iv) the viscous dissipation rate (V) of the fluid flow; and calculating the pressure differential based on the total flow velocity (Q), kinetic energy (K), advection energy rate (A), and viscous dissipation rate (V). Further embodiments are also described.

Description

Technical Field

[0001] The present invention relates to a method and system for estimating pressure drop through a tubing, and in a specific embodiment relates to the estimation of blood pressure drop through a blood vessel based on velocity measurements obtained from various medical imaging modalities. Background Technology

[0002] Pressure drop or pressure difference measured across a vascular segment (also referred to as pressure gradient in some clinical literature) is widely used clinically as a biomarker for a wide range of cardiovascular disorders (Baumgartner, Hung, Bermejo, Chambers, Evangelista, Griffin, Iung, Otto, Pellikka, etc.). Sawaya, Stewart, Babaliaros (2009), and Vahanian, Baumgartner, Bax, Butchart, Dion, Filippatos, Flachskampf, Hall, Iung, Kasprzak, Nataf, Tornos, Torracca, and Wenink (2007) are well-known examples. A known example is coarctation of the aorta (CoA), where pressure drop is used as a diagnostic metric for risk stratification of patients undergoing surgery (Jenkins, Ward, 1999) and for evaluating patients after stent placement (Tan et al., 2005). Other examples of pressure-based measures in clinical practice include transvalvular descent – ​​an acceptance measure for classifying the severity of aortic stenosis (Baumgartner, Hung, Bermejo, Chambers, Evangelista, Griffin, Iung, Otto, Pellikka). In 2009, De Bruyne, Manoharan, Pijls, Verhamme, Madaric, Bartunek, Vanderheyden, Heyndrickx, 2006 and Feldman, 2006), left ventricular outflow tract (LVOT) pressure drop - used to define guidelines for the treatment of hypertrophic cardiomyopathy (HCM) (Gersh et al., 2011), and transstenotic pressure gradient in the coronary arteries - used to quantify fractional flow reserve (FFR) (Deng et al., 2014).

[0003] Current clinical guidelines use catheter measurements (Feldman, 2006 and Konecny, Khanna, Novak, Jama, Zawadowski, Orban, Pressman, Bukartyk, Kara, Cetta, Borlaug, Somers, Reeder, 2014) or echocardiographic Doppler recordings (Bach, 2010; Firstenberg, Greenberg, Smedira, Prior, Scalia, Thomas, Garcia, Michael, Smedira, Thomas, 2000; Fyfe, Currie, Seward, Tajik, Reeder, Mair, Hagler, 1984; Labovitz, Ferrara, Kern, Bryg, Mrosek, Williams, 1986; and Zhang, Nitter-Hauge, 1985). Pressure catheterization has seen significant improvements in probe sensitivity (de Vecchi, Clough, Gaddum, Rutten, Lamata, Schaeffter, Nordsletten, Smith, 2014; Garcia, Carrozza, 2007; and Iwasaki, Kusachi, 2009) and surgical management, making it the gold standard for pressure drop measurement. However, despite its advantages, the application of pressure catheterization is limited to specific patient populations due to its inherent invasiveness and associated risks and costs. To expand the base of patients who can benefit from these assessments, non-invasive assessments using Doppler echocardiography have been developed. This approach estimates the pressure difference based on the magnitude of the peak velocity acquired along the direction of the ultrasound beam using a simplified Bernoulli equation (Hatle, Brubakk, Tromsdal, Angelsen, 1978; and Oshinski, Parks, Markou, Bergman, Larson, Ku, Mukundan, Pettigrew, 1997). Although useful for patient stratification, the accuracy of this approach is limited by operator dependence and mathematical assumptions that neglect transient effects and viscous losses on the flow (Holen & Simonsen, 1979; Laske, Jenni, Maloigne, Vassalli, Bertel & Turina, 1996; and Zhang & Nitter-Hauge, 1985).

[0004] When working with the same Doppler echocardiographic data, such as in the characterization of diastolic performance, pressure differential estimation has been improved by using the Euler equation (Bermejo, Antoranz, Yotti, Moreno, Garcia-Fernandez, 2001; Greenberg, Vandervoort, Firstenberg, Garcia, Thomas, Neil, Firstenberg, 2001; and Yotti, Bermejo, Antoranz, ...). Allue, Silva, Desco, Moreno, García-Fernández (2004). This scheme benefits from the high temporal resolution of the data but neglects effects related to advection acceleration beyond the acoustic ray and viscous dissipation. Doppler acquisition also depends on the operator's ability to detect the direction of blood flow. All these factors motivate ongoing research to improve robustness, accuracy, and operator independence.

[0005] Recent developments in magnetic resonance imaging (MRI) and echocardiography have allowed for the acquisition of velocity data in three dimensions of space and time (Deng, Fan, Xie, He, Natsuaki, Jin, Bi, An, Liu, Zhang, Fan, Li, 2014; Herment, Besson, Frouin, 2008; Markl, Wallis, Brendecke, Simon, Frydrychowicz, Harloff, 2010; and Nielsen, Powell, Gauvreau, Marcus, Prakash, Geva, 2005). Ongoing research has yielded a large number of different techniques for estimating pressure differentials using these images. Specifically, four-dimensional phase-contrast MRI (4D PC-MRI) data enables a solution to the Poisson pressure equation (PPE), in which pressure is explicitly derived as a function of the acquired velocity field (Bock, Frydrychowicz, Lorenz, Hirtler, Barker, Johnson, Arnold, Burkhardt, Hennig, Markl, 2011 and Krittian, Lamata, Michler, Nordsletten, Bock, Bradley, Pitcher, Kilner, Markl, Smith, 2012), which allows estimation of the contributions of convection effects and viscous dissipation in all spatial directions (Lamata et al., 2014). This approach has been successfully applied to the estimation of pressure in aortic coarctation (Riesenkampff et al., 2014). Based on these data-driven methods, reconstructions of the velocity field at the vessel wall have been proposed (Donati et al., 2014) to improve the accuracy of calculating viscous effects, and data assimilation techniques attempt to overcome the limitations of data acquisition based on physical principle simulations (de Hoon et al., 2014).

[0006] Alternative approaches to estimating pressure differentials in vascular anatomy are based on 3D computational fluid dynamics (CFD) simulations (Kim, Vignon-Clementel, Coogan, Figueroa, Jansen, Taylor, 2010; LaDisa, Figueroa, Vignon-Clementel, Jin Kim, Xiao, Ellwein, Chan, Feinstein, Taylor, 2011; Sankaran, 2012; and Vignon-Clementel, Figueroa, Jansen, Taylor, 2010). In this case, a patient-specific geometry is reconstructed from images such as computed tomographic angiography, and velocity boundary conditions are defined based on flow measurements. Thus, pressure and velocity are simulated using a cardiovascular model (Coogan, Humphrey, Figueroa, 2013; and Xiao, Alastruey, Figueroa, 2014), providing detailed measurements of flow, pressure differentials, wall shear stress, and others. While these detailed metrics are provided, forward cardiovascular modeling based on CFD requires robust multi-scale schemes (Formaggia, Gerbeau, Nobile, Quarteroni, 2002; Gresho, Sani, 1987; and Vignon-Clementel, Figueroa, Jansen, Taylor, 2006) for boundary conditions, accurate anatomical definitions, and expensive, parallel simulation solutions on computer clusters. Summary of the Invention

[0007] In one implementation described herein, we propose a non-invasive, semi-automatic method (i.e., the work-energy relative pressure (WERP) method) for estimating pressure differentials based on the work-energy principle. The setup offers the benefits of simplicity and computational efficiency, requiring integration and computation that can be performed directly from images acquired using 4D PC-MRI or echocardiography (ECG). The mathematical operations behind the method are introduced, and we detail its application to cardiovascular flow data. In the first accompanying appendix, we describe the testing of the method on a series of computer-based test cases with progressively increasing complexity, evaluating its robustness to segmentation variability and noise. Subsequently, the proposed method is compared in detail with other available methods on computer-based CFD solutions. Finally, satisfactory execution of the scheme is demonstrated on 4D PC-MRI acquisitions from a cohort of nine healthy patients by comparing the estimated aortic pressure differentials with previously reported results obtained using a PPE-based method (Lamata et al., 2014).

[0008] In another embodiment based on the above implementation, we also describe two simplified schemes for pressure estimation. These schemes require less information than the WERP method and can therefore be used with data from imaging modalities other than 4D PC-MRI. Specifically, 3D Doppler echocardiography (ECG) images or 2D PC-MRI are used in the first simplified scheme, and even Doppler ECG images from conventional 2D probes are used in the second simplified scheme. The technique can also be applied to reconstructing velocity fields from composite ECGs or from speckle / contrast tracking in ultrafast ECGs. Because 2D and 3D ECG imaging is more prevalent in clinical settings than 4D PC-MRI, the simplified schemes have significantly wider applications than the full WERP scheme.

[0009] In view of the foregoing, from a first aspect, a method is provided for determining a pressure difference in a pipe caused by fluid flow within the pipe, the method comprising the steps of: obtaining time-related three-dimensional fluid velocity data at multiple points along the pipe; processing the time-related three-dimensional fluid velocity data to determine: i) the flow velocity (Q) of the fluid through the pipe; ii) the kinetic energy (K) of the fluid flow through the pipe; iii) the advection energy rate (A) of the fluid flow through the pipe; and iv) the viscous dissipation rate (V) of the fluid flow; and calculating the pressure difference based on the flow velocity (Q), kinetic energy (K), advection energy rate (A), and viscous dissipation rate (V).

[0010] In another aspect, a method for determining the pressure difference in a pipe caused by fluid flow is also provided, the method comprising the steps of: obtaining fluid velocity data at the inlet and outlet planes of the pipe; processing the fluid velocity data to determine: i) the flow velocity (Q) of the fluid through the pipe; and ii) the advection energy rate (A) of the fluid flow through the pipe; and calculating the pressure difference based on both the flow velocity (Q) and the advection energy rate (A), without considering the viscous dissipation rate (V) and kinetic energy (K) of the fluid flow through the pipe.

[0011] Another approach provides a method for determining the pressure difference across a pipe caused by fluid flow within the pipe, the method comprising the steps of: obtaining fluid velocity data only at the outlet of the pipe (a valid assumption when the velocity at the inlet is negligible compared to the outlet velocity); processing the fluid velocity data to determine: i) the flow velocity (Q) of the fluid through the pipe; and ii) the advection energy rate (A) of the fluid flow through the pipe (but only considering the outlet); and calculating the pressure difference based on both the flow velocity (Q) and the advection energy rate (A), without considering the viscous dissipation rate (V) and kinetic energy (K) of the fluid flow through the pipe.

[0012] Another aspect provides a set of methods for determining the advection energy rate (A) in all previous embodiments, depending on the resolution and availability of the imaging modality-adapted data. These methods include: (a) using data from a complete cross-section of the inlet or outlet plane of the pipe, where each data point is a complete 3D velocity vector as derived from 4D PCMRI; (b) using data from a complete cross-section of the inlet or outlet plane of the pipe, where each data point is a projection of a 3D velocity vector in the direction of the acoustic ray as derived from 3D ECG; (c) using data from only the lines of the cross-section of the inlet or outlet plane of the pipe, where each data point is a projection of a 3D velocity vector in the direction of the acoustic ray as derived from 2D ECG; and (d) using data from a combination of lines of the cross-section of the inlet or outlet plane of the pipe, where each data point is a projection of a 3D velocity vector in the direction of the acoustic ray as derived from multiple 2D ECGs.

[0013] Finally, from another aspect, some embodiments of the present invention also provide a method for determining a pressure difference in a pipe caused by fluid flow within the pipe, the method comprising the steps of: obtaining time-dependent three-dimensional fluid velocity data at multiple points along the pipe; segmenting a region of interest in the pipe from which flow information is desired; defining a flow vector field w through the segmented region, the flow vector field being non-divergent and having zero values ​​in the sidewalls of the pipe; processing the flow vector field w and the time-dependent three-dimensional fluid velocity data to determine: i) the flow velocity (Qw) of the flow vector field w; ii) the alternative kinetic energy (Kw) of the fluid flow through the pipe; iii) the alternative advection energy rate (Aw) of the fluid flow through the pipe; and iv) the alternative viscous dissipation rate (Vw) of the fluid flow; and the total calculated pressure difference based on the flow velocity (Qw), and the alternatives of kinetic energy (Kw), advection energy rate (Aw), and viscous dissipation rate (Vw). Attached Figure Description

[0014] Embodiments of the invention will now be further described by way of example only and with reference to the accompanying drawings, in which the same reference numerals refer to the same parts, and in the drawings:

[0015] Figure 1 This is a set of figures illustrating various uses of embodiments of the present invention;

[0016] Figure 2 This is a set of graphs illustrating measurement results collected by embodiments of the present invention and the associated mathematical formulas used for calculating pressure drop;

[0017] Figure 3 This is a diagram illustrating a comparison between the processing of one embodiment of the present invention and a prior art alternative;

[0018] Figure 4 These are diagrams illustrating aspects of the theoretical basis of the first embodiment of the present invention;

[0019] Figure 5 This is a block diagram of the first embodiment of the present invention;

[0020] Figure 6 This is a flowchart relating to the operation of the first embodiment of the present invention;

[0021] Figure 7 This is a flowchart relating to the operation of the second embodiment of the present invention;

[0022] Figure 8 This is a flowchart relating to the operation of the third embodiment of the present invention;

[0023] Figure 9 This is a block diagram of the second embodiment of the present invention;

[0024] Figure 10 These are block diagrams of the second and third embodiments of the present invention; and

[0025] Figures 11 to 17(a) Figure 17(b) and Figure 17(b) are various figures illustrating the results and effectiveness of embodiments of the present invention. Detailed Implementation

[0026] Embodiments of the invention will now be described. Three embodiments will be described: the first embodiment relates to the estimation of pressure drop in the heart using a full WERP scheme, which models the pressure drop in detail over the entire length of the vessel of interest; the second embodiment relates to a so-called “full advection WERP” scheme, which models the pressure drop only via measurements at the vessel’s inlet and outlet; and the third embodiment is a so-called “simplified advection WERP” method, which uses only measurements at the vessel’s outlet. Figure 2 The required measurements and calculations for each option are illustrated, while Figure 11 Further details of the anatomical regions of interest are given (particularly in the second and third embodiments) (see Figure 1 (A) and different data that can be obtained from different imaging modes ( Figure 1 (B)).

[0027] exist Figure 1 In (A), the anatomical regions for calculating pressure drop are defined as follows: Transvalvular region (TVR): from the left ventricular outflow tract (LVOT) (plane 1) to the venous vasoconstriction (VC) (plane 2); Ascending aortic region (AAR): from the VC to the cephalobrachial artery (plane 3); Descending aortic region (DAR): from the left subclavian artery (plane 4) to the same height as the aortic valve (plane 5).

[0028] exist Figure 1 (B) illustrates a schematic of the velocity field at VC acquired during peak contraction using different techniques and improved data availability, as shown below. Top left: Single velocity value obtained from continuous 1D Doppler. Top right: 1D velocity curve along the acoustic plane available from 2D color Doppler. Bottom left: Complete 2D velocity curve available from 3D color Doppler. Bottom right: Complete velocity vector field above the entire plane (instead of simply its projection along the acoustic ray) available from 4D PC-MRI.

[0029] The first implementation of what is known as the complete WERP scheme can be summarized as follows: Given a fluid passing through a pipe-like structure, the pressure difference between the inlet and outlet is calculated using the work-energy relative pressure (WERP) method, derived from the work-energy principle from Neville-Stokes' equation. The pressure difference is calculated as the sum of three energy transfer rates (kinetic, lateral, and viscous) segmented by the net flow through the pipe. The kinematic term integrates the rate of change of kinetic energy of all particles within the pipe-like domain over time. The second term describes the energy transfer rate due to the physical movement of the fluid entering and leaving the domain. And the third term describes the energy dissipated due to the viscous friction of the fluid.

[0030] Regarding the second implementation, known as the complete advection WERP scheme, this can be summarized as follows: given a fluid passing through a pipe-like structure, the pressure difference between the inlet and outlet is calculated using the advection WERP equation, which only takes one of the three summation terms of energy (the advection term), thus simplifying the WERP equation. More specifically, this component of energy neglects the contribution of advection transport energy from the sidewalls, because the velocity perpendicular to the wall is small in the near-wall region, and the calculation of the pressure difference is reduced to the integral of the advection transport energy rate at the inlet and outlet planes of the pipe-like structure.

[0031] Finally, regarding the third implementation of the simplified advection WERP scheme, its overview is as follows: Given a fluid passing through a pipe-like structure, the pressure difference between the inlet and outlet is calculated using the simplified advection WERP equation, which only takes into account the contribution of the inlet or outlet plane of the pipe-like domain, thus simplifying the complete advection WERP equation. It is assumed here that the integral of the advection transport energy rate at the other end of the pipe-like domain is negligible.

[0032] First implementation method: Complete WERP

[0033] 2. Methods

[0034] Starting with the work-energy principle, we derive the formula for the pressure difference across vascular segments (Section 2.1). We then detail the discrete formulas (Section 2.2) and preprocessing steps required to work with 4D PC-MRI data (Section 2.3).

[0035] 2.1. Pressure difference from fluid work-energy

[0036] Pressure differences in a fluid system are related to the dynamics of the flow field. This relationship is described by the well-known Navel-Stokes equations, in which, in the absence of gravity, pressure changes are balanced by fluid acceleration and viscous stress. Using the conservation of mass and momentum for a closed system, work-energy production is performed on an incompressible isothermal Newtonian fluid with a boundary Γ:

[0037]

[0038] Where v represents velocity, p represents pressure, and n is the normal vector on Γ. And ρ and μ represent fluid density and dynamic viscosity, respectively. Here, A is the time derivative of the kinetic energy within Ω. e V is the advection transport energy rate, which describes the energy transfer caused by the physical movement of fluid into and out of Ω, and V e It is the viscous dissipation rate. H(p) and S e Let Γ represent the energy input to the fluid system, hydraulic power, and shear energy rate, respectively. Here, we assume that the boundary of Ω can be written as Γ = Γ. i ∪Γ o ∪Γ w Where i, o, and w indicate contributions from the blood vessel inlet, outlet, and wall. The mathematical details derived from the work-energy principle can be found in our published paper, Donati et al., Non-invasive pressure difference estimation from PC-MRI using the work-energy equation, Medical Image Analysis, Vol. 26, pp. 159-172, December 2015, the entire contents necessary for understanding this embodiment are incorporated herein by reference.

[0039] Starting with this work-energy balance, as the first approximation, we neglect the effect from the sidewall Γ. w Advection energy A eThe contribution is small in the near-wall region because the velocity is smaller compared to the core blood flow (Taylor, Figueroa, 2009; Xiao, Humphrey, Figueroa, 2013; and Xiao, Alastruey, Figueroa, 2014). Therefore, calculations are limited to inlet and outlet sections, for example,

[0040] Equation (1)

[0041]

[0042] Furthermore, we assume that the pressure is almost constant at both the inlet and outlet planes, which makes:

[0043] Equation (2)

[0044]

[0045] When there is little or no compliance, on the wall, |v·n|<<1, the global quality compatibility condition arises:

[0046] Equation (3)

[0047]

[0048] Assumption:

[0049] Equation (4)

[0050] H(p) = ΔpΛ, where Δp = p o -p i It is the pressure difference between the outlet and the inlet, and It considers the flux through the surface, that is, the term that can be expressed as a function of the inlet surface using only Equation 3.

[0051] Regarding the shear energy S e We assume that the contribution of each boundary segment—the inlet, outlet, and wall—is zero. On the inlet / outlet plane, this term contributes only when a significant gradient exists along the boundary normal. While these gradients may occur, particularly in tortuous or tapered vessels, they are extremely slight and effectively scaled down by the low viscosity of the blood. This independent variable regarding the flow gradient cannot be assumed near the vessel wall, where it causes significant wall shear stress. However, since this shear stress is primarily orthogonal to the wall velocity (which increases primarily along the boundary normal), we assume that these shear stresses affect S. e Its contribution is negligible.

[0052] Based on the above assumptions, the work-energy relative pressure (WERP) formula for estimating the pressure difference based on energy contribution is derived as follows:

[0053] Equation (5)

[0054]

[0055] From this equation, we observe that all RHS terms originate directly from flow data, which enables the calculation of pressure differentials. However, we also observe that this calculation requires |Λ|>0 (e.g., flow is observed through vascular segments).

[0056] 2.2. Calculations from 4D PC-MRI

[0057] Let Vt represent the velocity image acquired at time t, Vt(i,j,k) represent the velocity field evaluated at voxel (i,j,k) at time t, and Δt represent the discrete time step between two consecutive acquisitions. We discretize the derivative in Equation 5 using the central finite difference method and set the time... The pressure difference between the inlet and outlet planes is estimated as follows:

[0058] Equation (6)

[0059]

[0060] in, The speed is approximated by equation (7) with second-order accuracy O(Δt2).

[0061] Equation (7)

[0062]

[0063] The calculation of WERP terms is performed through the voxelized version I of Ω. ROI The upper derivative is used for execution. The area integral is used to define the entry point by clipping the 3D mask. and exports Section (see) Figure 3 The planar evaluation is obtained using the normal vector N2D(i,j). Then, based on the image-based velocity field estimation, this discrete term is:

[0064] Equation (8)

[0065]

[0066]

[0067]

[0068]

[0069] Where dS=Δx 2 and dV=Δx 3These are the pixel surface and voxel volume, respectively, based on the voxel length Δx. Discrete evaluations of all contributions rely on approximate velocity fields M(V) and M'(V) obtained by averaging over the 3D mask and the 2D plane defined above. 2D The definition of (V),

[0070] Equation (9)

[0071]

[0072]

[0073] In the above content, δ ij It is the Kronecker transformation function, and q is used to make O(Δx) the basis. 2 The approximate base data is smoothed into parameters for velocity values ​​(see [reference]). Figure 4 , Figure 4 A schematic diagram of the finite difference center template is shown. The velocity field V evaluation at P = (i,j,k) using the operator M(V) uses the standard (a) and the filtered scheme (b), and the velocity field V derivative evaluation at P = (i,j,k) using the operator D(V) uses the standard (c) and the filtered center difference scheme (d).

[0074] In the above, if q = 0, then M(V)(i,j,k) = V(i,j,k) and M 2D (V)(i,j) = V(i,j) returns the velocities measured at voxels (i,j,k) and (i,j), respectively. Alternatively, if q = 3, the velocity field measurement is treated as O(Δx) based on measurements of adjacent voxels. 2 The approximate weighted sum effectively averages out potential artifacts caused by noise.

[0075] Similarly, in Equation 8, the discrete tensor D(V) is calculated as:

[0076] Equation (10)

[0077] D(V)(i,j,k)=(G(V)(i,j,k)+G(V)T(i,j,k)),

[0078] Where G(V) is the velocity gradient tensor, which is defined as:

[0079] Equation (11)

[0080]

[0081] in,

[0082] Equation (12)

[0083]

[0084] as well as

[0085] Equation (13)

[0086]

[0087] Furthermore, if q = 0, the velocity gradient is approximated by the second-order central difference centered at voxel (i,j,k). By imposing q = 3, a filtering scheme is employed, where the velocity derivative is approximated by a weighted average of the derivatives calculated using the second-order central differences at adjacent voxels, thus reducing noise pollution (see [link to relevant documentation]). Figure 2 ).

[0088] 2.3. Required Preprocessing

[0089] Several pretreatment steps are required before application in clinical settings. Pretreatment tools outlined in Bock et al. (2011) are used to correct (1) field inhomogeneities and eddies (Chan, Von Deuster, Giese, Stoeck, Harmer, Aitken, Atkinson, Kozerke, 2014; Moussavi, Untenberger, Uecker, Frahm, 2014; and Rohde, Barnett, Basser, Marenco, Pierpaoli, 2004). Subsequently, based on the maximum velocity V... max Thresholding definition of calibration speed magnitude (2) Binary mask I ROI (including those with greater than) The velocity-sized voxels, S is the segmentation thresholding parameter). The inlet and outlet points are manually selected by the user (3) according to the clinical problem under the investigation. The inlet and outlet planes perpendicular to the blood vessel are then defined using the skeleton solution of the binary mask (4). As a result of this processing, the original 3D image and the binary mask of the inlet / outlet plane required for WERP calculation are defined. In this work, the image acquisition process is simulated in a computer for the validation tests proposed in Results 3.1, 3.2 and 3.3 of Donati et al., Non-invasive pressure difference estimation from PC-MRI using the work-energyequation Medical Image Analysis, Vol. 26, pp. 159-172, December 2015. The simulated PC MRI image is then processed after steps (2) to (4) before applying the WERP method.

[0090] 2.4. System Description

[0091] Figure 5 An example imaging system is illustrated, capable of applying the aforementioned techniques to calculate blood flow through blood vessels using a fully WERP method. Here, a 4D phase-contrast magnetic resonance imaging (4D PC MRI) system 60 is provided, comprising MR imaging coils in which a subject is located, and these coils are controlled by an MRI imaging control system 58, which includes an MRI controller processor 50. The MRI imaging control system 58, including the MRI controller processor 50, operates in a conventional manner to allow the acquisition of 4D phase-contrast magnetic resonance imaging data of the internal blood vessels of a subject, for example, to allow for the acquisition of pressure differential data. For example, the transvalvular pressure drop (TPD) in the transvalvular region of the heart between the left ventricular outflow tract and venous vasoconstriction can be expected. Of course, other blood vessels can also be monitored as desired. For example, secondary branches can be taken into account.

[0092] The 4D PC MRI systems 58 and 60 acquired 4D PC MRI data 52 of the imaging subjects. This data was saved for further processing and analysis. (As previously mentioned...) Figure 1 As can be seen, the complete velocity vector field above the imaging plane (rather than simply its projection along the sound wave line) is available from 4D PC-MRI.

[0093] The 4D PC MRI systems 58 and 60 also include a WERP calculation program 54, which processes the 4D PC MRI data 52 as described above according to the WERP processing method to obtain WERP pressure estimation data 56. As described above, the WERP processing method calculates the pressure difference along the vessel as the sum of three energy transfer rates (kinetic, lateral, and viscous) divided by the net flow through the vessel. The resulting WERP pressure estimation data 56 provides an estimate of the pressure at each point along the vessel between the measured inlet and outlet planes, taking into account the vessel wall between the inlet and outlet. It provides a complete solution for every point within the vessel between the inlet and outlet.

[0094] Figure 6 This is a flowchart summarizing some of the key elements of the complete WERP process. In step 6.2, 4D phase-contrast MRI data relating to the blood vessels of the subject to be pressured is obtained. Note that this can be obtained "on-site" (i.e., just before performing the calculation) or can be recorded or stored data from a previous scan. At this point, because WERP processing can be run on the recorded or stored MRI data, the presence of a human subject is not required to perform WERP processing to obtain the pressure drop.

[0095] In step 6.4, as Figure 3As shown, the 4D PC MRI data is segmented to define the region of interest corresponding to the vessel from which the pressure drop is to be obtained. Then, at step 6.6 and again as... Figure 3 As shown, the inlet and outlet planes of the blood vessel are defined.

[0096] Next, at step 6.8, for each WERP term, the image-based velocity field is integrated over the defined region of interest based on the MRI data. The precise calculation performed is detailed in Equation 8 and related equations above. As shown in 6.12, the result can then be combined into several independent terms to obtain the total pressure drop over the defined region of the blood vessel using Equation 5 above. The calculated WERP pressure drop data is then saved, and at step 6.14, this data is output to the user, for example, by displaying it on a screen (not shown).

[0097] Second and third implementation methods: Complete advection WERP and simplified advection WERP

[0098] The second and third embodiments involve simplifications of the full WERP processing, which mean that less complex imaging modalities that acquire less information can be used. Specifically, as described below, the full advection WERP protocol can be performed using any modal that renders velocity data in two anatomical planes (e.g., 2D PC MRI or 3D Doppler ECG), while the simplified advection protocol can be used with any modal that renders velocity data in a single anatomical plane (e.g., 2D PC MRI, 3D Doppler ECG, or 2D Doppler ECG data). Access to this partial information (i.e., velocity data in a single anatomical plane) is feasible using Doppler ECG imaging equipment that is very common in many clinical settings.

[0099] The discussion then proceeded to the presentation of the proposed solutions, the results of various experiments, and a comparison with existing Bernoulli-based solutions.

[0100] As shown in the first embodiment of the present invention, the WERP formula is used to reveal the contributions of pressure drop caused by time acceleration (motion descent), spatial transport of momentum (advection descent), and frictional losses (viscosity descent), in order to better understand the direction toward the patient. 22 Improved hemodynamics of the human aorta in stratified layers.

[0101] Based on this, we conducted a thorough comparison between WERP and Bernoulli's formula for calculating the drop on 3D PCMRI (or 4D flow MRI) data to assess the severity of AS in a cohort of 32 patients with BAV. The correlation between non-invasive drop estimates obtained using different methods was investigated, with consistent overestimation observed in Bernoulli's formula. Further investigation demonstrated that this inconsistency is a result of the fact that the Bernoulli method does not consider information about the velocity curve. Examining the WERP prediction of TPD, a clear dependence on advection drop was observed, particularly in patients with stenosis. Using this fact, we proposed a novel simplification of the WERP formula that improves the accuracy of the calculated TPD while limiting the dependence on integrated flow data, making estimation possible based on 2D PCMRI or 3D Doppler echocardiography data. The applicability of the method to 2D Doppler echocardiography data was also tested.

[0102] Materials and methods

[0103] Patient data

[0104] For these implementation methods, a cohort of 32 subjects with varying degrees of aortic stenosis was selected. Each subject underwent two cardiovascular MRI scans: one on a 1.5T system (Erlangen Siemens Avanto, Germany) for anatomical imaging, and the second on a 3T system (Erlangen Siemens Trio, Germany) for 4D PCMRI assessment, both systems using 32-channel cardiac coils. All images were gated electrocardiogram (ECG).

[0105] Subjects followed current clinical guidelines 23 The mean TPD was assessed in two groups: Group I (n=20) and Group II (n=12). Mild stenosis was diagnosed if the mean TPD was >20 mmHg. These pressure values ​​were calculated using Bernoulli's principle for peak transvalvular velocity values. Aortic dimension and hemodynamic data are shown in Table 1.

[0106]

[0107]

[0108] The value is the mean ± standard deviation.

[0109] Table 1 - Aortic dimension and hemodynamics in n=32 patients divided into two groups based on mean systolic blood pressure drop: Group I (Δp≤20mmHg, n=20) and Group II (Δp>20mmHg, n=12).

[0110] Definition of pretreatment and anatomical regions

[0111] Using Bock et al. 27 The developed preprocessing tool corrects field inhomogeneities and eddies in 4D PC-MRI images. 24–26 The lumen of the left ventricle and the aorta are identified using a thresholding criterion based on peak velocity magnitude, which defines a binary mask. A skeletonization algorithm is then used. 28 To extract the centerline of the aorta and its vertical plane.

[0112] exist Figure 1 Pressure drop was calculated on the three anatomical regions illustrated in Figure A, while Figure 2 Given Figure 12 The measurements and calculations required for each protocol are illustrated. The assessment of AS is then based on the TPD calculated along the transvalvular region (TVR) between the left ventricular outflow tract (LVOT) (plane 1) and the venous systole (VC) (plane 2), which is reported in the following results. The LVOT plane is located in accordance with the work of Garcia et al. 29 The defined aortic valve plane is 12 mm prior, and the VC is detected from the medical image as a plane containing the peak velocity magnitude, which is also the plane with the maximum narrowing of the aortic valve ejection.

[0113] For completeness, pressure drops evaluated in the ascending aorta region (AAR) from VC (plane 2) to the cephalobrachial artery (plane 3) and the descending aorta region (DAR) from the left subclavian artery (plane 4) to the same height at the aortic valve plane (plane 5) are also included in Supplementary Material B.

[0114] Simulated 3D, 2D and continuous (1D) Doppler echocardiography

[0115] To eliminate arbitrary observer- and mode-dependent dependencies, echocardiographic data were obtained by sampling 4D PC-MRI from each subject. Idealized conditions were adopted: perfect alignment between the blood flow jet and the sound wave line, and the absence of shadows. These conditions enabled optimal comparisons between the different mathematical formulas introduced in the next chapter.

[0116] Simulate echocardiography in planes 1 and 2 to calculate TPD (see...). Figure 2 The required measurements and calculations for each scheme are illustrated, given that Figure 12 B, and Figure 2 The required measurements and calculations for each scheme are illustrated, given that Figure 2The raw PC-MRI data were linearly interpolated onto a grid of 1mm × 1mm sampling points in each plane. 3D Doppler echocardiography was acquired by projecting the velocity along the acoustic wave direction, taking into account the funnel effect of the probe. To achieve this, the probe position was simulated 10cm upstream of the VC along the direction of the aortic jet, where the aortic valve jet direction was defined as the direction of the velocity vector at the pixel with the maximum velocity magnitude, thus defining an idealized 2D velocity curve by color Doppler. Similarly, a set of 1D velocity curves from the 2D color Doppler acquisition was defined by the intersection of the previously projected velocity field with the hypothetical acoustic wave plane. Because the velocity curves are not axisymmetric, a total of 12 curves containing peak velocities and with randomly oriented lines (in increments of 15°) were generated for each case (see Supplementary Material D for illustrations of 2D and 1D curves for each case). Finally, continuous (1D) Doppler echocardiography data was simulated using the size of the peak velocity pixels projected along the aortic jet direction at each time point.

[0117] Non-invasive pressure drop estimation

[0118] To assess patient-specific severity of AS, we will compare TPDs obtained using a range of protocols varying from the most complete (WERP) to the simplest (Bernoulli) equations. Taking several simplifications, as widely reported in Supplementary Material A, the resulting protocols can all be derived from the Navel-Stokes equation for Newtonian isotherms. For clarity, Figure 2 The diagram illustrates all the methods being compared and the assumptions made to obtain different formulas.

[0119] Bernoulli's principle is widely accepted in AS evaluations. It is derived from the Neville-Stokes momentum equation by considering the flow along the streamline, neglecting the components from motion and viscous pressure. 30,31 Any contribution. A series of additional assumptions will approximate the pressure drop Δp in mmHg. sB The definition is generated as follows:

[0120]

[0121] Among them, v MAX This is the peak velocity at VC estimated from 4D PC-MRI images. Assuming blood density ρ = 1060 kg / m³ 3 Then factor 4 allows for the conversion of pressure from Pa to mmHg (133 Pa – 1 mmHg). In this work, this is referred to as the simplified Bernoulli (sB) formula. 32 .

[0122] Based on this scheme and using the same imaging data, we also considered the proximal velocity v acquired at LVOT. PROXTherefore, the corrected Bernoulli (cB) voltage drop is defined. Clinical guidelines in v PROX >1.5m / s or v max For speeds <3.0 m / s, the corrected formula is used in patient evaluation. The result is that, using a Bernoulli-based formula, only a single velocity value or two are needed to estimate the pressure drop. This facilitates the application of continuous 1D Doppler echocardiography, thereby enabling the definition of pressure drop. and

[0123] The availability of the integrated velocity field from 4D PC-MRI also allows for the calculation of pressure drop using fewer assumptions about the flow. Assuming negligible compliance effects and nearly constant pressure at the inlet and outlet planes of the tubular region, the WERP method21 approximates the Navel-Stokes equation to calculate the pressure drop based on Δp from 4D PC-MRI data. w The pressure drop is calculated as follows:

[0124]

[0125] Where Q is the flow velocity calculated at the outlet. is the time derivative of the kinetic energy within the vascular region, A is the advection transport energy rate describing the energy transfer caused by the physical movement of fluid entering and leaving the region, and V is the viscous dissipation rate describing the energy loss due to friction. These quantities are estimated directly from images described by Donati et al.21.

[0126] The separation of the energy contribution and the permissible component in Equation 102 results in a kinematic pressure drop. Advection pressure drop Δp AW = -A / Q and viscous pressure drop Δp AW = -V / Q is defined. Because the calculation of K and V requires the entire velocity field (volume integral), unlike A and Q (surface integral), we focus on estimating their approximations to reduce data dependence. This is why the remainder of this section focuses on the advection term of TPD.

[0127] The complete advection WERP (cAW) scheme estimates the pressure drop Δp based on the velocity fields extracted at the inlet and outlet planes. cAW (This is the same as the previously defined Δp) AW Similar. Symbol Δp cAW It is emphasized that the descent in the complete advection WERP equation is evaluated using both the inlet and outlet planes. This takes into account the three-dimensional encoded velocity field over the 2D region, rather than the single peak velocity value of sB.

[0128] Similar to the sB scheme, the assumption that the outlet velocity is much greater than the inlet velocity triggers a simplified advection WERP (sAW) pressure drop Δp from 4D PC-MRI data. sAW It only considers the contribution of advection energy at the exit plane.

[0129] However, because the availability of 4D PC-MRI velocity fields is limited in clinical practice, the WERP-based formulas mentioned above can also be applied to more widely used echocardiographic data. Therefore, by applying the advection WERP method to 3D Doppler echocardiographic images, we estimate the decrease in velocity field based on the velocity field obtained along the beam direction in a plane perpendicular to the blood ejection at VC and LVOT. and

[0130] Specifically, the sSW scheme can be further simplified to calculate the advection transport energy rate A by relying solely on the velocity values ​​at VC sampled along the line, thus generating the pressure drop from 2D color Doppler echocardiography data.

[0131] In this work, we primarily focus on mean pressure drop because clinically accepted guidelines for echocardiographic assessment of AS are based on these mean pressure drops. Mathematically, in its simplest form, mean pressure drop (or mean pressure drop components)... The time average of the instantaneous drop Δp(t) during the contraction period is estimated. For completeness, the peak pressure drop (or peak pressure drop component) Δp is... MAX Instead, it is calculated as the maximum absolute value of the instantaneous decrease.

[0132] result

[0133] Analysis of the pressure component of TPD

[0134] The effects of motion, advection, and viscous pressure components on TPD were analyzed in two patient groups using the WERP formula. We combined the temporal transients of the systolic decline with their peak values ​​(see [link to WERP formula]). Figure 11 Together, the peak values ​​of the kinetic pressure drop were evaluated separately during acceleration and deceleration. The peak advection drop (P<0.001) and viscous drop (P<0.001) revealed a clear distinction between the groups, unlike the kinetic drop during acceleration (P<0.01).

[0135] Subjects in Group II had a mean advection TPD (28.74 ± 6.01 mmHg absolute peak, representing 96.9% of the total TPD, ranging from 91.5% to 98.5%), which was almost an order of magnitude higher than the motion component (4.76 ± 3.35 mmHg absolute peak during acceleration and 4.63 ± 2.96 mmHg absolute peak during deceleration) and more than two orders of magnitude higher than the viscous component (0.14 ± 0.07 mmHg absolute peak). Group I also showed a prevalence of the advection component (86.2% of the total TPD, ranging from 60.5% to 98.7%), but to a lesser extent (mean absolute peak values ​​of 4.83 ± 3.10 mmHg, 2.51 ± 1.67 / 0.97 ± 0.08 mmHg, and 0.02 ± 0.01 mmHg for the advection, motion, and viscous components during acceleration / deceleration, respectively).

[0136] Comparison of TPD using WERP and Bernoulli formulas

[0137] We compare the TPD calculated from 4D PC-MRI images using the cAW formula with results obtained using the sAW, sB, and cB formulas. We also compare the mean pressure drop estimated in TVR using different schemes, which are reported separately for group I and group II using linear regression and correlation coefficients. Figure 12 Consistent overestimation of the decrease was revealed using the sB formula (linear regression slope 0.492 for group I and 0.367 for group II, mean overestimation 99.1%, ranging from 49.2% to 145.1%), the cB formula (0.573 for group I and 0.291 for group II, mean overestimation 71.3%, ranging from 14.6% to 141.1%), and the sAW formula (0.887 for group I and 0.817 for group II). For patients in group I, a higher correlation was reported between the cAW regimen and each comparative method.

[0138] Improved TPD Calculation: Understanding the Velocity Field

[0139] Finally, we explore the differences between the different methods. Figure 12 The reasons behind the inconsistency between mid and high brightness were investigated, along with the applicability of WERP-based formulas to different imaging data. To this end, the velocity field (Δp) from the original 4D PC-MRI was analyzed. sAW The advection TPD and the simulated and idealized 3D Doppler are used. and 2D Doppler The TPD calculated from echocardiographic data was compared. Additionally, we compared it with simulated continuous Doppler echocardiographic images. The report is based on TPD estimates from the sB scheme.

[0140] Figure 13 An example of an overestimation is shown using the sB formula (regression slope of 0.496) and the sAW formula applied to 1D velocity curves obtained from idealized color Doppler (regression slope of 0.755 when compared with the average result from 12 velocity curves in each case). The specific choice of the curve determined by the acoustic plane introduces variability. A higher correlation is achieved using idealized color Doppler 3D acquisition with the sAW formula (regression slope of 1.076).

[0141] This section is supplemented in Supplementary Material C with a computer analysis of three velocity curves in an idealized narrow region, which demonstrates the overestimation of the velocity curves by means of Bernoulli's formula in a unique region.

[0142] discuss

[0143] Here we report the overestimation of TPD using Bernoulli's principle, explain its root cause, and propose a formula to correct it by considering the velocity curve in the cross section of the blood ejection at the venous vasoconstriction point.

[0144] TPD is driven by the force that causes the ventricle to accelerate spatially through a flow ejected via a narrow cavity, regardless of the other two components of the pressure drop, such as... Figure 11 As illustrated, referring to the Navel-Stokes addition that contributes to pressure, momentum transfer (i.e., advection force) is the major contributor to TPD, with minor roles played by time acceleration (or descent of motion) and the small influence of viscous dissipative stress. This finding confirms the rational choice of Bernoulli's principle to quantify clinical guidelines. 23,13,36,37 The pressure drop is recorded using continuous Doppler data because Bernoulli's principle is a simplification of the physical phenomena of flow through a pipe, considering only advection forces.

[0145] However, the results described here reveal an overestimation of the descent calculated based on Bernoulli's principle compared to the results obtained using a more complete formula for advection descent derived with the WERP method. This fundamental bias arises because Bernoulli's principle relies on simplifying blood vessels to a single streamline. 38–40 Just as blood vessels are one-dimensional tubes, the calculation of blood flow through 3D blood vessels, and therefore the calculation of advection energy and advection descent, should take into account the blood velocity in the cross-section of the vessel, not just a single peak velocity value. Only if the blood velocity curve is flat, where all particles in the cross-section of the vessel have the same velocity, can Bernoulli's principle be applied without any loss of accuracy (see Supplementary Material C below for a quantitative description of the errors for three idealized velocity curves).

[0146] As illustrated in Figure 17, the analysis of our cohort of 32 subjects reveals significant variability in the shape of the velocity curves. Therefore, the amount of overestimation by Bernoulli's principle is uneven: in the narrow case group, Bernoulli overestimated the TPD by an average of 99.1% (ranging from 49.2% to 145.1%) with the simplified formula, and by an average of 71.3% (ranging from 14.6% to 141.1%) with the corrected formula, leading to poor linear regression fit (see Figure 17). Figure 12 (Screens A and B). As an additional reference, the computer results in Supplementary Material C reveal how the shape of a perfect parabola with β=2 introduces an overestimation of 100.9%. Therefore, the proposed correction is very large and variable, thus justifying the need for further research to quantify its impact in the stratification of subjects with valvular stenosis.

[0147] Our mathematical analysis reveals that, as in the widely adopted Bernoulli formula, reasonable assumptions simplify the calculation of advection TPD to considering only the velocities at the inlet and outlet of the vascular domain. This finding is highly relevant for the clinical translation of our findings, as velocity data are needed only at two planes of the vascular anatomy, rather than across the entire lumen or ventricular pool. Furthermore, guidelines respond only to cases with minor stenosis that require data in the proximal (inlet) region, where the corrected Bernoulli formula should be used. 23,34 Our results confirm that the strategy improves accuracy for control cases, but illustrate an increase in inconsistency compared to using the adjusted Bernoulli formula in mild to severe illness cases (the correlation and slope in group II correspond to a decrease in group I with adjustment, compared to...). Figure 12 (Screens A and B in the text). These findings suggest that, as is currently being done in the most practical clinical situations, considering only the outlet velocity curve (at the venous vasoconstriction point) and ignoring the proximal velocity is a reasonable methodological strategy. The goal of a non-invasive approach to VS stratification was then set to robust and accurate assessment of advection energy at the venous vasoconstriction point.

[0148] Our results then reveal how the calculation of the advection descent depends on the quantity and quality of data available regarding velocity at the VC point. We present mathematical formulas suitable for different sources of blood velocity data and analyze the existence of fundamental biases among them. Our results reveal that 3D echocardiographic data introduce a small bias (1.076 for the regression line slope, see [link]). Figure 133D echocardiographic data provides an idealized 2D velocity profile at the VC position, which has artifacts from the funnel effect and from the projection of velocities alongside the acoustic ray. Considering the same artifacts, but only having information about a portion of the 2D velocity profile along one acoustic ray, as with 2D echocardiography, the advection energy is overestimated (the linear regression slope is 0.755, see [reference]). Figure 13 Furthermore, if only the peak value is available and the equation is then simplified to a Bernoulli equation, the slope is overestimated to be the maximum (0.496, see [reference]). Figure 13 The translation of theoretical research results makes it feasible for large-scale imaging acquisition protocols and modes, and further research is needed to define the optimal strategy for controlling the location of venous constriction to be imaged (the point of maximum advection descent) to identify the direction of the jet and its vertical plane, and to maximize the quantity and quality of velocity data at that plane.

[0149] Our findings also provide a fitting illustration of the overestimation of TPD using continuous Doppler echocardiography compared to catheter recording, which is currently attributed to pressure recovery following narrowing of the transvalvular ejection. 5,6,9 The root cause is the inherent assumption of Bernoulli's principle, which reduces vascular fluidization to a single streamline. This work demonstrates that when considering the physical principles governing human hemodynamics (Navel-Stokes momentum equation) to calculate the pressure drop that best explains the changes observed in a dense velocity field, the simplification by Bernoulli's equation introduces an overestimation of TPD.

[0150] Supplementary materials

[0151] The mathematical details behind A.WERP pressure drop estimation

[0152] The original form of the WERP formula is defined based on the Navel-Stokes equation, which is based on the work-energy principle proposed by Donati et al., and includes the vascular region Ω (the entrance plane Γ of this region). INLET and the exit plane Γ OUTLET Pressure drop Δp (defined from the segmentation of the lumen in a 4D PCMRI image) w The estimate is:

[0153]

[0154] The blood flow velocity Q, kinetic energy K, advection energy rate A, and viscous dissipation rate V can be evaluated by solving the digital surface and volume integrals into the following equations:

[0155]

[0156]

[0157]

[0158]

[0159] Where v is the time-dependent three-dimensional velocity field at the universal voxel, n is the normal direction on the inlet / outlet plane, and ρ = 1060 kg / m 3 v = 0.004 Pa·s are blood density and dynamic viscosity, respectively, and The complete advection pressure drop Δp was evaluated using the WERP method by separating the pressure components. cAW =-A / Q is derived from equation A.2.

[0160]

[0161] Therefore, the descent calculation is simplified to an area integral on the inlet and outlet planes, making it applicable to 2D PCMRI or 3D Doppler echocardiography data.

[0162] Equation A.3 can also be simplified by assuming an exit velocity much greater than the inlet velocity (which may remain in the crosslobe region defined from LVOT to VC, especially in narrow cases):

[0163]

[0164] The sAW scheme can also be simplified to estimating the advection energy rate from the velocity values ​​of VC along a single line (rather than in a complete vertical plane), thus making it applicable to 2D color Doppler echocardiographic images. By replacing the area integral at the outlet plane with a line integral along a line λ defined by the intersection of the assumed acoustic plane with the outlet plane of the aortic lumen, and by taking into account the fact that the velocity values ​​have been projected along the direction of the acoustic line, equation A.4 can be rearranged as:

[0165]

[0166] This makes it possible to use the WERP formula to calculate advection descent based on 2D color Doppler echocardiography.

[0167] It is worth noting that the advection WERP and Bernoulli equations are similar because they both use the advection effect to describe the characteristics of pressure drop, and the mathematical relationship between them is explained here. In the WERP method, the blood flow velocity Q can be neutrally estimated at the inlet or outlet plane defined from the image data as follows:

[0168] Q=∫ Γ v·ndx=v MAX Ψ (Equation A.6)

[0169] Here, v MAXIt is the maximum velocity at the entrance / exit plane, and Ψ=∫ Γ Φdx, where Φ is the normalized shape function in the normal direction of the inlet / outlet velocity curve. This is obtained using the cAW formula in Equation A.3:

[0170]

[0171] If we assume that the velocity is primarily aligned to the plane normal n, then substitution in equation A.6 (selectively evaluated at the inlet / outlet plane) to equation A.7 yields:

[0172]

[0173] in, and It is a function that depends only on the shape of the normalized curve. Therefore, under the assumption of a flat velocity curve (i.e., V as in Bernoulli's formula), OUTLET =V MAX，OUTLET and V INLET =V MAX，INLET And based on the blood density ρ = 1060 kg / m³ 3 Equation A.8 can be simplified to equation cB. From equation A.7, in V OUTLET >>V INLET Under the assumptions, the sAW expression in equation A.4 can be expressed as:

[0174]

[0175] This equation assumes that a flat velocity curve will again generate the sB formula.

[0176] B. Aortic pressure drop downstream of the aortic valve

[0177] Figure 14 and Figure 15 The temporal transients of the mean pressure drop components (total, kinematic, advection, and viscous) during systole, obtained in the AAR and DAR as defined above, are shown in the table. The results, presented in a table, report how the widening of the aortic flow jet downstream of the VC captured in the AAR caused a recovery of total TPD in Group II subjects, with the magnitude of the decrease comparable to that observed in the TVR (see [link to table]). Figure 11The values ​​were compared, but with opposite signs (mean absolute peak values ​​of 28.74 mmHg in AAR and 31.84 mmHg in TVR). Again, in group II, the spatial acceleration effect due to aortic morphological changes, by virtue of the increased effect of the motion term on the total pressure drop (mean absolute peak values ​​of 16.30 mmHg during acceleration and 14.14 mmHg during deceleration), was approximately two orders of magnitude more pronounced than the dissipative loss (mean absolute peak value of 0.51 mmHg). A clear distinction was shown between patients in groups I and II for all pressure terms.

[0178] In DAR, the total pressure drop was reduced in both groups due to abrupt changes in aortic geometry or the absence of obstructions in blood flow hemodynamics; the kinetic component was more prevalent than the others, and the advection component was reasonably reduced. These values ​​are similar to previously reported results in the healthy control group 22.

[0179] C. The impact of velocity curves on pressure drop estimation

[0180] The observed differences between WERP and Bernoulli's formula were experimentally verified using computer studies. Consider steady flow in a straight pipe with a varying diameter. The inlet and outlet velocity fields v(x, y) were analyzed using a general formula for poweroids.

[0181]

[0182] Among them, v MAX It is the peak speed, x c and y c Here, R is the center coordinate, R is the radius, and β is a coefficient that takes into account the curve shape. We use the pipe dimension and flow characteristics (such as cardiac output CO = 5 L / min, the ratio R between the outlet and inlet radii) to determine the curve shape. OUTLET / R INLET =0.25, density ρ = 1060 kg / m³ 3 We used a viscosity μ = 0.004 Pa·s to define the baseline case, representing the tubular size and flow characteristics in the human thoracic aorta with AS. Additionally, we chose spatial discretization dx = 0.5 mm and a velocity shape factor β = 4 to reproduce the quasi-parabolic curve. We then compared the pressure gradient ratio PGR = Δp estimated using the cB and cAW formulas. cB / Δp cAW These formulas selectively test: (1) the effect of cardiac output (CO = 4 L / min and CO = 6 L / min), and (2) the ratio between radii (R). OUTLET / R INLET =0.125 and R OUTLET / R INLET(3) Stenosis grade in terms of dx = 0.5 mm, spatial discretization (dx = 0.25 mm and dx = 1 mm), and (4) shape of the velocity curve in terms of shape factor (β = 2 and β = 10) to reproduce the configuration that may be found in the human aorta (from parabolic (β = 2) to blunt curve (β = 10)), see [reference] Figure 8 .

[0183] The results show that the Bernoulli scheme yields a global overestimation independent of spatial discretization, the inlet / outlet radius ratio, or the imposed velocity. Conversely, the difference between the WERP and Bernoulli estimates is highly dependent on the shape of the 3D velocity profile, with the minimum gap obtained using a blunt profile (PGR = 1.18).

[0184] D. Velocity curve at the venous vasoconstriction site from PCMRI data

[0185] Figure 17 shows the simulated 3D color Doppler echocardiographic velocity profiles from PCMRI data at VC. For each case, surface points of the 2D velocity field and 12 curves illustrate the 1D velocity profiles obtained as previously described. The velocity profiles are typically blunt, where v... MAX <2.5 m / s, for subjects in group 1, clearly shows lower variability in velocity curves compared to subjects in group II, where the peak velocity is v MAX ≥2.5 m / s. The velocity profile is highly asymmetric for Group II patients, thus exhibiting greater variability in excitation within this group.

[0186] System Description of Implementation Method 2

[0187] The second implementation involves performing only a subset of the WERP processing by considering only the translation terms of the complete WERP equation. This requires less information and allows the use of simpler imaging systems, particularly 2D PC MRI systems and 3D Doppler ECG systems. Figure 9 This is a block diagram of a 2D MRI system that can be used to perform calculations of pressure drop.

[0188] Here, a 2D phase-contrast magnetic resonance imaging (2D PC MRI) system 902 is configured, comprising MR imaging coils in which the subject is located. These coils are controlled by an MRI imaging control system 98, which includes an MRI controller processor 90. The MRI imaging control system 98, including the MRI controller processor 90, operates in a conventional manner to allow the acquisition of 3D phase-contrast magnetic resonance imaging data of the subject's internal blood vessels, for example, to allow for the acquisition of pressure differentials. For example, the transvalvular pressure drop (TPD) in the transvalvular region of the heart along the left ventricular outflow tract and venous vasoconstriction can be expected. Other blood vessels can also be monitored as desired.

[0189] The 2D PC MRI systems 98 and 902 acquire 2D PC MRI data 92 of the imaging subjects, which is saved for further processing and analysis.

[0190] The 2D PC MRI systems 98 and 902 also include a complete advection WERP calculation program 94, which processes the 2D PC MRL data 92 as described above according to the complete advection WERP processing method to obtain WERP pressure estimation data 96. The complete advection WERP processing method simplifies the WERP equation by taking only one of the three summation terms of energy (the advection term). The resulting pressure estimation data 96 gives an estimate of the pressure at the measured inlet and outlet planes, but ignores the influence of the vessel wall between the inlet and outlet. Therefore, it does not provide a complete solution for every point within the vessel between the inlet and outlet, but only for the inlet and outlet; however, the pressure drop across the vessel can then be obtained from these solutions.

[0191] Figure 10 (A) illustrates a 3D Doppler ultrasound ECG equivalent system. Here, transducer 1000 acquires 3D Doppler measurements from the subject under the control of a 3D Doppler ECG system 102, which includes a 3D Doppler ECG controller 104. The system generates 3D Doppler ECG data 106, which is then used as input to a cAW calculation program 108, which performs the necessary processing as described above to obtain a solution for the inlet and outlet planes of the monitored blood vessel, which is then stored and output as cAW output data 110.

[0192] Figure 7 The basic method of operation of the cAW protocol is illustrated. Here, at s.7.2, MRI or ECG data of the inlet and outlet planes of the subject's blood vessel from which the pressure drop is to be obtained are acquired. Then, based on this data, for example, the complete advection WERP values ​​for the inlet and outlet planes are obtained using the processing expressed by Equation A.3 above (s.7.4). Finally, the complete advection WERP data is stored and / or, for example, output to the user on a display screen (s.7.6).

[0193] Third Implementation System Description

[0194] The third implementation relates to a simplified advection WERP scheme, which can be performed using 2D Doppler ECG data, but it can certainly also be used with other imaging systems that acquire more information, such as 2D PC MRI, 3D Doppler ECG, etc. However, in this example, Figure 10(B) illustrates a 2D Doppler ultrasound ECG equivalent system. Here, the transducer 1000 acquires 2D Doppler measurements from the subject under the control of a 2D Doppler ECG system 152, which includes a 2D Doppler ECG controller 154. The system generates 2D Doppler ECG data 156, which is then used as input to an sAW calculation program 158, which performs the necessary processing as described previously to obtain a solution for the outlet plane of the monitored vessel, which is then stored and output as sAW output data 160.

[0195] Figure 8 The basic method of operating the sAW protocol is illustrated here. Here, at s.8.2, ECG data of the outlet plane of the subject's blood vessel from which the pressure drop is to be obtained is acquired. Then, based on this data, simplified advection WERP values ​​for the inlet or outlet plane are obtained, for example, using the processing expressed by Equation A.5 above (s.8.4). Finally, the simplified advection WERP data is stored and / or, for example, output to the user on a display screen (s.8.6).

[0196] For completeness, a brief additional description of the remaining figures is provided below.

[0197] Figure 11 The TPD estimated from 4D PC-MRI data using the WERP formula is shown. Top: Mean temporal transient of TPD during systole for the total component (top left), motor component (top right), advection component (bottom left), and viscous component (bottom right) on subjects in Group I (light gray filled area) and Group II (dark gray filled area). Bottom: Distribution of peak TPD (mean ± standard deviation) Δp, including the significance of the difference between Group I and Group II. MAX (mmHg) (Unpaired T test). Peak values ​​of kinematic pressure drop are reported separately for acceleration and deceleration (in parentheses, negative peak) contraction. Note that because peak kinematic, advection, and viscosity decreases do not occur simultaneously, the pressure component is not included in the total TPD.

[0198] Figure 12 The correlation between the mean TPD estimated from 4D PC-MRI data using cAW for the sB(A), cB(B), and sAW(C) formulas is shown. The case-specific values ​​and the gray dashed line in the regression line (A) for Group I subjects (blue) and Group II subjects (red) illustrate the relationship between Δp as defined by the simplified Bernoulli formula. sB =20 mmHg (Baumgartner, 2009) is the threshold for mild to severe stenosis.

[0199] Figure 13The results of WERP advection reduction calculations based on 4D PC-MRI and idealized echocardiographic data are shown. (A) Transient systolic advection TPD calculated from 4D PCMRI (black solid line), color Doppler 3D using the sAW formula (grey solid line), color Doppler 2D using the sAW formula on the 1D curve obtained at the peak systolic frame (grey circle), and continuous Doppler 1D using the sB formula (grey dashed line). (B) Linear regression and correlation factors between peak advection TPD calculated from 4D PC-MRI and Doppler echocardiographic data: color Doppler 3D using the sAW formula (blue solid line), color Doppler 2D using the sAW formula on 1D curve 2 (black solid line with error bar graphs for 12 sampled curves), and Doppler 1D using the sB formula (red solid line). Case-specific values ​​are shown for each Doppler-based acquisition technique.

[0200] Figure 14 This illustrates the pressure drop in AAR estimated from 4D PC-MRI data using the WERP formula. Top: Mean temporal transient of the pressure drop during systole, including the total component (top left), motion component (top right), advection component (bottom left), and viscous component (bottom right) on subjects in Group I (light gray filled area) and Group II (dark gray filled area). Bottom: Distribution of peak pressure drop (mean ± standard deviation) Δp, including the significance of the difference between Group I and Group II. MAX (mmHg) (Unpaired T test). Peak values ​​of kinetic pressure drop are reported separately for acceleration and deceleration contractions (in parentheses, negative peak value).

[0201] Figure 15 The pressure drop in DAR estimated from 4D PC-MRI data using the WERP formula is shown. Top: Mean temporal transient of the drop during systole, including the total component (top left), motion component (top right), advection component (bottom left), and viscous component (bottom right) on subjects in Group I (light gray filled area) and Group II (dark gray filled area). Bottom: Distribution of peak pressure drop (mean ± standard deviation) Δp, including the significance of the difference between Group I and Group II. MAX (mmHg) (Unpaired T test). Peak values ​​of total pressure drop and kinetic pressure drop are reported separately for acceleration and deceleration (in parentheses) contraction.

[0202] Figure 16 The results of a computer test on a 3D straight tube with a stable velocity field are shown. Top: Representation of velocity curves at the inlet / outlet plane obtained with different velocity shape factors. Bottom: Based on the heart output / outlet radius ratio R... OUTLET / R INLETThe pressure gradient ratio (PGR = Δp) between the spatial discretized dx and the velocity shape coefficient β estimated using the cB and cAW formulas. cB / Δp cAW ).

[0203] Figure 17 shows the 2D velocity curves of simulated 2D color Doppler echocardiographic images from the exit plane of Group I (Figure 17A) and Group II (Figure 17B).

[0204] Various modifications to the above-described embodiments (whether by adding, deleting, or replacing) will be apparent to those skilled in the art, in order to provide alternative embodiments.

[0205] References

[0206] 1. Cioffi G, Faggiano P, Vizzardi E, Tarantini L, Cramariuc D, Gerdts E, deSimone G. Prognostic effect of inappropriately high left ventricular mass inasymptomatic severe aortic stenosis. Heart. 2011; 97:301–307.

[0207] 2. Otto CM. Valvular Aortic Stenosis. Disease Severity and Timing of Intervention. J Am Coll Cardiol. 2006;47:2141–2151.

[0208] 3. Minners J, Allgeier M, Gohlke-Baerwolf C, Kienzle RP, Neumann FJ, Jander N. Inconsistencies of echocardiographic criteria for the grading of aortic valve stenosis. Eur Heart J. 2008;29:1043–8.

[0209] 4. Vahanian A, Alfieri O, Andreotti F, Antunes MJ, Barón-Esquivias G, Baumgartner H, Borger MA, Carrel TP, De Bonis M, Evangelista A, Falk V, Iung B, Lancellotti P, Pierard L, Price S, H-J, Schuler G, Stepinska J, Swedberg K, Takkenberg J, Von Oppell UO, Windecker S, Zamorano JL, Zembala M, Bax JJ, Ceconi C, Dean V, Deaton C, Fagard R, Funck-Brentano C, Hasdai D, Hoes A, Kirchhof P, Knuuti J, Kolh P, McDonagh T, Moulin C, Popescu BA, Reiner Sechtem U, Sirnes PA, Tendera M, Torbicki A, Von Segesser L, Badano LP, Bunc M, Claeys MJ, Drinkovic N, Filippatos G, Habib G, Kappetein AP, Kassab R, Lip GYH, Moat N, Nickenig G, Otto CM, Pepper J, Piazza N, Pieper PG, Rosenhek R, Shuka N, Schwammenthal E, Schwitter J, Mas PT, Trindade PT, Walther T. Guidelines on the management of valvular heart disease (Version 2012). Eur Heart J. 2012; 33:2451–96.

[0210] 5. Bach DS. Echo / Doppler evaluation of hemodynamics after aortic valve replacement: principles of interrogation and evaluation of high gradients. JACC Cardiovasc Imaging. 2010; 3:296–304.

[0211] 6. Bahlmann E, Cramariuc D, Gerdts E, Gohlke-Baerwolf C, Nienaber C a., Eriksen E, Wachtell K, Chambers J, Kuck KH, Ray S. Impact of pressure recovery on echocardiographic assessment of asymptomatic aortic stenosis: A SEAS substudy. JACC Cardiovasc Imaging. 2010; 3:555–562.

[0212] 7. Baumgartner H, Khan S, DeRobertis M, Czer L, Maurer G. Discrepancies between Doppler and catheter gradients in aortic prosthetic valves in vitro. A manifestation of localized gradients and pressure recovery. Circulation. 1990; 82:1467–1475.

[0213] 8. Baumgartner H, Stefenelli T, Niederberger J, Schima H, Maurer G. "Overestimation" of catheter gradients by Doppler ultrasound in patients with aortic stenosis: A predictable manifestation of pressure recovery. J Am Coll Cardiol. 1999; 33:1655–1661.

[0214] 9. Garcia D, Dumesnil JG, Durand LG, Kadem L, Pibarot P. Discrepancies between catheter and doppler estimates of valve effective orifice area can be predicted from the pressure recovery phenomenon: Practical implications with regard to quantification of aortic stenosis severity. J Am Coll Cardiol. 2003; 41: 435–442.

[0215] 10. Gardner RM. Direct arterial pressure monitoring. Curr Anaesth Crit Care. 1990; 1: 239–246.

[0216] 11. Niederberger J, Schima H, Maurer G, Baumgartner H. Importance of pressure recovery for the assessment of aortic stenosis by Doppler ultrasound. Role of aortic size, aortic valve area, and direction of the stenotic jet in vitro. Circulation. 1996; 94: 1934–1940.

[0217] 12. Baumgartner H, Hung J, Bermejo J, Chambers JB, Evangelista A, Griffin BP, Iung B, Otto CM, Pellikka PA, M. Echocardiographic assessment of valve stenosis: EAE / ASE recommendations for clinical practice. J Am Soc Echocardiogr. 2009; 22: 1–23; 101–102.

[0218] 13. Nishimura R A, Otto CM, Bonow RO, Carabello BA, Erwin JP, Guyton RA, O’Gara PT, Ruiz CE, Skubas NJ, Sorajja P, Sundt TM, Thomas JD. 2014 AHA / ACC guideline for the management of patients with valvular heart disease: A report of the American college of cardiology / American heart association task force on practice guidelines. J Am Coll Cardiol. 2014; 63:e57–e185.

[0219] 14. Markl M, Wallis W, Brendecke S, Simon J, Frydrychowicz A, Harloff A. Estimation of global aortic pulse wave velocity by flow-sensitive 4D MRI. Magn Reson Med. 2010; 63:1575–82.

[0220] 15. Markl M, Kilner PJ, Ebbers T. Comprehensive 4D velocity mapping of the heart and great vessels by cardiovascular magnetic resonance. J Cardiovasc Magn Reson. 2011; 13:7.

[0221] 16. Stankovic Z, Allen BD, Garcia J, Jarvis KB, Markl M. 4D flow imaging with MRI. Cardiovasc Diagn Ther. 2014; 4:173–92.

[0222] 17. Deng Z, Fan Z, Xie G, He Y, Natsuaki Y, Jin N, Bi X, An J, Liu X, Zhang Z, Fan Z, Li D. Pressure gradient measurement in the coronary artery using 4D PC-MRI: towards noninvasive quantification of fractional flow reserve. J Cardiovasc Magn Reson. 2014; 16: O55.

[0223] 18. Donati F, Nordsletten DA, Smith NP, Lamata P. Pressure Mapping from Flow Imaging: Enhancing Computation of the Viscous Term Through Velocity Reconstruction in Near-Wall Regions. In: EMBC, IEEE EMB. 2014.

[0224] 19. Krittian SBS, Lamata P, Michler C, Nordsletten D a, Bock J, Bradley CP, Pitcher A, Kilner PJ, Markl M, Smith NP. A finite-element approach to the direct computation of relative cardiovascular pressure from time-resolved MR velocity data. Med Image Anal. 2012; 16:1029–37.

[0225] 20. Meier S, Hennemuth A, Friman O, Bock J, Markl M, Preusser T. Non-invasive 4D Blood Flow and Pressure Quantification in Central Blood Vessels via PC-MRI. 2010; 903–906.

[0226] 21. Donati F, Figueroa CA, Smith NP, Lamata P, Nordsletten DA. Non-invasive pressure difference estimation from PC-MRI using the work-energy equation. Med Image Anal. 2015; 26:159–172.

[0227] 22. Lamata P, Pitcher A, Krittian S, Nordsletten D, Bissell MM, Cassar T, Barker AJ, Markl M, Neubauer S, Smith NP. Aortic relative pressure components derived from four-dimensional flow cardiovascular magnetic resonance. Magn. Reson. Med. 2013;

[0228] 23. Baumgartner H, Hung J, Bermejo J, Chambers JB, Evangelista A, Griffin BP, Iung B, Otto CM, Pellikka PA, Quinones M. Echocardiographic assessment of valve stenosis: EAE / ASE recommendations for clinical practice. Eur J Echocardiogr. 2009; 10:1–25.

[0229] 24. Chan RW, Von Deuster C, Giese D, Stoeck CT, Harmer J, Aitken AP, Atkinson D, Kozerke S. Characterization and correction of eddy-current artifacts in unipolar and bipolar diffusion sequences using magnetic field monitoring. J Magn Reson. 2014; 244:74–84.

[0230] 25. Moussavi A, Untenberger M, Uecker M, Frahm J. Correction of gradient-induced phase errors in radial MRI. Magn Reson Med. 2014; 71: 308–312.

[0231] 26. Rohde GK, Barnett AS, Basser PJ, Marenco S, Pierpaoli C. Comprehensive Approach for Correction of Motion and Distortion in Diffusion-Weighted MRI. Magn Reson Med. 2004; 51: 103–114.

[0232] 27. Bock J, Frydrychowicz A, Lorenz R, Hirtler D, Barker AJ, Johnson KM, Arnold R, Burkhardt H, Hennig J, Markl M. In vivo noninvasive 4D pressure difference mapping in the human aorta: phantom comparison and application in healthy volunteers and patients. Magn Reson Med. 2011; 66: 1079–88.

[0233] 28. Lee TC, Kashyap RL, Chu CN. Building Skeleton Models via 3-D Medial Surface Axis Thinning Algorithms. CVGIP Graph. Model. Image Process. 1994; 56: 462–478.

[0234] 29. Garcia J, Capoulade R, Le Ven F, Gaillard E, Kadem L, Pibarot P, Larose Discrepancies between cardiovascular magnetic resonance and Doppler echocardiography in the measurement of transvalvular gradient in aortic stenosis: the effect of flow vorticity. J Cardiovasc Magn Reson. 2013; 15: 84.

[0235] 30. Laske A, Jenni R, Maloigne M, Vassalli G, Bertel O, Turina MI. Pressure gradients across bileaflet aortic valves by direct measurement and echocardiography. Ann Thorac Surg. 1996; 61: 48–57.

[0236] 31. Zhang Y, Nitter-Hauge S. Determination of the mean pressure gradient in aortic stenosis by Doppler echocardiography. Eur Heart J. 1985; 6: 999–1005.

[0237] 32. Hatle L, Brubakk A, Tromsdal A, Angelsen B. Noninvasive assessment of pressure drop in mitral stenosis by Doppler ultrasound. Br Heart J. 1978; 40: 131–140.

[0238] 33. Chambers JB. Aortic stenosis. Eur J Echocardiogr. 2009; 10: i11–i19.

[0239] 34. Rijsterborgh H, Roelandt J. Doppler assessment of aortic stenosis: Bernoulli revisited. Ultrasound Med Biol. 1987; 13:241–248.

[0240] 35. Scantlebury DC, Geske JB, Nishimura RA. Limitations of Doppler echocardiography in the evaluation of serial stenoses. Circ Cardiovasc Imaging. 2013; 6:850–852.

[0241] 36. Gersh BJ, Maron BJ, Bonow RO, Dearani JA, Fifer MA, Link MS, Naidu SS, Nishimura RA, Ommen SR, Rakowski H, Seidman CE, Towbin JA, Udelson JE, Yancy CW. 2011 ACCF / AHA Guideline for the Diagnosis and Treatment of Hypertrophic Cardiomyopathy: a report of the American College of Cardiology Foundation / American Heart Association Task Force on Practice Guidelines. Developed in collaboration with the American As. J Am Coll Cardiol. 2011; 58:e212–60.

[0242] 37. Otto C, Burwash I, Legget M, Munt B, Fujioka M, Healy N, Kraft C, Miyake-Hull C, Schwaegler R. Prospective Study of Asymptomatic Valvular Aortic Stenosis. Circulation. 1997; 95:2262–70.

[0243] 38. Bermejo J, Antoranz JC, Yotti R, Moreno M, Garcia-Fernandez MA. Spatio-temporal mapping of intracardiac pressure gradients. A solution to Euler’s equation from digital postprocessing of color Doppler M-mode. Ultrasound Med Biol. 2001; 27:621–630.

[0244] 39. Garcia D, Pibarot P, Durand LG. Analytical modeling of the instantaneous pressure gradient across the aortic valve. J Biomech. 2005; 38:1303–1311.

[0245] 40. Heys JJ, Holyoak N, Calleja AM, Belohlavek M, Chaliki HP. Revisiting the simplified bernoulli equation. Open Biomed Eng J. 2010; 4:123–8.

[0246] 41. de Vecchi A, Clough RE, Gaddum NR, Rutten MCM, Lamata P, Schaeffter T, Nordsletten D a, Smith NP. Catheter-induced errors in pressure measurements in vessels: an in-vitro and numerical study. IEEE Trans Biomed Eng. 2014; 61:1844–50.

[0247] 42. Falahatpisheh A, Rickers C, Gabbert D, Heng EL, Stalder A, Kramer H-H, Kilner PJ, Kheradvar A. Simplified Bernoulli’s method significantly underestimates pulmonary transvalvular pressure drop. J Magn Reson Imaging. 2015; n / a–n / a.

Claims

1. An apparatus for determining a pressure difference across a pipe caused by fluid flow within the pipe using the work-energy relative pressure (WERP) method, the apparatus comprising: Processor, the processor being configured to: Time-correlated three-dimensional fluid velocity data were obtained at multiple points along the pipe; The region of interest from which flow information is desired is segmented within the pipeline; A flow vector field (w) is defined through the divided domain, which is non-divergent and has zero value in the sidewall of the pipe; Process the flow vector field (w) and the time-correlated three-dimensional fluid velocity data to determine: i) The velocity (Qw) of the flow vector field (w); ii) The alternative kinetic energy (Kw) of the fluid flow through the pipe; iii) The alternative advection energy rate (Aw) of the fluid flow through the pipe; and iv) The alternative viscous dissipation rate (Vw) of the fluid flow; as well as The pressure difference is calculated based on the entirety of the flow velocity (Qw), the substitution kinetic energy (Kw), the substitution advection energy rate (Aw), and the substitution viscous dissipation rate (Vw). Wherein, the flow velocity (Qw) depends on the area integral of the flow vector field (w) on the inlet or outlet plane of the pipe or on any other plane dividing the pipe into two subsections, or the flow velocity (Qw) is calculated as a weighted average of the aforementioned area integrals. The alternative kinetic energy (Kw) depends on: the flow vector field (w), the time-dependent three-dimensional fluid velocity data, and the fluid density. Wherein, the alternative advection energy rate (Aw) depends on the sum of the area integrals of the flow vector field (w) and the time-related three-dimensional fluid velocity data or data derived therefrom at the inlet and outlet planes of the pipe and the fluid density, or the sum of the area integrals of the flow vector field and the time-related three-dimensional fluid velocity data or data derived therefrom on all the outer surfaces of the domain, namely the inlet plane, outlet plane, and sidewalls of the cylinder, and The alternative viscous dissipation rate (Vw) depends on the flow vector field (w), the three-dimensional fluid velocity data, and the fluid dynamic viscosity.

2. The apparatus according to claim 1, wherein, where v is the time-dependent three-dimensional velocity field at a generic voxel, w is a non-divergent vector defined in the vessel domain, n is the normal direction to the inlet and outlet planes of the conduit, p and m are the fluid density and dynamic viscosity, respectively, Ω is the identified conduit region for which the pressure difference is to be determined and has an inlet plane Γ INLET and an outlet plane Γ OUTLET .

3. The apparatus according to claim 2, wherein, The pressure difference is given by the following formula:

4. The apparatus according to claim 1, wherein, The fluid is blood, and the tube is a blood vessel.

5. The apparatus according to claim 1, wherein, The time-correlated three-dimensional fluid velocity data is obtained using 4D phase-contrast magnetic resonance imaging or cardiac echocardiography with the aid of the Doppler effect, or by tracking the contrast agent during ultrafast acquisition.

6. A system for determining a pressure differential across a pipe caused by fluid flow within the pipe, the system comprising one of the following: a) A 4D phase-contrast magnetic resonance imaging (4D phase-contrast magnetic resonance imaging) device configured to acquire time-correlated three-dimensional fluid velocity data from the pipe, the system further comprising the means according to any one of claims 1 to 5; or b) An echocardiographic imaging apparatus configured to acquire time-related three-dimensional fluid velocity data from the pipe, the system further comprising means according to any one of claims 1 to 5.

Images