The present invention relates to a system and method for medical imaging, and more particularly, the present invention relates to a system and method for cardiac imaging using ultrasonic waves.
For decades automotive, aerospace, and energy industries have used advanced simulation technology to virtually design, test and validate their products before they are built. Built on these successes, there has been a profound interest in medical sciences to use virtual simulation for designing, testing, and validating new treatment modalities. Of particular interest are modelling cardiovascular disease as it represents the primary cause of mortality in the industrialized nations. The heart is the most vital and complex organ of the human body controlled by the interplay of anatomical, electrical, and biomechanical events.
In the past three decades, there has been a concerted effort in human heart modelling for a variety of reasons including facilitating the clinical decision-making process and guiding in the treatment planning and acceleration process. Despite these efforts, there remains a significant barrier in applying virtual human heart models in clinical applications. The primary reason is that the known virtual simulation models are engineering models that are based on medical images (anatomical data) combined with clinical measurement data. The clinical measurement data comes from variety of invasive and non-invasive modalities that capture the biomechanical events for multi-scale computational modelling. Most of these models have also been developed and validated using data from invasive measurements in controlled conditions in animal models. These models are useful for demonstrating the proof of concept and development of general models. However, such models are not useful in human heart modelling for individualized prediction of different treatment modalities with the goal of virtually selecting the most promising treatment within the paradigm of Personalized Medicine.
For patient-specific human heart modelling, the challenge is to fuse the patient-specific geometrical data retrieved from imaging modality with real-time in-plane biomechanical data of these geometrical points. All imaging platforms convert physical data into image data. Until now, scientists have focused on the acquisition of better images or on post-processing techniques to increase spatial and temporal resolution.
A need is appreciated for a system and method for retrieval and fusion of geometrical data with in-plane biomechanical data from any echocardiographic dataset through cardiac cycle of any anatomical point for patient-specific heart modelling.
The accompanying figures, which are incorporated herein, form part of the specification and illustrate embodiments of the present invention. Together with the description, the figures further explain the principles of the present invention and enable a person skilled in the relevant arts to make and use the invention.
Subject matter will now be described more fully hereinafter. Subject matter may, however, be embodied in a variety of different forms and, therefore, covered or claimed subject matter is intended to be construed as not being limited to any exemplary embodiments set forth herein; exemplary embodiments are provided merely to be illustrative. Likewise, a reasonably broad scope for claimed or covered subject matter is intended. Among other things, for example, the subject matter may be embodied as apparatus and methods of use thereof. The following detailed description is, therefore, not intended to be taken in a limiting sense.
The word “exemplary” is used herein to mean “serving as an example, instance, or illustration.” Any embodiment described herein as “exemplary” is not necessarily to be construed as preferred or advantageous over other embodiments. Likewise, the term “embodiments of the present invention” does not require that all embodiments of the invention include the discussed feature, advantage, or mode of operation.
The terminology used herein is to describe particular embodiments only and is not intended to be limiting to embodiments of the invention. As used herein, the singular forms “a”, “an” and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will be further understood that the terms “comprises”, “comprising,”, “includes” and/or “including”, when used herein, specify the presence of stated features, integers, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and/or groups thereof.
The following detailed description includes the best currently contemplated mode or modes of carrying out exemplary embodiments of the invention. The description is not to be taken in a limiting sense but is made merely to illustrate the general principles of the invention since the scope of the invention will be best defined by the allowed claims of any resulting patent.
The following detailed description is described with reference to the drawings, wherein like reference numerals are used to refer to like elements throughout. In the following description, for purposes of explanation, specific details may be outlined in order to provide a thorough understanding of the subject innovation. It may be evident, however, that the claimed subject matter may be practiced without these specific details. In other instances, well-known structures and apparatus are shown in block diagram form in order to facilitate describing the subject innovation.
In one aspect, disclosed are a system and method for applying mathematical and physical laws to reversal wave data obtained from echo datasets to predict the trajectories of anatomical points in a cardiac cycle for heart modelling. The disclosed system allows to retrieve an infinite number of cardiac phases or volumes between the original cardiac phases and volumes, enabling the system to predict the coordinates of adjusting anatomical points and thereby increased spatial resolution even better than cardiac tomography (CT) systems. The system has three modules: the algorithm module, the development module, and the testing module based on real clinical cases.
In one aspect, the disclosed system and method uses 2D or 3D echo datasets stored in a DICOM file format. The disclosed system can extract 2D or 3D echo images or video clips from ultrasound reversal waves of the 2D or 3D echo datasets. The disclosed system can then assign and track an arbitrary pixel or voxel on the extracted images per cardiac cycle (phase by phase). Thereafter the system can determine an original reversal wave equation ρp attached to the arbitrary pixel or voxel p. Thereafter, the system, using Lagrange-Euler equations, extracts the curve with passed from the selected pixel or voxel p within a cardiac cycle. The Lagrange-Euler equations are reformulated based on 2D and 3D echo data sets, which is called “fp” deformable map. The system can then generate reconstructed curvelets from the interior products ρp and fp that is called by cp=ρp·fp; wherein the cp's are new coefficients for the new 2D and 3D curvelets attached p. From the new coefficients, 2D and 3D images or frames are reconstructed. The new 2D/3D video clips are made vs. the original 2D/3D echo video clips. 2D and 3D pixel/voxel tracking program are coded and run per a cardiac cycle. New 2D and 3D quantitative data like motion (velocity and displacement of a voxel) and Deformation (longitudinal, radial, and circumferential strain component for each voxel) can then be calculated. A geometrical important index which is called “Curvature” is reformulated based on 3D echo datasets and then coded as distinct colors on the full volume for example whenever the curvature is positive, it can be coded as green color and so on.
In one aspect, any above structures can be distinguished with separate colors and can be tracked throughout the cardiac cycle, like mitral annulus, valve leaflets and chordae.
In one aspect, disclosed is an ultra-resolution mathematical 2D and 3D echocardiographic platform. Ultrasound is the most widely used imaging modality in medicine and especially in cardiovascular patients for diagnosis, treatment, follow-up, and guided interventions. The disclosed system and method can overcome the fundamental limitations with Ultrasound imaging modality i.e., low 3D volumes (coupled with moving intra-cardiac structures) and inability to capture all the mechanical events in the heart.
In one aspect, disclosed is an imaging platform based on physical laws, mathematics, and numerical algorithms on reversal wave data to retrieve the trajectories of all the anatomical points and retrieve infinite number of frames or volumes from the original dataset. The disclosed system and method, by converting the image data into mathematical data/pixel data, can provide a precise temporal vector tracking of any landmark position with precise in-plane mechanical properties including vector velocity and strain without needing a Doppler.
In one aspect, the disclosed platform can be universal and can process the input of any echocardiographic imaging platform and can in theory be built in any system for live-processing or be coupled with other techniques. In clinical settings, the frame rate could be increased for 2D echo to 1000 frames (compared to current clinical maximum around 50-60 frames/s) and for 3D to 500 volumes/s (compared to current clinical maximum around 35-40 volumes/s).
In one aspect, the disclosed system and method can be applied to other imaging modalities, such as CT/MRI. The disclosed system and method may also provide for development of ultra-resolution image fusion and simulation. For example, image fusion of CT-can and ultrasound. The disclosed platform, by converting image data into mathematical data of both CT and ultrasound, and creating new equations, can provide for creating new imaging fusion modality. The principles of artificial intelligence can be incorporated without departing from the scope of the present invention. It is also envisioned that the disclosed system and method can provide for fusion of auto quantitative indices (like motional and deformational indices) to images.
Disclosed are a system and method for medical imaging-based modeling of a heart for therapeutic and diagnostic applications. The disclosed system and method can use existing data from an imaging modality, such as echocardiography data to create a virtual heart model with high resolution and depicting all the movements in the heart.
Referring to
A cardiac point p at the phase t with the velocity vector vp,t and gradient vector ∇vp,t, Vt is the space of velocity vectors attached at time t 5. Based on the F-sheaves theory 6, to have the space of “3 by 3” matrices thorough the time in a cardiac cycle as follow:
The first row describes the velocity components, the second row shows the divergence at the point p and the third row states the vortices at the point p phase by phase within a cardiac cycle. Therefore, H is realized as a stack of smooth 3D manifolds (cross sections of H) through the cardiac cycle toward the velocity vector fields.
The 3D strains are presented by 3 by 3 matrices in the following way:
Force indices are formulated and calculated based on extracted motion and deformation parameters by the following tensor product:
General force of p per cardiac cycle is the summation of the above formula:
The Lagrangian equation for 3D cardiac segment (point p) is described by the following reformulation based on the acquired datasets:
Where: p* is the set of all fibers passing through p; δVolumep(t) is the change in volume of the point p from the first volume to n's volume in time tn; and δVolumep,n(tn) is the determinant of strain matrix:
Based on this, Lagrangian equations are reformulated:
By implementing these equations, can be provided curves lp's point by point. In fact, a fibre bundle of curves lp's can be constructed. This fibre bundle represents flow movements (optimized trajectories) of each cardiac point per cardiac cycle.
Addition of Gaussian curvatures, connections, Riemannian curvatures, and Hamilton-Ricci flow equation:
For a cardiac point ‘p’ was obtained the optimized trajectory lp. Was set the Gaussian curvature 8 point ‘p’ along the optimized trajectory lp by the following formula:
r
l
=The Gussian curvature at the point‘p’
tanαl
r
l
=∂p runs away on the cardiac pointsαl
H as a 4D smooth manifold is divided to H3 and H2. H1 is constructed based on Lagranjian equations and H2 is constructed by utilizing Hamilton-Ricci flow equations 9.
Whenever a cardiac point ‘p’ moves along the optimized trajectory lp for instance from phase t1 with the velocity vector v(p, t1) and stain εp,t
On the other hands, vector ‘v(p, t1)’ is connected to the vector ‘v(p, t2)’ by the stent εp,t
g(u,v)(w)=∇u∇vw−∇v∇uw
The Hamilton-Ricci flow equations
state the bending behavior of Heart (H) where r is the whole Gaussian curvature and R is the average of rl
Echocardiographic datasets can be acquired from any echocardiography machines like Philips (EPIQ CVx and iE33 xMATRIX), GE Healthcare (Vivid 3, Vivid 7 and LOGIQ E9) and Esaote (MyLab™ 60, MyLab™ 70). Original reversal equation waves are extracted for each anatomical point. The wave equations are partial differential equations, which are scalar functions w=w(x1, x2, t) of a time variable t and spatial variable x1, x2. The magnitude w is the displacement of vibrant cardiac segments away from their resting locations from end of systole to the end of diastole in one cardiac cycle. The equations are:
Wave equations are solved for each echocardiographic segment as linear combinations of simple solutions that are sinusoidal plane waves with various directions of propagation and wavelengths, but all with the same propagation speed c. By applying Fourier transform to the wave solution, can be obtained velocities of anatomical points in original phases per cardiac cycle. Therefore, the strain components and myocardial torsions can be quantified. Longitudinal strain (shortening and lengthening), radial strain (thickening) and torsion indices are extracted for each segment within the heart cycle by the formula below.
Original echocardiographic segments are acquired. These segments are mapped to the Cartesian plane as geometrical points. Was searched the regions of p's point at time to corresponding to n's frame and were labeled dn. If Jn,t
Velocity vectors are stretched as the calculated strain values. Therefore, motion and deformation are considered simultaneously.
Hadamard transform for fast running and high intensity resolution was used. K-theory in algebraic geometry was used to distinguish structures during the entire cardiac cycle by use of colors. Clinical verification tests were applied on echocardiographic datasets for different echocardiography machines. C++ coding for 3D cases to read original files and write new datasets is based on defining many commands to create 4D mathematical matrices attached to 3D cases.
Development of mathematical algorithms for 2D datasets:
STEP 1: A 2D image is formed by pixels, so the first step is to extract the coordinates of the original pixel positions in time and space from reversal waves and equations. For example, when using echocardiographic images based on the 2D original ultrasound reversal waves and Fourier coefficients, it is possible to extract the original cardiac segment position in space and time for each corresponding point within a cardiac cycle (
STEP 2: The second step is to extract and calculate mechanical parameters, such as motion and deformation indices, of the original pixels. This allows the formulation and calculation of force indices. The force formula based on velocity, displacement, strain, and strain rate is:
Wherein, p is a 2D cardiac segment; dn(p) is the displacement of p at n's frame; v(p, n, t) is velocity at the same frame; and ε(p, n, tn), and ε′(p, n, tn) are the strain and strain rate at n's frame. The general force of p per cardiac cycle is the summation of the formula (1).
STEP 3: The third step is to find the cardiac segment (pixel) trajectory based on Lagrangian equations. It is important to reformulate this system of equations based on echocardiographic datasets. Lagrangian equations for a 2D cardiac segment (point p) are described by the following reformulation based on acquired datasets:
p* is the set of all fibres p,r passing through p; δAp(t) is the change in area of point p from the first frame to n's frame in time tn; and δAp,n(tn) is the determinant of the generalised strain matrix:
There are four strain components: εp,1 (x-axis direction), εp,2 (y-axis direction), εp,3 (rotation around x-axis) and εp,4 (rotation around y-axis).
By rewriting the new data in the generalized Lagrangian equations, following was obtained:
the Lagrangian equations can also be written as:
Was applied kinetic and potential energetic formulas of the cardiac segment to the left side of the Lagrangian equations:
The fourth step is to find numerical solutions to the equation which is described below in step 4 (step 4 for 2D image and 3D are same).
Development of mathematical algorithms for 3D datasets:
STEP 1: Unlike the 2D image, a 3D image is formed with a voxel. Therefore, the first step in developing a mathematical algorithm for 3D is the extraction of voxel coordinates in time and space. (
STEP 2: The second step is to extract and calculate the mechanical parameters of the original voxel. Force indices are formulated, calculated, and coded based on the extracted mechanical parameters. The force formula based on the velocity, displacement, strain, and strain rate is:
Where: p is a 3D cardiac segment; dn(p) is the displacement of p at n's frame; v(p, n, t) is its velocity at the same volume; and ε(p, n, tn), and ε′(p, n, tn) are the strain and strain rate at n's volume. The general force of p per cardiac cycle is the summation of the formula (6).
STEP 3: The third step is to design equations to find the cardiac segment (voxel) trajectory based on Lagrangian equations. Lagrangian equations for a 3D cardiac segment (point p) are described by the following reformulation based on acquired datasets:
p* is the set of all fibres p,r passing through p; δVolumep(t) is the change in volume of point p from the first volume to n's volume in time tn; and δVolumep,n(tn) is the determinant of generalised strain matrix:
Here, there are nine strain components: εx (strain along the x-axis), εxy (rotation around the y-axis in the xy-plane), εxz (rotation around the x-axis in the xz-plane), εyx (rotation around the y-axis in the xy-plane), εy(strain along the y-axis), εyz (rotation around the y-axis in the yz-plane), εzx (rotation around the z-axis in the xz-plane), εzy (rotation around the zy-axis in the yz-plane) and εz (strain along the z-axis).
By rewriting the new data in the generalized Lagrangian equations, following was obtained:
the Lagrangian equation can be written as:
Were applied kinetic and potential energetic formulas of the cardiac segment to the left side of the Lagrangian equations:
Solutions for 2D and 3D equations:
STEP 4: The fourth step for both 2D and 3D datasets is to provide a numerical solution to the proposed equations. This solution parameterizes the cardiac segment (pixel/voxel). These new parameterizations provide smooth functions (in L2-norm functional spaces) attached to the cardiac segment (pixel/voxel) trajectory detections. Were designed equations corresponding to each pixel/voxel and based on Lagrangian mechanics. It could be numerically solving this algorithm of partial differential equations. A function (belonging to L2-norm functional space) was created for each pixel/voxel: fp, f sub pixel, fv, f sub voxel.
New 2D curvelets attached to each pixel are reconstructed. A pixel is then presented as a 2D wedge with considerations for longitudinal, radial, and angular tensions (rotations). New curvelet coefficients are introduced and quantified by inner products between fp and φp(Fourier coefficients of the original image datasets attached to the point p) fp·φp (
As in 2D, new 3D curvelets linked to each voxel are reconstructed. A voxel is presented as a 3D wedge and, as with 2D, longitudinal, radial, angular, and circumferential directions are calculated. New curvelet coefficients are introduced and quantified by inner products between fv and φv (Fourier coefficients of the original image datasets attached to the point v) fv·φv(
Once acquired from the 2D/3D echography system, the patient's echo datasets are loaded into the C++-coded image-processing program. Echo datasets are read and wave datasets, including Fourier coefficients and initial mechanical parameters, are stored.
The coefficients of the original waves and diagrams are read and calculated. The number of original cardiac phases/volumes is also calculated. Calculated coefficients translate to the mechanical parameters of a pixel or voxel per cardiac cycle. The force index resulting from motions and deformations was formulated and calculated. Original motion and deformation indices near the force data were entered into the Lagrangian equations to extract multiple locations for each pixel or voxel in different phases, from the end of diastole to the end of systole, within a cardiac cycle. Solutions are coded as functions in L2 norm spaces. These functions are referred to as fp for each echocardiographic segment. Fourier coefficients fp are calculated. Inner products between Fourier coefficients of original reversal waves φp with the coefficients of fp introduce a new series for each point p. By applying the inverse Fourier transform on the obtained series can be created new curvelet datasets (
Echocardiography emits high-frequency sound waves onto cardiac tissue, interpreting their reflection as images. In fact, the scattering ultrasonic waves, the reversal waves, carry a lot of information to create images. These reversal waves are input for mathematical equations. After solving these equations, echocardiographic images are formed based on the Fourier series. In other words, an echocardiographic image is a solution of these equations of reversal waves. By putting these solutions together side by side, the movement of the heart is observed in the cardiac cycle and a video file is created.
Therefore, conceptually, an image or a frame in a cardiac phase corresponds to a set of solutions from the equations of reversal waves. For a simple understanding of the basis of mathematical solution in this manuscript—which is based on ultrasonic reversal waves—it is consider a set of solutions of wave equations as the images (detailed mathematical equations is explained in methodology). Hence, in the following general description of the mathematical solution, instead of a set of solutions of wave equations, images are used as analogy that corresponds to these solutions.
A raw two-dimensional echocardiography video is acquired. 2D raw images or frames are extracted from the video. A typical 2D echocardiographic video file consists of different frames, and each frame is made of multiple pixels (
Based on the motion and deformation of anatomical points in the raw echocardiographic input data, the corresponding original velocity and strain of the pixels can be calculated on the Cartesian coordinate plane. According to Newton's second law, the calculation of the force that leads to motion and deformation is very important. Therefore, the force index in terms of velocity and strain is formulated (formulations with details were calculated in above referring to
Conversely, each frame consists of several pixels, for example N pixels. Therefore, each frame can easily be considered equivalent to an ordered N-tuples
Because the heart is elastic, the distance, angles, and deformation between the anatomical points in the heart do not change in a cardiac cycle. It can be defined a mapping “f” (fp's) from this N-dimensional space and the spheres drawn into the 3D Cartesian coordinate space so that the values of this mapping are compatible with the trajectories obtained by solving the Lagrangian equations. This mapping f (
For the 3D mode (
In another case was applied mathematical algorithms on 3D surgical view of the mitral valve. Original frames of 12 were increased to 120. Were traced 60 points on the annulus of the mitral valve and tracked the velocity vectors throughout the cardiac cycle. By using K-theory in algebraic geometry different structures of the mitral valve could be tracked throughout the cardiac cycle (
In another case, were applied mathematical algorithms on long-axis view of the heart on 2D TEE. Here was increased the number of frames from 66 to 660. Over 120 points were selected for which the velocity vectors were tracked throughout the cardiac cycle. By using K-theory methods in algebraic geometry, could track the mitral apparatus including the native chords throughout the cardiac cycle (
A voxel tracking in a single full heartbeat acquisition was detected for five selected points; Second row (a6-a10)
Pixle tracking were also studied to follow blood velocity vectors (green vectors) inside the left ventricular cavities (
For experimentation, the disclosed method was evaluated for several complicated cases as shown in
One of the advantageous of our methods is based on the combination of K-theory and Lagrangian equations. In this way, for example, two arbitrary points on the heart mitral valve annulus was selected. The distance to these two points is the shortest curve that connects these two points which is the critical point of the Lagrangian energy integral between those two points. With this method, one can calculate the length between any two traced points. Moreover, by differentiating the second order of the energy integral function, we enter to the Jacobi computational field. This makes it possible to calculate the smallest variation between two selected points while moving on the bend between the two points. This provides it possible to determine the actual bend that changes during the vibration between the two points and calculate its length which is particularly important during mitral valve repair.
While the foregoing written description of the invention enables one of ordinary skill to make and use what is considered presently to be the best mode thereof, those of ordinary skill will understand and appreciate the existence of variations, combinations, and equivalents of the specific embodiment, method, and examples herein. The invention should therefore not be limited by the above-described embodiment, method, and examples, but by all embodiments and methods within the scope and spirit of the invention as claimed.
This application claims priority from a U.S. Provisional Patent Appl. No. 63/344,778 filed on May 23, 2022, which is incorporated herein by reference in its entirety.
Number | Date | Country | |
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63344778 | May 2022 | US |