Reconstruction of 3D Left Ventricular Motion from ... - Semantic Scholar

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 14, NO. 4, DECEMBER 1995

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Reconstruction of 3D Left Ventricular Motion from Planar Tagged Cardiac MR Images: an Estimation Theoretic Approach Thomas S. Denney Jr. and Jerry L. Prince

Abstract | Magnetic resonance (MR) tagging has shown great potential for noninvasive measurement of the motion of a beating heart. In MR tagged images, the heart appears with a spatially encoded pattern that moves with the tissue. The position of the tag pattern in each frame of the image sequence can be used to obtain a measurement of the 3D displacement eld of the myocardium. The measurements are sparse, however, and interpolation is required to reconstruct a dense displacement eld from which measures of local contractile performance such as strain can be computed. In this paper, we propose a method for estimating a dense displacement eld from sparse displacement measurements. Our approach is based on a multidimensional stochastic model for the smoothness and divergence of the displacement eld and the Fisher estimation framework. The main feature of this method is that both the displacement eld model and the resulting estimate equation are de ned only on the irregular domain of the myocardium. Our methods are validated on both simulated and in vivo heart data. Keywords | Cardiac motion analysis, reconstruction, magnetic resonance tagging, left ventricle.

I. Introduction

A major problem in cardiac imaging is the measurement of cardiac motion for identi cation of ischemic and infarcted tissues. Magnetic resonance (MR) tagging [1], [2] has shown great potential for noninvasive measurement of the motion of a beating heart. Tagged images appear with a spatially encoded pattern that moves with the tissue and can be analyzed to reveal the motion of the myocardium and to extract measures of local contractile performance such as strain. In this paper we address the problem of reconstructing a dense motion eld from a collection of tagged MR images. Planar tags are at saturation planes which are applied to the left ventricle (LV) at end-diastole. As shown in Figure 1, images acquired on planes orthogonal to these tag planes show the tags as dark lines which are nearly straight in images taken shortly after end-diastole and are curved in later images. Points along these tag lines can be identi ed with a semi-automated tracking algorithm such as that proposed by Guttman et al. [3], Kumar and Goldgof [4], or Young et al. [5]. Because of the possibility of motion parThis research was supported by Whitaker Foundation Biomedical Engineering Research Grant 91-0108, NIH grant R01-HL45090, and National Science FoundationPresidential Faculty Fellow Award MIP9350336. T. S. Denney Jr is with the Department of Electrical Engineering, Auburn University, Auburn, AL, USA 36849. Email:[email protected] J. L. Prince is with the Department of Electrical and Computer Engineering, The Johns Hopkins University,The Johns Hopkins University, Baltimore, MD, USA 21218. E-mail: [email protected]

allel to the tag planes, these points give information only about the displacement in the direction normal to the tag planes. To measure the displacement in other directions, tag planes must be applied in dierent orientations and additional orthogonal images acquired. Several schemes for acquiring tagged images for the analysis of full 3-D, time-varying motion of the LV have been reported [6], [7], [8], [9]. All methods create tag planes and collect images in a coordinate system de ned by the longaxis of the LV. By convention, the z direction is taken to be along the long-axis and the x and y directions are de ned by the horizontal and vertical directions in a short-axis image (such as that in Figure 1a). Spatial arrays of images are collected in order to sample the entire LV. The strategy of O'Dell et al. [7], [8], which we use in this paper, acquires stacks of short-axis images, as depicted in Figure 2a, and a collection of long-axis images that are rotated about the long-axis, as depicted in Figure 2b. Imaging time imposes a severe limitation, so these image planes sparsely sample the LV (see Sections III and IV). Regardless of the particular imaging and tagging geometry that is used, the result (after analysis to determine the locations of the tag lines in all acquired images) is a sparse set of displacement measurements at discrete points in space and time. The next step is to compute a dense displacement eld from these sparse measurements. Previous research on this problem has focused on either polynomial or spline interpolation methods. Motivated by the geometry of the LV, O'Dell et al. [7], [8] proposed tting a prolate spheroidal interpolating polynomial to the displacement measurements. One diculty with this approach is that the correct order of the interpolating polynomial is not known. Young and Axel [9] proposed tting a nite element model to displacement measurements from grid-tagged images using heuristically chosen interpolation functions. Kumar and Goldgof [4] proposed a method for estimating 2-D LV motion from grid tagged images using a thin-plate spline interpolating function. In this paper, we develop a framework for computing a dense displacement eld from sparse measurements by formulating the problem as a stochastic estimation problem. A priori knowledge of the measurements is incorporated in a stochastic measurement model, and a priori knowledge about heart motion is incorporated in a stochastic process model. These two models are then used to derive an optimal estimate equation. The advantage of this approach is that the estimate equation, which implies an optimal in-

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(a)

(b)

Fig. 1. (a) Tagged LV shortly after end-diastole. (b) Tagged LV at end-systole.

2φ

y φ

3φ

z

φ

2p p

x

x -p -2p

(a)

left ventricle

(b)

left ventricle

Fig. 2. (a) Short-axis image planes. (b) Long-axis image planes.

terpolating function [10], is determined from a priori information about the behavior of heart muscle rather than by heuristic selection. Also, this a priori information does not come from a global model of the LV such as those in [11], [12], [13], but rather from two very general physiological characteristics of myocardium. First, we use the fact that the heart wall is composed of connected muscle bers [12], and as a result, the displacement vectors of two points close together in space are similar in magnitude and direction. We refer to this as the smoothness assumption . Second, since the myocardium is mostly water, it is approximately incompressible1. We refer to this as the isochoric assumption . There is considerable debate as to the validity of this

assumption [14]. Our experimental results, however, show an improvement in estimation error when the isochoric assumption is used in computing the displacement estimate. The process model derived from these two assumptions is similar to the continuous velocity eld models used in optical ow estimation of motion [15], [16]. Using the discretization approach developed by Chin et al. [17], [18] for optical ow, we can incorporate a 3-D segmentation of the LV (if available) thereby restricting the domain of the displacement eld model to the LV. With this setup, we derive the optimal estimate equation using the Fisher estimation framework [19]. We call this estimate the irregular domain displacement estimate (IDDE). This paper is organized as follows. We develop the IDDE 1 Small volume changes, perhaps up to 10 percent, are due to myocardial perfusion during the cardiac cycle. in Section II. In Sections III and IV we validate the IDDE

DENNEY AND PRINCE: RECONSTRUCTION OF 3D LEFT VENTRICULAR MOTION

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can be arbitrarily small. Since h is xed for a given study, the values of Nx , Ny , and Nz are adjusted to make sure the grid volume encloses the LV. (See Section II-D for a discussion on the choice of h). We assume that the segmentation of the LV can be obtained from image processing methods such as those described in [3], and the collection of grid points that intersect the segmented LV are denoted by LV . B. Measurement Model

Fig. 3.

N x Ny Nz

grid .

through simulations and in vivo experiments respectively. The nal section summarizes and discusses our future research direction. II. Irregular Domain Displacement Estimate

In this section we develop our irregular domain displacement estimate (IDDE). After describing notation and geometry in Section A, we develop the observation and process models in Section B. These models are written in matrix notation in Section C, and the Fisher estimation framework is used to de ne the IDDE in Section D. Finally, Section E brie y describes the last step, a transformation from spatial to material coordinates. A. Notation and Coordinate Systems

Scalars are designated by lower-case letters (e.g. x, a), and 3D vectors are designated by lower-case bold letters (e.g. u, r). The components of a 3D vector r are designated by rx , ry and rz . Lexicographically ordered vector elds and matrices are designated by upper-case letters (e.g. U, V ). Two coordinate systems are used to describe the location of points in space: a material coordinate system xed on the left ventricle (LV) with end-diastole as the reference state; and a spatial coordinate system xed in space so that the material and spatial coordinate systems coincide at end-diastole. The z-axis of the spatial coordinate system runs through the center of the LV at end-diastole from apex to base. The x and y axes are orthogonal to the z-axis and to each other, and the x-axis is parallel to the top and bottom edges of a short-axis image. Displacement vectors are estimated only at those points that are both within the LV and on an Nx Ny Nz grid

enclosing the LV, as depicted in Figure 3. Points outside the LV are not used in computing the displacement estimate. De ned in spatial coordinates for a deformed LV, the grid is taken to have a uniform grid spacing h, which

In O'Dell et al.'s imaging method [7], [8], three sets of 2D images are acquired. The rst two sets are short-axis images taken at the planes depicted in Figure 2a. The third set consists of long-axis images taken at the planes depicted in Figure 2b. Although these are three distinct data sets, all images correspond to the same phase of the cardiac cycle | i.e., the same instant of time. Additional sets can be acquired at dierent times to obtain sequences of image sets. In each image set, one component of the displacement eld is measured. To measure the x-component, a set of parallel tag planes orthogonal to the x-axis are applied to the heart at end-diastole, and a stack of short-axis images is acquired at a later time in the cardiac cycle. To measure the y-component, tag planes orthogonal to the y-axis are applied, and another set of short-axis images is acquired. To measure the z-component, tag planes orthogonal to the z-axis are applied, and a set of long-axis images is acquired. We refer to tag planes orthogonal to the x-axis as vertical tags, tag planes orthogonal to the y-axis as horizontal tags, and tag planes orthogonal to the z-axis as long-axis tags. Since the reference position of the tag planes is known from the image acquisition process, it is only necessary to obtain these image sets at that phase of the cardiac cycle in which a displacement eld estimate is desired. It should be noted that although we develop our measurement model in the context of the parallel tag plane technique of O'Dell et al. [7], [8], the concepts are the same for the tagging geometries of Zerhouni [1], [6] and Young and Axel [2], [20], [9]. The tag lines in each image are tracked within the segmented myocardium using a method such as that proposed by Guttman et al. [3] or Kumar and Goldgof [4]. A tracked tag de nes the intersection of a deformed tag plane within the image plane, as shown in Figure 4. A point r on a tracked tag is known to have originated from somewhere on a particular reference tag plane at end-diastole, but nothing more is known. Thus, as shown in Figure 4, only that component of displacement in the direction of the reference tag plane normal, e, can be measured. Further, these measurements can only be made at points lying on the tracked tags, which are in spatial coordinates. Thus, the fundamental measurements available from planar tagged images is a sparse collection of measurements of displacement eld components in spatial coordinates. We represent these measurements in a uni ed mathematical framework as follows. First, we let ?x denote the set of x-displacement measurements, and ?y and ?z be the sets of y and z-displacement measurements. We designate the cardinality of ?x , ?y , and ?z by Mx , My , and Mz re-

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IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 14, NO. 4, DECEMBER 1995 ri j+1k+1

rijk+1

e

ri+1j+1k+1 ri+1j k+1

p

z ri j+1k

rijk r

y x

? u ?

e⋅u

ri+1j k

r

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Reference Tag Plane

ri+1j+1k

Fig. 5. The eight grid points nearest to r.

the interpolation method to signi cantly aect the nal displacement estimate. Substituting (2) into (1) yields X y(rm ) = e(rm ) m;ijk u(rijk) + w(rm ) ; (3) Deformed Tag Plane

ijk2 rm

m = 1; : : :; M ; where the m subscript on indicates that the weights are generally dierent for each measurement. Equation (3) is our measurement model.

Fig. 4. Displacement eld measurements from planar tagged images. The materialpoint at the spatial point r on the tag line originated o the image plane at the point p on the referencetag plane. Only the component of the displacement vector u in the direction of the tag plane normal e is known. C. Displacement Field Model

Ideally, one would like to reconstruct the displacement spectively and de ne M Mx +My +Mz . We then model eld vector u(rijk) from the system of M equations given by (3). This system, however, is underdetermined, which each measurement as means that a priori knowledge about the displacement eld y(rm ) = e(rm ) u(rm) + w(rm ); m = 1; : : :; M (1) must be used to compute a unique estimate. In this section we develop stochastic models for the displacement eld where rm is a point on a tracked tag and based on the assumptions of smoothness and incompress8 ibility, which will be used in subsequent sections to derive < ex; if y(rm ) 2 ?x a displacement eld estimate equation. e(rm ) = ey ; if y(rm ) 2 ?y : : ez ; if y(rm ) 2 ?z C.1 Smoothness Assumption To yield a smooth displacement eld, we require that Here, ex , ey , and ez are unit vectors corresponding the x, y, and z axes, and w(rm ) is a white random process with u(r ) ? u(r ) = v (r ); 8r ; r 2

a Gaussian density given by N(0; w2 ). The measurement u(ri+1 j k ) ? u(ri j k ) = vx (ri j k ); 8ri j k ; ri+1 j k 2 LV i j +1 k ij k y ij k i j k i j +1 k LV noise w models the error inherent in tag tracking methods u ( r ) ? u ( r ) = v ( r ); 8 r ; r 2

i j k+1 ij k z ij k i j k i j k+1 LV ; [21]. In general, the points rm , m = 1; : : :; M, in (1) will (4) not be on grid points. To account for this we model the where for all rijk 2 , vx (rijk), vy (rijk) and vz (rijk) are displacement vector at r (where r can be any one of these independent white random vector processes each of which measured points rm ) as a function of its eight nearest grid has a joint Gaussian density given by N(0; h2s2 I), where points. We de ne r as the set of indices of the eight grid 0 is 3-vector of zeros, I is the 3 3 identity matrix. points nearest r, as depicted in Figure 5. Then u(r) can Note that Equation (4)andonly models dierences between be modeled as neighboring vectors that are both in LV . As a result, X the displacement eld inside the left ventricle is completely u(r) = ijku(rijk) ; (2) unrelated to the displacement eld outside. ijk2 r where the weights ijk are de ned in the Appendix. Es- C.2 Isochoric Assumption sentially, u(r) should be equal to a trilinear interpolation An isochoric assumption on the displacement eld can of its eight nearest neighbors. Note that for each measure- be speci ed in a straightforward manner. From continment, trilinear interpolation is only used over a single grid uum mechanics, an isochoric displacement eld satis es (cf. cell which is on the order of 1.0mm on a side (See Sec- [22]) tion II-D.). Over a region this small, we do not expect r u(r) = 0 (5)


at every point r, where r is the divergence operator. In practice, there are small changes in the volume of the heart wall during the heart cycle because of perfusion, and there will be some error from a nite dierence approximation to (5). We allow for this uncertainty by modeling the isochoric assumption as [ux (ri+1 j k ) ? ux(ri j k )] + [uy (ri j +1 k ) ? uy (ri j k )] + [uz (ri j k+1) ? uz (ri j k )] = vd (ri j k ) (6) for all ri j k ; ri+1 j k ; ri j +1 k ; ri j k+1 2 LV , where vd (ri j k ) is a white random process with a Gaussian density given by N(0; h2d2 ). Equation (6) is similar to Equation (4) in the sense that it models the value of the displacement vectors relative to each other, but unlike (4), Equation (6) couples the x, y, and z components of the vector eld. D. Fisher Estimate

In this section we develop an optimal estimate of the displacement eld at all grids points in LV . The Fisher estimation framework provides a natural avenue for this goal, as we now demonstrate. We begin by writing the model equations given by (3), (4), and (6) as matrix equations. We de ne NLV as the number of grid points in LV and ULV as a 3NLV 1 vector containing all the displacement vectors u(rijk) such that rijk 2 LV . The M measurement equations given by (3) can be stacked to form the matrix equation Y = EULV + W : (7) Next we stack the equations in (4) to form the matrix equation SULV = Vxyz ; (8) where S is a spatial dierence operator and Vxyz = [VxT VyT VzT ]T . The vectors Vx , Vy , Vz contain the ordered vector elds vx (rijk), vy (rijk), and vz (rijk) respectively. Equation (6) can be written as the matrix equation DULV = Vd ; (9) where Vd is the lexicographically ordered scalar eld vd (rijk), and D is a spatial dierence operator. Equations (8) and (9) can be stacked together to form a stochastic model for the displacement eld as follows S U = Vxyz : (10) Vd D LV Note that in this formulation the domain LV can be quite arbitrary, and is de ned on a pixel by pixel basis by the structure of S, D and ULV . As a result, the model de ned by (10) is a general model for incompressible elastic motion of irregularly shaped objects. Typically when a stochastic state model of the form (10) is available, the Bayesian estimation framework is used to compute a state estimate. A Bayesian estimate cannot be used here, however, because Equation (10) does not

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completely model the statistics of the displacement eld ULV . In particular, because the matrix A = [S T DT ]T in (10) is singular | note that AB = 0 when B is any nonzero constant vector | the mean and covariance of ULV cannot be computed from (10). In cases where the prior knowledge is probabilistic but incompletely speci ed, the Fisher estimation framework can be used [19]. In this framework, prior probabilistic knowledge is recast into what are known as a priori observations. In our case, Equation (10) is simply rearranged to look like an observation equation as follows: S V xyz (11) 0 = D ULV ? Vd : This equation, together with (7) forms the augmented system 2 3 2 3 2 3 Y E W 4 0 5 = 4 S 5 ULV + 4 ?Vxyz 5 ; (12) 0 D ?Vd which captures both the observations and the prior statistical information. The Fisher estimate is the linear estimator that minimizes the expected mean-square estimation error subject to the constraint that EfU^LV g = EfULV g, where Efg denoted expectation. Assuming constant noise variances s2 , d2 , and w2 , the optimal estimate U^LV is given by (13) U^LV = 12 E T Y ; w where = h212 S T S + h212 DT D + 12 E T E : (14) s w d We call U^LV the irregular domain displacement estimate (IDDE). The estimate equation (13) can be solved using a sparse linear system solver. We use the method of conjugate gradients [23]. The dimensions of depend on the number of grid points in LV , which depends on the grid spacing h. Smaller grid spacings increase the dimension of , which results in an exponential increase in computation time. Also in this type of problem, the numerical condition of degrades as its dimensions increase [24], which both increases the computation time and limits the accuracy of the solution. Larger spacings, however, unnecessarily decrease both the resolution and the accuracy of the displacement estimate. A good practical choice is to make h approximately equal to the pixel size in the MR image data. In practice, the variances s2 , d2 , and w2 are not known, and must be chosen empirically. When both sides of Equation (13) are multiplied by w2 , it becomes clear that the IDDE is only a function of the ratios w2 =s2 and w2 =d2 . In our experiments, we choose a value for w2 based on the tag tracker accuracy and vary s2 and d2 to minimize some measure of estimation error. In in vivo applications, s2 and d2 can be chosen to minimize the dierence between

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the actual and reconstructed deformed tag line locations (see Section IV.C). When s2 is decreased, less weight is given to the tag line measurements and the IDDE is more smooth. When d2 is decreased, more weight is given to the incompressibility constraint, which means that the average local volume change will be smaller. E. Transformation to material coordinates

The estimate U^LV found by solving (13) is an estimate of the displacement eld on a regularly spaced lattice in spatial coordinates. To calculate quantities such as strain, it is necessary to have a displacement eld in material coordinates. Thus, a transformation of coordinate systems is required. This is accomplished using an interpolation from an irregularly spaced lattice as illustrated in 2D in Figure 6. As shown in Figure 6a, our displacement estimate u^ (r) is initially de ned on a regularly spaced lattice. From the de nition of displacement, we know that there is a material point p that moves to the spatial point r and that the two are related by ^ (r) = r : p+u (15) Thus as depicted in Figure 6a, for each r on the lattice, we can determine a corresponding material point p = r ? u^ (r). The collection of these material points de ne an irregularly spaced grid of material points as shown in Figure 6b. Now consider the regularly spaced lattice of material points shown in Figure 6c. In general none of these points will coincide with the points in Figure 6b, and the material displacement u^ (p) for each regularly spaced lattice point p must be interpolated from the irregularly spaced grid. We decompose the irregular grid into tetrahedra and use linear interpolation on each tetrahedron to estimate the displacement in material coordinates on a regular grid. The accuracy of this interpolation depends on the size of the tetrahedra, which should stay approximately constant with time because the isochoric assumption was used in computing the spatial coordinate estimate. III. Simulation Experiments

A. IDDE Evaluation

To evaluate the IDDE we conducted a simulation experiment using an LV deformation simulator program written by Walter O'Dell and Chris Moore [8]. This program assumes the LV at end-diastole is a prolate spheroidal shell with inner radius 32mm and outer radius 40mmcentered at the origin. It simulates a typical deformation taking place at end-systole [11] and includes bulk translation, bulk rotation, and bulk shear, and torsion, transmural twist, transmural shear, and a gross incompressibility constraint. Our experiment consisted of the following steps. First we generated tag line position data from a simulated 3-D planar tag imaging experiment using the above model of LV deformation. The simulated experiment consisted of 11 short-axis image planes separated by 5.0mm and 11 longaxis image planes equally spaced over 180 degrees. We used 11 horizontal and vertical tag planes separated by

Fig. 7. Segmentation of simulated LV.

5.5mm, and 11 long-axis tag planes separated by 5.0mm. Next a 64 64 64 lattice with a uniform spatial increment of 1.125 mm was de ned around the deformed LV, and the segmentation shown in Figure 7 was obtained from the heart model parameters. Note that the segmentation is in spatial coordinates because the measurements are in spatial coordinates, and it is the true segmentation at this lattice resolution. The apex of the LV is not included in the segmentation because in practice, it is dicult to image [3] and is not typically included in motion analyses. Note that we enforce the isochoric assumption locally through the divergence-free condition [see Equations (5) and (6)], and the global volume of the heart does not factor into the estimation process. We next computed the IDDE assuming that s2 = 1, 2 d = 1 and w2 = 10?4. The values for s2 and d2 were determined by computing the IDDE for dierent values of these parameters, comparing the result to the true displacement and nding the values that resulted in the lowest RMS estimation error (see Section III-B for a study of IDDE accuracy versus s2 and d2 ). The IDDE was compared to the true displacement at each point in the irregular domain shown in Figure 7, and a 3D error map was constructed. A short-axis slice of the error volume is shown in Figure 8a. Plots of both the true and estimated displacement magnitude along the white line in this slice are shown in Figure 8b. The global mean, root-mean-square (RMS), root-median-square (RMdS), and median percent (Md%) error for this reconstruction are shown in Table I. These quantities were computed over 32,616 evenly spaced points inside LV. We note that RMdS errors are smaller than the RMS errors, which means that the RMS error is skewed upwards by a few particularly bad points near the boundary (See Figure 8). We also note from the median percent error that the estimated displacement eld is within 2% of the true displacement on average. We make three observations from the above results.


r

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r ^ u(p) p

^ -u(r) p

(a)

(b)

(c)

Fig. 6. (a) Spatial coordinate displacement eld de ned on a regularly spaced lattice.(b) Material coordinate displacement eld de ned on an irregularly spaced lattice. (c) Material coordinate displacement eld de ned on a regularly spaced lattice. TABLE I IDDE global errors.

Error y z Mag 0.015 0.028 0.160 0.105 0.247 0.076 0.044 0.155 2.437 1.208 2.476

First, from the 1D plots in Figure 8 we observe that the IDDE error is smallest in the middle of the LV wall and gets larger toward the endocardial and epicardial boundaries. This eect occurs because there are fewer displacement measurements near the boundaries. Second in Figure 8b we observe that the IDDE underestimates the true displacement magnitude. The mean errors in Table I, however, have dierent signs and are small in magnitude compared to the RMS errors, which means that we overestimate the displacement magnitude in other regions of the LV. Therefore, we conclude that underestimation is not a systematic problem with the IDDE. Finally we observe that the global errors in Table I are similar to the global errors presented in [7], [8] and [9]. B. Eect of IDDE Model Parameters

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RMS IDDE Error

x Mean (mm) -0.008 RMS (mm) 0.156 RMdS (mm) 0.075 Md % 3.986

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2

Fig. 9. IDDE error versus s2 = d2 .

The accuracy of the IDDE depends on the variances w2 , 2 s and d2 . The measurement noise variance, w2 is derived from the tag tracker accuracy. The other variances are chosen empirically. To study the aect of s2 and d2 on the estimation accuracy, we computed the IDDE for fore we conclude that incompressibility assumption reduces several values of s2 and d2 with w2 = 10?4. The RMS the IDDE error and that we should choose s2 = d2 . The errors between the magnitudes of the IDDE and actual RMS errors versus s2 = d2 are plotted in Figure 9, which displacement vectors are listed in Table II. We also com- reveals that the error is fairly constant for the last three puted the IDDE without the incompressibility assumption decades of s2 = d2 (although it increases again slightly at for the same values of s2 . The RMS errors for this case s2 = d2 = 10). Therefore, we conclude that the IDDE are listed in the \No Assumption" row in Table II. We see is robust to the choice of s2 = d2 provided that they are that the IDDE errors are minimized when s2 = d2 . There- chosen in the proper range.

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IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 14, NO. 4, DECEMBER 1995 TABLE II IDDE error versus

d2 ns2 10?3 10?2 10?1 100 101 No Assumption

10?3 0.26303 0.33212 0.34942 0.35133 0.35153 0.35155

10?2 0.49322 0.25598 0.32400 0.34094 0.34282 0.34303

s2

and

d2 .

10?1 0.77503 0.49009 0.25135 0.31725 0.33367 0.33569

100 0.83049 0.77682 0.48735 0.24714 0.30938 0.32699

101 0.84062 0.83471 0.78037 0.49111 0.25148 0.32113

In each image, points on the inner and outer contours of the LV and along the tag lines were identi ed using the method of Guttman et al. [3]. Each contour consisted of 32 points. The points identi ed along each tag line within the myocardium were spaced 1.0mm apart. In order to compute the IDDE we discretized the problem by de ning a 3D grid that encompassed the LV. A bounding box for the LV was determined from the contours identi ed in each image. A regular grid was constructed with a spacing of 1.25mm in each direction that encompassed the bounding box. The x-y face of the grid was de ned to be parallel with the short-axis image planes. A segmentation of the LV which gave LV for each time frame was obtained using the contour points in each image. B. Displacement Field Estimate

Fig. 10. Image plane orientation for in vivo experiment.

IV. In Vivo Heart Experiments

In this section we use the IDDE to estimate the displacement eld of an in vivo LV from actual planar tagged MR image data. A. Experimental Parameters

The LV of a normal human volunteer was imaged using a parallel planar tag imaging protocol [25]. In accordance with the 3D displacement measurement scheme of O'Dell et al. [7], [8] (see Section II), two short-axis sequences and one long-axis image sequence were acquired. The short-axis image sequences consisted of seven slices with a slice separation of 8mm. The long-axis image sequence consisted of six slices with an angular separation of 30 degrees. The image planes were oriented as shown in Figure 10, where the light planes represent short-axis image planes and the dark planes represent long-axis image planes. In each sequence, the tag planes were separated in the reference state (enddiastole) by 6mm, which resulted in 12 tags across the LV in both the short-axis and long-axis sequences. The resulting images are 256 256 with a pixel size of h = 1:25mm and are spaced 32.5ms apart for a total of 10 time frames through systole.

The displacement eld in spatial coordinates for each time frame was computed using the IDDE and then transformed to material coordinates using the procedure described in Section II-E. For all time frames, we assumed w2 = 10?4 . The divergence-free variance d2 was set equal to the smoothness variance s2 at each time frame (see Section III-B ), and s2 was chosen to linearly increase with time from s2 = 0:1 at time frame 0 to s2 = 1:0 at time frame 9. The rationale for increasing s2 with time is as follows. First, strain can be loosely de ned as the spatial variation in displacement (for a more precise de nition, see [22] ). Second, in the IDDE the spatial variation in displacement is controlled by s2 . Third, myocardial strain increases with time during systole. Finally, a linear increase in s2 was chosen as a rst step. More precise methods of chosing s2 will be explored in future work. C. Results In in vivo heart experiments the true displacement eld

of the LV is not known and the accuracy of the displacement eld estimate is dicult to determine. A qualitative indication of the accuracy can be obtained by using the displacement estimate to reconstruct the tag line data. Note that (1) the reconstruction of any given tag line requires all three components of the displacement eld, and (2) tag line reconstruction only gives an indication of error near the tag planes, where we expect the IDDE to be most accurate. Figure 11a shows an image sequence of one shortaxis horizontally tagged slice at end-systole (t = 340ms),


where points on the tag lines have been darkened for emphasis. Each white dot represents a point where a tag plane deformed in three dimensions by the IDDE intersects the image plane. If there were no error in the estimated displacement eld, the white dots would lie on the tag lines. Figures 11b and 11c show analogous images of a short-axis vertically tagged slice and a long-axis slice. The estimated tag points and the actual tag points match well in all three images. In some cases the dots lie next to the tag lines because both the actual and estimated tag line points are rounded to the nearest pixel. A quantitative measure of the IDDE accuracy was obtained by computing average distance between the actual tag lines and the reconstructed tag lines. A plot of average error versus time is shown in Figure 12. The error increases with time from a value of 0.22mm near end-diastole (t = 47ms) to 0.28mm at end-systole (t = 340ms). The increase in error with time is to be expected because both the average magnitude of the displacement eld and the strain increase with time. D. Eect of Incompressibility Constraint

We studied the eect of the incompressibility constraint on the IDDE estimate by computing the IDDE at the nal time frame (t = 340ms) for several values of s2 and d2 with w2 = 10?4. The average tag line error for each pair of values is shown in Table III. Two important observations can be made from Table III. First, the errors in Table III have the same order of magnitude as the errors in Table II. The errors do not vary as much with s2 and d2 as those in Table II because the error is only computed at the tag lines where we expect the IDDE to be most accurate. Second, the lowest errors ( 0:27mm) occur near the region where s2 = d2 . In contrast, the lowest error without the isochoric assumption is 0.28mm. Given the relatively small variation in error in Table 3 compared to that in Table 2, the slight decrease in tag line error observed with the isochoric assumption suggests that a greater decrease in error might be observed if points between tag lines were included in the analysis. The relationship between tag line error and total error will be explored more fully in future research. V. Discussion

633

(a)

(b)

In this paper we presented an estimation-theoretic framework for reconstructing a dense 3D displacement eld within the LV from planar tag correspondences. In this framework a priori information about the displacement measurements and heart motion is incorporated into a discrete stochastic measurement model and a discrete 3D (c) stochastic displacement eld model. Fisher estimation Fig. 11. Displacement eld estimation results: (a) vertically tagged techniques are then used to derive the optimal estimate short-axis. (b) horizontally tagged short-axis. (c) long-axis. called the IDDE from these two models. The IDDE assumes that a segmentation of the LV is available and restricts the displacement eld model to the domain of the myocardium. The IDDE also incorporates an incompressibility assumption. The accuracy of the IDDE depends on the choice of

634

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 14, NO. 4, DECEMBER 1995 TABLE III IDDE error versus

d2 ns2 10?5 10?4 10?3 10?2 10?1 100 101 No Assumption

10?3 0.4047 0.3717 0.3234 0.3156 0.3229 0.3221 0.3220 0.3220

s2

10?2 0.3287 0.3435 0.3179 0.2853 0.2838 0.2852 0.2918 0.2908

Average Tag Line Error (mm)

0.30

0.20

0.10

0.00

0.0

100.0

200.0 Time (ms)

300.0

Fig. 12. Average tag line error versus time.

and

400.0

d2

in vivo heart. 10?1 100

for

0.3437 0.3401 0.3483 0.3257 0.2780 0.2717 0.2799 0.2806

0.3374 0.3374 0.3388 0.3413 0.3206 0.2791 0.2703 0.2847

101 0.3363 0.3363 0.3361 0.3357 0.3308 0.3133 0.2784 0.2899

ment eld of an in vivo LV from actual MR image data. A displacement eld was estimated for each time frame in an image sequence. A quantitative error measure showed reconstruction errors of 0.28mm at end-systole. Our results also suggest that the incompressibility assumption may improve the IDDE accuracy relative to estimates computed with only the smoothness assumption. The ultimate goal of this research is to reconstruct a 3-D strain map of the left-ventricle. Strain can be computed from the material coordinate IDDE by using nite dierence approximations to the spatial derivatives. Future research in this area will include characterizing the local error and spatial resolution of this strain map as well as the eects of tag line and displacement estimation errors. In addition, a more complete evaluation of our methods on multiple normal and pathologic in vivo data sets is needed as well as a comparison of our results with those from other displacement and strain reconstruction methods.

global process noise variances, which are chosen empiriAppendix cally. We anticipate that with the development of more Appendix: Trilinear Interpolation Weights specialized displacement models, a more physiologicallybased approach will be used. For the present, however, the the point r = [rxry rz ]T and its eight nearest simulation results in this paper show that both algorithms gridConsider are robust to the choice of these parameters provided they ables:points as shown in Figure 5. De ne the following variare chosen in the proper range. Tag and contour detection errors will also aect the acq = (rx ? rijkx )=h curacy of the IDDE. The degree to which tag detection s = (ry ? rijky )=h errors are propagated through to the IDDE depends on 2 the amount of smoothing speci ed by the choice of s and t = (rz ? rijkz )=h : d2 . Smaller values of these parameters will result reduced sensitivity to tag detection errors at the expense of spatial The weights are given by resolution. Contour detection errors will aect the IDDE by changing the domain over which the IDDE is computed. i j k = (1 ? q)(1 ? s)(1 ? t) The speci c eects of these errors are not known but are i+1 j k = q(1 ? s)(1 ? t) expected to be most pronounced near the endocardial and i j +1 k = (1 ? q)s(1 ? t) epicardial boundaries. i+1 j +1 k = qs(1 ? t) The IDDE algorithm as presently implemented takes roughly 20 minutes per time frame to execute on a SUN i j k+1 = (1 ? q)(1 ? s)t SPARC 20, which is too long for clinical use. No attempt, i+1 j k+1 = q(1 ? s)t however, was made to optimize the algorithm for speed, i j +1 k+1 = (1 ? q)st and we expect to make considerable performance improve i+1 j +1 k+1 = qst : ments to the algorithm in future work. We used the IDDE algorithm to estimate the displace-


Acknowledgements

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