Improved Wall Motion Analysis in Two Dimensional Echocardiography. J.P. Hamers

Transcription

1 Improved Wall Motion Analysis in Two Dimensional Echocardiography J.P. Hamers BMTE7.2 Supervisor Eindhoven University of Technology: dr. ir. J.M.R.J. Huyghe Supervisor Columbia University: J.W. Holmes M.D. Ph.D Eindhoven University of Technology (TU/e) Department of Biomedical Engineering Division: Biomechanics and Tissue Engineering Group: Cardiovascular Biomechanics New York, December 26

2 Contents Introduction 2 2 Materials and Methods 4 2. General approach Mathematical model Simulations Results 3. Centroid shift Angle and distance estimation Ellipsoid Artificial data set Real data Fractional shortening Discussion 2 5 Conclusion 24 6 Future work 25 Appendix 26 References 27

3 Improved Wall Motion Analysis in Two Dimensional Echocardiography 2 Introduction It is well known that regional systolic function is impaired during regional ischemia [2]. Two dimensional stress echocardiography is one of the most frequently used techniques for diagnosing coronary artery diseases which can result in ischemia. High frequency ultrasound is used to evaluate the structural, functional and hemodynamic status of the cardiovascular system. Different two dimensional images made at different positions contain the information needed for diagnosis. One of the important parameters that can be characterized using stress echocardiography is wall motion of the heart, especially the motion of the left ventricle (LV). Typically, a qualitative analysis of the motion of the LV endocardial surface is employed. However quantitative measurement should improve the ability to diagnose a patient. Fractional shortening (FS) in two dimensions is used for quantification of wall motion. This measure reflects the extent of inward motion between end-diastole and end-systole which can be calculated from short-axis cross-sectional views []. The short-axis views of the LV normally appear slightly elliptical, exhibiting uniform contraction and wall thickening during ventricular systole resulting in a constant FS. With ischemia changes in the magnitude, timing and direction of regional wall motion are observed, depending on the severity of ischemia and the size of the affected region resulting in more elliptical short-axis views and therefore regions with a different FS [4]. Calculation of the FS requires definition of a centroid and tracing of the endocardial borders of end-diastole and end-systole []. One problem for two dimensional segmental wall motion analysis is defining an appropriate centroid. Because there is no anatomic marker for the center of the heart, many different approaches have been pursued to identify a centroid of the LV [3]. The centroid can either be fixed throughout the cardiac cycle or floating from image to image. Due to rotation and translation of the heart during the cardiac cycle the position of the transducer used for stress echocardiography is not constant. This results in short-axis views that are made at different angles and distances from the apex. Both centroid definitions are influenced by variations in angle and position and might influence the FS and might lead to a wrong diagnosis [3, 4]. Another method to analyse wall motion is real-time three dimensional (RT3D) ultrasound. An advantage of this method is that the whole heart is visualized at once and the data is stored in one three dimensional data set. The orientation of the short-axis slice is determined by the angle and the distance from the apex. The three dimensional data set of RT3D contains information about these parameters and compensations for these variations are easily applied. Therefore locally defined centroids, fixed or floating, can be replaced by a central axis from base to apex used for FS calculation []. Since no such information is available for two dimensional echocardiography, definition of a central base-apex axis is a difficult process and the angle and

4 Improved Wall Motion Analysis in Two Dimensional Echocardiography 3 distance should be estimated. Important is what the range can be in which the estimations of these paramters and therefore the central axis definition are reliable. The goal of this project is to determine a central axis using only two dimensional information that can be used for FS calculation. Therefore a procedure will be introduced in section 2.2 that estimates the rotation angle and the distance from the apex. Simulations using different geometries are performed to test and to determine the accuracy of the procedure and discussed in section 2.3. Finally the FS is calculated for various geometries and rotation angles to define the influence on the variations of these parameters.

5 Improved Wall Motion Analysis in Two Dimensional Echocardiography 4 2 Materials and Methods 2. General approach Cardiologists use three views from the heart to analyse the wall motion, namely two long-axis views and one short-axis view. Determination of a central axis that intersect the short-axis views in a certain point requires information of the orientation in space of this view relative to the long-axis views. The orientation is specified by the angle α with respect to the horizontal plane and the distance d from the apex (figure ). z d x Figure : This figure visualizes the short-axis view (blue) at angle α made on a distance z = d from the apex. The long-axis view is represented in red. The procedure to obtain the orientation is based on lookup tables both for α as well as for d. In general, cross sectional views of the LV appear mildly elliptical. Variations in angle and distance cause shape changes of the short-axis view and therefore the degree of ellipticity, which can be defined by several parameters. Only two of them are used for angle estimation and are discussed in section 2.2. It is assumed that there exists a relation, independent of d, between α and the ellipticity. This relation is the actual angle lookup table (ALT). The area of a cross-section perpendicular to the central axis of the LV on any distance from the apex will be different. Therefore the area can give an indication on what distance the short-axis view is made and can be used as a distance lookup table (DLT). Once these lookup tables are created the ellipticity of the short-axis view is calculated and the ALT is used to estimate α. Estimation of the distance from the apex requires an area calculation of the short-axis view perpendicular to the central axis, because the DLT is based upon this principle. Therefore the first thing that needs to be done is rotation of the intersection plane with an angle of α resulting in a plane that should be perpendicular to the central axis depending on the accuracy of the ALT. Subsequently, the area of the resulting plane is calculated and the DLT is employed to estimate d. Once α and d are known the real position of the short-axis view is determined by these two parameters only. Because of the assumption that the two long-axis

6 Improved Wall Motion Analysis in Two Dimensional Echocardiography 5 views are perpendicular a central base-apex axis can be specified, namely from the midpoint of the base plane to the apex. So the centroid of the intersection plane should coincide with this central axis and can be used for FS calculation. The total procedure for estimation of the above described paramters is discussed in detail in section Mathematical model Traced endocardial borders of end-diastole and end-systole are used for calculation of the radial FS. Once this data is extracted from the short-axis views the FS is calculated according to equations -3. EDR ESR FS = EDR () EDR = (x d xc d ) 2 + (y d yc d ) 2 (2) ESR = (x s xc s ) 2 + (y s yc s ) 2 (3) with EDR the end diastolic radius, ESR the end systolic radius, (x d, y d ) the coordinate of a point on the endocardial surface during end-diastole, (xc d, yc d ) the coordinate of the end diastolic centroid, (x s, y s ) the coordinate of a point on the endocardial surface during end-systole and (xc s, yc s ) the coordinate of the end systolic centroid. The centroids of the intersection plane are calculated according to equations 4 and 5. xc = yc = n x k (4) k= n y k (5) k= with (x k,y k ) the coordinate on the endocardial border. The following paragraphs will discuss the procedure to estimate the rotation angle and distance from the apex in detail.. Mesh The first step is to create a mesh that represents the LV using two long-axis views. Two points of every long-axis are used to fit a spline. Fitting is done for every z-value and eventually interpolation is done to create the mesh (figure 2). The long-axis views could have any shape so that the three dimensional geometry does not necessarily have to be an ellipsoid. The only assumption made is that the two long-axis views are perpendicular to each other.

7 Improved Wall Motion Analysis in Two Dimensional Echocardiography 6 Figure 2: View from above of the two long-axis views (blue and red). Two points of each view on a certain z-value are used to fit a spline. After interpolation a full three dimensional geometry is produced representing the left ventricle. z y x 6 After interpolation the data will be translated according to equations 6-8 so that the coordinates of the apex are (,, ). x n = (x int x int ), x int = y n = (y int y int ), y int = z n = ( z zint min ) n x int (6) k= n y int (7) with (x int, y int, z int ) the interpolated coordinates of the spline fit. k= 2. Lookup tables The second step is to create the ALT and DLT. It is assumed that the only angle involved during echocardiography is a rotation around one axis and the range of rotation is between 3 and 3. Two different parameters extracted from properties of ellipses (figure 3) are used to create ALT s (equations 9 and ). (8) a b Figure 3: Ellipse: a is defined as the long axis of the ellipse and b the short axis of the ellipse. ¹ diff = a 2 b 2 (9) ratio = a/b () Creating the ALT s requires intersection planes at different angles and distances from the apex. Because no interpolation is used, the coordinates between elements are unknown. Therefore a certain thickness for the intersection plane is

8 Improved Wall Motion Analysis in Two Dimensional Echocardiography 7 used. After trial and error a thickness of.5 was the best option. Intersection planes at different angles within the above specified region are obtained by translation of the ellipsoid so that the coordinate of rotation coincides with the centroid of the plane at z = d, rotation of the ellipsoid at an angle α and thereafter intersect the ellipsoid at z = and a zero degree angle (figure 4). z z d d x x Figure 4: The left figure visualize the short-axis view (blue) at angle α made on a distance z = d from the apex. After translation of a distance d and rotation of the ellipsoid at an angle α the intersection plane (blue) is taken at z = and α = (right figure). For every angle-distance combination both parameters are calculated. An important assumption made is that the value of these parameters only depend on the angle and are independent of the distance from the apex. For every angle the mean of the parameters are calculated and a polynomial is fitted through those data points and these polynomials are the actual ALT s. Creating the DLT requires area calculations. Areas are calculated for intersections at every z-value of the mesh and a polynomial is fit through the data. The area is calculated using a numerical integration method according to equation and this method is illustrated in figure 5. Because the data points can be irregularly spaced the calculation is split in a part where the y-coordinate and a part where the y-coordinate <. The total area is the sum of these two parts. For both parts the same numerical integration method is used. area = n k= dx k f (x k ) dx k = x k x k () x k =.5 (x k+ x k ) + x k 3. Estimate angle The next step is to determine the values of the parameters specified in equations 9 and for the short-axis data and use the ALT to estimate the rotation angle α. 4. Estimate distance Rotation of the short-axis view around an angle α result in a intersection perpendicular to the central axis. Calculation of the area of this intersection and

9 Improved Wall Motion Analysis in Two Dimensional Echocardiography 8 y f(x(k)) x k x k x k+ x k x k x Figure 5: Numerical integration method for calculation of the area of the intersection plane with x k the x-coordinate calculated by.5 (x k x k ) + x k and x k is the x-coordinate calculated by.5 (x k+ x k ) + x k. estimation of the distance is straightforward. The total procedure, implemented in Matlab (version 7.), is summarized in figure Simulations The first simulation that has been performed was finding the relation between the centroid of the intersection plane and the rotation angle α. For this simulation the left ventricle is assumed to be an ellipsoid defined by parametric equations (equations 2-4). x = asinφcos θ (2) y = b sin φsinθ (3) z = c cos θ (4) with a the radius in x direction, b the radius in y direction, c the half of the length of the ellipsoid, θ the angle in longitudinal direction and φ the angle in circumferential direction. The values for a, b and c are not physiological but arbitrarily chosen and the values of all parameters are summarized in table.

10 Improved Wall Motion Analysis in Two Dimensional Echocardiography 9 Long-axis data FIT MESH Short-axis data 3D GEOMETRY CALCULATE parameter* FIND ANGLE CREATE angle lookup table CREATE ROTATE COORDINATES rotated short-axis view distance lookup table CALCULATE FIND DISTANCE Area Figure 6: Flowchart of the procedure used to estimate the angle and the distance from the apex.*the parameters that are used to estimate the angle are defined in equations 9 and. Table : Values of the parameters used for specifying the ellipsoid. Parameters Value a 3 b 2 c 5 φ.5π φ.5π θ π θ π Data points of intersection planes at rotation angles 3π 8 α 3π 8 on constant distance d = 2 are determined and for each α the centroid difference is calculated using equation 5. centdiff = (xc r xc o ) 2 + (yc r yc o ) 2 (5) with (xc r, yc r ) the centroid of the rotated intersection plane and (xc o, yc o ) the point where the central axis intersects the short-axis view. All centroids are calculated according to equations 4 and 5. The next simulations concerned the estimation of the angle and distance. Three different geometries representing the left ventricle are used to test the procedure introduced in section 2.2: an ellipsoid, an artificial data set containing two long-axis views and one short-axis view and a real data set containing also

11 Improved Wall Motion Analysis in Two Dimensional Echocardiography two long-axis views and one short-axis view. It is assumed that the long-axis views are perpendicular. In case of the ellipsoid step one (fitting a mesh) can be skipped because a three dimensional geometry is already present defined by the parametric equations 2-4. For all simulations two different angle lookup tables are used: one for the difference and one for the ratio (equations 9 and ). Eventually the goal was to find out if the calculation of the FS is influenced by variation in rotation angle and distance. The last set of simulations concerned FS calculation which requires end-diastolic and end-systolic data. This is done for both the ellipsoid as well as for the real dataset. The systolic ellipsoid is defined by scaling the diastolic ellipsoid with scaling factor p. The percentage shortening is therefore similar in all three principal directions (equations 6-8). It is more realistic when the short-axis view is rotated in multiple directions. Two different angles are used to rotate the intersection plane, one that causes rotatation around the x-axis and one that causes rotation around the y-axis. The FS will be different in ischemic regions compared to healthy regions. Ischemia is simulated by shifting the systolic ellipsoid in a certain direction by a value of tr. (equations 6-8). x = a( p)sinφcos θ tr (6) y = b( p)sin φsinθ (7) z = c( p)cos θ (8) with p the percentage shortening and tr the translation in x-direction creating ischemic regions. For this simulation a fixed centroid is used which is equal to (, ). For the real dataset end-diastolic and end-systolic data from an unhealthy LV is used to find out if the FS calculation is influenced by rotation and different centroid definitions, namely a fixed and a floating centroid. The fixed centroid coincides with the central axis from base to apex which is parallel to the z-axis and intersect the origin so that the centroid used for FS calculation is (, ) on any distance. Floating centroids are calculated both for end-diastole and end-systole according to equation 4 and 5.

12 Improved Wall Motion Analysis in Two Dimensional Echocardiography 3 Results 3. Centroid shift Figure 7 shows the relation between the x and y coordinate of the centroid of the rotated short-axis view for several rotation angles. Due to a rotation angle around the y-axis only the x ( ) coordinate of the centroid will shift while the y ( ) coordinate remains constant. A linear relation between the rotation angle and the centroid shift in x direction can be observed. Figure 8 relates the disance between the centroid of an intersection perpendicular to the central axis of the LV and the centroid of a rotated short-axis view. The larger the rotation angle, the larger the distance will be..6.4 coordinate Figure 7: The relation between the rotation angle and the change in the x ( ) and y ( ) coordinate of the centroid of the rotated short-axis view. The intersection plane is made at a distance 2 from the apex..6.5 distance Figure 8: The change in distance between the point where the central axis intersects the short-axis view and the centroid of the rotated short-axis view for different rotation angles. The intersection plane is made at a distance 2 from the apex.

13 Improved Wall Motion Analysis in Two Dimensional Echocardiography Angle and distance estimation 3.2. Ellipsoid After the mesh of the ellipsoid is made two ALT s and one DLT is created according to step 2 in section 2.2. The range in which the estimations for the angle and distance are reliable is dependent on the accuracy of the method used. Due to the assumption that the value of the parameters calculated for the ALT s does not depend on the distance from the apex will result in variations in angle estimation. Figure 9 shows the standard deviations (red) and the mean ( ) of both parameters. Estimations of the angle are for both parameters reliable in a region where the real rotation was between.3 rad and.3 rad. Outside this region the standard deviations of both parameters increase rapidly. ratio [] difference Figure 9: Relations between the rotation angle and the parameters used for the ALT s: ratio (left) and difference (right). The mean ( ) and the standard deviation (red) are visualized. Results of the angle and distance estimations for the two different parameters are shown in figures - 3. The left graph represents the angle estimation while the right graph shows the distance estimation. For increasing distances the ratio based ALT gives a better estimation of the rotation angle since those datapoints are in closer proximity to the real angle (+). However, the distance is better estimated using the difference based ALT. Due to the larger standard deviations at larger rotation angles the estimation of the angle is for both parameters less accurate on a distance smaller than 3% (d =.5) and a distance larger than 7% (d = 3.5) of the apex-base axis. In between this region estimations are more reliable distance Figure : Estimation of the angle (left) and the distance from the apex (right). For both figures (+) represents the real angle and distance, ( ) the estimation using the difference of the two axis of the intersection and ( ) the estimation using the ratio of the two axis of the intersection.

14 Improved Wall Motion Analysis in Two Dimensional Echocardiography distance Figure : Estimation of the angle (left) and the distance from the apex (right). For both figures (+) represents the real angle and distance, ( ) the estimation using the difference of the two axis of the intersection and ( ) the estimation using the ratio of the two axis of the intersection distance Figure 2: Estimation of the angle (left) and the distance from the apex (right). For both figures (+) represents the real angle and distance, ( ) the estimation using the difference of the two axis of the intersection and ( ) the estimation using the ratio of the two axis of the intersection distance Figure 3: Estimation of the angle (left) and the distance from the apex (right). For both figures (+) represents the real angle and distance, ( ) the estimation using the difference of the two axis of the intersection and ( ) the estimation using the ratio of the two axis of the intersection.

15 Improved Wall Motion Analysis in Two Dimensional Echocardiography Artificial data set Similar observations can be made about the mean and standard deviations of the ALT s using the artificial dataset. Because the three dimensional geometry is not a real ellipsoid the variation in angle is even greater for both parameters although the largest standard deviation is observed for the difference (figure 4). ratio difference Figure 4: Relations between the rotation angle and the parameters used for the ALT s: ratio (left) and difference (right). The mean ( ) and the standard deviation (red) are visualized. The results of the angle and distance estimations are represented in the same manner as for the ellipsoid. A remarkable observation can be made regarding the angle estimation. For this artificial dataset the ratio based ALT results in better estimations on every distance from the apex, what is expected due to the smaller standard deviations. The difference based ALT completely fails for large rotation angles and at very small and large distances to estimate the rotation. However, distance estimation is not influenced by these unreliable α estimations distance Figure 5: Estimation of the angle (left) and the distance from the apex (right). For both figures (+) represents the real angle and distance, ( ) the estimation using the difference of the two axis of the intersection and ( ) the estimation using the ratio of the two axis of the intersection.

16 Improved Wall Motion Analysis in Two Dimensional Echocardiography distance Figure 6: Estimation of the angle (left) and the distance from the apex (right). For both figures (+) represents the real angle and distance, ( ) the estimation using the difference of the two axis of the intersection and ( ) the estimation using the ratio of the two axis of the intersection distance Figure 7: Estimation of the angle (left) and the distance from the apex (right). For both figures (+) represents the real angle and distance, ( ) the estimation using the difference of the two axis of the intersection and ( ) the estimation using the ratio of the two axis of the intersection distance Figure 8: Estimation of the angle (left) and the distance from the apex (right). For both figures (+) represents the real angle and distance, ( ) the estimation using the difference of the two axis of the intersection and ( ) the estimation using the ratio of the two axis of the intersection.

17 Improved Wall Motion Analysis in Two Dimensional Echocardiography Real data The variations in angle estimation for both end-diastole and end-systole are presented in figures 9 and 2. Obviously, large variations for both parameters in the two end stages of the cardiac cycle are present. It seems that variations for end-systole are less compared to end-diastole and again the ratio based ALT is less sensitive for distance changes ratio difference Figure 9: Relations between the rotation angle and the parameters used for the ALT s for end-diastole: ratio (left) and difference (right). The mean ( ) and the standard deviation (red) are visualized. 24 ratio difference Figure 2: Relations between the rotation angle and the parameters used for the ALT s for end-systole: ratio (left) and difference (right). The mean ( ) and the standard deviation (red) are visualized. For only two distances results of angle estimations are presented in figures 2-24 for both end-diastole and end-systole. Clearly, the procedure completely fails to predict the real α and d. No true distinction can be made between the ratio and difference based ALT. However angle estimation is most reliable for both procedures for real angles larger than. Distance estimations are for this particularly LV more sensitive for wrong angle estimations than geometries like an ellipsoid or the artificial dataset. Also distance predications are more stable for end-diastole because these are more parallel to the real distance.

18 Improved Wall Motion Analysis in Two Dimensional Echocardiography distance Figure 2: Estimation of the angle (left) and the distance from the apex (right) for end-diastole. For both figures (+) represents the real angle and distance, ( ) the estimation using the difference of the two axis of the intersection and ( ) the estimation using the ratio of the two axis of the intersection distance Figure 22: Estimation of the angle (left) and the distance from the apex (right) for end-systole. For both figures (+) represents the real angle and distance, ( ) the estimation using the difference of the two axis of the intersection and ( ) the estimation using the ratio of the two axis of the intersection distance Figure 23: Estimation of the angle (left) and the distance from the apex (right) for end-diastole. For both figures (+) represents the real angle and distance, ( ) the estimation using the difference of the two axis of the intersection and ( ) the estimation using the ratio of the two axis of the intersection distance Figure 24: Estimation of the angle (left) and the distance from the apex (right) for end-systole. For both figures (+) represents the real angle and distance, ( ) the estimation using the difference of the two axis of the intersection and ( ) the estimation using the ratio of the two axis of the intersection.

19 Improved Wall Motion Analysis in Two Dimensional Echocardiography Fractional shortening Results of FS calculation for normal left ventricles, in this case ellipsoids, are presented in figures 25 and 26 and show a constant FS both for short-axis views (red) as well as for rotated short-axis slices (blue) around multiple angles. The percent shortening (p) during systole is.45 which is exactly reproduced by FS calculation. fractional shortening angle [deg] fractional shortening angle [deg] Figure 25: Fractional shortening for a normal heart with p =.45, tr =, rotation angle in x direction is and the rotation angle in y direction is 5. short-axis views are represented in red and rotated short-axis slices in blue. Figure 26: Fractional shortening for a normal heart with p =.45, tr =, rotation angle in x direction is 5 and the rotation angle in y direction is 5. short-axis views are represented in red and rotated short-axis slices in blue. In ischemic regions of the LV contraction during the cardiac cycle and therefore the FS will be less compared to healthy regions. Due to the translation in x direction there are regions with a higher FS. Figures show that the influence on the FS due to rotation is negligible because the position of the ischemic region and the value of the FS are similar. fractional shortening angle [deg] fractional shortening angle [deg] Figure 27: Fractional shortening for an ischemic heart with p =.45, tr =., rotation angle in x direction is and the rotation angle in y direction is. short-axis views are represented in red and rotated short-axis slices in blue. Figure 28: Fractional shortening for an ischemic heart with p =.45, tr =., rotation angle in x direction is and the rotation angle in y direction is 5. short-axis views are represented in red and rotated short-axis slices in blue.

20 Improved Wall Motion Analysis in Two Dimensional Echocardiography 9.9 fractional shortening angle [deg] Figure 29: Fractional shortening for an ischemic heart with p =.45, tr =., rotation angle in x direction is and the rotation angle in y direction is 5. short-axis views are represented in red and rotated short-axis slices in blue. FS result for the real dataset for an unrotated short-axis slice is presented in figure 3. The top figures represent the end-diastolic and end-systolic border for different centroid definitions. For a fixed centroid (top left) the short-axis slice reveals a large ischemic region in the bottem since motion of this region during the cardiac cycle is negligible. However, when floating centroids for both end-diastole and end-systole are used this ischemic region is remarkable smaller. The systolic border tends to shift upward. The actual FS (bottem) shows that indeed a more constant FS is acquired for a floating centroid compared to a fixed centroid. The FS inside the ischemic region is larger when a floating centroid is defined. 25 y y x 2 x FS [ ] angle [deg] Figure 3: Top left: the end-diasolic border ( ), the end-systolic border ( ) and (*) the fixed centroid. Tor right: the end-diasolic border ( ), the end-systolic border ( ) and (*) the floating centroid. Bottom: the FS using a fixed centroid ( ) and for a floating centroid ( ). These results are obtained using a short-axis slice that is perpendicular to the central axis.

21 Improved Wall Motion Analysis in Two Dimensional Echocardiography 2 For rotation of the short-axis slice (α = 2 ) similar results are obtained (figure 3). A larger FS is predicted if the centroid is fixed, rather than floating. Due to a little gap near the left side of the end-diastolic border a peak in the FS (bottem) is present y y x x FS [ ] angle [deg] Figure 3: Top left: the end-diasolic border ( ), the end-systolic border ( ) and (*) the fixed centroid. Tor right: the end-diasolic border ( ), the end-systolic border ( ) and (*) the floating centroid. Bottom: the FS using a fixed centroid ( ) and for a floating centroid ( ). These results are obtained using a short-axis slice with a rotation angle α = 2. Comparison between FS calculations of unrotated and rotated short-axis slices result in an almost analogous FS except for a difference near α = 27 (figure 32). This is probably caused by the gap in the end-diastolic border. But for a large part FS calculation is not influenced by rotation of short-axis views FS [ ] angle [deg] Figure 32: Comparison of the FS for an intersection plane of rotation (blue) and the FS for an intersection plane of 2 rotation (red).

22 Improved Wall Motion Analysis in Two Dimensional Echocardiography 2 4 Discussion Based on the available information the procedure used for estimation is quite reliable. However, several steps have to be changed to improve the estimations. The only manner in which a three dimensional geometry can be created is through splines of the two long-axis views. Because the long-axis views may have any shape variations in geometry are obtained. This improves the accurary of the estimations. The only imperfection of creating the mesh is assuming that both long-axis views are perpendicular, which not necessarily have to be true in every case. Translations performed according equations 6-8 should coincide the apex with the origin but it results in a slightly difference from the origin. But the influence on the definition of the central axis is negligible. The most important step used in this procedure is the creation of the lookup tables, since the accuracy of the estimations is defined by these tables. Using only one rotation angle simplifies the problem, but in reality the transducer can rotate around three axes. The angle estimation is based on the relation between the rotation angle and the degree of ellipticity. When two or three rotation angles are involved it becomes even more complex and other parameters should be introduced to define a relation for α. What is even more important is the assumption that the relation between α and the degree of ellipticity is independent of d. In reality, when one looks at the geometry it can be seen that it is curved. Therefore calculations of degrees of ellipticities on different distances from the apex will vary resulting in large standard deviations even for true ellipsoids. Dependency is largest at larger rotation angles (figures 9 and 4). The accuracy of the ALT is influenced by the resolution of the mesh and the thickness of the intersection plane. The higher the resolution the smaller the thickness and the better it can estimate small rotation angles. Increased standard deviations for small rotation angles is caused by the resolution of the mesh (figure 9). Definition of the DLT is perfect and in all simulations the estimation of d is not influenced by a bad estimation of α even though those two are linked (figure 6). Results of the centroid shift simulation are obvious and it can be concluded that definition of a central axis is necessary instead of using locally defined centroids to obtain the right FS calculation. For a true ellipsoid the results of angle and distance estimation are quite accurate. One reason for the distinction in α between the two different ALT s could be that the difference based ALT is more sensitive to distant changes than the ratio based ALT. However, no explanation is found for the difference in distance estimation.

23 Improved Wall Motion Analysis in Two Dimensional Echocardiography 22 Remarkable oberservations are made for the results of the artificial dataset. At first the standard deviations of the ratio based ALT are smaller compared to the other ALT implying a less sensitive behaviour for geometry changes than the difference. Second, angle estimation using the difference ALT completely fails at small distances and calculate angles larger than 5 rad. An explanation could be that no boundary conditions are specified for the polynomials. If the calculated difference for the short-axis view is outside the region where the polynomial is defined it will try to find a solution by interpolation. It is better to try to find the best solution within the range instead of searching beyond this boundary. This holds for all the lookup tables even for the DLT although no surprising results are obtained for distance estimations. Several reasons for the inadequate angle and distance estimations for the real dataset can be found. First of all the mesh fitting process is only based on two long-axis views causing differences with the real LV. Therefore slices through the fitted mesh will differ from real short-axis slices. Secondly the parameters on which the ALT creation is based are only useful for intersections that are true ellipses. Normally a short-axis slice does not appear as a full ellipse, but have a different shape. The assumption that the parameters are independent on distance for the same rotation angle is not true. This causes the large standard deviations and so wrong estimations. The fourth reason is that the ALT s are split up in two parts, one that calculates positive angles and the other one negative angles. The criterion to use either one is based on a property of a true ellipse. If an intersection is taken at a larger angle the long axis of the ellipse (a, figure 3) and so the distance from the origin to one side of the long axis will increase. Rotation clockwise induces an increase in distance from the origin to the right side of a and a counterclockwise rotation induces the opposite. Because the short-axis slice is not a true ellipse the values for a and b can be wrong. This can lead to a wrong conversion of the ALT. FS calculations are performed both for the ellipsoid and for the real data set of an ischemic LV. For the ellipsoid ischemia is simulated by translation of the end-systolic shortaxis view in x direction. More realistic results are obtained by rotating the short-axis view around two axes instead of one. In general ischemia occurs only at a certain place within the myocardium so a better way to simulate ischemia is to reduce the parameter p (percentage shortening during systole) locally and thus make p coordinate dependent. This should give more realistic results in FS calculation. Nevertheless, rotation has negligible influence on the value of the FS and the site where ischemia occurs. This holds also for the FS determined for the real data set except a small range next to the ischemic region. Maybe it is a result of the gap in the end-diastolic border or the centroid calculation. Unevenly spaced datapoints can shift the origin when the mean is computed. But that does not explain similar FS in other regions.

24 Improved Wall Motion Analysis in Two Dimensional Echocardiography 23 Eventually the definition of the central axis needs some remarks. For both of the long-axis views an axis is defined from the apex to the valve ring in the baseplane. This normally results in central axes that will not be parallel to the z-axis. Angle and distance estimations become more important when this is true, because wrong estimations result in a wrong point of intersection of the axis with the short-axis view. As mentioned earlier rotation will shift the centroid and therefore can influence the FS calculation. In this project the central axis was parallel to the z-axis and unreliable estimations had no effect on the FS computation. In reality this will be the case so improvement of the procedure becomes necessary.

25 Improved Wall Motion Analysis in Two Dimensional Echocardiography 24 5 Conclusion Two dimensional stress echocardiography is a widely used technique to evaluate the structural, functional and hemodynamic status of the cardiovascular system. Important information such as wall motion of the LV are obtained using two dimensional short-axis cross-sectional views and is quantified by FS. Determination of a central apex-base axis used for FS calculation is a difficult process when two dimensional data is used. Because no information is available about the orientation in space of the short-axis view it is not known where a possible central axis will intersect this view. Due to rotation and translation of the heart during the cardiac cycle or the transducer short-axis views at several angles and distances from the apex could influence the FS calculation, because of wrong centroid definitions. A method to estimate the orientation of the short-axis view has been introduced so that a central axis can be defined for FS calculation. Although this method contains several limitations and for real data angle and distance estimations fail completely, the overall conclusion is that FS calculation is not influenced by rotation or translation. Another important conclusion is that using a central axis reveals a better approximation of the FS compared to floating centroids that differ from image to image and between end-diastole and end-systole.

26 Improved Wall Motion Analysis in Two Dimensional Echocardiography 25 6 Future work Due to the various limitations of the introduced procedure it is necessary to improve this method to estimate the right orientation of the short-axis view. Especially improvements have to be made for real datasets. The ALT creation is the most important but difficult step of this process. Several adaptations that can be made are summed below. Parameters that are less sensitive to distance changes should be used to find a relation between the angle and the ellipticity. Boundary conditions need to be specified in case the angle estimation exceeds the range in which the polynomial is defined. If the ALT is split into a positive and a negative part a better criterion should be determined. In reality the transducer can rotate around several axis so at least two angles should be included for simulations to get more realistic results. These recommendations should improve the ability to estimate the orientation of the short-axis view and so the definition of a central apex-base axis.

27 Improved Wall Motion Analysis in Two Dimensional Echocardiography 26 Appendix This appendix contains a list of the Matlab routines used for the simulations and provides information what the program does. During the simulations small changes are applied to some programs so only the general overview is presented here. area cal.m This program is based on the already build in routine trapz.m and calculates areas of intersection planes for unevenly spaced datapoints which is useful for creation of the DLT. centroid shift.m Centroid shift.m calculates the distance between the fixed centroid and the centroid of an intersection plane rotated around an axis. create.m The create.m routines use two long-axis views to produce a three dimensional geometry representing the LV by means of spline fitting and interpolation. ellipsoid fake.m This program uses parametric equations to define an ellipsoid. A scaling factor can create an end-systolic ellipsoid and a translation can simulate an ischemic region. final.m These routines estimate the angles and distances using the lookup tables created by the routines lookup diff.m, lookup ratio.m and lookup distance.m. frac short.m In this routine the actual FS calculation is performed only for an ellipsoid. intersect.m This routine determines datapoints of intersection planes with the ellipsoid for several rotation angles. The only thing that is changed for other intersect.m routines is the thickness of the intersection plane. lookup diff.m Difference based ALT s are created using several rotation angles and distances. lookup distance.m For every z-value of the mesh the area is calculated and eventuall a polynomial represents the DLT. lookup ratio.m Ratio based ALT s are created using several rotation angles and distances.

28 Improved Wall Motion Analysis in Two Dimensional Echocardiography 27 References [] S.L. Herz, C.M. Ingrassia, S. Homma, K.D. Costa, J.W. Holmes (25). Parameterization of Left Ventricalar Wall Motion for Detection of Regional Ischemia. Annals of Biomedical Engineering Vol 33, No. 7, pp [2] R. Mazhari, J.H. Omens, J.W. Covell, A.D. McCulloch (2). Structural basis of regional dysfunction in acutely ischemic myocardium. Cardiovascular Research Vol 47, pp [3] J.D. Pearlman, R.D. Hogan, P.S. Wiske, T.D. Franklin, A.E. Weyman (99). Echocardiographic Definition of the Left Ventricular Centroid. I. Analysis of Methods of Centroid Calculation From a Single Tomogram. The American College of Cardiology Vol 6, No 4, pp [4] P.S. Wiske, J.D. Pearlman, R.D. Hogan, T.D. Franklin, A.E. Weyman (99). Echocardiographic Definition of the Left Ventricular Centroid. II. Determination of the Optimal Centroid During Systole in Normal and Infarcted Hearts. The American College of Cardiology Vol 6, No. 4, pp