WO2000048509A1 - Procedes de traitement de donnees d'irm repere caracteristiques du mouvement tissulaire capables du suivi 4d du ventricule gauche - Google Patents

Procedes de traitement de donnees d'irm repere caracteristiques du mouvement tissulaire capables du suivi 4d du ventricule gauche Download PDF

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WO2000048509A1
WO2000048509A1 PCT/US2000/004265 US0004265W WO0048509A1 WO 2000048509 A1 WO2000048509 A1 WO 2000048509A1 US 0004265 W US0004265 W US 0004265W WO 0048509 A1 WO0048509 A1 WO 0048509A1
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spline
tissue
tag
planes
imaging data
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PCT/US2000/004265
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Amir A. Amini
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Barnes-Jewish Hospital
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T7/00—Image analysis
    • G06T7/20—Analysis of motion
    • G06T7/246—Analysis of motion using feature-based methods, e.g. the tracking of corners or segments
    • G06T7/251—Analysis of motion using feature-based methods, e.g. the tracking of corners or segments involving models
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T7/00—Image analysis
    • G06T7/20—Analysis of motion
    • G06T7/246—Analysis of motion using feature-based methods, e.g. the tracking of corners or segments
    • A—HUMAN NECESSITIES
    • A61—MEDICAL OR VETERINARY SCIENCE; HYGIENE
    • A61B—DIAGNOSIS; SURGERY; IDENTIFICATION
    • A61B5/00—Measuring for diagnostic purposes; Identification of persons
    • A61B5/05—Detecting, measuring or recording for diagnosis by means of electric currents or magnetic fields; Measuring using microwaves or radio waves 
    • A61B5/055—Detecting, measuring or recording for diagnosis by means of electric currents or magnetic fields; Measuring using microwaves or radio waves  involving electronic [EMR] or nuclear [NMR] magnetic resonance, e.g. magnetic resonance imaging
    • A—HUMAN NECESSITIES
    • A61—MEDICAL OR VETERINARY SCIENCE; HYGIENE
    • A61B—DIAGNOSIS; SURGERY; IDENTIFICATION
    • A61B5/00—Measuring for diagnostic purposes; Identification of persons
    • A61B5/103—Detecting, measuring or recording devices for testing the shape, pattern, colour, size or movement of the body or parts thereof, for diagnostic purposes
    • A61B5/11—Measuring movement of the entire body or parts thereof, e.g. head or hand tremor, mobility of a limb
    • A61B5/1107—Measuring contraction of parts of the body, e.g. organ, muscle
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T2207/00—Indexing scheme for image analysis or image enhancement
    • G06T2207/10—Image acquisition modality
    • G06T2207/10072—Tomographic images
    • G06T2207/10088—Magnetic resonance imaging [MRI]
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T2207/00—Indexing scheme for image analysis or image enhancement
    • G06T2207/30—Subject of image; Context of image processing
    • G06T2207/30004—Biomedical image processing
    • G06T2207/30048—Heart; Cardiac

Definitions

  • the present invention relates to methods for determining indicators of tissue motion from imaging data obtained from imaged tissue.
  • this invention relates to methods for determining a displacement vector field and other indicators of tissue motion from tagged magnetic resonance imaging (MRI) data obtained from imaging a left ventricle of a heart .
  • MRI magnetic resonance imaging
  • Noninvasive imaging techniques for assessing the dynamic behavior of the human heart are invaluable in the diagnosis of ischemic heart disease, as abnormalities in the myocardial motion sensitively reflect deficits in blood perfusion.
  • MRI Magnetic Resonance
  • Conventional Magnetic Resonance (MR) studies of the heart provide accurate measures of global myocardial function, chamber volume, ejection fraction, and regional wall motions.
  • MR tagging the magnetization property of selective material points in the myocardium are altered in order to create tagged patterns within a deforming body such as the heart muscle. The resulting pattern defines a time-varying curvilinear coordinate system on the underlying tissue.
  • the work of Park, Metaxas, and Axel considers geometric primitives which are generalization of volumetric ellipsoids, through use of parameter functions which allow for spatial variations of aspect ratios of the model along the long axis (LA) of the LV.
  • This model is specially useful for computing the twisting motion of the heart.
  • Prince and McVeigh, and Gupta and Prince developed optical flow based approaches for analysis of tagged MR images.
  • the approach of Guttman, Prince, and McVeigh for analysis of radial tagged images is to use a graph-search technique that determines the optimal inner and outer boundaries of the myocardium as well as tag lines by finding points one after the other in a sequence, using initial search starting points on the determined LV boundaries.
  • Denney and McVeigh presented a technique for reconstructing a three-dimensional myocardial strain map using the discrete model-free (DMF) algorithm, which decomposes the myocardial volume into a finely spaced mesh of points and reconstructs a three-dimensional displacement and strain fields based on local incompressibility and first order displacement smoothness.
  • Moulton, et al . developed a method for approximating continuous smooth distributions of finite strains in the left ventricle from the deformations of MRI tissue tags.
  • a 3-D displacement field on tag surfaces is extracted using sets of MR images and employing spline surface interpolation, followed by a global polynomial fit for determining 3-D displacements, and regional strains between the reference and deformed states .
  • O'Dell, et al . used images from three sequences of parallel tags from segmented k-space imaging obtained at different times.
  • the least squares fitting of a truncated power series in the prolate spheroidal coordinate system on the whole of the myocardium is performed in order to measure dense displacements.
  • a method which uses all of the extracted tag information There is also a need for a method which performs B-spline interpolation over 3-D space and time simultaneously.
  • the invention comprises a method for tracking motion of tissue in three or more dimensions by obtaining a model from imaging data from which a grid of control points may be defined.
  • the imaging data has tag planes.
  • the method comprises the steps of: calculating knot planes from the grid of control points of the imaging data; fitting the knot planes to the tag planes to obtain the model of the tissue; and representing motion of tissue in three dimensions with the model of the tissue.
  • the invention comprises a method for representing indicators of tissue motion with a B-spline model from imaging data.
  • the imaging data has tag planes.
  • the method comprises the steps of: defining displacement vectors corresponding to the tag planes; fitting the B-spline model to the defined displacement vectors; and deriving indicators of tissue motion from the fitted
  • the invention comprises a method for reconstructing tag surfaces with B-spline surfaces from imaging data having sets of image slices with tag data and calculating motion between the B-spline surfaces.
  • the method comprises the steps of : reconstructing at least a first B-spline surface from a first spatial stack of B-spline curves corresponding to a first tag surface from a first set of image slices; reconstructing at least a second B-spline surface from a second spatial stack of B-spline curves corresponding to a second tag surface from a second set of image slices; and calculating motion between B-spline surfaces.
  • the invention comprises a method for warping a first area in a first image slice of imaging data into a corresponding second area in a second image slice of imaging data successive in time to interpolate a dense displacement vector field using smoothing splines.
  • the imaging data contains tag lines.
  • the method comprises the steps of : finding coordinates of the tag lines in both slices of imaging data; and reconstructing a dense displacement vector field with smoothing splines using coordinates of the tag lines .
  • FIGs . 1A-1C illustrate three orthogonal sequences of tag planes representing the imaging protocol for SPAMM-MRI .
  • the directions for three orthogonal sequences of tag planes are represented by reference characters 1, 2, 3 on the cube of FIG. lC.
  • the set of tag lines referred to by reference character 1 are the intersections of tag planes in direction 1 with SA image planes.
  • the set of tag lines referred to by reference character 2 are the intersections of tag planes in direction 2 with the SA image planes.
  • the set of tag lines referred to by reference character 3 are the intersections of tag planes in direction 3 with LA image planes.
  • FIG. 2 is a side view illustration of one model knot plane fitting a group of tag lines belonging to one tag plane of the left ventricle of the heart.
  • FIGs. 3A-3D are side views of iterations in fitting a set of knot planes to a set of tag planes.
  • FIG. 3A illustrates the initial position.
  • FIG. 3B is an illustration after two (2) iterations.
  • FIG. 3C is an illustration after five (5) iterations.
  • FIG. 3D is an illustration of the final position.
  • FIG. 4 is an illustration of knot solids simultaneously fitting a temporal frame sequence of data where the data includes three orthogonal sequences of tag planes .
  • FIG. 5 is a perspective illustration of a B-spline surface representation of a long-axis tag plane reconstructed from a spatial stack of B-splines. "Out -of -plane" movement of the heart is visualized by deviation from flatness of the long-axis B-spline surface.
  • FIGs. 6A-6C are perspective illustrations of the movement of material points of the left ventricle of the heart from diastole to one-third through systole.
  • FIG. 6A shows the initial 3-D location of material points shown on every second slice of the MRI data.
  • FIG. 6B shows the location of the material points for every fourth slice of the MRI data.
  • FIG. 6C shows the new computed location of material points one- third through systole.
  • FIGs. 6A-6C illustrate the non-rigid motion of material points of the heart and indicate that points further up in slices (around the base) move downward, whereas points near the heart's apex are relatively stationary.
  • FIG. 7 illustrates one display of a sequence of displacement vector fields of the left ventricle from a sequence of 2D image slices with smoothing spline warps during systole.
  • FIG. 7 shows that segmental motion of all myocardial points can easily be quantitated and visualized from the location, direction and length of the displayed vectors. An akinetic area in the upper left area is indicated.
  • FIGs. 8A-8C illustrate images collected from a pig at baseline and after induction of posterolateral myocardial infarction in the short axis (SA) orientation.
  • FIG. 8A is an undeformed MRI image slice (slice 0, frame 0) of a pig's LV in short axis orientation at baseline.
  • FIG. 8B illustrates the deformed slice (slice 0, frame 11) corresponding to FIG. 8A.
  • FIG. 8C is a 2-D projection of the 3-D reconstructed motion field corresponding to the slice 0, frame 0 material points.
  • one method of the invention represents indicators of tissue motion with a B-spline model from imaging data by using displacement vectors.
  • the indicators of motion include a display of beads derived from tag planes corresponding to material points of the tissue, a display of the vector field, a display of the strain field, or a display of the B-spline surface.
  • Another method of the invention warps areas in the imaging data to interpolate dense displacement vector fields using smoothing splines.
  • one method of the invention includes the step of deriving indicators of tissue motion in three primary steps.
  • the "deriving" step includes interpolating, reconstructing, or representing.
  • the imaged tissue includes tissue of the heart, skeletal muscles, and other tissues which would benefit from the imaging methods and processing described herein.
  • the tissue motion can be represented discretely in three dimensional space or continuously in four dimensions (three dimensional space over time) .
  • imaging data containing tag data from an imaging apparatus as described above is obtained.
  • One imaging apparatus for obtaining tagged images is a magnetic resonance imaging (MRI) unit having a spatial modulation of magnetization (SPAMM) pulse sequence.
  • MRI magnetic resonance imaging
  • SPAMM spatial modulation of magnetization
  • a typical imaging protocol for obtaining tagged images uses a SPAMM pulse sequence for imaging the left ventricle of the heart. Multiple images in both short-axis and long axis views of the heart are collected to cover the entire volume without gaps.
  • rf tagging pulses are applied in two orthogonal directions.
  • the repetition time (TR) of the imaging sequence is approximately 7.1 msec
  • the echo time (TE) is 2.9 msec
  • the rf pulse flip angle is 15 degrees
  • the time extent of rf tag pulses is 22 msec. Echo sharing is used in collecting each time-varying image sequence for given slice position (called a Cine sequence) .
  • B-splines are suitable for representing a variety of industrial and anatomical shapes.
  • Three advantages of B- spline representations are as follows. (1) They are smooth, continuous parametric curves which can represent open or closed curves. For this application, due to parametric continuity, B-splines will allow for sub-pixel localization of tags, (2) B-splines are completely specified by few control points, and (3) individual movement of control points will only affect their shape locally. In medical imaging, local tissue deformations can easily be captured by movement of individual control points without affecting static portions of the curve .
  • the method of the invention for representing indicators of tissue motion models the imaging data in two, three, or four dimensions.
  • the imaging data contains tag lines and the B-spline model is a two- dimensional curve.
  • the imaging data contains tag planes and the B-spline model is a three-dimensional surface.
  • the imaging data contains tag planes deforming in time and the B- spline model represents a three-dimensional volume over time. Modeling in different dimensions employs the same overall methods described herein. For simplicity, the four dimensional method is described, followed by an explanation of how the two- and three-dimensional equations would differ in their final result.
  • One method of the invention represents indicators of tissue motion with a B-spline model by initially defining displacement vectors corresponding to the tag data in the imaging data. The imaging data is acquired over a time interval. The next step is fitting the B-spline model to the defined displacement vectors to obtain a displacement vector field for the entire time interval . Indicators of tissue motion are then derived from the fitted B-spline model.
  • a tensor product B-spline model is then fit to the defined displacement vectors using a least squares method.
  • the coefficients of the B- spline model equation correspond to the unknown control points for which the equation is to be solved.
  • the equation is optimized using numerical methods to obtain the coefficients. From this B-spline model, indicators of tissue motion can be derived by obtaining the displacement vectors for any instant in the sampled time interval.
  • This method of the invention represents indicators of tissue motion with a B-spline model fitted to initial displacement vectors corresponding to the tag data in the imaging data.
  • the imaging data is acquired over a time interval from tl to t2.
  • a B-spline model is then fit to the available displacement vectors to obtain a displacement vector field for the entire time interval.
  • Indicators of tissue motion are then derived from the B-spline model at any instant in the time interval beginning with time tl and ending with time t2.
  • Another method of the invention utilizes a knot solid to represent each frame of imaging data and utilizes three sequences of solid knot planes to detect three sequences of LV tag planes in order to obtain a 4-D object (3-D B-solid + ID B-spline interpolation over time) .
  • an adaptive conjugate gradient descent method optimizes an objective function encoding the distance between model knot planes and MRI tag planes leading to the minimum quickly and accurately.
  • material points can be displayed over time to generate a "movie" to illustrate the control points movements in three dimensions over time, which movements represent tissue motion.
  • the B-spline model can be displayed as a visual indicator of the shape of the tissue at any time instant during the interval in which the tissue is imaged.
  • the epicardial and endocardial boundaries of the left ventricle can be displayed.
  • knot lines and knot planes are temporal functions. The 3-D solid captured at each knot time instant is called a knot solid.
  • a tensor product 4-D B-spline model is expressed as :
  • ⁇ I x J x K x L is the total number of model control points and N 2 (u) , N 3 (v) , N k (w) , N 1 (t) are B-spline basis functions which blend control points p 1Jkl .
  • N 2 (u) , N 3 (v) , N k (w) , N 1 (t) are B-spline basis functions which blend control points p 1Jkl .
  • This procedure may be mathematically stated as:
  • FIGs 1A-1C illustrate three orthogonal sequences of tag planes representing the imaging protocol .
  • the directions for three orthogonal sequences of tag planes are represented by reference characters 1, 2, 3 on the cube of FIG. 1C.
  • the set of tag lines referred to by reference character 1 are the intersections of tag planes in direction 1 with SA image planes.
  • the set of tag lines referred to by reference character 2 are the intersections of tag planes in direction 2 with the SA image planes.
  • FIG. IB the set of tag lines referred to by reference character 3 are the intersections of tag planes in direction 3 with LA image planes.
  • the tag lines on SA and LA images are formed by intersecting image slices with one or two orthogonal sequences of tag planes. From the tag lines on SA and LA images, the shape of three orthogonal sequences of tag planes at each time frame can be reconstructed. The direction for each sequence of tag planes is shown in FIGs. 1A-1C.
  • the B-spline model there is one temporal sequence of knot solids and each knot solid contains three orthogonal sequences of knot planes. Since SA and LA volumetric frames of MRI data will include three orthogonal sequences of tag planes, if each knot solid is fit to each time frame by matching each knot plane to its corresponding tag plane, the model will then automatically interpolate the volumetric deformations of the LV continuously over time.
  • FIG. 2 is a side view illustration of one model knot plane fitting a group of tag lines belonging to one tag plane of the left ventricle of the heart.
  • the Chamfer distance potential is employed to build an objective function for fitting the tag planes. In general, if edge or line features can be extracted reliably, Chamfer distance potential is a good choice for obtaining fits.
  • Chamfer distance is the Euclidean distance from each model point to the nearest target point, denoted by C
  • C(S , t) Dist (S,c (t) ) where c is the closest target point to the model point S.
  • c ( ) means the target is a function of time, and C (S , t) is the 4-D Chamfer distance.
  • the Chamfer distance image also called the potential image or potential function
  • the Chamfer distance image is an image in which every voxel value stores that voxel's distance to the closest object target point. It is clear that in this image, target points have zero values, and hence are darkest.
  • the advantage of building the Chamfer distance potential image is that every voxel value is calculated only once, off line, prior to model fitting. When the model deforms during an iterative fitting process, the Chamfer distance of each model point is immediately available without any additional computations:
  • each B-spline model knot plane is interactively fit to its corresponding LV tag plane.
  • a potential problem in the mapping of an array of knot planes to array of tag planes is the possible confusion which may be brought about in the registration of plane pairs.
  • the potential function is split to surmount this problem.
  • n tag planes n separate 4-D potential functions are generated and the ith potential function for attracting the ith knot plane is computed solely from the ith tag plane. All three tag orientations in the model are processed in the same manner. The split potential functions attract corresponding knot planes. Hence, by splitting the potential function, errorless registration between model knot planes and LV tag planes is realized.
  • the total energy for the model is defined as the sum of the energy of each knot solid which is defined by summing the energy of each knot plane.
  • the energy of each knot plane is further defined as the integral of the corresponding split potential function over the knot plane surface.
  • E j " j * C, (S(u,v,w,t))dudv ⁇
  • C u , C v , C w are used to denote the 4-D split Chamfer distance potentials and U Titan, V m r W m , and T m are the maximum knot values. It should be noted that although the potential functions are split, all knot planes are simultaneously optimized.
  • the adaptive conjugate gradient descent method is used which shortens the step length prior to taking a step in the search direction that passes over the minimum point. The process halts if the step length is less than a threshold.
  • the model is based on a 4-D grid of control points and the total energy of the model is a function of all control points.
  • Each control point contributes to a few frames of data (in time) and a few nearby tag planes (in space), i.e., each control point determines the local position and movement of myocardial points in k knot interval (k being the order of the B-spline) . Conversely, each myocardial point and its movement over time are determined by k control points in each direction.
  • each tag plane applies a force to nearby control points in its normal direction
  • three sets of orthogonal and nearby tag planes determine the displacements of a given control point in 3-D space.
  • the sequence of image frames determines the temporal movement of control points, generating a 3-D time-varying grid of control points. In each fitting iteration, when the system moves all control points in four dimensions, the entire model is optimized simultaneously for all frames of data.
  • FIGs. 3A-3D are side views of iterations in fitting a set of knot planes to a set of tag planes.
  • FIG. 3A illustrates the initial position.
  • FIG. 3B is an illustration after two (2) iterations.
  • FIG. 3C is an illustration after five (5) iterations.
  • FIG. 3D is an illustration of the final position.
  • FIG. 4 is an illustration of knot solids simultaneously fitting a temporal frame sequence of data where the data includes three orthogonal sequences of tag planes.
  • FIG. 3A shows the initial position of tag planes and knot planes. Each vertical line is the intersection of a plane with the paper.
  • the set of shorter lines which appear textured are the knot planes.
  • Each fitting iteration for 3 sequences of knot planes takes about 2.8 seconds.
  • the solid fitting algorithm converges in about 30 iterations to FIG. 3D. Therefore, the total fitting process for this example approximately took 84 seconds on a 296Mhz Sun Ultra 30 platform.
  • V ( u , v, w) S ⁇ u , v, w, t ) - S ( u , v, w, t 0 ) (3)
  • V(u,v) S 1 ( u , v) - S 0 (u,v) (5)
  • a quadric-quadric-quadric-quadric B-spline model is adopted to perform validations.
  • the method discussed below is employed to detect tag lines by optimizing deformable B-spline grids.
  • the energy function whose minimum is sought, is a linear combination of intensity points along the parametrized grid in the image and the SSD (sum-of -squared-differences) values around tag intersections for tracking interframe motion.
  • FIGs. 8A-8C illustrate the results of applying the above method to images collected from a pig at baseline and after induction of posterolateral myocardial infarction in the short axis orientation.
  • FIG. 8A is an undeformed MRI image slice
  • FIG. 8B illustrates the deformed slice (slice 0, frame 11) corresponding to FIG. 8A.
  • FIG. 8C is a 2-D projection of the 3-D reconstructed motion field corresponding to the slice 0, frame 0 material points.
  • the displayed motion fields of FIG. 8C correspond to the optimized B-spline model at frame 11 and are compared relative to the model at frame 0.
  • the image acquisition strategy avoids the need for model fitting in frame 0.
  • a tag surface is represented by a B-spline surface. From the B-spline surface representations of the three intersecting tag surfaces, coordinates of material points can be determined. The coordinates of the material points can be determined for successive image frames and displayed in a temporal sequence to visualize motion of the tissue.
  • the method for reconstructing tag surfaces with B-spline surfaces from imaging data having sets of image slices with tag data and calculating motion between the B-spline surfaces comprises the steps of : reconstructing at least a first B-spline surface from a first spatial stack of B-spline curves corresponding to a first tag surface from a first set of image slices; reconstructing at least a second B-spline surface from a second spatial stack of B-spline curves corresponding to a second tag surface from a second set of image slices; and calculating motion between B-spline surfaces.
  • B-splines are suitable for representing a variety of industrial and anatomical shapes. Three advantages of B-spline representations are as follows.
  • B-splines are smooth, continuous parametric curves which can represent open or closed curves.
  • B-splines will allow for sub-pixel localization of tags
  • B-splines are completely specified by few control points
  • individual movement of control points will only affect their shape locally.
  • a B- spl me curve is expressed as
  • the second index may denote ordering along the x axis
  • the first index may denote ordering along the z axis (image slices)
  • B -/k takes on an identical form to B l k
  • uniform B-splines are considered so that the knots are spaced at consecutive integer values of parametric variables.
  • the above procedure provides a mechanism for tracking points within short -axis image slices by reconstructing only one B-surface per spatial stack of image slices at any one discrete sampling instant in time.
  • position of image slices are fixed relative to the magnet's coordinate system, and therefore this approach can only yield within short-axis-slice motion of material points.
  • a second sequence of images is acquired with the requirement that tag planes intersecting the new slices be in parallel to short axis images.
  • FIG. 5 illustrates a tag surface constructed from a spatial sequence of long-axis images.
  • FIG. 5 is a perspective illustration of a B-spline surface representation of a long-axis tag plane reconstructed from a spatial stack of B-splines.
  • Out-of- plane movement of the heart is visualized by deviation from flatness of the long-axis B-spline surface. Coordinates of material points may be obtained by computing intersections of three intersecting B-splme surfaces representing three intersecting tag surfaces. For each triplet of intersecting B-splme surfaces,
  • FIGs. 6A-6C show results of the intersection computation for a few material points and are perspective illustrations of the movement of material points of the left ventricle (LV) of the heart from diastole to one-third through systole.
  • LV left ventricle
  • FIG. 6A shows the initial 3-D location of material points shown on every second slice of the MRI data.
  • FIG. 6B shows the location of the material points for every fourth slice of the MRI data.
  • FIG. 6C shows the new computed location of material points one-third through systole.
  • FIGs. 6A-6C illustrate the non-rigid motion of material points of the heart and indicate that points further up in slices (around the base) move downward, whereas points near the heart's apex are relatively stationary.
  • a thin-plate spline is a type of smoothing spline.
  • the location of tag lines must be determined in two frames of imaging data with coupled B-snake grids. Then, dense deformations are obtained.
  • a method for warping a first area in a first image slice of imaging data into a corresponding second area in a second image slice of imaging data successive in time to interpolate a dense displacement vector field using smoothing splines, said imaging data containing tag lines comprises the steps of: finding coordinates of the tag lines in both slices of imaging data; and reconstructing a dense displacement vector field with smoothing splines using coordinates of the tag lines.
  • Coupled snake grids are a sequence of spatially ordered snakes, represented by B-spline curves, which respond to image forces, and track non-rigid tissue deformations from SPAMM data.
  • the spline grids are constructed by having the horizontal and vertical grid lines share control points.
  • a MN spline grid is defined by ⁇ ( x N) -4 ⁇ control points which are may be represented by the set :
  • the SSD function determines the sum- of-squared-differences of pixels in a window around point v 13 in the current frame (with intensity function J) with a window around the corresponding B-snake grid intersection in the previous frame (with intensity function J) . That is, when the location of the grid in I is sought
  • Tracking tissue deformations with SPAMM using snake grids provides 2-D displacement information at tag intersections and ID displacement information along other ID snake points.
  • the displacement measurement from tag lines however are sparse; interpolation is required to reconstruct a dense displacement field from which strain, torsion, and other mechanical indices of function can be computed at all myocardial points.
  • the location of all myocardial points between two frames of imaging data can be displayed to reveal motion of the myocardium. This describes an efficient solution for reconstructing a dense displacement vector field using localized coordinates of tag positions. It assumes only 2-D motion (as is roughly the case towards the apical end of the heart) .
  • intersections of two grids are "pulled” towards one another by minimizing
  • u int and v int are the x and y components of displacement at tag intersections as well as intersections of myocardial contours with tag lines.
  • any point on a snake in one frame must be displaced to lie on its corresponding snake in all subsequent frames.
  • This constraint is enforced by introducing a sliding spring.
  • One endpoint of the spring is fixed on a grid line in the first frame, and its other endpoint is allowed to slide along the corresponding snake in the second frame, as a function of iterations.
  • (x,y) are the coordinates of a point on the snake in the current frame
  • X,y) is the closest point to (x+u, y+v) on the corresponding snake in the second frame.
  • An optimization function can be obtained by a linear combination of the three terms in Eqs . (6) - (8) .
  • finite elements There are two fundamentally different approaches to the minimization of this function, namely, finite elements and finite differences.
  • the method of finite elements involves guessing the form of the solution (a continuous function or a combination of continuous functions) and then calculating the parameters of this function.
  • the implementation of a finite elements method is usually very fast since there are few parameters to be calculated. However, in many cases, the presumption about the form of the solution may be too restrictive.
  • the finite differences approach on the other hand needs no such initial guess. But this method yields the values of the solution only at selected grid points - values at points in between need to be interpolated.
  • the values have to be calculated at a large number of grid points and in these cases, the finite difference algorithms are slower. Only finite difference techniques are considered. Finite element methods are omitted for the following discussion. Again, there are a number of ways to proceed. Preferably, a quadratic approximation to the optimization function is found and then a conjugate gradient or quasi-Newton algorithm is used to minimize this quadratic approximation. The reason of course is that quasi -Newton algorithms have quadratic convergence properties for functions which are almost quadratic.
  • the discrete form of the function ⁇ 1 can be obtained by substituting the discrete derivatives into the first equation in Eq. (6) .
  • the partial derivatives of ⁇ x can be calculated using the computational molecule approach though special attention should be paid in computing the molecules near the endocardial and epicardial boundaries where the smoothness constraint should break in order not to smooth over the motion discontinuity.
  • the discretization of the function ⁇ 2 and calculation of its partial derivatives is almost trivial.
  • the function ⁇ 3 which is nonquadratic .
  • the partial derivatives of ⁇ 3 are
  • T hor and T ver are predicates equal to one if the snake point of interest lies on a horizontal, or a vertical grid line. Needless to say, the above functions can be discretized by replacing the continuous values by the corresponding values at the grid points.
  • x is the vector of variables
  • A is the constant Hessian matrix of second-order partial derivatives
  • Jb and c are constant vectors.
  • ⁇ 1 and ⁇ 2 can be cast in the above form.
  • ⁇ 3 the values x and y are dependent on x and y, respectively, which makes ⁇ 3 nonquadratic .
  • the discrete optimization function form of ⁇ 3 is given by
  • FIG. 7 illustrates one display of a sequence of displacement vector fields of the left ventricle during systole.
  • FIG. 7 shows that segmental motion of all myocardial points can easily be quantitated and visualized from the location, direction and length of the displayed vectors. An akinetic area in the upper left area is indicated.
  • the step of reconstructing the dense displacement vector field comprises the steps of: assigning an arbitrary vector field to the first grid of the first image slice; warping the second grid onto the first grid by applying the vector field to the second grid to create a warped grid from the second grid; comparing the warped grid to the first grid; and redefining the vector field to minimize any errors found in the comparing step; whereby the first grid of the first image slice is undeformed or deformed, and the second grid of the second image slice is deformed or undeformed, respectively.
  • the first algorithm investigated is the CG algorithm.
  • the CG algorithm For an order-N quadratic problem, the CG algorithm is guaranteed to converge in N iterations. Moreover, it does not store the Hessian matrix and requires o (N) storage for an order-N optimization problem.
  • the CG algorithm does not explicitly calculate or store the Hessian matrix A and can be adapted to the function ⁇ in Eq. (11) .
  • the Hessian matrix for ⁇ is not known.
  • the derivative V ⁇ (p) can be calculated using the derivatives for the functions ⁇ lf ⁇ 2 , and ⁇ 3 . This knowledge of the gradient of ⁇ is used in the implementation of the CG algorithm.
  • the CG algorithm is basically a form of steepest-descent algorithm, except for the fact that the descent directions are chosen very efficiently.
  • the following sequence of operations are performed for the objective function.
  • Third, calculate the vector g 1+1 -Vf ⁇ x 1+l ) .
  • Quasi-Newton algorithm is another different optimization method that has been investigated. It differs from CG in that it has higher memory requirements but better convergence properties for non-quadratic functions. Quasi-Newton methods means techniques which use an approximation to the inverse Hessian matrix in each iteration as opposed to Newton methods which use the exact inverse. A generic quasi-Newton algorithm calculates and stores an approximation to the inverse Hessian matrix in each iteration. Hence for an order-N optimization problem, this method needs o (N 2 ) storage.
  • the advantage of a quasi- ⁇ ewton algorithm lies in that it has quadratic convergence properties for general smooth functions (not necessarily quadratic) .
  • a specific quasi- ⁇ ewton algorithm is characterized by the approximation it uses for the Hessian matrix.
  • the quasi- ⁇ ewton method used in this paper is called the Davidon-Fletcher-Powell (DFP) algorithm which is described in the next paragraph.
  • DFP Davidon-Fletcher-Powell
  • the DFP is a quasi- ⁇ ewton method.
  • Second minimize the function along the current direction d i and calculate x 1+1 .

Abstract

L'invention concerne un procédé permettant le suivi 3 ou 4D du mouvement de tissu par réalisation d'un modèle à partir de données d'imagerie, selon des plans de repères (1, 2, 3) permettant de définir une grille de points de contrôle. Tout d'abord, on calcule des plans de noeuds à partir de la grille des données d'imagerie. Ensuite, on adapte ces plans de noeuds aux plans de repère, ce qui donne un modèle du tissu. Puis, on représente le mouvement 3 ou 4 D du tissu avec le modèle du tissu. Cette invention concerne également d'une part un procédé de reconstruction de surfaces de repères avec des surfaces de spline B à partir de données par imagerie, où des séries de coupes d'images comportent des données repères, et d'autre part un procédé de calcul du mouvement entre les surfaces de spline B.
PCT/US2000/004265 1999-02-19 2000-02-18 Procedes de traitement de donnees d'irm repere caracteristiques du mouvement tissulaire capables du suivi 4d du ventricule gauche WO2000048509A1 (fr)

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WO2003088150A1 (fr) * 2002-04-09 2003-10-23 University Of Iowa Research Foundation Reconstruction et analyse du mouvement d'un embryon
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WO2011070166A1 (fr) * 2009-12-12 2011-06-16 Bjaellmark Anna Système pour quantifier et visualiser le profil de rotation ventriculaire du cœur
CN113710187A (zh) * 2018-12-12 2021-11-26 豪迈帝凯奥斯泰纳克公司 软组织建模与用于整形外科手术程序的计划系统
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Cited By (14)

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WO2002037416A2 (fr) * 2000-10-31 2002-05-10 Koninklijke Philips Electronics N.V. Procede et systeme permettant la detection et le suivi de marqueurs dans des images a marqueurs irm
WO2002037416A3 (fr) * 2000-10-31 2002-07-04 Koninkl Philips Electronics Nv Procede et systeme permettant la detection et le suivi de marqueurs dans des images a marqueurs irm
US7215800B2 (en) 2001-01-23 2007-05-08 Koninklijke Philips Electronics N.V. Following the deformation of a structure per unit length defined on an image of a sequence of images of an organ which is deformable over time
EP1227441A1 (fr) * 2001-01-23 2002-07-31 Koninklijke Philips Electronics N.V. Suivi de la déformation d'une structure linéique définie sur une image d'une séquence d'images d'un organe déformable dans le temps
FR2819919A1 (fr) * 2001-01-23 2002-07-26 Koninkl Philips Electronics Nv Suivi de la deformation d'une structure lineique sur une image d'une sequence d'images d'un organe deformable dans le temps
WO2003088150A1 (fr) * 2002-04-09 2003-10-23 University Of Iowa Research Foundation Reconstruction et analyse du mouvement d'un embryon
US7194124B2 (en) 2002-04-09 2007-03-20 University Of Iowa Research Foundation Reconstruction and motion analysis of an embryo
US7463754B2 (en) 2003-11-13 2008-12-09 Honda Motor Co. Adaptive probabilistic visual tracking with incremental subspace update
US7369682B2 (en) 2004-07-09 2008-05-06 Honda Motor Co., Ltd. Adaptive discriminative generative model and application to visual tracking
US7650011B2 (en) 2004-07-09 2010-01-19 Honda Motor Co., Inc. Visual tracking using incremental fisher discriminant analysis
US7623731B2 (en) 2005-06-20 2009-11-24 Honda Motor Co., Ltd. Direct method for modeling non-rigid motion with thin plate spline transformation
WO2011070166A1 (fr) * 2009-12-12 2011-06-16 Bjaellmark Anna Système pour quantifier et visualiser le profil de rotation ventriculaire du cœur
CN113710187A (zh) * 2018-12-12 2021-11-26 豪迈帝凯奥斯泰纳克公司 软组织建模与用于整形外科手术程序的计划系统
CN114814963A (zh) * 2022-03-03 2022-07-29 吉林大学 一种基于b样条抽道的时间域电磁数据甚低频噪声压制方法

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